Can insights from Triple Evolution be Applied to Moons? # TRENDY3 Angel Bashyal 1 Aayush Gautam 1 Suman Satyal 2 Shree Krishna Bhattarai 3 1Tribhuvan University 2University of Texas at Arlington 3University of Carolina at Charlotte

INTRODUCTION Chaos Indicator MEGNO Orbital Evolution

Direct orbital simulations were performed to generate time series plots. This project studied the feasibility of two hypothetical exomoons, Earth-sized (ME) and Mean Exponential Growth of Nearby Orbits (MEGNO) criterion helps us predict the Moon-sized , around HD 100777 b (Laligurans), orbiting a Sun-like HD dynamical stability of the exomoon. (MM) Points taken within chaotic(red) and quasiperiodic(green) regions. 100777(Sagarmatha). MEGNO value Y = α × t + β determines the chaos present for the certain orbital The orbital integrations were performed with the suite of N-body integrators provided in configuration. REBOUND python library [3]. For a quasi-periodic initial condition, α ' 0.0 and β ' 2.0 (so Y → 2.0) for t → ∞. If the orbit is chaotic, then Y → λt/2 for t → ∞. [1] Here λ is the Maximum Lyapunov Exponent (MLE) of the orbit.

Orbital Simulations

The system was modelled as a Restricted Three Body Problem (RTBP) of the star, and moon. [2] First we performed MEGNO value calculations over a 100x100 grid for an integration period of 10,000 . The lower MEGNO value represents greater stability. MEGNO value of two implies quasi-periodicity while values near and greater than 4 imply chaos. [4]

Figure 1. Jupiter type exoplanet HD 100777b orbiting Sun-like star HD 100777. Source : SBS Nepali website.

. Time series of semi major axis and eccentricity of MM exomoons within stable regions, on the Orbital Stability left, and of ME within unstable region, on the right.

Hill sphere approximates the farthest gravitational influence a planet can have on its moons. Roche’s limit is the minimum distance from centre of the planet that satellite can maintain stable orbits. RESULTS and CONCLUSIONS Orbital Resonance is a numerical relationship between periods of different planetary bodies. Time series of semi-major axis and eccentricity of both moons show stable orbits within the green region for the full 10 Million years. Ejections have been shown in red region within . 10,000 years.The results for stability and instability(ejection/collision) are in line with predic- tions by MEGNO. MOTIVATION MEGNO maps for Earth-sized and Moon-sized exomoons. The moons show stable behavior from own Roche limits (0.001AU) upto the Hill’s radius of Total number of confirmed exoplanets: 4,695 as of March 17. But no exomoon yet! planet ≈ 0.021AU.Chaotic behavior starts from there onwards for both moons. Best config- Only one candidate currently, Kepler 1625b-I (Teachey & Kipping 2018) around a Jupiter uration for both exomoons to show quasiperiodic orbits was for semi-major axis ≈ 0.01 to planet around Sun-like star. 0.012 AU. Cyclotron radio emission, photometric transit timing are promising methods for their discovery, but still to achieve required precision. References The orbital stability of such exomoon candidates can be predicted with computational simulations. [1] K. Gozdziewski et.al. Global dynamics of planetary systems with the megno criterion. AA, November 2001. [2] C. D. Murray and S. F. Dermott. Solar System Dynamics. CUP, 1st edition, 2000. [3] H. Rein and S.-F. Liu. Rebound: an open-source multi-purpose n-body code for collisional dynamics. Astronomy Astrophysics, 537:A128z, Jan 2012. [4] B. Hinse T. C Satyal, S.and Quarles. Figure 2. Two-dimensional plot of Earth-sized exomoon provides further confirmation of stable orbits. Application of chaos indicators in the study of dynamics of s-type extrasolar in stellar binaries. MNRAS, May 2013.

This research was supported in part through computational resources provided by the Kathmandu University Supercomputer Centre, which was established with equipment donated by CERN.