Accretion-modified Stars in Accretion Disks of Active Galactic Nuclei: Observational Characteristics in Different Regions of the Disks

Stars and compact objects embedded in accretion disks of active galactic nuclei (AGNs), dubbed accretion-modified stars (AMSs), often experience hyper-Eddington accretion in the dense gas environment, resulting in powerful outflows as the Bondi explosion and formation of cavities. The varying gas properties across different regions of the AGN disk can give rise to diverse and intriguing phenomena. In this paper, we conduct a study on the characteristics of AMSs situated in the outer, middle, and inner regions of the AGN disk, where the growth of the AMSs during the shift inward is considered. We calculate their multiwavelength spectral energy distributions (SEDs) and thermal light curves. Our results reveal that the thermal luminosity of the Bondi explosion occurring in the middle region leads to UV flares with a luminosity of ∼1044 erg s−1. The synchrotron radiation of Bondi explosion in the middle and inner regions peaks at the X-ray band with luminosities of ∼1043 and ∼1042 erg s−1, respectively. The γ-ray luminosity of inverse Compton radiation spans from 1042–1043 erg s−1 peaked at the ∼10 MeV (outer region) and ∼GeV (middle and inner regions) bands. The observable flares of AMS in the middle region exhibit a slow rise and rapid Gaussian decay with a duration of months, while in the inner region, it exhibits a fast rise and slow Gaussian decay with a duration of several hours. These various SED and light-curve features provide valuable insights into the various astronomical transient timescales associated with AGNs.


INTRODUCTION
In light of high metallicity in broad-line regions (BLRs) of active galactic nuclei (AGNs; Hamann & Ferland 1999), stellar evolution and compact objects in AGN accretion disks were originally realized by Artymowicz et al. (1993) and Cheng & Wang (1999) for metal production in BLRs, γ-ray bursts (GRBs), and gravitational waves (GWs), respectively.There is fast-growing evidence for such a scenario of compact objects in the accretion disks from LIGO detections of GWs.The detection of GWs from the merger of binary black holes (BBHs) with masses heavier than typical stellar BHs, particularly the GW190521 event (Abbott et al. 2020), has potentially indicated that AGN disks could serve as sites for BBH mergers.Utilizing observations from the Zwicky Transient Facility, Graham et al. (2020) reported the first potential optical electromagnetic counterpart (EMC) candidate (ZTF19abanrhr) for the GW event S190521g, which could originate from a BBH merger within an AGN disk.Although the potential association between these two astronomical transients remains a subject of ongoing debate (e.g., Ashton et al. 2021;Palmese et al. 2021;Morton et al. 2023), there is emergence of more appealing evidence from Graham et al. (2023) who subsequently reported nine EMC candidates for BBH mergers detected by LIGO/Virgo during the O3 run.Additionally, Lazzati et al. (2023), Levan et al. (2023), andTagawa et al. (2023c) claimed several observed GRBs possibly occurring in an AGN disk.Li et al. (2023a) and Han et al. (2024) proposed that some GW events may originate from a BBH merger in AGN disks.Moreover, it would be a giant step to understand AGN phenomena of fueling the central supermassive black holes (SMBHs) and metallicity  We consider AGN disks of SMBHs with mass M p and accretion rates Ṁp .It is convenient to use the dimensionless accretion rates of Ṁp = Ṁp c 2 /L Edd, p , where L Edd, p = 4πGM p m p c/σ T is the Eddington luminosity of the SMBH, G is the gravitational constant, m p is the proton mass, c is the speed of light, σ T is the Thompson scattering cross section.In this paper, we discuss the case of Ṁp ∼ 1, namely, SMBH is accreting in the regime of Shakura-Sunyaev model (Shakura & Sunyaev 1973).We consider the simple case that the sMBH is trapped in and corotates with the disk, as shown in Figure 1.Then, its dimensionless Bondi accretion rate is described by the Hoyle-Lyttleton-Bondi formulation (Hoyle & Lyttleton 1939;Bondi 1952) and modified in thin disks (Kocsis et al. 2011), and the Bondi radius is given by where M s is the mass of the sMBH, ρ p is the gas density around the sMBH, c s is the speed of sound, H p is the half-thickness of the SMBH disk, R Hill = (M s /3M p ) 1/3 R is the Hill radius of the sMBH, R is the locus of the sMBH in the SMBH disk, is the relative velocity between the sMBH and the gas at the Hill radius or Bondi radius, Ω K = GM p /R 3 is the Keplerian angular velocity.The subscripts "s" and "p" refer to the secondary (i.e., sMBH) and primary BH (i.e., SMBH) of the binary systems.The second term of Equation (1) accounts for the geometry effect of the SMBH disk, indicating that the Bondi accretion will be suppressed by a factor of (H p /R Bon ) if the half-thickness of the SMBH disk is smaller than the Bondi radius R Bon .The third term represents the tidal effect of the central SMBH, suggesting that only the gas located within the Hill radius will undergo accretion onto the sMBH.Generally, Bondi accretion onto the sMBH in the standard disk is hyper-Eddington due to the dense gas environment (e.g., Wang et al. 2021a), resulting in the development of powerful outflows (e.g.Ohsuga et al. 2005;Jiang et al. 2014;Wang et al. 2021a).The outflow power is where η out is the conversion efficiency of channeling gravitational energy into outflows, η rad ≈ 0.1 denotes the radiation efficiency, f adv ≈ 0.9 is the advection fraction, and f acc is the fraction of the accretion rate onto the sMBH to the Bondi accretion rate (see Equation (A2)).In fact, f adv is quite uncertain in consideration of advection, photon trapping, and outflows(e.g., see the simulations in Takeuchi et al. 2009).
