Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 4 / i

Table of contents
- Solution i

Solution

Subtract the star's acceleration from that of

M_{1}

in an inertial frame. With

r_{i}

measured from the star,

\ddot{r}_{1} = - \frac{G ( M _{*} + M _{1} )}{r _{1}^{3}} r_{1} + G M_{2} [\frac{r _{2} - r _{1}}{∣ r _{2} - r _{1} ∣ ^{3}} - \frac{r _{2}}{r _{2}^{3}}] .

This has the requested form

\ddot{r}_{1} = \nabla_{1} (U_{1} + R_{1})

with

U_{1} = \frac{G ( M _{*} + M _{1} )}{r _{1}},

and disturbing function

R_{1} = G M_{2} [\frac{1}{∣ r _{2} - r _{1} ∣} - \frac{r _{1} \cdot r _{2}}{r _{2}^{3}}] .

The first term is the direct disturbing function, the attraction of

M_{2}

on

M_{1}

. The second is the indirect disturbing function, which subtracts the acceleration of the star-centered origin by

M_{2}

.

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