High-energy Emission from Tidal Disruption Events in Active Galactic Nuclei

Figure 1 combines a density slice from the actual simulations with a schematic depiction of the bigger picture. A stream with ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}\gtrsim 1$ punches through the disk at the first impact. Tidal compression redistributes energy within the stream; as a result, a dilute, wide-angle plume of stream material bursts out from the lower side of the disk. Our focus is on the fraction that returns to the disk.

Because stream material leaves the simulation domain with sound speeds much less than virial, $\lesssim 0.1\,{[{{GM}}_{{\rm{h}}}/{({R}^{2}+{z}^{2})}^{1/2}]}^{1/2}$ (Chan et al. 2019), we assume it follows ballistic trajectories from the boundaries of the simulation domain until it strikes the disk. For each time step in each run, we determine which boundary cells have outflowing gas with specific angular momentum $\geqslant 1.1{({{GM}}_{{\rm{h}}}{r}_{{\rm{p}}})}^{1/2}$ in the y-direction; these cells have outflowing stream material, as opposed to vertically expanding disk gas. For each of these cells, we assume that all the outflowing material over the time step is launched outward on a Keplerian trajectory. We calculate when, where, and with what velocity this trajectory hits the midplane. Even gravitationally unbound gas can be intercepted if its hyperbolic trajectory crosses the disk. Trajectory calculations are continued until the end of the simulations, at time $1400{({{GM}}_{{\rm{h}}}/{r}_{{\rm{p}}}^{3})}^{-1/2}$, which is more than double the simulation duration in Chan et al. (2019); for M_h = 3 × 10⁶ M_⊙ and M_⋆ = M_⊙, this translates to ∼10 days, or about a quarter of the orbital period of the most tightly bound debris when these parameters apply. The mass fallback rate as a function of time is a convolution of the outflow rate and the flight-time delay.

Figure 2 contrasts the rate at which stream material crashes back to the disk as a function of time with the rate at which it exits the first-impact region. Most of the material is unbound, and only ∼20% is involved in the second impact.

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Figure 3 is a map of where the material lands on the disk over the course of the simulation, and the left panel of Figure 4 displays the azimuthally integrated mass fallback rate. The trajectories followed by the bulk of the material from the first impact to the second have eccentricities of ≈1, and the planes of the trajectories are almost perpendicular to the disk. These trajectories dump the material along the projection of the unperturbed stream onto the disk. A tiny portion of the material is heavily deflected and put on mildly eccentric, highly inclined trajectories with a range of orientations; this material ends up at r ≲ 3 r_p. Although the detailed shape of the splash zone and the rate at which material is delivered to it vary with time and ${\dot{M}}_{{\rm{s}}}/{\dot{M}}_{{\rm{d}}}$, overall they are remarkably independent of these two variables. This means the properties of the second impact depend less on the AGN properties $({\dot{M}}_{{\rm{a}}})$ and more on the TDE properties (M_h, M_⋆).

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The right panel of Figure 4 shows the azimuthally integrated energy dissipation rate per logarithmic radius:

$$\begin{eqnarray}&&{R}^{2}\int d\phi \,{\rho }_{{\rm{f}}}{v}_{{\rm{f}},{\rm{z}}}K,\end{eqnarray} \tag{ 6 }$$

where ρ_f and v _f are the density and velocity of the material falling back, $K\equiv \tfrac{1}{2}\parallel {{\boldsymbol{v}}}_{{\rm{f}}}-R{\rm{\Omega }}{\hat{{\boldsymbol{e}}}}_{\phi }{\parallel }^{2}$ is the specific energy dissipation rate, ${\rm{\Omega }}={({{GM}}_{{\rm{h}}}/{R}^{3})}^{1/2}$ is the Keplerian orbital frequency, and the integral is over where the material hits. At radii greater than a few r_p, material hits the disk with ∥ v _f ∥≪ RΩ; as a result, the dissipated energy is mainly derived from the kinetic energy of the disk, that is, K ∝ 1/R. At all radii, the energy dissipation rate due to the second impact is orders of magnitude greater than that of the underlying disk; in fact, the energy dissipated per orbital time can reach ∼10⁻³ times the local binding energy at certain radii. The expedited inflow this entails could help feed the small-scale disk emptied by the shocks from the first impact (Chan et al. 2019).

