 Solution

ID: past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2021/section-b/5/c/solution

The particle count is the enclosed mass divided by the particle mass

m

N (ξ) = \frac{4 π ρ _{c} a ^{3}}{m} \int_{0}^{ξ} s^{2} e^{- Ψ (s)} d s .

\int_{0}^{ξ} s^{2} e^{- Ψ (s)} d s = [s^{2} Ψ^{'} (s)]_{0}^{ξ} = ξ^{2} Ψ^{'} (ξ),

where regularity makes the lower endpoint vanish. Thus

N (ξ) = \frac{4 π ρ _{c}}{m} (\frac{K}{4 π G ρ _{c}})^{3/2} ξ^{2} Ψ^{'} (ξ) .

The prefactor is a mass divided by a mass, and the remaining factor is dimensionless. In particular, at fixed

K

the dependence on central mass density is

N \propto ρ_{c}^{- 1/2} f (ξ)

, with

f (ξ) = ξ^{2} Ψ^{'} (ξ)

. Both changes must be retained when comparing equilibria at fixed

N

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