Black hole mass measurement using molecular gas kinematics: what ALMA can do

1.5 MB·

by  Ilsang Yoon

Mon.Not.R.Astron.Soc.000,1–19(2002) Printed23January2017 (MNLATEXstylefilev2.2)
Black hole mass measurement using molecular gas kinematics:
What ALMA can do
7
1 Ilsang Yoon1,2⋆
0
2 1NationalRadioAstronomyObservatory,520EdgemontRoad,Charlottesville,VA22903,USA
2LeidenObservatory,LeidenUniversity,P.O.Box9513,2300RALeiden,TheNetherlands
n
a
J
Accepted1988December15.Received1988December14;inoriginalform1988October11
0
2

] ABSTRACT
A Westudythelimitsofspatialandvelocityresolutionofradiointerferometrytoinferthemass
G ofsupermassiveblackhole(SMBH)ingalacticcentreusingthekinematicsofcircumnuclear
moleculargas,byconsideringtheshapesofgalaxysurfacebrightnessprofile,signal-to-noise
.
h ratios(S/N)ofposition-velocitydiagram(PVD)andsystematicerrorsduetospatialandve-
p locitystructureofthemoleculargas.WearguethatforfixedgalaxystellarmassandSMBH
- mass,thespatialandvelocityscalethatneedstoberesolvedincreasesanddecreasesrespec-
o
tivelywithdecreasingSérsic indexofthe galaxysurfacebrightnessprofile.We validateour
r
t argumentsusingsimulatedPVDsforvaryingbeamsizeandvelocitychannelwidth.Further-
s
a more,we consider the systematic effects to the inferenceof the SMBH mass by simulating
[ PVDsincludingspatialandvelocitystructureofthemoleculargas,whichdemonstratesthat

their impacts are not significant for the PVD with good S/N unless the spatial and velocity
1
scaleassociatedwiththesystematiceffectsarecomparabletoorlargerthantheangularres-
v
olutionandvelocitychannelwidthofthePVD frompurecircularmotion.Also,we caution
3
8 thatabiasingalaxysurfacebrightnessprofileowingtothepoorresolutionofgalaxyphoto-
6 metricimage can largelybiasthe SMBH mass byan orderofmagnitude.Thisstudy shows
5 thepromiseandthelimitofALMAobservationtomeasuretheSMBHmassusingmolecular
0 gaskinematicsandprovidesausefultechnicaljustificationforALMAproposalwithscience
. goalofmeasuringSMBHmass.
1
0
Keywords: blackholephysics–methods:observational–ISM:kinematicsanddynamics–
7
galaxies:nuclei
1
:
v
i
X
r 1 INTRODUCTION Measuring the dynamical mass of SMBH for wide range of
a blackholemassandhostgalaxytypeisrequiredtomakearobust
Supermassiveblackholes(SMBH)ingalacticcentresinfluencethe
inferenceoftheco-evolutionbetweenSMBHandhostgalaxyand
formationandevolutionofthehostgalaxies(Richstoneetal.1998;
alsoimportanttocalibrateotherempiricalmethods(e.g.,X-raylu-
Kormendy&Ho2013)assuggestedbywell-knownempiricalre-
minosity) for indirect mass measurement in order to increase the
lationsbetweentheSMBHmassandhostgalaxyproperties(e.g.,
samplesizeof SMBHs.However, dynamical measurement of the
Magorrianetal. 1998; Ferrarese&Merritt 2000; Gebhardtetal.
SMBHmass ischallenging due totherequirement of high angu-
2000; Tremaineetal. 2002; Marconi&Hunt 2003; Häring&Rix
larandvelocityresolution(Ferrarese&Ford2005).Exceptforthe
2004; Gültekinetal. 2009). Therefore understanding the forma-
rarepopulationofgalaxieswithnuclearmaser(e.g.,Miyoshietal.
tionandevolution of theSMBHisan integralpart of thecurrent
1995), stellar kinematics (mostly in early-type galaxies; e.g.,
galaxy formation theory (Silk&Mamon 2012). In observation, a
Kormendy 1988; Boweretal. 2001; McConnelletal. 2011) and
firststeptowardtheunderstandingoftheSMBHisademographic
ionisedgaskinematics(inspiralandsomeearly-typegalaxies;e.g.,
studyforwiderangeofblackholemasses,howeveritisfarfrom
Ferrarese,Ford&Jaffe 1996; Sarzietal. 2001; Barthetal. 2001)
complete: the number of SMBHs with dynamical mass measure-
havebeenusedtomeasurethedynamicalmass.Asaresult,most
mentiscurrently70-80primarilyduetothedifficultyofachieving
of theSMBHmass measurements havebeen obtained for asam-
therequireddepthandresolution(Shankaretal.2016)andthede-
pleofearly-typegalaxiesandspiralgalaxieswithprominentbulge
tectionofSMBHswithlow(e.g.,denBroketal.2015;Sethetal.
becausetheyhaveawelldevelopeddynamicallyrelaxedstellarsys-
2010, for MBH< 106M⊙) and high (e.g., McConnelletal. 2011, temandpowerfulenergysourceionisingthegas.
DRAforFMBTH>1 0 10-M-⊙ ) dyDnamRicalmAassFhasTbeen r a-rel-yr e poDrted.RAFT -- DRAFT -- DRAFT --
Forcomprehensiveunderstandingoftheconnectionbetween
theSMBHanditshostgalaxy,itisimportanttoexpandthemea-
⋆ E-mail:[email protected] surement of SMBH mass tothe regimeof small bulgeless galax-
2 IlsangYoon
ies hosting lower mass SMBH. A recent study based on a large ofthegasdensityprofileandtheprofileshape,andothercompli-
compilationofSMBHsshowsthattheempiricalrelationbetween cated velocity structures including inflow/outflow, random veloc-
SMBHmassandstellarvelocitydispersionknownasM σ re- itymotionanddiscwarpmaycontaminatetheobservedmolecular
−
lation(e.g.,Tremaineetal.2002)hasadifferentnormalizationfor gas kinematics and affect our inference of the SMBH mass. We
early-andlate-typegalaxies(McConnell&Ma2013).Inaddition, considertheseeffectsbysimulatingtheposition-velocitydiagrams
there are strong evidences of the low luminosity AGN in bulge- (hereafter PVD) for a range of observing parameters and proper-
less galaxies or dwarf galaxies hosting low mass SMBH (e.g., ties of the galaxy and the circum nuclear molecular gas. In par-
Filippenko&Ho 2003; Reinesetal. 2011; McAlpineetal. 2011; ticular,thelimitsofthemeasurableSMBHmassgiventhesesys-
ArayaSalvoetal.2012)whoseinferredmassesarelargelyuncer- tematiceffectsandtheresolutionofinterferometryhavenotbeen
tainwithoutaccuratedynamicalmassmeasurement.Furthermore, discussedindetailusingrealisticsimulationsof theobserved gas
alargenumberofgalaxiespresumablyhostingdustobscuredAGN kinematics,whichwillbeusefulfortheSMBHmassmeasurement
(e.g., Mateosetal. 2013; Satyapaletal. 2014) has been revealed usingALMAobservations.Inaddition,itwillalsoserveasause-
by red mid-IR colour (Sternetal. 2012). For those low mass or ful guideline for the technical justification of ALMA proposal to
dustobscuredgalaxies,itmaybeevenmoredifficulttoapplythe observemolecular gaskinematics ingalaxiesfor differentgalaxy
currentmethodsusingstellarandionisedgaskinematicsiftheydo surfacebrightnessprofilesandrealisticobservingconditions.
