Table Of ContentA universal minimal mass scale for present-day
central black holes
7
1
0 Tal Alexander1 and Ben Bar-Or2
2
n
a 1Department of Particle Physics & Astrophysics, Weizmann Institute of
J
Science, Rehovot 76100, Israel.
2
] 2Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
A
G
. Abstract
h
p Intermediate mass black holes (BHs) with masses in the range M ∼
•
- 102–105M arethelong-soughtmissinglink[1]betweenstellarmassBHs
o (cid:12)
born of supernovae, and supermassive BHs, which are tied to large-scale
r
t galactic evolution, as suggested by the yet unexplained empirical M/σ
s
a correlationbetweenthecentralBH’smassandthestellarvelocitydisper-
[ sionofthehostgalaxy’sbulge[2,3],M•(σ(cid:63))=Ms(σ(cid:63)/σs)β. Weshowthat
low-massBHseedsthatgrowbyaccretingstarsingalacticnucleithatfol-
1
low a universal M/σ relation[4, 5], all grow over the age of the universe
v
above a present-day mass scale M ∼2×105M (5% lower C.L.), inde-
5 0 (cid:12)
pendentlyoftheunknowninitialseedmassanditsformationprocess. M
1 0
4 depends only weakly on the uncertainties in the BH formation redshift,
0 and provides a universal minimal mass scale for BHs that grow also by
0 gasaccretionormergers. ThiscanexplainwhynointermediatemassBHs
. with mass M <M were found to date[6], and implies that present day
1 • 0
galaxies with velocity dispersion σ < S = σ (M /M )1/β ∼ 35kms−1
0 (cid:63) 0 s 0 s
7 (5% lower C.L.) do not have a central BH, or formed their seed BH only
1 recently. A dearth of BHs with mass below M0 has observable implica-
: tionsforthenatureandratesoftidaldisruptionofstarsbycentralBHs[7]
v
and for gravitational wave (GW) sources from BH–BH mergers and the
i
X inspiral of compact stellar remnants into BHs in galactic nuclei[8].
r TheexistenceofBHswithmassM <100M isnowconfirmedbythedetection
a • (cid:12)
of gravitational waves (GW) from stellar mass BH-BH mergers[9]. Massive
black holes (MBHs) with M (cid:38) 106M are detected in galactic nuclei by a
• •
variety of methods[6], and are often observed as luminous active galactic nuclei
(AGN). The theoretical interest and observational quest for intermediate mass
BHs (IMBHs) has been going on for many decades now[10], starting soon after
the identification of quasars as accreting MBHs[11]. However, there are no
definite detections of IMBHs to date[6]. Although it is harder to detect the low
1
mass IMBHs[12], the continuing absence of firm evidence for their existence is
increasingly puzzling.
ObservationsindicatethatAGNacquiremostoftheirmassbyluminousac-
cretionofgas[13,14]inapparentcorrelationwithcosmicstarformationhistory
from at least redshift[15] z ∼3. The existence of very high-redshift quasars[16]
impliesthatrapidBHgrowthandluminousaccretioncanoccurverysoonafter
the Big Bang. The early stages of MBH evolution are poorly understood, and
in particular the formation processes and masses of the initial BH seeds remain
unknown[17]. Suggested formation mechanisms typically rely on the extreme
conditions in the early universe (e.g. very low metallicity, very high gas or star
density), and lead to rapid BH seed formation. It is therefore plausible that
the seeds of present-day central BHs were formed at least a Hubble time ago
(z >1.8 for a look-back time of t =10 Gyra).
H
The M/σ relation that is observed in the local universe over (cid:38) 3 decades
in MBH mass, reveals a tight correlation between M and the stellar velocity
•
dispersion σ of the host galaxy’s bulge, well outside the MBH’s radius of di-
(cid:63)
rect dynamical influence[6], r ∼ GM /σ2. This suggests an evolutionary link
h • (cid:63)
between the growth of the central BH and its galaxy. Recent analyses of large
heterogeneous samples of MBHs and their host galaxies find that a universal
M/σrelationb holdsforallgalaxytypes[4,5],althoughthescopeofthisrelation
and its evolution with redshift remain controversial[15]. We assume here this
universal M/σ holds at all redshifts[19]. By fixing r , the M/σ relation im-
h
poses tight connections between the MBH mass and the dynamical properties
of its immediate stellar surroundings[20], and specifically the rate at which it
consumes stars (see Appendix).
