Editing Equipartition theorem (section)

===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>