1 The slope of the black-hole mass versus velocity dispersion correlation arxiv:astro-ph/ v1 26 Mar 2002 Scott Tremaine 1, Karl Gebhardt 2, Ralf Bende...

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The slope of the black-hole mass versus velocity dispersion correlation Scott Tremaine1 , Karl Gebhardt2 , Ralf Bender3 , Gary Bower4 , Alan Dressler5 , S. M. Faber6 , Alexei V. Filippenko7 , Richard Green8 , Carl Grillmair9 , Luis C. Ho5 , John Kormendy2 , Tod R. Lauer8 , John Magorrian10 , Jason Pinkney11 , and Douglas Richstone11 ABSTRACT Observations of nearby galaxies reveal a strong correlation between the mass of the central dark object M• and the velocity dispersion σ of the host galaxy, of the form log(M• /M⊙ ) = α + β log(σ/σ0 ); however, published estimates of the slope β span a wide range (3.75 to 5.3). Merritt & Ferrarese have argued that low slopes (. 4) arise because of neglect of random measurement errors in the dispersions and an incorrect choice for the dispersion of the Milky Way Galaxy. We show that these explanations, and several others, account for at most a small part of the slope range. Instead, the range of slopes arises mostly 1 2

Princeton University Observatory, Peyton Hall, Princeton, NJ 08544; [email protected]

Department of Astronomy, University of Texas, RLM 15.308, Austin, [email protected], [email protected] 3

Texas 78712;

geb-

Universit¨ats-Sternwarte, Scheinerstraße 1, M¨ unchen 81679, Germany; [email protected]

4

Computer Sciences Corporation, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218; [email protected] 5

The Observatories of the Carnegie Institution of Washington, 813 Santa Barbara St., Pasadena, CA 91101; [email protected], [email protected] 6

UCO/Lick Observatories, University of California, Santa Cruz, CA 95064; [email protected]

7

Department of Astronomy, University of California, Berkeley, CA 94720-3411; [email protected]

8

National Optical Astronomy Observatories, P. O. Box 26732, Tucson, AZ 85726; [email protected], [email protected] 9

SIRTF Science Center, Mail Stop 220-6, 1200 East California Blvd., Pasadena, CA 91125; [email protected] 10

Department of Physics, University of Durham, Rochester Building, Science Laboratories, South Road, Durham DH1 3LE, UK; [email protected] 11

Dept. of Astronomy, Dennison Bldg., Univ. of Michigan, Ann Arbor 48109; [email protected], [email protected]

–2– because of systematic diﬀerences in the velocity dispersions used by diﬀerent groups for the same galaxies. The origin of these diﬀerences remains unclear, but we suggest that one signiﬁcant component of the diﬀerence results from Ferrarese & Merritt’s extrapolation of central velocity dispersions to re /8 (re is the eﬀective radius) using an empirical formula. Another component may arise from dispersion-dependent systematic errors in the measurements. A new determination of the slope using 31 galaxies yields β = 4.02±0.32, α = 8.13±0.06, for σ0 = 200 km s−1 . The M• –σ relation has an intrinsic dispersion in log M• that is no larger than 0.3 dex, and may be smaller if observational errors have been underestimated. In an Appendix, we present a simple kinematic model for the velocity-dispersion proﬁle of the Galactic bulge.

1.

Introduction

Observations of the centers of nearby early-type galaxies (ellipticals, lenticulars, and spiral bulges) show that most or all contain massive dark objects (hereafter “black holes”). The masses of these objects are consistent with the density of quasar remnants expected from energy arguments (Soltan 1982; Fabian & Iwasawa 1999; Yu & Tremaine 2002). There appears to be a strong correlation between the mass M• of the black hole and the velocity dispersion σ of the host galaxy, of the form12 log(M• /M⊙ ) = α + β log(σ/σ0 ),

(1)

where σ0 is some reference value (here chosen to be σ0 = 200 km s−1 ). The ﬁrst published estimates of the slope β, 5.27±0.40 (Ferrarese & Merritt 2000a) and 3.75±0.3 (Gebhardt et al. 2000a), diﬀered by 3 standard deviations. Subsequently, Ferrarese & Merritt (hereafter FM) revised their slope downwards, to 4.8 ± 0.5 (Ferrarese & Merritt 2000b), 4.72 ± 0.36 (Merritt & Ferrarese 2001a), 4.65 ± 0.48 (Merritt & Ferrarese 2001b), and then 4.58 ± 0.52 (Ferrarese 2002). Although the discrepancy between the estimate by Gebhardt et al. (hereafter the Nukers) and the estimates by FM has declined monotonically with time, and is now only 1.4 standard deviations, it is still worthwhile to understand the reasons behind it. In particular, the slope is the most important point of comparison to theoretical models that attempt to explain the M• –σ relation (Adams, Graﬀ, & Richstone 2001; Burkert & Silk 2001; Haehnelt & Kauﬀmann 2000; Ostriker 2000). This paper has three main goals. (i) In §§2–4 we explore the reasons for the wide range 12

All logarithms in this paper are base 10.

–3– in estimated slopes of the M• –σ relation. In §2 we focus on the statistical techniques used to estimate slopes by the two groups; we shall argue that the estimator used by the Nukers is more accurate, but that the choice of estimator cannot explain most of the diﬀerences in slope between FM and the Nukers. In §3 we describe the data sets used by the two groups. In §4 we examine several explanations that have been proposed for the slope range, including the neglect of random measurement errors in the dispersions, the dispersion used for the Milky Way, and diﬀerences in sample selection, and show that none of these is viable. We argue instead that the slope range reﬂects systematic diﬀerences in the velocity dispersion measurements used by the two groups. (ii) In §5 we present a new analysis of the M• –σ relation using recent data. (iii) Finally, in the Appendix we model the velocity-dispersion proﬁle of the Milky Way bulge, which helps to ﬁx the low-mass end of the M• –σ relation.

2.

The fitting algorithm

The data consist of N galaxies with measured black-hole masses, velocity dispersions, and associated uncertainties. We assume that there is an underlying relation of the form y = α + βx,

(2)

where y = log(M• /M⊙ ), x = log(σ/σ0 ). We assume that the measurement errors are symmetric in x and y with root-mean-square (rms) values ǫxi and ǫyi for galaxy i. The goal is to estimate the best-ﬁt values of α and β and their associated uncertainties. The Nukers and FM use two quite diﬀerent estimators. In this section we review the assumptions inherent in the two estimators and their respective advantages and disadvantages. In subsequent sections we shall usually give results for both estimators; we shall ﬁnd that the diﬀerences are signiﬁcant, but not large enough to explain the slope range. The Nukers’ estimate is based on minimizing 2

χ ≡

N X (yi − α − βxi )2 i=1

ǫ2yi + β 2 ǫ2xi

,

(3)

(e.g., Press et al. 1992, whose procedures we use). The “1σ” uncertainties in α and β are given by the maximum range of α and β for which χ2 − χ2min ≤ 1. An attractive feature of this approach is that the variables x and y are treated symmetrically; in other words, if we set βe = 1/β, α e = −α/β, equation (3) can be rewritten in the form χ2 ≡

N X e i )2 (xi − α e − βy i=1

ǫ2xi + βe2 ǫ2yi

,

(4)

