🔗 Formula of the day: Jeans length

🔗 Physics

In stellar physics, the Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation, named after James Jeans. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter. For stability, the cloud must be in hydrostatic equilibrium, which in case of a spherical cloud translates to:

${\displaystyle {\frac {dp}{dr}}=-{\frac {G\rho (r)M_{enc}(r)}{r^{2}}}}$,

where ${\displaystyle M_{enc}(r)}$ is the enclosed mass, ${\displaystyle p}$ is the pressure, ${\displaystyle \rho (r)}$ is the density of the gas (at radius ${\displaystyle r}$), ${\displaystyle G}$ is the gravitational constant, and ${\displaystyle r}$ is the radius. The equilibrium is stable if small perturbations are damped and unstable if they are amplified. In general, the cloud is unstable if it is either very massive at a given temperature or very cool at a given mass; under these circumstances, the gas pressure cannot overcome gravity, and the cloud will collapse.