Virial mass

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

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

where ${\textstyle x=\Omega _{m}(z)-1}$, ${\displaystyle \Omega _{m}(z)={\frac {\Omega _{0}(1+z)^{3}}{E(z)^{2}}},}$ ${\displaystyle \Omega _{0}=\Omega _{m}(0)={\frac {8\pi G\rho _{0}}{3H_{0}^{2}}},}$ and ${\displaystyle E(z)={\frac {H(z)}{H_{0}}}}$.[3][4] Since it depends on the density parameter of matter ${\displaystyle \Omega _{m}(z)}$, its value depends on the cosmological model used. In an Einstein–de Sitter model it equals ${\displaystyle 18\pi ^{2}\approx 178}$. This definition is not universal, however, as the exact value of ${\displaystyle \Delta _{c}}$ depends on the cosmology. In an Einstein–de Sitter model, it is assumed that the density parameter is due to matter only, where ${\displaystyle \Omega _{m}=1}$. Compare this to the currently accepted cosmological model for the universe, ΛCDM model, where ${\displaystyle \Omega _{m}=0.3}$ and ${\displaystyle \Omega _{\Lambda }=0.7}$; in this case, ${\displaystyle \Delta _{c}\approx 100}$ (at a redshift of zero; with increased redshift the value approaches the Einstein-de Sitter value and then drops to a value of 56.65 for an empty de Sitter universe). Nevertheless, it is typically assumed that ${\displaystyle \Delta _{c}=200}$ for the purpose of using a common definition, also giving the correct one-digit rounding for a long period 1090 > z > 0.87, and this is denoted as ${\displaystyle r_{200}}$ for the virial radius and ${\displaystyle M_{200}}$ for the virial mass. Using this convention, the mean density is given by

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

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]

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

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

If the convention that ${\displaystyle \Delta _{c}=200}$ is used, then this becomes[1]

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

where ${\displaystyle H(t)}$ is the Hubble parameter as described above, and G is the gravitational constant. This defines the virial mass of an astrophysical system.

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

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

where ${\displaystyle \rho _{c}}$ is the critical density, and the overdensity ${\displaystyle \delta _{c}={\frac {200}{3}}{\frac {c_{200}^{3}}{\ln(1+c_{200})-{\frac {c_{200}}{1+c_{200}}}}}}$ (not to be confused with ${\displaystyle \Delta _{c}}$) and the scale radius ${\displaystyle r_{s}}$ are unique to each halo, and the concentration parameter is given by ${\displaystyle c_{200}={\frac {r_{200}}{r_{s}}}}$.[5] In place of ${\displaystyle \delta _{c}\rho _{c}}$, ${\displaystyle \rho _{s}}$ is often used, where ${\displaystyle \rho _{s}}$ is a parameter unique to each halo. The total mass of the dark matter halo can then be computed by integrating over the volume of the density out to the virial radius ${\displaystyle r_{200}}$:

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

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 ${\displaystyle r_{200}}$:

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

Then the circular velocity for the dark matter halo is given by

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

where ${\displaystyle x=r/r_{200}}$.[5]

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.

• Dark matter halo
• Jeans equations
• Navarro–Frenk–White profile
• Virial theorem