Gaseous Dynamical Friction in Presence of Black Hole Radiative Feedback

0.73 MB·

by  KwangHo Park

Draft version March 28, 2017
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
GASEOUS DYNAMICAL FRICTION IN PRESENCE OF BLACK HOLE RADIATIVE FEEDBACK
KwangHo Park and Tamara Bogdanovic´
CenterforRelativisticAstrophysics,SchoolofPhysics,GeorgiaInstituteofTechnology,Atlanta,GA30332,USA
Draft version March 28, 2017
ABSTRACT
Dynamicalfrictionisthoughttobeaprincipalmechanismresponsiblefororbitalevolutionofmassive
black holes (MBHs) in the aftermath of galactic mergers and an important channel for formation of
7 gravitationally bound MBH binaries. We use 2D radiative hydrodynamic simulations to investigate
1 theefficiencyofdynamicalfrictioninthepresenceofradiativefeedbackfromanMBHmovingthrough
0
a uniform density gas. We find that ionizing radiation that emerges from the innermost parts of the
2
MBH’s accretion flow strongly affects the dynamical friction wake and renders dynamical friction

r inefficient for a range of physical scenarios. MBHs in this regime tend to experience positive net
a acceleration, meaningthattheyspeedup, contrarytotheexpectationsforgaseousdynamicalfriction
M
in absence of radiative feedback. The magnitude of this acceleration is however negligibly small and
should not significantly alter the velocity of MBHs over relevant physical timescales. Our results
4 suggest that suppression of dynamical friction is more severe at the lower mass end of the MBH
2
spectrum which, compounded with inefficiency of the gas drag for lower mass objects in general,
] implies that < 107M(cid:12) MBHs have fewer means to reach the centers of merged galaxies. These
A findings provide formulation for a sub-resolution model of dynamical friction in presence of MBH
radiative feedback that can be easily implemented in large scale simulations.
G
Subject headings: accretion, accretion disks – black hole physics – hydrodynamics – radiative transfer
.
h
p
- 1. INTRODUCTION tobestrongerformassiveperturbersgiventhattheycan
o interact with a sufficiently dense pool of ambient gas.
Amassiveperturbermovingthroughsomebackground
r Because of its efficiency, gaseous drag has been exten-
t mediumcreatesinitadensitywake. Thewaketrailsthe
s perturberandexertsonitagravitationalforceinthedi- sively investigated in simulations that follow pairing of
a massiveblackholes(MBHs)ingasrichmergersofgalax-
rectionoppositetothemotion,thusactingasabrakeand
[ ies (see Mayer 2013, for review). The most important
earning this interaction a name: “dynamical friction.”
questions for this research area are: in which galaxies
2 In his seminal work, Chandrasekhar (1943) studied the
v efficiency of dynamical friction for a massive perturber does gaseous dynamical friction lead to successful gravi-
6 movingthroughastellarbackgroundandfoundthedrag tationalpairingofMBHsandonwhattimescales? Calle-
2 forcetobemaximizedwhenthevelocityoftheperturber garietal.(2009,2011),forexample,findthatMBHpairs
5 is comparable to the velocity dispersion of stars, v ≈ σ. with mass ratios q < 0.1 are less likely to form gravita-
0 tionallyboundbinarieswithinaHubbletime,largelydue
Ostriker (1999) later evaluated the dynamical friction
0 to the inefficiency of the gas drag on the smaller of the
force acting on a perturber traveling through a uniform
. two MBHs. When pairing is successful, gaseous dynam-
1 gaseousmediumandfoundthatthegaseousdragismore
ical friction is capable of transporting the MBHs from
0 efficient than stellar drag for transonic perturbers. This
galactic radii of a few hundred to ∼ 1pc on timescales
7 work also established that the gaseous drag is operating
1 most efficiently when 1 < M (cid:46) few, where the Mach as short as ∼107year and tens of times faster than stel-
: number M≡v/c is defined as the ratio between the lar dynamical friction (for e.g., Escala et al. 2005; Dotti
v s,∞ et al. 2006; Mayer et al. 2007).
perturber velocity v and the sound speed of the ambient
Xi gas“atinfinity”c ,i.e.,inadistantregionunaffected Interestingly, gravitational interaction of the MBH
s,∞ with the surrounding gas is quite local. For example,
ar bdyyntahmeipcaelrtfurircbteior.nIfonrctheitsamkeisldfloyrmsupersonic regime, the for a BH with mass ∼106M(cid:12) most of the gravitational
drag force is contributed by the gas that resides within
only a few parsecs of the MBH (Chapon et al. 2013).