These powerful outflows will significantly impact the local structures of the SMBH disk, generating a cavity (Wang et al. 2021a,b;Kimura et al. 2021) with a radius denoted as R cav , provided that the kinetic energies of the outflows exceed the local dissipation rates of gravitational energy within the SMBH disk (Wang et al. 2023c), i.e., L out ≳ 4πR 3 cav ε + /3, where ε + ≈ Q + /H p is the volume dissipation rates of the SMBH disk, Q + = 3GM p Ṁp /8πR 3 is surface heating rates (Frank et al. 2002), and we disregard the factor of the inner boundary condition of accretion disks.Based on this condition, the maximum of R cav can be determined as In the outer region of the self-gravitating disks, we demonstrate that the outflows are momentum-driven since the diffusion timescale t diff is much shorter than the expansion timescale t out , as shown in Table 2.So the outflow velocity v out and its expansion timescale t out satisfy the momentum equation M out v out /t out = L out /c (King 2003) and kinematic equation R out = v out t out , respectively, where R out = min{R Bon , R Hill , H p , R cav } is the outflow radius when the mass falling from SMBH disk is halted, and M out = 4πR 3 out ρ p /3 is the gas mass within R out .Then, we obtain While in the inner and middle regions, the energy-driven outflows are developed because the diffusion timescale of the photons is much longer than the expansion timescale.So the outflow velocity v out and its expansion timescale t out satisfy the energy equation L out t out = M out v 2 out /2, and the kinematic equation R out = v out t out .Then, we obtain Here, we approximate M out as the total mass of the outflow given that the initial outflow mass from the sMBH disk M ini = (1 − f acc ) Ṁs t out ≲ M out during the time interval of t out and can be ignored (see later calculations in § 3).The gravity of the gas within the Bondi radius may enhance the Bondi accretion (Wandel 1984;Kocsis et al. 2011), which depends on the ratio of M out /M s .The accretion onto the sMBH lasts for a timescale of t last = max{t out , t vis }, determined by t out and the viscosity timescale of the sMBH disk (t vis , see Equation A18).Here, we compare the model with the case of extremely low accretion rates (ADAF disk, Ṁp ≪ 1) discussed in Wang et al. (2023c) and point out the difference that the role of external pressure of the SMBH disk is unimportant in our model.In § 3, we demonstrate that only a small fraction of the kinetic energy is utilized to work against the external pressure of the SMBH disk.Because the gas density in the standard disk and self-gravitating disk (see Equations ( 16), ( 27), (38), and Figure 5) is much higher than that in ADAF disk (∼ 10 −14 g cm −3 , Wang et al. 2023c) while the temperature is much lower than that in the ADAF disk (∼ 10 10 K, Narayan & Yi 1995), the kinematic energy of outflows is much larger than the thermal energy (i.e., the work used to overcome the external pressure) after the outflows rush out of the SMBH disk.
If the velocity of the outflow exceeds the local sound speed, a shock will emerge and heat the gas in the SMBH disk (e.g., Blandford & Eichler 1987).The temperature of the shocked gas can be determined using the Rankine-Hugoniot jump conditions, where the adiabatic index Γ ad = 5/3, and k B is the Boltzmann constant.The heated gas will quickly expand as its gas pressure increases significantly after being shocked.Once the pressure equilibrium between the shocked gas and SMBH disk is attained, ρ sh k B T sh /m p = ρ p c 2 s , a cavity is formed (Wang et al. 2021b).We then obtain the density of the shocked gas LIU ET AL.
The hot gas in the cavity will lose its thermal energy through free-free cooling within a timescale of where j ff is the free-free emission coefficient.After cooling, the cavity will be refilled under the pressure of the SMBH disk on the rejuvenation timescale of (Wang et al. 2023c) where v turb = αc s is the turbulence velocity.After the cavity is refilled with cold gas from the SMBH disk, the above physical process will restart, indicating episodic accretion.This is very similar to the cases of BH seed growth at high redshift (Wang et al. 2006) and accretion onto intermediate mass BHs in dense protogalactic clouds (Milosavljević et al. 2009).Apart from the above episodic Bondi explosion, another important feature of the sMBH is the migration from the outer to inner regions due to the torque of the SMBH disk.Here, we consider two timescales: the viscosity timescale of the SMBH disk and the type I migration timescale given by respectively, where v ′ r is the radial velocity of the SMBH disk, (Tanaka et al. 2002), γ σ = −d ln Σ p /d ln R, and Σ p = 2ρ p H p is the surface density.The real migration timescale is t mig = min(t ′ vis , t Lind ).

AMS IN THE STANDARD DISK AND SELF-GRAVITATING DISK
In this paper, we employ the self-gravitating disk model in the outer region (Sirko & Goodman 2003) and standard model in the middle and inner regions (Shakura & Sunyaev 1973).The disk becomes self-gravity dominated beyond a critical radius R SG determined by the Toomre parameter Q = Ω K c s /πGρ p H p = 1 (Toomre 1964), which yields R SG /R g = 1.2 × 10 3 α 28/45 0.1 M −52/45 8 Ṁ −22/45 p (e.g., Wang et al. 2021a).

