Accretion-modified Stars in Accretion Disks of Active Galactic Nuclei: Slowly Transient Appearance

Considering that black holes formed from massive stars are kicked off with random velocities of ${v}_{\bullet }^{{\prime} }\approx {10}^{2}-{10}^{3}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ with respect to their remnant (the black holes are still tightly bound by the SMBHs), accretion onto the black holes in SMBH disks can be described by the Hoyle–Lyttleton–Bondi (HLB) formulation. Neglecting the weak dependence on adiabatic index and self-gravity of the accreted gas (Wandel 1984), we take the simplest form

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{Bon}}=\displaystyle \frac{4\pi {G}^{2}{m}_{\bullet }^{2}{\rho }_{{\rm{d}}}}{{\left({c}_{s}^{2}+{v}_{\bullet }^{2}\right)}^{3/2}},\end{eqnarray} \tag{ 3 }$$

where m_• is the black hole mass from supernova explosion and v_• is the kick-off velocity relative to the motion of the SMBH disk. The kick-off velocity of the black holes is uncertain, but it could be up to ${v}_{\bullet }^{{\prime} }\sim {10}^{3}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ if comparable with the case of neutron stars (e.g., Nakamura et al. 2019). We consider two extreme scenarios, in which the black hole is corotating and counterrotating with the SMBH disk. If ${v}_{\bullet }^{{\prime} }\sim {V}_{{\rm{K}}}\,=3000\,{r}_{4}^{-1/2}\,\mathrm{km}\,{{\rm{s}}}^{-1}$, with V_K the Keplerian velocity, we expect two populations of accreting black holes in the disk: corotating black holes with ${v}_{\bullet }\approx | {V}_{{\rm{K}}}-{v}_{\bullet }^{{\prime} }| \ll {c}_{s}$ and counterrotating ones with ${v}_{\bullet }\,\approx {V}_{{\rm{K}}}+{v}_{\bullet }^{{\prime} }\approx 2{V}_{{\rm{K}}}\gg {c}_{s}$ (see Section 2.2). It should be stressed that the current choice of kick-off velocities are just for the two possible regimes, and is not necessarily representative of the bulk population. The corresponding accretion rates are

$$\begin{eqnarray}&&\begin{array}{l}{\dot{m}}_{\mathrm{Bon}}=\displaystyle \frac{{\dot{M}}_{\mathrm{Bon}}}{{\dot{M}}_{\mathrm{Edd}}}\,\\ \approx \,\left\{\begin{array}{ll}8.9\times {10}^{8}\,{m}_{1}{\left({\alpha }_{0.1}{M}_{8}\right)}^{-2/5}{\dot{M}}^{1/10}{r}_{4}^{-3/4} & (\mathrm{for}\,\mathrm{corotation}),\\ 5.1\,{m}_{1}{\left({\alpha }_{0.1}{M}_{8}\right)}^{-7/10}{\dot{M}}^{11/20}{r}_{4}^{-3/8} & (\mathrm{for}\,\mathrm{counter}\,\mathrm{rotation}),\end{array}\right.\end{array}\end{eqnarray} \tag{ 4 }$$

and the Bondi radius ${R}_{\mathrm{Bon}}={{Gm}}_{\bullet }/\left({c}_{s}^{2}+{v}_{\bullet }^{2}\right)$,

$$\begin{eqnarray}&&\begin{array}{l}{R}_{\mathrm{Bon}}\\ =\,\left\{\begin{array}{ll}1.2\times {10}^{15}\,{m}_{1}{\left({\alpha }_{0.1}{M}_{8}\right)}^{1/5}{\dot{M}}^{-3/10}{r}_{4}^{3/4}\,\mathrm{cm} & (\mathrm{for}\,\mathrm{corotation}),\\ 3.7\times {10}^{9}\,{m}_{1}{r}_{4}\,\mathrm{cm} & (\mathrm{for}\,\mathrm{counter}\,\mathrm{rotation}),\end{array}\right.\end{array}\end{eqnarray} \tag{ 5 }$$

where m₁ = m_•/10M_⊙ is the mass of the black hole inside the SMBH disk. The Hill radius is ${R}_{\mathrm{Hill}}={({M}_{\mathrm{Bon}}/{M}_{\bullet })}^{1/3}R\,\approx 1.5\times {10}^{15}\,{M}_{\mathrm{Bon},2}^{1/3}{M}_{8}^{2/3}{r}_{4}\,\mathrm{cm}$, which constrains the size of the AMS. Since R_Bon ≈ R_Hill, tidal disruption of the AMS can be avoided. The validity of the HLB accretion is guaranteed by R_Bon ≪ R_ρ , where ${R}_{\rho }={\left|d\mathrm{ln}{\rho }_{{\rm{d}}}/{dR}\right|}^{-1}$ is the density scale of the SMBH disk (i.e., the length that the density changes by an e-folding factor). A schematic illustrating an AMS embedded in an SMBH disk is shown in the left panel of Figure 1.

