Mass and the Properties of Main Sequence
Stars
Mass is the most important properties of the
main-sequence stars. It determine their
luminosity, surface temperature, radius, and
lifetime.
• Nuclear fusion requires high
temperatures and densities in the core,
~10 Rsun
and the star’s internal conditions are
determined by the equilibrium condition
between the inward pull of gravity and
the outward push of pressure.
• In a star that has high mass, the greater
weight of its overlying layers means the
star must sustain a higher nuclear fusion ~3 R
sun
rate to generate the additional pressure
needed to maintain gravitational
equilibrium.
1 Rsun
− The higher nuclear fusion rate
makes the star more luminous.
− The high luminosity requires a star
to have either high temperature or
0.1 Rsun
large size, or both.
− The higher luminosity also means
that it will run out of fuel faster than
less massive stars.
The Lifetime of Main-Sequence Stars
The lifetime of a star is determined by how fast it burns its
supply of hydrogen…This hydrogen burning rate can be
inferred from the luminosity of the star.
The Mass-Luminosity Relation
Once we have observationally determined the luminosity and
mass of many main sequence stars, we find that the higher the
mass M of a star is, the higher is its luminosity (L).
L/L⊙ = (M/M⊙ )3.5
Note: The Mass-Luminosity relation applies to main-sequence
stars only!
For example,
 A 10 M⊙ star is roughly (103.5 ~ ) 3,000 brighter, or
burning its hydrogen times 3,000 faster.
 We know that the lifetime of the Sun is about 10
billion years.
 The more massive star would have a lifetime of about
10 × 10 billion years ÷ 3,000 ~ 30 million years.
Giant and Supergiants
Giants and supergiants are stars nearing the ends of their lives.
• Giants and supergiants do not follow the relationship between
surface temperature and luminosity for hydrogen-burning, mainsequence stars.
− The supply of hydrogen fuel in the core of the giants is
running out, and they respond to this fuel shortage by
releasing fusion energy at a furious rate. Thus, in order to
radiate away this huge amount of energy, the surface of a
dying star must expand to an enormous size (Chapter 12)
• Because giants and supergiants are so bright, we can see them
even if they are not especially close to us.
− Many of the brightest stars visible to the naked eye are giants
or supergiants.
− They are often identifiable by the reddish color produced by
their cool surfaces.
• Giants and supergiants are considerably rarer than mainsequence stars. When we look at the sky, most of the stars we
see are main sequence stars.
• Betelgeus: M2 I
Betelgeuse and R Doradus
White Dwarf
White dwarfs are the exposed core of the dead low-mass mainsequence stars, supported against gravity by electron degenerate
pressure (Chapter 12).
• Properties
− Hot surface (not long after the formation), comparable or higher
than the surface of the Sun.
− Low luminosity (0.0001L⊙ to 0.1L⊙ )
− High mass: comparable to the Sun
• White dwarfs have high surface temperature and low luminosity:
 Small size – comparable to the size of the Earth.
• White dwarfs are small in size, but high in mass:
 VerySirius
high density
For example,
B (DA2) is a white dwarf
with a diameter of 12,000 km and a mass of ~ 1
solar mass. Its surface temperature of 25,000 K
makes it brighter than the main star (A1, 9,900
K) in this Chandra X-ray image.
Sirius
Summary of Sizes of Stars – From
Supergiants to White Dwarfs
Supergiant ~ 100 – 1000 Rsun
Note that the sizes
in this figure are
not to scale!
• Orbit of Mars
• Orbit of
Mercury
Giant ~ 10 – 100 Rsun
Main-Sequence Star ~ 0.1 – 10 Rsun
White Dwarf ~ 0.01 Rsun
About the size of Earth!
• Properties of Stars
• Classifying Stars
• Star Clusters
– Open and Globular
Clusters
– Dating the Age of the
Universe by Globular
Clusters
Star Clusters
Most stars are formed from giant clouds of gas
with enough material to form many stars.
When we look into the sky, we often find
clusters of stars. There are two types of
clusters:
• Open Clusters
– Found in the disk of the galaxy.
– Contains a few thousand stars.
– Span about 30 light-years.
• Globular Clusters
– Found in the halo of the galaxy.
– Up to one million stars.
– Spans about 60 to 150 light-years.
Because
 Stars in the same cluster lie at about the
same distance from Earth
 Stars in the same cluster are formed
roughly at the same time.
They are useful as a cosmic clock…
The Pleiades
HR Diagram of Star Cluster
Pleiades is an open cluster that
contains thousands of stars…
• The H-R diagram of Pleiades
shows that most of the stars
fall in the main sequence
curve.
• However, it is missing the O
and B type stars.
