hw5 - Physics 160: Stellar Astrophysics Homework#5...

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Unformatted text preview: Physics 160: Stellar Astrophysics Homework #5 Due Tuesday November 8th at 5pm in SERF 340 Reading: Carroll & Ostlie sections 11.1-11.3, 12.1-12.3 Exercise [90 pts]: Your homework this week is to investigate the Lane-Emden equation computationally. (1) [50 pts] Employing a numerical integrator, compute and compare the density profiles for polytropes n = 1.5 (degenerate star) and n = 3 (star in radiative equilibrium). In both cases, plot the ratio of the density to the central density (ρ/ρc) as a function of the dimensionless quantity ξ = r/λn. Also compute the values of ξ1 and dD/dξ at ξ1, where the density goes to zero. Be sure to show both your plot(s) and the code (e.g., C, Mathematica, etc.) you used to calculate these profiles (see note on next page). (2) [20 pts] For both polytropes, determine the ratio of the average density <ρ> = 3M/4πR3 to central density ρc (remember R = λξ1 and see page 338 of Carroll & Ostlie for an expression for M). Explain qualitatively why one polytrope is more centrally concentrated. (3) [20 pts] For the n = 3 case, which has an equation of state P = Kρ4/3, use the expression for K on page 340 of Carroll & Ostlie to compute a star’s mass for µ = 0.5 and β = 0.1, 0.5 and 0.9 (β is the ratio of gas to radiation pressure). What do the differences in these values tell you about the importance of radiation pressure in the interiors of low and high mass stars? NOTE: If you are writing your own code for the numerical integration (it’s good for the soul!), here’s a useful hint: 2nd order differential equations can be integrated using the Euler method by solving two equations: one for the parameter and one for its derivative. In this case, the Lane-Emden equation can be written as: d2 D 2 dD + = −Dn dξ 2 ξ dξ which can be split into two first-order differential equations: f (ξi ) = dD (ξi ) dξ (the derivative of D) df 2 (ξi ) = − f (ξi ) − Dn (ξi ) dξ ξi (the Lane-Emden equation) You can then iterate these as ξi+1 = ξi + h D(ξi+1 ) = D(ξi ) + hf (ξi ) f (ξi+1 ) = f (ξi ) + h df (ξi ) dξ until you get to the first zero point ξ1, where h is some small number (if you’d like to use a different numerical integration scheme, you are welcome to but be sure to explain your method). Don’t forget your boundary values! ...
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This note was uploaded on 02/26/2012 for the course PHYS 160 taught by Professor Norman,m during the Fall '08 term at UCSD.

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