–4– e This symmetry which has the same form as equation (3) if x ↔ y, α ↔ α e, and β ↔ β. ensures that we are not assuming (for example) that y is the dependent variable and x is the independent variable in the correlation; this agnosticism is important because we do not understand the physical mechanism that links black-hole mass to dispersion. We shall call estimators of this kind “χ2 estimators” and denote them by αχ , βχ . One limitation is that this approach does not account for any intrinsic dispersion in the M• –σ relation (i.e., dispersion due to the galaxies themselves rather than to measurement errors). Thus, for example, one or two very precise measurements with small values of ǫxi and ǫyi can dominate χ2 , even though the large weight given to these observations is unrealistic if the intrinsic dispersion is larger than the measurement errors. There are two heuristic approaches that address this concern. (i) Simply set ǫyi ≡ ǫy = constant, corresponding to the same fractional uncertainty in all the black-hole mass estimates. The value of ǫy is adjusted so that the value of χ2 per degree of freedom is equal to its expectation value of unity. This approach was adopted by Gebhardt et al. (2000a). (ii) Replace ǫyi by (ǫ2yi +ǫ20 )1/2 , where the unknown constant ǫ0 , which represents the intrinsic dispersion, is adjusted so that the value of χ2 per degree of freedom is unity. The second procedure is preferable if and only if the individual error estimates ǫyi are reliable. We shall use both approaches in §5.2. FM use the estimator PN (yi − hyi)(xi − hxi) βAB = PNi=1 αAB = hyi − βAB hxi; (5) PN 2 , 2 i=1 (xi − hxi) − i=1 ǫxi P PN −1 here hxi ≡ N −1 N i=1 xi and hyi ≡ N i=1 yi are the sample means of the two variables. This estimator is described by Akritas & Bershady (1996), who also provide formulae for the uncertainties in α and β. The Akritas-Bershady (hereafter AB) estimator accounts for measurement uncertainties in both variables, and is asymptotically normal and consistent. When ǫxi = 0 and ǫyi ≡ ǫy = constant, the AB and χ2 estimators give the same estimates for α and β (but not their uncertainties). Despite its merits, the AB estimator has several unsettling properties. (i) The measureP 2 ment errors in velocity dispersion, ǫxi , only enter equation (5) through the sum ǫ . Thus, P 2 PxiN for example, a single low-precision measurement can dominate both ǫxi and i=1 (xi − 2 hxi) , rendering the estimator useless, no matter how many high-precision measurements are in the sample. (ii) The errors in the black-hole mass determinations ǫyi do not enter equation (5) at all: all observations are given equal weight, even if some are known to be much less precise than others. (iii) We have argued above that the variables x and y should be treated symmetrically, but this is not the case in equation (5). (iv) Even if the variables xi are drawn from a Gaussian distribution, there will occasionally be samples for which the denominator of equation (5) is near zero. In this case the estimator βAB will be very large. These occa-

–5–

Fig. 1.— The distribution of the estimators βχ (eq. 3; solid line) and βAB (eq. 5; dashed line) for 100,000 Monte Carlo simulations of a sample with β = 5 that resembles the actual sample FM1 (12 galaxies, distributed as a Gaussian with standard deviation 0.20 in x, and Gaussian measurement errors with standard deviations ǫx = 0.06, ǫy = 0.18). Values > 5.5 or < 3.3 are plotted in the outermost bins of the histogram. The sample means are marked by arrows.

–6– sional large excursions are frequent enough that the variance of βAB in a population of galaxy samples is inﬁnite, no matter how large the number N of data points may be. (v) Figure 1 shows the distribution of estimates of βχ (solid line) and βAB (dashed line) obtained from 100,000 Monte Carlo trials drawn from a population that has β = 4.5 and other parameters similar to the sample FM1 deﬁned below (for details see ﬁgure caption). The distribution of βχ is substantially narrower than βAB (note that values of either estimator outside the range of the histogram are plotted in the outermost bins). The estimator βχ has a mean of 4.52 and a standard deviation of 0.36. The distribution of βAB has a mean of 4.69, and as stated above the standard deviation of this mean is inﬁnite. Thus, in this example at least, βAB is both biased and ineﬃcient. In this paper, we sometimes use a third ﬁtting procedure, which is closely related to principal component analysis (Kendall, Stuart, & Ord 1983). Suppose that the intrinsic distribution of x and y (the distribution that would be observed in the absence of measurement errors) is a biaxial Gaussian, with major and minor axes having standard deviations σa and σb , respectively, and the major axis having slope β ≡ tan θ. If σb were zero, all of the points would lie exactly on a line of slope β; thus σb characterizes the intrinsic dispersion in the correlation between x and y. Let us also assume that the measurement errors are Gaussian, with standard deviations ǫx and ǫy that are the same for all galaxies. The observed distribution of x and y, which is obtained by convolving the intrinsic Gaussian with the measurement errors, is still Gaussian. The shape of this Gaussian is fully described by the three independent components of the symmetric 2 × 2 dispersion tensor σij ≡ h(xi − hxi i)(xj − hxj i)i,

i = 1, 2, j = 1, 2,

(6)

where (x1 , x2 ) ≡ (x, y) and h·i denotes a sample average. In this idealized but plausible model, at most three of the ﬁve parameters ǫx , ǫy , σa , σb , and β can be determined from the data, no matter how many galaxies we observe. For example, if ǫx and ǫy are known, the other parameters can be estimated using the formulae tan 2θ =

2σxy , σxx − σyy + ǫ2y − ǫ2x

σb2 = σxx − ǫ2x − σxy cot θ = σyy − ǫ2y − σxy tan θ, σa2 = σxx − ǫ2x + σxy tan θ = σyy − ǫ2y + σxy cot θ;

(7)

there are two solutions for θ diﬀering by π/2, and we choose the solution for which σa > σb > 0. These equations, which we call Gaussian estimators, are related to the AB estimator (5), which in this notation is simply tan θ = σxy /(σxx − ǫ2x ). However, the Gaussian estimators have the advantages that (i) they are symmetric in x and y, and (ii) they account naturally for the possibility that there is an intrinsic dispersion σb in the M• –σ correlation. The

–7– Gaussian estimators can easily be extended to include measurement errors that diﬀer from galaxy to galaxy, and to provide uncertainties in the estimators (e.g., Gull 1989), and with these extensions they are likely to provide a more reliable slope estimator than either the χ2 or AB estimators. We close this section with a general comment on ﬁtting linear relations such as (1). The choice of the reference value σ0 aﬀects the uncertainty in α and the covariance between the estimated values of α and β. A rough rule of thumb is that σ0 should be chosen near the middle of the range of values of σ in the galaxy sample to minimize the uncertainty in α and the correlation between α and β. As an example, Ferrarese & Merritt (2000b) use σ0 = 1 km s−1 and ﬁnd an uncertainty in α of ±1.3. However, most of this uncertainty arises because errors in α and β are strongly correlated at this value of σ0 (correlation coeﬃcient r = −0.998). Simply by choosing σ0 = 200 km s−1 the uncertainty in α is reduced by a factor of more than ten, to ±0.09.

3.

The data

The M• –σ relation has been explored in the literature using a number of distinct datasets: 1. Sample FM1 Much of FM’s analysis is based on a set of 12 galaxies with “secure” black-hole mass estimates (sample A, Table 1 of Ferrarese & Merritt 2000b). However, their deﬁnition of “secure” is not itself secure: in §5, we reject one of the galaxies in this sample (NGC 4374) because of concerns about the reliability of its mass estimate, and the best estimate of the mass of another (IC 1459) has recently increased by a factor of six. Half of the black-hole mass estimates in this sample come from gas kinematics, as determined by Hubble Space Telescope (HST ) emission-line spectra, and the remainder from stellar and maser kinematics. Unless otherwise indicated, when discussing this sample we shall use the upper and lower limits to the dispersion and black-hole mass given by Ferrarese & Merritt (2000b).13 The slope estimators then yield βχ = 4.47 ± 0.44,

βAB = 4.81 ± 0.55.

(8)

The minimum χ2 per degree of freedom is 0.69, which indicates an acceptable ﬁt; thus there is no evidence for any intrinsic dispersion in this sample. 13

The error bars in x and y are given by (log σupper − log σlower )/2 and (log M•,upper − log M•,lower )/2, respectively.

