The mass, M, contained within a star of radius r
is determined by the density of the stellar material,
_{}. Hence the quantities M, r and
_{} in the
equation of hydrostatic support
are not independent. We now need to derive an equation which relates
these three parameters.
Consider the mass of a thin spherical shell inside a star with
an inner radius of r and an outer radius of
r+r, as shown in
figure 7.
Figure 7:
A thin spherical shell inside a star.
Because the shell is thin, its volume is given by its surface area
times its thickness:
4 r^{2}r.
Recalling that mass = density x volume, the mass of the shell
is then given by:
4 r^{2}_{}r.
The mass of the shell is also given by the difference between the mass
of the star within radius r+r
and the mass of the star within
radius r, which for a thin shell can be written as:
M_{r+r } -
M_{r } = (dM / dr)
r
(in the limit r -> 0,
M /
r =
dM / dr, where
M =
M_{r+r } -
M_{r }).
Equating these last two equations gives:
dM / dr =
4 r^{2}_{}.
We shall call this the equation of mass conservation.
We now have two of the equations of stellar structure, but we
have four unknown quantities (P, M,
_{} and r). Clearly, we
are going to have to derive some additional equations. Before
we do this, however, we will briefly discuss the validity of
our two main assumptions - that stars are spherically symmetric
and in hydrostatic equilibrium - and then use the two equations
we have derived to set lower limits on the central pressures and
mean temperatures of stars. This will enable us to say something
about the physical state of stellar material and the source of
energy in stars.