Bondi Radius Research Articles

Late-time Radio Flares in Tidal Disruption Events

Radio monitoring unveiled late (hundreds to a thousand days) radio flares in a significant fraction of tidal disruption events. We propose that these late-time radio flares are a natural outcome if the surrounding density profile flattens outside the Bondi radius. At the Bondi radius, the outflow is optically thin (above a few GHz) to synchrotron self-absorption. As more and more material is swept up, the radio emission rises asymptotically as ∝ t 3 until the outflow begins to decelerate. A detection of such a rise and a late-time maximum constrains the black hole mass and the mass and energy of the radio-emitting outflow. We show that this model can give reasonable fits to some observed light curves, leading to reasonable estimates of the black hole and outflow masses. We also find that the slope of the density profile within the Bondi radius determines whether an early-time (∼102 days) radio peak exists.

How planets grow by pebble accretion

The characterization of super-Earth- to Neptune-sized exoplanets relies heavily on our understanding of their formation and evolution. In this study, we link a model of planet formation by pebble accretion to the planets’ long-term observational properties by calculating the interior evolution, starting from the dissipation of the protoplanetary disk. We investigate the evolution of the interior structure in 5–20 M⊕ planets, accounting for silicate redistribution caused by convective mixing, rainout (condensation and settling), and mass loss. Specifically, we have followed the fate of the hot silicate vapor that remained in the planet’s envelope after planet formation as the planet cools. We find that disk dissipation is followed by a rapid contraction of the envelope from the Hill or Bondi radius to about one-tenth of that size within 10 Myr. Subsequent cooling leads to substantial growth of the planetary core through silicate rainout accompanied by inflated radii, in comparison to the standard models of planets that formed with core-envelope structure. We examined the dependence of rainout on the planet’s envelope mass, on the distance from its host star, on its silicate mass, and on the atmospheric opacity. We find that the population of planets that formed with polluted envelopes can be roughly divided into three groups based on the mass of their gas envelopes: bare rocky cores that have shed their envelopes, super-Earth planets with a core-envelope structure, and Neptune-like planets with diluted cores that undergo gradual rainout. For polluted planets that formed with envelope masses below 0.4 M⊕, we anticipate that the inflation of the planet’s radius caused by rainout will enhance the mass loss by a factor of 2–8 compared to planets with unpolluted envelopes. Our model bridges the gap between the predicted composition gradients in massive planets and the core-envelope structure in smaller planets.

Prospects for AGN Studies with AXIS: AGN Fueling—Resolving Hot Gas inside Bondi Radius of SMBHs

Prospects for AGN Studies with AXIS: AGN Fueling—Resolving Hot Gas inside Bondi Radius of SMBHs

Jets with a Twist: The Emergence of FR0 Jets in a 3D GRMHD Simulation of Zero-angular-momentum Black Hole Accretion

Spinning supermassive black holes (BHs) in active galactic nuclei magnetically launch relativistic collimated outflows, or jets. Without angular momentum supply, such jets are thought to perish within 3 orders of magnitude in distance from the BH, well before reaching kiloparsec scales. We study the survival of such jets at the largest scale separation to date, via 3D general relativistic magnetohydrodynamic simulations of rapidly spinning BHs immersed into uniform zero-angular-momentum gas threaded by a weak vertical magnetic field. We place the gas outside the BH sphere of influence, or the Bondi radius, chosen to be much larger than the BH gravitational radius, R B = 103 R g. The BH develops dynamically important large-scale magnetic fields, forms a magnetically arrested disk (MAD), and launches relativistic jets that propagate well outside R B and suppress BH accretion to 1.5% of the Bondi rate, ṀB . Thus, low-angular-momentum accretion in the MAD state can form large-scale jets in Fanaroff–Riley (FR) type I and II galaxies. Subsequently, the disk shrinks and exits the MAD state: barely a disk (BAD), it rapidly precesses, whips the jets around, globally destroys them, and lets 5%–10% of ṀB reach the BH. Thereafter, the disk starts rocking back and forth by angles 90°–180°: the rocking accretion disk (RAD) launches weak intermittent jets that spread their energy over a large area and suppress BH accretion to ≲2% ṀB . Because the BAD and RAD states tangle up the jets and destroy them well inside R B, they are promising candidates for the more abundant, but less luminous, class of FR0 galaxies.

