Classification of orbits in three-dimensional exoplanetary systems

Abstract

The three-dimensional version of the circular restricted problem of three bodies is utilized to describe a system comprising a host star and an exoplanet. The third body, playing the role of a test particle, can be a comet or an asteroid, or even a small exomoon. Combining the grid classification method with two-dimensional color-coded basin maps, we determine the nature of the motion of the test particle by distinguishing between collision, escaping, and bounded motion. In the case of ordered bounded motion, we also obtain the orientation (retrograde or prograde) as well as the geometry (circulating around one or both of the two main bodies) of the trajectories of the third body, which starts from either the pericenter or apocenter. Following this approach, we are able to systematically explore the dependence of the motion type of the test particle on the initial values of the semimajor axis, eccentricity, and inclination of its orbit.

Two fast and accurate routines for solving the elliptic Kepler equation for all values of the eccentricity and mean anomaly

“…A common approach for solving Eq. ( 1) is using the Newton-Raphson root-finding scheme, which is an efficient choice for a single computation of E. However, in many applications, such as the search for exoplanets or modeling of their formation (e.g., Kane et al 2012;Brady et al 2018;Mills et al 2019;Sartoretti & Schneider 1999), Markov chain Monte Carlo sampling methods of multiplanetary systems (Gregory 2010;Ford 2006;Borsato et al 2014;Zotos et al 2021), large sky surveys (Leleu et al 2021;Worden et al 2017), or studies of high eccentricity orbits (Ciceri et al 2015;Sotiriadis et al 2017), KE must be solved an exceedingly large number of times (Eastman et al 2019;Makarov & Veras 2019). In such highly demanding computational tasks, the repetitive solution of KE may become the slowest step and therefore the bottleneck.…”