Mathematical methods and models for the Hera mission: average temperature and Mixed Yarkovsky effect for non-circular orbits

The accurate determination of thermal accelerations acting on small bodies orbiting the Sun requires knowing the surface temperature at any moment. In analytical methods, the computation of the temperature is often simplified by assuming that it varies slightly about a constant value. However, the mean temperature is constant only in the case of a circular orbit. In this work, a time-dependent temperature that closely represents the mean temperature of a spherical body revolving around the Sun on an eccentric orbit is defined. Through the introduction of a robust algorithm, the average temperature is computed for Didymos, the target asteroid of the Planetary Defence mission Hera. By computing the small deviations from the average temperature, a formulation for the Yarkovsky thermal effect is obtained, which includes not only the classical diurnal and seasonal variants, but also the coupling of these two contributions, extending the available theories which are valid only in the case of a body in a circular orbit. Results of numerical tests are presented and show that for a fast-rotating object in a non-circular orbit the diurnal and seasonal variants do not decouple and the resulting mixed terms have a long-term effect on the orbital elements.