Radiative hydrodynamical simulations of super-Eddington accretion flow in tidal disruption event: the accretion flow and wind Free

ABSTRACT

One key question in tidal disruption events theory is how much of the fallback debris can be accreted to the black hole. Based on radiative hydrodynamic simulations, we study this issue for efficiently ‘circularized’ debris accretion flow. We find that for a black hole disrupting a solar-type star, |$15{{\, \rm per\, cent}}$| of the debris can be accreted for a 10⁷ M_⊙ black hole. While for a 10⁶ M_⊙ black hole, the value is |$43{{\, \rm per\, cent}}$|⁠. We find that wind can be launched in the super-Eddington accretion phase regardless of the black hole mass. The maximum velocity of the wind can reach 0.7c (with c being the speed of light). The kinetic power of wind is well above 10⁴⁴ erg s⁻¹. The results can be used to study the interaction of wind and the circumnuclear medium around quiescent supermassive black holes.

1 INTRODUCTION

In galaxies, stars can move towards the supermassive black hole at the galaxy centre. If the pericentre of the orbit of a star is equal to or smaller than the tidal disruption radius R_T (Hills 1975), the star can be tidally disrupted, triggering the so-called tidal disruption events (TDEs; Rees 1988; Evans & Kochanek 1989). Roughly half of the stellar debris is unbound and can escape. The other bound half of the stellar debris falls back. The predicted fallback rate declines with time roughly as |$\dot{M}_{\rm fb} \propto t^{-5/3}$|⁠.

The TDEs were first detected in the soft X-ray bands by the ROentgen SATellite (ROSAT) X-ray All-Sky Survey (see Komossa 2015 for review). For these TDEs, the decline of their X-ray light curve is well consistent with the predicted t^−5/3 law. The X-ray is generated in the black hole accretion process. Thus, the consistency requires that the fallback rate roughly equals to the black hole accretion rate. This requirement is not obviously satisfied (Guillochon & Ramirez-Ruiz 2015; Shiokawa et al. 2015). The stellar debris falls back to the orbit pericentre, which is much larger than the black hole horizon radius. In order to be accreted to the black hole, a viscous torque is required to transfer the angular momentum of the fallback debris. The fallback debris supplies gas to the viscous accretion flow. It is not guaranteed that all the fallback debris can be transported to the black hole horizon by the viscous accretion flow. From the theoretical point of view, it is quite necessary to study whether and how the black hole accretion rate correlates with the debris fallback rate. This is important to understand the X-ray light curve of TDEs.

For the optical/ultraviolet (UV) selected TDEs, the main puzzle is its origin of the optical/UV emission (see van Velzen et al. 2020; Gezari 2021 for review). The inferred location of the optical/UV radiation is ∼10^14–16 cm (Hung et al. 2017; van Velzen et al. 2020; Gezari 2021). However, the accretion flow is predicted to have a size of several times of 10¹³ cm if one assumes that a solar-type star is disrupted by a black hole with 10⁶–10⁷ M_⊙. One proposed scenario is that the optical/UV emission is generated in the fallback debris colliding-induced shock process (Piran et al. 2015; Jiang, Guillochon & Loeb 2016; Steinberg & Stone 2022). The location of shock is consistent with the observation inferred optical/UV radiation location. In the alternative ‘reprocessing’ scenario, the soft X-ray/extreme ultraviolet (EUV) emission generated very close to the black hole is reprocessed into optical/near-ultraviolet (NUV) bands by a surrounding optically thick and geometrically vertically extended envelope (Loeb & Ulmer 1997; Coughlin & Begelman 2014; Roth et al. 2016; Liu et al. 2017, 2021; Metzger & Stone 2017; Metzger 2022; Wevers et al. 2022) or wind (Strubbe & Quataert 2009; Lodato & Rossi 2011; Metzger & Stone 2016; Piro & Lu 2020; Uno & Maeda 2020; Bu et al. 2022; Parkinson et al. 2022; Mageshwaran, Shaw & Bhattacharyya 2023). Recently, the presence of TDE winds has been directly confirmed by the UV and X-ray spectra (Yang et al. 2017; Kara et al. 2018; Parkinson et al. 2020).

