Inverting Phase Functions to Map Exoplanets

We describe how to generate a longitudinal brightness map for a tidally locked exoplanet from its phase function light curve. We operate under a number of simplifying assumptions, neglecting limb darkening/brightening, star spots, detector ramps, as well as time-variability over a single planetary rotation. We develop the transformation from a planetary brightness map to a phase function light curve and simplify the expression for the case of an edge-on system. We introduce two models--composed of longitudinal slices of uniform brightness, and sinusoidally varying maps, respectively--which greatly simplify the transformation from map to light curve. We discuss numerical approaches to extracting a longitudinal map from a phase function light curve, explaining how to estimate the uncertainty in a computed map and how to choose an appropriate number of fit parameters. We demonstrate these techniques on a simulated map and discuss the uses and limitations of longitudinal maps. The sinusoidal model provides a better fit to the planet's underlying brightness map, although the slice model is more appropriate for light curves which only span a fraction of the planet's orbit. Regardless of which model is used, we find that there is a maximum of ~5 free parameters which can be meaningfully fit based on a full phase function light curve, due to the insensitivity of the latter to certain modes of the map.