can write Here is the radius at which the velocity dispersion is being measured, and is the mass within that radius. The virial mass and radius are generally defined for the radius at which the velocity dispersion is a maximum, i.e. As numerous approximations have been made, in addition to the approximate nature of these definitions, order-unity proportionality constants are often omitted (as in the above equations). These relations are thus only accurate in an order of magnitude sense, or when used self-consistently. An alternate definition of the virial mass and radius is often used in cosmology where it is used to refer to the radius of a sphere, centered on a galaxy or a galaxy cluster, within which virial equilibrium holds. Since this radius is difficult to determine observationally, it is often approximated as the radius within which the average density is greater, by a specified factor, than the critical density where is the Hubble parameter and is the gravitational constant. A common choice for the factor is 200, which corresponds roughly to the typical over-density in spherical top-hat collapse (see Virial mass), in which case the virial radius is approximated as The virial mass is then defined relative to this radius as Stars The virial theorem is applicable to the cores of stars, by establishing a relation between gravitational potential energy and thermal kinetic energy (i.e. temperature). As stars on the main sequence convert hydrogen into helium in their cores, the mean molecular weight of the core increases and it must contract to maintain enough pressure to support its own weight. This contraction decreases its potential energy and, the virial theorem states, increases its thermal energy. The core temperature increases even as energy is lost, effectively a negative specific heat. This continues beyond the main sequence, unless the core becomes degenerate since that causes the pressure to become independent of temperature and the virial relation with equals −1 no longer holds. See also Virial coefficient Virial stress Virial mass Chandrasekhar tensor Chandrasekhar virial equations Derrick's theorem Equipartition theorem Ehrenfest's theorem Pokhozhaev's identity References Further reading External links The Virial Theorem at MathPages Gravitational Contraction and Star Formation, Georgia State University Physics theorems Dynamics (mechanics) Solid mechanics Concepts in physics Equations of astronomy