Unformatted text preview: radius in this star can be written as: P (r) = 2π 2 2
Gρ0 (R − r2 )
3
The equation of hydostatic equilibrium is: dP
M (r)
= −g (r)ρ(r) = −G 2 ρ0
dr
r
We can express the M(r) term as simply the mean density times the volume up to radius r: M (r) = 4π
ρ0 r3
3
Inserting this in: dP
4π
= − Gρ2 r
0
dr
3
Multiplying both sides by dr, we can integrate from radius r to R, with the assumption that P(R) = 0 (space) P (R )
P (r ) 4π
dP = 0 − P (r) = − Gρ2
0
3 ⇒ P (r) = R rdr = −
r 2π 2 2
Gρ0 (R − r2 )
3
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. This simply involves setting r = 0 in the equation above: 2π 2 2 2π
G ρ0 R =
G
P (0) = Pc =
3
3 3M
4π R3 2 M2
R = kG 4
R 2 as long as k= 3
= 0.119
8π
(c) [5 pts] Using the solar values for M and R, determine numerical values for ρ0 and Pc. Also determine the core...
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 Fall '08
 Norman,M
 Physics, Work, Classless InterDomain Routing, dτ, the00, Kap, lim ξ

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