Insolation on exoplanets with eccentricity and obliquity

Patterns of both mean and peak insolation display various symmetries with respect to latitude and longitude on the planet's surface. Most of these are relatively simple and easily understood; for example, a resonant planet whose orbital period is half of an odd multiple of its rotation period (as in Mercury's 3/2 resonance) experiences identical insolation patterns at longitudes 180° apart. However, such half-odd resonances also exhibit a totally unexpected symmetry of the time-averaged insolation with respect to the planet's equator, not shared by the peak insolation, or by any whole-number resonances. This emergent symmetry can be understood by Fourier analysis of the time-varying insolation.