4. A particle of unit mass moves with speed $v$ in the gravitational field of the Sun and is
influenced by radiation pressure. The forces acting on the particle are $\mu/r^2$ towards
the sun and $kv$ opposing the motion, where $\mu$ and $k$ are constants. Write down the
vector equation of motion and show that the vector $\mathbf{H}$, defined by
$$
\mathbf{H} = e^{kt} \mathbf{x} \times \dot{\mathbf{x}}
$$
is constant. Deduce that the particle moves in a plane through the origin.
Establish the equations
$$
r^2\dot{\theta} = he^{-kt} \quad \text{and} \quad \mu r = h^2e^{-2kt} - r^3(\ddot{r} + k\dot{r})
$$
where $r$ and $\theta$ are plane polar coordinates centred on the Sun and $h$ is a constant.
Show that, when $k = 0$, a circular orbit of radius $a$ exists for any value of $a$, and find
its angular frequency $\omega$ in terms of $a$ and $\mu$.
When $k/\omega \ll 1$, $r$ varies so slowly that $\dot{r}$ and $\ddot{r}$ may be neglected in the above
equations. Verify that in this case an approximate solution is
$$
r = ae^{-2kt}, \quad \dot{\theta} = \omega e^{3kt}
$$
Give a brief qualitative description of the behaviour of this solution for $t > 0$. Does
the speed of the particle increase or decrease?
