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

Subtract the star's acceleration from that of $M_1$ in an inertial frame. With ${\mathbf{r}}_i$ measured from the star,This has the requested form ${\ddot{\mathbf{r}}}_1={\nabla }_1(U_1+{\mathcal{R}}_1)$ withand disturbing functionThe 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 $M_2$.