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The Evolution of Black Hole Mass and Spin in Active Galactic Nuclei
A. R. King, J. E. Pringle, J. A. Hofmann
TL;DR
The paper addresses how AGN black holes can grow while avoiding rapid spin-up and excessive radiative efficiency. It models repeated randomly oriented, self-gravity-limited accretion episodes and finds that accretion drives spins toward modest values, while coalescences have little average long-term effect.
Problem
Aligned merger-driven growth spins black holes near unity, increasing radiative efficiency and making Eddington-limited growth difficult at early cosmic times.
Method
The paper combines statistical arguments and simulations of randomly oriented, self-gravity-limited disc episodes, including stable co- and counter-alignment and Eddington-rate accretion.
Results
Accretion rapidly drives black holes toward modest spin values, while repeated accretion and coalescences produce average spins with fluctuations of Δa = ±0.2 and little net coalescence effect.
Takeaways & Limitations
AGN black holes generally remain at fairly low spin, although prograde coalescences between similarly massive SMBHs can produce larger spins, especially in giant ellipticals.
Takeaways & Limitations
The treatment relies on simple arguments and thin discs with efficient cooling; conclusions may not hold when cooling is inefficient.
Abstract
from arXiv · showhide
We argue that supermassive black hole growth in AGN occurs via sequences of randomly--oriented accretion discs with angular momentum limited by self--gravity. These stably co-- or counter--align with the black hole spin with almost equal frequency. Accretion from these discs very rapidly adjusts the hole's spin parameter to average values $\bar a \sim 0.1-0.3$ (the precise range depending slightly on the disc vertical viscosity coefficient $α_2$) from any initial conditions, but with significant fluctuations ($Δa\sim \pm 0.2$) about these. We conclude (a) AGN black holes should on average spin moderately, with the mean value $\bar a$ decreasing slowly as the mass increases; (b) SMBH coalescences leave little long--term effect on $\bar a$; (c) SMBH coalescence products in general have modest recoil velocities, so that there is little likelihood of their being ejected from the host galaxy; (d) black holes can grow even from stellar masses to $\sim 5\times 10^9 \msun$ at high redshift $z\sim 6$; (e) jets produced in successive accretion episodes can have similar directions, but after several episodes the jet direction deviates significantly. Rare examples of massive holes with larger spin parameters could result from prograde coalescences with SMBH of similar mass, and are most likely to be found in giant ellipticals. We compare these results with observation. (abridged)
1 INTRODUCTION
The paper addresses how SMBHs grow despite merger-driven mass supply and the high efficiencies of rapidly spinning holes. It proposes that self-gravity limits accretion into chaotic, episodic discs whose random orientations preserve observed AGN properties.
- Growth problem: Major mergers can supply black holes with mass increases ΔM_merger of order their existing mass M.Assuming all accreted mass shares one angular-momentum direction would spin holes up to Kerr parameters near unity.
- Growth problem: Rapidly spinning holes have high radiative efficiency, so the Eddington limit restricts growth and complicates the existence of very massive black holes at early cosmic times.
- Growth problem: ∼10^-3 of the galaxy bulge mass is typically contained in the black hole, while cosmological simulations cannot resolve the relevant accretion hydrodynamics.At 1 pc, disc inflow times are ≳10^9 yr, over ten times the rapid-growth timescale, and simulations resolve scales at least 100 times larger.
- Self-gravity constraint: Accretion discs become self-gravitating when their mass exceeds M_sg ∼(H/R)M, with AGN self-gravity expected outside R_sg ∼0.01–0.1 pc.Gas outside this radius is expected to form stars or be expelled, while inner gas forms the accretion disc powering the AGN.
- Proposed picture: High-luminosity growth is modeled as rapid sequences of self-gravity-limited accretion episodes with masses ΔM_episode ∼M_sg, totaling ΔM_merger.Random episode orientations reproduce characteristic jet behavior and preserve broad agreement with AGN luminosity functions and expected merger rates.
- Scope: Simple arguments are used because a proper hydrodynamic treatment of the complex accretion event awaits advances in computing power.
2 SMBH SPINUP
The paper explains SMBH spin evolution through competing accretion and Lense–Thirring torques. Self-gravity limits disc angular momentum, making co- and counter-alignment comparably likely and driving holes toward modest spin.
- Motivation: Spin governs accretion efficiency and therefore the Eddington-limited maximum growth rate, with low-spin holes growing faster than holes with a ∼1.
- Alignment mechanism: The Lense–Thirring effect aligns or counter-aligns a randomly oriented disc before accretion torque substantially changes the hole spin.Its large lever arm makes it operate more quickly than the accretion torque.
- Model interpretation: In warped discs, J_d should be interpreted as the angular momentum passing through the warp radius during alignment, rather than necessarily the whole outer disc.
- Spin evolution: For significant Kerr spin, retrograde ISCO accretion carries specific angular momentum 11:3 relative to prograde accretion, making spindown more effective than spinup.
- Alignment mechanism: Self-gravity limits the disc angular momentum to J_d < 2J_h in almost all cases, so randomly oriented episodes produce nearly equal co- and counter-alignment frequencies.
- Spin evolution: Repeated episodes therefore drive holes toward low spin, with a mean value ā ∼0.2–0.3.
3 SMBH ACCRETION EPISODES
The accretion model treats AGN feeding as evolving, self-gravity-limited thin discs. Episode masses and durations follow from standard disc structure, while major mergers comprise chaotic sequences whose detailed time profile does not affect the conclusions.
- Disc model: The disc is modeled as a sequence of steady states using AGN disc properties derived by Collin–Souffrin and Dumont, essentially matching Shakura–Sunyaev steady discs.
