The Role of Outflow Feedback on Accretion of Compact Objects in Accretion Disk of Active Galactic Nuclei

As embedded in the AGN disk environment, COs (e.g. BH or NS) will capture and accrete gas within its sphere of gravity. AGN disk rotates differentially, the accreted gas hence holds shear velocity w.r.t the CO, and would form a rotating disk around the CO rather than fall directly into it. To intuitively understand this process, we take gas accretion onto a planet embedded in a proto-planetary disk as an analogy (e.g. Kimura et al., 2021b). As shown in simulations (Ayliffe & Bate, 2009; Tanigawa et al., 2012), a nearly Keplerian rotating circum-planetary disk forms around the planet; the majority of mass is concentrated in the outer region of the disk, mainly around the circularization radius $r_{\rm{cir}}$ of the accreted gas (see below). Back to the AGN disk case, the initially circular gas disk would then go through viscous evolution and effective accretion to the CO on viscous timescale $t_{\rm{vis}}$ (e.g. Kato et al., 2008)¹¹1When the initial mass inflow rate to CO is extremely large, as in the AGN disk environment we discuss, the infalling gas potentially follows an alternative structure called “zero-Bernoulli accretion” flows suggested by Coughlin & Begelman (2014); the inflow inflates nearly to the poles and forms a weakly bound, quasi-spherical structure, rather than a large-scale accretion disk. The criteria of which structure occurs under the specific environment parameters, and the distinctions of the relevant outflow feedback will be studied in another work..

The initial inflow mass rate around the disk outer boundary can be described by the Bondi-Holye-Lyttleton (BHL) formulation (see Edgar, 2004, for a review); under the corrections of the SMBH gravity effect in radial direction and the finite height of the AGN disk in vertical direction, the gas inflow rate is estimated as follows (Kocsis et al., 2011; Pan & Yang, 2021b):

and the outer boundary radius of the circum-CO disk $r_{\rm{obd}}$ can be approximate to $\sim r_{\rm{cir}}$, which can be estimated from the angular momentum conservation of the infalling gas, i.e.,

where $m_{\rm{CO}}$ is the mass of CO, $r_{\rm{rel}}:=\min\{r_{\rm{BHL}},r_{\rm{Hill}}\}$ is the radius of CO gravity sphere, $r_{\rm{Hill}}=(m_{\rm{CO}}/3M)^{1/3}R_{\rm{CO}}$ is the Hill radius, and $r_{\rm{BHL}}=Gm_{\rm{CO}}/(v_{\rm{rel}}^{2}+\tilde{c}_{s}^{2})$ is the BHL radius; the canonical equation of the BHL accretion rate is

where $\rho_{\rm{CO}}$ is the density of AGN disk gas surrounding the CO, and $v_{\rm{rel}}$ is the relative velocity between the CO and the gas, which contains components of the gas relative bulk motion and shear rotation. To estimate the shear velocity $v_{\rm{shear}}$, we linearize the velocity disparity at the location of CO $R_{\rm{CO}}$ and the boundary of the CO gravity sphere $(R_{\rm{CO}}\pm r_{\rm{rel}})$, i.e.,

where we consider the CO co-rotates with the Keplerian AGN disk at its midplane, $V_{\rm{K}}=\sqrt{GM/R}$, in which case the gas bulk motion can be neglected (Kocsis et al., 2011). In fact, the CO can hold large bulk velocity $v_{\rm{bulk}}$ w.r.t. the ambient disk gas, as briefly discussed in Section 4.6. Here we primarily investigate the long-term evolution of COs in the AGN disk, which are probably on the Keplerian co-rotating orbits; the COs with large $v_{\rm{bulk}}$ may escape from the disk, or the gas dynamical and accretion frictions reduce the relative velocity of CO, leading to the subsequent co-rotation (e.g. Ostriker, 1999; Bartos et al., 2017; Secunda et al., 2019; Fabj et al., 2020; Nasim et al., 2022).

When living in the AGN disk, the CO can interact with the surrounding disk through gravity, inducing gravitational torque to additional exchange energy and angular momentum with the AGN disk (the disk–satellite interactions would be analogous to a giant planet in a proto-planetary disk; see Armitage 2007, Baruteau et al. 2014, as the reviews). The ambient gas has the tendency to be repelled from the CO, and thus gas density is reduced in the disk annulus at $R_{\rm{CO}}$. The gas density and half-width of the reduced region are estimated as $\rho_{\rm d,gap}/\rho_{\rm d}=1/[1+0.04(m_{\rm{CO}}/M)^{2}(H/R_{\rm{CO}})^{-5}\alpha^{-1}]$ and $R_{\rm d,gap}/R_{\rm{CO}}=0.21(m_{\rm{CO}}/M)^{1/2}(H/R_{\rm{CO}})^{-3/4}\alpha^{-1/4}$ (Kanagawa et al., 2015, 2016; Tanigawa & Tanaka, 2016). We set the density used to calculate the BHL accretion rate, i.e. in Equation (4), as

For the circum-CO accretion disk with extremely high inflow mass rate and distant outer boundary, the outer disk region would be self-gravity unstable, accordingly the gas captured by the CO fragments, resulting in the reduction of mass inflow rate (e.g. Pan & Yang, 2021b; Tagawa et al., 2022). The Toomre parameter of the circum-CO disk can be approximatively expressed as $Q_{\rm{CO}}=\Omega_{\rm K}{c}_{s}/\pi G\Sigma\sim 2\alpha_{\rm{CO}}h^{3}v_{K}^{3}/G\dot{M}_{\rm{inflow}}$, where $\Omega_{\rm{K}}$ is the Keplerian angular velocity, ${c}_{s}\sim hv_{\rm{K}}$ is the sound speed, $\Sigma=\dot{M}_{\rm{inflow}}/2\pi rv_{\rm{r}}$ is the disk surface density and $v_{\rm{r}}=\alpha_{\rm{CO}}h^{2}v_{\rm{K}}$ is the inflow radial velocity, $\alpha_{\rm{CO}}$ is the viscosity parameter, $h=H_{\rm{CCOD}}/r$ is the disk height ratio, $H_{\rm{CCOD}}$ is the vertical scale height of the disk, and $v_{\rm{K}}=\sqrt{Gm_{\rm{CO}}/r}$; the stabilization of circum-CO disk requires $Q_{\rm{CO}}\geq 1$, so the modified mass inflow rate is expressed by

In short, when a CO accretes the AGN disk gas, a circum-CO disk would form, with the initial radius $r_{\rm{obd}}$ and the mass inflow rate $\dot{M}_{\rm{obd}}$ given by Equations (3) and (7).

