# Virial Theorem

StephenPrivitera
A protostar will begin gravitational collapse only if the total gravitational potential energy exceeds twice the thermal energy. In other words, a gas has to be sufficiently cool and sufficiently dense to collapse. Also, as the protostar collapses about half of the gravitational PE is converted into heat, and about half is radiated into space. So suppose we have a protostar cloud with a thermal energy of T. If it is to collapse its gravitational PE should be >2T, say 3T. After some time, the protostar loses 2T in gravitational energy so that its gravitational potential energy is T. But half of the lost energy is now in heat, so the total thermal energy is now 2T. Will this protostar stop collapsing? How much must it collape before nuclear fusion begins?

Staff Emeritus
Gold Member
How much must it collape before nuclear fusion begins?
The easy answer is "until the core is hot and dense enough". Are you looking for something specific, e.g. how hot? how dense? or perhaps "it first begins to burn its deuterium, and destroy what little lithium it has, when [answer goes here]"

Jeebus
If the motions are not random/isotropic, the virial theorem still applies, but its form changes a bit. Similarly, since our system is made up of many objects, we can gain some insight by seeing how the orbital velocities vary with radius from the center outward.

For example, in a spiral galaxy, the dominant motion of the stars in the disk is circular rotation in the plane of the disk. The variation in the orbital velocities with radius V(r) is called the rotation curve.

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Suppose you have a finite collection of point particles interacting gravitationally via good old Newtonian mechanics. And suppose that:

1. The time averages of the total kinetic energy and the total potential energy are well-defined.
2. The positions and velocities of the particles are bounded for all time.

Then we have:
<T> = -<V>/2

where <T> is the time average of the total kinetic energy, and <V> is the time average of the total potential energy.
I always found this to be a bit magical. It seems surprising at first that such a simple law could hold so generally. But in fact, it's just a special case of something called the "virial theorem", which also applies to forces other than gravity, and impacts everything from astronomy to the theory of gases.

For example, out in space, very often a bunch of particles will collapse to form a gravitationally bound system. If the system is roughly in equilibrium so the time averages of kinetic and potential energy are close to their current values, the virial theorem implies that T = -(1/2) V. we know that <T> = -<V>/2. This is a terrific thing, because it let's you find the masses of bound systems. In fact, it's really the reason we think that dark matter exists.

To be specific, suppose you measure the speeds of a bunch of visible objects in your system, and infer T. Then the virial theorem tells you V. If you find out that the potential well is deeper than what you'd get by adding up the contributions from the masses of everything you see, you know there's dark matter. People do this for spiral galaxies, elliptical galaxies, and galaxy clusters, getting strong evidence for dark matter in all cases, I guess.

StephenPrivitera
Originally posted by Nereid
or perhaps "it first begins to burn its deuterium, and destroy what little lithium it has, when [answer goes here]"
Yes, I suppose this is exactly what I'm looking for. What measurements imply that nuclear fusion is occurring? Mass? density? Temperature?

StephenPrivitera
Originally posted by Jeebus
For example, out in space, very often a bunch of particles will collapse to form a gravitationally bound system. If the system is roughly in equilibrium so the time averages of kinetic and potential energy are close to their current values, the virial theorem implies that T = -(1/2) V. we know that <T> = -<V>/2.

Does this mean that a protostar which has the initial property T < V/2 will collapse until it reaches equilibrium at T = V/2? Suppose we know the average temperature, mass and density of a protostar. Can we predict what size star will result from the collapse by finding when T = V/2?