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Stars Crushed by Black Holes. II. A Physical Model of Adiabatic Compression and Shock Formation in Tidal Disruption Events

We develop a Newtonian model of a deep tidal disruption event (TDE), for which the pericenter distance of the star, $r_{\rm p}$, is well within the tidal radius of the black hole, $r_{\rm t}$, i.e., when $β\equiv r_{\rm t}/r_{\rm p} \gg 1$. We find that shocks form for $β\gtrsim 3$, but they are weak (with Mach numbers $\sim 1$) for all $β$, and that they reach the center of the star prior to the time of maximum adiabatic compression for $β\gtrsim 10$. The maximum density and temperature reached during the TDE follow much shallower relations with $β$ than the previously predicted $ρ_{\rm max} \propto β^3$ and $T_{\rm max} \propto β^2$ scalings. Below $β\simeq 10$, this shallower dependence occurs because the pressure gradient is dynamically significant before the pressure is comparable to the ram pressure of the freefalling gas, while above $β\simeq 10$ we find that shocks prematurely halt the compression and yield the scalings $ρ_{\rm max} \propto β^{1.62}$ and $T_{\rm max} \propto β^{1.12}$. We find excellent agreement between our results and high-resolution simulations. Our results demonstrate that, in the Newtonian limit, the compression experienced by the star is completely independent of the mass of the black hole. We discuss our results in the context of existing (affine) models, polytropic vs.~non-polytropic stars, and general relativistic effects, which become important when the pericenter of the star nears the direct capture radius, at $β\sim 12.5$ (2.7) for a solar-like star disrupted by a $10^6M_{\odot}$ ($10^{7}M_{\odot}$) supermassive black hole.