Tidal Disruption Events in Active Galactic Nuclei

The stream pericenter distance of typical TDEs is only tens of ${r}_{{\rm{g}}}$ (Section 2). Given other approximations used, it is reasonable to neglect relativistic effects. The hydrodynamics equations are

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+{\rm{\nabla }}\cdot (\rho {\boldsymbol{v}})=0,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(\rho {\boldsymbol{v}})+{\rm{\nabla }}\cdot (\rho {\boldsymbol{vv}}+p{\mathsf{I}})=-\rho {\rm{\nabla }}{\rm{\Phi }},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial E}{\partial t}+{\rm{\nabla }}\cdot [(E+p){\boldsymbol{v}}]=-\rho {\boldsymbol{v}}\cdot {\rm{\nabla }}{\rm{\Phi }}.\end{eqnarray} \tag{ 11 }$$

Here ρ, ${\boldsymbol{v}}$, and p are density, velocity, and pressure, ${\rm{\Phi }}(R,z)=-{{GM}}_{{\rm{h}}}/{\left({R}^{2}+{z}^{2}\right)}^{1/2}$ is the gravitational potential of the black hole, and ${\mathsf{I}}$ is the isotropic rank-two tensor. We use an adiabatic equation of state. The total energy is $E=\tfrac{1}{2}\rho {v}^{2}+\rho e$, where $\rho e=p/(\gamma -1)$ is the internal energy. Disk pressure can be dominated by gas or radiation (Section 2); for simplicity, we choose the adiabatic index to be $\gamma =\tfrac{5}{3}$. As we shall see later in Section 5.3.1, the thermodynamic conditions of a realistic disk is likely more complicated than a simple choice between gas and radiation pressure.

Simulation quantities are expressed in code units of length r₀, time ${{\rm{\Omega }}}_{0}^{-1}\equiv {\left({{GM}}_{{\rm{h}}}/{r}_{0}^{3}\right)}^{-1/2}$, velocity ${v}_{0}\equiv {r}_{0}{{\rm{\Omega }}}_{0}$, and density ${\rho }_{0}$. Because Newtonian gravity, unlike relativistic gravity, is scale-free, the dimensionless versions of Equations (9)–(11) in this unit system are independent of r₀ and ${\rho }_{0}$. With an appropriate choice of these two quantities, our results can be scaled to the conditions of any particular TDE.

We set ${r}_{0}={r}_{{\rm{p}}}$, the characteristic length scale of the system. Time is reported as the number of disk orbits at $R={r}_{0}$, or disk orbits for short; one disk orbit is $2\pi {({r}_{{\rm{p}}}/{r}_{{\rm{t}}})}^{3/2}$ times the stellar dynamical time. The mass return time in Equation (2) is

$$\begin{eqnarray}&&{T}_{\mathrm{mb}}\sim 400\,{\left(\displaystyle \frac{{M}_{{\rm{h}}}}{{10}^{6}{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{-1/2}\times {\left(\displaystyle \frac{{r}_{{\rm{p}}}}{{r}_{{\rm{t}}}}\right)}^{3/2}\,\mathrm{disk}\ \mathrm{orbits}.\end{eqnarray} \tag{ 12 }$$

The value of ${\rho }_{0}$ will be determined in Section 3.5.

When translating simulation results from code units to physical units, we adopt the same fiducial parameters as in Section 2, to wit, ${M}_{{\rm{h}}}={10}^{6}{M}_{\odot }$, ${M}_{\star }={M}_{\odot }$, ${r}_{\star }={R}_{\odot }$, ${r}_{{\rm{p}}}={r}_{{\rm{t}}}$, ${R}_{{\rm{s}}}={r}_{\star }$, $\alpha =0.1$, and $\eta =0.1$.

The initial disk has a constant midplane density ${\rho }_{0}$ and a Gaussian scale height $\propto R$:

$$\begin{eqnarray}&&\rho (R,z)={\rho }_{0}\exp [-{\rm{\Delta }}{\rm{\Phi }}/({{ \mathcal H }}^{2}{v}_{{\rm{K}}}^{2})].\end{eqnarray} \tag{ 13 }$$

Here ${\rm{\Delta }}{\rm{\Phi }}(R,z)\equiv {\rm{\Phi }}(R,z)-{\rm{\Phi }}(R,0)$ is the gravitational potential difference from the midplane and is $\approx \tfrac{1}{2}{v}_{{\rm{K}}}^{2}{\left(z/R\right)}^{2}$ for our spherically symmetric Φ, where ${v}_{{\rm{K}}}^{2}(R)\equiv {\left(\partial {\rm{\Phi }}/\partial \mathrm{ln}R\right)}_{z=0}$ is the square of the midplane Keplerian orbital velocity. The aspect ratio of the simulated disk is ${ \mathcal H }=0.1;$ it is much larger than the aspect ratio of a Shakura & Sunyaev (1973) disk for numerical reasons, and the effect of this choice will be considered in Section 3.5. The pressure of the disk is $p(R,z)=\rho {{ \mathcal H }}^{2}{v}_{{\rm{K}}}^{2}$. Its orbital velocity, given by

$$\begin{eqnarray}&&{v}_{\phi }^{2}(R,z)={v}_{{\rm{K}}}^{2}+({\rm{\Delta }}{\rm{\Phi }}+{{ \mathcal H }}^{2}{v}_{{\rm{K}}}^{2})(\partial \mathrm{ln}{v}_{{\rm{K}}}^{2}/\partial \mathrm{ln}R),\end{eqnarray} \tag{ 14 }$$

is slightly sub-Keplerian to counteract pressure forces.

