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Homework 4 Solutions

# This simply involves setting r 0 in the equation

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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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