Sun, Stars and Planets1
Dr Yvonne Unruh, Prof Juliet Pickering
2020-21
1
These notes are based on lecture notes from Dr David Clements, Prof Juliet Pickering and Dr
Yvonne Unruh
Preface
The Sun, Stars and Planets course is an introduction to stellar and planetary physics. It is not
a pre-requisite for any future courses at Imperial that might deal with astrophysical topics
– such as Astrophysics, Cosmology or Space Physics – but it will provide context for those
courses.
The course is divided into two parts given by two different lecturers. The first deals with
the Sun and with the astrophysics of stars, including their structure. We shall be treating stars
as non-rotating, non-magnetic and essentially static objects; under these assumptions, their
structure can be described by relatively simple equations which we will derive and use during
this part of the course. Stellar structure is well understood, so there should be no major new
results arriving during this lecture course. This section will be taught by Yvonne Unruh.
Planets, in contrast, are rather more complex systems than stars, with a wide range of
different physical phenomena, from weather to volcanism and much more. While there will
certainly be a number of rigorous mathematical results in the planets half of the course, this
means that there will also be a range of qualitative results and open-ended discussions, especially towards the end of the course when we talk about the possibilities of life and intelligence
on other planets. The study of planets in our own and other solar systems is also a very active
field with space missions and observatories producing new results all the time. In particular
results from the Rosetta mission to a Comet and the New Horizons to Pluto and the Kuiper Belt
are still being analysed may produce new results while this course goes on, so there should be
some news flashes and updates as things develop. New results on exoplanets are also arriving
all the time. This section will be taught by Prof Juliet Pickering.
The course will consist of the following elements
• 22 recorded lectures – lectures will include slides and mathematical derivations. The
derivations will be included in these notes but the detailed steps in the mathematics are
best covered in the lectures.
• Four problem sheets and solutions.
• These lecture notes1 .
Lecture recordings, lecture slides, problem sheets, as well as their solutions, and other materials as necessary, will be available on Blackboard.
The office hours for this course will be on Tuesdays from 9-10am and on Thursdays from
4-5pm on MSTeams during weeks 1 to 5. In week 6 they will be from 4-5pm on Tuesday and
9-10am on Wednesday. Please email us (y.unruh or j.pickering at imperial.ac.uk) if these times
are not convenient and you would like to arrange alternative times.
1
Please send corrections to j.pickering or y.unruh at imperial.ac.uk
Textbooks
There is no single textbook for this course. The notes are reasonably comprehensive and buying a book should not be needed to follow the course. However, if you are interested in (stellar)
astrophysics, there is a whole plethora of standard astrophysics textbooks that cover the material taught in the first part of the course. I list two modern textbooks (that take very different
approaches) along with two classic texts. However, if you already own an astrophysics textbook (e.g., Zeilik, Caroll & Ostlie, Kartunnen et al, etc, etc) these should all serve you very
well.
1. An Introduction to the Sun and Stars, Simon F. Green & Mark H. Jones (eds), Cambridge
University Press, ISBN: 0 521 83737 5
This is a large colourful and well illustrated introductory book to the Sun and Stars part
of the course. It is a relatively ‘easy read’ that includes plenty of useful information
diagrams and figures, but it can be overly descriptive in places (at least to my taste).
2. Principles of Astrophysics, by Charles R. Keeton, Springer, ISBN: 978-1-4614-9235-1 (9781-4614-9236-8 for eBook)
A general (though still reasonably slim) undergraduate astrophysics textbook that also
covers a substantial amount of the third-year Astrophysics course material. This is definitely not a coffee table book, though it is kept at a very approachable level. It lacks the
more descriptive parts covered in this course (e.g., there is no section on comparative
planetology), but, together with the book by Frank Shu (see below), it is probably the
text that presents the most “physics-based” view, and constitutes a good longer-term
investment if you are interested in general astrophysics. It is not cheap (≈ £50). It is
available through the library, so you can check it out before you invest.
3. The Stars: their structure and evolution, by R.J. Taylor, Cambridge University Press, 2nd
edition (1996), ISBN: 0 521 45885 4
This is a shorter, less colourful book, but is one of the basic astrophysical texts in this
field. It covers the mathematical and quantitative side of the subject very well, and goes
deeper into many areas than is possible in this course.
4. The Physical Universe, by Frank Shu, University Science Books, ISBN: 978-0935702057
This is one of the classic textbooks in astrophysics that aims to cover “all” of traditional
astrophysics. Some of the material on planets and planetary systems is out of date
(it was written before the discovery of exoplanets!) in terms of results and our state of
knowledge, but the underlying physics is rock solid. Described by some as the ’Feynman
lectures for astrophysicists’.
Books recommended for the planetary and solar-system part include
5. An Introduction to the Solar System, edited by David A. Rothery, Neil McBride and Iain
Gilmour, published by Cambridge University Press
ISBN: 978 1 107 60092 8
As with Introduction to the Sun and Stars, this is a large, colourful and very well illustrated introductory text to the Solar System side of this course. It has lots of facts,
descriptions, diagrams and figures, but is rather light on mathematics. It covers basic
ideas well, but without the rigour that is usual for any Imperial College course.
6. Exploring the Solar System, by Peter Bond, published by Wiley-Blackwell
ISBN: 978 1 4051 3499 6
This is another well presented and illustrated introductory text much like the Rothery
book above. It covers a number of topics somewhat more deeply, and in addition includes
chapters on the Sun and on exoplanets. It is also rather lightweight on mathematics.
7. Planets & Planetary Systems, by Stephen Eales, published by Wiley-Blackwell
ISBN: 978 0 470 01693 0
This is a shorter text book than many of the others listed, and lacks many of the colourful
illustrations. However, it makes up for this in taking a much more mathematical point
of view of the subject material. It also includes sections on exoplanets and on life in the
universe. It is thus a very useful textbook for this part of the course. If you only get one
textbook, this should probably be it.
8. An Introduction to Astrobiology, edited by David A. Rothery, Iain Gilmour and Mark
Sephton, published by Cambridge University Press
ISBN: 978 1 107 60093 5
Astrobiology is a relatively young subject that brings together astronomy, physics, biology and much else besides. This book provides an excellent introduction to these
disparate fields and their contributions to astrobiology. As such, it goes rather further
than this course will in many areas, but will provide a lot of extra material for those who
are interested in this new and growing field.
There are, of course, many other textbooks, popular books and well illustrated coffee table
books available that cover these areas as well, and the library is well stocked with such texts.
Outline Syllabus and Lecture Contents
The lectures will be divided up as follows:
1. Modelling the Stellar Interior – Density & Pressure
2. Modelling the Stellar Interior – Pressure & Temperature
3. Energy Generation in Main-Sequence Stars – Nuclear fusion
4. Energy Transport in Main-Sequence Stars – Radiation & Convection
5. Stellar structure equations – the Sun
6. The Stellar Main Sequence – Scaling relations from the Stellar Structure Equations
7. Solar and Stellar Spectra
8. The Sun’s Atmosphere and Radiation
9. Stellar Astronomy: Putting the Sun in Context
10. The Hertzsprung-Russell Diagram
11. Binary Stars
12. An Overview of the Solar System and its formation
13. Terrestrial Planets: Heating, Cooling and Interiors
14. Terrestrial Planets: Surfaces and Surface Temperatures
15. Terrestrial Planet Atmospheres
16. Gas Giants: Structure and Atmospheres
17. Moons: Formation and Properties
18. Small Bodies: Comets, asteroids and the Outer Solar System
19. Exoplanets: Detection
20. Exoplanets: Properties and Characterisation
21. Astrobiology: Life on Other Planets
22. The Search for Extraterrestrial Intelligence
The Examination
In principle, all the material in these lecture notes, and the lectures, except where explicitly
noted, is examinable. In practice, less than that is easily examined. At the end of each chapter
we have highlighted things to remember, which are the central issues that may appear in the
examination.
The exam format has changed somewhat this year. An example paper illustrating this new
format is the department’s examinations website and also on the Blackboard course pages.
Contents
1 Modelling the Stellar Interior I - Density & Pressure
1.1 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . .
1.2 Mass Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The Dynamical or Free Fall Timescale . . . . . . . . . . . . . . . .
1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Solar units . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Mean Density and Pressure of the Sun . . . . . . . . . . .
1.4.3 Estimate of the Minimum Pressure at the Centre of a Star
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2 Modelling the Stellar Interior II - pressure & temperature
2.1 Stellar Plasma - what stars are made of . . . . . . . . . . .
2.2 Thermodynamic Equilibrium . . . . . . . . . . . . . . . . .
2.3 Local Thermodynamic Equilibrium . . . . . . . . . . . . .
2.4 Pressure in the Solar Interior . . . . . . . . . . . . . . . . .
2.4.1 Radiation Pressure . . . . . . . . . . . . . . . . . .
2.4.2 Gas Pressure . . . . . . . . . . . . . . . . . . . . .
2.5 Mean Molecular Weight . . . . . . . . . . . . . . . . . . .
2.6 Rough Estimate of Stellar Core Temperatures . . . . . . .
2.7 The Virial Theorem . . . . . . . . . . . . . . . . . . . . . .
2.8 The Contraction of a Star . . . . . . . . . . . . . . . . . . .
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3 Energy Generation in Stars
3.1 The Sun’s Energy Source . . . . . .
3.2 Nuclear Fusion . . . . . . . . . . . .
3.3 Stability . . . . . . . . . . . . . . .
3.4 Luminosity and Energy Generation
3.5 Solar Neutrinos – historical aside .
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4 Energy Transport in Stars
4.1 Radiative Heat Transport . . . . . . . . . . . .
4.1.1 A Photon’s Random Walk . . . . . . .
4.1.2 The Radiation Transport Equation . .
4.2 Opacity . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Heat Transport by Conduction . . . .
4.3 Convective Heat Transport . . . . . . . . . . .
4.3.1 The Schwarzschild Stability Criterion
4.3.2 Heat Transported by Convection . . .
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5 The Stellar Structure Equations
5.1 The Stellar Structure Equations . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Boundary Conditions for the Stellar Structure Equations and their Validity
5.3 Homology Transformations . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 The Internal Structure of the Sun . . . . . . . . . . . . . . . . . . . . . . .
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6 The Stellar Main Sequence
6.1 Scaling relations from the Stellar Structure equations
6.1.1 Density and Pressure at the Stellar Centre . .
6.1.2 Temperature at the Stellar Centre . . . . . . .
6.1.3 The Luminosity - Mass relation . . . . . . . .
6.1.4 The Radius - Mass relation . . . . . . . . . .
6.2 Stellar Surface Temperature . . . . . . . . . . . . . .
6.3 Lifetime of a Main Sequence Star . . . . . . . . . . .
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7 Stellar Spectra
7.1 Thermal Radiation . . . . . . . . . . . . . . . . . . .
7.2 Absorption Lines . . . . . . . . . . . . . . . . . . . .
7.2.1 Energy Levels and Transitions . . . . . . . .
7.2.2 The Occurrence and Strength of Lines . . .
7.2.3 Uses of Stellar Spectral Analysis . . . . . .
7.3 Spectral Classification of Stars . . . . . . . . . . . .
7.4 Luminosity and Luminosity Classification Systems
8 The Sun’s Atmosphere and Spectrum
8.1 Atmospheric Structure . . . . . . . . . . . .
8.1.1 The Photosphere . . . . . . . . . . .
8.1.2 The Chromosphere . . . . . . . . . .
8.1.3 The Transition Region and Corona .
8.2 The Solar Spectrum . . . . . . . . . . . . . .
8.3 Solar Activity and The Sun’s Magnetic Field
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9 Stellar Astronomy: Putting the Sun in Context
9.1 The Variety of Stars . . . . . . . . . . . . . . . . . . . .
9.2 Stellar Magnitudes . . . . . . . . . . . . . . . . . . . .
9.2.1 The Definition of Stellar Magnitudes . . . . . .
9.2.2 The Astronomer’s Unit of Distance: The Parsec
9.2.3 The Definition of Absolute Magnitude . . . . .
9.2.4 Observational Passbands . . . . . . . . . . . .
9.2.5 Bolometric Magnitude . . . . . . . . . . . . . .
9.3 Stellar Colours and Temperatures . . . . . . . . . . . .
9.4 Measuring Stellar Distances: Trigonometric Parallax .
9.5 Proper Motion . . . . . . . . . . . . . . . . . . . . . . .
9.6 Stellar Distance Indicators . . . . . . . . . . . . . . . .
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10 The Hertzsprung-Russell Diagram
10.1 Hertzsprung-Russell (HR) Diagrams . . . . . . . . . . . . . . . . .
10.2 HR diagrams: the Brightest and Nearest Stars . . . . . . . . . . .
10.3 The Distribution of Stars on the HR Diagram . . . . . . . . . . . .
10.4 Mass and the HR Diagram . . . . . . . . . . . . . . . . . . . . . .
10.4.1 The Range of Stellar Masses on the Main Sequence . . . .
10.5 Main Sequence Life . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Stellar Evolution Rates . . . . . . . . . . . . . . . . . . . . . . . .
10.6.1 Estimating Cluster Age from the Main-Sequence Turn-off
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11 Binary Stars
11.1 Binary Star Orbits . . . . . . . . . . . . . . . . . . . .
11.2 Measuring Stellar Masses . . . . . . . . . . . . . . . .
11.2.1 Visual Binaries . . . . . . . . . . . . . . . . .
11.2.2 Spectroscopic Binaries . . . . . . . . . . . . .
11.2.3 Eclipsing Binaries . . . . . . . . . . . . . . . .
11.3 Comments on interacting binaries – not examinable .
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12 An Overview of the Solar System
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Overall Inventory of the Solar System . . . . . . . . . . . . . . . . . . . . . . .
12.4 Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Venus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6 Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6.1 The Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.7 Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.7.1 Phobos and Deimos: The Moons of Mars . . . . . . . . . . . . . . . .
12.8 The Asteroid Belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.9 Jupiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.9.1 The Moons of Jupiter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.10 Saturn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.10.1 The Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.10.2 The Moons of Saturn . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.11 Uranus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.12 Neptune . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.13 Pluto, Trans-Neptunian Objects (TNOs) and the Kuiper Belt . . . . . . . . . .
12.14 Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.15 The Oort Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.16 Kepler’s Three Laws of Planetary Motion . . . . . . . . . . . . . . . . . . . . .
12.17 Formation of the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.17.1 Formation of Rocky/metallic planets versus gaseous planets . . . . . .
12.17.2 Outer solar system: Formation of giant planets: the standard model .
12.17.3 Solar system formation and formation of planetary systems around
other stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.18 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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13 Terrestrial Planets: Heating, Cooling Processes and Interiors
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 The Active Earth . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Primordial Heating . . . . . . . . . . . . . . . . . . . . . . . .
13.4 The Structure of the Earth . . . . . . . . . . . . . . . . . . . .
13.5 Long Duration Heat Sources . . . . . . . . . . . . . . . . . . .
13.6 The decay of long term heating sources . . . . . . . . . . . . .
13.7 Heat Loss from Planets . . . . . . . . . . . . . . . . . . . . . .
13.8 Cooling Processes . . . . . . . . . . . . . . . . . . . . . . . . .
13.9 Volcanism and Tectonics on Other Terrestrial Planets . . . . .
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14 Terrestrial Planet Surfaces and Temperatures
14.1 Introduction . . . . . . . . . . . . . . . . . .
14.2 Major Factors in Shaping Planetary Surfaces
14.3 Impact Cratering . . . . . . . . . . . . . . .
14.4 Volcanism and Tectonics . . . . . . . . . . .
14.5 Erosion . . . . . . . . . . . . . . . . . . . . .
14.6 The Surface Temperatures of Planets . . . .
14.7 The Greenhouse Effect . . . . . . . . . . . .
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15 Terrestrial Planet Atmospheres
15.1 Introduction . . . . . . . . . . . . . . .
15.2 Why do we have an atmosphere at all?
15.3 Atmospheric Density and Pressure . .
15.4 Temperature Variations with altitude .
15.5 Thermal Escape . . . . . . . . . . . . .
15.6 Current Atmospheric Composition . .
15.7 Origin of Atmospheres . . . . . . . . .
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16 Gas Giants: Structure and Atmospheres
16.1 Introduction . . . . . . . . . . . . . . . . . . .
16.2 Basic Properties of Gas Giants . . . . . . . . .
16.3 The Internal Structure of Jupiter and Saturn .
16.4 Excess Heat in Jupiter and Saturn . . . . . . .
16.5 The Internal Structure of Uranus and Neptune
16.6 Gas Giant Atmospheres . . . . . . . . . . . . .
16.7 Ring Systems . . . . . . . . . . . . . . . . . . .
16.7.1 Estimate of the tidal force . . . . . . .
16.7.2 Estimate of the Roche Limit . . . . . .
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109
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17 Moons: Formation and Properties
17.1 Introduction . . . . . . . . . . .
17.2 Orbits and Masses . . . . . . . .
17.3 Formation . . . . . . . . . . . .
17.3.1 The Moon . . . . . . . .
17.3.2 Titan . . . . . . . . . . .
17.4 Tidal Forces and Tidal Heating .
17.5 Tidal Locking . . . . . . . . . .
17.6 Circularisation . . . . . . . . . .
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17.7 Orbital Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.7.1 Stable Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.7.2 Unstable Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 Small Bodies: Comets, Asteroids and the Outer Solar System
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2.1 Ceres . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Kuiper Belt and Trans-Neptunian Objects . . . . . . . . . . .
18.4 Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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19 Detecting Exoplanets
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.3 What is a Planet Anyway? . . . . . . . . . . . . . . . . . . . . . . . .
19.4 Direct Detection: How Hard Can it Be? . . . . . . . . . . . . . . . . .
19.5 The Astrometric and Radial Velocity Methods of detecting exoplanets
19.6 Planetary Transit Searches . . . . . . . . . . . . . . . . . . . . . . . .
19.7 Other ways to detect planets . . . . . . . . . . . . . . . . . . . . . . .
19.7.1 Pulsar Planets . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.7.2 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . .
20 The Exoplanet Population
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
20.2 The Current State of Planet Searches . . . . . . . . . . .
20.3 Selection Effects . . . . . . . . . . . . . . . . . . . . . . .
20.4 Exoplanet Masses . . . . . . . . . . . . . . . . . . . . . .
20.5 Exoplanet Composition . . . . . . . . . . . . . . . . . . .
20.6 Exoplanet Orbits: Hot Jupiters and Planetary Migration
20.7 Host Star Metallicity . . . . . . . . . . . . . . . . . . . .
20.8 Exoplanets: A young Science . . . . . . . . . . . . . . . .
21 Astrobiology: Life on Other Planets
21.1 Introduction . . . . . . . . . . . . . . . .
21.2 Life on Earth: History . . . . . . . . . . .
21.3 Lessons from the History of Life on Earth
21.4 Life Elsewhere in the Solar System . . . .
21.4.1 Mars . . . . . . . . . . . . . . . .
21.4.2 Europa . . . . . . . . . . . . . . .
21.4.3 Enceladus . . . . . . . . . . . . .
21.5 Life Outside the Solar System . . . . . .
21.5.1 Host Star . . . . . . . . . . . . .
21.5.2 Gas Giant Moons . . . . . . . . .
21.6 The Galactic Habitable Zone . . . . . . .
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22 The Search for Extraterrestrial Intelligence
22.1 How to Find Life on Other Planets . . . . . . . . . . . . . . . . .
22.2 The Search for Extraterrestrial Intelligence (SETI) - introduction
22.3 The Drake Equation . . . . . . . . . . . . . . . . . . . . . . . . .
22.4 The Fermi Paradox . . . . . . . . . . . . . . . . . . . . . . . . .
22.5 SETI and CETI . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.6 The Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
Figure 1: The Milky Way Galaxy, as
it might be seen from above (from
NASA). The red lines indicate galactic
longitudes. (Non-examinable aside:
The galactic coordinate system has
the Sun at its centre; galactic longitude 0◦ is the direction from the Sun
to the galactic centre; galactic latitude
0◦ is in the plane of the Milky Way).
Preamble – The Local Neighbourhood
The Earth is a planet with a diameter of 12,756 km, orbiting the Sun at a distance of 1.5×1011 m,
or 1 Astronomical Units (AU). The Earth is just one of eight planets in the Solar System; we
will hear more about the planets and other bodies in the Solar System later in the course. The
Solar System has a radius of about 100 AU, though parts of it may stretch out to much greater
distances.
The nearest stars to the Sun, α Centauri A and B, and the slightly closer Proxima Centauri
are much further away - about 4 × 1016 m (or 1.3 parsec), and it takes light a little over four
years to travel to us from them. The local neighbourhood of the Sun, including stars that
you might recognise from the night sky like Sirius, Procyon and Tau Ceti, as well as fainter
stars that cannot be seen by the unaided eye like Wolf 359 and Barnard’s Star, lie in a region
about 200 lightyears. The stars in our local neighbourhood form just a very small part of the
Milky Way Galaxy. This is a spiral-shaped gravitationally bound and rotationally supported
collection of stars, gas and other material that is about 200,000 light years across (see Fig. 1).
Our galaxy lies in a group of other galaxies known as the Local Group. This is roughly 10
million lightyears (or 3 megaparsec) across and includes the galaxy M31 in the constellation
of Andromeda. This is the nearest large spiral galaxy to our own, and will collide with the
Milky Way in about 4 billion years. On still larger scales, the local group of galaxies is a small
offshoot from the Virgo galaxy cluster which itself is part of a larger supercluster of galaxies.
On the largest possible scales the Universe is thought to be highly uniform, but on smaller
scales – and those small scales are much much larger than the scale of a single galaxy – this
clearly is not the case. How this large scale structure, as is called, came to be is a subject for
the cosmology course, but the only reason that we can see these galaxies is that they contain
stars, so that understanding the nature, formation and evolution of stars is an essential step to
understanding how cosmologists reach their conclusions. Galaxies contain from a few 100,000
to 3×1012 stars. Our own galaxy contains about a couple hundred billion stars.
2
Sun, Stars and Planets 2020-21
Lecture 1
Modelling the Stellar Interior I Density & Pressure
For the next several lectures we will be building a model of a star’s interior. This may seem
like a complicated task, but during most of their life time, stars are fairly simple systems. By
applying standard laws of physics – conservation of mass, conservation of energy, Newtonian
gravity, and the ideal gas equation – and making simplifying assumptions about hydrostatic
and thermodynamic equilibrium, simplified stellar models can be put together. The first stages
of this do not require us to know anything about the generation of energy in stars. This mirrors
the original derivation of the equations of stellar structure which took place in ignorance of
the processes that power them.
In this lecture we will look at how pressure provides the necessary support against gravitational collapse.
What is a Star?
A star is a self gravitating mass of gas that radiates energy from an internal source.
Other astrophysical bodies, such as planets, contain more than just gas - rock, ices etc. as
we will see later in this course. They also do not radiate significant amounts of their internal
energy, but instead simply reflect light from the Sun.
1.1
Hydrostatic Equilibrium
A star is held together by gravity – by the gravitational attraction exerted on any given part of
the star by all the others. Gravity is attractive: in the absence of any other force the star would
rapidly collapse into a black hole. Something else acts against gravity to prevent collapse. This
is the pressure of the stellar material, resulting from the kinetic energy of the atoms, ions and
electrons inside it. This works in just the same way that pressure from the kinetic energy of
the molecules prevents the Earth’s atmosphere from collapsing.
The two forces of gravitational attraction and thermal pressure oppose each other and
govern stellar structure (and, as we will see later, atmospheric structure on planets). When
these forces are in exact balance then the system – be it Sun, star or planet – is in a state
known as hydrostatic equilibrium.
3
P(r+ δr)
A
r+δ r
P(r)
Fg
r
Figure 1.1: Coordinate system
and parameter definitions used
in the derivation of the hydrostatic equilibrium equation.
The arrows indicate the forces
acting on a cylindrical volume
element with mass δm = ρAδa
inside a spherical star.
Assumptions for Hydrostatic Equilibrium in Stars
The first basic assumptions we will make about the nature of stars that will allow us to solve
for their internal structure are that
i. stars are spherical and symmetric about their centres (i.e., we can assume spherical
symmetry)
ii. stellar properties change slowly with time, allowing us to neglect the rate of change of
these quantities with time.
The Balance between Pressure and Gravitational forces
Consider a small cylinder of matter inside a spherically symmetric star, as seen in Fig. 1.1.
The lower face of this cylinder is a distance r from the centre of the star, the upper face is a
distance r + δr from the centre of the star. Both faces have area A; the volume of the cylinder
is thus A δr. The mass of material in this volume element will be δm = ρ(r) A δr where ρ(r)
is the density of stellar material at a distance r from the centre of the star.
The forces acting on this volume element are twofold: gravity and pressure. Gravity acts
on the material in the volume element, attracting it towards the centre of the star. It is a well
known result that gravity inside a filled sphere acts in such a way that (a) only the material
inwards of your location in the sphere has any effect and (b) this acts as if all of that material
were gathered into a point at the centre of the sphere. The magnitude of the gravitational
force pulling the volume element towards the centre is thus
Fg =
G m(r) δm
G m(r) ρ(r) A δr
=
,
r2
r2
(1.1)
where G is Newton’s gravitational field constant, and m(r) is the mass of stellar material
contained within a radius of distance r.
The pressure force on the outer face due to the material outside r + δr acts inward, while
the pressure force on the inner element acts outward, resulting in a net outward force with
magnitude
dP
Fp = P (r + δr) A − P (r) A =
A δr,
(1.2)
dr
4
Sun, Stars and Planets 2020-21
where we have assumed δr r.
For equilibrium these two forces must be equal, i.e., Fp = Fg and thus
dP
G m(r) ρ(r)
.
= −
dr
r2
(1.3)
This is the equation of hydrostatic equilibrium, applicable to stellar interiors, planetary atmospheres and many other systems.
We can rewrite this equation slightly as
dP
= − g ρ(r).
dr
(1.4)
When r = R, the radius of the star, then g is the surface gravity of the star1 . In the case of
the Sun, the surface gravity, g, is 300 m s−2 , and it is of a similar order of magnitude for other
(main-sequence) stars. This notation is often used when considering thin atmospheres of stars
and planets where g can be assumed to be constant.
In deriving Eq. 1.3 we assumed that all the forces acting on any element of material in the
star were exactly balanced, with zero net forces being felt. If this were not the case, then the
star would expand or contract due to the acceleration produced by the net force (F = ma).
Exercise: Solar eclipse records show that the Sun’s radius has not changed appreciably (say
less than 10%) over at least a thousand years. Use this to estimate the maximum net acceleration (or force) on a fluid element at the surface of the Sun and compare this to the Sun’s
surface acceleration g ' 300 m s−2 .
1.2
Mass Continuity
The mass, density and radius are clearly not independent since the mass m(r) contained
within a sphere of radius r is determined by the density of the material ρ(r) at all points
within r. If we consider a spherical shell of thickness δr at a radius r from the centre of a
sphere with density ρ(r), the mass of the shell will be
δm = 4πr2 ρ(r) δr,
(1.5)
dm
= 4πr2 ρ(r).
dr
(1.6)
which implies
The mass-continuity equation is thus a second differential equation (alongside the equation
of hydrostatic equilibrium) that we can use to describe the internal structure of stars.
Note that m(r) in the mass continuity and hydrostatic
R r equilibrium equations is not the
total mass, but the mass enclosed within r, i.e., m(r) = 0 4πρ(r)r2 dr. The total mass of a
star M∗ is obtained by integrating over the whole radius R of the star, yielding (as expected)
total mass
Z
R
M∗ =
4πr2 ρ(r)dr.
(1.7)
0
As we have three unknowns, m(r), ρ(r) and P (r), we still need at least one further relation.
This will be the topic of the next lectures, but for now let us examine some of our assumptions
and look at what results we can already produce.
1
This is similar to the ‘little g’, the gravitational field strength of 9.8 m s−2 , on the surface of the Earth.
5
1.3
The Dynamical or Free Fall Timescale
How long would it take for a star to collapse if the supporting pressure forces disappeared?
There are several different ways of estimating this – an alternative is included in problem
sheet 1 – but a simple approach is to look at the time taken for material at the surface of the
star to travel the distance to its centre under the influence of the surface gravity g. Thus
g =
GM
R2
s =
and
1 2
gt .
2
(1.8)
Setting s equal to R, the radius of the star, we find
1 GM 2
τ
R =
2 r2 ff
⇒
τff =
2R3
GM
1/2
∼
R3
GM
1/2
.
(1.9)
Exercise: evaluate the free-fall timescale of the Sun (see below for the solar radius
and mass).
1.4
1.4.1
Applications
Solar units
In stellar physics, it is often more intuitive to measure stellar properties relative to solar values.
The basic properties of the Sun are its mass M , its radius R , and its luminosity L . Together
with the mean Sun-Earth distance, defined to be 1 astronomical unit (AU), these are called
‘solar units’ and are frequently used in astrophysics. Their values are
• M = 1.99 × 1030 kg,
• R = 6.96 × 108 m,
• L = 3.83 × 1026 W, and
• 1 Astronomical Unit (AU) = 1.50 × 1011 m.
1.4.2
Mean Density and Pressure of the Sun
The mean density ρ of the Sun is similar to the density of water as
ρ =
M
∼ 103 kg m−3 .
4
3
3 πR
(1.10)
To obtain a rough estimate of the solar mean pressure, we can use Eq. 1.3 and substitute approximate mean values for mass and radius. The mean mass as a function of radius can be
guessed to be M /2 and the mean radius can be guessed to be R /2. Using the mean density
from Eq. 1.10, the equation of hydrostatic equilibrium implies
dP
Gm(r)ρ(r)
= −
dr
r2
⇒
P
R
∼
GM /2
ρ
(R /2)2
⇒
P ∼
G M2
.
2 R4
Exercise: calculate the value of the mean pressure in the Sun using Eq. 1.11.
(1.11)
6
Sun, Stars and Planets 2020-21
1.4.3
Estimate of the Minimum Pressure at the Centre of a Star
The two equations describing the internal structure of a star allow us to estimate the lower
limit for the core pressure of a star of known mass and radius. Recall that
dP
Gm(r)ρ(r)
dm
= −
and
= 4πr2 ρ(r).
2
dr
r
dr
Dividing the hydrostatic equilibrium equation by the mass continuity equation, we obtain a
single differential equation for the pressure with respect to mass
dP/dr
Gm(r)ρ(r)
= − 2
dm/dr
r 4πr2 ρ(r)
⇒
dP
Gm
= −
.
dm
4πr4
(1.12)
I have dropped the explicit dependence of the mass and pressure on r; strictly speaking, we
now ought to write r(m), P (m), etc. Integrating both sides of this equation from the centre
of the star to the surface yields
−
Z
0
Ms
dP = −
[P ]PPsc
= Pc − Ps =
Z
0
Ms
Gm
dm,
4πr4
(1.13)
where Pc and Ps are the core and surface pressure, respectively.
We cannot evaluate the right hand side of this equation since we do not know what r is
as a function of m. We can, however, derive a lower limit. For all parts of a stellar interior r
will be less than the star’s radius R, hence
Z M
Z M
Gm
Gm
GM 2
dm
>
dm
=
.
(1.14)
4πr4
4πR4
8πR4
0
0
Rearranging, and taking Ps Pc gives a minimum value for the central pressure of a star,
Pc > Pmin =
GM 2
.
8πR4
(1.15)
For the Sun, setting M = M and R = R we find Pmin, = 4.5 × 1013 N m−2 . This result
requires no knowledge about the chemical composition or physical state of the material at the
core of the Sun. In comparison, atmospheric pressure on Earth at sea level is about 105 N m−2 ,
so pressure at the centre of the Sun is at least 108 times greater than this2 .
Things to Remember
• How to derive the equations of hydrostatic equilibrium and mass continuity
• The definition and derivation of the dynamical (free-fall) timescale
• The minimum core pressure of a star
2
You may have noticed that this estimate is lower than that for the mean pressure (Eq. 1.11). The value in
Eq. 1.15 is a strict lower limit, though perhaps not a useful one given that for most of the star R−4 r−4 . The
solar core pressure is in excess of 1016 N m−2 , more than two orders of magnitude larger than our estimate.
7
Lecture 2
Modelling the Stellar Interior II pressure & temperature
As part of our quest to model the internal structure of stars, we arrived at two equations,
Eqs 1.3 and 1.6, that link density, mass and pressure as functions of radius. This is not enough
to solve for the structure of stars – we have two equations and three unknowns. To proceed
further, we need to consider the equation of state that provides the link between the pressure
and density.
While the derivations up to now hold for self-gravitating objects in general (including
brown dwarfs, gas-giant planets, ‘dead stars’ – such as white dwarfs and neutron stars), these
next lectures focus on stars and assume that these stars are fusing hydrogen in their cores.
This is the case for most of a star’s lifetime. Stars in the hydrogen-fusion stage are said to be
on the main sequence (see lecture 6).
2.1
Stellar Plasma - what stars are made of
The material that makes up a star is an ionised gas or plasma, and can be assumed to behave
like an ideal gas. The high temperature of stars means that all but the most tightly bound
electrons will be separated from the atoms. This allows stellar material to reach greater compression without deviation from the perfect gas law since, for the most part, the constituents
of the gas have a size comparable to nuclear scales, i.e., ∼ 10−15 m, rather than comparable
to atomic scales, ∼ 10−10 m (this is the case, for example, for H+ ions which are just single
protons). The stellar plasma can thus reach higher densities than would be possible for a neutral gas before the size of the constituents becomes important and behaviour deviates from a
perfect gas. Plasma also differs from an ordinary gas in that forces between electrons and ions
have a much longer range than forces between neutral atoms.
2.2
Thermodynamic Equilibrium
You will already be familiar with the equation of state for ideal gases, which links pressure
and density via temperature. For an equation of state like this to work, the system must be in
a state of thermodynamic equilibrium.
Any system left in isolation for sufficiently long will settle into a state of thermodynamic
equilibrium. In this state, the overall properties of the system do not vary from point to point
8
Sun, Stars and Planets 2020-21
and do not change over time. Individual particles within the system do have changing properties - for example any given hydrogen atom in the plasma of a stellar atmosphere might be
stripped of its electron, forming an H+ ion, or an ion might gain an electron and return to a
neutral state. However there is a statistical steady state in which any process and its inverse
occur at the same rate, balancing each other out. Also, since the properties of the system
do not vary from point to point in thermodynamic equilibrium, then all parts have the same
temperature.
The key point about a system in thermodynamic equilibrium is that all physical properties, such as pressure, specific heat and internal energy, can be calculated in terms of density,
temperature and chemical composition alone.
2.3
Local Thermodynamic Equilibrium
The density of stellar plasma is so high that there is a short mean-free-path, and thus many collisions between electrons, ions and photons. The timescale between collisions is much shorter
than the timescale for changes in pressure, temperature and composition, so one might expect
the plasma to be in a state of thermodynamic equilibrium.
However, in a star, pressure, density and temperature change with radius, so the thermodynamic equilibrium of the stellar plasma is not global. Instead, the plasma is in a state of
local thermodynamic equilibrium (LTE). LTE implies that electrons, ions and photons all
have the same temperature at a given position in the star, and this temperature is equivalent
to the kinetic temperature of the gas1 . Since the mean-free-path for photons in the stellar
plasma is small, and the radiation is in LTE with the matter, the intensity of the radiation is
given by the Planck blackbody function2 .
2.4
Pressure in the Solar Interior
Given the assumption of LTE and the fact that the stellar plasma acts like an ideal gas we
are one step closer to being able to determine the structure of the Sun. The next thing we
must look at are sources of pressure inside a star. This can come from two possible sources:
radiation pressure and gas pressure.
2.4.1
Radiation Pressure
The interactions of photons with gas particles produces radiation pressure, given by
Prad =
1 4
aT ,
3
(2.1)
where Prad is the radiation pressure, T is the temperature of the gas and radiation in LTE and
a is the radiation constant, with a value of 7.55×10−16 J m−3 K−4 .
1
For many low density space plasmas, e.g., the solar wind, this is not the case, and the electrons and ions have
different temperatures in the same material
2
We will revisit blackbody (or cavity) radiation in Lecture 7, but you are already familiar with Planck’s radiation
law and Stefan-Boltzmann’s law from the Statistical Physics course, L12, Sec 10.
9
2.4.2
Gas Pressure
Gas pressure comes from the interactions of the ions and electrons. This can be calculated
from the kinetic theory of gases, assuming that the plasma behaves like an ideal gas. The
usual version of the ideal gas equation is
Pgas = nkB T,
(2.2)
where Pgas is the gas pressure, n is the number density of particles, kB is the Boltzmann
constant and T is the temperature. In stellar evolution it is usual to express the number density
n in terms of the mean molecular weight. The mean molecular weight is the mean mass of
gas particles in units of the mass of hydrogen, mH . The mean particle mass is thus µmH . The
mass density of the gas, ρ, can then be expressed in terms of the number density, n, and the
mean molecular weight, µ,
ρ = µmH n
⇒
µ =
ρ
.
nmH
(2.3)
The ideal gas equation (our equation of state for this course) can then be rewritten as
Pgas = nkB T =
ρkB T
ρRT
,
=
µmH
µ
(2.4)
where the specific gas constant for hydrogen is given by R = kB /mH ' 8300 J kg−1 K−1 .
2.5
Mean Molecular Weight
The mean molecular weight µ encodes information about the chemical composition of the gas,
which in turn is a function of the density, temperature and chemical abundance. Calculating
µ (i.e., ρ/(nmH )) is generally time consuming and complicated since to find n, the number
density of each species, one needs to calculate the fractional ionisation of all the elements (for
all relevant ρ − T pairings). To simplify matters, we will assume that the gas is fully ionised
for all species so that µ depends only on the chemical abundance. We will further assume that
the plasma consists of hydrogen, helium and a single ‘metal’ representing all elements heavier
than helium3 .
Consider first the total number density for all species in the gas n which is the sum of
the number density of all ions (or nuclei) and electrons. Fully ionised hydrogen contributes
one proton and one electron to the number density; fully ionised helium, contributes a helium
nucleus and two electrons, i.e., three particles. For a ‘metal’ with electron number `, a fully
ionised atom will contribute ` electrons and one nucleus to the number density. The total
number density is hence
n = 2 nH+ + 3 nHe2+ + (` + 1) nmetal`+ .
(2.5)
To calculate µ, we need a way to relate n to the mass density ρ. The chemical composition
of a gas is usually specified in terms X, Y and Z, the mass fractions for hydrogen, helium
and the metals (elements heavier than helium), respectively. X is thus the ratio of the mass
of gas in the form of hydrogen to the total mass of gas, i.e., X = M (H)/Mtot . Given that
the electron mass is much less than that of a proton, the total mass of hydrogen M (H) can
3
Elements heavier than helium are generically referred to as ‘metals’ by astronomers.
10
Sun, Stars and Planets 2020-21
be approximated as the the total number of protons N (H+ ) multiplied by the mass of the
hydrogen atom, mH (we do not need to distinguish between proton, hydrogen or atomic mass
given me mp ). Thus
X = H mass fraction =
M (H)
N (H+ ) mH
n +
'
= H mH ,
Mtot
Mtot
ρ
(2.6)
where we have divided numerator and denominator by the volume for the final step. Similar
arguments apply to helium and metals giving
N (He2+ ) mHe
n 2+
M (He)
'
' He 4 mH ,
Mtot
Mtot
ρ
N
n
`+ mmetal
`+
M (metal)
Z =
' metal
' metal 2` mH ,
Mtot
Mtot
ρ
Y =
(2.7)
(2.8)
where we have used that heavier nuclei contain roughly the same number of protons and
neutrons. Substituting these expressions into Eq.2.5 for the number density yields
n = 2 nH+ + 3 nHe2+ + (` + 1) nmetal`+
Xρ
1
ρ
3 Yρ
` + 1 Zρ
3
' 2
+
+
' 2X + Y + Z
.
mH
4 mH
2` mH
4
2
mH
(2.9)
Since Z contains all the metals of different types we also have
X + Y + Z = 1.
(2.10)
Recalling the definition of the mean molecular weight, µ = ρ/(nmH ), we can thus obtain an
expression for µ. For a pure hydrogen and helium gas (i.e., Z = 0 and Y = 1 − X, see Eq. 2.10)
we find
1
4
ρ
=
=
.
(2.11)
µ=
3
mH n
5X
+3
2X + 4 Y
For the Sun, the values for X and Y at the surface are about 75% and 24% respectively,
giving µ ∼ 0.6. In the Sun’s core the value of Y increases to more than 60% and µ ∼ 0.8.
Exercise: there is a more accurate calculation for µ when Z 6= 0 on problem sheet 1
which you should look at and add to your notes.
2.6
Rough Estimate of Stellar Core Temperatures
Since the equation of state links pressure and temperature, we can use it, together with the
equation of hydrostatic equilibrium (1.3), to estimate the central temperature. From the hydrostatic equilibrium equation, dP/dr = −Gm(r)ρ(r)/r2 , we infer that the central pressure
Pc is related to the total mass M , the radius R and mean density ρ according to
Pc
GM ρ
∼
.
R
R2
(2.12)
Taking the equation of state, P = ρRT /µ with µ ∼ 1 and ρ ∼ ρ yields
Pc ∼ R ρ Tc ∼
GM ρ
R
⇒
Tc ∼
GM
.
RR
(2.13)
11
Eq. 2.13 implies that the core temperature increases for higher-mass stars. The argument
behind this is that ρ is almost independent of stellar mass4 . If this is the case, then M =
(4/3) πR3 ρ implies that R ∝ M 1/3 , i.e., we expect mass to increase more rapidly than radius
so that M/R is larger for stars with a greater mass: the bigger a star is, the higher its core
temperature.
