Week 3-- Formation of a solar system
I. Gravitational contraction and the Jeans mass
If you look out in the galaxy, you will see a lot of H gas in one of two forms: either it is all spread out everywhere like the air around us (so is contained by external pressure as used in the ideal gas law, and that's what we call the interstellar medium, or ISM), or it is highly condensed into strongly self-gravitating balls (the stars). And if you look close to those stars, you will see tiny chunks of matter, the asteroids and comets, which are mostly not H, they are made of what astronomers loosely call "metals." These three types of objects are generally systems in force balance ("hydrostatic equilibrium"), and can be interpreted, in the above order, in terms of the three applications of the virial theorem that we have discussed: containment by external pressure (the mechanical part of the ideal gas law, = 3/2 PV), containment by self-gravity (the gravitational version of the virial theorem, = -1/2 ), and containment by internal bonding (the spring-like virial theorem, = ). But how did they get that way?

Since we regard the universe as starting in a very spread-out and uniform state, we must regard the first of the above types of state as the "original condition", and we must then understand how the other two derived from the first. Not surprisingly, this is a story of gravity, because although gravity is very weak, it is also purely attractive, so builds up a dominant influence over the very large distances encountered in astronomy. This pervasive attraction can usher in one of the few examples in astronomy of a violation of force balance: gravitational instability, at the scale of what is called the "Jeans mass."

II. Gravitational Instability

There are more rigorous ways to explain gravitational instability, but the essence of them all is that containment by external gas pressure gives over to gravitational collapse when the scale of the gravitational potential energy in some subsystem exceeds the scale of the thermal kinetic energy in that subsystem. In other words, when a subsystem obeying the ideal-gas-law form of the virial theorem starts to also satisfy the conditions of the self-gravitating form of that theorem, it suffers from an excess of containment, i.e., it is strongly compressed by two separate effects at once, and goes into a state of free-fall. This free fall only proceeds faster as the system contracts and its mass density rho increases, because the timescale for free-fall (as seen in class) is always 1/(G*rho)^(1/2) (you can check that expression has units of time, so what other time could that be?).

In other words, when one asserts the ideal gas law and says = 3/2 PV, one is considering a small scale where self-gravity is unimportant (like the air in the room), and one asserts the gravitational virial theorem, = -1/2 , one is considering large scales where internal gravity dominates external pressure (like in a star). But if a subsystem starts out in the former situation and experiences an increase in density (like mass being added to a giant molecular cloud, GMC, which is where star formation occurs), it can cross over from the first to the second, in the process experiencing both effects and initiating collapse. This collapse, once started, proceeds more and more rapidly (the free-fall timescale gets shorter and shorter as rho rises), which is why this is an instability and is not close to force balance.

The eventual end of the instability is when the system becomes so dense that the free-fall time is shorter than the energy transport timescale, which means the thermal kinetic energy (associated with temperature via the thermodynamic part of the ideal gas law) is trapped inside the system as it contracts. This is called "adiabatic contraction", and it implies that the must be due to the work done on the system, and not by the force balance. But the work done is -, while the force balance only requires -1/2 , so adiabatic contraction always leads to stabilization and return of force balance. At this stage we have a ball of gas that is trapping its heat inside as it contracts, and as it re-establishes force balance, it must continue its slow contraction only at the rate allowed by the heat loss due to its luminosity. We have a "protostar."

III. The Jeans Mass

The way to find the mass scale where this collapse initiates is to equate the required for force balance due to external pressure to the required for force balance due to internal gravity. This is the point where the two team up with nearly equal influence to overcome the internal that would otherwise prevent collapse. So we say - ~ PV where P is the external pressure, V is the volume of the system (and note PV has units of energy, so makes sense to compare to the PE), and PE ~ GM^2 / R. Since V~R^3, and P~rho*k*T/m where m is the mean mass per particle and rho ~ M/R^3, putting all this together (try it yourself) ultimately gives M ~ (kT / G*m*rho^3)^(3/2) . Note that at higher T, a larger M is required to get collapse (which makes sense is it will require more self-gravity when the KE per particle is high), and at larger initial rho, a smaller M is required (which makes sense because higher rho means self gravity is more important). This explains why star formation happens in a GMC, because that is where T is low and rho is high. (For fun, estimate the Jeans mass for the air in the room around you. See why we treat it as an ideal gas?)