The spatial distribution and energy dissipation of the second impact depend on the strength of the pericentric tidal compression. This in turn depends on the kinematic properties of the unperturbed stream, which cannot be accurately determined without simulating the full disruption process. Therefore, the results presented here should be understood as capturing the qualitative, not the quantitative, aspects of the second impact.

Figure 5 shows an idealized picture of the second impact in the inertial frame. Falling stream material is slowed down, compressed, and heated by a strong, standing reverse shock. The layer of shocked material, sitting on the disk, is ferried away by disk rotation. This material cools quite swiftly; with the loss of pressure support, presumably it sinks into the disk within one orbit. As is evident from the narrowness of the splash zones displayed in Figure 3, the time for the disk to rotate through the stream is much shorter than an orbital period. The structure in Figure 5 is thus time-steady: the unperturbed disk enters from the left side and exits to the right topped with cooled shocked material.

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Over longer timescales, this time-steady picture changes gradually as the rate at which stream material descends on the disk varies. The mass current of the second impact slowly builds up in the early stages of the TDE because the stream takes time to fly from the first impact to the second. In addition, on timescales comparable to the mass-return time of the TDE, the rate at which stellar material returns to the pericenter declines, and so does the mass current going to the second impact.

Our simulations, which last for ∼10 days when M_h = 3 × 10⁶ M_⊙ and M_⋆ = M_⊙, are long enough to cover only the buildup period. To study how the second impact changes over this period, we divide the simulation duration into five equal intervals. The fallback material is characterized by its speed, mass flux, and energy dissipation rate; for simplicity, we assume these three quantities are time-independent within each interval and ϕ-independent across the stream, so we can consider their averages 〈v_f,z〉, 〈ρ_f v_f,z〉, and 〈ρ_f v_f,z K〉.

To calculate the cooling emission, we follow a point on the disk at radius R from the moment it enters the stream, which we take to be t = 0. A column of shocked material accumulates above the point as it moves through the stream; meanwhile, the column cools by radiating away the dissipated energy. The point leaves the stream at t_dep = Δϕ/Ω, where Δϕ is the characteristic azimuthal width of the second impact at radius R; we nevertheless continue tracking the cooling over the entire orbit to ensure the column has largely cooled off by the time it returns to the stream. The observed cooling emission is the sum from columns all over the disk, each in a different cooling stage.

The dominant cooling mechanism is inverse Compton scattering of seed photons from the disk. In this sense, the shocked layer behaves similarly to an AGN corona (e.g., Haardt & Maraschi 1991), except that the shocked layer can be quite Compton-thick. Synchrotron emission should not contribute greatly because it is strongly self-absorbed, and we shall see later that free–free cooling is also unlikely to be important.

We assume the shocked layer is vertically homogeneous, and ions and electrons are thermal at the same temperature. Thermal electrons at the temperatures we find are mildly relativistic; therefore, we ignore electron–positron pair production and the Klein–Nishina reduction in the cross section, both affecting only a small fraction of photons.

The spectrum of the Comptonized photons is taken to be a power law with a cutoff. Seed photons from the disk carry a radiative flux

$$\begin{eqnarray}&&{F}_{{\rm{d}}}=\displaystyle \frac{3{{GM}}_{{\rm{h}}}{\dot{M}}_{{\rm{a}}}}{8\pi {R}^{3}},\end{eqnarray} \tag{ 7 }$$

and their spectrum is blackbody at temperature ${T}_{{\rm{d}}}={({F}_{{\rm{d}}}/{\sigma }_{\mathrm{SB}})}^{1/4}$, where σ_SB is the Stefan–Boltzmann constant. Because the seed spectrum is much narrower than the Comptonized spectrum, we approximate the former as a delta function. We use F_d of the unperturbed disk here, but spiral shocks extending outward from the first impact (Chan et al. 2019) could modify the disk and thus F_d.