Exercise: Evaluate Tc for solar parameters and compare it to the value of 1.6 × 107 K
derived from a more accurate solar model.
2.7
The Virial Theorem
We are next going to look at another consequence of the equations of hydrostatic equilibrium
and mass continuity that will produce a result that relates thermal kinetic energy and potential
energy. This is known as the (scalar)5 virial theorem, and is usually written as 2hEk i+hEp i = 0,
where Ek is the thermal kinetic energy and Ep is the gravitational potential energy of objects
sitting in a potential well. The virial theorem describes average properties of a system, here
indicated by the angled brackets.
While we come across this result here in the context of gas in a star where we might think
of the thermal kinetic energy being determined by looking at the motions of gas particles in
the stellar plasma, we can also apply it to much larger systems. On galaxy scales, the mass
of elliptical galaxies can be determined by looking at the motions of the stars inside them –
essentially treating stars as individual thermal particles moving around in a potential well.
On still larger scales, the mass of galaxy clusters can be determined via the virial theorem by
treating galaxies as individual particles moving in the cluster potential well6 .
To prove the virial theorem for a system in hydrostatic equilibrium we start with the hydrostatic equilibrium equation expressed in terms of mass (see Eq. 1.12 and recall that the radius
r(m) is a function of mass)
Gm
dP = −
dm.
(2.14)
4π r4
Multiplying Eq. 2.14 by the volume V = 34 πr3 and integrating over the whole star yields
Z
Ps
Pc
V dP = −
Z
0
Ms
4πr3 Gm
1
dm = −
4
3 4π r
3
Z
0
Ms
Gm
1
dm = Ep .
r
3
(2.15)
Indices c and s indicate centre and surface values, respectively; Ep is the gravitational potential
energy of the star. The left-hand side of Eq. 2.15 can be integrated by parts
Z Ps
Z Vs
surface
V dP = [P V ]centre −
P dV = Ps V − hP iV ' − hP iV,
Pc
0
where we have used that the surface pressure Ps is much smaller than the mean pressure, i.e.,
Ps hP i. The equation thus reduces to
1
hP iV = − Ep .
3
4
(2.16)
This is not strictly true; in fact, for main sequence ρ generally decreases as mass increases.
Traditionally, the virial theorem is derived for a stable system of particles bound by a potential and considers
the forces on and positions of the particles. For power-law forces F ∝ r−(n+1) it takes on a particular simple
form, 2hEk i = nhEp i.
6
It was such measurements of the motions of galaxies in the nearest large galaxy cluster to us, the Coma Cluster
that provided the first hints of the existence of dark matter.
5
12
Sun, Stars and Planets 2020-21
This is one representation of the virial theorem and gives the average pressure hP i needed to
support a self-gravitating system with total gravitational energy Ep and volume V .
The virial theorem can be written in a more familiar form using the relation between the
pressure and the (internal) energy density. For an ideal gas, the internal (or kinetic) energy
density u = Ek /V is given by u = P/(γ − 1), where γ is the ratio of the specific heat at
constant pressure to the specific heat at constant volume. For a monatomic gas this has a
value of γ = 5/37 . The expression hP iV in Eq. 2.16 is equivalent to h(γ − 1)uiV = 32 huiV
and we find
2
1
hui V = − Ep
⇒
2Ek + Ep = 0.
(2.17)
3
3
This is the virial theorem, stating that for a system like a star in hydrostatic equilibrium the
negative gravitational energy equals twice the thermal energy.
2.8
The Contraction of a Star
An important result of the virial theorem, Eq. 2.17, concerns the contraction of self gravitating
spheres of gas, like a star, in the absence of any other source of energy.
We can write the total energy of such as sphere as Etot = Ek + Ep (where I have dropped
the angled brackets for the averages). If the star radiates energy away into space then its total
energy must decrease. We use the virial theorem to find an expression for the average total
energy
1
1
Etot = Ek + Ep = − Ep + Ep = Ep = − Ek .
2
2
(2.18)
The total energy of a star is thus negative and equal to half the gravitational energy or equal
to, but of opposite sign, its thermal energy. Thus the loss of (total) energy due to radiation
leads to a decrease in the potential energy, but an increase in the kinetic (or internal) energy. A
star or other body composed of a perfect gas, with no other energy supply, thus contracts and
heats up as it radiates energy away. Remember that Ep is always negative for a self-gravitating
body and becomes more negative as a body of fixed mass contracts.
This perhaps seems rather paradoxical: any attempt to lose energy causes the star to contract and thus release energy at a rate that not only supplies the energy being radiated away
from the surface but also heats the material of the star. This assumes a fully ionised monatomic
gas, with γ = 5/3, but this result applies as long as γ > 4/38 .
Contrast this with a lump of hot metal. This cools, radiating away energy until it reaches
thermodynamic equilibrium with its surroundings. By contrast, a self-gravitating sphere becomes hotter as it collapses and radiates away energy. This does not break the laws of thermodynamics – heat is still flowing from a hotter body (the star) to a colder body (the rest of
the universe). In the next lecture we discuss the process of internal energy generation which
stops main-sequence stars from contracting.
An alternative way to remember these relations is via the kinetic energy per particle which is 23 kB T in a
monatomic gas. The energy density is hence u = 32 nkB T . From the ideal gas equation, P = nkB T , we see that
u = 23 P .
8
Note that γ = 4/3 for a relativistic gas. Thus a photon gas, i.e., radiation, obeys urad = 3Prad = aT 4 .
7
13
Things to Remember
• The definition and use of the equation of state.
• The definition and calculation of the mean molecular weight
• Estimation of the central temperature of a star
• The derivation and use of the virial theorem
Things to Do
• Calculate µ for non-negligible metal abundances, see Q2 on PS1
14
Sun, Stars and Planets 2020-21
Lecture 3
Energy Generation in Stars
To this point, our analysis of stellar structure has been done in the absence of any knowledge
about what might power them. This matches the progress of the historical study of the nature
of the Sun and other stars. In this chapter we will look at possible energy sources for the Sun,
given that we know it has been shining for ∼4.5 billion years with a luminosity (power) of
3.9×1026 W. Like the astronomers of the early 20th century, we will find that an internal source
of energy other than gravitational collapse is required. Unlike our predecessors, though, we
are equipped with a knowledge of modern nuclear physics and so can tell that nuclear fusion,
converting hydrogen into helium, is the source of the Sun’s power.
3.1
The Sun’s Energy Source
Energy Sources and Timescales
Stars lose energy by radiation at a rate L which is the stellar luminosity. We can thus construct
a timescale τ = Eavail /L that reflects how long a particular energy reservoir will last at a given
rate of energy release. Let us first consider such a timescale in the case where the energy is
provided by cooling (i.e., its present internal energy is exhausted) or by contraction.
It follows from the virial theorem that, for a system in hydrostatic equilibrium (such as a
star), the thermal energy is ∼ GM 2 /R (we can neglect factors of 2 here). We can thus define
the thermal (or Kelvin-Helmholtz) timescale as
τth =
GM 2
.
RL
(3.1)
This time period indicates the time needed for a star to passively cool. In the case of the Sun,
τth, ∼ 107 years which is far less than the time over which we know the solar luminosity to
have been constant1 .
1
In the mid 19th century, Kelvin, Helmholtz and others assumed that the Sun’s luminosity was provided by
accretion of, e.g., meteors and subsequent cooling. Despite the name, the timescale resulting from gravitational
contraction was first calculated correctly in the 1880s by Ritter (Shaviv 2008, New Astr. Rev. 51, 803). The contraction
timescale was taken as the best estimate of the solar age, in contradiction to estimates of about a billion years
for the Earth’s age that were put forward by geologists and biologists (e.g., Darwin asserted that the solar age
calculated by Kelvin and Helmholtz was too short for evolution to produce the variety of species). In the early
20th century, most physicists and astronomers accepted that the solar energy source was ‘subatomic’, though the
theory of nuclear fusion, was only developed in the 1930s.
15
Matter to Energy
If the Sun is radiating away neither gravitational or thermal energy then its energy must be
released through the conversion of matter from one form to another. The Sun’s luminosity
is 4 × 1026 W. Using E = mc2 we can calculate that the Sun is converting mass to energy
at a rate of 4.4 × 109 kg s−1 . This has been pretty much constant over the Sun’s life time
of ∼ 4.5 billion years, indicating that mass loss over this time will be ∼ 3 × 10−4 M . This
level of rest mass energy release from the Sun is not possible from chemical reactions, which
can release a maximum of only 5 × 10−10 of the rest mass energy of the reacting material.
At the time these calculations were originally made, around the start of the 20th century, no
known mechanism was capable of this. We now know that nuclear reactions can release a
much greater fraction of the mass energy of the matter taking part in these reactions. Fission
reactions, such as those that power modern nuclear reactors, can release about 5×10−4 of rest
mass energy when heavy nuclei split apart. Fusion reactions where light nuclei join together
can release as much as 1% of rest mass energy.
3.2
Nuclear Fusion
Nuclear fusion is the process by which low mass nuclei combine to form larger mass nuclei.
It is energetically favourable for this to occur since nuclei of atomic mass up to that of iron
(atomic number 56) are more tightly bound than lower mass nuclei. Fusion reactions that
produce nuclei up to and including those of iron are thus exothermic – they release energy.
The most important reactions during the majority of a star’s lifetime are those that convert
hydrogen to helium. This process is known as hydrogen burning, and stars powered by this
process are known as main-sequence stars. Hydrogen burning converts four hydrogen nuclei
(protons) into one helium-4 nucleus (α particle) and releases energy amounting to the restmass difference between the two nuclei. The rest-mass difference between four protons and
an α particle is 0.03 mH . A hydrogen fusion reaction thus releases 0.03 mH c2 (about 27 MeV)
which is 0.03/4 = 0.00775 of the initial mass converted to energy.
For how long could such reactions power the Sun? The nuclear timescale τnuc is given
by the amount of nuclear energy available, ∆M c2 , divided by the rate at which the energy is
radiated away, i.e., the luminosity of the star, L. Thus
τnuc =
∆M c2
0.00775 M c2
=
L
L
(3.2)
for hydrogen burning (and assuming that all of the hydrogen will be converted to helium). For
the Sun we find τnuc ∼ 1011 yr. This nuclear timescale is much greater than the dynamical
(or free-fall) timescale τff (Eq. 1.9) and also far exceeds2 the timescale needed for the Sun to
passively cool from a high temperature, i.e., the thermal timescale, τth (Eq. 3.1).
Fusion reactions take place through a series of nuclear reactions in which colliding nuclei
combine and fragment in a chain of reactions that eventually produce new stable nuclei together with the release of other particles. The difference in the binding energy of the ‘fuel’
and the final products is released and eventually radiated away from the Sun. The released
energy also maintains the high temperature at the core of the Sun, allowing fusion reactions
to continue.
2
You should convince yourself that τnuc τth τff .
16
Sun, Stars and Planets 2020-21
Figure 3.1: The proton-proton chain, with its three branches. From An Introduction to the Sun
and Stars, ed. Green & Jones
There are several routes by which hydrogen can be converted to helium, leading to different nuclear reaction chains. All of these, of course, have to obey the standard conservation
laws of physics: conservation of charge, energy, baryon and lepton number. The two main
reaction chains converting H to He in stars are the proton-proton chain (the pp chain, see
Fig. 3.1) and the carbon-nitrogen-oxygen cycle (the CNO cycle, see Fig. 3.2) in which nuclei
of carbon and nitrogen act as catalysts for the conversion of hydrogen to helium3 . The CNO
cycle requires higher temperatures to operate than are present in the core of the Sun, so for
our own star the pp chain dominates, while for higher-mass stars (and hence higher core temperatures, see Sec. 2.6) H fusion will be via the CNO cycle. The relative importance of the pp
chain and CNO cycle as a function of temperature is shown in Fig. 3.3.
The pp chain has three branches. The simplest of these goes through an intermediate
stage that produces 3 He. This is known as pp branch i. At higher temperatures intermediate
routes going via the production of beryllium and either lithium (branch ii) or boron (branch
iii) become available. In the Sun branch i dominates and is responsible for 85% of the energy
generation. Branch ii produces about 15% and branch iii produces < 1%. As a star’s mass,
3
The details of the pp chain and CNO cycle are not examinable; Figs 3.1 and 3.2 are provided for completeness.
17
Figure 3.2: The CNO (bi)cycle (from
Dina Prialnik’s stellar structure and
evolution book, CUP).
Each fusion reaction of a nucleus and
a proton produces a heavier nucleus
(as indicated) and emits a photon (not
shown). Unstable nuclei decay, emitting positrons and electron neutrinos.
Starting with 15
7 N and following the
right-hand cycle, fusion with a proton results in the production of an α
particle (42 He nucleus) and a 12
6 C nucleus. Subsequent reactions with pro13
tons result in 13
7 N (decaying to 6 C)
14
and 7 N. In the left-hand cycle, the re16
action of protons with 15
7 N yields 8 O
17
and 17
9 F (decaying to 8 O). Further fu14
sion leads to 7 N and the emission of
4 He. Both cycles follow the same path
2
15
from 14
7 N back to the initial 7 N catalyst.
and thus its core temperature, increases, pp branches ii and iii become more important. The
net effect of each pp chain is to convert four protons into a 4 He nucleus. This reaction can be
summarised as
4 11 H −→ 42 He + 2e+ + 2νe + 2γ,
(3.3)
where e+ is a positron, νe is an electron-neutrino and γ is a gamma ray photon.
The rate of energy production is a function of density and temperature4 . Reasonable
approximations for the rate of energy generation rate are5
pp ∝ ρT 4
20
and CNO ∝
∼ ρT .
(3.4)
Energy production rates and thus also stellar properties are very sensitive to temperature. If
you increase the mass of a star by a factor of 10, for example, then its luminosity will increase
by a factor of 104 when the pp chain is operating, and even more dramatically if the star is
massive enough to operate the CNO cycle.
3.3
Stability
The most direct experience of fusion we have on Earth are hydrogen bombs, which clearly are
not stable when they go off, and controlled fusion experiments, which have yet to achieve a
stable state for more than a few seconds. Why does the Sun not simply explode like a bomb?
The Sun is confined by self-gravity, and the solar plasma behaves essentially like an ideal
gas. The Sun is in a state of hydrostatic equilibrium (the inward force of gravity on each layer
is balanced by the net outward force of pressure), meaning that the virial theorem applies.
4
It also depends on the mass fraction of the reactants/catalysts; for the pp chain this introduces a X 2 dependence, for the CNO cycle the dependence is X · (MCNO /Mtot ).
5
Exponents between 16 and 20 can be found in the literature for the temperature dependence of the CNO cycle.
18
Sun, Stars and Planets 2020-21
Figure 3.3: The relative importance of
the pp and CNO cycles as the stellar
core temperature rises. Rates are relative to the solar energy release rate.
From Green & Jones, An Introduction to
the Sun and Stars.
Let us now assume that there were a perturbation leading to an increase in the Sun’s energy production. This increase in total energy leads to an increase in potential energy (recall
that according to the virial theorem Etot = 21 Ep = −Ek ). The Sun would thus expand and
cool. As the energy production rate through fusion depends on a positive power of the temperature, the cooling results in a drop in the energy production rate.
Conversely, if we take a perturbation that leads to a decrease in energy production, the
associated decrease in the total energy results in a contraction of the star that is also associated with an increase in temperature, and hence a ramping up of the energy production. The
perturbations are thus reversed and thermal equilibrium will be recovered6 .
Laboratory fusion experiments try to keep the plasma stably confined using either inertia
or magnetic fields - they use magnetic confinement or inertial confinement. In the case of the
Sun, the outward pressure coming from the heat and energy generated by fusion reactions is
balanced by gravity. Stars are in effect gravitationally confined fusion reactors.
3.4
Luminosity and Energy Generation
The next step is to derive the energy equation that relates the rate of energy generation to
the rate of energy transport. This is effectively the energetic equivalent of the mass continuity
equation. We will assume that a star is spherically symmetric and that energy is transported
solely in the radial direction, r. We further assume that the rate of energy generation in the
star is proportional to the mass at a given position in the star. We set to be the energy
produced per unit time and per unit mass at a given stellar radius.
Consider a thin spherical shell of thickness δr at distance r from the stellar centre. The
luminosity entering and leaving the shell is L(r) and L(r + δr) as indicated in Fig. 3.4. The
difference between the energy crossing the top and bottom boundary of the shell is given by
the energy released in the shell. This assumes that the energy released is not used to heat up
the material or change the volume of this shell which is reasonable under the assumptions
of hydrostatic equilibrium. We have also neglected any changes of the stellar properties with
time. This will be true if the timescale for changes to the nuclear burning processes (e.g., the
supply of hydrogen fuel running out), is much greater than the thermal timescale of the star.
This is the case during the hydrogen-burning phase (and indeed most of a star’s lifetime).
6
The stability depends crucially on the link between the pressure and the temperature. In gases that are supported by degeneracy pressure this link is broken (higher temperature does not lead to an increase in pressure)
and thus the onset of fusion in degenerate matter will typically lead to thermonuclear runaway
19
r+δr
r
L(r)
L(r+ δ r)
Figure 3.4: Propagation and generation of luminosity through a shell of thickness δr. The
luminosity entering the shell at r is given by
L(r), the luminosity emerging at r + δr is
given by L(r + δr).
The amount of energy produced in the shell per unit time is given by the mass of the shell,
δm = 4πρ r2 δr multiplied by the energy production rate per mass, , so that
L(r + δr) − L(r) =
⇒
dL
δr = δm = (4πρr2 δr)
dr
dL
= 4πr2 ρ .
dr
(3.5)
Eq. 3.5 provides one more equation for the internal structure of a star, but it comes at the
cost of introducing two more quantities, and L. We earlier saw how the energy generated
per unit mass and time, , is a function of density, temperature and chemical abundance. In
the next lecture we will look at energy transport throughout the star; this will provide a link
between the luminosity and the temperature gradient.
3.5
Solar Neutrinos – historical aside
The only product of the pp chain fusion reactions that can easily escape the Sun are the neutrinos. These interact with matter so weakly that they can stream freely from the core of the
Sun to the outside. Photons, by contrast, scatter off the ionised plasma in the Sun and take a
considerable time to emerge (see Sec. 4.1. Observing solar neutrinos is thus a direct test of our
models for the internal structure and power source of the Sun. Huge numbers of neutrinos are
produced – 100 billion pass through your thumbnail every second – but their interaction rate
is so low that they are very hard to detect.
The probability of detecting a neutrino is proportional to the square of the neutrino’s energy. The neutrinos emitted by the β-decay of 85 B in the pp iii chain are much more energetic
than other neutrinos emitted in the pp chain and are thus easiest to detect. Their reaction
rates are proportional to T 18 . A measurement of the number of 85 B β-decay neutrinos from
the Sun would thus allow the temperature of the core to be very accurately determined and
allow comparison to theoretical calculations.
In 1964 Davis and Bahcall proposed an experiment to detect solar neutrinos. Their idea
was that a stable nucleus of 37
17 Cl might absorb a neutrino and be converted to an unstable
37
−
argon nucleus through 17 Cl + νe −→ 37
18 Ar + e . By measuring the subsequent argon-37
decay (back to chlorine, with a half-life of 35 d), it would be possible to determine the neutrino
20
Sun, Stars and Planets 2020-21
flux. The reaction rates per chlorine atom are small, so a large amount of chlorine was needed
and the experiment had to done in a place shielded from other particles that might produce an
unstable nucleus. The experiment was eventually set up in a deep mine to provide shielding
from cosmic rays; it 400 000 litres of the cleaning fluid perchloroethylene, C2 Cl4 . At the end of
a typical 80-day experimental run the number of 37
18 Ar nuclei was counted. There were usually
31
only about 50 of these in a tank containing 10 other nuclei.
However, the number of 37
18 Ar nuclei found was only about one third of the number expected. This result, termed the Solar Neutrino Problem, was later confirmed by a Japanese
experiment (and honoured with the 2002 Physics Nobel Prize). Initially it was thought that our
models of the interior of the Sun were flawed, and that some unexpected factor was producing
lower core temperatures than expected. But other, independent studies, using helioseismology - examining the acoustic oscillations of the Sun - showed that our models of the interior
of the Sun were in fact correct.
It turned out that one of the assumptions about the underlying particle physics of neutrinos was wrong. There are three types of neutrino, each associated with a type of lepton, so
we have electron, muon and tauon neutrinos, νe , νµ and ντ . In the then accepted model of
particle physics neutrinos had zero mass, an effect of which is that a neutrino’s type is fixed
when it is produced. Later experiments have shown that there are neutrino oscillations that
imply that neutrinos are not massless and can change flavour, e.g., an electron neutrino can
change into a muon or tau neutrino. This explains why only 1/3 of the expected solar neutrinos
are detected as electron neutrinos.
Things to Remember
• Thermal and Nuclear timescales
• Hydrogen fusion as the power source of main-sequence stars
• The pp chain as the dominant fusion reaction in low-mass stars; the CNO cycle
dominating for higher mass stars
• Stability of the Sun
• The energy equation (dL/dr) and how to derive it
21
Lecture 4
Energy Transport in Stars
In this lecture we examine how energy is released and then transported to the surface. There
are three ways of transporting energy: radiation, conduction and convection. Radiation
and conduction both depend on the collision of energetic particles with less energetic particles,
leading to the exchange of energy. In the case of radiation the energy is carried by photons,
in the case of conduction the energy is carried by particles, electrons and ions.
4.1
Radiative Heat Transport
Radiative heat transport is the transfer of heat by photons.
If photons could stream freely from the centre of the Sun they would reach the surface
in only R /c ' 2 s. In actual fact, the energy released as radiation at the centre of a star
slowly diffuses outwards, with photons being scattered, absorbed and re-emitted in random
directions many times before reaching the surface and escaping. This process can be described
as a random walk.
4.1.1
A Photon’s Random Walk
The temperature at the centre of the Sun is around 107 K, and the blackbody radiation photons
associated with this temperature would be in the X-rays. The light from the Sun’s surface is in
the visual range, corresponding to a temperature of about 6000K. A photon emerging from the
surface of the Sun thus has an average energy about 104 times less than the average energy of
a photon in the Sun’s core. This reduction in energy must come as a result of coupling between
radiation and matter, as the photons diffuse outwards.
During the diffusion process a given photon will travel, on average, a distance defined as
the mean free path, `, before it is scattered or absorbed by matter, and re-emitted in a random
direction. This process matches the statistical problem known as a random walk where for a
given number of steps, N , at each of which a random direction is chosen, the resulting net
displacement is a distance N 1/2 ` from the starting point (see Fig. 4.1a).
The total net displacement for a random walk after N steps with mean distance ` is r =
r1 + r2 + . . . rN . The root mean square of the radial
√ displacement after N steps, i.e., the
2
1/2
2
1/2
average distance travelled, is hr i = hN ri i = N `.
For the Sun, the average mean free path of a photon is ∼1 mm. The typical number of
steps taken to travel from the core to the surface is N = (R /`)2 , and the time for each step
is `/c. The diffusion timescale τdiff , i.e., the time taken for a photon to diffuse to the surface of
22
Sun, Stars and Planets 2020-21
distance
Origin
energy density
r+l
r1
r5
r2
flux
r4
r3
u (r + l)
_1 v u (r + l)
6
r
_1 v u (r − l)
6
r−l
(a)
u (r)
u (r − l)
(b)
Figure 4.1: Cartoons illustrating the concepts of a random walk from the origin (a) and energy
diffusing a distance of ` (b), where ` is the mean-free-path length.
the Sun and escape is
R2
N`
=
∼ 105 years,
(4.1)
c
`c
after which time the photon will have undergone ∼ 1024 scattering events, travelling roughly
1012 times longer than by free streaming. Each interaction with matter redistributes energy
from the photon to the matter until photons and matter are in thermal equilibrium, with the
photon energy distribution characterised by a blackbody spectrum matching the temperature
of the material with which it is in thermal equilibrium. This process is known as thermalisation. It is worth noting that if the mean free path were much larger, then a photon could
move from a region of high temperature to one o temperature without losing any energy.
τdiff =
4.1.2
The Radiation Transport Equation
If heat is being transported by radiation in the diffusive, random-walk process discussed above,
and the mean free path ` is much smaller than the scales on which temperature T and density
ρ vary, how does temperature T change with radius r?
Let us consider how energy diffuses between regions separated by a mean free path distance `. Taking material with energy density u(r) diffusing at a velocity v, only 1/6 of the
velocity leads to travel in a specific direction (think of the number of faces on a cube, which
is 6, only one of which is pointed in our direction of interest, see Fig. 4.1b). The net flux out of
the core is then the difference between the energy diffusing upwards and the energy diffusing
downwards, thus
F =
1
1
1
du
v u(r − `) − v u(r + `) = − v ` .
6
6
3
dr
(4.2)
For photons v = c and u = aT 4 (see Sec. 2.7) and thus
du
dT
= 4aT3
.
dr
dr
(4.3)
Substituting this into Eq. 4.2 and using F = L/(4πr2 ), we find
4 a c ` 3 dT
L
= −
T
.
(4.4)
2
4πr
3
dr
We define the opacity κ = 1/(ρ`) (see Sec. 4.2) and solve for dT /dr to obtain the radiative
transport equation
dT
3κρL
= −
.
(4.5)
dr
16πacr2 T 3
23
photon
photon
electron
nucleus
electron
electron
nucleus
nucleus
unbound
electron
unbound electron
at higher velocity
free−free absorption
nucleus
bound−free absorption
bound−bound absorption
photon
unbound
electron
photon
photon
unbound electron
unbound electron
electron scattering
Figure 4.2: The various different types of scattering process.
4.2
Opacity
The radiative transport equation, Eq. 4.5, relates the rate of energy transport to the temperature gradient and the opacity. The opacity of a material is a measure of its resistance to the
passage of radiation. If we consider the passage of light through a material, then the chance
of a photon being absorbed by that material in traveling a given distance will depend on the
density of the material and some constant, which we call the opacity, κ. If the distance traveled through the material is the mean free path, `, then the probability of absorption is (on
average) 1, leading to 1 = κρ`, i.e.,
1
κ =
(4.6)
ρ`
as defined earlier. The opacity is a function of the material’s density, temperature and chemical
abundance and has units of (area)−1 .
The calculation of the opacity for a stellar atmosphere is a complicated process since the
properties of all atoms and ions present in the star have to be considered. The sources of
opacity lie in the detailed microscopic absorption and scattering responses of the material
in a star to radiation at all wavelengths. In principle, though, there are four basic processes
involved. These are summarised below and in Fig. 4.2.
• Bound-bound absorption occurs when a photon of incident light is absorbed by an atom
or ion through the excitation of a bound electron to a higher energy level.
• Bound-free absorption occurs when a photon of incident light is absorbed by an electron
that then escapes from its parent atom or ion. This is the equivalent of photoionisation
and is, in fact, the most important source of opacity in the Sun.
• Free-free absorption is when a (free) electron or ion gains energy through the absorption
of a photon.
• Electron scattering describes an electron and photon scattering elastically off each other.
Although no energy is exchanged between the particles, and thus this isn’t true absorption, this process does slow down the rate at which energy escapes from a star because
the direction of travel of the scattered photon changes.
24
Sun, Stars and Planets 2020-21
Surroundings
r+ δ r
Figure 4.3: Definitions used to derive the
conditions for convection where a body of
material is transported adiabatically from
one part of the star to another.
The initial density and pressure of the
fluid element and of the surroundings (at
position r) are labelled ρ0 and P0 . The element is then displaced by δr and adjusts
its density and pressure to ρf and Pf . The
pressure and density of the surrounding
fluid at r + δr is P1 and ρ1 .
4.2.1
ρ 1, P1
ρ f , Pf
ρ 0, P0
ρ 0, P0
δr
r
fluid
element
Heat Transport by Conduction
Radiative heat transport refers to heat transferred by photons, heat transport by conduction
refers to heat transported by the individual motions of particles (ions and electrons). The
diffusion equation, Eq. 4.2 applies to conduction just as much as to radiation, though with
different values for the particle speed v, the mean free path, l and the energy density, u. Heat
transport by conduction is negligible in main-sequence stars like the Sun, though it plays a
role for the densest stellar remnants, such as, e.g., white dwarfs.
4.3
Convective Heat Transport
Convection is heat transport through the bulk motion of material.
We know that the temperature at the core of the Sun is several million K, but the surface
has a temperature of only about 6000 K. If the outer parts of the Sun are in radiative equilibrium, which is what we expect, then there must be a temperature gradient to maintain the
requisite energy flow L, given the equation of radiative heat transport (equation 4.5). If the
opacity in these regions, κ, becomes high, or if L is high, then a steep temperature gradient
may be necessary to maintain the energy flow. A steep temperature gradient is potentially
unstable to convection if the gradient is steeper than the temperature gradient that would be
produced by matter rising adiabatically. If this happens then convection will take place.
4.3.1
The Schwarzschild Stability Criterion
Let us determine when an atmosphere is stable against the onset of convection by considering
what happens to a fluid element that is displaced with respect to its surrounding atmosphere.
We assume that the fluid element is embedded in an atmosphere as pictured in Fig. 4.3. At
position r, the density and pressure of the fluid element and the surrounding gas are equal with
P (r) = P0 and ρ(r) = ρ0 . We now assume that the element is displaced upwards sufficiently
slowly that it remains in pressure balance with its surroundings, but quickly enough that no
heat is transferred to its surroundings, i.e., the displacement is adiabatic.
25
At position r + δr, the pressure and density of the surrounding atmosphere are (by definition)
dP
dρ
and
ρ1 = ρ(r + δr) = ρ0 + δr .
(4.7)
dr
dr
The adiabatic change of the fluid element means that its density ρf and Pf at r + δr obey
1/γ
1/γ
Pf
P1
= ρ0
,
(4.8)
ρf = ρ0
P0
P0
P1 = P (r + δr) = P0 + δr
where we have used that the element remains in pressure balance with its surroundings (i.e.,
Pf = P1 ) for the final equality.
If the fluid element at r + δr is denser than its new surroundings it will sink back down.
Stability (against random displacements) thus requires ρf > ρ1 , or
1/γ
P1
dρ
ρ0
> ρ0 + δr .
(4.9)
P0
dr
Inserting the expression for P1 and expanding yields
δr dρ
δr dP
> ρ0 1 +
ρ0 1 +
γP dr
ρ dr
(4.10)
and
1 dP
1 dρ
>
.
(4.11)
γP dr
ρ dr
This is the Schwarzschild stability criterion. A medium that satisfies Eq. 4.11 will be stable
to convection and heat transport will be by radiation.
The Schwarzschild criterion is often expressed in terms of T and P using the equation of
state, P = RρT /µ. As P ∝ ρT , we have ln ρ = ln P − ln T + const. Differentiating the
equation of state and rearranging gives
1 dρ
1 dP
1 dT
=
−
.
ρ dr
P dr
T dr
(4.12)
Substituting this into equation 4.11 we get
1 dP
1 dT
1 dP
>
−
.
γP dr
P dr
T dr
(4.13)
Rearranging and writing this in terms of the absolute value (recall that the pressure, density
and temperature decrease outwards with r and their gradients are thus negative) we find that
regions are stable against the onset of convection when
dT
1 T dP
< 1−
.
(4.14)
dr
γ P dr
The expression on the right-hand side of the equation is the adiabatic temperature gradient. In an atmosphere where the temperature gradient is steeper than the adiabatic temperature gradient, the Schwarzschild criterion will not be satisfied and we will get convection.
Steep temperature gradients (and thus convection) typically arise in cooler regions with high
opacity and in the hottest regions of the core1 .
1
As the criterion also depends on the adiabatic coefficient γ, we also expect ionisation to play a role. Typical
regions where one might expect to see convection is where there is a phase transition between ionised gas and
neutral gas (or also in regions where molecules form). In these cases γ is closer to 1, decreasing the pre-factor on
the right-hand side (and making it more difficult to satisfy the stability condition).
26
Sun, Stars and Planets 2020-21
4.3.2
Heat Transported by Convection
A phenomenological picture of convection is that fluid elements rise a distance lmix and then
merge with their surroundings, where lmix is called the mixing length (which is typically of the
order of the pressure scale height2 ). Convective elements are assumed to rise or fall through a
distance comparable with their size before their excess heat is exchanged with their surroundings.
Convection is a very efficient process for transporting heat and it will be the dominant
process in regions unstable to convection and we can change the Schwarzschild instability
criterion from an inequality to an equality to obtain the heat transport equation. Expressed in
terms of dT /dr (and using the ideal gas equation), the heat transport equation for convection
will be
dT
1 T dP
= 1−
.
(4.15)
dr
γ P dr
Things to Remember
• Heat transport by radiation is a diffusive process
• Photons escape very slowly by a random walk
• The equation for dT /dr if radiation is the heat transport mechanism
• Sources of opacity
• Schwarzschild criterion for convective instability
• The equation for dT /dr for convection
2
The increase in height after which the pressure of an atmosphere decreases by a factor of e.
27
Lecture 5
The Stellar Structure Equations
We now have all the equations needed to describe the (internal) structure of stars. In this
lecture we will discuss these in the context of the Sun’s interior structure.
5.1
The Stellar Structure Equations
We derived four differential equations governing stellar structure. These are the equations of
dP
dr
dm
dr
dT
dr
dT
dr
dL
dr
hydrostatic equilibrium
mass continuity
heat transport
energy generation
= −
Gmρ
,
r2
(5.1)
= 4πr2 ρ,
(5.2)
3κρL
= −
,
2 3
16πacr
T
1 T dP
= 1−
,
γ P dr
(radiation)
(5.3)
(convection) (5.4)
= 4πr2 ρ .
(5.5)
Here we have used the following symbols
T:
ρ:
L:
:
κ:
c:
temperature
density
luminosity
energy generated per second per unit of mass
opacity
speed of light
P:
r:
m:
µ:
G:
a:
pressure
radius
mass
mean molecular weight
gravitational field constant
radiation constant
There are seven dependent variables, though only four differential equations. We thus need
three additional (closure) relations. These are the equation of state (lecture 2), and equations
for the energy generation rate (lecture 3) and opacity (lecture 4).
Equation of state:
1
aT4
3
RρT
=
µ
Prad =
Pgas
for radiation pressure,
(5.6)
for gas pressure.
(5.7)
28
Sun, Stars and Planets 2020-21
Energy generation is given by
(ρ, T, X) ∝ ρα T η .
(5.8)
We can take α and η as constants, though they are in fact slowly varying functions of ρ and
T . For most fuels α=1 while η=4 for hydrogen burning via the pp chain, α is of the order of 16
to 20 for the CNO cycle and even higher for higher-mass burning stages.
Opacity is typically parameterised as
κ(ρ, T, X) = κ0 ρα T β .
(5.9)
The dominant opacity mechanisms and the exponents change depending on stellar mass – we
will revisit this later when we determine the luminosity-mass relationship.
For a star in steady state close to thermodynamic equilibrium, the closure equations will
depend on ρ, T and the chemical composition, X 1 .
5.2
Boundary Conditions for the Stellar Structure Equations and
their Validity
To solve any set of differential equations we require appropriate boundary conditions. Simple
versions of the boundary conditions can be fairly easily derived. The values of m and L will be
zero at r=0, while at the surface of the star, r = R, the mass and luminosity will simply be the
stellar mass and luminosity, i.e., m = M and L = L∗ . We can also set P = T = 0 at r = R.
How valid are such simple boundary conditions? By definition m = L = 0 are clearly
valid at r = 0. But the assumed surface boundary conditions (ρ = P = T = 0) are more
approximate; there is no sharp edge to a star. In the case of the Sun, the density and temperature at the visible surface are estimated to be ∼ 10−4 kg m−3 and ∼ 6000 K, respectively.
Both of these values are much less than their respective mean values, but they are clearly
non-zero. Luckily the solutions to the stellar structure equations for the interior of a star are
not significantly affected by our choice of boundary conditions2 .
5.3
Homology Transformations
The stellar structure equations for a given set of closure relations (5.9 to 5.7)3 are subject to a
homology transformation. This means that given one solution to the equations, with P, T, L, ρ
stated as functions of r for a given total mass M and chemical composition, we can find new
solutions for new masses simply by multiplying the other physical variables by appropriate
scaling factors. A simple example of a homology transformation would be the pressure at the
bottom of a column of bricks. If you know what that pressure is for a column of height h you
can then say that the pressure for a column of height 2h will be twice that value, since the
weight of bricks being support is doubled.
For stars in a mass range where the input physics of are the same, we expect that the solutions to the stellar structure equations will be the same in terms of a dimensionless coordinate
1
We will assume that the chemical composition is fixed and homogeneous throughout the star here.
Note that we set T = 0 at r = R to solve for the stellar interior structure and to obtain the luminosity. We
can then derive an (non-zero) effective temperature for the star from the luminosity and the radius.
3
We will typically also assume that energy transport is through radiation only. Homology generally does not
hold if there is a transition between radiative and convective transport as the radius where this transition occurs
is mass dependent.
2
29
f x = f(r)/f
1
*
0.5
0
0
0.5
1
x = r/R
Figure 5.1: Cartoon illustrating how a homology solution might work. The x-axis denotes the fractional radius x = r/R, so the
core is at x = 0 while the stellar surface
is at x = 1. The blue solid line the functional behaviour for a quantity fx where the
outer boundary condition is zero (e.g., the
pressure). The dashed green line is for a quantity where the boundary condition is 0 at the
centre, e.g., the green line could apply to the
mass or luminosity.
x = r/R, where R is the radius of the star and r is the distance from the centre of the star to
a position within it. We then use
r
x =
(5.10)
R
to give a generic position x within a star4 . The homology solution implies that conditions at a
position r1 /R1 = x would be the same as those in a star with different radius R2 at a position
x = r2 /R2 . We can thus produce a series of curves for the different properties that will give
a solution for any given star once an appropriate scaling is made. A diagrammatic example of
this is shown in Fig. 5.1.
We write each property f (r) in the form f (r) = f∗ fx (x) = f∗ fx , where where fx will
have the same functional form for all stars5 , while f∗ is specific to a particular star. Thus we
define
P (r) = Pc Px ,
ρ(r) = ρc ρx ,
m(r) = M mx
L(r) = L Lx ,
T (r) = Tc Tx
(5.11)
where x = r/R and Pc , ρc and Tc are the core pressure, density and temperature, while M and
L are the stellar (surface) mass and luminosity. We will then obtain a set of (dimensionless)
differential equations for Px , mx , ρx , Lx and Tx that need to be solved if we want to recover
the internal structure of all stars that follow the same homology relations. In parallel, we
obtain a set of scaling relations for ρc , Pc , Tc , L, M and R that prescribe the global properties
of the stars in the homology group.
In practise, the complex nature of some of the closure relations, in particular of the opacity
equation, and the presence of convective energy transport in some stellar layers, means that
the stellar structure equations need to be solved numerically. However, the scaling relations
derived either using the approach outlined above, or simple dimensional arguments (as we
will use in the next lecture) still offer useful insights.
Worked Example – not examinable
The following example illustrates the homology approach for the trivial example of the mass
continuity equation, dm/dr = 4πr2 ρ(r).
4
Usually the stellar structure equations are expressed in terms of mass rather than radius as this is more intuitive when considering stellar evolution. Then x set to be m/M ; the homology argument remains the same
otherwise.
5
Note that I am using fx instead of fx (x) here for clarity and ease of notation
30
Sun, Stars and Planets 2020-21
We define the dimensionless variable x = r/R and rewrite quantities that are functions
of r in terms of scaling factors and functions of x, thus
m(r) = M mx
and
ρ(r) = ρc ρx ,
where the scaling factors and x-dependent functions have been colour-coded in red and blue,
respectively. Note that the scaling factor for the mass is M (i.e., the mass at the outer boundary; at r = x = 0 the mass is zero), while the scaling factor for the density is the core density
ρc (and the outer boundary condition is ρ(r = R) = ρx (x = 1) = 0).
We now re-express the mass continuity equation in the new variables and use dx/dr =
1/R to find
dm(r)
d [M mx ]
dmx dx
M dmx
=
= M
=
= 4π R2 ρc x2 ρx .
dr
dr
dx dr
R dx
We thus find that (as expected!) the central density ρc is proportional to the mass divided by
the cube of the radius. The internal density profile can be obtained from solving the differential
equation
dmx
= 4π x2 ρx .
dx
Departures from Homology
The homology approach to solving the stellar structure equations means that these equations
need only be solved once for the solutions to be known for an entire set of stars of different
mass but the same chemical composition. Nevertheless, there may be deviations from homology scalings even among chemically homogeneous stars. These will occur when some of the
assumptions behind the homology solutions break down. Two obvious flaws in our approach
are
• Radiation pressure becoming significant in comparison to thermal pressure. This will
happen in high mass stars.
• Stars where convection becomes important. This can occur in the core or the outer parts
of the star, or both.
5.4
The Internal Structure of the Sun
Before turning to scaling relations for hydrogen-fusing stars, we look at the internal structure
of the Sun in more detail.
The internal structure of the Sun is sketched in Fig. 5.2. The central region, the core, is
where the bulk of energy generation takes place. Surrounding it is the radiative zone where
energy transport takes place through radiation diffusion (see Sec. 4.1). Beyond that, about 75%
of the way from the centre of the Sun to the surface, the material of the Sun becomes unstable
to bulk motions and convection becomes the dominant energy transport process. The region
where this takes place is called the convection zone.
The plots shown in the following are mainly from the standard solar model by Bahcall,
Serenelli & Basu (2005, ApJ 621, L85); if you are keen to explore solar and stellar structure
further, an easy to use interface for the state-of-the art MESA stellar evolution code is available
at http://mesa-web.asu.edu/.
