2013MNRAS.434.2465C

Distant stars and planets will remain spatially unresolved for the foreseeable future. It is nonetheless possible to infer aspects of their brightness markings and viewing geometries by analysing disc-integrated rotational and orbital brightness variations. We compute the harmonic light curves, F_l^m(t), resulting from spherical harmonic maps of intensity or albedo, Y_l^m(θ,φ), where l and m are the total and longitudinal orders. It has long been known that many non-zero maps have no light curve signature, e.g. odd l > 1 belong to the nullspace of harmonic thermal light curves. We show that the remaining harmonic light curves exhibit a predictable inclination dependence. Notably, odd m > 1 are present in an inclined light curve, but not seen by an equatorial observer. We therefore suggest that the Fourier spectrum of a thermal light curve may be sufficient to determine the orbital inclination of non-transiting short-period planets, the rotational inclination of stars and brown dwarfs, and the obliquity of directly imaged planets. In the best-case scenario of a nearly edge-on geometry, measuring the m = 3 mode of a star's rotational light curve to within a factor of 2 provides an inclination estimate good to ±6°, assuming that stars have randomly distributed spots. Alternatively, if stars have brightness maps perfectly symmetric about the equator, their light curves will have no m = 3 power, regardless of orientation. In general, inclination estimates will remain qualitative until detailed hydrodynamic simulations and/or occultation maps can be used as a calibrator. We further derive harmonic reflected light curves for tidally locked planets; these are higher-order versions of the well-known Lambert phase curve. We show that a non-uniform planet may have an apparent albedo 25 per cent lower than its intrinsic albedo, even if it exhibits precisely Lambertian phase variations. Finally, we provide low-order analytic expressions for harmonic light curves that can be used for fitting observed photometry; as a general rule, edge-on solutions cannot simply be scaled by sini to mimic inclined light curves.