We have already seen that it is often more convenient to
write the equations of stellar structure
in terms of M instead of r. Dividing each of the
equations of stellar structure by the equation
of mass conservation, and assuming that no energy is carried by
convection, we can write:
I. dP / dM = - GM /
4r^{4},
II. dr / dM = 1 /
4r^{2}_{},
III. dL / dM = ,
IV. dT / dM = - 3L /
64^{2}acr^{4}T^{3},
where we have also inverted the equation of mass conservation. The
corresponding boundary conditions become:
r = L = 0 at M = 0 and
_{} = T = 0 at
M = M_{s}.
These differential equations must be supplemented by the three additional
relations for P, and
. We derived approximate forms for these
parameters in
the physics of stellar interiors part
of the course; if we assume that radiation pressure is negligible,
that stellar material behaves as an ideal gas, and that
the laws of opacity and energy generation can be approximated by
power laws, we can write:
P =
R_{}T /
_{},
=
_{0}_{}^{}T^{}, and
=
_{0}_{}T^{},
where , _{}
and _{} are constants and
_{0} and
_{0} are constants for a given
chemical composition.
Using the above approximations in conjunction
with the differential equations and associated boundary conditions,
it is possible to solve for the structure of a star of any given
total mass, M_{s}. If we then wish to consider
the structure of a star of a different mass, the equations must be
solved again for the new M_{s}. If we then
wish to consider the structures of a large number of stars covering a
wide range of M_{s}, such as when attempting to
explain the observed
mass-luminosity relation of
main-sequence stars, we must solve the
equations of stellar structure many times. This is clearly very tedious
and time-consuming. Fortunately, there is a much simpler way of
formulating the equations of stellar structure which ensures that,
once the structure of a star of any one mass has been determined, the
structure of a star of any mass (but with the same homogeneous
chemical composition) can be obtained by a simple
scaling of variables. Such a sequence of stellar models is known as
a homologous series and we shall now describe how such a
sequence can be obtained.
Our aim is to formulate the equations of stellar structure
in such a way that they are independent of the total mass,
M_{s}. In so doing, we are assuming
that the way in which any physical quantity such as luminosity
varies from the centre of the star to the surface is the same for
stars of all masses and only the absolute value of the luminosity
varies from star to star. This is
illustrated schematically in
figure 21, where the ratio of luminosity
to surface luminosity (L /L_{s}) is plotted
against the fractional mass, m, defined as the
ratio of mass to total mass:
m = M / M_{s}.
Figure 21:
Fractional luminosity as a function of fractional mass.
The curve shown in
figure 21 is the same for all stars
with the same laws of opacity and energy generation, but the value
of L_{s} depends on M_{s} and
it is proportional to some power of M_{s}, which
depends on the values of ,
_{} and
_{}. The same is also true of the other
physical quantities such as radius (r_{s}) and
temperature (T).
Expressing this mathematically, we can write:
V.r = M_{s}^{a1}r^{*}(m)
VI._{} = M_{s}^{a2}_{}^{*}(m)
VII.L = M_{s}^{a3}L^{*}(m)
VIII.T = M_{s}^{a4}T^{*}(m)
IX.P = M_{s}^{a5}P^{*}(m)
where a_{1 }, a_{2 }, a_{3 },
a_{4} and a_{5} are constants and
(as indicated) r^{*}, _{}^{*},
L^{*}, T^{*} and P^{*}
depend only on the fractional mass, m.
If we now substitute the expressions V, VI, VII, VIII and IX
for r,
_{}, L, T and
P into equations I, II, III and IV, we can eliminate
M_{s} provided that the values of
a_{1 }, a_{2 }, a_{3 },
a_{4} and a_{5} are chosen
correctly. So, for equation I,
We have now obtained 5 equations for the five constants a_{1
}, a_{2 }, a_{3 },
a_{4} and a_{5}. We have also
obtained 5 new equations of stellar structure which are independent of
M_{s}. They are only independent of
M_{s}, however, if the 5 equations for a_{1
}, a_{2 }, a_{3 },
a_{4} and a_{5} have consistent
solutions. It can be shown that this is indeed the case for all
reasonable values of , _{} and _{}
(see page 114 of Tayler). Note that the general solution for the
a's is too complicated to be given here, but we will be
considering solutions for special values of , _{} and _{}later.
Collecting things together, the 5 new equations which govern the structure
of a star of any mass are:
These equations can now be solved to find
r^{*}, _{}^{*},
L^{*}, T^{*} and P^{*}
in terms of m, using the boundary conditions:
r^{*} = L^{*} = 0 at m = 0 and
_{}^{*} = T^{*} = 0 at
m = 1,
where the centre and surface of the star are at m = 0 and
m = 1, respectively. The above equations must be solved
on a computer and, after the solution has been obtained, the quantities
r^{*}, _{}^{*},
L^{*}, T^{*} and P^{*}
can be converted into r,
_{}, L, T and
P for a star of any given mass M_{s} by using
the relations V, VI, VII, VIII and IX and the values of the constants
a_{1 }, a_{2 }, a_{3 },
a_{4} and a_{5} previously found.
Hence the equations of stellar structure have only to be solved once
in this manner and the properties of stars of all masses can then be
obtained.