Lecture 14:
The Sun and energy transport in stars
Astronomy 111
Energy transport in stars
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What is a star?
What is a star composed of?
Why does a star shine?
What is the source of a star’s energy?
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Laws of Stellar Structure I:
The Gas Law
•  Most stars obey the Perfect Gas Law:
•  Pressure = Density × Temperature
•  In words:
Compressing a gas results in higher P & T
Expanding a gas results in lower P & T
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Laws of Stellar Structure II:
The Law of Gravity
•  Stars are very massive & bound
together by their Self-Gravity.
•  Gravitational binding increases as 1/R2
•  In words:
Compress a star, gravitational binding gets stronger.
Expand a star, gravitational binding gets weaker.
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Hydrostatic Equilibrium
•  Gravity wants to make a star contract.
•  Pressure wants to make a star expand.
•  Counteract each other:
–  Gravity confines the gas against Pressure.
–  Pressure supports the star against Gravity.
•  Exact Balance = Hydrostatic Equilibrium
The star neither expands nor contracts.
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Hydrostatic Equilibrium
Gravity
Gas Pressure
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Core-Envelope Structure
•  Outer layers press down on the inner
layers.
The deeper you go into a star, the greater the
pressure.
•  The Gas Law says:
“More pressure= hotter, denser gas”
•  Consequences:
–  hot, dense, compact CORE
–  cooler, lower density, extended ENVELOPE
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Core-Envelope Structure
Compact
Core
Extended
Envelope
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Example: The Sun
•  Core:
Radius = 0.25 Rsun
T = 15 Million K
Density = 150 g/cc
•  Envelope:
Radius = Rsun = 700,000 km
T = 5800 K
Density = 10-7 g/cc
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The Essential Tension
•  The Life of a star is a constant tug-of-war
between Gravity & Pressure.
•  Tip the internal balance either way, and it
will change the star’s outward appearance.
•  Internal Changes have External
Consequences
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Why do stars shine?
•  Stars shine because they are hot.
–  emit thermal (~blackbody) radiation
–  heat “leaks” out of their photospheres.
•  Luminosity = rate of energy loss.
•  To stay hot, stars must make up for the
lost energy, otherwise they would go
out.
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Case Study: The Sun
•  Question:
How long can the Sun shine?
•  Answer:
Consider the internal heat content of the Sun.
Luminosity = rate of heat loss.
Lifetime = Internal Heat ÷ Luminosity
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What if no source of energy?
•  The Sun’s Luminosity losses would not be
balanced by input of new internal heat.
•  The Sun would steadily cool off & fade out.
•  18th Century:
–  Assumed a solid Sun (iron & rock)
–  Found Lifetime ~ 10 Million Years
•  No Problem:
Earth was thought to be thousands of years old.
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The Age Crisis: Part I
•  Late 1800s:
–  Geologists: Earth was millions of years old.
•  How can Earth be older than the Sun?
–  Astronomers: The Sun is a big ball of gas in
Hydrostatic Equilibrium.
•  Kelvin & Helmholtz proposed Gravitational
Contraction as a source of energy.
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Kelvin-Helmholtz Mechanism
•  Luminosity radiates away heat
–  Outer layers of the Sun cool at little,
lowering the gas pressure.
–  Lower Pressure means Gravity gets the
upper hand and the star contracts a little.
–  Contraction compresses the core, heating
it up a little, adding heat to the Sun.
•  Sun could shine for ~100 Million years
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Gravity & Pressure in
Equilibrium
G P
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Luminosity radiates away
heat & Pressure
Drops
G P
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Balance tips in favor of
gravity, Sun shrinks.
G P
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Contraction makes core heat up,
increasing the internal Pressure.
G P
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Balance restored, but with higher gravity,
pressure & temperature than before...
G P
Starts the cycle all over again...
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The Age Crisis: Part II
•  Early 1900s:
•  Geologists show that the Earth is >2
Billion years old.
•  Kelvin Says:
The Geologists are wrong.
•  Nature Says:
Kelvin is wrong... There is new physics.
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Nuclear Fusion
•  1905:
Einstein demonstrates that Mass and Energy
are equivalent: E=mc2
•  1920s:
Eddington noted that 4 protons have 0.7%
more mass than 1 Helium nucleus (2p+2n).
