Lecture 16 - Stellar Evolution (3/16/99)
Stellar Energy --- | ---
Chapter 16-3, 16-5, Prelude P-5 (ZG4)
M57, the Ring Nebula, in the consellation Lyra. This is
an example of a planetary nebula, the ejected envelope of the
post-main-sequence white dwarf remnant at the center of the ring.
||What determines the path of a
star on the H-R diagram?
||The uncertainty principle
||Follow the life-cycle of a
- Main Sequence Revisited
- What is the zero-age main sequence (ZAMS)?
- How does the main sequence lifetime of a star scale with mass?
- How does the H-R locus for a star change with composition?
- What is a subdwarf?
- What happens to the core of a star when it has exhausted a given
fuel for a particular fusion reaction?
- Is there any pressure other than gas pressure available to stop
the collapse of a stellar core?
- If light has energy and momentum, does a particle have a wavelength?
What is it?
- What is the de Broglie wavelength of a relativistic electron?
- What is the relation between electron waves and Bohr orbitals?
- What is the smallest "orbital radius" r into which you can squeeze an
electron (or proton) of momentum p?
- What is the Heisenberg Uncertainty principle for position and
- What is the uncertainty relation for energy and time?
- What is the Pauli exclusion principle and what is its
implication for electrons?
- What is a Fermion?
- What is the degeneracy pressure of an electron gas?
- What is the equation of state for a degenerate electron gas?
- Because the degeneracy pressure is independent of temperature,
what happens in the contraction of a degenerate helium core after
hydrogen burning ends?
- What is the mass-radius relation for a degenerate core?
- What is the Chandrasekhar limit?
- Post-Main Sequence Evolution
- How does the luminosity of a star evolve while it is on the
main sequence and why?
- What happens to the core after hydrogen exhaustion?
- What happens to the stellar envelope when hydrogen shell burning
- What is a subgiant?
- What is the red giant branch (RGB)?
- What is the helium flash?
- How is degeneracy lifted after the helium flash?
- What happens after core helium burning ends?
- What is the horizontal branch (HB)?
- What is the asymptotic giant branch (AGB)?
- What is a planetary nebula?
- What is a white dwarf and what is its ultimate fate?
- Late Stages of Stellar Evolution
- Why are the post-main-sequence phases shorter?
- Why is the helium shell burning stage characterized by variability?
- Why are there large stellar winds in late stages of evolution?
- What are typical mass loss rates on the AGB?
- Massive Stars
- Why do massive stars have convective cores?
- What does a convective core imply for the interior composition?
- How do heavy elements form in a massive star?
- What happens when a iron/nickel core forms?
Stellar Remnants: Planetary Nebulae
The Ring Nebula M57, courtesy HST/STSCI
Planetary nebulae are the outer layers of a star, blown away during the
thermal pulsing of the helium shell-burning phase of its evolution. The
Hubble Space telescope has snagged a number of spectacular images of planetary
nebulae, showing the variety of rings blown by these dying stars.
Mass loss is a general phenomenon, with both AGB stars and hot main
sequence stars losing significant amounts of their envelopes during various
stages in their lives.
Stellar Evolution in Outline:
- The Life Cycles of Stars
- Stars have "lives" in that they are born out of dust and gas,
grow under gravity, start burning nuclear fuel and become full-fledged
stars, go through stages as different fuel sources are found,
exhaust their energy and die.
- Stars return much (in fact sometimes most) of their mass back
to the interstellar medium, from which new generations of
stars are born.
- The Sun is 4.5 billion years old, while the oldest stars known
appear to be 18 billion years old. The Sun is not a first
generation star, it was formed from material that had been
processed through at least one generation of stars that lived
and died previously!
- Almost all the elements heavier than helium were "created" from
fusion in the cores of stars. The material that our bodies are
made of was synthesized in the center of stars now long dead!
- The understanding of the stages of a star's life comes from
understanding the Hertzsprung-Russell (H-R) diagram of L vs. T.
- A star like the Sun spends 90% of its life burning hydrogen
on the main sequence. Eventually the hydrogen runs out and
the star must change its ways. This is when things get
- The Tale of the Core
- When hydrogen is exhausted in the center (only about the innermost
10% of the stars mass is available for fusion as we will see),
the core, which is now made of mostly helium, must contract to
derive gravitational energy to keep up the pressure against gravity.
