Tidal Disruption Events in Active Galactic Nuclei: Thermal Radiation and Jets

Introduction

Tidal disruption events (TDEs) occur when a star passes close enough to a massive black hole to be torn apart by tidal forces. Roughly half the stellar debris becomes unbound, while the other half remains bound and returns to the black hole on highly eccentric orbits [38, 37]. The returning debris produces a luminous flare lasting from months to years (see e.g. Gezari [16]). Most observed TDEs appear to be dominated by thermal optical/UV [16, 18], but a small subset show luminous hard X-ray emission, rapid variability, and other signatures associated with relativistic jets [5, 7, 8, 6, 1, 36].

Most TDE studies consider an isolated black hole, but roughly $\sim10\%$ of galaxies host an AGN (e.g., [21, 26, 43, 14]). At least this fraction of TDEs is therefore expected to occur in AGNs. A number of mechanisms have been suggested that may enhance the rate per AGN [25, 27, 24, 29, 52, 35, 51], increasing the overall fraction of such events relative to the total number of TDEs.

When a TDE takes place within an AGN, the disruption phase is more or less unchanged as the AGN’s disk is extremely dilute in comparison to a star. The situation changes when the returning debris stream collides with the pre-existing accretion disk, shocking the gas and driving rapid inflow in the region inside the debris-disk impact point [11]. The collision drastically changes both the debris stream and the disk dynamics, as well as the radiative output and how magnetic flux is carried toward the black hole. Following these effects requires radiation-MHD simulations in which the disk structure, magnetic stresses, shocks, and radiative losses evolve together.

A small but growing number of TDE candidates have recently been identified within AGN [4, 32, 46, 13] exhibiting behavior not well explained by disruptions around isolated black holes. It has been speculated that the Bowen fluorescence lines seen in some TDEs are powered by the AGN host [49], but this is an ambiguous signal because ordinary TDEs may also produce them [30]. This uncertainty further underscores the need for a theoretical framework for such TDEs, tailored to the AGN environment.

Chan et al. [11] carried out the first study of a TDE debris stream colliding with an AGN disk. Their hydrodynamic simulations considered an idealized Shakura-Sunyaev disk. They showed that within a fraction of the debris return time, the stream-disk collision excites shocks, drives rapid inflow, and depletes the disk interior to the impact radius. In these simulations, the energy dissipation and mass inflow rates often exceeded those of the unperturbed disk by orders of magnitude, although much of the dissipated energy was advected into the black hole without escaping as radiation. An Eddington-scale luminosity is expected, as in any case in which an optically thick source is powered by gravitation ([30]). Those runs were purely hydrodynamic, so they could not follow radiative transport, magnetic stresses, or magnetic flux accumulation.

Subsequent hydrodynamic studies extended this picture. Ryu et al. [42] simulated in-plane, prograde, and retrograde disruptions in an AGN disk and found that continuous stream-disk interaction can fully mix the debris into the disk before it returns coherently. Zhang et al. [53] explored the dependence on the star’s orbital inclination and included Schwarzschild apsidal precession, identifying a two-phase light curve. Both studies, like Chan et al. [11], adopt a thin, gas-pressure-dominated disk with adiabatic or locally approximated cooling, and neither evolves the radiation field or includes dynamically important magnetic pressure.

Tripto et al. [50] gave analytic estimates for magnetic-flux transport driven by the stream-disk collision. Assuming that the local flux density was of order the local field intensity, they argued that for a wide range of parameters, and under conservative assumptions, the sharply-increased mass accretion when a TDE occurs in an AGN could supply enough horizon-threading flux to power a transient jet.

Here we report results derived from a radiation-MHD simulation of the stream-disk interaction at pericenter. Our simulation simultaneously gives a good description of both the radiation transfer and the transport of magnetic flux toward the black hole. It is distinguished from previous work by using an initial structure drawn from a recent radiation-MHD simulation of a stationary accretion disk ([22]). This structure differs substantially from the Shakura-Sunyaev models considered in Chan et al. [11, 9] (2021), or the semi-analytic thin AGN disk solutions used in Ryu et al. [42]; Zhang et al. [53].

Methods

Athena++ with radiation and magnetic fields

We carry out 3D radiation magnetohydrodynamic (RMHD) simulations using Athena++ ([45]). The code evolves the equations of ideal magnetohydrodynamics together with a time-dependent 3D radiation transport module. The simulations are performed in spherical-polar coordinates $(r,\theta,\phi)$ centered on the black hole. We use the pseudo-Newtonian Paczynski & Wiita [34] gravitational potential,

where $r_{\mathrm{g}} \equiv G M_{\mathrm{BH}}/c^2$. This approximation reproduces the main relativistic effect relevant for the inner disk — the existence of an innermost stable circular orbit — while keeping the calculation Newtonian.

Magnetic fields are evolved with the constrained-transport algorithm in Athena++, which preserves $\nabla\cdot\mathbf{B} = 0$ to numerical precision. That property matters when we ask whether the debris impact changes the magnetic flux near the black hole. Magnetic pressure and Maxwell stresses also enter the disk structure and angular-momentum transport.

The radiation field is described by a large number of angle-groups evolved simultaneously with the gas, including energy and momentum exchange between radiation and matter. The transport equations are “gray”, i.e., both opacity and emissivity are averaged over photon frequency assuming a blackbody spectrum. Treating radiation transfer explicitly is essential because radiation affects the subsequent evolution through cooling, radiation pressure forces are comparable to gravity, and radiation viscosity is significant ([22]).

Specifics

Initial AGN disk

The initial state of the accretion flow is taken from a late-time ($t \simeq4 \cdot10^4 M_{\mathrm{BH}}$) snapshot of the “AGN0.07” simulation of Jiang et al. [22].¹ This simulation followed a sub-Eddington radiation-MHD accretion disk around a supermassive black hole ($M_{\mathrm{BH}} = 5 \cdot10^8 M_{\odot}$), with a time-averaged accretion rate $\simeq0.08\dot{M}_{\mathrm{Edd}}$ during roughly the second half of the simulation². Throughout this paper, we define $\dot{M}_{\mathrm{Edd}} \equiv10 L_{\mathrm{Edd}}/c^2$ with a fiducial radiative efficiency $\eta= 0.1$. The initial magnetic field was axisymmetric, with a pair of nested quadrupolar field loops above and below the disk’s midplane.

In the initial state of the Jiang et al. [22] simulation, a thick torus at $40-100r_{\mathrm{g}}$ feeds a central region whose accretion flow is determined dynamically. Regrettably, this simulation ran long enough to reach inflow equilibrium only inside $\simeq15r_{\mathrm{g}}$, a distance comparable to our impact radius. Dynamics driven by the stream collision cannot be completely separated from the continued relaxation of the $r > 15r_{\mathrm{g}}$ region. However, the abrupt changes we observe occur on timescales much shorter than the undisturbed disk’s inflow timescale, and can therefore be attributed to the stream collision. The principal limitation created by the lack of inflow equilibrium outside $\simeq15r_{\rm g}$ is that the mass stored in the initial torus ($\simeq40-100r_{\rm g}$) may contribute excess mass to the disk’s atmosphere at larger radii. This is unlikely to significantly influence our results, for two main reasons. First, as shown in § 3.1.2, the late-time mass inside $30r_{\rm g}$ is split roughly equally between debris and gas that began outside $30r_{\rm g}$; thus, even if mass inflow from outside $30r_{\rm g}$ is overestimated, its contribution remains constrained. Second, our simulation is not run for a long duration, so the slower effects of matter inflowing from further-out regions remain limited. However, the existence of this torus prevents us from exploring the longer-term interaction of the debris stream with outer regions of the disk (Chan et al. 2021).

We rescaled this system to be consistent with $M_{\rm BH}=5\cdot10^{6}M_{\odot}$ while maintaining $\dot{M}\approx0.07\dot{M}_{\rm Edd}$. This was necessary because $M_{\rm BH}\sim5\cdot10^{8}M_{\odot}$ is excluded for TDEs involving main-sequence stars by the Hills criterion (Hills [20]). The conversion is described in Appendix A. The rescaled disk was evolved for several orbits before the debris stream was introduced to reduce transients and assess its stability.

