We have seen that stars are composed of highly ionized gases known as
plasmas. Hence, although the density is so high that the typical
inter-particle spacing is of the order of an atomic radius, the effective
particle size is more like a nuclear radius, i.e. 10^{5} times
smaller. This means that, despite the high pressures and densities in
the interior, the stellar material behaves like an ideal gas with a
gas pressure given by:
P_{gas} = nkT,
where k is Boltzmann's constant and n = N / V, i.e. the number of particles, N,
per cubic metre. We need to
rewrite this equation in the form
P = P (_{}, T,
composition),
which means that we require an expression for n in terms of
density and chemical composition. Noting that density = mass /
volume, we can immediately write:
_{} = nm,
where m
is the mean particle mass. This can be rewritten in
terms of the mass of the hydrogen atom, m_{H},
as follows:
_{} = nm_{H}_{},
where _{} is known as the
mean molecular weight of the stellar material and is the
mean mass of the particles in the gas in terms of the mass of the
hydrogen atom, m_{H} = 1.67 × 10^{-27} kg.
Hence the equation for the gas pressure becomes:
P_{gas} = kT_{} / m =
kT_{} /
m_{H}_{}.
If we now define the gas constant, R, as
R = k / m_{H},
we obtain:
P_{gas} =
R_{}T /
_{}.
This is known as the equation of state of an ideal gas and
has been written in the form that we shall use for much of the
remainder of this course.
We calculated earlier that we can ignore the
effects of radiation pressure in stars like the Sun and hence the gas
pressure is equal to the total pressure. This is not true of all
stars, however, in which case we can generalize the above equation as
follows:
P = P_{gas} + P_{rad}
and hence
P =
R_{}T /
_{}
+ (aT^{4}/3),
where a is the radiation density constant and we have introduced
no new variables.