[1] It causes the collapse of interstellar gas clouds and subsequent star formation.
It occurs when the internal gas pressure is not strong enough to prevent the gravitational collapse of a region filled with matter.
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 gradient cannot overcome gravitational force, and the cloud will collapse.
The Jeans instability likely determines when star formation occurs in molecular clouds.
[4] Contrary to the writing of Halley, Isaac Newton, in a 1692/3 letter to Richard Bentley, wrote that it's hard to imagine that particles in an infinite space should be able to stand in such a configuration to result in a perfect equilibrium.
[5] [6] James Jeans extended the issue of gravitational stability to include pressure.
In 1902, Jeans wrote, similarly to Halley, that a finite distribution of matter, assuming pressure does not prevent it, will collapse gravitationally towards its center.
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 bigger 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.
for sound waves to cross the region and attempt to push back and re-establish the system in pressure balance.
It was later pointed out by other astrophysicists including Binney and Tremaine[8] that the original analysis used by Jeans was flawed: in his formal analysis, although Jeans assumed that the collapsing region of the cloud was surrounded by an infinite, static medium, the surrounding medium should in reality also be collapsing, since all larger scales are also gravitationally unstable by the same analysis.
Remarkably, when using a more careful analysis taking into account other factors such as the expansion of the Universe fortuitously cancel out the apparent error in Jeans' analysis, and Jeans' equation is correct, even if its derivation might have been dubious.
[9] An alternative, arguably even simpler, derivation can be found using energy considerations.
Consider a homogeneous spherical gas cloud with radius
When this energy equals the amount of work to be done on the gas, the critical mass is attained.
The critical mass is attained as soon as the released gravitational energy is equal to the work done on the gas:
Assuming the cloud to consist of atomic hydrogen, the prefactor can be calculated.
It is named after the British astronomer Sir James Jeans, who concerned himself with the stability of spherical nebulae in the early 1900s.
[10][11] Perhaps the easiest way to conceptualize Jeans' length is in terms of a close approximation, in which we discard the factors
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.
It is also the distance a sound wave would travel in the collapse time.
Jeans instability can also give rise to fragmentation in certain conditions.
To derive the condition for fragmentation an adiabatic process is assumed in an ideal gas and also a polytropic equation of state is taken.
However, in astrophysical objects this value is usually close to 1 (for example, in partially ionized gas at temperatures low compared to the ionization energy).
[13] More generally, the process is not really adiabatic but involves cooling by radiation that is much faster than the contraction, so that the process can be modeled by an adiabatic index as low as 1 (which corresponds to the polytropic index of an isothermal gas).
[citation needed] So the second case is the rule rather than an exception in stars.