 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-55/3/iv/solution

M = 4 π ω_{3} (\frac{K}{π G})^{3/2}, ω_{3} = - ξ_{1}^{2} θ^{'} (ξ_{1}) ≃ 2.01824.

Substituting the preceding constant

K

gives

M = \frac{4 ω _{3}}{π G ^{3/2}} (\frac{3}{a _{r}})^{1/2} (\frac{R}{μ})^{2} \frac{1 - β}{β ^{2}} .

For fixed composition this is $M \propto 1 - β / β^{2}$ , as required. Squaring and rearranging yields the Eddington quartic relation,

(1 - β) / β^{4} \propto μ^{4} M^{2}

. The larger masses in this model have smaller gas fractions and relatively more radiation support.

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