solving the laneemden equation 
Figure 22: 
Numerical solutions to the LaneEmden equation for (lefttoright)
n = 0, 1, 2, 3, 4 and 5.

Table 2:^{ }  Numerical solutions to the LaneEmden 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 _{}). 
n 
_{}_{R} 
 _{}
d / d_{}
_{}
_{ =
R
}

0  2.45  3.33 x 10^{1} x _{}_{R} 
1  3.14  1.01 x 10^{1} x _{}_{R} 
2  4.35  2.92 x 10^{2} x _{}_{R} 
3  6.90  6.14 x 10^{3} x _{}_{R} 
4  15.00  5.33 x 10^{4} x _{}_{R} 
Figure 23: 
Comparison of numerical solution for an n = 3 polytrope of the
Sun versus the Standard Solar Model.
