Basic Problems in Astrophysics (4)

We now look at one of the most basic problems in terms of the "virial theorem" and how it can be used to show the total energy of a star is negative and equal to one half the gravitational potential energy (or the negative of the gas kinetic energy.

According to the virial theorem:

2K + W = 0

for any spherical system in hydrostatic equilibrium, where K is the gas kinetic energy:

K = 3/2[y - 1] U

y = the ratio of specific heats (c_p/c_v)

and U is the internal energy, W is the gravitational potential energy.

From this one can obtain the binding (or total energy) of a star as:

E(S) = K + W

Combining the above with:

2K + W = 0

we obtain: 2K = -W or K = -W/ 2

E(S) = -W/2 + W = W/2

But W = -2 K

so E(S) = W/2 = (-2 K)/ 2 = -K

Thus, the total energy of a star E(S) is equal to half the gravitational potential energy (e.g. W/2) or to the negative of the gas kinetic energy:

E(S) = - K = - 3/2[y - 1] U

This is the putative basis for how a collapsing gas cloud eventually emerges as a star with a total energy equal to the negative of the gas kinetic energy.