Outer Region
In the outer region beyond R SG = 1.2 × 10 3 R g , the disk becomes self-gravitating.As described by Equations (B13), (B14), (B15), and (B17) in Appendix B, the half-thickness, gas density, sound speed of the SMBH disk, and radial velocity are where α 0.1 = α/0.1 is the viscosity parameter, r 5 = R/10 5 R g , and M 8 = M p /10 8 M ⊙ denotes the mass of the central SMBH.By substituting the disk parameters into Equations (1) and (2), we can estimate the dimensionless Bondi accretion rate and Bondi radius as follows: where q6 = q/10 −6 , q = M s /M p = 10 −6 m 2 /M 8 , and m 2 = M s /10 2 M ⊙ .First, the Bondi radius is much smaller than the disk half-thickness, so the geometry effect can be ignored.Second, the Hill radius of the sMBH is R Hill = (M s /3M p ) 1/3 R ≈ 1.0 × 10 16 cm, much larger than the Bondi radius, so the tidal effect of the SMBH is also marginal.Third, the differential velocity ∆v = Ω K R Bon /2 = 2.0 × 10 5 cm s −1 , is about 1 order of magnitude lower than the sound speed, indicating that the modification to the spherical accretion can also be ignored.Finally, the sphere mass within the Bondi radius is M Bon = t(days) The spectral energy distributions of the powerful outflows developed by the AMS in the outer region of the SMBH disk and the blackbody radiation from the SMBH disk.The integrated blackbody radiation from the SMBH disk is shown in green.AGN disk gas is shocked by accretion-powered outflows and emits thermal radiation (purple) with an effective temperature of T eff .The shock formed from the strong outflows accelerates electrons in AGN disk, emitting synchrotron emissions (red) and generating high-energy photons by IC scattering (blue).Right: The thermal light curve of the Bondi explosion, described by Equation ( 50), which behaves like a Heaviside step function since the inflow and outflow coexist (see Figure 1).The rise timescale and the duration are t rise ≈ t eff and t last ≈ tout, respectively.The related parameters are listed in Table 2.
4πR 3 Bon ρ p /3 = 5.0 M ⊙ , much smaller than the sMBH mass, implying that its gravity only has a marginal effect on enhancing the accretion process.
By replacing size ∆R in Equation (A1) with R Bon , we obtain the outer radius of the sMBH disk as corresponding to 7.9 × 10 6 r g , where r g = GM s /c 2 is the gravitational radius of the sMBH.This gives an accretion fraction f acc ≈ 8.7 × 10 −4 .However, the formation of the sMBH disk can be quenched by the turbulence with a size scale of l turb ∼ αc s /Ω K ∼ αH p ∼ 2.3 × 10 15 cm, comparable with R Bon .The chaotic turbulence leads to a low AM material accretion onto the sMBH (Chen & Lin 2023) and gives rise to a small-sized sMBH disk as indicated by its AM expression ∆ℓ s = √ X out GM s .The simulations of Bondi-Hoyle accretion in a turbulent medium show that turbulence reduces the accretion rate by a factor of a few, possibly dependent on the magnetic field and the Mach number of the accreting gas (e.g., Lee et al. 2014;Burleigh et al. 2017).In consideration of the above uncertainty of the turbulence, we set f acc ∼ 0.1 and define a parameterized conversion efficiency η out .With Equation (4), we obtain the conversion efficiency η out ≈ 10 −3 .Based on Equations ( 3) and ( 17), the sMBH experiences a hyper-Eddington accretion process and forms powerful outflow with where η3 = η out /10 −3 .With the condition that the power of the outflow exceeds the local heat production rate of the AGN disk, the cavity radius is given by Equations ( 5) and ( 20) as R cav = 1.7 × 10 18 cm, much larger than the disk half-thickness.This indicates that the outflow can rush out of the disk and significantly change the local disk structure.In this case, the maximum cavity radius is The gas mass within the cavity is given by M cav = 4πR 3 cav ρ p /3 = 2.4×10 2 M ⊙ .When the accretion is halted, the outflow radius is R out = min{R Bon , R Hill , H p , R cav } = R Bon = 6.2 × 10 15 cm and the outflow mass is M out = 4πR 3 out ρ p /3 = 5.0 M ⊙ .Based on Equations ( 6), ( 7), and (20), we derive the velocity of the outflow and its expansion timescale as The energy of outflows is E out = L out t out = 3.4 × 10 51 erg.We calculate the energy required to work against the external pressure and the AMS gravity.First, the energy used to overcome the pressure of the SMBH disk is E p = 4πP R 3 out /3 = 1.0 × 10 48 erg, much smaller than E out , indicating that it is easy to overcome the disk pressure and the effect can be ignored.Second, the energy used to overcome the gravity of the AMS is E g = G(M out + M s )M out /R out = 2.2 × 10 46 erg.These two considerations verify that almost all of the dissipated energy of the sMBH disk is transformed into the kinetic energy of the outflow E out .We also calculate the initial mass of outflows M ini ≈ (1 − f acc ) Ṁs t out = 1.7 M ⊙ ≲ M out , validating the previous approximation of ignoring it in Equations ( 22) and ( 23).
The velocity of the outflow is about 1 order of magnitude larger than the sound speed, inevitably leading to shock formation and the generation of relativistic electrons and nonthermal radiation, as shown in Figure 2.Meanwhile, based on Equations ( 10) and ( 22), the gas will be heated to a high temperature of T sh = 3.0 × 10 5 K by the strong shock.Utilizing the pressure equilibrium condition between the shocked gas and SMBH disk in Equation ( 11), we derive the density of shocked gas ρ sh = 8.5 × 10 −16 cm −3 , much lower than the density of the SMBH disk gas.After the outflow rushes out of the SMBH disk, it will form a cavity of shocked gas with high temperature and low density.The hot gas loses its thermal energy mainly by free-free cooling with a cooling timescale of t cool = 2.2 d given by Equation ( 12), much shorter than the cavity expansion timescale t out .