4π(GM )2ρ (cid:34) (cid:18) 1 (cid:19)0.5(cid:35) This proximity implies that the dynamical friction wake
F =− BH ∞ ln Λ 1− (1)
DF v2 M2 can be strongly affected, and possibly obliterated, by ir-
radiation and feedback from an accreting MBH. Indeed,
where M is the mass of the perturber (hereafter as- studies of dynamical evolution of MBHs find that dy-
BH
sumed to be a black hole; BH) and ρ is the density of namical friction can be significantly reduced due to the
∞
the gas at infinity. lnΛ=ln(r /r ) is the Coulomb wake dispersal caused by purely thermal feedback from
max min
logarithm and r and r represent the smallest and MBHs in simulations that follow gravitationally recoiled
min max
largest spatial scales in the gas wake that contribute to MBHs (Sijacki et al. 2011) and BH pairing (Souza Lima
dynamical friction, respectively. Based on Equation (1), etal.2016)wheretheterm“wakeevacuation”isdubbed.
itfollowsthattheeffectofdynamicalfrictionisexpected These results bring into question the assumed efficacy
of gaseous dynamical friction and call for further explo-
[email protected],[email protected] ration of the effects of radiative feedback powered by
2 Park and Bogdanovi´c
MBH accretion. solution (Hoyle & Lyttleton 1939; Bondi & Hoyle 1944).
Here, we investigate this question by considering in- While PR13 have not explicitly investigated the proper-
teraction of matter and radiation in the gravitational tiesofthegaswakeintheirsimulations,itfollowsthatin
potential well of a moving MBH. The most important thisregimetheefficiencyofdynamicalfrictionisrestored
findings of this work are that there are regimes, set by to its classically derived value.
the properties of the MBH and its ambient medium, in Wethereforefocusonscenariosdescribedby1≤M<
which gas dynamical friction is rendered inefficient by 4, when gas dynamical friction is classically expected to
the MBH feedback. We lay out the relevant physical bemostefficient,and(1+M2)M n <109M cm−3,
BH ∞ (cid:12)
regimesandscalesinSection2andevaluatetheefficiency when the influence of the BH radiative feedback on dy-
ofdynamicalfrictionusinglocalradiationhydrodynamic namical friction wake is expected to be significant. Hav-
simulations in Section 3. We discuss implications of our ing established the relevant regime we estimate the ex-
findings and conclude in Section 4. tentofthedynamicalfrictionwakeandcompareittothe
size of the Hii region.
2. REGIMESOFDYNAMICALFRICTIONINTHE
PRESENCEOFRADIATIVEFEEDBACK
2.1. Relevant scales
Photoionization and radiation pressure exerted by ra-
The extent of the dynamical friction wake that would
diation escaping from the innermost parts of the BH ac-
form behind the MBH in absence of radiative feedback
cretionflowcancreatetheionizedregion(theStr¨omgren
sphereorHiiregion)aroundtheaBHandstronglyalter canbeestimatedasaradiusofthegravitationalinfluence
of the MBH
the properties of surrounding gas. This radiation “re-
sponse”, also referred to as the BH radiative feedback, GM (cid:18) T (cid:19)−1(cid:18) M (cid:19)
has been found to lower the accretion rate by orders of R ≈ BH =52.1pc ∞ BH . (2)
DF c2 104K 106M
magnitude relative to a fiducial case in which radiative s,∞ (cid:12)
feedback is neglected (Milosavljevi´c et al. 2009; Li 2011; (cid:112)
Herec = γk T /µm =9.1kms−1(T /104K)1/2,
Park&Ricotti2011,2012;Pacucci&Ferrara2015;Park s,∞ B ∞ p ∞
assuming isothermal gas with hydrogen composition
et al. 2016). In the context of dynamical friction, this
characterized by the adiabatic index γ = 1 and mean
accretion regime is also likely to correspond to reduced
atomic weight µ = 1. The constants have their usual
efficiency of the gas drag due to the impact of ionizing
meaning. As mentioned in Section 1, most of the gravi-
radiation on the BH’s density wake.
tational drag force is contributed by the gas that resides
Park et al. (2014a) and Inayoshi et al. (2016) however
withinanorderofmagnitudesmallerregionthanthates-
findaphysicalregimeathighergasdensitiesinwhichthe
timated in equation (2), so this value can be considered
nature of accretion onto the BH fundamentally changes
from this picture, as the Hii region collapses under the a conservative upper limit.
