Selection bias in dynamically-measured super-massive black hole samples: dynamical masses and dependence on S\'ersic index

0.8 MB·

by  Francesco Shankar

Mon.Not.R.Astron.Soc.000,1–12(2016) Printed10January2017 (MNLATEXstylefilev2.2)
Selection bias in dynamically-measured super-massive black hole
samples: dynamical masses and dependence on Se´rsic index
1⋆ 2 2
Francesco Shankar , Mariangela Bernardi , Ravi K. Sheth
7
1 1DepartmentofPhysicsandAstronomy,UniversityofSouthampton,Highfield,SO171BJ,UK
0 2DepartmentofPhysicsandAstronomy,UniversityofPennsylvania,209South33rdSt,Philadelphia,PA19104
2

n
a
J

6
ABSTRACT

] We extend the comparison between the set of local galaxies having dynamically measured
A
black holes with galaxies in the Sloan Digital Sky Survey (SDSS). We first show that the
G mostup-to-datelocalblackholesamplesofearly-typegalaxieswithmeasurementsofeffec-
tiveradii,luminosities,andSe´rsicindicesofthebulgesoftheirhostgalaxies,havedynami-
.
h calmassandSe´rsicindexdistributionsconsistentwiththoseofSDSSearly-typegalaxiesof
p
similar bulgestellar mass. The hostgalaxiesoflocalblackholesamplesthusdonotappear
-
o structurally different from SDSS galaxies, sharing similar dynamical masses, light profiles
r andlightdistributions.Analysisoftheresidualsrevealsthatvelocitydispersionismorefun-
st damental than Se´rsic index nsph in the scaling relations between black holes and galaxies.
a Indeed, residuals with n could be ascribed to the (weak) correlation with bulge mass or
sph
[ even velocity dispersion. Finally, targetted Monte Carlo simulations that include the effects

1 of the sphere of influence of the black hole, and tuned to reproduce the observed residuals
v andscaling relationsin termsof velocitydispersionandstellar mass, show that, at leastfor
2 galaxieswithMbulge & 1010M⊙ andnsph & 5,theobservedmeanblackholemassatfixed
3 Se´rsicindexisbiasedsignificantlyhigherthantheintrinsicvalue.
7
1 Keywords: (galaxies:)quasars:supermassiveblackholes–galaxies:fundamentalparame-
0 ters–galaxies:nuclei–galaxies:structure–blackholephysics
.
1
0
7
1 1 INTRODUCTION detectedanysignificantcorrelation,whilemorerecentlySavorgnan
:
v (2016),bycompilingalargergalaxysamplewithaccurateanduni-
Thescalingrelationsbetweensupermassive blackholesand their
i formphotometricdecompositions,hasclaimedasignificantcorre-
X hostgalaxieshavebeenaveryhottopicinthelastthirtyyears(see,
lationcharacterizedbyaslopeof3.39 0.15andanintrinsicscatter
e.g.,Ferrarese&Ford2005;Shankar2009;Graham2016,forre- ±
r of 0.25dex.Thescatteriscomparableto,orevensmallerthan,
a views).Thisisbecausesuchscalingsmaybethesmokinggunof ∼
theoneinthescalingwithvelocitydispersion,pavingthewayfor
a“co-evolution” between the twosystems (e.g., Silketal.2013),
itsuseasablackholemassindicatoringalaxies(e.g.,Grahametal.
although the physical processes involved arestill highly debated,
2007;MutluPakdiletal.2016).
ranging from quasar feedback to black hole mergers, clumpy ac-
Unveilingtheactualexistenceoftheblackhole-Se´rsicindex
cretion,and/orgalaxy-scalegravitationaltorques(e.g.,Silk&Rees
relationcouldbeakeypieceofevidenceforsomeimportantgalaxy
1998;Vittorinietal.2005;Jahnke&Maccio`2011;Bournaudetal.
evolutionary patterns. For example, more or less violent disc in-
2011b; Angle´s-Alca´zaretal. 2015). Besides the well-known cor-
stabilitiesingas-rich,high-redshiftdiscscouldfeedbothaninner
relations with velocity dispersion σ (Ferrarese&Merritt 2000;
bulgeandacentralblackhole(e.g.,Bournaudetal.2011a).Apro-
Gebhardt&etal. 2000) and (bulge) stellar mass M (e.g.,
bulge
gressively more prominent bulge component, possibly character-
Marconi&Hunt 2003; Laueretal. 2007; Kormendy&Ho 2013;
izedbyaproportionallyincreasinggalaxySe´rsicindex,maythen
La¨skeretal. 2014; Sagliaetal. 2016), correlations with the light
beabletohaltstarformationinthehostgalaxy(e.g.,Martigetal.
concentration and Se´rsic index have also been measured (e.g.,
2009;Dekel&Burkert2014).Aninitialcorrelationbetweenblack
Grahametal.2001;Graham&Driver2007;Savorgnan2016,and
hole mass and Se´rsic index could have thus been established by
referencestherein).
these high-redshift dissipative processes. If galaxy mergers have
ThecorrelationbetweenblackholemassandSe´rsicindex,in
been the actual drivers behind the origin of the large sizes and
particular,hasbeenthesubjectofnumerousstudiesinrecentyears.
highSe´rsicindicesinpresent-daymassivegalaxies(e.g.,Hilzetal.
Somegroups (e.g.,Sanietal. 2011; Beifiorietal.2012) havenot
2013;Nipoti2015),thenblackholesshouldhavenecessarilyfol-
lowedinsomedegreetheirhostgalaxymergerstopreserveacor-
⋆ E-mail:[email protected] relationwithSe´rsicindex.
