## 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.

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:

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

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

where,
, is the mean molecular weight of the gas and
is the mass of the proton. In terms of the mass density,
,

Combining equations 77, 78 and 79;

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

At the era of decoupling,
and
. Inserting into equation 80 yields:

Thus, the smallest possible mass capable of collapse at the time of decoupling
was
. 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
and
the Jeans' Mass is
. 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.