Our initial disk is not a Shakura & Sunyaev (1973) disk. For example, its surface density profile is

$$\begin{eqnarray}&&{\rm{\Sigma }}(R)={\int }_{0}^{{z}_{\max }}{dz}\,\rho \approx {\left(\tfrac{1}{2}\pi \right)}^{1/2}{ \mathcal H }{\rho }_{0}R,\end{eqnarray} \tag{ 15 }$$

where ${z}_{\max }$ is the distance of the vertical boundaries of the simulation domain from the midplane. This surface density profile is neither the ${\rm{\Sigma }}\propto {R}^{-3/5}$ profile of a gas-dominated disk nor the ${\rm{\Sigma }}\propto {R}^{3/2}$ profile of a radiation-dominated disk, although it is close to the latter. The advantage of our initial disk is that it is scale-free, so we can scale our results to fit any TDE of interest. As long as we pick ${\rho }_{0}$ such that ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}$ is the same for our initial disk and for a Shakura & Sunyaev (1973) disk, our results are not qualitatively affected.

Ideally, we would like our stream to travel on a parabolic trajectory vertical to the midplane: The stream would approach pericenter from $\phi =\pi$, cross the axis, reach pericenter at $(R,\phi ,z)=({r}_{0},0,0)$, and return to infinity along $\phi =\pi$. Unfortunately, the trajectory would then cross the cylindrical cutout in the center of the simulation domain. We solve this problem in two ways. For lighter streams that are effectively stopped by the disk, we can simply lower the upper-vertical boundary of the simulation domain until the stream intersects the boundary at $\phi =0;$ if the stream is injected as a boundary condition from the intersection, it will not encounter the cutout before it terminates at pericenter. For heavier streams that can punch through the disk, we additionally rotate the trajectory 0.15 rad from the vertical to make it avoid the cutout altogether. The sense of the rotation is to make the trajectory prograde with respect to the disk, increasing the likelihood that the outgoing material will miss the cutout. Since the vertical and inclined streams differ so little in inclination, we treat them as directly comparable. We leave the exploration of streams of other spatial orientations to future work.

We choose the stream boundary condition so that, if the stream traveled ballistically to pericenter, its cross section there would be circular and its transverse density profile would be a Gaussian of radius ${ \mathcal W }{r}_{0}$, where ${ \mathcal W }=0.02$. The simulated stream width is slightly wider than in typical TDEs. The density at the center of the Gaussian is set by matching the desired value of ${\dot{M}}_{{\rm{s}}}$, and the pressure is ${10}^{-6}\,{v}_{0}^{2}$ times density. Tracing orbits back to the upper-vertical boundary determines the stream density, velocity, and pressure there.

The disk mass current is ${\dot{M}}_{{\rm{d}}}\approx 0.01\,{\rho }_{0}{r}_{0}^{2}{v}_{0}$. We use ${\dot{M}}_{{\rm{s}}}\in \{0.5,1,2,4,8\}\times {10}^{-3}\,{\rho }_{0}{r}_{0}^{2}{v}_{0}$ for the vertical stream and ${\dot{M}}_{{\rm{s}}}\in \{1,2,4\}\times {10}^{-2}\,{\rho }_{0}{r}_{0}^{2}{v}_{0}$ for the inclined stream. Altogether, we have ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\in \{\approx 0.05,\approx 0.1,\approx 0.2,\approx 0.4,\approx 0.8,\approx 1,\approx 2,\approx 4\}$. Combinations of ${M}_{{\rm{h}}}$ and ${L}_{{\rm{a}}}$ giving such ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}$ can be read off from Figure 1. Only the heaviest stream pertains to our fiducial parameters (Section 3.3); the lighter streams are also valid if ${M}_{{\rm{h}}}$ is larger or α is smaller.

The initial disk and the stream have respectively $\left|j\right|\,\lesssim 0.2\,{r}_{0}{v}_{0}$ and $j\gtrsim 1.4\,{r}_{0}{v}_{0}$, where j is the specific angular momentum in the $\phi =\tfrac{1}{2}\pi$ direction. For the purpose of separating disk-like gas from stream-like gas when analyzing our results in Section 4, we adopt the conservative cutoff $j=1.1\,{r}_{0}{v}_{0}$.

The aspect ratio of a Shakura & Sunyaev (1973) disk at stream pericenter, ${\left(H/R\right)}_{{\rm{d}}}\sim 0.003$, and the stream width, ${R}_{{\rm{s}}}\sim {r}_{\star }$, are both difficult to resolve spatially; this is why the simulated disk height and stream width are artificially increased to ${ \mathcal H }{r}_{0}$ and ${ \mathcal W }{r}_{0}$ respectively, with ${ \mathcal H }=0.1$ and ${ \mathcal W }=0.02$ (Section 3.4). To preserve the mass currents under such a modification, we simultaneously adjust the midplane density of the disk and the central density of the stream. We demand that the simulated disk mass current, $4{ \mathcal W }{r}_{0}{\rm{\Sigma }}({r}_{0}){v}_{0}$, be equal to ${\dot{M}}_{{\rm{d}}}$ of a realistic disk. This, together with Equations (3) and (15), yields

$$\begin{eqnarray}&&{\rho }_{0}=\displaystyle \frac{1}{6{\pi }^{3/2}{ \mathcal H }{ \mathcal W }}\displaystyle \frac{{M}_{\star }}{{r}_{\star }^{3}}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{{\rm{h}}}}\right)}^{3/2}{\left(\displaystyle \frac{{r}_{{\rm{p}}}}{{r}_{{\rm{t}}}}\right)}^{-9/2}{\left({\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\right)}^{-1}.\end{eqnarray} \tag{ 16 }$$

Similarly, the central density of the stream is determined by matching the desired ${\dot{M}}_{{\rm{s}}}$. Because the mass currents involved in the collision are independent of the particular values of ${ \mathcal H }$ and ${ \mathcal W }$ provided both are $\ll 1$, the outcome of the collision should not be severely affected by our choices for these quantities. Volume-integrated quantities in our simulations, such as the inflow rate (Section 4.2.2) and the energy dissipation rate (Section 4.3), are likewise genuine. Although the simulated disk is unrealistically thick at the beginning, any structure created in the course of the simulations with aspect ratio $\gg { \mathcal H }$ is realistic because its thickness is due to the injection of internal energy much greater than the artificial internal energy in the initial condition.