nothaveawell-developedbulgeforreliableradialvelocitydisper- Inthiswork,westudytheparameterspaceoftheangularand
sionmeasurementorhavedustobscuringtheionisedgasandalso velocityresolutionofradiointerferometryforrangesofthegalaxy
affectingthegalaxyimageanalysisformassmodeling. types and the structure of circum nuclear molecular gas in order
Recentlythehighangularandvelocityresolutionradiointer- to characterise how the information of molecular gas kinematics
ferometry enables to make a use of the observation of molecular canbebestutilisedfortheSMBHmassmeasurement.InSection
gaskinematicsformeasuringthedynamicalmassofSMBH(e.g., 2,weprovideasimpleargumenttoderiverelevantspatialandve-
Davisetal.2013a;Onishietal.2015;Barthetal.2016).Molecu- locityresolutionforgivengalaxysurfacebrightnessprofileshape
largasasakinematictraceroftherotationvelocityingalacticcen- and discuss the systematic effects due to the spatial and velocity
tre has advantages; in principle it is possible in any galaxy type structureofthemoleculargasandtheeffectofpoorresolutionof
with associated molecular gas if it exists, and the high angular galaxyimagetodeterminethesurfacebrightnessprofileshapeand
resolutions routinelyachievable bynew (sub)-millimetreinterfer- itsimpacttotheinferenceofSMBHmass.InSection3,wesim-
ometry(e.g.theAtacamaLargeMillimetre/sub-millimetreArray; ulateobservedPVDsandmeasuretherotationvelocitytoconfirm
ALMA) mean it may be possible to probe larger volumes of the ourargumentsandtesttheimpactofsystematicerrorsinthePVD
universethaneverbefore,enablingacompletemasslimitedcensus analysis.InSection4,wediscussthespatialresolutionandvelocity
ofSMBH(Davis2014). channel width tomeasure theSMBH massas aconvenient guid-
Withincreasingimportanceandpotentialofthemoleculargas anceforALMAproposal.InSection5wesummarisetheresults.
kinematicsforSMBHmassmeasurement,afigureofmeritofthe Inthiswork,weusetheconcordanceLCDMcosmologicalmodel:
moleculargaskinematicshasbeendiscussedinDavis(2014)which Ωm = 0.3, ΩΛ = 0.7 and H0 = 100h kms−1Mpc−1 where
shows that the difference of rotation velocity with and without h=0.7.
SMBHcanbedetectedbytheobservationofmoleculargaskine-
maticsoutsidetheblackholesphereofinfluenceusingthesimple
argumentbasedonanassumedgalaxyrotationvelocityatdesired
angularresolutionoftheinterferometryandthattherequiredangu- 2 IMPACTOFSUPERMASSIVEBLACKHOLETOTHE
larresolutionbecomeslargerwithdecreasingconcentration(mea- ROTATIONVELOCITY
suredbylogarithmicslopewithin0.1′′radius)ofthegalaxysurface
To measure the SMBH mass using gas kinematics, one needs to
brightnessprofile.
accuratelymeasuretherotationvelocityatthegalacticcentreand
Although some useful formulae and discussions using
decomposeintotwocomponents: onefromthepointmassdueto
galaxy samples with velocity measurement from ATLAS3D sur-
theSMBHandtheotherfromtheextendedmatterdistributiondue
vey (Krajnovic´ 2013) have been provided in Davis (2014), their
tothegalaxystellarcomponent.Thetechniqueisnotformallyvalid
arguments are based on a single molecular gas parcel rotating
inthecaseofdynamicallyhot,warpedorinflowing/outflowinggas
around SMBH with assumed rotation velocity (i.e., 100 km/s) at
(Davis2014).HoweverlikeinDavis(2014),inthissection,weas-
thedesiredangularradius,withoutconsideringobservationaleffect
sumethatthemotionofcircumnucleargasispurelycircularand
(e.g. beam smearing), systematic motions (e.g., non-circular mo-
onlygovernedbytheSMBHandgalaxystellarmass,andtheaddi-
tion)andspatialdistributionof moleculargas.Inaddition,unlike
tionalsystematiceffectswillbediscussedlater.
ATLAS3D survey,detailkinematicinformationisnotavailablefor
For a flat gas disc sharing the same inclination (i) as the
mostofgalaxies,whichisinconvenientforadoptingsimplerelation
galaxy,rotationvelocityofagasparcelatradiusris
inDavis(2014)todeterminetherelevantphysicalscaleforagiven
galaxytoresolvethegaskinematicsfortheSMBHmassmeasure- GM
V(r) = V(r) 2+ BH sin(i) (1)
ment.ItisusefultogeneraliseandexpandtheworkinDavis(2014) rot r gal r
tothewiderangeofobservingandgalaxyparameterspacetopro-
vide aconvenient and useful guideline for estimatingtheangular whereV(r)galisacircularvelocityofthegalaxyduetothegalaxy
andvelocityresolutionoftheproposedinterferometryobservation stellarmassdistributionandMBHistheSMBHmassinthegalaxy
withoutrelyingonapriorknowledgeofgalaxyrotationvelocity. centre.V(r) = GM(<r) whereM(<r)istheenclosedmass
Indeed,moleculargaskinematicsingalacticcentreiscompli- atradiusr angadl estqimatedr byconvertinggalaxysurfacebrightness
DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
cated.ItisgovernedbytheSMBHandgalaxystellarmassdistri- distribution to stellar mass distribution, using a mass-to-light ra-
butiondeterminedbygalaxysurfacebrightness profileshapethat tio(M/L)attheobservingband.RotationvelocityV(r) isthen
rot
varieswithdifferentgalaxytypes.Alsosignal-to-noiseratio(S/N) measuredfromthePVD.
SMBHmass measurementusingALMA 3
High-resolutionPVDisrequiredtoaccuratelymeasurethero-
tationvelocityandALMAcaneasilyachieveasub-arcsecresolu-
tion which resolves the black hole sphere of influence (SOI) for
m(Doasvtisof20la1r4g)e,wSMhoBseHrsadwiuitshisMcBomHm>onl4y×defi1n0e8dMa⊙s and i > 30◦ 103 MMgBaHl==11.0.0··1100171[[MM⊙⊙]]
i=60◦
r = GMBH (2) Distance=15.29[Mpc]
SOI σ∗2 s]
whereσ∗ isthestellarvelocitydispersion.Davis(2014)findsthat km/ 102
[
tmheasrseiqsuairpepdroaxnigmualaterlyretswolouttiiomnestoladregteerctthtahnereffecbtyocfablcluaclaktinhgolea ocity Vdiff
SOI el
maximum angular radius where the velocity difference with and V
withoutSMBHisdetectedbyafactorof5timeslargerthantheve- 101
Total
locitychannelwidth,usingatypicalgalaxyrotationvelocity(i.e., Star
BH
V(r)gal = 100 km/s)measured bytheATLAS3D survey atanas- n=1 reqv
n=2
sumed angular radius resolved by Combined Array for Research n=3
inMillimeter-waveAstronomy(CARMA).However,asalsomen- 100 n=140−3 10−2 10−1 100 101 102
tionedintheintroduction,estimatinggalaxyrotationvelocityasa
Radius[arcsec]
functionofradiususingmoreeasilyavailablephotometricinforma-
tionismoreusefulandconvenienttodeterminetheangularresolu- Figure1.Rotation velocities ofgalaxies withM∗ = 1011M⊙ contain-
tionandvelocitychannelwidthfordetectingSMBHingalaxiesfor ingcentralSMBHwithMBH =107M⊙.DifferentcolourindicatesSérsic
awiderangeofmorphologywithoutpriorknowledgeofthegalaxy indexofthegalaxyandKeplerianrotationduetotheSMBHisshownby
kinematics. theblackdottedline.Theradius,wherethegalaxyrotationandtheSMBH
Inpractice,observedrotationvelocityV(r)obsisdifferentfrom rotationvelocityareequal(reqv)foreachgalaxysurfacebrightnessprofile,
V(r) inequation(1)duetotheuncertaintiesofpositionandve- isindicatedbythearrowwiththesamecolourasthecorrespondingrota-
rot
tioncurve.VelocitydifferenceV betweenthetotal(galaxy+SMBH)and
locitymeasurementsandthecomplicatedvelocitystructureinthe diff
galaxyrotationvelocity atr isalsoshownbythearrowwiththesame
gas.However,inthissection,weassumeV(r) =V(r) byne- eqv
obs rot colourscheme.
glectingcomplicatedvelocitystructureandsimplifyingtheuncer-
taintiesinPVDbyconsideringonlyinterferometrybeamsizeand
velocitychannelwidthandweconsiderothersystematiceffectsby profileres givesagood approximation withmuchlesserror than
simulatingPVDsinSection3.4. theuncertainty inre fromthefittingprocessaslongasrb re
≪
(Trujilloetal.2004).Sointhiswork,weuseequation(4)forcore-
Sérsicgalaxymodel.