(cid:13)c 2016RAS
2 F.Shankaret al.
On the other hand, both disc instabilities and repeated results are not affected by the removal of these sources. The er-
black hole mergers should also induce the build-up of a rorsquotedbySavorgnanetal.(2016)onthephotometricparam-
closer link between black hole mass and stellar mass (e.g., etersincludesystematics(e.g.,fromcomparisonwithdifferentau-
Jahnke&Maccio` 2011), at variance with the recent results by thorsandanalysismethods).However,sincewewillbeinterested
our group (Shankaretal. 2016, Paper I hereafter) and others inscalingrelations-theestimateofwhichincludesaccountingfor
(Blucketal. 2016; vandenBosch 2016). In Paper Iwe showed errors - we do not include the systematic contribution to the er-
that, following a number of previous claims (e.g., Bernardietal. roronn atthispoint.Specifically,weonlyaccountforrandom
sph
2007;vandenBoschetal.2015),thelocalsampleofgalaxieswith errors when estimating the intrinsic slope, zero-point and scatter.
dynamically-measured supermassive black holes is highly biased Weassesstheinfluenceofsystematicsasfollows.Whenadifferent
withrespecttoanunbiasedlargesampleofgalaxiesofsimilarstel- analysis method is used to estimate the photometric parameters,
larmass.Inparticular,blackholegalactichostsappeartohavesig- then we use these new values to estimate scaling relations in the
nificantly higher velocity dispersion (and slightly lower sizes) at same way as before (i.e., accounting only for the random errors
fixedstellarmass.PaperIusedMonteCarlosimulationsandresid- associatedwiththesenewvalues).Thedifferencesbetweenthein-
ual analysis to show that such biases can result if the sample of ferredscalingrelationscontributetothesystematicerroronthein-
local galaxies is preselected with the requirement that the black ferredscalingrelation.Inpractice,weusedasthe“othervalues”the
hole sphere of influence must be resolved to measure black hole sampleof La¨skeretal.(2014),whichalsoincludes accuratepho-
masses with spatially resolved kinematics. The same simulations tometricanalysisfromtheWIRcamimager attheCanadaFrance-
and statistical analysis clearly point to velocity dispersion being HawaiiTelescope, with Se´rsic-based light profile fitting routines.
more fundamental than stellar mass or effective radius, and pre- We retain 28 galaxies from their original sample, containing the
dictsignificantlylowernormalizationsfortheintrinsicscalingre- most secure dynamical black hole mass measurements according
lations.Thelatterpartlysolvesthesystematicdiscrepancybetween toKormendy&Ho(2013).
dynamically-basedblackhole-galaxyscalingrelationsversusthose To represent the full galaxy sample, we use objects in the
of active galaxies (e.g., Reines&Volonteri 2015), favouring pro- Sloan Digital Sky Survey (SDSS) DR7 spectroscopic sample
portionallylowervirialcalibrationfactorsfvirforestimatingblack (Abazajianetal.2009)intheredshiftrange0.05 < z < 0.2with
holemassesinactivegalaxies(e.g.,Ho&Kim2014). the photometric measurements from Meertetal. 2015. Through-
However,itispossiblethatsomeofthebiasmaybeinduced out this paper, we restrict the analysis to galaxies whose prob-
by real structural differences, i.e., physical effects could also be ability of being elliptical or lenticular p(E–S0) is greater than
playing a role. One of the two aims of this Letter is to address 0.80, based on the Bayesian automated morphological classifier
thequestion of structural differences between local galaxies with by Huertas-Companyetal. (2011); we refer to this as the SDSS
dynamically-measured black holesandtheir counterparts inlarge E-S0sample2.Whendealingwithtotalstellarmasseswewillin-
unbiased samples of galaxies. After briefly introducing the data steadrefertoonlyellipticalswithp(E)>0.8.Stellarmassesarede-
adoptedinthisworkinSection2,wefocusondynamicalmasses rivedbycombiningtheSEREXPestimatesoftheluminosityfrom
and(bulge) Se´rsicnsph distributionsinSection3.Wethenmove Meertetal. (2015) with mass-to-light ratios Mstar/L detailed in
to the second aim of this work, which is to compare the impor- Bernardietal.(2010,2013)andChabrier(2003)InitialMassFunc-
tanceofSe´rsicindexwithothervariablesintheblackholescaling tion (IMF). Systematic differences in Mstar/L can be of order
relations, inorder to determine if n plays afundamental role. 0.1dex(e.g.Bernardietal.2016).SDSSvelocitydispersions are
sph
WeusededicatedMonteCarlosimulationstointerpretourresults convertedfromRe/8tothe0.595kpcapertureoftheHyperleda3
andpresentourconclusionsinSection4.TwoAppendicesprovide database(Patureletal.2003),usingthemeanaperturecorrections
detailsofouranalysis.AppendixAdescribeshowouranalysisac- (e.g.,Jorgensenetal.1996;Cappellarietal.2006)
counts for statisticalmeasurement errors,and Appendix B shows
howtheslopesofcorrelationsinvolvingthreevariablesarerelated σR =(R/Re)−0.066 . (1)
toslopesofpairwiseregressions. (cid:18)σe(cid:19)
We note that blurring by seeing effects could potentially re-
ducecentralvelocitydispersionmeasurements(e.g.,Grahametal.
1998), however we do not foresee any major difference in the
2 DATA
seeing affecting ground-based measurements of σ in SDSS and
Following Paper I, we use the Savorgnanetal. (2016) sample thosecataloguedinHyperleda.Strictlyspeaking,theSe´rsicindex
of galaxies having dynamically measured black holes, with self- n we will adopt in this work is always referred to the galaxy
sph
consistent1measurementsofSe´rsicluminosities,effectiveradiiand spheroidal component extracted from a SEREXPluminosity pro-
Se´rsicindicesofthespheroidalcomponents,aswellasestimatesof file fitting in both the SDSS and the Savorgnanetal. (2016) and
thetotalhostgalaxyluminositiesandeffectiveradii.Centralveloc- La¨skeretal.(2014)samples.Thehalf-lightradiiRe,bulge andRe
ity dispersions are from Hyperleda, while stellar masses are ob- aredefinedastheradiicontaininghalfofthebulgeandtotalgalaxy
tainedbyapplyingtothe3.6µm(Spitzer)luminositiesaconstant luminosity, respectively. In the following, we will label the total
mass-to-lightratioof(M/M )/(L/L ) = 0.6fromMeidtetal.