Zoom In Zoom Out Reset image size

It should be noted that such a hyper-Eddington accretion is much higher than the case in the early universe, where seed black holes only attain a dimensionless accretion rate of ${\dot{m}}_{\bullet }$ ∼ a few hundred (Volonteri & Rees 2005; Takeo et al. 2020; Toyouchi et al. 2021). Radiative feedback to the seed growth dominates in this case (Wang et al. 2006; Milosavljević et al. 2009a, 2009b; Regan et al. 2019). It depends on the Compton temperature, which in turn relies on the hard X-ray spectral slope. The slopes get steeper with increases of accretion rates (Wang & Netzer 2003; Wang et al. 2004), lowering the Compton temperature. Moreover, luminosity saturation of the slim disk decreases the relative emission in hard X-rays (Abramowicz et al. 1988; Wang & Zhou 1999). Considering this fact, radiative feedback becomes weaker with increases of accretion rates. Radiative feedback could be still important for ${\dot{m}}_{\bullet }\sim {10}^{3}$ from numerical simulations (Takeo et al. 2020), but mechanical feedback of outflows could dominate if ${\dot{m}}_{\bullet }\sim {10}^{9}$ discussed in the current cases.

The Bondi mass, which is defined as gas mass within the Bondi radius, can be approximated by ${M}_{\mathrm{Bon}}\approx 4\pi {R}_{\mathrm{Bon}}^{3}{\rho }_{{\rm{d}}}/3$

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}{M}_{{\rm{Bon}}}\\ =\,\left\{\begin{array}{ll}2.4\times {10}^{2}{M}_{\odot }\,{m}_{1}^{3}{\left({\alpha }_{0.1}{M}_{8}\right)}^{-1/10}{\dot{M}}^{-7/20}{r}_{4}^{3/8} & ({\rm{for}}\,{\rm{corotation}}),\\ 7.3\times {10}^{-15}{M}_{\odot }\,{m}_{1}^{3}{\left({\alpha }_{0.1}{M}_{8}\right)}^{-7/10}{\dot{M}}^{11/20}{r}_{4}^{9/8} & ({\rm{for}}\,{\rm{counterrotation}}).\end{array}\right.\end{array}\end{array}\end{eqnarray} \tag{ 6 }$$

We note that the corotating AMSs are much more massive than those counterrotating. According to Equation (2) in Artymowicz et al. (1993), the drag force on an AMS parallel to its path is given by ${F}_{{\rm{d}}}=4\pi {G}^{2}{M}_{\mathrm{AMS}}^{2}{\rho }_{{\rm{d}}}{v}_{\bullet }^{-2}$ for v_• < v_esc, where ${v}_{\mathrm{esc}}\,=\sqrt{2{{GM}}_{\mathrm{AMS}}/{R}_{\mathrm{Bon}}}$ is escaping velocity from the AMS surface, where M_AMS = M_Bon + m_• ≈ (M_Bon, m_•) for corotating and counterrotating AMSs, respectively. Therefore, the slow-down timescale is given by ${F}_{{\rm{d}}}\times ({v}_{\bullet }{t}_{\mathrm{slow}})={M}_{\mathrm{AMS}}{v}_{\bullet }^{2}/2$, we have

$$\begin{eqnarray}{t}_{\mathrm{slow}}=\left\{\begin{array}{cc}14.2\,{M}_{\mathrm{Bon},2}^{-1}{\rho }_{\bar{10}}^{-1}{v}_{100}^{3}\,\mathrm{yr} & (\mathrm{for}\,\mathrm{corotation}),\\ 3.8\times {10}^{6}\,{m}_{1}^{-1}{\rho }_{\bar{10}}^{-1}{v}_{3000}^{3}\,\mathrm{yr} & (\mathrm{for}\,\mathrm{counterrotation}),\end{array}\right.\end{eqnarray} \tag{ 7 }$$

where v_100,3000 = v_•/(100, 3000) km s⁻¹, ${\rho }_{\bar{10}}={\rho }_{{\rm{d}}}/{10}^{-10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. We find that the counterrotating AMSs need much longer times to slow-down than the corotating ones, and they have to spend a significant fraction of the typical AGN lifetime (${t}_{\mathrm{AGN}}\sim R/{v}_{r}\,\sim {10}^{7}\,{r}_{4}{v}_{2}^{-1}\,\mathrm{yr}$, where v₂ = v_r /10² cm s⁻¹ from Equation 1). The randomly kicked BHs may leave the SMBH disk for some cases if v_• t_slow ≳ H, but they are still bound by the SMBH so that they are finally trapped by the disks. As fully random kick-off from supernova explosion, black holes, whose velocities perpendicular to the corotation direction will be dramatically damped according to Equation (7), and finally trapped by the SMBH disks.

Finally, we point out that the spin and angular momentum distribution of the AMSs are complicated issues. On the one hand, the accreted gas spins up the AMSs with an opposite direction to rotation of the SMBH disk since the AMS side near to the SMBH is rotating faster than its outside, namely AMS spins are from differential rotation of the disk. On the other hand, the tidal torque of the SMBH is still sufficiently strong to drive the AMS spin (at the characteristic radius 10⁴ R_g), although the AMSs avoid the tidal disruption. The tidal torque drives the Bondi sphere to corotate with the disk when the Bondi sphere is large enough (like the case of the moon, whose spin follows its orbit around the Earth through the tidal torque of the Earth). Therefore, the outer part of the Bondi sphere generally has low spin because of the action of the negative tidal torque; however, the inner part may have some angular momentum where the torque is insufficiently strong. This could be the reason why there are slim accretion disks around the central black holes in the AMSs. This needs some detailed calculations for the AMSs as done for trapped stars in the SMBH disks by Jermyn et al. (2021), but it is beyond the scope of this paper.

