Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 320 / 2 / a

Table of contents
- Solution a

Solution

Write

μ = GM

and

s = b^{2} + r^{2}

. The gravitational potential of the spherical isochrone model obeys

\frac{d Φ}{d r} = \frac{μ r}{s ( b + s ) ^{2}} .

The spherical Poisson equation therefore gives

ρ (r) = \frac{1}{4 π G r ^{2}} \frac{d}{d r} (r^{2} \frac{d Φ}{d r}) = \frac{M}{4 π s ( b + s ) ^{2}} (3 - \frac{r ^{2}}{s ^{2}} - \frac{2 r ^{2}}{s ( b + s )}) .

Its small-radius asymptotic expansion is

ρ (r) = \frac{3 M}{16 π b ^{3}} - \frac{5 M r ^{2}}{16 π b ^{5}} + O (r^{4}), r ≪ b,

so the model has a finite-density core. At large radius,

ρ (r) = \frac{M b}{2 π r ^{4}} - \frac{3 M b ^{2}}{4 π r ^{5}} + O (r^{- 6}), r ≫ b .

The

r^{- 4}

envelope has finite total mass

M

.

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