⊙ ⊙
(e.g. 2014). As detailed in Paper I, from the original sample of
66galaxiesweremove18objectswithuncertainblackholemass
2 FollowingPaperI,whendealingwithbulgeswepreferentiallyadoptE-
and/or surface brightness, or unavailable central velocity disper-
S0galaxieswithp(E–S0)> 0.8asourreferenceSDSScomparisonsam-
sion, or because they are ongoing mergers. We checked that our
ple,becausedeterminingthecentralvelocitydispersionofspiralsfromthe
SDSSspectra(whicharenotspatiallyresolved)isnotpossible.Wechecked,
however,thatnoneofourresultsdependsontheexactcutinp(E–S0).
1 Thesamesurfacebrightnessprofilefittingprocedurehasbeenadopted 3 Fromhereonwards,unlessotherwisestated,velocitydispersionsσwill
foreachofthe66galaxiesinthesample. alwaysbedefinedattheapertureofHyperleda.
(cid:13)c 2016RAS,MNRAS000,1–12
SMBHs:selectionbiasandSe´rsicindices 3
Figure1.Left:Meandynamicalmass,Mdyn = K(nsph)Reσ2/G,asafunctionofstellarmass.Right:Sameformatastheleftpanelsbutforthebulge
component:Mdyn,bulge=K(nsph)Re,bulgeσ2/GasafunctionofMbulge.SolidlinesineachpanelshowthemeanrelationdefinedbytheSDSSofonly
E(left)orE-S0(right)samples,withtheSEREXPstellarmassesandphotometricparametersfromMeertetal.(2015);greybandsmarkthe1σdispersion
aroundthemean.SymbolsshowtheSavorgnanetal.(2016,toppanels)andLa¨skeretal.(2014,bottompanels)samples.Filledredcircles,greentriangles,and
bluestarsshow,respectively,ellipticals,lenticulars,andspirals,thelattertworeportedonlyintherightpanels.Dottedlinesineachpanelmarktheone-to-one
relations.TheagreementwiththeSDSSgalaxiesisgood.
galaxystellarmass,galaxybulgestellarmass,totalgalaxydynam- from Prugniel&Simien (1997). It is clear that the bulge dynam-
ical mass, and galaxy bulge dynamical mass as Mstar, Mbulge, ical mass of all galaxy types in the Savorgnanetal. (2016) and
M , and M , respectively. In the next sections, unless La¨skeretal.(2014)samplesbroadlyagreewiththoseofSDSSE-
dyn dyn,bulge
otherwise noted, we will compute median instead of mean quan- S0sgalaxiesofsimilarstellarmass.Thedatatendtoshowslightly
tities. While this makes little difference when dealing with stel- largerdynamicalbulgemassesatlowerstellarbulgemasses(right
lar/dynamicalmassesorvelocitydispersions,itmattersmorewith panelsinFigure1),mostprobablyinducedbytheverylargeveloc-
the (non-Gaussian) Se´rsic distributions at fixed stellar mass, for itydispersionscharacterizingthelowmassgalaxieswithdynami-
whichmediansaremoreappropriate. calmeasurementsofblackholes,asemphasizedinPaperI.How-
ever, most of theSavorgnanetal.(2016) andLa¨skeretal. (2014)
data are still broadly consistent with SDSS galaxies within the
quoted uncertainties. In line with a number of previous studies
3 RESULTS
(e.g.,Forbesetal.2008;Shankar&Bernardi2009;Bernardietal.
To test the hypothesis that galaxies with dynamically-measured 2011b;Cappellarietal.2013,andreferencestherein),itisalsoin-
black holes are a structurally different subset of the full galaxy terestingtonotethatinboththeSDSSandSavorgnanetal.(2016)
population–representedbytheSDSS–Figure1showsthemean samples all ellipticals have a dynamical mass a factor of 2
∼
dynamical mass (solid lines), along with its 1σ dispersion (grey higher than their total stellar mass (left); this ratio is smaller but
bands),fortheSDSSE-S0galaxiesasafunctionoftotal(leftpan- stillgreaterthanunityifonlythebulgecomponent isused(right;
els)andbulge(rightpanels)stellarmass.TheSDSSE-S0sarecom- compare solid and dotted lines, thelatter marking the one-to-one
paredtotheSavorgnanetal.(2016)andLa¨skeretal.(2014)sam- relations).
ples(topandbottompanels,respectively),dividedintoellipticals,
lenticulars/S0,andspirals,aslabelled.Heredynamicalmassisal- Figure2showsthecorrelationbetweenSe´rsicn andtotal
sph
ways computed for both samples as Mdyn = K(nsph)Reσ2/G, (left)orbulge(right)stellarmass.Solidlinesandgreyregionsmark
with the Se´rsic index-dependent virial constant K(n ) taken themedianand1σdispersionsfortheSDSSonlyE(left)orE-S0s
sph
(cid:13)c 2016RAS,MNRAS000,1–12
4 F.Shankaret al.
Figure2.Se´rsicindexnsphasafunctionofgalaxytotalstellarmass(left)andbulgestellarmass(right).SymbolsshowtheSavorgnanetal.(2016,toppanels)
andLa¨skeretal.(2014,bottompanels)samples,dividedintoellipticals,lenticulars,andspirals,aslabelled.Solidlineandgreyshadedregionshowtherelation
definedbySDSSonlyE(left)orE-S0(right)samples(blacklineswithgreyareas).Thepurplelong-dashedlineintherightpanelsshowsthemedianSe´rsic
indexforSDSSSabgalaxies.ThereisnosignificantmismatchbetweenSDSSgalaxiesandblackholesamples.
