Jeans instability

Jeans instability

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. For stability, the cloud must be in hydrostatic equilibrium, :frac{dp}{dr}=-frac{G ho M_{enc{r^2},where M_{enc} is the enclosed mass, p is the pressure, G is the gravitational constant and 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 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. 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. 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. 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 R, mass M, and with a gaseous sound speed c_s. Imagine that we compress the region slightly. It takes a time,

:t_{sound} = frac{R}{c_s} simeq (5 imes 10^5 mbox{ yr}) left(frac{R}{0.1 mbox{ pc ight) left(frac{c_s}{0.2 mbox{ km s}^{-1 ight)^{-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_{ m ff} = frac{1}{sqrt{G ho simeq (2 mbox{ Myr})left(frac{n}{10^3 mbox{ cm}^{-3 ight)^{-1/2}

where G is the universal gravitational constant, ho is the gas density within the region, and n = ho/mu is the gas number density for mean mass per particle mu = 3.9 imes 10^{-24} g, appropriate for molecular hydrogen with 20% helium by number. 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. However, when the free-fall time is less than the sound-crossing time, gravity wins, and the region undergoes gravitational collapse. The condition for gravitational collapse is therefore:

:t_{ m ff} < t_{sound}

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 ho simeq (0.4 mbox{ pc})left(frac{c_s}{0.2 mbox{ km s}^{-1 ight)left(frac{n}{10^3 mbox{ cm}^{-3 ight)^{-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 M_J is just the mass contained in a sphere of diameter the Jeans length:

:M_J = left(frac{4pi}{3} ight) holeft(frac{R_J}{2} ight)^3 = left(frac{pi}{6} ight) frac{c_s^3}{G^{3/2} ho^{1/2 simeq (2 mbox{ M}_{odot}) left(frac{c_s}{0.2 mbox{ km s}^{-1 ight)^3 left(frac{n}{10^3 mbox{ cm}^{-3 ight)^{-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.

ee also

* Bonnor-Ebert mass

References

* [http://www.jstor.org/pss/90845 "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., "Galaxy Formation" 1998.


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • 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 the cloud to collapse. It is named after the British astronomer Sir James… …   Wikipedia

  • James Hopwood Jeans — Infobox Scientist name = James Hopwood Jeans box width = image width = 240px caption = giving the 1933 Royal Institution Christmas Lectures birth date = birth date|1877|09|11 birth place = Ormskirk, Lancashire, England death date = death date and …   Wikipedia

  • Nebular hypothesis — Star formation Interstellar medium Molecular cloud …   Wikipedia

  • Equipartition theorem — [ Thermal motion of an α helical peptide. The jittery motion is random and complex, and the energy of any particular atom can fluctuate wildly. Nevertheless, the equipartition theorem allows the average kinetic energy of each atom to be computed …   Wikipedia

  • Universe — For other uses, see Universe (disambiguation). Physical cosmology …   Wikipedia

  • Star — For other uses, see Star (disambiguation) …   Wikipedia

  • T Tauri star — Drawing of a T Tauri star with a circumstellar accretion disc …   Wikipedia

  • Structure formation — refers to a fundamental problem in physical cosmology. The universe, as is now known from observations of the cosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately 13.7 Gyr ago. [cite journal |author=D.… …   Wikipedia

  • Dark matter — Not to be confused with antimatter, dark energy, dark fluid or dark flow. For other uses, see Dark Matter (disambiguation) …   Wikipedia

  • Dark nebula — Dark cloud redirects here. For other uses, see Dark Cloud (disambiguation). Star formation …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”