Periodic boundary conditions are used for the azimuthal direction. Outflow boundary conditions are used for the radial and vertical directions: Velocity is copied from the last physical cell into the ghost zone and inward-pointing velocity components are zeroed, while density and pressure are isothermally extrapolated such that the pressure gradient balances gravitational and centrifugal forces in the ghost zone. On the upper-vertical boundary, the stream injection condition (Section 3.4) supersedes this boundary condition wherever the former predicts a larger density.

The numerical vacuum is an axisymmetric hydrostatic torus. Its density is determined by the constraint

$$\begin{eqnarray}&&\mathrm{constant}\,=\,-{\rm{\Phi }}(R,z)+\displaystyle \frac{{\left({v}_{\phi }^{2}\right)}_{{\rm{v}}}}{2-2q}-K\displaystyle \frac{{\rm{\Gamma }}}{{\rm{\Gamma }}-1}{\rho }_{{\rm{v}}}^{{\rm{\Gamma }}-1},\end{eqnarray} \tag{ 17 }$$

its velocity is ${{\boldsymbol{v}}}_{{\rm{v}}}={v}_{0}{\left(R/{r}_{0}\right)}^{1-q}\,{\hat{{\boldsymbol{e}}}}_{\phi }$, and its pressure is ${p}_{{\rm{v}}}=K{\rho }_{{\rm{v}}}^{{\rm{\Gamma }}}$. The parameters in these equations are q = 1.75, $K=0.5\,{\rho }_{0}^{1-{\rm{\Gamma }}}{v}_{0}^{2}$, and ${\rm{\Gamma }}=0.9;$ the constant on the left-hand side of Equation (17) follows from requiring that ${\rho }_{{\rm{v}}}$ attain a maximum of ${10}^{-6}{\rho }_{0}$ at $(R,z)=({r}_{0},0)$. As the simulation progresses, we keep the density and pressure of every cell greater than or equal to its vacuum value at all times. In addition, whenever the density of a cell drops to $0\lt \xi \lt 1$ times vacuum, we simultaneously modify its velocity to $\xi {\boldsymbol{v}}+(1-\xi ){{\boldsymbol{v}}}_{{\rm{v}}}$ so as to prevent velocities from erroneously growing in low-density regions.

The simulation domain spans $[0.15\,{r}_{0},5\,{r}_{0}]\times [-\pi ,\pi ]\,\times [-3.2\,{r}_{0},1.6\,{r}_{0}]$ in $(R,\phi ,z)$. The inner-radial boundary at $R=0.15\,{r}_{0}$ encloses a cylindrical cutout in the center of the simulation domain, introduced to exclude the coordinate singularity at R = 0. The radius of the cutout in physical units is $\approx 7\,{r}_{{\rm{g}}}\,{[{M}_{{\rm{h}}}/{({10}^{6}{M}_{\odot })]}^{-2/3}({M}_{\star }/{M}_{\odot })}^{-1/3}({r}_{\star }/{R}_{\odot })({r}_{{\rm{p}}}/{r}_{{\rm{t}}})$, which is just outside the innermost stable circular orbit of a nonrotating black hole; therefore, we are simulating the entire disk interior to the stream pericenter, and we may regard gas entering the cutout as falling into the black hole. The lower-vertical boundary is twice as far below the midplane as the upper-vertical boundary so that we can follow as much of the gas leaving the lower side of the disk as feasible.

The number of grid cells is $200\times 200\times 300$. We employ a power-law grid in the radial direction for which ${\rm{\Delta }}{R}_{i+1}/{\rm{\Delta }}{R}_{i}=1.01;$ this policy gives us cells with approximately square poloidal cross sections at $R={r}_{0}$. The pressure scale height of the disk at stream pericenter, $\approx \sqrt{2}{ \mathcal H }{r}_{0}$, is resolved with around nine vertical cells, and the stream width, ${ \mathcal W }{r}_{0}$, is barely resolved with around two azimuthal cells at the injection point.

The simulations run to 100 disk orbits which, according to Equation (12), is $\sim 0.3\,{T}_{\mathrm{mb}}\,{[{M}_{{\rm{h}}}/{({10}^{6}{M}_{\odot })]}^{-1/2}({M}_{\star }/{M}_{\odot })}^{1/2}{({r}_{{\rm{p}}}/{r}_{{\rm{t}}})}^{-3/2}$. This means our simulations focus on the time when the mass return rate is greatest, and the mass return rate varies little over the course of our simulations.

We use the ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 0.2$ simulation to illustrate how the disk in any simulation evolves in general (see movie). The left column of Figure 2 displays an early time, 10 disk orbits. The incoming stream is visible above the disk in the poloidal slice. The vertical structure near the center of the cylindrical slice is the part of the stream closest to pericenter; it is bent because the disk deflects the stream in the direction of disk rotation. The stream opens an annular gap in the disk at R ≈ r₀, manifest in the poloidal and midplane slices; the gap separates the inner disk (Section 4.2) at R ≤ r₀ from the outer disk.

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Nonaxisymmetric features appear in the midplane slice. The most salient one is the curved bow shock with its tip at $(R,\phi )\approx ({r}_{0},0.1\pi );$ here $\phi \gt 0$ because of stream deflection. The inner half of the bow shock extends inward, forming a prominent spiral shock in the inner disk. There are typically multiple spiral shocks; together, they drive an extremely powerful inflow that is orders of magnitude stronger than in the unperturbed disk (Section 4.2.1).

