Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 316 / 3 / i

Table of contents
- Solution i

Solution

Let

r = r_{2} - r_{1}

and

r = ∣ r ∣

. Newton's second law for the two planetesimals gives

\ddot{r}_{1} = G m_{2} \frac{r}{r ^{3}} + \frac{F _{1}}{m _{1}}, \ddot{r}_{2} = - G m_{1} \frac{r}{r ^{3}} + \frac{F _{2}}{m _{2}} .

Subtracting yields the forced relative two-body problem

\ddot{r} + G (m_{1} + m_{2}) \frac{r}{r ^{3}} = \frac{F _{2}}{m _{2}} - \frac{F _{1}}{m _{1}} .

The internal gravitational forces cancel from the centre of mass equation but add in the relative equation; external forces affect the internal orbit through their difference in acceleration.

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