LaneEmden Equation ***
Problem: Use the LaneEmden equation to create a basic model for the white dwarf Sirius B. Calculate and provide plots of density and pressure. Note that a polytropic index of 3 applies to all white dwarfs. (A method devised by Larson (2012) is employed in the calculations.)
Hints:
Solve the LaneEmden equation for n = 3.
Boundary conditions for the LaneEmden equation for Sirius B: at the centre of the star, ξ = 1 and its first derivative is 0.
Data: (SI units)
 (1) 
 (2) 
 (3) 
 (4) 
 (5) 
 (6) 
Define variables in the LaneEmden equation:
Useful Equation:
Solution:
The LaneEmden equation may be written as.
 (7) 
Boundary conditions. At the centre of Sirius B, ξ = 1, and its first derivative is 0.
 (8) 
Solve the equation:
 (9) 
 (10) 
Since the variable x (see above) ranges from 0 to 1, we can try 0.7 to test the solution:
 (11) 
We are concerned here with the value of ξ(x). Select this value and plot it from 0 to 150.
 (12) 
Calculate the density (ξ(r) = ξ(x*and the pressure (from the equation of state) and plot them:
 (13) 
 (14) 
Plot rho over a range to determine where the value at the surface will be found:
The surface value (0) will be found between x = 0 and x = 10. Calculate the value and assign it to Xsurf:
 (15) 
Plot the density from the centre of Sirius B to the surface:
Plot the pressure from the centre of Sirius B to the surface:
Now adjust the plot so that density and pressure appear as functions of Sirius B's radius (R):

Reference
Larson, S. (2012). Stellar Structure with Maple. http://sciencejedi.com/professional/classes/astrophysics/lectures/maple_laneEmden.pdf (Accessed 20160829).