At each moment, the model is characterized by E_g and E_r, which are, respectively, the vertically integrated gas and radiation energy densities; by E_∞, which is the energy per area that has escaped from the top of the column; and by H_sh, which is the height of the column of shocked material. They obey the equations

$$\begin{eqnarray}&&\,\frac{{{dE}}_{{\rm{g}}}}{{dt}}= \langle {\rho }_{{\rm{f}}}{v}_{{\rm{f}},{\rm{z}}}K \rangle \vartheta ({t}_{\mathrm{dep}}-t)-(A-4{{\rm{\Theta }}}_{{\rm{C}}})\frac{{E}_{{\rm{r}}}}{{t}_{\mathrm{sca}}},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\,\displaystyle \frac{{{dE}}_{{\rm{r}}}}{{dt}}={F}_{{\rm{d}}}+(A-4{{\rm{\Theta }}}_{{\rm{C}}})\displaystyle \frac{{E}_{{\rm{r}}}}{{t}_{\mathrm{sca}}}-\displaystyle \frac{{E}_{{\rm{r}}}}{{t}_{\mathrm{esc}}},\,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{\infty }}{{dt}}=\displaystyle \frac{{E}_{{\rm{r}}}}{{t}_{\mathrm{esc}}},\,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{dH}}_{\mathrm{sh}}}{{dt}} & = & \left\{\begin{array}{ll}{v}_{\mathrm{sh}} & \mathrm{if}\ t\lt {t}_{\mathrm{dep}},\\ ({H}_{\mathrm{eq}}/{H}_{\mathrm{sh}}-1){c}_{{\rm{s}}} & \mathrm{if}\ t\geqslant {t}_{\mathrm{dep}},\end{array}\right.\,\end{array}\end{eqnarray} \tag{ 11 }$$

where ϑ is the step function and the other symbols are as defined in the rest of the section. The initial condition is (E_g, E_r, E_∞, H_sh) = (0, 0, 0, 0); the results are insensitive to small perturbations to the initial condition.

Equation (8) characterizes the accrual of dissipated energy in the column and its conversion to radiation through Compton scattering. Equation (9) follows the increase in radiation energy as a result of disk injection and Compton scattering and its decrease due to radiative losses. Equation (11) describes how the column grows during deposition while mass and heat are added to it and how it collapses after deposition as it cools. We ignore the internal energy increase during this collapse due to adiabatic compression and release of gravitational energy, both being secondary effects.

In Equations (8) and (9), A is the fractional photon energy gain due to inverse Compton scattering, and Θ_C is the Compton temperature in units of m_e c²/k_B, with m_e being the electron mass and k_B the Boltzmann constant. The Compton temperature is the temperature of an electron gas in thermal balance with photons through Compton scattering, and it is equal to one-fourth the intensity-weighted mean photon energy.

Three timescales appear in the equations:

$$\begin{eqnarray}&&\,{t}_{{\rm{dep}}}\,=\,{\rm{\Delta }}\phi /{\rm{\Omega }},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\begin{array}{l}\,{t}_{\mathrm{sca}}\,=\,1/({n}_{{\rm{e}}}{\sigma }_{{\rm{T}}}c),\end{array}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\begin{array}{l}{t}_{\mathrm{esc}}\,=\,({H}_{\mathrm{sh}}/c)\max (1,{\tau }_{{\rm{T}}}),\end{array}\end{eqnarray} \tag{ 14 }$$

where n_e is the electron number density, σ_T is the Thomson cross section, and τ_T is the Thomson thickness. These timescales are, respectively, the duration of deposition, the mean timescale for a photon to scatter off an electron, and the timescale for radiation to escape. We also define

$$\begin{eqnarray}&&\begin{array}{l}\,{v}_{\mathrm{sh}}\,=\,[(\gamma -1)/(\gamma +1)]\langle {v}_{{\rm{f}},{\rm{z}}}\rangle ,\end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\begin{array}{l}\,{H}_{\mathrm{eq}}^{2}\,=\,2{c}_{{\rm{s}}}^{2}{R}^{3}/({{GM}}_{{\rm{h}}}),\end{array}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\begin{array}{l}\,{c}_{{\rm{s}}}^{2}\,=\,\left(\tfrac{2}{3}{E}_{{\rm{g}}}+\tfrac{1}{3}{E}_{{\rm{r}}}\right)/{{\rm{\Sigma }}}_{\mathrm{sh}},\end{array}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\begin{array}{l}\,{n}_{{\rm{e}}}\,=\,{{\rm{\Sigma }}}_{\mathrm{sh}}/(2\bar{m}{H}_{\mathrm{sh}}),\end{array}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\begin{array}{l}\,{\tau }_{{\rm{T}}}\,=\,{\sigma }_{{\rm{T}}}{{\rm{\Sigma }}}_{\mathrm{sh}}/{m}_{{\rm{H}}},\end{array}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Sigma }}}_{\mathrm{sh}}\,=\,\langle {\rho }_{{\rm{f}}}{v}_{{\rm{f}},{\rm{z}}}\rangle \min ({t}_{\mathrm{dep}},t).\end{array}\end{eqnarray} \tag{ 20 }$$