31
Core
.25
.7
Radiative Zone
Convection
Zone
Figure 5.2: Sketch of the internal structure of
the Sun. The interior structure of the Sun can
be probed using helioseismology which also allows us to recover the internal rotation of the
Sun. Neutrinos act as a probe of the core of the
Sun.
Composition
Throughout most of its interior, the Sun is made up of ∼ 73% by mass of hydrogen, 25% helium
and ∼ 2% of metals (elements heavier than helium), see Fig. 5.3a. Within the core, hydrogen
is increasingly depleted as a result of 4.5 billion years of fusion processes. About half of the
hydrogen originally present in the core of the Sun has so far been converted to helium.
Pressure, Temperature and Density variation
The results of the standard solar model show temperature, pressure and density all increasing
with depth, as we would expect, see Fig. 5.3(b-d). At the centre, where T ∼ 15.6 × 106 K,
the density is predicted to be about 1.5 × 106 kg m−3 (roughly fourteen times that of lead);
the central pressure is 1010 times the atmospheric pressure on the surface of the Earth. As
illustrated on the logarithmic plot on panel (d) of Fig. 5.3, the pressure and density fall off very
steeply near the surface6 .
Luminosity and Energy Generation Rate
The energy generation in the Sun is concentrated in the core. Once outside the core, the energy
generation rate drops to zero and energy is transported to the surface with no additional
energy being produced. At that point the luminosity (see dashed line and right-hand axis in
Fig. 5.3b) reaches the constant luminosity we see at the surface.
Comment: Internal Structure of Other Stars
Main-sequence stars of similar masses to the Sun will have a comparable internal structure,
though the relative size of the convection zone increases for lower-mass stars and shrinks for
higher-mass stars. As we have seen in Lecture 3, hydrogen fusion will be via the CNO cycle
for high mass stars (with higher core temperatures). The steeper temperature sensitivity of
the energy release rate for the CNO cycle leads to the core of more massive stars becoming
convective. High-mass stars thus have convective cores and radiative envelopes.
6
This sort of justifies our earlier use of the boundary conditions (ρR = PR = 0) at the stellar surface
Sun, Stars and Planets 2020-21
hydrogen
0.6
0.4
density [105 kg/m3]
H and He fractions
15
(a)
0.8
(b)
1.0
0.8
10
0.6
0.4
5
helium
luminosity [L(sun)]
32
0.2
0.2
0.2
0.4
0.6
radius / R
0.8
(c)
10
core
radiative zone
0.2
0.4
0.6
radius / R
convective
envelope
5
(d)
0.2
0.4
0.6
radius / R
0.8
1.0
105
104
1014
103
1012
102
101
1010
0
0.0
0.0
1.0
0.8
1016
pressure [Pa]
Temperature [106 K]
15
0
0.0
1.0
0.0
density [kg/m3]
0.0
0.2
0.4
0.6
radius / R
0.8
100
1.0
Figure 5.3: The four panels show (a) solar H (solid line) and He (dashed line) abundance, (b)
density and luminosity, (c) temperature, (d) pressure and density. All quantities are plotted
as a function of radius (in units of the solar radius R ). The data used are from the solar
model by Bahcall, Serenelli & Basu. On panel (b), the density is shown as the red solid line
(left-hand axis) while the luminosity is the orange dashed line and refers to the right-hand y
axis. Panel (d) shows the pressure (left-hand axis, solid red line) and density (right-hand axis,
dashed orange line) using logarithmic scales. The orange dotted line indicates the mean solar
density.
Things to Remember
• The stellar structure equations as a homologous set of equations
• Boundary conditions for the stellar structure equations
• Solar internal structure: core, radiative zone and convection zone
33
Lecture 6
The Stellar Main Sequence
In this lecture we will look at the properties of hydrogen-fusing stars, and how they form
the so-called stellar main sequence. We will see how this naturally emerges from the stellar
structure equations. We will determine how stellar luminosity scales with stellar mass, and
thence predict the main sequence lifetime for stars of any given mass.
6.1
Scaling relations from the Stellar Structure equations
In this section, we will derive scaling relations for the non-zero boundary values of the stellar
structure equations. Specifically, these are the stellar radius R, mass M , and luminosity L∗
as well as the core values for the density ρc , pressure Pc and temperature Tc . Of these the
“observables” M , R and L are of particular interest in later chapters. To derive the scalings,
we will assume that radiation pressure is negligible and that energy transport is by radiation
only.
6.1.1
Density and Pressure at the Stellar Centre
The mass density is mass per unit volume and we thus expect the central density to scale as
the total mass M divided by the cube of the radius
ρc ∝
M
.
R3
(6.1)
To derive the scaling for the central pressure we use the equation of hydrostatic equilibrium
dP (r)
Gmρ
= − 2
dr
r
−→
Pc
M ρc
M2
∝
∝
R
R2
R5
−→
Pc ∝
M2
.
R4
(6.2)
As we are only interested in the scaling we can drop all numerical and physical constants.
6.1.2
Temperature at the Stellar Centre
To see how central temperature Tc scales with M and R we use the equation of state (the ideal
gas equation) along the scalings we found for the central density and pressure above. We will
here assume that the mean molecular weight µ is the same for all stars (i.e., that this does not
depend on M and R). Then
P =
RρT
µ
−→
Tc ∝
Pc
M 2 R3
∝ 4
ρc
R M
−→
Tc ∝
M
.
R
(6.3)
34
Sun, Stars and Planets 2020-21
This scaling is for the core temperature and does not tell us anything about the (observed)
surface temperature (recall that we set the surface temperature to zero for the homology, and
thus also scaling, argument).
6.1.3
The Luminosity - Mass relation
The relationship between mass and luminosity is one of the most important results to emerge
from homology or scaling arguments. We reach it by starting with the equation of heat transport and adopting the general opacity parameterisation introduced in Sec. 5.1, thus
dT
3κρL
,
= −
dr
16πacr2 T 3
with κ(ρ, T, X) = κ0 ρα T β .
(6.4)
This implies
Tc
ρc L∗ ραc Tcβ
∝
R
R2 Tc3
−→
L∗ ∝
Tc4−β R
ρ1+α
c
−→
L∗ ∝ M 3−β−α R3α+β .
(6.5)
The luminosity-mass-radius relation thus depends on the assumed opacity parameterisation.
This is not too surprising as the opacity regulates the ease with which energy can escape.
Unfortunately, there is no single opacity law that is valid at all masses, and for most stars the
dominant mechanism changes throughout the star.
Here we will assume that opacity is independent of density and temperature, i.e., α = β =
0. This prescription is appropriate for electron scattering which holds for the most massive (and
thus hottest) stars1 . For high-mass stars with constant opacity Eq. 6.5 becomes
L∗ ∝ M 3 .
(6.6)
Cooler stars follow a trend that is better described by L∗ ∝ M 4 . For the overall dependence
of L∗ on M a reasonable choice based on observations (see also Fig. 6.1a) is
L∗ ∝ M 3.5 .
(6.7)
This is a key relation that we will refer back to frequently2 .
6.1.4
The Radius - Mass relation
We note that two of the stellar structure equations involve the luminosity; this will allow us
to find R (and hence all other quantities) in terms of the stellar mass alone. Starting with
equation of energy generation, dL = 4πr2 ρ dr with = 0 ρT η , we find
L∗
∝ R2 ρ2c Tcη
R
−→
L∗ ∝
M 2+η
.
R3+η
(6.8)
We thus have two equations involving the luminosity, mass and radius and we can eliminate the luminosity to obtain a mass-radius relation. Here we will take our previous scaling for
1
At intermediate masses, a popular choice is to use Kramers’ opacity (α ∼ 1; β ∼ − 3.5). You will find
plenty of worked examples using this prescription in textbooks (it leads to a very unrealistic mass-radius relation,
but a reasonable mass-luminosity relation). For the outer layers of cool stars (including the Sun) H− , the negative
hydrogen ion, is the main opacity source; in this case α ∼ 0.5 and β ∼ −7.7 which is very different from Kramers’
law.
2
You should memorise L ∝ M 3.5 and be able to reproduce the derivation of Eq. 6.6. I do not expect you to
remember any of the opacity parameterisations – these would be given explicitly.
35
−3
− 3/4
− 3.5
(a)
(b)
Figure 6.1: Overplot of derived scaling relations (in colour) and observations (original data
plots taken from G. Torres et al. (2010) A&ARev. 18, 67). On both plots the x axis shows stellar
mass (in units of M ) decreasing to the right. On plot (a), the y axis indicates log(L) in solar
units; filled symbols are for stars on the main sequence. The magenta line has a slope of
−3.5 as adopted for the average mass-luminosity relation, L ∝ M 3.5 ; the dashed line is for
L ∝ M 3 . Plot (b) shows the stellar radius in units of R ; the blue line indicates the derived
R ∝ M 3/4 relationship; the black dashed line is the theoretical mass-radius relation from
Girardi et al. (2000), A&AS 159, 371.
L∗ from the radiation transport equation with the constant opacity (recall that this was only
valid for constant opacity and strictly speaking is thus only appropriate for high-mass stars3 ),
i.e., L∗ ∝ M 3 . We note that the energy generation through the CNO cycle and the pp chain
have a very different temperature dependence, we now need to consider the two regimes that
arise due to different values of η.
Low-mass stars with η = 4. Substituting this as well as our earlier scaling relations (Tc ∝
M/R, ρc ∝ M/R3 ) into Eq. 6.8 and equating the luminosities we get
M 3 ∝ R3
M2 M4
R6 R4
⇒
R ∝ M 3/7 .
(6.9)
High-mass stars with η = 16 have4
R ∝ M 15/19
or approximately R ∝ M 3/4 .
(6.10)
A comparison between these estimates and observations is shown in Fig. 6.1. The agreement
for high-mass stars is surprisingly good. For intermediate-mass stars that is indeed an initial
flattening, but low-mass stars show a scaling close to R ∝ M .
6.2
Stellar Surface Temperature
Now that we have an expression for the luminosity and radius of the star, we can also estimate
the surface temperature of a star (that we had set to zero in the original homology argument).
3
The mass-luminosity relation is steeper for lower-mass stars, though not as steep as is often quoted from a
derivation using Kramers’ opacity. As an exercise, you might want to derive the mass-radius relation for low-mass
stars in the case where L∗ ∝ M 4 .
4
Taking η = 17 (as in the recorded lecture) yields R ∝ M 0.8 for high-mass stars.
36
Sun, Stars and Planets 2020-21
We will do this by assuming that the star radiates as a blackbody. From statistical physics, we
know that the flux F emitted by a blackbody of temperature T is given by Stefan-Boltzmann’s
law, i.e., F = σT 4 . For a blackbody with radius R, the luminosity is thus
L∗ = 4πR2 σT 4 .
(6.11)
We can then use the earlier proportionality relations to find R in terms of the luminosity and
hence derive a theoretical luminosity-temperature relationship.
In Lecture 10 we will compare such a theoretical estimate to observations when we consider
the Hertzsprung-Russell (HR) diagram; this is a plot of log(luminosity) against log(effective
temperature) – the concept of effective temperature will be introduced in Lecture 7.
Exercise: Use mass-radius relations derived in the previous section (Eqs 6.9 and 6.10) to derive a scaling for the luminosity with blackbody temperature. Adopt L ∝ M 3 for the massluminosity equation as this was also used to derive the mass-radius relationships.
6.3
Lifetime of a Main Sequence Star
The main sequence corresponds to the hydrogen-burning life of a star. The total amount of
energy available in this phase is proportional5 to the mass of the star, M . The rate at which
this energy is radiated away (i.e., the luminosity) determines the lifetime of a star. As the
luminosity L∗ ∝ M 3.5 , the scaling for the main-sequence lifetime τMS of a star is given by
τMS ∝
energy
M
∝
L∗
L∗
⇒
τMS ∝ M −2.5 .
(6.12)
Massive stars are thus the quickest to exhaust their fuel supplies and leave the main sequence.
Exercise: Estimate the main-sequence lifetime of a 5M and a 0.5M star given that the
Sun’s lifetime is approximately 10 billion years.
Things to Remember
• How to obtain stellar scalings from the stellar-structure equations
• The form of the main sequence and its physical origin
• Luminosity increases with mass on the main sequence
• Massive stars have shorter main-sequence lifetimes
To Do
• Derive a scaling for luminosity with the blackbody temperature as suggested in
Sec. 6.2
5
We have assumed here that all stars fuse about the same fraction of their original mass. This is not strictly
true, but the effect is relatively small given the level of approximation.
37
Lecture 7
Stellar Spectra
The last several lectures have concentrated on the theory of how the Sun and other stars work,
but this theory was derived from observations of the light emitted by the Sun and other stars.
In this lecture we will look at some of the emission processes to try and see how they are used
in understanding the properties of stars.
7.1
Thermal Radiation
Most of the radiation we see from the Sun is from the (lower) photosphere. In this part of the
Sun the photons are (almost) in thermodynamic equilibrium with the matter. The resulting
spectrum is close to a blackbody spectrum (see Statistical Physics Notes, Sec. 10). This means
that the intensity distribution is given by Planck’s function (with units of power per area, per
solid angle and per frequency bin)
Bν (T ) =
2hν 3
1
,
c2 ehν/kB T − 1
(7.1)
where h = 6.6 × 10−34 J s−1 is Planck’s constant, kB = 1.4 × 10−23 J/K−1 is Boltzmann’s
constant, c is the speed of light and ν and T are frequency and temperature, respectively.
When expressed in terms of wavelength, it is given by
Bλ (T ) =
1
2hc2
.
λ5 ehc/λkB T − 1
(7.2)
The peak of the Planck function obeys Wien’s displacement law which is gives the peak wavelength1 λmax of a blackbody Bλ as a function of temperature2
λmax T = 2.898 × 106 nm K.
(7.3)
To obtain the flux (power per surface area) emitted from the surface of a blackbody we integrate over solid angle3 and find that the (monochromatic) flux is given by Fν (T ) = πBν (T )
The equivalent expression for the peak frequency is νmax T −1 = 5.88 × 1010 Hz K−1 .
A useful aide memoire for this is to remember that human body temperature, at about 300K, leads to a peak
blackbody spectrum at 10µm wavelength. This is why the thermal imagers used by the police and emergency
services work in the mid-infrared band, at about 8-14µm wavelength.
3
In the integral, a factor of 21 arises due to considering outward radiation only, an additional factor of 12 is due
to the cosine dependence that accounts for tilt between the surface normal and the line of sight (Lambert’s cosine
law).
1
2
38
Sun, Stars and Planets 2020-21
Figure 7.1: Monochromatic fluxes (πBν and πBλ ) for blackbodies at typical stellar temperatures. The left-hand plot shows the monochromatic flux per unit frequency bin, the right-hand
plot is for wavelength units. The solid black line on the right-hand graph shows the Sun’s spectrum.
and Fλ (T ) = πBλ (T ) for the frequency and wavelength distributions, respectively. Fig. 7.1
shows the blackbody fluxR distributions
R for a range of typical stellar temperatures.
The total flux F = Fν dν = Fλ dλ represents the power emitted per unit area of a
blackbody. It is given by the Stefan-Boltzmann law
F = σT 4 ,
(7.4)
where T is the temperature of the blackbody and σ is Stefan-Boltzmann’s constant. Note the
steep increase in flux with increasing temperature (as can also be seen by the steep increase in
the area under the curves plotted in Fig. 7.1. The total power emitted by a blackbody, i.e., the
luminosity of the object, will be the power emitted per unit area multiplied by its total area.
Thus for a spherical blackbody emitter of temperature T and radius R, its luminosity would
be L = 4πR2 σT 4 .
Blackbody radiation has its origin in the internal, i.e., thermal, energy of the source of
the emission. Such sources are called thermal sources of radiation4 . For a radiation source to
be thermal the photons that are eventually emitted in a blackbody spectrum must be more
likely to interact with the matter of the source than to escape. When they do escape this will
only happen after a considerable number of interactions internal to the material. A common
feature of sources of blackbody radiation is that they are opaque. We see this quite clearly
in the Sun, for example, but it is also the case that many astronomical sources that produce
continuous spectra can be reasonably approximated by a blackbody.
Effective Temperature and Stellar Colours
Real stars are only approximate blackbodies and their spectra spectra deviate from the Planck
function (see, e.g., Fig. 7.1 for the Sun). We can, however, define a stellar effective temperature,
Teff , by comparing the luminosity of the star to that of a blackbody. For a star with luminosity
L∗ and radius R∗ its effective temperature is defined by
4
L∗ = 4 πR∗2 σTeff
.
4
(7.5)
There is also non-thermal emission, such as, e.g., synchrotron emission emitted by charges accelerated in a
magnetic field. We shall not discuss these further in this course.
39
Figure 7.2: Part of the solar spectrum showing absorption lines (in black) superimposed on
blackbody emission.
This means that as the effective temperature of a star rises, its luminosity rises very rapidly.
From Wien’s displacement law we know that the peak of the blackbody spectrum depends
on the temperature. A measurement of the wavelength of this peak can determine the temperature of the emitting source. This is one way (rather approximate way) to measure the
temperatures of stars.
For the solar photosphere at a temperature of ∼6000 K, Bλ peaks in the visible part of the
spectrum at a wavelength of about 500 nm, in what we would call the yellow part of the spectrum. This is why we see the Sun as yellow. The visible covers the range from violet at about
400 nm to red at about 700 nm5 . Cooler stars appear redder, with their blackbody radiation
peaking at longer wavelengths, while hotter stars appear bluer, with their blackbody spectrum
peaking at shorter wavelengths. This is actually rather difficult to see with the unaided eye in
the night sky since the most light-sensitive cells in the eye, the rods, are in fact insensitive to
colour. If you can get to look through a reasonable size telescope, though, stars start to appear
to have the most brilliant jewel like colours, including deep reds and vibrant blues.
Exercise: A star has a luminosity five times that of the Sun and from its spectrum it has an
effective temperature of 30000 K. What is its radius? At what wavelength does its radiation
peak?
7.2
Absorption Lines
Observations of stellar spectra do not show a perfect blackbody. Instead we see a blackbodylike spectrum onto which a series of absorption lines are superimposed. These arise because
of the presence of cooler material in the star’s outer atmosphere, i.e., the layers above the
region where the continuum emission is produced. Light passing through this material may
be absorbed at wavelengths that would put electrons into excited states in its constituent
atoms (see Fig. 7.3). This cuts out light from the otherwise thermal spectrum at the specific
wavelengths of these transitions.
7.2.1
Energy Levels and Transitions
A photon passing through a medium can be absorbed if its energy matches the difference
in energy between two energy levels in an atom in that medium (bound-bound transition),
or if it is of high enough energy to ionise the atom (bound-free transition). For bound-bound
absorption where an electron moves from a lower energy state En to a higher energy state En0
(see Fig. 7.4), the energy of the photon, hν, must equal the energy difference ∆E = En0 − En
between the energy levels, i.e., hν = ∆E.
5
This corresponds to 4000 Å to 7000 Å if one uses Ångstroms which are traditional units still used by a lot of
astronomers; 1Å= 10−10 m).
40
Sun, Stars and Planets 2020-21
Figure 7.3: The process by which absorption lines are imposed on the underlying blackbody
7.2.2 Energy levels and transitions
spectrum of the photosphere by the stellar atmosphere above it (taken from An Introduction
A photon can be absorbed if its energy matches the difference between two energy lev
to the Sun and Stars edited by S.F. Green and M.H. Jones).
atom. An electron is excited to a higher lying energy level or atom is ionized:
Photon of energy h is absorbed:
Figure 7.4: A photon of energy hν =
∆E = En0 − En is absorbed in exciting an
electron from one energy level to another.
---------------------------------- En’
h = ΔE
------------------------------------ En
ΔE = hc/, =hc/ΔE, h = Planck Constant (Js), = wavelength, ħ =h/2, me(e- mass)= 9.11 x
oenergy
= electric
of free
The
levelspermittivity
En of hydrogen,
thespace.
most abundant element in stellar atmospheres, are
given by
For hydrogen:
2
me e2 /4π0
1
hc R
1
En = −
= − 2 = − 2 13.6 eV,
(7.6)
2
2
2~
n
n
n
where ~ = h/2π, me is the electron mass, 0 is the permittivity of free space and e is the
2 (J) = -13.6/
2 (eV)
= -h c R R
/ n=
chargeOr
on anEnelectron.
1.097 × 107 m−1 nis the
Rydberg constant, and in4the last2step
we
where
constant,
R = mhydrogen
e /(8 oand
h3 so
c),isand 1eV=1.6 x
have converted to electron volts. 13.6
eV isthe
theRydberg
energy required
to ionise
known as the hydrogen ionisation potential.
The most prominent hydrogen lines are from lower-level transitions. Fig. 7.5 shows some
of the transitions and the corresponding wavelengths from the first five energy levels for hydrogen, from n = 1 to n = 5. Different series of absorption (or, in appropriate circumstances,
emission) lines are produced when electrons are excited (or de-excited) to different levels. For
example, lines produced by excitations from the n = 1 state give rise to the Lyman series of
lines seen in the ultraviolet (e.g., Lyman-α is seen at a wavelength of 121.6 nm and comes from
the n = 1 to n = 2 transition), while transitions from the n = 2 state give rise to the Balmer
series, which are seen in the optical part of the spectrum. The most prominent of the Balmer
lines is called Hα at a wavelength of 656.3 nm; it comes from the n = 2 to n = 3 transition.
41
Balmer
series
Hα
Hβ
Ly α Ly β
Ly γ
Hγ
Lyman
series
Ly δ
n=1
n=2
Hα 656 nm – red
n=3
n=4
Hβ 486 nm – blue
n=5
Hγ 434 nm – blue
E= -h c R / n2 (J)
Hδ,Ηε 410, 397 nm violet
= -13.6/ n2 (eV)
Figure 7.5: The first five energy levels of hydrogen and some of the spectral lines they produce. The Lyman series of lines result from photons being excited from the n = 1 state and
are found in the ultraviolet, while the Balmer series are found in the optical and result from
electrons being excited form the n = 2 state. These lines can also be seen in emission when
the circumstances are appropriate.
7.2.2
The Occurrence and Strength of Lines
The presence and strength of any given absorption line of any given species in the spectrum
of a star depends on a number of things
• The amount of the element present (ie, the relative abundance)
• the probability that an electron is in the appropriate energy level (this depends on temperature; the higher the temperature the more likely that energy levels corresponding
to higher energies will be occupied)
• the probability that a photon of a given energy will be absorbed
A study of the absorption lines in the spectrum of a star provides information on the chemical
abundances and temperature of the star, and hence on its density and pressure. The observation of stellar spectra is thus a crucial tool in understanding their properties.
The number of different lines observed depends on the complexity of an atom or ion’s energy level structure. That of hydrogen is quite simple. Helium is more complicated, while iron
group elements can have hundreds of energy levels leading to many thousands of transitions.
Astronomers use very large databases of atomic and molecular data to identify and analyse
the transitions seen in stellar spectra. These databases are based on laboratory and observed
spectra and on theoretically calculated values, and may amount to many millions of lines of
data.
7.2.3
Uses of Stellar Spectral Analysis
The examination of the spectra of stars has many uses, as hinted at above. These include
the the classification of the spectral type and luminosity class of a star (see below); the measurement of photospheric chemical abundances; the measurement of radial velocities of a star
42
Sun, Stars and Planets 2020-21
Figure 7.6: The strength of
various absorption lines as a
function of photospheric temperature and spectral type
(from An Introduction to the
Sun and Stars edited by S.F.
Green and M.H. Jones).
from the Doppler shift of a spectral line; the measurement of stellar rotation from the additional line broadening; the measurement of mass inflow or outflow from asymmetries in line
profiles, and the measurement of photospheric magnetic fields through the Zeeman effect.
7.3
Spectral Classification of Stars
The first step in understanding the physics of stars is to systematically classify them into
different types. This is the astrophysical equivalent of taxonomy in biology, which was the
first step to understanding the evolution of different species. The original classification system
used spectral characteristics to subdivide stars into different types that are given the names O,
B, A, F, G, K and M for normal ‘main-sequence’ stars (see Table 7.1)6 , with O being the hottest
and M the coolest. In recent years these have been joined by Y, T and L spectral types, which
are used for brown dwarfs, substellar objects with masses too low to sustain hydrogen fusion.
The letter classification is supplemented by numerical subclasses, (usually) ranging from
0 to 9, with 0 being the hottest. An F9 star and a G0 star are thus quite similar. The presence
and strength of different absorption lines in a stellar spectrum can be used to determine which
species (atoms, ions or molecules) are a dominant or significant source of opacity (see Fig. 7.6).
This allows the temperature of the stars to be roughly determined. In simple terms, hot stars
will show hydrogen and helium absorption lines, while cooler stars will show more lines from
neutral atoms and molecules since these species can only survive in cooler stellar atmospheres.
The Sun is a G2 star while Rigel is a B8 star. From the spectral type we can determine a
star’s average photospheric temperature to an uncertainty of about 5%. There are a number
of mnemonic phrases that can be used to remember the OBAFGKM sequence.
7.4
Luminosity and Luminosity Classification Systems
As well as having different photospheric temperatures, stars can also have different luminosities. A cool star with a large radius, for example, might have a higher luminosity than a hot
6
This method of classifying stellar types based on their photospheric temperatures as derived from their spectral
properties was devised by Annie Jump Cannon at Harvard College Observatory and applied to the Henry Draper
catalog of 400 000 stars from 1918 to 1924.
43
Table 7.1: The spectral classification of stars into stellar types.
Type
O
B
A
F
G
K
M
Colour
Blue
Blue-white
White
Yellow-white
Yellow
Orange
Red
Teff (K)
>25000
11000-25000
7500-11000
6000-7500
5000-6000
3500-5000
<3500
Main characteristics
He+ lines; strong UV
Neutral He lines
Strong H lines
Weak metal lines
Sun-like spectrum
Metal lines dominate
Molecular bands noticable
Examples
Mintaka
Rigel, Spica
Sirius, Vega
Procyon
Sun, Capella
Arcturus, Aldebaran
Betelgeuse, Antares
star with a small radius. We thus need a classification system for luminosity in addition to the
OBAFGK temperature classification system described above.
The width of a spectral absorption line in the spectrum of a star is affected by luminosity.
A large star has lower density in its outer layers than a smaller star of the same temperature
since the mass of the outer layers is spread over a larger volume. The higher the density, and
thus pressure, in a stellar atmosphere, the more frequent are the interactions between atoms,
ions and molecules. These interactions lead to distortions in the energy levels of the interacting atoms so that the affected species can interact with a wider range of photon wavelengths.
The absorption lines thus get broader in higher pressure and density environments in a process
known as collisional or pressure broadening. The increased rate of interactions in higher density stellar atmospheres can also lead to recombination of ionised species, resulting in weaker
absorption lines in lower luminosity stars at the same temperature.
These effects lead to the following result: at a given temperature, larger, higher luminosity
stars have narrower spectral absorption lines and stronger absorption lines for certain ionised
species, while smaller, lower luminosity stars at the same temperature have broader and, for
some ionised species, weaker absorption lines. The correlation of luminosity and absorption
line width and line strength is calibrated using stars of known distance, and thus known luminosity. This is then applied to larger samples of stars whose distance is not known. The
correlation is not precise, but it allows stars to be classified into different luminosity classes,
allowing us to distinguish different stellar types, and to compare relative luminosities.
The most luminous stars have luminosity classes Ia and Ib; these are supergiants, evolved
stars with very inflated radii (∼ 100R ). Luminosity classes II and III apply to giant stars, IV
are subgiants, and V, finally, are dwarf stars which is the term used for all main sequence stars
regardless of their mass. (so even a 20M star on the main sequence is termed a dwarf star).
The full designation of a star’s classification thus includes its spectral type and its luminosity
class. The Sun is described as G2V, a G2 dwarf, while Betelgeuse is M2 Ia – an M2 supergiant.
Things to Remember
• Thermal radiation: Planck function, Stefan-Boltzmann and Wien’s displacement
laws
• Stellar absorption lines determined by energy levels in electrons of atoms / ions
• The origin of the Balmer and Lyman series in Hydrogen
• Stellar spectral classification
44
Sun, Stars and Planets 2020-21
Lecture 8
The Sun’s Atmosphere and Spectrum
The surface of the Sun as we see it is called the photosphere, and while most of the radiation
is produced in the photosphere, there is a much larger region extending further away and
merging into the solar wind that can can be thought of as the atmosphere of the Sun (see
Fig. 8.1). Other stars also have similar properties, leading to the concept of stellar atmospheres.
8.1
Atmospheric Structure
The Sun’s atmosphere is composed of four main layers: the photosphere, chromosphere, transition region and the corona. Fig. 8.1 shows a model for the Sun’s temperature structure along
with a rough indication of where the different layers are located. The upper layers of the atmosphere are very dynamic and show large spatial variations; the location of the transition
region can vary by a few hundred km and the temperature in the corona can reach 2 million
K in some coronal regions.
8.1.1
The Photosphere
The light we see from the Sun comes from the photosphere (‘sphere of light’). This is a layer
of the stellar atmosphere just 500 km thick - very thin compared to the size of the Sun. In
physical terms the photosphere is more like an atmospheric layer than the surface of the
Earth, and defining where exactly the ‘surface’ is located is not obvious1 . The photosphere
has quite a low density – about 1000 times less dense than the Earth’s atmosphere at sea level.
Its temperature decreases from about 9000 K at the lower boundary to about 4500 K at the top
of the photosphere. We usually quote the ‘effective temperature’, though radiation is emitted
over a range of temperatures (see Chap. 7)
The temperature decreases outwards in the photosphere down to about 4500 K at the socalled temperature minimum which marks the transition to the chromosphere. It is due to the
cooler photospheric layers that we see the plethora of absorption lines in the Sun’s spectrum.
8.1.2
The Chromosphere
The chromosphere is a region extending from the top of the photosphere about 500 km above
the convective region of the Sun, to about 2000 km above it; it is thus about 1000 to 2000 km
thick, though as Fig. 8.1 indicates, its extent and and temperature can vary. It can only be easily
seen from Earth during a total eclipse when the Moon blocks out the light of the photosphere
1
There are similar issues with defining the surface of gas giant planets like Jupiter and Saturn.
45
Figure 8.1:
Model of the
solar atmosphere showing
temperature as a function of
height above the solar surface.
Some of the more prominent
upper-atmosphere diagnostic
lines are indicated. Taken from
Yang et al (2009) A& A 501, 745.
Figure 8.2: An image of part of the solar chromosphere obtained during the 2002 total eclipse
in Australia. Credit: EuroAstro
and the chromosphere appears as a narrow region with a reddish colour Fig. 8.2). This eclipse
photograph also shows that the chromosphere is inhomogeneous and structured; in general,
as we go to higher and less dense atmospheric layers, the role of magntic fields in structuring
the atmosphere becomes more important.
The faint glow of the chromosphere (’sphere of colour’) comes from the emission spectrum
of gas in the solar atmosphere. In the hot, low density material of the chromosphere, energy continues to be transported by radiation. Hydrogen atoms in the solar atmosphere that
absorb this energy then re-radiate at the specific wavelengths of spectral lines, as electrons
decay from the transitions made during bound-bound radiation absorption. In the visible, the
predominant emission line is the Balmer Hα line at 656.3 nm, which gives the chromosphere
its red colour. Other lines are also present from other species, the most common of which is,
of course, helium. It was through spectroscopic observations of the solar chromosphere that
helium2 was first discovered, during an eclipse in 1886, before it was found on Earth in 1895.
2
… and named after the Sun, ‘helios’.
46
Sun, Stars and Planets 2020-21
Figure 8.3: An image
of the solar corona,
extending far beyond
the disc of the Sun.
8.1.3
The Transition Region and Corona
Above the chromosphere is a very thin layer roughly 100 km thick across which the temperature of the solar atmosphere rises very steeply from about 20 000 K in the upper chromosphere
to roughly a million degrees in the solar corona (see Fig. 8.1). This region is called the transition
region3 .
The corona is the outermost layer of the Sun’s atmosphere. It gets its name form the crownlike appearance that can be seen during a solar eclipse where parts of the corona appear as
long streamers stretching away from the Sun, as can be seen in Fig. 8.3. The corona spans a
huge range of radii and stretches far beyond the visible disc of the Sun. Particles from the
solar corona in fact reach Earth. As well as being extensive, the hot coronal gas (∼ 106 K)
is very tenuous. The extended faint coronal emission seen during an eclipse or when using
a coronograph (a device that covers the light of the Sun with an opaque disc, simulating the
effect of an eclipse) is mainly due to scattered light: photons emitted by the photosphere scatter
off free electrons in the corona (Thomson scattering). The lower layers of the corona produce
X-ray and EUV emission lines. An image taken with SDO/AIA at 19.3 nm (due to Fe11+ ) is
shown in Fig. 8.4.
The low density in the corona means that it does not emit as a blackbody since it is not
opaque. We can still get an idea of typical photon energies by equating the average photon
energy hν to the thermal energy ∼ kB T . As coronal temperatures are of the order of one
million Kelvin or more, we infer that most of the corona’s emission is at X-ray wavelengths.
The Sun’s magnetic field that permeates the corona shapes its appearance.
3
The large temperature rise is unexpected, and the question of how to heat the corona, the tenuous hot layer
above the transition region, is still unanswered. It is thought that at least some of the heating is due to magnetic
fields being twisted by motions in the photosphere/convection zone, and then releasing the stored energy in the
corona.
47
Figure 8.4: Image of the Sun’s pole
taken with SDO/AIA in the Fe xii line
at 19.3 nm and showing a very large
coronal hole. Also visible are X-ray
bright points and active-region loops.
The Solar Wind4
The influence of the Sun spreads far beyond the regions of the corona that can easily be seen
by optical observations. Particles from the Sun stream out along solar magnetic field lines
into interstellar space, forming the solar wind. The solar wind transports particles at speeds
of a few hundred km/s, but with very low densities (at Earth the number density of protons in
the solar wind is usually about 7 × 106 m−3 ).When some of these particles become trapped by
the Earth’s magnetic field they can excite the atoms they interact with, producing the light
displays known as aurorae (or auroras).
The dynamic solar atmosphere
The Sun’s upper atmosphere is shaped by the magnetic field and can undergo rapid changes.
Some of the magnetic phenomena that can affect the Earth directly include solar flares and
coronal mass ejections.
Solar Flares are rapid bursts of electromagnetic radiation of duration 100–1000 s, with as
much as 1025 J released in this time. The radiation emitted by flares is predominantly in the
X-ray and extreme UV.
Coronal Mass Ejections (CMEs) are violent ejections of coronal material. Typically 5 ×
1012 − 5 × 1013 kg of material is ejected at temperatures ∼ 107 K. CMEs are often associated
with solar flares. The rate of CMEs is typically one a day, though this rate rises and falls with
the solar activity cycle. CMEs probably result from a rapid reconfiguration of the magnetic
field in lower parts of the corona. Sometimes CMEs hit the Earth producing both spectacular
auroral events but also having an impact on everything from orbiting satellites to electrical
power grids.
8.2
The Solar Spectrum
The Sun’s spectrum is shown in Fig. 8.5. It appears largely as a blackbody spectrum but with
a number of absorption features (in the visible and near-UV) and emission features (mainly
in the far UV and X-ray range) superimposed upon it. Radiation above ∼200 nm is predominantly formed in the photosphere. The portion below ∼400 nm where the flux is below the
blackbody spectrum (dotted line) originates from the cooler upper photospheric layers where
the gas is less transparent to radiation. Due to the temperature inversion above the temperature minimum (see Fig. 8.1), we can see a plethora of chromospheric and coronal emission lines
for wavelengths below about 160 nm. The Lyman continuum and the distinct emission lines
between about 40 nm to 160 nm are mostly of chromospheric origin. Emission below 40 nm
originates in the transition region and corona.
4
This section (indicated by the gray font) is not examinable
48
Sun, Stars and Planets 2020-21
Figure 8.5: The solar
flux spectrum.
The
dotted line shows a
Planck function with
temperature
5780 K.
The spectrum shown in
black was taken when
there was little sign of
solar activity; the red
spectrum was taken
two weeks earlier when
three very small active
regions were present.
Data from Tom Woods,
lasp.colorado.edu/lisird.
The solar spectrum varies with time thanks to the solar cycle and the influence of sunspots
and associated phenomena (see Sec. 8.3. Indeed, the spectrum plotted in red in Fig. 8.5 was
taken 2 weeks before the spectrum shown in black.
8.3
Solar Activity and The Sun’s Magnetic Field
Sunspots
Sunspots are small features on the surface of the Sun that appear as dark spots. All sunspots
are characterised by higher magnetic field strengths than are found elsewhere in the photosphere (see Fig. 8.6). These intense magnetic fields disrupt convection, reducing the rate of
heat transport in the sunspot. Sunspots form in pairs that are linked by a loop of magnetic
field lines which arch above the photosphere before returning to re-enter the Sun at the second
sunspot in the pair.
Sunspots are indicators of solar activity which follows a roughly 11 year cycle from low
numbers of sunspots, and thus low activity at solar minimum, to a peak at solar maximum,
and then back down to solar minimum. The first sunspot sightings were recorded in China
and we have systematic records dating back to ∼ 1600. Sunspot numbers and distributions are
shown in Fig. 8.8. The top panel depicting the distribution of sunspots is called the ‘butterfly’
diagram and shows that sunspots emerge in two latitudinal bands. Their emergence latitude is
around ±20◦ at the beginning of a cycle before migrating to lower latitudes later in the cycle.
Sunspots are parts of active regions and are accompanied with other magnetic features
such as faculae and plage formed by smaller-scale flux concentrations. A close-up of an active
region is shown in Fig. 8.7. The smaller-scale magnetic features are slightly brighter than the
average solar photosphere. As their area and number increases much more dramatically than
the sunspot area during a solar maximum, the total solar flux increases by about 0.1 % between
solar activity minimum and maximum.
49
Figure 8.6: A diagram of the magnetic fields associated with sunspots (NASA).
The magnetic activity cycle
Observations of the Sun reveal changes in the number of sunspots, in the occurrence of solar
flares and coronal mass ejections, as well as subtle changes in the Sun’s spectral emission.
All of these changes happen over a roughly 11-year long solar activity cycle. The underlying
driver of the activity cycle is the Sun’s magnetic field that renews itself on an 11-year time
scale. In fact, the solar field switches polarity after 11 years, so a full magnetic cycle takes
about 22 years.
The regeneration of the Sun’s magnetic field is driven by dynamo processes, i.e., flows in
the electrically conducting solar plasma that induce magnetic fields. The Sun’s dynamo is
not yet fully understood5 . A map of the Sun’s surface magnetic field is shown in the bottom
panel of Fig. 8.8. This mirrors the sunspot ‘butterfly diagram’ (top panel in Fig. 8.8), but also
highlights the magnetic polarity reversal half way through the ∼22-year cycle.
Things to Remember
• Solar atmosphere structure: photosphere, chromosphere, transition region and
corona
• Solar spectrum and main deviation from blackbody spectrum
5
For an overview, see David Charbonneau’s 2010 article, (Living Rev. in Sol. Phys˙ 7:3).
50
Sun, Stars and Planets 2020-21
Figure 8.7: Active-region observed in the G-band (∼430 nm) representing photospheric emission (left), and in Ca ii H showing chromospheric emission (right). Image taken with the Dutch
Open Telescope; credit Rob Rutten.
Figure 8.8: Top panel: The solar ‘butterfly diagram’ shows sunspot areas and locations between approximately 1875 and 2015. Bottom panel: The magnetic butterfly diagram showing
the Sun’s surface magnetic field. Blue and yellow show magnetic fields of opposite polarity.
Figures taken from D. Hathaway (2015), Living Rev. in Sol. Phys. 12:4.
51
Lecture 9
Stellar Astronomy: Putting the Sun
in Context
In this chapter we will look at some of the observational aspects of astronomy. Astronomy is
one of the oldest sciences and comes with quite a lot of baggage in terms of standard ways
of doing things. There are also some specific problems that come from the fact that the objects studied by astronomers are a very long way away – so far away, in fact, that measuring
distances to them becomes a challenge. Here we look at some of the units, problems and
limitations of astronomical observations.
9.1
The Variety of Stars
Main-sequence (i.e., hydrogen burning) stars, have a wide range of parameters. Their masses
range from just below 0.1 M to around 100 M ; their radii range from approximately 0.1 R
to 15 R ; their effective temperatures range from (very approximately) 2000 K to 50 000 K.
Consequently their luminosities span about 10 orders of magnitude from approximately 10−4
to 106 L . The maximum mass that main-sequence stars can reach is not firmly established,
and while there is quite a precise lower-mass limit for the lowest-mass stars, the temperature
and radii for such low-mass stars has not been observationally confirmed.
9.2
Stellar Magnitudes
Observations of a star measure the amount of light from that star that reaches us. When
we measure its brightness we are measuring its apparent brightness. To study and compare
stars we have to look at their intrinsic brightness, with the effects of different stellar distances
taken into account so that we can determine luminosities [in Watts] rather than flux received
at Earth [in Watts/m2 ]. And yet, when we look out onto the night sky we see in only two
dimensions – we have no idea how far away anything might be. A bright point of light in the
sky might be a faint star very close to us or a bright star the other side of our Galaxy. It might
even be a galaxy itself and much much more distant than any star.
The first step towards quantitative studies of stars were made by Hipparchus, a Greek
astronomer who worked around 100 BCE. He produced the first star catalogs, in which he
classified stars into six classes according to their visual brightness, with stars of the first magnitude being the brightest, and sixth-magnitude stars being the faintest stars visible with the
naked eye. A 1st magnitude star is about a hundred times brighter than a 6th magnitude star.