The initial mass inflow rates of COs are extremely hyper-Eddington, i.e. $\dot{M}_{\rm{obd}}\gg\dot{M}_{\rm{Edd}}$, as shown in Figure 1; a specific property of the accretion is photon trapping, i.e. the photons and radiation energy are trapped and advected inward with the dense gas inflow because of the photon diffusion timescale exceeding the disk accretion timescale. The radiation-effective flow converts to be radiation-ineffective nearly at the trapping radius (e.g. Kato et al., 2008; Kitaki et al., 2021):

where $\dot{m}\equiv\dot{M}_{\rm{obd}}/\dot{M}_{\rm{Edd}}$ and $r_{g}=Gm_{\rm{CO}}/c^{2}$. Inside $r_{\rm{tr}}$, the disk radiation pressure efficiently drives outflow, leading to reduction of the circum-CO disk mass inflow rate; the radius-dependent rate $\dot{M}_{\rm{in}}$ can be written as (Blandford & Begelman, 1999):

where the power-law index $s$ is a free parameter and its value is taken to be $0<s\leqslant 1$; numerical simulations show that the value lies within $\sim 0.4-1.0$ (Yang et al., 2014), but $\sim 0$ out to $O(10^{2})r_{g}$ for the case of modestly circularized gas inflow (Kitaki et al., 2021). Under the uncertainty, we set three values $s=0.2$, $0.5$, $0.8$ to study the outflow feedback. For $r_{\rm{tr}}<r_{\rm{obd}}$, the circum-CO disk is advection-dominated within $r_{\rm{tr}}$, and is radiation-dominated between $r_{\rm{tr}}$ and $r_{\rm{obd}}$; but when $r_{\rm{tr}}>r_{\rm{obd}}$, the disk is totally advective accompanied with the outflow formation²²2But the mass inflow rate is not large enough to become neutrino-cooling dominated, see Kohri et al. (2005) and Equation (20) in Kumar et al. (2008)., and the mass inflow rate is written as

The radiation-pressure-driven outflow can take away a considerable fraction of viscous heating of the circum-CO disk, as discussed in Appendix A. The local viscous heating rate per unit area is given by Equation (A7), so the total disk generated heat is

where $\dot{M}_{\rm{out}}$ is the mass inflow rate at $r_{\rm{out}}$; $r_{\rm{out}}=r_{\rm{obd}}$ for $r_{\rm{tr}}>r_{\rm{obd}}$ case, and $r_{\rm{out}}=r_{\rm{tr}}$ for $r_{\rm{tr}}<r_{\rm{obd}}$ case, for which we ignore the heat generated in the outer radiation-dominated region $r_{\rm{tr}}<r<r_{\rm{obd}}$ because only tiny gravitational energy is released via accretion at large radius. We take $r_{\rm{in}}\sim 10r_{g}$ as the inner radius outside which the self-similar solutions hold and the outflow cooling is efficient, because when $r\lesssim 10r_{g}$, the relativistic effects of the central CO can not be neglected (Pan & Yang, 2021b), the outflow is dragged by the strong gravitational force to be weak (Jiao et al., 2015) and the cooling is dominated by advection rather than outflow (Wu et al., 2022). We roughly assume that a constant fraction $f_{w}$ of the heat is taken away by the disk wind and radiation, i.e.,

where $f_{w}$ contains the constant coefficients reflecting the wind property in Equation (11); we set $f_{w}=0.5$ as the fiducial value.

The accretion features of the NS are sightly different from those of the BH due to the existence of a stellar magnetic field and hard surface (Takahashi et al., 2018). A strong magnetic field may truncate the circum-NS disk at a radius $r_{\rm{m}}$ where the magnetic stress of NS magnetosphere matches the disk stress (e.g. Zhang & Dai 2009; and Lai 2014, Romanova & Owocki 2015, as reviews). For a dipole field, equaling magnetic pressure $B(r_{\rm{m}})^{2}/8\pi$ to disk pressure $\rho_{\rm{CNSD}}(r_{\rm{m}})c_{s}(r_{\rm{m}})^{2}$, where the density $\rho_{\rm{CNSD}}$ of the circum-NS disk is given by Equation (B1), $r_{\rm{m}}$ can be estimated as

where $k=0.4[(\alpha_{\rm{CO}}/0.1)^{2}(h/0.5)^{2}/4]^{1/(7+2s)}$, which is slightly lower than the typical value $0.5-1$ (e.g. Ghosh & Lamb, 1979; Chashkina et al., 2019), the distinction may derive from the properties of hyper-Eddington accretion disk; for simplicity, we leave the specifically analysis of the disk structure and truncation for a future work, and take $k=0.5$ briefly. When magnetic field is weak, the disk can extend to the NS surface $r_{\rm{NS}}$; so in the NS accretion case, we take $r_{\rm{in}}=max\{r_{\rm{m}},r_{\rm{NS}},10r_{g}\}$ to determine the rate of mass eventually flowing onto the NS, where $10r_{g}$ is set due to the effects of NS’s gravity as explained above (but we ignore the potential effects of the energy released near the NS surface on the accretion structure and properties, e.g. Takahashi et al. 2018). In addition, the accretion flow would release additional energy with luminosity $L_{\rm{acc}}\simeq G\dot{M}_{\rm{in}}(r_{\rm{in}})m_{\rm{NS}}/r_{\rm{NS}}$ when ultimately hitting the NS hard surface, where we crudely ignore the neutrino cooling processes around the NS surface. So the total energy injected into the wind outflow contains the disk and the accretion component (Chashkina et al., 2019), i.e.,

The direction of the outflow significantly affects its feedback on the environment. Many numerically simulations and analytical solutions of the super-critical accretion flow around CO predicted that the emitted winds are focused in the high latitude region, i.e. the outflow is anisotropic (e.g. Jiao et al., 2015; Sa̧dowski & Narayan, 2015), the feedback as well (e.g. Takeo et al., 2020; Tagawa et al., 2022). These works mainly investigate disk parameter regions of the mass inflow rate $\lesssim O(10^{3})\dot{M}_{Edd}$ and the radial dimension $\lesssim O(10^{4})r_{g}$ (see Table 1 in Kitaki et al. 2021, which lists the recent simulation studies), where the outflow is automatically collimated by the geometrically thick disk and the optically thin polar funnel, but these parameters are much smaller than the case of CO accretion in the AGN disk. Also, when studying the feedback, only the kinetic energy of the outflow has been considered and the winds are set to spread over a limited solid angle, thereby, the initial setup of outflow is anisotropic.

