A Fully General, Non-Perturbative Treatment of Impulsive Heating

Impulsive encounters between astrophysical objects are usually treated using the distant tide approximation (DTA) for which the impact parameter, $b$, is assumed to be significantly larger than the characteristic radii of the subject, $r_{\mathrm{S}}$, and the perturber, $r_{\mathrm{P}}$. The perturber potential is then expanded as a multipole series and truncated at the quadrupole term. When the perturber is more extended than the subject, this standard approach can be extended to the case where $r_{\mathrm{S}} \ll b < r_{\mathrm{P}}$. However, for encounters with $b$ of order $r_{\mathrm{S}}$ or smaller, the DTA typically overpredicts the impulse, $\Delta \mathbf{v}$, and hence the internal energy change of the subject, $\Delta E_{\mathrm{int}}$. This is unfortunate, as these close encounters are the most interesting, potentially leading to tidal capture, mass stripping, or tidal disruption. Another drawback of the DTA is that $\Delta E_{\mathrm{int}}$ is proportional to the moment of inertia, which diverges unless the subject is truncated or has a density profile that falls off faster than $r^{-5}$. To overcome these shortcomings, this paper presents a fully general, non-perturbative treatment of impulsive encounters which is valid for any impact parameter, and not hampered by divergence issues, thereby negating the necessity to truncate the subject. We present analytical expressions for $\Delta \mathbf{v}$ for a variety of perturber profiles, apply our formalism to both straight-path encounters and eccentric orbits, and discuss the mass loss due to tidal shocks in gravitational encounters between equal mass galaxies.