Lecture #7 -- Stellar Interiors, Feb 2016


I. Stellar Interiors
To understand how a star evolves over time, it is necessary to understand what is happening inside the star. We can only see the surface of stars using light, so most of what we know about the interiors must come from theoretical modeling. (Actually, there are two ways to directly probe the interior, one is to look at the neutrinos which believe it or not can escape directly from the nuclear processes in the core and occasionally be observed at Earth, and the other is to look for the evidence of waves that penetrate into the interior and leave their mark at the surface like water waves. Both of these approaches are hard to observe in stars other than the Sun, though asteroseismology is a growing field since we can see low-order modes that affect the stellar brightness as a whole). Amazingly, we now have a great deal of information about stellar interiors, even though we only see the surface. Indeed, so much progress has been made that when the Sun appeared to be emitting way too few neutrinos compared to the theory of its structure, it is now thought to be the neutrinos that are doing something unexpected, not the Sun!
II. Main-sequence timescales and the need for nuclear burning
The maximum time a star can remain bright without undergoing substantial change must be proportional to how much energy is already in there to be released, which is proportional to mass of the star, and inversely proportional to the rate that it emits energy, which is its luminosity. Since luminosity generally increases with star mass, these two effects are in competition, but the luminosity effect wins-- since massive stars are spectacularly luminous, they also must spend less time on the main sequence. Everything they do is in "fast motion". But even the Sun could not have remained bright for as long as the Earth has been here, 4.6 billion years, unless it had an internal energy source other than gravitational contraction. It is easy to show that the luminosity of the Sun would require that it contract noticeably on million-year timescales, and that is not supported by the fossil record of life on Earth, nor by the geological record of underwater sediments. Thus we seek a source of energy locked up in the content of the Sun, energy that can be released gradually by the Sun.
Chemical reactions are not sufficient, it has to be nuclear energy. This was something of a crisis in astronomy in the early 1900s before we knew there was such a thing as nuclear energy! But when Einstein argued that mass could be converted to energy (E=mc^2), it all fell into place. By 1939, the exact process was understood, and you all know what dramatic "spinoff technologies" that led to, which are rarely discussed any more but still exist.
III. Interior Models
To model the interior of a star, you need to understand where the mass is (and mass can't just disappear, it has to go somewhere or be converted to energy in the core), what is pushing on it (and those forces must be in balance for stars that are not expanding or contracting), and how the energy gets from the nuclear sources in the center to the emitted luminosity at the surface. The first relates to the density, the second to the pressure, and the third to the temperature. But there's still the overall radius of the star that you have to figure out, so you need a fourth condition, which is the connection between temperature and pressure called the "equation of state" (usually the ideal gas law: density is proportional to pressure and inversely proportional to temperature.) There's actually one other thing you need to know, which is the composition, and to get that you have to include the nuclear burning, because it will gradually change the composition, but that is something of a detail until later stages of evolution.
IV. The Virial Theorem
As shown in class, the virial theorem is a globally integrated version, expressed in energy units, of the local force balance in a self-gravitating ball of gas. So it is a good way to talk about force balance when you want to talk about the whole star as if it was just one thing, and it tells us that the total kinetic energy equals half the (negative of) the gravitational potential energy. This also means the average kinetic energy of a proton-electron pair equals half the average gravitational potential energy of a proton-electron pair, if you like per-particle thinking. There is no way to overstate the profound importance of the virial theorem, it is basically the first thing to know about stars.

V. Stellar Interiors
Stellar interiors are a story of transport of momentum and energy. The momentum transport story is the local force balance, and the energy transport story is what connects the core temperature to the surface temperature. The virial theorem gives us the global version of the momentum transport story, and either radiative diffusion, or convection (or both, depending on the star) tells us the story of energy transport. Since stars emit light (and therefore heat) from their surface, the energy transport story is an accounting of where this heat comes from and how it gets to the surface. It is easiest to separate radiative diffusion and convection and imagine stars that transport their energy mostly by either one or the other. So we talk about fully convective stars, or fully radiatiave stars, as conceptual anchors, even though most stars are some kind of combination.
Va. Fully Convective Stars (Hayashi Track)

