Week 7-- Solar Evolution II

I. A star as one thing
To understand pre-main-sequence and the arrival at the main sequence, one can treat an entire star as if it was "just one thing," for conceptual purposes (not counting the surface T that we see in the H-R diagram, and of course there needs to be an internal T gradient to get the heat out, but these are details). So we use global pedagogical tools like the virial theorem, the history of radius contraction, and the radiative diffusion rate to track the evolution of the star until it settles on the main sequence. We have talked a lot about this already, the story includes the Hayashi track for fully convective stars (which always have a "red" surface T), and the Henyey track for radiative interiors (which have a M-L relation based on radiative diffusion). The timescales are controlled by E / L, where E is the internal energy GM^2 / R, and L is the luminosity (this is called the Kelvin-Helmholtz timescale).

There is also some evolution on the main sequence, where the star fuses H into He, which keeps a similar M but reduces the number of particles N (since H-->He turns 8 particles into 3, think about it). Many textbooks incorrectly claim this causes contraction because fewer particles at the same fusion T must have higher density to hold up the weight of the star, but this is the same tiresome error that treats "weight" like it depends only on M. It does not, it also depends on R, so what actually happens is the R increases to reduce the weight, so that fewer particles can hold it up! Thus, reducing the number of particles in a star undergoing fusion (which regulates the core T) causes expansion of the star, not contraction. This expansion implies a higher L because the "leaky bucket of light" is becoming a bigger, leakier bucket! So stars on the main sequence generally expand a little and become a little more luminous.
However, textbooks have noticed that the core of MS stars do actually contract a bit, because of the He gradient-- there is more He and less H toward the center, and the star must adjust for this in ways that the global considerations we have used so far cannot accommodate. This is a kind of detail in the main sequence, but it is responsible for the end of the main sequence (there is too little H near the center), and more importantly, it ushers in a whole new era for the star where the H gradient becomes so important it completely alters how stars evolve. The upshot is that stars reverse their course on the H-R diagram after the main sequence is over, for reasons we will now explore.

II. A star as three things: core, shell, envelope

As H runs out near the center, fusion shifts to a shell around the core. This is the end of the main sequence because it completely changes the star's internal structure. Now we must think of the star as three things, an inert core with no fusion, a shell of about the same size which does undergo fusion, and a large envelope whose attributes are ruled by the internal fusion engine. The fusion is so capable of producing energy that it transfers any energy it needs to the envelope to produce a self-regulated R and T structure throughout the envelope. As we will see, this can lead to a very expanded envelope indeed.

First we must understand what happens to the core. To do that, we must understand the physics of fusion (which self-regulates to maintain no net heat loss, and this generally acts to keep the core T in the range 10-20 million K, so fairly fixed), the physics of net heat loss (which holds when fusion is not present in the core, and leads to core contraction in the usual way), and the physics of degeneracy (which means the core is approaching its quantum mechanical ground state, so its ability to lose heat is being interdicted, which is the same as saying its T is not as high as we'd expect if it were an ideal gas).

III. Evolution of the core
When H fuel runs out, the core suffers net heat loss, causing contraction. Even more importantly, net heat loss causes any system to approach closer to its quantum mechanical ground state. Normally this is of no consequence, because the system is so far from its ground state that there is essentially no quantum mechanics to worry about. But the core of the Sun will eventually lose so much heat that quantum mechanics does become important, and this gets called "degeneracy" (for reasons that I frankly have no idea why, it always seemed like the opposite of degeneracy to me because the core is going from a vast number of states that we are not bothering to distinguish, into fewer states that are approaching the single state that is the ground state).

The fate of the core of the Sun is related to the discovery of stars called white dwarfs. These stars have similar surface T to many of the main-sequence stars, but are much less luminous (a classic example is Sirius A and B, where A is the bright main-sequence star and B is the dim white dwarf). This means the white dwarfs must be much smaller (indeed about 100 times smaller R), but the white dwarfs are generally not much less massive (they are evolved versions of stars that were not much less massive), so must have extremely high densities, perhaps a million times the Sun. This seemed very odd to the astronomers of the day, but given that a star losing net heat for a really long time is going to shrink really a lot, the real surprise about white dwarfs is that they have not shrunk a lot more than they have. To understand why a star slows down in its shrinking, we need to understand the effects of quantum mechanics on the thermodynamics (the T) of the white dwarf-- we need to understand degeneracy.