Here v_sh is the upward speed of the reverse shock in the corotating frame, H_eq is the scale height in hydrostatic equilibrium, c_s is the isothermal sound speed, Σ_sh is the column density, $\gamma =\tfrac{5}{3}$ is the adiabatic index, $\bar{m}=\tfrac{8}{13}{m}_{{\rm{H}}}$ is the mean particle mass, and m_H is the hydrogen mass.

Although the electrons are mildly relativistic at the beginning, the radiative output is dominated by later stages when the gas is cooler; we therefore have

$$\begin{eqnarray}&&A\approx 4{\rm{\Theta }},\end{eqnarray} \tag{ 21 }$$

where

$$\begin{eqnarray}&&{\rm{\Theta }}\equiv {k}_{{\rm{B}}}{T}_{{\rm{g}}}/({m}_{{\rm{e}}}{c}^{2}),\end{eqnarray} \tag{ 22 }$$

with

$$\begin{eqnarray}&&{T}_{{\rm{g}}}=(\gamma -1)\bar{m}{E}_{{\rm{g}}}/({k}_{{\rm{B}}}{{\rm{\Sigma }}}_{\mathrm{sh}})\end{eqnarray} \tag{ 23 }$$

being the gas temperature.

We compute the Compton temperature in two different ways, distinguished by whether the column is Compton-thin or Compton-thick. The dividing line τ_T = 1/e is chosen to be as large as possible while being small enough that the probability of n scatters is ≈τ_T ⁿ . If τ_T < 1/e, most disk photons escape without being scattered, so the spectrum of the radiation emerging from the column consists of a delta-function disk component in addition to a Comptonized component:

$$\begin{eqnarray}&&\begin{array}{l}F(\epsilon )d\epsilon =\left[(1-{\tau }_{{\rm{T}}}){F}_{{\rm{d}}}\delta (\epsilon -{k}_{{\rm{B}}}{T}_{{\rm{d}}})\right.\\ \ \ +\,\left.\displaystyle \frac{{F}_{{\rm{C}}}}{{k}_{{\rm{B}}}{T}_{{\rm{g}}}}{\left(\displaystyle \frac{\epsilon }{{k}_{{\rm{B}}}{T}_{{\rm{g}}}}\right)}^{p}\exp \left(-\displaystyle \frac{\epsilon }{{k}_{{\rm{B}}}{T}_{{\rm{g}}}}\right)\right]\,d\epsilon ,\end{array}\end{eqnarray} \tag{ 24 }$$

where is the photon energy and F_C is the normalization of the Comptonized component. In this regime, repeated Compton scattering leads to a spectral index $p=\mathrm{ln}{\tau }_{{\rm{T}}}/\mathrm{ln}(1+A)$ (Rybicki & Lightman 1979; Krolik 1999), and energy conservation and the definition of Θ_C can be written as

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{E}_{{\rm{r}}}}{{t}_{\mathrm{esc}}} & = & {\int }_{{k}_{{\rm{B}}}{T}_{{\rm{d}}}^{-}}^{\infty }d\epsilon \,F(\epsilon )\\ & = & (1-{\tau }_{{\rm{T}}}){F}_{{\rm{d}}}+{\rm{\Gamma }}(p+1,{T}_{{\rm{d}}}/{T}_{{\rm{g}}}){F}_{{\rm{C}}},\end{array}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}\begin{array}{rcl}4{{\rm{\Theta }}}_{{\rm{C}}}{m}_{{\rm{e}}}{c}^{2} & = & \displaystyle \frac{{\int }_{{k}_{{\rm{B}}}{T}_{{\rm{d}}}^{-}}^{\infty }d\epsilon \,\epsilon F(\epsilon )}{{\int }_{{k}_{{\rm{B}}}{T}_{{\rm{d}}}^{-}}^{\infty }d\epsilon \,F(\epsilon )}\\ & = & \displaystyle \frac{(1-{\tau }_{{\rm{T}}})({T}_{{\rm{d}}}/{T}_{{\rm{g}}}){F}_{{\rm{d}}}+{\rm{\Gamma }}(p+2,{T}_{{\rm{d}}}/{T}_{{\rm{g}}}){F}_{{\rm{C}}}}{(1-{\tau }_{{\rm{T}}}){F}_{{\rm{d}}}+{\rm{\Gamma }}(p+1,{T}_{{\rm{d}}}/{T}_{{\rm{g}}}){F}_{{\rm{C}}}}{k}_{{\rm{B}}}{T}_{{\rm{g}}},\end{array}\end{eqnarray} \tag{ 26 }$$

with Γ being the incomplete gamma function. Solving these equations yields F_C and Θ_C; in cases where F_C < 0, we set Θ_C = 0.