After the hot gas in the cavity is cooled quickly by free-free emission, the cavity is refilled with the surrounding cold gas turbulence.Based on Equations ( 13) and ( 21), the cavity can be refilled in a rejuvenation timescale of The rejuvenation is much slower than the cavity formation and leads to episodic accretion.Although the sMBH disk size is much smaller than the result of Equation ( 19), we calculate the lower limit of the radial velocity of the sMBH disk and the upper limit of the viscosity timescale based on Equations (A17), (A18), and ( 19), which is smaller than the expansion timescale t out .This means that the inflow and outflow coexist, as shown in Figure 1.The sMBH disk gas will keep accreting onto the sMBH for a timescale of t last = max{t out , t vis } = t out , until the outflow expands to the Bondi radius.The corresponding duty cycle is δ = t last /(t last + t rej ) = 3.5 × 10 −3 .The average mass growth timescale of the sMBH is t grow = M s /f acc δ Ṁs ṀEdd,s = 2.6 × 10 6 yr.Using Equations ( 14) and ( 15), we obtain the viscosity timescale of the SMBH disk t ′ vis = 1.4 × 10 7 yr, and type I migration timescale t Lind = 1.1 × 10 7 yr, respectively, where γ σ = −d ln Σ p /d ln R = 3/2 is used.The real migration timescale of the sMBH is t mig = t Lind .For an exponentially growing case, the sMBH mass can grow up to M ′ s = exp(t Lind /t grow )M s = 69.0M s .

Middle Region
In the middle region of the standard disk, gas pressure prevails over radiation pressure, and the electron scattering optical depth is significantly larger than the absorption.According to the standard model, the radius of the middle region is 3.0 (Svensson & Zdziarski 1994), and the half-thickness, density, midplane temperature, sound speed, and radial velocity of the SMBH disk are (Kato et al. 2008) where r 3 = R/10 3 R g .
As discussed in § 3.1, the sMBH grows heavier due to the episodic accretion during the migration.We, therefore, consider an sMBH with 10 3 M ⊙ mass in the middle region.By substituting the disk parameters into Equations ( 1) and (2), we can estimate the dimensionless modified Bondi accretion rate and Bondi radius as follows: R Bon = 2.7 × 10 15 r 3 M 8 q where q5 = q/10 −5 , q = M s /M p = 10 −5 m 3 /M 8 , and m 3 = M s /10 3 M ⊙ .First, the Bondi radius is significantly larger than the SMBH disk half-thickness, leading to a suppression of the Bondi accretion rate by about 2 orders of magnitude.Second, the Hill radius, given by R Hill = (M s /3M p ) 1/3 R ≈ 2.2×10 14 cm, is also smaller than the Bondi radius, resulting in a suppression of the Bondi accretion rate by 1 order of magnitude.Third, we calculate the differential velocity ∆v = Ω K R Hill /2 = 7.1 × 10 6 cm s −1 , which is approximately three times larger than the sound speed.This suggests that the accretion deviates from spherical Bondi accretion.So we ignore the sound speed when calculating Ṁs and R Bon .Finally, the gas mass within the Hill radius is M Bon = 2πH p R 2 Hill ρ p /3 = 6.4 M ⊙ , much smaller than the sMBH mass, indicating that the enhancement effect on the accretion rate from the gas self-gravity within the Bondi sphere is marginal.
By substituting ∆R in Equation (A1) with R Hill , we can determine the outer radius of the sMBH disk as This corresponds to 1.2 × 10 5 r g , resulting in an accretion fraction of f acc ≈ 6.9 × 10 −3 .Utilizing Equation (4), we derive a conversion efficiency of η out ≈ 6.9 × 10 −5 .Here, we would like to point out that the turbulence scale l turb ∼ αH p ∼ 0.1H p is much smaller than the size of accretion region min{R Bon , R Hill , H p } = H p , so the turbulence will not affect the formation of sMBH disk and can be ignored.Based on Equation ( 3) and ( 28), the sMBH forms outflows with a power given by where η4 = η out /10 −4 .Based on Equations ( 5) and ( 31), the powerful outflows interact with the SMBH disk and form a cavity with a radius of R cav = 6.5 × 10 15 cm, which is much larger than the half-thickness of the SMBH disk.This indicates that the outflow can rapidly rush out of the SMBH disk and significantly alter its local structure.Therefore, the maximum cavity radius is (32) The gas mass within the cavity is M cav = 4πR 3 cav ρ p /3 = 0.34 M ⊙ .When the accretion is halted, the outflow radius is R out = min{R Bon , R Hill , H p , R cav } = H p = 3.6 × 10 13 cm and the outflow mass is M out = 4πR 3 out ρ p /3 = 0.34 M ⊙ .With Equations ( 8), (9), and (31), the velocity of the outflow and its expansion timescale are derived as The energy of outflows is E out = L out t out = 2.8 × 10 49 erg.We calculate the energy required to work against the external pressure and the AMS gravity.First, the energy used to overcome the pressure of the SMBH disk is E p = 4πP R 3 out /3 = 3.7 × 10 45 erg.Second, the energy used to overcome the gravity of the AMS is E g = GM out M s /R out = 2.5 × 10 48 erg.These two kinds of energies are much smaller than the outflow energy E out , indicating that the influence of SMBH disk pressure and AMS gravity on the outflow expansion could be ignored.We also calculate the initial mass of outflows M ini ≈ (1−f acc ) Ṁs t out = 0.15 M ⊙ ≲ M out , which verifies the validity of the former approximation of ignoring it in Equations ( 33) and (34).
The outflow velocity is about 2 orders of magnitude larger than the local sound speed, which gives rise to powerful shocks when the outflows collide with the SMBH disk gas.The shocks would accelerate the electrons and lead to nonthermal emissions, as can be seen in Figure 3.Meanwhile, the shock will heat the SMBH disk gas to an extremely high temperature T sh = 1.9 × 10 8 K given by Equations ( 10) and ( 33).Utilizing the pressure equilibrium condition between the shocked gas and SMBH disk based on Equation ( 11), we derive the density of the shocked gas, ρ sh = 1.2 × 10 −12 g cm −3 , about 3 orders of magnitude lower than the density of the SMBH disk gas.With Equation ( 12), the cooling timescale of the shocked gas is given by t cool = 3.4 × 10 3 s, much shorter than the cavity expansion timescale, indicating that the free-free cooling mechanism is very efficient.
