Jeans instability - Citizendia

The Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface Gravitational collapse in Astronomy is the inward fall of a massive body under the influence of the force of Gravity. For stability, the cloud must be in hydrostatic equilibrium,

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

where $M e n c$ is the enclosed mass, p is the pressure, G is the gravitational constant and r is the radius. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass 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 for gravity to overcome the gas pressure.

Jeans mass

The Jeans mass is named after the British physicist Sir James Jeans, who considered the process of gravitational collapse within a gaseous cloud. See also Kingdom of Great Britain Great Britain (Breatainn Mhòr Prydain Fawr Breten Veur Graet Breetain is the larger of the two main islands Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Sir James Hopwood Jeans OM FRS MA DSc ScD LLD ( September 11 1877 in Ormskirk, Lancashire &ndash September Gravitational collapse in Astronomy is the inward fall of a massive body under the influence of the force of Gravity. 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. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface Gravitation is a natural Phenomenon by which objects with Mass attract one another Remarkably, 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. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater 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. Gravitational collapse in Astronomy is the inward fall of a massive body under the influence of the force of Gravity.

The approximate value of the Jeans mass may be derived through a simple physical argument. One begins with a spherical gaseous region of radius $R$ , mass $M$ , and with a gaseous sound speed $c s$ . Sound is a vibration that travels through an elastic medium as a Wave. Imagine that we compress the region slightly. It takes a time,

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

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,

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

where $G$ is the universal gravitational constant, $ρ$ is the gas density within the region, and $n = ρ / μ$ is the gas number density for mean mass per particle $\mu = 3.9 \times 10^{-24}$ g, appropriate for molecular hydrogen with 20% helium by number. The free-fall time is the characteristic Time that would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse Now, when the sound-crossing time is less than the free-fall time, pressure forces win, and the system bounces back to a stable equilibrium. The free-fall time is the characteristic Time that would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse However, when the free-fall time is less than the sound-crossing time, gravity wins, and the region undergoes gravitational collapse. The free-fall time is the characteristic Time that would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse Gravitational collapse in Astronomy is the inward fall of a massive body under the influence of the force of Gravity. The condition for gravitational collapse is therefore:

t ff < t s o u n d

With a little bit of algebra, one can show that the resultant Jeans length $R J$ is approximately:

$R_J = \frac{c_s}{\sqrt{G \rho}} \simeq (0.4 \mbox{ pc})\left(\frac{c_s}{0.2 \mbox{ km s}^{-1}}\right)\left(\frac{n}{10^3 \mbox{ cm}^{-3}}\right)^{-1/2}$

This length scale is known as the Jeans length. 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 All scales larger than the Jeans length are unstable to gravitational collapse, whereas smaller scales are stable. Gravitational collapse in Astronomy is the inward fall of a massive body under the influence of the force of Gravity. The Jeans mass $M J$ is just the mass contained in a sphere of diameter the Jeans length:

$M_J = \left(\frac{4\pi}{3}\right) \rho\left(\frac{R_J}{2}\right)^3 = \left(\frac{\pi}{6}\right) \frac{c_s^3}{G^{3/2} \rho^{1/2}} \simeq (2 \mbox{ M}_{\odot}) \left(\frac{c_s}{0.2 \mbox{ km s}^{-1}}\right)^3 \left(\frac{n}{10^3 \mbox{ cm}^{-3}}\right)^{-1/2}$

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 in fact 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". Later analysis by Hunter corrects for this effect.

The Jeans instability likely determines when star formation occurs in molecular clouds. Star Formation is the process by which dense parts of Molecular clouds collapse into a ball of plasma to form a Star. See also Solar nebula A molecular cloud, sometimes called a stellar nursery if Star formation is occurring within is a type of Interstellar

References

J.H. Jeans. The Stability of a Spherical Nebula. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 199, (1902), pp. 1-53
Longair, Malcolm S. In Astrophysics, the Bonnor-Ebert mass is the largest mass that an Isothermal gas sphere embedded in a pressurized medium can have while still remaining in Hydrostatic , "Galaxy Formation" 1998.

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