The disk structure computed by Jiang et al. [22] is quite different from what an $\alpha$-model would predict, beginning with the fact that the ratio of stress to total pressure is $\propto r^{-1}$, rather than constant. Moreover, the AGN0.07 disk is magnetically supported and has a surface density (and therefore optical depth) a factor $\simeq10$ below that of a thin $\alpha$-disk with the same accretion rate. At the impact radius, $r\simeq15r_{\rm g}$, the disk prior to stream injection has midplane density $\rho_{\rm d}\simeq10^{-9}\ {\rm g\,cm^{-3}}$, surface density $\Sigma_{\rm d}\simeq4\cdot10^{2}\ {\rm g\,cm^{-2}}$, and Rosseland-mean optical depth $\tau_{\rm R}\simeq100$. The photosphere lies only $\simeq6^{\circ}$ from the midplane. Very much unlike an $\alpha$-disk, the gas pressure is smaller than both the magnetic and radiation pressures by more than a factor of $10^{3}$.

Debris stream

Our injection scheme follows the same general approach as that of Chan et al. [11], as depicted in Figure 1. We inject the stream $\sim4r_{\rm g}$ above the equatorial plane, so that it crosses the plane at $r=15r_{\rm g}$. For a star with $M_{\star}\approx0.8M_{\odot}$ this is approximately the physical tidal radius ([39]). It is injected with a velocity corresponding to an almost parabolic trajectory (App. § 9). The orbit is nearly perpendicular to the disk, with inclination $i\simeq86^{\circ}$. The injected mass current is $\dot{M}_{\rm s}=8.3\cdot10^{25}\ {\rm g\,s^{-1}}=12\dot{M}_{\rm Edd}$, which is $\sim170$ times the unperturbed disk accretion rate of $0.07\dot{M}_{\rm Edd}$. The initial inner disk mass is $\sim2.5\cdot10^{-4}M_{\odot}$ within $r<30r_{\rm g}$ and $\sim4.5\cdot10^{-5}M_{\odot}$ within $r<15r_{\rm g}$. Thus, the stream injects comparable masses in just $\sim0.7t_{\rm orb}$ and $\sim t_{\rm orb}/8$, respectively, where $t_{\rm orb}\approx365GM_{\rm BH}/c^{3}\simeq0.1\ {\rm d}$ is the

Keplerian orbital period at $15r_{\rm g}$. The stream has a transverse width of $R_{\odot}\simeq0.1r_{\rm g}$ by $3R_{\odot}\simeq0.3r_{\rm g}$ at injection, i.e. $\sim1.2-4R_{\odot}$.

Throughout the simulation $\dot{M}_{\rm s}$ is held constant, corresponding to the peak fallback rate. This is consistent because our simulation ran for 43.5 orbits at $15r_{\rm g}$, or 4.4 d, which is only $\simeq0.06t_{0}$, where $t_{0}\simeq75$ d is the debris fallback time for $M_{\rm BH}=5\cdot10^{6}M_{\odot}$ and $M_{\star}\simeq0.8M_{\odot}$ ([39 c]). A more detailed description of the stream injection parameters and procedure is given in App. § 9.

Comparison with Chan et al. (2019)

The controlling parameter of this simulation, as in Chan et al. [11], is the mass-current ratio $\dot{M}_{\rm s}/\dot{M}_{\rm d}$, where $\dot{M}_{\rm d}=\Sigma_{\rm d}R_{\rm s}\sqrt{GM_{\rm BH}/r_{\rm imp}}$ is the azimuthal mass current of the disk under the stream footprint, with $r_{\rm imp}$ being the impact radius. With $\Sigma_{\rm d}\simeq4\cdot10^{2}\ {\rm g\,cm^{-2}}$ and $R_{\rm s}\simeq R_{\odot}\simeq0.1r_{\rm g}$, one finds $\dot{M}_{\rm d}\simeq2\cdot10^{23}\ {\rm g\,s^{-1}}$, so $\dot{M}_{\rm s}/\dot{M}_{\rm d}\sim400$.

A Shakura & Sunyaev disk with our black hole mass and impact radius already gives $\dot{M}_{\rm s}/\dot{M}_{\rm d}\sim30$ ([44]), a factor of a few above the maximum of $\dot{M}_{\rm s}/\dot{M}_{\rm d}\sim4$ considered by Chan et al. [11] even before any change in disk model. The AGN0.07 disk’s surface density is $\simeq11$ times lower than an $\alpha$-disk at the same parameters, pushing $\dot{M}_{\rm s}/\dot{M}_{\rm d}$ a further order of magnitude to $\sim350$, consistent with the $\sim400$ found above and well outside the range $[0.05,4]$ explored by Chan et al. [11]. Thus, their hydrodynamic results remain a qualitative guide, but with such a large contrast in the mass-current ratio, it is difficult to quantitatively compare between the simulations.

One of the notable results of the difference in $\dot{M}_{\mathrm{s}}/\dot{M}_{\mathrm{d}}$ is that the stream does not merely disrupt the inner disk: unlike in Chan et al. [11], it also replenishes it with a comparable or greater mass of fresh material. This point is further discussed in § 3.1.2.

Numerical details

In our spherical coordinate system, the radial coordinates of cell-centers are logarithmically spaced, with $r_{\mathrm{in}}=4r_{\mathrm{g}}$ and $r_{\mathrm{out}}=1.6\cdot10^{3}r_{\mathrm{g}}$. The grid covers the full polar and azimuthal ranges, $0\le\theta\le\pi$ and $0\le\phi<2\pi$. The base grid has $64\times32\times64$ cells in $r$, $\theta$, $\phi$, respectively. Cell sizes are self-similar, with $\Delta r/r=\Delta\theta=\Delta\phi$ at every refinement level: they are uniform in logarithmic measure but not in physical size. The base values are $\Delta r/r=\Delta\theta=\Delta\phi\simeq9.8\cdot10^{-2}$ ($\Delta\theta=\pi/32$).

We use static mesh refinement, following the AGN0.07 setup of Jiang et al. [22], to increase the resolution in the disk midplane and around the stream-disk interaction region. This midplane refinement reaches $\Delta r/r=\Delta\theta=\Delta\phi=6.1\cdot10^{-3}$. We additionally refine the grid around the stream injection point and its trajectory with two rectangular regions in $(r,\theta,\phi)$. The first encloses the injection site with boundaries defined by $15r_{\mathrm{g}}\le r\le17r_{\mathrm{g}}$, $1.25\le\theta\le1.97$, and $1.42\le\phi\le1.62$. The second extends the refined zone along the incoming stream, bounded by $16r_{\mathrm{g}}\le r\le21r_{\mathrm{g}}$, $1.97\le\theta\le2.3$, and $1.38\le\phi\le1.62$. In the two stream regions the finest cells reach $\Delta r/r=\Delta\theta=\Delta\phi=1.5\cdot10^{-3}$, corresponding to $0.025r_{\mathrm{g}}$ per cell at the impact radius. We define the northern hemisphere as the one from which the stream flows, and the southern hemisphere as the one into which it flows.

The radial boundary conditions are outflow conditions both at the inner and outer boundaries. The polar boundaries use the standard coordinate-singularity treatment, in which ghost zones are filled from the corresponding cells across the pole with vector components appropriately sign-flipped. The azimuthal direction is periodic. No matter is allowed to enter the simulation domain except for the source terms that inject the debris stream.

The radiation field is discretized using the default angular quadrature of Davis et al. [12] at fourth order, corresponding to 80 discrete ray directions per cell. The radiation boundary conditions are very similar to the hydrodynamic boundary conditions: pure outflow on the radial boundaries, polar in the $\theta$ direction, and periodic in $\phi$. The opacity is computed locally from the gas density and temperature using tabulated opacity (Zhu et al. [55]). Atomic features (both scattering and absorption), electron scattering (with energy exchange), and free-free are all included. The emissivity is defined in terms of the usual thermal reciprocity.

