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The Jeans instability is a concept in
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
that describes an instability that leads to the
gravitational collapse Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of gravity. Gravitational collapse is a fundamental mechanism for structure formati ...
of a cloud of gas or dust. It causes the collapse of interstellar gas clouds and subsequent
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
formation. It occurs when the internal gas
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
is not strong enough to prevent the gravitational collapse of a region filled with matter. It is named after
James Jeans Sir James Hopwood Jeans (11 September 1877 – 16 September 1946) was an English physicist, mathematician and an astronomer. He served as a secretary of the Royal Society from 1919 to 1929, and was the president of the Royal Astronomical Soci ...
. For stability, the cloud must be in hydrostatic equilibrium, which in case of a spherical cloud translates to \frac=-\frac, where M_\text(r) is the enclosed mass, ''p'' is the pressure, \rho(r) is the density of the gas (at radius ''r''), ''G'' is the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
, 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; under these circumstances, the gas pressure gradient cannot overcome gravitational force, and the cloud will collapse. This is called the "Jeans Collapse Criterion". The Jeans instability likely determines when
star formation Star formation is the process by which dense regions within molecular clouds in interstellar space—sometimes referred to as "stellar nurseries" or "star-forming regions"—Jeans instability, collapse and form stars. As a branch of astronomy, sta ...
occurs in
molecular cloud A molecular cloud—sometimes called a stellar nursery if star formation is occurring within—is a type of interstellar cloud of which the density and size permit absorption nebulae, the formation of molecules (most commonly molecular hydrogen, ...
s.


History

In 1720,
Edmund Halley Edmond (or Edmund) Halley (; – ) was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720. From an observatory he constructed on Saint Helena in 1676–77, H ...
considered a universe without edges and pondered what would happen if the "system of the world", which exists within the universe, were finite or infinite. In the finite case, stars would gravitate towards the center, and if infinite, all the stars would be nearly in equilibrium and the stars would eventually reach a resting place. Contrary to the writing of Halley,
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
, 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. 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. For an infinite distribution of matter, there are two possible scenarios. An exactly homogeneous distribution has no clear center of mass and no clear way to define a gravitational acceleration direction. For the other case, Jeans extends what Newton wrote about: Jeans demonstrated that small deviations from exact homogeneity lead to instabilities.


Jeans mass

The Jeans mass is named after the
British British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories and Crown Dependencies. * British national identity, the characteristics of British people and culture ...
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
Sir
James Jeans Sir James Hopwood Jeans (11 September 1877 – 16 September 1946) was an English physicist, mathematician and an astronomer. He served as a secretary of the Royal Society from 1919 to 1929, and was the president of the Royal Astronomical Soci ...
, who considered the process of
gravitational collapse Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of gravity. Gravitational collapse is a fundamental mechanism for structure formati ...
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 (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
support to balance the force of
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
. 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 is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
as a function of its
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
and
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
. 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. One begins with a spherical gaseous region of radius ''R'', mass ''M'', and with a gaseous sound speed ''c''S. The gas is compressed slightly and it takes a time t_\text=\frac\approx0.5\text\cdot\frac\cdot\left(\frac\right)^ 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_\text=\frac\approx2\text\cdot\left(\frac\right)^, where ''G'' is the universal gravitational constant, \rho is the gas density within the region, and n = \rho/\mu is the gas number density for mean mass per particle ( is appropriate for molecular hydrogen with 20% helium by number). When 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 Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of gravity. Gravitational collapse is a fundamental mechanism for structure formati ...
. The condition for gravitational collapse is therefore t_\text. The resultant Jeans length \lambda_\text=\frac\approx0.4\text\cdot\frac\cdot\left(\frac\right)^. All scales larger than the Jeans length are unstable to gravitational collapse, whereas smaller scales are stable. The mass contained in a sphere of radius R_\text=\frac\lambda_\text is the Jeans mass M_\text=\frac\rho R_\text^3=\frac\cdot\frac\approx2\text_\odot\cdot\left(\frac\right)^3\left(\frac\right)^ .