If τ_T ≥ 1/e, we assume every photon is scattered at least once on its way out, so we retain only the Comptonized component:

$$\begin{eqnarray}&&F(\epsilon )d\epsilon =\displaystyle \frac{{F}_{{\rm{C}}}}{{k}_{{\rm{B}}}{T}_{{\rm{g}}}}{\left(\displaystyle \frac{\epsilon }{{k}_{{\rm{B}}}{T}_{{\rm{g}}}}\right)}^{p}\exp \left(-\displaystyle \frac{\epsilon }{{k}_{{\rm{B}}}{T}_{{\rm{g}}}}\right)d\epsilon .\end{eqnarray} \tag{ 27 }$$

The spectral parameters F_C, p, and Θ_C are determined in this regime from photon conservation, energy conservation, and the definition of Θ_C:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \,\frac{{F}_{{\rm{d}}}}{{k}_{{\rm{B}}}{T}_{{\rm{d}}}} & = & {\int }_{{k}_{{\rm{B}}}{T}_{{\rm{d}}}}^{\infty }d\epsilon \,{\epsilon }^{-1}F(\epsilon )={\rm{\Gamma }}(p,{T}_{{\rm{d}}}/{T}_{{\rm{g}}})\displaystyle \frac{{F}_{{\rm{C}}}}{{k}_{{\rm{B}}}{T}_{{\rm{g}}}},\end{array}\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \,\frac{{E}_{{\rm{r}}}}{{t}_{\mathrm{esc}}} & = & {\int }_{{k}_{{\rm{B}}}{T}_{{\rm{d}}}}^{\infty }d\epsilon \,F(\epsilon )\,={\rm{\Gamma }}(p+1,{T}_{{\rm{d}}}/{T}_{{\rm{g}}}){F}_{{\rm{C}}},\end{array}\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&4{{\rm{\Theta }}}_{{\rm{C}}}{m}_{{\rm{e}}}{c}^{2}\,=\,\displaystyle \frac{{\,\int }_{{k}_{{\rm{B}}}{T}_{{\rm{d}}}}^{\infty }d\epsilon \,\epsilon F(\epsilon )}{{\int }_{{k}_{{\rm{B}}}{T}_{{\rm{d}}}}^{\infty }d\epsilon \,F(\epsilon )\,}=\displaystyle \frac{{\rm{\Gamma }}(p+2,{T}_{{\rm{d}}}/{T}_{{\rm{g}}})}{{\rm{\Gamma }}(p+1,{T}_{{\rm{d}}}/{T}_{{\rm{g}}})}{k}_{{\rm{B}}}{T}_{{\rm{g}}}.\end{eqnarray} \tag{ 30 }$$

Figure 6 shows one solution of the model; other solutions are qualitatively similar, differing only in the durations of the various stages of evolution. The first stage is so optically thin that $(A-4{{\rm{\Theta }}}_{{\rm{C}}}){t}_{\mathrm{esc}}/{t}_{\mathrm{sca}}=(A-4{{\rm{\Theta }}}_{{\rm{C}}}){\tau }_{{\rm{T}}}{m}_{{\rm{H}}}/(2\bar{m})\ll 1;$ thus, the net amplification of the disk emission is small. The third term on the right-hand side of Equation (9) largely offsets the first term, so E_r grows slowly and T_g remains high. Indeed,

$$\begin{eqnarray}&&\begin{array}{l}\quad {E}_{{\rm{g}}}\,\approx \, \langle {\rho }_{{\rm{f}}}{v}_{{\rm{f}},{\rm{z}}}K \rangle t,\end{array}\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&\begin{array}{r}\,{E}_{{\rm{r}}}\,\approx \,{F}_{{\rm{d}}}t/(1+c/{v}_{\mathrm{sh}}),\end{array}\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&\begin{array}{l}\,{E}_{\infty }\,\approx \,{F}_{{\rm{d}}}t.\end{array}\end{eqnarray} \tag{ 33 }$$

