Knowing how non-Keplerian forces affect the eccentricity and angular momentum vectors, let us study the dynamical consequences of a real life effect.

  1. Compute the dynamical effects that the external force

\begin{align} {\bf f}_{\rm ext} =\frac{GM}{r^3}{\bf r} \left(\frac{4GM}{r}-\dot{\bf r}\cdot\dot{\bf r}\right) +\frac{4GM}{r^2}\dot{r}\dot{\bf r} \end{align}

has on \({\bf h}\) and \({\bf e}\).

  1. Assume that \({\bf r}\) is instantaneously a Keplerian orbit at all times and that the orientation of the orbit is given by the orientation vectors \(\hat{\bf u}={\bf e}/e\), \(\hat{\bf n}={\bf h}/h\) and \(\hat{\bf v}={\bf n}\times\hat{\bf u}\). Then use the conic section solution

\begin{align} {\bf r} = r(\cos f \hat{\bf u} + \sin f \hat{\bf v});\;\;\; \text{with}\;\;\; r=\frac{a(1-e^2)}{1+e\cos f} \end{align}

to write \(\dot{\bf h}\) and \(\dot{\bf e}\) explicitly in terms of the true anomaly \(f\)