In analytical wind ‘reprocessing’ model, the properties of winds are arbitrary given (e.g. Strubbe & Quataert 2009) due to the lack of knowledge of the properties of TDEs wind. Efforts have been made to explore the properties of TDEs wind. By assuming that circularization is efficient, the properties of winds from a ‘circularized’ super-Eddington accretion flow have been investigated by numerical simulation works (Dai et al. 2018; Curd & Narayan 2019). However, we note that these works just give the properties of winds at a snapshot around peak fallback rate. The time evolution of winds is not studied. Thomsen et al. (2022) perform several discrete simulations with different accretion rates to study the time evolution of TDEs wind. For this method, the winds from an early stage of the TDEs accretion flow have nothing to do with those from a later stage of the flow. To what extent this method can represent the real time evolution of wind is not clear. Curd & Narayan (2023) studied the circularized accretion flow in TDEs. In order to be consistent with the situation of TDEs, the fallback debris is injected into the computational domain with an injection rate declining as (t/t_fb)^−5/3 law (t_fb is the debris fallback time-scale). However, we note that in Curd & Narayan (2023), an unrealistic shorter fallback time-scale is employed in order to shorten the simulation time. There are also works studying winds from the shock process (Jiang et al. 2016; Lu & Bonnerot 2020).

In addition to optical/UV emission, the TDEs wind may also be responsible for the radio emission of TDEs (see Alexander et al. 2020 for review). The winds from TDEs can interact with the circumnuclear medium (CNM; Barniol Duran, Nakar & Piran 2013; Matsumoto & Piran 2021) or the dense clouds surrounding the central black hole (Mou et al. 2022; Bu et al. 2023), which can result in the formation of shocks. The power-law electrons that are responsible for the radio emission can be accelerated in the shock process. The shock models are used to constrain the properties of winds, such as the velocity of the wind.

Despite the importance of winds in understanding the electromagnetic radiation of TDEs, the detailed properties of TDEs winds are still poorly known. Although there are many analytical and simulation works focusing on the winds from active galactic nuclei (AGNs), the results cannot be directly applied to TDEs. The reason is that the accretion flow in TDEs is quite different from that of an AGN. For example, the size of the accretion flow in TDEs is quite smaller than that of an AGN. Also, the accretion flow in TDEs has no quasi-steady state due to the fact that the gas supply rate to the flow declines as t^−5/3 law.

In order to study winds in TDEs, one needs to take into account the specific conditions for TDEs. Bu et al. (2022, hereafter BU22) performed a hydrodynamic simulation with radiative transfer to study the ‘circularized’ accretion flow in TDEs. In that paper, we take into account the specific conditions for TDEs. For example, we inject gas at two times of the pericentre of the orbit of the star, which is predicted to be the location of the accretion flow. The gas injection rate is set to declin as (t/t_fb)^−5/3 law to mimic the gas supply rate to the accretion flow due to the fallback of stellar debris. In BU22, we adopt the typical values for t_fb, which is important for matching the special conditions for TDEs.

In this paper, based on the simulations in BU22, we study two important issues. The first one is the relationship between the black hole accretion rate and the fallback rate of the stellar debris. The second one is the property of the TDEs winds. The structure of this paper is as follows. In Section 2, we briefly introduce the numerical simulations of BU22. In Section 3, we introduce the black hole accretion rate and properties of wind in TDEs. We summarize and discuss the results in Section 4.

2 NUMERICAL SIMULATIONS

We briefly introduce the simulations in BU22. Two-dimensional axisymmetric hydrodynamic simulations with radiative transfer are performed in BU22. The flux-limited diffusion approximation (Levermore & Pomraning 1981) is used to deal with radiation transfer.

The disrupted star is assumed to be solar type with radius R_* = R_⊙ and mass M_* = M_⊙, with R_⊙ being the solar radius. We consider the case in which the disrupted star moves on a parabolic trajectory towards the central black hole. We also assume that the orbital pericentre (R_p) of the star is equal to the tidal disruption radius. Therefore, the penetration factor β = R_T/R_p = 1. We simulate circularized accretion flow by assuming that when the disrupted stellar debris falls back, it can be very quickly circularized to form an accretion flow. Because of the angular momentum conservation, the circularized accretion flow forms at the circularization radius R_C, which is two times the disruption radius R_T. We use an anomalous stress tensor to mimic the angular momentum transfer by Maxwell stress.

