The virial mass of a halo is defined as the mass enclosed by a sphere with a radius \( R_{\rm vir} \) within which
the enclosed mass has a over density of \(\Delta_{\rm c}(z)\) of the critical
density of the universe as the solution of the collapse of a spherical top-hat overdensity at virialized (Peebles 1980).
\( M_{\rm vir}(z) = {4\pi \over 3} \Delta_{\rm c}(z) \rho_{\rm crit}(z) R_{\rm vir}(z)^3 \),
where \( \rho_{\rm crit}(z) \) is the critical density of the universe at redshift z.
The value of \( \Delta_{\rm c}(z) \) depends on redshift and cosmology
through the parameter \(\Omega(z) = \Omega_0(1+z)^3/E(z)^2 \).
The Calculator adopts the fitting formulae proposed by Bryan & Norman (1998) for
the solution of the cases when \( \Omega_R= 0 \) (Eke et al. 1996),
\( \Delta_{\rm c} = 18\pi^2+82x-39x^2 \),
and for the solution of the cases when \( \Omega_\Lambda= 0 \) (Lacey & Cole 1993),
\( \Delta_{\rm c} = 18\pi^2+60x-32x^2 \),
where \( x=\Omega(z)-1 \). These are accurate to 1% in the range \( \Omega(z)=0.1-1\).
The virial velocity of a halo is defined as the circular velocity of the halo at the virial radius,
\( V_{\rm vir} = \sqrt{ G M_{\rm vir} \over R_{\rm vir}} \).
The virial temperature is defined as
\( T_{\rm vir} = {\mu m_p V_{\rm vir}^2 \over 2k} \approx 35.9 \left({V_{\rm vir} \over km\,s^{-1}}\right)^2, \)
where \(\mu\) is the mean molecule weight, which is taken to be 0.55,
\( m_p \) is proton mass,
\(k\) is the Boltzmann constant.