The size of the elongated Hii region sensitively de-
gravity of the surrounding gas, and accretion continues
pends on the MBH accretion rate and velocity. For
unaffected by radiation at supper Eddington rates. The
the purpose of the estimates presented here, we neglect
two studies differ in their treatment of radiation trans-
the elongation of the Hii region and estimate the ra-
port: the latter considers the effects of radiation trap-
dius of a spherically symmetric ionization sphere that
pingathighaccretionrates(Begelman1978,1979)while
wouldformaroundastationaryMBHaccretinghydrogen
the former neglects them. Both nevertheless indicate
that conditions for hyper-accretion arise for a BH im- gas,RHII =(3N˙/4παrec)1/3n−∞2/3(Osterbrock&Ferland
mersed in neutral gas with number density n , when 2006). The number of ionizing photons per unit time
∞
n M > 109 − 1010M cm−3. This finding implies emitted from the innermost parts of a BH’s accretion
∞ BH (cid:12)
thatthedensitydistributionofgasinthehyper-accretion flow is
regimeisonlyweaklyaffectedbyradiationpressure,and N˙ =(cid:90) ∞ Lνdν ≈ α−1(cid:18)Lbol(cid:19) (3)
thus the efficiency of dynamical friction is likely to be hν α hν
restored to that predicted classically, i.e., in absence of ν0 0
radFioartimveovfeinegdbBaHcks.,accretionrate,andthusaccretionlu- w(cid:82)0∞hicLhνdisνchanardacLtνeri∝zedν−bαy.boHloemree,trαic=lum1.i5noissittyheLbsople=c-
minosity, also depend on BH velocity. We draw on the troscopic index representative of the spectral energy dis-
results of Park & Ricotti (2013, hereafter PR13), who tribution (SED) of active galactic nuclei (AGN) in the
used radiation hydrodynamic simulations to investigate high accretion state, and hν0 = 13.6eV for hydrogen.
thegrowthandluminosityofBHsmovingthroughauni- αrec = 4×10−13cm3s−1 corresponds to the Case A re-
form gaseous medium with temperature T = 104K. combination coefficient for hydrogen gas at the temper-
∞
PR13 show that in this case the BH radiative feedback ature T∞ = 104K, used here so to match simulations
causes a formation of an Hii region elongated along the describedinthenextsection. NotethattheCaseBcoef-
direction of the BH motion and filled with ionized hy- ficientisamoreappropriatechoiceforthehighgasdensi-
drogen (or H–He) gas of temperature, T ≈ 4×104K tiesinthisproblem,whichwouldresultinasmallchange
in
(6×104K). A shell of gas with increased density, corre- Str¨omgren radius being proportional to α−1/3. We con-
rec
sponding to (1+M2)n , forms in front of the BH as sider accretion powered BH luminosity, L = ηM˙ c2,
∞ bol
a consequence of the “snowplow” effect caused by radia- andassumeradiativeefficiencyη =0.1(Shakura&Sun-
tionpressure. Forawiderangeofsimulatedgasdensities yaev 1973).
the shell becomes gravitationally unstable and collapses These ingredients provide an estimate for the size of
when M (cid:38) 4, restoring the properties of the gas flow the Hii region, given a known MBH accretion rate. In
around the BH to the classical Bondi–Hoyle–Lyttleton the case of moving MBHs with the Mach number in the
Gaseous Dynamical Friction in Presence of BH Radiative Feedback 3
TABLE 1
Simulation Parameters
ID Nr×Nθ ∆r/r M tend(Myr)
HR 400×128 0.026 0.5,1.0,2.0 201.0,74.9,74.9
MR 300×96 0.035 0.5,1.0,2.0 201.0,201.0,201.0
LR 200×64 0.053 0.5,1.0,1.5,2.0 201.0,201.0,201.0,201.0
10-2
10-3
B
˙M 10-4
/
˙M
10-5 M=0.5
=1.0
M
=2.0
M
10-6
0 20 40 60 80 100 120 140
time (Myr)
Fig. 2.— Accretion rate onto the MBH as a function of time
shown in units of Bondi accretion rate defined in Equation (5).
