Lightcurves of Stars & Exoplanets: Estimating Inclination, Obliquity, and Albedo

N. B. Cowan, P. A. Fuentes, and H. M. Haggard

MNRAS 434 (2465), 2013.

Exoplanets will remain spatially unresolved for the foreseeable future. It is in principle possible to determine a planet's brightness markings by analyzing its disk-integrated brightness variations, in either thermal or reflected light. In order to identify the degeneracies inherent in these inverse problems, we develop analytic solutions to the associated forward problems. Our approach is to compute the "harmonic lightcurves" resulting from spherical harmonic maps of intensity or albedo. The convolutions often contain a nullspace: a class of non-zero maps that have no lightcurve signature. We derive harmonic thermal lightcurves for both equatorial and inclined observers. The nullspace for these two situations is significantly different, with odd modes being present in the latter case, but not the former. We therefore suggest that the Fourier spectrum of a thermal lightcurve is 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 rotator, factor-of-two measurements of the amplitudes of odd modes in the thermal lightcurve provide an inclination estimate good to a few degrees. In general, however, inclination estimates will remain qualitative until detailed hydrodynamic simulations and/or occultation maps can be used as a calibrator. We further derive harmonic reflected lightcurves for tidally-locked planets; these are higher-order versions of the well-known Lambert phase curve. We show that a non-uniform diffusely-reflecting planet with a precisely Lambertian phase curve may have planetary and Bond albedos significantly different from that inferred if the planet is assumed to be uniform. Lastly, we provide low-order analytic expressions for harmonic lightcurves 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 lightcurves.

Full text: CoFuHaLightcurvesStarsExoplanets.pdf