For the hyper-Eddington accretion process of CO, winds initially take along the radiation energy when just launching. Because the outflow and the AGN disk surroundings (the optical depth from the disk midplane $\tau_{d}\sim\kappa_{ff}\rho_{d}H=1.7\times 10^{4}\alpha_{-1}^{-4/5}M_{8}^{1/5}\dot{\mathcal{M}}_{0.5}^{1/5}\gg 1$) are both extremely dense, photons can not effectively diffuse away but are trapped in the outflow, coupled with and continually accelerating the optically thick gas (Hashizume et al., 2015; Sa̧dowski & Narayan, 2015). Thus, the outflow contains not only kinetic but also thermal energy. We can extrapolate the properties of winds generated by the large-scale hyper-Eddington accretion disk from the simulations and the self-similar solutions shown in Appendix A. As discussed in Begelman (2012) and can be seen by Equations (9) and (12), most of the inflow mass is turned into the outflow at large radius comparable to $r_{\rm out}$, conversely, most accretion energy is released closer to $r_{\rm in}$, carried by winds over a wide solid angle (Sa̧dowski et al., 2016; Sa̧dowski & Narayan, 2016); winds successively loaded at different radius of the circum-CO disk would go through interactions and internal collisions (Metzger, 2012; Kremer et al., 2019), causing the gas thermalization during the outward propagation and mixture. So, we predict that the outflow kinetic energy is close to the radiation-domination thermal energy within the circum-CO disk scale. Winds moving radially at bulk velocity $v_{\rm w}$ would press and heat the circum-CO disk as well, where the thermal pressure $\sim\rho_{\rm w}c_{\rm s,w}^{2}\sim\rho_{\rm w}v_{\rm w}^{2}$. When the sound speed of the heated disk gas exceeds the CO local escape velocity, the gas outside is ejected and hence the disk is truncated (e.g. Tagawa et al., 2022). The truncation radius $r_{\rm{tru}}$ can be estimated as

where $\rho_{\rm{CCOD}}$ is the density of circum-CO disk, whose expression is shown in Appendix B; the density of the outflow $\rho_{\rm{w}}$ is set as a simplified spherical distribution, i.e.,

in other words, we assume that the heavy gas would distribute roughly uniformly over a wide solid angle at large radius because of the gas thermal mixture, and we neglect the spatial existence of the circum-CO disk compressing the wind region when calculating the disk truncation, which may increase $\rho_{\rm{w}}$ but would not significantly affect the result because of the disk thickness $h\sim O(10^{-3})-O(10^{-1})$ not too large; the velocity of the mixed disk wind can be obtained from $\dot{M}_{\rm{w}}v_{\rm{w}}^{2}/2\simeq L_{\rm{w}}/2$, i.e.,

in the NS case we replace the energy term with $L_{\rm{wNS}}$. After getting $r_{\rm{tru}}$, we can decide the disk outer boundary $r_{\rm{obd}}=\min\{r_{\rm{cir}},r_{\rm{tru}}\}$, then calculate the modified $\dot{M}_{\rm{obd}}=\dot{M}_{\rm{in}}(r_{\rm{obd}})$ and the mass accretion rate of CO $\dot{M}_{\rm{CO}}=\dot{M}_{\rm{in}}(r_{\rm{in}})$ using Equations (9) and (10). As examples shown in Figure 2, we find that the mass accretion rate of CO is well hyper-Eddington when ignoring the effects of outflow feedback on the environment; and we confirm a comprehensible tendency that the mass rate increases with the decrease of index $s$, which controls the strength of disk wind.

Even if the outflow is initially anisotropic (collimated by the geometrically thick disk) when leaving the local circum-CO disk system, because

where the double-greater-than symbol holds (we check numerically and find that the inequality assuredly holds for the regions $R_{\rm{CO}}\gtrsim O(10^{2})R_{g}$ we interested) as $v_{\rm{w}}/v_{\rm{K}}(r_{\rm{obd}})\sim f_{w}^{1/2}(r_{\rm{obd}}/r_{\rm{in}})^{(1-s)/2}\gg 1$ with the supersonic accretion $v_{\rm{K}}(r_{\rm{obd}})>\tilde{c}_{s}$ and $\rho_{\rm{w}}\sim\rho_{\rm{CO}}$ due to $\dot{M}_{\rm{w}}\sim\dot{M}_{\rm{in}}$, the pressure of the environment gas is unable to prevent the outflow becoming asymptotically isotropic and spherically symmetric (though there may still have anisotropy, with lower wind mass rate and larger wind velocity at higher latitude, e.g., Kitaki et al. 2021) before sweeping a large amount of the AGN disk gas; the equatorial region’s gas captured by the CO will also be pushed away. During the adiabatic expansion, most of heat in the outflow would be converted to bulk kinetic energy and accelerate the winds with initial velocity of Equation (17) (Kremer et al., 2019; Piro & Lu, 2020), so the asymptotic velocity, kinetic energy and momentum of the outflow are

For the moment, we ignore the detailed time-evolution of the circum-CO accretion disk, which may follow the works of Kumar et al. (2008) and Shen & Matzner (2014), determining the specific evolution of mass rate accreted onto CO and the variation of outflow generated from the accretion system. For now, the structure of the hyper-Eddington accretion disk, much less the disk evolution, has not been well studied, as discussed in Appendix B; at the same time, the qualitative properties of CO accretion can be approximatively described using the viscous timescale. So, for simplicity, we put aside the complicated time-evolution of the piecewise disk, instead we assume that the efficient accretion proceeds within $t_{\rm{vis}}$, after which the efficient accretion stops.

In analogy with the interaction between the stellar wind and interstellar medium (Weaver et al., 1977), the wind from the circum-CO disk interacts with the surrounding AGN disk gas via a forward shock, forming a shell of shocked disk gas separated by the shocked outflow material; inside the shell, a low-density outflow cavity forms. We ignore the radial and vertical structure of the AGN disk, and roughly set the disk with uniform density $\rho_{\rm{d}}$ and a concrete height $H$ at each $R_{\rm{CO}}$. A schematic diagram of outflow evolution in the AGN disk is shown in Figure 3.

During the efficient accretion of CO sustaining timescale of $t_{\rm{acc}}=t_{\rm{vis}}(r_{\rm{obd}})=\alpha_{\rm{CO}}^{-1}h^{-2}\Omega_{\rm{K}}^{-1}$, we approximately assume the properties of winds remain unchanged. The radius and velocity of the expanding shell evolve over time as (Weaver et al., 1977):

and

When $r_{\rm{shell}}(t_{\rm{bre}})=H$, the shell will vertically break out from the AGN disk and propagate in the above lower density region (e.g. broad-line region); meanwhile, along the AGN disk, the outflow continually interacts and pushes the shocked gas shell laterally, which now shows a ringlike structure, until the efficient accretion stops. Because the vertical expansion of the shell is much more rapid ($\rho_{\rm{BLR}}\sim 10^{-17}\,\text{g}\,\text{cm}^{-3}\ll\rho_{\rm{d}}$), we assume that the shocked outflow undergoes an overall rapid depressurization after the breakout (e.g. Schiano, 1985) ³³3Moranchel-Basurto et al. (2021) studied the supernova explosions in the AGN disk, and found that the analytic evolution (vertically and laterally) of the SNe-driven bubble under the assumption of bubble receiving rapid depressurization after the breakout matches well with the numerical simulations., and then the ring moves driven by the momentum of the long-lasting outflow ($t_{\rm{acc}}\gg t_{\rm{bre}}$ holds within wide parameter regions, as shown in Figure 4), i.e.,

