hw4 - Physics 160: Stellar Astrophysics Homework #4...

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Unformatted text preview: Physics 160: Stellar Astrophysics Homework #4 Due Tuesday November 1st at 5pm in SERF 340 Reading: Carroll & Ostlie sections 10.1-10.6 Exercises [70 pts + 30 pts bonus]: (1) (a) [10 pts] Show that the equation of hydrostatic equilibrium can be combined with the definition for optical depth to derive: dP g = dτ κ (b) [5 pts] Estimate the pressure of the Sun’s photosphere (τ = 2/3) assuming κ = 0.1 m2/kg. (2) Consider a star with mass M, radius R and uniform density, ρ0 = 3M 4π R 3 (a) [10 pts] Using the equation of hydrostatic equilibrium, show that the pressure as a function of radius in this star can be written as: P (r) = 2π 2 2 Gρ0 (R − r2 ) 3 (b) [10 pts] Show that the core pressure of this star can be written in the form M2 Pc = kG 4 R and find the value of the constant k. (c) [5 pts] Using the solar values for M and R, determine numerical values for ρ0 and Pc. Also determine the core temperature, Tc, assuming an ideal, fully ionized hydrogen gas. (3) Derivations! (a) [10 pts] Derive the Maxwell-Boltzmann distribution in energy (Equation 10.28) starting with the Maxwell-Boltzmann distribution in velocity (Equation 8.1). (b) [10 pts] Derive the Gamow peak energy (Equation 10.34). (c) [10 pts] Show that for an ideal gas, the equation of state is P = Kρ5/3 (note: you’ll have to show this by writing down the specific heats for an ideal gas). (4) Polytropes! [BONUS +30 PTS] (a) [10 pts] Show that for an n=1.5 polytrope, the combination MR3 is a constant. (b) [10 pts] Show that for an n=3 polytrope, the total mass of a star is independent of the central density. (c) [10 pts] Show that for an n=5 polytrope, even though ξ1 →∞, the total mass remains finite. ...
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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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