After the hot gas in the cavity is rapidly cooled, the cavity, based on Equations ( 13) and (32), will be refilled in a rejuvenation timescale of t rej = 4.9 α −1 0.1 r This rejuvenation is much slower than the cavity expansion.Based on Equations (A17), (A18), and (30), we can derive the radial velocity and the viscosity timescale of the sMBH disk, The viscosity timescale is much longer than the expansion timescale t out .This means that after the outflows rush out of the SMBH disk, the accretion onto AMS still continues and lasts for a timescale of t last = max{t out , t vis } = t vis .The duty cycle of the accretion process is δ = t last /(t last + t rej ) = 3.4 × 10 −2 .The average net mass accretion rate of the sMBH is Ṁave = f acc δ Ṁs ṀEdd,s = 4.2 × 10 3 ṀEdd,s .The average mass growing timescale of the sMBH is t grow = M s / Ṁave = 1.1 × 10 5 yr.
Using Equations ( 14) and ( 15), we obtain the viscosity timescale of the SMBH disk, t ′ vis = 7.9 × 10 5 yr, and type I migration timescale, t Lind = 6.4 × 10 2 yr, respectively, where γ σ = −d ln Σ p /d ln R = 3/5 is used.The real migration timescale of the sMBH is t mig = t Lind ≪ t grow , indicating that the sMBH can hardly grow up before migrating to the inner region.

Inner Region
In the inner region of a standard disk, radiation pressure prevails over gas pressure, and the electron scattering depth is significantly greater than absorption.According to the standard model, the inner region radius is R/R g ≤ 3.0 × 10 2 (α 0.1 M 8 ) 2/21 Ṁ 16/21 p (Equation ( 13) or (25) in Svensson & Zdziarski 1994), and the half-thickness, density, midplane temperature, sound speed, and radial velocity of the SMBH disk are (Kato et al. 2008) where r 1 = R/10 R g is the radius of the disk from the SMBH.As discussed in § 3.2, the sMBH hardly grows, due to its quick migration.We, therefore, still consider an sMBH with 10 3 M ⊙ mass in the inner region.Substituting the disk parameters into Equations ( 1) and ( 2), the Bondi accretion rate and Bondi radius can be estimated as Ṁs = 3.9 × 10 5 α −1 0.1 r 1 q To derive Equations ( 39) and ( 40), four conditions are considered.First, the Hill radius R Hill = (M s /3M p ) 1/3 R ≈ 2.2×10 12 cm, which is about 1 order of magnitude smaller than the Bondi radius.This indicates that the tidal effect of SMBH must be considered.Second, the Bondi radius is about 1 order of magnitude larger than the SMBH disk half-thickness, which means that the SMBH disk geometry suppresses the accretion rate.Third, the differential velocity between the gas in the Hill radius and the sMBH is expressed as ∆v = Ω K R Hill /2 = 7.1×10 7 cm s −1 , which is comparable with the sound speed and not considered in this case just for a rough estimation.Finally, the gas mass within the Hill radius is M Bon = 2πH p R 2 Hill ρ p /3 = 1.3×10 −4 M ⊙ ≪ M s , indicating that the enhancement effect from the self-gravity of Bondi sphere gas is marginal.
The outer radius of sMBH disk X out can be estimated through the conservation of AM based on Equation (A1).By setting ∆R = R Hill , we derive This corresponds to 1.2 × 10 3 r g .With Equation (A2), the accretion fraction f acc ≈ 6.9 × 10 −2 is obtained.With Equation ( 4), the conversion efficiency η out ≈ 6.9×10 −4 is derived.Like the middle region case, the turbulence scale l turb ∼ αH p ∼ 0.1H p is much smaller than the size of accretion region min{R Bon , R Hill , H p } = H p , so the uncertainty of f acc introduced by turbulence can be ignored.Based on Equation ( 3) and ( 39), the power of the outflow is given by where η3 = η out /10 −3 .Based on Equations ( 5) and ( 42), the cavity radius is derived R cav = 1.3 × 10 13 cm.The derived cavity radius is much larger than the SMBH disk half-thickness, which indicates that the outflow can significantly change the local disk structure.In this case, the maximum cavity radius is The gas mass within the cavity volume is M cav = 4πR 3 cav ρ p /3 = 8.0 × 10 −5 M ⊙ .When the accretion is halted, the outflow radius is R out = min{R Bon , R Hill , H p , R cav } = H p = 1.2 × 10 12 cm and the outflow mass is M out = 4πR 3 out ρ p /3 = 8.0 × 10 −5 M ⊙ .Using equations ( 8), (9), and (42), we obtain the velocity of the outflow and the cavity expansion time The energy of outflows is E out = L out t out = 6.6 × 10 46 erg.We calculate the energy required to work against the external pressure and the AMS gravity.First, the energy used to overcome the pressure of the SMBH disk is E p = 4πP R 3 out /3 = 1.0 × 10 45 erg.Second, the energy used to overcome the gravity of the AMS is E g = GM out M s /R out = 1.7 × 10 46 erg.These two kinds of energies are much smaller than the outflow energy E out , indicating that the influence of SMBH disk pressure and AMS gravity on the outflow expansion could be ignored.We also calculate the initial mass of outflows M ini ≈ (1−f acc ) Ṁs t out = 3.4 × 10 −5 M ⊙ ≲ M out , which verifies the validity of former approximation of ignoring it in Equations ( 44) and ( 45).