where we roughly assume that the ring is cylindrical and only moves laterally. Because $\rho_{\rm{w}}\propto r_{\rm{shellb}}^{-2}$, the ringlike shell evolves after the breakout as:

and

where we simply ignore the specific depressurization process and link the outflow-energy and outflow-momentum driven phase directly. Lastly, when the CO accretion stops, the evolution of the ringlike shell along the AGN disk translates from the outflow-momentum driven phase to its own momentum conservation snowplow phase, i.e.,

where $r_{\rm{sh}}$ and $\dot{r}_{\rm{sh}}$ represent the radius and velocity of shell during the snowplow phase; on both sides of the equation we have eliminated $\pi\Sigma_{\rm{d}}=2\pi\rho_{\rm{d}}H$, which approximatively represents the mass of the swept disk gas $\pi r_{\rm{sh}}^{2}\Sigma_{\rm{d}}$. With the continuous sweep, the shell will decelerate to a velocity similar to the environment sound speed, i.e., $\dot{r}_{\rm{sh}}\sim\tilde{c}_{\rm{s}}$, which ends up being transonic and is unable to propagate further as a shock. So the horizontal half-width of the outflow cavity in the AGN disk can be estimated by

The timescale of the cavity formation can be calculated from Equation (26), i.e.,

which is the sum of the efficient accretion timescale and the evolution timescale of the subsequent snowplow phase.

But if $t_{\rm{bre}}>t_{\rm{acc}}$, the CO would stop strong accretion before the successful shell breakout. In this case we roughly take the whole energy of the CO-accretion-generated winds, i.e., $E_{\rm{w}}=L_{\rm{w}}t_{\rm{acc}}$, as injection; the shell thus expands driven by $E_{\rm{w}}$, the radius and velocity are given as (e.g. Ostriker & McKee, 1988):

and

where we ignore the mass of wind for simplicity. We can also achieve the breakout time from $r_{\rm{shellE}}(t_{\rm{breE}})=H$. After the breakout accompanied with the rapid cavity depressurization, the shell propagating along the AGN disk enters the snowplow phase, i.e.,

the half-width $r_{\rm{cavE}}$ and the formation timescale $t_{\rm{cavE}}$ of the cavity can be solved similar to Equation (27) and (28).

The accretion rate of CO in the underdense cavity is too low to continuously produce powerful outflow maintaining the cavity structure (see below); conversely the relatively dense AGN disk gas will refill the cavity roughly on a timescale given by (e.g. Wang et al., 2021a):

Then the efficient CO accretion will be activated in the refilled cavity again; subsequently the cavity formation and refilling would take place alternately.

We take $m_{\rm{BH}}=10M_{\odot}$ and $s=0.5$ as a typical case to study the influence of outflow feedback on the BH accretion in the AGN disk environment, of which the specific processes have been discussed in Section 3.1. We can calculate the $r_{\rm{cav}}$(or $r_{\rm{cavE}}$) under variable AGN disk parameters, and then estimate the density of outflow cavity as

where we treat the cavity as spherical, which overestimates the density of cavity because the shell expands much faster above the AGN disk causing a much larger volume. Then we use Equation (7) to get the mass accretion rate of BH in the cavity, replacing $\rho_{\rm{CO}}$ with $\rho_{\rm{cav}}$. Relevant properties of the cavity and BH accretion are shown in Figure 4. We find that the size of outflow-induced cavity is much larger than the BHL and Hill radius, i.e., the size of BH gravity sphere, which suppresses the BH further capturing the AGN disk gas. The density of cavity is significantly reduced compared to the initial unperturbed environment. We calculate the trapping radius $r_{\rm{tr,cav}}$ of BH accretion in the cavity using Equation (8), and find that it is well smaller than the circum-BH disk outer boundary, reflecting that the BH mass accretion rate is significantly reduced, and hence the induced outflows are not powerful enough to notably affect or maintain the cavity structure. For the weak accretion in cavity, we simply neglect the specific evolution of the BH mass accretion rate and set a factitious value of $\dot{M}_{\rm{BH,cav}}=10\dot{M}_{\rm{Edd}}$, which is the approximate minimum mass inflow rate at the inner boundary of the advection-dominated disk (e.g. Pan & Yang, 2021b).

The timescale $t_{\rm{acc}}$ of efficient accretion is well shorter than the duration time $t_{\rm{cav}}+t_{\rm{ref}}$ of the cavity evolution, as shown in Figure 5. A smaller index $s$ results in a overall more powerful outflow feedback, accordingly larger size and longer duration time of the cavity. The variation trend of the contour map relies on various effects ($s=0.5$ case as an example in Figure 5); first, a general trend of $t_{\rm{acc}}/(t_{\rm{cav}}+t_{\rm{ref}})$ being larger for smaller BH location $R_{\rm{BH}}$ comes from the AGN disk gas environment being denser when closer to SMBH, which hinders the expansion of cavity; second, the varied outer boundary radius of the circum-BH disk results in different features of the major disk region, thereby different dependence of the disk accretion timescale variation on the AGN disk parameters; third, the gap-depth is larger for lighter and closer SMBH cases, or the self-gravity instability in circum-BH disk inhibits the mass inflow rate, leading to lower mass accreted onto BH and weaker outflow feedback, accordingly, the size and evolution timescale of the cavity are smaller; in addition, the relative size of BHL radius, Hill radius and AGN disk height, and the unsuccessful breakout also affect the accretion and feedback properties of the BH.

Because the accretion of BH is inefficient in the outflow cavity, though hyper-Eddington during $t_{\rm{acc}}$, the averaged mass accretion rate onto the BH is well reduced in a whole circulation $t_{\rm{cav}}+t_{\rm{ref}}$. We can estimate the reduced mass accretion rate as:

some specific cases are shown in Figure 5. Indeed, the BH accretion rate receives a markedly overall reduction, which can be directly indicated by comparing the cases ignoring circum-BH disk outflow feedback as shown in Figure 2. The general variation trend of the reduced mass rate depending on the AGN disk parameters has no significant change from the omitting-feedback cases. The reduction effect is more prominent for smaller index $s$, which produces more powerful disk winds; but meanwhile, the initial mass accretion rate is higher, the combined effects bring about a relatively larger mass rate compared with the larger $s$ cases.

In a word, the circum-BH disk winds significantly impact the accretion processes of BH located in the AGN disk, the mass accretion rate of BH is markedly reduced, and the BH’s surroundings is observably changed in the form of a long-standing, underdense cavity, or to say, feedback of the outflow generated by the hyper-Eddington disk can not be neglected. Moreover, the mass rate accreted onto BH relies heavily on parameters $M$ and $R_{\rm{BH}}$, and thus varies during the BH evolution, rather than remains as a fixed value.

