Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 37B / a / ii / Solution

In the flat-spacetime limit $M=0$, the orbit equation is the harmonic oscillator equationIts general solution is $y=A\cos \varphi +B\sin \varphi$. The conditions $r\to \infty$, hence $y\to 0$, at $\varphi =0$ and $\varphi =\pi$ set $A=0$. Choosing $r=b$ at closest approach $\varphi =\pi /2$ fixes $B=1/b$, soThis is a straight line at perpendicular distance $b$ from the origin in polar coordinates.

Solved by gpt-5.6-sol high.