IV. Degeneracy and "Degeneracy Pressure"
For historical reasons, most astronomy textbooks repeat an incorrect analysis of what degeneracy pressure is and how it responds to the addition of heat, so we will clarify this here in hopes that new students will not need to "unlearn" the misconceptions later on. What is quite common for textbooks and lecture notes to state are words to the effect that degeneracy pressure is some kind of weird pressure that "emerges" from quantum mechanics and "augments" the normal "thermal pressure." Any suggestion that it is some kind of "additional" pressure that comes from quantum mechanics is a completely wrong way to think about "degeneracy pressure", as is demonstrated by the well-known fact that if you waved a magic wand over a white dwarf that labelled all of its electrons, such that they were no longer indistinguishable and would not obey the Pauli exclusion principle, the particles would need to redistribute their kinetic energy from a quantum mechanical Fermi-Dirac distribution into a classical Maxwell-Boltzmann distribution. Guess what would nstantaneously happen to the pressure when this occurs?

Nothing. Absolutely nothing, no loss of "additional" pressure stemming from the quantum mechanics! The reason is, nonrelativistic particles always produce a gas pressure that depends only on their kinetic energy density, so anything that redistributes kinetic energy but does not change it has no effect at all on gas pressure. So what does it affect?

Quantum degeneracy is a thermodynamic effect, not a mechanical effect that could change pressure. Thus, degeneracy affects temperature, period. The fact that no two electrons can be in the same state redistributes the same amount of kinetic energy (which is typically quite large, just as it is for the electrons in a metal fork) into a surprisingly low temperature configuration. Indeed, for complete degeneracy, the system is in its ground state, which means it would be in thermal equilibrium with a zero temperature reservoir (informally we might say the system would be at zero temperature). So that's what "degeneracy" does, and is actually the only thing it does-- it takes a given average kinetic energy per particle and associates it with a much lower temperature than you would get for an ideal gas (the latter has 3/2 kT as the average kinetic energy par particle).

So if degeneracy has no direct effect on pressure, what does it affect, and what does the term "degeneracy pressure" mean? Let us be very clear, so we get this right the first time-- degeneracy is a thermodynamic effect, not a mechanical effect, so it affects temperature not pressure. It is about how a given amount of kinetic energy is distributed over a collection of indistinguishable Fermions, such that no two particles can be in the same state. This can lead to a much lower kT than the average kinetic energy per particle, but the pressure is completely normal, given the amount of kinetic energy present due to the system's history of energy acquisition and loss. Lowering kT, for given kinetic energy, reduces heat transport, and in the limit of complete degeneracy where T would be zero, there can be no net heat loss at all. (Zero T cannot actually be achieved, as correctly speculated by Nernst as a corollary to his famous theorem, but the system can come close and we idealize the limit.)

So how does the interdiction of net heat loss connect with pressure? The pressure comes from the kinetic energy, and the virial theorem in a self-gravitating gas says that the latter equals the total amount of net heat lost from the system in its history. Hence when no more net heat can be lost, we have an end point in the increase in the kinetic energy and the decrease in the radius. "Degeneracy pressure" is simply the garden variety gas pressure that happens to be present when this endpoint is reached. It can be calculated by noting this endpoint occurs when the volume associated with the deBroglie wavelength of each electron matches the volume per electron, so this will involve quantum mechanics via the Heisenberg uncertainty principle and the Pauli exclusion principle. The formula for this that is seen everywhere is correct, the problem is in the way it is explained, because although we need quantum mechanics to understand where this endpoint occurs (just as we need fusion physics to understand where the main sequence occurs in the initial contraction of the core), we do not need it to understand where this pressure comes from, and it produces no "augmentation" of any pressure anywhere.

The only time degeneracy would "augment" the pressure is if the system temperature is being controlled, and then the low ratio of kT/KE caused by degeneracy would induce a large kinetic energy KE and hence a large pressure. But the KE in degenerate stellar cores comes from a history of gravitational contraction, not some sort of external temperature control, so degeneracy does not augment the pressure in any way.