2.1 RotationVelocity To compute the enclosed mass within r, we integrate I(r)
up to r being normalised by the total luminosity ∞I(r)dr and
WecomputeV(r) usingM andV(r) inequation(1).Tocom- 0
rot BH gal multiplythebolometricluminosityandmass-to-ligRhtratio.Weuse
puteV(r) ,weadoptSérsicmodelsforgalaxysurfacebrightness
gal a fiducial mass-to-light ratio, M/L = 1.0 and adjust an abso-
profiletoestimatetheenclosedmassataradiusr.Surfacebright-
lute magnitude parameter to obtain galaxy stellar mass that we
nessprofileofSérsicmodel(Sersic1968)varieswithashapepa-
want to control. Given galaxy stellar mass M∗ and Sérsic index
rameternandproducesaprofilecontinuouslychangingfromdisc
n,weassignthehalf-lightradiusofthegalaxyusingtherelationin
toellipticalgalaxies.ItisalsopossibletoextendSérsicmodel to
Shenetal.(2003):
incorporateacore-deficitofellipticalgalaxiesbyusingcore-Sérsic
msuorfdaecle(eb.gri.g,hGtnraehssampreotfialle.2is00m3)e.aAsulrthedoufgrhomintphreachtiicgeh,trheesoglaultaixoyn res =0.1(cid:16)MM⊙∗(cid:17)0.14(cid:16)1+ 3.98×M10∗10M⊙(cid:17)0.25 ,ifn<2.5 (5)
galaxy image with additional modeling of the details of the pro- kpc 2.88×10−6(cid:16)MM⊙∗(cid:17)0.56 ,ifn>2.5
filesuchasnuclearstarcluster(e.g.,Emsellem,Monnet&Bacon IfgalaxystellarmassM∗andSérsicindexnaregiven,theenclosed
1994),weuseSérsicparametricmodel inthisstudybecause itis
stellarmassM∗(<r)is
flexibletodescribeawiderangeofgalaxysurfacebrightnesspro-
filesverywellandthuseffectivetoshowtheimpactofgalaxysur- rI(r)dr γ(2n,x)
facebrightnessprofileshapeontheSMBHmassmeasurement. M∗(<r)=M∗ R∞0 I(r)dr =M∗ Γ(2n) (6)
0
SurfacebrightnessprofileofSérsicmodelis R
where x = b ( r )1/n. For n > 0.36, b is approximated as
I(r)=I(0)exp[ bn(r/res)1/n] (3) follows(Ciottin&rBesertin1999). n
−
withI(0)beingthecentralintensity,res thehalf-lightradius,and b 2n 1+ 4 + 46 + 131 2194697 +O(n−5).
ntheshapeparameter. n≈ −3 405n 25515n2 1148175n3−30690717750n4
Surfacebrightnessprofileofcore-Sérsicmodelis (7)
I(r)=I [(rb)γu(r r)+eb(rb/re)1/n−b(r/re)1/nu(r r )] (4) Figure 1 shows the rotation velocity of a galaxy with incli-
b r b− − b nationi = 60◦ and M∗ = 1011M⊙ containing 107M⊙ SMBH,
withbeingu(r a)theHeavisidestepfunction,incaseofsharp fordifferentsurfacebrightnessprofileshapescharacterisedbySér-
DRAFT -−- DRAFT -- DRAFT -- DRAFT -- DRAFT --
transitionatr (Trujilloetal.2004).Althoughthisisdifferentfrom sic index n. In x-axis, we use angular size at a given luminos-
b
theoriginalcore-Sérsicmodelprofile(Grahametal.2003),replac- itydistance of the galaxy (roughly at a distance of Virgocluster,
ingbwithb andhalf-lightradiusr withthatoftheouterSérsic 15 Mpc) instead of physical size for easy comparison to the
n e
≈
4 IlsangYoon
angular resolution of ALMA. Each colour in Figure 1 represents gionsmallerthantheFWHMofHSTWFC3(0.15′′ 1)andJWST
theSérsicindexrangingfrom1to4.Rotationvelocityshownby NIRCam(0.068′′ 2) in J-band are shown by light and dark grey
the solid lines are decomposed into the rotation velocity due to area to show the limit of angular resolution beyond which the
the SMBH and the galaxy stellar mass distribution. Dotted lines galaxy surface brightness profile can be accurately measured us-
and dot-dashed lines are the rotation velocity due to the SMBH inghighresolutionnear-infraredimage.Althoughthecurrentstate-
mass and galaxy stellar mass respectively. We also annotate the of-the-art resolution image for SMBH mass measurement using
angular radius r where the rotation velocity due to the SMBH HST/ACSI-bandimagehasslightlysmallerFWHM( 0.1′′),the
eqv ≈
andduetothegalaxystellarmassdistributionbecomeequal(i.e., importantquestioniswhethertheveryinnerregionwherethero-
M∗γΓ(2(2nn,x)) = MBH) and the velocity difference Vdiff at reqv be- tationvelocityisdominatedbytheSMBH,isresolvedbothbythe
tween the total rotation velocity and the rotation velocity due to photometricPSFandtheinterferometrebeamornot,andtherefore
the galaxy stellar mass. If the beam sizeislarger than r or the theactualsizeofFWHMisnotrelevantforthediscussioninthis
eqv
velocitychannel widthislargerthan V for thegiven beamsize section.
diff
resolving r , the velocity difference V is not resolved. So r InFigure2,r forallgalaxyprofilesislargerthanr and
eqv diff eqv eqv SOI
and Vdiff determines the required beam size and velocity channel all colour solid lines except for the galaxy with n = 4, M∗ =
width∆V (definedas∆V = 31VdifffordetectingVdiffat3σ)ofthe 1011M⊙ and MBH> 106M⊙ are larger than rmax by a factor of
radiointerferometry. few for galaxy with n = 4 and by more than an order of mag-
Figure1indicatesthatatfixedgalaxystellarmass,theimpact nitude for galaxy withn = 1, and this deviation becomes larger
of Keplerian motion due to the SMBH extends toa larger radius for thelower SMBH mass. Thebest agreement between r and
max
asmsgaalllearxSyésrusrifcaicnedbexri)g.hFtonresesxpamropfillee,btheecormoteastiolenssvecloonccietyntdrauteedto(tih.ee. rfoeqrvMisBsHe>en1f0o7rMth⊙ewgahliacxhyiswsiitmhiMlar∗to=th1e0g1a1lMax⊙iesanindDnav=is3(20∼144)
SMBHat0.5′′ scaleis5timessmallerthanthatduetothegalaxy usedforderivingr .However,r wasdeterminedfor5σstatis-
max max
stellarmassifthegalaxysurfacebrightnessprofilefollowstheel- ticalsignificance(Davis2014)andthusr canbelargerbylinear
max
liptical galaxy’s (n = 4 shown by red curve) while the rotation factor of the inverse of significance level in case of small veloc-
velocityduetotheSMBHissameastherotationvelocitydueto ityerror(i.e.,channel width)comparedwiththerotationvelocity
the galaxy stellar mass at the same 0.5′′ angular scale if the sur- (seeequation(9)inDavis(2014)),ifloweringthesignificancelevel.