–8– 2. Sample G1 The sample used by Gebhardt et al. (2000a) contains 26 galaxies. Of these, the majority (18) of the mass estimates are from axisymmetric dynamical models of the stellar distribution function, based on HST and ground-based absorption-line spectra. All of the galaxies in sample FM1 are contained in this sample except for NGC 3115. The stated rms fractional uncertainty in the black-hole masses is 0.22 dex, but following Gebhardt et al. (2000a), we shall adopt ǫy = 0.30, which yields a minimum χ2 per degree of freedom equal to unity. Gebhardt et al. (2000a) take ǫx = 0, corresponding to negligible uncertainties in the dispersions; this approximation is discussed in §4.1. The slope estimators then yield βχ = 3.74 ± 0.30,

βAB = 3.74 ± 0.23.

(9)

A maximum-likelihood estimate of the intrinsic dispersion in black-hole mass at constant velocity dispersion for this sample is 0.22 ± 0.05 dex. 3. Sample FM2 Merritt & Ferrarese (2001b) supplement sample FM1 with 10 additional galaxies, mostly taken from Kormendy & Gebhardt (2001), for a total of 22 galaxies. The stated rms fractional uncertainty in the black-hole masses is 0.24 dex. The slope estimators yield βχ = 4.78 ± 0.43,

βAB = 4.65 ± 0.49.

(10)

The minimum χ2 per degree of freedom is 1.1, and there is no evidence for any intrinsic dispersion in the black-hole mass. 4. Sample G2 These are the 22 galaxies listed by Kormendy & Gebhardt (2001) that are also in sample FM2. By comparing samples FM2 and G2 we can isolate the eﬀects of diﬀerent treatments of the same galaxies. We shall assume 20% uncertainty in the dispersion of the Milky Way, and 5% uncertainty in the velocity dispersions of external galaxies (see §§4.1 and 4.3). Using G2’s stated uncertainties in the black-hole masses, the slope estimators yield βχ = 3.70 ± 0.20, βAB = 3.61 ± 0.31. The minimum χ2 per degree of freedom is 2.8, which suggests that either the uncertainties in the black-hole masses are underestimated or there is an intrinsic dispersion in black-hole mass. Adding an intrinsic dispersion of 0.17 dex decreases the value of χ2 per degree of freedom to unity, and reduces the best-ﬁt slope to βχ = 3.61 ± 0.29,

βAB = 3.61 ± 0.31.

(11)

A maximum-likelihood estimate of the intrinsic dispersion in black-hole mass at constant velocity dispersion for this sample is 0.16 ± 0.05 dex.

–9– 4.

Why are the slopes different?

Our goal is to determine why diﬀerent investigations yield such a wide range of slopes. In particular, the two samples from FM give slopes & 4.5 (“high” slopes) with both the χ2 and AB estimators, while the two samples from the Nukers give slopes . 4.0 (“low” slopes) with both estimators. In §§4.1–4.4 we describe several explanations for the slope range that have been proposed in the literature, all of which are found to be inadequate. In §4.5 we suggest that systematic diﬀerences in the dispersions used by FM and the Nukers are responsible for most of the slope discrepancy.

4.1.

Measurement errors in velocity dispersion

Merritt & Ferrarese (2001a) argue that random measurement errors in the velocity dispersion can have a signiﬁcant eﬀect on the slope of the M• –σ regression. In particular, they claim that the Nukers’ assumption of zero measurement error in σ leads them to underestimate the slope. To test this claim, we plot in Figure 2 the slope β derived from the G1 sample using both the AB and χ2 estimators, as a function of the assumed rms error ǫx in the log of the velocity dispersion. For nearly all of the galaxies in sample G1, the data typically have signal-to-noise ratio around 100, and the formal uncertainties in the dispersions are around 2–3% (ǫx = 0.009– 0.013). However, at this level, stellar template variations, assumptions about the continuum shape, ﬁtting regions used, and atmospheric seeing conditions all can have a noticeable eﬀect on the estimated dispersion. To account crudely for these systematic errors, we double the uncertainties in the dispersions, to 5% (ǫx = 0.021). The uncertainty is larger in the Milky Way (see §4.3), and in a few galaxies that we have not observed ourselves and that do not have accurate dispersion proﬁles in the literature. The statement of Merritt & Ferrarese (2001a) that velocity-dispersion errors are “easily at the 10% level” is indeed correct for the sample FM1, where the rms fractional error in the dispersions is 14% (ǫx = 0.057), but their dispersions are mostly based on heterogeneous data that are 20–30 years old (Davies et al. 1987). Figure 2 shows that the eﬀect of random errors in the dispersions is negligible: at the 5% level, the change in β for sample G1 is only 0.03 or 0.04 for the χ2 and AB estimators respectively, and even at the 10% level the corresponding changes are only 0.12 and 0.16.

– 10 –

Fig. 2.— The dependence of the slope β on the assumed measurement uncertainty in velocity dispersion, for the sample G1. The abscissa is the rms measurement error in log σ. Solid and dashed lines show the slopes derived from the AB and χ2 estimators, respectively. The error bars show the computed uncertainty in the slope at zero error. The formal uncertainty in the dispersion measurements of G1 is ǫx ≃ 0.01; allowing for possible systematic errors in the stellar template and continuum subtraction increases ǫx to ∼ 0.02 (5%).

– 11 – 4.2.

Measurement errors in black-hole mass

We next ask whether the combined eﬀects of varying assumptions about measurement errors in both velocity dispersion and black-hole mass can explain the discrepancy between the high and low slopes. As usual, we parametrize these uncertainties by ǫx and ǫy , the rms measurement error in the log of the velocity dispersion and black-hole mass. For simplicity, in this subsection these errors are assumed to be the same for all galaxies in each sample. The eﬀects of these uncertainties on the slope β can then be explored using the Gaussian estimators (7). These estimators have two advantages over the χ2 or AB estimators for this purpose: (i) the slope estimator depends only on the diﬀerence ǫ2y − ǫ2x and hence is a function of only one variable; (ii) the condition that the derived intrinsic dispersion σb2 be positive-deﬁnite provides an upper limit to the allowable errors. The left and right panels of Figure 3 show the slope β and the maximum allowed value of ǫx for each of the galaxy samples in §3. For each sample there is a minimum slope β and a maximum value of ǫ2y −ǫ2x , beyond which the intrinsic dispersion σb2 is negative. In particular, for sample FM1 the minimum allowable slope is β = 4.39; thus there are no assumptions about the measurement errors that can lead to a slope in the low range. The slope vs. error lines in the left panel of Figure 3 are approximately parallel for all four samples; thus there is no single set of measurement errors that could remove the discrepancy between the high slopes found by FM and the low slopes found by the Nukers. Consistent slopes would require that (ǫ2y − ǫ2x )Nuker ≃ (ǫ2y − ǫ2x )FM − 0.07. This relation, combined with the constraint σb2 > 0, cannot be satisﬁed with any plausible combination of measurement errors—note in particular that ǫy should be similar for the two groups since they rely on many of the same black-hole mass determinations, and ǫx should be smaller for the Nuker samples than the FM samples, since the Nukers employ high signal-to-noise ratio slit spectra while FM rely on central velocity dispersions from the pre-1990 literature. We conclude that random measurement errors cannot explain the slope discrepancy.

4.3.