Nonaxisymmetric features in the outer disk can be understood with the help of the cylindrical slice. The incoming stream pushes stream and disk gas out of the other side of the disk. Part of this gas reaches as far as $z\approx -{r}_{0}$ before gravity pulls it back down to strike the disk at $\phi \approx -\tfrac{3}{4}\pi ;$ the impact compresses the disk and launches a spiral shock stretching outward from $(R,\phi )\approx (1.2\,{r}_{0},0.8\pi )$. The impacting gas glances off the disk and falls back to it once again at $\phi \approx \tfrac{3}{8}\pi$, launching another, much weaker, spiral feature. At the time shown, the weak spiral feature is hidden from view by the similarly located and much stronger bow shock, but it becomes more conspicuous at late times when the bow shock is weaker. Both spiral features are stationary in space.

The center column depicts an intermediate time, 25 disk orbits. The gap widens and the spiral shocks deplete the inner disk, as evidenced by the poloidal and midplane slices. The reduction of disk gas at $R\lesssim 1.2\,{r}_{0}$ is the reason why the stream suffers less deflection in the cylindrical slice. The collision heats the inner disk, causing its gas to puff up and move to larger radii, easily seen by comparing the poloidal slices of early and intermediate times.

The intermediate time is taken during an episode of disk evolution in which the outer edge of the gap, as witnessed in the midplane slice, becomes highly nonaxisymmetric; as the outer edge orbits around, acoustic waves are sent propagating outward at the same frequency as its orbital frequency. These waves have large enough amplitudes to obscure the spiral features in the outer disk.

The right column presents a late time, 65 disk orbits. The inner disk is largely cleared out; as a result, spiral shocks in the inner disk are barely visible. The outer edge of the cavity returns to approximate axisymmetry, so waves are no longer launched and the two spiral features in the outer disk re-emerge. The spiral shock extending outward from $(R,\phi )\,\approx (1.5\,{r}_{0},0.8\pi )$ remains well-defined, but the weak spiral feature starting from $(R,\phi )\approx (2.6\,{r}_{0},0.4\pi )$ is merely a diffuse density enhancement. The outer half of the bow shock appears as a spur joining the weak spiral feature.

The incoming stream in the ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 0.2$ simulation is not heavy enough to produce a perceptible stream of outgoing material (Section 4.4); for a better view of the outgoing material, we turn to the ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 1$ simulation. The slices shown in Figure 3 roughly follow the orbital plane of the outgoing material. The outgoing material exits the disk not as a dense, collimated structure, but as a clumpy plume orders of magnitude more dilute than the incoming stream. The plume is more spread out in the orbital plane than perpendicular to it.

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4.2.1. Dynamics

Figure 4 displays three snapshots from three simulations. The snapshots are selected from early times when the inner disk is still largely intact, but our comments below hold for all simulations at all times. At the times shown, the stream has delivered the same amount of mass, momentum, and energy across all simulations.

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The top row demonstrates how the collision excites spiral shocks in the inner disk. Spiral shocks vary in number, position, and strength over time, but there is often a dominant pair, one connected to the bow shock, the other located almost directly opposite in the other half of the inner disk. The former spiral shock is typically stronger and is marked with black arcs in the figure.

Spiral shocks deflect orbiting gas. The stronger spiral shock tends to deflect gas inward, giving rise to a region of inward mass flux immediately after the shock in the middle row. The net effect of the multiple spiral shocks is to remove angular momentum from the orbiting gas; as a result, gas falls toward the black hole on gradually shrinking, tightly wound trajectories. As is evident in the bottom row, rotation departs more and more from Keplerian as gas moves inward. Gas ultimately plunges into the cutout, most of it doing so over only a small fraction of the circumference, and then into the black hole.

4.2.2. Mass

The spiral shocks are extremely efficient at destroying the inner disk. For a radiation-dominated, ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 4$ disk with our fiducial parameters (Section 3.3), the inflow rate across the inner-radial boundary given by Figure 5 is ∼1.2 × 10³ to 1.2 × 10⁴ times the unperturbed level, or ∼6 to 50 times Eddington! In effect, the coherent spiral shocks exert an extremely strong stress on the flow, leading to an inflow rate much faster than ordinary disk processes, such as correlated MHD turbulence, could cause.

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For ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\lesssim 0.2$, the inflow rate has a spike at early times lasting a few disk orbits, then relaxes at late times to $\sim {\dot{M}}_{{\rm{s}}}$. For ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\gtrsim 0.4$, the inflow rate rises in the first ∼1 to 20 disk orbits, then decays gradually as the inner disk is depleted (Section 4.1); the inflow rate at ≳50 disk orbits is nearly independent of ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}$ because the stream appears essentially as a solid obstacle to the disk, deflecting a fixed fraction of the inner disk toward the black hole every disk orbit.

For ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\lesssim 1$, the inflow rate is quasiperiodically modulated over at least part of the simulation. The quasiperiodic variation is particularly strong for $0.1\lesssim {\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\lesssim 0.2$ and is visible in the top panel throughout the simulation. This quasiperiodicity is due to the interaction between the stationary spiral shocks and some orbiting nonaxisymmetric feature, such as the lopsided outer edge of the gap (Section 4.1).

Figure 6 shows the inflow time, or the time it takes a gas packet in the inner disk to move into the cutout on its way to the black hole. Thanks to the spiral shocks, the inflow time is orders of magnitude shorter than that of an unperturbed Shakura & Sunyaev (1973) disk measured at stream pericenter, which is $\sim 1.8\,\times {10}^{5}\,{(\alpha /0.1)}^{-1}{[{\left(H/R\right)}_{{\rm{d}}}/(0.003)]}^{-2}$ disk orbits, but the multistep nature of the shock-driven inflow mechanism (Section 4.2.1) means the inflow time is still at least a few disk orbits.