facebrightnessprofileisexponential(n=1shownbybluecurve). Asthegalaxysurfacebrightnessprofilebecomeslessconcentrated,
Thisimpliesthat theeffect of SMBH for fixedmassM can be therequiredbeamsizefordetectingSMBHbecomeslargeratfixed
BH
detectedbylargerangularresolutionasthegalaxysurfacebright- M andthedetectableSMBHmassbecomessmalleratfixedbeam
BH
nessprofilebecomeslessconcentratedwhichisconsistentwiththe size.Thisisanadvantagefordetectinglowermassblackholere-
finding in Davis (2014). If lowering the SMBH mass, the dotted sidinginasmallbulgelessgalaxy.
linemoves downwards and thus, for fixedangular resolution, the Inthesamemanner,wealsoshowthevelocitychannelwidth
reqv oflargerSérsicindexgalaxystartstobeunresolvedfirst.This ∆V (= 13Vdiff)fordifferentSMBHmass,foragalaxywithM∗ =
impliesthattheeffectofSMBHforthegivengalaxystellarmass 1010M⊙ (Figure 3(a)) and M∗ = 1011M⊙ (Figure 3(b)). Ob-
andangular resolution, canbemoreeasilydetected forthelower servedrotationvelocitydepends ongalaxyinclination(i)andwe
black hole mass if the galaxy surface brightness profile becomes assumei = 60◦ inFigure3.Therequiredvelocitychannelwidth
less concentrated. This will be discussed further in the following for detecting SMBH becomes smaller at fixed M and the de-
BH
section. tectable SMBH mass becomes larger at fixed velocity channel
width,asgalaxysurfacebrightnessprofileislessconcentrated.This
isadisadvantageforthesmallbulgelessgalaxy.Howevertherange
2.2 SpatialandVelocityResolution
ofvariationofthevelocitychannelwidthfordifferentgalaxySérsic
Theradiusr defined inDavis(2014) wheretheKeplerianmo- indexiswithinafactoroffewthatismuchsmallerthanthevaria-
max
tionduetoSMBHshowsastatisticallysignificantdeviationfrom tionofangularresolution(i.e.,orderofmagnitude)andALMAhas
therotationvelocityduetothegalaxystellarmassdistribution,is sufficientvelocityresolution(6 1km/s)tocovertherangeofthis
largerthanthedefinitionoftheblackholesphereofinfluencer channelwidthinFigure3.Sothepracticalbenefitisthespatialres-
SOI
(equation(2)).UsinggalaxysamplesintheATLAS3D,Davis(2014) olution. However wenote that smallvelocity channel widthdoes
findsthatr =1.92r at5σstatisticalsignificancelevel. notimprovethesignificanceofthedetectionifvelocityuncertainty
max SOI
ForgivenM wecomparethreedifferentradii,r ,r and includingsystematicandrandommotionislargerthanthevelocity
BH SOI max
r forthesamestellarmassgalaxieswithdifferentsurfacebright- channelwidthasalsonotedbyDavis(2014).
eqv
nessprofiles.Blackholer iscomputed bycombining equation
SOI
(2) and the relation between the SMBH mass and the stellar ve-
locitydispersioninMcConnell&Ma(2013).Figure2showsthese 2.3 EffectsofSpatialandVelocityStructureofMolecular
Gas
radii(i.e.,angularsizesforgivengalaxydistance)fortherangeof
SMBH masses, for a galaxy with M∗ = 1010M⊙ (Figure 2(a)) Theaboveargumentregardingthespatialandvelocityresolutionis
andM∗ = 1011M⊙ (Figure2(b)).SolidblacklineisrSOI,dashed fortherotatingmoleculargasfollowingthegravitationalpotential
blacklineisrmaxandcolouredsolidlinesarereqvfordifferentSérsic from the central SMBH and the galaxy stellar mass. In a realis-
indices(n=1,2,3and4).Thefollowingrelationprovidesacon- ticsituation,weneedtounderstand how significanttheeffectsof
venientwaytodeterminereqvforagivenratiobetweentheSMBH spatialandvelocitystructureofmoleculargasaretotheobserved
massandthegalaxystellarmass
rotationvelocity.Itisnottrivialtoquantifyhow thesesystematic
DRAMBFHT= γ ( 2n-,x-) . DRAFT -- DR(8)AFT -- DRAFT -- DRAFT --
M∗ Γ(2n)
1 http://www.stsci.edu/hst/wfc3/documents/handbooks/currentIHB/c07_ir0
wherex=bn(rers)1/nandbnisestimatedbyequation(7).There- 2 http://www.stsci.edu/jwst/instruments/nircam/PSFs/
SMBHmass measurementusingALMA 5
109 109
RSOI JWSTF150W RSOI JWSTF150W
Rmax Rmax
Reqv,n=1 Reqv,n=1
108 Reqv,n=2 HSTF150W 108 Reqv,n=2 HSTF150W
Reqv,n=3 Reqv,n=3
Reqv,n=4 Reqv,n=4
107 107
]⊙ ]⊙
M M
[H106 [H106
B B
M M
105 105
104 Mgal=1.0·1010[M⊙] 104 Mgal=1.0·1011[M⊙]
Distance=15.29[Mpc] Distance=15.29[Mpc]
103 10−3 10−2 10−1 100 101 102 103 10−3 10−2 10−1 100 101 102
Radius[arcsec] Radius[arcsec]
(a) M∗=1010M⊙ (b) M∗=1011M⊙
Figure2. Relation between theSMBH massand thethree different angular scales: rSOI,rmax andreqv foragiven galaxy stellar massM∗.ForrSOI,we
combineequation(2)andtherecentcompilation ofM −σrelation (McConnell&Ma2013).Forr ,weusethebestfitrelation betweenr andr
max SOI max
(r =1.92r )inDavis(2014).Forr differentcoloursindicategalaxysurfacebrightnessprofilescharacterisedbydifferentSérsicindexn.
max SOI eqv
109 109
n1.0 n1.0
n2.0 n2.0
n3.0 n3.0
108 n4.0 108 n4.0
107 107
]⊙ ]⊙
M M
[H106 [H106
B B
M M
105 Mgal=1.0·1010[M⊙] 105 Mgal=1.0·1011[M⊙]
i=60◦ i=60◦
104 104
10130−1 100 101 102 10130−1 100 101 102
∆V (Vdiff/3)[km/s] ∆V (Vdiff/3)[km/s]
(a) M∗=1010M⊙ (b) M∗=1011M⊙
Figure3.SameasFigure2,butforthevelocitychannelwidth∆V.
effects influence the detection of SMBH mass using analytic ar- velocitytracersatthecentreandthePVDanalysiswillnotprobe
guments.Howeverwewillquantitativelydiscusstheireffectshere theregionwiththelargeststatisticalsignificanceinvelocityspace
and show their significance using simulations of PVD in Section for detecting SMBH. In addition, for the same background noise
3.4. and the same amount of total flux from the molecular gas, if the
gas profile shape in central region is homogeneous across many
First,weconsiderthesystematiceffectduetothespatialdis-
synthesised beams, a peak flux at the centre where the impact of
tribution of the molecular gas. Recent ALMA observations have
SMBHmassisobservedthemostbecomesrelativelylowandthe
revealed detail structure of molecular gas at a 10-100 pc scale.