The dispersion of the Milky Way

Merritt & Ferrarese (2001a) also argue that the slope is strongly aﬀected by the assumed dispersion for the Milky Way Galaxy, for which the Nukers’ estimated dispersion σ = 75 km s−1 should be increased to σ = 100 km s−1 . We show in Figure 4 how the derived slope depends on the Milky Way dispersion, for both samples G1 and FM1. We see that in fact β is quite insensitive to the Milky Way dispersion used in the G1 sample: increasing the dispersion from 75 km s−1 to 100 km s−1 as suggested by Merritt & Ferrarese (2001a) increases β only by 0.13. The corresponding slope change is substantially larger for sample

– 12 –

Fig. 3.— The dependence of the slope β on the assumed rms errors in black-hole mass and velocity dispersion. The rms errors in log M• and log σ are ǫy and ǫx , respectively (assumed the same for all galaxies). The left panel shows the slope derived from the Gaussian estimator (7), for samples FM1 (solid line), FM2 (short-dashed line), G1 (dotted line), and G2 (longdashed line). The lines stop where the intrinsic dispersion σb2 < 0. The right panel shows the maximum allowed value of ǫx ; for larger values the intrinsic dispersion is negative. The ﬁlled circles denote the locations corresponding to the estimated values of ǫx and ǫy in each survey; in the right panel these are connected by vertical lines to the curves for the corresponding survey.

– 13 – FM1—0.27 for the χ2 estimator and 0.44 for the AB estimator—but this strong sensitivity reﬂects the small size of that sample and is not relevant to conclusions drawn by Gebhardt et al. (2000a) from sample G1. Despite this conclusion, it is worthwhile to determine a more accurate value for the Milky Way dispersion to use in the M• –σ relation. We review the data on the dispersion of the Galactic bulge in the Appendix, where our results are summarized in the dispersion proﬁle of Figure 9 and equation (A3). We stress that the dispersion proﬁle of the Milky Way is determined from a heterogeneous set of tracers with uneven spatial coverage, and by very diﬀerent methods than the dispersions of the external galaxies discussed in this paper. We therefore assign our estimates of the Milky Way dispersion an uncertainty of 20%, much larger than the formal uncertainty, and much larger than the 5% uncertainty that we assume for the dispersions of external galaxies. The conversion of the dispersion proﬁle in equation (A3) to a characteristic dispersion is diﬀerent for FM and the Nukers. FM deﬁne their dispersion to be the luminosity-weighted rms line-of-sight dispersion within a circular aperture of radius re /8, where re is the eﬀective radius. For re = 0.7 kpc as derived in the Appendix, we ﬁnd σ = 95 km s−1 . Because the bulge is triaxial, we correct the dispersion that we measure from our particular location to the average over all azimuths in the Galactic plane. Binney, Gerhard, & Spergel (1997) model the bulge as a triaxial system with axis ratios 1 : 0.6 : 0.4 and long axis at an angle φ0 = 20◦ from the Sun-center line. If the density is stratiﬁed on similar ellipsoids, the ratio r 2 ≡ σ 2 (φ0 = 20◦ )/hσ 2 (φ0 )i depends only on the axis ratios (Roberts 1962). For the axis ratios given by Binney et al., r = 1.07. Thus our best estimate for the dispersion within re /8 is σFM = 90 ± 18 km s−1 ; if we use this instead of FM’s estimate of σ = 100 ± 20 km s−1 , the slope derived from sample FM1 is reduced from βAB = 4.81 ± 0.55 to βAB = 4.66 ± 0.42, and for sample FM2 from βAB = 4.65 ± 0.49 to 4.54 ± 0.40. In contrast, the Nukers use the luminosity-weighted rms line-of-sight dispersion within a slit aperture of half-length re . This dispersion depends weakly on the slit width, which we take to be 70 pc (corresponding to 1 arcsec at Virgo). In this case we ﬁnd σ = 110 km s−1 ; reducing this by a factor r to account for triaxiality, we have σ = 103 ± 20 km s−1 , close to the value advocated by FM. This change increases the slope derived by Gebhardt et al. (2000a) from βχ = 3.74 ± 0.15 only to βχ = 3.88 ± 0.15. Thus, improved estimates of the velocity dispersion of the Milky Way bulge reduce the slope discrepancy only slightly.

– 14 –

Fig. 4.— The dependence of the slope β on the assumed velocity dispersion for the Milky Way, in samples G1 (left panel) and FM1 (right panel). The ﬁlled circles and error bars show the assumed dispersion and the corresponding slope and error bars. Solid and dashed lines show the slopes derived from the AB and χ2 estimators, respectively; these are the same for the G1 sample because ǫx = 0 and ǫy is the same for all galaxies.

– 15 – 4.4.

Different samples

Merritt & Ferrarese (2001b) argue that the shallower slope obtained by the Nukers arises in part from the inclusion of galaxies in which the black-hole sphere of inﬂuence is not well resolved. However, the samples FM2 and G2 contain exactly the same 22 galaxies, all of which are claimed by Merritt & Ferrarese (2001b) to have a well-resolved sphere of inﬂuence, and the diﬀerence in slope βχ (eqs. 10 and 11) is actually larger than between the samples FM1 and G1.

4.5.

Aperture and effective dispersions

Why, then, are the slopes diﬀerent, particularly in the samples FM2 and G2, which contain the same galaxies? If we ﬁt the dispersions in these samples to a relation of the form log σG2 = γ + (1 + δ) log σFM2 ,

(12)

we ﬁnd δχ = 0.13 ± 0.10,

δAB = 0.23 ± 0.10,

(13)

signiﬁcantly diﬀerent from the value δ = 0 that should obtain if there were no systematic diﬀerences between the dispersions (see Figure 5). A relation of this kind implies that the slopes β determined from the FM2 and G2 samples will be related by βFM2 /βG2 = 1 + δ. Then if δ is in the range 0.15–0.20, and the Nuker sample gives β = 4, the sample from FM will give β = 4.6–4.8, well inside the high range. Thus it appears that the major cause of the range of slopes is systematic diﬀerences in the dispersions: FM’s dispersions lead to high slopes, and the Nukers’ dispersions lead to low slopes. This possibility was suggested by Gebhardt et al. (2000a), but was later rejected by Merritt & Ferrarese (2001a), who argued that systematic diﬀerences between dispersions are unimportant because there was “remarkably little diﬀerence on average” between the dispersions (they quote a mean ratio of 1.01 and a correlation coeﬃcient in the logs of 0.97). However, these statistics have no bearing on the slope 1 + δ in equation (12). There are several possible explanations of the diﬀerence in dispersions: 1. The Nukers use the rms dispersion within a slit aperture of length 2re (hereafter σ1 ), while FM’s results are based on the rms dispersion within a circular aperture of radius re /8 (hereafter σ8 ). The ratio σ8 /σ1 could depend systematically on the velocity dispersion of the galaxy, for example, σ1 ∝ σ81+δ1 .

(14)

– 16 –

Fig. 5.— Comparison of the velocity dispersions in the samples FM2 and G2. The dispersion ratio σFM2 /σG2 is plotted against σG2 . Solid circles denote power-law galaxies and open circles denote core galaxies. We take the uncertainties in the FM2 dispersions from Merritt & Ferrarese (2001b), and assume that the uncertainties in the G2 dispersions are 20% for the Milky Way and 5% for external galaxies. The uncertainties in dispersion ratio are computed by assuming that the errors in the FM2 dispersions and the G2 dispersions are independent. The plot shows that FM’s dispersions are higher than Nuker dispersions at low dispersion, and lower at high dispersion.

– 17 – In this case the diﬀerence in slope estimates would reﬂect the structural properties of the galaxies. 2. FM do not actually measure σ8 . Instead, they use the central velocity dispersion (hereafter σc ), typically measured in an aperture of radius rap ≃ 2 arcsec (Davies et al. 1987), and correct this to a circular aperture of radius re /8 using the relation (Jørgensen, Franx, & Kjaergaard 1995) σ8′ = σc (8rap /re )0.04 (the prime is used to distinguish this approximation to σ8 from the actual value of σ8 ). The ratio σ8′ /σ8 could depend systematically on the velocity dispersion of the galaxy, for example, σ8 ∝ (σ8′ )1+δ2 .

