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,
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
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
III. The Jeans Mass
The way to find the mass scale where this collapse initiates is to
equate the
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.