# Exoplanet surface mapping

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## Abstract

In this thesis we consider the reconstruction of albedo maps of exoplanets. This is done with a new variant of spin-orbit tomography that has been described in [Cowan and Agol, 2008] and more in depth in [Fujii and Kawahara, 2012]. This method reconstructs the albedo map from the reﬂected-light curve, the total intensity of the light that originates from the host star and is reﬂected by the planet. In the mentioned papers, the surface map of the planet is modeled as a sum of ﬁnite sized surface elements with constant albedo, and the relation between this approximation of the map and the light-curve in the time domain is determined. In this report, we use that the signal is quasi periodic due to diurnal and annual motion, and work with the Fourier peaks of the light-curve. We also approximate the map in a diﬀerent way, writing it as the sum of spherical harmonics, and neglecting spherical harmonics with high spatial frequencies. This has the advantage that the relation can be worked out analytically (for edge-on and face-on observations) without the use of complex mathematics, and that both the surface map and the light-curve contain a daily frequency. We derive an equation for the reﬂective light-curve under the assumption that the surface map is not a function of time (no clouds), and that the reﬂection is Lambertian (equal in magnitude in all directions). This transformation is found to be a linear function of the surface map. This equation is worked out for edge-on and face-on observations with arbitrary axial tilt, which describes the orientation of the spin axis with respect to the observer and the orbital plane. Furthermore, we describe how to invert this relation if the axial tilt is known to the observer. We also aimed at recovering the map if the axial tilt is unknown to the observer, since this would make sure that the reconstruction does not rely on other observations. In contrast to what was found in papers like [Fujii and Kawahara, 2010] and [Fujii and Kawahara, 2012], we did not succeed in this. A number of methods were used for this. The ﬁrst two looked at the problem from a mathematical perspective: the minimization of the distance between the measured light-curve and the light-curve from the reconstructed map, and Tikhonov regularization. The two failed because both the column space and the singular values respectively are not a function of the axial tilt. The third method that has been treated and tested involved the maximization of the ‘amount’ of positive albedo on the reconstructed map, but a test showed that the distinction that this method makes is in the same order of magnitude as the numerical error, thus proving that this method was not useful as well. Further study might show what causes the results of the two methods to diﬀer in this respect