As shown by Equation (4), the black holes of AMSs have extremely high accretion rates ($\sim {10}^{9}\,{\dot{M}}_{\mathrm{Edd}}$), which have never been discussed in the literature (except for accretion of neutrons onto black holes for γ-ray bursts). There are two schools of models of super-Eddington accretion onto black holes: (1) the classical model with so-called slim disks without outflows (Abramowicz et al. 1988; Wang & Zhou 1999) and (2) the model with accretion onto black holes with strong outflows (e.g., Ohsuga et al. 2005). Actual physics could fall between these two schools: photon trapping and outflows coexist (Kitaki et al. 2018). The first generally shows a cold disk with strong photon trapping. Simulations show that super-Eddington accretion produces powerful outflows strongly influencing its surroundings, but the midplane still continues accretion with rates from 300 to 10³ L_Edd/c². The current accretion rates are higher than the cases discussed in Takeo et al. (2020) by five orders. We introduce two parameters in the BH accretion: (1) f_a as a fraction of the Bondi rates contributing to BH growth (Takeo et al. 2020) and (2) f_out(∼1) as a fraction of the Bondi rates to outflows. We have ${\dot{m}}_{\mathrm{grow}}={f}_{{\rm{a}}}{\dot{m}}_{\mathrm{Bon}}$.

The hyper-Eddington accreting black hole of the AMS has never been studied. In principle it should be a self-consistent system under the government of accretion and (radiative and kinematic) feedback. When accretion rates are intermediately super-Eddington compared with the current context, such as 10³ L_Edd/c², radiative feedback to its surroundings dominates and drives the accretion to be episodic (Wang et al. 2006; Milosavljević et al. 2009a, 2009b). If the hyper-Eddington accretion develops very powerful outflows with kinematic luminosity (L_kin), for example, with a power of $\sim \dot{m}{L}_{\mathrm{Edd}}$, the kinematic momentum-driven feedback will dominate. The swept shell of the outflows follows (King 2003)

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}\left({M}_{r}\dot{r}\right)=\displaystyle \frac{{L}_{\mathrm{kin}}}{{v}_{\mathrm{out}}},\end{eqnarray} \tag{ 8 }$$

where M_r is the shell mass inside r and v_out is the velocity of the outflows. The kinematic luminosity during a single accretion episode is given by

$$\begin{eqnarray}&&{L}_{\mathrm{kin}}=\eta {f}_{\mathrm{out}}{\dot{m}}_{\mathrm{Bon}}{L}_{\mathrm{Edd}}=1.3\times {10}^{47}\,{\eta }_{0.1}{m}_{1}{\dot{m}}_{9}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 9 }$$

where ${\dot{m}}_{9}={\dot{m}}_{\mathrm{Bon}}/{10}^{9}$, η is the dissipating efficiency rather than the radiative efficiency and determined by the last stable orbit around black holes. Usually η = 0.1 is taken in the literature. The post-shock gas has a temperature ${{kT}}_{\mathrm{sh}}\,\approx 9{m}_{{\rm{p}}}{v}_{\mathrm{out}}^{2}/16$, and accretion onto the BH is halted when T_sh is higher than the virial temperature ${{kT}}_{\mathrm{vir}}\approx {m}_{{\rm{p}}}{c}^{2}{(r/{r}_{{\rm{g}}})}^{-1}$. We have the condition

$$\begin{eqnarray}&&{v}_{\mathrm{out}}=\displaystyle \frac{4}{3}\displaystyle \frac{c}{\sqrt{{r}_{{\rm{a}}}/{r}_{{\rm{g}}}}},\end{eqnarray} \tag{ 10 }$$

to quench the accretion onto the black hole at a radius r_a of the slim disks. Integrating Equation (8), we have

$$\begin{eqnarray}&&{M}_{r}{v}_{\mathrm{out}}\approx \left(\displaystyle \frac{{L}_{\mathrm{kin}}}{{v}_{\mathrm{out}}}\right){t}_{{\rm{a}}},\end{eqnarray} \tag{ 11 }$$

where t_a is the accretion timescale. Here, we neglect the initial condition in the integration. Since the radial velocity of the hyper-Eddington accretion is ${v}_{r}=\alpha {V}_{{\rm{K}}}/\sqrt{5}$ (see Equation 11 in Wang & Zhou 1999), the accretion may be halted beyond r_a after

$$\begin{eqnarray}&&{t}_{{\rm{a}}}=\displaystyle \frac{{r}_{{\rm{a}}}}{{v}_{r}}.\end{eqnarray} \tag{ 12 }$$

