TYCHO -- Realistically Simulating Exoplanets within Stellar Clusters I:
  Improving the Monte Carlo Approach
  To fully understand the diverse population of exoplanets, we must study their
early lives within open clusters, the birthplace of most stars with masses
$>0.5M_\odot$ (including those currently in the field). Indeed, when we observe
planets within clustered environments, we notice highly eccentric and odd
systems that suggest the importance of dynamical pathways created by
interactions with additional bodies (as in the case of HD 285507b). However, it
has proven difficult to investigate these effects, as many current numerical
solvers for the multi-scale $N$-body problem are simplified and limited in
scope. To remedy this, we aim to create a physically complete computational
solution to explore the role of stellar close encounters and interplanetary
interactions in producing the observed exoplanet populations for both open
cluster stars and field stars. We present a new code, TYCHO, which employs a
variety of different computational techniques, including multiple $N$-body
integration methods, close encounter handling, modified Monte Carlo scattering
experiments, and a variety of empirically informed initial conditions. We
discuss the methodology in detail, and its implementation within the AMUSE
software framework. Approximately 1$\%$ of our systems are promptly disrupted
by star-star encounters contributing to the rogue planets occurrence rate.
Additionally, we find that close encounters which perturb long-period planets
lead to 38.3$\%$ of solar-system-like planetary systems becoming long-term
unstable.
