virial

Virial is an adjective used in physics, astronomy, and thermodynamics to denote concepts tied to the virial theorem and related constructs such as virial coefficients, radii, and temperatures. It appears in contexts from mechanical systems to the structure of gaseous matter and the dynamics of astronomical halos.

The virial theorem is a foundational result in mechanics that relates the average kinetic energy T of a stable, bound system to its average potential energy U. For many common force laws, the relation is written as 2⟨T⟩ + ⟨U⟩ = 0, which implies ⟨T⟩ = -⟨U⟩/2. In gravitational and other inverse-square systems, this form is widely used. The theorem provides a bridge between microscopic motions and macroscopic properties such as total energy, size, and mass distributions, and it is applied in stellar and galactic dynamics as well as in molecular and plasma simulations. A system that satisfies the relation within a timescale is described as virialized.

In thermodynamics, the virial expansion expresses deviations of a real gas from ideal behavior. The equation of state is written as PV = NkT [1 + B2(T)(N/V) + B3(T)(N/V)^2 + ...], where the Bn are virial coefficients dependent on temperature and intermolecular potentials. The coefficients quantify the influence of interactions on pressure at finite density, reducing to the ideal gas law when all Bn vanish.

In cosmology and astrophysics, virial concepts describe dark matter halos and large-scale structure. The virial radius R_vir defines a region within which the average density matches a specified multiple Δ_vir(z) of the critical density; the enclosed mass is M_vir. The corresponding virial temperature T_vir characterizes the thermal state of gas in the halo, and systems are described as virialized when the virial theorem applies on the relevant timescales.