Selection bias in the M_BH-sigma and M_BH-L correlations and its consequences

0.47 MB·English

by  M. Bernardi

ApJ in press
PreprinttypesetusingLATEXstyleemulateapjv.12/14/05
SELECTION BIAS IN THE M σ AND M L CORRELATIONS AND ITS CONSEQUENCES
• •
− −
Mariangela Bernardi1, Ravi K. Sheth1, Elena Tundo1,2 and Joseph B. Hyde1
ApJ in press
ABSTRACT
Itiscommontoestimateblackholeabundancesbyusingameasuredcorrelationbetweenblackhole
mass and another more easily measured observable such as the velocity dispersion or luminosity of
the surrounding bulge. The correlation is used to transform the distribution of the observable into
an estimate of the distribution of black hole masses. However, different observables provide different
7 estimates: the M σ relation predicts fewer massive black holes than does the M L relation.
• •
0 This is because the−σ L relation in black hole samples currently available is inconsiste−nt with that
0 in the SDSS sample, f−rom which the distributions of L or σ are based: the black hole samples have
2
smaller L for a given σ or have larger σ for a given L. This is true whether L is estimated in the

n opticalorintheNIR.Theσ Lrelationinthe blackholesamplesissimilarlydiscrepantwiththatin
−
a the ENEAR sample of nearby early-type galaxies. This suggests that current black hole samples are
J biased towards objects with abnormally large velocity dispersions for their luminosities. If this is a

3 selectionratherthan physicaleffect, then the M σ andM L relationscurrentlyin the literature
• •
− −
are also biased from their true values. We provide a framework for describing the effect of this bias.

2 Wethencombineitwithamodelofthebiastomakeanestimateofthetrueintrinsicrelations. While
v we do not claim to have understood the source of the bias, our simple model is able to reproduce
0 the observedtrends. Ifwe have correctlymodeledthe selectioneffect, then ouranalysis suggeststhat
0
the bias in the M σ relation is likely to be small, whereas the M L relation is biased towards
3 h •| i h •| i
predicting more massive black holes for a given luminosity. In addition, it is likely that the M L
9 •−
relation is entirely a consequence of more fundamental relations between M and σ, and between σ
0 •
and L. The intrinsic relation we find suggests that at fixed luminosity, older galaxies tend to host
6
more massive black holes.
0
/ Subject headings: galaxies: elliptical — galaxies: fundamental parameters — black hole physics
h
p
o- 1. INTRODUCTION M• (Yu & Tremaine2002;Aller & Richstone2002;Mar-
coni et al. 2004; Shankar et al. 2004; McLure & Dunlop
r Several groups have noted that galaxy formation and
t 2004). Unfortunately,combiningtheM σrelationwith
s supermassive black hole growth should be related, and •−
a many have modeled the joint evolution of quasars and the observed distribution of velocity dispersions results
: in an estimate of the number density of black holes with
v galaxies (e.g., Monaco et al. 2000; Kauffmann &
M >109M which is considerably smaller than the es-
i Haehnelt 2001; Granato et al. 2001; Cavaliere & Vit- • ⊙
X timate basedoncombiningthe M L relationwith the
torini 2002; Cattaneo & Bernardi 2003; Haiman et al. •
−
observed luminosity function (Tundo et al. 2006).
r 2004; Hopkins et al. 2004; Lapi et al. 2006; Haiman et
a The fundamental cause of the discrepancy is that the
al. 2006andreferencestherein). However,thenumberof
luminosityandvelocitydispersionfunctionsareobtained
black hole detections to date is less than fifty; this small
from the SDSS (Blanton et al. 2003; Sheth et al. 2003),
numberpreventsadirectestimateoftheblackholemass
buttheσ LrelationintheSDSS(Bernardietal. 2003b)
function. On the other hand, correlations between M
• −
is different from that in the black hole samples (e.g. Yu
and other observables can be measured quite reliably, if
&Tremaine2002). InSection2,wearguethattheSDSS
one assumes that the M observable relation is a sin-
•− σ L relation is not to blame, suggesting that it is the
gle power-law. M is observed to correlate strongly and
• −
black hole samples which are biased.
tightly with the velocity dispersion of the surrounding
If black hole samples are indeed biased, then this bias
bulge (Ferrarese & Merritt 2000; Gebhardt et al. 2000;
may be physical or it may simply be a selection effect.