Compared to the BH accretion case, the NS accretion is relatively more efficient because the magnetosphere potentially truncates the circum-NS disk at a larger radius $r_{\rm{m}}$, Equation (13), to bring more mass accreted onto NS; meanwhile, feedback from the NS accretion system is more powerful because the existence of hard surface releases additional energy, Equation (14), to inject into the outflow.

The specific properties of NS accretion are shown in Figure 6. We find that only the NS with a strong magnetic field (e.g. $\sim 10^{12}G$) can give rise to magnetospheric disk truncation, on account of the large mass inflow rate of circum-NS disk leading to large disk pressure, which can only be resisted by large magnetic pressure; by contrast, when NS magnetic field is weak, the disk extends to the NS surface instead (comparing first and second panels in Figure 6). And we find that the NS-accretion energy $L_{\rm{acc}}$ is generally larger than the disk-generating energy $L_{\rm{w}}$ (seeing all three panels in Figure 6), indicating that the NS surface liberation dominates the outflow energy injection, which is a remarkable feature in comparison to the BH accretion. Also, the nature of the circum-NS disk affects the accretion and the induced outflow feedback. The effects of NS mainly work in the cases of larger $R_{\rm{NS}}$ and smaller $M$, where the NS mass inflow rate is relatively small, and thereby the disk is weak to confront the magnetosphere. Larger index $s$ results in more disk mass losing into wind and hence weaker disk pressure at the inner region, leading to a more outside magnetospheric truncation; but even so, fewer mass eventually flows onto NS surface, releasing less accretion energy compared with the smaller $s$ case (comparing first and third panels in Figure 6).

Similar to BH, the outflow cavity also forms in the NS accretion system. As shown in Figure 7, NS spends most of time in the underdense cavity going through inefficient accretion, and hence the averaged mass accretion rate is markedly reduced, of which the general trend is akin to the BH case. As discussed above, more energy is injected into the outflow, which results in more powerful feedback on the AGN disk environment and longer duration of the cavity evolution. Accordingly, the mass rate reduction is indeed more prominent with the existence of a NS hard surface and a strong magnetic field (comparing all three columns in Figure 7). The enhanced reduction is more effective at larger $R_{\rm{NS}}$, where the AGN disk density is lower to block the cavity expansion, i.e., an additional energy injection would lead to a punchier expansion; on the contrary, when the NS is closer to SMBH, the environment is denser to block the cavity expansion, hence the cavity size would not change significantly though injecting additional energy. In addition, the more powerful $L_{\rm{acc}}$ for the cases of small $M$ and large $R_{\rm{NS}}$ causes the outflow feedback and the mass accretion rate of NS with strong magnetic fields a remarkable change. (comparing first and second columns in Figure 7).

In Section 3.1, we assume that the shocked shell and outflow undergo rapid depressurization after the breakout from the AGN disk, and then the ringlike shell evolves driven by the momentum of wind and itself successively. However, the AGN disk usually holds plane-parallel stratified structure (e.g. Grishin et al., 2021), rather than a steep cutoff at the height $H$; even so, the shocked gas would break out and reduce pressure at $\sim H$ (e.g. Schiano, 1985; Mac Low & McCray, 1988; Olano, 2009), hence we think our assumption can roughly describe the processes of wind-surroundings interaction. The uncertainty lies at the duration of the hot gas pushing the shell, where we simplistically link the outflow-energy and -momentum driven phase directly at $t_{\rm{bre}}$. In fact, during the depressurization, the shell is still pushed laterally by the thermal pressure. On the other hand, the shocked gas perhaps experiences effective radiation cooling soon before or after the shell breakout, or even when the shell still expanding deeply in the AGN disk. Since the actual structure of the AGN disk is complex, and the shell evolution needs to contain the concrete cooling mechanisms of the shocked gas, which requires precise hydrodynamical simulations and is beyond the scope of this work, we instead consider two simplified and extreme cases, evolution following the adiabatic or momentum conservation phase in the spherically symmetric medium all the while, to study the effects of the duration of depressurization and the gas cooling. Corresponding descriptions of the evolution are shown in Appendix C.

We choose $t_{\rm{acc}}/(t_{\rm{cav}}+t_{\rm{ref}})$ to typically reflect the features of CO accretion and outflow evolution, as shown in Figure 8. The cavity still comes into being in the two extreme cases, so we confirm that the formation of cavity and the reduction of averaged CO accretion rate are common and inevitable, regardless of the cooling efficiency. Comparing different treatments, the pure momentum conservation case leads to the largest $t_{\rm{acc}}/(t_{\rm{cav}}+t_{\rm{ref}})$, i.e., the weakest outflow feedback; correspondingly, cavity expansion is strongest in the entirely adiabatic case. Namely, the longer duration of the thermal energy pushing the shocked shell, the more powerful expansion of the cavity.

When estimating the strength of the disk winds, i.e. Equation (12), we set $r_{in}\sim 10r_{g}$ on account of the theoretical studies; but various numerical simulations show different inner radii where the outflow is observably produced, ranging $10-100r_{g}$ (e.g. Jiang et al., 2014; Yang et al., 2014; Sa̧dowski et al., 2015; Kitaki et al., 2021). Setting the larger inner radius, the disk winds would be weaker, with $L_{w}^{\prime}/L_{w}\sim(r_{in}^{\prime}/r_{in})^{1-s}$. To study the selection of a larger $r_{in}$, we decrease $f_{w}$ as a substitution, as shown in Figure 8, finding that the feedback is assuredly weaker, but still the cavity forms and the averaged CO accretion rate is significantly reduced.

We also verify the validity of our description on the outflow cavity evolution. We have assumed that the relevant environment parameters, mainly $\rho_{\rm{d}}$, are fixed during the outflow-AGN disk interaction; but if the size of cavity is too large, the AGN disk properties would vary with radius $R$ to affect the cavity evolution, our description are thus unfaithful. Safely, the cavity is always much smaller than the AGN disk radius, i.e. $r_{\rm{cav}}\ll R_{\rm{CO}}$, as shown in left panel of Figure 9, so the fixed-environment assumption holds and our description is a feasible estimation. Moreover, since the half-width of the outflow cavity exceeds the CO gravity radius, as shown in Figure 4, the shear motion of the AGN disk gas would exert ram pressure to affect the shell evolution (e.g. Rozyczka et al., 1995), which limits the horizontal expansion width of the ring when its velocity decreases to the shear velocity, i.e., $\dot{r}_{\rm{sh}}\simeq(3/4)\tilde{\Omega}_{\rm K}r_{\rm{sh}}$ (Moranchel-Basurto et al., 2021). Using Equation (26), the maximum half-width of the cavity is given by

where the width is multiplied by 2 from the results of numerical simulation (Moranchel-Basurto et al., 2021). As shown in Figure 9, we find that $r_{\rm{shear}}\sim r_{\rm{cav}}$, hence we think taking the shear into account can markedly deform the cavity structure, but would not significantly change the results in Section 3.