We have two models. In model M7, the black hole mass M_BH = 10⁷ M_⊙. In model M6, we have M_BH = 10⁶ M_⊙. The tidal radius for a 10⁷ M_⊙ black hole is R_T = 5R_s (R_s is the Schwarzschild radius), while for a 10⁶ M_⊙ black hole, R_T = 47/2R_s. We inject the circularized stellar debris around R_C = 2R_T. The injection rate is set according to the theoretically predicted debris fallback rate |$\dot{M}_{\rm inject} = \dot{M}_{\rm fb} = \frac{1}{3}(M_\ast /t_{\rm fb})(1+t/t_{\rm fb})^{-5/3}$|⁠, with t_fb being the debris fallback time-scale. The fallback time-scale |$t_{\rm fb} \approx 40 \, {\rm d} \, \left(\frac{M_{\rm BH}}{10^6\,\mathrm{ M}_\odot } \right)^{1/2} \left(\frac{M_\ast }{\mathrm{ M}_\odot }\right)^{-1} \left(\frac{R_\ast }{\mathrm{ R}_\odot } \right)^{3/2}$|⁠. Note that in our simulations, t = 0 corresponds to the one fallback time-scale of the most bounded debris, at which the accretion begins rather than the point at which the star is disrupted. We define the Eddington accretion rate as |$\dot{M}_{\rm Edd} = 10 L_{\rm Edd}/ c^2$|⁠, with L_Edd being Eddington luminosity. The peak injection rate at t = 0 for model M6 is |${\sim} 133 \dot{M}_{\rm Edd}$| and |${\sim} 4.7 \dot{M}_{\rm Edd}$| for model M7. The injected stellar debris is assumed to have a local Keplerian rotational velocity. The internal energy of the injected debris is assumed to be |$1{{\, \rm per\, cent}}$| of the local gravitational energy. The simulations have computational domain in radial direction 2R_s ≤ r ≤ 10⁵R_s and in the θ direction 0 ≤ θ ≤ |$\pi$|/2. The resolution is N_r × N_θ = 768 × 128. Outflow boundary conditions are applied at the inner and outer radial boundary. At θ = 0, we use the axisymmetric boundary conditions. At θ = |$\pi$|/2, we use reflecting boundary conditions. The more details of the simulations are given in BU22.

3 RESULTS

The mass accretion rate onto the black hole is calculated at the inner radial boundary of the simulations (2R_s). Because we just simulate the region above the mid-plane, the accretion rate is two times the value above the mid-plane,

where ρ and v_r are gas density and radial velocity, respectively.

We now introduce the method to calculate the mass flux of wind. Turbulence is present in our simulations. Thus, we cannot judge the fluid element as wind only by v_r > 0. Because the outward moving portion of a turbulence eddy also has positive velocity. As given in Curd & Narayan (2023), we define gas that has positive Bernoulli parameter Be > 0 and v_r > 0 as wind. Following Curd & Narayan (2023), before defining the Bernoulli parameter Be, we define the electron scattering optical depth first. Along a viewing angle θ, the electron scattering optical depth is integrated from the outer radial boundary inwards |$\tau (\theta , r)=\int _{10^5R_\mathrm{ s}}^r \rho \kappa _{\rm es} \, \mathrm{ d}r^{\prime }$|⁠. The electron scattering opacity κ_es = 0.34 cm² g⁻¹. In the optically thick regions (τ_es > 1), the radiation is well coupled with gas and can contribute to the acceleration of gas, so we treat it as contributing to the Bernoulli parameter. In the optically thin region, the radiation is decoupled from gas, so we do not include it in the calculation of the Bernoulli parameter. As given in Curd & Narayan (2023), the Bernoulli parameter is calculated as follows:

where v, e_gas, E_rad, and G are gas velocity, gas internal energy density, radiation energy density, and gravitational constant, respectively. We set a specific heat ratio for gas γ_gas = 5/3; for radiation we set γ_rad = 4/3. The calculation of the Bernoulli parameter with optically thick radiation is also referred to Yoshioka et al. (2022).

The mass flux of wind is calculated as follows:

The kinetic power of wind is calculated as follows:

3.1 Model M7

In model M7, the central black hole mass is 10⁷ M_⊙. The circularized stellar debris is injected into the computational domain around 10R_s, which is two times the stellar orbital pericentre. In the presence of viscosity, an accretion flow forms, which transports gas towards the central black hole. The simulation covers 205 d since the peak fallback rate. The fallback rate is super-Eddington and the flow is radiation pressure dominated.

