Editing Virial theorem (section)

== In astrophysics ==
The virial theorem is frequently applied in astrophysics, especially relating the [[gravitational energy|gravitational potential energy]] of a system to its [[kinetic energy|kinetic]] or [[thermal energy]]. Some common virial relations are {{Citation needed|date=December 2019}}
<math display="block">\frac35 \frac{GM}{R} = \frac32 \frac{k_\mathrm{B} T}{m_\mathrm{p}} = \frac12 v^2 </math>
for a mass {{mvar|M}}, radius {{mvar|R}}, velocity {{mvar|v}}, and temperature {{mvar|T}}. The constants are [[Gravitational constant|Newton's constant]] {{mvar|G}}, the [[Boltzmann constant]] {{math|''k''<sub>B</sub>}}, and proton mass {{math|''m''<sub>p</sub>}}. Note that these relations are only approximate, and often the leading numerical factors (e.g. {{sfrac|3|5}} or {{sfrac|1|2}}) are neglected entirely.

=== Galaxies and cosmology (virial mass and radius) ===
{{Main|Virial mass}}
In [[astronomy]], the mass and size of a galaxy (or general overdensity) is often defined in terms of the "[[virial mass]]" and "[[virial radius]]" respectively. Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models, such as an [[singular isothermal sphere|isothermal sphere]]), it can be hard to define specific, finite measures of their mass and size. The virial theorem, and related concepts, provide an often convenient means by which to quantify these properties.

In galaxy dynamics, the mass of a galaxy is often inferred by measuring the [[rotation velocity]] of its gas and stars, assuming [[circular orbit|circular Keplerian orbits]]. Using the virial theorem, the [[velocity dispersion]] {{mvar|σ}} can be used in a similar way. Taking the kinetic energy (per particle) of the system as {{math|1=''T'' = {{sfrac|1|2}}''v''<sup>2</sup> ~ {{sfrac|3|2}}''σ''<sup>2</sup>}}, and the potential energy (per particle) as {{math|''U'' ~ {{sfrac|3|5}} {{sfrac|''GM''|''R''}}}} we can write

<math display="block"> \frac{GM}{R} \approx \sigma^2. </math>

Here <math>R</math> is the radius at which the velocity dispersion is being measured, and {{mvar|M}} is the mass within that radius. The virial mass and radius are generally defined for the radius at which the velocity dispersion is a maximum, i.e.

<math display="block"> \frac{GM_\text{vir}}{R_\text{vir}} \approx \sigma_\max^2. </math>

As numerous approximations have been made, in addition to the approximate nature of these definitions, order-unity proportionality constants are often omitted (as in the above equations). These relations are thus only accurate in an [[order of magnitude]] sense, or when used self-consistently.

An alternate definition of the virial mass and radius is often used in cosmology where it is used to refer to the radius of a sphere, centered on a [[galaxy]] or a [[galaxy cluster]], within which virial equilibrium holds. Since this radius is difficult to determine observationally, it is often approximated as the radius within which the average density is greater, by a specified factor, than the [[Critical density (cosmology)|critical density]]
<math display="block">\rho_\text{crit}=\frac{3H^2}{8\pi G}</math>
where {{mvar|H}} is the [[Hubble's law|Hubble parameter]] and {{mvar|G}} is the [[gravitational constant]]. A common choice for the factor is 200, which corresponds roughly to the typical over-density in spherical top-hat collapse (see [[Virial mass]]), in which case the virial radius is approximated as
<math display="block">r_\text{vir} \approx r_{200}= r, \qquad \rho = 200 \cdot \rho_\text{crit}.</math>
The virial mass is then defined relative to this radius as
<math display="block">M_\text{vir} \approx M_{200} = \frac43\pi r_{200}^3 \cdot 200 \rho_\text{crit} .</math>

=== Stars ===
The virial theorem is applicable to the cores of stars, by establishing a relation between gravitational potential energy and thermal kinetic energy (i.e. temperature). As stars on the [[main sequence]] convert hydrogen into helium in their cores, the mean molecular weight of the core increases and it must contract to maintain enough pressure to support its own weight. This contraction decreases its potential energy and, the virial theorem states, increases its thermal energy. The core temperature increases even as energy is lost, effectively a negative [[specific heat]].<ref name="BASUCHATTOPADHYAY2010">{{cite book|author1=BAIDYANATH BASU|author2=TANUKA CHATTOPADHYAY|author3=SUDHINDRA NATH BISWAS|title=AN INTRODUCTION TO ASTROPHYSICS|url=https://books.google.com/books?id=WG-HkqCXhKgC&pg=PA365|date=1 January 2010|publisher=PHI Learning Pvt. Ltd.|isbn=978-81-203-4071-8|pages=365–}}</ref> This continues beyond the main sequence, unless the core becomes degenerate since that causes the pressure to become independent of temperature and the virial relation with {{mvar|n}} equals −1 no longer holds.<ref name="Rose1998">{{cite book|author=William K. Rose|title=Advanced Stellar Astrophysics|url=https://books.google.com/books?id=yaX0etDmbXMC&pg=PA242|date=16 April 1998|publisher=Cambridge University Press|isbn=978-0-521-58833-1|pages=242–}}</ref>