The Jeans Mass and Gravitational Stability

Primordial density fluctuations are constrained to have a minimum mass because the conditions at decoupling are such that thermal pressure of matter can balance gravitational collapse. That is the equilibrium of the force of gravity (GM^2/R) and the force exerted by the thermal movement, or kinetic energy (3/2NkT) of the particles inside a cloud of gas.

In term of the total energy we have the following three cases that define dynamical stability.


\begin{displaymath}2 E_k + E_G = 0 \quad\quad {\rm static\,\, equilibrium}\end{displaymath}


\begin{displaymath}E_k < -{1\over 2} E_G \quad\quad {\rm collapse}\end{displaymath}


\begin{displaymath}E_k > -{1\over 2}E_G \quad\quad {\rm expansion}\end{displaymath}

In the case of galaxy clusters the kinetic energy refers to the motion of individual galaxies. In the case of a clump of gas, it refers to the motion of the individual gas particles, the atoms. Thus, for a parcel of gas, assumed to be ideal,  we can write the condition for collapse as:

\begin{displaymath}E_k = {3\over 2} NkT < -{1\over 2} E_G\end{displaymath}


\begin{displaymath}\Rightarrow 3 NkT < {GM^2\over \langle R\rangle }\quad\quad {\rm Jeans'
Condition}\eqno(77)\end{displaymath}

From the Jeans' condition we see that there is a minimum mass below which the thermal pressure prevents gravitational collapse.

\begin{displaymath}M_{min} = M_J = \left({3NkT\langle R\rangle\over G}\right)^{1/2}\end{displaymath}

The number of atoms corresponding to the Jeans' mass is given by:

\begin{displaymath}N = {M_J\over m_p\mu}\eqno(78)\end{displaymath}

where, $\mu$ , is the mean molecular weight of the gas and $m_p$ is the mass of the proton. In terms of the mass density, $\rho$ ,

\begin{displaymath}M_J = {4\pi\over 3}\rho \langle R\rangle^3\eqno(79)\end{displaymath}

Combining equations 77, 78 and 79;


\begin{displaymath}M_J = \rho^{-1/2} T^{3/2}\left({5k\over G\mu m_p}\right)^{3/2}\left(
{3\over 4\pi}\right)^{1/2}.\end{displaymath}

As expected, high density favors collapse while high temperature favors larger Jeans' mass. In units favored by astronomers the above condition becomes:

\begin{displaymath}M_J \approx 45 M_\odot T_k^{3/2} n_{cm^{-3}}^{-1/2}\eqno(80).\end{displaymath}

At the era of decoupling, $T_k \approx 3000$ and $n \approx 6\times 10^3$ . Inserting into equation 80 yields:

\begin{displaymath}M_J(t_D) \approx 10^5\, M_\odot .\end{displaymath}

Thus, the smallest possible mass capable of collapse at the time of decoupling was $10^5$ $M_\odot$ . That is about the mass of a present day globular cluster. Nothing smaller could have formed. By contrast in the interstellar medium of our Galaxy where $T\approx 50 K$ and $n\approx 10^3$ the Jeans' Mass is $\approx 500\, M_\odot$ . Today, far smaller parcels of gas can collapse. That is why molecular clouds can collapse and form individual stars. The Jeans' mass at recombination is an important constraint on models attempting to explain how galaxies formed.