In astrophysics, the virial mass is the mass of a gravitationally bound astrophysical system, assuming the virial theorem applies. In the context of galaxy formation and dark matter halos, the virial mass is defined as the mass enclosed within the virial radius ${\displaystyle r_{\rm {vir))}$ of a gravitationally bound system, a radius within which the system obeys the virial theorem. The virial radius is determined using a "top-hat" model. A spherical "top hat" density perturbation destined to become a galaxy begins to expand, but the expansion is halted and reversed due to the mass collapsing under gravity until the sphere reaches equilibrium – it is said to be virialized. Within this radius, the sphere obeys the virial theorem which says that the average kinetic energy is equal to minus one half times the average potential energy, ${\displaystyle \langle T\rangle =-{\frac {1}{2))\langle U\rangle }$, and this radius defines the virial radius.

Virial radius

The virial radius of a gravitationally bound astrophysical system is the radius within which the virial theorem applies. It is defined as the radius at which the density is equal to the critical density ${\displaystyle \rho _{c))$ of the universe at the redshift of the system, multiplied by an overdensity constant ${\displaystyle \Delta _{c))$:

$${\displaystyle \rho (<r_{\rm {vir)))=\Delta _{c}\rho _{c}(t)=\Delta _{c}{\frac {3H^{2}(t)}{8\pi G)),}$$

where ${\displaystyle \rho (<r_{\rm {vir)))}$ is the halo's mean density within that radius, ${\displaystyle \Delta _{c))$ is a parameter, ${\displaystyle \rho _{c}(t)={\frac {3H^{2}(t)}{8\pi G))}$ is the critical density of the Universe, ${\displaystyle H^{2}(t)=H_{0}^{2}[\Omega _{r}(1+z)^{4}+\Omega _{m}(1+z)^{3}+(1-\Omega _{tot})(1+z)^{2}+\Omega _{\Lambda }]}$ is the Hubble parameter, and ${\displaystyle r_{\rm {vir))}$ is the virial radius.^[1][2] The time dependence of the Hubble parameter indicates that the redshift of the system is important, as the Hubble parameter changes with time: today's Hubble parameter, referred to as the Hubble constant ${\displaystyle H_{0))$, is not the same as the Hubble parameter at an earlier time in the Universe's history, or in other words, at a different redshift. The overdensity ${\displaystyle \Delta _{c))$ is approximated by

$${\displaystyle \Delta _{c}\approx 18\pi ^{2}+82x-39x^{2},}$$

${\textstyle x=\Omega _{m}(z)-1}$

${\displaystyle \Omega _{m}(z)={\frac {\Omega _{0}(1+z)^{3)){E(z)^{2))},}$

${\displaystyle \Omega _{0}=\Omega _{m}(0)={\frac {8\pi G\rho _{0)){3H_{0}^{2))},}$

${\displaystyle E(z)={\frac {H(z)}{H_{0))))$

${\displaystyle \Omega _{m}(z)}$

${\displaystyle 18\pi ^{2}\approx 178}$

${\displaystyle \Delta _{c))$

${\displaystyle \Omega _{m}=1}$

${\displaystyle \Omega _{m}=0.3}$

${\displaystyle \Omega _{\Lambda }=0.7}$

${\displaystyle \Delta _{c}\approx 100}$

${\displaystyle \Delta _{c}=200}$

${\displaystyle r_{200))$

${\displaystyle M_{200))$

$${\displaystyle \rho (<r_{200})=200\rho _{c}(t)=200{\frac {3H^{2}(t)}{8\pi G)).}$$

Other conventions for the overdensity constant include ${\displaystyle \Delta _{c}=500}$, or ${\displaystyle \Delta _{c}=1000}$, depending on the type of analysis being done, in which case the virial radius and virial mass is signified by the relevant subscript.^[2]

Defining the virial mass

Given the virial radius and the overdensity convention, the virial mass ${\displaystyle M_{\rm {vir))}$ can be found through the relation

$${\displaystyle M_{\rm {vir))={\frac {4}{3))\pi r_{\rm {vir))^{3}\rho (<r_{\rm {vir)))={\frac {4}{3))\pi r_{\rm {vir))^{3}\Delta _{c}\rho _{c}.}$$

${\displaystyle \Delta _{c}=200}$

$${\displaystyle M_{200}={\frac {4}{3))\pi r_{200}^{3}200\rho _{c}={\frac {100r_{200}^{3}H^{2}(t)}{G)),}$$

${\displaystyle H(t)}$

Applications to dark matter halos

Given ${\displaystyle M_{200))$ and ${\displaystyle r_{200))$, properties of dark matter halos can be defined, including circular velocity, the density profile, and total mass. ${\displaystyle M_{200))$ and ${\displaystyle r_{200))$ are directly related to the Navarro–Frenk–White (NFW) profile, a density profile that describes dark matter halos modeled with the cold dark matter paradigm. The NFW profile is given by

$${\displaystyle \rho (r)={\frac {\delta _{c}\rho _{c)){r/r_{s}(1+r/r_{s})^{2))},}$$

${\displaystyle \rho _{c))$

${\displaystyle \delta _{c}={\frac {200}{3)){\frac {c_{200}^{3)){\ln(1+c_{200})-{\frac {c_{200)){1+c_{200))))))$

${\displaystyle \Delta _{c))$

${\displaystyle r_{s))$

${\displaystyle c_{200}={\frac {r_{200)){r_{s))))$

${\displaystyle \delta _{c}\rho _{c))$

${\displaystyle \rho _{s))$

${\displaystyle \rho _{s))$

${\displaystyle r_{200))$

${\displaystyle M=\int \limits _{0}^{r_{200))4\pi r^{2}\rho (r)dr=4\pi \rho _{s}r_{s}^{3}[\ln({\frac {r_{200}+r_{s)){r_{s))})-{\frac {r_{200)){r_{200}+r_{s))}]=4\pi \rho _{s}r_{s}^{3}[\ln(1+c_{200})-{\frac {c_{200)){1+c_{200))}].}$

From the definition of the circular velocity, ${\displaystyle V_{c}(r)={\sqrt {\frac {GM(r)}{r))},}$ we can find the circular velocity at the virial radius ${\displaystyle r_{200))$:

$${\displaystyle V_{200}={\sqrt {\frac {GM_{200)){r_{200)))).}$$

$${\displaystyle V_{c}^{2}(r)=V_{200}^{2}{\frac {1}{x)){\frac {\ln(1+cx)-(cx)/(1+cx)}{\ln(1+c)-c/(1+c))),}$$

${\displaystyle x=r/r_{200))$

Although the NFW profile is commonly used, other profiles like the Einasto profile and profiles that take into account the adiabatic contraction of the dark matter due to the baryonic content are also used to characterize dark matter halos.

To compute the total mass of the system, including stars, gas, and dark matter, the Jeans equations need to be used with density profiles for each component.