Tremaine et al. 2002), as well as with bulge luminosity
If physical, then only a special set of galaxies host black
(McLure & Dunlop 2002) and bulge stellar mass (Mar-
holes, and the entire approach above (of transforming
coni & Hunt 2003; Ha¨ring & Rix 2004): in all cases, a
theluminosityorvelocitydispersionfunction)iscompro-
single power law appears to be an adequate description.
mised. On the other hand, if it is a selection effect, then
Since it is considerably easier to measure bulge prop-
oneisledtoaskiftheM observablerelationscurrently
ertiesthanM•,blackholeabundancesarecurrentlyesti- •−
in the literature are biased by this selection or not. The
mated by transforming the observedabundances ofsuch
main purpose of this work is to provide a framework for
secondaryindicatorsusingtheobservedcorrelationswith
describingthese biases. Section 3describes Monte-Carlo
1Dept. of Physics and Astronomy, Universityof Pennsylvania, simulations of the selection effect. It shows an example
209South33rdSt,Philadelphia,PA19104, U.S.A. ofthebiaswhichresults,andprovidesananalyticmodel
2Dipartimento di Astronomia, Universita’ di Padova, vicolo of the effect. Our findings are summarized in Section 4,
dell’Osservatorio3/2I-35122, Padova,Italy
whereestimates ofthe intrinsicblack holemassfunction
2
Fig. 1.—Comparisonof the σ−Lrelationinthe black holesampleof Ha¨ring& Rix(2004) (symbols) withthat from local early-type
galaxy samples (lines). We provide two estimates of the bulge luminosity for the Ha¨ring & Rix compilation: the open circles come from
rescaling the bulge luminosity to the SDSS r-band, and the filled circles from assuming a mass-to-light ratio to scale the bulge masses
Mbulge toluminosities. Dottedlineshowsafittothefilledcircles. Solidandshortdashedlinesshowhσ|LiintheSDSS-B06andENEAR
samples. Dot-dashedlineshowsthebiasedrelationobtainedfromtheENEARsampleifonefailstoaccount forthefactthatthevelocity
dispersionsplayedanimportantroleindeterminingthedistancesfromwhichtheluminositieswereestimated. Theblackholecompilation
is clearlyoffset fromthe SDSS-B06, the ENEAR and even the biased ENEAR relations. Appendix A gives the fits shown, and describes
exactlyhowtheblackholesamplewascompiled.
φ(M ), and the intrinsic M σ and M L relations 2.1. Remarks on SDSS photometry and spectroscopy
• • •
− −
are provided.
Since 2002, when the Bernardi et al. (2003a) sam-
Throughout, we assume a spatially flat background
plewascompiled,anumber ofsystematic problemswith
cosmology with Ω = 0.3 and a cosmological constant.
0 SDSS photometry and spectroscopy have been discov-
All luminosities and masses have been scaled to H =
0 ered. The initial (SDSS-DR1 and earlier) photometry
70 km s−1 Mpc−1.
overestimatedthe luminosities and sizes of extended ob-
jects. Althoughlaterreleasescorrectfor these problems,
2. MOTIVATION: THE σ L RELATION they still overestimate the sky level in crowded fields
− (e.g. Mandelbaum et al. 2005; Bernardi et al. 2006;
This section shows that the σ L relation of early-
− Lauer et al. 2006), and so they tend to underestimate
typegalaxiesissignificantlydifferentfromthatincurrent
the luminosities of extended objects in such fields. This
black hole samples. This difference exists whether L is
problem is particularly severe for early-type galaxies in
computed in a visual band, or in the near-infrared.
3
nearby clusters. In addition, the SDSS slightly overes- 2.3. The σ L relations in the near-infrared
−
timates the velocity dispersions at small σ (the velocity
We have considered the possibility that the discrep-
dispersions in Bernardi et al. 2003a differ from the val-
ancy between the σ L relations may depend on wave-
ues reported in the SDSS database and are not biased −
length. Figure2 presentsa similaranalysis,but now the
at smallσ). Bernardi(2006)discusses ananalysis of the
luminosities are from the K-band. In the case of the
σ Lwhichresultsonceallthesesystematiceffectshave
early-type sample, the SDSS-B06 sample was matched
−
been corrected-for: the net effect does not significantly
to 2MASS, and then the 2MASS isophotal magnitudes
change the σ L relation reported by Bernardi et al.
werebrightenedby 0.2 magsto make them morelike to-
−
(2003b). Nevertheless,theresultswhichfollowarebased
tal magnitudes (Kochanek et al. 2001; Marconi & Hunt
onluminositiesandvelocitydispersionswhichhavebeen
2003). Thesolidlineshowstheresulting σ L relation.
