Virial mass

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 r v i r {displaystyle r_{ m {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 virial 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, ⟨ T ⟩ = − 1 2 ⟨ V ⟩ {displaystyle langle T angle =-{frac {1}{2}}langle V angle } , and this radius defines the virial radius. 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 r v i r {displaystyle r_{ m {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 virial 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, ⟨ T ⟩ = − 1 2 ⟨ V ⟩ {displaystyle langle T angle =-{frac {1}{2}}langle V angle } , and this radius defines the 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 ρ c {displaystyle ho _{c}} of the Universe at the redshift of the system, multiplied by an overdensity constant Δ c {displaystyle Delta _{c}} : where ρ ( < r v i r ) {displaystyle ho (<r_{ m {vir}})} is the halo's mean density within that radius, Δ c {displaystyle Delta _{c}} is a constant, ρ c ( t ) = 3 H 2 ( t ) 8 π G {displaystyle ho _{c}(t)={frac {3H^{2}(t)}{8pi G}}} is the critical density of the Universe, H 2 ( t ) = H 0 2 [ Ω r ( 1 + z ) 4 + Ω m ( 1 + z ) 3 + ( 1 − Ω t o t ) ( 1 + z ) 2 + Ω Λ ] {displaystyle H^{2}(t)=H_{0}^{2}} is the Hubble parameter, and r v i r {displaystyle r_{ m {vir}}} is the virial radius. 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 H 0 {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 Δ c {displaystyle Delta _{c}} is given by Other conventions for the overdensity constant include Δ c = 500 {displaystyle Delta _{c}=500} , or Δ c = 1000 {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. Given the virial radius and the overdensity convention, the virial mass M v i r {displaystyle M_{ m {vir}}} can be found through the relation Given M 200 {displaystyle M_{200}} and r 200 {displaystyle r_{200}} , properties of dark matter halos can be defined, including circular velocity, the density profile, and total mass. M 200 {displaystyle M_{200}} and r 200 {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 M = ∫ 0 r 200 4 π r 2 ρ ( r ) d r = 4 π ρ s r s 3 [ ln ⁡ ( r 200 + r s r s ) − r 200 r 200 + r s ] = 4 π ρ s r s 3 [ ln ⁡ ( 1 + c 200 ) − c 200 1 + c 200 ] . {displaystyle M=int limits _{0}^{r_{200}}4pi r^{2} ho (r)dr=4pi ho _{s}r_{s}^{3}=4pi ho _{s}r_{s}^{3}.}