Combining Equations (9)–(12), we have

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{r}_{{\rm{a}}}}{{r}_{{\rm{g}}}}={\left(\displaystyle \frac{16\alpha }{9\sqrt{5}}\displaystyle \frac{{M}_{\mathrm{Bon}}{c}^{3}}{{r}_{{\rm{g}}}{L}_{\mathrm{kin}}}\right)}^{2/5}=1.3\\ \,\times {10}^{5}\,{\eta }_{0.1}^{-2/5}{\alpha }_{0.1}^{13/25}{M}_{8}^{3/25}{\dot{M}}^{-9/50}{r}_{4}^{9/20}\end{array}\end{eqnarray} \tag{ 13 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}{t}_{{\rm{a}}}={\left(\displaystyle \frac{16\sqrt[3]{5}}{9}\displaystyle \frac{{c}^{4/3}{r}_{{\rm{g}}}^{2/3}{M}_{\mathrm{Bon}}}{{\alpha }^{2/3}{L}_{\mathrm{kin}}}\right)}^{3/5}=5.1\\ \,\times {10}^{4}\,{m}_{1}{\eta }_{0.1}^{-3/5}{\alpha }_{0.1}^{-11/50}{M}_{8}^{9/50}{\dot{M}}^{-27/100}{r}_{4}^{27/40}\,{\rm{s}},\end{array}\end{eqnarray} \tag{ 14 }$$

where we take M_r = M_Bon. The results show the necessary accretion timescale to quench the Bondi accretion through the outflows from slim accretion disks. t_a and r_a are not sensitive to the density distribution of the Bondi sphere. Figure 1 (right panel) shows a cartoon of the Bondi explosion driven by the powerful outflows from slim accretion disks with hyper-Eddington rates.

The momentum-driven outflows are able to push the Bondi sphere within t_a, but the cumulative kinematic energies during the period are much larger than the self-gravitational energy of the Bondi sphere so that it is undergoing explosion driven by cumulative energy of the outflows (${E}_{\mathrm{SG}}\approx {{GM}}_{\mathrm{AMS}}^{2}/{R}_{\mathrm{Bon}}\,\approx {10}^{48}{M}_{\mathrm{AMS},2}^{2}{R}_{\mathrm{Bon},15}^{-1}\,\mathrm{erg}$ ≪ E_kin). Figure 2 shows a cartoon of the Bondi explosion from an SMBH disk to the BLR. The cumulative kinematic energies within the sphere is

$$\begin{eqnarray}&&{E}_{\mathrm{kin}}={L}_{\mathrm{kin}}{t}_{{\rm{a}}}=1.3\times {10}^{52}\,{\eta }_{0.1}{m}_{1}{\dot{m}}_{9}{t}_{{\rm{a}},5}\,\mathrm{erg},\end{eqnarray} \tag{ 15 }$$

where t_a,5 = t_a/10⁵ s. We denote this as the Bondi explosion. Energies of one Bondi explosion are equivalent to that of about 10 supernovae. The explosion in the BLR (with medium density ρ_BLR) as a quasi-sphere can be approximately described by the Sedov self-solution. However, the SMBH disk is not a sphere, so we use the adiabatic approximation for the expansion in the disk. Taking the SMBH disk as a slab, its opening solid angle to the Bondi sphere is about ${\epsilon }_{{\rm{d}}}\approx 0.5{\left(H/{R}_{\exp }\right)}^{2}$, where ${R}_{\exp }$ is the expanding radius of the Bondi explosion in the SMBH disk. Using the Sedov solution in BLR and the adiabatic approximation of $4\pi {R}_{\exp }^{2}{{Hn}}_{{\rm{d}}}{{kT}}_{{\rm{d}}}\,={\epsilon }_{{\rm{d}}}{E}_{\mathrm{kin}}$ in the SMBH disk

$$\begin{eqnarray}{R}_{\exp }=\left\{\begin{array}{ll}{\left(\tfrac{{E}_{\mathrm{kin}}}{{\rho }_{\mathrm{BLR}}}\right)}^{1/5}{t}^{2/5}=55.1\,{E}_{52}^{1/5}{n}_{\mathrm{BLR},7}^{-1/5}{t}_{10}^{2/5}\,\mathrm{ltd} & (\mathrm{for}\,\mathrm{BLR}\,\mathrm{medium}),\\ {\left(\tfrac{{{HE}}_{\mathrm{kin}}}{8\pi {n}_{{\rm{d}}}{{kT}}_{{\rm{d}}}}\right)}^{1/4}=9.0\times {10}^{15}\,{E}_{52}^{1/4}{\alpha }_{0.1}^{1/5}{M}_{8}^{9/20}{\dot{M}}^{-7/40}{r}_{4}^{15/16}\,\mathrm{cm} & (\mathrm{for}\,\mathrm{SMBH}-\mathrm{disk}),\end{array}\right.\end{eqnarray} \tag{ 16 }$$

the explosion velocity is

$$\begin{eqnarray}{V}_{\exp }=\left\{\begin{array}{ll}\tfrac{2}{5}{\left(\tfrac{{E}_{\mathrm{kin}}}{{\rho }_{\mathrm{BLR}}}\right)}^{1/5}{t}^{-3/5}=1.8\times {10}^{3}\,{E}_{52}^{1/5}{n}_{\mathrm{BLR},7}^{-1/5}{t}_{10}^{-3/5}\,\mathrm{km}\,{{\rm{s}}}^{-1} & (\mathrm{for}\,\mathrm{BLR}\,\mathrm{medium}),\\ {\left(\tfrac{2{{HL}}_{\mathrm{kin}}}{\pi {n}_{{\rm{d}}}{{kT}}_{{\rm{d}}}{t}_{a}^{3}}\right)}^{1/4}=1.7\times {10}^{6}\,{\alpha }_{0.1}^{1/5}{M}_{8}^{9/20}{\dot{M}}^{-7/40}{r}_{4}^{15/16}{L}_{47}^{1/4}{t}_{{\rm{a}},5}^{-3/4}\,\mathrm{km}\,{{\rm{s}}}^{-1} & (\mathrm{for}\,\mathrm{SMBH}-\mathrm{disk}),\end{array}\right.\end{eqnarray} \tag{ 17 }$$

