A small probe of mass $m$ is in low orbit about a planet of mass $M$. If there is no drag on the probe then its orbit is governed by

$$\ddot{\mathbf{r}}=-\frac{G M}{|\mathbf{r}|^{3}} \mathbf{r}$$

where $\mathbf{r}$ is the location of the probe relative to the centre of the planet and $G$ is the gravitational constant. Show that the basic orbital trajectory is elliptical. Determine the orbital period for the probe if it is in a circular orbit at a distance $r_{0}$ from the centre of the planet.

Data returned by the probe shows that the planet has a very extensive but diffuse atmosphere. This atmosphere induces a drag on the probe that may be approximated by the linear law $\mathbf{D}=-A \dot{\mathbf{r}}$, where $\mathbf{D}$ is the drag force and $A$ is a constant. Show that the angular momentum of the probe about the planet decays exponentially.