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The inflow time is not the characteristic timescale on which the inner disk mass decreases; this is because, as we shall see in Section 4.4, the inner disk captures enough of the stream to resupply itself. Figure 7 demonstrates this considerably slower depletion of the inner disk. Although the surface density declines monotonically with time, by the end of the simulations at 100 disk orbits, which is $\sim 0.3\,{T}_{\mathrm{mb}}$ for our fiducial parameters (Section 3.3), the surface density is only about an order of magnitude lower than its unperturbed value. Because a radiation-dominated, ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 4$ disk with our fiducial parameters (Section 3.3) starts out with a Thomson optical depth of ∼8000, it stays optically thick to electron scattering throughout the simulations.

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Figure 8 tracks the time evolution of the energy dissipation rate. For a radiation-dominated, ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 4$ disk with our fiducial parameters (Section 3.3), the energy dissipation rate is ∼300 to 2000 times the unperturbed disk accretion luminosity, or ∼1.5 to 10 times the Eddington luminosity.

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For ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\lesssim 0.2$, the energy dissipation rate is always close to the stream kinetic energy current; therefore, energy dissipation is well described as due to a perfectly inelastic collision between the stream and an immovable disk. For ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\gtrsim 0.8$, the energy dissipation rate initially rises but eventually falls; the energy dissipation rate shows little dependence on ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}$ at ≳15 disk orbits. Because the disk cannot fully stop the stream in these simulations, less of the stream kinetic energy is dissipated. Instead, the heavy stream acts as a stationary barrier to orbiting disk gas, and disk kinetic energy is dissipated when disk gas runs into the stream or the shocks created by the collision. The falloff in the energy dissipation rate over time is the result of the disk density decreasing (Section 4.1). For ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 0.4$ lying between the two extremes, the behavior of the energy dissipation rate switches from resembling lighter-stream simulations in the beginning to resembling heavier-stream simulations later on, suggesting that the stream takes time to clear out a gap in the disk before it can go through with little impediment.

The top panel of Figure 9 plots the outgoing material mass current, showing only the simulations for which the outgoing material has a mass current ≥1% that of the incoming stream throughout most of the simulation. A greater fraction of the stream passes through at later times because the disk density is lower (Section 4.1), but there are large and rapid fluctuations about the overall rising trend.

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The bottom panel displays the mass current of the incoming stream stopped by the disk. For ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\lesssim 0.2$, the stopped mass current is $\approx {\dot{M}}_{{\rm{s}}}$, which in turn is approximately the inflow rate in Figure 5. We may therefore picture the stream as being absorbed into the disk and deflected straight toward the black hole; the latter part is consistent with the fact that the disk removes all kinetic energy from lighter streams (Section 4.3). For ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\gtrsim 0.8$, the disk shaves off only a fraction of the incoming stream because the inner disk is cleared out early on (Section 4.2.2).

In all cases, the stopped mass current is almost always larger than the inflow rate in Figure 5. Mass loss from the inner disk (Section 4.2.2) slows down significantly if just a fraction of this mass current is diverted to resupply the inner disk. The inner disk mass still decreases over time because the outer disk receives part of the stopped mass current, because gas splashes back from the upper side of the disk, and because the collision heats up the inner disk (Section 4.1), causing it to expand out through the upper-vertical boundary or outward in radius.

Since the outgoing material is rather cold when it reaches the lower-vertical boundary, with sound speed $\lesssim 0.1\,{v}_{0}$, its dynamics beyond the simulation domain is dictated by its mechanical energy. Figure 10 shows the specific mechanical energy distribution of the outgoing material, again only for simulations with substantial outgoing material. The most bound part has specific binding energy $\sim 0.05\,{{GM}}_{{\rm{h}}}/{r}_{0}$, hence it flies out on an elliptical orbit to ∼19 times the pericenter distance, then returns to pericenter after $\sim 0.09\,{T}_{\mathrm{mb}}\,{[{M}_{{\rm{h}}}/{({10}^{6}{M}_{\odot })]}^{-1/2}({M}_{\star }/{M}_{\odot })}^{1/2}{({r}_{{\rm{p}}}/{r}_{{\rm{t}}})}^{-3/2}$.

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Most of the outgoing material, having received a kick from the disk during the collision, is unbound with specific mechanical energy $\sim 0.03\,{{GM}}_{{\rm{h}}}/{r}_{0}$. The outgoing material carries much less energy than is dissipated in the collision (Section 4.3), but is much more energetic than the debris unbound during the initial disruption. The disk is less dense at later times (Section 4.1) and imparts a smaller force on the stream, thus the outgoing material is less and less unbound after ∼40 disk orbits.

Two caveats must be noted. First, we use a marginally bound parabolic incoming stream in the simulations, whereas a realistic incoming stream has specific binding energy $\sim 0.01\,{{GM}}_{{\rm{h}}}/{r}_{0}\,\times {[{M}_{{\rm{h}}}/{({10}^{6}{M}_{\odot })]}^{-1/3}({M}_{\star }/{M}_{\odot })}^{1/3}{({r}_{{\rm{p}}}/{r}_{{\rm{t}}})}^{-1}$. We therefore expect the specific mechanical energy distribution of realistic outgoing material to be shifted toward lower energies by a similar amount, changing the fraction of bound and unbound material. Second, the bound outgoing material may not return to pericenter if it is intercepted on the way by the large-scale disk (Kathirgamaraju et al. 2017).

TDEs in AGNs are governed by several timescales. Here we group them together in ways that allow meaningful comparisons.

