solving the lane-emden equation
It is possible to solve the Lane-Emden equation analytically, i.e.
to derive a simple mathematical expression for the dimensionless density
as a function of
dimensionless radius , for only three
values of the polytropic index, n. These are
n = 0, = 1 -
(2 / 6),
n = 1, =
n = 5, = 1 /
[1+(2 / 3)]0.5.
Solutions for all other values of n must be obtained numerically,
i.e. using a computer to plot
versus , as will now be described.
We start by expressing the Lane-Emden equation in the form
d2 / d2 = - [ (2 / )
d / d ] -
The numerical integration technique we will adopt consists of stepping
outwards in radius from the centre of the star and evaluating the density
at each radius. At each radius, the value of density, i+1, is given by the value of density at the
previous radius, i , plus the
amount the density changes by over the step. Denoting the step length as
(d / d)i+1].
The rate of change of density with radius,
d / d ,
is an unknown in the above equation. To determine its value at each
radius, we can use the same technique and write
(d / d)i+1 =
(d / d)i + ( d2 /
The d2 /
d2 term in the above
equation is given by the rearranged form of the Lane-Emden equation given
above, so we obtain
(d / d)i+1 =
(d / d)i -
[ (2 / i) .
(d / d)i ] +
The numerical integration for a particular value of n can now
proceed as follows. Starting at the
centre, at which the values of ,
d / d
and are known (because
the boundary conditions tell us that d /
d = 0 and =
1 at = 0), determine
(d / d)i+1. This value can then be used to
determine i+1. The radius
is then incremented by adding
to and the process repeated until the
surface of the star is reached (when becomes
negative). The results of such a numerical integration for
five different values of n are shown in
figure 22. Note that in this integration,
set to 0.001 and the starting value for
was 0.00001 rather than 0
(in order to avoid an infinity in the calculations).
Numerical solutions to the Lane-Emden equation for (left-to-right)
n = 0, 1, 2, 3, 4 and 5.
Figure 22 shows that decreasing the polytropic
index results in a stellar model in which the mass is more and more
centrally condensed. In addition, it can be seen that the density
of the n = 5 polytrope never falls below zero and, by inspection
of the analytical solution given above, will do so at an infinite radius.
The n = 5 polytrope (and n 5 polytropes in general) thus represents the non-physical
case of an infinite star. For the
n < 5 polytropes, the solution drops below zero at a finite value
of , and hence the root
(i.e. the radius of the polytrope,
R) can be
determined using a linear interpolation between the points immediately
before and after becomes negative. The roots
for a range of polytropic indices are listed
in table 2.
Numerical solutions to the Lane-Emden equation, showing the
polytropic index n and the values of
and d / d at the surface of the polytrope. It can be seen that
the roots of the n = 0 and 1 solutions are in agreement
with the roots of the analytical solutions listed above (which can be
derived by setting =0 and rearranging for
d / d
3.33 x 10-1 x R
1.01 x 10-1 x R
2.92 x 10-2 x R
6.14 x 10-3 x R
5.33 x 10-4 x R
How do these polytropic models compare to the results of a detailed solution
of the equations of stellar structure? In order to make
this comparison, which we shall do by comparing an n=3 polytropic model
of the Sun (known as the Eddington Standard Model)
with the so-called Standard Solar Model (SSM) [Bahcall 1998; Physics Letters B, 433,
1], we must convert the dimensionless radius,
, and density, ,
to actual radius (in m) and density (in kg m-3). We must also
determine how the mass, pressure and temperature vary with radius.
We do this as follows.
First, we must determine the value of the scale factor, . At the surface of the n=3 solar polytrope, where
= 0, we can write
R / R ,
where R is the radius of the star (in this case, the radius of
the Sun) and R is the
value of at the surface (i.e.
the root of the Lane-Emden equation for n=3, as listed in
Next, we determine the mass as a function of radius.
The rate of change of mass with radius is given by the equation of
dM / dr =
By integrating and substituting
the above equation, we obtain
To evaluate this equation we need to rewrite the Lane-Emden equation in the form
d / d
Combining the last two equations, we obtain an equation for the mass
of the star as a function of radius
M = - 4
d / d.
The value of d / d as a function of is known from
the numerical integration procedure outlined above. The value
of c, however, is not
known. To determine it, we combine the equation for M above with a
formula for the mean density of the Sun, i.e.
- 3 c
(1 / )
d / d
where the last term (between the vertical lines) means that it is evaluated
at the surface of the star and can be calculated from the figures for
n=3 given in table 2.
We can therefore use this equation to determine
c, which in turn allows
us to determine the variation of M with
. This can be transformed to the variation
of M with r using the formula
. The result
is plotted in the second panel from the top in
Comparison of numerical solution for an n = 3 polytrope of the
Sun versus the Standard Solar Model.
It is now straightforward to determine the variation of density with
radius using the equation
The result is plotted in the top panel of
We can determine the variation of pressure with radius using the polytropic
equation of state
P = K
To evaluate this equation we only require K, which we can obtain
from the definition of scale length we adopted when deriving the
The result is plotted in the third panel from the top in
We can determine the variation of temperature with radius by equating the
equation of state of an ideal gas with the
polytropic equation of state
which upon rearrangement and substitution of =
The result is plotted in the bottom panel of figure 23. For comparison, we also plot the
profiles of temperature, density, pressure and mass from the SSM,
which is the most up-to-date solution to the equations of stellar
structure currently available.
It can be seen that the polytrope model does remarkably
well, considering how simple the physics is - we have used only the
mass and radius of the Sun and an assumption about the relationship
between internal pressure and density as a function of radius.
The agreement is particularly good at
the core of the star, where the polytrope gives a
central density of 7.65 x 104 kg m-3, a central
pressure of 1.25 x 1016 N m-2 and a central
temperature of 1.18 x 107 K,
in comparison with the SSM values of
1.52 x 105 kg m-3,
2.34 x 1016 N m-2
and 1.57 x 107 K. Only in
the outer regions of the Sun, where convection takes place, do the two
solutions deviate significantly from one another.
©Vik Dhillon, 13th November 2012