A central BH grows by (1) The accretion of stars, compact remnants and
darkmatterparticlesthataredeflectedtowarditonnearlyradialorbits. These
either cross directly the event horizon if they are dense enough, or else are first
tidally disrupted, and then part of the debris is accreted on the BH[7]. (2) The
accretion of unbound gas from the interstellar medium, which must first lose
almost all of its angular momentum by viscous processes to circularize down
to the innermost stable circular orbit and fall through the event horizon. (3)
A merger with another BH, by the formation of a binary system that initially
decays dynamically and finally decays by the emission of GWs.
Ofthesethreegrowthchannels,onlytheaccretionofstarsisanunavoidable
consequence of the existence of a central BH in a stellar system. Furthermore,
the stellar tidal disruption event (TDE) rate in steady state can be estimated
from first principles, for given boundary conditions[21, 22]. It has been noticed
early on that the natural scale for the steady state TDE rate on low-mass
IMBHs/MBHs (M < 106M ), Γ ∼ 10−4yr−1 (see appendix) implies that
• •
these may acquire a substantial fraction of their mass from TDEs, and that
this also implies, in the linear growth approximation, a natural BH mass scale,
aFortheconcordancecosmologicalmodel[18]: h=0.696,Ω=0.286,ΩΛ=0.714.
bHere we adopt the empirical fit[5] M• = 108.32±0.04±(cid:15)(σ(cid:63)/200kms−1)(5.35±0.23)M(cid:12),
where(cid:15)=0.49±0.03istheintrinsicscatter.
2
MTDE ∼Γt ∼106M [23, 24]. However, previous studies typically focused on
• H •
the rates and prospects of TDE detection, and not on long-term BH growth.
Although it was recently argued that MTDE arises as a minimal BH mass in
•
a specific formation model[25], the commonly held assumption remains that
IMBHs with M (cid:28)MTDE do exist, and therefore this constrains the formation
• •
scenarios, or sets an upper bound on the efficiency of TDE accretion, rather
than a lower bound on the mass of IMBHs[26].
Here we derive a universal lower bound on the present-day mass scale of
MBHs, M , that is independent of the unknown initial seed mass and its for-
0
mation process. We argue that quasi-steady state around a low-mass BH is
a plausible and self-consistent assumption, and apply the empirical M/σ rela-
tion to set the boundary conditions. We then show that the exact non-linear
BH growth equation can be bound from below by a simple inequality that
includes only growth by TDEs, and derive the universal lower mass bound
M ∼ few×105M . Intriguingly, M is just below the lowest-mass MBHs
0 (cid:12) 0
discovered to date[6], min(M )(cid:46)106M .
• (cid:12)
Stars around a central BH are constantly scattered in angular momentum
√
to nearly radial orbits below a critical (“loss-cone”) value, j = 1−e2 (e
lc
is the orbital eccentricity), that takes them closer to the BH than the tidal
disruption radius, r (cid:39) Q1/3R , where they are destroyed (Q = M /M is the
t (cid:63) • (cid:63)
BH to star mass ratio, R is the stellar radius). The tidal radius for a typical,
(cid:63)
Solar type star is outside the BH’s event horizon for M <108M . Such stars
• (cid:12)
are therefore first disrupted by an IMBH, rather than swallowed whole. Only
a fraction f ∼ 1/4–1/2 of their mass is then accreted by the BH[27]. The
a
TDE rate of stars originating at a distance r from the BH depends on the
number of stars there, and on the competition between the 2-body relaxation
time and the orbital time in supplying and draining loss-cone orbits. In steady
state, the integrated contribution from all radii is a function of M and the
•
boundary conditions at r , fixed by σ . The TDE rate (see Appendix) is well-
h (cid:63)
approximated by two simple power-laws Γ (cid:39) Γ Qb(β) (Eqs A-13, A-19) whose
(cid:63)
indexchangesacrossacriticalBHmassM ∼106M . Generally,thedynamics
c (cid:12)
leading to tidal disruption are dominated by stars originating from ∼ r (M )
h •
for M ≥ M , and by stars originating from a critical radius r (M ) < r (M )
• c c • h •
for M < M [21] (see appendix). For the empirically-determined range of the
• c
M/σ relation index[6, 15], 4(cid:46)β <6, the power-law index is always b(cid:28)1.