(right).SymbolsshowtheM hostsfromSavorgnanetal.(2016, ifnotlowerthanthoseofSDSSgalaxiesofsimilarbulgemassor
bh
toppanels)andLa¨skeretal.(2014,bottompanels).Thepanelson velocitydispersion.
theleftshowthatellipticals(redcircles)matchtheSDSSSe´rsicin- Figure3showsthatthemeanvelocitydispersionasafunction
dexdistributions.Thematchisextendedtolenticulars(greentrian- of Se´rsicindex n for early-type galaxies in our SDSS sample
sph
gles)whenswitchingtobulgestellarmasses(rightpanel).Spirals (long-dashed purple line) israther flat4 at n & 5. A direct fit
sph
(bluestars)intheSavorgnanetal.(2016)sample(top,right)tend tothedatabySavorgnanetal.(2016),reportedintheleftpanelof
tofallslightlybelowthemediantracedbytheSDSSE-S0galaxies, Figure 3 and labelled per morphological type, yields a systemat-
butarewithinthemedianSe´rsicdistributionsofE-S0andconsis- icallyhigher and steeper correlation with σ n0.3 (black thick
∝ sph
tentwiththeSab(purplelong-dashed line)SDSSgalaxies.Thus, dotted line). We interpret this as another sign of existing biases
the top panels of Figure 2 suggest that local galaxies with black in the local sample of galaxies with dynamical measurements of
holemassmeasurementsarenot,onaverage,structurallydifferent blackholes, inlinewithPaperI.TheLa¨skeretal.(2014) sample
fromSDSSgalaxiesofsimilarstellarmass. instead (right panel of Figure 3) appears broadly consistent with
SDSS data, with a negligible dependence on Se´rsic index, espe-
ciallyathighn ,asinourSDSSdata.InFigure3weonlyshow
sph
ThebottompanelsshowasimilaranalysisoftheLa¨skeretal. galaxies with logM /M > 10, to make a fair comparison
bulge ⊙
(2014) sample. In both panels, the correlations are much noisier withour(selectionbiased)SDSSE-S0mocksample,describedin
thanbefore.SpiralstendtoliesomewhatabovethemedianSDSS the next section, which can reliably probe only above this lower
Se´rsicindexofSDSSgalaxies.Infact,thesymbolsinthebottom limitinbulgemass.
rightpanelsuggestthatnsphdecreasesasMbulgeincreases;thisis Figure4showsthecorrelationbetweenblackholemassMbh
oppositetothetrendintheSavorgnanetal.(2016)sample,andwill and bulge Se´rsic index n . Symbols show the galaxies in the
sph
beimportantinwhatfollows.Thisdifferenceshowshowchalleng-
ing accurate determinations of Se´rsic indices can be. Finally, we
alsoverifiedthat,fortheearly-typegalaxiesinourblackholemass 4 Incontrast,themeanSe´rsicindexisasteeperfunctionofvelocitydis-
samples,theprojectedmassdensitywithinafewkpcaresimilarto, persion,thoughthescatterislarge.
(cid:13)c 2016RAS,MNRAS000,1–12
SMBHs:selectionbiasandSe´rsicindices 5
Figure3.CorrelationbetweenvelocitydispersionandSe´rsicindexnsph.Long-dashedpurplelineisthemedianrelationinSDSSearly-typegalaxies,while
symbolsmarkthegalaxiesintheSavorgnanetal.(2016,leftpanel)andLa¨skeretal.(2014,rightpanel)sampleshavinglogMstar/M⊙ >10,dividedper
morphologicaltype,aslabelled.Theblackthickdottedlinesarethedirectfitstothesedata.TheSavorgnanetal.(2016,leftpanel)sample,inparticular,hasa
highernormalizationandasteeperslopethantheSDSSrelation.
Savorgnanetal.(2016)andLa¨skeretal.(2014)samples(leftand ficient is r = 0.81. In contrast, the upper right panel shows that
rightpanelsrespectively)havinglogMstar/M⊙ > 10.Bluedot- residualsfromtheMbh-σrelationshowamuchweakercorrelation
dashed and purple dotted lines are the curved relations described withthosefromthen -σ correlation(r = 0.48). Together, the
sph
by Graham&Driver (2007) and Savorgnan (2016), respectively. twoupperpanelsimplyM σ4.1±0.1n0.8±0.1.
bh ∝ sph
We describe the grey regions and other curves later. A direct fit
Similarly, the two lower panels imply M
totheSavorgnanetal.(2016)andLa¨skeretal.(2014)datayields M0.7±0.1n0.9±0.1. However, the correlation with bulbghe ma∝ss
M n1.8 and M n0.1, respectively. The Appendix ad- star sph
drebshse∝sthespqhuestionobfhw∝hethesprhornotsuch(different)behaviours atfixednsph (lowerleftpanel)tendstobetighterthantheonein
Se´rsic index at fixed M (lower right panel has r . 0.31).
wouldbeexpectedifblackholemassiscloselycorrelatedwithve- bulge
BothslopeandPearsoncorrelationcoefficient droptoabout zero
locitydispersion,asemphasizedinPaperI,buttheσ-n trends
sph when considering only E-S0 galaxies, suggesting that most of
forthetwosamplesareerydifferent(asshowninFigure3).
the correlation in Figure 4 between black hole mass and Se´rsic
Forthispurpose,wenowtestifthecorrelationbetweenblack
index could be induced by the relation between Se´rsicindex and
holemassandSe´rsicindex,evidentatleastintheSavorgnanetal.
stellar (bulge) mass. If barred galaxies are excluded from the
(2016)sample,isfundamental,ormerelyaconsequenceofothers.