K
corrected for all these effects. Because they are not eas- h | i
It is very similar to that in the r band if r K =2.8.
ily obtained from the SDSS database, in what follows, − −
Marconi&Hunt(2003)matchedtheirblackholecom-
we refer to the corrected SDSS sample as the SDSS-B06
pilationto2MASS,andthenrecomputedthephotometry
sample.
to determine L . The open triangles show the σ L
K K
Bernardi (2006) shows that the σ L relation in the −
relationforthe subsetofobjectswhichareinthe Ha¨ring
−
SDSS-B06sampleisconsistentwiththatinENEAR,the
& Rix sample. The filled circles are obtained by tak-
definitive sample of nearby (within 7000 km s−1) early-
ing these objects (i.e. those in both Marconi & Hunt
typegalaxies(daCostaetal. 2000;Bernardietal. 2002;
and Ha¨ring & Rix), and then shifting the M based
bulge
Alonso et al. 2003; Wegner et al. 2003), provided that
r bandluminositiestoK assumingr K =2.8. Dashed
one accounts for the bias which arises fromthe fact that − −
and dotted lines show fits to these relations. Compari-
σ playedanimportantroleindeterminingL. Hence,the
son with the solid line (2MASS-B06) shows a significant
local (ENEAR) and not so local determinations (SDSS-
offset—theblackholesamplesaredifferentfromthepop-
B06)oftheσ Lrelationareconsistentwithoneanother.
ulation of normal early-type galaxies.
−
2.2. The σ L relations in the optical
− 3. BIASESINTHE M• σ AND M• LRELATIONS
Figure 1 compares the correlationbetween σ and L in − −
If black hole samples are indeed biased, then this bias
ENEARand SDSS-B06 with that in the recentcompila-
maybe physicalorsimply aselectioneffect. One is then
tions ofblack holes by Ha¨ring& Rix (2004). The dotted
led to ask if the relations currently in the literature are
line shows a fitto σ L definedby the filled circles. Ap-
h | i biasedby this selectionor not. To addressthis, we must
pendixAdescribesthisandotherblackholecompilations
be able to quantify how much of the M σ correlation
wehaveconsiderinthefollowingfigures. Italsogivesthe •−
arisesfromtheM LandL σ relations,andsimilarly
fits shown in this and the other figures. •− −
for the M L relation.
The solid line shows σ L in the SDSS-B06 sample. •−
h | i
The dot-dashed line shows a determination of the EN-
3.1. Demonstration using mock catalogs
EAR σ L relation which attempts to correct for pe-
culiar ve−locity effects in the distance estimate. This Our approach is to make a model for the true joint
correction is known to bias the σ L relation towards distribution of M•, L and σ, and to study the effects
a steeper slope (Bernardi 2006).hT|heishort-dashed line of selection by measuring the various pairwise correla-
shows the result of using the redshift as distance indica- tions before and after applying the selection procedure
tor when computing the ENEAR luminosities; while not to this joint distribution. In what follows, we will use
ideal, comparison of this relation with the dot-dashed V = log(σ/km s−1), and with some abuse of notation,
line provides some indication of the magnitude of the M to denote the logarithmofthe blackhole mass when
•
bias. Note in particular that using the redshift as dis- expressed in units of M and L to denote the absolute
⊙
tanceindicatorbringstheENEARrelationsubstantially magnitude(since,inthecurrentcontext,M forabsolute
closertothe SDSS-B06relation,forwhichtheredshiftis magnitudeistooeasilyconfusedwithmass). Weassume
an excellent distance indicator. thatcorrelationsbetweenthelogarithmsofthesequanti-
The agreement between ENEAR and SDSS-B06 sug- ties define simple linear relations, so our model has nine
gests that the difference between the σ L relation in free parameters: these may be thought of as the slope,
−
the black hole sample (dotted) and in SDSS-B06 (solid) zero-point and scatter of each of the three pairwise cor-
cannot be attributed to problems with the SDSS deter- relations,or as the means and variancesofL, V andM
•
mination. and the three correlation coefficients r , r and r .