5.1.1. Timescales of Unperturbed Disk Flows

5.1.2. Timescales of Flows Caused by the TDE

5.1.3. Timescales of Energy Release

The cooling time is the time it takes radiation originating from the midplane to diffuse out of the geometrically (Section 4.1) and optically (Section 4.2.2) thick inner disk; it is ${\tau }_{{\rm{T}}}H/c$, where H is the height of the perturbed inner disk measured at stream pericenter. For an inner disk of aspect ratio unity, the cooling time is

$$\begin{eqnarray}&&\sim 30\,\,{\rm{d}}{\rm{a}}{\rm{y}}{\rm{s}}\times {\left(\displaystyle \frac{{M}_{{\rm{h}}}}{{10}^{6}{M}_{\odot }}\right)}^{1/3}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{-1/3}\left(\displaystyle \frac{{r}_{\star }}{{R}_{\odot }}\displaystyle \frac{{r}_{{\rm{p}}}}{{r}_{{\rm{t}}}}\right)\left(\displaystyle \frac{{\tau }_{{\rm{T}}}}{{10}^{4}}\right).\end{eqnarray} \tag{ 18 }$$

Here ${\tau }_{{\rm{T}}}$ is the Thomson optical depth at stream pericenter (Figure 7). For a radiation-dominated, ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 4$ disk with our fiducial parameters (Section 3.3), the cooling time drops from ∼21 to ∼1.4 days over the course of ≈12 days. The cooling time is so long because ${\tau }_{{\rm{T}}}$ is ∼0.1 times the unperturbed Thomson optical depth (Section 4.2.2) but H rises greatly due to sudden heating. How quickly ${\tau }_{{\rm{T}}}$ declines depends on the amount of resupply (Section 5.1.2).

Because the inner disk is optically thick to scattering, the inflow time (Section 5.1.2) is also the time radiation has to escape before being swallowed. The ratio of the inflow time to the cooling time is a rough estimate of the fraction of internal energy generated in the collision that is released as radiation. For a radiation-dominated, ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 4$ disk with our fiducial parameters (Section 3.3), the ratio rises from ∼0.06 to ∼0.3 over $\approx$ 12 days. Most of the energy is therefore advected into the black hole, suppressing the collision luminosity. The situation may change if the inner disk is further depleted and its optical depth diminished accordingly.

5.1.4. Hierarchy

We present an inventory of the time-integrated amount of energy we may expect over the course of the entire event, including the energy from processes not directly simulated. All items except the last are concerned with internal energy production in the disk; the energy emitted as light from the disk could be much smaller because radiation could be trapped in the inflow and swallowed by the black hole (Section 5.1.3).

For lighter streams, the stream loses all of its kinetic energy at the collision (Section 4.3); its gas assimilates into the disk (Section 4.4) and moves inward to the black hole within an inflow time. The energy dissipated in this whole process is the same as if the stream were accreted directly onto the black hole, that is, ${E}_{\star }\sim 9\times \,{10}^{52}\,\,\mathrm{erg}\times ({M}_{\star }/{M}_{\odot })(\eta /0.1)$. The energy dissipated may be smaller because shocks in the inner disk may send gas straight into the black hole (Section 4.2.1).

In the meantime, shocks excited by the collision inner disk (Section 4.1). The energy produced is ${E}_{{\rm{d}}}\sim 6\times {10}^{50}\,\mathrm{erg}\times [{M}_{{\rm{h}}}/$ ${{({10}^{6}{M}_{\odot })]}^{-1/3}({M}_{\star }/{M}_{\odot })}^{-7/6}{({r}_{\star }/{R}_{\odot })}^{7/2}{({r}_{{\rm{p}}}/{r}_{{\rm{t}}})}^{7/2}{(\alpha /0.1)}^{-1}\,{(\eta /0.1)}^{2}$ ${[({L}_{{\rm{a}}}/{L}_{{\rm{E}}})/0.005]}^{-1}$ for a radiation-dominated inner disk. It can be smaller if shocked gas plunges into the black hole (Section 4.2.1) or if only part of the pre-existing inner disk is accreted.

After the event is over, magnetic stresses gradually replenish the inner disk with gas from the outer disk (Section 5.1.3). Energy of order the binding energy of the unperturbed inner disk is released, which is ${E}_{{\rm{r}}}\sim 9\times {10}^{49}\,\mathrm{erg}\times {[{M}_{{\rm{h}}}/{({10}^{6}{M}_{\odot })]}^{1/3}({M}_{\star }/{M}_{\odot })}^{-5/6}{({r}_{\star }/{R}_{\odot })}^{5/2}{({r}_{{\rm{p}}}/{r}_{{\rm{t}}})}^{5/2}{(\alpha /0.1)}^{-1}(\eta /0.1){[({L}_{{\rm{a}}}/{L}_{{\rm{E}}})/0.005]}^{-1}$ if the inner disk is radiation-dominated. Should magnetic stresses be strong enough to keep the inner disk refilled even while the event is in progress (Section 5.1.2), the continuous inflow of gas from the outer disk to the black hole during the event may lead to a total amount of energy dissipated exceeding ${E}_{{\rm{d}}}$ + ${E}_{{\rm{r}}}$.

For heavier streams, the situation is more complicated. The inner disk is similarly flushed out by shocks, generating ${E}_{{\rm{d}}}$ of internal energy in the process, and the inner disk is likewise resupplied by the outer disk, yielding ${E}_{{\rm{r}}}$. The difference is that the disk now captures only a fraction ${f}_{{\rm{c}}}\sim 0.5$ of the incoming stream (Section 4.4), and the rest of the stream emerges on the other side as outgoing material. The energy available from the initial dissipation at the impact point and from the subsequent inflow of stream gas is therefore only $\approx {f}_{{\rm{c}}}{E}_{\star }$. The value of ${f}_{{\rm{c}}}$ decreases with ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}$ and time, the decrease being the sharpest around ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\sim 1$ (Section 4.4).