- Self-gravity limit: The disc mass inside radius R and its semi-thickness H determine when the disc becomes self-gravitating.
- Episode properties: The self-gravitating mass M_sg is identified with the mass ΔM_episode accreted in one feeding episode.
- Episode properties: τ_sg = M_sg/Ṁ sets the disc evolution timescale and characteristic duration of an accretion episode.
- Merger implementation: A major merger is represented as a chaotic sequence of possibly overlapping or slightly separated episodes, with conclusions depending only on total accreted mass and angular momentum.
4 THE EVOLUTION OF BLACK HOLE MASS AND SPIN
Self-gravity-limited, randomly oriented accretion episodes drive black-hole spins toward moderate mean values while preserving substantial fluctuations. Simulations show that the mean depends weakly on α2 and that spin evolution affects the mass-growth rate.
- Jd remains below 2Jh except when a is small, so retrograde episodes can stably counter-align and spin the hole down.
- Randomly oriented episodes decrease a until expected prograde spinup balances retrograde spindown as Jh falls.
- The simulations assign each episode mass Msg, Eddington-rate accretion, self-gravity-limited angular momentum, isotropic orientation, and co- or counter-alignment by the stated criterion.
- Numerical Simulations: The accretion-time simulations omit non-accreting epochs and therefore measure the shortest possible time for accretion to build the hole's mass.
- Numerical Simulations: Changing α2 from 0.03 to 1 lowers the mean spin, while both simulations use the same random orientation sequence.
- Numerical Simulations: For M≈10^6–10^9 M⊙, mean a is ≈0.3–0.2 at α2=0.03 and ≈0.2–0.1 at α2=1, with excursions Δa≈±0.2.
5 SMBH COALESCENCES
Accretion rapidly restores a coalesced black hole toward the accretion-driven mean as its mass doubles, so coalescences have little long-term effect on the mean spin.
- Accretion drives a coalescence product back toward the mean spin trend as the hole doubles its mass.
6 SMBH RECOIL VELOCITIES
The paper predicts modest recoil from SMBH coalescences because the holes generally have relatively low spins. These recoil velocities are below the escape velocity of the merged host.
- Recoil velocities from comparable-mass SMBH coalescences are expected to be ≲200 km s−1.
- These velocities lie below the merged host's escape velocity, consistent with most massive galaxies retaining nuclear SMBH.
- Unlike an alternative aligned-flow explanation, this model attributes modest recoil to self-gravity-limited accretion and nearly equal co- and counter-alignment.
7 MAXIMUM MASS GROWTH RATE FOR SMBH
The paper concludes that randomly oriented accretion episodes keep spin and radiative efficiency sufficiently low for rapid SMBH growth. This permits growth from stellar seeds to the largest observed high-redshift masses.
- Black holes can grow from stellar initial masses to ≈5 × 10^9 M⊙ by z≈6 when a≲0.5 keeps accretion efficiency sufficiently low.
8 SMBH SPIN DIRECTIONS
The paper tracks how accretion episodes change SMBH spin directions and therefore jet orientations. Episodes can preserve jet alignment briefly, but several episodes erase the original spin-axis direction and host-galaxy correlation.
- 8 SMBH SPIN DIRECTIONS: The inner accretion disc and any resulting jet are aligned or anti-aligned with the black-hole spin axis.Observations cited in the paper find radio, [O III], dust-disc, and optical-jet orientation relationships.
- 8 SMBH SPIN DIRECTIONS: Accretion aligns the hole spin with the total angular-momentum vector Jt = Jh + Jd, while the disc angular momentum remains smaller than the hole's.Near the mean spin, Jd/Jh is <= 0.26 or 0.16 for α2 = 0.03 or 1.
- 8 SMBH SPIN DIRECTIONS: Successive accretion episodes tend to produce jets in similar directions, especially when the spin is larger than average.This may account for double-double radio sources with apparently common projected axes.
- 8 SMBH SPIN DIRECTIONS: After a few episodes, the spin axis loses memory of its original direction and shows no correlation with host-galaxy structures; coalescences can also disorient it.
9 DISCUSSION
The discussion argues that randomly oriented, self-gravity-limited accretion episodes drive SMBH toward modest spins with fluctuations, while coalescences have little long-term effect on the mean. Observational spin constraints are sensitive to efficiency systematics and selection bias, with larger individual spins possible in special cases.
- Coalescences: Coalescences can rapidly reduce spin, with mass doubling lowering a by a factor ∼5.3, but subsequent accretion pushes it back toward the mean.Retrograde coalescences reduce spin more effectively than prograde coalescences increase it, so coalescences have little net long-term effect on mean spin.
- Observational constraints: The effective accretion efficiency for the Soltan argument is the symmetrized efficiency ǫ± = 1/2[ǫ(a) + ǫ(−a)] when retrograde and prograde accretion are equally probable.Figure 5 compares efficiency against spin and distinguishes retrograde accretion by negative a.
- Observational constraints: The symmetrized efficiency spans only 0.057–0.230, so only a very accurate efficiency estimate can constrain spin reliably.
- Observational constraints: Broad-iron-line analyses have suggested a > 0.78 in one AGN group, but selection and line-strength effects make the fraction of such high-spin SMBH uncertain.
- Exceptions and scope: Spin-sensitive observations are biased toward a ∼1 because prograde accretion is especially efficient near maximal spin.
- Exceptions and scope: The statistical treatment does not exclude larger individual spins, which may arise from prograde coalescences of similarly massive SMBH and may occur in giant ellipticals.The thin-disc, efficient-cooling assumption may fail in rare situations, so the general conclusions may not apply there.