Quarantine Days: The Dumb Kid
The collapse of the nebular cloud into a disk brings the grains into closer proximity, where they can grow by contact and sticking. Over a few hundred thousand years they can grow to centimeter size or a bit larger, but after they reach ten centimeters or so, the problems begin. Probably the worst problem is that these pebbles are still coupled to the much larger mass of gas but not strongly coupled enough to be carried with it. The gaseous disk, being partially supported by pressure, moves at sub-Keplerian speeds, but the pebbles, which are not held up by pressure, need to move faster. Since gas friction continually slows them down and steals angular momentum, they spiral into the star, and not slowly, about 100 years for the worst case at about 1 meter. So anything bigger than a few centimeters gets sucked into the star before it can grow much more by accretion.
So how are you going to get these dust grains to accrete to the ten kilometer plus size where gravity can help, or even to the 10 meter plus size where gas friction is less effective at causing them to spiral in?
Planetary scientists, being clever fellows, have thought of a number of possibilities (all of them probably quite familiar to that master planetary former, Slartbartifast of Megarathea). Most of them involve instabilities (gravitational, hydrodynamic, or magnetohydrodynamic) of the disk. How does one analyze such instabilities? The main tool is linear stability analysis.
The idea is to construct a steady state, add a small perturbation, plug same into the relevant equations (Navier-Stokes, Laplace's eq for gravity, suitable magnetohydrodynamic stuff, equation of continuity) and turn the crank. Throw away quadratic terms and anything else that offer a plausible excuse and guess a solution in the form of a Fourier Series in terms of Exp[i*omega*t - i*k*x]. Next find the dispersion relation for omega and figure out when the solutions blow up. Then you have an instability that might produce rapid mass accumulation.
My problem. After I have read a couple of pages of stability analysis, my eyelids get heavy and I start to yawn. I have become that dumb kid I always pitied. The one who always fell asleep when he tried to learn math or physics.
I read somewhere that people on death row sleep maybe 16 hours per day. I'm approaching that. In fact, I might head for the bed right now. My afternoon nap was interrupted by somebody who had the nerve to ring my doorbell at three pm.