It is worth mentioning that the late evolution phase of ring driven by momentum is simplified, i.e., we ignore the ambient pressure $\sim\rho_{\rm d}\tilde{c}_{s}^{2}$ and assume that the efficient accretion stops after $t_{\rm{acc}}$ in Equation (23) and (26); in practice, the accretion of CO would last for a few $t_{\rm{acc}}$ and launch winds to continuously push the shell. Meanwhile, there exists a stand-off radius defined by the pressure balance $\rho_{\rm w}v_{\rm w}^{2}=\rho_{\rm d}\tilde{c}_{s}^{2}$ (e.g. Schiano, 1985), i.e.,

when the half-width of the cavity becomes larger than $r_{\rm{sta}}$, the ambient pressure would be the dominant external force to decelerate the shell. We compare the stand-off radius with the cavity half-width, as shown in Figure 9, and find that $r_{\rm{sta}}\sim r_{\rm{cav}}$, which implies that the wind is strong enough to push the shell, i.e., Equation (23), and the ring evolution is mainly driven by its own momentum at around $r_{\rm{cav}}(r_{\rm{sta}})$, i.e., Equation (26); in other words, our simplified model can roughly describe the shell evolution. As an aside, the dropping disk wind would hinder the ambient inflow and maintain a cavity with decreasing $r_{\rm{sta}}$ if the CO hyper-Eddington accretion still persists during the cavity refilling phase, which can lengthen $t_{\rm{ref}}$ to further reduce the averaged accretion rate of CO. All in all, the long-term evolution of the circum-CO disk and the cavity are vital for the understanding of the CO accretion in the AGN disk, further researches are therefore needed.

We have considered the effect of underdense gap when calculating the BHL inflow rate; as shown in left panel of Figure 10, the gas density is deeply reduced by the CO gravity for smaller SMBH mass and closer CO location cases. When studying the outflow feedback we briefly ignore the gap structure, instead we assume the shell expanding within the unperturbed AGN disk environment, which may underestimate the size of cavity and the reduction of averaged mass accretion rate because the shell can expand more easily in the underdense gap. Nevertheless, since the size of cavity $r_{\rm{cav}}$ is general larger than the gap half-width $R_{\rm{d,gap}}$ (the pink line in Figure 10), the assumption is roughly suitable except for the very light SMBH and very close CO location cases.

Note that the structure of underdense gap is commonly studied in a stable state, but the evolution of outflow cavity is dynamical and $r_{\rm{cav}}>R_{\rm{d,gap}}$ generally holds, so the properties of the gap environment around the CO may be affected by the outflow. The timescale of gap opening can be evaluated by the viscous timescale, i.e. $t_{\rm{gap,v}}\simeq 0.1R_{\rm{d,gap}}^{2}/(\alpha\tilde{c}_{s}H)$ (where we estimate the deep gap opening timescale being one-tenth of the timescale for the whole gap to attain the steady state, as shown by Figure 13 in Kanagawa et al. 2017), and the deep gap half-width can be set to $\Delta r\simeq 0.08(m_{\rm{CO}}/{M})^{1/2}(H/R_{\rm{CO}})^{-3/4}\alpha^{-1/4}R_{\rm{CO}}$ (Kanagawa et al., 2018). Tagawa et al. (2022) alternatively suggests the gap opening timescale can be calculated via the angular momentum of gas within the annular gap removed by the gravitational torque of CO, i.e. $t_{\rm{gap,g}}=\Delta J/T_{\rm{CO}}$, where $\Delta J\sim 2\pi R_{\rm{CO}}^{2}\Delta r^{2}\tilde{\Omega}_{\rm{K}}\Sigma_{\rm{d}}$ and $T_{\rm{CO}}\sim(m_{\rm{CO}}/{M})^{2}(H/R_{\rm{CO}})^{-3}R_{\rm{CO}}^{4}\tilde{\Omega}_{\rm{K}}^{2}\Sigma_{\rm{d}}$. Setting the cavity refilling timescale as $t_{\rm{ref}}=\Delta r/\tilde{c}_{s}$, we find that $t_{\rm{ref}}<t_{\rm{gap}}$ holds within wide parameter space, i.e., $t_{\rm{gap,v}}/t_{\rm{ref}}=10(m_{\rm{CO},1}/M_{8})^{1/2}(H/R_{4})^{-7/4}(\alpha_{-1})^{-5/4}$ or $t_{\rm{gap,g}}/t_{\rm{ref}}=110(m_{\rm{CO},1}/M_{8})^{-3/2}(H/R_{4})^{13/4}(\alpha_{-1})^{-1/4}$, so we boldly speculate that the deep gap around the CO has no enough time to be completely rebuilt to affect the efficient accretion of CO and $\dot{M}_{\rm{obd}}$ would be enhanced, because after the faster refilling of gas with density $\rho_{\rm{d}}$ from the AGN disk to the cavity within $t_{\rm{ref}}$, the circum-CO disk has already being built by the denser gas before the density being completely reduced to the lower $\rho_{\rm{d,gap}}$ within $t_{\rm{gap}}$; conversely, the deep gap would be markedly rebuilt for the very light SMBH and very close CO location cases with $t_{\rm{gap,g}}<t_{\rm{ref}}$, as shown by the purple line in Figure 10.

Not only the outflow would impede the build of deep gap around CO, the gap itself may not open for different AGN disk models. So we show the cases of which no gap opens in right panel of Figure 10. The most prominent feature is that the outflow feedback is more effective when $M$ is smaller, and hence more dramatically reduces the averaged CO mass accretion rate, which is in contrast to the gap-opening case as shown in Figure 5, because the gap effect is more noteworthy in these parameter regions. Overall, the outflow feedback likely impacts the gap around CO; we suggest that the CO accretion system in the AGN disk carried by lighter SMBH are useful to investigate whether the gap opens and the properties of the potential gap.

Since the density of environment is extremely large, the mass rate captured by the CO set in the AGN disk is hyper-Eddington, as shown in Figure 1, which even would be higher than the accretion rate of the SMBH itself; in other words, just one embedded BH (NS) can exhaust the mass supply of AGN disk if the BH (NS) accretion persists the AGN lifetime. Tagawa et al. (2022) highlighted a trouble as “ Depletion problem ”, i.e., the AGN disk gas can be depleted by COs if there is no feedback.