DifferentlinestylesmarkMRrunswithM=0.5,1.0,and2.0.
(1+M2)M n <109M cm−3. Itfollowsthatinthis
BH ∞ (cid:12)
regime the dynamical friction wake is likely to be fully
Fig. 1.—Snapshotsofgasnumberdensityatdifferenttimesfor
ionized and dispersed by the MBH radiative feedback,
theHRrunwithM=0.5. Theflowofgasisfromtoptobottom
and the BH is located at (x,y)=(0,0) in each panel. The shape especiallygiventheproximityofthewakementionedbe-
of the Hii region and dynamical friction wake reach steady state fore.
after150Myrinthissimulation. In the next section we examine the detailed structure
range 1 < M < 4, PR13 find that the accretion rate of the wake and the Hii region, and numerically evalu-
mediated by radiation feedback can be expressed as
ate the resulting dynamical friction force from radiation
(cid:18)T (cid:19)5/2 hydrodynamic simulations.
M˙ ≈1.2×10−2M2M˙ ∞ (4)
B Tin 3. RADIATION-HYDRODYNAMICSIMULATIONS
whereM˙ isthenominalBondiaccretionrateforisother- 3.1. Numerical Setup
B
mal gas (Bondi & Hoyle 1944) We run a set of 2D radiation hydrodynamic simula-
tions using a parallel version of the non-relativistic code
M˙ = πe3/2ρ∞(GMBH)2 ZEUS-MP (Stone & Norman 1992; Hayes et al. 2006).
B c3 Simulations are carried out in a polar coordinate sys-
s,∞
tem defined by coordinates (r, θ) and assuming axisym-
=8.7×10−3(cid:16) n∞ (cid:17)(cid:18) T∞ (cid:19)−32(cid:18) MBH (cid:19)2 M yr−1. metry with respect to the MBH’s direction of motion.
1cm−3 104K 106M (cid:12) The extent of the computational domain is given by
(cid:12)
(5) r ∈ (0.1pc,3.0kpc) and θ ∈ (0,π) and other relevant
quantitiesareshowninTable1. Inthecalculationofgas
Combining Equations (3)–(5), we obtain the estimate dynamics we consider only the gravitational potential of
for the radius of the Hii sphere the MBH and neglect the self-gravity of the gas.
An MBH with mass M = 106M , located at the
R ≈430pcM2/3(cid:16) n∞ (cid:17)−13(cid:18) Tin (cid:19)−12(cid:18) MBH (cid:19)23 . origin of the coordinate syBsHtem (r = (cid:12)0), is placed in a
HII 1cm−3 4×104K 106M “wind tunnel.” In this setup, the gas of uniform density
(cid:12)
(6) n∞ =1.0cm−3 andtemperatureT∞ =104Kisassumed
Considering the ratio of Equations (6) and (2) we get to flow into the computational domain in direction θ =
π. We evaluate the accretion rate onto the MBH by
RRHII ∼8M23 (cid:18)1T04∞K(cid:19)(cid:18)4×T1i0n4K(cid:19)−12 (cid:18)106MMBHncm∞−3(cid:19)−13 cthalecucloamtipnugttahtieomnaalssdoflmuxaitnh,rdouefignhetdhebyinntheer bspouhnerdearwyitohf
DF (cid:12)
(7) radius rmin = 0.1pc, and convert it to MBH luminosity
where we find that R (cid:38) R for all values of the as L = 0.1M˙ c2. The SED of ionizing radiation is
HII DF bol
Mach number in the range 1 < M < 4 and when described as L ∝ ν−1.5 using 50 energy bins ranging
ν
4 Park and Bogdanovi´c
Fig. 3.—Top: 2Dsnapshotsofthenumberdensityofgasatt=150MyrforMRrunsM=0.5,1.0,and2.0(lefttoright). Allsnapshots
show steady state gas distributions. Bottom: contribution to dynamical friction from the gas surrounding the MBH, evaluated as the
gravitationalaccelerationperunitvolumealongthedirectionoftheMBHmotion.