Bridging Scales in Black Hole Accretion and Feedback: Magnetized Bondi Accretion in 3D GRMHD

Fueling and feedback couple supermassive black holes (SMBHs) to their host galaxies across many orders of magnitude in spatial and temporal scales, making this problem notoriously challenging to simulate. We use a multi-zone computational method based on the general relativistic magnetohydrodynamic (GRMHD) code KHARMA that allows us to span 7 orders of magnitude in spatial scale, to simulate accretion onto a non-spinning SMBH from an external medium with a Bondi radius of R B ≈ 2 × 105 GM •/c 2, where M • is the SMBH mass. For the classic idealized Bondi problem, spherical gas accretion without magnetic fields, our simulation results agree very well with the general relativistic analytic solution. Meanwhile, when the accreting gas is magnetized, the SMBH magnetosphere becomes saturated with a strong magnetic field. The density profile varies as ∼r −1 rather than r −3/2 and the accretion rate is consequently suppressed by over 2 orders of magnitude below the Bondi rate . We find continuous energy feedback from the accretion flow to the external medium at a level of . Energy transport across these widely disparate scales occurs via turbulent convection triggered by magnetic field reconnection near the SMBH. Thus, strong magnetic fields that accumulate on horizon scales transform the flow dynamics far from the SMBH and naturally explain observed extremely low accretion rates compared to the Bondi rate, as well as at least part of the energy feedback.

Easy to Use DFT Approach for Computational pKa Determination of Carboxylic Acids.

In pKa computational determination, the challenge in exploring and fostering new methodologies and approaches goes in parallel with the amelioration of computational performances. In this paper a "ready to use methodology" has been compared to other strategies, such as the re-shaping in solvation cavity (Bondi radius re-shaping), wanting to assess its reliability in predicting the pKa of a broad list of carboxylic acids. Thus, the functionals B3LYP and CAM-B3LYP have been selected, using SMD as continuum solvation model. Exploiting our previous results, two water molecules were made explicit on the reaction centre. Data show that our model (CAM-B3LYP/2H2 O) is capable to accurately predict pKa, leading to mean absolute error (MAE) values lower than 0.5. Noteworthy, good results were achieved in computing the pKa of substituents bearing nitro and cyano groups. Focusing on B3LYP, eventually remarkable outputs were obtained only when Bondi correction was applied to the complex with two water molecules. Hence, massive outcomes were obtained in foreseeing the trichloro and trifluoro acetic acid pKa. These findings demonstrated that no complex level of theory nor external factor is required to accurately predict carboxylic acids pKa, with MAE well below 0.5 units.

Kinematics and Collimation of the Two-sided Jets in NGC 4261: VLBI Study on Subparsec Scales

We report multifrequency VLBI studies of the subparsec scale structure of the two-sided jets in the nearby radio galaxy NGC 4261. Our analyses include new observations using the source frequency phase referencing technique with the Very Long Baseline Array at 44 and 88 GHz, as well as archival data at 15 and 43 GHz. Our results show an extended double-sided structure at 43/44 GHz and provide a clear image of the nuclear region at 88 GHz, showing a core size of ∼0.09 mas and a brightness temperature of ∼1.3 × 109 K. Proper motions are measured for the first time in the two-sided jets, with apparent speeds ranging from 0.31 ± 0.14 c to 0.59 ± 0.40 c in the approaching jet and 0.32 ± 0.14 c in the receding jet. The jet-to-counterjet brightness ratio allows us to constrain the viewing angle to between ∼54° and 84° and the intrinsic speed to between ∼0.30 c and 0.55 c. We confirm the parabolic shape of the upstream jet on both sides of the central engine, with a power-law index of 0.56 ± 0.07. Notably, the jet collimation is found to be already completed at subparsec scales, with a transition location of about 0.61 pc, which is significantly smaller than the Bondi radius of 99.2 pc. This behavior can be interpreted as the initial confinement of the jet by external pressure from either the geometrically thick, optically thin advection-dominated accretion flows or the disk wind launched from it. Alternatively, the shape transition may also be explained by the internal flow transition from a magnetically dominated to a particle-dominated regime.