The velocity of outflow is about 1 order of magnitude larger than the sound speed, which inevitably leads to the formation of shock when the outflow collides with the surrounding gas medium.The powerful shock accelerates the thermal electrons to relativistic state, which gives rise to nonthermal emissions due to the synchrotron radiation and IC scattering processes, as can be seen in Figure 4.Meanwhile, the shock will also heat the SMBH disk gas with an extremely high temperature of T sh = 1.9 × 10 9 K, given by Equations ( 10) and ( 44).This can lead to the generation of relativistic jet if the sMBH is rapidly spinning (Rees et al. 1982).Using the pressure equilibrium condition between the shocked gas and SMBH disk based on Equation ( 11), we derive the density of shocked gas ρ sh = 8.0 × 10 −10 g cm −3 , much lower than the density of the SMBH disk gas.With Equation ( 12), the cooling timescale of the shocked gas can be derived t cool = 16.4 s, much shorter than the cavity expansion timescale, which implies that the free-free cooling mechanism is very efficient.
After the hot gas in the cavity is cooled quickly, the cavity, based on Equations ( 13) and ( 43), will be refilled in a rejuvenation timescale of much longer than the cavity formation timescale.Based on Equations (A17), (A18), and (41), the radial velocity and the viscosity timescale of the sMBH disk are derived, The viscosity timescale is much longer than the expansion timescale t out .This means that after the outflows rush out of the SMBH disk, the accretion onto AMS still continues and lasts for a timescale of t last = t vis .The duty cycle of the accretion process is δ = t last /(t last +t rej ) = 4.2×10 −2 .The average dimensionless net mass accretion rate of the sMBH is Ṁave = f acc δ Ṁs = 1.1×10 3 , which indicates that mass growing timescale of sMBH is much shorter than the Salpeter time t Salp = M s / ṀEdd = 0.45 Gyr.However, the sMBH in the inner region will merge with the SMBH quickly due to the GW radiation (see Equation ( 24) in Wang et al. 2023c), which prevents the sMBH mass from growing to be too large.After the sMBH disk is consumed, the cavity will quickly cool in a timescale of t cool and the accretion process restarts.

Thermal emission
Now, we calculate the thermal light curves of the Bondi explosion.The diffusion timescale of the photons is given by t diff = κM ej /β 0 cR cav , where M ej = M cav is the ejecta mass, and a value of β 0 = 13.7 is utilized to accommodate various density profiles of the diffusion mass (Arnett 1982).For AMS in the outer region, we set κ = 1.0 for simplicity, ignoring its complex dependence on the radius, as shown in Figure 5.For AMSs in the middle and inner regions, the opacity is determined by electron scattering and κ = 0.4.Since the post-shock medium is quite hot (T sh ∼ 10 5 , 10 8 , 10 9 K in the outer, middle, and inner regions, respectively), free-free absorption, bound-free and bound-bound absorption could not be important in the main phase of the Bondi explosion.We may include these effects for the later phase of the explosion.The cavity expands with a timescale of t out , which is much shorter than the diffusion timescale, as indicated in Table 2. Therefore, an effective light-curve timescale t eff = √ 2t diff t out is introduced (Chatzopoulos et al. 2012).The electromagnetic emissions from the Bondi explosion will emerge within the timescale of t eff .We calculate the thermal light curves for a simple form of the input luminosity L inp = L out θ(t last − t), where θ(t last − t) denotes the Heaviside step function.More detailed interaction processes between the outflows and SMBH disk gas, such as the reverse and forward shock, are beyond the scope of this study.The output luminosity of the homologously expanding photosphere is described by Equation (3) in Chatzopoulos et al. (2012), The initial radius is set as the outflow radius R out , and the initial internal energy can be disregarded.We then obtain the light curves of the thermal emission, which exhibit an increase over a timescale of t last and a Gaussian decay over the effective light-curve timescale t eff .The luminosity reaches its peak after a time interval of t last as , the detailed calculated results of which are listed in Table 2.
Due to the highly effective free-free cooling of the hot gas within the cavity, the expanding shell loses its energy in the form of blackbody radiation with an effective temperature T eff = (L peak /4πR 2 cav σ SB ) 1/4 , where σ SB is Stefan-Boltzmann constant.The blackbody spectra are plotted in purple (see Figures 2-4).The thermal luminosity L peak ≈ 10 43 erg s −1 of Bondi explosion in the outer region peaks in the infrared band, slightly higher than that of the AGN disk.In the middle region, the thermal luminosity of L peak ≈ 10 44 erg s −1 of the Bondi explosion greatly exceeds that of the SMBH disk in the UV band, while the Bondi explosions occurring in the inner region could lead to soft X-ray flares with a luminosity of L peak ≈ 10 43 erg s −1 .

Nonthermal Emissions
As discussed in § 3, the velocity of the outflows exceeds the local sound speed, leading to the development of shocks (Blandford & Eichler 1987) and the generation of nonthermal electrons and emissions (Amano et al. 2022).Following the approach of Inoue & Takahara (1996) and utilizing a simplified homogeneous one-zone model framework, we compute the broadband SED of synchrotron radiation and IC scattering.It is worth noting that, unlike the jet case, there is no beaming effect in the case of outflows, and thus, the beaming factor is set to unity.