52
Sun, Stars and Planets 2020-21
Human perception of brightness is a logarithmic quantity, so that equal steps of perceived
brightness correspond to equal ratios of flux. This means that the magnitude scale, which we
have inherited from Hipparchus and the Greeks, is a logarithmic scale. The ordering system
adopted by Hipparchus, with 1st magnitude being brighter than 6th magnitude, also means
that the system works in the opposite direction to what we are used to. A lower magnitude
implies a brighter object, while fainter objects have higher magnitudes.
The modern magnitude system is defined so that a magnitude difference of 5 magnitudes
corresponds to a factor of 100 change in flux. A 10th magnitude star is thus 100 times fainter
than a 5th magnitude star, and a 20th magnitude star is 10 000 times fainter than a 10th magnitude star. Most star catalogs provide observed (i.e., apparent) stellar magnitudes rather than
a physical value like a luminosity, L.
9.2.1
The Definition of Stellar Magnitudes
Hipparchus’ rather qualitiative division of stars into different magnitude classes was formalised
in 1865 by Norman Pogson as follows. First he defined the difference in magnitudes between
two stars
F1
m1 − m2 = − 2.5 log10
,
(9.1)
F2
where mi refers to the magnitude of object i, and Fi refers to its flux. Note that the more positive the magnitude difference between object 1 and 2 becomes, the fainter object 1 becomes
compared to object 2.
To go from this definition of magnitude difference to a magnitude scale for all objects, a
fiducial flux must be chosen that represents a specific value of magnitude. There are a number
of such systems, but the simplest, and the scheme that matches the original classifications of
Hipparchus, is what is called the Vega magnitude system. In this, the magnitude of the A0
star Vega, one of the brightest stars on the sky, is defined to be zero. The magnitude system
thus becomes
F1
m1 = − 2.5 log10
.
(9.2)
FVega
This magnitude system can be applied to anything on the sky since it just compares the apparent brightness of objects relative to the brightness of the star Vega. The magnitudes of
some objects you might be familiar with are shown in Tab. 9.1. The Sun and the faintest objects detectable by the next generation of large telescopes are about 62 magnitudes apart. This
corresponds to a difference in brightness of over 6 × 1024 , so magnitudes are a fairly compact
way of looking at objects over a very wide brightness range.
The flux of a star received at Earth is given by
F =
L
,
4πd2
(9.3)
where L is the luminosity of the star and d is the distance of the star from the Earth. The
apparent magnitude of a star thus depends both on the star’s intrinsic luminosity, L, but also
on its distance d, which has nothing to do with the physics of the object, but everything to do
with its chance positioning in the galaxy. If we want to examine the intrinsic properties of stars
to see if our calculations concerning the properties of stars are correct we need a measure that
is independent of distance. One way to do this is to define an absolute magnitude, which is
the magnitude that the star would have if it were placed at some standard distance from the
Earth.
53
Table 9.1: The magnitudes of various objects.
Object
The Sun
The Full Moon
Venus at its brightest
The Brightest star
Dimmest naked eye star in London
$(*-."#+&/("*.&*,&.#"*%0*+*1+$"(/ Dimmest naked eye star, dark skies
Faintest object detectable by HST
Faintest object detectable by the E-ELT
Magnitude
-26.73
-12.7
-4.1
-1.46
+3
+6
+31
+36
&/(*+#**56./6*+*7(&8#6*%0*2*!9:9*5%,7-*",;#(&-*
"(/%&-
?!" '
!"'(+)*+
!"#$%$
Figure 9.1: The definition of a parsec.
!"&'()*+
*7.86#<D(+$*=*@9E*A*2?!# '4*
9.2.2
The Astronomer’s Unit of Distance: The Parsec
Astronomers most commonly use the parsec as their unit of choice. This is the distance at
which a length of 1 astronomical unit (the distance from the Sun to the Earth, about 150 million
km) subtends an angle of 1 arc second (= 1/3600 degrees; see Fig. 9.1). This distance is thus
(3600 · 180/π) AU = 206265 AU = 3.09 × 1016 m, or 3.26 light years. This definition might
seem a bit contrived, but we will see in Sec. 9.4 that it arises quite naturally when measuring
distances to stars.
9.2.3
The Definition of Absolute Magnitude
We define the absolute magnitude to be the apparent magnitude an object would have if it
were placed a standard distance of 10 pc from the Earth.
To determine the absolute magnitude of a star $of luminosity L at a known distance d1
from us we rewrite its apparent magnitude m and absolute magnitude M as
m = − 2.5 log10 (F1 ) + k
and
M = − 2.5 log10 (F2 ) + k,
(9.4)
where k represents the calibration factor for the photometric system. Here F1 = L/(4πd2 ) is
the flux received from the star at distance d and F2 is the flux received from the (same) star
at a distance of 10 pc. As the star’s luminosity L is the same for both F1 and F2 , it follows
that F1 d21 = F2 d22 . Thus
F1 d21
M = − 2.5 log10 F2 + k = − 2.5 log10
+k
d2
2
d1
d1
= − 2.5 log10 F1 + k − 5 log10
= m − 5 log10
,
(9.5)
d2
d2
where we have used the definition of m above. Since d2 is (by definition) 10 parsec, we have
M = m − 5 log10 (d) + 5,
(9.6)
54
Sun, Stars and Planets 2020-21
Table 9.2: The Johnson filter system.
Filter
U
B
V
R
I
Name
Ultraviolet
Blue
Visual
Red
Infrared
Central wavelength
365 nm
440 nm
550 nm
640 nm
800 nm
where d is the distance to the star in parsecs.
Exercise: what is the absolute magnitude of a star whose distance is 60 pc and apparent
magnitude is 3.61?
9.2.4
Observational Passbands
The flux of a star is usually measured with detectors and filters that are sensitive to a specific
range of wavelengths. There are a number of different such filter systems, but one of the most
commonly used is the Johnson system that has five filters that covers the range of wavelengths
to which the eye is sensitive, plus a bit more. The five different filters in this system are given
the names U , B, V , R and I with properties as described in Tab. 9.212 . Most stars have absolute
visual magnitudes in the range −6 to 16. The Sun is quite an average star with an absolute
visual magnitude MV = 4.83.
9.2.5
Bolometric Magnitude
The total magnitude, apparent or absolute, of a star represents the flux of the star summed
over all wavelengths. This is termed the bolometric magnitude, mbol or Mbol for apparent
or absolute3 . The difference between the bolometric magnitude of a star and its magnitude in a
given passband is called the bolometric correction, BC. For a given stellar type and luminosity
class you can go from the magnitude measured in a given passband, say V, to the bolometric
magnitude by adding the bolometric correction. Thus
mbol = mV + BC.
(9.7)
The BC values for each type of star (OBFGKM) and luminosity class are tabulated for use.
They are generally negative since there is more energy in the whole of the spectrum than there
is in a limited part of it.
9.3
Stellar Colours and Temperatures
As the spectral distribution of the energy output of a star is temperature dependent, the effective temperature of a star can be estimated by measuring its magnitude in two different
1
This filter system also extends into the near-infrared with filters named J, H, K, L, M at increasing wavelengths. This explains why infrared astronomers have problems with the alphabet: to them H comes after J.
2
There are a large number of different filter systems; apart from the Johnson system, the ‘Sloan’ ugriz filters
are also commonly used.
3
The Sun’s absolute bolometric magnitude is 4.74.
negative, since there is more energy in the whole spectrum than in a limited part of it.
Mbol = 0 for a main-sequence star with L = 3 x 10
Exercise:
28
W.
Show that Mbol = -2.5 log ( L /(3 x 1028))

and calculate Mbol and mbol


55
Recall that stars radiate approximately as black bodies,
9.3 Stellar colours and temperatures
Figure 9.2: Sketch illustratso their intensity is given by the Planck function:
B
V
U
log B
T
log
ing the concept of colour indices for blackbodies of differNote: x axis is
ent
temperatures.
The vertical
frequency
in this plot,
lines
location of the
unlikeindicate
the Planck
Vdistribution
, B and Uplots
bands. As tempershown in lecture 7,
ature
which increases
are againsta larger fraction
of
the emission is detected
wavelength.
in bluer filters. Note that
this plot shows frequency on
the x-axis, so shorter (bluer)
wavelengths, corresponding to
higher frequencies, are to the
right.

Measuring a star's brightness in (say) U, B, V bands gives a measure of its effective (surface)
temperature. Table 9.3: The colour indices in U − B and B − V for stars of four different temperatures.
Define colourColours
indices:
U (or
– Bvery
= nearly
mU – m
B – V =asm
are zero
zero)
700 K by definition
this
the
B at 9and
B –ism
V effective temperature
adopted for the A0 star Vega.
Teff
U–B
B–V
40 000 K
-1.15
-0.35
9 900 K
0.0
0.0
Cooler stars are redder ;
U-B, B-V positive
4 900 K
0.47
Teff
0.89
40 000 K 9 700 K
U −B
−1.18
0.00
B
−V
−0.32
−0.01
hotter stars are bluer
5 770 K
0.13
0.65
4 900 K
0.47
0.89
U-B, B-V negative
We can therefore determine a star’s temperature using colour indexes, known as the photometric
method of temperature
this is are
independent
of distance.)
passbands. determination.
These magnitude(Note:
differences
termed ‘colour
indices’. For example, the U −B
However, a more accurate measurement of photospheric temperature can be obtained for
index
is
simply
m
−m
(or
indeed
M
−M
),
i.e.,
the
difference
between an object’s magniU
B
U
B
individual stars by the spectroscopic method (explained in lecture 7) based
on analysis of the
4 . Cooler stars emit more strongly
tude
in
the
U
and
in
the
B
band
(this
is
sketched
in
Fig.
9.2)
spectral absorption lines in the observed stellar spectra.
at longer wavelengths and are thus redder, with U − B and B − V positive. Hotter stars emit
more strongly at shorter wavelengths and are therefore bluer, with U −B and B−V 4negative
– see Tab. 9.3 for some examples.
A star’s effective temperature can thus be determined using colour indices. This is known
as the photometric method of temperature determination and is independent of distance.
This is a very efficient way to determine temperatures, though it is less accurate than using
stellar spectra (see Sec. 7.3).
9.4
Measuring Stellar Distances: Trigonometric Parallax
Determination of the distance to a star is very useful. It allows us to determine the object’s
luminosity, to determine their physical size, and, if the star has a companion, to find masses
from the details of the orbital motions. Some stars are close enough that they have small
apparent motions against the background of more distant stars due to the orbit of the Earth
around the Sun (see Fig. 9.3). This is not true motion but is a stellar parallax. Distances to
these stars can be measured geometrically using trigonometric parallax.
Given the parallax angle p in arcseconds, and the definition of the parsec the distance to
4
The convention is to subtract the ‘redder’ from the ‘bluer’ magnitude.
9.4 Measuring stellar distances: Method of trigonometric p
Distance is measured directly geometrically by this method. It is im
order to: be able to estimate the luminosity of an object; to find ma
motions (using Kepler’s 3rd law); and for estimating physical sizes o
have small periodic apparentSun,
motions
(with
respect
to more distant
Stars and
Planets
2020-21
Earth’s motion in its orbit around the Sun. This is not a “true” motio
56
Figure 9.3: Measurement
of distance using trigonometric parallax. The line of
sight to a nearby star differs from June to December thanks to the Earth’s
orbit around the Sun. Half
of the difference in angle is
the parallax, p.
In the d
the star
Decemb
other sid
Sun. Th
Earth, a
the angl
is the pa
d=1/p
d
Naturally as we observe the parallax of more distant stars the para
the star is thus simply
distance to the star. As the nearest stars are still far from Earth, th
are of the order of less than an arcsecond. Our nearest star, alpha
1
d =
for d in
parsec and pare
in arcsec.
(9.8)
0.76, arcsec.
Parallaxes
measured by both photography
and d
p
from space to avoid blurring and problems observing through the a
Since even the closest stars to Earth are still a long way away, parallaxes are usually less than
00 .
p =aparallax
angle
, d = distance fr
1 arsec (or 100 ). Our nearest d=1/p
star, Alpha Where:
Centauri, has
parallax angle
of in
0.76arcseconds
Measurements of parallax angles form the surface of the Earth are hampered by the blur[ Reminder: Parallax Second= Parsec (pc) Fundamental unit of dis
ring effects of turbulence in the Earth’s atmosphere – a process known as atmospheric seeing.
1 arcsecond
has
a distance
of The
1 Parsec."
] measurePrecise parallax measurements
are thus best
done
from space.
best parallax
1 parsec
(pc) isofequivalent
to: 206,265
AU, distances
3.26 Light
ments possible from Earth have
an accuracy
about 0.01 arcseconds,
allowing
to Years, 3.086
be measured out to 100 pc. As this does not
many
ESA launched
thearcsec
Hipparcos
eg:cover
a star
hasstars,
a parallax
of 0.02
– what is its dis
satellite in the 1990s, which could measure parallaxes of 0.001 arcseconds (thus reaching a
distance of 1000 pc). The current Gaia mission is a successor to Hipparcos and is measuring
If p = 1 arc-sec, d = 1 parsec and If p = N arc-secs, d =
parallaxes of a few microarcseconds, allowing the distances to stars across the entire Milky
Way Galaxy to be measured for the first time5 .
Limitations: if the stars are too far away, the parallax is too smal
Smallest measurable parallax from the ground is ~ 0.01 arcsec, so
9.5 Proper Motionparallaxes is limited: although it is a good method to distances up
that
arecan
this
close.
However,
measurements
(Hippar
Stars are not fixed in space.stars
Instead,
they
move
relative
to the Sun,satellite
though their
motion
accuracy of 0.001 arcsec. Hipparcos measured parallaxes for abou
as seen from Earth is very small.
The motion of a star has1000pc
two components
(see Fig.
9.4): one
parallel
to mission
the line ofwill
sightmeasure
–
for brighter
stars.
Gaia
space
paralla
towards or away from us – the other perpendicular to the line of sight. This later component,
which changes the star’s angular position on the sky over time, is called proper motion. It
9.5 Proper
Motion
fixed
in space:
they move relative
is given this name as it is intrinsic
to the star
and not aStars
result not
of the
motion
of the observer,
however
are
very
tiny
seen
from
the
Earth.
as is the case for trigonometric parallax. Proper motions are usually measured in terms of
Two components
arcseconds per year.
Motion along line o
The transverse speed of a star can be determined from its proper motion and its distance
– which might be determined through trigonometric parallax. If µ is the proper motion
of a
position
on sky.
star in arcseconds per year, and d is the star’s distance from us, then the transverse speed
will
Motion perpendicu
be
position in the sky
vt = d sin(µ),
(9.9)
Such a change in p
motion, so
to the star, and not
observer or a movi
motion is usually e
per year (3600 arc
Its 2020 data release reported colour and parallax measurements for about 1.5 million stars as well asproper
precise
magnitude and position determinations for about 1.8 million stars.
5
μ
d
stars that are this close. However, satellite measurements (Hipparcos) measure parallaxes
accuracy of 0.001 arcsec. Hipparcos measured parallaxes for about 120,000 stars, out as f
1000pc for brighter stars. Gaia space mission will measure parallax accurate to a few  arc
9.5 Proper Motion
Stars not fixed in space: they move relative to the
57 Sun. The motions
however are very tiny seen from the Earth.
Two components of the motion:
Motion along line of sight does not change position on sky.
Motion perpendicular to it does change star
position in the sky (by tiny amount).
Such a change in position of a star is called
proper motion, so called because it is intri
to the star, and not a result of the motion of
observer or a moving reference point. Prop
μ
Figure 9.4: The motion of a star relative to the
motion is usually expresses in seconds of a
Earth (indicated with the standard ‘Earth’ symd
year
(3600 arc
= 1 degree).
bol ⊕). There per
are two
components,
thesec
radial
velocity that is parallel to the line of sight, and
proper motion that is perpendicular to it.
and for small angles sin(µ) = µ. Barnard’s Star has the largest known proper motion, at
10.3 arcsecond per year. It has a trigonometric parallax of 0.5500 and is thus 1.8 pc from Earth.
Stellar proper motions generally decrease with increasing distance from Earth, so the proper
motion of a star can be used as a rough indicator of the star’s distance.
The radial component, vr of a star’s motion relative to us can be obtained via the Doppler
effect, which shifts the central wavelength of a spectral line. If we know the rest wavelength
for this line, λ, from laboratory measurements, and the observed wavelength of the line, λ0 ,
then the radial velocity of the star vr is
vr = c
λ0 − λ
λ
.
(9.10)
Lines that are shifting to longer wavelengths because an object is moving away from us are
called redshifted, while lines shifting to shorter wavelengths because an object is moving towards us as are termed blueshifted.
The radial velocity vr and the transverse velocity vt together specify the overall motion v
of a star through space relative to the Earth as
v=
q
vt2 + vr2 .
(9.11)
The motions of stars are not simply random, but are related to large-scale motions of stars in
our Galaxy – for example as they orbit in the Galaxy’s disc6 , or if they are part of stellar clusters.
Whether a star is a member of a given star cluster can often be determined by comparing its
motion through space with that of other cluster members.
9.6
Stellar Distance Indicators
We have discussed above a couple of indicators for the distance of a star. These are summarised
here together with two further important distance indicator methods.
6
The Sun completes an orbit of the Milky Way Galaxy in about 225 million years.
58
Sun, Stars and Planets 2020-21
• Parallax – the measurement of a stars motion against background stars that results from
the Earth’s motion around the Sun. Usable for only the most nearby stars unless you
are observing with specialised instrumentation in space.
• Proper motion – the proper motion will be greatest for the nearest stars, so the smaller
the proper motion, the more distant a star is likely to be.
• Variable stars – some types of star have periodically varying luminosity that is directly
related to the luminosity through the stellar structure equations7 . A measurement of
the period of one of these stars thus yields its luminosity which, together with the flux
received from it, provides the distance.
• Colour and spectral type – these can give Teff and g ∝ M/R2 of a star and thus L =
4 . If one assumes that all stars of the same spectral type have the same size8
4πσR2 Teff
then a measurement of Teff can predict L and with this and a measurement of apparent
magnitude you can find the distance. However, you need to know the radius R for the
appropriate spectral types. This can be calibrated with stars of the same spectral type
whose distance is already known from, for example, trigonometric parallax.
Things to Remember
• Definition of apparent and absolute magnitudes
• Relation between apparent magnitude, distance and luminosity
• Appreciate the difference between bolometric and passband magnitudes
• Stellar temperature measurement from colour indices
• Stellar distance measurements from the trigonometric parallax
• Definition of the proper motion and relation to true space velocity of a star
7
This is not covered in this course, but if you are interested you can look up Cepheid variable stars in many
textbooks; they are an important method for establishing the distance scale of the Universe.
8
If spectroscopic measurements are available they can be used to set a relatively tight limit on R through g,
see Sec. 7.4
59
Lecture 10
The Hertzsprung-Russell Diagram
The Hertzsprung-Russell (HR) diagram is a way of looking at the variety of stars in the sky and,
in conjunction with the stellar structure equations, can be used to tell us about the physical
properties of these stars and their evolution during and after their hydrogen-burning lifetime
on the main sequence.
10.1
Hertzsprung-Russell (HR) Diagrams
We can systematise our observations of stars, and theoretical predictions for them, using HR
diagrams. The suitable physical properties for comparing stars, as we have already seen, are
effective temperature, Teff , luminosity L and radius R. However we do not need all three,
4 .
since they are related through the equation L = 4πR2 σTeff
What actually gets plotted in an HR diagram, though, depends. A theorist would want
to plot temperature against luminosity, while an observer will likely plot colour index1 , say
B − V , against absolute magnitude. This was first done independently by Eljnar Hertzsprung
in 1911 for stars in clusters and by Henry Norris Russell in 1913 for nearby stars. A rather
splendid modern HR diagram is shown in Fig. 10.1 using data for about 100 000 stars from the
Hipparcos satellite. You can see the main sequence forming the wiggly diagonal across the
centre of the plot, but there are also other objects, off the main sequence. We will discuss
these later.
An important point to note is the way the axes are arranged in an HR diagram. The yaxis is usually the absolute magnitude, so the luminosity increases along y (even though the
numerical values decrease). In Fig. 10.1 the x-axis shows the colour index, B − V . As seen in
Chap. 9, a low value of this index means that an object is bluer (it is brighter in the B (blue)
band than in the V (visible) band, and thus hotter. The x-axis in effect has temperature rising
to the left, the opposite way round to what we might expect.
For absolute magnitude to be plotted we must know the distance to a star. An alternative
way of plotting an HR diagram for a set of stars all thought to be at the same, but unknown,
distance (e.g., in a star cluster), is to plot apparent magnitude rather than absolute magnitude.
Of course any stars that are not part of the star cluster will not appear at the correct location
when this is done.
The Theorist’s HR Diagram and Stellar Radii
In terms of stellar parameters, plotting the luminosity against effective temperature (rather
1
You might also see the spectral type on the x axis. Hertzsprung in fact plotted magnitude against ‘effective
wavelength’ (∼ Wien peak).
60
Sun, Stars and Planets 2020-21
Figure 10.1: The HR diagram as
obtained from Hipparcos satellite
data. The two standard axes on
this plot are B − V colour on the
x-axis, and absolute magnitude on
the y-axis. Recall that lower (‘more
negative’) values in magnitude indicate brighter objects, so points towards the top of the diagram represent more luminous objects as indeed indicated on the right-hand y
axis. Objects with a smaller B −
V colour index are bluer, and thus
hotter stars; temperature hence increases to the left (see top x axis).
Diagram courtesy of ESA.
than magnitude and colour index, or spectral type) gives more immediately accessibly informa4 ,
tion. Since the luminosity L and effective temperature Teff are related through L = 4πR2 σTeff
each point on the HR diagram corresponds to a unique stellar radius. A star that is very luminous but very cool has to be very large, and is thus given the name red giant. A star that is hot
but has a low luminosity has to be very small, and is thus termed a white dwarf. A sketch of a
luminosity-temperature diagram is shown in Fig. 10.2; the diagonal red lines indicate lines of
constant radius.
10.2
HR diagrams: the Brightest and Nearest Stars
If we plot the HR diagram of the brightest stars on the sky, the objects that appear on it will
be a mixture of stars that are common enough and bright enough that they are both nearby
and visible, and stars that are so bright that they can be seen from a very great distance. As
main-sequence stars (luminosity class V) are the most common stars, we expect to see these
along with a small number of the most luminous stars in luminosity classes I and II. This is in
fact what is seen in Fig. 10.3.
Conversely, the HR diagram of the nearest stars (Fig. 10.3, right) includes only the most
common stellar types, missing out rare high-mass and high-luminosity main-sequence stars,
and the rarer and even higher luminosity red giants. It also includes a large number of lowluminosity systems like white dwarfs.
61
M
ai
n
log L
R
Se
qu
en
ce
Red
Giants
White Dwarfs
log Teff
10.3
Figure 10.2: A theorist’s version of the
HR diagram, with luminosity increasing upwards and temperature increasing to the left. The red lines indicate how radius changes across the diagram; the main sequence, red giants
and white dwarf regions indicated.
The Distribution of Stars on the HR Diagram
Why are stars distributed on the HR diagram in the way that we see? We know that stars do
not have an infinite lifetime since they will, at the very least, run out of fuel at some point.
Given that stars have a finite lifetime we can surmise that there are distinct stages between a
star’s birth and death, and that each star is characterised by some specific range of luminosity
L and surface temperature, Teff . The star will thus move around the HR diagram as it evolves.
We can also surmise that not all stars we see today are at the same stage in their evolution.
We know this, at least in part, because we can see young stars forming in places like the Orion
Nebula. Finally, we can assume that the Galaxy holds a large population of stars. Starting
from these assumptions, we can infer a number of things:
• In a large population of stars, the longer a particular evolutionary stage lasts, the greater
the number of stars that we will observe in that stage. This means that only a few stars
will be observed going through any particularly brief phases in their evolution.
• The parts of the HR diagram where there are concentrations of stars must be regions
where stars spend a comparatively large fraction of their liftetimes. On this basis a star
must spend most of its life on the main sequence (90% of stars are found on the main
sequence).
• Similarly, since stars are found in the giant, supergiant and white dwarf regions, we
must expect some stars to spend some part of their lifetims in them.
Note that the concentration of stars on the HR diagram depends on the fraction of stars that
pass through a region and on how quickly they pass through it. Hence some parts of the
HR diagram might appear to have few stars, or none at all, simply because they correspond
to stages of stellar evolution that we can’t see because the stars are shrouded in obscuring
material (e.g., dust), and are thus not directly detectable.
10.4
Mass and the HR Diagram
Absolute magnitude and colour index give us temperature and radius from observations, but
there is one further stellar parameter that is very important: mass. How does stellar mass vary
from one position on the HR diagram to another?
This is shown in Fig. 10.4. Stellar masses are measured from 0.08 M to about 100 M ,
though there may be a handful of stars observed to have still greater masses. The Sun, at
62
Sun, Stars and Planets 2020-21
–8
Lum. Class I
Lum. Class II
MV
Lum. Class V
4
B0
G0
M0
Figure 10.3: HR diagrams for the brightest and nearest stars. The sketch on the left indicates
the location of the brightest 100 stars. These include main-sequence stars that are common
and bright enough for us to see, along with a few rare but very luminous stars. The figure on
the right shows stars within 25 pc of the Sun (GAIA collaboration 2018, A&A 616, A10) and
includes the most common stellar types (low-mass main-sequence stars and white dwarfs).
GAIA colour indices (GBP -GRP ) of 1 and 3 correspond roughly to spectral types K0 and M3
(Teff ∼ 5000 K and 3000 K, respectively).
1 M is a very average star. We find that the lower the mass, the more common the star; there
are very many low-mass stars, but high-mass stars are very rare. The function that determines
how many stars of a given mass are born is called the Initial Mass Function, and its shape is
intimately linked to the processes behind star formation.
Examination of where various mass stars lie on the HR diagram gives us our first hints as
to the processes of stellar evolution.
i. Along the main sequence mass correlates with luminosity and hence with Teff . As mass
increases, L and Teff increase, as expectated from the homology solutions to stellar
structure. We see a 500 times increase in mass, from the lowest to highest masses on
the main sequence, giving a 1010 increase in luminosity.
ii. Within the classes of supergiants, red giants and white dwarfs, however, there is no
correlation of mass or luminosity with photospheric Teff – it is as if the nice ordered
masses along the main sequence have become jumbled.
iii. Supergiants have greater masses than red giants which have greater masses than white
dwarfs. Supergiant masses are comparable with masses of stars on the upper main
sequence, while the masses of red giant are comparable to stars on the lower end of the
main sequence.
10.4.1
The Range of Stellar Masses on the Main Sequence
Observationally, there are far more low mass stars than high mass stars, and since low mass
stars can be hard to find – they are going to be much lower luminosity since L ∝ M 3.5 – their
numbers are easy to underestimate.
63
Figure 10.4: Typical masses of
stars at various positions on the HR
diagram. Masses are given in solar
mass units (An Introduction to the
Sun & Stars eds. Green & Jones).
There is, however, a lower mass limit to what we describe as a star. The lower the mass,
the lower the temperature of the core. As temperature declines the energy generated from
hydrogen fusion drops, eventually becoming insignificant. Stars are powered by hydrogen
fusion, so an object that cannot generate power this way is no longer a star. This happens
for masses roughly < 0.08M , when the rate of internal energy generation by fusion is no
longer sufficient to match the rate of energy radiated form the surface. Such objects are called
brown dwarfs. They are sufficiently low in surface temperature that they emit predominantly
in the infrared. They are more massive than the most massive planets like Jupiter, and, unlike
planets, seem to form from the interstellar medium in the same way as stars.
The upper mass limit of stars is a different issue. Stars clearly cannot have a mass greater
than the mass of interstellar medium material from which they formed, but this is not a significant limitation since such cloud masses are typically in the several thousand M range.
However, as a star becomes more massive its temperature also rises leading to an extra pressure term – radiation pressure – becoming significant. We know that radiation pressure
Prad = 13 (aT 4 ) while gas pressure Pgas = nkT so
Prad
aT 3
.
=
Pgas
3nk
(10.1)
As temperature rises for the highest mass stars, radiation pressure contributes much more
strongly. On PS2 there is an estimate of the Eddington luminosity where the force due to the
photons matches that of gravity and a star is not able to accrete extra mass, setting an upper
mass limit (thought to be around 100 M )2 . The instabilities from radiation pressure and their
short life may in fact be racing each other in the very highest mass stars such as Eta Carinae,
an HST image of which is shown in Fig. 10.5. What you actually see in this image are lobes of
material blown off the star, which is losing mass at a significant rate. At the same time it is
evolving towards the end of the main sequence as it burns its hydrogen supply very rapidly.
2
Observationally there is good evidence for stars with masses up to about 150 M ; claims of stars with masses
greater than 300 M are not widely accepted, though it is likely that the earliest very metal-poor stars were able
to reach higher masses.
64
Sun, Stars and Planets 2020-21
Figure 10.5: An HST image of Eta
Carinae, a multiple star system the
most massive of which has a mass
about a hundred times the mass of
the Sun. Image courtesy of NASA.
10.5
Main Sequence Life
The curve defined by values of L and Teff corresponding to static, homogeneous stars that
have just commenced hydrogen burning forms what is called the zero age main sequence
(ZAMS). ZAMS stars are essentially static, neither expanding, contracting nor pulsating, and
their energy losses are supplied by nuclear burning in the core. The equations of stellar structure (see earlier chapters) hold well.
As we have seen in Chap. 3, the nuclear fusion reactions take place in the core. Nuclear
reaction rates vary with T . As we move from the centre of the star, there is a boundary demarking the core within which nuclear reactions occur. The size of the core varies with stellar
mass, as do the details of the nuclear reactions and the energy transport mechanisms through
which the energy produced at the core reaches the outer layers and is radiated away.
In lower main-sequence stars (M <1.5M hydrogen burning in the core is through the
p-p chain. Energy transport from the core is radiative, with a convective envelope further out.
In upper main-sequence stars (M >1.5M ) core temperatures are sufficient for hydrogen
burning via the CNO cycle. This releases energy at a much higher rate than the p-p chain once
sufficiently high core temperatures are reached. This triggers convective instability leading
to energy transport in the central regions of higher mass stars being convective rather than
radiative. This is summarised in Fig. 10.6 where the energy transport mechanisms throughout
the stellar interior are shown for stars with masses between 0.1 and 10 M .
During the main sequence phase a star does not significantly change its luminosity or
photospheric Teff . If it did it would move along the main sequence and this would not be
consistent with the ordered range of masses that is observed. However, as stars fuse hydrogen,
their luminosities increase slightly, and they drift slightly upwards in the HRD, making the
main sequence the band that is observed rather than a narrow line.
Stages of Stellar Evolution in the HR Diagram
The Post-Main Sequence Phase: When the hydrogen burning phase of a star’s life is over,
65
Figure 10.6: The fraction of stellar
mass in the convective core, radiative region and convective envelope
as a function of stellar mass. From
An Introduction to the Sun & Stars
eds. Green & Jones.
less massive stars become red giants while more massive stars become supergiants. This
is consistent with the masses seen in the giant/supergiant regions of the HR diagram, and
with the rarity of supergiants, since there are very few stars massive enough to become a
supergiant.
Final Stages of Stellar Evolution: red giants evolve to a point where the shed their
outer layers and become a planetary nebula. The stellar remnant left behind then evolves to
become a white dwarf. The end point of the evolution of a supergiant is a supernova, which
destroys the original star in a massive explosion.
At these later stages of stellar evolution mass loss becomes important. Images of planetary
nebulae – these are transitionary stages at the end of life for low-mass stars – show impressive
examples of mass loss, with shells of material being flung off the central star. Up to 50% of
mass-loss can also occur towards the end of a star’s life on the so-called ‘asymptotic giant
branch’.
10.6
Stellar Evolution Rates
In Sec. 6.3 we argued that stars of different masses evolve at different rates and that the mainsequence lifetime should scale roughly as M −2.5 . We can test this observationally by looking
at the HR diagrams of star clusters if we assume (a) that all the stars in a star cluster form at
the same time and (b) the compositions of all the stars in a star cluster are very similar since
they formed from the collapse of the same gas cloud.
These two points are vitally important to this observational test. If the stars form at the
same time from gas of the same composition then the only differences between them will be
their mass – they will otherwise be a single, coherent homologous group. This allows us to
look at the HR diagram of a star cluster to look for differences in evolution rate that result
solely from a star’s mass.
Open clusters are groups of stars found within our own Galaxy. They typically contain
2
10 −105 stars and are relatively young systems, aged 100 Myr to a few Gyr, since they are not
strongly gravitationally bound and will thus be gradually disrupted as they orbit the Galaxy.
The most obvious example of an open cluster is the Pleiades, which can be seen with the naked
eye even from London on good nights. Another example is a cluster known as NGC188. The
HR diagrams for these two clusters are shown in Fig. 10.7.
The comparison between the Pleiades and NGC188 HR diagrams shows that the stellar
populations in these two clusters are quite different. NGC188 must be older than the Pleiades
66
Sun, Stars and Planets 2020-21
Figure 10.7: Observational HR diagrams for two different open clusters: the Pleiades (Kamai
et al, 2014, AJ 148, 30) on the left, and NGC188 (Gondoin, 2005, A&A, 438, 291) on the right.
Note that the distribution of stars on the main sequence for these two clusters is very different.
The squares and triangles on the NGC188 HR diagram indicate likely non-members of the
cluster.
since the Pleiades main sequence is populated to much higher masses than that of NGC188.
In NGC188 there has been enough time for all but the lowest mass stars to leave the main
sequence3 . The Pleiades cluster is too young (about 100Myr) for this to have happened.
Globular Clusters are more tightly bound clusters of stars, some of which are found in
the outer reaches of our own Galaxy. They can also be seen in nearby companion galaxies
to our own. They are far older than open clusters, with ages up to ∼ 14 Gyrs and typically
including about 106 stars. The HR diagram of the globular cluster M13 is shown in the righthand panel in Fig. 10.8. In this HR diagram the more massive stars have all evolved off the
main sequence to become red giants. The age of this globular cluster can be estimated from
the main sequence turnoff point. In this case it is about 14 Gyr, meaning that this cluster must
have formed quite soon after the Big Bang.
10.6.1
Estimating Cluster Age from the Main-Sequence Turn-off
The point at which the main sequence ends, known as the main sequence turn off, can be
used as an indicator of cluster ages. The left-hand panel of Fig. 10.8 illustrates the way the
main sequence ‘peels off’ (i.e., the turn-off point moves to lower masses) with age. At the
turn-off point from the main sequence a star has ended its hydrogen burning phase. This
means that it has released all of the energy available to it from this power source. Given a
luminosity L, the turnoff point should occur at a time τt when the total energy available to a
star from hydrogen burning on the main sequence, EMS , has been radiated away at a rate L,
3
In addition there are a considerable number of stars in the red giant region of the HR diagram of NGC188.
Most stars leaving the main sequence become red giants.
67
Figure 10.8: HR diagrams of clusters. Left-hand panel: HR diagram combining data from 32
open clusters observed with GAIA and colour coded according to their age with blue for the
youngest and red for the oldest clusters (GAIA collaboration, 2018, A&A 616, A10). Right-hand
panel: HR diagram of the globular cluster M13, with different parts of the diagram, including
the main sequence turnoff, indicated (data from Rey et al 2001, AJ 122, 3219).
thus τt = EMS /L.
EMS is given by the mass hydrogen ‘burnt’, multiplied by the fusion energy released per
kg of hydrogen, η ∼ 6.58 × 1014 J/kg. The total mass of hydrogen that is available for fusion
is given by f XM , where X ∼ 0.6 is the hydrogen mass fraction of the star, and f ∼ 0.2 is the
fraction of hydrogen that the star will fuse before it leaves the main sequence. Thus EMS =
ηf XM , or EMS = ξM if we absorb all the prefactors into a single factor ξ.
From the homology scaling relations for main-sequence stars we obtained a mass-luminosity
relation with L ∝ M α , where we adopted α =3.5 as a compromise value for all stars on the
main sequence. We thus have M ∝ L2/7 and hence M = M (L/L )2/7 . Thus the cluster
age is given by
EMS
M
M
τ =
= ξ
= ξ
L
L
L
L
L
2/7
M
= ξ
L
L
L
−5/7
.
(10.2)
The value of log L/L for the main-sequence turnoff can be read off the y-axis of a properly
calibrated HR diagram, and ξ can be estimated from the values given above. We can make our
lives a bit easier if we assume that ξ is essentially the same for all stars and scale the equation
above to the Sun’s lifetime, τ which we know to be about 10 billion years. Then we simply
find τ /τ = (L/L )−5/7 .
Things to Remember
• Hertzsprung-Russell diagrams: understand, sketch explain and discuss
• The main sequence on the HR diagram and how to derive expected luminositytemperature scalings
• HR diagrams of clusters and how to estimate cluster ages from the mainsequence turnoff
68
Sun, Stars and Planets 2020-21
Lecture 11
Binary Stars
Approximately 2/3 of all stars are in binary or more complicated systems1 . Stars in a binary
orbit their common centre of mass and, potentially, can interact with one another.
Some binary stars are easy to spot, appearing as two separate stars that orbit around each
other. Two of the nearest stars to the Sun, α and β Centauri are an example of such a binary system2 . Other binary stars are so close together that the angular separation between
them cannot be resolved by our telescopes. The light from the two stars is thus blended together in imaging observations, leading potentially to the measurement of unusual colours or
luminosities in the HR diagram.
Binary star systems provide one of the few ways in which we can directly measure the
mass of stars. The basis of this method to observe how the star moves under the action of
gravity as it orbits around its companion.
11.1
Binary Star Orbits
In general, binary stars orbit around a common centre of mass on elliptical orbits. A line going
from one of the stars to the other always passes through the centre of mass. For semi-major
axes a1 and a2 and masses M1 and M2 , we have M1 a1 = M2 a2 (see Fig. 11.1 for definitions).
Thus for a1 < a2 as in the sketch here, we also have M1 > M2 .
For simplicity we will assume in this lecture that the two stars in a binary system have
circular orbits about a common centre of mass as seen in Fig. 11.2. The two stars have masses
M1 and M2 and they orbit at distances r1 and r2 from the centre of mass of the system. The
distance between the stars r is thus r1 + r2 . The orbital periods of the stars, the time taken
for each to complete an orbit, will be equal, with a value P . Both stars will thus have the same
angular velocity ω (with units of radians per unit time) given by
2π
.
(11.1)
P
Gravity provides the centripetal force to maintain a circular orbit. Recalling that the centripetal acceleration is a = ω 2 ri where ri is the distance to the centre of mass, we find
ω =
for star 1 :
1
GM1 M2
= M1 r1 ω 2 ;
r2
for star 2 :
GM1 M2
= M2 r2 ω 2 .
r2
(11.2)
This is not surprising as stars form in giant molecular clouds where several stars may be forming at the same
time.
2
It is in fact very likely that they are in a triple system with the star nearest to the Sun, Proxima Centauri. Prox
Cen, in turn, is host to two known exoplanets, including an Earth-mass planet in its (theoretical) habitable zone.
69
Figure 11.1: Binary stars orbit around a common centre of mass. Positions for the stars (S1
and S2 with masses M1 and M2 , respectively) are given at two times (primed and unprimed).
The imaginary line linking the stars goes through the centre of mass (red cross) at all times.
S1’
S1
r1
com
r2
S2
Figure 11.2: Binary stars in circular
orbits around their common centre
of mass. The distance between the
two stars is given by r = r1 + r2 .
S2’
The LHS of these equations are identical, so the RHS must also be equal, implying M1 r1 =
M2 r2 which is indeed the centre-of-mass definition. Adding the equations in Eq. 11.2 yields
G
4π 2
2
2
(M
+
M
)
=
ω
(r
+
r
)
=
ω
r
=
r.
2
1
1
2
r2
P2
(11.3)
If we then write M = M1 + M2 where M is the total mass of the system we find
M =
4π 2 r3
.
G P2
(11.4)
While we have not shown this here, Eq. 11.4 also holds for elliptical orbits where the radius r
is replaced by the sum of the semi-major axes a = a1 + a2 . We see that as M1 or M2 increase,
the period P decreases, and as r (or a) increases, P increases.
ng the stellar masses
s for the two components of a binary system can be measured in three
nd astrometric binaries
70
Sun, Stars and Planets 2020-21
scopic binaries
ng binaries 11.2 Measuring Stellar Masses
The stellar masses of the two components of a binary system can be measured in three sets
of circumstances: visual binaries, spectroscopic binaries and eclipsing binaries. We will
look at each of these in turn.
aries: here we can see both stars separately as 2 distinct points of light
n measure each star’s orbit on the sky.
ure the ratio of
the orbit sizes and use [13.3] 
=
In visual binaries we can see both stars separately as two distinct points of light and can
11.2.1
Visual Binaries
measure each star’s orbit as it changes position on the sky. From this we can measure the
ratio of orbital sizes and determine the mass ratio of the stars via the centre-of-mass definition,
a1 M1 = a2 M2 . (I am using the semi-major axes here as the approach in this section is valid
for elliptical orbits.)
eparation of the stars, α, is measured.
om the Earth to the binary star system is found from the measured para
nα /d
α
a
d
𝑀 = 𝑀 + 𝑀 = and knowing the stars’ separation, a , and m
gives M (=M1 + M2 ).
...and thus, having the ratio of the stellar masse
To determine the mass, we need to measure the angular separation between the stars,
can find M1 and
Mthe2 distance
individually.
α, and
d from the Earth to the binary system. We can then determine a, the
Figure 11.3: Sketch of a visual binary system showing the (physical) separation a and the
angular separation α between the two components. The distance of the binary system from
the observer is denoted d.
semi-major axis, from the angular separation α and distance d (see Fig. 11.3) as3
a = d sin (α) ' d α.