• The high-luminosity end of
the curve moves away from
the main-sequence curve…
If the stars in Pleiades were all
formed at the same time,
then higher mass stars would
move off the main sequence
curve first. Therefore, the
theoretical lifespan of the
most massive star of the
cluster remaining in the main
sequence tells us about the
H-R Diagram of Pleiades
Dating the Age of Star Clusters
When a star cluster is born, it contains stars spanning the entire range of the H-R
diagram.
• As the cluster ages, the high-luminosity, hot, blue stars move away from the
main sequence curve first.
• The point where the curve of the H-R diagram deviates from the main
sequence curve (the main-sequence turn-off point) indicates the age of the
cluster.
Evolution of the H-R Diagram of Star Cluster
New-born
100 million
years
10 billion
years
Main sequence
curve
Temperature
Temperature
Time
Temperature
Examples of H-R Diagram of Star
Clusters
We have only being plotting the H-R
diagrams for about 100 years. Therefore,
we do not have a time sequence of H-R
diagrams to show the evolution of any
cluster. However, if we plot the H-R
diagrams of several star clusters with
different age, we should see the
evolutionary effect…
Dating the Age of the Universe
with Globular Cluster
The age of the oldest star cluster should give us an lower limit of the age of the
universe, since no star can form before the universe was born!
• Most of the open clusters are relatively young. Very few are older than 5
billion years.
• The age of some of the oldest globular
cluster, such as M5 below, is about 13
billion years. Therefore, the age of the
universe must be more than 13 billion
years.
Image of M5, in Constellation Serpentis, with
apparent brightness magnitude of mv = 12
H-R Diagram of M4
Age: ~ 10 billion years.
Chapter 12 Star Stuff
• Star Formation
• Evolution of Low-Mass Stars
• Evolution of High-Mass Stars
From Clouds to Protostar
Stars form in cold (10-30 K), dense
(although still very low density compared
with the density we are used to) molecule
clouds composed of mostly hydrogen
and helium.
• The low temperature allows the
formation of hydrogen molecule H2 –
hence molecule clouds.
• Low temperature and ‘high density’
allow gravity to compress the clouds
without resistance from thermal
pressure.
• Because of the low density, the gas can
radiate away its thermal radiation
quickly. The temperature of the gas
remain low (~ 100 K), and emits in the
infrared wavelengths.
• As the cloud undergoes gravitational
contraction, density increases, making
it increasingly difficult for radiation to
escape.
• The gas heats up as the density
increases, eventually forms a dense,
hot protostar!
Molecule cloud glows in the infrared, but
is dark in the visible light image!
Disks and Jets
The random motion of the molecule can
contain a net angular momentum, as
the cloud contract,
• this angular momentum is
conserved, and results in the fast
rotation of the protostar and the
subsequent formation of a disk and
jets
• Details of how the jets are
formed is still unknown. Magnetic
field probably plays an important
role!
Image of jet and disk of a protostar!
Jet in Neutron Stars
•
•
•
•
•
Similar to the core of the low-mass stars,
electrons degeneracy pressure will resist the
gravitational pressure. However, because of the
high mass, it cannot hold off the gravitational
collapse like in the case of the white dwarfs.
As gravity overcomes electron degeneracy
pressure, and the core collapse rapidly, the
electrons and protons recombine to form
neutrons, and releasing neutrinos and energy at
the same time  Supernova explosion.
Eventually the neutron degeneracy pressure
will balance the gravitational pressure (if the
star is not too massive) to form a neutron star.
The estimated of the neutron stars are about 10
km in diameter, with a mass of about 1 M⊙ 
Too small to be directly observed!
However, the strong gravity of the neutron stars
pull surrounding materials in, forming an rapidly
rotating accretion disk. The high speed
collisions between the materials and the
neutron stars generate strong X-ray, as the
image of crab nebula from Chandra X-ray
Conbined Hubble’s visible (red) and
Chandra’s X-ray (blue) images.
More Example of Astronomical Jets
Jets are found in many different
spatial scales. In this
composite picture of x-ray
(blue) picture from Chandra Xray Observatory, visible (white)
image from Hubble Space
Telescope, and radio (red)
image from the Very Large
Array radio telescope, jets
(seen in radio emission in red)
are ejected from a
supermassive black hole in
galaxy cluster MS
0735.6+7421 in constellation
Camelopardus.
http://chandra.harvard.edu/photo/2006/ms0735/
Examples of Star Forming
Molecular Clouds and EGGs
The Eagle Nebula is a star
forming region in the
constellation Serpens.
• Evaporating Gaseous
Globules (EGGs) are
dense regions of
molecular hydrogen (H2)
clouds that have
gravitationally collapsed
to form stars.
• UV radiation from hot
bright star (off the
image) evaporates the
outer layer of the dense
H2 cloud, revealing the
denser regions that are
forming stars.
