A young star is contracting under gravity and radiating energy into space. According to the virial theorem, what happens to the star's core temperature as it contracts and loses total energy?
AIt cools down, because energy is being radiated away and the star has less energy available for heat
BIt stays constant, because energy radiated out is exactly compensated by gravitational contraction energy
CIt increases, because the virial theorem for gravity gives 2⟨K⟩ = −⟨V⟩, so as ⟨V⟩ becomes more negative, kinetic energy — and hence temperature — must increase
DIt depends entirely on whether the star is in hydrostatic equilibrium
For gravity (n = −1), the virial theorem gives 2⟨K⟩ = −⟨V⟩. As the star contracts, gravitational potential energy ⟨V⟩ becomes more negative. By the virial theorem, ⟨K⟩ = −⟨V⟩/2 must increase — and since kinetic energy is proportional to temperature, the star heats up even as it loses total energy. Half the liberated gravitational energy is radiated away; the other half heats the gas. This 'negative heat capacity' is counterintuitive but follows directly from the virial theorem.
Question 2 Multiple Choice
For an ideal gas (no interparticle interactions), the virial theorem reduces to which standard result?
AThe van der Waals equation of state, which corrects for molecular interactions
BThe Carnot efficiency limit for heat engines
CThe equipartition theorem: each translational degree of freedom carries (1/2)k_BT of kinetic energy
For an ideal gas, there are no pairwise interactions — all virial coefficients B_k = 0. The only forces are from the container walls. Applying the virial theorem to these wall interactions recovers the equipartition result: total kinetic energy ⟨K⟩ = (3/2)Nk_BT for a monatomic gas in three dimensions. The virial expansion P = nk_BT(1 + B₂n + …) then provides correction terms for real gases, where each virial coefficient encodes k-body interaction contributions.
Question 3 True / False
For a gravitationally bound system in equilibrium, the total energy E equals the negative of the time-averaged kinetic energy: E = −⟨K⟩.
TTrue
FFalse
Answer: True
From the virial theorem with n = −1: 2⟨K⟩ = −⟨V⟩, so ⟨V⟩ = −2⟨K⟩. Total energy E = ⟨K⟩ + ⟨V⟩ = ⟨K⟩ − 2⟨K⟩ = −⟨K⟩. Since ⟨K⟩ > 0, E < 0 — gravitationally bound systems always have negative total energy. This also implies negative heat capacity: adding energy (making E less negative) decreases ⟨K⟩ and therefore cools the system, while removing energy increases ⟨K⟩ and heats it.
Question 4 True / False
When a cloud of gas collapses under gravity, it cools because it radiates most of the released gravitational potential energy into space.
TTrue
FFalse
Answer: False
By the virial theorem (n = −1): 2⟨K⟩ = −⟨V⟩. As the cloud collapses, ⟨V⟩ becomes more negative, so ⟨K⟩ — and thus temperature — increases. Roughly half the released gravitational potential energy is radiated away, but the other half goes into heating the gas. This is why collapsing gas clouds heat up rather than cooling, and why proto-stellar nebulae eventually ignite as stars. Losing energy → getting hotter is the signature of gravitational negative heat capacity.
Question 5 Short Answer
Explain in your own words why gravitationally bound systems have 'negative heat capacity' — why removing energy from such a system causes it to heat up.
Think about your answer, then reveal below.
Model answer: The virial theorem for gravity requires 2⟨K⟩ = −⟨V⟩ in equilibrium. Total energy is E = ⟨K⟩ + ⟨V⟩ = ⟨K⟩ − 2⟨K⟩ = −⟨K⟩. If the system radiates energy (E decreases, becomes more negative), then −⟨K⟩ must decrease, meaning ⟨K⟩ must increase. Since temperature is proportional to kinetic energy, the system heats up when it loses energy. This is the opposite of ordinary systems like an ideal gas, where adding energy increases temperature. The virial theorem's constraint — that gravity forces kinetic and potential energies to maintain a fixed ratio — is what produces this counterintuitive behavior.
The result has profound astrophysical consequences: stars heat up as they radiate and contract, proto-stellar clouds ignite as they collapse, and gravitationally bound clusters cannot 'cool' in the ordinary sense. It also explains the stability of stars: as they radiate, they contract and heat their cores, eventually reaching pressures sufficient to sustain fusion — a self-regulating process driven by the virial theorem.