Density, pressure, and temperature floors are imposed in low-density regions to maintain numerical stability. In the fiducial run these are $\rho_{\mathrm{floor}}=10^{-14}\ \mathrm{g\,cm^{-3}}$, $P_{\mathrm{floor}}=2.7\cdot10^{-3}\ \mathrm{dyn\,cm^{-2}}$, and $T_{\mathrm{floor}}=3.2\cdot10^{3}\ \mathrm{K}$. For photosphere detection, because floor-density gas alone would give $\tau\sim1$ quickly and produce a spurious photosphere, cells with $\rho<2\rho_{\mathrm{floor}}$ are assigned a negligible density, $10^{-4}\rho_{\mathrm{floor}}$, in these integrals.

Results

Dynamic disk response

Figure 2 and Figure 3 depict density slices and stream lines, and the corresponding surface density evolution, before stream injection and at several later times. At $t=0$ the gas occupies a thin, nearly equatorial disk, densest near the midplane between roughly $6r_{\mathrm{g}}$ and $20r_{\mathrm{g}}$, with photospheres at $|z|\simeq3$–$5r_{\mathrm{g}}$. After impact, spiral arms and a dense incoming stream appear in the equatorial cut, matching the same spiral feature in the surface density. In the midplane annulus $6r_{\mathrm{g}}\le r\le25r_{\mathrm{g}}$, the ratio of the azimuthally-averaged density $\rho_{\mathrm{mid}}(r,t)$ to $\rho_{\mathrm{mid}}(r,t=0)$ diminishes to $\sim0.6$ at $t=15t_{\mathrm{orb}}$ in both the inner and outer parts of that range; by $t=43t_{\mathrm{orb}}$, the inner part is only $\sim0.4$, while the density in the outer part has returned to $\sim1$. Closer in, inside the ISCO ($r\simeq6r_{\mathrm{g}}$), the midplane density rises by a factor $\sim15$ by $t=15t_{\mathrm{orb}}$ and then stays roughly steady. The meridional slices show where midplane gas goes: it is driven off the equator, thickening the flow and filling high latitudes. The surface density over the same radial range does not drop correspondingly, and exceeds its pre-impact value by late times.

The stream drills through the disk, and because it cuts across the rotating gas, it carves out a gap spanning the full azimuthal extent of the disk radius, elongated along the direction of rotation. This resembles the largest values of $\dot{M}_{\mathrm{s}}/\dot{M}_{\mathrm{d}}$ used in Chan et al. [11], which is not surprising given the value we use (§ 2.2.3). Within the inner $r=30r_{\mathrm{g}}$ domain of Figure 2, the north photosphere rises from $|z|\simeq3$–$5r_{\mathrm{g}}$ to $|z|\simeq10$–$20r_{\mathrm{g}}$, while the south photosphere has already left that frame. Figure 4 shows the same epoch on a larger scale: both photospheres are then visible, the south following the outgoing stream to $|z|\simeq200$–$600r_{\mathrm{g}}$.

At $t=0$ the equatorial streamlines follow nearly circular orbits, while off-plane streamlines are radial, consistent with the non-rotating sparse material there. After impact, the equatorial streamlines deviate from this circular structure, manifesting the enhanced accretion rate and disturbed disk structure; near the stream impact site the streamlines point more directly inward, tracing the funneling of debris toward smaller radii. The meridional streamlines change more markedly, showing an unorganized structure in the disk, consistent with gas driven off the equator to high latitudes.

The morphology matches the hydrodynamic picture of Chan et al. [11] in several respects: a bow-like disturbance at the stream-disk contact, spiral shocks in the inner flow, and inflow concentrated in non-axisymmetric lanes. In their runs, however, the disk interior to the impact radius thinned by roughly an order of magnitude in surface density after the impact. Here, the midplane density in $6$–$25r_{\mathrm{g}}$ falls by only a factor of a few, while gas density rises near the black hole (factor $\sim15$ inside the ISCO), gas is pushed to high latitudes, and undergoes continuous infall into the black hole. The surface density itself increases monotonically over most radii after an initial cavity-driven dip, reaching $\gtrsim10^{3}\ \mathrm{g\ cm^{-2}}$ by late times.

In 3D renderings (Figure 5), the initial thin disk becomes a thick, lopsided, vertically extended body. Dense stream material remains visible as a coherent plume, especially below the midplane, while the former equatorial layer is distorted and inflated.

Accretion rate and enclosed mass

The mass accretion rate onto the inner cutout (Figure 6, panel (a)) rises from the unperturbed value $0.07\ \dot{M}_{\mathrm{Edd}}$ to $\simeq10$–$13\ \dot{M}_{\mathrm{Edd}}$ over 43.5 orbits at $15r_{\mathrm{g}}$ (4.4 d), comparable to the injected stream mass current $\dot{M}_{\mathrm{s}}=12\dot{M}_{\mathrm{Edd}}$. The corresponding bolometric luminosity is shown in the same figure in panel (d) and remains near Eddington (§ 3.2). Relative to the unperturbed disk, $\dot{M}$ and $L$ separate by at least an order of magnitude in Eddington units, the photon trapping-dominated behavior anticipated by Chan et al. [11].

To quantify how the inner disk is depleted and resupplied, Figure 6 (panel (b)) plots the enclosed disk mass within $r < 12r_{\mathrm{g}}$ as a function of time. We choose $12r_{\mathrm{g}}$ since it is the outermost radius in the inner disk confidently free of outflowing stream matter. The enclosed mass rises by a factor $\sim10$, from $\sim10^{-5}\,M_{\odot}$ to $\sim10^{-4}M_{\odot}$ over the first $\sim15$ orbits. It then falls to $\sim5 \cdot10^{-5}M_{\odot}$ and subsequently fluctuates around that value. A midplane cavity opens (Figure 3), but the enclosed mass within a sphere of radius $30r_{\mathrm{g}}$ does not decrease, because much of the gas is pushed out of the midplane (Figure 2 and Figure 8) and the inner disk is continuously resupplied from the stream and mass at larger radii.

Figure 6 panel (a) shows us that more mass is accreted by the black hole than is brought from large radii, and stream mass is the only other source of mass in the simulation. At late times ($t \gtrsim35\,t_{\mathrm{orb}}$), stream material supplies $\simeq8\,\dot{M}_{\mathrm{Edd}}$ of the $\simeq12\,\dot{M}_{\mathrm{Edd}}$ accreted through $r = 4r_{\mathrm{g}}$, i.e. about two-thirds of the injected $\dot{M}_{\mathrm{s}}$; the remainder is pre-existing disk material. Thus, although the stream punctures the disk and part of it continues

past the impact point, most of it is eventually accreted. Little of it falls in promptly: the bulk is deflected at the collision, swings past pericenter, and falls inward only afterward. This is dramatically different from ordinary TDEs, where most of the stream mass continues to orbit at $\sim10^3 r_{\rm g}$ for at least several $t_0$ ([30]). We discuss this further in the following § 3.1.2.

These results differ from the hydrodynamic picture of Chan et al. [11], who found a surface-density depletion of roughly an order of magnitude interior to the impact radius. The difference is most plausibly attributable to a small fraction of the stream mass joining the disk during the collision, as discussed above: because our $\dot{M}_{\rm s}/\dot{M}_{\rm d}$ is much larger than theirs (§ 2.2.3), even this small fraction is enough to offset the depletion seen in Chan et al. [11].

Mass sources in the disk

Figure 7 tags the two components and tracks their evolution. Panel (a) shows the enclosed mass of each component, and panel (b) the corresponding mass fractions. By the end of the run ($t \sim43$ orbits), the enclosed mass within $30r_{\rm g}$ is composed of $\sim50\%$ stream material and $\sim50\%$ material that was outside $30r_{\rm g}$. The total enclosed mass itself also grows steadily over this period (panel a), and the persistent rise of the $r > 30r_{\rm g}$ contribution shows that mass from well outside the $15r_{\rm g}$ impact radius continues to be advected inward long after the initial collision. We return to this as effect (3) in § 3.3.

