Application Center - Maplesoft

# Lane-Emden Equation

You can switch back to the summary page by clicking here.

Lane-Emden Equation ***

Problem: Use the Lane-Emden 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 Lane-Emden equation for n = 3.

Boundary conditions for the Lane-Emden 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 Lane-Emden equation:

Useful Equation:

Solution:

The Lane-Emden 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 2016-08-29).