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 rk, 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

McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.

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, 2E 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.
Collins Dictionary of Astronomy © Market House Books Ltd, 2006

virial theorem

[′vir·ē·əl ‚thir·əm]
(statistical mechanics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
(13) a linear term with respect to gravitational potential, [phi], is zero, which is a consequence of the quantum virial theorem. Taking into account that the Hamiltonian is a Hermitian operator, it is possible to show that for n [not equal to] 1:
A role of the classical virial theorem in establishing of the equivalence between averaged over time gravitational mass and energy is discussed in detail in Refs.
More strictly speaking, we show that the expectation value of the mass is equal to E/[c.sup.2] for stationary quantum states due to the quantum virial theorem. Our second result is a breakdown of the equivalence between passive gravitational mass and energy at a microscopic level for stationary quantum states due to the fact that the mass operator does not commute with energy operator, taken in the absence of gravitational field.
Eventually, Eddington [10] came to use the virial theorem when addressing the general theory of star formation.
The point can be made by first examining the ideal gas law and then, a result from the inappropriate application of the virial theorem.
When the virial theorem is applied to a self-gravitating gaseous mass, wherein the kinetic theory of gases has been used to insert temperature dependence [10-12], it is well-established ([13], [14, see Eq.
Ladera, Eduardo Aloma y PilarLeon, The Virial Theorem and its Applications in the Teaching of Modern Physics, Lat.
The virial theorem only considers the potential energy between the bodies of the system.
From the virial theorem, which applies to systems of bodies, we find that the net energy resulting from the gain in potential energy and loss in kinetic energy remains unchanged, meaning that the orbitals of stars in galaxies remain unaffected by time contraction.