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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 Fall '08
 Norman,M
 Physics, Work, Classless InterDomain Routing, dτ, the00, Kap, lim ξ

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