(15)

Such a trend could arise if the correction factor depends on galaxy luminosity or type. In this case the diﬀerence in slope estimates would reﬂect a shortcoming in FM’s analysis rather than a real physical eﬀect. 3. The dispersion measurements used by one or both of the two groups could be subject to dispersion-dependent systematic errors (e.g., one set of measurements is systematically low at high dispersions). In this case the diﬀerence in slope estimates would reﬂect problems with the data reduction. To explore these possibilities, we have examined a sample of 40 early-type galaxies for which Faber et al. (1997) have compiled eﬀective radii and central velocity dispersions, and have ﬁtted HST photometry to a ﬁve-parameter “Nuker law” proﬁle. Each galaxy is assumed to contain a central black hole, with mass given by the M• –σ relation in the form derived below (eq. 19). We use the Nuker law and the assumptions of spherical symmetry, constant mass-to-light ratio, isotropic velocity dispersion, and 1 arcsec slit width to compute the ratios σ8 /σ1 and σ8′ /σ8 . This approach is model-dependent, but has the advantages that (i) the discussion is independent of observational errors in the dispersions, since the dispersion ratios are determined by a dynamical model; (ii) the sample is larger, since more galaxies have HST photometry than dispersion proﬁles. We ﬁnd σ8 σc log = (0.004 ± 0.002) + (0.021 ± 0.010) log , σ1 200 km s−1 ′ σc σ8 = −(0.012 ± 0.003) − (0.056 ± 0.014) log . (16) log σ8 200 km s−1 The ﬁrst of these equations suggests that there is a small but signiﬁcant systematic trend in the ratio σ8 /σ1 with dispersion, of the form (14) with δ1 ≃ −0.02. However, this trend has the wrong sign and only a small fraction of the amplitude required to explain the

– 18 – systematic diﬀerences in equations (12) and (13); thus explanation 1 in the list above does not appear to be important. The second of these equations suggests that there is a larger systematic trend in the ratio with dispersion, of the form (15) with δ2 ≃ 0.06. This trend is suﬃcient to explain about one-third of the systematic diﬀerences seen in equations (12) and (13). The origin of this trend, explanation 2 in the list above, is clear. Gebhardt et al. (1996) and Faber et al. (1997) show that the shape of the surface-brightness proﬁle in the central parts of early-type galaxies depends on the galaxy luminosity (and hence on its velocity dispersion). Thus, the use of a single empirical formula to correct from σc to σ8 will lead to systematic errors that are correlated with velocity dispersion. It is always better to use the actual kinematic observations, as was done in G1, than to apply empirical correction factors.

σ8′ /σ8

Explanation 3, dispersion-dependent systematic measurement errors, is more diﬃcult to assess. Hudson et al. (2001) compare dispersion measurements from 27 sources, including the catalog used by FM (Davies et al. 1987), and in most cases ﬁnd no evidence for dispersion-dependent errors of the amplitude found in equation (13) (see Hudson et al.’s Figure 3). Nevertheless, it is striking that the data points in Figure 5 appear sharply lower for dispersions & 300 km s−1 than for smaller dispersions. Measuring large dispersions is particularly diﬃcult because the spectral lines blend together. The principal conclusion is that we badly need a systematic campaign of accurate HST and ground-based measurements of the radial velocity-dispersion proﬁles of early-type galaxies with black-hole candidates. A second conclusion is that the slope of the M• –σ relation should be estimated only from dispersion measurements at or within well-deﬁned metric radii, rather than central velocity dispersions measured within apertures of a given angular radius. The same sample of 40 galaxies can also be used to explore the degree of contamination of the dispersions by the dynamical inﬂuence of the central black hole. We computed spherical, isotropic dynamical models with and without a black hole of mass given by equation (19). We denote the ratio of the dispersion σ1 with and without the black hole by f1 , and the analogous ratio for the dispersion σ8′ by f8′ . The results are shown in Figure 6. In most galaxies the addition of the black hole raises either dispersion no more than 3–4%. However, in a few cases the contamination is much larger, more than 15%. In such galaxies the dispersion measures σc , σ1 , σ8 , or σ8′ are all misleading. Future versions of the M• –σ relation should be based on dispersion measures that are less strongly weighted to the center.

– 19 –

Fig. 6.— Eﬀect of central black holes on measured dispersions. For 40 early-type galaxies listed in Faber et al. (1997) we have computed isotropic, spherical dynamical models that match the surface-brightness distributions and mass-to-light ratios given in that paper. For each galaxy two models are computed: one with no central black hole, and one with a black hole of mass given by equation (19). The abscissa represents the ratio of the dispersions σ1 in these two models, where σ1 is the luminosity-weighted rms line-of-sight dispersion within a slit aperture of half-length re and width 1 arcsec (used by the Nukers). The ordinate represents the ratio of the dispersions σ8′ ; here σ8′ is an approximation to the dispersion within a circular aperture of radius re /8, obtained from the central dispersion using an empirical correction formula (used by FM). The galaxies in the upper right corner are M32 and NGC 3115.

– 20 – 5.

Black-hole mass vs. velocity dispersion: a new estimate

In this section we present a new analysis of the M• –σ relation using the 31 galaxies in Table 1; over half of these have new or revised black-hole mass or dispersion determinations since the analysis by Gebhardt et al. (2000a).

5.1.

Comments on individual galaxies

Milky Way: We use the black-hole mass estimate by Chakrabarty & Saha (2001), For comparison, Ghez et al. (1998) ﬁnd (2.4 ± 0.2) × 106 M⊙ , and Genzel et al. (2000) ﬁnd (2.6–3.3)×106M⊙ . The dispersion and its uncertainty are discussed in §4.3. The bulge mass-to-light ratio is taken from Kent (1992).

(1.8 +0.4 ) × 106 M⊙ . −0.3

M32: The velocity dispersion is obtained from van der Marel et al. (1994) and the black-hole mass, M• = (2.5 ± 0.5) × 106 M⊙ , is from Verolme et al. (2002). In estimating the dispersion, we have excluded the region near the center that is strongly perturbed by the black hole (cf. Figure 6). Other recent mass estimates, by van der Marel et al. (1998) and Joseph et al. (2001), give similar results: (3.9 ± 0.8) × 106 M⊙ and (3 ± 1) × 106M⊙ , respectively. M31: The modeling is complicated by the double nucleus. Kormendy & Bender (1999) ﬁnd M• = (3.0 ± 1.5) × 107 M⊙ , although this result relies heavily on the small displacement between the center of light of the nucleus and bulge. Tremaine (1995) and Bacon et al. (2001) ﬁnd M ≃ 7 × 107 M⊙ , but without detailed model ﬁtting. We adopt the range (2.0–8.5)×107M⊙ . NGC 1068: The black-hole mass is taken from Greenhill & Gwinn (1997); the error estimates are our own and are very approximate. The dispersion (Kobulnicky & Gebhardt 2000) is somewhat uncertain because of contamination from the bright nucleus. NGC 3115: The black-hole mass is based on stellar kinematics (Kormendy et al. 1996a). Although NGC 3115 does not have three-integral axisymmetric dynamical models, it does have a compact, high-contrast stellar nucleus, and the mass of the nucleus plus black hole can be estimated from the virial theorem. In estimating the dispersion, we have excluded the region near the center that is strongly perturbed by the black hole (cf. Figure 6). NGC 3245: The velocity dispersion is obtained from Simien & Prugniel (1998). NGC 4258: The velocity dispersion is obtained from H´eraudeau & Simien (1998).