The top panel of Fig. 1 shows the time evolution of the black hole accretion rate (black line) and the stellar debris fallback rate (red line) in the unit of Eddington accretion rate. The bottom panel of Fig. 1 shows the time evolution of the black hole accretion rate in unit of the stellar debris fallback rate. The black hole accretion rate is highly variable. The variability is due to the fact that the flow is quite turbulent. Yang et al. (2014) performed hydrodynamic radiation pressure-dominated accretion flow. They also find that the flow is quite turbulent. Our result is consistent with that in Yang et al. (2014). The turbulence is due to the flow is convectively unstable. On average, the black hole accretion rate is significantly smaller than the stellar debris fallback rate (see the bottom panel). We quantitatively calculate the ratio of mass accreted to the black hole to the mass that falls back,

Only |$15{{\, \rm per\, cent}}$| of the fallback debris mass is accreted to the black hole.

For radiation pressure-dominated accretion flow, winds are a common phenomenon (e.g. Dai et al. 2018; Curd & Narayan 2019). We also find that strong winds are present in our simulations. In Fig. 2, we show the radial profiles of the wind mass flux at four snapshots. At each snapshot, there is a bump in the wind mass flux in the region 10–100R_s. We inject the fallback debris in this region, the calculation of the wind mass flux is quite affected. So the bumps should not be taken seriously. We pay attention to the region r > 100R_s. This region is far away from the gas injection region and initially there is no gas at all. All of the wind in the region r > 100R_s is from the region inside 100R_s. We can see that the wind head moves outwards with time. At t = 10 d, the wind head is located roughly at 2000R_s. At t = 50 d, the wind head has arrived at roughly at 10⁴R_s. Finally, at t = 200 d, the wind head arrives at 6 × 10⁴R_s. At the final point of our simulation, the wind head has not arrived at the outer boundary of the simulation. In the future, it is interesting to study how the winds evolve at a much larger scale.

It is interesting to ask how much of the fallback debris mass is taken away by the wind. We quantitatively calculated the ratio of the time-integrated mass taken away by the wind to the mass of the debris falls back,

The result is shown in Fig. 3. It can be seen that the value of f_wind is a function of radius. It roughly increases with a radius inside 500R_s. In the region r > 500R_s, it decreases with radius. The results can be understood as follows. We inject gas in the region 8R_s < r < 12R_s. An accretion flow forms inside 8R_s. The mass flux of the wind from the accretion flow is quite small (see Fig. 2). Outside the injection region, the wind mass flux is large. The reason for the increase of the value of f_wind with radius is as follows. We define winds to have a positive Bernoulli parameter. There should be such outflows that have negative Bernoulli parameters. Such outflows have not been recorded as wind. With the outwards motion, the Bernoulli parameter of such outflows increases. Finally, the Bernoulli parameter of some portion of these outflows becomes positive. Then the recorded mass flux of winds increases with radius. Therefore, the wind mass flux increases with radius. We take the Bernoulli parameter along the mid-plane at t = 200 d as an example to illustrate this point. Fig. 4 plots the radial profile of the Bernoulli parameter along the mid-plane at t = 200 d. The contribution of the radiation energy to the Bernoulli parameter is zero as shown in this figure. This is because at this snapshot, the photosphere τ_es = 1 at the mid-plane is located inside 100R_s. However, we find that even in the optically thin region r > 100R_s, the radiation pressure is |${\sim} 15{{\, \rm per\, cent}}$| of the gravity. The continue acceleration of gas by radiation pressure makes the Bernoulli parameter having a transition from a negative value to a positive value at 230R_s. We note that numerical simulations for hot accretion flow (Yuan et al. 2015) and super-Eddington accretion flow (Yang et al. 2023) all find that with the outward motion of wind, the Bernoulli parameter can increases. The negative Bernoulli parameter of some outward moving gas can become positive at some larger location due to the acceleration of gas. We find that at ∼500R_s, f_wind reaches its maximum value of 0.813. Therefore, more than |$80{{\, \rm per\, cent}}$| of the fallback debris escapes. This is consistent with the above conclusion that roughly |$15{{\, \rm per\, cent}}$| of the fallback debris mass goes to the black hole horizon (f_BH = 0.15). In the region r > ∼500R_s, f_wind decreases with radius. The reason is as follows. All of the gas is from the injection region. It takes time for wind to arrive at large radii as shown in Fig. 2. Therefore, at large radius, for a period since the beginning of the simulation, there is no wind at all. The larger the radius is, the longer the period will be. Therefore, the value of f_wind at larger radii decreases outwards.