Assume that a BH seed is formed in a stellar system of mass M with a
sys
large enough initial mass M to dominate its radius of influence, M (cid:28) M (cid:28)
i (cid:63) i
min(M ) (cid:28) M . Consider first the case where the BH grows only by the
• sys
accretion of stars. The BH growth equation is then
Q˙ =f Γ Qb, Q(0)=Q =M /M , (1)
a (cid:63) i i (cid:63)
and its solution is
(cid:40)
[Q 1−b+(1−b)t/t ]1/(1−b) b(cid:54)=1
Q(t)= i a , (2)
Qiet/ta b=1
3
where t = (f Γ )−1 is the accretion timescale. The growth solution has three
a a (cid:63)
branches. The solution for b = 1 is exponential, which diverges to infinity in
infinite time. When b>1, Q diverges on a finite timescale,
t =t Q1−b/|1−b| , (3)
∞ a i
and is supra-exponential. The b ≥ 1 solutions are functions of the initial BH
seedmassQ onalltimescales. Exponentialgrowthdescribesradiationpressure-
i
regulated accretion of gas at the Eddington limit. Supra-exponential growth
describes Hoyle-Lyttleton wind accretion[28], spherical Bondi accretion[29], or
their generalization of accretion on an accelerating BH[30].
In contrast, when b < 1, which is the relevant case for the M/σ relation,
Q grows sub-exponentially and asymptotes at t (cid:29) t to a power law that is
∞
independent of initial seed mass Q
i
Q(t)(cid:39)[(1−b)t/t ]1/(1−b). (4)
a
Since t (cid:28) t for BH seeds with mass M (cid:46) 105M (Eq. 3), all BH seeds
∞ H • (cid:12)
reach the same present-day mass after t∼O(t ) (Figure 1).
H
Consider next the realistic case where the BH grows in parallel by the typi-
cally dominant channels of gas accretion and/or mergers, Q˙ =Q˙ +Q˙ , where
(cid:63) +
Q˙ = f Γ Qb (Eq.1) is the mass accreted from stars, and Q˙ ≥ 0 is the mass
(cid:63) a (cid:63) +
accreted from the other channels. The full growth equation is then
Q˙ =Q˙ +Q˙ ≥f Γ Qb, Q(0)=Q . (5)
(cid:63) + a (cid:63) i
The solution of the stars-only growth Eq. (1) then provides a lower limit on
the actual mass Q(t) of the growing BHc. The universal minimal mass scale of
central BHs at t , and the corresponding minimal velocity dispersion scale are
H
then (Eq. 4)
M =[(1−b)f Γ t ]1/(1−b)M , S =(M /M )1/βσ , (6)
0 a (cid:63) H (cid:63) 0 0 s s
withtheindexbfortheM <M branchofEq. (A-14). Galaxieswithσ <S
• c (cid:63) 0
do not have a central BH, or have formed it only recently, for otherwise the
co-evolution of the BH and nucleus over t would have driven σ to a much
H (cid:63)
larger present-day value.
M increases with the power-law index α of the stellar cusp around the
0
BH (n ∝ r−α) and with the accreted mass fraction f . Assuming Solar type
(cid:63) a
stars, M extends over the relevant range in α and f from 1.2×105M for
0 a (cid:12)
the shallowest possible cusp of unbound stars and a low accreted mass fraction
(α = 1/2, f = 1/4) to 1.3×106M for a steep isothermal cusp with a high
a (cid:12)
accreted mass fraction (α = 2, f = 1/2). M = 8.7×105M for the fiducial
a 0 (cid:12)
parameters we adopt here: a dynamically relaxed cusp[31] where α=7/4, and
an intermediate accreted mass fraction f = 3/8. Note that the agreement
a
cThe stellar mass contribution Q(cid:63) to the final BH mass Q is related to the stars-only
growthsolutionQbyQ(cid:63)<Q≤Qforb<0,butbyQ≤Q(cid:63)≤Qforb≥0.
