Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 316 / 4 / iii / Solution

Iterating Kepler's equation once gives $E=M+e\sin M+O(e^2)$. The exact radius expression then yieldsExpansion of the true anomaly conversion gives $f=E+e\sin E+O(e^2)$, soThe factor of two includes both the nonuniform relation between $M$ and $E$ and the change from center angle $E$ to focus angle $f$.