(11.5)
Eq. 11.4 (with r = a) and a measurement of the period P yields M = (M1 + M2 ). But since
we already know M1 /M2 we can simply calculate the individual stellar masses.
11.2.2
Spectroscopic Binaries
Spectroscopic binaries are known to be binaries because of the periodic shift in spectral lines
observed as a result of the Doppler effect as the two stars orbit around each other – the objects
are too close for them to be resolved in imaging observations. This situation is shown diagrammatically in Fig. 11.4. The velocity components along the line of sight can be measured but we
do not know r, the distance between the two binary components.
The orbital period P can easily be measured from spectroscopic observations. Note that
spectroscopic Doppler shifts only measure the ‘radial velocity’, i.e., the line-of-sight component of the velocity. We measure the inclination of a binary system using the angle i between
3
Beware the units here: a ' dα is correct for α in radians, and a and d in m (or other ‘standard’ distance
units – as long as they are consistent). Given the curious definition of the parsec, you can also use a[AU] =
d[parsec] α[arcsec].
71
3
4
positive
redshifted
2
7
to observer
5
S1
1
com
r2
S2
2
3
3
4
1
1
6
8
5
5
2
4
2
6
1 v
2
5
r1
v1
7
6
radial velocity
8
6
4
8
time
7
3
8
negative
blueshifted
7
Figure 11.4: Orbits and resulting velocity shifts for a spectroscopic binary.
n
l.o.s.
Figure 11.5: The inclination
angle i of a binary system is
the angle between the line of
sight and the normal to the orbital plane.
i
the line of sight from us to the binary system and the normal to the orbital plane of the binary
system (see Fig. 11.5). The radial velocities, v1 and v2 of stars S1 and S2 are then given by
v1 =
2πr1 sin i
P
and
v2 =
2πr2 sin i
.
P
(11.6)
As before, we want to use Eq. 11.4 to obtain M , though we have no measurement of r.
Indeed, often only either v1 or v2 are accessible. We thus first find an expression for r in terms
of one of the ri and then use the expressions in Eq. 11.6 to find an expression for the mass in
terms of the radial velocity. From the centre-of-mass definition
M1
M
r = r1 + r2 = r1 1 +
= r1
.
(11.7)
M2
M2
To replace the orbital radius with radial velocity, we multiply 11.4 by (sin i)3 and substitute
the results from Eqs 11.6 and 11.7. So
4π 2 M 3
M 3 4π 2
v1 P 3
4π 2 3
3
3
3
(r1 sin i) =
r sin i =
,
(11.8)
M sin i =
GP 2
GP 2 M23
2π
M23 GP 2
which implies that
M (sin i)3 =
M3 P 3
v
M23 2πG 1
⇒
(M2 sin i)3
P 3
=
v .
2
M
2πG 1
(11.9)
This expression is called the (binary) mass function. An equivalent equation connects M1 and
v2 ; this follows straightforwardly from the centre-of-mass equation with r1 /r2 = v1 /v2 (see
Eq. 11.6).
72
Sun, Stars and Planets 2020-21
flux
detected lightcurve
secondary eclipse
primary eclipse
period
time
Figure 11.6: Cartoon of an eclipsing binary where two stars orbit each other with the orbital
plane seen ‘edge on’. Here the secondary star (orange) has a smaller radius and is fainter than
the primary star (yellow).
In a ‘single-lined binary’, one of the two stars is so faint that only the lines (and hence
velocities) of one star can be measured. In this case we can measure P and v1 and derive the
mass function from these. For a ’double-lined binary’ P , v1 and v2 can all be measured, and
the masses can be determined, as long as the inclination angle i is known. If i is not known,
then a minimum mass can be found by setting sin i = 1. This is an important result not just
for stars but also for the detection of exoplanets.
11.2.3
Eclipsing Binaries
If the orbit of the stars is seen nearly edge on then one star can pass in front of the other along
our line of sight (see Fig. 11.6). This leads to regular variations in the light received from the
system as the stars alternately pass in front of each other, reducing the total amount of light
seen4 . This means that the angle of inclination i has to be close to 90 degrees. If the velocities
of the stars can be measured in the usual way, their masses can be calculated. The duration
of the eclipses can also be used to estimate the radii of both stars, and it is often possible to
derive the temperatures of the stars.
This configuration is also important in the study of exoplanets, as we will see later.
11.3
Comments on interacting binaries – not examinable
Since the radii of stars expand with increasing age, a stage may be reached where the outer
envelope of the more massive star in a binary system can be attracted onto the companion
binary star. Mass transfer from one star to the other then follows, and angular momentum
can be transferred as well. Such a system is called an interacting binary and the interactions
can have dramatic effects on the evolution of the two stars. Below are just three examples out
of a whole zoo of interacting binaries.
• Algol (β Per:) this was the first interacting binary to be discovered. It has an orbital
period of 2.87 days and is made up of a high-mass main-sequence star and a low-mass
red giant. We know that there has to have been mass transfer between the two sources
4
Usually, the brighter of the two stars is termed the “primary” and the fainter the “secondary”.
73
since the high-mass main-sequence star should have evolved off the main sequence long
before the low-mass star could become a red giant. The star that we are now seeing as a
low-mass red giant must once have had much higher mass, and when it entered its red
giant phase it lost material to its companion.
• Dwarf Novae: these are a class of object that brighten by 2 to 5 magnitudes for a few
days each month. They are thought to be made up from an interacting binary where
one star is losing mass onto an accretion disc around a white dwarf companion. The
matter in this accretion disc builds up until it reaches a critical temperature at which
its viscosity changes and all the material in the disc collapses onto the white dwarf.
The resulting conversion of potential energy into heat provides the energy for the nova
outburst.
• Millisecond Pulsars: These are neutron stars rotating hundreds of times a second,
spun up by the transfer of mass from a binary companion. These objects have been
used to test the theory of General Relativity as they are amongst the best clocks in the
observable Universe.
Things to Remember
• Derivation of relations between binary stars – masses, separations, orbital period
• Using binaries (visual, spectroscopic and eclipsing) to obtain stellar masses
74
Sun, Stars and Planets 2020-21
Lecture 12
An Overview of the Solar System
When you look at the stars and the galaxy, you feel that you are not just from any
particular piece of land, but from the solar system.
- Kalpana Chawla
12.1
Introduction
Much of this half of the course will deal with the objects in our own Solar System. This first
chapter of the planets section provides an overview of those objects, planets and otherwise,
where they are, what their key properties are and how they differ, and what the key physical
drivers were behind their formation. There will also be some discussion about how we arrived
at our present knowledge of the Solar System, both theoretical and observational.
When looking at the properties of extrasolar planets and planetary systems it is also useful to
see how these compare and contrast with the local example of the Solar System. A broad idea
of what our solar system contains is thus a necessary first step in our study of planets.
12.2
Units
In many circumstances, astronomers do not use standard SI units since the numbers involved
are, literally, astronomically large. We thus use, in addition to SI, units that might be called
‘astronomer’s units’ which are based on scalings from known objects or places. We may thus
talk about numbers of solar masses or solar luminosities, scaling to the mass and luminosity
of the Sun. Such quantities are denoted with or Sun eg. M or LSun . Similarly we also scale
to the Earth using ⊕ or E and Jupiter using Jup or J , for example: M⊕ , MJ , R⊕ , RJ .
We also have the special unit of distance called the Astronomical Unit, or AU. This is the
distance between the Earth and the Sun, which is 149.6×106 km.
12.3
Overall Inventory of the Solar System
The Solar System consists of the following objects:
• The Sun - a star with a surface temperature of ∼5780 K, mass of 2 × 1030 kg., and radius
7 × 108 km, with a rotational period of ∼ 27 days and magnetic activity. The first part
75
of this course will have told you much more about the Sun.
• 8 planets. Four of these are terrestrial planets, like the Earth, four are gas giants like
Jupiter. Many of these planets have moons.
• Asteroids
• Kuiper-Belt objects and Trans-Neptunian objects
• All of the above are in the same orbital plane, known as the ecliptic, and are on mostly
circular, prograde (ie. in the same direction as the Sun) orbits. A small number of KBOs
and comets are exceptions to this.
• The Oort cloud - which is roughly spherical, contains about 1011 nascent comets, and
possibly extends out to 10000 AU.
The age of the Solar System is roughly equal to the age of the Sun and the age of the
Earth, which are found to be ∼ 4.6 × 109 years. This is relatively young compared to the
Universe, 13.8 billion years. This means there was gas and dust from previous generations of
stars available when the Sun and Solar System formed.
We shall now look at each of these objects in turn.
12.4
Mercury
Mercury is the closest planet to the Sun. It has no atmosphere and images reveal that its surface is heavily cratered. Surface temperatures range from 740 K in the powerful glare of the
Sun, to 80 K on the far side of Mercury from the Sun. It is clearly a very harsh place. The NASA
satellite Messenger operated around Mercury between 2011 and 2015, and the ESA mission
BepiColombo was launched in October 2018, due to arrive in 2025.
Mercury has a weak magnetic field about 1.1% as strong as Earth’s. This has implications, as
we shall see later, about the internal structure of the planet.
The heavily cratered surface implies that the surface is very old. Some regions are less cratered,
suggesting that they have been resurfaced at some point in the distant past by geological
activity. Other surface features suggestive of tectonic activity exist (eg. faults), but there are
no indications of recent geological activity.
12.5
Venus
Venus is the next planet as we travel outwards from the Sun. Unlike Mercury, its surface features cannot easily be studied since it has a thick, opaque atmosphere, mostly made up of
carbon dioxide. Clouds can be seen in the atmosphere - they are made from tiny droplets of
sulphuric acid. Venus is actually a more hostile environment than Mercury. Apart from the
sulphuric acid rain. the atmospheric pressure is 100 times that of Earth, and the surface temperature is 670 K, hot enough to melt lead.
Despite the challenges of these conditions, several Russian probes have managed to land on
the surface of Venus and beam back images during their brief lifetimes, while the NASA Magellan satellite used radar to see through the obscuring atmosphere and map the surface. These
76
Sun, Stars and Planets 2020-21
missions have revealed a surface with few impact craters and, instead, signs of lava planes and
volcanoes.
Venus lacks a magnetic field, making it different from the Earth and Mercury, and implying
that its internal structure may be rather different to the Earth, which, given that they have
similar mass, is surprising. The young surface, with an absence of cratering, and the absence
of plate tectonics suggest the interesting possibility that Venus goes through periodic total
resurfacing events, where the entire surface is covered by layers of lava. Counting impact
craters suggests that the most recent major resurfacing event could have occurred 300-500 Myr
ago.
12.6
Earth
The Earth is the planet we are most familiar with. In the context of this survey its most important aspects are that it has an atmosphere that is ∼80% Nitrogen and 20% Oxygen and has a
surface temperature of 288 K. The presence of oxygen in the atmosphere is unique in the Solar
System and is something we will discuss later on.
Why do you think oxygen is so unusual as an atmospheric constituent?
Earth has few visible impact craters, indicating that the surface is young. It has a strong
magnetic field, and active volcanoes and tectonic plates. These combined, as we shall see,
provide information on the internal structure of the planet. Water is common, with oceans
covering 70% of the surface.
12.6.1
The Moon
The Earth also has an unusually large moon, the Moon, which has no atmosphere. It shows
many impact craters but also signs of historic lava flows, the mare or seas. The Moon is in
a synchronous orbit with the Earth, so that the same face of the Moon always points to the
Earth. The Moon does not have a significant global magnetic field. The Earth-Moon system
is thought to have been formed through a huge impact between the proto-Earth and a Mars
sized body about 100-150 Myr after the formation of the Solar System.
Since the Moon is so close to Earth it is quite well studied, and it is so far the only extraterrestrial body to have been physically visited by people. However, much remains to be understood.
For example, orbital studies have shown that the far side of the Moon, the side that never
points towards the Earth (sometimes incorrectly called the dark side of the Moon), is quite
different geologically from the near side, being much more heavily cratered. This is thought
to be due to this side of the Moon being older. In situ studies of the far side are underway
with the Chinese Chang’e 4 spacecraft which landed there early in 2019, with operational
rover Yutu-2. A sample return mission, Chang’e 5, was successful in December 2020.
12.7
Mars
Mars is smaller than the Earth or Venus, but larger than Mercury. It has a very thin atmosphere, with a pressure only about 0.6% of Earth’s and mostly made up of carbon dioxide. The
mean surface temperature is 233 K. The surface of Mars has a distinct orangey-red colour due
77
to the colour of the rocks and dust on its surface. There are many huge geological features on
Mars, including the largest volcano in the Solar System, Olympus Mons, which is 24 km high,
and a huge canyon system, Valis Marineris, that extends 400 km across the surface.
Despite the thin atmosphere Mars has strong weather systems with seasons, and dust storms
that can last for weeks and that can cover a significant fraction of the planet’s surface.
The current central question about Mars is whether it was once hospitable for life, and whether
life ever formed there. Mounting evidence for the past existence of liquid water, and the current existence of substantial quantities of water ice adds credence to these ideas, and the new
generation of Mars rovers and orbiting satellites are gathering large volumes of data about
the role of water on the surface of Mars in its distant past. New results from the Perseverance
rover, Tianwen-1 and Hope missions will continue to emerge during the course of these lectures.
12.7.1
Phobos and Deimos: The Moons of Mars
Mars also has two small moons, Phobos and Deimos. They are likely asteroids that have been
captured by Mars’s gravitational field.
12.8
The Asteroid Belt
The asteroid belt lies between the orbits of Mars and Jupiter and is made up of a large number
of rocky and metallic bodies ranging in size from Ceres, with a diameter of 950 km, downwards,
with many, many more small bodies than large. Those that have been studied in detail have
plentiful impact craters. The NASA Dawn mission visited the asteroid belt in the last few
years, studying the asteroids Vesta and Ceres in detail, so we are leaning much more about
these objects.
12.9
Jupiter
Jupiter is the largest planet in the Solar System and the first ‘gas giant’. It is composed mostly
of hydrogen (90%) and helium(10%) with traces of methane, ammonia and water vapour. The
features we see on Jupiter are not those of a solid ‘surface’ but are in fact ever-changing cloudscapes that lie at the top of its deep atmosphere. The different colours of the cloud bands represent detailed differences in composition, chemistry and depth. An example of one of these
weather systems is the Great Red Spot, a storm system larger than the Earth, that has persisted for several hundred years. The temperature of the cloud tops that we see is ∼120 K,
but the temperature and pressure will rise going deeper into the atmosphere. At a depth of
10000 km, the temperature should be ∼6000 K with a pressure 106 times that on the surface
of the Earth. Jupiter also has a large and powerful magnetic field, 20000 times stronger than
that of Earth.
12.9.1
The Moons of Jupiter
Jupiter has a large number of moons, dominated by the four large ‘Galilean’ satellites, socalled because they were first observed by Galileo, called Io, Europa, Ganymede and Callisto,
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Sun, Stars and Planets 2020-21
in order outward from Jupiter.
Io is the most volcanically active body in the Solar System, with a large number of active
volcanoes and a surface covered by sulphur deposits from the eruptions - its surface visibly
resembles a giant pizza! The heating necessary to maintain this level of volcanic activity comes
from ‘tidal heating’, something we will discuss later. Io is largely made up of rocky material,
not dissimilar from the Earth.
Europa, the next moon outwards from Jupiter, is very different from Io, with a surface that is
made up of ice. Despite this, Europa is a largely rocky body, but the icy layer is expected to be
about 100 km deep. The icy surface shows few impact craters, and instead appears to be made
up of broken ice packs and fractured plains. This suggests that liquid water may at times
reach the surface through ‘cryovolcanoes’. Possible signs of Water vapour plumes escaping
from these may have been detected by the Hubble Space and Keck Telescopes, but we await
the JUICE and Europa Clipper missions’ visits later this decade. This possible water vapour
detection suggests the possibility that a subsurface ocean of liquid water might lie beneath
the surface, kept liquid through similar tidal heating processes to those on Io, but operating
at lower temperatures.
The remaining Galilean moons, Ganymede and Callisto are broadly similar, made predominantly of ice and with heavily cratered, old surfaces. Ganymede is the largest moon in the
Solar System, with a diameter larger than that of Mercury, but, since it is largely made of ice
rather than rock, it has a substantially smaller mass.
All four of Jupiter’s Galilean moons lie within its magnetosphere, and are thus bombarded
by charged particles. Io is especially strongly affected, and suffers from an especially harsh
radiation environment as a result. Sulphur and oxygen atoms released by Io’s volcanism are
heated by the charged particles in Jupiter’s magnetosphere and escape the moon’s gravity
to eventually form a ring of plasma around Jupiter. Ions streaming from this ‘plasma torus’
are picked up by the magnetic field and accelerated into Jupiter’s ionosphere, producing an
electrical current of several million amps and leading to spectacular aurorae around Jupiter’s
poles.
12.10
Saturn
Saturn is the second biggest gas giant in our Solar System, having a radius about 15% smaller
than that of Jupiter. Its atmosphere also has a banded appearance similar to that of Jupiter.
Storm systems have been observed in Saturn’s atmosphere by the Cassini spacecraft, but nothing on the scale of Jupiter’s Great Red Spot. Saturn has such a rapid rotation speed, with a day
only 10.7 hours long, that there is significant atmospheric bulging at Saturn’s equator. Similar
to Jupiter, Saturn also has a large magnetic field.
12.10.1
The Rings
The most distinctive feature of Saturn, of course, is the ring system. While all the gas giants
have ring systems of some kind, Saturn’s is the most visible. It is not solid, but is made up
of many small icy and rocky particles, ranging in size from 1 cm to a few metres. The rings
are most likely the result of the break up of a moon following a catastrophic impact. The ring
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particles all orbit in the equatorial plane of Saturn, creating the disk we see. The rings themselves are surprisingly thin, only 10 m thick, but the reflectivity of the particles makes the ring
system highly visible in reflected sunlight.
The rings are structured into many smaller subrings as a result of the gravitational influence
of small ‘shepherd’ moons that are among Saturn’s 61 satellites.
12.10.2
The Moons of Saturn
Saturn’s moons are generally small, with only seven having radii greater than 200 km. However, the largest of these, Titan, is one of the most interesting objects in the entire Solar System.
Titan is one of the largest moons in the Solar System, and is roughly half the size of the Earth.
It has a thick atmosphere that is predominantly nitrogen and methane. The remainder of
the atmosphere, <1%, is made up of complex hydrocarbons. These make Titan’s atmosphere
opaque, but also indicate that complex hydrocarbon chemistry is taking place. The Cassini
spacecraft and the Huygens lander have examined Titan in detail, and have revealed a surface
of ice beneath the smoggy atmosphere, together with lakes and seas of liquid hydrocarbons,
filled by a rain of ethane and methane.
Another moon of Saturn that has aroused considerable interest of late is Enceladus. Its surface
has few impact craters, but is instead covered by many cracks, suggesting that it has been
resurfaced through cryovolcanic activity at some time in its past. Direct evidence for this was
found by our Head of Department Prof Michele Dougherty and her team, who discovered
geysers of water vapour, mixed with other compounds, being vented into space from cracks
in Enceladus’ surface.
12.11
Uranus
Uranus, like Neptune, is smaller than Jupiter or Saturn, but still has a mass 15 times that of
Earth. It appears as a rather featureless blue-green planet. It is unusual in the Solar System in
that its axis of rotation is tipped ∼98o away from being ‘vertical’ to the plane of the ecliptic.
It is suspected that this is due to a major impact in its earlier history that knocked the planet
onto its side. One pole of Uranus thus always points towards the Sun, while the other always
points away. This results in unusual atmospheric flows with one side of the planet always
warmer than the other. Uranus has a ring system, the second most prominent in the Solar
System, but its ring particles are much darker than those found in Saturn’s rings. Like all the
other gas giants, Uranus has a large magnetic field.
Uranus has at least 27 moons, but only five are larger than 200 km in radius. Some of these
show evidence for cryovolcanism in their past.
12.12
Neptune
Neptune is the last planet in our Solar System, and is the last of the four gas giants. It has a
mass about 17 times that of Earth. Its atmosphere is a distinct blue colour resulting from the
small amount of methane it contains absorbing light at the red end of the spectrum while the
rest is reflected. More features are observable in Neptune’s atmosphere than in Uranus’, with
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Sun, Stars and Planets 2020-21
banding and pale clouds being detectable by both flypast missions such as Voyager and by remote observation from the Hubble Space Telescope. Giant storm systems also occur, although
the ‘Great Dark Spot’ detected by Voyager 2 in 1989 had gone away by the time HST observed
the planet in 1994.
Neptune has at least 13 moons, but only three have a radius larger than 200 km. The largest
of these, Triton, is an unusual object that orbits in the opposite direction (a retrograde orbit)
to all the other Neptunian moons around Neptune. This suggests that it did not form at the
same time as Neptune and the rest of its moons, but was instead captured by the planet at a
later date. Such a capture would have been associated with impacts and other activity that
would leave their mark on the surface of Triton, and indeed we find that Triton has a strange
divided surface, with geyser-like plumes evident in one area, and a rough, resurfaced, geography elsewhere. The geysers are likely responsible for Triton’s tenuous nitrogen atmosphere.
12.13
Pluto, Trans-Neptunian Objects (TNOs) and the Kuiper
Belt
Beyond Neptune, there are no single dominant mass planets. Instead, there is a plethora of
small bodies that form a belt of objects, known as the Kuiper Belt, extending outwards from
the orbit of Neptune. The first of these to be discovered was Pluto, long regarded as a planet
in its own right, but the discovery of other, similarly sized, if not larger, TNOs starting in the
1990s has led to a re-evaluation of Pluto’s status. The discovery of Eris, which is larger than
Pluto, tipped the balance, and Pluto was demoted to being a ‘minor planet’ by the International Astronomical Union in 2006.
Kuiper belt objects are left overs from the formation of the Solar System and, since they represent relatively pristine material from the formation epoch, they are of great interest. Their
distance from the Sun makes them difficult to study, but the NASA New Horizons mission
flew past Pluto in 2015, and told us much more about these objects. It also visited a small
Kuiper Belt object (Arrokoth, also known as ”Ultima Thule”) which was found to be a strange
shaped object, formed from a slow collison between two separate objects.
12.14
Comets
Comets are small bodies from the outer Solar System whose orbits take them close to the Sun.
When this happens, their surface heats up, and volatiles boil off, forming the distinctive tail
that, in the case of bright comets, can even be visible during daylight. Comets come in two
different types, defined by whether they are short or long period. Short period comets are
thought to come from the Kuiper Belt, while long period comets, with periods greater than
about 100 years, come from further away. They come from the last, and most distant, part of
the Solar System to be discussed here: The Oort Cloud.
The ESA Rosetta mission led to the rendezvous with the comet 67P/Churyumov-Gerasimenko
in 2014, dropping the lander Philae on the comet, the first landing onto a comet’s surface.
Rosetta then followed 67P as its orbit took the comet on its closest approach past the Sun. The
mission ended in 2016 with the Rosetta orbiter descending to the surface and being turned
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off. This comet, with ”rubber duck” appearance, was found to consist of two bodies that had
gently collided and fused.
12.15
The Oort Cloud
The Oort Cloud is named after Jan Oort, the astronomer who first suggested its existence.
Oort’s idea was that a large population of cometary bodies, that formed in the inner Solar
System at the same time as the rest of the planets, would be thrown out of the Solar System
by gravitational interactions with giant planets like Jupiter and Saturn. The resulting cloud
of comets could include as many as 1011 objects and extend to tens of thousands of AU in
distance, forming a roughly spherical cloud surrounding the Solar System. Long period comets,
with high inclinations relative to the ecliptic, many of which have retrograde orbits, fall into
the inner Solar System from the Oort Cloud.
12.16
Kepler’s Three Laws of Planetary Motion
Looking at the motion of the objects of the Solar System, Johannes Kepler derived three laws
of planetary motion using observations made by Tycho Brahe:
i. Planets follow an elliptical orbit with the Sun at one focus
ii. The line joining the planet and the Sun sweeps out area at a constant rate
iii. The square of the time a planet takes to go round the Sun, P, is proportional to the cube
of the semi-major axis of its orbit ie.
P 2 ∝ a3
Kepler’s explanation for these laws involved Platonic solids and the harmony of the celestial
spheres. It was not until Newton turned his attention to planetary orbits that we arrived at
something we would recognise today as a full physical explanation of Kepler’s laws, using
Newton’s laws of motion and gravitation. We do not cover the full formal derivation of these
laws in this course, however you have seen a simple derivation of the third law in the lecture
on binary stars.
Note that for the ellipse of the orbits, there are then four cases, depending on the value of
eccentricity of the ellipse e.
• For e = 0 we simply get a circular orbit.
• For 0 < e < 1 we have an ellipse with eccentricity e - this is in fact the situation for the
orbits of all the planets, proving Kepler’s first law.
• For e = 1 we get a parabola.
• For e > 1 we have a hyperbolic trajectory.
The first two of these options are bound orbits, so is what we see for the planets. The eccentricity of Earth’s orbit is currently e = 0.017, and comet 67P has e = 0.64. The last two
options are unbound orbits, and are what is seen for objects that achieve escape velocity. In
the case of the Solar System, the Pioneer and Voyager satellites have managed this, and will
travel forever between the stars of our Galaxy. Mysterious visitor to our Solar System in 2017,
Oumuamua, has e = 1.2 and is unbound.
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12.17
Sun, Stars and Planets 2020-21
Formation of the Solar System
The physics of star and planetary formation is a large and complex topic. The core concepts,
though, emerge from several observational facets of our Solar System, which allow us to get
some basic idea for these processes without going into details. The first key observation is that
the orbits of most bodies in the Solar System are roughly circular, and they are all prograde
ie. orbiting in the same direction as the Sun’s rotation. Another observation is the pattern of
density of planets decreasing with distance from the Sun, rocky inner planets, gas giant outer
planets. These observations suggest that the Sun and planets all formed as part of the collapse
of a solar nebula. This broad picture was first developed by Laplace in the late 18th century.
The starting point is a slowly rotating molecular cloud, consisting mainly of hydrogen, but
also traces of dust – ices, carbon, metallic and silicate substances. This cloud starts to collapse
under the force of its own gravity - more on this will be covered in the Astrophysics course.
As this happens, thanks to conservation of angular momentum, the cloud contracts and spins
faster. The collapse then continues, perpendicular to the rotation axis. The pre-stellar material, made up of dust and gas, settles into the rotation plane and the collapse proceeds fastest
at the centre, where the Sun will eventually condense. Away from the centre, clumps of dust
and gas start to coalesce as the diffuse disk material breaks up. Once these planetesimals
reach 10 km in size they begin to accrete material themselves through runaway gravitational
attraction, ”gravitational focusing”, eventually forming a few 100 or so planetary ”embryos”,
with size about 1000 km. At the same time the infant Sun ignites, starting fusion in its core,
and the resultant powerful stellar wind clears away gas and dust that is not already gravitationally bound into condensed objects. Chance collisions between planetary embryos cause
giant impacts, resulting in fragmentation, melting and reforming of the material into a new
combined heated mass, which would then slowly cool. This led to formation of the 9 planets
we are familiar with. The eventual composition of a planet will depend on where in the solar
nebula the planet formed.
12.17.1
Formation of Rocky/metallic planets versus gaseous planets
It is important to remember the temperature distribution of the protoplanetary disc. The
material was hotter nearer the young Sun, and temperature fell towards the edges of the disc.
In the inner hotter regions (with T>500 K), small ice particles could not exist, and so accretion
of rocky metallic material dominates. In the outer regions however (T<300 K), ices could
remain frozen and go to form planets – the ices could be melted in planetary formation, but
would be retained as gas due to the gravitational field of the planet. So rocky metallic planets
formed in inner solar system, gaseous planets in outer solar system.
12.17.2
Outer solar system: Formation of giant planets: the standard model
The larger volume of space occupied by planet forming material in the outer Solar system
meant it produced fewer but larger planetary embryos. An embryo of mass 5 − 10M⊕ would
form around 5 AU after approximately 4 × 105 years (ten times as long as it took to form an
embryo at 1 AU). This hypothetical body would act as a kernel that by gravitational focusing would sweep up planetesimals, smaller debris and nebula gas leading to the formation of
Jupiter. As the planet grew it captured hydrogen gas and helium from the solar nebula and
this gas became the Jovian atmosphere. The kernels for Saturn, Uranus and Neptune took
progressively longer to form, maybe 2, 10 and 30 million years. This increasing timescale with
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distance from the Sun is probably the key to understanding the outward trend in composition
among the giant planets. Jupiter has approximately 300 M⊕ of H2 and He, Saturn 70 M⊕
and Uranus and Neptune 1 M⊕ each – why the diminishing amounts of gas? - The kernels of
Saturn, Uranus and Neptune formed later so maybe most of the gas in their neighbourhood
had been removed before they grew massive enough to attract much gas gravitationally. The
removal of the gas in the neighbourhood of the forming gas giants would be caused by the
young Solar wind, so that no more gas could be captured by the giant planets. Calculations
would predict that the Earth should have been able to capture an atmosphere ≈ 0.03M⊕ from
the solar nebula, ≈ 3 × 105 times the mass of its present atmosphere (which has a totally different composition). We would conclude that the Earth’s primitive atmosphere (and those of
other terrestrial planets) was probably lost during the phase of the early Sun’s strong Solar
wind.
12.17.3
Solar system formation and formation of planetary systems around
other stars
This is the broad picture of star and planet formation that astronomers work with, but the
details are still uncertain. Observations of planets in other star systems, ie. exoplanets, are
prompting rapid development in this field. For example, it seems that gas giant planets often
migrate to the inner regions of a star system, even though they have to form at distances from
their parent star comparable to those of Jupiter and Saturn.
12.18
Summary
This lecture has provided a brief tour of the Solar System, showing both the variety of objects
in it, and looking at some of the features common to all of them. Solar System science is a
very rapidly moving field, with active research going on from the ground and in space.
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Things to Remember
• The names of the planets
• Their order going out from the Sun and that they lie in the same plane - the
ecliptic
• The basic geography of the Solar System including the asteroid belt, the most
famous moons/rings, the Kuiper Belt and the Oort Cloud
• Kepler’s 3 laws as stated
• How to explain the consequences of Kepler’s laws with respect to circular, elliptical, parabolic and hyperbolic orbits
• Be able to show the 3rd law in the context of a circular orbit (refer back to Binary
Stars lecture)
• The basic principles behind our model of the formation of the Solar System
• The natural consequences of this model with respect to prograde orbits, and variation of planetary composition with distance from the Sun
To Do
• Problem Sheet 3, Question 1. This question looks at densities of Solar System
bodies, and how these relate to the nature of these bodies and their formation.
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Lecture 13
Terrestrial Planets: Heating, Cooling
Processes and Interiors
”We take it that, as before, the Earth consists of a core and a mantle, but that inside
the core there is an inner core in which the velocity is larger than in the outer one.”
- Inge Lehmann, Geologist (1936)
13.1
Introduction
The Earth is a terrestrial planet, along with Mercury, Venus and Mars. There are commonalities
between them, but also substantial differences. In this chapter we will look at the internal
structure of terrestrial planets and the factors that drive that structure. This will provide some
key insights into how and why the four terrestrial planets differ. Along the way we’ll also
uncover some of the forces that have shaped the Earth and its geography over its 4.5 billion
year history.
13.2
The Active Earth
In London, it is easy to think of the Earth as fixed and unchanging, but we know that this
is not in fact the case. Earthquakes and volcanic eruptions are just two reminders that our
planet is a dynamic system, even if much of that dynamism operates on timescales far longer
than that of a human life.
The Earth beneath us is in fact a lot more dynamic even than that, as can be seen when material
from deep beneath the surface bursts out during a volcanic eruption. Where did the energy
for that heat come from and how has this driven the large scale structure of the Earth and the
surface features we see today?
13.3
Primordial Heating
The formation of the Earth was a violent process, with large impactors peppering the forming
planet and with smaller bodies accreting at a high rate. The kinetic energy of these impactors
is largely turned to heat during the collision, and the accretion of smaller bodies also leads to
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Sun, Stars and Planets 2020-21
heating of the young Earth. In general, the potential energy of all the mass that falls onto a
planet during its formation, which is converted to heat, is given by:
PE ∼
GM 2
R
(13.1)
where M is the mass of the planet, G is the Gravitational constant, and R is the radius of the
planet.
The immediate consequence of all this energy being deposited into the young planet as it is
forming, is to make much of the material molten, and to keep it that way for many thousands
of years.
13.4
The Structure of the Earth
The material that made up the forming Earth includes substances with a range of densities. If
a mixture of liquids with different densities is allowed to settle, the densest material will end
up at the bottom, in this case at the centre of the Earth. The constituents of the young Earth
can be determined by looking at the constituents of meteorites. These include Al, Si, Ti, Fe, Ni,
Mg, Ca, with Fe and Ni being quite abundant. The densest of these materials are Fe and Ni, so
these constituents separated out, and fell towards the core of the Earth, leading to the formation of a solid, largely iron core, surrounded by a liquid nickel-iron outer core. Above this is the
mantle, made of a material called peridotite which includes minerals containing Mg, Ca, Fe,
Al, Si, Na, O, Cr, but which is essentially 40-60% SiO2 . The minerals that make up peridotite
include feldspar, olivine, pyroxene, spinel, garnet and others. Above the mantle lies the crust,
much of which is made of basalt, ∼75% SiO2 . On top of this crust are the sedimentary rocks
produced by erosion processes which make up most of the landscapes that we can see on the
surface (see Fig. 13.1).
The separation of the Earth into core, mantle and crust is based on the results of seismology
- essentially looking at how the speed of sound changes as seismic waves travel through the
Earth. Figure 13.2 shows how the physical properties that affect the transmission of seismic
waves change with depth, and their effect on the passage of seismic waves.
An alternative way of thinking about the structure of the Earth is based not on the constituents
of the material but on its physical state. This leads to a different classification that we will find
useful later on. In this approach the core is the same, but the mantle is then divided not into
the upper and lower mantle, which is based on composition, but into the region where the
rock is molten or under sufficient pressure that it can flow — the asthenosphere — and the
region where the rock is rigid — the lithosphere — where flow is not possible. The upper parts
of the mantle and the crust make up the lithosphere. This is also shown in Fig. 13.1.
13.5
Long Duration Heat Sources
As well as the initial heat input from the formation of the planet, there are two long duration
sources of heat that help to keep the Earth’s interior hot. The most important of these is radiogenic heating from long lived unstable isotopes. The radiation given off by their decay is
absorbed by their surroundings, leading to an increase in temperature. Table 13.1 summaries
87
Figure 13.1: Structure of the Earth from Fig. 2.1 of Rothery, McBride & Gilmour.
the most important long lived isotopes in the Earth for long term radiogenic heating. The
Earth’s age is 4.5 billion years, which is comparable to the half lives of these isotopes.
Isotope
235 U
238 U
232 Th
40 K
Half-life (109 ) yrs
0.71
4.5
13.9
1.3
Present Rate of Heat Generation (10−12 W kg−1 )
0.125
2.91
3.27
1.08
Table 13.1: Half-lives of the most important radiogenic heat sources in the Earth’s crust and
mantle today, and present rate of heat generation per kg of crust and mantle.
Tidal heating is the other potential source of long term heating. Tidal heating comes from
the effects of a nearby orbiting massive body - in the case of the Earth, the Moon produces
tidal effects. The most noticeable are the tides in the Earth’s oceans, but there is also a ∼1 m
maximum rise and fall of the Earth’s rocky surface due to the Moon. This deformation of
the Earth imparts energy which appears as heating. It is thought that this heating is largely
deposited in the crust and mantle. The amount of energy imparted to the Earth from the
Moon by this tidal interaction is small, nearly two orders of magnitude less than the energy
input from radiogenic heating, but tidal heating is very important for other bodies in the Solar
dout: Structure and Atmospheres of Planets
88
4: Interiors of the Earth
and terrestrial planets
Sun, Stars and Planets 2020-21
e Earth’s structure
sitional:
rich)
60% SiO2)
% SiO2)
d outer shell
: convecting part of the mantle
bs.usgs.gov/gip/dynamic/inside.html Interior structure of the Earth, taken from
Karttunen et al Fundamental Astronomy
Figure 13.2: Internal structure of the Earth and how this changes the results of seismology
eg. speed of seismic
vs. depth.of
Taken
Karttunen et al. Fundamental Astronomy.
Left:waves
Comparison
the from
composition
of the terrestrial planets and the
Moon
(McBride & Gilmour)
System, as we will
see later.
Below: relative sizes & core masses
13.6
The decay
of long
heating
(in %) taken
fromterm
Karttunen
et al sources
While radiogenic heating and tidal heating persist today, they are not an infinite resource.
With time the radioactive species responsible for heating will decay away. The tidal heating
rate will also fall as angular momentum is transferred from the rotation of the Earth to the
orbit of the Moon, and as the Moon moves away from the Earth. Ultimately, therefore, the
interiors of all terrestrial planets cool.
13.7
Heat Loss from Planets
The Earth cools by radiating heat away into space. Volcanic eruptions are the most obvious
example of this, but there are many other, less dramatic ways in which the heat from the upper layers of the asthenosphere travels through the lithosphere. The amount of heat that the
Earth, or any other planet, can thus expel is determined by its surface area. In contrast, the
amount of heat, and, indeed, the amount of active radiogenic heating underway, is dependent
terrestrial planets
89
zes and core
planet and terrestrial moon structure
%
me
McBride
& Gilmour
Introduction
SolarNote
System
Figure 13.3: Comparison
of the interiors
of the 4 terrestrial
planets to
andthe
the Moon.
that
the smaller the object, the thicker the crust. The Moon, for example, has a crust that is 1000 km
deep. From McBride & Gilmour.
on the volume of the Earth, or other object.
Heat loss rate ∝
Surface area
4πR2
1
=
∝
3
Volume
4/3πR
R
(13.2)
Thus smaller planets cool more rapidly than larger planets. The long term result of cooling is
that the lithosphere thickens and the asthenosphere becomes thinner. Mars, for example, has
a much thicker crust than the Earth, and Mercury has an even thicker crust. See Fig 13.3 for
comparisons of the interior structures of the terrestrial planets.
13.8
Cooling Processes
How does the heat travel from the core of the Earth to the surface?
There are four key processes that allow the Earth to cool:
• Conduction
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Figure 13.4: Convection transferring heat from the core to the surface. Fig 2.14 of Rothery,
McBride & Gilmour.
This is the most familiar process, whereby heat transfers from a hotter to a cooler region
through thermal conduction. In the lithosphere, where rocks are rigid and cannot flow,
this is the main method of heat transference.
• Convection
In the asthenosphere, where material is able to flow, convection operates, and is the
most efficient way that heat is transferred. Hotter material expands, and is thus less
dense, so rises, while cooler material contracts, becomes more dense, and falls. The
cooler material then warms up and the process continues. Large scale convection cells
exist in the asthenosphere, where this process can operate. Solid state convention, in
which rocks flow by a few cm/year, drives this process. See Fig 13.4 for a diagram of
how this operates.
• Eruption/Advection
The lithosphere is too rigid to allow convection, so the last stage of the process of heat
transference from the core to the surface takes place when molten rock, or magma,
spreads over the surface and cools, or as it is injected into the lithosphere and cools
beneath the surface, and the heat is conducted away to the surrounding crust.
• Plate tectonics
The surface crust of the Earth is made up of a series of plates that essentially float on the
surface of the asthenosphere. Some of these plates are thicker, and form the continents,
91
Figure 13.5: The key features of plate tectonics: sea floor spreading, continental plates, subduction zones, and arcs of volcanoes around the edges of continental plates. From USGS
while others are thinner, and form the floor of the oceans. The plates move relative
to each other, with new material being produced by hot magma emerging from the
asthenosphere at mid-ocean ridges, leading to sea floor spreading, and with old, cold,
material sinking into the asthenosphere at the edges of continents in subduction zones.
13.9
Volcanism and Tectonics on Other Terrestrial Planets
Given that the cooling rate of a planet is set by its surface area to volume ratio, we would
expect smaller planets, and similar objects like the Moon, to have thicker lithospheres and
thus be less tectonically active. The smallest terrestrial planet in our Solar System is Mercury.
Studies show that its surface is old, as evidenced by heavily cratering, but that there are regions where some resurfacing has taken place. Our best estimate is that this resurfacing took
place roughly a billion (109 ) years ago (1 Gyr). The Moon, while not a planet, shares many of
the properties of a terrestrial planet, so is another useful check of our ideas about planetary
volcanism. Like Mercury, there are regions of the Moon that are old and heavily cratered, but
others, the maria, or seas, that are younger and appear to have been resurfaced by more recent
lava flows. These are also old, about 1 Gyr old, and are thought to be related to impact events
that punched holes through thinner portions of the Moon’s lithosphere, allowing lava to flow
over the surface.
Venus, with a similar size and mass to Earth, would be expected to have a similar lithosphere
thickness and thus similar tectonic activity. However, observations, conducted using the radar
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mapping instruments of the Magellan spacecraft, have found no evidence for tectonic activity.
The surface of Venus, though, is young, showing none of the extensive cratering that is seen
on the older surfaces of the Moon or Mercury. If there are no tectonic plates on Venus, how
does its interior cool?
One idea is that the lithosphere of Venus acts like the lid on a huge pressure cooker. Instead of
the continuous leaking of internal heat that we see on Earth, the idea is that Venus occasionally
blows its top, with the entire planet being resurfaced through periodic volcanic catastrophes.