signatureofSMBHwillbelesssignificantthanthatforthemolec-
Mostofthemshowdisclikemorphologywithandwithoutholeat
ulargaswithmoreconcentratedprofile,whichimpliesthatthegas
thecentre(e.g.Izumietal.2013;Combesetal.2014;Onishietal.
discprofileshapemightaffecttheuncertaintyofrotationvelocity
2015; Xuetal. 2015). If the gas distribution is continuous and
measurement and the resulting SMBH mass. Since the proposed
smooth, the central region of PVD is sampled well and the im-
DRAFT -- DRAFT -- DRAFsenTsitiv it y-re-qu ir emDentoRfALAMAFisdTefine d p-er-be am D,theflRuxvAari-FT --
pact of SMBH to the rotation velocity, if significant, can be de-
ationduetothespatialgeometryofgasdiscwithinALMAbeam
tected.However,ifthemoleculargasdistributionhasaholeatthe
may impact modeling gas kinematics given that the geometry of
centreandthesizeoftheholeislargerthanr ,therewillbeno
eqv
6 IlsangYoon
circum nuclear molecular gas is not known a priori. Also if the bution (not by the SMBH) and the difference between the rota-
geometryofmoleculargasdiscbecomesclosertotheface-on,the tionvelocitiesdue toslightlydifferent surface brightness profiles
projectedlineofsightvelocitydecreasesandsmallervelocitychan- of the galaxy can not be discriminated. The core-Sérsic galaxy
nelwidthisrequiredtoresolvethevelocitydifferencebetweenthe has M∗ = 1011M⊙ and MBH = 106M⊙, and follows the sur-
pure galaxy rotation and the galaxy-SMBH rotation. In contrast, face brightness profile with Sérsic index n = 3 and core radius
ifthe galaxyisedge onand thebeam sizeistoolargetoresolve r = 0.01r (r r inequation(4)).Weconsidertwodifferent
b e b e
≪
the kinematic structure along the minor axis, the velocity profile coreprofiles: nearly flatcore using γ = 0.1 and power law core
issmearingduetothevelocitycomponentsmigratedfromthemi- usingγ = 0.5,(seeequation(4)),representingthetypicalvalues
noraxis(Barthetal.2016).Anotherpotentialproblemmightbea forellipticalgalaxiesseeninTrujilloetal.(2004).
warp in the gas disc. Warp is prominent in HI distribution (e.g., Figure4(a)and4(b)showstherotationvelocitiesduetodiffer-
García-Ruiz,Sancisi&Kuijken2002)butalsoseeninthenuclear entgalaxysurfacebrightnessprofilesandSMBHmassesandtheir
molecular gas disc (Sofue&Rubin 2001). It introduces a bias in residualsfromthetruerotationvelocity.AsinFigure2,theregion
theinclinationcorrectionandunderestimatestherotationvelocity smallerthanthePSFFWHMforHSTWFC3andJWSTNIRCam
depending onthelocationofthemajoraxis(Verganietal.2003), isshownbylightanddarkgreyarearespectively.Thehighestangu-
whichcanbemitigatedbymodelingfull3Ddatacube. larresolutionofALMArangesfrom0.006′′at675GHzto0.037′′
Second, we consider the systematic effect due to the veloc- at110GHz.Weassumea0.01′′beamsizeasatypicalALMAres-
ity structure of molecular gas. In real, the velocity structure in olution for the most extended array configuration and show it by
molecular gasismorecomplicatedthanthepurecircularmotion. verticaldashedlineinFigure4.
This velocity structure which we call non-circular motion in this IntheupperpanelofFigure4(a)and4(b),weshowthreedif-
workincludesthegasinflowduetoangular momentumloss(e.g. ferentrotationvelocities.First,theredlineshowsthetruerotation
Combesetal.2014)andthegasoutflow duetostrongstellarand velocityofthecore-Sérsicgalaxywithnearlyflatcore(γ = 0.1,
AGNfeedback(e.g.García-Burilloetal.2014),bothofwhichhave Figure4(a))andpowerlawcore(γ =0.5,Figure4(b)).Second,the
beenobservedbyALMA.Forasetofmoleculargasparcelswithin blacklineshowstherotationvelocityofthegalaxythatfollowsthe
radiobeamalongthelineofsight atacertainprojecteddistance, surfacebrightnessprofilewiththesamenasthecore-Sérsicgalaxy
thisbulkmotion(infloworoutflow)contributestothelineofsight andthesameSMBHmassbutwithoutthecore.Last,theblueline
velocity of the gas parcels positively or negatively depending on shows the galaxy rotation velocity that shows a better agreement
wherethegasparcelsarelocated.Thiswillcauseavelocityspread with the true rotation velocity (red) for the scale larger than the
inPVDandatagivenradius,makethePVDthickeralongtheve- ALMAangularresolution,byusingverysmall(orno)adjustment
locityaxis.Asaresult,thesignature ofKeplerian rotationowing oftheSérsicindexandmass-to-lightratiowhileusinganorderof
totheSMBH willnot besignificant. Althoughthedriving mech- magnitudedifferentSMBHmass.Indetail,thebluelineinFigure
anismsaredifferent,bothinflowandoutflowhavethesameeffect 4(a)showsthecasewherethegalaxyhas5%smallerSérsicindex
tothePVDswiththeonlydifferenceinthevelocitysign.Inaddi- (n= 2.85)and3%largermass-to-lightratio(M/L = 1.03)than
tion,thereisarandommotioninthecircumnuclearmoleculargas thetruevaluesbuthasa10timessmallerSMBHmass(105M⊙).
whichalsointroducesadispersioninthevelocityaxisinPVD. ThebluelineinFigure4(b)showsthecasewherethegalaxyhasthe
sameSérsicindex(n=3.0)andmass-to-lightratio(M/L=1.0)
as the model galaxy shown by the black line, but has a 12 times
2.4 GalaxySurfaceBrightnessProfileBias
moremassiveSMBH(1.2 107M⊙).
×
TomeasuretheSMBHmassusingrotationvelocityofthecircum InthelowerpanelofFigure4(a)and4(b),weshowtheresid-
nuclear molecular gas,thesurfacebrightness profileat thegalac- ualvelocitiesfromthetruerotationvelocity(redcurvesintheup-
ticcoreneedstobeaccuratelydetermined(e.g.,Ferrarese&Ford per panels), for the model galaxies in the upper panels using the
2005),whichrequireshighangularresolutiontoresolver .How- same colour. For both cases of the biased galaxy surface bright-
eqv
ever,theangularresolutionofgalaxyimagemightnotbesufficient ness profile with largely biased SMBH mass that shows a better
for distant galaxies toaccurately characterise thecore luminosity agreementwiththetruerotationvelocity,thevelocitydifferenceat
profilewhileradiointerferometrystillhasasufficientresolutionto angularscalelargerthantheassumedALMAresolution(0.01′′)is
resolver . smallerthanthetypicalvelocitychannelwidthadoptedformeasur-
eqv
Detail analysis of the high resolution HST images of early- ingSMBHmassusingmoleculargaskinematics(e.g.,10km/sin
type galaxies reveals that the galaxy core surface brightness pro- Davisetal.(2013a))andtheirvelocityresidualatthespatialscale
files deviate from the Sérsic model fit and have a flat core resolvedbyHSTandALMAissmallerthanthevelocityresiduals
or a power law slope (e.g., Trujilloetal. 2004) although Sér- ofthegalaxywiththesameglobalSérsicindexasthecore-Sérsic
sic model is a good fit to the overall profile in general (e.g., galaxy(bluelines).
