Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 316 / 4 / v

Solution

ρ_{p}

R_{p} = (3 M_{p} /4 π ρ_{p})^{1/3}

χ^{2} = \frac{2 G M _{p} / R _{p}}{G M _{⋆} / a _{p}} = \frac{2 M _{p} a _{p}}{M _{⋆}} (\frac{4 π ρ _{p}}{3 M _{p}})^{1/3} .

Raise this equation to the power

3/2

and solve for

M_{p}

M_{p} = \frac{3}{32 π} χ^{3} M_{⋆}^{3/2} a_{p}^{- 3/2} ρ_{p}^{- 1/2} .

3/ (32 π)

, and the gravitational constant cancels. Dimensional checking gives mass on the right:

M_{⋆}^{3/2} a_{p}^{- 3/2} ρ_{p}^{- 1/2}

has that dimension.

This uses the planet's own escape velocity. If instead

χ

is defined at finite-comet contact, an additional factor

(1 + R / R_{p})^{3/2}

appears on the right, and

R_{p}

itself depends on

M_{p}

; the quoted pure-planet scaling is then only the

R / R_{p} \to 0

limit.

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