from $4\pi {R}_{\exp }H({n}_{{\rm{d}}}{{kT}}_{{\rm{d}}}){V}_{\exp }={\epsilon }_{{\rm{d}}}{L}_{\mathrm{kin}}$ for SMBH disk, and the explosion timescale is

$$\begin{eqnarray}{t}_{\exp }=\left\{\begin{array}{ll}{\left(\tfrac{3}{4\pi }\right)}^{5/6}\displaystyle \frac{{M}_{\mathrm{Bon}}^{5/6}}{{\rho }_{\mathrm{BLR}}^{1/3}{E}_{\mathrm{kin}}^{1/2}}=9.9\,{M}_{\mathrm{Bon},2}^{5/6}{n}_{\mathrm{BLR},7}^{-1/3}{E}_{52}^{-1/2}\,\mathrm{yr} & (\mathrm{for}\,\mathrm{BLR}\,\mathrm{medium}),\\ {t}_{{\rm{a}}}/2 & (\mathrm{for}\,\mathrm{SMBH}-\mathrm{disk}),\end{array}\right.\end{eqnarray} \tag{ 18 }$$

from the condition of ${M}_{\mathrm{Bon}}=4\pi {R}_{\exp }^{2}{V}_{\exp }{\rho }_{\mathrm{BLR}}{t}_{\exp }$, and ${t}_{\exp }\,={R}_{\exp }/{V}_{\exp }$ for the SMBH disk, where n_BLR,7 = n_BLR/10⁷ cm⁻³, n_BLR = ρ_BLR/m_p, E₅₂ = E_kin/10⁵² erg, L₄₇ = L_kin/10⁴⁷ erg s⁻¹, t₁₀ = t/10 yr, n₁₄ = n_d/10¹⁴ cm⁻³, and n_d = ρ_d/m_p is the number density of the SMBH disk. The temperature of the shock-swept medium is

$$\begin{eqnarray}&&T\approx \displaystyle \frac{2\left({{\rm{\Gamma }}}_{\mathrm{ad}}-1\right){m}_{{\rm{p}}}}{{\left(1+{{\rm{\Gamma }}}_{\mathrm{ad}}\right)}^{2}k}{V}_{\exp }^{2}=2.3\times {10}^{7}\,{V}_{\exp ,3}^{2}\,{\rm{K}},\end{eqnarray} \tag{ 19 }$$

where ${V}_{\exp ,3}={V}_{\exp }/{10}^{3}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and we take the adiabatic index Γ_ad = 5/3. It should be noted that the expansion velocity of the SMBH disk is relativistic and equivalent to a Lorentz factor ${{\rm{\Gamma }}}_{\exp }\approx {V}_{\exp }/c\approx 5$. A cavity with a radius ${R}_{\exp }$ is formed by the Bondi explosion in the SMBH disk. The small fraction _d(∼10%) of E_kin could be thermalized in the SMBH disk, so we leave the relativistic blast waves as an open topic for the future. The Bondi explosion expands into the BLR medium.

Zoom In Zoom Out Reset image size

Here we would like to stress that the above scenario of the Bondi explosion is the most conservative. The Bondi spheres are actually streaming toward the central black holes, leading to the possibility that the shocks can be enhanced through collision with the outflow. If we include radial self-gravity of the Bondi sphere (e.g., Wandel 1984), the shocks could be enhanced further. The streaming kinetic energy could be of the same order of magnitude as E_SG; therefore, the characteristic features of the explosion remains. Additionally, it is interesting to compare the long γ-ray bursts (GRBs) and the Bondi explosion for the similarities and differences between them. Woosley (1993) pioneered an idea about the long GRBs. They originate from failed type Ib supernovae of massive stars; however, highly relativistic jets are developed from disk accretion of neutrons onto the central black holes with neutrino cooling. Their accretion rates, $\sim 1\,{M}_{\odot }\,{{\rm{s}}}^{-1}\approx {10}^{15}{\dot{M}}_{\mathrm{Edd}}$, are much higher than the AMS cases. The similarity is that both kinds of explosions are driven by accretion onto black holes, but the differences rest on the fact that not only are cooling mechanisms distinguished but the GRBs are more violent in the extremely compact regions than the Bondi explosion. Unlike the GRBs dominated by relativistic jets, the Bondi explosions are driven by powerful outflows appearing as slow transients.

