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Homework 4 Solutions

# 5 13 1 5 25k c 4 g 1 2 13 1 5 25k c 4 g 1

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Unformatted text preview: , average velocity <v> per particle, and pressure <P> (remember our derivation of <P> for an ideal gas in class). These are all just moments of various forms of energy: 1 f (E )Ed3 pd3 x N 13 −3/2 3 /2 =n (2m F ) 2V 2π (2m) N 8π E= 3 2 3 = 5 = −3/2 F F × 2 5 5 /2 F F E 3/2 dE 0 Velocity: v= =√ = 3 2m 3 4 21 mN 2E/m = −3/2 F × 1 2 2 F f (E )E 1/2 d3 pd3 x 2F 3 = vF m 4 Pressure: P= 2 2 1 nm v 2 = n E = n 3 3 5 F (d) [5 pts] As derived in section 16.3, the Fermi energy is F ¯2 h = 3π 2 n 2m 2 /3 Use this to show that the pressure of a nonrelativistic degenerate gas scales as ρ5/3. ¯ 2 h2 P= n 3π 2 n 5 2m 2 /3 ∝ n5/3 ∝ ρ5/3 assuming n = ρ/µmH. (4) Some useful polytrope relations [30 pt] (a) [10 pts] Show that for an n=1.5 polytrope, the combination MR3 is a constant. We can simply write down the expressions for M and R for the polytrope model from pages 337 and 338: M R3 = 2 −4πλ3 ρc ξ1 n dDn dξ 3 (λn ξ1 ) ξ1 Nearly all of the values in this expression are constants except for λn and ρc, so just looking at the proportionalities: M R3 ∝ λ6.5 ρc 1 λn is itself a function of ρc; evaluating for the case of n = 1.5: −1/3 λ1 . 5 2.5K ρc = 4π G 1/ 2 −1/3 λ1 . 5 2.5K ρc = 4π G 1/ 2 ∝ ρ−1/6 c hence the combination of λ6 and ρc is indepe...
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