VL •V •L
The agreement between ENEAR and SDSS-B06, and Toillustrate,let V andσ denotethemeanandrms
V
h i
the disagreement with the black hole samples strongly values of V, and use similar notation for the other two
suggests that the σ L relation in black hole samples is variables. Then the mean V as a function of L satisfies
−
biasedtolargerσ forgivenL,orto smallerLforagiven
V L V L L
σ. Similar conclusions are obtained if we use Kormendy h | i−h i = −h i rVL, (1)
σ σ
& Gebhardt (2001) or Ferrarese & Ford (2005) compi- V L
lations. At a given luminosity, the black hole samples and the rms scatter around this relation is
have logσ larger than the ENEAR or SDSS-B06 rela-
tions by∼0.1dex. As a resultofthis difference,smaller σV|L =σV 1−rV2L. (2)
luminositiescanapparentlyproduceblackholesoflarger
q
mass than do their associated velocity dispersions (e.g., The combination aV|L rVLσV/σL is sometimes called
≡
Yu & Tremaine 2002). Tundo et al. (2006) show this in theslopeofthe V L relation,and V aV|L L is the
h | i h i− h i
some detail. zero-point. Similarly, the slopes of M V and M L
• •
h | i h | i
4
Fig. 2.—SameasFigure1butnowusingtheK-bandluminosity. TrianglesshowthesubsetoftheHa¨ring&RixsampleforwhichK-band
luminosityestimatesareavailable(fromMarconi&Hunt2003). SolidcirclesshowtheresultofrescalingtheMbulge-basedluminositiesof
these objects to the K-bandusingr−K =2.8. Dotted and dashed lines show fits to the hσ|LKi relations defined by these points. Solid
lineshowshσ|LKiobtainedbymatchingtheSDSS-B06early-typesampletothe2MASSdatabase,andthenusingtheK-bandluminosities
withtheSDSS-B06velocitydispersions. Thisfitisverysimilartothesolidlineshowninthepreviouspanelsifr−K=2.8. Thesefitsare
giveninAppendixA.
are a r σ /σ and a r σ /σ respec- al. (2003) show explicitly that the resulting mock cata-
•|V •V • V •|L •L • L
≡ ≡
tively. Noticethatthezeropointsaretrivialifoneworks logprovidesluminosityandvelocitydispersionfunctions
withquantitiesfromwhichthemeanvaluehasbeensub- which are consistent with those seen in the SDSS.
tracted. We will do so unless otherwise specified. We turn this into a catalog of black hole masses by
Our analysis is simplified by the fact that five of the assuming that, like V at fixed L, the distribution of M
•
ninefreeparametersareknown: themeanandrmsvalues at fixed V and L is Gaussian. Hence, for each V and L,
of L and V as well as the correlation between V and wegenerateavalueofM thatisdrawnfromaGaussian
•
L have been derived from the SDSS-B06 sample (these distribution with mean
values are similar to those reported by Bernardi et al.
2003b,who also show that the distribution of V given L
is Gaussian). Hence, we use M• L,V M• L L r•L r•VrVL
h | i=h i + −h i −
σL =0.84, σV =0.11, and rVL =−0.78, σ• σ• σL 1−rV2L
to make a mock catalog of the joint V L distribution V V r•V r•LrVL
− + −h i − , (3)
in the SDSS-B06 early-type galaxy sample. Sheth et σ 1 r2
V − VL
5
Fig. 3.—ComparisonofthejointdistributionofM•,σandL,before(opensymbols)andafter(filledsymbols)applyingaselectioncut.
Dashed and solid lines show the intrinsic and observed correlations. The probability of selection is assumed to increase as one moves to
therightofthedashedlineinthebottom leftpanel.
and rms galaxy+blackholecatalogasdescribedabove,modelthe
selectioneffect,andthenmeasurethethreepairwisecor-
σ =σ 1−rV2L−r•2V −r•2L+2rVLr•Lr•V (4) relations in the selection-biased catalog. Figure 3 shows
•|LV •s 1−rV2L an example of the result when
σ =0.49, r =0.88, and r = 0.70.