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The second stage commences when $(A-4{{\rm{\Theta }}}_{{\rm{C}}}){t}_{\mathrm{esc}}/{t}_{\mathrm{sca}}\,=(A-4{{\rm{\Theta }}}_{{\rm{C}}}){\tau }_{{\rm{T}}}{m}_{{\rm{H}}}/(2\bar{m})\sim 1$. Although the column is still Compton-thin, the scattered photons gain so much energy that the luminosity of a cohort of injected photons is at least doubled. This drives up E_r, which in turn accelerates Compton cooling, resulting in an exponential surge in E_r. The oscillations in this stage are due to the mutual feedback between E_g and E_r in our model, in which a column evolves independently of its neighbors; if adjacent columns interacted by photon diffusion, these oscillations would likely be damped.

The growth of E_r comes to an end when the system finds a new equilibrium. In this equilibrium, dissipated energy is efficiently converted to radiation energy, which in turn is lost rapidly to outward streaming. This means the two terms of Equation (8) are comparable, while the second term of Equation (9) is counteracted by the third. The effective temperature of the emergent radiation is then

$$\begin{eqnarray}&&{T}_{{\rm{eff}}}={\left(\displaystyle \frac{{E}_{{\rm{r}}}}{{t}_{{\rm{esc}}}{\sigma }_{{\rm{SB}}}}\right)}^{1/4}\sim {\left(\displaystyle \frac{ \langle {\rho }_{{\rm{f}}}{v}_{{\rm{f}},{\rm{z}}}K \rangle }{{\sigma }_{{\rm{SB}}}}\right)}^{1/4}.\end{eqnarray} \tag{ 34 }$$

The third stage goes from the moment when τ_T ∼ 1, should it happen, to the end of deposition. Compton cooling is faster in the Compton-thick column, but the gas is kept warm by continual energy dissipation. The balance of terms is the same as in the second stage; hence, $(A-4{{\rm{\Theta }}}_{{\rm{C}}}){t}_{\mathrm{esc}}/{t}_{\mathrm{sca}}\,=(A-4{{\rm{\Theta }}}_{{\rm{C}}}){\tau }_{{\rm{T}}}^{2}{m}_{{\rm{H}}}/(2\bar{m})\sim 1$. The close match between dissipative gains and radiative losses means that once deposition ceases, the gas promptly cools off and little further emission occurs.

Because all quantities as functions of time are approximately power-law, later stages of cooling tend to outweigh earlier stages in terms of contribution to the overall radiative output.

On the whole, both p and Θ_C increase over time in all three stages as long as material continues to be deposited, but p never rises to above 3, its value in the Rayleigh–Jeans limit. By contrast, Θ is almost constant in the first stage but falls gradually in the next two stages, bounded below by Θ_C. This observation can be further generalized: a Compton-thicker layer tends to have a larger p and a smaller Θ. Consequently, the spectrum emerging from the column transforms from a soft power law with a high-energy cutoff to a hard power law with a low-energy cutoff as cooling progresses.

Free–free cooling is more efficient for small M_h, large M_⋆, and large R/r_p. It can be orders of magnitude faster than Compton cooling at converting E_g to E_r in the first phase, but for most of the cases we considered, Compton cooling quickly catches up in the second and third phases and even monopolizes the cooling budget toward the end of deposition. Therefore, the effect of early-time free–free cooling is likely limited to creating extra seed photons for Compton cooling. Free–free cooling can also dominate after the end of deposition; however, in most cases only a small fraction of the total dissipated energy remains in the column by the time stream material stops arriving, so the contribution of late-time free–free cooling is minor.

Another source of seed photons is the disk interior to the first impact, which could be emitting at Eddington levels (Chan et al. 2019, 2020). Although this could, in principle, be a competitive source of seed photons, the luminosity and angular distribution of this light remain so uncertain that we do not include it here.