Savorgnanetal. (2016) sample, then the Pearson coefficients in
Correlationsbetweentheresidualsofscalingrelationsareaneffi-
thetworighthandpanelsofFigure5decreasetor 0.33(top)
cientwayofaddressingthisquestion(Sheth&Bernardi2012,and ∼
and r 0.14 (bottom). Our analysis thus strongly suggests that
PaperI). ∼
velocitydispersionismorefundamentalthanSe´rsicindex,further
TheoriginalerrorsassignedtotheSavorgnan(2016)sample supportingandextendingtheresultsinPaperI.
includeboththestatisticalaswellasthesystematicerrorsthataf-
AsimilaranalysisoftheLa¨skeretal.(2014)sample,reported
fectphotometricdecompositions.Thisisaparticularlyrelevantis-
inFigure6,alsoyieldsatightcorrelationwithvelocitydispersion
sueforSe´rsicindices.Thequotederrorsinn areinfactofthe
sph
(r = 0.89 in upper left panel), and extremely weak correlations
orderof 35%,whiletypicalstatisticalerrorsamounttoatmost
∼ withSe´rsicindex(r < 0.3intopandbottomrightpanels).Using
. 20 25%,i.e.,.0.1dex(Bernardietal.2014).Asdiscussed
− only E-S0 galaxies yields even stronger dependence on velocity
in Section 2, when computing residuals with respect to n , we
sph
dispersionandnearlynodependenceonSe´rsicindex.Evenassum-
willalwaysconsideronlythestatistical 0.1dexerrors.Thedif-
∼ ingsubstantiallylargerstatisticaluncertaintiesinn stillyields
ferenceinthemeasuredslopesfromdifferentsamplesshouldthen sph
veryweakcorrelationsinthepanelsontheright.Finally,notethat
provideanindicationoftheimpactofadditionalsystematicuncer-
La¨skeretal.(2014)alsoprovideSe´rsicindicesderivedallowingfor
tainties.Wenotethattheimpactofsystematicuncertaintiesshould
acoreinsomegalaxies(seeLa¨skeretal.2014,fordetails).Using
notbeincludedinanysinglemeasurementsimplybyinflatingthe
theseinsteadyieldsresultsconsistentwithFigure6.
measuredstatisticaluncertainties.Forsimilarreasonsweadopttyp-
icalaverageerrorsforthebulgestellarmassesof0.13dex,i.e.,30% Intheanalysesabove,theerrorsonvelocitydispersionswere
(see,e.g.,Meertetal.2013),insteadoftheirreportedaveragevalue takentobe5%(e.g.,Tremaineetal.2002;Graham&Scott2013),
of 0.17 dex.Appendix Adescribesinsomedetailhow weac- inlinewithwhat isquoted intheHyperleda database. However,
∼
count for statistical measurement errors, and assign error bars in larger errors in velocity dispersion for these same galaxies have
theanalysiswhichfollows. been reported in the literature (e.g., Ferrarese 2002), in linewith
The upper left panel of Figure 5 shows that residuals in the those measured for SDSS galaxies (e.g., Bernardietal. 2011a).
Savorgnan (2016) sample from the M -n relation correlate Largererrorsinvelocitydispersionwouldstrengthenourmainre-
bh sph
very well with those from the σ-n relation: the Pearson coef- sultthatvelocitydispersionismorefundamentalthanSe´rsicindex.
sph
(cid:13)c 2016RAS,MNRAS000,1–12
6 F.Shankaret al.
Figure4.Correlation between black hole massMbh andbulgeSe´rsic indexnsph.Symbols showthe galaxies inthe Savorgnanetal. (2016,leftpanel)
and La¨skeretal. (2014, right panel) samples having logMstar/M⊙ > 10. Blue dot-dashed and purple dotted lines are the curved relations described
byGraham&Driver (2007)andSavorgnan (2016),respectively. Black solid lines and greybands show the selection biased relation inthe Monte Carlo
simulationsdescribedinthenextsectionwhentheintrinsicrelationisgivenbyModelIofShankaretal.(2016)(dashedblackline).This(selectionbiased
relation)isbroadlysimilartothatobserved,suggestingthatthemeanblackholemassatfixednsphcanbeseverelyoverestimated,atleastfornsph&5.The
blackthickdottedlinesarethedirectfitstothedata.TheLa¨skeretal.(2014,rightpanel)sample,inparticular,showsnodependenceonSe´rsicindexanditis
broadlyinlinewiththepredictionsoftheMonteCarlosimulations.
retainonlythoseobjectsfor whichthegravitationalsphere ofin-
Table1.SlopesoflinearrelationsinourSDSSgalaxysample.
fluenceisgreater than thetypical resolution oftheHubble Space
Telescope,i.e.,r GM /σ2 >0.1′′.
X infl bh
≡
First, we note that the selection-biased mock residuals pre-
logMstar logσ lognsph dictedbyour MonteCarlos(graybandsinFigures5and6),pre-
logMstar 2.05 0.36 dictstrong correlations, especially invelocity dispersion, at fixed
Y logσ 0.33 0.16 Se´rsicindex(leftpanels),andweakcorrelationswithSe´rsicindex,
lognsph 0.19 0.55 inagreementwiththeLa¨skeretal.(2014)sample,butnotwiththe
Savorgnanetal. (2016) one. It is interesting to note that the pre-
dictions of the Monte Carlos without selection bias (purple long
4 DISCUSSION dashedlines)wouldpredictsignificantlysteeperresidualsatfixed
Se´rsicindex(seetheAppendixforfurtherdetails).
