After the Main Sequence
Laws of Stellar Structure
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Hydrostatic Equilibrium
Energy Transport
Conservation of Mass
Conservation of Energy
Limits of the Main Sequence
Upper Limit: High mass stars
Lower Limit: Low Mass Stars
Why do heavier elements fuse at higher temperatures?
The electrostatic force that two charged particles exert on each other is called the Coulomb
force and is given by the following equation
F
kq1q 2
d2
k is a constant, q1 and q2 are the charges, and d is the distance between them.
For fusion to occur, the particles must get very close to each other. However, the closer
they get, the more strongly they repel each other. They must collide at very high speeds
in order to get close enough for the strong nuclear force to bind them. Since the average
speed of a particle of an ideal gas is proportional to the square root of the temperature,
fusion will only occur at high temperatures.
A combination of high stellar core temperatures and quantum mechanical tunneling
makes fusion possible in stellar cores.
The heavier elements have more nuclear charge than the light elements. Since the
Coulomb force is proportional to the product of the charges, the heavier elements undergo
fusion at higher temperatures than the light elements.
Energy Production in Stars
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Proton-proton chain (M  1.1 MSun)
This is the main source of a star’s energy for spectral classes cooler than F0
It dominates at core temperatures between 3106 and about 2107 K
CNO cycle (M  1.5 MSun)
Same result as the PP chain, but carbon acts as a catalyst; i.e., it facilitates the
fusion of H to He but doesn’t get used up. This produces energy faster than the
PP chain in spectral classes hotter than F0.
It dominates at core temperatures greater than about 2 2107 K
Triple a process 3 2He4  6C12 + energy
Requires core temperatures  108 K
Carbon fusion
Tcore  6 108 K
Neon fusion
Tcore  1.5109 K
Oxygen fusion
Tcore  2109 K
Silicon fusion
Tcore  3109 K
Only stars with ZAMS masses greater than 20 solar masses will undergo
silicon fusion.
The most tightly-bound element is 26Fe56. It can release energy by neither
fusion nor fission.
What causes a star to become a red giant?
As hydrogen fuses, the helium nuclei fall toward the center of the star and accumulate
there to form a helium core.
Inert He
As more He rains down into the
(not hot
core, the conversion of
enough for
gravitational energy into heat
the triple a
H fusion
and light increases its
process)
He
H and He
temperature and pressure.
envelope (not
The hot He core heats up the
hot enough
hydrogen in a shell outside
for fusion)
the region that was the core
Eventually, the combination of radiation
of the main sequence star.
pressure and thermal pressure from the
This shell becomes hot
shell causes the star’s envelope to expand
enough for H fusion to begin.
and cool. The star becomes a red giant.
The temperature of the core is high enough to cause the H shell to fuse rapidly, resulting in
a dramatic increase in the star’s luminosity.
Hubble Space Telescope Image of a Red Supergiant
This scale implies that Betelgeuse is about 1000 times larger than the Sun!
The Pauli Exclusion Principle
• Electrons, neutrons, protons, and neutrinos are examples of
fermions. They are particles with an odd multiple of one-half the
fundamental unit of angular momentum.
• The Pauli exclusion principle is a physical law obeyed by all
fermions.
In a bound sample of fermions of a given type, no two
particles can have both the same energy and the same spin
orientation.
• The condition in which all of the electrons in an object are in
their lowest possible energy states is called electron
degeneracy.
Energy Diagrams for Degenerate and Non-degenerate
Electron Gases
If the electron density is low, as it is in a
“normal” star, most energy levels are empty
and an electron can easily acquire enough
energy to jump to a higher energy level. Under
these conditions, the electron gas is called
non-degenerate. Its pressure is proportional to
its temperature (PV = NkT)
If the electron density is greater than about
109 kilograms per cubic meter, there will be
electrons in all of the lowest levels. Under
these conditions, the electron gas is called
degenerate. Only the few electrons with the
highest energies can easily acquire enough
energy to jump to higher energy levels. For
most electrons, the nearby levels are
already filled with electrons.
Non-degenerate
Degenerate
An electron bound to an atom can only have
one of a set of discrete energies. This is also
true for electrons bound inside a star.
The Properties of Degenerate Matter
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In order to compress it, we must change the energy of large numbers of
electrons. However, only a few electrons (those in the highest occupied levels)
can have their energies changed by small amounts. Therefore, the degenerate
matter resists compression; it is extremely rigid.
It easily conducts both heat and electricity.
In contrast to an ordinary gas, its pressure depends only on its density – not on
its temperature.
The “free” electrons in a metal form a degenerate electron gas; that’s why a
metal is a good conductor of both electricity and heat.