The analysis so far concerns a single column. Figure 7 shows the overall spectra resulting from the second impact for various values of M_h and M_⋆. These are obtained by adding together the contribution of every column on the disk. In all cases, it can be roughly represented by a broken power law whose break coincides with the peak in L . Depending on M_h and M_⋆, the photon energy of the peak can be anywhere between ∼10 keV and 1 MeV, as expected from Compton cooling of mildly relativistic electrons. Generally speaking, with increasing M_h and decreasing M_⋆, the peak broadens and shifts toward higher energies. Additional cooling at t ≳ t_dep modifies the spectrum, the most prominent effect being the creation of a secondary $\sim$keV peak for small M_h and large M_⋆.

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The spectra in Figure 7 have bolometric luminosities ranging from $\sim$10⁴² to 10⁴⁴ erg s⁻¹, on the level of weak AGNs. Because the shocked material cools efficiently, these luminosities are close to the total energy dissipation rate, which is $\sim 0.2\,{\dot{M}}_{{\rm{s}}}$ times the specific kinetic energy of the disk at the second impact (Section 4.2). Second-impact radii are multiples of r_p, so the luminosity scales as

$$\begin{eqnarray}\begin{array}{rcl}L\propto {{GM}}_{{\rm{h}}}{\dot{M}}_{{\rm{s}}}/{r}_{{\rm{p}}} & \propto & {M}_{{\rm{h}}}^{1/6}{M}_{\star }^{7/3}{r}_{\star }^{-5/2}{{\rm{\Psi }}}^{-1}{{\rm{\Xi }}}^{3/2}\\ & \propto & {M}_{{\rm{h}}}^{1/6}{M}_{\star }^{0.13}{{\rm{\Psi }}}^{-1}{{\rm{\Xi }}}^{3/2}.\end{array}\end{eqnarray} \tag{ 35 }$$

As seen in Figure 8, this equation best describes the luminosities from the model when the constant associated with the first proportionality is ∼4 × 10⁻³:

$$\begin{eqnarray}\begin{array}{rcl}L & \sim & 1.8\times {10}^{43}\,{\rm{erg}}\,{{\rm{s}}}^{-1}\\ & & \times \,{{\rm{\Psi }}}^{-1}{{\rm{\Xi }}}^{3/2}{\left(\displaystyle \frac{{M}_{{\rm{h}}}}{3\times {10}^{6}{M}_{\odot }}\right)}^{1/6}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{0.13}.\end{array}\end{eqnarray} \tag{ 36 }$$

The proportionality constant is so small because merely ∼20% of the bound debris participates in the second impact and because, according to Figure 3, the second impact happens at several tens of r_p.

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In view of the weak "nominal" dependence of L on M_h and M_⋆, the correction factors Ψ and Ξ actually control how L scales with M_h and M_⋆. The combination Ψ⁻¹Ξ^3/2 decreases with M_h and increases with M_⋆, counteracting the nominal M_h-dependence but augmenting the nominal M_⋆-dependence. As is apparent from Figures 7 and 8, L is most sensitive to M_h for M_h ≳ 3 × 10⁶ M_⊙ and to M_⋆ for 0.1 M_⊙ ≲ M_⋆ ≲ M_⊙.

TDEs in AGNs have observational signatures that are emphatically unlike those of TDEs in vacuum. They do not have optical/UV light curves that follow the mass-return rate, as is commonly expected for their vacuum counterparts (e.g., Rees 1988). Instead, the destruction of the inner disk by the first impact sustains an Eddington-level luminosity plateau over tens of days (Chan et al. 2019, 2020). On top of that, the second impact begets a longer-lasting signal whose luminosity does follow the mass-return rate thanks to efficient cooling, but its typical photon energy is not in the optical/UV: most of the spectra in Figure 7 peak between ∼10 keV and 1 MeV. Such spectra bear no resemblance to the tens-of-electronvolt thermal spectrum originally expected from the accretion disk formed by TDEs in vacuum (Cannizzo et al. 1990; Ulmer 1999) and sometimes observed in soft X-ray TDEs (see Saxton et al. 2020 for a review).

The hard second-impact spectrum could mean that TDE searches focused on softer spectra would miss TDEs in AGNs altogether. The hunt is made difficult by the fact that, unlike jetted TDEs with isotropic-equivalent X-ray luminosities of $\sim$10⁴⁶ to 10⁴⁸ erg s⁻¹ (Bloom et al. 2011; Burrows et al. 2011; Levan et al. 2011; Cenko et al. 2012), the second impact emits at rates four orders of magnitude lower.

