THE NAME OF JEANS

The Jeans mass is named after the British physicist Sir James Jeans, who considered the process of gravitational collapse within a gaseous cloud. He was able to show that, under appropriate conditions, a cloud, or part of one, would become unstable and begin to collapse when it lacked sufficient gaseous pressure support to balance the force of gravity. The cloud is stable for sufficiently small mass (at a given temperature and radius), but once this critical mass is exceeded, it will begin a process of runaway contraction until some other force can impede the collapse. He derived a formula for calculating this critical mass as a function of its density and temperature. The greater the mass of the cloud, the smaller its size, and the colder its temperature, the less stable it will be against gravitational collapse.

The approximate value of the Jeans mass may be derived through a simple physical argument. One begins with a spherical gaseous region of radius ${\displaystyle R}$, mass ${\displaystyle M}$, and with a gaseous sound speed ${\displaystyle c_{s}}$. The gas is compressed slightly and it takes a time

${\displaystyle t_{sound}={\frac {R}{c_{s}}}\simeq (5\times 10^{5}{\mbox{ yr}})\left({\frac {R}{0.1{\mbox{ pc}}}}\right)\left({\frac {c_{s}}{0.2{\mbox{ km s}}^{-1}}}\right)^{-1}}$

for sound waves to cross the region and attempt to push back and re-establish the system in pressure balance. At the same time, gravity will attempt to contract the system even further, and will do so on a free-fall time,

${\displaystyle t_{\rm {ff}}={\frac {1}{\sqrt {G\rho }}}\simeq (2{\mbox{ Myr}})\left({\frac {n}{10^{3}{\mbox{ cm}}^{-3}}}\right)^{-1/2}}$

where ${\displaystyle G}$ is the universal gravitational constant, ${\displaystyle \rho }$ is the gas density within the region, and ${\displaystyle n=\rho /\mu }$ is the gas number density for mean mass per particle ${\displaystyle \mu =3.9\times 10^{-24}}$ g, appropriate for molecular hydrogen with 20% helium by number. While the sound-crossing time is less than the free-fall time, pressure forces temporarily overcome gravity, and the system returns to a stable equilibrium. However, when the free-fall time is less than the sound-crossing time, gravity overcomes pressure forces, and the region undergoes gravitational collapse. The condition for gravitational collapse is therefore:

${\displaystyle t_{\rm {ff}}<t_{sound}.}$

The resultant Jeans length ${\displaystyle \lambda _{J}}$ is approximately:

${\displaystyle \lambda _{J}={\frac {c_{s}}{\sqrt {G\rho }}}\simeq (0.4{\mbox{ pc}})\left({\frac {c_{s}}{0.2{\mbox{ km s}}^{-1}}}\right)\left({\frac {n}{10^{3}{\mbox{ cm}}^{-3}}}\right)^{-1/2}.}$

This length scale is known as the Jeans length. All scales larger than the Jeans length are unstable to gravitational collapse, whereas smaller scales are stable. The Jeans mass ${\displaystyle M_{J}}$ is just the mass contained in a sphere of radius ${\displaystyle R_{J}}$ (${\displaystyle R_{J}}$ is half the Jeans length):

${\displaystyle M_{J}=\left({\frac {4\pi }{3}}\right)\rho R_{J}^{3}=\left({\frac {\pi }{6}}\right){\frac {c_{s}^{3}}{G^{3/2}\rho ^{1/2}}}\simeq (2{\mbox{ M}}_{\odot })\left({\frac {c_{s}}{0.2{\mbox{ km s}}^{-1}}}\right)^{3}\left({\frac {n}{10^{3}{\mbox{ cm}}^{-3}}}\right)^{-1/2}.}$

It was later pointed out by other astrophysicists that in fact, the original analysis used by Jeans was flawed, for the following reason. In his formal analysis, Jeans assumed that the collapsing region of the cloud was surrounded by an infinite, static medium. In fact, because all scales greater than the Jeans length are also unstable to collapse, any initially static medium surrounding a collapsing region will also be collapsing. As a result, the growth rate of the gravitational instability relative to the density of the collapsing background is slower than that predicted by Jeans' original analysis. This flaw has come to be known as the "Jeans swindle".

The Jeans instability likely determines when star formation occurs in molecular clouds.

An alternative, arguably even simpler, derivation can be found using energy considerations. In the interstellar cloud, two opposing forces are at work. The gas pressure, caused by the thermal movement of the atoms or molecules comprising the cloud, tries to make the cloud expand, whereas gravitation tries to make the cloud collapse. The Jeans mass is the critical mass where both forces are in equilibrium with each other. In the following derivation numerical constants (such as π) and constants of nature (such as the gravitational constant) will be ignored. They will be reintroduced in the end result.

Consider a homogenous spherical gas cloud with radius R. In order to compress this sphere to a radius R – dR, work must be done against the gas pressure. During the compression, gravitational energy is released. When this energy equals the amount of work to be done on the gas, the critical mass is attained. Let M be the mass of the cloud, T the (absolute) temperature, n the particle density, and p the gas pressure. The work to be done equals p dV. Using the ideal gas law, according to which p=nT, one arrives at the following expression for the work: end part 1

Jeans' length the nameof jeans by science.