The Triple Alpha Process and the “Helium Flash”
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2
He is called the alpha particle, and the fusion of three helium-4 nuclei to produce a carbon
nucleus is called the triple alpha process. The triple alpha process actually takes place in two steps
4
2
He  42 He  48 Be  
8
4
Be  42 He  126 C  
In stars with masses between 0.4 and 4 solar masses, the helium core becomes degenerate
before the temperature is high enough to ignite helium. This results in an explosion called
the helium flash.
Helium ignition
 temperature increase without increase of pressure (the gas is degenerate)
 increase of helium fusion rate throughout the core
 further temperature increase without increase of pressure
 further increase of helium fusion rate throughout the core 
After a few minutes, the temperature is so high that the core becomes non-degenerate.
Although the peak luminosity may be as high as 1014 times that of the Sun, all of the
energy is absorbed by the red giant’s envelope. This, combined with the short duration of
the event, makes the helium flash virtually unobservable.
White Dwarfs
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When gravity compresses a star so much that a mass comparable to the mass
of the Sun is squeezed into a volume comparable to the mass of Earth, the
density is about 109 kilograms per cubic meter.
This compact object, supported by electron degeneracy pressure, is called a
white dwarf.
Estimate of the density of a white dwarf
R
7000km  7  106 m,
M
M
2  1030 kg
density 
mass
volume
volume 
4 3
R
3
3
4
6
volume    7  10 m   1.4  1021 m3
3
2.0  1030 kg
density 
 1.4  109 kg / m3
21 3
1.4  10 m
A cubic inch of this material would weigh
about 10 tons!
White Dwarfs in Globular Cluster M4
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M4 is 7000 light years away.
The image on the right is the HST image of the small region indicated in the groundbased image on the left. Circles are drawn around the white dwarfs.
Globular clusters are old, so they are expected to contain many white dwarfs. Of M4’s
100,000 or so stars, about 40,000 are expected to be white dwarfs.
Low
Mass
Star
Low mass stars are called red dwarfs
Masses between 0.08 and 0.4 solar masses
The lifetime of a 0.4 solar mass star is about 100109 years.
White
Dwarf
A low mass star is convective throughout its entire
volume. The helium created by fusion is mixed with
the material in the rest of the star. Because of this, it
never has a dense helium core surrounded by a shell
in which hydrogen is undergoing fusion, which is the
condition that results in a red giant.
Consequently, these stars never become red giants.
Black
Dwarf
They will gradually cool to become black dwarfs
composed mainly of a mixture of H and He.
HST Near Infrared Image of a Red Dwarf
The white streak is a result of
overexposure of GL 105A.
Gliese 105C
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27 light years away in constellation Cetus.
GL 105C is 25,000 times fainter than GL 105A.
The temperature of GL 105C may be as low as 2600 K.
Mass is about 0.08 to 0.09 times the mass of the sun.
Since this is an infrared image, the colors you see are false colors.
Medium
Mass
Star
Red Giant
Masses between 0.4 and 8 solar masses.
The envelope of the red giant becomes unstable (thermal pulses)and
is expelled. The nebula that results is called a planetary nebula.
Fluorescence occurs only if the temperature of the star inside the
expelled material is least 25,000 K.
Eventually, a star will be incapable of generating energy by nuclear
fusion. If its mass is then less than the Chandrasekhar limit (1.4 M)
the star will be a hot white dwarf that slowly cools by emitting
electromagnetic radiation.
WhiteDwarf
+
Planetary
Nebula
Black
Dwarf
These white dwarfs are composed primarily of carbon and oxygen.
The carbon and oxygen are ionized and embedded in a degenerate
gas of electrons.
When the temperature is low enough, the carbon and oxygen
crystallize.
A Planetary Nebula
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The hot relic of the dying star’s interior emits ultraviolet radiation, which
causes fluorescence of the expelled envelope.
NGC 2392 is 5000 light years away in Gemini.
Gravitational Equipotential Surfaces – Accretion Disks in
Binary Star Systems
Inner Lagrangian Point
1 = black (low potential)
2 = brown (Roche lobes)
3 = yellow
4 = green
5 = blue
6 = red (high potential)
L5
M1
M2
L1
L4
What is a Nova?
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Always found in binary star systems with one of the stars being a white dwarf.
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Material from the companion passes through the inner Lagrangian point,
forming an accretion disk around the white dwarf.
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Material from the accretion disk, rich in hydrogen, falls onto the surface of the
white dwarf.
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The hydrogen-rich surface layer gets thicker, hotter, and more dense,
eventually becoming degenerate.
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When the temperature reaches millions of degrees, the hydrogen detonates in a
thermonuclear explosion that blasts away much of the surface layer.
The process is repeated, so these are called “recurrent novae”.
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Recurrent Nova T Pyxidis
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~ 6000 ly from Earth
Recurs with a period of about 20 years.
Consists of a couple of thousand bright knots
Diameter ~ 1 ly at the time of these pictures