The Balancing Act — Hydrostatic Equilibrium

Section 5 of 7

The Virial Theorem

Part 4: The Virial Theorem for Stars

Force balance tells us what support a star needs at each radius, but not how gravity and thermal energy are linked as the star contracts, radiates, and evolves. For that we need an energy argument. The virial theorem is the key tool: it connects the star’s thermal energy to its gravitational binding energy and explains why contraction heats a self-gravitating object instead of cooling it.

Virial theorem

For a bound, self-gravitating system in equilibrium, $2K_{\text{th}} + U_{\text{grav}} = 0$ , so the total thermal energy is fixed at half the magnitude of the (negative) gravitational potential energy. It implies that gravitational contraction heats a star.

Energy balance in self-gravitating systems

Hydrostatic equilibrium is a force-balance statement; the virial theorem is an energy-balance statement relating the star’s total thermal energy to its gravitational potential energy. For a bound star, the gravitational potential energy has a characteristic scale.

The minus sign matters: a bound self-gravitating object has less energy than the same mass dispersed to infinite separation. Contraction (smaller $R$ ) makes $|U_{\text{grav}}|$ larger.

For a star in hydrostatic equilibrium, the virial theorem links this to the thermal energy.

Rearranging, $K_{\text{th}} = -\tfrac{1}{2}U_{\text{grav}}$ , and the total energy $E_{\text{tot}} = K_{\text{th}} + U_{\text{grav}} = \tfrac{1}{2}U_{\text{grav}} = -K_{\text{th}} < 0$ — the star is bound. In quasi-static contraction, roughly half the released gravitational energy increases the thermal energy of the gas, while roughly half must be radiated away.

Numeric answer

Using $U_{\text{grav}} \sim -G M^2/R$ , if a star contracts to one-third its radius at fixed mass, by what factor does $|U_{\text{grav}}|$ change?

Your answer

The negative heat capacity paradox

A star radiates energy from its surface, so its total energy $E_{\text{tot}}$ becomes more negative. Because $E_{\text{tot}} = \tfrac{1}{2}U_{\text{grav}}$ , this means $U_{\text{grav}}$ also becomes more negative — the star contracts into a more tightly bound state. But the virial theorem also says $K_{\text{th}} = -\tfrac{1}{2}U_{\text{grav}}$ , so if $U_{\text{grav}}$ becomes more negative, $K_{\text{th}}$ becomes larger. More thermal energy means higher typical particle speeds and a higher temperature.

So the star gets hotter as it loses energy. This is negative heat capacity, a defining feature of self-gravitating systems.

Negative heat capacity

The property of a self-gravitating system whereby losing total energy raises its temperature: radiation drives contraction, contraction deepens the gravitational well, and the virial theorem converts that into a higher thermal energy. It is how gravitational contraction heats a protostar toward fusion.

Two-panel generated figure with a virial energy ledger bar chart for gravitational, thermal, and total energy on the left and a causal chain from radiating energy to core heating on the right. — Figure 10The virial theorem fixes the bookkeeping: K_th = -(1/2) U_grav and E_tot < 0. That is why a star can lose energy, contract, and still get hotter.ASTR 201 (generated)

Observable

Protostars shrink and heat instead of free-falling

Protostars radiate energy while gradually shrinking, instead of collapsing in free fall.

Model

A bound self-gravitating gas obeys the virial theorem

A bound self-gravitating gas obeys $2K_{\text{th}} + U_{\text{grav}} = 0$ .

Inference

Losing energy makes the core hotter

As radiation removes total energy, the star contracts, $U_{\text{grav}}$ becomes more negative, and the thermal energy increases. The core gets hotter as the star loses energy.

Think first — energy logic

A protostar loses energy by radiation. Before using any equations, reason physically: does it expand or contract? Does the temperature rise or fall?

Predict the direction of both changes (size and temperature), using physical reasoning rather than formulas.

Then open the reasoning below to check.

Quick check

Check your prediction: as a protostar radiates energy away, does it expand or contract, and does its core temperature rise or fall? Reason physically.