Tilted, reorienting inner flow

Figure 8 maps the gas density in the $\theta$-$\phi$ plane at radii $r = 6$, $10$, and $20r_{\rm g}$. Prior to stream injection ($t = 0$), the pre-existing disk appears as a thin, axisymmetric equatorial band with nearly $\phi$-independent density. The band is almost absent at $r \leq6r_{\rm g}$, the plunging region at and within the ISCO of the pseudo-Newtonian potential, and becomes progressively thicker and denser toward $r = 20r_{\rm g}$. After the debris stream is introduced, the disk becomes tilted with respect to its initial midplane and appears in these images as a band with a crudely sinusoidal and non-axisymmetric wiggle.

Figure 9 shows the direction of the total angular momentum of all gas with $r < 14r_{\rm g}$ and $\theta\in[\pi/4, 3\pi/4]$ on a polar projection with radius equal to the tilt from the initial disk axis and a rotation angle equal to the azimuth, color-coded by time. The tilt grows to $30$–$35^\circ$ within $\simeq15$ orbits ($\simeq1.5$ d). The azimuth sweeps from $\phi\simeq180^\circ$ through $270^\circ$ to $\phi\simeq0$–$45^\circ$, then wanders randomly in the range $\phi\simeq340^\circ\pm20^\circ$. The driver for this tilt is most plausibly momentum transfer from the debris stream to the inner disk.

!Direction of the total angular momentum of all gas with $r < 14r_{\rm g}$ and $\theta\in[\pi/4,3\pi/4]$ bbox=0.08,0.49,0.48,0.82

This tilted flow could generate disk variability on the same timescale, particularly the wiggling of the inner radiation-flux direction (as seen in Figure 12). In addition, aspect-dependent signatures (obscuration of the corona, or the orientation of an emerging jet) could track the reorientation of the inner flow.

Preferred inflow trajectory

Figure 10 maps radial accretion rate in g s$^{-1}$ cm$^{-2}$ at $t = 15.35t_{\rm orb}$; positive (blue) values correspond to inflow. In the midplane (left panel), inflow is concentrated in a few broad spiral lanes that cover only a limited fraction of the azimuth and are interleaved with outflow regions. This morphology matches that seen by Chan et al. [11].

To further assess the impact of disk tilt on the inflow trajectory, we show in Figure 11 the inflow on the inner cutout, in units of g s$^{-1}$ cm$^{-2}$. Prior to stream injection, the accretion through $r = 4r_{\rm g}$ is confined to a thin belt at the equatorial midplane, $\theta= \pi/2$. After impact the high-$\dot{M}$ band warps into a sinusoidal track in the $\theta$–$\phi$ plane, displaced by $\sim25^\circ$–$45^\circ$ from the equator by $t = 15.35t_{\rm orb}$, closely following the tilted midplane of Figure 8 and Figure 9. By $t = 43.50t_{\rm orb}$ the band is broader and more filamentary, spanning much of $|\theta- \pi/2| \lesssim60^\circ$, indicating that all mass at large tilt angles, and not only the tilted disk, is contributing to the rising accretion rate. This mass could have originated from both the stream and disk matter that was shocked, heated, and lifted to these trajectories. Despite this, most of the mass inflow still takes place through the band occupied by the tilted disk.

Dynamical response summary

In short, $\dot{M}$ rises from $0.07\dot{M}_{\rm Edd}$ to $\simeq12\dot{M}_{\rm Edd}$ in $4.4$ d while the enclosed mass within $30r_{\rm g}$ grows by a factor $\simeq5$. Roughly $\sim50\%$ of that late-time mass is disk matter that was outside $30r_{\rm g}$ a few orbits earlier, and the rest is stream mass. In addition, the inner flow tilts by $30$–$35^\circ$ with a wandering azimuth. The disk puffs up to a thick, lopsided, vertically extended body.

Radiation

Bolometric luminosity

Figure 6, panel (d) shows the bolometric luminosity in units of $L_{\rm Edd}$, measured by integrating the vertical radiation flux over the sky-plane footprint of the scattering photosphere (Figure 2 and Figure 4), i.e. the luminosity seen by an observer along the $z$ line of sight in each hemisphere. After the stream is introduced, the north-hemisphere luminosity rises from $\sim0.02L_{\rm Edd}$ to $\sim L_{\rm Edd}$ over the run. The south side peaks near $\sim0.1L_{\rm Edd}$ within a few orbits and then falls to $\sim0.01L_{\rm Edd}$: the optically thick outgoing stream obscures thermal photons from escaping and carries them with it. As the stream moves away from the disk, it expands, draining the photon energy by adiabatic expansion work. The escaping thermal output is therefore dominated by the northern hemisphere and remains $\sim L_{\rm Edd}$.

Over the same interval $\dot{M}$ grows by more than two orders of magnitude relative to its initial value while $L$ stays near Eddington, so the radiative efficiency $\eta\equiv L/\dot{M}c^2$ falls steadily: with $\dot{M}_{\rm Edd}\equiv10L_{\rm Edd}/c^2$, late-time values $L\sim L_{\rm Edd}$ and $\dot{M}\simeq12\,\dot{M}_{\rm Edd}$ imply $\eta\simeq0.01$, an order of magnitude below the fiducial thin-disk value $0.1$. This is the trapping-dominated regime anticipated by Chan et al. [11], who argued that when the inflow time is shorter than the photon diffusion time, much of the dissipated energy is advected with the gas rather than radiated. In fact, Eddington luminosity is a general result valid for any optically thick system powered by the conversion of gravitational energy into radiation (e.g., accretion). We can check whether this optical thickness criterion holds at the impact radius.

Equating the diffusion time, $\sim\tau H/c$, with the inflow time, $\sim r/v_r$, defines a trapping radius $r_{\rm tr}\simeq2(\dot{M}c^2/L_{\rm Edd})(H/r)r_{\rm g}\approx240(H/r)r_{\rm g}$ for $\dot{M}\simeq12\,\dot{M}_{\rm Edd}$. For any $H/r\gtrsim0.1$, this exceeds the impact radius, so the collision region lies inside the trapping zone, and most of the heat dissipated there is carried inward before it can escape.

Radiation flux and radiation temperature

Figure 12 (top) displays the radiation temperature $T_{\rm rad}=(E_{\rm rad}/a)^{1/4}$ overlaid with radiation flux streamlines $F_{\rm rad}$ at $t=15.35t_{\rm orb}$. The innermost disk ($\lesssim10\,r_{\rm g}$) near the tilted midplane is the hottest, reaching $T_{\rm rad}\gtrsim10^6\ {\rm K}$. Across most of the southern hemisphere, out to $r \lesssim30\,r_{\rm g}$, the gas maintains $T_{\rm rad} \simeq5 \times10^{5}\,\mathrm{K}$. In the northern hemisphere, while comparable temperatures ($T_{\rm rad} \sim5 \times10^{5}\,\mathrm{K}$) persist out to $\simeq5\,r_{\rm g}$ above the midplane, farther into the northern hemisphere the temperature drops sharply to $T_{\rm rad} \lesssim10^{5}\,\mathrm{K}$.

The streamlines are colored by the isotropic-equivalent luminosity $L_{\rm iso}=4\pi r^2(\mathbf{F}_{\rm rad}\cdot\hat{\mathbf{r}})$. Its absolute value spans the range $\simeq10^{44}$ to $\simeq10^{46}\,\mathrm{erg\,s^{-1}}$, and is largest along the outgoing stream in the southern hemisphere, where it approaches $\sim10^{46}\,\mathrm{erg\,s^{-1}}$. It is important to note that the entirety of the southern hemisphere inside $30r_{\rm g}$ is inside the photosphere, and there are significant portions of that region where the flux is directed inward. Inside the photosphere, the optical depth is large, so radiation is advected with the gas and $\mathbf{F}_{\rm rad}$ runs parallel to the flow; only above the photosphere does the flux turn and escape. The flux is strong in the outgoing stream, but those photons do not immediately escape: they remain trapped and are carried south with the debris. Similarly, the regions where the nominal $L_{\rm iso}$ is negative have inward-directed fluid velocity (see Figure 2).