The angular (θ) direction distribution of the mass flux of wind is shown in Fig. 5. In order to eliminate the fluctuation, we do time average to the wind mass flux. The black line, blue line, green line, and red line correspond to average period of 15–45, 46–75, 76–105, and 106–135 d, respectively. The top panel is for 500R_s. The bottom panel is for 2000R_s. It is clear that close to the rotational axis, the mass flux of wind is lowest. The mass flux of wind close to the rotational axis is more than two orders of magnitude lower than that in the angular region of θ > 40°. The mass flux of wind increases from θ = 0° to θ = 40°. Recent numerical simulations of super-Eddington accretion flow also found the similar angular distribution of mass flux of winds (Yang et al. 2023). The low gas density close to the rotational axis results in the low mass flux of wind there. In the region 40° < θ < 90°, the mass flux of the wind is roughly a constant with θ angle.

TDE winds are probably responsible for the radio emission in some TDEs (see Alexander et al. 2020 for review). The winds can interact with the CNM (Barniol Duran et al. 2013) or dense cloud surrounding the black hole (Mou et al. 2022; Bu et al. 2023), which induces shock. The power-law electrons responsible for the radio emission can be accelerated in the shock. In the shock model, the very important two parameters are the velocity and the kinetic power of winds. Therefore, it is very important to give the velocity and kinetic power of TDEs winds by simulations.

In Fig. 6, we plot the radial profile of the radial velocity of the wind along several viewing angles. It is clear that generally the velocity of the wind decreases from the rotational axis towards the mid-plane. Close to the rotational axis, the maximum velocity of the wind can achieve 0.7c. At the mid-plane, the velocity of the wind is roughly 0.1c. The decrease of wind velocity from the rotational axis towards the mid-plane is a common phenomenon in both radiation pressure-dominated super-Eddington accretion flow (e.g. Yang et al. 2023) and low accretion rate hot accretion flow (e.g. Yuan et al. 2015). At t = 10 d, the wind along the viewing angles of θ < 30° arrives at ∼2000R_s, which is the distance the wind moving with a velocity of ∼0.5c in 10 d. Along a fixed viewing angle (especially in the region θ < 30°), the velocity of winds is roughly a constant with radius. This means that the velocity of the wind is much larger than the escape velocity, the gravity can hardly decelerate the wind. With the roughly constant velocity, at the end of the simulation 200 d, the wind arrives at r ∼ 6 × 10⁴R_s ∼ 1.8 × 10¹⁷ cm. We note that in our simulations, we do not consider the deceleration of wind by the CNM or dense cloud. In the future, it is interesting to simulate a more realistic case in which the CNM or dense cloud is properly considered.

In Fig. 7, we plot the radial profile of the kinetic power of wind at four snapshots. There are bumps in the region 10R_s < r < 100R_s. We also find bumps in the radial profile of mass flux of the wind above (see Fig. 2). As introduced above, the bumps are related to the wind injection in this region. We pay attention to the wind at much larger radii r > 100R_s, where it is hardly affected by the wind injection. At t = 10 d, the wind moves to roughly r ∼ 2000R_s (see the top left-hand panel of Fig. 6), therefore, we can see that for this snapshot, the kinetic power of wind outside 2000R_s is cut-off. With the increase of time, the cut-off radius of the kinetic power of wind increases due to the outwards movement of wind. The kinetic power of wind can be >0.5L_Edd ∼ 6.5 × 10⁴⁴ erg s⁻¹. The kinetic power of the wind is enough to account for most of the radio emissions in radio TDEs (Mou et al. 2022; Bu et al. 2023).

