A gas cloud that is neither expanding nor contracting at a constant temperature is in hydrostatic equilibrium. In this equilibrium the force exerted by thermal movement of the gas is balanced by the force of gravity. Mathematically,
\begin{equation} 2E_k+E_G=0 \end{equation}where Ek is the thermal energy due to motions of the gas and EG is the gravitational energy. Each can be written as:
\begin{equation} E_k=\frac{3}{2}Nk_bT \end{equation} \begin{equation} E_G=\frac{GM^2}{R} \end{equation}where T is the temperature of the gas, N is the number of atoms, M is the mass of the cloud, R is the radius of the cloud, G is the gravitational constant, and kb is Boltzmann’s constant. Solving for M we get,
\begin{equation} M_{Jeans}=\sqrt{\frac{3Nk_bT}{G}} \end{equation}Examining this equality, we see that if Ek happens to be smaller than EG, the cloud will collapse. This means the Jeans’ Mass (MJeans) is the lower limit below which thermal pressure would prevent gravitational collapse. In more useful quantities of temperature in kelvin, density in number per cubic centimeter, and mass in solar masses:
\begin{equation} M_J \approx 45 M_{Sun}T^{3/2}n^{-1/2} \end{equation}We can see here that high density favors smaller Jeans’ masses while high temperatures require much larger Jeans’ masses. The Jeans’ mass is named after James Jeans (1877–1946), who first considered this scenario.