Caon,Capaccioli&D’Onofrio1993;Graham2001).Ifthegalaxy Therotationvelocityofthetruecore-Sérsicgalaxy(redline)
coreprofileisnotresolved,thestellarmassdistributionatthegalac- andthesameSérsicgalaxyprofilewithoutcore(blackline)arein-
ticcentreissystematicallybiasedandasaresult,thederivedrota- distinguishableatthescaleofHSTandJWSTresolution( 0.1′′)
≈
tionvelocityisalsobiased.Thisbiasintroducesalargesystematic witha1-2 km/s velocity difference as seen in thelower panel of
errorintheSMBHmassmeasurement.InSection3.5,weillustrate Figure4(a)and4(b)asshownbyblackline.Thereforeifthecore
howthebiasinthegalaxysurfacebrightnessprofilecanimpactthe ofthegalaxysurfacebrightnessprofileisnotresolved,theSérsic
SMBHmassmeasurement. galaxyprofileswiththeunresolved core(blacklinesintheupper
In Figure 4, we show the rotation velocity of a galaxy with panels of Figure 4) may be the best determination of the galaxy
SMBH using core-Sérsic galaxy surface brightness profile with stellarmassdistribution.Howevertherotationvelocityofthisbi-
DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
corebeingunresolvedbyputtingthisgalaxyatlargedistancesuch ased profile is not a good match to the true rotation velocity of
that at a scale of the HST and JWST seeing, the galaxy rotation thecore-SérsicgalaxyatangularscalesimilartotheALMAbeam
velocity is largely dominated by the galaxy stellar mass distri- (0.01′′)andthebluelinewithslightlyadjustedgalaxyparameters
SMBHmass measurementusingALMA 7
10−5 10−4 10−3 10−2 10−1 100 101 10−5 10−4 10−3 10−2 10−1 100 101
103 103
Distance=222.29[Mpc] Distance=222.29[Mpc]
i=60.0◦ i=60.0◦
ALMA M∗=1.0·1011M⊙ ALMA M∗=1.0·1011M⊙
JWSTF150W JWSTF150W
s] HSTF150W s] HSTF150W
m/ m/
k k
y[ 102 y[ 102
ocit ocit
el el
V V
Total:Core-Sersic(true),n=3.0γ=0.1 Total:Core-Sersic(true),n=3.0γ=0.5
BH:MBH=1.0·106[M⊙] BH:MBH=1.0·106[M⊙]
Total:n=3.0MBH=1.0·106[M⊙] Total:n=3.0MBH=1.0·106[M⊙]
101 Total:n=2.85MBH=1.0·105[M⊙] 101 Total:n=3.0MBH=1.2·107[M⊙]
40 HSTF150W 40 HSTF150W
m/s] 2300 ALMA JWSTF150W m/s] 2300 ALMA JWSTF150W
k 10 k 10
y[ 0 y[ 0
ocit −−2100 ocit −−2100
el −30 el −30
V −−5400 10−nn5==32..085MMBBH1H=0=1.1−0.0·41·0160[5M[M⊙⊙]]10−3 10−2 10−1 100 101 V −−5400 10−nn5==33..00MMBBHH1==011..−20··4110076[[MM⊙⊙]]10−3 10−2 10−1 100 101
Radius[arcsec] Radius[arcsec]
(a) nearlyflatcore (b) powerlawcore
Figure4.RotationvelocityofagalaxywithM∗ = 1011M⊙ containing 106M⊙ SMBH.Thesurfacebrightnessprofileofthegalaxyisthecore-Sérsic
profilewithrb =0.01reandn=3inequation(4).Panel(a):thegalaxyhasanearlyflatcore(γ =0.1).Theredlineshowsthetotalrotationvelocityof
thetruegalaxymodelincludingSMBH.TheblacklineshowstherotationvelocityofthesamegalaxywiththesameSérsicindexandthesameSMBHmass
butwithoutthecore.ThebluelineistherotationvelocityoftheSérsic(withoutcore)modelgalaxywith5%smallerSérsicindex(i.e.,n = 2.85)and3%
largermass-to-lightratio(i.e.,M/L=1.03)thanthegalaxyshownbytheblackline,howeverwitha10timessmallerSMBHmass(105M⊙)thanthetrue
value(106M⊙).Panel(b):thegalaxyhasacorewithpowerlawprofile(γ=0.5).TheredandblacklineindicatesthesamecaseinPanel(a).Bluelineisthe
rotationvelocityoftheSérsic(withoutcore)modelgalaxywiththesameSérsicindex(i.e.,n=3.0)andmass-to-lightratio(i.e.,M/L=1.0)asthegalaxy
shownbytheblackline,howeverwitha12timesmoremassiveSMBHmass(1.2×107M⊙)thanthetruevalue(106M⊙).
(nand M/L) but withorder of magnitude different BH massisa 3 DETECTINGTHEIMPACTOFSMBH
bettermatchtothetruerotationcurveattheangularresolutionof
Wehavediscussedtherequiredangularandvelocityresolutionto
ALMA. This results in a biased SMBH mass in model fitting as
measuretheimpactofSMBHbasedontherotationvelocitywith-
demonstratedinSection3.5.
out considering theobservational measurement processes. Wear-
guethat,ifthegalaxystellarmass,surfacebrightnessprofileshape
and inclination are known, there is a required spatial resolution
correspondingtor andvelocitychannelwidthcorrespondingto
eqv
1V todetect theSMBHmassthatoneaimstomeasure. Inthis
3 diff
section,weconfirmthisargumentandshowhowtheangularreso-
Lowering ALMA resolution less than 0.01′′ in thisexample
lutionandvelocitychannelwidthaffecttheSMBHmassdetection
will remove the bias, however achieving 0.01′′ resolution is very
bysimulatingPVDswithmodelrotationvelocityincludingtheob-
difficult(ifnot feasible) inpracticefor manyof galaxiesbecause
servational measurement processes. Thenweincorporatethesys-
ofalongintegrationtime.Sofornearbygalaxies,ALMAhassuf-
tematiceffectsinthePVDsimulationtotesthowsignificanttheir
ficient spatial resolution to resolve r and thus even though the
eqv impactsaretothemeasurementofrotationvelocityforseveralrep-
galaxyrotationvelocityisbiasedduetotheunresolvedgalaxycore
resentativecases.
in the photometric image, ALMA can distinguish the difference
betweenslightlydifferentgalaxyrotationvelocities.However,for
distant galaxies, the ALMA spatial resolution becomes poor and
cannotdifferentiatethevelocitiesduetothebiasedgalaxysurface 3.1 SimulationsofPosition-VelocityDiagram
brightnessprofiles.ThisimpliesthatatthecurrentbestALMAres-
WeuseKinMS(Davisetal.2013b)tosimulatePVD.KinMS3 is
olution,itisdifficulttobreakthedegeneracybetweendifferentro-
publicly available IDL code tosimulate gas kinematics by incor-
tationvelocitiesduetothebiasedcoresurfacebrightnessprofiles
poratingtheobservationaleffects:beamsizeandvelocitychannel
notresolvedbyHSTorJWST,ifthegalaxyisatadistancesimilar
widthandthepropertiesofmoleculargas:userdefinedgasdensity
orlargerthan 200Mpc.Inprinciple,toavoidabiasoftheSMBH
≈ profile,randomvelocitydispersion,bulkmotions(inflow/outflow),
massowingtothebiasedgalaxysurfacebrightnessprofile,theres-
DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
olutionofgalaxyimageshouldbesimilartotheALMAbeamsize
andideally,thebothneedstobecomparabletor assumingthata
eqv
sufficientvelocityresolutionisachievedbyALMA. 3 https://github.com/TimothyADavis/KinMS
8 IlsangYoon
warp and blobs inthe molecular gas. Although ithas been origi- Wecaution thatthevelocitydeterminedbytracingtheouter
nallydeveloped tomodel thegaskinematicsinellipticalgalaxies envelope of PVD is biased toward the material along the line of
inATLAS3Dsurvey(Davisetal.2013b),itisalsodirectlyapplica- sight and does not very well trace the Keplerian rise in the PVD
bletotheanalysisofmoleculargaskinematicsfortheSMBHmass seeninFigure5.Someoftheissuesdiscussedinthissectioncanbe
measurement(Davisetal.2013a;Onishietal.2015).Fordetailin- removedbyfittingthe2DPVDintensitymapitselfortheentire3D
formationofthecode,werefertoDavisetal.(2013b). datacube.Thedetailedcomparisonoftheperformanceofdifferent
InPVDsimulation,wegeneratethenoiseasfollows.Wesam- methodsisbeyondthescopeofthiswork.However,wenotethat
plethespatialdistributionofmoleculargasusingafinitenumber ourapproachtracingtheouterenvelopeofPVDisconservativeand
ofrandomsamplesandmakePVD.Aftersamplingthedistribution thereforeensuresthatmoresophisticatedfittingmethodsprovidea
100 timesusing different random numbers, weestimatethestan- betterconstrainttotheSMBHmass.
