  We analyse a simple model of the heat transfer to and from a small satellite
orbiting round a solar system planet. Our approach considers the satellite
isothermal, with external heat input from the environment and from internal
energy dissipation, and output to the environment as black-body radiation. The
resulting nonlinear ordinary differential equation for the satellite's
temperature is analysed by qualitative, perturbation and numerical methods,
which show that the temperature approaches a periodic pattern (attracting limit
cycle). This approach can occur in two ways, according to the values of the
parameters: (i) a slow decay towards the limit cycle over a time longer than
the period, or (ii) a fast decay towards the limit cycle over a time shorter
than the period. In the first case, an exactly soluble average equation is
valid. We discuss the consequences of our model for the thermal stability of
satellites.