– 21 –

Table 1. Galaxy sample Galaxy

Type

MB

M• (low,high) M⊙

Milky Way N221=M32 N224=M31 N821 N1023 N1068 N2778 N2787 N3115 N3245 N3377 N3379 N3384 N3608 N4258 N4261 N4291 N4342 N4459 N4473 N4486=M87 N4564 N4596 N4649 N4697 N4742 N5845 N6251 N7052 N7457 IC1459

SBbc E2 Sb E4 SB0 Sb E2 SB0 S0 S0 E5 E1 S0 E2 Sbc E2 E2 S0 S0 E5 E0 E3 SB0 E1 E4 E4 E3 E2 E4 S0 E3

−17.65 −15.83 −19.00 −20.41 −18.40 −18.82 −18.59 −17.28 −20.21 −19.65 −19.05 −19.94 −18.99 −19.86 −17.19 −21.09 −19.63 −17.04 −19.15 −19.89 −21.53 −18.92 −19.48 −21.30 −20.24 −18.94 −18.72 −21.81 −21.31 −17.69 −21.39

1.8 × 106 (1.5, 2.2) 2.5 × 106 (2.0, 3.0) 4.5 × 107 (2.0, 8.5) 3.7 × 107 (2.9, 6.1) 4.4 × 107 (3.9, 4.9) 1.5 × 107 (1.0, 3.0) 1.4 × 107 (0.5, 2.2) 4.1 × 107 (3.6, 4.5) 1.0 × 109 (0.4, 2.0) 2.1 × 108 (1.6, 2.6) 1.0 × 108 (0.9, 1.9) 1.0 × 108 (0.5, 1.6) 1.6 × 107 (1.4, 1.7) 1.9 × 108 (1.3, 2.9) 3.9 × 107 (3.8, 4.0) 5.2 × 108 (4.1, 6.2) 3.1 × 108 (0.8, 3.9) 3.0 × 108 (2.0, 4.7) 7.0 × 107 (5.7, 8.3) 1.1 × 108 (0.31, 1.5) 3.0 × 109 (2.0, 4.0) 5.6 × 107 (4.8, 5.9) 7.8 × 107 (4.5, 12) 2.0 × 109 (1.4, 2.4) 1.7 × 108 (1.6, 1.9) 1.4 × 107 (0.9, 1.8) 2.4 × 108 (1.0, 2.8) 5.3 × 108 (3.5, 7.0) 3.3 × 108 (2.0, 5.6) 3.5 × 106 (2.1, 4.6) 2.5 × 109 (2.1, 3.0)

Method

σ1 km s−1

Distance Mpc

M/L, band

Ref

s,p s,3I s s,3I s,3I m s,3I g s g s,3I s,3I s,3I s,3I m,a g s,3I s,3I g s,3I g s,3I g s,3I s,3I s,3I s,3I g g s,3I s,3I

103 75 160 209 205 151 175 140 230 205 145 206 143 182 130 315 242 225 186 190 375 162 152 385 177 90 234 290 266 67 340

0.008 0.81 0.76 24.1 11.4 15.0 22.9 7.5 9.7 20.9 11.2 10.6 11.6 22.9 7.2 31.6 26.2 15.3 16.1 15.7 16.1 15.0 16.8 16.8 11.7 15.5 25.9 93.0 58.7 13.2 29.2

1.0,K 1.85,I 5,V 5.8,V 5.0,V ··· 6.4,V ··· 6.9,V 3.7,R 2.7,V 4.6,V 2.8,V 3.7,V ··· 5.0,V 4.4,V 6.3,I ··· 6.3,V 4.0,V 1.9,I ··· 9.0,V 4.8,V ··· 4.8,V 8.5,V 6.3,I 3.4,V 3.1,I

1 5 2,3,4 6,7 8 9 6,7 10 11 12 6,13 14 6,7 6,7 15 16 6,7 17 10 6,7 18,19 6,7 10 6,7 6,7 20 6 21 22 6,7 23

Note. — Distances are taken from Tonry et al. (2001) for most of the galaxies; where these are not available the distance is determined from the recession velocity, assuming a Hubble constant of 80 km s−1 Mpc−1 . Absolute magnitudes are for the hot component of the galaxy only. The massto-light ratios M/L are usually determined from the same dynamical models that are used to derive the black-hole masses; they are given here for reference but play no role in our analysis. Methods: s=stellar radial velocities; p=stellar proper motions; m=maser radial velocities; a=maser accelerations; g=rotating gas disk from emission-line observations; 3I=axisymmetric dynamical models, including three integrals of motion. References for the black-hole masses: (1) Chakrabarty & Saha (2001); (2) Tremaine (1995); (3) Kormendy & Bender (1999); (4) Bacon et al. (2001); (5) Verolme et al. (2002); (6) Gebhardt et al. (2002); (7) Pinkney et al. (2002); (8) Bower et al. (2001); (9) Greenhill & Gwinn (1997); (10) Sarzi et al. (2001); (11) Kormendy et al. (1996a); (12) Barth et al. (2001a); (13) Kormendy et al. (1998); (14) Gebhardt et al. (2000b); (15) Herrnstein et al. (1999); (16) Ferrarese, Ford, & Jaffe

– 22 – NGC 4486: The mass is the average of the values given by Harms et al. (1994) and Macchetto et al. (1997), corrected to a distance of 16.1 Mpc. NGC 2787, NGC 4459, NGC 4596: The black-hole masses are based on Space Telescope Imaging Spectrograph (STIS) measurements of ionized-gas disks by Sarzi et al. (2001). The disk inclinations are determined from dust-lane morphology. Note that the distance and dispersion for NGC 2787, 7.5 Mpc and 140 km s−1 , are much smaller than the values assumed by Sarzi et al. (2001). Our distance is from Tonry et al. (2001) and the dispersion was measured by one of us (Gebhardt). For NGC 4459 and NGC 4596 we have used the dispersions σ8′ from Sarzi et al. (2001), since on average these are close to σ1 (eq. 16). IC 1459: The mass estimate that we use (Cappellari et al. 2002), based on stellar kinematics, is much larger than an earlier estimate by the same group from gas kinematics, (2–6)×108 M⊙ (Verdoes Kleijn et al. 2000). The mass estimate from stellar kinematics is much more reliable, since the gas rotation curve is asymmetric and non-Keplerian. We do not include the following galaxies in our sample: NGC 4594 (Kormendy et al. 1996b), NGC 4486B (Kormendy et al. 1997), NGC 4350 (Pignatelli, Salucci, & Danese 2001), NGC 3031=M81, and NGC 3998 (Bower et al. 2000) exhibit strong evidence from stellar dynamics for a black hole, but do not yet have threeintegral dynamical models. NGC 4374 (M84) has strong evidence for a black hole from gas dynamics, but the published estimates of the black-hole mass diﬀer by far more than the stated errors: Bower et al. (1998) ﬁnd (0.9–2.6)×109M⊙ ; Maciejewski & Binney (2001) ﬁnd 4 × 108 M⊙ , and Barth et al. (2001b) ﬁnd 109 M⊙ . The mass assigned to this galaxy is a factor of four larger in sample FM2 than in sample G2, which is by far the largest discrepancy between the two samples. NGC 4945 has a mass estimate from maser emission (Greenhill, Moran, & Herrnstein 1997) but no reliable dispersion. NGC 5128 has a mass estimate from ground-based observations of a rotating gas disk (Marconi et al. 2001) but no HST spectroscopy; moreover, the galaxy has peculiar morphology, presumably because of a recent merger, and thus may not follow the same M• –σ relation as more normal galaxies. Our sample contains eight galaxies with black-hole mass estimates based on gas kinematics. We have some concern that these results may have large systematic errors, due in part to uncertainties in the the spatial distribution of the gas (e.g., ﬁlled disk or torus conﬁguration; uncertain inclination and thickness) and the large but uncertain correction

– 23 – for pressure support. We therefore urge caution when interpreting results from samples in which a large fraction of the black-hole mass estimates are based on gas kinematics. Eventually, galaxies with black-hole mass determinations from more than one technique will be invaluable for disentangling the systematic errors in diﬀerent methods.