dard deviation of the ensemble PVDs from the ‘noise-less’ PVD For illustration, we select a typical example of SMBH and
generatedfromthedensitydistributionsampledbyalargenumber hostgalaxy.TheassumedSMBHmass(108M⊙)andgalaxystel-
(106) of random samples. Then thesignal-to-noise ratio(S/N)of lar mass (1011M⊙) are not in the extreme range (e.g., dynami-
PVDisdefinedbythepeaksignalinthePVDandthestandardde- cally measured 1010M⊙ SMBH in 1011M⊙ galaxy reported by
viationoftheensemblePVDs.Wevarythenumberofrandomsam- vandenBoschetal.2012).IntheupperrightpanelofFigure5,we
ples(N )toobtainarangeofS/Ninthiswork.HoweverS/N areabletodetectasignatureoftheSMBHasrevealedbythehigh
samp
does not onlydepend onthenumber of random samplesbut also velocitytipofblackcontouratthecentreindicativeoftheKeple-
onthebeamsize,velocitychannelwidthandpixelresolution;the rianrotation.Iflookingattheresidualrotationvelocityshownby
largerthevalues,thelargertheS/Nforthesamenumberofrandom the blue line in the lower right panel of Figure 5, the maximum
samples.Wefindthatformostofourexperimentsinthiswork,ap- deviationatthecentreismuchlargerthan3σvelocityuncertainty
proximately10000randomsamplesgivesagoodS/Nrangingfrom andthedeviationislargerthan1σrangefortheentirerangeofthe
60to100.However,wenotethattherealnoiseinPVDisoriginated spatialscale.Inthiscase,itisrelativelyeasytoclaimadetectionof
by the spatially correlated interferometer noise and thus depends theSMBHmass.HoweverforgivenM andgalaxystellarmass,
BH
onthespatialscaleoftheobservation.Thisrequiresamorecom- the significance of this deviation will be smaller with increasing
plicatedsimulationofthePVDusingrealisticarrayconfiguration, beamsizeandvelocitychannel widthoftheradiointerferometry.
whichisbeyondthescopeofthecurrentwork. SinceitisalreadyobviousthattheSMBHwithlargeenoughmass
Figure 5 shows the simulation of a 1011M⊙ galaxy with willbeeasilydetectedasshowninthisexample, wewilltestthe
n = 3 and i = 60◦ containing 108M⊙ SMBH, at 15 Mpc caseof thelowest detectable SMBH masses for thegiven galaxy
≈
distance.Therequiredbeamsize(r )andvelocitychannelwidth parametersinferredfromFigure2and3.
eqv
(1V ) for this galaxy is 0.5′′ and 12 km/s based on the discus-
3 diff
sioninSection2.2.Molecular gasdistributionaround theSMBH
3.2 EffectofNoise
issimplyassumed tobeanexponential discwith1′′ scaleradius
andsameinclinationangleasthegalaxy.ThepeakS/Nis 100 Beforeweinvestigatetheimpact ofsystematiceffect tothePVD
≈
using10000randomsamples. analysis, we discuss the effect of noise first. We take a fiducial
Thetopleftpanelshowsthevelocityintegratedfluxdistribu- galaxy with M∗ = 1011M⊙ including the SMBH with MBH =
tion(i.e.moment0map)with0.3′′beam.Thetoprightpanelshows 107M⊙ ofwhichsurfacebrightnessfollowsaSérsicprofilewith
PVDofthegalaxyusing0.3′′beamsizeand10km/svelocitychan- n = 3 and i = 60◦. Then we assume that the gas surface den-
nelwidth.Weuseslightlysmallerbeamsizeandvelocitychannel sity distribution is an exponential disc with 1′′ scale radius and
widththantherequiredvalues(0.5′′ and12km/s),forthepurpose sample the density distribution using a finite number of random
of clear demonstration of the impact of SMBH. Yellow contours samplesasdiscussedabove.Forthisgalaxy,theminimumrequired
on top of the background black contours are the PVD due to the beamsizeandvelocitychannelwidthis0.15′′ and8km/srespec-
galaxystellarmasswithouttheSMBHandthebackground black tivelyinferredfromFigure2and3.Forthisbeamsizeandchan-
contours are the PVD for the entire system (galaxy and SMBH). nel width, wemake PVDswith4 different signal-to-noise ratios;
Thelowerleftpanelshowstheinputrotationvelocitytogenerate S/N=10,30,60and120determinedby150,1400,6000and24000
thePVDshowingthecontributionfromtheSMBHandthegalaxy randomsamplesrespectively.NotethatS/Nisroughlyproportional
stellarmassdistribution.Thelowerrightpanelshowstherotation to N asexpectedfromstatistics.
samp
velocityofthegalaxywithandwithoutSMBHmeasuredfromthe pFigure6showstherotationvelocityofthegalaxywithSMBH
PVD and the difference of the two velocities using blue line en- (black), without SMBH (red) and the residual velocity between
closedby3σand1σlevelvelocityuncertainty(definedby3 and the two (blue). For each S/N, we generate 100 ensemble PVDs
×
1 velocitychannelwidth)asshownbythedashedanddottedline. and each solid line with error bars is the mean of this ensemble
×
Rotationvelocitiesarederivedfromtheyellowandblackcon- measurementswiththestandarddeviationsoftheensemblePVDs.
tours by using ‘envelop-tracing’ methods (Sofue&Rubin 2001), WhenS/Nislow(S/N=10),boththeblackandredlinehaslarge
which makes use of the terminal velocity in PVD along the ma- errorsandthustheresidualvelocityislargelyuncertain;thestan-
jor axis. The terminal velocity is defined by a velocity at which dard deviation in the central region is larger than 3∆V. As S/N
the intensitybecomes equal toI = (ηI )2+I2 on PVD, increases,theuncertaintyoftherotationvelocitymeasurementbe-
t max lc
whereImaxandIlcarethemaximumipntensityandintensitycorre- comessmaller.WhenS/N=60,itbecomesmuchsmallerthan∆V
spondingtothelowestcontourlevel.Usingη =0.5,thisdefinesa intheouterregionandcomparabletothe∆V inthecentre.This
velocityat50%leveloftheintensityprofileatafixedpositionfor noise simulation confirms our intuition that the rotation velocity
sufficientlylargesignal-to-noiseratiooravelocityalongthelow- differencewithandwithoutSMBHbecomeslesssignificantasS/N
DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
estcontourlevelifthesignal-to-noiseratioissmall(Sofue&Rubin decreases and itwill increase the uncertainty of the SMBH mass
2001).Thiswillcapturethecircularvelocitythatliesattheouter measurement.