With fully convective stars, the internal temperature gradient is maintained close to the "adiabatic lapse rate," which just means if you imagine some pressure stratification due to the force balance (in essence, from the weight of the overlying layers), you can insert the correct temperature structure by simply imagining taking a parcel of gas and moving it adiabatically all around in that pressure structure. So you keep the pressure of the parcel to equal the pressure of its surroundings (sound crossing times tend to be by far the fastest processes, and that is the timescale for force equilibration if a parcel is moving around), and let its temperature come to whatever adiabatic expansion or contraction would dictate. The result for a simple monatomic gas with no internal degrees of freedom to hide energy is dT/dP = T/P * 2/5, and that constrains the T structure. Also, the luminosity carried by this T structure is pretty much anything you need, without much affect on the structure, because convecting gas parcels are usually extremely efficient at carrying a net heat flux. This presents both a benefit and a problem-- the benefit is, we can neglect any need to iterate the temperature structure to be self-consistent with the luminosity, but the problem is, this also means we don't yet have a constraint on the luminosity-- it must be constrained some other way.
There are two very different types of stars that are mostly fully convective: pre-main-sequence stars (and the earlier protostar phase, during which mass accretion is still going on), and red giants. A protostar is still forming in some sense, but it is also a star in its own right because it is in global force balance, and a pre-main-sequence star is very much a star in force balance, whose mass is not going to change much. It is just gradually contracting as it loses heat, but at any snapshot of the process, it looks like a star like any other, and will obey the virial theorem and have some kind of internal structure controlled by convection. The radius of such a star is set by its history of contraction, and its surface temperature is set by the need to maintain a strong opacity near the surface. This is the key assumption of the "Hayashi track", that the star has a very outside-in flavor to its structure, where conditions at the surface are presenting an important constraint to the whole rest of the star. The surface physics dictates that the opacity must peak there, which requires maintaining the surface temperature of around 3000 K, so such stars are red. This also controls the luminosity of the star, by the Stefan-Boltzmann law (the L ~ T^4 R^2 law, for T the surface T), which is about the only time that the surface of a star controls its luminosity-- in all other cases the surface is slave to the luminosity that is determined in the interior.
The second type of star that is fully convective is a red giant, whose huge envelope looks just like a Hayashi-track pre-main-sequence star, so it also has a surface T of about 3000 K, but its luminosity is set by constraints on the rate of fusion in a very tiny region near the center of the star, called the shell of H fusion (these stars have run out of H in their cores). Since the surface T must maintain the opacity peak at around 3000 K, and the luminosity is determined, here the L ~ T^4 R^2 rule is actually a constraint on R, and since L is huge and surface T is low, this means R must be really huge. The self-regulated shell fusion acts a lot like the self-regulation of core fusion in a main-sequence star, but the light has in some sense already leapfrogged the core, so it escapes more quickly and this is what causes the luminosity to be so high. Even though the envelope is so huge, convection is so efficient at carrying this huge luminosity that it does not provide any constraints on that luminosity-- the bottleneck is radiative diffusion through this very narrow shell. The width of this shell is about 100,000 times smaller than the radius of the star, making red giants truly remarkable objects.
Vb. Fully Radiative Stars (Henyey Track)



VI. Stellar Evolution
Stellar evolution is basically a story with segments of time that follow these rules:
1) The star loses heat to the blackness of space.
2) If this heat is not replaced by fusion, the star contracts, and the rate of gravitational energy release is linked to the luminosity by the virial theorem.
3) If fusion is occurring, whether in the core or in a shell or shells, it will self-regulate to replace the lost heat, mitigating the need for contraction.
4) When fusion fuel runs out, the cycle repeats until one of two things eventually happens: the star can no longer lose heat because it approaches a quantum mechanical ground state like some gigantic gravitationally bound molecule, or the star can no longer maintain a stable force balance and it catastrophically collapses. The first possibility leads to a white dwarf, the latter to a supernova. There is a case where you can get both-- a core collapse supernova can lead to the creation of a neutron star (which will be a pulsar if we can see its radio beam), which is also close to a quantum mechanical ground state that cannot lose more heat. Whether the star reaches a ground state, or loses dynamical stability and collapses, is a battle between quantum mechanics and relativity, because quantum mechanics is responsible for the existence of a ground state, and relativity is responsible for the reason that stars lose dynamical stability when relativistic effects reduce the efficiency of their pressure support. What determines which of these happens to a star is its mass, where lower mass stars are higher density (right?) and hit their quantum mechanical ground states before they lose dynamical stability, whereas higher mass stars are lower density and go relativistic and lose dynamical stability before they hit their quantum mechanical ground state. Stars in the middle, with masses about 10 times solar, do both-- they lose dynamical stability and collapse, but still manage to find a quantum mechanical ground state in the form of a neutron star. Stars of solar-like mass never collapse and neutronize, they reach the ground state of their electrons, and the final stages of that process is called a white dwarf. The highest mass stars completely give up dynamical stability and fall into a black hole, oblivious to the possibility that quantum mechanics could have saved them that fate had their potential to create gravity been a bit weaker (i.e., had they had a lower mass).