The cavity formed by the Bondi explosion causes the BH to have very low accretion rates, and thus they are hardly detectable individually. However, the cold gas of the SMBH disk replenishes the cavity rejuvenating AMSs in a timescale of

$$\begin{eqnarray}&&\begin{array}{l}{t}_{\mathrm{rej}}=\displaystyle \frac{{R}_{\exp }}{{c}_{s}}\\ \,=\,268.9\,{\alpha }_{0.1}^{3/10}{M}_{8}^{11/20}{\dot{M}}^{-13/40}{r}_{4}^{21/16}{E}_{52}^{1/4}\,\mathrm{yr}.\end{array}\end{eqnarray} \tag{ 20 }$$

The duty cycle of the BH accretion is δ_• = t_a/(t_a + t_rej) ≈ 6 × 10⁻⁶. The episodic accretion efficiently constrains the BH growth. With the duty cycle, growth timescale is about ${t}_{\mathrm{grow}}={({\delta }_{\bullet }\dot{m}{f}_{{\rm{a}}})}^{-1}{t}_{\mathrm{Salp}}\,=10\,{({\delta }_{\bar{5}}{\dot{m}}_{9}{f}_{{\rm{a}},\bar{3}})}^{-1}\,{t}_{\mathrm{Salp}}$, where ${t}_{\mathrm{Salp}}\,={m}_{\bullet }/{\dot{M}}_{\mathrm{Edd}}=0.45\,\mathrm{Gyr}$ is the Salpeter time, ${\delta }_{\bar{5}}={\delta }_{\bullet }/{10}^{-5}$, and ${f}_{{\rm{a}},\bar{3}}={f}_{{\rm{a}}}/{10}^{-3}$ is the fraction of the outflow to the accretion rates. The most uncertainty is in regards to f_a, but it is easy for the BHs to grow up to 10² M_⊙ from 10M_⊙. This requires numerical simulations of hyper-Eddington accretion at a small scale of ≲10³ r_g.

It should be noted that most BHs are quiescent because of the extremely low duty cycles in SMBH disks. If they are in the sub-Eddington accretion status, they are radiating in X-ray bands like X-binaries. However, emissions from this kind of AMS (with a low accretion rate) depend on the details of its surroundings and its mass function. Total emissions of these accreting BHs may significantly contribute to the observations. This would be an interesting topic to explore in the future.

The Bondi explosion can be divided into two phases. First, internal shocks due to the collision between the outflows and the Bondi sphere. The dissipated energies could be of the orders of E_SG and channeled into thermal energy because of the very large optical depth ${\tau }_{\mathrm{es}}^{0}\approx {\kappa }_{\mathrm{es}}{\rho }_{{\rm{d}}}R\sim {10}^{4}$ for initial Bondi sphere with R_Bon ∼ 10¹⁵ cm and ρ_d ∼ 10⁻¹⁰ g cm⁻³, where κ_es is electron scattering opacity. The optical depth of the expanding Bondi sphere is τ_es ≈ κ_es(3M_Bon/4π R³)R ∝ R⁻², we find ${\tau }_{\mathrm{es}}={\tau }_{\mathrm{es}}^{0}{({R}_{\mathrm{Bon}}/R)}^{2}$. Since the Bondi explosion is mostly in the BLR medium, we neglect the part in the SMBH disk. When R ∼ 10² R_Bon ∼ 10¹⁷ cm, this phase ends within an interval of $\delta {t}_{{\tau }_{\mathrm{es}}}\sim {10}^{6}\,{\rm{s}}$ from Equation (16). We then have a variation of luminosity ${\rm{\Delta }}L\sim {E}_{\mathrm{SG}}/\delta {t}_{{\tau }_{\mathrm{es}}}\approx {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. This is too weak to observe for a quasar.

When R = 100R_Bon, the explosion begins to nonthermally radiate. For the simplest estimation, we assume that electrons accelerated by shocks will generate nonthermal emissions with a fraction about ξ of E_kin. For a giant flare of nonthermal emission with a luminosity of about ${\rm{\Delta }}L\propto {V}_{\exp }^{2}$, we have

$$\begin{eqnarray}&&{\rm{\Delta }}{L}_{\mathrm{AGN}}\approx {L}_{0}{\left(\tfrac{t}{{t}_{\mathrm{es}}}\right)}^{-6/5}\,(\mathrm{for}\,t\geqslant {t}_{\mathrm{es}}),\end{eqnarray} \tag{ 21 }$$

from the self-similar expansion in the SMBH disk, where ${L}_{0}=\xi {E}_{\mathrm{kin}}/5{t}_{\mathrm{es}}=1.0\times {10}^{44}\,{\xi }_{0.05}{E}_{52}{t}_{\mathrm{es},6}^{-1}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, ξ_0.05 = ξ/0.05 (e.g., Blandford & Eichler 1987), and t_es,6 = t_es/10⁶ s. L₀ is determined by the total energy of ξ E_kin. Spectral energy distributions (SEDs) depend on relativistic electrons and surrounding photons (synchrotron radiation and inverse Compton scattering), but this is a significant contribution to the steady luminosity of quasars.