Intheprevioussectionweshowedthatvelocitydispersionismore
fundamentalthanSe´rsicindexn fordeterminingM .Indeed, The long-dashed black lines in Figure 4 shows the intrinsic
sph bh
theMbh-nsphcorrelationseemstobemostlyinducedbythensph- Mbh-nsph relationinour SDSSE-S0sample predictedby Equa-
M and M -M relations. However, because the M tion2.Itisremarkablyflat,becausevelocitydispersionisaweak
bulge bh bulge bh
sampleisbiased(tolargeσ)bythewayinwhichthesamplewasse- function of Se´rsic index (Figure 3; see Appendix for more dis-
lected,wemustmakesurethattherelationsdefinedbythesymbols cussion.) The solid black line and associated grey region show
inFigure5arenotaffectedbytheselectioneffect.Weusetargetted themeanand1σdispersioninthepredictedMbh-nsph relationof
Monte-Carlosimulationstodoso:detailsaregiveninPaperI,so theselectionbiasedsample(i.e.,afterselectingobjectswithlarge
herewebrieflysummarizethemainpoints. enoughrinfl).Noticethatitliesalmostanorderofmagnitudeabove
To each SDSS galaxy in our sample5 we associate a super- theintrinsicrelationatnsph &5.
massiveblackholefollowingthefavouredmodelinPaperI For completeness, blue dot-dashed and purple dotted lines
in Figure 4 show fits to the observed M -n relation from
M σ M bh sph
log bh =γ+βlog +αlog bulge , (2) Graham&Driver (2007) and Savorgnan (2016), respectively. At
M 200kms−1 1011M
⊙ (cid:16) (cid:17) (cid:18) ⊙(cid:19) least for relatively massive, large nsph early-type galaxies, these
with (γ,β,α) = (7.7,5.0,0.5) and a total (Gaussian) scatter of fits and the measurements are in broad agreement with the grey
0.25 dex (inclusive of observational errors). We repeat the above region defined by our selection-biased Monte Carlos. Hence, we
procedure several times tocreate a “full”black hole sample, and concludethatatleastsomeofthedifferencebetweentheintrinsic
relation(blacklong-dashedline)andthedataatlargen canbe
sph
ascribedtoselectioneffects.
5 ThesimulationsarebasedontheSDSSsamplefromMeertetal.(2013)
whichismagnitudelimited,thoughallmockresidualsareweightedthrough AtsmallernsphandlowerMstar thedatabySavorgnanetal.
(2016)tendtocurvedownwards asindicated bythefits,whereas
Vmax.Wehavefurtherverifiedthatnoneofourconclusionsarechangedif
weadoptedafullmockcaseextractedfromthestellarmassfunctionandto our Monte Carlos do not. Including an intrinsic dependence be-
whichvelocitydispersions,bulgefractionsandSe´rsicindicesareassigned tweenMbh andnsph,despitenotbeingfavouredbytheresiduals
viaempirically-basedcorrelations. inFigure5,stillproducesaflatbiasedMbh-nsph relation.Itmay
(cid:13)c 2016RAS,MNRAS000,1–12
SMBHs:selectionbiasandSe´rsicindices 7
Figure5.Correlations betweenresidualsfromtheobservedscalingrelations,asindicated ineachpanel.Redcircles,greentriangles,andbluestarsshow
ellipticals,lenticulars,andspiralgalaxiesintheSavorgnanetal.(2016)sample.Thebluesolidanddottedlinesmarkthebest-fitscalingrelationandthe1σ
uncertaintyintheslope(best-fitslopesarereportedintheupper,rightcorners).ThePearsoncorrelationcoefficientrisreportedinthetop,leftcornerofeach
panel.Thegreybandsandpurplelong-dashedlinesshowtheresidualsextractedfromtheMonteCarlosimulationsdescribedinthetextwithandwithout
selection intheblackholegravitational sphereofinfluence. Theresidualcorrelations withSe´rsicindexatfixedvelocity dispersion(toprightpanel)and,
especially,with(bulge)stellarmass(bottomrightpanel),areweak.
bethatother,possiblymass-dependent,selectioneffectsshouldbe over-predicted at thehigh-mass end, as was also revealed for the
includedinourMonteCarlostoaccountfortheSe´rsicindexdistri- M -M andM -σrelations(PaperI).
bh bulge bh
butionofthelogM /M .10galaxiesinthelocalsamplesof
bulge ⊙
galaxieswithdynamicallymeasuredblackholes.SeetheAppendix
forfurtherdiscussionoftheexpectedslopesofthegreyregionsin
Figures4–6. ACKNOWLEDGMENTS
To summarize, inthiswork we havecompared SDSSearly- Wewarmlythank AlisterGraham, GiuliaSavorgnan,andRonald
typegalaxieswiththelocalsampleofgalaxieswithdynamically- La¨skerforprovidingtheirdatainelectronicformatandforuseful
measured black holes from the Savorgnanetal. (2016) and discussions.
