The Jeans Mass.
The following discussion is all about the equilibrium of the forces of gravity and the force excerted by the thermal movement of the particles inside a cloud of gas. It's pretty mathematical, but understandable for the patient reader.
Consider a spherical cloud of gas, having a uniform desity, rho. The potential (gravitational) energy can be found by integrating the potential energy of a shell of width dr over the radius of the sphere. The total mass is de density times the volume of the sphere. The potential goes as 1/r.
The potential energy is then given by:
Here R is the outer radius, M(r) the mass as a funtion of the distance from the center of the sphere and G the gravitational constant.
After evaluation of the integral, this leads to:
So, this is the gravitational energy of the spherical cloud. What is the kinetic energy the cloud has, because of the thermal motion of the particles? Of course, this is just the average kinetic energy of one particle times the amount of particles in the cloud. The amount of particles can be estimated by dividing the total mass of the cloud by the average mass of a particle. This average mass can be taken to be some constant times the mass of a proton. (normally gas clouds in space consist of neutral hydrogen, thus one proton plus an electron, but the mass of an electron is very small, especially compared to the mass of the proton.)
This leads to an amount of particles of:
The kinetic energy for an ideal gas (which is a gas with a density much smaller than the so called "quantum concentration", see a good textbook on thermal physics for this. Obviously some gas in space can be seen as an ideal gas.) is:
So, the total kinetic energy is:
Which can be written as:
Now, according to the "virial theorem" (this is discussed quantum-mechanically in Quantum-mechanis by Bransden and Joachain, or mechanically in any good book on classic mechanics, like Jackson J.D. Classical mechanics) in a closed system the negative potential energy in equilibrium equals two times the thermal energy. One gets, after filling in the equations derived above for both ernergies:
The conclusion is that a cloud of density rho at temperature T will expand if it has a mass smaller than this and collapse if it has a mass bigger than this. Writing the density as mass over volume, (where volume goes as r^3) one can also transform this equation as an equation for the Jeans-Radius. I'll leave this to you, for a rainy sunday afternoon.(it's easy, really)
Back to home.
Back to planet-formation.