Jeans mass

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In
stellar physics Astrophysics is a science that employs the methods and principles of physics in the study of astronomical object In astronomy, an astronomical object or celestial object is a naturally occurring physical entity, association, or structure t ...
, 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 A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, a ...
. It occurs when the internal gas
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

is not strong enough to prevent
gravitational collapse Gravitational collapse is the contraction of an astronomical object In , an astronomical object or celestial object is a naturally occurring , association, or structure that exists in the . In , the terms ''object'' and ''body'' are often ...
of a region filled with matter. 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 , 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.

# 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. ** Britishness, the British identity and common culture * British English, ...

physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at leas ...

Sir
James Jeans Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, a ...
, who considered the process of
gravitational collapse Gravitational collapse is the contraction of an astronomical object In , an astronomical object or celestial object is a naturally occurring , association, or structure that exists in the . In , the terms ''object'' and ''body'' are often ...
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 In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

support to balance the force of
gravity Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ...

. 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 the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
as a function of its
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

and
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

. 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 is the contraction of an astronomical object In , an astronomical object or celestial object is a naturally occurring , association, or structure that exists in the . In , the terms ''object'' and ''body'' are often ...
. 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 elasticity (solid mechanics), elastic medium. At , the speed of sound in air is about , or a kilometre in or a mile in . It depends strongl ...
$c_s$. The gas is compressed slightly and it takes a time :$t_\text = \frac \approx 0.5 \mbox \cdot \frac \cdot \left\left(\frac\right\right)^$ 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 timeThe free-fall time is the characteristic time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparently irreversible process, irreversible succession from the past, through the pr ...
, :$t_ = \frac \approx 2 \mbox \cdot \left\left(\frac\right\right)^$ where $G$ is the universal gravitational constant, $\rho$ is the gas density within the region, and $n = \rho/\mu$ is the gas
number densityThe number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration In chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds ...
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 In , an astronomical object or celestial object is a naturally occurring , association, or structure that exists in the . In , the terms ''object'' and ''body'' are often ...
. The condition for gravitational collapse is therefore: :$t_ < t_\text.$ The resultant Jeans length $\lambda_\text$ is approximately: :$\lambda_\text = \frac \approx 0.4 \mbox \cdot \frac \cdot \left\left(\frac\right\right)^.$ This length scale is known as the Jeans length. All scales larger than the Jeans length are unstable to
gravitational collapse Gravitational collapse is the contraction of an astronomical object In , an astronomical object or celestial object is a naturally occurring , association, or structure that exists in the . In , the terms ''object'' and ''body'' are often ...
, 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): :$M_\text = \frac \rho R_\text^3 = \frac \cdot \frac \approx 2 \mbox_\odot \cdot \left\left(\frac\right\right)^3 \left\left(\frac\right\right)^.$ 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 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". 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, star fo ...

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 hydrogen, ...
s. 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 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. :$M_\text = \left\left(\frac\right\right)^\frac$ If during the derivation all constants are taken along, the resulting expression is :$M_\text = \left\left( \frac \right\right)^\frac \left\left(\frac\right\right)^\frac$ 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 :$M_\text = 3 \times 10^4 \left\left(\frac\right\right)^\frac$

# 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 e ...
, who concerned himself with the stability of spherical nebulae in the early 1900s. The formula for Jeans length is: : $\lambda_\text = \left\left(\frac\right\right)^\frac,$ where $k_\text$ is
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
, $T$ is the temperature of the cloud, $\mu$ is the mean molecular weight of the particles, $G$ is the , $m_p$ 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: : $\lambda_\text \approx \left\left(\frac\right\right)^\frac.$ 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. :$\lambda_\text = \frac = c_\text\left\left(\frac\right\right)^\frac,$ where G is the , $c_\text$ 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 elasticity (solid mechanics), elastic medium. At , the speed of sound in air is about , or a kilometre in or a mile in . It depends strongl ...
, and $\rho$ is the enclosed mass density. It is also the distance a
sound wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

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 process In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qu ...

es, $PV^\gamma = \text \rightarrow V \sim P^.$ : For an
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
, $PV = nRT \Rightarrow P\cdot P^ = P^ \sim T \Rightarrow P \sim T^\frac.$ : Polytropic
equation of state In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
, $P = K \rho^\gamma \rightarrow T \sim \rho^.$ : Jeans mass, $M_\text \sim T^\frac \rho^ \sim \rho^\rho^.$ : Thus, $M_\text \sim \rho^.$ If the
adiabatic index In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume ...
$\gamma > \frac$, 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.