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# Lane-Emden Equation

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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). 