Editing Equipartition theorem (section)

==Applications==

===Ideal gas law===
{{See also|Ideal gas|Ideal gas law}}

[[Ideal gas]]es provide an important application of the equipartition theorem. As well as providing the formula
<math display="block">
\begin{align}
\langle H^{\mathrm{kin}} \rangle &= \frac{1}{2m} \langle p_{x}^{2} + p_{y}^{2} + p_{z}^{2} \rangle\\
&=  \frac{1}{2} \left(
\left\langle p_{x} \frac{\partial H^{\mathrm{kin}}}{\partial p_{x}} \right\rangle + 
\left\langle p_{y} \frac{\partial H^{\mathrm{kin}}}{\partial p_{y}} \right\rangle +
\left\langle p_{z} \frac{\partial H^{\mathrm{kin}}}{\partial p_{z}} \right\rangle \right) = 
\frac{3}{2} k_\text{B} T
\end{align}
</math>
for the average kinetic energy per particle, the equipartition theorem can be used to derive the [[ideal gas law]] from classical mechanics.<ref name="pathria_1972" /> If '''q''' = (''q<sub>x</sub>'', ''q<sub>y</sub>'', ''q<sub>z</sub>'') and '''p''' = (''p<sub>x</sub>'', ''p<sub>y</sub>'', ''p<sub>z</sub>'') denote the position vector and momentum of a particle in the gas, and
'''F''' is the net force on that particle, then
<math display="block">
\begin{align}
\langle \mathbf{q} \cdot \mathbf{F} \rangle &= \left\langle q_x \frac{dp_x}{dt} \right\rangle + 
\left\langle q_y \frac{dp_y}{dt} \right\rangle + 
\left\langle q_z \frac{dp_z}{dt} \right\rangle\\
&=-\left\langle q_x \frac{\partial H}{\partial q_x} \right\rangle -
\left\langle q_y \frac{\partial H}{\partial q_y} \right\rangle - 
\left\langle q_z \frac{\partial H}{\partial q_z} \right\rangle = -3k_\text{B} T,
\end{align}
</math>
where the first equality is [[Newton's second law]], and the second line uses [[Hamilton's equations]] and the equipartition formula. Summing over a system of ''N'' particles yields
<math display="block">
3Nk_\text{B} T = - \left\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \right\rangle.
</math>
[[Image:Translational motion.gif|frame|right|Figure 5. The kinetic energy of a particular molecule can [[thermal fluctuations|fluctuate wildly]], but the equipartition theorem allows its ''average'' energy to be calculated at any temperature. Equipartition also provides a derivation of the [[ideal gas law]], an equation that relates the [[pressure]], [[volume]] and [[temperature]] of the gas. (In this diagram five of the molecules have been colored red to track their motion; this coloration has no other significance.)]]

By [[Newton's third law]] and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure ''P'' of the gas. Hence
<math display="block">
-\left\langle\sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k}\right\rangle = P \oint_{\text{surface}} \mathbf{q} \cdot d\mathbf{S},
</math>
where {{math|''d'''''S'''}} is the infinitesimal area element along the walls of the container. Since the [[divergence]] of the position vector {{math|'''q'''}} is
<math display="block">
\boldsymbol\nabla \cdot \mathbf{q} =
\frac{\partial q_x}{\partial q_x} + 
\frac{\partial q_y}{\partial q_y} + 
\frac{\partial q_z}{\partial q_z} = 3,
</math>
the [[divergence theorem]] implies that
<math display="block">P \oint_{\mathrm{surface}} \mathbf{q} \cdot \mathbf{dS} = P \int_{\mathrm{volume}} \left( \boldsymbol\nabla \cdot \mathbf{q} \right) \, dV = 3PV,</math>
where {{math|''dV''}} is an infinitesimal volume within the container and {{mvar|V}} is the total volume of the container.