La¨skeretal.(2014)sampleswithself-consistentestimatesofbulge
luminosities,effectiveradii,andSe´rsicindices.Wefindthelatter
sample to be consistent with SDSS galaxies in terms of dynami-
calmassandSe´rsicindexdistributions.Analysisoftheresidualsin APPENDIXA: ACCOUNTINGFORMEASUREMENT
Figures5and6,revealsthatvelocitydispersionismorefundamen- ERRORS
tal than Se´rsic index n in the scaling relations between black
sph
To include errors in the determination of the correlations, espe-
holesandgalaxies. Indeed, residualswithn couldbeascribed
sph
ciallythosebetweenresiduals,wefollowBernardietal.(2003)and
totheunderlyingcorrelationswithσandM .Ourconclusions
bulge
aresupportedbytargettedMonteCarloteststhatincludetheeffects Sheth&Bernardi (2012).Foranysetofmeasurements xi,yi and
ofthesphereofinfluenceoftheblackhole.Theyshowthat,atleast (normalized)weightswi,wefirstcomputethelinearrelationswith
forgalaxieswithM &1010M andn &5,theobserved slopemy|xandzeropointzpy|xgivenby
bulge ⊙ sph
msicedviaalnuebblayckuphtooleanatoardgeirvoenfmnaspghniitsudbeia,sie.ed.,hbilgahcekrhthoalenmthaessienstrainre- my|x= SSxxxy−EExxxy (A1)
−
(cid:13)c 2016RAS,MNRAS000,1–12
8 F.Shankaret al.
Figure6.SameasFigure5butfortheLa¨skeretal.(2014)sample.TheresidualcorrelationswithSe´rsicindexatfixedvelocitydispersionandstellarmassare
extremelyweak.
and oftheresidualforeachsetofvariablesweproceedasfollows.Sup-
posewehavethreevariables,say,x=logM ,y=logn ,and
zp = y m x , (A2) bh sph
y|x h i− y|xh i z = logσ.Wefirstcalculatethecorrelationcoefficientrforeach
withtheweightedaverages y and x .Theotherquantitiesare pairas
h i h i
Sxx = (xi−hxi)2wi, Syy = (yi−hyi)2wi, (A3) rxy = Sxy−Exy (A7)
Xi Xi √Sxx−Exx Syy−Eyy
andthencomputethepslopem andcorrelationcoefficientr
xy|z xy|z
Sxy = (xi x )(yi y )wi, (A4) oftheresidualas
−h i −h i
i
X
Exx= he2xiiwi, Eyy = he2yiiwi, (A5) mxy|z= rxy1−−rrxy2zzryzsSSxyxy , (A8)
i i
X X and (cid:2) (cid:3)
and
Exy = i hexeyiiwi ≈k ExxEyy. (A6) rxy|z= [1r−xyr−x2z]rx1zr−yzry2z . (A9)
X p q
Thetermsex andey inEquationA5representtheunknownmea- ForeachpanelinFig(cid:2)ures5an(cid:3)d6weran200iterationsfollowing
surementerrorsinthevariablesxandy;onlytheirvariances e2 thestepsoutlinedaboveand,inabootstrapfashion,eachtimeelim-
and e2 areknown.ThefactorkinEquationA6accountsforhcoxri- inatingthreeobjectsatrandomfromtheoriginalsamples.Fromthe
h yi
relation between the measurement errors ex and ey. We will al- full ensemble of realizations wethen compute the mean slope of
wayssetk = 0except whencalculatingtheslopesandresiduals thecorrelationandits1σuncertainty,whichwereportintheright,
in the n and M correlations, for which we set k = 0.9 uppercornerofeachpanel,whiletheupperleftcornerreportsthe
sph bulge
(Meertetal.2013),astheSe´rsicindexandgalaxyluminosityare meanvalueofthePearsoncoefficientr.Theanalyticmethodology
derivedfromthesamefittingprocedure. describedabove ismainlyintended forsymmetricerrors.Totake
Inordertodeterminethefinalslopeandcorrelationcoefficient into account the asymmetry in black hole mass uncertainties, for
(cid:13)c 2016RAS,MNRAS000,1–12
SMBHs:selectionbiasandSe´rsicindices 9
eachcorrelationweran100iterationsconsideringonlythepositive σ nδσ|n∗Mασ|n∗ whichmattersinthebottompanels.Averag-
∝ sph star
error,and100iterationsconsideringonlythenegativeone.Consid- ingEquationB5overσatfixedn yields
sph
eringinsteadtheaverageorsquarederrorinblackholemassyields
consistentresultswithintheuncertainties.
δtot =α(δ∗|nσ+β∗|nσδσ|n)+βδσ|n, (B7)
andthisequalstheresultofaveragingEquationB6overMstar at
APPENDIXB: RELATIONBETWEENCOEFFICIENTSIN fixednsph:
PAIRWISECORRELATIONSANDCORRELATIONS
BETWEENRESIDUALS δtot =β(δσ|n∗+ασ|n∗δ∗|n)+αδ∗|n. (B8)
Themaintextaddressesthequestionofwhetherornot theM -
bh
n correlationshowninFigure4isfundamental.Wedosofol- These final expressions, show how the slope δtot of the
sph
logM logn relationdependsontheblack-holeparameters
lowingSheth&Bernardi(2012).Namely,westartwithEquation2 bh sph
h | i
inthemaintext,with(α,β)=(0.5,5),andaroundwhichthereis α,β,andonthescalingrelationsbetweenMstar,nsphandσ.The
latter are reported in Figures B1 and B2 for the Savorgnanetal.
0.25dexscatterthatdoesnotdependonn .