If 4 protons fuse into 1 Helium nucleus, the
remaining 0.7% of mass is converted to
energy.
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Fusion Energy
•  Fuse 1 gram of Hydrogen into 0.993
grams of Helium.
•  Leftover 0.007 grams converted into
Energy:
•  E = mc2 = 6.3x1018 ergs
•  Enough energy to lift 64,000 Tons of
rock to a height of 1 km.
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Hydrogen Fusion
•  Question:
How do you fuse 4 1H (p) into 4He (2p+2n)?
•  Issues:
Four protons colliding at once is unlikely.
Must turn 2 of the protons into neutrons.
Must be hot: >10 Million K to get protons
close enough to fuse together.
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Proton-Proton Chain
3-step Fusion Chain
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neutrino
positron photon
2H
3He
3He
2H
positron photon
neutrino
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4He
The Bottom Line
•  Fuse 4 protons (1H) into one 4He
nucleus.
•  Release energy in the form of:
–  2 Gamma-ray photons
–  2 neutrinos that leave the Sun
–  2 positrons that hit nearby electrons,
creating two more Gamma-ray photons
–  Motions (heat) of final 4He and 2 protons.
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The Age Crisis: Averted
•  Luminosity of the Sun is ~4x1033 erg/sec
–  Must fuse ~600 Million Tons of H into He
every second.
–  ~4 Million tons converted to energy per
second.
–  Sun contains ~1021 Million Tons of Hydrogen
–  Only ~10% is hot enough for fusion to occur.
•  Fusion Lifetime is ~10 Billion Years.
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Test: Solar Neutrinos
•  Question:
How do we know that fusion is occurring in
the core of the Sun?
•  Answer:
Look for the neutrinos created in Step 1 of
the Proton-Proton chain.
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What are Neutrinos?
•  Massless, weakly interacting neutral particles.
•  Travel at the speed of light.
•  Can pass through a block of lead 1 parsec
thick!
•  Any neutrinos created by nuclear fusion
in the Sun’s core would stream out of
the Sun at the speed of light.
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Solar Neutrinos: Observed!
•  Detection of neutrinos is very hard:
–  Need massive amounts of detector materials
–  Work underground to shield out other radiation
•  Answer:
–  We detect neutrinos from the P-P chain in all
the experiments performed to date!
•  Success! But...
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The Solar Neutrino Problem
•  We detect only ~1/3 of the number
expected...
•  Why?
–  Is our solar structure model wrong?
–  Is the subatomic particle theory of neutrinos
wrong?
•  The last seems most plausible
–  recent evidence neutrinos have mass
–  question for the physics of 21st century!
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Putting Stars Together
•  Physics needed to describe how stars
work:
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Law of Gravity
Equation of State (“gas law”)
Principle of Hydrostatic Equilibrium
Source of Energy (e.g., Nuclear Fusion)
Movement of Energy through star
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Hydrostatic Equilibrium
•  Balance between Pressure & Gravity.
–  If Pressure dominates, the star expands.
–  If Gravity dominates, the star contracts.
•  Sets up a Core-Envelope Structure:
–  Hot, dense, compact core.
–  Cooler, low-density, extended envelope.
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Hydrostatic Equilibrium
Gravity
Gas Pressure
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Energy Generation
•  Stars shine because they are hot.
•  To stay hot stars must make up for the
energy lost by shining.
•  Two Energy sources available:
–  Gravitational Contraction (KelvinHelmholtz)
•  Only available if star is NOT in equilibrium
–  Nuclear Fusion in the hot core.
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Main-Sequence Stars
•  Generate energy by fusion of 4 1H into 1
4He.
•  Proton-Proton Chain:
–  Relies on proton-proton reactions
–  Efficient at low core Temperature (TC<18M K)
•  CNO Cycle:
–  Carbon acts as a catalyst
–  Efficient at high core Temperature (TC>18M K)
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Proton-Proton Chain
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CNO Cycle:
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Controlled Nuclear Fusion
•  Fusion reactions are Temperature
sensitive:
–  Higher Core Temperature = More Fusion
–  ε(PP) ~ T4
–  ε(CNO) ~ T18
•  BUT,
–  More fusion makes the core hotter,
–  Hotter core leads to even more fusion, …
•  Why doesn’t it blow up like an H-Bomb?