(see, e.g., Bernardi et al. 2003b). The combination • •V •L −
r r measures how much of the correlation between We have tried a variety of ways to specify a selec-
•V VL
M and L is simply a consequence of the correlations tion effect. In Figure 3, we supposed that no objects
•
between M and V and between V and L. The combi- withluminosities brighterthan LV were selected,and
• h | i
nationr r canbeinterpretedsimilarly. Noticeagain that the fraction of fainter objects which are selected is
•L VL
thatthemeanvaluesofthethreequantitiesmainlyserve erf(x/√2), where x = (L LV )/σ . The open di-
L|V
−h | i
to set the zero-points of the correlations. Since we are amonds in each panel show the objects in the original
working with quantities from which the mean has been catalog,andthefilledcirclesshowtheobjectswhichsur-
subtracted, our model for the true relations is fixed by vived the selection procedure. The panels also show the
specifying the values of σ , r and r . slopes of the relations before (dashed) and after (solid)
• •V •L
To study how selection effects might have biased the selection. Note that because the selection was made at
observedM σandM Lrelationswegenerateamock fixed V, the slope of M V in the selected sample is
• • •
− − h | i
6
Fig. 4.— Residuals from the M•−σ relation do not correlate with residuals from the σ−L correlation (top left); whereas residuals
from the M•−L relation do (top right). They also correlate with residuals from the L−σ relation (bottom right), and weakly with σ
(bottom left). Inallpanels,opencirclesshowtheintrinsiccorrelationsandfilledcirclesshowcorrelationsafterapplyingtheselectioncut.
ComparisonwithFigureA2showsthatthefilledcirclestrace similarlociitothoseseenintheHa¨ring&Rix(2004) compilation.
not changed from its original value. However, M L trivial: forexample,itismoredifficulttomatchallthree
•
h | i
and V L are both steeper than before. The exact val- observed correlations if the selection is based on V L
h | i h | i
ues of the change in slope depend upon our choices for rather than LV . Changes in the parameter values of
h | i
the intrinsic model, for reasons we describe in the next more than five percent produce noticable differences.
subsection. Thecorrelationsshowninthetworighthandpanelsof
The parameter choices given above, with the selection Figure 4 are similar in the full and in the bias selected
described above, result in zero-points, slopes (given at sample,suggesting that the selectionhas notaltered the
top of each panel) and scatter (0.22 dex, 0.33 dex and shape of this correlation. This suggests that the cor-
0.05dexinthetopleft,right,andbottompanels,respec- relations seen in the right hand panel of Figure A2 are
tively)whichareinexcellentagreementwiththoseinthe real–theyarenotduetoselectioneffects. Wecommentin
Ha¨ring&Rix(2004)compilation(seeAppendixA).Fig- the final section on what these correlationsmight imply.
ure 4 shows that residuals from these relations exhibit The result of using the biased M L relation (both
•
h | i
similar correlations to those seen in the real data (com- slope and scatter) in the mock luminosity function to
pare Figure A2; residuals from the M σ relation do predict black hole abundances is shown as the dashed
•
−
notcorrelatewithσ,LorL σ residuals,sowehavenot line in Figure 5. It overpredicts the true abundances in
−
shown these correlations here). This agreement is non- themockcatalog,whichareshownbythesolidline. The
7
Fig. 5.— Comparison of the intrinsic (solid) and selection-biased counts based on using velocity dispersion (dotted) and luminosity
(dashed)astheindicatorofM•. HashedregionsshowtheabundancesinferedbyTundoetal. (2006)bycombiningtheM•−LandM•−σ
relationsinthecompilationofHa¨ring&Rix(2004) withthe SDSS-basedluminosityandvelocityfunctions (Bernardietal. 2003b; Sheth
etal. 2003).
