Homework 5 Solutions

Explain qualitatively why one polytrope is more

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Unformatted text preview: entral 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). (b) [15 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. (c) [15 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? From p340: 3(1 − β ) a K= = f (β ) 1/ 3 k β µmH 31 / 4 k a1/4 µmH 4/ 3 4/ 3 = f (β ) × (6.7 × 1010 N m2 kg−4/3 ) where I’ve defined 1−β β4 f (β ) = 1/ 3 The polytropic mass is given by 2 M = −4πρc λ3 ξ1 3 dD dξ ξ1 where...
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This document was uploaded on 02/26/2014 for the course PHYS 160 at UCSD.

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