EGGs in Eagle Nebula in constellation Serpens
http://antwrp.gsfc.nasa.gov/apod/ap061022.html
Star-Forming Region in W5
This picture of the star
forming region W5 in
constellation Cassiopeia
was obtained by the Spitzer
Space Telescope. The insert
at the lower-left-hand corner
is the same region taken in
the visible wavelength.
Dusts and dense H2 cloud
blocks visible radiation, and
the region looks dark in the
visible image.
• Infrared radiation are
emitted by the cold and
dense H2 clouds.
• Additionally, infrared
radiation can propagates
through the gas and
dust, allowing us to see
http://www.spitzer.caltech.edu/Media/releases/ssc200523/index.shtml
Star Forming Region in NGC 2467
This picture of NGC 2467
shows stars at different
stages in star formation
process.
• The bright stars on the
left of the image are
stars that have already
formed and the winds
probably have
dispersed the planetary
nebulae around them.
• The star at the lower
left is emerging from its
planetary nebula.
• The deep dark lanes
near the center are
dense regions that are
probably forming new
stars inside.
• The bright walls of gas
on the right are gases
been evaporated by
http://antwrp.gsfc.nasa.gov/apod/ap050131.html
The Mass Limits of Main Sequence Stars
Usually a single group of molecular clouds can
give birth to a star cluster containing
thousands of stars. The mass distribution of
the stars is such that there are a whole lot
more low mass stars than high mass stars.
• Upper limit of stellar mass: ~ 100 Msun
The core temperature becomes so high
that radiation pressure (pressure exerted
by photons) upsets the equilibrium
between the thermal pressure and the
gravitational pull. The star becomes
unstable…
– No star with mass greater than 100
Msun has been observed.
• Lower limit of stellar mass: ~ 0.08 Msun
The core temperature of objects with
mass less than 0.08 Msun is not hot
enough to trigger hydrogen burning.
– Jupiter is 0.001 Msu
Brown Dwarfs
Brown dwarfs are objects that does
not have enough mass to maitain
core hydrogen fusion, with mass
less than 0.08 Msun.
• Brown dwarfs are supported by
electron degenerate pressure
(like white dwarfs).
• Brown dwarfs and large planets
are similar in size
• Distinction between brown
dwarfs and planets is fussy:
– Support mechanism?
– Deuterium fusion (>13
Mjupiter)?
The Origin of Degenerate Pressure
1. Fermions and Bosons.
In quantum physics, particles are divided into two types: fermions and
bosons. In quantum physics, one of the intrinsic properties of particles are
called spin. Spin is associate with the angular momentum of the particle
around its center of mass. In quantum physics, spin can only have values
equal to multiple of 1/2, such as ½, 1, 1 ½ , 2, …it is a quantized quanty.
Fermions are particles with half-integer spin, such as
• Electrons,
• Protons,
• Neutrons
Bosons are particles with integer spin, such as
• Deuterium: isotope of hydrogen, containing one proton and one
neutron in its nuclei.
• Helium-4 (superconductivity).
• Photons
The Origin of Degenerate Pressure
2. Pauli’s Exclusion Principle and
Heisenberg’s Uncertainty Principle
Degenerate pressure arises from two fundamental laws of quantum physics:
1.
Pauli’s Exclusion Principle for the fermions:
No two particles (fermions) can occupy the same quantum mechanical
state simultaneously.
2.
Heisenberg’s Uncertainty Principle:
The product of the uncertainty in the position of a particle and its
momentum is always greater than the Planck constant
x p ≥ h
where x is the uncertainty in the position of the particle, p is the
uncertainty in the momentum of the particle, and h = 6.626  10-27 gm
cm2/sec is the Planck’s constant.
Pauli’s Exclusion Principle
Under normal conditions, electrons in
atoms can occupy a large number of
energy states, like students in a
mostly-empty class room: there are
more seats available than people. In
this situation, we do not need to
worry about the exclusion principle.
• When atoms are compressed,
like in a white dwarf where
thermal pressure is no longer
able to resist the gravitational
force of the matter, the number
of available energy states is
reduced, similar to a packed
classroom…in which only one
person is allowed in each seat
(the exclusion principle).
• The reduced number of energy
level available in the
compressed atoms is equivalent
to confined space allowed for the
electrons, or small x in the
uncertainty principle.
Uncertainty Principle and
Degenerate Pressure
According to Heisenberg’s Uncertain Principle,
x p ≥ h
very small x requires that p ≥ h / x be very large.
– Very large uncertainty in the momentum of the electrons means
that their velocity varies over a very large range (recall the
definition of momentum: p = mv)
– A very large range in the possible range of velocity of a large
collection of particles is equivalent to saying that this collection of
particles have a very high temperature (Recall the definition of
temperature in Chapter 5.)
– High ‘temperature’ means high pressure!