5.2.

Slope estimation

We use the sample of galaxies and black-hole masses in Table 1 to estimate the logarithmic slope β in the M• –σ relation. We assume 20% uncertainty in the dispersion of the Milky Way (cf. §4.3) and 5% uncertainties in the dispersions of external galaxies (cf. §4.1), although the uncertainties in the dispersions of a few galaxies that we have not observed ourselves may be larger. Initially we assume 0.33 dex rms uncertainties in the black-hole masses, which yields χ2 per degree of freedom of unity. Using the χ2 and AB estimators deﬁned in §2, we ﬁnd βχ = 4.03 ± 0.33,

βAB = 4.12 ± 0.34.

(17)

This approach does not account for the varying precision of the mass estimates for diﬀerent galaxies. Therefore we have also computed the slope using the estimated errors in the black-hole masses in Table 1, adding to these in quadrature a common intrinsic dispersion with rms value ǫ0 (i.e., ǫyi → (ǫ2yi + ǫ20 )1/2 ). We ﬁnd that ǫ0 = 0.27 gives βχ = 4.00 ± 0.31,

βAB = 4.12 ± 0.34,

(18)

with minimum χ2 per degree of freedom of 1.00. A maximum-likelihood estimate of the intrinsic dispersion in black-hole mass at constant velocity dispersion for this sample is ǫ0 = 0.23 ± 0.05 dex. In both equations (17) and (18) the AB estimator for the slope is larger than the χ2 estimator by about 0.1; since we have shown in §2 that the AB estimator may be biased, we prefer to rely on the χ2 estimator. For our ﬁnal answer we simply average βχ from equations (17) and (18). Including results for the parameter α obtained in the same way, we have α = 8.13 ± 0.06,

β = 4.02 ± 0.32;

(19)

the parameter α is evaluated for σ0 = 200 km s−1 , for which the correlation coeﬃcient between α and β is only −0.09. Thus our best estimate (19) is just at the edge of the low range, β . 4.0. To test the robustness of this result, we have tried culling the sample in several ways:

– 24 –

Fig. 7.— The data on black-hole masses and dispersions for the galaxies in Table 1, along with the best-ﬁt correlation described by equations (1) and (19). Mass measurements based on stellar kinematics are denoted by circles, on gas kinematics by triangles, and on maser kinematics by asterisks; Nuker measurements are denoted by ﬁlled circles. The dashed lines show the 1σ limits on the best-ﬁt correlation.

– 25 – • If we consider only the 21 galaxies from Table 1 with masses determined from stellar kinematics, we ﬁnd α = 8.13 ± 0.09, β = 4.02 ± 0.44; (20) the close agreement in the parameters in equations (19) and (20) implies that there is no signiﬁcant systematic bias between masses determined by stellar kinematics and other methods. • The dispersions for the Milky Way and for external galaxies are determined by quite diﬀerent methods. The Milky Way also has one of the smallest and most accurate black-hole masses in our sample, and therefore has an unusually strong inﬂuence on the slope of the M• –σ relation. If we remove the Milky Way from our sample, the slope is reduced to β = 3.88 ± 0.32, a change of 0.14 (0.4 standard deviations) towards even lower slopes. • We have argued in §4.5 that high velocity-dispersion measurements may be subject to systematic errors. Thus we also estimate the slope using only the 25 galaxies in the sample with dispersion less than 250 km s−1 . We ﬁnd β = 3.77 ± 0.49; once again the slope is even lower than our best estimate (19). • The galaxy sample with the most homogeneous observations and analysis consists of the 10 galaxies analyzed by Pinkney et al. (2002) and Gebhardt et al. (2002). These all have HST spectra acquired with STIS as well as ground-based spectra, HST photometry, and axisymmetric orbit-based dynamical models, and were all reduced and analyzed in the same way. For this sample we ﬁnd β = 3.67 ± 0.70; once again the slope is consistent with and even lower than our best estimate. • We have removed 9 galaxies from the sample which were subject to criticism: the Milky Way (uncertain dispersion), M31 (no accurate models of the double nucleus), NGC 1068 (both the dispersion and the interpretation of the maser kinematics are uncertain), NGC 2778 (the lowest signal-to-noise ratio in the Gebhardt et al. 2002 sample and a correspondingly large uncertainty in the black-hole mass), NGC 3115 (no three-integral dynamical models), NGC 3379 and NGC 5845 (these have only a single FOS pointing rather than STIS slit spectra at HST resolution; while there is no obvious problem with either measurement, other galaxies in the Gebhardt et al. 2002 sample have superior spatial coverage of the kinematics), NGC 4459 (the inclination of the gas disk is uncertain because the kinematic data comes from a single long-slit spectrum; also, the dispersion is uncertain because it is obtained from low-resolution data), and NGC 6251 (the most distant galaxy with a black-hole mass measurement; the sphere of inﬂuence of the black hole is poorly resolved and in addition there are

– 26 – the usual uncertainties—uncertain disk orientation, inﬂuence of random motions in the gas—associated with mass measurements from gas kinematics). The reduced sample of 22 galaxies has a slope β = 3.79 ± 0.32, once again lower than our best estimate. Since most of these culled samples have slopes that are smaller than our best ﬁt (19), we suspect that our best ﬁt may slightly overestimate the true slope by 0.1–0.3. The data from Table 1 and the ﬁt (19) are shown in Figure 7. In Figure 8 we show the residuals to the best-ﬁt correlation. The two largest residuals in Figure 7 belong to NGC 2778 (−0.75 dex) and the Milky Way (−0.72 dex); the largest positive residual belongs to NGC 3115 (+0.63 dex). The poor ﬁt of the Milky Way probably arises because its dispersion proﬁle has been determined by diﬀerent methods than the other galaxies, using heterogeneous tracers and a variety of surveys; we have allowed for this by assigning the Milky Way dispersion an uncertainty of 20%, compared to 5% for external galaxies. The large residual in NGC 3115 may arise because its mass has been estimated by simply applying the virial theorem to its nucleus, rather than by dynamical modeling. The large residual in NGC 2778 may reﬂect the low signal-to-noise ratio of its kinematic data (Pinkney et al. 2002).

6.

Conclusions

The masses M• of dark objects (“black holes”) in the centers of nearby early-type galaxies are related to the velocity dispersion σ by the log-linear relation (1). We have used the sample of 31 galaxies in Table 1 to determine the parameters in this relation, where σ is deﬁned to be the luminosity-weighted rms velocity dispersion in a slit extending to the eﬀective radius. Our best estimate for the slope of this relationship is 4.0 ± 0.3 (eq. 19), although several culled, and perhaps higher quality, samples give slopes that are lower by 0.1–0.3. There is no evidence for systematic diﬀerences in either slope or normalization between black-hole mass measurements based on stellar kinematics and gas kinematics. If the stated measurement errors in the black-hole masses are correct, or if they are underestimated because of systematic errors, the intrinsic dispersion in the M• –σ relation is no larger than about 0.3 dex in black-hole mass (i.e., less than a factor of two). The range of slopes for the M• –σ relation found in the literature appears to arise mostly from systematic diﬀerences in the velocity dispersions used by diﬀerent groups. We do not believe that these diﬀerences reﬂect the diﬀerent deﬁnitions of dispersion used by the groups (FM use the dispersion within a circular aperture of radius re /8, and the Nukers use the dispersion within a slit aperture of half-length re ). It appears that part of the diﬀerence results

– 27 –

Fig. 8.— The residuals between the black-hole masses and dispersions for the galaxies in Table 1 and the best-ﬁt correlation described by equation (1) with β = 4.02 (eq. 19). Mass measurements based on stellar kinematics are denoted by circles, on gas kinematics by triangles, and on maser kinematics by asterisks; Nuker measurements are denoted by ﬁlled circles.