Putting these equalities together yields
<math display="block">3Nk_\text{B} T = -\left\langle \sum_{k=1}^N \mathbf{q}_k \cdot \mathbf{F}_k \right\rangle = 3PV,</math>
which immediately implies the [[ideal gas law]] for ''N'' particles:
<math display="block">PV = N k_\text{B} T = nRT,</math>
where {{math|1=''n'' = ''N''/''N''<sub>A</sub>}} is the number of moles of gas and {{math|1=''R'' = ''N''<sub>A</sub>''k''<sub>B</sub>}} is the [[gas constant]]. Although equipartition provides a simple derivation of the ideal-gas law and the internal energy, the same results can be obtained by an alternative method using the [[partition function (statistical mechanics)|partition function]].<ref name="configint">
Vu-Quoc, L., [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008. this wiki site is down; see [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 this article in the web archive on 2012 April 28].
<!---
L. Vu-Quoc, [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008.
--->
</ref>

===Diatomic gases===
{{See also|Two-body problem|Rigid rotor|Harmonic oscillator}}

A diatomic gas can be modelled as two masses, {{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}}, joined by a [[spring (device)|spring]] of [[Hooke's law|stiffness]] {{math|''a''}}, which is called the ''rigid rotor-harmonic oscillator approximation''.<ref name="mcquarrie_2000c">{{cite book | last = McQuarrie | first = DA | year = 2000 | title = Statistical Mechanics | edition = revised 2nd | publisher = University Science Books | isbn = 978-1-891389-15-3 | pages = [https://archive.org/details/statisticalmecha00mcqu_0/page/91 91–128] | url = https://archive.org/details/statisticalmecha00mcqu_0/page/91 }}</ref> The classical energy of this system is
<math display="block">H = 
\frac{\left| \mathbf{p}_1 \right|^2}{2m_1} + 
\frac{\left| \mathbf{p}_2 \right|^2}{2m_2} + 
\frac{1}{2} a q^2,</math>
where {{math|'''p'''<sub>1</sub>}} and {{math|'''p'''<sub>2</sub>}} are the momenta of the two atoms, and {{mvar|q}} is the deviation of the inter-atomic separation from its equilibrium value. Every degree of freedom in the energy is quadratic and, thus, should contribute {{math|{{frac|1|2}}''k''<sub>B</sub>''T''}} to the total average energy, and {{math|{{frac|1|2}}''k''<sub>B</sub>}} to the heat capacity. Therefore, the heat capacity of a gas of ''N'' diatomic molecules is predicted to be {{math|7''N''·{{frac|1|2}}''k''<sub>B</sub>}}: the momenta {{math|'''p'''<sub>1</sub>}} and {{math|'''p'''<sub>2</sub>}} contribute three degrees of freedom each, and the extension {{mvar|q}} contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be {{math|1={{sfrac|7|2}}''N''<sub>A</sub>''k''<sub>B</sub> = {{sfrac|7|2}}''R''}} and, thus, the predicted molar heat capacity should be roughly 7&nbsp;cal/(mol·K). However, the experimental values for molar heat capacities of diatomic gases are typically about 5&nbsp;cal/(mol·K)<ref name="Wueller_1896" /> and fall to 3&nbsp;cal/(mol·K) at very low temperatures.<ref name="Eucken_1912" /> This disagreement between the equipartition prediction and the experimental value of the molar heat capacity cannot be explained by using a more complex model of the molecule, since adding more degrees of freedom can only ''increase'' the predicted specific heat, not decrease it.<ref name="maxwell_1875" /> This discrepancy was a key piece of evidence showing the need for a [[Quantum mechanics|quantum theory]] of matter.

[[Image:Chandra-crab.jpg|thumb|left|upright=1.25|Figure 6. A combined X-ray and optical image of the [[Crab Nebula]]. At the heart of this nebula there is a rapidly rotating [[neutron star]] which has about one and a half times the mass of the [[Sun]] but is only 25 km across. The equipartition theorem is useful in predicting the properties of such neutron stars.]]