The angular distribution (or opening angle) of wind is an important parameter for the study of interaction between wind and CNM. In Fig. 8, we show the angular distribution of the kinetic power of wind. We do time average to eliminate the fluctuation. The black line, blue line, green line, and red line correspond to average period of 15–45, 46–75, 76–105, and 106–135 d, respectively. The top panel is for 500R_s. The bottom panel is for 2000R_s. It can been seen that at both radii, the kinetic power is largest in the region of 30° < θ < 50°. Also, in this region, the kinetic power of the wind is almost a constant with θ. In the region θ < 30°, with the decrease of θ, the kinetic power decreases very quickly. The velocity of the wind in this region is highest (see Fig. 6). However, the mass flux of wind is lowest (see Fig. 5). The low mass flux in this region results in the low kinetic power. In the region of θ > 50°, the kinetic power of wind decreases quickly with increase of θ due to both the quick decrease of velocity with θ (see Fig. 6) and the slow decrease of mass flux with θ (see Fig. 5).

3.2 Model M6

In model M6, the central black hole mass is 10⁶ M_⊙. We inject the fallback stellar debris around the circularization radius 47R_s. The simulation covers 32 d since the peak fallback rate. The reason for the much shorter simulated period compared to model M7 is as follows. The time-step (Δt) of integration of the simulation is determined by the conditions in the innermost region of the grids. The value Δt ∼ Δr_min/c, with Δr_min being the smallest grid at the inner radial boundary. The value of Δr_min in model M6 is 10 times smaller than that in model M7. Therefore, Δt in model M6 is 10 times smaller. The CPU time needed for simulating 32 d for model M6 is longer than that for simulating 200 d for model M7. The fallback rate is super-Eddington. A viscous radiation pressure-dominated accretion flow develops and strong winds are found.

We first study the mass accretion rate onto the black hole. The top panel of Fig. 9 shows the time evolution of the black hole accretion rate (black line) and the stellar debris fallback rate (red line) in the unit of Eddington accretion rate. The ratio of black hole accretion rate to the debris fall back rate is shown in the bottom panel of Fig. 9. As in model M7, the accretion rate fluctuates with time due to the turbulent motions induced by convective instability. The ratio of accretion rate to the debris fallback rate is higher in model M6 compared to that in model M7 (see Fig. 1). We also quantitatively calculate the ratio of mass be accreted to the black hole to the mass falls back by using equation (1). The time integration is from t = 0 to 32 d. We find that in this model,

|$43{{\, \rm per\, cent}}$| of the fallback debris mass is accreted to the black hole. As a note that in model M7, f_BH = 0.15. In super-Eddington accretion flow, the high scattering optical depth results in that the photons comoving with the gas in the optically thick region. Ohsuga, Mineshige & Watarai (2003) found that for super-Eddington accretion flow, the higher the accretion rate, the easier the photons can be trapped. In other words, with the increase of accretion rate, the photons are easier to be advected into the black hole horizon rather than be advected to lager radii by winds. The wind is relatively weaker in higher accretion rate flow. In model M6, the accretion rate is much higher than that in model M7. Therefore, the wind (in the sense of ratio of wind mass flux to accretion rate) is relatively weaker in model M6.

We show the radial profiles of the mass flux of wind at four snapshots in Fig. 10. In this model, the injection radii are 47R_s. A viscous accretion flow forms inside this radius and wind is generated. Inside this radius, we can see that the mass flux of wind increases with radius. This can be understood as follows. In an accretion flow, except the region very close to the black hole, the wind can be generated at any radii. The mass flux of wind at a given radius includes both the flux of wind from the smaller radii and that generated locally. Therefore, we can find that the wind mass flux increases with radius inside ∼47R_s. However, outside this radius, all the wind comes from the smaller radii and winds cannot be generated locally. Therefore, the mass flux of wind is roughly a constant with radius as shown in Fig. 10. We also see the cut-off of the mass flux of wind as in model M7. The cut-off radius increases with time. At the end of the simulation, the wind arrives at 6 × 10⁴R_s.

We calculate the mass taken away by wind using equation (6). The result is shown in Fig. 11. It is clear that from 10R_s to 100R_s, the mass flux of wind increases quickly. The reason is same as the case for model M7. The wind is defined as outflow with positive Bernoulli parameter. There is outflow with negative Bernoulli parameter. With the outwards motion, the negative Bernoulli parameter of some portion of such outflows becomes positive. Thus, the mass flux of wind increases with radius. Outside 100R_s, the mass flux of wind decreases with radius. As explained for Model M7, the wind needs to spend time to arrive at large radius. The larger the radii, the longer the period that there is no wind. Thus, the value of f_wind at larger radii decreases outwards. Note that there is a small bump around 1500R_s. The small bump is due to the variation of Bernoulli parameter of wind.

