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-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 similar to 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. This is sufficient to determine the longitudes of primary equatorial hot spots and cold spots, as well as the presence of secondary maxima/minima.