Distant TNO Inclinations as a Constraint on Primordial Cluster Perturbations in the Presence of Planet Nine

The Sun was almost certainly born in a stellar cluster, implying some degree of external forcing from neighboring stars during the Solar System's infancy. Published estimates of the strongest relevant stellar encounter, however, span a broad range, from relatively gentle perturbations to violently disruptive flybys. The modest inclination dispersion of the distant trans-Neptunian population has previously been used to argue that strong primordial encounters were unlikely and that the outer Solar System was not violently stirred at birth. In this work, we independently examine this constraint and ask whether this conclusion holds in the presence of Planet Nine. Because Planet Nine can reshape the distant trans-Neptunian population over gigayear timescales, the extent to which the present-day inclination distribution preserves a direct record of primordial cluster excitation is a priori unclear. To address this question, we compare long-term N-body integrations in which distant test particles begin either in strongly cluster-perturbed configurations or in comparatively quiescent ones, and are then evolved under the influence of the known giant planets, the Galactic tide, the effects of passing stars, and Planet Nine. We find that Planet Nine does not transform a highly excited primordial population into one resembling the observed low-inclination distant sample. Instead, comparisons between the simulated and the observed distant TNO populations indicate that the observed distant TNOs are most consistent with a relatively mildly perturbed birth environment, even in the presence of Planet Nine. If this comparatively low inclination dispersion is confirmed to be an intrinsic feature of the distant TNO population, rather than an artifact of observational bias, it constitutes a robust constraint on the severity of primordial cluster perturbations.