Important Properties of Degenerate
Pressure
Degenerate pressure becomes appreciable only when the atoms are
compressed by a tremendous pressure. This is because the Planck
constant is a very small number…
h
p 
x
,
and
h = 6.626  10-27 gm cm2/sec
• Thermal pressure depends on the temperature. A gas cloud at a
temperature of 0 K does not posses any thermal pressure. However,
degenerate pressure does not depend on temperature. The
temperature of the white dwarfs can be at absolute zero, its electron
degenerate pressure will be the same as it is at 25,000 K.
• There are different kind of degenerate pressure:
– Electron degenerate pressure (in white dwarfs and brown dwarfs
– chapter 12).
– Neutron degenerate pressure (in Neutron Stars – Chapter 13).
• Star Formation
• Evolution of Low-Mass Stars
• Evolution of High-Mass Stars
Evolution of Low Mass Stars – I
Low Mass Stars: M < 8 – 10 M⊙
Evolutionary History for a typical low-mass star like the Sun
1.
2.
3.
4.
5.
6.
7.
During the main-sequence phase, helium produced by the proton-proton chain
(hydrogen burning) accumulates at the core. As a main sequence star exhausts its core
hydrogen supply, its energy output is reduced.
Without the thermal pressure of the hydrogen fusion, gravitation contraction continue,
and the core temperature rises.
Because the temperature required to start helium burning is much higher (~ 100 million
degrees), there isn’t enough thermal pressure at the core to resist the gravitational
contraction (just yet).
The core temperature rises, as well as the outer layer of the star where there are still
substantial supply of hydrogen, triggering shell hydrogen burning, at a much higher
temperature than the core temperature in the main sequence stars.
The high temperature shell hydrogen burning produces more energy than the same
star in its main sequence core hydrogen burning stage  Higher luminosity.
The high thermal pressure of the shell hydrogen burning push the envelop of the star
outward, much larger than its size at the main sequence stage  giant.
The large surface area of the giant cools off fast  red giant.
•
From sub giant to red giant: few hundred million years.
Structure of Red Giants
•
•
•
•
Inert Helium core  Most
of the mass of the star is
concentrated at the helium
core.
The electron degeneracy
pressure of the inert
helium core balance the
gravitational contraction.
Hydrogen-burning shell.
Hydrogen envelop.
Evolution of Low-Mass Star – II
The time it takes to reach the red giant state depends on the mass of
the star
• For star with lower mass then the Sun, it takes longer.
• As the shell hydrogen fusion stops, the helium core of the low
mass stars may never a temperature high enough for helium
fusion to start.
• As fusion stops, the gravitational collapse continue,
eventually stopped by the electron degenerate pressure of
the helium core.
• The star become a helium white dwarf.
• For star more massive than the Sun, it takes less than 10 billion
years.
• As the shell hydrogen fusion exhausts its fuel, gravitational
collapse continue. However, the high mass of the star means
that the core temperature can reach 100 million degrees,
sufficient for helium fusion to start.
Evolution of Low-Mass Star– III
Triple alpha process in helium burning stars
• Helium fusion converts three helium atoms into one carbon, and
generating energy.
• Theoretical model suggests that before core helium fusion
phase, the star is supported by the electron degenerate pressure
of the helium core. This degenerate pressure does not increase
with the increasing core temperature as the star contracts.
• However, once helium fusion starts, it releases a large amount of
energy in a short time, causing the star to expand rapidly. This is
referred to as the Helium Flash.
Evolution of Low-Mass Star – IV
After helium flash, the star settles
into a helium burning stage, the
energy of the star decreases…
– The helium burning stars are
smaller, hotter, and less
luminous than the star in the
red giant state.
– The helium core of the lowmass stars fuse helium into
carbon at about the same
rate. Therefore, they appears
on the HR diagram as a
horizontal line.
– This state is represented in
the HR diagram as the
horizontal branch.
– Low-mass stars spend
about 100 million years in
this stage.
Evolution of Low-Mass Star – V
The helium fuel in the core eventually runs out, and core fusion ceases.
• The carbon core will begin to contract due to gravity.
• The increased temperature due to the contraction will cause shell helium
burning around the carbon core.
• Further out, a shell hydrogen burning continue on top of the helium shell
– double-shell buring, ~ 1 million years.
• Both shells contract with the carbon core, driving the increase in
temperature and fusion rate.
• The star expands
further, becomes
larger and more
luminous than its
red giant phase.
• Fusion of carbon
requires high
temperature, ~ 600
million degrees.
This is unlikely to
happen for lowmass stars.
Click to start animation
The end of Low-Mass Stars:
Planetary Nebulae
As the star’s luminosity and radius
increase, its wind will grow stronger as
well. The star ejects its outer layer to
form the beautiful planetary nebula.
• The exposed core will be hot for a
long time, emitting UV radiations.
• The UV radiation will ionize the gas
in the expanding shell, making it
grows brightly.