===Extreme relativistic ideal gases===
{{See also|Special relativity|White dwarf|Neutron star}}

Equipartition was used above to derive the classical [[ideal gas law]] from [[Newtonian mechanics]]. However, [[special relativity|relativistic effects]] become dominant in some systems, such as [[white dwarf]]s and [[neutron star]]s,<ref name="huang_1987" /> and the ideal gas equations must be modified. The equipartition theorem provides a convenient way to derive the corresponding laws for an extreme relativistic [[ideal gas]].<ref name="pathria_1972" /> In such cases, the kinetic energy of a [[relativistic particle|single particle]] is given by the formula
<math display="block">H_{\mathrm{kin}} \approx cp = c \sqrt{p_x^2 + p_y^2 + p_z^2}.</math>
Taking the derivative of {{mvar|H}} with respect to the {{math|''p<sub>x</sub>''}} momentum component gives the formula
<math display="block">p_x \frac{\partial H_{\mathrm{kin}}}{\partial p_x}  = c \frac{p_x^2}{\sqrt{p_x^2 + p_y^2 + p_z^2}}</math>
and similarly for the {{math|''p<sub>y</sub>''}} and {{math|''p<sub>z</sub>''}} components. Adding the three components together gives
<math display="block">\begin{align}
\langle H_{\mathrm{kin}} \rangle
&= \left\langle c \frac{p_x^2 + p_y^2 + p_z^2}{\sqrt{p_x^2 + p_y^2 + p_z^2}}  \right\rangle\\
&=
\left\langle p_x \frac{\partial H^{\mathrm{kin}}}{\partial p_x} \right\rangle + 
\left\langle p_y \frac{\partial H^{\mathrm{kin}}}{\partial p_y} \right\rangle + 
\left\langle p_z \frac{\partial H^{\mathrm{kin}}}{\partial p_z} \right\rangle \\
&= 3 k_\text{B} T
\end{align}</math>
where the last equality follows from the equipartition formula. Thus, the average total energy of an extreme relativistic gas is twice that of the non-relativistic case: for {{mvar|N}} particles, it is {{math|3 ''Nk''<sub>B</sub>''T''}}.

===Non-ideal gases===
{{See also|Virial expansion|Virial coefficient}}

In an ideal gas the particles are assumed to interact only through collisions. The equipartition theorem may also be used to derive the energy and pressure of "non-ideal gases" in which the particles also interact with one another through [[conservative force]]s whose potential {{math|''U''(''r'')}} depends only on the distance {{mvar|r}} between the particles.<ref name="pathria_1972" /> This situation can be described by first restricting attention to a single gas particle, and approximating the rest of the gas by a [[spherical symmetry|spherically symmetric]] distribution. It is then customary to introduce a [[radial distribution function]] {{math|''g''(''r'')}} such that the [[probability density function|probability density]] of finding another particle at a distance {{mvar|r}} from the given particle is equal to {{math|4''πr''<sup>2</sup>''ρg''(''r'')}}, where {{math|1=''ρ'' = ''N''/''V''}} is the mean [[density]] of the gas.<ref name="mcquarrie_2000b">{{cite book | last = McQuarrie | first = DA | year = 2000 | title = Statistical Mechanics | edition = revised 2nd | publisher = University Science Books | isbn = 978-1-891389-15-3 | pages = [https://archive.org/details/statisticalmecha00mcqu_0/page/254 254–264] | url = https://archive.org/details/statisticalmecha00mcqu_0/page/254 }}</ref> It follows that the mean potential energy associated to the interaction of the given particle with the rest of the gas is
<math display="block">\langle h_{\mathrm{pot}} \rangle = \int_0^\infty 4\pi r^2 \rho U(r) g(r)\, dr.</math>
The total mean potential energy of the gas is therefore <math> \langle H_\text{pot} \rangle = \tfrac12 N \langle h_{\mathrm{pot}} \rangle </math>, where {{mvar|N}} is the number of particles in the gas, and the factor {{frac|1|2}} is needed because summation over all the particles counts each interaction twice.
Adding kinetic and potential energies, then applying equipartition, yields the ''energy equation''
<math display="block">H = \langle H_{\mathrm{kin}} \rangle + \langle H_{\mathrm{pot}} \rangle = 
\frac{3}{2} Nk_\text{B}T + 2\pi N \rho \int_0^\infty r^2 U(r) g(r) \, dr.</math>
A similar argument,<ref name="pathria_1972" /> can be used to derive the ''pressure equation''
<math display="block">3Nk_\text{B}T = 3PV + 2\pi N \rho \int_0^\infty r^3 U'(r) g(r)\, dr.</math>

