virial theoremtheorem
Stars in galaxies whirl around the center like children on a carousel. The virial theorem links the speed of stars to the force of gravity: if a galaxy weighed exactly as much as what is visible, the stars would have to fly apart. Fritz Zwicky first noticed the discrepancy and proposed invisible 'dark' mass.
How it works
The theorem is indispensable in astrophysics: from measured velocities of stars or gas particles one calculates the total mass of galaxies and clusters. In plasma physics it connects pressure and gravity inside stars. Even the dynamics of globular clusters and X-ray gas in clusters follow virial equilibrium.
💡 If you compress a gas ball by half, according to the virial theorem its temperature will rise — this is how protostars heat up until they ignite thermonuclear fusion.
2 \langle T \rangle = \langle \sum_k \mathbf{F}_k \cdot \mathbf{r}_k \rangle
⟨T⟩ — time-averaged kinetic energy of the particle system, F_k — net force acting on the k-th particle, r_k — radius vector of the k-th particle, sum over all particles of the system; angle brackets denote averaging over a sufficiently long time interval.
2 \langle T \rangle = - \langle U \rangle
⟨T⟩ — average kinetic energy, ⟨U⟩ — average potential energy of gravitational (or Coulomb) interaction; the relation holds for stable bound systems obeying the inverse-square law.