envelopeofPVD. Inthefollowingexperimentstoinvestigatetherequiredspatial
SMBHmass measurementusingALMA 9
Figure5.PVDsimulationwithS/N=100foragalaxywithM∗ = 1011M⊙ i = 60◦,n= 3andMBH = 108M⊙.Weassumeasimpleexponentialdisc
withscaleradiusof1′′forthemoleculargassharingthesameinclinationanglewiththegalaxy.0.3′′beamsizeand10km/svelocitychannelwidthisused
togeneratethisPVD.ThetopleftpanelshowstheintegratedlineintensitymapandthetoprightpanelshowsthesimulatedPVDforthegalaxystellarmass
(yellow)ontopofthePVDoftheentiresystem(black:galaxyandSMBH).ThelowerleftpanelshowstheinputrotationvelocitytogeneratethisPVDand
thelowerrightpanelshowstherotationvelocitiesmeasuredfromthePVD(blackfortheentiresystemandredforthegalaxywithoutSMBH)andtheresidual
ofthetworotationvelocities(blueline).Dashedanddottedlineis3and1σsignificancefortheresidualrotationvelocity,definedby3×and1×velocity
channelwidth.
andvelocityresolution(Section3.3)andtodemonstratetheimpact largerthan3∆V.Althoughthe1.0r beamsize(redline)shows
eqv
ofsystematiceffectstothePVDanalysis(Section3.4),weensure smallermaximumresidualthanthatforthe0.5r beamsize,the
eqv
thatPVDhasareasonablygoodS/N(S/N 60)bychoosingthe residualisstillsignificant;largerthanorsimilarto3∆V.However
≈
appropriate number of random samples for given beam size and as the beam size increases (i.e., > 2r ), the maximum residual
eqv
velocitychannelwidthforeachexperiment. decreasesandthesignificanceoftheresidualrotationvelocitydoes
notstronglyindicatearotationvelocityduetotheSMBH.
EachpanelinFigure8showsthecasewheredifferentveloc-
3.3 SpatialandVelocityResolution itychannelwidthshavebeenusedtogeneratethePVDofthesame
WesimulatePVDsofthesamegalaxy(i.e.,agalaxywithM∗ = galaxyfor thefixed1.0reqv beamsize. LikeinFigure7,theerror
1011M⊙,MBH=107M⊙andi=60◦)inFigure6forfourdiffer- bar has been determined bythe standard deviation of therealisa-
tionof100ensemblePVDs.Theresidualvelocitiesforeachpanel
entSérsicindices(n=1,2,3and4)andinvestigateasignificance
arescaledbythevelocitychannelwidthbeingused.Thevelocity
ofthedifferencebetweentherotationvelocityofthegalaxywith
andwithoutSMBH,usingdifferentbeamsizesandvelocitychan- channelwidth∆V hasbeenadjustedwithrespecttotherequired
nelwidths. velocity channel width for each panel (∆V0 = 13Vdiff). Similarly
to Figure 7, as the velocity channel width increases from 0.5 to
EachpanelinFigure7showstheresidualrotationvelocityof
thegalaxywiththecorrespondingSérsicindex,fordifferentbeam 2.0∆V0,theresidualvelocitybecomeslesssignificant.Wefindthat
sizes.Thevelocitychannelwidth∆V isfixedto 1V .Beamsize thechoiceof∆V 1.0∆V0 showsasignificant(> 3σ)velocity
3 diff differenceformost≈typesofgalaxiesthatwesimulatedinthiswork.
variesfrom0.5to2.0r (asshownbydifferent colours) forthe
eqv
fixedvelocitychannelwidthusedineachpanel.Ineachpanel,the In summary, regarding the effect of spatial resolution, if the
widthof ∆V and3∆V areshown by thedottedand dashed line radio interferometry beam size is smaller than or similar to r ,
eqv
respectivelytoshowthe1σand3σsignificanceoftheresidualro- thePVDisclearlyresolvedandtheexcessoftherotationvelocity
tationvelocity.Errorbarhasbeendeterminedbythestandardde- due to the SMBH is detected above 3σ level when the velocity
viationoftherealisationof100ensemblePVDs. channelwidthresolves 1V .Thenregardingtheeffectofvelocity
DRAFThTere si du-al-ve lo cDitywRith0A.5r FbeaTmsi ze s-ho-w n byDtheRblueAFresTolut io n,-if-th e vDelociRt3ycdihfAfanneFlwiTdth is s-ma-lle r thDan 1RV ,AtheFT --
eqv 3 diff
symbolalwaysshowsasignificantdifferenceatthecentralregion effect of SMBH is detected above 3σ level if the scale of r is
eqv
in all panels of Figure 7; the maximum residual is significantly resolvedbytheinterferometrybeam.
10 IlsangYoon
(a) S/N=10 (b) S/N=30
(c) S/N=60 (d)S/N=120
Figure6.RotationvelocitiesandtheirresidualsofthesimulatedgalaxywithM∗ =1011M⊙andMBH =107M⊙measuredusing0.15′′beamand8km/s
velocitychannelwidth.Eachpanelshowsthesamegalaxywithdifferentsignal-to-noiseratio,S/N=10,30,60and120.Theblackandredsolidlineshow
therotationvelocityofthegalaxywithandwithoutSMBHandthebluelineisthedifference ofthetworotationvelocities. Rotationvelocity(solidline)
andassociatederrorbarhasbeendeterminedbymeanandstandarddeviationoftherealisationof100ensemblePVDs.Thevelocityuncertaintiesduetothe
velocitychannelwidthcorrespondingto1and3σsignificance, 1V andV areshownbythedottedanddashedline.
3 diff diff
3.4 SystematicEffects PVDandtheycanbeappliedtoothergalaxieswithdifferentnand
SMBHmasses.
For investigating the systematic effects, we note that one of
thedisadvantagesofusingPVDisthatdiscinclination,kinematic
Intheprevioussection,wediscussedthespatialandvelocityreso-
positionangleandcentroidparametersareneededtobewellcon-
lutionforthePVDanalysisusingrotationvelocityofpurecircular
strained(Barthetal.2016).Usingmoresophisticatedmethodsin-
motionwithoutsystematiceffectandconfirmedourargumentthat
cludingfitting3Ddatacubemightbemorereliableandcanresolve
r andV settherequiredbeamsizeandvelocitychannelwidth.
eqv diff someoftheissuesrelatedwiththegasdiscgeometry.However,we
Inthissection, wedemonstrate theimpact of possible systematic
alsonotethatfittingPVD,ontheotherhand,hasbenefitedfromthe
effectstothePVDanalysis:spatialstructure(i.e.,gasdensitydistri-
bettersensitivitytothecentralvelocityupturnandmaygivebetter
bution)andvelocitystructure(i.e.,inflow/outflow,randommotion
constraintsonM thanfitting3Ddatacube(Barthetal.2016).
andwarp)asdiscussedinSection2.3.Wedonotperformanexten- BH
sivesearchintheparameter space forallthesesystematiceffects
andonlyshowthePVDsimulationswithS/N 60ofagalaxywith
≈
M∗ =1011M⊙thatfollowsaSérsicprofilewithn=3andhosts 3.4.1 Geometryofmoleculargasdistribution
107M⊙ SMBH,usingtherequiredangularresolution(0.15′′)and
velocitychannelwidth(8km/s).Weshowtheresultsofsimulations Weconsider threedifferenttypesofdensityprofilesofmolecular
DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
using selective parameters for each effect. However our findings gas:thesameexponential discprofileastheprevioussimulations
anddiscussionsregardingthesesystematiceffectsarevalidforthe butwithdifferentscaleradius,thesame1′′scaleradiusexponential
given angular resolution and velocity channel widthof simulated disc but with different inclination angles and the density profile