Another challenge in the identification of second-impact signatures is distinguishing them from common AGN variability. Two aspects may provide the means to do so: spectral hardness and pattern of time variation.

The second impact has much harder spectra than ordinary AGNs. The coronal component of unobscured AGN spectra typically has $-1\lesssim d\mathrm{log}{L}_{\epsilon }/d\mathrm{log}\epsilon \lesssim -0.7$ from ∼1 to 100 keV (e.g., Liu et al. 2016; Ricci et al. 2017). Judging by Figure 7, the X-rays from the second impact should have $-1\lesssim d\mathrm{log}{L}_{\epsilon }/d\mathrm{log}\epsilon \lesssim 0$. The contrast between unobscured AGN spectra and the significantly harder second-impact spectra is enhanced to the degree that the corona is disrupted by the first impact, an outcome suggested in our earlier work (Chan et al. 2019; Ricci et al. 2020).

In addition to unobscured AGNs, there is a roughly comparable number of obscured AGNs (e.g., Huchra & Burg 1992; Reyes et al. 2008; Brightman & Nandra 2011; Wilkes et al. 2013; Oh et al. 2015). Our sightlines to these AGNs are obscured by neutral gas with hydrogen column densities N_H of ∼10²² to 10²⁴ cm⁻² or more. A column with N_H ∼ 10²² cm⁻² absorbs essentially all photons below ∼2 keV, and one with N_H ∼ 10²⁴ cm⁻² absorbs all photons under ∼7 to 10 keV. A column with still higher N_H scatters all photons with energies up to ≳100 keV. A significant fraction of all obscured AGNs, from ∼20% to 50%, may belong to this last category (Ricci et al. 2015; Georgantopoulos & Akylas 2019; Kammoun et al. 2020). Because this obscuration is mostly located parsecs away from the black hole, if it blocks X-rays from an AGN, it will equally block those due to a TDE. The only photons we can see are relatively high-energy, which makes the contrast between the second impact and the unperturbed AGN especially high.

The smooth brightening and fading of the second impact could provide another method of distinction. The X-ray light curves of normal AGNs generally exhibit "red noise," that is, their Fourier power spectra are power laws declining from timescales of months or years to timescales of hours (e.g., Lawrence et al. 1987; McHardy & Czerny 1987; Lawrence & Papadakis 1993), and individual modes of the power spectra have little or no phase coherence (Krolik et al. 1993). By contrast, the second impact should brighten and fade smoothly over a period of months, in response to the variation of the mass-return rate. The existence of a dominant timescale and the phase coherence implied by the smoothness of the light curve may help make a second-impact flare distinct.

Our treatment of light production at the second impact depends on a number of approximations and simplifications. A few warrant identification as possible starting points for future improvements.

The simulations underlying our cooling model injected the stream on a parabolic orbit (Chan et al. 2019) when, in fact, the debris is weakly bound. Therefore, the stream's kinetic energy at the pericenter is overestimated by an amount comparable to its typical binding energy. After correcting for this offset, the second impact would happen at smaller radii. This would increase the energy dissipated per unit stream mass somewhat, raise the luminosity, and change the spectrum. A greater fraction of the stream could also end up in the second impact, making the cooling emission from it softer and more thermal (Section 5.3).

As in Chan et al. (2019, 2020), we consider here a specific stream configuration: the stream hits the disk perpendicularly while passing through its pericenter at radius ${{ \mathcal R }}_{{\rm{t}}}$. Our results might change depending on the stream orientation and pericenter, but preliminary estimates suggest that orientation does not qualitatively affect our conclusions.

Our model ignores the production of electron–positron pairs for simplicity (Section 5.2), but many pairs should be produced in freshly shocked gas with temperatures k_B T_g ≳ m_e c². In pair equilibrium, some of the thermal energy is held in the rest mass of pairs, while pair annihilation adds to the number density of photons. The increase in scattering opacity due to pairs combined with the presence of additional photons tends to promote lower gas and radiation temperatures and more nearly thermal spectra.

Lastly, our simulations cover merely a fraction of the mass-return time, during which the stream mass current can be taken to be approximately constant. However, as the mass-return rate rises, reaches a peak, and then falls over the course of the TDE, we expect the bolometric luminosity of the second impact to roughly track that rate, while the spectrum should evolve from harder at the initially low mass-return rate to softer and more thermal near the peak and then back to harder as the mass-return rate decays.