The mass taken away by wind is only |$12{{\, \rm per\, cent}}$| of the injected mass. As introduced above, the mass be accreted to the black hole is |$43{{\, \rm per\, cent}}$| of the injected mass. We find that there is gas with negative Bernoulli parameter, which is just doing turbulent motions around 100R_s. We note that the simulation of model M6 just covers 32 d since the peak fallback rate. The wind may have not sufficiently developed. In the future, it is very necessary to run simulation with much longer physical period to study the further evolution of wind.

The angular distribution of the mass flux of wind is shown in Fig. 12. We do time average to eliminate the fluctuation. The top panel is for 500R_s and the bottom panel is for 2000R_s. As in the case of model M7, the mass flux of wind is lowest close to the rotational axis due to the low density there. The mass flux increases from θ = 0° to 20°. In the region 20° < θ < 90°, the mass flux of wind is roughly a constant with θ angle.

In Fig. 13, we plot the radial profile of the radial velocity of the wind along several viewing angles for model M6. As in the case of model M7, the velocity of the wind is highest around the rotational axis. The velocity decreases from the rotational axis towards the mid-plane. The velocity around the rotational axis can be as high as 0.7c. The velocity at the mid-plane is one order of magnitude lower than that around the rotational axis. At small viewing angle, the velocity of the wind is roughly a constant with radius. The reason is as follows. At small viewing angle, the velocity of the wind is significantly higher than the local escape speed. Or in other words, the kinetic energy of wind is significantly larger than the gravitational energy, the gravity can hardly decelerate the wind.

The radial distributions of the kinetic power of wind are shown in Fig. 14. In the region r < 200R_s, the kinetic power increases quickly with the radius. This is due to the fact that both the mass flux (see Fig. 10) and the velocity (see Fig. 13) of wind increase with radius. Outside 200R_s, the kinetic power keeps roughly a constant with radius until the cut-off. The roughly constant behaviour indicates again that the wind can hardly be decelerated by the gravity of the black hole. With the increase of time, the cut-off radius of the kinetic power increases. The kinetic power of wind can achieve 2 × 10⁴⁴ erg s⁻¹, which is enough to account for the radio emission in radio TDEs.

We show the angular distribution of the kinetic power of wind in Fig. 15. The power is highest in the angular region very close to the rotational axis 5° < θ < 20°. The reason is as follows. From Fig. 13, we can see that the velocity around the rotational axis is highest. The mass flux in this angular region is comparable to that in other angular region (Fig. 12). In the region θ > 20°, the kinetic power decreases very quickly with θ. The kinetic energy flux at the mid-plane can be more than two orders of magnitude lower than that in the region 5° < θ < 20°. The reason is that in this region θ > 20°, the mass flux is roughly a constant with θ (see Fig. 12). However, the velocity of the wind decreases very quickly with θ (see Fig. 13). The velocity at the mid-plane can be one order of magnitude lower than that around the rotational axis. The kinetic power is ∝v³. Therefore, the kinetic power at the mid-plane is significantly low.

4 SUMMARY AND DISCUSSIONS

We study the black hole accretion and wind of circularized accretion flow in TDEs based on the radiative hydrodynamic simulations of BU22. We assume that a solar-type star is disrupted. We have two models with M_BH = 10⁶ and 10⁷ M_⊙. We also assume that the orbital pericentre of the disrupted star equals to the tidal radius. When the debris falls back, we assume that it can be very quickly circularized. An accretion flow is formed in the presence of viscosity.

The first issue we study is the relationship between the black hole accretion rate and the debris fallback rate. We find that only a part of the fallback debris can be accreted to the black hole. Specifically, for a 10⁷ M_⊙ black hole, |$15{{\, \rm per\, cent}}$| of the fallback debris is accreted by the black hole; for a 10⁶ M_⊙ black hole, |$43{{\, \rm per\, cent}}$| of the fallback debris is accreted by the black hole.