Elsewhere, especially in most of the northern hemisphere, $L_{\rm iso}$ is consistently positive, but lower in magnitude, typically $\sim10^{44}$–$10^{45}\,\mathrm{erg\,s^{-1}}$, and is at the higher end of that range for most of the northern hemisphere outside the photosphere (shown at this time in the middle row of Figure 2). Figure 13 shows, for a number of other times during the simulation, the azimuthally averaged $L_{\rm iso}(\theta)$ at $r=30\,r_{\rm g}$ (outside the photosphere). We see that each curve is nearly flat in polar angle and then drops at the angle where this radius passes inside the photosphere. After impact, the cutoff first moves out to $\theta\sim80^\circ$, then retreats as the photosphere extends vertically (Figure 4), so that by $t \simeq43\,t_{\rm orb}$ the escaping flux is confined to a cone of opening angle $\sim40^\circ$–$45^\circ$. Across that cone $L_{\rm iso}$ is roughly flat at $\simeq5L_{\rm Edd}$. The polar value rises from $\simeq0.3L_{\rm Edd}$ before impact to $\simeq5L_{\rm Edd}$ at the end of the run.

Because the overall optical depth of the southern hemisphere is greater than in the north, the photon-bright hemisphere is the north (Figure 6, panel d). On the other hand, the south is the jet-bright side, carrying the larger polar-cap magnetic flux (§ 3.3).

Photosphere and emitted photon energies

The photospheres marked in Figure 2 and Figure 4 provide a direct estimate of the emitting surface. After stream injection, the south photosphere follows the outgoing stream to $|z|$ of several hundred $r_{\rm g}$ (Figure 4): that surface is larger than the northern photosphere, but it is the obscured, thermally dim side.

Figure 14 reconstructs a spectral luminosity from the thermalization-photosphere flux: each sky-plane patch is assigned an effective temperature from its vertical flux and contributes at the corresponding Wien peak, so $\int L_\nu\,d\nu$ equals the bolometric $L$. The resulting $L_\nu$ peaks between $\sim10$ and $\sim100\,\mathrm{eV}$, in the EUV. The thermally photon-bright north side hardens over time (peak shifting from $\sim20$–$30\,\mathrm{eV}$ to $\sim50$–$80\,\mathrm{eV}$) while the south remains softer ($\sim20$–$50\,\mathrm{eV}$).

Radiation summary

The escaping luminosity, measured on the scattering photosphere, is dominated by the thermally photon-bright north hemisphere and approaches $\sim L_{\rm Edd}$ while $\dot{M}$ climbs to $\simeq12\dot{M}_{\rm Edd}$. The radiative output stays near Eddington even though the inflow is highly super-Eddington, corresponding to a radiative efficiency $\eta\simeq0.01$ under our $\dot{M}_{\rm Edd}=10L_{\rm Edd}/c^2$ convention. The south is thermally dim because the stream obscures photons and carries them with it, and is the jet-bright side. $L_\nu$ reconstructed from the thermalization-photosphere flux peaks at $10$–$100$ eV, in the EUV.

Magnetic-flux transport toward the black hole

Can the stream-disk interaction deliver enough horizon-threading flux, fast enough, to power a transient jet? As we argued in Tripto et al. [50], a viable mechanism must (i) supply magnetic flux of order $\Phi\sim10^{29}\ {\rm G\,cm^2}$ when $M_{\rm BH} \sim10^6 M_\odot$ in order to generate a jet luminosity $\sim10^{44}\ {\rm erg\,s^{-1}}$, and (ii) act within a time $\sim t_0$, typically several weeks, of the onset of mass return.

The quantity relevant to jet launching is the ordered flux threading the polar region, as seen in Figure 12 (bottom). Because these images show that the flux is very coherent within the black hole’s polar caps, we define $\Phi$ as the flux on the inner cut-out at angles $< \pi/4$ from the polar axis. The AGN0.07 run, initialized with a quadrupolar field, wholly lacks such order at the midplane (Figure 15 demonstrates the disorder on the midplane at the moment the debris is first injected). Figure 15 also shows that the $\phi$-averaged midplane $-B_z$ changes from a sign-alternating profile with $|B_z| \lesssim500$ G before impact to a single-signed inner profile peaking at $\simeq2200$ G afterward. Thus, our simulation shows that the debris impact also organizes the vertical field through the disk midplane, as discussed in Tripto et al. [50].

Figure 6, panel (c) shows the polar-cap flux as a function of time. Prior to stream injection, the flux on the two poles is comparable, as expected, and flat at $\Phi\simeq6 \cdot10^{27}\ {\rm G\,cm^2}$ over the preceding 13 orbits. Within 1–3 orbits (0.1–0.3 d), the mean flux jumps by a factor $\simeq4$ to $\simeq2.5 \cdot10^{28}\ {\rm G\,cm^2}$, and the fluxes on the two poles begin to depart from each other. The southern hemisphere flux then grows slowly, eventually reaching $\simeq2 \cdot10^{29}\ {\rm G\,cm^2}$ by orbit 30 (3 d) and then saturating; meanwhile, the north-hemisphere flux also rises slowly until $\simeq20$ orbits and thereafter fluctuates in the range $\sim2 \times10^{28}$–$10^{29}$ G cm$^{2}$.

Three mechanisms appear to contribute to that growth.

Effect (1): collapse of pre-existing flux interior to the impact radius. The rapid inflow discussed in § 3.1.1 carries the disk’s associated field inward. This alone can account for the initial factor of $\simeq4$ increase, to a mean flux $\simeq2 \cdot10^{28}$ G cm$^{2}$. The first few orbits are exactly the time during which the material originally in the inner disk is swept toward the horizon. The pre-impact flux in the disk, estimated by the cone integral described below, is $\simeq10^{28}$ G cm$^{2}$, matching this initial rise. It is the sole effect we considered in our baseline analytic estimate (Tripto et al. [50]).

Effect (2): conversion of toroidal field to poloidal field. Figure 16 illustrates this mechanism, in which the poloidal flux on the inner cut-out grows when toroidal field-lines are bent into poloidal fieldlines while leaving the poloidal flux in the remainder of the domain untouched. The initial (pre-disk) quadrupolar field is wound by orbital shear into a toroidal field of coherent sign over an extended radial range (loop 1); because such a loop closes on itself without leaving the disk plane, this winding does not add poloidal flux. The impact of the debris stream converts part of this toroidal reservoir into poloidal flux through two geometric deformations of field lines. First, matter attached to these field lines moves inward, pulling its portion of the field lines inward toward the black hole and stretching the field lines across a wide range of $r$ (loop 2). Next, the debris pushes matter in the inner disk out of the disk plane, in most cases moving it into the southern hemisphere, while matter on the same field lines that remains near its original radius stays approximately in the disk plane (loop 3). As a result, the attached field lines are bent by $\sim90^\circ$, and poloidal flux is created in the black hole polar caps. The direction of the debris’s motion therefore accounts for the observed upper/lower-cone flux asymmetry (Figure 6, panel (c)). This description is qualitatively consistent with the simulation’s field morphology (Figure 12, bottom). It also explains why $B_z$ in the inner disk becomes predominantly negative, as seen in Figure 15. Figure 16

shows that field on a polar cap with a given sign requires vertical field of the opposite sign somewhere in the disk; Figure 12 (bottom) shows that the majority of the polar cap flux, the part on the south side, has $B_z > 0$; the mean $B_z$ in the inner disk must therefore be negative.

Effect (3): advection from outside the impact radius. Figure 7 shows that $\simeq50\%$ of the late-time mass inside $30r_{\rm g}$ originated from outside $30r_{\rm g}$, well beyond the $15r_{\rm g}$ impact point. Mass advected from outside the impact point carries additional magnetic flux, further increasing the flux available for advection onto the black hole. We

Tripto et al. [50] noted this possibility as well, but excluded it from that analysis to obtain a conservative estimate.