In order to show the characteristic of light curves, we assume that the shocks accelerate electrons to have a power law as ${N}_{\gamma }\propto {\gamma }^{-{n}_{e}}$ (${\gamma }_{\min }\leqslant \gamma \leqslant {\gamma }_{\max }$) with an index n_e , where γ is the Lorentz factor of relativistic electrons. The maximum energy of an electron is determined by the balance between the acceleration and cooling. For the equipartition with the BLR hot phase, we have a magnetic field of $B\approx 0.6\,{({n}_{7}{T}_{7})}^{1/2}\,{\rm{G}}$ from u_B = n_BLR kT_hot, where u_B = B²/8π is energy density of the magnetic field and T_hot = 10⁷ T₇ is the temperature of the hot phase of the BLR. To illustrate the flare generated by the Bondi explosion, we consider a radio-quiet quasar with M_• = 10⁸ M_⊙ and $\dot{M}=1$. Its bolometric luminosity is ${L}_{\mathrm{bol}}=1.3\times {10}^{45}\,{\eta }_{0.1}{M}_{8}\dot{M}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Photon energy density in the BLR is about ${u}_{\mathrm{ph}}={L}_{\mathrm{bol}}/4\pi {R}^{2}c\,\approx 0.16\,{L}_{45}{R}_{50}^{-2}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ much higher than the magnetic fields, where L₄₅ = L_bol/10⁴⁵erg s⁻¹, R₅₀ = R/50 ltd. This leads to external inverse Compton (IC) scattering as the dominant cooling of the relativistic electrons accelerated by shocks. Acceleration timescale is given by ${t}_{\mathrm{acc}}={R}_{{\rm{L}}}c/{V}_{\exp }^{2}\approx 5.8\times {10}^{2}\,{\gamma }_{5}{B}_{0}^{-1}{T}_{7}^{-1}\,{\rm{s}}$ (Blandford & Eichler 1987), while the inverse Compton cooling timescale is ${t}_{\mathrm{IC}}=3{m}_{e}c/4{\sigma }_{{\rm{T}}}\gamma {u}_{\mathrm{ph}}=1.9\,\times {10}^{3}\,{\gamma }_{5}^{-1}{u}_{0.16}^{-1}\,{\rm{s}}$, where R_L is the Larmor's radius, γ₅ = γ/10⁵, and B₀ = B/1 G. We have ${\gamma }_{\max }=1.8\times {10}^{5}\,{u}_{0.16}^{-1/2}{B}_{0}^{1/2}{T}_{7}^{1/2}$ from t_acc = t_IC, and the maximum frequency of synchrotron and IC radiation are given by

$$\begin{eqnarray}\begin{array}{rcl}{\nu }_{\mathrm{syn}}^{\max } & \approx & 0.17\,{B}_{0}{\gamma }_{\max ,5}^{2}\,\mathrm{keV},\\ {\nu }_{\mathrm{IC}}^{\max } & \approx & {\gamma }_{\max }^{2}{\nu }_{\mathrm{UV}}\approx 41.7\,{\gamma }_{\max ,5}^{2}{\nu }_{15}^{\mathrm{UV}}\,\mathrm{GeV},\end{array}\end{eqnarray} \tag{ 22 }$$

with the peak luminosities of

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{syn}} & = & 2.5\times {10}^{43}\,\left(\tfrac{{u}_{{\rm{B}}}/{u}_{\mathrm{ph}}}{0.25}\right)\left(\displaystyle \frac{{L}_{\mathrm{IC}}}{1\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right)\mathrm{erg}\,{{\rm{s}}}^{-1},\\ {L}_{\mathrm{IC}} & \approx & 1\times {10}^{44}\,{\xi }_{0.05}{E}_{52}{t}_{\mathrm{es},6}^{-1}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 23 }$$

for a spectral index of ${L}_{\nu }\propto {\nu }^{(1-{n}_{e})/2}$, where ${\nu }_{15}^{\mathrm{UV}}={\nu }_{\mathrm{UV}}/{10}^{15}\,\mathrm{Hz}$ is the UV photon frequency from the SMBH disk. In Figure 3, we show the mean SED of a typical quasar in order to compare with multiwaveband light curves of the Bondi explosion.

Zoom In Zoom Out Reset image size

From Figure 3, several remarkable features are found: (1) the synchrotron radiation peaks between soft X-rays (contributes to 1/3) and has significant contribution to optical and UV (about ∼0.1); (2) the Bondi explosion drives a radio-quiet quasar to become radio-intermediate in radio bands (n_e = 2); (3) the external inverse Compton peaks at ∼40 GeV, and with about 5 × 10⁴³ erg s⁻¹, which can be detected in nonblazar AGNs by Fermi-LAT; (4) the transient appearance with a temporal profile as t^−6/5 and decaying a factor of 10 within about 6.8t_es, namely, 3 months for 10⁸ M_⊙ quasars and the 10M_⊙ BH; (5) the intra-band emissions have no delays since they originate from the same regions. Moreover, for one BH with a few tens of solar mass, the above features will be more prominent. These characteristics are unique characteristics when identifying flares in AGNs.

Expansion of the Bondi explosion exhausts a small fraction of the total energy (∼10%) in the SMBH disk, and most energy is released outside the disk as slow transients from radio to γ-rays. This is very different from that of the relativistic expansion driven by jets from neutron stars or black holes in SMBH disks discussed in Perna et al. (2021) and Zhu et al. (2021) (actually for GRBs in the disks). Future detections of AGN light curves are useful to distinguish the nature of the AMSs in the SMBH disks.