The influence of a static planetary atmosphere on spin transfer during pebble accretion

Context. Pebble accretion has been used to explain the small size of Mars, the heavy element contents of the gas giants, and the size distribution of asteroids. More recently, pebble accretion has been proposed as a means to explain not only the growth but also the prograde spin preference of most larger bodies in the Solar System. Pebble accretion could induce planetary and asteroid spin equal to or exceeding the spins currently measured. However, as these planetesimals grow, they start condensing the gas of the disc around them, forming an atmosphere within their Bondi radius. Aims. We study the effect an atmosphere has on the pebble orbits and spin build-up on the planet’s surface during pebble accretion in the extreme case of a static atmosphere. Pebble feedback to the gas is not taken into account. Methods. The equations of motion for pebbles in a planar, global frame with a planet and a central star were integrated using the AIS15 integrator of REBOUND. An adiabatic atmosphere was then placed around the planet, and the spin deposited on the planet’s surface was measured. These simulations were evaluated for different distances to the star, Stokes numbers, and planet masses. Results. In general, an atmosphere dampens the spin the planet’s surface receives by absorbing part of the angular momentum of the pebbles and circularising their orbits. This could prevent the excessive spin values predicted in some 3D pebble accretion simulations without an atmosphere. For planets larger than 0.5 M⊕, a stationary atmosphere absorbs all angular momentum, leaving no spin for the surface. Significant quantities of angular momentum are stored in the inner and intermediate atmosphere (<0.3 Bondi radii). Depending on the atmospheric and disc model, this spin could be transported either to the disc through atmospheric recycling or to the planet through drag between the surface and the atmosphere. Further research is required to quantify the spin transfer within the atmosphere.

Slowly Rotating Accretion Flow Around Supermassive Black Holes in Elliptical Galaxies: Case With Outflow

Observational evidence and many numerical simulations show the existence of winds (i.e., uncollimated outflows) in the accretion systems of elliptical galaxy centers. One of the primary aims of this study is to investigate the solutions of slowly rotating accretion flows around supermassive black holes with outflows. This paper presents two distinct physical regions, supersonic and subsonic, which extend from the outer boundary to the black hole. In our numerical solution, the outer boundary is chosen beyond the Bondi radius. Due to strong gravity, we ignore outflow (i.e., s = 0) in the inner region (within ∼10r s ). The radial velocity of the flow at the outer region is significantly increased due to the presence of the outflow. Compared to previous works, the accretion mode, namely, the slowly rotating case, that corresponds to low accretion rates that have general wind output is carefully described, and the effect of the galaxy potential and feedback by the wind in this mode are taken into account. As the power-law form of the mass accretion rate is mathematically compatible with our equations, we consider a radius-dependent mass accretion rate (), where s is a free parameter and shows the intensity of outflow. There is an unknown mechanism for removing the mass, angular momentum, and energy by outflows in this study. The effects of the outflow appear well on the outer edge of the flow.

Spin of protoplanets generated by pebble accretion: Influences of protoplanet-induced gas flow

Context. In the pebble accretion model, protoplanets accrete millimeter-to-centimeter-sized particles (pebbles). When a protoplanet grows, a dense gas envelope forms around it. The envelope affects accretion of pebbles and, in particular, the spin angular momentum transfer at the collision to the planet. Aims. We aim to investigate the spin state of a protoplanet during the pebble accretion influenced by the gas flow in the gravitational potential of the protoplanet and how it depends on the planetary mass, the headwind speed, the distance from the host star, and the pebble size. Methods. We performed nonisothermal three-dimensional hydrodynamical simulations in a local frame to obtain the gas flow around the planet. We then numerically integrated three-dimensional orbits of pebbles under the obtained gas flow. Finally, assuming uniform spatial distribution of incoming pebbles, we calculated net spin by summing up specific angular momentum that individual pebbles transfer to the protoplanet at impacts. Results. We find that a protoplanet with the envelope acquires prograde net spin rotation regardless of the planetary mass, the pebble size, and the headwind speed of the gas. This is because accreting pebbles are dragged by the envelope that commonly has prograde rotation. As the planetary mass or orbital radius increases, the envelope is thicker and the prograde rotation is faster, resulting in faster net prograde spin. When the dimensionless thermal mass of the planet, m = RBondi/H, where RBondi and H are the Bondi radius and the disk gas scale height, is larger than a certain critical mass (m ≳ 0.3 at 0.1 au or m ≳ 0.1 at 1 au), the spin rotation exceeds the breakup one. Conclusions. The predicted spin frequency reaches the breakup one at the planetary mass miso,rot ~ 0.1 (a/1 au)−1/2 (where a is the orbital radius), suggesting that the protoplanet cannot grow beyond miso,rot. It is consistent with the Earth’s current mass and could help the formation of the Moon with a giant impact on a fast-spinning proto-Earth.

