Returning to the derivation of the equations of stellar structure,
we can immediately write down one equation which relates the rate
of energy release to the rate of energy transport.
Consider a spherically symmetric star in which energy transport is
radial and in which time variations are unimportant.
L_{r} is the rate of energy flow across a sphere
of radius r and
L_{r+r}
is the rate of energy flow across a sphere of radius
r + r, as shown in
figure 9.
Figure 9:
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 energy released in the shell can then be written as
4 r^{2}_{}r
,
where is defined as the energy release
per unit mass per unit time (W kg^{-1}).
The conservation of energy leads us to
equate the energy crossing the sphere at
r + r to the sum
of the energy crossing the sphere at r and the
energy released in the shell, as follows:
L_{r+r} =
L_{r} +
4 r^{2}_{}r
,
For a thin shell we can also write:
L_{r+r } -
L_{r } = (dL / dr)
r
(in the limit r -> 0,
L /
r =
dL / dr, where
L =
L_{r+r } -
L_{r }).
Equating these last two equations gives:
dL / dr =
4 r^{2}_{}
.
We shall call this the equation of energy production.
We now have three of the equations of stellar structure.
This is not yet a complete set of equations as we
have five unknowns (P, M,
_{},
L and ). In order to make
further progress, we need to consider energy transport in stars.