The Jeans Mass
The study of collapsing gas clouds in space is important because they may collapse to form stars. A gas cloud has a self gravity which tends to cause the cloud to contract.This is opposed by gas pressure inside the cloud which will tend to make the cloud expand. This gas pressure increases with kinetic energy. The net effect is that if the total energy, gravitational plus kinetic of the cloud is negative, the cloud will tend to contract.
Consider a spherical cloud of gas of radiushaving a uniform density,The gravitational potential energy can be found by integrating the potential energy of a shell of thicknessover the surface of the sphere. If the mass of the sphere is and the mass of the shell isthen the gravitational potential energy is given by
If the gas cloud contains particles with average mass bar m then the number of particles isFrom the kinetic theory of gases the average kinetic energy of a monotonic gas iswhereis Boltzmann's constant andis the absolute temperature in Kelvins.
The total kinetic or thermal energy of the atoms of the gas is
According to the virial theorem, in a stable closed system the negative potential energy in equilibrium equals two times the thermal energy.
from above so
The conclusion is that a cloud of densityat temperature T will expand if it has a mass smaller than this and collapse if it has a mass bigger than this. The mass (1) above is called the Jeans mass.