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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 10.110.6
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 MaxwellBoltzmann distribution in energy (Equation
10.28) starting with the MaxwellBoltzmann 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.
 Fall '08
 Norman,M
 Physics, Work

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