4
withtheobservedlowerlimitonBHmasses,M (cid:46)min(M )∼106M ,follows
0 • (cid:12)
frombasiclocalphysics(tidaldisruptionandloss-conedynamics)andempirical
global properties of the universe (a universal M/σ relation and the age of the
universe). It is not introduced implicitly by any of the assumptions.
The scatter around the universal M/σ relation is thought to reflect intrin-
sic physical differences between individual galaxies, which are well beyond the
measurementerrors(footnoteb). Therefore,itinducesroughlyGaussianintrin-
sic probability distributions for M and S : M = (1.0±0.7)×106M and
0 0 0 (cid:12)
S =75±29kms−1. Thelowest-σ galaxiesknowntoharborAGN[32,33](and
0 (cid:63)
hence BHs, with roughly estimated masses in the range 105 (cid:46) M (cid:46) 106M ),
• (cid:12)
haveσ ∼30−40kms−1. Thiscorrespondstothe5%lowerC.L.S (cid:46)35kms−1
(cid:63) 0
and M (cid:46)2×105M , which we adopt here as typical lower mass and velocity
0 (cid:12)
dispersion limits for BHs and their host galaxies. Lower mass BHs are rarer
yet: an IMBH with mass ≤ 104M is at the 10−4 C.L., and one with mass
(cid:12)
≤103M is below the 10−6 C.L.
(cid:12)
This derivation of a universal minimal mass scale for present-day central
BHs rests on four assumptions. (1) Most BH seeds were formed in the early
universe, at a look-back time t ∼O(t ). (2) The growth is typically not mass
i H
limited, that is, the growing BH is embedded in a massive-enough system with
M (cid:29) M at all times. (3) The accretion of stars proceeds efficiently[34]
sys •
(f (cid:38) few×0.1) in a relaxed stellar cusp. (4) The boundary conditions at the
a
radius of influence are set by a universal M/σ relation at all times.
The assumption of an early start for BH seeds is plausible given the early
appearance of AGN, and since suggested BH seed formation mechanisms typ-
ically require the extreme conditions of the early universe[17]. The require-
ment of enough stellar mass for accretion is also plausible, given that the
formation and retention of a BH seed typically require a substantially more
massive host system, and given the modest mass scale of M ∼ O(105M ).
0 (cid:12)
The growth rate of the BH by the accretion of stars in an isolated system,
dM /dt∼f M N /(log(1/j )T ) (Eq. A-7), where M N is the stellar mass
• a (cid:63) (cid:63) lc R (cid:63) (cid:63)
intheradiusofinfluenceandT therelaxationtime,isbydefinitionslowenough
R
to allow the system to adjust and remain in equilibrium, since the timescale for
the BH to grow by order of the stellar mass, (dM /dt)/(M N ), is longer by
• (cid:63) (cid:63)
a factor log(1/j )/f (cid:29)1 than the timescale for the cluster’s return to steady
lc a
state, T (cid:39)T /4 (Eq. A-3).
SS R
The least secure assumption is that a universal M/σ relation holds near its
present-day value as the BH grows. However, this is broadly consistent with
observations of quasars up to[35, 19] z ∼1, and with simulations of large scale
structure evolution up to[36, 37] z ∼4.
BH–BHmergersnotonlyincreasetheBHmass,butalsoaffectthedynamics
of the stellar system around it. Although mergers initially increase the plunge
rate by randomizing the orbits and / or by the Kozai resonance[38], on longer
timescales they may strongly suppress it by ejecting stars from the radius of
influence. Such suppression is not effective around IMBHs, because the return
to steady state is fast for M (cid:46) 105M , T < O(108yr) (Eq. A-3)[39, 40].