Now when we put in the T and rho of a GMC, the Jeans mass comes out very large, thousands to a million solar masses. This is not the mass of a star, this is the mass of the whole GMC when star formation initiates. So the whole GMC (or more typically, the dense "core" of the GMC) begins to collapse, but it does not collapse into one giant star. As the density rises, the Jeans mass falls (right?), which means that smaller and smaller parts of the collapsing core go gravitationally unstable. If by chance these smaller parts have a higher density then their immediate surroundings, their free-fall time is shorter (right?), so these pockets contract faster and hence fragment into smaller Jeans-unstable balls within the larger collapse. It is these smaller fragments that create stars, over a range of masses that is a bit like the crumbs create when you drop a cookie-- lots of tiny bits and a few larger ones. So we have the "initial mass function" (IMF) of the stars-- a range of masses with lots of low-mass stars and a few high-mass ones. The Sun is kind of in the middle, perhaps at the large end of the smaller "crumbs."

IV. The planets

How about those tiny rocks that orbit the stars? Some are surrounded in giant balls of hydrogen (the gas giants), but all are expected to have a rocky/metallic core that is the nucleus of the planet, and all are expected to orbit in the same plane (Pluto is likely to have been perturbed at some point). This is because to form solid objects like that, you need very high density, so that dust can form and the dust can stick together by van der Waals type forces, eventually building up enough mass to be self-gravitating and to collect even more rock, metal, and if cool enough, the copiously available hydrogen gas. How is such a high density environment achieved? It comes from the creation of a disk, which is a consequence of angular momentum-- but exactly how the disk comes about is a subject of modern research.

The presence of angular momentum in the collapsing ball has important consequences. In principle, it could cease the contraction before it even becomes adiabatic, because a system losing heat contracts to a radius that is inversely proportional to the heat lost (consider the virial theorem again), but it cannot reach a radius smaller than what is required by its angular momentum J. To estimate that minimum radius, write J ~ M*v_rot*R, where v_rot is the characteristic rotational speed (and note that v_perp cannot be less than the characteristic v of the gas, which is of course (GM/R)^(1/2) due to the virial theorem). Put that together, and you see that J must be greater than order (G*M^3*R)^(1/2). That means there is an R_min that scales like J^2 / GM^3, where a system with J cannot contract further. Since if it ever reached that R_min, it would have v~v_rot, meaning all the gas is going around in the same way, i.e., in a disk.

But this creates a paradox-- if any significant initial J produces an R_min in which everything is orbiting itself in a giant disk, then there are no stars, just disks! It's not clear if we would be able to see disks like that, but we surely do see lots of stars, so we know that contracting stars must have a way to shed a lot of their initial J (since it is hard to avoid having initial J in those swirling fragments, just by chance). So we need to understand both how contracting stars lose J, but at the same time, keep enough of it to create orbiting disks that ultimately become planets.

An important clue comes from the fact that if you add up the total orbital angular momentum in the planets, you find it dominates the total angular momentum in the solar system (it is about 100 times the angular momentum in the slow rotation of the Sun). But even more importantly, it is about the amount of angular momentum needed to make the Sun a "critical rotator", which is the rotation rate at which the equator of the Sun would literally be in orbit around the Sun. So the planets contain the maximum amount of angular momentum that the Sun could contain, were it in the Sun, and still be able to contract enough to initiate fusion. This is likely not a coincidence, presumably it is saying that the Sun shed angular momentum completely from the system (and other forming stars suggest this happens in the form of polar jets, initiated by uncertain mechanisms likely involving magnetic fields) until it contracted to a form not too much different from what it is now. At that point, instead of ejecting angular momentum completely from the system, the angular momentum went into orbit around the Sun, creating a protoplanetary disk. How that occurred is still a subject of modern research, as is pretty much every other aspect of the uncertain dynamics or our solar system. But one question that has been answered, and will be continued later in the course, is that this seems to be a fairly typical outcome-- protoplanetary disks, and the exoplanets they create, seem fairly ubiquitous throughout the galaxy.