sph
(2016)andLa¨skeretal.(2014)samples,respectively.IneachFig-
AveragingthisexpressionoverallσatfixedMstaryields
ure the residual correlations of velocity dispersion (top panels),
logMbh logMstar α logMstar+β logσ logMstar . (B1) bulgestellarmass(middlepanels),andSe´rsicindex(bottompan-
h | i∝ h | i
els)areplottedagainstthetheothertwovariables.Thegrayband
Ifασ|∗istheslopeofthehlogσ|logMstarirelation,thenwehave ineachpanelmarkstheresultsfromtheMonteCarlosimulations
that
basedontheMeertetal.(2013)SDSSsampleinclusiveofbiason
hlogMbh|logMstari∝(α+βασ|∗) logMstar, (B2) theblIancskerhtionlgeδgravit=atio0n.1a6lsfprhoemreToafbilnefl1ueinncEeq.uationB7,andthe
σ|n
whichsuggestsdefining slopesoftheSDSSresidualsδ =0.37andβ =2.14from,
∗|nσ ∗|nσ
respectively,themiddlerightandmiddleleftpanelsofFiguresB1
αtot =α+βασ|∗. (B3) andB2,wewouldgetδtot =0.5(0.37+2.14 0.16)+5 0.16
× × ≈
Similarly,averagingoverallMstaratfixedσinsteadyields 1.2,implyingasignificantcorrelationbetweenblackholemassand
Se´rsicindex,eventhoughEquation2doesnotexplicitlydependon
βtot =β+αβ∗|σ, (B4) Se´rsic index. On the other hand, setting δ∗|n = 0.36 (Table 1),
δ = 0.06(upperrightpanels)andα =0.31(upperleft
swhhoewres eβx∗p|σliciistlythtehastloαpteotof6=thαe ahnlodgβMtostta6=r|loβg,σbuitrtehlaattiothne. rTehlais- p0σa.5n|ne∗ls)0.i3n−6Equa0t.i5o.nTBh8isyiiselwdseaδkteotrt=han5(t−he0σ.|e0nx6∗pe+ct0ed.3v1a×lue0.o3f61).2+;
tionbetweenthetwodependsonthetwoprojectionsoftheMstar- thed×iscrepa≈ncymaybeaconsequence ofthefactthatδ isso
σ correlation.InourSDSSsample,σ ∝ Ms0t.a3r andMstar ∝ σ2, closetozero. σ|n∗
making(αtot,βtot) (2,6)when(α,β) = (0.5,5).Theseval-
≈ Exceptforthis,alloftheotherself-consistencyconditionsare
uesof(αtot andβtot)areinagreementwiththosereportedinthe
satisfiedinthemocksbeforeweapplythesphereofinfluencese-
leftpanelsofFigures5and6(long-dashedpurplelines).
lection.However,thereisnoguaranteethattheywillbesatisfiedin
Ofcourse,theserelationsshouldholdinthefullsample:se-
theselection-biasedmocksorinthe(selection-biased)data.
lectioneffectsmaymodifytheserelationsandintroducecurvature.
Nevertheless, the top panels of Figure 5 suggest M
ThisisindeedwhatweobserveintheresidualsatfixedSe´rsicindex σ4.1n0.8 in the selection biased sample. Using these valubehs ∝in
(leftpanelsofFigures5and6).OurMonteCarlos,inclusiveofthe sph
Equation B7, along with the fact that δ 0.3 (left panel of
selectionbiasintheblackhole’sgravitationalsphereofinfluence, σ|n ∼
predict significantly flatter, and in fact curved, residuals, roughly cFliogsuereto3t)hseaMysthatwne1.e8xwpeecsteδetoitn≈the0l.e8ft+pa4n.e1l(o0f.3F)ig≈ure24..TUhsisinigs
consistentwith(αtot,βtot) (1,4). bh ∝ sph
≈ Equation B8 instead means we should use the values in the bot-
Similarly,ifthe0.25dexscatteraroundEquation2doesnot
tompanelsofFigure5alongwithδ 0.36(notethatFigure2
depend on nsph, we expect correlations such as those in the top ∗|n ≈
showstheinverserelation,α ).Thisyields0.7+0.9(0.4) 1.1,
panelsofFigures5and6tosatisfy n|∗ ≈
whichissomewhatlowerthantheslopeof1.8,perhapsagainbe-
logMbh lognsph,logσ causethecorrelationwithnsphissoweak.Sincethesescalingsare
h | i
satisfiedinthefullmocks,weconcludethatthesedifferencesare
βlogσ +α logMstar lognsph,logσ
∝ h | i duetotheselectionbias.
(αδ ) logn +(β+αβ ) logσ, (B5)
∝ ∗|nσ sph ∗|nσ Ifweusethevaluesinthetoppanel ofFigure6instead,we
whereasthoseinthebottompanelsshouldscaleas find 0.4+3.9( 0.1) 0.01, where we have used the fact that
− ≈
logσ logn 0.1 for this sample (right panel of Fig-
sph
hlogMbh|lognsph,logMstari uhre3).|Thisiscilos≈e∼tot−heMbh ∝ n0sp.1h scalingofthedirectrela-
∝(βδσ|n∗) lognsph+(βασ|n∗+α) logMstar. (B6) tthioisnsslhoopwenisinvethryerdiigfhfetrheanntdfrpoamnetlhaotfiFnigthuerele4f,tdheasnpditepathneelfaocftFthiga-t
Theseexpressionsshowthat,iftheMbh-nsphcorrelationisdriven ure4.Weconcludethattheseverydifferentscalingsareindicating
bythecorrelationbetweenσandMstar,andtheircorrelationswith that systematics in the determination of nsph prevent a definitive
nsph, then the coefficients of correlations between residuals de- determinationofsomeaspectsoftheMbh-nsph-σ relation.How-
pend both on the black-hole parameters α,β, and on the Mstar- ever,themainuncertaintiesarerelatedtothefactthatcorrelations
σ-nsphcorrelations.Specifically,inthetoppanels,theparameters withnspharenotstrong:ourfindingthattheMbh-σcorrelationis
which matter are those for Mstar ∝ nδsp∗h|nσσβ∗|nσ, whereas it is strongerisverylikelytobecorrect.
(cid:13)c 2016RAS,MNRAS000,1–12
10 F.Shankaret al.
FigureB1.Residualcorrelationsofvelocitydispersion(toppanels),bulgestellarmass(middlepanels),andSersicindex(bottompanels)againstthetheother
twovariables.GraybandsaretheresultsfromtheMonteCarlosimulationsbasedontheMeertetal.(2013)SDSSsampleinclusiveofbiasontheblackhole
gravitationalsphereofinfluence.
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