Stellar Structure and the Lane-Emden Function:
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Here we want to numerically integrate Equation 1 and use the result
to find the value of .
Chandrasekhar presents the following table, where the numerical
results were obtained from a variety of sources.
| Table 1
First try to use NDSolve
Here the numerical integration was not able to leave the starting
gate. The same problem happens when
(as well as in other cases).
there is the simple analytic solution presented above.
The reason for the difficulty is seen in the expanded form of
From this we can see that, due to the potentially singular term,
there might be difficulty with a numerical algorithm that starts
However, if we can move the boundary condition away from the origin,
we can use NDSolve
from that point on. We can use the differential equation, Equation
1, along with the boundary conditions at the origin, Equation 2,
to make a series expansion for
We will create a general function for doing this, but will develop
it by working with a particular case: a test case for .
First define the series in terms of unknown parameters.
In this expression the boundary conditions were imposed by choosing
Now define a table of the unknown coefficients
The differential equation defines an expression for the unknown
Collect the coefficients in this expression.
Now solve the equations that result from setting the coefficients
to zero (the
creates the list of equations from the list of coefficients).
Here is the resulting series.
Now we put all of this together into a function. We make the function
self-documenting in the conventional Mathematica fashion
through the device of a usage message.
The usage message allows us to access the definition of the function.
It also enables the template creation feature the for the function
menu item, or its keyboard equivalent).
Here is an example of the use of LaneEmdenSeries.
(Chandrasekhar shows the result to ;
but, in his case we are sure the he could calculate it out to a
significantly higher order: presumably he did. For mortals like
us, we can do the same - without error - with
Mathematica as a tool.)
Using the Series Solution in NDSolve
Since we designed the function LaneEmdenSeries
to automatically incorporate the boundary conditions of Equation
2, it can be used to move the boundary condition to any other point
away from the origin where the series is convergent. A prudent approach
is to choose a point close to 0, say
and explore the result.
First compute the new boundary conditions at .
Now attempt to use NDSolve
with the new boundary conditions. We can experiment with machine
arithmetic (the default) versus higher precision arithmetic by using
various options to NDSolve.
We find that the quoted results (of Table 1) are achieved in the
latter case (for the series expansion point and degree that we have
picked) if we set the option .
Here is the result for
(we use Module
to localize the parameters).
Let's take a look at the result.
Find the first real zero using FindRoot
(with the option ).
These numbers agree with the results quoted in Table 1. The open
question here is "how many of the digits of this answer are
correct?" This can be investigated by changing the various
options to NDSolve
and FindRoot as
well as the degree and starting point of the Lane-Emden series used
to move the boundary conditions. The advantage of having access
to arbitrary precision arithmetic is that you can investigate the
numerical properties of a solution with much greater versatility
than when you are under the constraints of machine arithmetic.
The procedure that we just used for the
case can be put together into a function so that you can experiment
with and generate results for any case of interest. In this way
you can reconstruct all of the entries of Table 1.