Linking long-term planetary $N$-body simulations with periodic orbits: application to white dwarf pollution

Abstract

Mounting discoveries of debris discs orbiting newly-formed stars and white dwarfs (WDs) showcase the importance of modeling the long-term evolution of small bodies in exosystems. WD debris discs are in particular thought to form from very long-term (0.1-5.0 Gyr) instability between planets and asteroids. However, the time-consuming nature of $N$-body integrators which accurately simulate motion over Gyrs necessitates a judicious choice of initial conditions. The analytical tools known as \textit{periodic orbits} can circumvent the guesswork. Here, we begin a comprehensive analysis directly linking periodic orbits with $N$-body integration outcomes with an extensive exploration of the planar circular restricted three-body problem (CRTBP) with an outer planet and inner asteroid near or inside of the $2$:$1$ mean motion resonance. We run nearly 1000 focused simulations for the entire age of the Universe (14 Gyr) with initial conditions mapped to the phase space locations surrounding the unstable and stable periodic orbits for that commensurability. In none of our simulations did the planar CRTBP architecture yield a long-timescale ($\gtrsim 0.25$% of the age of the Universe) asteroid-star collision. The pericentre distance of asteroids which survived beyond this timescale ($\approx 35$ Myr) varied by at most about 60%. These results help affirm that collisions occur too quickly to explain WD pollution in the planar CRTBP $2$:$1$ regime, and highlight the need for further periodic orbit studies with the eccentric and inclined TBP architectures and other significant orbital period commensurabilities.

Accepted for publication in MNRAS

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• Numerical Simulations of Collisional Cascades at the Roche Limits of White Dwarf Stars
• Explaining the variability of WD 1145+017 with simulations of asteroid tidal disruption
• The Fate of Exomoons in White Dwarf Planetary Systems
• Numerical Simulations of Gaseous Disks Generated from Collisional Cascades at the Roche Limits of White Dwarf Stars
• Post-main-sequence debris from rotation-induced YORP break-up of small bodies II: multiple fissions, internal strengths and binary production
• Binary star influence on post-main-sequence multi-planet stability
• The dynamical history of the evaporating or disrupted ice giant planet around white dwarf WD J0914+1914
• The critical binary star separation for a planetary system origin of white dwarf pollution
• Constraining planet formation around 6$M_{\odot }$-8$M_{\odot }$ stars
• The unstable fate of the planet orbiting the A-star in the HD 131399 triple stellar system