from 13.6eV to 100keV. ing M , n , and T fixed. Each simulation is run
BH ∞ ∞
The composition of the H–He gas is evolved by fol- for several sound crossing timescales, defined as t =
cross
lowing the species of Hi, Hii, Hei, Heii, Heiii, and R /c , which corresponds to 46Myr for the size of
HII s,∞
e−. The radiation transport is coupled with hydro- the Hii region defined by equation (6). This ensures
dynamics and includes photo-heating, photo-ionization, that the accretion rate onto the MBH and density dis-
Compton heating by UV and X-ray photons from the tribution of the gas in the dynamical friction wake reach
BH, and gas cooling (Park & Ricotti 2011, 2012; Park steady state. The length of each simulation is recorded
et al. 2014a,b). We assume that BH radiation is the as t in Table 1. We test numerical convergence with
end
only heating source and adopt a simple analytic form of the set of high (HR), medium (MR), and low resolution
atomic cooling function Λ(T) due to neutral and ionized (LR) runs. The HR runs are carried out with resolution
HandHe,whichincludescoolingbyrecombination,colli- of (N ×N ) = (400×128), MR with (300×96), and
r θ
sionalionization/excitation,free-freetransitions,anddi- LR with (200×64). The radial bins are logarithmically
electric recombination of Heii (Shapiro & Kang 1987). spaced so that ∆r/r is constant everywhere on the grid
Molecularcoolingisneglectedandthecoolingrateisset while the bins in polar angle are evenly spaced and have
to zero below T = 104K. The calculation of radiation size ∆θ = π/N . The outer boundary of the computa-
θ
transport accounts for the radiation pressure and thus tionaldomainisdefinedasinflowwhere0≤θ ≤π/2and
allowsustoaccuratelycapturetheeffectsofbothenergy outflowforπ/2<θ ≤π. Weapplytheoutflowboundary
and momentum feedback (see Park & Ricotti 2012, for conditions at the inner domain boundary.
implementation details). Hydrodynamic and radiative
transfer equations are solved at every time step defined 3.2. Evolution of the Hii region and overdensity wake
as ∆t =min(∆ ,∆ ), where ∆ is the hydro-
hydro chem hydro The snapshots in Figure 1 illustrate the evolution of
dynamicaltimestepand∆ isthetimesteprequired
chem the gas density in the HR run with M = 0.5. A low
to calculate the change of chemical abundances (Ricotti
density Hii region with T ≈6×104K forms promptly
et al. 2001; Whalen & Norman 2006). in
around the MBH, while the distribution of gas outside
We investigate the effect of radiative feedback on the
of it reaches steady state 150Myr after the beginning
dynamicalfrictionwakebyvaryingtheMBHMachnum-
of the simulation. This timescale is consistent with ∼ 7
ber from 0.5 (subsonic) to 2.0 (supersonic), while keep-
soundcrossingtimesfortheHiiregionofthesize200pc.
Gaseous Dynamical Friction in Presence of BH Radiative Feedback 5
10-7
10-8 =1.0
=1.0 (acceleration) M
22~ ()a=Gm(<r)/rkm/yrgas||111111000000----1111--983210 Mdeceleration Upstream Bow Shock (D) up Downstream HII tail (D)down 22~ ()a=Gm(<r)/rkm/yrgas||1111100000----1111-93210 MMM===120...505
10-14 10-14
10-1 100 101 102 103 10-1 100 101 102 103
radius(pc) radius (pc)
Fig. 4.— Left: magnitude of the MBH acceleration along the direction of motion (either positive or negative) contributed by the gas
withinpolarradiusr fortheLRM=1.0run. TheMBHislocatedattheoriginr=0. Dashedlinesmarkthelocationsoftheupstream
bowshockandthetailofthedownstreamHiiregion,respectively. Right: magnitudeofthepositivecomponentofacceleration(resulting
inMBHspeed-up)contributedbythegaswithinr. DifferentlinesmarkLRrunswithM=0.5,1.0,1.5,and2.0.