- This contraction is slow, as the Sun could actually generate
its current luminosity by contraction of 40 meters per year!
Q: How long can the Sun keep this up before contracting
- As the helium core contracts, it heats up, and this ignites the
unfused hydrogen in a shell around the core. This further liberates
new energy which flows out into the envelope of the star.
- The envelope structure was set up to be in equilibrium with
the luminosity of the hydrogen burning core. Now it has extra
energy pumped in at the bottom, and it is too dense to let
all this new heat and radiation pass through.
To compensate it will expand
in radius, dropping in density, and thus in surface temperature
at the outside.
- The radius will increase greatly, by as much as 25 times! The
star moves to the right (becomes cooler) in the H-R diagram.
- Eventually the envelope will drop in density enough to allow the
new luminosity to come out through the surface, and the star
will increase in luminosity, moving upward on the H-R diagram.
- Thus, after hydrogen core burning on the main sequence, a star
move upward and to the right (higher L, lower T).
- The star is much larger in radius, and is called a giant
star. It is cooler, and thus redder in color, so it is commonly
called a red giant.
- Some stars are even larger than these giants, with radii almost
a thousand times the radius of the Sun. These are called
- Supergiants are high mass stars that have evolved off the main
sequence. Betelgeuse in the constellation Orion
is a red supergiant with mass 15 Msun and radius of nearly 1000 Rsun
(almost 4 AU!).
- Blue giants and supergiants usually refer to high-mass (and thus
high temperature) main sequence stars.
- Low Mass Stars
- Stars less than 1/12 solar mass are called brown dwarfs.
The central temperature of these stars never gets high enough
to fuse hydrogen. These are "failed stars" and are very faint.
They can be thought of as large "Jupiters".
- There may be very many brown dwarfs in our galaxy, though we
wouldn't see them easily. Recently, the first brown dwarfs were
discovered in binary systems, where their masses could be deduced.
- Stars between 1/12 and 0.4 solar masses are called red dwarfs.
These are main sequence hydrogen burning stars, but cooler, smaller,
and fainter than the Sun (hence "red" and "dwarf").
- Red dwarfs are much less luminous than the Sun. The approximate
mass-luminosity relation (L/Lsun) = (M/Msun)^3.5 leads to the
lifetime (t/tsun) = (M/L)/(Msun/Lsun) = (Msun/M)^2.5, so a star
with mass 0.1 Msun would have a lifetime of 300 times the Sun's
lifetime of 10 billion years, or 300 billion years! Red dwarf
stars burn their fuel so slowly that they live a very long time
(the ultimate conservationists).
- Red dwarf stars, because of their low luminosities, are fully
convective - that is they transport heat from their cores to
the surface by gas currents. This also mixes the stars fully
so that they can use all of the hydrogen in the star to burn,
and the helium made in the core will be mixed throughout the
- When a red dwarf has used up all its hydrogen, it will simply
collapse under gravity until it becomes a helium white
dwarf (see below). Then it will fade away fainter and
cooler until it is a cold dead ball of gas.
- Medium Mass Stars
- Stars from about 0.4 to 3 solar masses are normal hydrogen
fusing main sequence stars like our Sun for most of their
- Stars with masses less than about 1.1 solar masses have no
convection in their cores. Heat transport is purely radiative
out into the envelope. Most importantly, there is therefore
no mixing of unburnt envelope hydrogen into the core. Once
the core uses up its fuel, it has no recourse.
- Only about the inner 10% to 13% of the mass of the star is available
for fusion in the hot core. This is why the lifetime of the
Sun on the main sequence is 10 billion years, not 80 billion
years. Gas outside the core is not available for fuel.
- Stars from 1.1 to 3 solar masses have some convection in their
core region. However, this can only mix in around a few percent
of the mass near the core, so this does not make much difference
in the lifetimes.
- In all these cases, when the core hydrogen has been fused to
helium, we are left with an inert helium core.
- The helium core cannot maintain pressure against the weight of
the star above it, and to generate heat and thus pressure it
contracts releasing gravitational energy. This is slow contraction
by about 40 meters per year to maintain 1 solar luminosity of
- The increase in temperature from the gravitational contraction
of the core raises the temperature of the hydrogen containing
layers just outside the core, causing fusion to occur. At this
point hydrogen fusion is going on in a shell around the contracting
core. This stage is called hydrogen shell burning.