• (cid:12) SS
Additional growth channels thus only increase present day BH masses, and
5
Look-back time [Gyr]
00..22 00..55 11 22 33 44 55 66 77 88 99 1100 1111 1122 1133
66
1100
M = 105.6 ± 0.5 M
0 O•
]O•
M 55
[z1100
M
110044 M = 2 103 M
i O•
4
M = 10 M
i O•
4
M = 5 10 M
i O•
00..0011 00..11 11 1100
z
Figure 1: The growth of the minimal BH mass M from the formation redshift
z
z at look-back time t , as function of redshift or time (Eq. 4) for 3 values
i i
of the BH seed mass, and for 3 seed formation redshifts spanning an order
of magnitude in cosmic time (for the universal M/σ relation without scatter,
assuming an α = 7/4 cusp of Solar type stars and an accreted mass fraction
f = 3/8). The convergence to evolution that is independent of initial mass
a
is rapid in redshift. The earlier the formation, the higher is M . The range
0
z =0.1−10offormationredshiftstranslatetoalowermasslimitonpresent-day
MBHs in the range M ∼10(5.6±0.5)M (gray band).
0 (cid:12)
reinforcetheconclusionthatgalacticnucleiwithM <M BHsshouldberare.
• 0
Figure (1) shows the redshift evolution of the lower mass limit M , which
z
increasesrapidlywithdecreasingredshifttoitspresent-dayvalueM . Present-
0
dayIMBHsmayexistinrecentlyformed,orinlower-masssystemswheregrowth
is supply-limited (e.g. globular clusters with M <106M ). However, it is not
(cid:12)
clear whether such low-mass systems can form a BH seed.
Early TDE-driven growth and the suppression of the cosmic BH mass func-
tion below M has implications for BH seed evolution, the cosmic rates and
0
properties of TDEs and of GWs from intermediate mass ratio inspiral events
(IMRIs). We conclude by listing these briefly.
A high rate of TDEs can allow BH seed growth to continue despite the
depletion of gas available for accretion by supernovae feedback[41]. The lack
of IMBHs at low redshifts means that electromagnetic searches will have to
6
reach high redshifts to detect TDEs from an IMBH (jetted TDEs may provide
an opportunity[42]). The prospects of detecting exotic processes related to
IMBHs, such as a tidal detonation of a white dwarf in the steep tidal field of
a low-mass BH[43], will be low. The mean observed rate of TDEs per galaxy,
Γ ∼ 10−5yr−1gal−1, is substantially lower than predicted rates[44]. A dearth
of BHs below M may resolve, at least in part, the rate discrepancy.
0
IMBHs produce GWs by IMRIs and by IMBH-IMBH mergers. IMRI de-
tection by planned space-borne GW observatories[8, 45, 46] is limited to z <
few×0.1, and is therefore unlikely. In contrast, IMBH mergers can be detected
to very high redshifts. A search for GWs from IMBH mergers and IMRIs can
reveal the formation history of BH seeds. We predict that BH seeds are driven
earlyontohighermassbytheaccretionofstars, andthereforeIMBHsarevery
rare in the present-day universe, but will be found near their high formation
redshifts.
Acknowledgments WearegratefulforhelpfuldiscussionswithY.Alexander,
J. Gair, A. Gal-Yam, J. Green, M. MacLeod, N. Neumayer, T. Piran, E. Rossi,
A. Sesana, N. Stone and B. Trakhtenbrot. TA acknowledges support by the
I-CORE Program of the Planning and Budgeting Committee and The Israel
Science Foundation (grant No 1829/12). BB acknowledges support by NASA
(grant NNX14AM24G) and by the NSF (grant AST-1406166).