Jeans' swindle

It was later pointed out by other astrophysicists, including Binney and Tremaine, 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. The influence of this medium was completely ignored in Jeans' analysis. This flaw has come to be known as the "Jeans' swindle". When using a more careful analysis, 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.


Energy-based derivation

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 The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
) will be ignored. They will be reintroduced in the result. Consider a homogeneous 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 Gravitational energy or gravitational potential energy is the potential energy an object with mass has due to the gravitational potential of its position in a gravitational field. Mathematically, it is the minimum mechanical work that has to be do ...
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 ''pdV''. Using the ideal gas law, according to which p=nT, one arrives at the following expression for the work: dW=nTR^2\,dR. The gravitational potential energy of a sphere with mass ''M'' and radius ''R'' is, apart from constants, given by the following expression: U=\frac. The amount of energy released when the sphere contracts from radius ''R'' to radius R-dR is obtained by differentiation this expression to ''R'', so dU=\frac\,dR. The critical mass is attained as soon as the released gravitational energy is equal to the work done on the gas: \frac=nTR^2. Next, the radius ''R'' must be expressed in terms of the particle density ''n'' and the mass ''M''. This can be done using the relation M=nR^3. A little algebra leads to the following expression for the critical mass: M_\text=\left(\frac\right)^. If during the derivation all constants are taken along, the resulting expression is M_\text=\left(\frac\right)^\left(\frac\right)^, where ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
, ''G'' the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
, and ''m'' the mass of a particle comprising the gas. Assuming the cloud to consist of atomic hydrogen, the prefactor can be calculated. If we take the solar mass as the unit of mass, and use units of ''m''−3 for the particle density, the result is M_\text=3\times10^4\left(\frac\right)^.


Jeans' length

Jeans' length is the critical radius of a cloud (typically a cloud of interstellar molecular gas and 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 Jeans, who concerned himself with the stability of spherical nebulae in the early 1900s. The formula for Jeans' length is: \lambda_\text=\left(\frac\right)^ where ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
, ''T'' is the temperature of the cloud, ''μ'' is the mean molecular weight of the particles, ''G'' is the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
, and ''ρ'' is the cloud's mass density (i.e. the cloud's mass divided by the cloud's volume). A way to conceptualize Jeans' length is in terms of a close approximation, in which the factors 15 and 4π are discarded and in which ''ρ'' is rephrased as M/r^3. The formula for Jeans' length then becomes \lambda_\text\approx\left(\frac\right)^, where ''r'' is the radius of the cloud. It follows immediately that \lambda_\text=r when k_\textT=GM\mu/r; i.e., the cloud's radius is the Jeans' length when thermal energy per particle equals gravitational work per particle. At this critical length, the cloud neither expands nor contracts. 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.


Jeans' length as oscillation wavelength

The Jeans' length is the oscillation wavelength (respectively, Jeans' wavenumber, ''k''J) below which stable oscillations rather than gravitational collapse will occur. \lambda_\text=\frac=c_\text\left(\frac\right)^, where ''G'' is the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
, ''c''S is the sound speed, and ''ρ'' is the enclosed mass density. It is also the distance a
sound wave In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
would travel in the collapse time.


Fragmentation

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. The derivation is shown below through a dimensional analysis: If the adiabatic index \gamma>\frac, the Jeans mass increases with increasing density, while if \gamma<\frac the Jeans mass decreases with increasing density. During gravitational collapse density always increases, thus in the second case the Jeans mass will decrease during collapse, allowing smaller overdense regions to collapse, leading to fragmentation of the giant molecular cloud. For an ideal monatomic gas, the adiabatic index is 5/3. 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). latzmaier G.A. lecture notes, University of California, Santa Cruz, https://websites.pmc.ucsc.edu/~glatz/astr_112/lectures/notes6.pdf/ref> 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). So the second case is the rule rather than an exception in stars. This is the reason why stars usually form in clusters.


See also

* Bonnor–Ebert mass * Langmuir waves (similar waves in a plasma)


References

* * * {{Star formation navbox Concepts in astrophysics Effects of gravity Fluid dynamic instabilities Interstellar media Star formation