Black hole feeding and feedback in a compact galaxy

ABSTRACT We perform high-resolution hydrodynamical simulations using the framework of MACER to investigate supermassive black hole (SMBH) feeding and feedback in a massive compact galaxy, which has a small effective radius but a large stellar mass, with a simulation duration of 10 Gyr. We compare the results with a reference galaxy with a similar stellar mass but a less concentrated stellar density distribution, as typically found in local elliptical galaxies. We find that about 10 per cent of the time, the compact galaxy develops multiphase gas within a few kpc, but the accretion flow through the inner boundary below the Bondi radius is always a single phase. The inflow rate in the compact galaxy is several times larger than in the reference galaxy, mainly due to the higher gas density caused by the more compact stellar distribution. Such a higher inflow rate results in stronger SMBH feeding and feedback and a larger fountain-like inflow-outflow structure. Compared to the reference galaxy, the star formation rate in the compact galaxy is roughly two orders of magnitude higher but is still low enough to be considered quiescent. Over the whole evolution period, the black hole mass grows by ∼50 per cent in the compact galaxy, much larger than the value of ∼ 3 per cent in the reference galaxy.

AGN feeding and feedback in M84: from kiloparsec scales to the Bondi radius

ABSTRACT We present the deepest Chandra observation to date of the galaxy M84 in the Virgo Cluster, with over 840 ks of data provided by legacy observations and a recent 730 ks campaign. The increased signal-to-noise ratio allows us to study the origins of the accretion flow feeding the supermassive black hole in the centre of M84 from the kiloparsec scales of the X-ray halo to the Bondi radius, RB. Temperature, metallicity, and deprojected density profiles are obtained in four sectors about M84’s active galactic nucleus (AGN), extending into the Bondi radius. Rather than being dictated by the potential of the black hole, the accretion flow is strongly influenced by the AGN’s bipolar radio jets. Along the jet axis, the density profile is consistent with ne ∝ r−1; however, the profiles flatten perpendicular to the jet. Radio jets produce a significant asymmetry in the flow, violating a key assumption of Bondi accretion. Temperature in the inner kiloparsec is approximately constant, with only a slight increase from 0.6 to 0.7 keV approaching RB, and there is no evidence for a temperature rise imposed by the black hole. The Bondi accretion rate $\dot{M}_{\rm B}$ exceeds the rate inferred from AGN luminosity and jet power by over four orders of magnitude. In sectors perpendicular to the jet, $\dot{M}_{\rm B}$ measurements agree; however, the accretion rate is >4σ lower in the North sector along the jet, likely due to cavities in the X-ray gas. Our measurements provide unique insight into the fuelling of AGN responsible for radio mode feedback in galaxy clusters.

Self-regulation of black hole accretion via jets in early protogalaxies

ABSTRACT The early growth of black holes (BHs) in high-redshift galaxies is likely feedback regulated. While radiative feedback has been extensively studied, the role of mechanical feedback has received less scrutiny to date. Here, we use high-resolution parsec-scale hydrodynamical simulations to study jet propagation and its effect on 100 M⊙ BH accretion in the dense, low-metallicity gas expected in early protogalaxies. As the jet propagates, it shocks the surrounding gas forming a jet cocoon. The cocoon consists of a rapidly cooling cold phase at the interface with the background gas and an overpressured subsonic phase of reverse shock-heated gas filling the interior. We vary the background gas density and temperature, BH feedback efficiency, and the jet model. We found that the width of the jet cocoon roughly follows a scaling derived by assuming momentum conservation in the jet-propagation direction and energy conservation in the lateral directions. Depending on the assumed gas and jet properties, the cocoon either stays elongated to large radii or isotropizes before reaching the Bondi radius, forming a nearly spherical bubble. Lower jet velocities and higher background gas densities result in self-regulation to higher momentum fluxes and elongated cocoons. In all cases, the outward cocoon momentum flux balances the inward inflowing gas momentum flux near the Bondi radius, which ultimately regulates BH accretion. The time-averaged accretion rate always remains below the Bondi rate, and exceeds the Eddington rate only if the ambient medium is dense and cold, and/or the jet is weak (low velocity and mass loading).

