Unformatted text preview: der 1. (a) [5 pts] What is the polytrope index (n) for this equation of state in the case η << 1? For η << 1, the polynomial in η -‐> 1, so P ∝ ρ5/3, which is an n=1.5 polytrope (b) [15 pts] Using the results from section 10.5 in Carroll & Ostlie, show that the radius of this object can be written as: η2
R = R0 (1 + η +
1 + η and that R0 = 2.8×10 7 M
M 1/3 m HINT: think about the relationship between core and average density for this particular polytrope model. We start with the expression for radius for the polytrope model: R = λ1.5 ξ1 where ξ1 = 3.65 is the first zero of the n=1.5 polytrope and 1/2
2.5Kρc λ1.5 = 4πG where 7 K = 10 η2
1+η m4 s−2 kg−2/3 = 107 f (η) m4 s−2 kg−2/3 Remember that the polytrope models have relationships between core and average densities; for n = 1.5, this is: ρc ≈ 6 ρ = 6 3 M
4π R3 su...
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