The thermal electrons are accelerated to a relativistic state due to shock acceleration (see, e.g., Drury 1983;Blandford & Eichler 1987).The acceleration timescale is given by t acc = 20ξ acc R L c/3v 2 out (Inoue & Takahara 1996), where R L = γm e c 2 /eB denotes the Larmor radius, m e is the electron mass, and ξ acc is a factor characterizing the acceleration efficiency that depends on the acceleration environment.For instance, ξ acc ∼ 1 (referred to as the Borm limit) in supernova remnant (Uchiyama et al. 2007), while in blazars, ξ acc can reach up to 10 7 (Inoue & Takahara 1996).In the context of AMS, where the shock is not relativistic, we take ξ acc ∼ 1.The magnetic field is derived as B = (32πa/3) 1/2 T 2 c , under the assumption of equipartition with the radiation energy density (Burbidge 1956), where a = 4σ T /c is the blackbody radiation constant.Note that the midplane temperature T c in the outer region of AGN disk is approximately a constant 10 4 K (see Figure 5).The nonthermal electrons lose their energy, due to IC scattering in a cooling timescale of t IC = 3m e c/4σ T γu ph , where γ is the Lorentz factor of the nonthermal electrons, and u ph = aT 4 c is the energy density of the local seed photons radiated by the SMBH disk.By utilizing the condition t acc = t IC , we derive the maximum Lorentz factor γ max .The energy spectrum of the nonthermal electrons is given by dN/dγ = (1 − p e )N 0 γ −pe /(γ 1−pe max − γ 1−pe min ), where N 0 is the total number density obtained by integrating dN/dγ from γ min to γ max .Here, we set γ min = 1 and the spectral index p e = 2. Now, we calculate the SED of the nonthermal radiation.First, we consider the IC scattering, where the photons from the SMBH disk are scattered by the nonthermal electrons and transformed into γ-rays.By setting the IC scattering luminosity as L IC = ζL out , we can derive the number density of the nonthermal electrons N 0 .Here, the parameter ζ representing the typical fraction of nonthermal emissions generated by shocks is quite uncertain (e.g., Blandford & Eichler 1987), so we set ζ ∼ 0.1.The number density of the seed photons is estimated as n ν = 4πB ν /hνc, where B ν = 2hν 3 /c 2 (exp(hν/k B T c ) − 1) is the blackbody spectrum.Subsequently, we calculate the SEDs of the synchrotron radiation with the magnetic field B and number density of nonthermal electrons N 0 .The calculated SEDs of the IC scattering and synchrotron radiation are plotted in Figures 2-4, with parameters listed in Table 2.

DISCUSSION
AMSs in the inner (middle) regions display quasi-periodic eruptions (QPEs) with a duration of several hours (months) and a period of several days (years), providing valuable insights into the QPE with various periods (e.g., Evans et al. 2023;Guolo et al. 2024).In fact, the AMS model has been applied to Sgr A * (Wang et al. 2023c), successfully explaining its quasi-periodic flickerings observed in the near-infrared band.
The light curves of thermal luminosity for Bondi explosions occurring in different regions of SMBH disk exhibit a broad range of duration timescales, spanning from several hours to decades at different bands from infrared to soft X-ray band.In the outer region, the diffusion timescale of Bondi explosion is much shorter than the expansion timescale, causing the light curve to behave like the Heaviside step function, as shown in Figure 2. The thermal luminosity can be estimated as L bol (t) ≈ L out and lasts for a time interval of decades.The thermal luminosity of Bondi explosions occurring in the middle region exhibits a slow rise and rapid decay (see Figure 3).Conversely, in the inner region, the thermal luminosity of the Bondi explosion experiences a rapid rise and slow decay (see Figure 4).These diverse and intriguing features show that AMSs could be accountable for astronomical transients of varying durations.Indeed, various atypical transients have been documented in recent years.For example, Ofek et al. (2021) reported an optical transient, AT 2018lqh, with a duration on the scale of days attributed to an explosion of lowmass ejecta (≈ 0.07 M ⊙ ), which is comparable to those of the AMS in the middle region if the sMBH mass and its location are appropriately adjusted.While in the X-ray band, Khamitov et al. ( 2023) presented an SRG/eROSITA X-ray catalog with significant proper motions explained by the presence of transient events, supporting the idea of AMSs being the possible physical origin of these transient events.
The SEDs of Bondi explosions occurring in different regions of the SMBH disk exhibit various features.The synchrotron radiations of AMS in the outer region could span from ∼ 10 GHz to optical bands.The ratio of radio to optical luminosity is ∼ 10 −5 , which could explain part of the radio emissions of radio-quiet AGNs.Recent radio sky surveys, such as the Very Large Array Sky Survey (Gordon et al. 2021) at 2-4 GHz, and the LOw-Frequency ARray Two-metre Sky Survey (Shimwell et al. 2022) at 120-168 MHz, can be helpful in the search for radio transients from Bondi explosion.These catalogs are compared with older sky surveys, such as the NRAO VLA Sky Survey (Condon et al. 1998), the Sydney University Molonglo Sky Survey (Mauch et al. 2003), and Faint Images of the Radio Sky at Twenty Centimeters (Helfand et al. 2015), to study the long timescale radio variability (Nyland et al. 2020).In fact, long-term radio variability (e.g., Hovatta et al. 2008;Park & Trippe 2017;Zhang et al. 2022) and some radio transients from, such as the Caltech-NRAO Stripe 82 Survey (Mooley et al. 2016) with variability timescales between 1 week and 1.5 yr, the Variables and Slow Transients Survey on the Australian Square Kilometer Array Pathfinder (Murphy et al. 2021) with variability timescales from 5 s to 5 yr, have been reported for a considerable number of AGNs.These radio transient events are very helpful to search for the AMSs.
