the name of jeans part 2

The name of  jeans

 

The gravitational potential energy of a sphere with mass M and radius R is, apart from constants, given by the following expression:

${\displaystyle U={\frac {M^{2}}{R}}}$

The amount of energy released when the sphere contracts from radius R to radius R – dR is obtained by differentiation this expression to R, so

${\displaystyle dU={\frac {M^{2}}{R^{2}}}dR}$

The critical mass is attained as soon as the released gravitational energy is equal to the work done on the gas:

${\displaystyle {\frac {M^{2}}{R^{2}}}=nTR^{2}}$

Next, the radius R must be expressed in terms of the particle density n and the mass M. This can be done using the relation

${\displaystyle M=nR^{3}}$

A little algebra leads to the following expression for the critical mass.

${\displaystyle M_{J}=\left({\frac {T^{3}}{n}}\right)^{\frac {1}{2}}}$

If during the derivation all constants are taken along, the resulting expression is

${\displaystyle M_{J}=\left({\frac {375k^{3}}{4\pi m^{4}G^{3}}}\right)^{\frac {1}{2}}\left({\frac {T^{3}}{n}}\right)^{\frac {1}{2}}}$

where k is Boltzmanns constant, G the gravitational constant, and m the mass of a particle comprising the gas. Assuming the cloud to consist of atomic hydrogen, the prefactor can be calculated. If we take the solar mass as the unit of mass, the result is

${\displaystyle M_{J}=3\times 10^{4}\left({\frac {T^{3}}{n}}\right)^{\frac {1}{2}}}$

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