Jeans length

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Jeans' Length is the critical radius of a cloud (typically a cloud of interstellar dust) where thermal pressure, which causes the cloud to expand, is balanced by the gravitational pressure that causes the cloud to collapse. It is named after its discoverer, James Hopwood Jeans.

The formula for Jeans Length:

$\lambda_J=\sqrt{\frac{15k_{B}T}{4\pi Gm\rho}}$

where $k B$ is Boltzmann's constant, $T$ is the temperature of the cloud, $r$ is the radius of the cloud, $m$ is the cloud's mass, $G$ is the Gravitational Constant and $ρ$ is the cloud's mass density (i.e. the cloud's mass distributed across the cloud's volume).

Perhaps the easiest way to conceptualize Jeans' Length is in terms of a close approximation, in which we discard the factors $15$ and $4π$ and in which we rephrase $ρ$ as $\frac{m}{r^3}$ . The formula for Jeans' Length then becomes:

$\lambda_J=\sqrt{\frac{k_B Tr^3}{Gm^2}}$

It is then immediately obvious that $λ J = r$ when $k_{B}T=\frac{Gm^2}{r}$ i.e. the cloud's radius is the Jeans' Length when thermal energy equals gravitational work. At this critical length the cloud neither expands nor contracts. It is only when thermal energy is not equal to gravitational work that the cloud either expands and cools or contracts and warms, a process that continues until equilibrium is reached.