According to the Kennicutt-Schmidt (KS) law of ${\dot{{\rm{\Sigma }}}}_{* }=2.5\,\times {10}^{-4}{{\rm{\Sigma }}}_{0}^{1.4}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$, star formation timescale is ${t}_{* }={{\rm{\Sigma }}}_{0}/{\dot{{\rm{\Sigma }}}}_{* }=4\times {10}^{7}\,{{\rm{\Sigma }}}_{5}^{-0.4}\,\mathrm{yr}$, where Σ_0,5 = Σ_gas/(10⁰, 10⁵) M_⊙ pc⁻² is gas surface density of star formation regions. We note that t_* is comparable with t_AGN. From Equation (1), we have surface density of SMBH disks of ${{\rm{\Sigma }}}_{\mathrm{disk}}=2.8\,\times {10}^{8}\,{\alpha }_{0.1}^{-4/5}{M}_{8}^{1/5}{\dot{M}}^{7/10}{r}_{4}^{-3/4}\,{M}_{\odot }{\mathrm{pc}}^{-2}$, the KS law results in a rate of ∼2.5M_⊙ yr⁻¹ in the SMBH disk. This means that the KS star formation exhausts all disk gas in a timescale much shorter than t_AGN by a factor of 0.04. This is obviously inconsistent with observations of AGN lifetime. One way to overcome this inconsistency is to decrease the star formation efficiency or only a small fraction of SMBH disk mass converting into stars. Star formation in this region is poorly constrained by observations (e.g., Collin & Zahn 2008). For concreteness, we assume that the maximum star formation is that all SMBH disk gas is converted into stars with a rate of about M_disk/t_AGN, and black hole numbers are of N_• ∼ M_disk/〈m_*〉 over the AGN episode, where ${M}_{\mathrm{disk}}\approx 2\pi {R}^{2}{{\rm{\Sigma }}}_{\mathrm{disk}}=4.4\times {10}^{6}\,{\alpha }_{0.1}^{-4/5}{M}_{8}^{11/5}{\dot{M}}^{7/10}{r}_{4}^{5/4}\,{M}_{\odot }$. For a conserved consideration of star formation efficiency of ζ = 0.1 in 10⁸ yr (Kennicutt 1998), ζ M_disk will be converted into stars M_⋆, and a fraction of about ${f}_{* }={10}^{2(1-{n}_{* })}\sim 0.1$ is massive stars for star formation with a top-heavy IMF (the maximum and minimum masses are (100, 1)M_⊙, n_* = 0.5 much more flattened than the Salpeter n_* = 2.35), where initial mass function ${{dN}}_{* }\propto {m}_{* }^{-{n}_{* }}$. According to stellar evolution theory (e.g., see Figure 16 in Woosley et al. 2002), about 10% stellar mass is converted into BHs (even high fraction to BHs) for 〈m_*〉 ∼ 100M_⊙ stars (evolved within t_AGN). The black hole numbers are about N_• ≈ 4.4 × 10² ζ_0.1 f_0.1 M_disk,6/〈m_*〉₂, where 〈m_*〉₂ = 〈m_*〉/10² M_⊙, ζ_0.1 = ζ/0.1 and f_0.1 = f_*/0.1. Bondi explosion rates are about

$$\begin{eqnarray}&&{\dot{{ \mathcal R }}}_{\mathrm{Bondi}}\approx \displaystyle \frac{{N}_{\bullet }}{{t}_{\mathrm{rej}}}\approx 1.6\,{N}_{\bullet ,440}{t}_{\mathrm{rej},270}^{-1}\,{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 24 }$$

where N_•,440 = N_•/440 and t_rej,270 = t_rej/270 yr. The Bondi explosion drives one transient appearance per year. Considering the uncertainties of N_•, such an event rate is also comparable to AGN giant flare rates. It would be very interesting to observationally test the real AGN flares. Additionally, AMSs with neutron stars or ∼1M_⊙ black holes have fainter flares than that of 10M_⊙ black holes; however, their numbers could be more and thus explosion rates could be higher than a few per year generating relatively stationary emissions. The correct answer to this problem depends on more sophisticated studies of the IMF and stellar evolution in the SMBH disks.

The Caltech-NRAO Stripe 82 Survey (CNSS), a dedicated radio transient survey (Mooley et al. 2016) to systematically explore the radio sky for slow transient phenomena on timescales between one day and several years, is very useful to detect the Bondi explosions suggested in this paper. The Very Large Array Sky Survey (VLASS) and Faint Images of the Radio Sky at Twenty (FIRST) cm survey, which have detected some variable objects (Nyland et al. 2020), are also very useful to test the present scenario. Moreover, the Fermi-LAT monitoring campaign of nonblazar AGNs also provides unique opportunities for testing the Bondi explosion mechanism proposed here. Indeed, several radio-quiet AGNs clearly show variability at >100 MeV (Sahakyan et al. 2018). Future joint radio and γ-ray observations of radio-quiet quasars have the potential to discover high-mass stellar black holes in SMBH disks.

Bondi explosions could not affect the broad-line region, which is composed of discrete clouds keeping a pressure balance with its surrounding medium. The shocked clouds will reach temperatures of up to ∼10⁸ K, but its free–free cooling timescale is about ${t}_{\mathrm{ff}}(\mathrm{BLR})\approx {10}^{5}\,{T}_{8}^{1/2}{n}_{10}^{-1}\,{\rm{s}}$, where n₁₀ = n_cloud/10¹⁰ cm⁻³ is the cloud density in the BLR. Since t_ff(BLR) is much shorter than the timescale of the shock crossing the BLR (${t}_{\mathrm{BLR}}={R}_{\mathrm{BLR}}/{V}_{\exp }\sim {10}^{8-9}\,{\rm{s}}$), the shocked clouds will recover rapidly. Thus, the BLR itself is not strongly affected by Bondi explosions.