- The energy from the core contraction and the shell burning flows
out into the envelope. The structure of the envelope was
set up to deal with the main sequence luminosity which was
approximately constant. Now it cannot handle this extra luminosity
so it expands outward becoming larger in radius, less dense, and
thus lower temperature ( T^4 proportional to L/R^2 with R increasing).
- At first only a little of the extra luminosity gets out, with most
of the energy going into expanding the envelope. Eventually the
stare grows in size by around 25 times its initial radius, and
it becomes low enough density so that the radiation can flow out
at the rate it is being generated. The luminosity now has grown
by a factor of 10 or so.
- The star is now cooler ("redder") and larger ("a giant"), and
is called a red giant.
- Red giants are located to the right and up from the main sequence.
They form a locus of stars called the giant branch.
- On the giant branch, the helium core is contracting and getting
hotter and hotter. These stars are massive enough that the core
eventually gets hot enough (100 million K) to fuse helium into
carbon (and nitrogen and oxygen) in the triple-alpha reaction.
- How this fusion begins depends upon the state of the core at this
time. Stars with masses below about 2 solar masses have cores
that are degenerate (see below) when helium fusion begins. This
means that the helium fusion begins all at once in a big flash
called the helium flash. In about 1 second the core emits
the equivalent luminosity of 10 billion suns! You dont notice
anything from the outside because this energy is absorbed by
- Stars with more the 2 solar masses do not have degenerate cores
at the time of helium fusion, so they start fusion gradually.
- In any event, helium fusion in the core stops the contraction,
and expands the core slightly, and the envelope contracts a little
increasing the temperature. The star moves to the left a little
and slightly down on the HR diagram after reaching the top of
the giant track at the helium flash. This is called the
horizontal branch, since it extends to the left
horizontally from the giant branch.
- The helium is burning in the core and hydrogen is burning in the
shell around the core for stars on the horizontal branch.
- After the helium is done burning in the core, the core, which is
now made of carbon, nitrogen and oxygen, contracts
again heating up and starting helium burning in a shell (with
a hyrogen shell further out).
- Thus, with helium and hydrogen burning in concentric shells, the
star moves up the giant branch again, becoming even larger.
This track parallels but is slightly to the left (hotter) than
the giant branch, and is called the asymptotic giant branch.
This branch extends a factor of 10 to 100 times the luminosity of
the giant branch, and stars on this branch are called red
- For these medium mass stars with less than 3 Msun, the core never
gets hot enough for the carbon to fuse. The helium core contracts
until it becomes degenerate.
- Degenerate Matter
- We have talked about degenerate matter as if it is some
special state that stuff turns into at high density. What is
- Upon collapse, the helium and C-N-O cores collapse to about
0.01 of their initial radius, thus increasing the density by
a factor of 10^6.
- The electrons and nuclei of the ionized gas are squeezed together
tighter and tighter. Do you remember what happened when we tried
to confine an electron into a small orbit around the nucleus?
- We found that we had to deal with the wave nature of the electron.
Interference effects allowed us to have only integer numbers of
waves in the orbit -> energy levels.
- The requirement that we be able to distinguish electron led us
to allowing only one electron of a given spin (either spin "up"
or spin "down") in each wavelength (orbital).
- The same thing happens here. If you confine an electron into
a radius r, the smallest you can make this is 2pi r = L
where L is the wavelength: L = h/ mv.
- Thus, we are limited to confining the electron into a space
L = 2pi r = h/ mv
- The rms velocity of the particles in the gas is proportional to the
< v^2 > = 3kT/m
thus for a given velocity, the electron can be confined to
a space no smaller than
r v = h/2pi m
- This relation is usually written as
DX DP > h/2pi
where DX represents the localization in space, and DP the
localization in momentum (velocity). Note that this also means that
if you know the position of a particle to some DX, you cannot
know its velocity to an accuracy better than DV = DP/m
(and visa versa).
This is known as the Heisenberg Uncertainty Principle and
is one of the foundations of quantum mechanics. This shows that
there is an inherent unpredictability at the wavelength scale
of any system.
- Now the distinguishability enters: you can have only two electrons
confined to a given wavelength-sized space, one of each allowed
spin. You can think of the gas as a huge parking lot, with lots
of available spaces (wavelengths = energy levels) for cars
(electrons). Its sort
of a strange parking lot, since you can put two cars in a given
space, as long as one is parked forward and the other backward
(spin s = +1/2 and spin s = -1/2)! As you compress
the gas, it is like filling up the parking lot until it is
completely full (all available energy levels below some level
have 2 electrons). To add more electrons they have to go on the
outskirts of the lot (higher energy levels) where there is
free space (emptly "orbitals"). A full lot (energy levels all
full) is said to be degenerate.