dotted line shows that the analogous σ based predictor zero-point may be strongly affected. The magnitude of
−
is, essentially unbiased. This is not surprising in view the offset depends on how much of the M σ relation
•
−
of the results shown in Figure 3—the observed M L is due to the σ L and M L relations. If n is posi-
• •
h | i − −
relationis biased towardspredicting more massive black tive (as Figure 1 suggests) and r > r r , then the
•V •L VL
holes for a given luminosity, whereas the M σ is al- observed relation predicts larger black hole masses for a
•
h | i
mostcompletely unbiased. The hashed regionsshow the givenluminositythanwouldthe truerelation. The asso-
L and σ based predictions derived by using the rela- ciated M σ relation is obtained by transforming L to
•
− − h | i
tions in the Appendix (taken from Tundo et al. 2006) V using equation (5):
withtheBernardietal. (2003b)luminosityfunctionand
M V M V V r
the Sheth et al. (2003) velocity function. They bracket h •| i=h •i + −h ir•V •L
theresultsfromourmockcatalogquiteaccurately. Note, σ• σ• σV (cid:18)r•VrVL(cid:19)
however, that in our mock catalog, the true abundances 1 (r /r r )
•L •V VL
are given by the solid line: the much larger abundances +nr•V − . (7)
1 r2
predictedfromtheluminosity-basedapproacharedueto − VL
the selection effect. For reference,this intrinsic distribu- Boththeslopeandzero-poinptofthisrelationareaffected,
tion is well-described by equation (8). and the magnitude of the effect depends on whether or
not r > r r . If the correlation between M and
•L •V VL •
3.2. Toy model of the bias L is stronger(weaker) than can be accounted for by the
If currentblack hole samples only select objects which M• V and V L relations, then the observed M• V
− − h | i
have logσ which is exactly n-standard deviations above hasasteeper(shallower)slopethanthetruerelation,and
the V L relation, then ashiftedzero-point,suchthatitpredictssmaller(larger)
h | i masses for a given σ than the true relation. Note that
V −σhVi = L−σhLi rVL+n 1−rV2L, (5) pthriosamcheeasntsoitoivsepreosstsiimblaetfeotrhbeotthrutheeaLbu−nadnadncσe−obfamseadssaivpe-
V L
q black holes.
and inserting this in the expression above yields
If the selection chooses objects with small L for their
M• L M• r•V r•LrVL L L σ, then the resulting relations are given by exchang-
h σ•| i = hσ•i +n 1−−rV2L + −σLh ir•L. (6) cinagseL, sVfoLr VansdinMtheLprbevoitohuhsatvweobieaxspedresslsoiopness.anIndztehrios
•
The slope of this relationpis still a•|L; because the selec- pointsh,w|heireas hM•|σihasthecorrectslopebutashifted
h | i
tionwasdoneatfixedL,theslopeoftheobserved M L zero-point. Once again, the sign and magnitude of the
•
h | i
relation is not biased from its true value. However, the bias depends on how much of the M observable corre-
•
−
8
lationisduetotheothertwocorrelations. Forthechoice withscatterof0.38and0.23dex,respectively. Abisector
of parameters used to make the Figures in the previous fittotheM Lrelationwouldhaveslope(σ /σ )(r +
• • L •L
−
subsection, r r = 0.69 r : the M L relation 1/r )/2 = 0.62; a similar fit to the M V relation
•V VL •L • •L •
≈ − − −
is almost entirely a consequence of the other two cor- wouldhaveslope(σ /σ )(r +1/r )/2=4.49. Notice
• V •V •V
relations. Thus, this toy model shows why the M σ thatthe slopessatisfya a a . This isbecause,
• •|L •|V V|L
h | i ≈
relationshownin Figure3 was unbiasedby the selection forthischoiceofparameters,thecorrelationbetweenM
•
(which was done at fixed σ). and L is almost entirely a consequence of the other two
correlations.
4. DISCUSSION If we have correctly modelled the selection effect, and
we do not claim to have done so, then our analysis sug-
Compared to the ENEAR and SDSS-B06 early-type
gests that the intrinsic joint distribution of M , lumi-
galaxy samples, the σ L correlationin black hole sam- •
− nosity and σ is similar to that between color,luminosity
ples currently available is biased towards large σ for a
and σ: of the three pairwise correlations, the two corre-
given L (Figures 1–2). If this is a selection effect rather
lations involving σ are fundamental, whereas the third
than a physical one, then current determinations of the
is not (Bernardi et al. 2005). Since residuals from the
M L and M σ relations are biased. We provided
• •
− − σ L relation are correlated with age (Bernardi et al.
a simple toy model of the effect which shows how these
h | i
2005), the correlations shown in the top right panels of
biases depend on the intrinsic correlations and on the
Figures4 andA2 suggestthat, atfixedluminosity, older
selection effect.
galaxies tend to host more massive black holes than less
To illustrate its use, we constructed simple models of
massive galaxies. Understanding why this is true may
the intrinsic correlation and of the selection effect. We
provide important insight into black hole formation and
then identified a particular set of parameters which we
evolution.
showedcanreproducealltheobservedtrends: thecorre-
lations between M and L or σ, the σ L correlationin
•
−
blackholesamples,aswellascorrelationsbetweenresid-
uals fromthese fits (Figures3 and4) areallreproduced.