The surface of Venus appears to be between 700Myr and 500Myr old, which would set the
date of the most recent catastrophic resurfacing. Volcanoes are seen on Venus, often showing
a strange, flat topped appearance suggesting slow growth. Historical lava flows up to 2000
km have been found, and there is some evidence of ongoing volcanic activity. However, the
continuous recycling of surface rocks into the mantle through subduction zones, and gradual
volcanic resurfacing at the edge of tectonic plates seen on Earth does not appear to be taking
place on Venus. Many volcanoes on Venus are presumably awaiting the next epoch of catastrophic volcanism.
Mars is intermediate in mass between Mercury and the Earth, so might be expected to have
an intermediate level of tectonic activity. There is indeed some evidence of tectonics on Mars,
with significant differences between the northern and southern hemispheres. The southern
highlands are similar to the thick crust of Earth’s continents, while the northern lowlands
are similar to the thinner crust of the oceans. However, the lithosphere of Mars is now much
thicker than that of Earth, so any ancient tectonic activity is likely to have stopped long ago.
Age estimations using cratering statistics suggest that the southern highlands are older, at
about 4.5 Gyr, while the northern lowlands and the Tharsis Bulge, home to Mars’ giant volcanoes, are younger at 3.7 Gyr.
Mars has the largest volcanoes in the Solar System, including the giant Olympus Mons. These
are all found in the Tharsis Bulge region, which appears to be similar to the ‘hot spots’ found
in several locations on the Earth. These hot spots seem to be the result of upwellings in the
asthenosphere at certain positions in the mantle. These mantle plumes bring heat into the
lithosphere in a way that is largely separate from tectonic activity. The Hawaiian Islands on
Earth are a result of a hot spot located in the middle of the Pacific Ocean plate, well away from
any region of sea floor spreading. Since the Pacific plate is moving, the volcanic islands, that
grow around the location of the hot spot, are gradually dragged away from the hot spot, leading to the production of a chain of volcanic islands and, further away, a series of sub-surface
sea mounts. On Mars, there is no tectonic activity, so the large ‘shield’ volcanoes that grow
from them just continue growing. This is why Olympus Mons is the largest volcano in the
Solar System.
93
Things to Remember
• The structure of the Earth including the names, constituents and properties of
different layers
• Heat sources for terrestrial planets, long and short duration including primordial
heating
• Heat loss processes, including conduction, convection, advection/eruption &
plate tectonics
• Dependence of heat sources and heat losses on the size of a body
• Consequences of this dependence of heat loss rate on object size, for the internal
structure and surface volcanism on other planets
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Sun, Stars and Planets 2020-21
Lecture 14
Terrestrial Planet Surfaces and
Temperatures
Touchdown confirmed. Perseverance is safely on the surface of Mars ready to begin
seeking signs of past life
- Swati Mohan, Flight Controller, Perseverance Team, NASA, February 2021
14.1
Introduction
In the previous lecture we found that the surface properties of planets are far from typical of
their interiors. The vast majority of the Earth’s volume is made up of molten or warm rock,
flowing, albeit slowly, in giant convection currents. The continental plates that make up the
surface of our planet are just low density material floating on this ocean of rock. However,
the surface of the Earth is something that we are intimately concerned with, and the surfaces
of other planets are, by and large, all that we can see of them. Understanding the processes
responsible for the planetary surfaces that we see, and that determine the basic properties of
planetary environments, including surface temperature, are thus a key ingredient to understanding the nature of planets in our own and other solar systems.
14.2
Major Factors in Shaping Planetary Surfaces
While the four terrestrial planets are all very different in appearance, they are all shaped
by similar physical processes. The relative importance of these different processes, however,
varies between planets, and this will apply just as much in other star systems as it does in our
own solar system. There are four central processes that determine the surface geography of
planets:
• Impact cratering
The majority of impact events occurred during the earliest stages of the Solar System,
but impacts continue to happen today, but at a much lower rate. Examples on Earth
include Meteor Crater in Arizona, a crater 170 m deep and about 1.2 km across that
was produced by an impact about 50,000 years ago, and the Tunguska event, likely an
airbursting small meteor, that levelled 2150 sq. km of forest in Siberia in 1908. In February 2013, a somewhat smaller meteor passed over the Russian city of Chelyabinsk. The
shockwave as it passed through the atmosphere causing moderate damage over a large
95
Planet
Cratering
Tectonics
Volcanism
Erosion
Comments
Mercury
Heavily cratered,
very old surface (∼4.5 Gyr)
No (some in past)
No
No
Most geologically inactive
terrestrial planet
Venus
Few craters, surface
only ∼500 Myr old
No tectonics
seen
Past volcanoes
seen
Yes
Catastrophic
resurfacing possibility
Earth
Few craters, geologically
active, erosion effects
Yes
Yes
Yes
Interior not yet solidified
Mars
Heavily cratered in parts
No current
tectonics
Past giant
volcanoes seen
Yes
Probably almost
solidified
Table 14.1: Summary of role of shaping processes in terrestrial planets
area and injuring roughly 1500 people. With an estimated mass of 12,000-13,000 tonnes
and a size of 20 m, the Chelyabinsk meteor is probably the largest natural object to enter
the Earth’s atmosphere since Tunguska.
• Volcanism
As discussed in the previous lecture, volcanoes are where the heat contained within a
planet can escape to the surface and, over geological timescales, these allow the planet’s
interior to cool. The effects of lava flows can be seen in many places in the Solar System,
and there are giant volcanoes on Mars.
• Tectonics
The surfaces of some terrestrial planets are made up of tectonic plates that float on top
of the hot, molten, interior. The interactions of these plates, and their movements on
the surface, are a driving force for shaping the surface of planets.
• Erosion
In the presence of a fluid, whether gas or liquid, surface features are eroded and modified
over time.
The importance, or otherwise, of each of these factors varies from planet to planet. Those
planets that have cooled rapidly, for example, so that their lithospheres are thick, are less likely
to experience tectonic or volcanic activity, while those with no atmosphere will not experience
extensive erosion. The importance of these factors for each of the terrestrial planets is listed
in Table 14.1.
14.3
Impact Cratering
Cratering is ubiquitous throughout the Solar System, caused by the impact of small bodies
with larger objects. Impacts can be thought of as the process of accretion of planetary material that is continuing to this day, though at a much lower rate than during the epoch of
planet formation. Younger surfaces on planets and moons have fewer craters, and this can be
used to date the surfaces that are seen. Where large scale resurfacing has occurred, through
volcanic activity, for example, signs of cratering are erased. Erosion, also, can erase the signs
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Sun, Stars and Planets 2020-21
of cratering given sufficient time.
During an impact, rocks are heated and subjected to very high pressures. Rocks melt and
fracture as a result. Material will be ejected from the impact crater, in both solid and molten
form, and a cavity is excavated leading to the familiar crater shape. A sufficiently powerful
impact will expel large quantities of material, leading to large area effects. Ejected rocks can
sometimes be expelled from the atmosphere and can even be given sufficient kinetic energy
to achieve escape velocity. In fact some meteorites found on Earth were ejected from Mars by
past impacts.
Impacts on water, which will be common in the case of Earth since it is 70% covered by water,
would produce massive tsunamis.
A large enough impact will produce significant environmental damage. The extinction of the
dinosaurs has been linked to the Chicxulub crater, found beneath the Yucatan Peninsula in
Mexico. Even bigger impacts can change the nature of the objects involved. The Moon, for
example, is thought to have been formed as a result of an impact between the young Earth
and a body roughly the size of Mars.
The impact rate in the Solar System has been in decline for at least the past ∼4 Gyr, but there
are suggestions, based on crater counting studies on the Moon, that there was a brief increase
in the impact rate about 3.8 - 4 Gyr ago. This phase in the development of the Solar System
has been termed the Late Heavy Bombardment, and may be linked to broader aspects of the
evolution of the Solar System.
14.4
Volcanism and Tectonics
The physical background to volcanism and plate tectonics was discussed in the previous lecture. Both can have a considerable effect on shaping planetary surfaces. Evidence for historical
large scale resurfacing events involving volcanic activity can be seen on the Moon and Mercury,
but these appear to have ended 3 Gyr ago. Volcanoes are clearly present on Mars, including
the massive Olympus Mons. The surface of Venus appears to be geologically young, less than
0.5 Gyr, and volcanoes have been seen on its surface in radar mapping observations by the
Magellan satellite, but the lack of tectonic activity on Venus has led to the idea that its surface
is periodically subject to catastrophic volcanism, where it is completely resurfaced from time
to time.
On Earth, there is evidence for large scale resurfacing during events known as flood basalts.
Examples of these include the Deccan Traps in India, about 65 million years old, the Columbia
River flood basalts, about 15 million years old, and the Siberian Traps, about 248 million years
old. These flood basalts are made up of tens to hundreds of separate lava flows stacked on top
of each other, reaching thicknesses of 1 to 3 km and covering thousands of square kilometres
of the surface. They are the result of the production of lava volumes up to 2 million cubic kilometres in size that erupted over timescales of 1 to 5 million years. This represents an annual
eruption rate over twenty times greater than those observed for present day hot spots such
as Hawaii. Many of the historical flood basalts are associated with mass extinctions, and the
volume of lava, ash and gas they produced was certainly enough to cause major environmental effects.
97
Planet
Venus
Earth
Mars
Mercury
Albedo
0.77
0.3
0.25
0.10
Mean Surface Temperature (K)
733
288
223
443
Surface Atmospheric Pressure (bar)
92
1.0
6 × 10−3
10−15
Table 14.2: Albedos of terrestrial planets. From Rothery McBride & Gilmour
Plate tectonics are responsible for the shape and distribution of continents and oceans across
the surface of the Earth, driven by the convection currents in the upper parts of the mantle.
There is evidence for historical tectonic activity on Mars. As with volcanism, tectonic activity is
expected to decline with time as a planet cools, and the lithosphere extends to greater depths.
Smaller planets, such as Mars or Mercury, will cool much faster than the Earth, leading to
the cessation of tectonic activity, and the geologically quiescent state we see today on these
planets.
14.5
Erosion
Where fluids are able to flow on a planet’s surface, erosion can take place. The flowing fluids
may be gas or liquid, leading to two different types of erosion: fluvial erosion where liquids
are involved - this can be seen as water erosion on Earth and, possibly, on Mars; and aeolian
erosion where the flowing fluid is the gas of an atmosphere - this can be seen in dry environments on Earth, on the surface of Mars and, to some extent, on Venus. These two different
erosive processes leave different signatures on the environment, allowing us to get some idea
of the presence, or absence, of water on the surface of Mars in the past. This is something
being investigated currently by the NASA Perseverance Mission that touched down on Mars
in February. Both fluvial and aeolian erosion leads to the formation of stratified sedimentary
rocks, like sandstone.
14.6
The Surface Temperatures of Planets
The presence, or absence, of liquid water on the surface of a planet is of great interest in the
context of exobiology - the search for life on other planets. In the case of Mars we are interested in whether water flowed on its surface in the past, and in the case of planets around
other stars - exoplanets - we are interested in determining whether life, or the conditions for
life, might persist today.
The temperature of a planet can be estimated, under certain assumptions, quite simply by
looking at the energy balance of incoming to outgoing radiation.
The energy input to a planet is the radiation received from its parent star, minus the fraction
of that radiation that is reflected away. The reflection fraction is given by the planet’s albedo,
a, where a = 1 means total reflection and a = 0 means total absorption. The total energy
received by a planet can then be calculated as follows:
Flux density in Wm−2 received = F =
L
4πd2
(14.1)
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Sun, Stars and Planets 2020-21
where L is the total luminosity of the star and d is the distance from the star to the planet.
We can approximate the area of the radiation that the planet intercepts as a circle of radius
R, the radius of the planet. The planet intercepts a total power of:
πR2 F = πR2
L
4πd2
(14.2)
Some of this energy is reflected by the planet’s albedo so the total power absorbed by the
planet is then:
L
Total Power Received = PR = πR2
(1 − a)
(14.3)
4πd2
If we then assume that the planet radiates this heat away as a perfect black body we can
find its no-atmosphere temperature TN A . The reason why we have to specify that this is a
no-atmosphere temperature will become apparent shortly. The total power emitted by a black
body at temperature TN A and radius R, PE is:
PE = 4πR2 .σTN4 A
(14.4)
from the Stefan-Boltzman equation, and where σ is the Stefan-Boltzman constant, σ = 5.67×
10−8 Wm−2 K−4 .
By setting the power received equal to the power radiated away we can determine the temperature at which these balance, and find TN A .
L
(1 − a)
4πd2
L (1 − a) 1/4
=
16 π σ d2
4πR2 .σTN4 A = πR2
⇒ TN A
(14.5)
(14.6)
Note that this is independent of R, the radius of the planet.
Given the albedo values for the various planets found in table 14.2 we can calculate the noatmosphere temperatures of a variety of planets in our Solar System. The albedo values given
in Table 14.2 are global averages. Different surface materials can have very different albedos.
On the Earth, for example, open ocean has an albedo of 0.06, while sea ice can have an albedo
as high as 0.7. This is one reason why the melting of arctic sea ice due to global warming is
such a concern.
How well does this equation work?
Exercise: Calculate TN A for the planets listed in Table 14.2.
When you do this, you will find that the TN A values for Mars and Mercury are a good match
to the results of the energy balance equation, but the Earth and Venus are anything but, with
Venus having a surface temperature 500 K higher than that estimated by TN A . Why is this?
14.7
The Greenhouse Effect
The answer to this is that Venus and the Earth both have significant atmospheres, and that
those atmospheres contain gases that allow more heat to be retained than is assumed by the
99
wavenumber, ν/cm−1
ultraviolet
104
2×
104 5 × 103 2 × 103 103
visible
infrared
517701K
0
0.1
0.2
0.5
1
2
5
10
−6
wavelength, λ/10 1m
(a)
absorption/%
(b)
100
80
60
40
20
0
0.1
O3
O2
O2
500
200
100
20
50
100
2881K
CO 2
CH 4
CO 2
H2O
N 2 O/CH 4
O3
CO 2
5×
104
H2O
relative spectral flux density
105
H 2 O (rotation)
opaque
0.2
0.5
1
2
5
10
wavelength, λ/10 −6 1m
20
50
transparent
100
© The Open University
Figure 14.1: (a) Spectra of blackbody sources at the temperatures of the Sun’s surface (5770 K)
and the Earth’s surface (288 K). The vertical scales for the two spectra are not the same; the
Sun’s radiation is much more intense than that of the Earth. (b) The absorption spectrum of
the Earth’s atmosphere: the wavelengths at which some atmospheric gases absorb energy are
indicate. Fig 5.22 of McBride & Gilmour, An Introduction to the Solar System.
no-atmosphere approximation. Solar radiation peaks in the optical part of the electromagnetic
spectrum. The atmosphere is (largely) transparent at these wavelengths, so the light of the
Sun passes straight through, allowing us to see, and allowing its radiation to heat the planet.
The re-emitted thermal radiation, however, peaks at longer wavelengths, in the mid and farinfrared (you can determine this using the Wien Displacement law). See Figure 14.1. The
atmosphere is not as transparent at these wavelengths as in the optical, due to the presence
of CO2 , H2 O, methane and other so-called greenhouse gases. These gases in the atmosphere
absorb some of the mid and far-IR radiation from the surface through their vibrational and rotational transitions, are excited (warm up a little), and re-radiate the energy, again as thermal
emission, in all directions. A reduced fraction of the radiation emitted from the surface thus
reaches space, and the planet therefore retains more of the energy received from the Sun than
it would without an atmosphere. The with-atmosphere temperatures, as observed for Venus
and the Earth, are thus higher than the calculated no-atmosphere temperatures, TN A .
The overall climate system on the Earth is of course more complex than this simple analysis
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Sun, Stars and Planets 2020-21
suggests, with the presence of clouds leading to local increases in albedo, the detailed content
of the atmosphere changing the albedo further, modifying the degree of the greenhouse effect,
and other factors such as the condensation of water vapour into rain providing other inputs
of heat. The details of these and other factors are studied in the Atmospheric Physics course.
One thing, though, is clear - if the fraction of greenhouse gases, such as CO2 and methane, in
the atmosphere increases, there will be more mid-IR absorption, and greater retention of heat.
Things to Remember
• The major factors shaping planetary surfaces: impacts, volcanism, tectonics and
erosion
• The dependence of impact rate on time
• The presence or absence of volcanism on other Solar System bodies and the reasons for this
• How to calculate the no-atmosphere surface temperature of planets
• What the Greenhouse effect is, and how it can change these no-atmosphere temperatures
Things to Do
• Calculate the TN A for planets listed in Table 14.2.
• Do Problem Sheet 3, Question 2 and Question 3(a).
101
Lecture 15
Terrestrial Planet Atmospheres
No water, no life. No blue, no green.
- Sylvia Earle
15.1
Introduction
In the last lecture we saw how important the presence, or absence, of an atmosphere is for the
surface temperatures of terrestrial planets. Atmospheres are also important for many other
processes, including erosion and, of course, biological. In this lecture we will look at the origin
of terrestrial planet atmospheres in the Solar System, how atmospheres escape from a planet,
how atmospheres are structured, and what they contain.
15.2
Why do we have an atmosphere at all?
Venus and the Earth have significant atmospheres. Mars has only a thin atmosphere, while
Mercury, the Moon and smaller rocky bodies in the asteroid belt and elsewhere usually have
little or no atmosphere at all. What makes Venus and the Earth so different? The answer to
this is that Venus and the Earth are more massive bodies than the others, and the atmosphere
is held onto their surface by the effects of gravity. We can examine the effects of gravity on the
structure of the atmosphere of a planet using the principle of hydrostatic equilibrium, whereby
the downward force due to gravity on each part of the atmosphere, must be balanced by the
pressure gradient in the atmosphere.
15.3
Atmospheric Density and Pressure
If the pressure gradient is to balance the force of gravity then:
dP
= − ρg
dz
(15.1)
where P is the atmospheric pressure, z is height above the surface, ρ is the atmospheric density
and g is the gravitational acceleration. Pressure and density are connected by the gas law:
P V = N kT
(15.2)
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Sun, Stars and Planets 2020-21
where P is pressure, V volume, k is Boltzmann’s constant, N is the number of atoms or
molecules, and T is temperature. We can rewrite this in terms of the density of a gas of mean
molecular weight hµA i and the atomic mass unit mamu as follows:
P hµA i mamu
= kT
ρ
Since V =
we get:
mass
ρ
(15.3)
and the mass of the gas = N hµA i mamu . Combining equations 15.1 and 15.3
dP
P hµA i mamu g
= −
dz
kT
(15.4)
To be able to solve this equation we need to make two simplifying assumptions. The first is
that g does not vary with height. This is a good assumption since the atmospheres of terrestrial
planets are thin compared with the sizes of the planets. The second is that T does not vary with
height. This is not as good an assumption, but is adequate since pressure changes more rapidly
with height than temperature. Equation 15.4 can then be solved by separation of variables,
and we get:
P = P0 e −
hµA i mamu
gz
kT
(15.5)
and there is a similar exponential fall off for density as a function of height. The quantity
(kT / hµA i mamu g) is known as the scale height and gives the height at which the pressure
and density fall to 1/e times their surface values. The scale height is actually quite small on
Earth, with pressure falling to about 60% of the sea level value at the summit of Mauna Kea, a
height of 4200m, and to just 33% of the sea level value at the height of Everest, about 8800 m.
This is why mountaineers on Everest often develop serious breathing and respiration related
problems, and why it takes so long to boil water at the camp sites on the climb. This also
shows that our approximation that g does not vary with height is a good one.
15.4
Temperature Variations with altitude
The atmosphere is largely transparent to solar radiation, so does not absorb energy in the
visible spectral region from the Sun. Instead, solar radiation is absorbed by the surface of
the Earth, which heats up, radiates in the IR, and this heats the atmosphere around it. The
temperature of the Earth’s atmosphere, and that of other terrestrial planets, thus decreases
with altitude (height). The behaviour of temperature with altitude divides the atmosphere
into different regions:
• Troposphere
This is the lowest layer of the atmosphere, in contact with the surface of the planet.
Temperature decreases rapidly with altitude in the troposphere
• Stratosphere
Only the Earth has a stratosphere, characterised by temperature rising with altitude.
This is a result of ozone molecules absorbing ultra-violet radiation from the Sun, leading
to an injection of energy, and thus heat, in this layer, causing the temperature to rise.
Other planets, such as Mars or Venus, do not have any ozone in their atmospheres, and
thus lack a stratosphere
103
160
140
140
80
100
80
60
40
120
altitude/km
altitude/km
altitude/km
120
60
40
60
20
0
(a) Venus
80
40
20
20
0
100
200
400
600
temperature/K
800
thermosphere
0
100
(b) Earth
150 200 250
temperature/K
mesosphere
300
100 150 200 250
temperature/K
300
(c) Mars
stratosphere
troposphere
© The Open University
Figure 15.1: Temperature vs. height for Venus, Earth and Mars, showing the separate regions
of the atmosphere. Only Earth has a stratosphere as only Earth has ozone in its atmosphere
to absorb UV radiation. From McBride & Gilmour.
• Mesosphere
Temperature decreases with altitude in the mesosphere, though at a slower rate than in
the troposphere
• Thermosphere
Temperature increases with altitude in the thermosphere as a result of several energy
injection processes, including the absorption of extreme-UV photons from the Sun, and
interactions with the Solar Wind.
15.5
Thermal Escape
If gravity keeps an atmosphere around a planet, how do low mass planets like Mercury and
Mars, lose their atmospheres?
For an atom or molecule of a gas to be able to escape from a planet, it must have a speed
greater than the escape velocity, vesc , of the planet. Formally the escape velocity is the speed
that is just sufficient for a particle to reach infinity from the gravitational potential well of
the planet. In practical terms, escape velocity is such that the kinetic energy of the particle is
equal to the change in potential energy required to climb out of the planet’s potential well ie.
∆P E = KE. This is given by:
r
1
GM m
2GM
2
mv =
⇒ vesc =
(15.6)
2 esc
R
R
where m is the mass of the particle, M is the mass of the planet and R is the planet radius,
G is the Gravitational constant. Note that escape velocity is independent of the mass of the
object trying to escape the planet’s gravity well.
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Sun, Stars and Planets 2020-21
Figure 15.2: The shape of the Maxwell-Boltzmann velocity distribution for three different
temperatures. Note that it has a tail towards higher velocities. These higher velocity particles
are those most likely to escape from a planetary atmosphere. (From Tanner McCarron and
Weston McCarron.)
The velocity distribution of particles in a gas at a temperature T is given by the MaxwellBoltzmann distribution:
4 m 32 2 − mv2
F (v)dv = √
v e 2kT dv
π 2kT
(15.7)
where T the temperature, v the velocity and F (v) is the probability of that velocity. This distribution has a high velocity tail, shown in Fig. 15.2. These are the particles most likely to
escape from a planetary atmosphere.
The most probable and rms velocities in a Maxwell-Boltzmann velocity distribution are:
r
r
2kT
3kT
vp =
; vrms =
(15.8)
m
m
We can set an approximate requirement for atmospheric retention, for example, that the most
probable thermal velocity should be less than a sixth of the escape velocity, ie. vp < (1/6)vesc .
Therefore, we can estimate that over the lifetime of the solar system, a given planet will lose
a particular species from its atmosphere if:
r
r
2kT
1 2GM
≥
(15.9)
m
6
R
One common factor in all these equations is that, at a given temperature, lighter particles
will have a higher velocity. We would thus expect that lighter species, such as hydrogen and
helium, will be more likely to escape from terrestrial planets, and this is in fact what we find
in Figure 15.3. The gas giants Jupiter, Saturn, Neptune and Uranus, are all massive enough to
retain hydrogen and helium, the most abundant elements in the universe. Earth and Triton
velocity and thermal velocity
105
ep its
y has to
~ factor 6)
ape
n sustain
pheres;
can keep
can
2 and N2 in
heres
Figure 15.3: Escape velocity plotted against surface temperature for a number of planets
and other Solar System bodies. The colourful lines represent approximate estimated upper
boundaries for retention of particular gas species.
can keep water in their atmospheres, while Mars, Venus and Titan can retain carbon dioxide.
Mercury and the Moon cannot even retain carbon dioxide, and thus have absent or extremely
thin atmospheres.
15.6
Current Atmospheric Composition
The current atmospheric composition of the terrestrial planets with appreciable atmosphere
is shown in Fig 15.4. The atmospheres of Venus and Mars can be classed as ‘oxidised’ in that
there is no free oxygen. Instead, oxygen is bound into compounds, mostly CO2 but also water,
H2 O, and sulphur dioxide, SO2 . The gas giants, as we will see in the next lecture, have atmospheres rich in hydrogen, since they are massive enough to retain this light species, and are
dominated by hydrogen, helium and compounds like methane (CH4 ), ammonia (NH3 ), water
and hydrogen sulphide (H2 S). They thus have atmospheres classed as ‘reducing’. Earth is the
only planet to have an ‘oxidising’ atmosphere, containing free oxygen. It also contains ozone,
O3 , which shields the surface from UV light and, due to the heating from this UV absorption,
a stratosphere exists in Earth’s atmosphere.
The presence of free oxygen in the Earth’s atmosphere is rather special since oxygen is a reactive element that would usually be bound into compounds, as seen on Venus and Mars. The
relative lack of carbon dioxide in Earth’s atmosphere, a major constituent in the atmospheres
of Mars and Venus, is also rather odd. Oxygen was not always a major constituent of Earth’s
atmosphere. In fact, the latest estimates suggest that it was only a major constituent for the
past billion years. Oxygen, of course, is produced by photosynthesis, so the presence of life
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Sun, Stars and Planets 2020-21
Ar
O2
Ar
SO 2
N2
N2
H2O
(variable)
O2
CO 2
N2
CO 2
Venus
Earth
Mars
© The Open University
Figure 15.4: Current composition of terrestrial planet atmospheres (Mercury essentially has
no atmosphere). Note that CO2 dominates both on Venus and Mars.
on the Earth is the reason that we have oxygen in the atmosphere. If life were to suddenly
disappear, the oxygen levels would gradually decrease as it combines chemically with other
elements, like carbon.
The relative absence of CO2 in the Earth’s atmosphere is another issue. This is the result
of various processes, an important one being the Urey weathering process, whereby carbon
dioxide dissolved in water reacts with silicates in rocks, leading to the deposition of calcium
carbonate (CaCO3 ). An example of this reaction is:
CaSiO3 + 2CO2 + H2 O → Ca2+ + SiO2 + 2HCO−
3
Ca2+ + 2HCO−
3 → CaCO3 + CO2 + H2 O
(15.10)
The CaCO3 produced by this reaction is dissolved in the water and eventually precipitates
out to form sedimentary rocks such as limestone and chalk. These same rocks, on Earth at
least, can also be produced by biological processes, that also serve to remove CO2 from the
atmosphere.
The Earth’s slow carbon cycle removes and recycles CO2 on long time scales, see Fig 15.5.
15.7
Origin of Atmospheres
Where did planetary atmospheres come from originally?
Any primordial atmosphere is likely to have been lost, since the planets were all very hot after
formation, the solar wind would have been quite powerful in the early Sun, and the bulk of the
107
Earth’s CO2 cycle
CO2 in atmosphere
CO2 dissolves in ocean
Silicate minerals react with CO2
to form carbonate rocks
Release of CO2
by volcanism
Subduction of carbonate rocks
Note – critical role of presence of liquid water
most CO2 bound up in carbonate rocks
Figure 15.5: Schematic diagram of the Earth’s slow carbon cycle.
content would have been the most common gases, hydrogen and helium, which the terrestrial
planets are too low in mass to retain. Instead, what we see now are likely to be secondary
atmospheres, the content of which will be controlled by the balance between gas sources and
gas sinks:
• Sources
– Outgassing from planetary interiors eg. volcanoes
– Evaporation and sublimation of material on the surface (eg. water, solid CO2 )
– Bombardment by bodies rich in volatiles
• Sinks
– Condensation and chemical reactions (temporary)
– Stripping by the solar wind
– Impacts
– Thermal escape
The key difference between Earth and Venus, which are otherwise quite similar planets, may
be that Venus never acquired a significant amount of water - forming from planetesimals that
lacked it, missing out on bombardment by material rich in water ice, or because it lost its water
through photoionisation of water molecules through being closer to the Sun, with the resulting hydrogen being lost to thermal escape. A lack of water would have meant that the Urey
weathering process and slow CO2 cycle were never able to leach CO2 out of the atmosphere.
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Sun, Stars and Planets 2020-21
Outgassing from the core would then not have been counteracted by chemical reactions, and
so the thick atmosphere could build up and make Venus the hot and inhospitable place it is
today. Mars, conversely, had too little outgassing to counteract the increased rate of thermal
escape due to its lower mass, at least in recent times, and thus lost its atmosphere, becoming
the cold, low pressure environment currently being explored by the Mars rovers.
Things to Remember
• That atmospheric pressure drops exponentially with increasing altitude, and the
definition of atmospheric scale height
• The structure of Earth’s atmosphere, with troposphere, stratosphere, mesosphere
and thermosphere, and the way temperature varies with altitude in these four
regions
• The atmospheric structures of Venus and Mars, and why they differ from the
Earth
• Know what escape velocity is and how to derive it
• Be able to derive a simple formula (using a constant multiplied by vp ) to show
whether a given molecular species can be estimated to be retained by a planet of
given mass and temperature
• The consequences of this for retention or loss of the atmospheres of planets in
the Solar System, and indeed for any planet or moon in orbit around any star.
• The compositions of the atmospheres of the terrestrial planets and how they
relate to other properties. The Urey weathering process, and the slow carbon
cycle.
• The sources and sinks of terrestrial planet atmospheres
To Do:
• Problem sheet 3, question 3 (b)
109
Lecture 16
Gas Giants: Structure and
Atmospheres
It’s like searching for several needles that change colour and shape unpredictably
- Prof Michele Dougherty, Imperial College, on the length of Saturn’s day
16.1
Introduction
The planets Jupiter, Saturn, Uranus and Neptune are collectively known as the Gas Giants.
They are the four largest planets in the Solar System and have properties that are very different
from the terrestrial planets. In this lecture we will examine their properties, including their
structure and atmospheres, and look at the differences between Jupiter and Saturn, which
share a number of properties, and Uranus and Neptune, which are similar to each other but
subtly different from the other gas giants. We will also examine the origins of a feature that
is common to all the gas giants, but which is most distinctive in Saturn - ring systems.
16.2
Basic Properties of Gas Giants
The gas giants are much lower density than the terrestrial planets, and have much deeper
atmospheres. Saturn, for example, has a sufficiently low density that it would float on water if you could find a big enough bucket. This means that they cannot be dominated by the kind
of rocky and metallic material that makes up most of the structure of the terrestrial planets.
Instead, the bulk of their mass comes from light elements, and they have much higher hydrogen and helium relative abundancies. Consideration of the mass and temperature of the gas
giants in the context of the thermal escape of gases from an atmosphere (see Fig. 15.3) shows
that hydrogen and helium can be retained by them.
Since the gas giants are effectively large balls of gas, it is difficult to define a ‘surface’ for
them, or to probe very far beneath the cloud layers that we see from outside, to determine
what their internal structure might be. The ‘surface’ issue is solved by arbitrarily defining the
surface of these planets to be where their atmospheric pressure equals that of Earth i.e. the
radius of these planets is defined as the radius at which P = 1 bar. What we know of their internal structure has largely been determined by examinations of their gravitational effect on
the passage of spacecraft that fly past or orbit them, measurements of their magnetic fields,
and the combination of these results with models of their deep interior. These results indicate
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Sun, Stars and Planets 2020-21
Figure 16.1: The internal structure of Jupiter and Saturn (from Rothery, McBride & Gilmour).
that the gas giants have rocky and icy cores that are at high temperature and pressure - in this
case we use the term ice to refer to volatile materials such as H2 O, CH4 and NH3 rather than
something that is actually frozen. Observations and flypasts also reveal that the gas giants
are all somewhat flattened in shape - they are prolate, bulging a little at their equators as a
result of their high rotation speeds. All of the gas giants have magnetic fields.
The gas giant planets formed further out in the Solar System than the terrestrial planets,
and the formation process took longer, especially for the outermost planets. The increased
distance from the Sun allowed them to accrete more hydrogen and helium, leading to their
greater mass. Some models of the early Solar System suggest that the gas giants likely formed
somewhat closer to the Sun than we see them now, and then proceeded to migrate outwards,
clearing the Solar System of debris as they did so. The asteroid belt and the Kuiper belt are
what they left behind.
Examination of the temperatures of the gas giants compared to the energy they receive from
the Sun reveals that all except Uranus are emitting excess heat ie. they are warmer than the
energy they receive from the Sun would suggest. The origin of this excess heat will be discussed
in the next sections below.
16.3
The Internal Structure of Jupiter and Saturn
The internal structure of the two largest gas giants is shown in Fig 16.1. Their cores are thought
to be a mixture of rock and ices (ie. volatiles), surrounded by a layer of ices. This material is at
temperatures and pressures of up to ∼16000 K and 50 Mbar (ie. 50 million Earth atmospheres)
111
in Jupiter, and ∼10000 K and 18 Mbar in Saturn. The rocky/icy cores in both planets have
masses of about 3 Earth masses in total. There may be further differentiation in the cores,
leading to a metallic iron centre, as in Earth, but we do not have sufficient data to be sure of
this.
Outside the rocky/icy cores is a region made up of helium and metallic hydrogen. The latter
is a form of hydrogen that only arises under intense pressure, where the hydrogen nuclei are
pressed together so hard that their electrons become delocalised from their parent nuclei, and
form a fermi gas of free electrons that can flow throughout the volume of the metallic hydrogen. This material is conductive and liquid at the temperatures and pressures prevalent in the
cores of these planets. You can think of this material as being somewhat like mercury at room
temperature and pressure.
Further out from the centre, the pressure subsides to values below 2 Mbar, where hydrogen
returns to its more familiar molecular form as H2 . This is a gradual process so there is no sharp
boundary between metallic and molecular hydrogen layers. Further out still, a similar smooth
transition occurs between liquid and gas. While hydrogen and helium are the dominant materials in these outer layers, they are also mixed with other icy material and some small amount
of rocky material.
16.4
Excess Heat in Jupiter and Saturn
Jupiter and Saturn are both warmer than the energy they receive from the Sun would suggest.
This is known as having excess heat. The origin for this excess is probably different for each
planet.
• Jupiter
There are three likely explanations for excess heat in the case of Jupiter. Firstly, as the
largest planet in the Solar System, it may still be radiating away the residual heat from
its formation - the cooling rate for this primordial heating, as for all heat loss in planets,
goes as the inverse of the radius. Secondly, there is the possibility that Jupiter is still
slowly contracting, converting potential energy to thermal energy as it does so. Finally,
and uniquely for Jupiter in the Solar System, there is the possibility that there may be a
low rate of deuterium fusion in the hottest densest regions.
• Saturn
Saturn is too small to have significant residual heat from its formation. Instead, the
best idea for how it generates its excess heat is that it comes from the separation of
helium from hydrogen in the metallic hydrogen layer. The helium then rains downwards,
releasing potential energy as it does so. This process cannot work in Jupiter as the
metallic hydrogen layer there is hotter, allowing helium to be dissolved in it, and stirred
by convection, keeping the materials well mixed.
16.5
The Internal Structure of Uranus and Neptune
The internal structure of the smaller two gas giants, Uranus and Neptune, is shown in Fig.
16.2. There are some similarities between them and Saturn and Jupiter, but also some striking
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Figure 16.2: The internal structure of Uranus and Neptune (from Rothery, McBride &
Gilmour).
differences. The first difference is that Uranus and Neptune have less hydrogen and helium
than the larger gas giants. This leads to the second key difference, which is that they are too
small to be able to produce the conditions necessary for the formation of metallic hydrogen.
The two outer gas giants have more volatiles in them than hydrogen or helium - roughly 20%
of their mass comes from these gases while they account for about 90% of the mass of Jupiter
and Saturn. Some people therefore classify them as ‘ice giants’, rather than the more generic
gas giants.
Apart from the absence of metallic hydrogen and the reduced amount of H and He, their structure is broadly similar to that of the other two gas giants, with a rocky and icy core, and inner
region of icy material, and then an outer region of hydrogen and helium. The rocky/icy cores
of Uranus and Neptune, at one Earth mass, are smaller than those of Jupiter and Saturn, which
is why the planets are smaller as they attracted less material when forming.
Neptune is found to produce excess heat. This is likely to be the result of continuing differentiation in its internal structure, with denser material falling towards the centre and releasing
potential energy as heat. Uranus, in contrast, releases no excess heat. This is a puzzle since
the two planets are very similar. Where one has an internal heat source, one would thus expect the other to have one as well. The only significant difference between the two is that the
orbital axis of Uranus is parallel to the ecliptic plane rather than perpendicular to it, and for
the length of some seasons (21 Earth years) points towards the Sun (see Fig. 16.3). This leads
to a very different distribution of heat within its atmosphere and could, in principle, lead to
113
Figure 16.3: How the ∼90 degree tilt of Uranus’ rotation axis relative to the other planets
affects its seasons as it orbits the Sun.
the disruption of the convection flows that would otherwise allow internally generated heat
to reach the surface and be radiated away. Further research is needed to determine whether
this explanation is correct.
16.6
Gas Giant Atmospheres
When we observe a gas giant planet, we are looking at their atmospheres not their surfaces.
These have many colours and structures in them, arising from the various processes, chemical, physical and meteorological that drive them. The temperature profile of the atmospheres
show similarities to some aspects of the terrestrial planets, with a troposphere, a convective
layer, where the temperature falls with height, and then a thermosphere where the temperature rises with height. Jupiter and Saturn (see Fig. 16.4) have highly reflective clouds in their
topmost layers, with the constituents of lower layers not yet fully identified. The colouring in
the atmospheres is not yet fully understood, but is likely due to trace elements, including, in
the case of Jupiter, sulphur.
A similar division between troposphere and thermosphere is seen in the atmospheres of Uranus
and Neptune (see Fig. 16.5), but the temperature increase in the atmosphere of Uranus with
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(a) Jupiter
(b) Saturn
150
150
thermosphere
thermosphere
50
–50
troposphere
0
hydrocarbon haze
NH3 ice clouds
NH4HS ice clouds
H2O ice clouds
1
10
0
–50
haze layers
pressure/bar
0.1
0.01
troposphere
50
altitude/km
100
pressure/bar
0.01
altitude/km
100
NH3 ice clouds
1
NH4HS ice clouds?
–100
–150
–100
300
100
temperature/K
–150
500
H2O ice clouds?
300
100
temperature/K
10?
500
© The Open University
Figure 16.4: The atmospheric structure of Jupiter and Saturn (from Rothery, McBride &
Gilmour).
+200
+200
+100
0
−100
40
(a) Uranus
thermosphere
altitude/km
altitude/km
thermosphere
CH 4 clouds
troposphere
60
80
100
T/K
120
140
+100
C2 H 2 clouds
C2 H 4 clouds
C2 H 6 clouds
0
CH 4 clouds
−100
40
(b) Neptune
troposphere
60
80
100
120
140
T/K
© The Open University
Figure 16.5: The atmospheric structure of Uranus and Neptune (from Rothery, McBride &
Gilmour).
115
height in the thermosphere is very slow, indicating that there is little or no convection and
that energy transport is very inefficient. This relates to the issue of there being no excess heat
detected in Uranus, as discussed above. Methane in the upper layers of both these planets
give them their bluish colour.
All the gas giants have banded structures in their atmospheres which are related to variations
in wind speeds, with different bands traveling at different speeds, and with turbulence occurring at the interfaces between different bands. The differing colours are related to different
materials, with dark bands, called belts, coming from rising material and light bands, called
zones, from sinking material. This can be explained in two ways - either the planets can be
seen as a series of coaxial rotating cylinders, or as a number of convection cells.
Long duration weather systems also appear within this banded structure, the most obvious
of which is the Great Red Spot on Jupiter, a 14000 x 26000 km storm that has been raging for
at least 170 years. Smaller storms have been seen on the other gas giant planets, but none as
persistent as this.
16.7
Ring Systems
All of the gas giant planets have ring systems. Saturn has the most obvious, partly because its
ring material has a high ice content and is thus highly reflective, but rings have been detected
for all of the others. The rings are composed of orbiting debris, with particle sizes ranging
from 1 cm to 10’s of metres. Ring systems have a wide range of structures. In Saturn we see
gaps and divisions in the rings due to small moons clearing their orbit, moons that shepherd
material, and to the effects of orbital resonance (see later lecture). Formation scenarios for
rings include the idea of satellites shattered by an impact, that they are made up of material
left over from the formation of the solar system that never coalesced to form a planet, and the
suggestion that they are the remains of a moon that migrated towards the plant and was then
disrupted.
Central to all these ideas is the result that the rings of all the gas giants are within their Roche
Limit. This is the radius around a planet within which an object that would otherwise be
held together by its self-gravity, will be torn apart by tidal forces. We will now look at how
to calculate an estimate of the tidal forces, and use this to derive an expression for the Roche
Limit.
16.7.1
Estimate of the tidal force
Tidal forces arise because of differential gravitational forces, the gravitational forces are not
equal across the planet or moon. Let us estimate the tidal forces (difference in gravitational
forces) on a small body due to a large body around which it is orbiting. Let us consider a small
body, of mass m and diameter d in orbit, at a distance r, around a body of larger mass M and
radius R, where M m and r d, see Fig 16.6.
Let us suppose that there is a 1 kg test mass at the centre of the small body. The gravitational
force produced by the large body on this test mass is:
F =
GM
r2
(16.1)
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2R
d
M
m
r
Figure 16.6: Diagram showing a small body, mass m, orbiting a larger body of mass M.
The gravitational force will be a little greater if the test mass was located on the nearside of
the small object to the larger object, and conversely the gravitational force will be a little less
if the test object is on the far side of the small object to the larger object.
