Time Variation of Kepler Transits Induced By Stellar Rotating Spots—a Way to Distinguish between Prograde and Retrograde Motion. I. Theory

摘要

Some transiting planets discovered by the Kepler mission display transit timing variations (TTVs) induced by stellar spots that rotate on the visible hemisphere of their parent stars. An induced TTV can be observed when a planet crosses a spot and modifies the shape of the transit light curve, even if the time resolution of the data does not allow the detection of the crossing event itself. We present an approach that can, in some cases, use the derived TTVs of a planet to distinguish between a prograde and a retrograde planetary motion with respect to the stellar rotation. Assuming a single spot darker than the stellar disk, spot crossing by the planet can induce measured positive (negative) TTV, if the crossing occurs in the first (second) half of the transit. On the other hand, the motion of the spot toward (away from) the center of the stellar visible disk causes the stellar brightness to decrease (increase). Therefore, for a planet with prograde motion, the induced TTV is positive when the local slope of the stellar flux at the time of transit is negative, and vice versa. Thus, we can expect to observe a negative (positive) correlation between the TTVs and the photometric slopes for prograde (retrograde) motion. Using a simplistic analytical approximation, and also the publicly available SOAP-T tool to produce light curves of transits with spot-crossing events, we show for some cases how the induced TTVs depend on the local stellar photometric slopes at the transit timings. Detecting this correlation in Kepler transiting systems with high enough signal-to-noise ratio can allow us to distinguish between prograde and retrograde planetary motions. In upcoming papers we present analyses of the KOIs and Kepler eclipsing binaries, following the formalism developed here.