Appendix
Summary of results from loss-cone theory
Thetidaldisruption(plunge)ratecanbeapproximatedbythefluxofstarsinto
the BH from from the boundary between the inner region, where stars slowly
diffuse into the loss-cone (the empty loss-cone) and the outer region, where
stellar scattering is strong enough that the loss-cone is effectively full (the full
loss-cone)[21]. The boundary is at a critical radius that satisfies
[J (a )/J ]2P(a )
q = c c lc c =1, (A-1)
log(J (ra )/J )T (a )
c c lc R c
√
(cid:112)
where P =2π a3/GM is the orbital period, M is MBH mass, J = GM a
• • c •
is the circular angular momentum at a, J is angular momentum of the loss-
(cid:112) lc (cid:112)
cone (J (cid:39) 2GM Q1/3R for tidal disruption, so J /J (cid:39) (a/R )/2Q1/3).
lc • (cid:63) c lc (cid:63)
T is the 2-body (non-resonant) relaxation time [40],
R
5 Q2P(a)
T (a)= , (A-2)
R 8N (a)logQ
(cid:63)
where Q = M /M , N (a) = µ Q(a/r )3−α is the number of stars enclosed in
• (cid:63) (cid:63) h h
r and r = η GM /σ2 is the radius of influence. The numeric prefactors are
h h • (cid:63)
conventionally assumed to be µ =2 and η =1.
h h
7
Thetimeforagalacticnucleustorecoverfromaperturbationandreturnto
steady state is[40]
T =T (r )/4. (A-3)
SS R h
The exact solution for the critical radius can be written as
(cid:18)5 R c2 (cid:16)σ(cid:17)2 (cid:19)1/(4−α)
δ =a /r = (cid:63) Q1/3
c c h 4µ Λ η GM c
h lc h (cid:63)
= (cid:0)A s2(cid:1)1/(4−α)Q(1+6/β)/(12−3α), (A-4)
c
where
5 R c2
A = (cid:63) , (A-5)
c 4µ Λ η GM
h lc h (cid:63)
Λ (Q,a)=logQ/log(J (a)/J ) , (A-6)
lc c lc
and where the last equality in Eq. (A-4) assumes the M/σ relation σ =
(cid:63)
(M /M )1/βσ , in terms of the dimensionless velocity dispersion scale s =
• s s
(M /M )−1/βσ /c.
s (cid:63) s
When δ > 1, the rate is estimated at[47] r . The transition occurs above
c h
M ∼ O(106M ), independently of α and almost independently of β. The
1 (cid:12)
plunge rate can then be conservatively approximated by the empty loss-cone
rate at a =min(a ,r ),
e c h
N (a ) 8 Λ µ2 (cid:18)a (cid:19)(9−4α)/2
Γ(cid:39) (cid:63) e = lc h e . (A-7)
log(J (a )/J )T (a ) 5P (r ) r
c e lc R e h h
The actual rate, including the contribution from the full loss-cone regime, can
be up to ×2 higher [47].
Defining r =GM /c2 and t =GM /c3, Γ can be represented as
(cid:63) (cid:63) (cid:63) (cid:63)
Γ(δ )= 1 (cid:40)γc(Q,ae)Q(2α−15)/6(4−α)(cid:0)σc(cid:63)(cid:1)7(3−α)/(4−α) δc ≤1 , (A-8)
c πt(cid:63) γh(Q,ae)Q−1(cid:0)σc(cid:63)(cid:1)3 δc >1
where
(cid:18)4Λ (cid:19)2α−1(cid:32) µ (cid:33)7(cid:18)R (cid:19)9−4α1/2(4−α)
γc(Q,ae)= 5lc η(3−hα) r(cid:63)(cid:63) , (A-9)
h
and
4µ2Λ
γ (Q,a )= h lc . (A-10)
h e 5η3/2
h
When σ is given by the M/σ relation, the rate can be expressed as
Γ(δ )=ΓΛ(Q,δ )Qb(δc), (A-11)
c (cid:63) c
8
where
(cid:40)
1 γ (Q,a )s7(3−α)/(4−α) δ ≤1
ΓΛ(Q,δ )= c e c . (A-12)
(cid:63) c πt γ (Q,a )s3 δ >1
(cid:63) h e c
ThenotationΓΛ denotestheweakfunctionaldependenceonQviathelogarith-
(cid:63)
mic term Λ (Q,a). The power-law index is
lc
(cid:40)
b(δ )= b< =7(3−α)/β(4−α)−(15/2−α)/3(4−α) δc ≤1 . (A-13)
c
b> =3/β−1 δc >1
For an n ∝r−α steady state stellar density cusp [31] with α=7/4, the power-
(cid:63)
law index is
(cid:40)
(105−23β)/27β M ≤M
b(α=7/4)= • c . (A-14)
3/β−1 M >M
• c
Note that b (cid:28) 1 for the empirically-determined range of the M/σ relation
index[6, 15], 4(cid:46)β <6.
