Thawing the frozen-in approximation: implications for self-gravity in deeply plunging tidal disruption events

The tidal destruction of a star by a massive black hole, known as a tidal disruption event (TDE), is commonly modeled using the "frozen-in" approximation. Under this approximation, the star maintains exact hydrostatic balance prior to entering the tidal sphere (radius $r_{\rm t}$), after which point its internal pressure and self-gravity become instantaneously negligible and the debris undergoes ballistic free fall. We present a suite of hydrodynamical simulations of TDEs with high penetration factors $\beta \equiv r_{\rm t}/r_{\rm p} = 5-7$, where $r_{\rm p}$ is the pericenter of the stellar center of mass, calculated using a Voronoi-based moving-mesh technique. We show that basic assumptions of the frozen-in model, such as the neglect of self-gravity inside $r_{\rm t}$, are violated. Indeed, roughly equal fractions of the final energy spread accumulate exiting and entering the tidal sphere, though the frozen-in prediction is correct at the order-of-magnitude level. We also show that an $\mathcal{O}(1)$ fraction of the debris mass remains transversely confined by self-gravity even for large $\beta$ which has implications for the radio emission from the unbound debris and, potentially, for the circularization efficiency of the bound streams.