Some of the outgoing material remains bound, with a range of specific binding energy from $\sim 0.05\,{{GM}}_{{\rm{h}}}/{r}_{{\rm{p}}}$ to zero (Section 4.4). Different parts of the bound material fly out to their respective apocentric distances of $\gtrsim 19\,{r}_{{\rm{p}}}$, at which point they run into the disk again. Because the bound material is much more diffuse than the incoming stream was at pericenter (Section 4.1), we expect the disk to capture all of the bound material. The kinetic energy dissipated is $\lesssim 2\times \,{10}^{49}\,\mathrm{erg}\times {({M}_{\star }/{M}_{\odot })}^{2}{({r}_{\star }/{R}_{\odot })}^{-1}{({r}_{{\rm{p}}}/{r}_{{\rm{t}}})}^{-3}({f}_{{\rm{b}}}/0.5)$, with ${f}_{{\rm{b}}}\leqslant 1-{f}_{{\rm{c}}}$ the fraction of the incoming stream ending up in the bound material; therefore, the second collision may be too dim to be seen against the first collision and the background AGN. The captured bound material accretes along with the disk, but the accretion energy can be neglected because the accretion timescale is much longer than the mass return time.

The previous paragraph pertains to the particular stream configuration used in our simulations, one where the stream plane is perpendicular to the disk plane and the stream pericenter is in the disk plane (Section 3.4). In the general case where the two planes are oriented randomly, it is more likely that the first and second collisions both occur near pericenter. Such a second collision dissipates $\sim 5\times {10}^{51}\,\mathrm{erg}\,\times$ ${[{M}_{{\rm{h}}}/{({10}^{6}{M}_{\odot })]}^{2/3}({M}_{\star }/{M}_{\odot })}^{4/3}{({r}_{\star }/{R}_{\odot })}^{-1}{({r}_{{\rm{p}}}/{r}_{{\rm{t}}})}^{-1}({f}_{{\rm{b}}}/0.25)$, much greater than if the second collision were at apocenter. This energy is a sizable fraction of that obtained by accreting the bound material directly, ${f}_{{\rm{b}}}{E}_{\star }$, because ${r}_{{\rm{t}}}/{r}_{{\rm{g}}}$ is typically only a few times ${\eta }^{-1}$, according to Equation (1). The dissipation occurs over large areas of the disk and possibly at high altitudes, both of which have implications for how much of the dissipated energy would emerge as radiation. The remainder of ${f}_{{\rm{b}}}{E}_{\star }$ is liberated when the bound material falls toward the black hole.

Some of the outgoing material is unbound by a kick from the disk and carries specific kinetic energy $\sim 0.03\,{{GM}}_{{\rm{h}}}/{r}_{{\rm{p}}}$ at infinity (Section 4.4). The total energy is $\sim 1.4\times {10}^{50}\,\mathrm{erg}\times [{M}_{{\rm{h}}}/$ ${{({10}^{6}{M}_{\odot })]}^{2/3}({M}_{\star }/{M}_{\odot })}^{4/3}{({r}_{\star }/{R}_{\odot })}^{-1}{({r}_{{\rm{p}}}/{r}_{{\rm{t}}})}^{-1}({f}_{{\rm{u}}}/0.25)$, where ${f}_{{\rm{u}}}\,=1-{f}_{{\rm{c}}}-{f}_{{\rm{b}}}$ is the fraction of the incoming stream unbound by the collision. Compared to the primary ejecta unbound as an immediate consequence of the disruption (Guillochon et al. 2016; Krolik et al. 2016; Yalinewich et al. 2019), this secondary ejecta has a fraction $\approx {f}_{{\rm{u}}}$ of the mass, higher velocity (Section 4.4), and a larger opening angle (Section 4.1). The ejecta drives a bow shock while running into the surrounding medium, and relativistic electrons accelerated in this shock can produce synchrotron radiation (Krolik et al. 2016; Yalinewich et al. 2019). At later times, the shock driven by the ejecta may mimic a supernova remnant (Guillochon et al. 2016). The denser medium around AGNs could mean that TDEs are more radio-bright in AGNs than in vacuum. The prompt emission due to the secondary ejecta could be more luminous than the primary ejecta because the fastest material has higher velocity and a wider interaction area (Krolik et al. 2016; Yalinewich et al. 2019). All these may explain why the radio transient Cygnus A-2 (Perley et al. 2017), if interpreted as a thermal TDE happening in an AGN (de Vries et al. 2019), is brighter in radio than typical thermal TDEs in vacuum.

5.3.1. Thermodynamics

Our simulations do not correctly track temperatures because the adiabatic index used corresponds to gas pressure and not radiation pressure. To estimate the gas and radiation temperatures of the inner disk, we observe that shocks raise the sound speed of the simulated inner disk to ∼0.2 to 0.5 times the local Keplerian orbital velocity; this corresponds to a gas temperature of

$$\begin{eqnarray}&&\sim 2\times {10}^{10}\,{\rm{K}}\times {\left(\displaystyle \frac{{M}_{{\rm{h}}}}{{10}^{6}{M}_{\odot }}\right)}^{2/3}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{1/3}{\left(\displaystyle \frac{{r}_{\star }}{{R}_{\odot }}\displaystyle \frac{R}{{r}_{{\rm{t}}}}\right)}^{-1}.\end{eqnarray} \tag{ 19 }$$

If gas and radiation in the inner disk had enough time to thermalize, they would come into an equilibrium temperature of

$$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{eq}} & \sim & 6\times {10}^{5}\,{\rm{K}}\times {\left(\displaystyle \frac{{M}_{{\rm{h}}}}{{10}^{6}{M}_{\odot }}\right)}^{1/6}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{1/12}\\ & & \times {\left(\displaystyle \frac{{r}_{\star }}{{R}_{\odot }}\displaystyle \frac{R}{{r}_{{\rm{t}}}}\right)}^{-1/4}{\left(\displaystyle \frac{\rho }{2\times {10}^{-10}{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{1/4},\end{array}\end{eqnarray} \tag{ 20 }$$

where ρ is the inner disk density during the collision, and its fiducial value is the typical density at the end of the ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 4$ simulation for a radiation-dominated disk with our fiducial parameters (Section 3.3). In thermodynamic equilibrium, the internal energy is dominated by radiation and varies $\propto {T}_{\mathrm{eq}}^{4}$, hence ${T}_{\mathrm{eq}}$ is rather insensitive to the specific values of the parameters on the right-hand side.