The Three Regimes of Atmospheric Evaporation for Super-Earths and Sub-Neptunes

A significant fraction of super-Earths and sub-Neptunes are thought to experience an extreme loss of volatiles because of atmospheric evaporation in the early stages of their life. Though the mechanisms behind the extreme mass loss are not fully understood, two contenders have been widely discussed: photoevaporation from X-ray and ultraviolet irradiation and core-powered mass loss. Here, it is shown that both mechanisms occur, but with different timescales, and that atmospheric loss can take place over three regimes. In the first regime, a planet has very high internal temperatures arising from its high-energy formation processes. These high temperatures give rise to a fully convecting atmosphere that efficiently loses mass without much internal cooling. The second regime applies to planets with lower internal temperatures, so a radiative region forms, but the photosphere still remains outside the Bondi radius. Hence, mass loss continues to depend only on the internal temperatures. Planets with the lowest internal temperatures are in the third regime, when the photosphere forms below the Bondi radius and mass is lost primarily because of X-ray and ultraviolet irradiation. This paper provides the first unifying framework for modeling atmospheric evaporation through the life span of a planet.

Dust processing in protoplanetary envelopes as the origin of hot minerals in comets

ABSTRACT Crystalline silicates are found in a large number of comets. These pose a long-standing conundrum for Solar system formation models as they can only be created in the inner hot disc at temperatures higher than 800 K, and there is no obvious mechanism to transport them out into the comets formation region. Here, we propose that these particles could have formed inside the hydrostatic envelopes surrounding young protoplanets still embedded in the protoplanetary disc. Using a simplified 1D model we investigate the thermal structure of these envelopes, and find that for core masses ranging from 0.08 to 1.5 M⊕, located anywhere between 1 and 30 au, the temperature and pressure at the base of the envelopes are high enough to quickly vapourize silicate particles of various sizes. Moreover, if the grain abundance is atleast solar, these envelopes become fully convective, allowing for dust ejection across the Bondi radius back into the disc. Amorphous silicates are hence thermally processed into crystalline particles in these envelopes, and then transported back to disc through convective diffusion to be finally incorporated into the cometary building blocks.

Accretion from Winds of Red Giant Branch Stars May Reveal the Supermassive Black Hole in Leo I

A supermassive black hole (SMBH) of ∼3 × 106 M ⊙ was recently detected via dynamical measurements at the center of the dwarf galaxy Leo I. Standing ∼2 orders of magnitude above standard scaling relations, this SMBH is hosted by a galaxy devoid of gas and with no significant star formation in the last ∼1 Gyr. This detection can profoundly impact the formation models for black holes and their hosts. We propose that winds from a population of ∼100 evolved stars within the Bondi radius of the SMBH produce a sizable accretion rate, with Eddington ratios between 9 × 10−8 and 9 × 10−7, depending on the value of the stellar mass loss. These rates are typical of SMBHs accreting in advection-dominated accretion flow mode. The predicted spectrum peaks in the microwaves at ∼0.1–1 THz (300–3000 μm) and exhibits significant variations at higher energies depending on the accretion rate. We predict a radio flux of ∼0.1 mJy at 6 GHz, mildly dependent on the accretion properties. Deep imaging with Chandra, the Very Large Array, and the Atacama Large Millimeter/submillimeter Array can confirm the presence of this SMBH and constrain its accretion flow.

Numerical Simulation of Hot Accretion Flow around Bondi Radius

Previous numerical simulations have shown that strong winds can be produced in the hot accretion flows around black holes. Most of those studies focus only on the region close to the central black hole; therefore it is unclear whether the wind production stops at large radii around Bondi radius. Bu et al. (2016) studied the hot accretion flow around the Bondi radius in the presence of nuclear star gravity. They find that, when the nuclear stars gravity is important–comparable to the black hole gravity, winds cannot be produced around the Bondi radius. However, for some galaxies, the nuclear stars gravity around Bondi radius may not be strong. In this case, whether winds can be produced around Bondi radius is not clear. We study the hot accretion flow around Bondi radius with and without thermal conduction by performing hydrodynamical simulations. We use the virtual particles trajectory method to study whether winds exist based on the simulation data. Our numerical results show that, in the absence of nuclear stars gravity, winds can be produced around Bondi radius, which causes the mass inflow rate to decrease inwards. We confirm the results of Yuan et al. (2012), which indicate this is due to the mass loss of gas via wind rather than convectional motions.