V. Shell fusion and envelope expansion
The importance of degeneracy is not so much that it happens, but that it does not happen before the core has contracted so much that the temperature in the shell around the core (which still contains hydrogen) goes to very high fusion temperature. Remember that H core fusion is able to regulate the temperature to always be around 15 million Kelvin, but fusion in a shell cannot do that, its T is set by the gravity of the core and so gets very high as the core contracts. Since fusion is very T sensitive, this means the shell fusion is essentially going nuts, which dumps an enormous amount of heat into the envelope, which puffs it out. Since the weight of the envelope depends not only on its mass, but also its radius, puffing out the envelope reduces its weight. This in turn reduces the pressure in the shell, such that when the envelope is of sufficiently large radius, the fusion in the shell can be reduced to match the rate that radiation diffuses out, and this creates the new equilibrium in a star undergoing shell fusion. Notice the contrast here: core fusion achieves thermal equilibrium by controlling its temperature, shell fusion does it by controlling its density (since its temperature is handed to it), and this requires puffing out the envelope.

The next thing to recognize is that, since the core is basically a white dwarf, the core radius is set by the mass-radius relation of degenerate gas, which means R drops as M rises. Hence the gravity in the shell rises as helium "ash" is added to the core, and since that sets the T in the shell, the T also rises. So the luminosity also rises, and so the envelope puffs out even more (recall that the envelope looks like a fully convective star, so has a surface T that is regulated to always be about 3000 Kelvin, so higher L requires larger R). Thus as the star ages and its core mass rises, the star gets very large and very luminous-- it becomes a red giant.

You might think the star will just continue to expand as its core mass continues to rise, but something else eventually happens-- when the core mass reaches about half a solar mass, the kT in the core will be high enough to fuse He (even though the core is partially degenerate, it is not completely degenerate, so its kT can still rise if the KE per particle is rising even more). At this point, the Sun will have fusion going on in its degenerate core, but the combination of fusion and degeneracy is thermally unstable for reasons that are rarely described correctly even in advanced textbooks. Disabuse yourself of the drastically wrong idea that adding heat to a degenate gas will not cause pressure rise and expansion (since you now understand where gas pressure comes from, you know that adding heat produces a completely normal pressure rise in degenerate gas). Instead, there is expansion, but it does not lower the kT, it raises it. Ideal gases that are self-gravitating will lower their kT when heat is added (as this is the opposite of the heat loss that caused kT to rise in the first place), but degenerate gases do the opposite-- they have the less counterintuitive behavior of actually raising their kT when heat is added. This is because the degeneracy was caused by losing heat, so returning heat reduces the degeneracy and raises the temperature. The more heat added, the more the T rises, and that is a thermal runaway. Since it is He undergoing fusion in the core, this is called the "helium flash."

The helium flash does not burn all the core helium, nor does it explode the star, it merely removes the degeneracy of the core until it acts more like an ideal gas, as the Sun's core is doing right now. That returns the core to thermal stability, and we have a core that is much like it is now, only burning He instead of H. Sometimes this is called the "helium main sequence", and it involves a somewhat higher luminosity than the Sun has now (in part because there is still going to be H fusion going on in a shell, it's just not a shell sitting on top of a very compressed and high-gravity degenerate core). Importantly, without high T in the fusion shell, there is no need for the shell to lower its density by puffing out the envelope, so the envelope sinks back down like a cake falling in the oven. The star is no longer a red giant.

VI. The Asymptotic Branch, or Red Giant for the second time
Eventually the core He is also used up, but He fusion continues in a shell, and the shell again sits on a degenerate high-gravity core and makes its fusion rate go nuts. So the envelope puffs out for the second time, but this time it puffs out so far that it never comes back-- it eventually drifts out into space and becomes what is strangely termed a "planetary nebula." Meanwhile the core never gets hot enough to fuse the C and O into anything else, so it just gradually cools as it loses heat (recall that self-gravitating degenerate gas does have its T drop as it loses heat, unlike a self-gravitating ideal gas). This is called a white dwarf, and will be the final resting place of our Sun.