A full solution of the equations of stellar structure requires a complex
computer code. Fortunately, by adopting various simplifying assumptions,
it is possible to solve the equations of stellar structure using a much
less complicated code, or even in some cases do without a computer
altogether and solve the equations analytically. For example, if we
adopt a simple relation between
pressure and density which is valid throughout the star, the equations
of hydrostatic support and
mass conservation can be solved
independently of the other 4 equations of
stellar structure, resulting in a model in which the structure of the
star is independent of the heat flowing through it. Such
polytropic models played an important role in the development of
stellar structure theory, particularly before the advent of powerful
computers, and we will now look at them in detail.
Let us proceed by multiplying the equation of
hydrostatic support by
r^{2} / _{}
and differentiating with respect to r, giving
d / dr [(r^{2} / _{}) dP / dr] = - G dM / dr
.
Substituting the equation of mass conservation on the right-hand side, we obtain
(1 / r^{2}) d / dr [(r^{2} / _{}) dP / dr] =
- 4 G
_{}.
Let us now adopt an equation of state of the form
P = K _{}
^{(n+1)/n},
where K is a constant and n is known as the
polytropic index. Combining the last two equations, we
obtain the following non-linear second-order differential equation
for the density inside the star
(1 / r^{2}) d / dr [(r^{2} / _{}) . [(n+1)K / n] .
_{}^{1/n}
d_{}
/ dr] = - 4 G _{},
which can be rearranged to
[(n+1)K / 4 Gn] .
(1 / r^{2}) d / dr [( r^{2} / _{}^{(n-1)/n}) d_{}
/ dr] = - _{}.
It is convenient at this point to replace r and
_{} in the above equation by
dimensionless variables. We therefore define a
dimensionless variable, _{}, where
_{} = r /
,
and is a constant scale factor. We also
define a dimensionless variable, , where
^{ n}
= _{} /
_{}_{c}
and _{}_{c} is the central
density. Substituting these dimensionless variables into the second-order
differential equation derived above, we obtain
[(n+1)K / 4 Gn] .
(1 / ^{2}_{}^{2}) d / d_{}
[(^{2}_{}^{2}
/ _{}_{c}^{(n-1)/n}
^{n-1})
d_{}_{c}^{n}
/ d_{}]
= - _{}_{c}^{n},
which simplifies to
[(n+1)K /
4 G
_{}_{c}^{(n-1)/n}]
. (1 / ^{2}_{}^{2}) d / d_{}
(_{}^{2}
d / d_{})
= - ^{n}.
The coefficient in square brackets on the left-hand side of this equation
is a constant having the dimension length squared. The square root of this
coefficient is hence an appropriate choice for
, i.e.
=
[(n+1)K /
4 G
_{}_{c}^{(n-1)/n}]
^{0.5}.
Combining the last two equations, we obtain the Lane-Emden equation:
(1 / _{}^{2}) d / d_{}
(_{}^{2}
d / d_{})
= - ^{n}.
To solve the Lane-Emden equation, we need two boundary conditions. At
the centre of the polytrope, i.e. when _{} = 0,
_{} =
_{}_{c}
and hence = 1. A second central boundary
condition follows from the equation of
hydrostatic support, in which
the M/r^{2} term tends to zero as r
tends to zero. This means that dP / dr = 0 at r=0
and, from the polytropic equation of state,
d / d_{} = 0
at _{} = 0.
In the next section we will look at how these boundary conditions are
used to solve the Lane-Emden equation and how the solutions compare to
more sophisticated models of stellar structure.
©Vik Dhillon, 27th September 2010