THE VIRIAL THEOREM — a dart

THE VIRIAL THEOREM twice the motion balances the binding

For a stable, self-gravitating system averaged over time, 2⟨T⟩ + ⟨U⟩ = 0 — twice the kinetic energy exactly cancels the (negative) potential. It means half the energy released by collapse goes into motion and half is radiated; it lets you weigh a galaxy cluster from how fast its members move. Fritz Zwicky used it in 1933 to find the first evidence of dark matter.

THE TECHNIQUE 2⟨T⟩ + ⟨U⟩ = 0

The demo takes a bound system’s U and shows T = −U/2 makes 2T+U vanish: live demo

HISTORY & CREDIT Clausius; Zwicky 1933

“The virial theorem is only for gases.” — it holds for any stable system bound by an inverse-square force: stars, clusters, galaxies. cited

the balance · time-averaged, 2⟨T⟩ = −⟨U⟩.
the weight · measure velocities (T), infer the mass (U) — even the mass you cannot see.
1933 · Fritz Zwicky applied it to the Coma cluster and inferred dark matter (after Clausius).

How fast it moves tells how much it weighs. physics

RECOMMEND FOR I-13 the balance, on the compiler

On i-13, for U=−100, T=50 makes 2T+U = 0 exactly:

$ i13 run gw_virial.i13 # 2T + U = 0 RUN OK · 20 step(s) U = -100 T = 50 -- -U/2 virial = 0 balanced = 1

Recommend as a NULL — a theorem (B39). 2⟨T⟩+⟨U⟩=0 is an exact identity for inverse-square-bound systems; the balance is forall-pinned. NULL — twice the motion balances the binding.