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stellar physics 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 said, Astrophysics "seeks to ascertain the nature of the hea ...
, the Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation, named after
James Jeans Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist, astronomer and mathematician. Early life Born in Ormskirk, Lancashire, the son of William Tulloch Jeans, a parliamentary correspondent and author. Jeans was ...
. 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 ...
is not strong enough to prevent
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 formatio ...
of a region filled with matter. For stability, the cloud must be in
hydrostatic equilibrium In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planeta ...
, 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 (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in th ...
, 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 cannot overcome gravity, and the cloud will collapse. The Jeans instability likely determines when
star formation Star formation is the process by which dense regions within molecular clouds in The "medium" is present further soon.-->interstellar space
occurs in
molecular cloud A molecular cloud, sometimes called a stellar nursery (if star formation is occurring within), is a type of interstellar cloud, the density and size of which permit absorption nebulae, the formation of molecules (most commonly molecular hydro ...
s.


Jeans mass

The Jeans mass is named after the British
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 caus ...
Sir
James Jeans Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist, astronomer and mathematician. Early life Born in Ormskirk, Lancashire, the son of William Tulloch Jeans, a parliamentary correspondent and author. Jeans was ...
, 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 formatio ...
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 ...
support to balance the force of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...
. 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 property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
as a function of its
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
and
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
. 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 The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
c_s. The gas is compressed slightly and it takes a time :t_\text = \frac \approx 0.5 \mbox \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 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. As such, it plays a fundamental role in setting the timescale for a wide var ...
, :t_ = \frac \approx 2 \mbox \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 The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects ( particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
for mean mass per particle (μ = 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 formatio ...
. The condition for gravitational collapse is therefore: :t_ < t_\text. The resultant Jeans length \lambda_\text is approximately: :\lambda_\text = \frac \approx 0.4 \mbox \cdot \frac \cdot \left(\frac\right)^. 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_\text is just the mass contained in a sphere of radius R_\text (R_\text = \frac\lambda_\text is half the Jeans length): :M_\text = \frac \rho R_\text^3 = \frac \cdot \frac \approx 2 \mbox_\odot \cdot \left(\frac\right)^3 \left(\frac\right)^.


The "Jeans swindle"

It was later pointed out by other astrophysicists including Binney and Tremaine that the original analysis used by Jeans was flawed, for the following reason. In his formal analysis, although Jeans assumed that the collapsing region of the cloud was surrounded by an infinite, static medium, the influence of this static medium was completely ignored in Jeans' analysis. This flaw has come to be known as the "Jeans swindle". 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.


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) will be ignored. They will be reintroduced in the result. Consider a homogenous spherical gas cloud with radius ''R''. In order to compress this sphere to a radius ''R'' – d''R'', work must be done against the gas pressure. During the compression,
gravitational energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
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 ''p'' d''V''. 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'' – d''R'' is obtained by differentiation this expression to ''R'', so :dU = \fracdR 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)^\frac If during the derivation all constants are taken along, the resulting expression is :M_\text = \left( \frac \right)^\frac \left(\frac\right)^\frac where ''k'' is Boltzmann's constant, ''G'' the gravitational constant, 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, the result is :M_\text = 3 \times 10^4 \left(\frac\right)^\frac


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 Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist, astronomer and mathematician. Early life Born in Ormskirk, Lancashire, the son of William Tulloch Jeans, a parliamentary correspondent and author. Jeans was ...
, who concerned himself with the stability of spherical nebulae in the early 1900s. The formula for Jeans length is: : \lambda_\text = \left(\frac\right)^\frac, where k_\text is Boltzmann's constant, T is the temperature of the cloud, \mu is the mean molecular weight of the particles, G is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in th ...
, m_p is the mass of a proton, and \rho is the cloud's mass density (i.e. the cloud's mass divided by the cloud's volume). Perhaps the easiest way to conceptualize Jeans' length is in terms of a close approximation, in which we discard the factors 15 and 4\pi and in which we rephrase \rho as \frac. The formula for Jeans' length then becomes: : \lambda_\text \approx \left(\frac\right)^\frac. where r is the radius of the cloud. It follows immediately that \lambda_\text = r when k_\text T = \frac; 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_\text) below which stable oscillations rather than gravitational collapse will occur. :\lambda_\text = \frac = c_\text\left(\frac\right)^\frac, where G is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in th ...
, c_\text is the
sound speed The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
, and \rho 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: : For adiabatic processes, PV^\gamma = \text \rightarrow V \sim P^. : For an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is am ...
, PV = nRT \Rightarrow P\cdot P^ = P^ \sim T \Rightarrow P \sim T^\frac. :
Polytropic A polytropic process is a thermodynamic process that obeys the relation: p V^ = C where ''p'' is the pressure, ''V'' is volume, ''n'' is the polytropic index, and ''C'' is a constant. The polytropic process equation describes expansion and com ...
equation of state In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or inter ...
, P = K \rho^\gamma \rightarrow T \sim \rho^. : Jeans mass, M_\text \sim T^\frac \rho^ \sim \rho^\rho^. : Thus, M_\text \sim \rho^. 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 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 equilibrium. Clouds of gas with masses greater than the Bonnor–Ebert mass mus ...
*
Langmuir waves Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability ...
(similar waves in a plasma)


References

* * * {{Star formation navbox Astrophysics