Jeans length

Jeans' length is the critical radius of a cloud (typically a cloud of interstellar dust) where thermal energy, which causes the cloud to expand, is counteracted by gravity, which causes the cloud to collapse. It is named after the British astronomer Sir James Jeans, who concerned himself with the stability of spherical nebula in the early 1900s.

The formula for Jeans Length is:

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

where $k_B$ is Boltzmann's constant, $T$ is the temperature of the cloud, $r$ is the radius of the cloud, $\mu$ is the mass per particle in the cloud, $G$ is the Gravitational Constant and $\rho$ is the cloud's mass density (i.e. the cloud's mass divided by the cloud's volume).[1]

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

$\lambda_J\approx\sqrt{\frac{k_B Tr^3}{GM \mu}}$

It is then immediately obvious that $\lambda_J=r$ when $k_{B}T=\frac{GM \mu}{r}$ i.e. the cloud's radius is the Jeans' Length when thermal energy per particle equals gravitational work per particle. 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.

Jeans' Length as oscillation wavelength

The Jeans' Length is the oscillation wavelength below which stable oscillations rather than gravitational collapse will occur.

$\lambda_J=\frac{2\pi}{k_J}=c_s\left(\frac{\pi}{G\rho}\right)^{1/2}$

Where G is the gravitational constant, $c_s$ is the sound speed, and $\rho$ is the enclosed mass density.

It is also the distance a sound wave would travel in the collapse time.

See also

• Langmuir waves (similar waves in a plasma)
• Jeans instability