We do not claim to have found the unique solution to
this problem. However, if we have correctly modeled
the selection effect, then our analysis suggests that the This work is partially supported by NASA grant
bias in the M• σ relation is likely to be small, whereas LTSA-NNG06GC19G, and by grants 10199 and 10488
h | i
the M• L relation is biased towards having a steeper from the Space Telescope Science Institute, which is op-
h | i
slope (Figure 3). This has the important consequence erated by AURA, Inc., under NASA contract NAS 5-
that estimates of black hole abundances which combine 26555.
the M• L relation and its scatter with an external de- Funding for the SDSS and SDSS-II has been pro-
h | i
termination of the distribution of L will overpredict the vided by the Alfred P. Sloan Foundation, the Partici-
true abundances of massive black holes. The analogous pating Institutions, the NSF, the US DOE, NASA, the
approach based on σ is much less biased (Figure 5). JapaneseMonbukagakusho,the MaxPlanckSocietyand
The choice of parameters which reproduces the theHigherEducationFundingCouncilforEngland. The
selection-biasedfitsreportedinAppendixAhasintrinsic SDSS website is http://www.sdss.org/.
distribution φ(M•) that is well described The SDSS is managed by the Astrophysical Research
Consortium (ARC) for the Participating Institutions:
φ dM
φ(M )dM = ∗ µα exp( µβ)β • (8) The American Museum of Natural History, Astrophys-
• •
Γ(α/β) − M
• ical Institute Postdam, the University of Basel, Cam-
where φ = 0.002/Mpc3, (α,β) = (2.0,0.3) and µ = bridge University, Case Western Reserve University, the
∗ University of Chicago, Drexel University, Fermilab, the
M /M∗, with M∗ a characteristic mass. This mass is
• • • Institute for Advanced Study, the Japan Participation
related to the mean mass 108M by M =M∗Γ[(α+
1)/β]/Γ(α/β), so M∗ = 105M⊙. Shince•ithis φ•(M ) is Group,theJohnsHopkinsUniversity,theJointInstitute
• ⊙ • for Nuclear Astrophysics, the Kavli Institute for Parti-
based on the SDSS-B06 luminosity and velocity func-
cle Astrophysics and Cosmology, the Korean Scientist
tions, it is missing small bulges, so it is incomplete at
Group, the Chinese Academy of Sciences (LAMOST),
small M .
• Los Alamos National Laboratory,the Max Planck Insti-
The intrinsic correlations in the sample are
tute for Astronomy (MPI-A), the Max Planck Institute
for Astrophysics (MPA), New Mexico State University,
M M =9.16 0.41(M +22) (9)
• r r
| − the Ohio State University, the University of Pittsburgh,
and D E the University of Portsmouth, Princeton University, the
U.S. Naval Observatory,and the University of Washing-
M logσ =8.85+3.82 log(σ/200 km s−1) (10) ton.
•
|
D E
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TABLE A1
Parametersof theblackhole sampleusedin this work. HR=Ha¨ring &Rix (2004);MH=Marconi&Hunt(2003).
MrHR isfromobserved absolutemagnitude;MrHR−Mbulge is inferredfromMrHR−Mbulge relation.
name σHR Log10 MH•R MrHR MrHR−Mbulge σMH MKMH
[km s−1] [MSun] [mag] [mag] [km s−1] [mag]
M87 375 9.54 −23.08 −22.86 375 −25.60
NGC3379 206 8.06 −20.85 −20.94 206 −24.20
NGC4374 296 8.69 −22.22 −22.41 −− −−
NGC4261 315 8.77 −21.90 −22.41 315 −25.60
NGC6251 290 8.78 −22.69 −22.80 290 −26.60
NGC7052 266 8.58 −22.57 −22.22 266 −26.10
NGC4742 90 7.20 −19.75 −18.83 90 −23.00
NGC0821 209 7.63 −21.43 −21.51 209 −24.80
IC1459 323 9.46 −22.37 −22.22 340 −25.90
M32 75 6.46 −16.99 −17.02 75 −19.80
NGC2778 175 7.20 −20.74 −21.04 175 −23.00
NGC3115 230 9.06 −21.12 −21.44 230 −24.40
NGC3245 205 8.38 −20.85 −20.94 205 −23.30
NGC3377 145 8.06 −20.06 −19.67 145 −23.60
NGC3384 143 7.26 −20.17 −19.86 143 −22.60
NGC3608 182 8.34 −21.24 −21.26 182 −24.10
NGC4291 242 8.55 −21.24 −21.51 242 −23.90
NGC4473 190 8.10 −21.18 −21.21 190 −23.80
NGC4564 162 7.81 −20.31 −20.56 162 −23.40
NGC4649 375 9.36 −22.50 −22.69 385 −25.80
NGC4697 177 8.29 −21.44 −21.37 177 −24.60
NGC5845 234 8.44 −20.11 −20.41 234 −23.00
NGC7332 122 7.17 −20.28 −19.61 −− −−
NGC7457 67 6.60 −18.85 −18.94 67 −21.80
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APPENDIX
THE σ L CORRELATION
−
Themaintextcomparestheσ Lrelationforthebulkoftheearly-typegalaxypopulationwiththatfortheblackhole
sample. This was done by using−luminosities transformedto the SDSS r band and scaledto H =70km s−1 Mpc−1.