The second issue we study is the wind. We find that wind can be launched by radiation pressure in the super-Eddington accretion phase of TDEs. For a 10⁷ M_⊙ black hole, more than |$81{{\, \rm per\, cent}}$| of the fallback debris is taken away by wind; for a 10⁶M_⊙ black hole, |$12{{\, \rm per\, cent}}$| of the fallback debris is taken away by wind. The velocity of the wind decreases from the rotational axis to the mid-plane. Close to the rotational axis, the maximum velocity of the wind can reaches 0.7c. At the mid-plane, the velocity of the wind is ∼0.1c. The kinetic power of wind is in the range of (2–6.5) × 10⁴⁴ erg s⁻¹.

In model M6, the simulation just covers 32 d since the peak fallback rate. From the bottom panel of Fig. 9, we can see that the ratio of the black hole accretion rate to the debris fallback rate has not settled to a constant value. It is unknown, what will this value evolve further. It is very necessary to run the simulations further to cover several hundred days. In that case, a more solid conclusion about black hole accretion rate and wind can be made.

In our simulations, magnetic field is not included. It is well known that wind can be launched from an accretion flow by the magnetocentrifugal force (Blandford & Payne 1982). If magnetic field is taken into account, the specific results about the properties of wind may be changed. In future, it is very necessary to study the wind by taking into account both magnetic field and the specific conditions of TDEs.

In our simulations, we use a viscous stress to transfer angular momentum. The value of α is set to be 0.1. Sadowski et al. (2015) found that in their super-Eddington simulations, the ratio of magnetic pressure to total pressure p_mag/p_tot ∼ 0.1. The value of α is defined as α = −B_rB_ϕ/(4|$\pi$|p_tot), with B_r and B_ϕ being the radial and toroidal components of the magnetic field. Because B_rB_ϕ/(4|$\pi$|⁠) ∼ p_mag, we have α ∼ p_mag/p_tot. Thus, the value of α of a super-Eddington accretion flow can be ∼0.1. Therefore, the value of α used in our simulations is similar to that given by magnetohydrodynamic simulations. The α viscosity in our work seems to be sufficient to drive accretion.

In Dai et al. (2018), the authors find strong winds in their simulations. However, they mainly pay attention to the radiation properties of the flow. There are no detailed descriptions of the properties of winds (e.g. kinetic power and mass flux). They plot a figure (right-hand panel of their fig. 3) to show the velocity of winds. Their results are as follows. First, generally, they find that the velocity of the wind decreases from the rotational axis towards the mid-plane. Second, the maximum velocity of the wind is around 0.7c. Third, the minimum velocity of the wind around the mid-plane is below 0.1c. The three properties of wind found in Dai et al. (2018) are consistent with that found in our work.

We assume that ‘circularization’ of fallback debris is efficient. However, the efficiency of ‘circularization’ of the fallback debris is still under debate (Kochanek 1994; Bonnerot et al. 2016; Hayasaki, Stone & Loeb 2016; Bonnerot, Rossi & Lodato 2017; Bonnerot & Lu 2020; Rossi, Servin & Kesden 2021). For less bound or less circularized gas, the mechanical energy (gravitational energy plus kinetic energy) is higher than that of well-circularized Keplerian flow. In this sense, the injected gas in our simulations has artificially lower mechanical energy compared to less circularized gas. It seems that gas with higher energy is much easier to produce winds. If in reality the fallback debris can be accreted by the black hole before well circularization, we may underestimate the strength of wind by our simulations. If the circularization process of the fallback debris can be finished in a much shorter time-scale compared to the accretion time-scale. The debris will be first circularized and then be accreted to the black hole. The results found in our simulations should be applicable to the accretion phase. However, the winds that may be launched in the prior circularization process need further investigations. However, we note that for super-Eddington accretion flow, the presence of radiation pressure-driven wind should be very solid, regardless of whether the flow is circularized. For completeness, it is very necessary in future to study the wind from a not fully circularized accretion flow.

ACKNOWLEDGEMENTS

D-FB is supported by the Natural Science Foundation of China (grants 12173065,12133008,12192220,12192223) and the science research grants from the China Manned Space Project (No. CMS-CSST-2021-B02). EQ is supported by the National Natural Science Foundation of China (grant 12173048) and NAOC Nebula Talents Program. X-HY is supported by the Natural Science Foundation of China (grant 11973018). This work made use of the High Performance Computing Resource in the Core Facility for Advanced Research Computing at Shanghai Astronomical Observatory.

DATA AVAILABILITY

The data underlying this paper will be shared on reasonable request to the corresponding author.

References