Effect (1) delivers $\simeq10^{28}\ {\rm G\ cm}^{2}$; effects (2) and (3) raise the collected flux by about an order of magnitude to $\simeq2 \cdot10^{29}\ {\rm G\ cm}^{2}$. Unfortunately, we cannot distinguish the contributions of these two mechanisms in the simulation.

Adopting a slightly modified version of the conventions of Tchekhovskoy et al. (2011) and Kelley et al. [28] (we discuss these conventions more carefully in [50], the Blandford-Znajek scaling gives $L_{\rm BZ}=c\xi_{\rm BZ}\Phi^2/(2\pi^2r_{\rm g}^2)$. With $\xi_{\rm BZ}=0.1$, which corresponds to a rapidly spinning black hole,

The polar flux is not symmetric between the two hemispheres: during the middle of the simulation, the flux threading the southern pole reaches $\sim2\cdot10^{29}\ {\rm G\ cm^2}$, raising that side’s isotropic-equivalent luminosity to $\sim10^{43}\ {\rm erg\ s^{-1}}$. These values are obtained with a quadrupolar (zero-net-flux in each hemisphere) initial field. Even higher values could be achieved with a more favorable magnetic configuration (e.g., dipolar) initial field whose inner region carries net poloidal flux [2]. The magnetic flux on the south polar cap exceeds that in the north by a factor of 3–10, corresponding to a Poynting flux ratio $\sim10$–100. The south is therefore the jet-bright hemisphere, opposite the thermally photon-bright north. There is a hint of decline on the favored (south) side after $\simeq3$ d that is not seen on the north; confirming this would require a longer run with a time-dependent fallback rate. The selection-effect implication of the hemisphere asymmetry is deferred to § 4.

These specific quantitative results can be compared with the approximate predictions of our analytic scheme (Tripto et al. [50]), which used the vertical magnetic flux on the midplane within the impact radius, here $r = 15r_{\rm g}$. Because the quadrupolar initial state makes the vertical field on the midplane disordered, we choose to measure the pre-impact flux on a narrow cone around the polar axis. In Figure 6, we plotted the flux on the cut-out, the end-product, so to speak; here we measure the amount of “raw materia”. Integrating

gives an estimate of the flux in the disk $\simeq10^{28}\ {\rm G\,cm^2}$ before impact, roughly twice the flux on the cut-out at the same time. This is an order of magnitude smaller than the flux achieved through the action of the debris stream. In other words, the impact of the tidal debris enlarges the available flux by a factor $\sim10$ relative to the simple analytic estimate of the available flux.

Magnetic flux summary

The simulation reaches $\Phi\simeq2\cdot10^{29}\ {\rm G\,cm^2}$ in the south after $\sim30$ orbits at $15r_{\rm g}$ (3 d), which is higher than the pre-existing flux by a factor of $\sim33$. For our black hole mass, $5\cdot10^6\,M_\odot$, that flux can lead to a jet luminosity $\sim10^{43}\ {\rm erg\,s^{-1}}$, which would be sufficient for some jetted TDEs. This flux is delivered to the black hole horizon by three mechanisms. Our analytic estimate (Tripto et al. [50]) used only one: the accumulation of flux from inside $r_{\rm imp}$; including toroidal-poloidal conversion and outer-radius advection, the flux grows by about an order of magnitude. The underlying quadrupolar field geometry of our simulation did not yield a consistent vertical-field sign; if it had, for example, a dipolar structure, the flux might have been even greater ([2]).

Discussion

Where the Matter Goes; Comparison with Previous Work

Chan et al. [11] reported a set of purely hydrodynamic simulations illustrating what would happen when a TDE debris stream strikes an accretion disk. In addition to omitting both radiation transfer and magnetic fields, these simulations used the $\alpha$-model to determine the initial state of the disk. Replacing that model with the AGN0.07 disk raises $\dot{M}_{\rm s}/\dot{M}_{\rm d}$ by about two orders of magnitude (to $\simeq400$, versus $\leq4$). Even so, our results, which incorporate radiation transfer for gas cooling and radiation forces, as well as magnetic fields, while also using a more physically realistic initial disk, confirm several of the key results of the earlier work. In only a few local disk orbits, the mass contained in the disk at radii inside $r_{\rm imp}$ is swept rapidly toward the black hole’s event horizon at a rate that can be $\sim O(10\dot{M}_{\rm Edd})$. Throughout the debris-disk interaction, spiral shocks are driven outward from the impact points, and the gas inflow is concentrated in azimuthally localized channels. However, there are several ways that the additional physics we have introduced changes the outcome of the event.

First, the ratio $\dot{M}_{s}/\dot{M}_{d}$ is so large that the stream’s momentum completely dominates the collision between it and the disk gas. The result is that the disk gas is strongly deflected, while the stream is deflected only slightly.

Second, two new mechanisms copiously resupply the disk’s inner region with matter. Magnetic stresses and radiation forces push matter from as far away as $\simeq2r_{\rm imp}$ to well inside $r_{\rm imp}$. In addition, a significant fraction of the mass brought to the disk in the debris stream is diverted so that it, too, finds itself at $r<r_{\rm imp}$ en route to the horizon; because $\dot{M}_{s}/\dot{M}_{d}$ is so large, the diverted debris mass is significant relative to the disk mass. Between the two of these sources, the mass in the inner disk at $\simeq40$ local orbits after the debris first arrived is $\approx3\times$ what it was at the start (as shown in Figure 3 (bottom)).

Third, the force of the debris stream striking the disk from the “north” pushes the disk gas “southward”, causing the disk to tilt. Over time, the irregularity of the mass distribution in the impact region is so great that, as shown in Figure 8, the inner disk is better described as a collection of mutually-tilted streams.

Fourth, the collision barely changes the stream’s momentum, but it does reshape its orbit. After the impact, about a third of the debris keeps roughly its original energy, nearly two-thirds becomes more tightly bound (a fifth of the debris by more than a factor of two), and only a few percent becomes less bound. Thereafter, the debris moves outward nearly ballistically beneath the bulk of the disk, turning around at $\sim50$–$400r_{\rm g}$; the most bound fifth returns within a few orbital periods at $r_{\rm imp}=15r_{\rm g}$, while the rest returns to the disk plane at larger radii ($\gtrsim O(100)r_{\rm g}$) at later times.

We expect that the debris that has spread out will be stopped by the outer disk, and that its fate will resemble that discussed by Chan et al. (2021). An exact calculation of this phase lies beyond the extent and duration of our simulation.

Photon Emission

The thermal luminosity in both hemispheres first jumps by a factor $\sim10$ within a few hours of the first interaction, to $\simeq0.1L_{\rm Edd}$, and then grows more slowly, reaching $\sim L_{\rm Edd}$ by $\sim4$ d. The luminosity from the northern hemisphere grows more slowly thereafter, reaching $\sim L_{\rm Edd}$ by $\sim4$ d. Because $L_{\rm Edd}$ is the characteristic luminosity for any gas mass suddenly heated by dissipation of kinetic energy obtained from its gravitational infall and then cooled by photon diffusion regulated by electron scattering within the same mass (Krolik [31]), the peak luminosity is unlikely to vary substantially for systems with different parameters. Meanwhile, the luminosity from the southern hemisphere falls, dropping to $\simeq1/2$ its initial value over that same time period. As a result, the north side is $\simeq150\times$ brighter than the south side. The cause of this sharp contrast is the debris stream matter, just noted, which makes the southern hemisphere considerably more optically thicker than the northern.

The spectrum reconstructed from the photosphere peaks in the EUV, at $10$–$100$ eV (Figure 14). So many astrophysical environments have large opacity to photons in this band that any detection of this continuum is likely to be through reprocessing within the galactic nucleus. Because we are considering TDEs in AGN, the nearest such reprocessor would be the AGN’s broad emission line region. If the greatest emission line response from the broad line gas comes from gas with the optimal ionization parameter (i.e., ratio of ionizing flux to density), the distance to the BLR adjusts $\propto L^{1/2}$. That this might be so is suggested by the strong $r\propto L^{1/2}$ correlation found when distance is estimated from the variability time lag. Applying a bolometric correction of $5$ to the standard luminosity normalization of this correlation (Bentz et al. [3]) indicates $r\simeq13L_{44}^{1/2}$ light-days. Strictly speaking, this correlation applies to Balmer lines, and it is possible that the spectrum we expect from these TDEs would more efficiently produce higher-ionization lines, so the lag might be somewhat shorter. In any event, one might expect a lag of roughly this magnitude, which is only a factor of a few shorter than typical values of the characteristic TDE timescale $t_{0}$.