- If you try to press the electrons closer together than this,
they resist very strongly, exerting a stiff pressure against
further compression. This is degeneracy pressure.
- When the core contracts to the point of degeneracy, the pressure
is no longer supplied by the temperature but by the electron
degeneracy. Furthermore, if you add heat into this degenerate gas
the pressure does not increase since the heat goes into the nuclei
motions, not the electrons (which are jammed into the parking lot
and moving one means moving all in the way or blasting it all the
way to the free spaces at the edge).
- To repeat: you can heat up a degenerate gas without increasing
- If the helium core is degenerate or nearly degenerate when its
reaches the threshold for helium fusion to occur, then something
catastrophic occurs. When the fusion starts, it releases energy
and heats the core further, but this does not increase the pressure
which would expand the core lowering the temperature, which is the
usual negative feedback control. If the core is degenerate, the
increase in temperature causes the core to grow hotter, which increases
the fusion rate, which releases more energy, which increases the
temperature, and so on. This is a case of positive feedback
which leads to uncontrolled fusion - the helium flash.
- In a few seconds the helium core ignites uncontrolled by thermal
pressure, releasing huge amounts of energy. For this short time
the core produces the luminosity of a billion suns. However, this
energy goes into removing the core degeneracy (unjamming the parking
lot by throwing cars from the inside way out to the edge where there
is space) and the rest gets absorbed in the envelope.
- The helium flash does not disrupt the star, and there is no outward
sign that it occured. The star continues now with a non-degenerate
helium burning core (and hydrogen burning shell).
- Planetary Nebulae
- While the star has a contracting C-N-O core with
helium and hydrogen burning in shells, it is ascending
the giant branch again. These two shells are pumping
out lots of energy into the envelope distending it even
further than before. This begins to blow off the outermost
layers compeletely, beginning the process of mass loss
of a large fraction of the stars mass.
- Also, the helium shell burning is somewhat unstable, with
sudden flares of burning causing shells of material to blow
off the star.
- This process accelerates as the mass is lost, until the entire
envelope is blown off and we are left with the bare core!
- All this envelope is blown away into shells around the naked
core, forming a planetary nebula.
- During the planetary nebula phase, the remaining star becomes
hotter and hotter as lower high temperature layers are exposed.
The luminosity remains the same, since that is generated in
the core. The star moves rapidly to the left in the H-R diagram,
until the naked core is exposed at 10^7 K.
- White Dwarfs
- After all burning has stopped in the core, and the envelope
has been stripped away, we are left with a degenerate C-N-O
core with a temperature of millions of degrees. This star
is very hot, "white-hot", and is called a white dwarf.
- The white dwarfs produced by low mass stars are made of helium
since only hydrogen fusion occured -> He white dwarf
- The white dwarfs produced by medium mass stars are made of
carbon, nitrogen and oxygen from helium fusion -> CNO white dwarfs
- A white dwarf has the mass of 0.1 to 1.4 Msun and radii about the
size of the Earth (10^4 km). The densities are about 10^6 times
that of the Earth.
- The companion star to Sirius, called Sirius B, is a white dwarf
with a mass of 0.98 Msun, but a radius of about 2.7 Rearth.
Q: Calculate the mean density and surface gravity of
- White dwarfs slowly cool down, radiating their vast heat through
their small surface areas. Eventually they fade away to become
- When you add mass to a white dwarf, it shrinks! This is because
adding mass increases the gravity, and the spacing between electrons
decreases (just like increasing the charge of the nucleus for
the ground state orbit). When the white dwarf mass reaches 1.4 Msun,
the electron degeneracy pressure can no longer withstand gravity,
and the core will collapse again.
- The limit of 1.4 Msun is known as the Chandrasekhar Limit.
You cannot have a white dwarf with a mass above this limit.
- We believe that stars with initial of about 8 solar masses or
less can get rid of enough of their mass to form a white dwarf
with mass 1.4 Msun or less. This limit is uncertain, and at least
some stars with masses in range of 8 to 10 Msun can form a white
dwarf also (these would be made of oxygen-magnesium-neon as the
products of carbon fusion).
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Steven T. Myers