Oncesteadystateisachieved,theshapeoftheHiiregion downstream (π/2 ≤ θ ≤ π) decelerates it. The shade of
shows minor deviation from spherical symmetry. thecolorindicatesthattheMBHmoststronglyinteracts
Figure2showstheaccretionrateasafunctionoftime with the gas in its immediate vicinity, within a radius of
fortheMRrunswithM=0.5,1.0,and2.0. Inallsimu- a few parsecs, but this distribution appears front–back
lationsMBHinitiallyexhibitsarelativelyhighaccretion symmetric and is thus not expected to significantly con-
rate, which decreases as the Hii region expands into the tribute to the acceleration of the MBH. Similarly, the
background medium as a consequence of the MBH ra- fluid elements perpendicular to the MBH’s line of mo-
diative feedback. The expansion of the Hii sphere is tion at θ ≈ π/2 make a negligible contribution. The
broughttoahaltat∼15−30Myrafterthebeginningof overdensity of gas that forms at the front and tail of the
the simulation, at which point the accretion rate reaches Hiiregionisthusexpectedtocontributethemosttothe
a turning (minimum) point and readjusts to a steady net acceleration of the MBH. In the run with M = 0.5
state value after ∼ 20−35Myr. This happens first in this distribution is approximately front-back symmetric,
theLRrunwithM=2.0,followedbyM=1.0and0.5. indicating a lower magnitude of acceleration. The M
This hierarchy of timescales is determined by the inflow = 1.0 and 2.0 runs, on the other hand, show enhanced
rate of the gas into the Hii region as seen by the MBH. contribution to MBH acceleration from the front of the
Figure 2 also illustrates the dependence M˙ ∝ M2 Hii region, while contribution from the tail appears less
captured by Equation (4) and applicable in the range significant. In order to determine whether the MBH ac-
1 < M < 4. This dependence of accretion rate on the celerates or decelerates, once the steady state distribu-
Mach number was first reported by PR13, who noted tion of gas is achieved, we integrate contributions to the
that accretion onto the MBH is mediated by a dense dynamical friction force from all fluid elements in the
shell,whichformsupstreamfromtheMBH,attheinter- computational domain.
face of the Hii region and ambient gas. This dense shell
3.3. Efficiency of dynamical friction
andtheassociatedbowshockarediscernibleinFigure3,
intheMRrunswithM=1.0and2.0,andareabsentfor TheleftpanelofFigure4showsthesteadystatemag-
the subsonic run with M = 0.5. The same figure shows nitude of the MBH acceleration (either positive or nega-
that the Hii region becomes more elongated in the di- tive) due to the gas enclosed within the sphere of radius
rection of the MBH motion with the increasing Mach r for the LR run with M=1.0. Positive acceleration is
number, engulfing a larger volume in the region where defined to be in the direction of the MBH motion, thus
the dynamical friction wake is supposed to form. De- resulting in the speed-up of the MBH. Note that the
spite this effect, a noticeable overdensity of gas persists assumption of azimuthal symmetry guarantees that the
at the tail of the Hii region in all simulations, indicat- components of acceleration perpendicular to the MBH’s
ing that some fraction of the dynamical friction wake lineofmotioncancelout,leavingtheparallelcomponents
remains present. as the only contribution.
ThebottompanelsofFigure3illustratethestrengthof The magnitude of acceleration is relatively low in the
gravitational interaction between the MBH and fluid el- immediate vicinity of the MBH (r < 3pc), consistent
ements in the computational domain, where we evaluate with the earlier observation of the front–back symme-
the magnitude of the acceleration per unit volume along try in this region. At r (cid:38) 3pc the magnitude of accel-
the direction of the BH motion as GM n|cosθ|/r2. eration gradually increases as the contrast between the
BH
The net effect of a given fluid element on the MBH de- upstream and downstream distribution of gas increases
pends on its location. The gas in the upstream (0≤θ ≤ with radius. At r ≈100pc, acceleration shows a sudden
π/2) acts to accelerate the MBH, while the gas in the jumpcoincidingwiththeupstreamoverdensityandasso-
ciated bow shock. The acceleration magnitude levels off
6 Park and Bogdanovi´c
1e 8
1.0
1.2 HR MR LR HR
K
x 6.0e-9 C MR
1.0 A
0.8 M=1.0 ]∞ DB LR
2)m/yr 00..46 M=2.0 2/cs,∞ 0.5 AD FEE
k ρ R
~ (a 0.2 =0.5 2)H w/
M B 0.0
0.0 M
K
0.2 G C
( A
π B
0.4 4 D
0 50 100 150 200 [ E
/ 0.5 E
time (Myr) DF D F
Fig. 5.—Evolutionofthenetacceleration,integratedovertheen- F RA
tirecomputationaldomain,fortheHR(solidlines),MR(dashed), x 3.0 O
andLR(dotted)runswithM=0.5,1.0,and2.0. Positiveaccel- N
erationimpliesspeed-upoftheMBH. 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
beyond400pc,whichmarksthespatialextentofthetail
M
oftheHiisphere. Evidently,thedynamicalfrictionwake
doesnotcontributetotheMBHaccelerationbeyondthis Fig. 6.—DynamicalfrictionforceasafunctionoftheMachnum-
point. ber. Thereddashedlineillustratesthemagnitudeofthedynamical
frictionforceintheabsenceofradiativefeedbackascalculatedby
TherightpanelofFigure4showsthemagnitudeofthe
Ostriker (1999). The blue dashed line is the same as the red one
positive component of MBH acceleration contributed by but plotted with a positive sign and arbitrarily scaled in magni-
the gas enclosed within radius r. The figure illustrates tude to match data points. Different symbols mark results of the
runs with radiative feedback at t=201.0Myr. Note the different
that in all simulations the largest contribution to the
magnitudesofthepositiveandnegativey-axis.
