# virial theorem

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## Virial theorem

A theorem in classical mechanics which relates the kinetic energy of a system to the virial of Clausius, as defined below. The theorem can be generalized to quantum mechanics and has widespread application. It connects the average kinetic and potential energies for systems in which the potential is a power of the radius. Since the theorem involves integral quantities such as the total kinetic energy, rather than the kinetic energies of the individual particles that may be involved, it gives valuable information on the behavior of complex systems. For example, in statistical mechanics the virial theorem is intimately connected to the equipartition theorem; in astrophysics it may be used to connect the internal temperature, mass, and radius of a star and to discuss stellar stability. The virial theorem makes possible a very easy derivation of the counterintuitive result that as a star radiates energy and contracts it heats up rather than cooling down. *See* Statistical mechanics

The virial theorem states that the time-averaged value of the kinetic energy in a confined system (that is, a system in which the velocities and position vectors of all the particles remain finite) is equal to the virial of Clausius. The virial of Clausius is defined to equal -½ times the time-averaged value of a sum over all the particles in the system. The term in this sum associated with a particular particle is the dot product of the particle's position vector and the force acting on the particle. Alternatively, this term is the product of the distance, *r*, of the particle from the origin of coordinates and the radial component of the force acting on the particle.

In the common case that the forces are derivable from a power-law potential, *V*, proportional to *r*^{k}, where *k* is a constant, the virial is just -*k*/2 times the potential energy. Thus, in this case the virial theorem simply states that the kinetic energy is *k*/2 times the potential energy. For a system connected by Hooke's-law springs, *k* = 2, and the average kinetic and potential energies are equal. For *k* = 1, that is, for gravitational or Coulomb forces, the potential energy is minus twice the kinetic energy. *See* Coulomb's law, Gravitation, Harmonic motion

## virial theorem

The energy of a system in equilibrium is distributed between the kinetic energy*E*

_{K}and the potential energy

*E*

_{P}such that, when averaged over time, 2

*E*

_{K}= –

*E*

_{P}. Application of the virial theorem to a cluster of galaxies or stars allows an evaluation of its total mass from observations of its size and velocity dispersion. The mean temperature at which the system would satisfy the virial theorem is the

*virial temperature*, and the system is said to be in

*virial equilibrium*.