===Anharmonic oscillators===
{{See also|Anharmonic oscillator}}

An anharmonic oscillator (in contrast to a simple harmonic oscillator) is one in which the potential energy is not quadratic in the extension {{mvar|q}} (the [[canonical coordinate|generalized position]] which measures the deviation of the system from equilibrium). Such oscillators provide a complementary point of view on the equipartition theorem.<ref name="tolman_1927">{{cite book | last = Tolman | first = RC | author-link = Richard C. Tolman | year = 1927 | title = Statistical Mechanics, with Applications to Physics and Chemistry | url = https://archive.org/details/statisticalmecha00tolm | publisher = Chemical Catalog Company | pages = [https://archive.org/details/statisticalmecha00tolm/page/76 76–77]}}</ref><ref name="terletskii_1971">{{cite book | last = Terletskii | first = YP | year = 1971 | title = Statistical Physics | edition = translated: N. Fröman | publisher = North-Holland | location = Amsterdam | isbn = 0-7204-0221-2 | pages = 83–84 | lccn = 70157006}}</ref> Simple examples are provided by potential energy functions of the form
<math display="block">H_{\mathrm{pot}} = C q^{s},\,</math>
where {{mvar|C}} and {{mvar|s}} are arbitrary [[real number|real constants]]. In these cases, the law of equipartition predicts that
<math display="block">
k_\text{B} T = \left\langle q \frac{\partial H_{\mathrm{pot}}}{\partial q} \right\rangle = 
\langle q \cdot s C q^{s-1} \rangle = \langle s C q^{s} \rangle = s \langle H_{\mathrm{pot}} \rangle.
</math>
Thus, the average potential energy equals {{math|''k''<sub>B</sub>''T''/''s''}}, not {{math|''k''<sub>B</sub>''T''/2}} as for the quadratic harmonic oscillator (where {{math|1=''s'' = 2}}).

More generally, a typical energy function of a one-dimensional system has a [[Taylor expansion]] in the extension {{mvar|q}}:
<math display="block">H_{\mathrm{pot}} = \sum_{n=2}^\infty C_n q^n</math>
for non-negative [[integer]]s {{mvar|n}}. There is no {{math|1=''n'' = 1}} term, because at the [[equilibrium point]], there is no net force and so the first derivative of the energy is zero. The {{math|1=''n'' = 0}} term need not be included, since the energy at the equilibrium position may be set to zero by convention. In this case, the law of equipartition predicts that<ref name="tolman_1927" />
<math display="block">
k_\text{B} T = \left\langle q \frac{\partial H_{\mathrm{pot}}}{\partial q} \right\rangle = 
\sum_{n=2}^{\infty} \langle q \cdot n C_{n} q^{n-1} \rangle = 
\sum_{n=2}^{\infty} n C_{n} \langle q^{n} \rangle.
</math>
In contrast to the other examples cited here, the equipartition formula
<math display="block">
\langle H_{\mathrm{pot}} \rangle = \frac{1}{2} k_\text{B} T - 
\sum_{n=3}^{\infty} \left( \frac{n - 2}{2} \right) C_{n} \langle q^{n} \rangle
</math>
does ''not'' allow the average potential energy to be written in terms of known constants.

===Brownian motion===
[[Image:Wiener process 3d.png|thumb|upright=1.5|Figure 7. Example Brownian motion of a particle in three dimensions.]]
The equipartition theorem can be used to derive the [[Brownian motion]] of a particle from the [[Langevin equation]].<ref name="pathria_1972" /> According to that equation, the motion of a particle of mass {{mvar|m}} with velocity {{math|'''v'''}} is governed by [[Newton's laws of motion|Newton's second law]]
<math display="block">\frac{d\mathbf{v}}{dt} = \frac{1}{m} \mathbf{F} = -\frac{\mathbf{v}}{\tau} + \frac{1}{m} \mathbf{F}_{\mathrm{rnd}},</math>
where {{math|'''F'''<sub>rnd</sub>}} is a random force representing the random collisions of the particle and the surrounding molecules, and where the [[time constant]] τ reflects the [[drag (physics)|drag force]] that opposes the particle's motion through the solution. The drag force is often written {{math|1='''F'''<sub>drag</sub> = −''γ'''''v'''}}; therefore, the time constant {{math|''τ''}} equals {{math|''m''/''γ''}}.

