Jeans Instability

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Jeans Instability

:

rac{dp}{dr}=-rac{Garrho M_{(

M_{(is the enclosed mass, p is the pressure, G is the gravitational constant, r is the radius of the region

It occurs once the enclosed matter exceeds the Jeans Mass or the region grows beyond the Jeans Length . If the gravitational instability is governed by waves of type

\Delta=\Delta_0 e^{\Gamma t + i \mathbf{k\cdot r}}

, a value of gamma
:

\Gamma=\left G arrho_0\left(1-rac{\lambda^2_J}{\lambda^2}
ight)
ight^{1/2}

represents an exponentially growing instablity, i.e. collapse.

\lambda_J

is the Jeans Length , G, is the Gravitational Constant , and

arrho_0

is the mass density.
The time scale for collapse is
:

au=\Gamma^{-1}=(4\pi Garrho_0)^{-1/2}

.

The Jeans instability is of central importance in the process of Star Formation in Giant Molecular Clouds

References:
Longair, Malcolm S. "Galaxy Formation" 1998