We argue that the AGN disk would not be starved by the accretion of embedded COs after the outflow feedback processes as described in Section 3.1 taken into account. First, due to the existence of outflow, the mass accretion rate onto CO, as shown in Figure 2, which is the actual loss of the AGN disk mass, is far below the initial mass inflow rate of circum-CO disk and the AGN accretion rate. Meanwhile, the existence of outflow cavity reduces the averaged circum-CO disk mass inflow rate akin to the reduction of averaged CO mass accretion rate, i.e.,

As shown in left panel of Figure 11, $\dot{M}_{\rm{in,red}}$ is generally lower than AGN accretion rate with nearly all inflow ejected into outflow $\dot{M}_{\rm{w}}\sim\dot{M}_{\rm{in,red}}$, thereby even though all the circum-CO disk winds escape from the SMBH, the AGN disk is still capable to feed the SMBH. Besides, as shown in right panel of Figure 11, the wind velocity $v_{\rm{w}}$ just leaving the circum-CO disk, i.e., Equation (19), is already generally lower than the escape velocity of the SMBH, $v_{\rm{esc}}=\sqrt{2GM/R_{\rm{CO}}}$. The parts of outflow moving along the AGN disk are eventually re-injected into the disk; the parts moving vertically would be dramatically decelerated by the dense environment, though breaking out, velocity of which is conceivable to be lower than $v_{\rm{w}}$, thus would eventually fallback into the AGN disk. To sum up, CO accretion with induced outflow feedback would just result in few mass loss, thus the AGN disk can live for a long time.

An additional process extracting the spin energy of an accreting BH via the formation of a jet would come into being when the BH is rotating, i.e. the Blandford-Znajek mechanism (Blandford & Znajek, 1977), the power of which $P_{\rm{BZ}}\propto B_{\perp}^{2}a_{\rm{BH}}^{2}$, where $B_{\perp}$ is the magnetic field crossing the BH horizon and $a_{\rm{BH}}$ is the BH’s dimensionless spin parameter. By two valid methods, we can estimate the magnetic field strength around the BH: one is the winds produced by hyper-accreting circum-BH disk stretch the disk local magnetic fields to make large-scale poloidal components, or the AGN disk environment directly provides the ordered poloidal magnetic fields, both of which are continuously advected inwards and accumulate at the vicinity of BH, ultimately leading to a magnetically arrested disk (Kimura et al., 2021a; Kaaz et al., 2022), the maximum jet power is $L_{\rm{MAD}}\sim 2.2a_{\rm{BH}}^{2}\dot{M}_{\rm{BH}}c^{2}$ for $h=0.5$ (e.g. Davis & Tchekhovskoy, 2020; Curd & Narayan, 2023); another is via building pressure balance $P_{\rm{mag}}=\alpha_{\rm{CO}}P_{\rm{disk}}$ at the inner boundary of the circum-BH disk with the assumption that magnetic fields are local, the jet power is $L_{\rm{BZ}}\sim 1/64\ a_{\rm{BH}}^{2}\dot{M}_{\rm{BH}}c^{2}$ (e.g. Wang et al., 2021b). The relative strength between the wind, Equation (12), and the jet is

the ratio of which is nearly independent of AGN disk parameters $M$ and $R_{\rm{CO}}$, and mainly dependent on index $s$; for instance, $L_{\rm{w}}/L_{\rm{MAD}}\sim 0.03,\ 0.06,\ 0.18$ or $L_{\rm{w}}/L_{\rm{BZ}}\sim 4.3,\ 8.5,\ 25.6$ for $a_{\rm{BH}}=0.7$ and $s=0.2,\ 0.5,\ 0.8$ cases. Two estimations of the jet power cause inversely relative strength to the wind, so a more physical description of the magnetic field around BH is important to determine the jet power and the relevant feedback strength, and needs to be detailedly studied.

The jet-cocoon evolution can also generate mechanical feedback to suppress the accretion of BH in the AGN disk (Kaaz et al., 2021; Tagawa et al., 2022). Assuredly, $L_{\rm{MAD}}>L_{\rm{w}}$ may cause jet other than wind dominants the feedback. However, the jet is highly anisotropic with narrow opening angle, and thus the size of cocoon cavity is smaller than $r_{\rm{Hill}}$ (Figure 1 in Tagawa et al. 2022); as a contrast, we confirm that the wind outflow is nearly isotropic, and open a cavity much larger than $r_{\rm{Hill}}$, as shown in Figure 4. Meanwhile, for the accretors being a BH with low spin $a_{\rm{BH}}$ or a NS and when the magnetic field around BH is weak, the jet is feeble or even absent, leading to the feedback dominated by wind outflow. Therefore, we suggest that the jet feedback would play a secondary but non-negligible role, and for the high spin BH case, a more complete feedback mechanism containing both jet and wind needs to be built.

We also estimate the radiation features of the wind-outflow-driven shocked shell during breakout, which may provide an opportunity to observe the cavity evolution. Following Kimura et al. (2021b), the luminosity during shell breakout can be achieved from the photon energy released within the photon diffusion timescale, and the blackbody temperature of the breakout photon can be estimated from the shock jump condition, i.e.,

and

where $a$ is the radiation constant; the variations of $L_{\rm{bre}}$ and $T_{\rm{rad}}$ under various parameters are shown in Figure 12. As can be seen, the breakout radiation emerges as a soft X-ray/UV/optical transient depending on the CO location; meanwhile, the AGN emission is also powerful in X-ray/UV/optical bands (Shang et al., 2011). By comparing $L_{\rm{bre}}$ with the AGN bolometric luminosity $L_{\rm{AGN}}\simeq 0.1\dot{M}_{\rm{AGN}}c^{2}$, we loosely infer that the breakout events are general dimmer than the background emission, even though the extreme $s=0.2$ case, so the AGN disk would not be outshone. We confirm that the outflow cavity breakout events taking place in the AGN disk are generally unobservable. As in Wang et al. (2021a), the subsequent interaction between the outflow bubble and the broad-line region gas may generate $\rm{TeV}$ flare, which can be detected in non-blazar AGNs.

There are two noteworthy features of the outflow cavity surrounding CO: CO is commonly harbored in the cavity environment because of the always satisfied $t_{\rm{acc}}\ll(t_{\rm{cav}}+t_{\rm{ref}})$, and the gas density of cavity is much lower than the otherwise unperturbed AGN disk. These features would significantly affect the CO evolution and the CO-related events in AGN disk.