However, it may be difficult for gas raised to $T\sim {10}^{10}\,{\rm{K}}$ to reach thermodynamic equilibrium. Because the cross section per mass for free–free absorption is $\propto \rho {T}^{-7/2}$, gas this hot would have essentially zero absorptivity and therefore, by Kirchhoff's law, essentially zero emissivity. The post-shock gas could cool by inverse Compton scattering off photons already present in the gas, but this, too, is problematic. The cooling rate is proportional to the local radiation energy density, which depends strongly on whether the gas originates from the colder stream or the hotter unperturbed inner disk. Even if Compton cooling is rapid, it creates no new photons, so the temperature decrease it achieves may be limited. Substantial Compton cooling may be possible only at high latitudes, where the post-shock gas is exposed to photons radiated by the outer disk. Close to the midplane, Compton cooling may create photons energetic enough to produce pairs, which could hasten thermalization. For all these reasons, without detailed calculations, we cannot make a firm statement about how rapidly, or through what radiation mechanisms, the inner disk may approach thermodynamic equilibrium.

5.3.2. Radiative Transfer

The collision converts a sizable fraction of the stream kinetic energy to internal energy. For a radiation-dominated, ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 4$ disk with our fiducial parameters (Section 3.3), the shocks dissipate energy ∼300 to 2000 times faster than internal stresses in the disk, corresponding to ∼1.5 to 10 times the Eddington rate (Section 4.3). How the large amount of internal energy created translates to emission, however, is highly uncertain.

The critical comparison here is between the inflow time and the cooling time (Section 5.1.3). Since the cooling time depends on the inner disk mass, its value can be properly determined only with MHD simulations that self-consistently model the amount of resupply from the stream and the outer disk (Section 5.1.2). But if the inflow time is indeed shorter than the cooling time over a significant fraction of the mass return time, as our hydrodynamics simulations suggest, then radiative transfer in the inner disk is inherently time-dependent.

The relative shortness of the inflow time means radiation near the midplane is trapped, so energy dissipation deep inside the inner disk is completely hidden from view. The only regions that can effectively cool are those so close to the photosphere that radiation can diffuse out before being swept into the black hole; consequently, the outgoing luminosity is a small fraction of the total energy dissipation rate. Even in regions from which radiation can escape, fluctuations shorter than the local diffusion time do not imprint themselves on the light curve.

A more careful treatment calls for three-dimensional, time-dependent radiative transfer calculations capable of handling the high degree of asymmetry of the system: gas near shocks is denser and hotter than gas elsewhere, and the outgoing material has high enough optical depth to partially obscure one side of the inner disk.

More radiation could escape if vertical advection, for example due to magnetic buoyancy (Jiang et al. 2014), is important in transporting radiation outward. Exploring this possibility requires time-dependent radiative MHD simulations.

The interaction among the processes above and thermalization (Section 5.3.1) may depend sensitively on the specific parameters of the system. The resulting emission may not resemble a regular TDE or AGN, and may have complex temporal and spectral behavior; different TDEs in AGNs could look entirely different.

Having laid out all these complications, we may nevertheless crudely estimate the bolometric collision luminosity ${L}_{{\rm{c}}}$ as proposed in Section 5.1.3: We take the ratio ${t}_{\mathrm{infl}}/{t}_{\mathrm{cool}}$ of inflow to cooling time to be roughly the fraction of energy that escapes as radiation, and we scale the energy dissipation rate Q (Section 4.3) by it. For heavy streams, Figure 11 shows that ${L}_{{\rm{c}}}$ could be briefly very super-Eddington, then settle at a near-Eddington level. For a radiation-dominated, ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\approx 4$ disk with our fiducial parameters (Section 3.3), the time to steady state is ∼1 day. The fact that ${L}_{{\rm{c}}}\sim {L}_{{\rm{E}}}$ is not surprising in retrospect. If the inner disk had enough time to thermalize (Section 5.3.1), then it would be supported vertically against gravity by radiation pressure, and the characteristic luminosity of such systems is ${L}_{{\rm{E}}}$ (Krolik 2010).

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The corona of the unperturbed inner disk may be destroyed along with the inner disk itself (Section 4.1) if MHD turbulence is suppressed in the shock-driven inflow replacing it. This could explain the dip in soft X-rays lasting for a few months in the nuclear transient observed by Trakhtenbrot et al. (2019a).

The super-Eddington inflow (Section 4.2.2) may be connected to the launching of a jet (Giannios & Metzger 2011; Krolik & Piran 2012), which has been invoked to explain the hard X-ray (Bloom et al. 2011; Burrows et al. 2011; Cenko et al. 2012) and radio (van Velzen et al. 2016) emission of several TDEs. This possibility can be tested using future MHD simulations.

Acoustic waves (Section 4.1) are launched when shocks in the outer disk push orbiting gas radially outward. Bending waves are generated when the stream delivers misaligned angular momentum and exerts a torque on the disk. These waves may have observable consequences on the outer disk long after the stream has ended.

As the mass return rate dwindles toward the end of the event, the shocks would become weaker, inner-disk gas would move inward more slowly, and the gas may have enough time to completely cool off before reaching the black hole (Section 5.1.3).