MBH acceleration originates from the region r (cid:38) 10pc.
This acceleration is net positive, thus speeding up the of error in net acceleration due to the finite numerical
MBH.Inallcases,theMBHdecelerationduetothewake resolution of our simulations.
beyondthetailoftheHiiregionisnegligible. Theaccel- Figure 6 shows the values of the dynamical friction
eration originating from r <10pc can be either positive force measured at the end of each simulation for dif-
ornegativebutisordersofmagnitudesmallerrelativeto ferent values of M. As a comparison, the red dashed
the larger spatial scales and can be neglected. line shows the dynamical friction force in the absence of
Figure5showstheevolutionofthenetaccelerationin- radiative feedback, as predicted by Ostriker (1999) for
tegratedovertheentirecomputationaldomainasafunc- ln(c t/r )=4. Asnotedbefore,thedynamicalfric-
s,∞ min
tion of time. In all cases acceleration initially peaks at tion force calculated from simulations presented in this
15−30Myrs. This is the same point in time where the study is net positive for 1.0 ≤ M < 4 and negative for
accretion rate reaches minimum (Figure 2), indicating a M = 0.5. The magnitude of this force is however over
buildup of the dense shell of gas in the upstream of the ∼9 orders of magnitude lower compared to that experi-
MBH, which exerts gravitational influence and initially encedbytheMBHsinabsenceofradiativefeedbackand
suppresses accretion onto the MBH. Gravitational influ- is therefore negligible for all values of the Mach number
ence of the upstream gas shell is counterbalanced by the M<4. Interestingly, thedynamicalfrictionforceinthe
gas in the downstream of the MBH, where overdensity presenceofradiativefeedbackappearstomirroritsclas-
wakedevelopswithadelaycorrespondingtoafewsound sical counterpart: it peaks around M≈1 and decreases
crossing times across the elongated Hii region. Conse- at larger values of the Mach number.
quently, the MBH acceleration reaches steady state with
a delay of ∼ 150Myr, long after the accretion rate has 4. DISCUSSIONANDCONCLUSIONS
settledintosteadystate. AninspectionofFigure5shows We investigate the efficiency of dynamical friction in
that the net acceleration for the scenario M = 1.0 con- presence of radiative feedback from an MBH moving
tinues to gently increase even after 200Myr. through a uniform density gas. Ionizing radiation that
For M = 1.0 and 2.0 the net end value of accelera- emerges from the innermost parts of the MBH’s accre-
tion is positive, indicating a speed-up of the MBH. As tionflowresultsintheformationoftheHiiregion,which
expected, the magnitude of acceleration is smallest for strongly affects the dynamical friction wake and renders
the M = 0.5 scenario, where the gas flow around the dynamical friction inefficient for a range of physical sce-
MBH exhibits a large degree of front–back symmetry. narios. We summarize our main findings below:
Figure 5 illustrates that the sign of net acceleration in
this case is resolution dependent, where the final out- • We identify a physical regime in which the dy-
come of the LR run is MBH speed-up and in the MR namical friction wake is likely to be fully ionized
and HR runs, deceleration. The net accelerations in the and dispersed by the MBH radiative feedback as
MR and HR runs track each other closely and differ by a regime in which the radius of the Hii sphere is
only∼0.05kmyr−2,indicatingnumericalconvergenceof larger than the extent of the dynamical friction
the MR run. We therefore use this value as a measure wake. This condition is fulfilled when M < 4
Gaseous Dynamical Friction in Presence of BH Radiative Feedback 7
and(1+M2)M n <109M cm−3. Outsideof in inhomogeneous and clumpy medium which can per-
BH ∞ (cid:12)
this regime the formation of the Hii region is sup- turbtheirotherwisesmoothorbitaldecay(Fiacconietal.
