Now, noting that dM = _{} dV,
the virial theorem can be rewritten as follows:
- =
3_{}P
dV =
3_{}(P / _{}) dM.
The pressure, P, in this equation is given by
P = P_{gas} + P_{rad} ,
where P_{gas} is the gas pressure and
P_{rad} the radiation pressure. If we assume
for the moment that stars are composed of an ideal gas with negligible
radiation pressure, we can write:
P = nkT = kT _{} / m,
where n is the number of particles per cubic metre, m
is the average mass of the particles
in the stellar material and k is Boltzmann's constant.
This gives for the virial theorem:
- =
3_{}(P / _{}) dM =
3_{}(kT / m)
dM.
We may now rewrite the inequality derived above as follows:
- =
3_{}(kT / m)
dM >
GM_{s}^{2} / 2r_{s}.
Rearranging gives:
_{}T
dM >
GM_{s}^{2}m
/ 6kr_{s}.
We can think of the integral on the left-hand side of the above equation
as the sum of the temperatures of all of the infinitesimal mass elements
dM which make up the star. The mean temperature of the star,
T_{av}, is then given by this integral divided by the
total mass of the star, M_{s}, i.e.
M_{s}T_{av} =
_{}T
dM,
Combining the last two equations, we
obtain an inequality which gives the minimum mean temperature of a
star: