Stars Crushed by Black Holes. I. On the Energy Distribution of Stellar
  Debris in Tidal Disruption Events
  The distribution of orbital energies imparted into stellar debris following
the close encounter of a star with a supermassive black hole is the principal
factor in determining the rate of return of debris to the black hole, and thus
in determining the properties of the resulting lightcurves from such events. We
present simulations of tidal disruption events for a range of $\beta\equiv
r_{\rm t}/r_{\rm p}$ where $r_{\rm p}$ is the pericentre distance and $r_{\rm
t}$ the tidal radius. We perform these simulations at different spatial
resolutions to determine the numerical convergence of our models. We compare
simulations in which the heating due to shocks is included or excluded from the
dynamics. For $\beta \lesssim 8$ the simulation results are well-converged at
sufficiently moderate-to-high spatial resolution, while for $\beta \gtrsim 8$
the breadth of the energy distribution can be grossly exaggerated by
insufficient spatial resolution. We find that shock heating plays a
non-negligible role only for $\beta \gtrsim 4$, and that typically the effect
of shock heating is mild. We show that self-gravity can modify the energy
distribution over time after the debris has receded to large distances for all
$\beta$. Primarily, our results show that across a range of impact parameters,
while the shape of the energy distribution varies with $\beta$, the width of
the energy spread imparted to the bulk of the debris is closely matched to the
canonical spread, $\Delta E = GM_\bullet R_\star/r_{\rm t}^2$, for the range of
$\beta$ we have simulated.
