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

TL;DR

This paper develops a harmonic analysis framework for interpreting brightness variations of unresolved stars and exoplanets, enabling estimation of their inclination, obliquity, and albedo from lightcurve data.

Contribution

It introduces a method to infer viewing geometries and surface properties of stars and exoplanets using harmonic lightcurve analysis, including new insights into the effects of inclination and surface symmetry.

Findings

Fourier spectrum of lightcurves can determine orbital inclination of non-transiting planets.

Measuring the m=3 mode can estimate stellar inclination within +/- 6 degrees.

Non-uniform planets can appear to have 25% lower albedo than their true value.

Abstract

[Abridged] 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 analyzing disk-integrated rotational and orbital brightness variations. We compute the harmonic lightcurves, F_l^m(t), resulting from spherical harmonic maps of intensity or albedo, Y_l^m(theta,phi), where l and m are the total and longitudinal order. Notably, odd m>1 are present in an inclined lightcurve, but not seen by an equatorial observer. We therefore suggest that the Fourier spectrum of a thermal lightcurve 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 lightcurve to within a factor of two provides an inclination estimate good to +/- 6 degrees, assuming stars have randomly distributed spots. Alternatively, if stars have brightness maps perfectly symmetric about the equator, their lightcurves 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 lightcurves 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% lower than its intrinsic albedo, even if it exhibits precisely Lambertian phase variations. 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.