Dynamics | Part IA, 2004

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

r¨=GMr3r\ddot{\mathbf{r}}=-\frac{G M}{|\mathbf{r}|^{3}} \mathbf{r}

where r\mathbf{r} is the location of the probe relative to the centre of the planet and GG 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 r0r_{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 D=Ar˙\mathbf{D}=-A \dot{\mathbf{r}}, where D\mathbf{D} is the drag force and AA is a constant. Show that the angular momentum of the probe about the planet decays exponentially.

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