Light curves of stars and exoplanets: estimating inclination, obliquity and albedo

Abstract

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, Fm l (t ), resulting from spherical harmonic maps of intensity or albedo, Y m l (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 higherorder 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 sin i to mimic inclined light curves.