Jeans length

Remarks

Example: if the density is multiplied by $4$ at fixed sound speed, then $\lambda_J$ is halved and $M_J = \frac{\pi}{6} \rho_0 \lambda_J^3$ is also halved, since $M_J \propto \rho_0 \cdot \rho_0^{-3/2} = \rho_0^{-1/2}$. For an isothermal ideal gas $c_s^2 = k_B T / (\mu m_H)$, where $k_B$ is the Boltzmann constant, $T$ the temperature, $\mu$ the mean molecular weight and $m_H$ the hydrogen mass, so $\lambda_J \propto T^{1/2} \rho_0^{-1/2}$. Non-example: the radius of a given cloud is not its Jeans length; the cloud is unstable to collapse when its size exceeds $\lambda_J$. Several conventions differ by factors of order one, for example using the radius instead of the diameter in the Jeans mass.