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Hydrostatic Thermostat
•  If fusion reactions run too fast:
–  core heats up, leading to higher pressure
–  pressure increase makes the core expand.
–  expansion cools core, slowing fusion.
•  If fusion reactions run too slow:
–  core cools down, leading to lower pressure
–  pressure drop makes the core contract.
–  contraction heats core, increasing fusion.
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Thermal Equilibrium
•  Heat always flows from hotter regions
into cooler regions.
•  In a star, heat must flow:
–  from the hot core,
–  out through the cooler envelope,
–  to the surface where it is radiated away as
light.
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Energy Transport
•  There are 3 ways to transport energy:
•  1) Radiation:
Energy is carried by photons
•  2) Convection:
Energy carried by bulk motions of gas
•  3) Conduction:
Energy carried by particle motions
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Radiation
•  Energy is carried by
photons.
–  Photons leave the core
–  Hit an atom or electron
within ~1cm and get
scattered.
–  Slowly stagger to the
surface (“random walk”)
–  Break into many lowenergy photons.
•  Takes ~1 Million years
to reach the surface.
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Convection
•  Energy carried by bulk motions of the
gas.
•  Analogy is water boiling:
Hot blob rises
cooler
water
sinks
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Conduction
•  Heat is passed from atom-to-atom in a
dense material from hot to cool regions.
•  Analogy:
Holding a spoon in a
candle flame, the handle
eventually gets hot.
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The Sun
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Energy Transport in Stars
•  Normal Stars:
–  A mix of Radiation & Convection transports
energy from the core to the surface.
–  Conduction is inefficient (density is too
low).
•  White Dwarfs:
–  Ultra-dense stars
–  Conduction dominates energy transport.
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Summary:
•  Stars shine because they are hot.
–  need an energy source to stay hot.
•  Kelvin-Helmholtz Mechanism
–  Energy from slow Gravitational Contraction
–  Cannot work to power the present-day Sun
•  Nuclear Fusion Energy
–  Energy from Fusion of 4 1H into 1 4He
–  Dominant process in the present-day Sun
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Summary:
•  Energy generation in stars:
–  Nuclear Fusion in the core.
–  Controlled by a Hydrostatic “thermostat”.
•  Energy is transported to the surface by:
–  Radiation & Convection in normal stars
–  Conduction in white dwarf stars
•  With Hydrostatic Equilibrium, these
determine the detailed structure of a star.
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Summary:
•  Stars are held together by their selfgravity
•  Hydrostatic Equilibrium
–  Balance between Gravity & Pressure
•  Core-Envelope Structure of Stars
–  Hot, dense, compact core
–  cooler, low-density, extended envelope
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Lecture 8: Telescopes
•  Refracting vs. reflecting
•  Angular resolution of a telescope
•  Properties of different types of
telescopes
•  What is “seeing”?
•  What are CCDs?
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Lecture 9: Formation of the
Solar System
•  How did solar system form?
•  Characteristics of inner and outer Solar
System
•  How did angular momentum play a role?
•  Define: planetesimals, radioactive age
dating, terrestrial, jovian
•  Name the four techniques astronomers
use to find extrasolar planets
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Lecture 10: Terrestrial planets
•  Characteristics of Earth, the Moon, and
terrestrial planets
•  How do we know the age of the Earth?
•  Where did the Moon come from?
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Lecture 11: Jovian planets
•  Characteristics of Jovian planets, dwarf
planets, and other solar system objects
•  Define dwarf planet, asteroid, Kuiper
belt, comet
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Lecture 12: Distances to stars
•  Calculate distance to a star given its
parallax
•  Define: apparent brightness, luminosity,
true binary, visual binary, spectroscopic
binary, eclipsing binary
•  What quantities can we measure using
binary stars?
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Lecture 13: Colors of stars
•  Why are stars different colors? Why do
stars have absorption lines in their
spectra?
•  Give the order of the spectral sequence
•  Be able to calculate luminosity and know
its units (energy/sec)
•  Be able to use the Stefan-Boltzman law
and know its units (energy/sec/area)
•  Know the main components and axes of
an H-R diagram
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Lecture 14: Energy transport in
the Sun
•  Define: hydrostatic equilibrium, P-P
chain, CNO cycle, neutrinos
•  Know what the Perfect Gas Law means
•  Describe how the P-P chain makes
energy in the Sun through fusion
•  List the three modes of energy transport
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