The dot product of this equation with the position vector {{math|'''r'''}}, after averaging, yields the equation
<math display="block">
\left\langle \mathbf{r} \cdot \frac{d\mathbf{v}}{dt} \right\rangle + 
\frac{1}{\tau} \langle \mathbf{r} \cdot \mathbf{v} \rangle = 0
</math>
for Brownian motion (since the random force {{math|'''F'''<sub>rnd</sub>}} is uncorrelated with the position {{math|'''r'''}}). Using the mathematical identities
<math display="block">
\frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{r} \right) = 
\frac{d}{dt} \left( r^{2} \right) = 2 \left( \mathbf{r} \cdot \mathbf{v} \right)
</math>
and
<math display="block">\frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{v} \right) = v^{2} + \mathbf{r} \cdot \frac{d\mathbf{v}}{dt},</math>
the basic equation for Brownian motion can be transformed into
<math display="block">
\frac{d^{2}}{dt^{2}} \langle r^{2} \rangle + \frac{1}{\tau} \frac{d}{dt} \langle r^{2} \rangle = 
2 \langle v^{2} \rangle = \frac{6}{m} k_\text{B} T,
</math>
where the last equality follows from the equipartition theorem for translational kinetic energy:
<math display="block">
\langle H_{\mathrm{kin}} \rangle = \left\langle \frac{p^2}{2m} \right\rangle = \langle \tfrac{1}{2} m v^{2} \rangle = \tfrac{3}{2} k_\text{B} T.
</math>
The above [[differential equation]] for <math>\langle r^2\rangle</math> (with suitable initial conditions) may be solved exactly:
<math display="block">\langle r^{2} \rangle = \frac{6k_\text{B} T \tau^{2}}{m} \left( e^{-t/\tau} - 1 + \frac{t}{\tau} \right).</math>
On small time scales, with {{math|''t'' ≪ ''τ''}}, the particle acts as a freely moving particle: by the [[Taylor series]] of the [[exponential function]], the squared distance grows approximately ''quadratically'':
<math display="block">\langle r^{2} \rangle \approx \frac{3k_\text{B} T}{m} t^2 = \langle v^{2} \rangle t^{2}.</math>
However, on long time scales, with {{math|''t'' ≫ ''τ''}}, the exponential and constant terms are negligible, and the squared distance grows only ''linearly'':
<math display="block">\langle r^{2} \rangle \approx \frac{6k_\text{B} T\tau}{m} t = \frac{6 k_\text{B} T t}{\gamma}.</math>
This describes the [[diffusion]] of the particle over time. An analogous equation for the rotational diffusion of a rigid molecule can be derived in a similar way.

===Stellar physics===
{{See also|Astrophysics|Stellar structure}}

The equipartition theorem and the related [[virial theorem]] have long been used as a tool in [[astrophysics]].<ref>{{cite book | last = Collins | first = GW | year = 1978 | title = The Virial Theorem in Stellar Astrophysics  | url = http://ads.harvard.edu/books/1978vtsa.book/ | publisher = Pachart Press| bibcode = 1978vtsa.book.....C }}</ref> As examples, the virial theorem may be used to estimate stellar temperatures or the [[Chandrasekhar limit]] on the mass of [[white dwarf]] stars.<ref>{{cite book | last = Chandrasekhar | first = S | author-link = Subrahmanyan Chandrasekhar | year = 1939 | title = An Introduction to the Study of Stellar Structure | publisher = University of Chicago Press | location = Chicago | pages = 49–53 | isbn = 0-486-60413-6}}</ref><ref>{{cite book | last = Kourganoff | first = V | year = 1980 | title = Introduction to Advanced Astrophysics | publisher = D. Reidel | location = Dordrecht, Holland | pages = 59–60, 134–140, 181–184}}</ref>

