Fourier series for eclipses on exoplanet binaries

Context. A double planet system or planet binary undergoes eclipses that modify the reflective light curve. In the time domain, the eclipse events are fast and weak. This would make their signal difficult to find and recognize in the phase light curve, even for small inclinations when eclipses happen frequently. However, due to the quasiperiodic nature of the phenomenon, the Fourier transform of the direct reflection signal consists of a double sum of sharp peaks. These peaks can be resolved for large close binaries and sufficiently long observation times with a star coronagraph. Aims. Eclipses modulate the phase curve, having an orbital period 2 π ∕ ω , with a contribution from the relative motion in the binary plane of a period 2 π ∕Ω. This leads to a spectral structure with basis frequencies ω and Ω. We aim to characterize these spectra. Methods. We studied the regime of short eclipses that occur when the planet radii are small compared to the planet separation. We derived formulas for the peak amplitudes applicable to homogeneous (Lambertian) planet binaries in circular orbit with small inclination. Results. The effects of an eclipse and of double reflection appear as first- and second-order contributions (in planet radius over separation) in the reflection signal, respectively. Small peaks appear as observable side bands in the spectrum. Identical structures around m Ω are characteristic of short-duration eclipses. Deceasing side bands could indicate double reflection between companions. Conclusions. Fourier analysis of the light curve of non-transiting planets can be used to find planets and their moons. Difficulties in interpreting the structures arise for small planet separation and when there are several moons in mean-motion resonance.