Itisusefultoapproximatetherateasatruepower-lawinordertoapplythe
analyticsolutionforthegrowthequation(Eq. 2). Thiscanbedonebychoosing
a fixed typical value for the logarithmic term, and then
Γ(δ )(cid:39)Γ (δ )Qb(δc), (A-15)
c (cid:63) c
where the normalization Γ is not a function of Q. Averaging the logarithmic
(cid:63)
termoverQ=103to107forα=7/4,µ =2η =1andβ =5,andnormalizing
h h
the M/σ relation by requiring that the radius of influence satisfy
r =r (Q )=GM2/β(Q M )1−2/β/σ2, (A-16)
0 h 0 s 0 (cid:63) s
(for the universal M/σ relation[5], r (cid:39) 0.79pc for Q = 106 and M = 1M )
0 0 (cid:63) (cid:12)
yields a constant B (cid:39)0.7 such that
c
(cid:20) 5 R (cid:21)1/(4−α)
δ (cid:39)B (cid:63)Q1−2/β Q(6+β)/3β(4−α). (A-17)
c c 4µ r 0
h 0
The value of Q where δ =1 is then
c
(cid:20)4µ r (cid:21)3β/(6+β)
Q =B3β(α−4)/(6+β) h 0 Q2/β−1 . (A-18)
c c 5 R 0
(cid:63)
Similarly, the logarithmically-averaged plunge rate can be approximated with a
constant B (cid:39)2.25 and expressed by the normalization
p
(cid:40)
γ 1 δ ≤1, (Q≤Q )
Γ (δ )= p c c , (A-19)
(cid:63) c πt Qb<−b> δ >1, (Q>Q )
(cid:63) c c c
9
σ [km/s] (β=5)
20 30 40 50 60 80 100 200
-3
10
Q /(f t )
a H
α=7/4 β=4 exact
α=7/4 β=5 exact
α=7/4 β=4 PL b= 0.12
α=7/4 β=5 PL b=-0.07
L > L
yr] TD E
e [1/ 10-4
at
R
-5
10
3 4 5 6 7 8 9
10 10 10 10 10 10 10
M [M ]
• O•
Figure 2: Examples of the exact plunge rate (Eq. A-7) as function of the
BH mass M or σ (for a β = 5 M/σ relation), compared to the power-law
• (cid:63)
approximation (Eq. A-15), for M =1M , R =1R , f =3/8, α=7/4 and
(cid:63) (cid:12) (cid:63) (cid:12) a
β =4, 5,normalizedtor (106M )=0.79pcandN (r )=2Q. Thetwomatch
h (cid:12) (cid:63) h
quite well. The maximal steady state stellar accretion rate that is consistent
with the mass of the BH, maxΓ = M /(M f t ) is compared to the plunge
• (cid:63) a H
rates. This simple linear estimate already indicates that BH growth by the
accretion of stars is important for low mass MBHs [23, 24], and is inconsistent
(too fast) for BHs with present-day mass M (cid:46)5×105M , which should have
• (cid:12)
grownabovetheirassumedmass. Radiationback-reactionabovetheEddington
limitL bytheaccretionluminosityL =ηf M c2Γ(η =0.1assumedforthe
E TD a (cid:63)
radiative efficiency) is only relevant for the lower BH masses, for some values of
the M/σ relation, where it may slow down the initial BH growth. For example,
for β = 5, L /L ∼ 6 at M = 103M , but falls below L /L = 1 for
TD E • (cid:12) TD E
M >5×103M .
• (cid:12)
where
(cid:18)r (cid:19)3/2
γ (cid:39) B B(9/2−2α) (cid:63) ×
p p c R
(cid:63)
(cid:34)(cid:18)4(cid:19)2α−1 (cid:18)R (cid:19)7(3−α)(cid:35)1/2(4−α)
µ7 (cid:63)Q1−2/β . (A-20)
5 h r 0
0
10
