virial theorem
/ VEER-ee-al or VY-ree-al /
Picture a swarm of stars buzzing inside a globular cluster, or atoms jiggling in a star — countless bodies pulling on one another, never settling down, yet collectively in a steady, lasting state. The virial theorem is the remarkable rule that, for such a self-gravitating system in equilibrium, the average energy of motion and the average gravitational energy stay locked in a fixed ratio. It is the balance sheet of a bound system that has settled into a long-lived state.
Stated simply, for a stable, gravitationally bound system the time-averaged kinetic energy K and the gravitational potential energy U obey 2K + U = 0, that is, U = -2K. The motion (kinetic energy) is exactly half as large, in magnitude, as the depth of the gravitational well (potential energy). A surprising corollary is that gravitating systems behave as if they have a 'negative heat capacity': drain energy from a star cluster and its members actually speed up, because shrinking the system deepens U faster than it cools K.
This one relation is among the most powerful weighing tools in astrophysics. Because the speeds of stars or galaxies are measurable (from Doppler shifts) and the size is measurable, the virial theorem turns motion into mass: the faster things move for a given size, the more gravitating mass must be present. Applied to galaxy clusters in the 1930s, it revealed far more mass than the visible stars could account for — the first strong evidence for dark matter, and still a cornerstone of how we 'weigh' clusters today.
Fritz Zwicky measured how fast galaxies dart around inside the Coma cluster, applied the virial theorem, and found the cluster needed hundreds of times more mass than its glowing galaxies provided — the 1933 hint of 'missing mass' we now call dark matter.
Motion plus size yields mass — and the answer exceeded the visible by far.
The clean 2K + U = 0 holds only for a relaxed system in equilibrium with inverse-square gravity; a cluster that is still collapsing, expanding, or being disturbed is not virialized, and assuming it is can badly mis-estimate the mass.