# Jeans Collapse

One of the reasons astronomers study the ISM is to learn how stars form. Broadly speaking, in order for a star to form, a cloud of material must collapse to high enough densities and temperatures that nuclear fusion can begin. Additionally, the cloud must collapse more quickly than it can be dispersed (by stellar winds and ionizing radiation from nearby stars, for example). We know that star formation begins in giant molecular clouds, which consist of cold, diffuse molecular hydrogen, some helium and other gases, and some dust. The lifetime of such clouds is thought to be about 10 million years. We would like a way of quantifying how and when such a cloud will collapse to form a star, and when it wont.

As a simplified case, we consider a spherical cloud of uniform temperature $T$ and density $\rho$ consisting of hydrogen and other elements, such that the average mass of a particle is $\mu m_H$ (where $m_H$ is the mass of a hydrogen atom and $\mu$ is an enrichment factor). Typically $\mu\approx2.33$ for interstellar clouds. There are two main forces acting on the cloud: its own self-gravity, pulling it inward, and the thermal kinetic energy of the particles, which supports the cloud and on average acts to prevent collapse. In order for collapse to occur — i.e. for the cloud to be a bound object — we must have $E_{grav}+E_{therm} \leq 0$, which gives

$\frac{3}{5} \frac{GM^2}{R} + \frac{3}{2} \frac{M}{\mu m_H} kT \leq 0$
$\frac{kT}{2 \mu m_H} \leq \frac{GM}{5R}$

using the assumption that the cloud has uniform density to write $M=\frac{4}{3}\pi \rho R^3$, we find:

$\frac{4}{15} \pi G \rho R^2 \geq \frac{kT}{2\mu m_H}$
$R \geq \left( \frac{15kT}{8\pi G\rho\mu m_H}\right)^{1/2}$

So the cloud will collapse if its radius is larger than the Jeans Radius, $R_J = \left( \frac{15kT}{8\pi G\rho\mu m_H}\right)^{1/2}$. We can also write this criterion in terms of the mass of the cloud; the cloud will collapse if its mass exceeds the Jeans Mass, $M_J=\left(\frac{3}{32\pi\rho}\right)^{1/2}\left(\frac{5kT}{G\mu m_H}\right)^{3/2}$. The constants are not exact, as we have made some glaringly wrong assumptions about the cloud. A more careful derivation from treating small perturbations to the equations of hydrostatic equilibrium (found here) produces a Jeans Length $\lambda_J$ of $\sqrt(2)$ times the Jeans radius I derived above.

There is a glaringly obvious flaw in the derivation of the Jeans mass, which is that it assumes that the gravitational potential gradient is zero everywhere. This implies that $\nabla^2\phi=0$, which can only be true if $\rho=0$ everywhere, which is obviously incorrect. Furthermore, once collapse begins our starting assumptions of uniform temperature no longer holds and the density is probably not strictly uniform throughout the entire cloud. Additionally, this analysis does not consider the effects of turbulence or magnetic fields, both of which are thought to play key roles in star formation. A slightly more accurate treatment accounting for the fact that there is typically a nonzero external pressure on a collapsing cloud tells us that a cloud will collapse if its mass exceeds the Bonnor Ebert mass,

$M_{BE} = \frac{c_{BE}v_T^4}{P_0^{1/2}G^{3/2}}$
where $c_BE \approx 1.18$ is a dimensionless constant, $v_T=\sqrt{\frac{kT}{\mu m_H}}$ is the average thermal speed of the particles in the cloud, $P_0$ is the external pressure, and $G$ is Newton’s gravitational constant. For more about Bonnor Ebert spheres and modified Bonnor Ebert spheres, see the module by Zachary Slepian [link].

Despite these flaws, observations have shown that the Jeans analysis produces approximately correct results. For a typical molecular cloud, we obtain $M_J \approx 70$ solar masses. We can approximate the time that it takes such a cloud to  collapse as the gravitational free-fall time for a pressureless gas, $t_{ff} \approx \left(\frac{3\pi}{32G\rho}\right)^{1/2}$. For densities of $n_H \approx 1000$ particles per cubic cm, typical of giant molecular clouds, $t_{ff} \approx 1.4$ Myr, which is less than the estimated lifetime of such clouds (about 10 Myr), so this is reasonable.