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Virial Theorem

(average kinetic energy equals half the average negative potential energy)

The Virial Theorem is a theorem relating the kinetic and Potential Energies of a stable system as averaged over time.

2<T> = - Sum(over k=1..N) < F(k) * r(k) >
  • <X> - X averaged over time
  • T - Kinetic Energy
  • F(k) - force on the kth object
  • r(k) - position of the kth object

In a Gravity-bound system, this amounts to the time-averaged kinetic energy equaling the negative of the time-averaged potential energy of the objects.

The theorem helps calculation of the average total kinetic energy in complex systems.

The terms virial radius and virial mass are used in astrophysics in a couple of ways in regards to a system such as a Galaxy or Galaxy Cluster:

  • The radius and mass within that radius at which the Velocity Dispersion is maximal.
  • The radius and mass within that radius within which virial equilibrium holds.


Referenced by:
Carbon Monoxide (CO)
Velocity Dispersion (σ)
Gravitational Collapse