pressed and the properties of the gas flow around 2013). Furthermore, the geometry and properties of the
theMBHarerestoredtotheclassicalBondi–Hoyle– HiiregionwillbeaffectedinscenarioswheretheHiire-
Lyttleton solution, thus restoring the effect of dy- gionisnotconfinedbythegaseousmedium. Anexample
namicalfriction. Thesecriteriacanbeutilizedasa of this would be an Hii region around the MBH orbit-
sub-resolutionmodelfordynamicalfrictioninlarge ing within the galactic gas disk. When the radius of the
scalesimulationsthatdonotresolvethescaleofthe Hii region exceeds the half thickness of the disk, the ra-
Hii region or MBH density wake. diation can escape outside of the disk plane, making the
problemfullythree-dimensional. Eveninsuchscenarios,
• Based on radiation hydrodynamic simulations we theproximityofthedynamicalfrictionwaketotheMBH
find that the net acceleration experienced by the islikelytoresultinthewakeobliterationbyhighenergy
MBHsinthisregimetendstobepositive,meaning radiation. Indeed, 3D hydrodynamic simulations that
thattheyspeedup,contrarytotheexpectationsfor capture the dynamics of MBHs in proper galactic setups
gaseous dynamical friction in absence of radiative also find a weakening of the dynamical friction force in
feedback. This reversal happens because the dom- thepresenceofMBHfeedback(Sijackietal.2011;Souza
inant contribution to the MBH acceleration comes Lima et al. 2016). The high resolution, local simulations
from the dense shell of gas that forms in front of presentedheresupportthesefindingsandprovideasetof
the MBH as a consequence of the snowplow effect criteria, formulated in terms of the properties of the gas
caused by radiation pressure. and MBH, under which the wake evacuation is efficient.
Along similar lines, we consider an isolated MBH on
• ThemagnitudeofMBHaccelerationpeaksatM≈ a linear trajectory, whereas the front–back asymmetry
1 and decreases for larger values of the Mach num- of the density wake for a perturber on a circular orbit
ber, similar to the dynamical friction force in ab- has been shown to cause small differences in the dynam-
sence of radiative feedback. The magnitude of ical friction force in absence of radiative feedback (Kim
acceleration is however negligibly small, implying & Kim 2007). A system consisting of multiple MBHs
that MBHs in this regime will not significantly would further add to the complexity in cases when dy-
change their velocity over timescales that deter- namical friction wakes can mutually affect one another
mine their motion and properties of their environ- (e.g., Kim et al. 2008). This type of study may require
ment. a transition from 2D to 3D simulations in the future in
order to relax the assumption of azimuthal symmetry
and properly capture the interplay of inhomogeneities,
• Our results suggest that suppression of dynami-
radiative feedback, and dynamical friction.
cal friction by radiative feedback should be more
severe at the lower mass end of the MBH spec-
trum, because these BHs must reside in regions This work is supported in part by the National
of relatively high gas density in order to experi- Science Foundation under the Theoretical and Com-
enceefficientdynamicalfrictionintheregimewhen putational Astrophysics Network (TCAN) grant AST-
(1+M2)MBHn∞ > 109M(cid:12)cm−3. Compounded 1333360. T.B. acknowledges support from the Research
with inefficiency of the gas drag for lower mass ob- CorporationforScienceAdvancementthroughaCottrell
jects in general this implies that < 107M(cid:12) MBHs ScholarAward. T.B.isamemberoftheMAGNAproject
have fewer means to reach the centers of merged (http://www.issibern.ch/teams/agnactivity) supported
galaxies. by the International Space Science Institute (ISSI) in
Bern, Switzerland. Numerical simulations presented in
This study makes several idealized assumptions, such this paper were performed using the high-performance
as the uniform gas density and infinite background computing cluster PACE, administered by the Office of
medium. In reality however, MBHs in the aftermath of Information and Technology at the Georgia Institute of
galactic mergers are likely to find themselves immersed Technology.
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