The relation between accretion rate and jet power in early-type galaxies with thermally unstable hot atmospheres

ABSTRACT We use Chandra X-ray data and Very Large Array radio observations for a sample of 20 nearby, massive, X-ray bright, early-type galaxies to investigate the relation between the Bondi accretion rates and the mechanical jet powers. We find a strong correlation ($\rho = 0.96^{+0.03}_{-0.09}$; BF10 > 100) between the Bondi accretion power, PBondi, and the mechanical jet power, Pjet, for a subsample of 14 galaxies, which also host cool H α+[N ii] line emitting gas and thus likely have thermally unstable atmospheres. The relation between the Bondi accretion power and the mechanical jet power for this subsample is well described by a power-law model $\log \frac{P_{\mathrm{Bondi}}}{{10^{43} \, \mathrm{erg \, s^{-1}}}} = \alpha + \beta \log \frac{P_{\mathrm{jet}}}{{10^{43} \, \mathrm{erg \, s^{-1}}}}$, where α = 1.10 ± 0.25 and β = 1.10 ± 0.24 with an intrinsic scatter $\sigma = 0.08^{+0.14}_{-0.06}$ dex. The results indicate that in all galaxies with thermally unstable atmospheres the cooling atmospheric gas feeds the central black holes at a similar jet-to-Bondi power ratio. For the full sample of 20 galaxies, the correlation is weaker and in a subset of galaxies with no signs of H α+[N ii] emission, we see a hint for a systematically lower jet-to-Bondi power ratio. We also investigate the dependence of jet power on individual quantities in the Bondi formula such as the supermassive black hole mass (M•) and the specific entropy of the gas (K) at the Bondi radius. For the subsample of H α+[N ii] emitting galaxies, we find a very tight correlation of Pjet with M• ($\rho = 0.91^{+0.06}_{-0.11}$; BF10 > 100) and, although poorly constrained, a hint of an anticorrelation for Pjet and K ($\rho = -0.47^{+0.60}_{-0.37}$; BF10 = 1.1).

Global structure and dynamics of slowly rotating accretion flows

ABSTRACT We study the global solutions of slowly rotating accretion flows around the supermassive black hole in the nucleus of an elliptical galaxy. The velocity of accreted gas surrounding the black hole is initially subsonic and then falls on to the black hole supersonically, so accretion flow must be transonic. We numerically solve equations from the Bondi radius to near the black hole. The focus of our discussion will be on the properties of slightly rotating accretion flows in which radiative losses have been ignored. This study discusses how outer boundary conditions (the temperature and specific angular momentum at the outer boundary) influence accretion flow dynamics. We investigate two physically discontinuous regimes: the Bondi-like type accretion and the disc-like type accretion. A Bondi-like accretion occurs when the specific angular momentum at the Bondi radius ℓB is smaller than the specific angular momentum at the marginally stable orbit ℓms. In comparison, a disc-like accretion occurs when the specific angular momentum at the Bondi radius ℓB is larger than the specific angular momentum of the marginally stable orbit ℓms. We also keep the assumption of hydrostatic equilibrium and compare our results with the case in which it is not considered. According to this study, considering the assumption of hydrostatic equilibrium reduces the mass accretion rate. Additionally, we find our solution for different ranges of the viscosity parameter α. Finally, we study the effect of galaxy potential on slowly rotating accretion flows.

Self-interacting dark scalar spikes around black holes via relativistic Bondi accretion

We consider the spike mass density profile in a dark halo by self-consistently solving the relativistic Bondi accretion of dark matter onto a non-spining black hole of mass M. We assume that the dominant component of the dark matter in the halo is a Standard model gauge-singlet scalar. Its mass m ≃ 10-5 eV and quartic self-coupling λ ≲ 1019 are constrained to be compatible with the properties of galactic dark halos. In the hydrodynamic limit, we find that the accretion rate is bounded from below, M min = 96πG 2 M 2 m 4/λħ 3. Therefore, for M = 106 M⊙ we have M min ≃ 1.41 × 10-9 M ⊙ yr-1, which is subdominant compared to the Eddington accretion of baryons. The spike density profile ρ 0(r) within the self-gravitating regime cannot be fitted well by a single-power law but a double-power one. Despite that, we can fit ρ 0(r) piecewise and find that ρ 0(r) ∝ r -1.20 near the sound horizon, ρ 0(r) ∝ r -1.00 towards the Bondi radius and ρ 0(r) ∝ r -1.08 for the region in between. This contrasts with more cuspy ρ 0(r) ∝ r -1.75 for dark matter with Coulomb-like self-interaction.