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Homework 1

Homework 1 - Physics 223 Stellar Astrophysics...

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Physics 223: Stellar Astrophysics Homework #1 [REVISED] Due Friday February 3 rd at 5pm in box in front of SERF 340 Reading : HKT Chapters 1, 7 & 9 Exercises: [130 pts total] (1) Lane-Emden Equation and Polytropes : In chapter 7 of HKT, the Lane- Emden equation is derived for polytrope models of stellar interiors: where For a stellar model, this equation must satisfy the boundary conditions θ n = 1, d θ n /d ξ = 0 at ξ = 0. (a) [10 pts] Derive Equation (7.48) in HKT, showing that close to the origin ( ξ = 0): (b) [10 pts] Show that the enclosed mass at ξ = r/r n can be expressed as: (c) [10 pts] Show that for an n=1.5 polytrope, the combination MR 3 is a constant. 1 ξ 2 d d ξ ξ 2 d θ n d ξ = - θ n n ρ ( r ) = ρ c θ n ( r ) r = r n ξ r n = ( n + 1) P c 4 π G ρ 2 c P c = K ρ 1+1 /n c θ n ( ξ ) = 1 - 1 6 ξ 2 + n 120 ξ 4 + ... m ( ξ ) = - 4 π r 3 n ρ c ξ 2 d θ n d ξ

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(d) [10 pts] Show that for an n=3 polytrope, the total mass of a star is independent of its central density. (e) [10 pts] Prove the analytic solution for n=1: and solve for ξ 1 and (d θ n /d ξ ) ξ 1
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