0
−
Specifically, we convert from B, V, R, I and K-band luminosities to SDSS r band using
−
B r=1.25, V r =0.34 r R=0.27, r I =1.07, and r K =2.8;
− − − − −
these come from Fukugita et al. (1995)for S0s and Es. We describe the samples and required transformations below.
The black hole sample
The main properties of the black hole sample shown in Figure 1 are listed in Table 1. The sample was compiled as
follows. We selected all objects classified as E or S0 by Ha¨ring & Rix (2004) in their compilation of black holes. We
10
Fig. A1.—TheM•−LandM•−σrelationsinthesubsetoftheHa¨ring&Rix(2004)samplewhichwedescribeinthemaintext. Solid
linesshowthefitstothefilledcirclesgiveninequations (A2)and(A3).
excluded NGC 4342 as they recommend, as well as NGC 1023 which Ferrarese & Ford (2005) say should be excluded
because it is not sufficiently well resolved. Ha¨ring & Rix provide estimates of the bulge luminosities and masses of
these objects; these assume H = 70 km s−1 Mpc−1. For consistency we also rescaled the black hole masses from
0
H = 80 km s−1 Mpc−1 to H = 70 km s−1 Mpc−1. We convert from the V, R and I-band luminosities the report
0 0
to SDSS r using the transformations given above. We use the bulge masses to provide an alternative estimate of the
bulge luminosities by setting
log (M /M ) 11.52
Mr = 22.34 10 bulge ⊙ − ; (A1)
− − 0.492
this was obtained from a linear regression of M on M . Table 1 also lists estimates of the K-band near-infrared
r bulge
luminosity from Marconi & Hunt (2003) which are shown in Figure 2.
A word on the velocity dispersions is necessary. Ha¨ring & Rix (2004) and Marconi & Hunt (2003) use the velocity
dispersions provided by Kormendy & Gebhardt (2001) (two galaxies in Marconi & Hunt have a different value). The
two estimates correspond to different apertures (r and central, which probably means r /8). Tremaine et al. (2002)
e e
provide a detailed discussion of aperture effects. However, it is not clear that their assumptions about the scale
dependence of the stellar velocity dispersion are consistent with more recent measurements (Cappellari et al. 2006).
Since the difference in aperture is not sufficient to account for the discrepancy between black hole samples and the
normal early-type galaxy samples we have not made any correction for this difference.
The M L, M σ, and σ L correlations in the Ha¨ring & Rix (2004) sample, with luminosities estimated from
• •
− − −
M are
bulge
(1.30 0.10)
logM• Mr =(8.57 0.10) ± (Mr+22) (A2)
| ± − 2.5
D E
with scatter of 0.33 dex,
σ
logM logσ =(8.21 0.05)+(3.83 0.10) log (A3)
•| ± ± 200kms−1
D E (cid:18) (cid:19)
with intrinsic scatter of 0.22 dex, and
(0.34 0.03)
logσ M =(2.41 0.10) ± (M +22) (A4)
r r
| ± − 2.5
D E
(Tundo et al. 2006). Figure A1 shows the M L and M σ relations and these fits. The dotted line in Figure 1 of
• •
− −
the main text shows equation (A4).
Figure A2 shows that residuals from the M σ relation do not correlate with residuals from the σ L correlation
•
− −
(top left), whereas residuals from the M L relation do (top right). Residuals from the M σ relation do not
• •
− −