The average temperature of a star can be estimated from the equipartition theorem.<ref>{{cite book | last = Chiu | first = H-Y | year = 1968 | title = Stellar Physics, volume I | publisher = Blaisdell Publishing | location = Waltham, MA | lccn = 67017990}}</ref> Since most stars are spherically symmetric, the total [[Newton's law of universal gravitation|gravitational]] [[Potential energy#Gravitational potential energy|potential energy]] can be estimated by integration
<math display="block">H_{\mathrm{grav}} = -\int_0^R \frac{4\pi r^2 G}{r} M(r)\, \rho(r)\, dr,</math>
where {{math|''M''(''r'')}} is the mass within a radius {{mvar|r}} and {{math|''ρ''(''r'')}} is the stellar density at radius {{mvar|r}}; {{math|''G''}} represents the [[gravitational constant]] and {{math|''R''}} the total radius of the star. Assuming a constant density throughout the star, this integration yields the formula
<math display="block">H_{\mathrm{grav}} = - \frac{3G M^{2}}{5R},</math>
where {{math|''M''}} is the star's total mass. Hence, the average potential energy of a single particle is
<math display="block">\langle H_{\mathrm{grav}} \rangle = \frac{H_{\mathrm{grav}}}{N} = - \frac{3G M^{2}}{5RN},</math>
where {{math|N}} is the number of particles in the star. Since most [[star]]s are composed mainly of [[ion]]ized [[hydrogen]], {{mvar|N}} equals roughly {{math|''M''/''m''<sub>p</sub>}}, where {{math|''m''<sub>p</sub>}} is the mass of one proton. Application of the equipartition theorem gives an estimate of the star's temperature
<math display="block">\left\langle r \frac{\partial H_{\mathrm{grav}}}{\partial r} \right\rangle = \langle -H_{\mathrm{grav}} \rangle = k_\text{B} T = \frac{3G M^2}{5RN}.</math>
Substitution of the mass and radius of the [[Sun]] yields an estimated solar temperature of ''T''&nbsp;= 14 million kelvins, very close to its core temperature of 15 million kelvins. However, the Sun is much more complex than assumed by this model—both its temperature and density vary strongly with radius—and such excellent agreement (≈7% [[approximation error|relative error]]) is partly fortuitous.<ref>{{cite book | last = Noyes | first = RW | year = 1982 | title = The Sun, Our Star | publisher = Harvard University Press | location = Cambridge, MA | isbn = 0-674-85435-7 | url = https://archive.org/details/sunourstar00robe }}</ref>

===Star formation===
The same formulae may be applied to determining the conditions for [[star formation]] in giant [[molecular cloud]]s.<ref>{{cite book |last1=Carroll |first1=Bradley W. |last2=Ostlie |first2=Dale A. |year=1996 |title=An Introduction to Modern Stellar Astrophysics |publisher=Addison–Wesley |location=Reading, MA |isbn=0-201-59880-9 }}</ref> A local fluctuation in the density of such a cloud can lead to a runaway condition in which the cloud collapses inwards under its own gravity. Such a collapse occurs when the equipartition theorem—or, equivalently, the [[virial theorem]]—is no longer valid, i.e., when the gravitational potential energy exceeds twice the kinetic energy
<math display="block">\frac{3G M^{2}}{5R} > 3 N k_\text{B} T.</math>
Assuming a constant density {{math|''ρ''}} for the cloud
<math display="block">M = \frac{4}{3} \pi R^{3} \rho</math>
yields a minimum mass for stellar contraction, the Jeans mass {{math|''M''<sub>J</sub>}}
<math display="block">M_\text{J}^{2} = \left( \frac{5k_\text{B}T}{G m_{p}} \right)^{3} \left( \frac{3}{4\pi \rho} \right).</math>
Substituting the values typically observed in such clouds ({{math|1=''T'' = 150 K}}, {{math|1=''ρ'' = {{val|2e-16|u=g/cm3}}}}) gives an estimated minimum mass of 17 solar masses, which is consistent with observed star formation. This effect is also known as the [[Jeans instability]], after the British physicist [[James Hopwood Jeans]] who published it in 1902.<ref>{{cite journal | last = Jeans | first = JH | author-link = James Hopwood Jeans | year = 1902 | title = The Stability of a Spherical Nebula | journal = [[Philosophical Transactions of the Royal Society A]] | volume = 199 | issue = 312–320 | pages = 1–53 | doi = 10.1098/rsta.1902.0012 | bibcode=1902RSPTA.199....1J| doi-access =  }}</ref>