The tidal force from the body of mass M that is working to pull the test masses located on the
near and far side of the small body apart is the difference in gravitational attraction from the
large body on them. Assuming r d, the tidal force, δF is:
δF ≈
dF
δr
dr
(16.2)
So substituting in for F from 16.1 and differentiating, gives the tidal force:
δF ≈ 2
dGM
r3
(16.3)
Note that we can just as easily calculate the tidal forces acting on the larger body due to the
smaller body using the same method.
16.7.2
Estimate of the Roche Limit
We can use our estimate of the tidal force in Eqn 16.3 now to derive an expression for the
Roche limit. If a moon is too close to a planet it will be ripped apart by the planet’s tidal
forces. Consider now the forces on a test mass of 1 kg on the far side of a moon from a large
planet. The large planet has mass Mp , radius R, and the moon of mass m and radius Rm
orbits at a distance of r as shown in Fig 16.7.
We will now consider the balance of the tidal force from the planet and the gravitational force
of the moon itself on a test mass placed on the far side of the moon. Here we estimate the
tidal force as the difference between the gravitational force from the planet on the test mass
on the moon’s far side and the test mass if it were placed at the centre of the moon. This will
allow us to calculate the distance at which the tidal force dominates and the moon is ripped
apart assuming it is just held together by its own gravitational force.
117
2R
2Rm
Mp
m
r
Figure 16.7: Diagram showing a small moon, mass m, orbiting a large planet of mass Mp .
The tidal force on the test mass of 1 kg situated on the far side of the moon is given by the
difference between the gravitational force due to the planet on the far side of the moon and
the centre of the moon. Using our expression from Eqn 16.3 we can write that the tidal force
Ft is:
G Rm Mp
Ft = 2
(16.4)
r3
The Roche Limit rL is then defined as the distance at which this tidal force from the large
mass of the planet Mp is balanced by the gravitational attraction between the 1 kg test mass
and the moon. Thus:
G Rm Mp
Gm
=2
2
Rm
r3
⇒
rL =
Mp
2
m
1/3
Rm
(16.5)
The Roche Limit is usually expressed in terms of densities, with ρp for the planet and ρm for the
moon, and using Rp for the radius of the planet and Rm for the radius of the moon. Looking
at things this way we find that:
ρp =
Mp
;
4
3
3 πRp
ρm =
m
(16.6)
4
3
3 πRm
Substituting this into equation 16.5 we get:
1/3
rL = 2
Rp3 ρp
3 ρ
Rm
m
!1/3
⇒
Rm
1/3
rL = 2
ρp
ρm
1/3
Rp
(16.7)
A more detailed calculation by Edouard Roche in 1848 leads to the actual value of Roche Limit:
rL = 2.456
ρp
ρm
1/3
Rp
(16.8)
We find that in practice not every moon within a planet’s Roche limit is ripped apart by these
tidal forces, and this is because the moon will be held together by forces other than just its
self gravitational force, which we have not considered here.
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Things to Remember
• The internal structures of the gas giants and the reasons for differences between
them
• The atmospheric structures of gas giant planets and their constituents
• The properties of gas giant ring systems
• The simple derivation of the Roche Limit that leads to equation 16.7
• How to apply the Roche Limit to moons and ring systems
To Do:
• Problem Sheet 3, questions 4 and 5
119
Lecture 17
Moons: Formation and Properties
My team on Cassini was responsible for discovering an atmosphere on one of Saturn’s
moons. We saw some observations on 2 of the flybys past the moon and it looked like
an atmosphere. We weren’t sure, but we thought it was important that we try to go
really close on the next flyby. And so on the next fly-by instead of being a thousand
km away, we persuaded the project team to take the spacecraft really close, at 170 km
away from the moon
- Prof Michele Dougherty, on the discovery of an atmosphere on Enceladus
17.1
Introduction
Nearly all of the planets in the Solar System have moons, with some of these moons, such as
Titan or Ganymede, being larger than the planet Mercury. Even minor planets, such as Pluto
and Eris, have their own moons. Eris has one known moon, called Dysnomia, while Pluto
has five detected moons - Charon, Hydra, Nix and the more recently discovered, and recently
named Styx and Kerberos. The presence of these moons around Pluto, and the likelihood that
there are other, smaller, and thus harder to detect, companions, caused difficulties in planning
for the NASA New Horizons flyby mission to Pluto.
The number and range of properties of moons around planets is summarised in Table 17.1.
The history and formation of moons can provide extra information about the formation of the
Solar System and about the planets around which they orbit.
17.2
Orbits and Masses
The mass of a planet can be determined if we know the details of the orbit of a moon around
it. If we make the simplifying assumption that the orbit of a moon around a planet is circular,
and that the planet’s mass M is much greater than that of the moon m, then we can balance
the gravitational and centrifugal forces on the moon, as:
GM m
= m a ω2
a2
where a is the distance between the planet and moon, and ω is the angular velocity of the
moon.
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Planet
Mercury
Venus
Earth
Mars
Moons
0
0
1
2
Jupiter
53 + >26
Saturn
53 +>29
Uranus
Neptune
27+
14+
Comments
The Moon, 1700 km radius
Phobos and Deimos, both irregular and small
(11 and 8km radius respectively)
Ganymede (2600 km), Callisto (2400 km)
Io (1800 km), Europa (1600 km)
Titan (2600 km) earth-like atmosphere!
Rhea (800 km), Iapetus (700 km)
Enceladus (250 km; water)
7 large + other Trojans, co-orbiting moons, rings…
Titania (800 km), Oberon (800 km)
Triton (1400 km)
Table 17.1: Summary of the number of moons around each planet in the Solar System, with
comments on some of their properties.
We can then derive the planet’s mass. All we need to know is the distance between the planet
and the moon, and the moon’s orbital period. These can all be determined observationally.
Thus:
GM
ω 2 a3
aω 2 = 2
⇒
M=
(17.1)
a
G
For the Earth-Moon system the parameters are: a = 386 × 106 m while the orbital period
of the Moon is 27.3 days, which converts to ω = 2π/2.36 × 106 s. Put these numbers into
equation 17.1 and we get the mass of the Earth = 6 × 1024 kg.
17.3
Formation
One of the reasons why the moons of the Solar System have such a wide range of properties
is that they have formed in a variety of ways. The three principle ways that moons are formed
are:
• Condensation
In a process similar to that which formed the planets around the Sun in a protoplanetary disk, smaller bodies are thought to be able to form in orbit around a larger planet
in a circumplanetary disk. Moons formed through this process will be prograde ie. orbiting the planet in the same direction that the planet rotates, and, since they formed in
relatively dense material around a forming planet, from a proto-satellite condensation
disk, they will be relatively higher mass objects. The four Galilean moons of Jupiter, Io,
Europa, Ganymede and Callisto, are likely to have formed through condensation.
• Capture
Gravitational interactions between planets and smaller, free floating, bodies can lead
to the smaller bodies becoming gravitationally bound to planets. Such captured moons
may have retrograde orbits. Examples of these include the moons of Mars, Phobos and
Deimos. Larger retrograde moons, such as Triton, are likely to be protoplanetary cores
that were captured by a larger planet during the later stages of planet formation.
121
• Collision and fragmentation
Small prograde moons, especially those close to a planet, are likely to have been formed
by collisions between moons that were once larger. These larger bodies are then split
into smaller, separate moons. This process takes place predominantly near to a planet
since the orbital velocities will be higher and thus the collisions more energetic, leading
to increased chances of fragmentation during a collision. Saturn’s moon Hyperion, for
example, is likely to have formed this way.
Smaller moons in general, whether formed through fragmentation or capture, are likely
to have irregular shapes since their gravitational fields are too weak to produce a spherical surface.
17.3.1
The Moon
Our own Moon is something of an exception among this range of formation methods, since it
appears to have formed as the result of a giant impact between the young Earth and a Mars
size body roughly 50-100 Myrs after the formation of the proto-Earth. Denser material in the
impactor would have remained with the young Earth, while the Moon subsequently formed
from the lighter ejecta that resulted from the collision. This explains why the Moon has a
smaller nickel-iron core, relative to its size, than other terrestrial-type bodies. The energy of
this collision would have re-melted the surfaces of both bodies.
17.3.2
Titan
Aside from our own Moon, Titan is perhaps the most unusual moon in the Solar System. It is
the only moon in the Solar System with its own substantial atmosphere since it is both cold
enough and massive enough to retain nitrogen and heavier gases against thermal escape. Its
surface temperature is only about 100 K so water ice serves as its bedrock, but it also has
complex chemistry in its atmosphere and on its surface. The temperature and pressure at the
surface are close to methane’s triple point, meaning that methane can exist as both a solid,
liquid or gas. Methane and ethane lakes have been observed and methane rain is likely to fall
as part of a methane cycle similar to the water cycle on Earth.
Titan’s atmosphere, like that of the Earth, includes a stratosphere region where temperature
rises with height (see Fig. 17.1). This is produced by layers in the atmosphere where solar UV
light drives chemical reactions between methane and nitrogen that produces a photochemical smog made of hydrocarbons that further absorbs solar radiation. The heaviest of these
hydrocarbons rain down onto the surface.
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Figure 17.1: The atmospheric structure of Titan is similar to that of the Earth, including a
stratosphere
17.4
Tidal Forces and Tidal Heating
We have already looked at one aspect of tidal forces when we examined the Roche limit, but
tidal effects also apply that are less dramatic than the production of ring systems, through
objects being broken apart. Where bodies orbit each other there will be a difference in forces
from one side of the object to the other. As discussed above (equation 16.4), the tidal force, on
a body of mass m a distance r from its centre of mass, due to its orbit at a distance d from a
larger body of mass M is:
r
Ft = 2GM m 3
(17.2)
d
If some part of the object is liquid, then this fluid will flow in response to the tidal force and
you get what we see in the seas of Earth - tides. We see tidal bulges.
Exercise: Using the orbital parameters of the Earth, Sun and Moon, compare the tidal forces
on the surface of the Earth due to the Moon and due to the Sun. (Problem Sheet 4, Question
2). This is what gives rise to ‘spring tides’ when the Moon is full or new, and thus aligned with
the Sun.
These forces will also result in the deformation of the moon or planet subject to the tidal forces
- the solid body will bulge very slightly, obviously far less than for a liquid bulge! This leads
to heating in just the same way that repeatedly squashing a squash or tennis ball produces
123
heating. In the case of the Earth and Moon there is not very much tidal heating - the effect
is very minor. There is a loss of energy of the Earth’s rotation, due to friction. The energy is
converted to a slight internal heating of the Earth. The loss of rotational energy means the
Earth is slowly slowing down its rotation rate, approximately by 0.002 s per century.
However, in other systems around massive planets, like Jupiter, there can be a considerable
amount of heat generated. This is what powers the volcanoes of Io, leads to the water geysers found on Saturn’s moon Enceladus, and which may maintain a liquid ocean beneath the
surface of Europa. Indeed water geysers may have recently been found on Europa, similar to
those already found on Enceladus. Titan too is thought to have a liquid water-ammonia ocean
beneath its icy surface, based on observations conducted by the Cassini spacecraft.
17.5
Tidal Locking
A rotating moon (or planet) will, in general, tend to pull the bulge produced by tidal forces
away from perfect alignment with the centre of mass of the body around which it is orbiting,
producing a force that acts to bring the tidal bulge back into alignment. There is therefore
a net torque opposing the direction of rotation, see fig 17.2. Over time, this force will act to
synchronise the orbital period and rotational period of the objects. The end point of this effect,
known as tidal locking, can be seen in the Earth-Moon system, where the Moon always points
the same face at the Earth, the Moon is said to be tidally locked. The Moon’s orbital period
equals its rotational period. (A one-to-one ratio in this kind of tidal locking is not always the
end point. Mercury, for example, has 1.5 rotation periods for each orbital period.) In the case
of the Moon, at this moment the Moon is tidally locked with the Earth, but the Earth is not
tidally locked to the Moon, although it may be in the distant future. The slowing Earth might
eventually become tidally locked with the Moon and no further evolution of the system will
occur. At that point the torque acting on the Earth and dissipation by tidal forces will cease. It
should be noted that at the moment, due to conservation of angular momentum, the rotational
angular momentum being lost by the Earth is gained by the Moon, and as a result the Moon
is gradually moving to a larger orbital radius, it is slowly getting further from the Earth.
17.6
Circularisation
Our discussion of tidal effects so far has assumed that the orbits are circular, but this is not,
in general, the case. Instead, most orbits are elliptical to some extent, with the orbital speed
varying over the period of the orbit - a moon travels fastest relative to its parent body at
closest approach, and slowest when at its greatest distance. The rotational period of the moon,
however, remains constant. If the moon were of perfectly uniform density this would make
little difference. However, most bodies are not perfectly uniform, so tidal forces can have an
effect on the orbit. The end result is that over long periods of time the orbits of small bodies
around larger bodies will tend to be circularised.
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Figure 17.2: Tidal torque is exerted on a moon orbiting in a prograde direction with a period
longer than the planet’s rotation period. The planet’s rotation means the bulge leads ahead of
the position of the moon. The far side bulge exerts retarding torque T2 on the moon, but near
side bulge exerts a larger positive torque T1 on the moon, so the moon gains a net positive
torque, and its orbit evolves outwards. The planet’s rotation is slowed. From de Pater and
Lissauer 2019, Fundamental Planetary Science.
17.7
Orbital Resonances
One object orbiting around another in isolation from everything else is a simple physical system, where the orbital parameters can be calculated analytically. However, in the real world,
there are always other bodies involved which can add complexity to the orbital mechanics. In
the case of a planet with many moons, the orbit of one moon can be affected by contributions
from other moons. In the Solar System more broadly, as we will see in the next lecture, planets
like Jupiter or the Earth can influence the orbits of other objects around the Sun.
In many circumstances, the gravitational interactions between orbiting bodies will occur at
random intervals and will average out over time. However, if an orbital configuration repeats
regularly and with a period that is a small integer number of orbits, then the small perturbations from these interactions will not average out. This is a process known as orbital resonance
and it occurs quite often in complex systems of orbiting bodies. Such resonances are described
in terms of the number of orbits of the inner body to the number of orbits of the outer body
(or bodies, for more complex interactions) eg. the Galilean moon Io is in a 4:2:1 resonance with
the moons Europa and Ganymede. So Io orbits Jupiter 4 times for every 2 orbits that Europa
makes and for every one orbit that Ganymede makes.
There are two possible effects from such an orbital resonance:
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17.7.1
Stable Resonances
The body (or bodies) are locked into its orbit, and cannot, for example, move outwards as a
response to tidal forces. This effect happens typically when the orbits of the objects concerned
never approach each other very closely, and are called stable resonances.
This is the situation for Io in its orbit around Jupiter. In isolation, the tidal effects that squeeze
Io as it orbits around Jupiter would have led it to move outwards, in the same way that the
Earth’s Moon has moved away over millions of years. Orbital angular momentum is thus
transferred to the Moon from the Earth, and the Moon climbs out of the Earth’s gravity well.
This cannot happen to Io because of the orbital resonance it is in with Ganymede and Europa.
The energy that would otherwise move Io up the gravity well from Jupiter instead goes into
tidal heating of Io’s interior, leading to the rampant volcanism that we can see on its surface.
Europa, too, is involved with this orbital resonance and, like Io, is also locked into its orbit. It
is further from Jupiter, so there is less tidal heating as a result, but this is still enough to melt
some of Europa’s icy interior, leading to a layer of liquid water beneath its surface, and producing cryovolcanism. Saturn’s moon Enceladus is in a 2:1 orbital resonance with the moon
Dione. Tidal heating here is likely to be responsible for the cryovolcanism that produces the
geysers seen on this moon.
17.7.2
Unstable Resonances
The body gets accelerated or decelerated in its orbit until it is no longer in resonance. In effect
this means that the orbital configuration affected is cleared of objects subject to the resonance.
These are called unstable resonances.
Observation of the rings of Saturn show a variety of gaps. The most obvious of these, observable from Earth and discovered originally in 1675 by Giovanni Cassini and named after him,
is the Cassini Division. This gap in the rings is produced by a 2:1 orbital resonance between
any particles in this ring and the moon Mimas.
Orbital resonances and other effects in fact make the structure of Saturn’s rings very complex,
with a wide variety of different gaps, sub-rings, and sub-structures such as spokes, braids and
shepherd moons.
Orbital resonances are not restricted to the orbits of moons around planets, but also apply in
the broader Solar System. They produce the Kirkwood Gaps seen in the distribution of orbits
in the asteroid belt, and, in the outer Solar System, lead to the class of objects called Plutinos
(see the next lecture for more details).
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Sun, Stars and Planets 2020-21
Things to Remember
• Use of the orbital period of an orbiting body to calculate the mass of its parent
planet
• The names and general properties of the most famous moons in the Solar System
(the Moon, Mars’ moons, the Galilean moons of Jupiter, Titan, Enceladus, Triton)
• Atmospheric structure of Titan
• The main formation mechanisms for moons
• Tidal forces, tidal heating, tidal locking and circularisation
• Orbital resonances in moon systems and how this can lead to tidal heating, especially in the example of Io
To Do:
Problem Sheet 4, questions 1 and 2
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Lecture 18
Small Bodies: Comets, Asteroids and
the Outer Solar System
The glow is one of a kind
- Marina Galand (Imperial College) on her team’s discovery of the first ever detected UV aurora on a comet, which lead to Daily Mail headline ‘Comets have their
own northern lights’ (21 Sept 2020)
18.1
Introduction
A full analysis of the orbital dynamics of the Solar System shows that the regions between
most of the planets lack stable orbits because of gravitational resonance effects. The only regions in the Solar System where this is not the case are between Mars and Jupiter, and outside
the orbit of Neptune. Unsurprisingly, we find plentiful small objects orbiting in these regions,
forming the Asteroid Belt, between Mars and Jupiter, and the Kuiper Belt, beyond the orbit
of Neptune. Further out, there is also the Oort cloud, where proto-comets kicked out of the
young Solar System, live. There are also other, shorter lived, populations of objects, such as
the Centaurs, and comets, which occasionally visit the inner Solar System. All together, these
objects form the small bodies and minor planets of the Solar System.
While they are not a significant constituent of the Solar System by mass, small bodies have not
been through the reprocessing involved in planet formation that the rocks and gases in larger
bodies have endured. The small bodies can thus provide us with clues about what material in
the early Solar System might have been like. Asteroids also occasionally collide with planets,
including the Earth, so monitoring them is not only useful scientifically, it may provide early
warning for major disasters.
18.2
Asteroids
There are estimated to be between 1.1 and 1.9 million asteroids in the main asteroid belt between Mars and Jupiter, and just over 500,000 of these have reasonably well determined orbits,
with another 500,000 or so known less well. (The Minor Planet Center has a detailed catalogue
and web site for these.) The total mass of these asteroids only amounts to about 0.001 Earth
masses. The largest asteroid in the main belt is Ceres, with a diameter of 900 km. The vast
majority are much smaller than this, with their size distribution matching the size distribution
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Sun, Stars and Planets 2020-21
orbital period / yr
number of asteroids per 0.01 AU
2.5
3.0
3.5
4.0
4.5
5
6
7 8 9 10 12
50
7:3
3 :1
5:2
2:1
25
0
1.9
2.1
2.3
2.5
2.7 2.9 3.1 3.3
4.0
5.0
semimajor axis / AU
© The Open University
Figure 18.1: The distribution of main belt asteroids as a function of their orbital radius. The
gaps in the distribution are known as the Kirkwood Gaps. Also noted are the orbital resonances
with Jupiter, which coincide with most of the Kirkwood Gaps. From McBride & Gilmour,
Introduction to the Solar System.
of impact craters on objects like the Moon and Mercury. This is expected since the impacts
were produced by asteroids.
While most asteroids lie between Mars and Jupiter, some exist in the inner Solar System. Near
Earth Asteroids are bodies that come close to the Earth. About 25,000 of these are currently
known. Potentially Hazardous Asteroids (PHAs) are those that come very close to Earth and
might collide with it at some point in the future. About 2173 of these are currently known, of
which 158 have diameters greater than 1 km. Thankfully only 32 PHAs are on the Sentry Risk
Table at the CNEOS (Center for Near Earth Object Studies) with a risk of impact with Earth
in the next 100 yrs.
Most asteroids are too small to have gone through any surface differentiation themselves.
Rather then being solid bodies, like Earth or Mercury, they are thought to be ‘rubble piles’
made up of lots of separate sub-fragments bound together by mutual gravity. The different
fragments can move relative to one another, leading to a somewhat ‘molten’ appearance, with
finer, dusty regolith material settling to lower points in the local gravitational potential, and
larger fragments moving upwards in a manner similar to the motion of brazil nuts in museli
when it is shaken. Data taken by the two Hyabusa spacecraft on their missions to the nearEarth asteroids Itokawa and Ryugu are consistent with this idea.
The distribution of Main Belt asteroids as a function of their semi-major axis (in AU) is shown
in Fig. 18.1. As you can see, the distribution is not uniform, but is characterised by several
distinct gaps. These were discovered in 1857 and are still known as Kirkwood Gaps in honour
129
of the astronomer who found them. Most of the Kirkwood Gaps are easily explained by orbital
resonances with Jupiter. The 2:1, 3:1, 5:2 and 7:3 resonances are clearly seen.
Asteroids come in a range of classes, largely determined by their reflectance spectrum which
allows an estimation of the material on their surface. Classes include:
• C class: surface dominated by carbon (carbonaceous), with reflectance ∼5%. These are
the most common type, representing 40% at 2 AU and 80% at 4 AU.
• S class: surfaces dominated by silicates (stony material), with reflectance ∼16%, and
with a distinct spectral absorption signature at ∼1 µm. These are the second most common class.
• M class: these asteroids are almost entirely metal, containing Ni and Fe. They are rarer,
but have reflectance ∼15%.
• D-class: these asteroids are very dark, with reflectance only ∼3%. They are increasingly
common at greater distances from the Sun. Their surfaces may include organic material.
There are also other classes including E and P. The overall numbers of different classes, especially the low reflectance ones, are difficult to judge since different classes are detected with
differing efficiencies, so the statistics are dominated by selection effects.
The size distribution of asteroids is a power law, with roughly equal amounts of mass in each
logarithmic mass bin, so there are many small asteroids, but only a few very large ones.
One dynamically interesting subclass of asteroid are the Trojan asteroids. They share the
same orbit as Jupiter, but lie 60 degrees ahead and 60 degrees behind the planet’s orbital
position. These points are the so-called L4 and L5 Lagrange points, where the gravitational
and centrifugal forces of two orbiting masses cancel out for a third, smaller, orbiting body. The
L4 and L5 points are stable in the gravitational potential, so that objects that arrive there will
stay there.
18.2.1
Ceres
Ceres, with a diameter of 945 km, is the largest of the main belt asteroids and was studied
by the NASA Dawn mission. Ceres has an oblate spheroid shape, suggesting that it is at least
partially differentiated and is in hydrostatic equilibrium. Current models of Ceres’ interior
include a rocky core, an icy mantle, and a surface layer made up of a mixture of water ice and
rocky material.
The Dawn mission led to a detailed study of Ceres. One of the most unusual surface features
found by Dawn are white spots at the centre of some of the impact craters (see Fig. 18.2).
These are thought to be conglomerations of salts, more reflective than the surrounding material. The origin of these salts is unclear, but one theory is that they are the remains of liquid
brine, brought to the surface from a subsurface liquid ocean through impacts or cryovolcanoes.
The brine then evaporates leaving behind the reflective salt deposit. If this idea is correct then
Ceres would be another place in the Solar System with liquid water beneath its surface. Possible support for this idea has come from the Herschel Space Observatory which found signs
of water vapour emerging from a number of regions around the equator. About 3 kg of water
is emerging per second. Whilst sublimation of water ice that has collected on the surface of
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Figure 18.2: A close up image of one of the white spots at the centre of an impact crater
on Ceres. By NASA / JPL-Caltech / UCLA / Max Planck Institute for Solar System Studies /
German Aerospace Center / IDA / Planetary Science Institute.
Ceres remains a possible explanation, it is more likely that this water is coming from Ceres’
interior and is a confirmation of the existence of a subsurface ocean.
18.3
Kuiper Belt and Trans-Neptunian Objects
The formation, and a hypothesized possible migration of the giant planets (Jupiter inwards
towards the Sun, and Saturn, Uranus, Neptune possibly outwards further from the Sun), has
had the effect of largely clearing small bodies from the region of the orbit of Jupiter out to the
region of the orbit of Neptune. Beyond the orbit of Neptune, though, small bodies can persist
undisturbed. As early as the 1950s, Gerard Kuiper proposed the existence of a belt of small
bodies beyond the orbit of Neptune. Pluto, and its largest moon Charon, were already known
at this point, discovered in 1930, but it was not until 1992 that any further trans-neptunian
objects (TNOs) were discovered. We now know of at least 70,000 such objects, with diameters
>100 km, forming what has been called the Kuiper belt, which lies 30-50 AUs or more from
the Sun (see Fig. 18.5). The total mass of objects in the Kuiper Belt has been estimated to be
∼0.1 Earth mass, meaning that the Kuiper Belt actually includes more mass than the ‘main’
asteroid belt between Mars and Jupiter.
Kuiper Belt objects (KBOs) are thought to be the left overs from earlier stages of the formation
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Figure 18.3: Pluto as seen by New Horizons, with a region blown up to show the widely
differing types of terrain. By NASA.
of the Solar System, made up of material with a high fraction of ices and volatiles. Reflectance
spectra of KBOs have a wide range of properties, which may be the result of long term changes
in their surface properties resulting from exposure to UV light from the Sun, but also resulting
from more abrupt surface changes coming from impacts between KBOs. Our knowledge of
the outer solar system is still very incomplete. The NASA New Horizons mission is helping
with this following its encounter with Pluto in 2015 and study of other KBOs since then.
New Horizon’s results on Pluto show a much more dynamic and interesting world than the
frozen, dead, cratered landscape that many expected (see Fig. 18.3). There has clearly been
active geology on Pluto, driven, perhaps, by its periodic warming and cooling that comes from
its elliptical orbit. Since Pluto is so cold, some of these features are driven by the movements
of frozen gases - there are, for example, glaciers of frozen nitrogen on its surface.
New Horizons went on to visit Arrokoth, nicknamed Ultima Thule at the time, in the outermost
close encounter of any Solar System object, at 43.4 AU. It revealed it to be a contact binary 36
km long, composed of two planetesimals that are joined along their major axes, see Fig. 18.4.
It has a low reflectance and an unexpected reddish brown colour, thought to be due to organic
compounds on its surface interacting with sunlight.
While we await future detailed studies using ground and space based telescopes, the classifications of KBOs are largely based on their orbital properties, and in particular on their orbital
eccentricity and semi-major axis. When these are plotted together (see Fig: 18.6) you can see
three separate groups of orbital characteristics which leads to the division of KBOs into separate classes:
• The majority are classified as Classical KBOs, and have low eccentricity orbits with radii
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Sun, Stars and Planets 2020-21
Figure 18.4: Arrokoth (Ultima Thule) as seen by New Horizons, NASA.
of ∼44 AU.
• The second largest group have a range of eccentricities and all lie at a radius close to
39.4 AU. Pluto is one of these objects, leading them to be termed Plutinos. If the orbital
period of the Plutinos is compared with that of Neptune, you find that Neptune orbits
the Sun three times for every two orbits that a Plutino makes: they are in a 3:2 orbital
resonance with Neptune, keeping them in this orbital position. Pluto is in many ways
indistinguishable from the other Plutinos, a result which eventually led to the reclassification of Pluto as a dwarf planet.
• The final group of KBOs have high eccentricities and large semi-major axis. They are
classified as Scattered Disk Objects and are likely the source of short period comets.
An additional class of small solar system body that is likely associated with KBOs are the socalled Centaurs. These are objects whose orbits cross the orbits of one or more major planet.
Such orbits will not be long lasting because they will eventually encounter the gravitational
field of a major planet and be captured or scattered into a different orbit. They may even hit
one of the giant planets. An example of this was the impact of Comet Shoemaker-Levy 9 with
Jupiter in 1994. The discovery of Centaurs predates that of KBOs other than Pluto by a number
of years. The first Centaur, called Chiron, was found in 1977. The fact that it was in an orbit
that was not stable in the long term hinted at the existence of a larger body of similar objects
in more stable orbits that would be able to feed Centaurs into the Solar System. The reservoir
for the Centaurs, and for short period comets, is the Kuiper Belt, which was discovered 15
years later.
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Figure 18.5: The orbits of a selection of Kuiper Belt Objects compared to the orbits of Jupiter,
Saturn, Uranus, Neptune and Pluto (J, S, U, N, P). From Rothery, McBride & Gilmour.
Figure 18.6: The orbital eccentricity plotted against semi-major axis for KBOs. From Rothery,
McBride & Gilmour.
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18.4
Sun, Stars and Planets 2020-21
Comets
The final class of small body in the Solar System that we will discuss are perhaps the most
spectacular: comets.
Comets are objects on highly eccentric orbits that come from the outer to the inner parts of
the Solar System. They are objects rich in ices and volatiles which are released when they heat
up in the inner parts of the Solar System, giving rise to the spectacular tails (one of dust and
one of ionised material, which interact differently with the solar wind). They have very short
lives, only a few 104 years before mass loss and dynamical interactions with planets lead to
their destruction. Their internal structure is thought to resemble a ‘dirty snowball’, where dust
and rock is mixed with ices, including water ice and other frozen volatiles. The dirty snowball
forms the comet’s nucleus, which is small (10-20km) and very porous. When they approach
the Sun they heat up and volatiles boil off, leading to the familiar shape of the comet tails.
There are two classes of comet based on their orbital period - short period (<200 years) and
long period. The short period comets, like Halley, have low orbital inclination and are usually
prograde. These comets are thought to originate in the Kuiper Belt. Long period comets have
much higher orbital inclination and are as often in retrograde as prograde orbits. They are
thought to come from the Oort cloud, a spherical reservoir of comets believed to lie at much
greater distances from the Sun than the Kuiper Belt, out to as far as 50000 AU, about a quarter
of the way to the nearest star. So far, no definitive detection of an object in the Oort Cloud
has been made. Instead, its existence is currently inferred from the presence of the long period
comets, in much the same way that the existence of the Kuiper Belt was once inferred from
the discovery of Centaurs. Sedna, a TNO with a very eccentric orbit with an aphelion of ∼1000
AU is our best current candidate for a member of the Oort cloud.
Figure 18.7: Rosetta image of Comet 67P/Churyumov-Gerasimenko showing the famous double lobed ‘rubber duck’ structure and plumes of gas and dust being ejected as it warms in the
Sun.
135
Towards the end of 2014 the Rosetta mission had a rendezvous with the periodic comet 67P/
Churyumov-Gerasimenko. The Philae lander was sent down to the surface in November 2014,
suffering a rather bumpy landing but still providing valuable data. Over the following two
years, the spacecraft followed the comet as it fell into the inner Solar System, heating up and
becoming more active with jets and streamers of gas and dust erupting from its surface. One
key discovery of this mission was the ‘rubber duck’ structure of the comet (Fig 18.7), which
seems to be made up of two separate bodies that fused together to form the object we see
today. The Rosetta mission ended at the end of September 2016 with the spacecraft itself
landing on the surface of the comet and turning itself off. It will now become part of the
comet, orbiting the Sun with it into the future.
Things to Remember
• The basic parameters of the asteroid belt and the classes & constituents of asteroids
• The Kirkwood gaps and their origin in orbital resonances. The Trojan asteroids.
• Properties of Kuiper Belt objects. The Centaurs.
• The internal structure and origin of comets, including the orbital properties of
long & short period comets
• How these relate to the Kuiper Belt and Oort Cloud
To Do:
Problem Sheet 4, question 3
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Sun, Stars and Planets 2020-21
Lecture 19
Detecting Exoplanets
It was a very old question which was debated by philosophers: are there other worlds
in the Universe? We look for planets which are the closest, which could resemble
Earth. Together with my colleague we started this search for planets, we showed it
was possible to study them
- Michael Mayor on the nobel prize winning discovery of the first exoplanet orbiting a Sun-like star, Prize 2019 Mayor and Queloz.
19.1
Introduction
We have spent the last 8 lectures, looking at the properties of objects in our own Solar System.
Not so long ago, that would be where things stopped, since we knew of no other planetary
bodies in the Universe. Whole theories of planet and planetary system formation were developed on the basis of the 8 planets and many minor bodies of our own Solar System, but there
was lack of other places where these models could be tested.
Over the last 25 years, though, there has been a revolution in our understanding of planetary
systems, resulting from a series of technological breakthroughs that have allowed planets in
other solar systems - exoplanets - to be discovered in ever greater numbers. Specific observatories on the ground and in space are completely dedicated to planet searches, and over 4720
planets in other systems are now known 1 . More planet discoveries are announced every day,
so some of the raw numbers in these notes will already be out of date. You can keep track of
the latest results through dedicated websites such as exoplanet.eu.
19.2
Units
For the rest of this course we will be looking at objects far away from the Solar System, and
will thus have to use astronomer’s units. Many of these are based on scalings to known objects
eg. the mass of the Sun (M ), the astronomical unit, the Solar luminosity (L ), but there are
other more specialised units that we will use.
1
Discoveries continue at a fast pace - previous years’ lecture notes have said ’over a hundred’ and later ’nearly
a thousand’ only to be updated as new discoveries are announced. You can keep up to date using websites like
exoplanet.eu
137
Figure 19.1: The number of exoplanets discovered each year. Plot produced using the tools
and database available at exoplanet.eu
The first, and most important, of these is the parsec. This is a distance of 3.26 light years,
3.08×1016 m, or 2.1 ×105 AU. The parsec is derived from the distance at which an object must
be for it to have a proper motion on the sky of 1 arcsecond when the Earth moves by 1 AU.
Essentially this means that a parsec is the length of the adjacent side of a triangle which has
an angle of 1 arcsecond and an opposite length of 1 AU. See Fig. 19.2 for a diagram.
The other thing that we will be using that could be termed an astronomer’s unit are magnitudes. These will be used to express the brightness of stars and changes in the flux received
from them. Magnitudes were introduced to you in the first (Stars) part of this course, but as
a reminder, the difference in magnitude between two objects is given by:
F1
m1 − m2 = −2.5 log
(19.1)
F2
where m1 and m2 are the magnitudes of the two objects, and F1 and F2 are the fluxes of the
two objects. This just deals with how to compare two objects. For more on magnitudes see
the Stars section of the course.
19.3
What is a Planet Anyway?
Within our own Solar System, there is a formal definition of a planet. It is a body orbiting the
Sun, whose gravity is strong enough to make it spherical, and which has cleared its neigh-
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Figure 19.2: Diagram showing how the parsec is derived
bourhood of smaller bodies. The latter part of this definition, agreed by the International
Astronomical Union (IAU) in 2005, is what demoted Pluto to minor planet status.
Outside our Solar System the definition of what is a planet is less clear. The IAU definition
states that to qualify as an exoplanet a body must be orbiting a star and have a mass below the
threshold at which thermonuclear fusion of deuterium can take place. This sets the maximum
mass for a planet at ∼ 13 MJ . No consideration is given to how these bodies formed, and
the minimum mass should match the minimum mass to qualify as a planet in our own Solar
System.
Things that are not considered exoplanets in this scheme include objects above the deuterium
burning mass limit, which are defined as brown dwarfs (these are essentially failed stars) and
free floating bodies that are low enough mass to qualify as an exoplanet, but which do not
orbit around a star. These are termed sub-brown dwarfs, and recent results suggest that they
may be more numerous than stars in our galaxy.
A physical definition of a planet based on formation history and/or composition, which might
be a more scientific approach, is still lacking.
19.4
Direct Detection: How Hard Can it Be?
In the current era of space telescopes and large, 8-10m, telescopes on the ground, one might
think that directly detecting an exoplanet orbiting around another star would be easy. Unfortunately, this is far from true, mainly because of the huge contrast between the light that
comes from the star and that which is reflected from the planet, and because of the small
angular separation between any planet and its parent star.
139
Consider a planet with an albedo of 1, visible only because of light reflected from its parent
star. The planet’s radius is Rp , and it is orbiting a distance d from a star of luminosity L∗ . The
stellar flux received by the planet will be:
F =
L∗
4πd2
(19.2)
The planet’s luminosity, Lp then comes from the total power it intercepts, which we will assume is across a disc of area πRp2 . Assuming that it has an albedo of 1:
Lp = F πRp2
Substituting in for F from 19.2 then gives:
L∗
Lp =
4
Rp
d
2
(19.3)
If we put in numbers appropriate for Jupiter into this equation - Rp ∼ 7 × 107 m and d ∼ 5
AU - and calculate its relative luminosity to the Sun, we find:
LJ
L
=
L
4
RJ
d
2
×
1
1
=
L
4
7 × 107
5 × 1.5 × 1011
2
⇒
LJ
∼ 2 × 10−9
L
(19.4)
So, to directly detect a planet like Jupiter orbiting another star you will have to remove the
light of that star to an accuracy of about 1 part in a billion - the star outshines the planet by
that much. This is very difficult to achieve. The situation is somewhat better in the infrared,
where the Black Body spectrum of the hot star is declining, but where that of the cooler planet
is peaking, but this still requires better than 1 part in a million exclusion of stellar light. There
are ways that this can be achieved, using techniques such as choronography and nulling interferometry, but this all means that direct detection is not an efficient way to search for planets.
However, it can be, and has been, used to follow up planets that have already been detected
by indirect methods, so as to better characterise the objects. For example, the first direct spectrum of an exoplanet was obtained in 2010 by Bowler et al using such an approach.
If we cannot search for exoplanets directly, how can they be found?
Fortunately there are a range of indirect methods that look for the effects of any planets that
may be present on their parent star.
19.5
The Astrometric and Radial Velocity Methods of detecting
exoplanets
A family of detection methods are based on studying the dynamical effects of an orbiting
planet on the parent star.
Just as with binary stars, a star and planet actually orbit around a common centre of mass, but
with the planet mass much smaller than the stellar mass. Viewing this in the centre of mass
frame it looks like Fig. 19.3. From the stars part of the course we know that for binary stars:
M1
r2
4π 2 r3
=
and M1 + M2 =
M2
r1
G P2
(19.5)
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Sun, Stars and Planets 2020-21
a_p
x
a_s
Figure 19.3: A star orbits around the common centre of mass (CoM) with orbital radius a s,
while the planet orbits around the CoM with a radius a p.
where P is the orbital period. So, for a planet of mass mp orbiting a star of much larger mass
Ms :
mp
as
4π 2 a3p
=
and Ms =
(19.6)
Ms
ap
G P2
where as and ap are the distances from the star and planet to the common centre of mass
respectively.
The first result from this analysis, as can be seen from Fig. 19.3 is that the position of a star
being orbited by a planet will appear to wobble around on the sky because of its reflex motion.
While the planet cannot be directly seen, its existence can be inferred if we can measure the
star’s regular displacement as . This leads to the astrometric method of detecting exoplanets.
In fact the angular displacement β = as /d, where d is the distance to the star from the
observer, is what is usually measured. From the above we can find that
since
β=
⇒
as
d
and
β=
mp
as
=
Ms
ap
mp ap
Ms d
(19.7)
Note that if ap is in AU and d is in parsec, then β will be in arcsec units.
So you get a larger, and thus more detectable, angular displacement β for large mass planets
in wide orbits around low mass stars that are ideally not too far from us.
For our own Solar System viewed from a distance of 10 pc, you would see a displacement lower
than 0.4 milliarcsec per year from the effect of Jupiter orbiting the Sun. This is a very small
angular shift, corresponding to the width of a finger at 5000 km, so it is not a particularly
141
viable method of planet detection.
EXERCISE: Calculate the displacement you get from the Earth’s orbit around the Sun, when
viewed from 10 pc. (Answer in lecture recording).
However, an alternative method based on this same idea comes from looking at the motion
of the star along the line of sight, rather than in the plane of the sky. What can be measured
here are changes in the velocity of the star in the line of sight, which can be found by looking
at Doppler shifts from spectral lines. Recall that vc = ∆λ
λ . This leads to the Radial Velocity
method of detecting exoplanets.
Recalling the results for binary stars (see Eq. 11.6):
M23 sin3 (i)
P
= v13
2
M
2πG
(19.8)
where i is the inclination angle between the orbital plane and the line of sight, and P is the
orbital period.
For planets, adapting Eqn 19.8 using M2 = mp , Ms mp so M = Ms + mp = Ms and
v1 = vs , gives:
2 1/3
Ms P
vs
(19.9)
mp sin(i) =
2πG
Without knowing the inclination angle i, this method only allows you to calculate the minimum mass for a planet.
The largest values of vs come for large mass planets orbiting low mass stars with short orbital
periods.
If you were observing along the plane of the ecliptic, the velocity shifts from the Sun-Jupiter
system amount to 12 m/s, and from the Sun-Earth system amount to 0.1 m/s. Velocity shifts
as low as 0.5 m/s have been measured, and the first clear detection of exoplanets around main
sequence stars were obtained using this method.
19.6
Planetary Transit Searches
If a planetary system’s orbital plane lies along our line of sight, planets will from time to time
pass in front of their star, absorbing some of the light from the star that would otherwise reach
us. This kind of thing can be seen in our own Solar System where Venus or Mercury can be
seen to pass in front of the Sun. The last transit of Venus was in June 2012. Planetary transits
will cause a small, but potentially measurable, dip in the brightness of a distant star observed
from Earth (see Fig. 19.4).
What flux decrease will a planetary transit produce?
