Orbital obliquity sampling in the Kepler-20 system using the 3D animation software Blender

Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany
e-mail: hmueller@hs.uni-hamburg.de

Received: 24 June 2021
Accepted: 3 August 2021

Abstract

Context. The mutual orbital alignment in multiple planetary systems is an important parameter for understanding their formation. There are a number of elaborate techniques to determine the alignment parameters using photometric or spectroscopic data. Planet–planet occultations (PPOs), which can occur in multiple transiting systems, are one intuitive example. While the presence of PPOs constrains the orbital alignment, the absence at first glance does not.

Aims. Planetary systems, for which the measurement of orbital obliquities with conventional techniques remains elusive, call for new methods whereby at least some information on the alignments can be obtained. Here we develop a method that uses photometric data to gain this kind of information from multi-transit events.

Methods. In our approach we synthesize multi-transit light curves of the exoplanets in question via the construction of a grid of projected orbital tilt angles α, while keeping all transit parameters constant. These model light curves contain PPOs for some values of α. To compute the model light curves, we use the 3D animation software Blender for our transit simulations, which allows the use of arbitrary surface brightness distributions of the star, such as limb darkening from model atmospheres. The resulting model light curves are then compared to actual measurements.

Results. We present a detailed study of the multi-transiting planetary system Kepler-20, including parameter fits of the transiting planets and an analysis of the stellar activity. We apply our method to Kepler-20 b and c, where we are able to exclude some orbital geometries, and find a tendency of these planets to eclipse in front of different stellar hemispheres in a prograde direction.

Conclusions. Despite the low statistical significance of our results in the case of Kepler-20, we argue that our method is valuable for systems where PPO signals larger than the noise can occur. According to our analysis, noise ≤ 2 × 10⁻⁴ for planets like Kepler-20 b, or a planet radius ≥ 3 R_Earth for the smaller component and Kepler-20-like photometry, would be sufficient to achieve significant results.