There could likewise be a reprocessed signal from absorption in the dusty obscuring torus. The distance between its inner edge and the black hole should likewise scale as $L^{1/2}$ because it is most likely determined by dust sublimation. Calibrating this relation to NGC 1068 (Gámez Rosas et al. [15]), we find $r\sim80L_{44}^{1/2}$ light-days. This reprocessing could therefore create a mid-IR flare echo with a delay of a few months.

Two interesting questions about the detectability of photons from TDEs in AGN remain open. One is the shape of the lightcurve over timescales of a few $t_{0}$; our simulation was far too short to shed any light on this question. The other is that both our simulation and that of Chan et al. (2021) suggest that a significant fraction of the debris mass falls back to the accretion disk at radii $\sim10$–$100\times r_{\rm imp}$, or $\sim10^{2}$–$10^{3}r_{\rm g}$. Some of this mass might therefore land on the region where the undisturbed AGN disk radiates its optical/UV continuum, the photons we most easily observe. Chan et al. (2021) argued that the energy released when debris falls back on the disk would emerge as hard X-rays; it is possible that there may also be an augmentation of the optical/UV luminosity.

Magnetic Flux and the Implied Jet Luminosity

As estimated in § 3.3, the generic simulation we conducted brought $\sim10^{29}\ \mathrm{G\ cm^2}$ of magnetic flux to the black hole in a time only a fraction of $t_0$. On the basis of this simulation, which was essentially Newtonian, we cannot speak to the luminosity of any resulting jet. However, if the black hole spins rapidly enough to provide a reasonable jet efficiency ($\xi_{\mathrm{BZ}} \sim0.1$), the expected jet power would be $\sim10^{43}\ \mathrm{erg\ s^{-1}}$, comparable to several of the observed jetted TDEs. Recall that Tripto et al. [50] argued that the presence of a magnetized accretion disk is the most favorable mechanism for providing the magnetic field needed to power jets in TDEs.

Like the thermal photon luminosity, the magnetic flux on the black hole polar caps is strongly asymmetric, but in the opposite sense: it is larger on the “south” pole than on the “north” by a factor of $\sim3$–$10$. With the expected jet luminosity scaling as $\Phi^2$, the luminosity asymmetry should be $\sim10$–$100$. This effect is also due to symmetry breaking arising from the debris stream’s direction of travel, a feature that distinguishes TDEs in AGN from ordinary AGN.

Very much unlike the thermal photon luminosity, the jet luminosity could be sensitive to a great many parameters. It increases rapidly with black hole spin (McKinney [33]; Hawley & Krolik [19]; Tchekhovskoy et al. [47] (2011)). It should also vary with the magnetic field geometry: as shown by Beckwith et al. [2], dipolar fields support jets better than the quadrupolar field of our simulation. Inclination of the stellar orbit relative to the disk material’s orbit (presumably aligned with the black hole spin) should also matter. The magnetic flux in the disk can be expected to rise with accretion rate and depend upon $M_{\mathrm{BH}}$, while the interaction with the debris stream will also change with $M_\star$. Consequently, observed jet luminosities are likely to deviate considerably from this single example.

The magnetic flux delivered to the black hole increases with the magnetic flux in the pre-existing disk inside $r_{\mathrm{imp}}$, but is not strictly limited to it because the action of the TDE itself can raise the flux given to the black hole by an order of magnitude or more: in the fraction of $t_0$ covered in our simulation, flux originally located as far away as $\simeq2r_{\mathrm{imp}}$ can be brought in; in addition, toroidal field can be rotated so as to be poloidal when carried to the horizon. However, the fact that the flux in the disk is limited does imply that the magnetic field strength on the black hole horizon does not necessarily rise to the limiting level associated with magnetically arrested disk (MAD) states. In our example, the normalized flux $\phi\equiv\Phi/\sqrt{\dot{M}r_g^2c} \propto\Phi(\dot{M}/\dot{M}_{\mathrm{Edd}})^{-1/2}M_{\mathrm{BH}}^{-3/2}$ remains of order $\sim0.1$. It changes by only a factor of a few while the flux grows by a factor $\sim33$, because $\dot{M}$ grows at the same time. This value of $\phi$ is more than two orders of magnitude below the prerequisite for MAD. Nonetheless, the associated jet luminosity can be quite substantial. The limited variation of $\phi$ also implies both that the magnetic pressure just outside the black hole is regulated to have somewhat less than the ram pressure of infalling matter and that the associated jet’s efficiency with respect to the accretion rate stays of the same order.

Observed Population

Nearly all the power in thermal photons emitted at the photosphere of the inner disk region is radiated at energies in the range $\sim10$–$100\ \mathrm{eV}$. Through the combined action of the many ionization edges found in that band and dust absorption, it is very difficult for such photons to travel more than a short distance through either the galactic nucleus in which they are generated or the ISM of our own galaxy. Consequently, even when our view is toward the bright side (the “north” side) of the AGN disk, this spectrum will only rarely be visible. In contrast, the spectra of jets are much broader, and large parts of their radiated power are often in the range $10$–$100\ \mathrm{keV}$, in which only the lowest end may have significant absorption opacity. This fact makes the jet radiation much more readily observable, especially when we are on the “south” side of the disk.

On the other hand, as already remarked, the absorption of thermal photons that renders them unobservable directly can create new photons that are easier to detect, either atomic emission lines or mid-IR thermal radiation from dust grains. Combining these several facts suggests that in observed samples, jetted TDEs will in most instances be seen from the “south” side of the AGN disk, and only through their jet radiation and possibly emission lines or mid-IR. The reprocessed signals are not subject to the same close-in obscuration as the thermal photon continuum, so they should be observable from either the north or the south side. In the mid-IR, the flare should be delayed by several months relative to the jet radiation flare. We caution, however, that our simulation was designed to mimic the period of peak mass-return rate in the TDE. It therefore cannot predict any specific time-dependence for either the thermal photon luminosity or the jet luminosity.

Finally, because the fraction of galaxies possessing AGN is $\sim10^{-1}$ and the fraction of AGN having strong jets is also $\sim10^{-1}$, the fraction of all TDEs resulting in strong jets should be $\sim10^{-2}$, consistent with the empirical estimate (Andreoni et al. [1]).

Summary

As first shown by Chan et al. [11], the debris dynamics of a TDE in an AGN differ drastically from those of an ordinary TDE; as we show here, these altered dynamics produce somewhat greater observed luminosities, radiate much more asymmetrically, and have quite different spectra than ordinary TDEs. They can also supply the magnetic flux needed to launch a jet. The stream–disk collision excites spiral shocks that drive rapid, super-Eddington accretion. At first, the collision carves a midplane cavity. This is followed by mass build-up in the innermost disk, reaching eventually a steady plateau of the surface density. Ultimately, the inflow into the black hole is about two-thirds captured stream material and one-third native disk gas fed from the outer region. Furthermore, interaction with the incoming stream rapidly tilts the inner disk out of its original plane on dynamical timescales, inducing irregular azimuthal precession.

The source of the photons is also entirely different from that in ordinary TDEs. In the ordinary case, the optical/UV light draws its energy from shocks at $\sim10^{3}r_g$ and is radiated from this shock region. When the TDE occurs in an AGN, the energy comes from shocks within a few tens of $r_g$ and is radiated from a photosphere of complex geometry, whose distance from the black hole ranges from $10$–$20r_g$ in the north hemisphere to several hundred $r_g$ in the south; the photosphere at $10$–$20r_g$ is much brighter thermally. The escaping luminosity in the brighter (northern) direction reaches $\sim L_{\mathrm{Edd}}$ or more, and $L_\nu$ peaks at 10–100 eV, in the EUV.