When embedded in the AGN disk, CO can not only capture the gas within its gravity sphere but also gravitational interact with the large-scale disk gas; correspondingly, the whole AGN disk gas exerts gravitational torques onto CO from the Lindblad and corotation resonances, leading to the CO migration (e.g. Ward, 1997), of which the timescale is expressed as (e.g. Kanagawa et al., 2018):

where $K=(m_{\rm{CO}}/M)^{2}(H/R_{\rm{CO}})^{-5}\alpha^{-1}$. Though the cavity with large size and long duration must modify migration, the Lindblad resonance primarily originates from the whole background gas (e.g. Armitage, 2007), most of which are unaffected to provide torque, and during the cavity refilling process $t_{\rm{ref}}$, AGN disk gas can enter the cavity to gravitationally interact with the CO, again producing torque; so we ignore the effect of outflow cavity on migration for simplicity. Because CO mainly stays in the underdense cavity, the averaged mass accretion rate of CO reduces significantly, and thereby the mass growth rate of CO reduces as well. We estimate the CO mass growth timescale containing outflow feedback as $t_{\rm{grow}}=m_{\rm{CO}}/\dot{M}_{\rm{CO,red}}$ and the timescale omitting outflow as $t_{\rm{nowind}}=m_{\rm{CO}}/\dot{M}_{\rm{obd}}$. By comparing relevant timescales as shown in Figure 13, we find that the outflow feedback markedly alters the nature of CO evolution. If not bringing in outflow, the extremely large mass accretion rate results in a violent mass growth, $t_{\rm{nowind}}\ll t_{\rm{mig}}$, thus the CO evolves in situ without migration. Conversely, the formation of outflow and the concomitant feedback lead to a moderate mass growth, basically $t_{\rm{grow}}>t_{\rm{mig}}$, so the CO can observably migrate before significantly becoming heavier. In a word, the accretion-induced outflow strangles the mass growth of CO.

As have already been mentioned, a variety of EM events involving CO are likely to take place in the AGN disk. Since the striking difference between the cavity and AGN disk environment (e.g. density, temperature, opacity), EM events would possess distinguishable features when they separately occur in the two environments. Propagating within the unperturbed dense AGN disk, the jet, ejecta, and outflow generated by the EM events undergo rapid deceleration, which can produce characteristic radiations. On the other hand, as the cavity is much more tenuous than the AGN disk but still denser than the interstellar medium, these jet, ejecta, and outflow would not be significantly decelerated but still interact with the cavity medium (e.g. Yuan et al., 2022), the produced EM emissions would propagate within the cavity environment as well (e.g. Kimura et al., 2021b), which may thus lead to characteristic signals differing from the ones in an unperturbed AGN disk or in a classical interstellar medium. Also, as the underdense cavity has a limited size, matter penetrating along the AGN disk will firstly collide with cavity and then AGN disk gas, causing a latter effective loss of the kinetic energy and thereby latter brightening, which may apply to the events with wide-angle ejecta and outflow, as a star tidally disrupted by a stellar mass BH (e.g. Perets et al., 2016; Yang et al., 2022), or BH-NS and NS-NS mergers (e.g. Zhu et al., 2021a; Ren et al., 2022). A more precise description of the cavity structure and the properties of various EM events occurring in the cavity-AGN disk environment are left in a follow-up work.

Tagawa et al. (2020a) showed that single-single gas-capture channel dominates the binary formation in the AGN disk, where the gas dynamical friction acting on CO, which is approximately proportional to the gas density surrounding CO, i.e., $a_{\rm{GDF}}\propto\rho_{\rm{CO}}$ (Ostriker, 1999), effectively removes the binary binding energy during the two bodies encountering in their mutual Hill radius, and thereby a CO binary forms. However, as COs are mainly harbored in the cavity environment, of which the gas density is much lower than the otherwise unperturbed AGN disk and the size is much larger than the Hill radius, as shown in Figure 4, the gas dynamical friction can be significantly weakened, $a_{\rm{GDF,cav}}\sim O(10^{-4}-10^{-2})\ a_{\rm{GDF}}$, hence the efficiency of single-single gas capture channel may be markedly reduced. In brief, the outflow feedback plays an important role and should not be neglected when studying the binary formation in the AGN disk.

CO can possess a large bulk velocity relative to the AGN disk gas, i.e. $v_{\rm{bulk}}\gg\tilde{c}_{s}\ and\ v_{\rm{shear}}$, e.g., the orbiting CO in the galactic nucleus holding nonzero inclination w.r.t. the disk plane will periodically cross the AGN disk with considerable perpendicular velocity (e.g. Fabj et al., 2020; Nasim et al., 2022), or the remnant of a binary CO merger will be ejected at large recoil velocity (e.g. Campanelli et al., 2007), or the CO can leave triple system with large escape velocity after the chaotic interaction (Valtonen & Karttunen, 2006), and so on. In the case of supersonic motion, CO triggers the formation of a bow shock, producing a downstream overdense shocked gas wake, which drags the CO and is accreted onto it (e.g. Antoni et al., 2019). Meanwhile, the BHL mass inflow rate is still hyper-Eddington, e.g. $\dot{M}_{\rm{BHL}}=1.6\times 10^{3}\dot{M}_{\rm{Edd}}(m_{\rm{CO}}/M_{\odot})(\rho_{\rm{d}}/10^{-10}\,{\rm g\,cm^{-3}})(v_{\rm{bulk}}/10^{8}\,{\,\text{cm}\text{ s}^{-1}})^{-3}$, so the radiation-driven outflow should emerge, the feedback of which results in the redistribution of bow shock separating the outflow and the surrounding gas, and an underdense wake region may be built, depending on the outflow geometry (Li et al., 2020). The accretion and dynamical friction on CO, which are important for the evolution of CO supersonically moving within AGN disk and the capture of CO by AGN disk (Fabj et al., 2020; Nasim et al., 2022), would be significantly affected by the potential outflow (Gruzinov et al., 2020; Li et al., 2020; Bosch-Ramon, 2022; Kaaz et al., 2022).

However, whether an outflow really forms and the concrete mechanism of its ejection from the supersonically moving CO have not been well studied, much less in the particular AGN disk environment. For BHL accretion with large $v_{\rm{bulk}}$, the inflow would be highly quasi-spherical (e.g. El Mellah & Casse, 2015), so no region is left to release outflow, and photons are trapped and advected inward with inflow at hyper-Eddington rate (e.g. Begelman, 1979; Blondin, 1986). Vast mass and energy ultimately flow into the horizon for the case of BH, or eventually hit the hard surface and potentially release huge energy for the case of NS (e.g. Shakura et al., 2015). On the other hand, the initial infalling gas may carry a small but unnegligible amount of angular momentum depending on the AGN disk properties, and thus an accretion disk may form around CO to launch winds via the mechanism discussed in Section 2.2; a self-consistent evolution process should come into being (probably alike Wang et al. 2021a).

Since the accretion feedback of CO with a large bulk velocity is vital but poorly understood, we would prepare another separate work to study the launch mechanisms and the properties of outflow; the interaction between the outflow and the AGN gas medium, the effects of outflow on CO accretion process and dynamical drag with the induced acceleration/deceleration of CO, the ability of AGN disk on capturing the crossing CO when considering accretion feedback, and the relevant EM radiations would be studied as well.