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 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 .