Time variation of Kepler transits induced by stellar rotating spots - a way to distinguish between prograde and retrograde motion I. Theory

TL;DR

This paper proposes a method to determine whether a transiting planet moves in a prograde or retrograde orbit relative to its star's rotation by analyzing transit timing variations caused by stellar spots.

Contribution

It introduces a novel approach linking TTVs and stellar brightness slopes to distinguish planetary orbital directions, supported by analytical and simulation models.

Findings

TTVs correlate with stellar flux slopes differently for prograde and retrograde orbits.

Spot crossing can induce measurable positive or negative TTVs.

The method can be applied to Kepler data with sufficient signal-to-noise ratio.

Abstract

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 to detect 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 disc, 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 towards (away from) the center of the stellar visible disc 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 coming papers we present analyses of the KOIs and Kepler eclipsing binaries, following the formalism developed here.