Jeans instability
Encyclopedia
In physics, 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, which in case of a spherical cloud translates to:
,
where
is the enclosed mass, 
is the pressure, 
is the density of the gas at 
, 
is the gravitational constant
and
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.
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
, mass 
, and with a gaseous sound speed
. Imagine that we compress the region slightly. It takes a time,
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
,
where
is the universal gravitational constant, 
is the gas density within the region, and 
is the gas number density for mean mass per particle 
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:
With a little bit of algebra, one can show that the resultant Jeans length
is approximately:
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
is just the mass contained in a sphere of radius 
(
is half the Jeans length):
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 cloud
s.
Michael K.-H. Kiessling believes that Jeans' results can be vindicated without engaging in a mathematical swindle on the premise that "as long as we obtain a sensible dynamics in some sensible limit, we should not worry too much if some potential ceases to exist in the same limit" concluding that in his treatment "All the ad hoc steps of the 'Jeans swindle' have materialized in a mathematically clean way." Mathematical Vindications of the "Jeans Swindle"
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
is not strong enough to prevent gravitational collapse
Gravitational collapse
Gravitational collapse is the inward fall of a body due to the influence of its own gravity. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body, in the opposite direction to the force of gravity...
of a region filled with matter. For stability, the cloud must be in hydrostatic equilibrium, which in case of a spherical cloud translates to:
,where
is the enclosed mass, is the pressure, is the density of the gas at , is the gravitational constantGravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
and
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 BritishGreat Britain
Great Britain or Britain is an island situated to the northwest of Continental Europe. It is the ninth largest island in the world, and the largest European island, as well as the largest of the British Isles...
physicist
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
Sir James Jeans
James Hopwood Jeans
Sir James Hopwood Jeans OM FRS MA DSc ScD LLD was an English physicist, astronomer and mathematician.-Background:...
, who considered the process of gravitational collapse
Gravitational collapse
Gravitational collapse is the inward fall of a body due to the influence of its own gravity. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body, in the opposite direction to the force of gravity...
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
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
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
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
as a function of its density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
and temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
. 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
Gravitational collapse is the inward fall of a body due to the influence of its own gravity. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body, in the opposite direction to 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
, mass , and with a gaseous sound speedSpeed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
. Imagine that we compress the region slightly. It takes a time,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
Free-fall time
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...
,
where
is the universal gravitational constant, is the gas density within the region, and is the gas number density for mean mass per particle g, appropriate for molecular hydrogen with 20% helium by number. Now, when the sound-crossing time is less than the free-fall timeFree-fall time
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...
, pressure forces win, and the system bounces back to a stable equilibrium. However, when the free-fall time
Free-fall time
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...
is less than the sound-crossing time, gravity wins, and the region undergoes gravitational collapse
Gravitational collapse
Gravitational collapse is the inward fall of a body due to the influence of its own gravity. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body, in the opposite direction to the force of gravity...
. The condition for gravitational collapse is therefore:
With a little bit of algebra, one can show that the resultant Jeans length
is approximately:This length scale is known as the Jeans length
Jeans length
Jeans' length is the critical radius of a cloud where thermal energy, which causes the cloud to expand, is counteracted by gravity, which causes the cloud to collapse...
. All scales larger than the Jeans length are unstable to gravitational collapse
Gravitational collapse
Gravitational collapse is the inward fall of a body due to the influence of its own gravity. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body, in the opposite direction to the force of gravity...
, whereas smaller scales are stable. The Jeans mass
is just the mass contained in a sphere of radius ( is half the Jeans length):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
Star formation
Star formation is the process by which dense parts of molecular clouds collapse into a ball of plasma to form a star. As a branch of astronomy star formation includes the study of the interstellar medium and giant molecular clouds as precursors to the star formation process and the study of young...
occurs in molecular cloud
Molecular cloud
A molecular cloud, sometimes called a stellar nursery if star formation is occurring within, is a type of interstellar cloud whose density and size permits the formation of molecules, most commonly molecular hydrogen ....
s.
Michael K.-H. Kiessling believes that Jeans' results can be vindicated without engaging in a mathematical swindle on the premise that "as long as we obtain a sensible dynamics in some sensible limit, we should not worry too much if some potential ceases to exist in the same limit" concluding that in his treatment "All the ad hoc steps of the 'Jeans swindle' have materialized in a mathematically clean way." Mathematical Vindications of the "Jeans Swindle"