Another interesting feature of the AMS SED is the high-energy γ-ray photons (peaked from approximately 10 MeV to GeV).For AMSs located in the middle regions of the SMBH disk, the γ-ray luminosity is estimated to be around 10 43 erg s −1 , corresponding to a flux of 3.7 × 10 −13 erg s −1 cm −2 for a nearby AGN with redshift z = 0.1, which is comparable with the sensitivity of the Fermi/LAT at the GeV band with 10 years of observation 1 .It is worth noting that the γ-ray luminosity is determined by several physical parameters, for example, as described in Equations ( 30) and ( 31), a higher sMBH mass results in a shorter size of the sMBH disk and more powerful outflows, leading to higher γ-ray luminosity and flux.On the other hand, the stacking technique of γ-rays presented in Paliya et al. (2019) and Ajello et al. (2021) will be useful for detecting fainter γ-rays below the Fermi/LAT sensitivity.In fact, significant γ-ray detections have been reported for some nearby low-luminosity active galactic nuclei (LLAGN; de Menezes et al. 2020).It is easy for the AMSs to outshine from the LLAGN, although SMBHs in LLAGN generally accrete with low, sub-Eddington accretion rates (Ho 2008), which influence the AMS luminosity (see Equations ( 20), (31), and ( 42)).

CONCLUSION
In this study, we investigate the AMSs embedded in different regions of the disk surrounding the SMBHs, namely, the inner region (typical radius R = 10 R g ), the middle region (typical radius R = 10 3 R g ), and the outer region(typical radius R = 10 5 R g ).In the inner and middle regions, the Toomre parameter, Q > 1, and the standard model (Shakura & Sunyaev 1973) is utilized.In the outer region, Q ≈ 1, the self-gravitating disk model (Goodman 2003;Sirko & Goodman 2003) is employed.Bondi explosions in these regions exhibit both similarities and distinct characteristics, depending on various gas environments, primarily including gas density, sound speed, and half-thickness of the SMBH disk.The main findings are summarized below: (1) The main physical processes are similar for AMSs embedded in the inner and middle regions of the SMBH disk.The AMS experiences hyper-Eddington accretion (∼ 10 6 − 10 7 L Edd,s /c 2 ), resulting in the development of strong outflows that collide with the SMBH disk, generating shocks, heating the SMBH disk gas, accelerating electrons to relativistic state, and emitting nonthermal radiation.In the inner and middle regions, after the powerful outflow rushes out of the SMBH disk, a cavity is formed, but it quickly cools and is then refilled by the surrounding cool gas of the SMBH disk.While in the outer region, the inflows of Bondi accretion and outflows from Bondi explosion coexist and can last for several decades, as shown in Figure 1.
(2) We compute the thermal light curves for a constant input luminosity within the lasting timescale of the outflows.The results show that the flare of the Bondi explosion in the inner region displays fast rise and slow Gaussian decay, with a timescale of several hours and a luminosity of ∼ 10 43 erg s −1 peaked at the soft X-ray band.While in the middle region, the Bondi explosion exhibits slow rise and rapid Gaussian decay, with a timescale of months and a luminosity of ∼ 10 44 erg s −1 peaked at the UV band.In the outer region, the light curve of Bondi explosion resembles the Heaviside step function, lasting for decades and contributing a slightly higher luminosity of ∼ 10 43 erg s −1 than the AGN disk itself in the infrared band.These light curves provide valuable insights into the diverse astronomical transient events associated with AGNs.
(3) We calculate the multiwavelength SEDs of the Bondi explosion from radio to γ-ray bands.The γ-rays luminosity of IC ranges from 10 42 −10 43 erg s −1 in the GeV (middle and inner regions) and 10 MeV (outer region) bands, respectively.Moreover, the radio emission due to synchrotron radiation from Bondi explosion occurring in the outer region can contribute to that of the radio-quiet AGNs or result in long-term radio variability over a timescale of several decades.The synchrotron radiation of Bondi explosion in the middle and inner regions peaks at the X-ray band with luminosities of ∼ 10 43 and ∼ 10 42 erg s −1 , respectively.IHEP AGN Group members are acknowledged for useful discussions.Many thanks to Fu-Lin Li for useful discussions about the supernova light-curve calculations.We acknowledge financial support from the National Key R&D Program of China (2021YFA1600404), the National Natural Science Foundation of China (NSFC-11991050, -11991054,-12333003).
LIU ET AL.
(B5)-(B11).The key physical quantity, opacity κ determined by the temperature and density of the disk gas, can not be expressed analytically (Alexander & Ferguson 1994).So, we derive their numerical solutions and plot them in Figure 5.

Figure 1 .
Figure 1.Bondi explosion of AMSs embedded in an AGN disk.AMSs in the inner and middle regions generate powerful outflows and form hot cavities, whereas for the AMS in the outer region, outflows and inflows coexist.

LFigure 3 .
Figure3.The same as the Figure2but for the middle region in the SMBH disk.Left: The blackbody radiation of the SMBH disk is shown in green.The shocked SMBH disk gas emits thermal radiation (purple) with an effective temperature of T eff .Nonthermal electrons accelerated through shocks emit synchrotron emissions (red) and generate high-energy photons by IC scattering (blue).Right: The thermal light curve experiences slow rise and rapid decay with timescales of t rise ≈ t last and t decay ≈ t eff , respectively.

Table 1 .
Studies on the Stars and Compact Objects in AGN Disks.
) The same as Figure3but for the outer in the SMBH disk.Left: The blackbody radiation of the SMBH disk is shown in green.The shocked SMBH disk gas emits thermal radiation (purple) with an effective temperature of T eff .Nonthermal electrons accelerated through shocks emit synchrotron emissions (red) and generating high-energy photons by IC scattering (blue).Right: The thermal light curve increases rapidly within a timescale of t rise ≈ t last and decreases in a timescale of t decay ≈ t eff , in contrast to the middle region case.

Table 2 .
Explanations and typical values of some key parameters.