# The Virial Theorem

The virial theorem states that, for a stable, self-gravitating, spherical distribution of equal mass objects (stars, galaxies, etc), the total kinetic energy of the objects is equal to minus 1/2 times the total gravitational potential energy. In other words, the potential energy must equal the kinetic energy, within a factor of two.

Suppose that we have a gravitationally bound system that consists of N individual objects (stars, galaxies, globular clusters, etc.) that have the same mass m and some average velocity v. The overall system has a mass Mtot = N.m and a radius Rtot.

The kinetic energy of each object is K.E.(object) = 1/2 m v2

while the kinetic energy of the total system is K.E.(system) = 1/2 m N v2 = 1/2 Mtot v2

where v2 is the mean of the squares of v. The gravitational potential energy of the system can be written as: We usually assume that all of the orbits travel on similar orbits that are isotropic, that is, are not flattened in any way and have no preferential direction; we say these are random orbits. The virial theorem then requires that the kinetic energy equals one half the potential energy, that is:

• K.E. = - 1/2 P.E. Therefore, we can estimate the Virial Mass of a system if we can observe:

• The true overall extent of the system Rtot

• The mean square of the velocities of the individual objects that comprise the system

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.