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Virial theorem
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For power-law potentials, average kinetic and potential energies are related.
2⟨K⟩ = n⟨U⟩For a bound system with potential U ∝ rⁿ, the long-time averages obey 2⟨K⟩ = n⟨U⟩. For gravity (n = −1) ⟨K⟩ = −½⟨U⟩, so E = −⟨K⟩.
A star that loses energy heats up: negative heat capacity.
It applies to time averages in stable systems.
In more detail
The virial theorem relates the average kinetic energy of a system to its average potential energy. For an inverse-square force like gravity, the average kinetic energy is half the magnitude of the potential energy.
Real-world uses
- Astronomers estimate the mass of galaxy clusters from the velocities of their members.
- Models of stars use the virial theorem to estimate internal energy balance.
- Astrophysicists apply the virial theorem to estimate the mass of a star cluster from its measured velocities.
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Frequently asked questions
What does the virial theorem say about stars?
That gravitational contraction releases energy, which heats the star, balancing its internal pressure.
Is it a general result?
It holds for many bound systems in equilibrium, under the assumption of a power-law force.
What does the virial theorem say in words?
For a bound system in a power-law potential, the average kinetic energy is set by the average potential energy.
Related topics in Work, Energy & Power
Sources of energy · Renewable energy · Conservation of energy in collisions · Principle of virtual work · Work–energy with variable mass · Hamiltonian as total energy