We have shown that the polar cap magnetic flux can grow by a factor $\sim33$ and then saturate during a small fraction of a characteristic mass-return time. Three effects contribute: the collapse of pre-existing flux interior to the impact radius, the organization of toroidal field into poloidal flux, and advection of flux from outside the impact radius. Our earlier estimate [50] retained only the first; the present run strengthens that estimate by about an order of magnitude. The resulting jet luminosity, $\sim10^{43}\,\mathrm{erg\,s^{-1}}$, is consistent with some jetted TDEs.

Both the thermal emission and the jet are highly anisotropic, but they favor opposite hemispheres. The north (the side from which the stream arrives) is thermally photon-bright and dominates the escaping luminosity. The south (the direction of stream motion) is thermally dimmer because the stream obscures photons, but its polar-cap flux exceeds the north’s by a factor of $\sim3$–$10$ near the flux peak, leading to a factor of $\sim10$–$100$ in Poynting flux, making the south the jet-bright side.

Acknowledgments

We thank Yan-Fei Jiang for providing the input data for our simulation and for extensive consultations about Athena++. We also thank him, Chi-Ho Chan, and Scott Noble for helpful discussions. This work was supported by the Simons Foundation SCEECS collaboration (MP-SCMPS-00001470) (NT,TP), by an ERC advanced grant MultiJets (NT,TP), and by NSF Grant 2110339 and NASA TCAN grant 80NSSC24K0100 (JHK).

Unit conversion

We initialize the ambient disk using a snapshot from the “AGN0.07” global RMHD simulation of Jiang et al. [22], which was carried out for a black hole mass $M_{\mathrm{BH},0}=5\cdot10^{8}\,M_\odot$. In this work we require a similar disk solution around $M_{\mathrm{BH}}=5\cdot10^{6}\,M_\odot$.

In Newtonian hydrodynamics one can often rescale mass/length/time units with multiple free parameters, but in radiation-MHD, the presence of universal dimensional constants and fixed microphysics reduces or eliminates this freedom. More generally, the number of free rescaling parameters is $N_f=\max(0,3-N_{\mathrm{udc}})$, where $N_{\mathrm{udc}}$ counts independent universal dimensional constants appearing in the equations and in the prescribed microphysics/boundary conditions (Granot [17]). Therefore, in RMHD with gravity there is no non-trivial unit rescaling that preserves the full set of equations; one must instead choose a mapping that keeps a selected set of parameters fixed. We keep (i) radii in units of $r_g$, (ii) the Eddington ratios $\dot{M}/\dot{M}_{\mathrm{Edd}}$ and $L/L_{\mathrm{Edd}}$, and (iii) the RMHD dimensionless parameters used by Athena++ equations ($\mathcal{C}\equiv c/v_\ast$ and $\mathcal{P}\equiv aT_\ast^4/(\rho_\ast v_\ast^2)$) invariant. A similar procedure, followed for much the same reasons, is described in Zhu et al. [54].

Let $M\equiv M_{\mathrm{BH}}$ denote the target mass and $M_0\equiv M_{\mathrm{BH},0}$ the original one. For a generic physical quantity $X$ given in cgs units, we denote its code value equivalent as $\bar{X}$, and the conversion factor from code units to physical units as $X_\ast=X/\bar{X}$. We choose the characteristic units $(r_\ast,v_\ast,\rho_\ast)$ to scale as

which implies

With these choices, the luminosity unit scales as $L_\ast\sim T_\ast^4r_\ast^2\propto M$, so $L/L_{\mathrm{Edd}}$ stays constant by construction, and the electron-scattering optical depth scale $\kappa_{\mathrm{es}}\rho_\ast r_\ast$ is also invariant.

Unit conversion

The mass ratio is $M/M_0=10^{-2}$. The implied conversion factors are therefore

This mapping preserves the electron-scattering opacity, but not the absorption opacity, which depends explicitly on $\rho$ and $T$. The AGN0.07 disk was deliberately placed near the iron opacity bump ($T\approx1.8\cdot10^5$ K; Jiang et al. [23]), whereas after rescaling ($T_\ast=10^{1/2}T_{\ast,0}=(M/M_0)^{-1/4}T_{\ast,0}$) the gas lies at $T\approx6\cdot10^5$ K, where the Rosseland-mean absorption opacity is substantially lower. The rescaled disk is therefore not an exact thermal equilibrium of our equations and evolves on its own even without stream injection. This drift is slow compared with the stream-driven evolution: over the $\approx13$ orbits preceding injection (Figure 6, $t<0$), the accretion rate, magnetic flux, luminosity, and mass within $30\,r_g$ remain nearly constant, and the mass within $12\,r_g$ grows by $\approx60\%$, roughly an order of magnitude slower than its growth after injection.

Stream Injection Scheme

Following the general prescription of Chan et al. [11], the TDE debris is introduced through a source term in the hydrodynamic equations. The injection region is described in parabolic coordinates $(\ell,m,n)$, in which the spherical radius is $r=(\ell+m)/2$ and $n$ is an azimuthal angle. This stream-aligned frame is rotated into the simulation frame by the angles $(\theta_{\mathrm{pitch}},\theta_{\mathrm{roll}},\theta_{\mathrm{yaw}})$.

The injected density has a Gaussian profile centered at $(\ell_0,m_0,n_0)$,

where $(\Delta\ell,\Delta m,\Delta n)$ set the size of the injection region and $\rho_c$ is normalized to the prescribed mass current $\dot{M}_s$. The corresponding physical widths at the injection point are $\simeq0.1\,r_g$ and $\simeq0.3\,r_g$ transverse to the stream, i.e. $\simeq R_\odot$ and $\simeq3R_\odot$ ($\sim1.2$–$4R_\odot$).

The injected velocity is directed along the tangent of a Newtonian parabolic orbit, with magnitude $v_s=(\eta_vGM_{\mathrm{BH}}/r)^{1/2}$, $\eta_v=2$ is the marginally bound value in a Newtonian potential. At the center of the injection region, this places the stream at the pericenter of its orbit, $r_p\simeq15\,r_g$, with specific angular momentum $\simeq5.7\,r_gc$ and inclination $i\simeq86^\circ$ to the disk. In the Paczyński–Wiita potential, $\eta_v=2.1$ leaves the stream mildly bound, $E\simeq-6\cdot10^{-3}c^2$, compared with

$|\Delta E|\sim4\cdot10^{-4}c^2$ for debris near peak return. This affects only gas that crosses the disk without strong interaction, since the collision changes the specific energy by $\sim GM_{\mathrm{BH}}/r_{\mathrm{p}}\sim0.07c^2$; we find no stream material returning from the ballistic apocenter $(\sim160r_{\mathrm{g}})$ during the run.

The stream pressure is set by an isothermal sound speed, $P=\rho c_{\mathrm{s}}^2$, so the source term injects mass, momentum, and gas internal energy. No magnetic field is injected with the debris, and the source term does not directly add radiation energy. Changes in the magnetic flux near the black hole therefore come from the response of the pre-existing AGN field to the impact, and radiation associated with the stream appears only after the injected gas shocks, heats, and couples to the radiation field. The parameter values are listed in Table 1.

Parameter	Value
$(\theta_{\mathrm{pitch}},\theta_{\mathrm{roll}},\theta_{\mathrm{yaw}})$	$(0,-1.9,1.5)$
$(\ell_0,m_0,n_0)$	$(7.8,23,1.7)$
$(\Delta\ell,\Delta m,\Delta n)$	$(0.1,0.05,0.025)$
Injection point $(r,\theta,\phi)$	$(15.4,1.3,1.5)$
$\eta_{\mathrm{v}}$	$2.1$
$c_{\mathrm{s}}$	$2.6\cdot10^{6}\ \mathrm{cm\,s^{-1}}$
$\dot{M}_{\mathrm{s}}$	$8.3\cdot10^{25}\ \mathrm{g\,s^{-1}}$
Derived orbit at the injection center:
Pericenter $r_{\mathrm{p}}$	$15.6$
Inclination to the disk $i$	$86^\circ$