The stellar flux Fes is the power emitted per unit area of the star. Therefore the luminosity of
the star Ls is given by this flux multiplied by the emitting area, which we will approximate to
a disc of area πRs2 where Rs is the radius of the star. Let us consider what happens when an
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parameters:
* orbital period
* planetary radius RP/RS
* planetary mass (need MS)
* inclination of orbit
Sun, Stars and Planets 2020-21
flux decrease ≈ (RP/RS)2
Sensitivity/bias:
• easier to detect larger
planetary radii and
• small semi-major axes
→ short periods
→ `hot Jupiters’
ΔF
Figure 19.4: A diagram showing schematically what happens to the light received form a star
as a planet transits along our line of sight.
orbiting planet, of radius Rp transits in front of this star on our line of sight. This will lead to
the luminosity of the star appearing to decrease due to the light from the star being blocked
by the planet - effectively the area of the stellar emitting surface has been decreased by the
area of the planet’s disc.
Ls ≈ πRs2 Fes
before transit
πRs2 − πRp2 Fes
0
during transit Ls ≈
So the drop in observed stellar luminosity is:
0
∆L = Ls − Ls ≈ πRp2 Fes
giving an observed fractional luminosity drop of:
∆L
=
Ls
Rp
Rs
2
=
∆F
F
(19.10)
where F is the flux we measure. For the Solar System, Jupiter would cause a 1% drop in the
light seen from the Sun, which is large enough to be measurable from the ground, while the
Earth would produce a 0.01% drop, which can be measured from space.
If you know that the planet is transiting then Doppler measurements can determine the planet’s
mass. The star’s radius can be determined from the duration of the transit, leading to the radius of the planet. That, combined with the mass, allows the density to be calculated, which
is the first step towards understanding what the planet is made of.
For a transit to be detected the planet’s orbital plane must be quite closely aligned with the
line of sight to the star. Assuming a random orientation of orbital inclinations for planetary
systems, and considering the diameter of the Sun, it can be shown that there is a chance
143
of about 1 in 200 for the transit of an Earth-like planet around a Sun-like star to be visible.
Such transits would happen only once a year for the Earth, and you would need to observe
at least two such transits to be sure that it was detected, and to measure the orbital period.
The Kepler satellite was thus designed to monitor a total of 105 stars for a period of 3 to 5
years in search of, among other things, Earth like planets. If they are common, it should be
able to detect several such systems. Sadly, a technical failure on the satellite led to the end of
its main planet hunting mission after only about 3 years, but a revised strategy allowed it to
continue work for a further 6 years as the K2 mission. Operation of Kepler ceased in 2018 as
it had run out of fuel. Kepler has so far discovered over 2662 planets and its data is still being
analysed to find more. A new transiting planet mission, TESS (the Transiting Exoplanet Survey
Satellite) was launched in 2018 and is expected to discover 20,000 new exoplanets during its
two year mission, as of March 2021 over 2000 exoplanet candidates from TESS have been
reported. Kepler and TESS were both NASA missions, but there are ESA exoplanet missions
as well. These include CoRoT, the first planet transit satellite that operated between 2006 and
2013, and the new future ChEOPS (launched in 2019), and future PLATO and ARIEL missions
which will better characterise transiting exoplanets.
19.7
Other ways to detect planets
While the transit and radial velocity methods are responsible for the detection of most of the
exoplanets that we know, there are a couple of other methods that have proven useful.
19.7.1
Pulsar Planets
The first of these is the timing of pulsars, which led to the detection of the very first exoplanets. Pulsars are rotating neutron stars, remnants of supernova explosions, which emit beams
of electromagnetic radiation from their magnetic poles. These beams act like lighthouses,
producing regular pulses which can be timed to picosecond accuracies. Regular deviations in
these pulses, produced by the same centre-of-mass shifts seen above for the transit method,
can be measured to accuracies better than 1m/s in velocity and 1000 km in distance. This has
allowed a small number of planets, with masses down to 0.0004 Earth masses, to be discovered.
None of them are likely to be particularly nice places to live, though, since these systems have
survived a supernova and are now bathed in hard radiation from the pulsar.
19.7.2
Gravitational Lensing
Gravitational lensing is the process by which light is bent and focussed as it passes close to a
large mass. Stars and planets are both large enough to produce a measurable magnification
of the light of a background star if they pass close enough to our line of sight. Large scale
monitoring projects like OGLE, originally intended to search for Baryonic Dark Matter, can
in principle detect the lensing amplification produced by a planet orbiting a star responsible
for lensing, and there are a small number of cases where this has been found. The advantage
of this approach is that it is sensitive to essentially all possible planetary masses, but the
disadvantage is that the lensing signal is not repeatable, so one can never be absolutely certain
what has produced it, or determine the full characteristics of any planetary system the lensing
has revealed.
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Things to Remember
• The definitions of parsec, AU, magnitude
• The definition of a planet, and how to distinguish it from a brown dwarf and a
minor planet
• The problems of direct detection of exoplanets; derivation of luminosity ratios
• The derivation of reflex motion, the astrometric method of finding exoplanets
• Derivation of the radial velocity signal, and radial velocity method for finding
exoplanets
• The derivation of the change in the flux from a star due to a planetary transit
• The use of the above methods in detecting planets and other planet detection
methods
To Do:
Problem Sheet 4, questions 4, 5, 6 and 7
145
Lecture 20
The Exoplanet Population
We are the first generation capable of studying planets around other stars
- Prof Giovanna Tinetti, UCL
20.1
Introduction
At the time of writing, over 4700 planets are now known and confirmed outside our own Solar
System, with there being at least 772 multiple-planetary systems. More exoplanets are being
discovered all the time thanks to ongoing survey programmes such as the TESS satellite, the
SuperWASP survey continuing analysis of data from the Kepler mission, looking for planetary
transits, and the HARPS radial velocity survey. We have reached the point where we can
draw conclusions about some aspects of the exoplanet population. However, the methods of
exoplanet detection all have limitations, so our view of the population as a whole is necessarily
incomplete, and biased by what are known as ‘selection effects’. In this lecture we will look at
what is known about the exoplanet population, try to deconvolve some of the selection effects,
and draw some conclusions about the overall population of planets in our galaxy.
20.2
The Current State of Planet Searches
New planet discoveries are announced all the time, so any attempt to describe the current
state of planet searches is doomed to become rapidly out of date. However, the broad picture
is that we now (May 2021) have discovered over 4720 confirmed planets, and of these, 3491
are in multiple planet systems, of which there are 772 known. While exoplanets were first
discovered in significant numbers by the radial velocity method, more recently most of the
exoplanets are being discovered using the transit technique, from the Kepler mission, which
recently ceased operation, and TESS which has been in operation over the last two years. TESS
has recently released a list of over 2000 exoplanet candidates that need to be followed up for
confirmation. About 15% of discovered planets are found in multiple planet systems. The parent stars of the planets discovered so far are largely F, G and K type main sequence stars.
This does not, however, mean that other stellar types do not have stars, since the majority of
planet searches have been targeted at F, G and K type stars. This is because these stars are
sufficiently long lived that life might have developed on planets around them, they are relatively bright, compared to the more common low mass M stars, and they are well suited to
the radial velocity method, since they have many well defined spectral lines, and have stable
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Sun, Stars and Planets 2020-21
stellar atmospheres. The nature of planets around other stellar spectral types is thus largely
unconstrained.
On the basis of current results we can say that at least 20-50% of F, G and K-type stars have
at least one giant planet, comparable in mass to Jupiter, in an orbit whose semi-major axis is
<20 AU.
20.3
Selection Effects
The issue of host stellar type is the first in a series of ‘selection effects’ that constrain and
bias what we are able to say about the exoplanet population. Selection effects arise in a wide
variety of sciences, especially observational ones like astrophysics, where one does not have
full control over what you find. Selection effects are often quite subtle and can require careful
consideration, but they can also be quite obvious once the observational problem is understood. The preponderance of F, G and K-type stars in the radial velocity method searches is
a case in point. Another is the sensitivity, or lack of it, of various methods to various types of
planets. The radial velocity method, for example, is not sensitive enough to discover an Earth
mass planet in an Earth-like orbit around any other stars. The radial velocity changes that the
Earth produces on the Sun have an amplitude of ∼ 0.1 m/s. The most accurate radial velocity
measurements so far achieved are of the order of 0.3 - 1.0 m/s, so we are still some way from
being able to search for Earth-like planets easily using the radial velocity method.
While some selection effects will exclude some classes of planet from what we can detect,
other selection effects will lead to other classes of planet being much easier to detect. High
mass planets that are in orbits very close to their parent stars produce the largest radial velocity shifts. Such ‘Hot Jupiters’ are thus very well represented in the current results of exoplanet
searches.
These and other selection effects must be carefully considered when using the current set of
known exoplanets to derive conclusions about the overall population of exoplanets. Nevertheless, this is what we are about to do.
20.4
Exoplanet Masses
Figure 20.1 shows a histogram of currently known planet masses, measured in terms of Jupiter
masses. As can be seen, the vast majority of currently known exoplanets have masses that
would class them as gas giants if they were in our own Solar System, with masses > 0.05 MJ ,
the mass of Uranus. In fact, most of these masses are not actual masses but are lower limits
to the mass of planets detected using the radial velocity method, and thus are in fact measurements of mp sin i where i is the angle of inclination of the orbit to the line of sight. To go
from such a radial velocity minimum mass to an actual mass measurement, a determination
of the inclination angle is needed. If a transit observation is available then we know that the
inclination angle i is high, close to 90o . An example of this is the planet HD209458b, where a
transit observation and a radial velocity shift is seen.
If a transit is not seen, then estimates for the inclination angle can sometimes also be obtained
by other observations. In the case of Epsilon Eridani, a dust ring around the star is observed.
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Figure 20.1: Histogram of planet masses. The x-axis shows log(planet mass/Jupiter mass).
Earth’s mass is 0.003 times that of Jupiter. As can be seen, nearly all currently known planets
have masses in the range of gas giants, with hardly any planets known that have masses
comparable to that of Earth. This is because current detection techniques make it very difficult
to detect and confirm a planet with mass comparable to that of Earth. Generated using data
from exoplanet.eu
This appears elliptical in the observations, but such rings are expected to be circular. From
the dust ring ellipticity, an inclination angle of 46o can be estimated. The orbital plane of the
planets in this system should match that of the dust ring, so we can then estimate the true mass
of the planet Epsilon Eridani b. From radial velocity measurements we have M sin i = 0.86MJ ,
thus:
0.86MJ
M sin i = 0.86MJ ; i = 46◦ ⇒ M =
= 1.2MJ
(20.1)
sin 46◦
The fact that most of the planets known have masses in the gas giant range is not surprising.
Our current detection techniques are largely insensitive to lower mass planets. We thus would
not expect to have found many, or in fact any, Earth mass planets in our studies to date. The
few Earth and lower mass planets that appear in Fig. 20.1 are the result of pulsar timing or
gravitational lensing detections which are not subject to the selection biases in favour of large
mass planets that apply to the radial velocity and, to some extent, transit methods that are
responsible for the majority of planet detections. Nevertheless, a statistical analysis of Kepler planet candidates, including many that are potentially Earth-mass but which cannot be
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Sun, Stars and Planets 2020-21
confirmed by radial velocity measurements (Petigura et al., 2013), had concluded that 22% of
Sun-like stars harbour planets in orbits such that liquid water is possible on their surfaces (a
region known as the habitable zone - see next lectures). The nearest star with such a planet
could be as close as 12 light years away.
Two very recent estimates, based on Kepler results are very striking. Bryson et al (The Astronomical Journal, 161, 36, 2021) suggest that between 0.4 and 0.8 of all solar type stars in the
Galaxy have Earth-like rocky worlds in orbit in their habitable zone. Given there are probably
around 17 billion solar type stars in our Galaxy, this means there are likely to be a vast number of Earths out there! A handful of these would be expected to be within a few light-years
of Earth. This result assumes that the section of the sky Kepler monitored for four years is
representative of the whole galaxy. Another Kepler study (Kunimoto et al, AJ, 159 (2020) ) also
predicts a large number of Earth analogs waiting to be discovered.
20.5
Exoplanet Composition
Detailed analysis of the composition of an exoplanet is not something we can yet achieve.
However, simply being able to measure the density of an exoplanet would be a big step towards
understanding what it might be made of, especially bearing in mind the range of densities of
planets in our own Solar System, with the terrestrial planets being much denser than the gas
giants. Masses and planetary radii are available for over a thousand planets so far. The vast
majority of these turn out to have low densities, comparable to those of our own gas giants.
There are a handful of exoplanets that have higher densities, though, and these can be considered candidate terrestrial planets. In most of these cases their mass estimates are currently
rather uncertain, so there are large uncertainties on their derived densities.
New observations will change this situation significantly over the next few years. The CHEOPS
satellite, launched in 2019 by the European Space Agency (ESA), will measure the size for as
many as 500 exoplanets through precise transit observations. Combining these size measurements with mass measurements from radial velocity observations will allow the density of the
planets to be precisely determined.
The best way to determine the composition of an exoplanet, or at least its atmosphere, is to
obtain spectroscopy, but this is an even more difficult job than direct detection of the continuum light of a planet. However, for transiting exoplanets, a number of tricks are possible. Time
resolved spectroscopy allows us to look at the effect of the exoplanet on the light of the star as
it passes through the planet’s atmosphere. Comparison of stellar spectra before, during and
after the transit allow the size and some aspects of the planet’s composition to be measured.
This was first achieved on the planet HD209458b, a gas giant orbiting 0.045 AU from its parent
star. Absorption in the Lyman α line coming from hydrogen in the planet’s atmosphere was
found to cover 15% of the stellar disk, rather than the 1.5% covered by the opaque core of the
planet, implying that the atmosphere of this planet is very extended, resulting from the fact
that it is being heated to high temperatures by the star. Since these initial observations, similar spectroscopic transit studies have revealed water vapour, carbon dioxide and methane in
HD209458b’s atmosphere, as well as hydrogen. These are all things you would expect to find
in a gas giant’s atmosphere.
The James Webb Space Telescope (JWST), to be launched hopefully later this year, will be used
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to study the atmospheres of the most interesting of objects using transit spectroscopy. Targets will be chosen from known planets that have been well characterised by CHEOPS and
by other observations, some of which will have come from the ground. Meanwhile, the TESS
mission, launched by NASA in 2018, is scanning the whole sky for transiting planets around
bright and nearby stars, and has just announced 2200 candidate planets. It is capable of finding Earth sized planets around low mass stars, and has already found some. These will also
be prime candidates for JWST follow-up studies of their atmospheres. The TESS project lists
objects of interest in a publicly accessible database so you can look at the data yourselves:
https://exofop.ipac.caltech.edu/tess/. In 2026, ESA will launch PLATO, a mission which aims
to look for transits around a sample of a million nearby bright stars. It will also be capable
of determining the sizes of these planets to an accuracy of about 3%. In some ways it is a
combination of the best characteristics of both CHEOPS and TESS.
While JWST and some ground based telescopes will be able to study the atmospheres of small
samples of exoplanets, or specific objects of special interest, in 2029 ESA will launch the ARIEL
mission which will use spectroscopy of planetary transits to study the atmospheres of over
1000 exoplanets. This will be the first large scale study of exoplanet atmospheres and should
produce a revolution in our understanding of the composition of these atmospheres and how
this relates to the formation and evolution of exoplanets.
20.6
Exoplanet Orbits: Hot Jupiters and Planetary Migration
One of the big surprises when exoplanets started to be discovered was that there are a large
number of ‘hot Jupiters’ - gas giant planets that orbit very close to their parent stars (see Fig.
20.2). These are in fact the easiest objects for both radial velocity and transit studies to detect,
but there was no expectation at all, before their detection, that such things would exist. The
reason for this is that gas giants are expected to form much further out in their star-planetary
systems since the young star will, on first ignition, heat and boil off all the volatiles in the inner
regions of the protoplanetary disk. This is why we see terrestrial planets close to our own Sun
and gas giants further out.
Hot Jupiters must therefore migrate inwards, from their formation location, to where they are
seen by our exoplanet observations. The best current idea for how this occurs is that there is
an interaction between the forming gas giant and the protoplanetary disk during the process
of formation that causes it to move inwards. The infall cannot proceed too far or the gas giant
will end up in the star, so some other process has to terminate the migration, possibly as a
result of the young star boiling away the protoplanetary disk. As a gas giant moves inwards
in its system, smaller terrestrial planets will be scattered out of their systems or be pushed
inwards and fall into their stars.
If this is common, how did our own solar system and planet stay as they are? There is some
evidence from our asteroid belt that Jupiter moved inwards by about 0.2 AU, but then this
motion stopped. It turns out, from modelling studies, that interactions between gas giants
when there is more than one in a system can slow or halt any inward migration. Perhaps our
existence on Earth is a result of such an interaction between Jupiter and Saturn.
The orbital eccentricities of exoplanets are often much larger than those seen in our own Solar
System (see Fig. 20.3). This may come about through orbital resonances between two gas
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Figure 20.2: Mass and orbital semi-major axis for non-pulsar planets. Many more gas giants
close to their parent star are found than expected. From Rothery, Gilmour & Sephton.
giants, or through close encounters between gas giants, which would result in one gas giant
being expelled from the system and the other acquiring a high eccentricity orbit, passing close
to its parent star.
20.7
Host Star Metallicity
One other result that has emerged from studies of exoplanets and their host stars is that it
appears that planets are more likely to be found orbiting stars with higher metallicities - ie.
that contain more enriched material. The origin of this effect is currently unclear, and it may
be that this is actually the result of a subtle selection effect and not a genuine signal. If real,
two possible explanations are:
• That an inherently more metal rich star will have more metals in its protoplanetary
disk, possibly enhancing the condensation of dust into planetessimals and increasing
the likelihood of planet formation.
• Alternatively, it might be that inner, rocky terrestrial planets often have their orbits
disrupted and end up falling into their parent star, enriching its atmosphere
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Figure 20.3: Orbital eccentricities vs. semi-major axis for exoplanets compared to those of
Jupiter and Saturn. From Rothery, Gilmour & Sephton.
20.8
Exoplanets: A young Science
The study of exoplanets is still very young. We have seen many surprises so far, including the
discovery of hot Jupiters and more broadly that planetary systems cover a much wider range
of properties, such as orbital eccentricities, than was once expected. It seems possible that
our own Solar System is rather more stable gravitationally than many of the other systems
uncovered so far. Whether we are lucky in living in a solar system where the young Earth
could survive or not is unclear. However, we still cannot easily detect terrestrial plants in other
systems, and the observations we do have are potentially subject to a wide range of selection
effects and biases. There is much work to be done in this field in exploring the properties of
exoplanets and how they are related. You can do some of this yourselves with exoplanet.eu,
which collects data on all exoplanets as they are discovered and provides tools for analysing
their properties.
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Sun, Stars and Planets 2020-21
Things to Remember
• The current state of exoplanet searches
• The results of selection effects in exoplanet searches
• The calculation & observed distribution of exoplanet masses
• The determination of exoplanet composition
• Exoplanet orbits, hot Jupiters and planetary migration
To Do:
Problem Sheet 4, Question 8
153
Lecture 21
Astrobiology: Life on Other Planets
The universe is very big - there’s about 100,000 million galaxies in the universe, that
means an awful lot of stars. And some of them, I’m pretty certain, will have planets
where there was life, is life, or maybe will be life. I don’t believe we are alone.
- Jocelyn Bell Burnell, Astronomer
21.1
Introduction
Not so long ago, the quest for life elsewhere in the universe could be regarded as speculation
that would remain impossible to test. A range of discoveries of the last 20 years, however, have
drawn this topic into the scientific mainstream, and there are now many people working in the
general area of astrobiology. This includes astronomers and physicists, but also biologists and
geologists, since studies of the history and diversity of life on Earth can inform our searches
for life elsewhere.
21.2
Life on Earth: History
The Earth formed roughly 4.5 Gyr ago, and the Late Heavy Bombardment ended about 4 Gy
ago. The earliest clear signs of life on Earth are structures called stromatolites, which are
built up by the action of a thin layer of photosynthesising blue green algae. The oldest known
stromatolite fossils until recently were 3.46 Gyr old, with even more recently discovered stromatolite fossils 3.7 Gyr old found in Greenland. Evidence for life arising even earlier than
this is provided by carbon isotope ratios in Earth’s oldest sediments. These suggest that autotrophic organisms that fixed atmospheric carbon were well established 3.8 Gyr ago, though
it has also been suggested that simple chemical processes might be mimicking this signature
of life. Some controversial studies take the beginnings of life back even further to within the
time frame of the Late Heavy Bombardment (H.Betts et al, Nature Ecol & Evol (2) 1556 (2018)
and M.Dodd et al, Nature (543) 61, (2017)). If life was genuinely well established 3.8 Gyr ago
that is only a very short time, compared to the age of the Earth, after the planet became inhabitable at all, following the Late Heavy Bombardment. Life would appear to have developed
relatively rapidly.
How life arose is a subject of great debate. One well established scenario suggests that life
started with simple, self-sustaining chemical reactions which gradually increased in complexity. These chemical reactions likely took place where there was a rich mix of chemicals and
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Sun, Stars and Planets 2020-21
plentiful available energy. Hydrothermal vents in the deep ocean are one possible site for
the first emergence of these processes. The self sustaining chemical networks require catalysts to operate. These may originally have been mineral catalysts, such as the iron sulphides
available in hydrothermal vents, but other organic materials, proteins and RNA (Ribonucleic
acid), are also capable of such catalysis. In addition to catalysis, RNA is also capable of selfreproduction, which would have given it such an advantage over the other processes operating
at the time, that it likely took over and the earliest biology on Earth was based on RNA. Later,
DNA (Deoxyribonucleic acid) came to dominate since, as long as the proteins necessary for its
reproduction are around, since it is more stable and less subject to reproduction errors.
The early history of life on Earth may have thus moved from mineral catalysed chemistry, to
a simple RNA-world, which then suffered a genetic takeover as the more stable and efficient
DNA came to dominate.
While photosynthesis is the dominant energy generation mechanism on Earth today, this is
dependent on the availability of sunlight. Deep ocean hydrothermal vents, while possessing
a rich chemistry, are well away from sunlight, and thus life there must feed itself not through
photosynthesis but through chemosynthesis, deriving energy from chemical processes rather
than from light. These organisms would have lived in what we consider extreme environments,
and we can see their descendants today, a class of single cell organism known as archaea, in
similarly extreme environments like hydrothermal vents and hot springs.
Figure 21.1: The oxygen content of the Earth’s atmosphere over time. The Berkner-Marshall
Point is the stage at which there is enough oxygen in the atmosphere for the ozone layer
to form. From Paumann et al., Biochimica et Biophysica Acta (BBA) - Bioenergetics, 1707,
231-253 (2005)
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Figure 21.2: Key dates in the history of life on Earth. From Rothery, Gilmour & Sephton.
As life developed and spread, photosynthesis started, and began producing oxygen as a byproduct. This was not a significant constituent of the atmosphere until very recently in geological terms. It was only about 500 Myr ago that oxygen levels approached those of today (see Fig
21.1) and were high enough to allow the ozone layer to form. Up until this point most of life on
Earth was anaerobic - ie. operated in the absence of oxygen. In fact, oxygen is toxic to anaerobic life, so the first mass extinction we know about was the result of photosynthetic organisms
polluting the Earth with the deadly poison that is oxygen. Anaerobic life is, of course, still with
us in the oxygen free slime at the bottom of oceans and in stagnant pools of water. Many of
the waste products of anaerobic life are in fact toxic to us, which is why pond gas, a product of
anaerobic bacteria, and the waste products of many other anaerobic processes smell bad to us.
For most of the history of life on Earth, life was made up of single celled organisms. Multicellular life, like us, only emerged about 1 Gyr ago, first as multicellular algae, then as the
first attempts at more complex multicellular life, the still poorly understood species of the
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Ediacaran Period. It was not until about 550 Myr ago that what we would regard as modern
multicellular life emerged during the Cambrian explosion, a sudden huge increase in the diversity in forms of life, the results of which we can still see today. Key dates in the history of
life on Earth are shown in Fig 21.2.
21.3
Lessons from the History of Life on Earth
What lessons can be drawn for the search for life elsewhere from this rapid overview of the
history of life on Earth?
Firstly, we can look at what appear to be the essentials for life:
• A supply of energy of some kind (photosynthesis dominates currently, chemosynthesis,
in the absence of light, probably dominated the early stages of life on Earth).
• The presence of liquid water. This is necessary to allow chemical reactions to take place
at all.
All other things that we might think are essential, such as DNA or the presence of oxygen, are
likely to be beneficial to specific types of life, but not to the existence of life in general.
Secondly, the world as we know it today is in fact a relatively recent occurrence. For much of
the history of life on Earth there were only unicellular species existing in a largely anaerobic
environment.
21.4
Life Elsewhere in the Solar System
Using the lessons gained from examining the history of life on Earth, what can we say about
the potential for life elsewhere in our own Solar System?
21.4.1
Mars
The place in the Solar System most likely to have once had an environment fairly similar to the
Earth is the planet Mars. While it is currently a cold, dry place with a very thin atmosphere,
there is now a growing body of evidence that suggests that liquid water once flowed on the
surface of Mars during a warm wet phase as recently as 3 Gyr ago. Other observations suggest
that small amounts of water may have flowed on the surface much more recently.
If liquid water existed, or exists today, on Mars, is there any evidence for life? As yet, there
is nothing unambiguous - if there was you would have heard about it - but there are some
interesting hints. The presence of methane in the atmosphere of Mars suggests that there
is something on the planet producing this gas. It is one of the byproducts of anaerobic life
on Earth, but it can also be produced by geological processes. Examination of isotope abundances in any methane detected by the Mars Rovers should be able to determine the origin
of this gas, since biological processes operate differently for different isotopes. Meanwhile,
possible evidence for historical life on Mars may have emerged in meteorites from Mars that
have landed on Earth. In 1996 the discovery of martian microfossils was claimed in the meteorite ALH84001, which originated on Mars. This result is far from agreed, and appears to
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be discredited, but the possibility of finding fossil martian lifeforms, whether in martian meteorites or in situ on Mars using rovers such as Curiosity, Perseverence, or Zhurong that has
just landed this month (May 2021), is one way in which the presence of ancient life on Mars
could be confirmed.
21.4.2
Europa
As discussed under the section on the Solar System’s moons, there is evidence for a liquid
ocean beneath the icy surface of the Galilean moon Europa. The conditions in such a subsurface ocean are very uncertain, but it is possible that the tidal heating of the moon by its orbit
around Jupiter, could lead to the presence of hydrothermal vents in this ocean, similar to those
thought to have been the cradle for life on Earth. Similar processes within Europa could lead
to the same kind of primitive life that emerged on the young Earth. Future missions to the
Jovian moons such as ESA’s JUICE project or NASA’s Europa Clipper will be looking for signs
of this ocean and any biological processes that might be taking place within it. The discovery
of a water plume on Europa, similar to that found on Enceladus, means that we may be able
to get an idea of subsurface conditions on this moon through observations from JUICE or even
from observatories closer to home. There may also be a similar subsurface ocean on the moon
Ganymede as well, though this would be buried under an even thicker layer of ice since it is
subject to less tidal heating than Europa.
21.4.3
Enceladus
The one other place in the solar system where there is clear evidence for the presence of liquid
water is Enceladus, the moon of Saturn, where jets of water vapour emerge from cracks in parts
of its surface. Enceladus, like Europa, is tidally heated, so here too there may be a subsurface
ocean and hydrothermal vents that could host biological systems. There is also some evidence
of a subsurface water layer in Saturn’s moon Titan.
21.5
Life Outside the Solar System
Having looked at possible homes for life in our own Solar system it is now time to look for
it elsewhere. The requirements for a habitable planet outside our own Solar system will be
broadly similar to what we have found locally, with the presence of liquid water being of
paramount importance. For an Earth-like planet to be capable of supporting life the following
conditions would have to hold:
• Large enough mass so that the atmosphere can provide sufficient pressure for water to
be a liquid on its surface. Surface atmospheric pressure is given by:
P = mc
GMp
R2
(21.1)
where P is the pressure, mc is the mass of a column through the atmosphere, Mp is the
mass of the planet and R is the radius of the planet.
• The planet must be large enough to have geological activity so that volatiles can be
incorporated into the crust, as seen in the carbon cycle on Earth. It should be noted that
Venus is an interesting exception to this rule, since it is similar in mass to Earth, but
We thus want planetary masses between 0.5 and 10 Earth masses; these are sometimes
called ‘Earth-mass planets’.
2. Planet positions: the habitable zone
For carbon-based life, (and carbon cycle) need liquid water, and thus temperatures be158
Sun, Stars and Planets 2020-21
tween 273 and 373 K. For naive calculation of habitable zone, see PS 4 where we found
a distance of 0.6 to 1.1 AU for the habitable zone.
from http://www.geosc.psu.edu/ kasting/PersonalPage/Kasting.htm
Figure 21.3: The location of the Continuously Habitable Zone for a range of stellar types
compared to the position of the planets in our own Solar System. From www.geosc.psu.edu.
lacks a carbon cycle, so while this is a necessary condition it is not sufficient for there
to be an active geological cycle.
• The planet must be large enough to retain an atmosphere: remembering the thermal
escape of atoms from atmospheres discussed in Section 15.5. This implies that the planet
must have a mass ≥ 0.5 M⊕ .
• The planet must be small enough not to have accreted an extended hydrogen rich atmosphere, and to have become a gas giant. This implies a mass ≤ 10 M⊕
For liquid water to exist the planet must also be at an appropriate temperature, between
273 and 373 K. These surface temperature limits define what is called the Habitable Zone
for planets in any given system. A simple calculation of the width of the Sun’s habitable
zone, using the considerations discussed in section 14.6, would estimate it to lie from 0.6
to 1.1 AU in distance from the Sun. Lower mass stars will have smaller habitable zones
closer to them, while higher mass stars will have them further out.
The power output of stars changes over the course of geological time. The Luminosity
of a star on the MS tends to increase slightly during its main sequence life. For life to
have time to evolve, we are actually interested in a narrower region where liquid water
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can persist for the entire history of the planet. For our own Solar System this extends
from 0.95 to 1.15 AU. Figure 21.3 shows where the continuously habitable zone lies for
a range of stellar types.
21.5.1
Host Star
The host star has influence beyond just keeping a planet’s temperature at the right level for
liquid water to exist. High mass spectral types, such as O, B and A stars, evolve too quickly for
life to have time to evolve - recall that stellar life time decreases as M −3 on the main sequence.
High mass stars also have high surface temperatures and would thus emit copious amounts
of UV light which might be harmful to life.
Very low mass stars such as M stars have their CHZ closer than the tidal locking radius. This
would mean that one side of the planet has permanent day and the other permanent night.
This might be a problem, with the two sides being respectively very hot and very cold, so the
cold side could act as a trap, freezing out the atmosphere over time. Recent modelling work,
though, suggests that a sufficiently dense atmosphere can circulate heat from one side of a
tidally locked planet to the other, avoiding this problem.
Many stars lie in binary systems which may lead to instability in planetary orbits. Orbits may
also be affected by other bodies in the same star-planetary system as we have seen with the
inward evolution of hot Jupiters in Lecture 20. Conversely it may be that a gas giant in a
stable orbit further from the star than a terrestrial planet, as is the case for Jupiter in the Solar
System, might limit the number of impacts from asteroids or comets experienced by terrestrial
planets in the inner planetary system.
21.5.2
Gas Giant Moons
The considerations given above apply to life on the surface of a terrestrial planet. As discussed
in section 21.4, life might also exist beneath the icy surfaces of gas giant moons, like Europa
and Enceladus, in our own Solar System. Gas giant moons elsewhere might also be capable of
harbouring life in this way, or, if they are warm enough and have a dense enough atmosphere
for the presence of surface water, they might also harbour life on their surface.
21.6
The Galactic Habitable Zone
The large scale geography of our galaxy may also influence where it is most likely to find life
bearing planets. The formation of terrestrial planets requires high-metallicity stars. These are
most likely to be found in the thin disk of the Galaxy. The outer regions of the disk have low
metallicity so would have fewer terrestrial planets, while regions closer to the centre of the
Galaxy would suffer from two disadvantages: firstly the stars are closer together, leading to
overcrowding and the possible gravitational disruption of a stellar system by a passing star;
secondly, a given star will be more likely to be close to an energetic event, such as a supernova,
which could wipe out life in nearby star systems. The core of our galaxy is thus not thought
to be hospitable for life.
The orbit of a star around the Galaxy is also important. The Sun’s orbit is nearly circular, so it
is less likely to stray into crowded regions in the core. The Sun’s orbit also avoids crossing the
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spiral arms of the Galaxy, which are also regions of high stellar density and thus hazardous to
life.
Things to Remember
• The history of life on Earth
• The requirements for life and the likely sites elsewhere in the Solar System
• The requirements for and potential sites of life around other stars.
Habitable, and Continuously Habitable Zone.
• The Galactic Habitable Zone
To Do:
Simple calculation of the limits in AU of the Habitable Zone of our Solar System, mentioned in the recorded lecture.
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Lecture 22
The Search for Extraterrestrial
Intelligence
It’s not enough to say, ”Oh goodie, I have oxygen”. Can you interpret it in the context of the environment? Can you prove that the oxygen didn’t come from planetary
processes, rather than life?
- Kikki Meadows, Astrobiologist, head of the Virtual Planetary Laboratory, NASA
It is surely unreasonable to credit that only one small star in the immensity of the
universe is capable of developing and supporting intelligent life. But we shall not get
to them and they will not come to us.
- P.D. James, in The Children of Men.
22.1
How to Find Life on Other Planets
We have discussed the considerations for an extrasolar planet to be able to support life, but
would we be able to detect such life if it were present?
We have already seen how exoplanet studies are beginning to be able to determine various
parameters for the atmospheres of hot Jupiters. Our observational capabilities are improving
and there are now plans for instruments that will eventually be able to take spectra of terrestrial exoplanet atmospheres. There are a number of ‘biomarkers’ that could appear in these
spectra if life is present. The infrared spectrum of an exoplanet would be a good place to look,
and Fig 22.1 shows the Earth’s infrared spectrum measured from space for reference. Chief
among these biomarkers is ozone, which has a prominent absorption feature in the infrared at
about 10 µm. This would be a clear sign of the presence of life since oxygen is a highly reactive
molecule, which, unless constantly replenished, would soon be locked up in other compounds
like CO2 . The only process we are aware of that can keep oxygen levels high enough for an
ozone absorption layer to exist is photosynthesis in plants. Other possible biomarkers include
methane and spectral features associated with chlorophyll in plants.
However, as we have seen, the ozone layer in the Earth’s atmosphere is relatively recent in geological terms, and chlorophyll, while common on Earth, will not necessarily be the molecule of
choice for photosynthesis on other planets. A more general signature of life will be signs of any
chemistry which is out of equilibrium - the abundance of oxygen in the Earth’s atmosphere
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Figure 22.1: Earth’s infrared spectrum, obtained in the daytime by the Nimbus-4 satellite over
a cloud-free part of the Pacific in the 1970s. From Rothery, Gilmour and Sephton, Introduction
to Astrobiology.
is an example of this - since the action of biological processes are the only way we know that
can maintain such a disequilibrium over time. Quite what we might find in the atmospheres
of biologically active exoplanets remains to be seen.
22.2
The Search for Extraterrestrial Intelligence (SETI) - introduction
Assuming that life does exist on other planets, the next great question is whether intelligent
life exists elsewhere. We do not yet have any evidence for extraterrestrial intelligence (ETI), but
absence of evidence is not evidence of absence. There are many issues surrounding the search
for extraterrestrial intelligence - how it should be conducted, whether we should try to make
contact ourselves, what to do if we ever do find evidence for it - but few hard and fast results.
The lecture for this section of the course would usually largely take the form of a discussion
about how we can guesstimate the number of extraterrestrial intelligences in the Galaxy, of
what uncertainties there are in such a prediction, and about broader issues concerning the
search and possible discovery of ETI. This year, instead of this discussion, the recorded video
lecture gives an overview of some of the issues - you yourselves might think of things that I
have not included in the lecture.
The two key results in this area are the Drake Equation and the Fermi Paradox, which will be
described here in turn.
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22.3
The Drake Equation
The Drake equation (devised by Frank Drake in 1961) encompasses the terms needed to predict
the number of intelligent civilisations in the Galaxy that at any given time are interested in
communication with other civilisations. At the time the equation was devised, there were few
constraints on any of the terms, but an extra 60 years of astrophysics has begun to tie some
of them down.
N = R × fp × nE × fL × fi × fc × Lc
(22.1)
where:
N = the number of technological civilisations in the Galaxy that are interested in communication
R = the average rate of star formation in the Galaxy (in stars per year)
fp = the fraction of those stars with planetary systems
nE = the average number of habitable planets in each system
fL = the fraction of those habitable planets on which life develops
fi = the fraction of those planets on which intelligent life develops
fc = the fraction of intelligent species interested and able to communicate with other species
Lc = the lifetime of a communicating civilisation
At this point the only terms in the Drake Equation where we have accurate values are R and
fp , which have values of roughly 10 to 20 for R, and about 0.5 - 1 for fp . nE is a term that we
should have better constraints on fairly soon, from long term transit studies using instruments
like Kepler and TESSA. Our best guess at it so far is that it is likely to be close to nE = 1. fL is,
essentially, the goal of the whole field of exobiology, but that study is currently in its infancy
and current estimates are highly uncertain.
That leaves fi , fc & T , which are not easily determined, and which are controlled by factors
that are biological and sociological.
What values do you think are reasonable for fi , fc & Lc , and what are your justifications for
these estimations?
22.4
The Fermi Paradox
The next important consideration in this field is known as the Fermi Paradox. This arises from
a question that the famous physicist Enrico Fermi asked in an informal discussion in 1950.
Fermi made the observation that, given the great age of the Universe (about 13.5 Gyr) and the
very large number of stars in our galaxy (about 1011 ), then we should be able to see evidence of
intelligent life through interstellar probes or spacecraft, unless intelligent life capable of interstellar travel and/or communication is very rare. His key question was ‘Where are they?’ since
it can be shown quite easily that a civilisation capable of interstellar travel and colonisation
can spread throughout the galaxy in what is, in geological and cosmological terms, a relatively
short time - 1-100 Myr.
This issue is also known as the ‘Great Silence’ since it applies just as much to communication
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as it does to physical contact with alien intelligences.
What possible resolutions can you come up with for the Fermi paradox, and what is your
justification for these solutions?
22.5
SETI and CETI
The Search for Extraterrestrial Intelligence (SETI) and attempts at Communication with Extraterrestrial Intelligence (CETI - confusingly pronounced the same way) are terms for various
attempts to observationally test the idea that extraterrestrial intelligences exist and might be
contacted by us. Much of the work has focussed on radio observations looking for narrow
band, artificial signals coming from Sun-like stars, directed towards us, that would appear as
sudden, unnatural brightening of the radio signal of the star system they are sent from. But
there are many other possible ways in which extraterrestrials might communicate with each
other, technologies that we do not yet have, for example by sending gravitational wave signals.
Within CETI, a small number of attempts have been made to transmit powerful radio signals towards certain locations in the Galaxy. The most significant of these was the use of
the Arecibo interplanetary radar to send a message coded as an image directed towards the
globular cluster M13, see Fig 22.2. The message will take 22000 years to reach the cluster and
any reply would take 22000 years to come back, so we do not expect a snappy conversation!
Meanwhile, normal radio and TV broadcasts from Earth are propagating through the nearby
regions of our galaxy. While cable and satellite TV mean that less power has been expended
on such transmissions over the last few decades, powerful transmissions of previous years are
still heading out into space.
Given the experience on Earth of what happens when two cultures with very different levels
of technological development interact, is it a good idea to be advertising our presence on the
Galactic stage?
22.6
The Future
Developments in radio astronomy currently underway will allow us to have far greater sensitivity to narrow band signals in the next decade. The Square Kilometer Array project (SKA) in
particular will be able to detect signals at the level of our airport radars but positioned up to
50 to 60 light years away, while more powerful early-warning type radars could be detected at
even greater ranges. At the same time, large ground based telescopes such as the E-ELT, and
space-based projects such as JWST, and planned ARIEL, amongst others, will be able to detect
terrestrial planets and search for signs of life in their atmospheric spectra.
In the next few decades we will thus be able to not only start filling in some more of the astrophysical and astrobiological terms in the Drake Equation, but might also be able to conduct
the first studies in SETI that could detect nearby civilisations like our own.
SETI, and CETI, might soon stop being the preserve of scientific speculation, and become actual
observational and practical studies in their own right.
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Figure 22.2: The Arecibo Message from 1974, sent to M13 (Sagan & Drake, Scientific American, 232(5), 80-89 (1975). Carl Sagan later said about the Arecibo message: “The decoded
message forms a kind of pictogram that says something like this: ‘Here is how we count from
one to ten. Here are five atoms that we think are interesting or important: hydrogen, carbon,
nitrogen, oxygen and phosphorus. Here are some ways to put these atoms together that we
think interesting or important - the molecules thymine, adenine, guanine and cytosine, and
a chain composed of alternating sugars and phosphates. These molecular building blocks are
put together to form a long molecule of DNA comprising about four billion links in the chain.
The molecule is a double helix. In some way this molecule is important for the clumsy looking
creature at the center of the message. That creature is 14 radio wavelengths or 5 feet 9.5 inches
tall. There are about four billion of these creatures on the third plant from our star. There are
nine planets altogether, four big ones toward the outside and one little one at the extremity.
This message is brought to you courtesy of a radio telescope 2,430 wavelengths or 1,004 feet
in diameter. Yours truly.’”
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Things to Remember
• How we might detect life on an exoplanet, biomarkers, exoplanet spectra
• The Drake Equation
• The Fermi Paradox
• The current status and potential for SETI observations