– 28 – from Ferrarese & Merritt’s analysis, in which central velocity dispersions are extrapolated to re /8 using an empirical formula. However, another—and possibly larger—component appears to arise from poorly understood systematic errors in the dispersion measurements. In a few galaxies, the inﬂuence of the central black hole may signiﬁcantly aﬀect the velocity dispersions—both the central dispersions used by FM and the slit dispersions used by the Nukers. Future analyses of the M• –σ relation should be based on velocity-dispersion measures that are less strongly weighted to the center. Other improvements in the analysis would include the use of statistical estimators that are more robust and that explicitly include an intrinsic dispersion in the black-hole mass, accounting properly for the asymmetric error bars in black-hole mass determinations, and estimating more accurately the uncertainties in individual dispersion measurements. The investment of the astronomy community in the diﬃcult task of measuring blackhole masses has not yet been matched by a commensurate investment in the much easier task of obtaining high-quality kinematic maps of galaxies containing black holes. A complete set of high-quality dispersion and rotation proﬁles for the galaxies in Table 1 would allow us to explore more deeply how the black-hole mass is related to the kinematic structure of its host galaxy. We thank Michael Hudson and Tim de Zeeuw for discussions, and Tim de Zeeuw for communicating results in advance of publication. Support for proposals 7388, 8591, 9106, and 9107 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. This research was also supported by NSF grant AST-9900316.

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This preprint was prepared with the AAS LATEX macros v5.0.

– 33 – A.

The effective dispersion for the Milky Way

The Milky Way has one of the most accurate black-hole masses and anchors the lowmass end of the M• –σ relation. Therefore it is important to have an accurate value for the dispersion of the Milky Way bulge. The ﬁrst task is to estimate the eﬀective or half-light radius re of the bulge. Kent (1992) models Spacelab K-band observations of the bulge with a major-axis emissivity proﬁle of the form −1.85 ji a for a < 0.94 kpc (A1) j(a) = jo K0 (a/a0 ) for a > 0.94 kpc, where K0 is a modiﬁed Bessel function, a0 = 0.67 kpc, and the constants ji and jo are chosen so that the emissivity is continuous. In a spherical galaxy described by equation (A1), the eﬀective radius is 1.50a0 or 1.0 kpc; Kent’s model is oblate and axisymmetric, with axis ratio 0.6, so the geometric mean of the three eﬀective semi-axes is smaller by (0.6)1/3 , yielding re = 0.84 kpc. Dwek et al. (1995) ﬁt COBE measurements in several bands to a wide variety of triaxial models for the emissivity. Their best-ﬁt model at K-band has a Gaussian emissivity proﬁle with an eﬀective semi-major axis of 1.86 kpc; the axis ratios are 1 : 0.4 : 0.3, so our best estimate for the eﬀective radius is re = 1.86 kpc (0.4 × 0.3)1/3 = 0.92 kpc. Their second-best model (E3) has j(a) ∝ K0 (a/a0 ) and an eﬀective radius re = 0.56 kpc. Binney, Gerhard, & Spergel (1997) use COBE L-band photometry to perform a disk/bulge decomposition. Their equation (1b) describes an analytic model for the bulge emissivity that ﬁts the data “very well”: 2

2

e−a /am j(a) = j0 , (1 + a/a0 )1.8

(A2)

where a is the semi-major axis and am = 1.9 kpc. They quote a0 = 100 pc, but this value reﬂects the fact that the data have been smoothed to an angular resolution of 1.5◦ or 200 pc, and photometry at higher resolution suggests that a0 is less than 1 pc (e.g., Genzel et al. 1996). The eﬀective semi-major axis for equation (A2) is 0.48am or 0.91 kpc; the corresponding geometric mean of the eﬀective semi-axes (1 : 0.6 : 0.4) is re = 0.57 kpc. Based on these estimates, we shall adopt re = 0.7 ± 0.2 kpc. The much larger estimate re = 2.7 kpc given by Merritt & Ferrarese (2001a) is based on a table in Gilmore, King, & van der Kruit (1990), which in turn appears to be based on the galaxy model of Bahcall & Soneira (1980), which in turn is based on comments by G. de Vaucouleurs in the 1970s that re is about one-third of the distance of the Sun from the Galactic Center.

– 34 – The next task is to estimate the velocity dispersion as a function of radius. We are 2 1/2 interested in the rms line-of-sight velocity hvlos i measured relative to the Local Standard of Rest, since this is the closest analog to the dispersions used in the M• –σ relation for external galaxies. This quantity diﬀers from the usual dispersion quoted in bulge studies, which is relative to the local mean velocity, σ = h(vlos − v)2 i1/2 , where v = hvlos i. When 2 i = v 2 + σ 2 . We use the following sources: papers quote values for σ and v we set hvlos 1. Due to the interest in the black hole in our Galaxy, the kinematics in the central few parsecs have been investigated much more thoroughly than the kinematics at larger radii (Genzel et al. 2000, especially their Figure 16). The entries at radii < 5 pc in Table 2 are taken from Genzel et al.’s Table 4; at these radii corrections for rotation are negligible. 2. OH/IR stars are mass-losing asymptotic giant branch stars, which are detected by hydroxyl maser emission from their circumstellar envelopes. They are old enough to represent a phase-mixed population and are unaﬀected by obscuration, and hence should be good tracers of the kinematics of the bulge. The survey by Lindqvist et al. (1992a,b) lists 133 OH/IR stars within 1◦ or 140 pc of the Galactic Center. We have divided these into three equal groups by projected distance from the center and computed the dispersion for each group. One limitation of this survey is that its radialvelocity coverage was relatively small, |vlos | ≤ 217 km s−1 , so that high-velocity OH/IR stars might have been missed. We have corrected for this cutoﬀ, assuming that the distribution of line-of-sight velocities is Gaussian, in the two bins where the correction is less than 10%, and have discarded the third bin. At larger distances, Sevenster et al. (1997) have located 307 OH/IR stars in the region |ℓ| < 10◦ , |b| < 3◦ . The minimum velocity range in this survey was −330 km s−1 < vlos < 402 km s−1 so velocity selection eﬀects are negligible. We have discarded all sources not having a standard doublepeaked proﬁle and all sources with expansion velocity > 17 km s−1 , which appear to represent a younger, more rapidly rotating population (Winnberg et al. 1998). The remaining 208 stars were divided into ﬁve equal groups by projected distance, and the mean projected distance and dispersion were computed for each group. 3. Beaulieu, Dopita, & Freeman (1999) have conducted an Hα survey for new planetary nebulae and remeasured the velocities of many known planetary nebulae. Their databases contain 183 planetary nebulae within 10◦ of the Galactic Center. We have divided these into four equal groups by projected distance, and the mean projected distance and dispersion were computed for each group. Beaulieu et al. estimate that their velocity errors are ±11 km s−1 , which is negligible.

– 35 –

Table 2. Velocity dispersion measurements in the inner bulge (r < 1 kpc) radius (pc) 0.085 0.33 0.34 0.39 0.67 0.78 1.2 3.9 15.3 38.5 117 160 171 288 299 314 527 562 612 789 851 989 1220 1284

2 1/2 hvlos i

Source

195 ± 34 164 ± 74 102 ± 8 99 ± 10 72 ± 5 85 ± 15 68 ± 13 54 ± 6 70 ± 7 101 ± 11 126 ± 14 156 ± 18 128 ± 14 129 ± 14 148 ± 19 130 ± 14 101 ± 11 110 ± 10 117 ± 12 88 ± 9 102 ± 12 100 ± 10 89 ± 9 79 ± 8

Genzel et al. (2000) ” ” ” ” ” ” ” ” ” ” ” ” ” ” Lindqvist et al. (1992a) ” ” Sevenster et al. (1997) Blum et al. (1995) ” ” ” ” ” ” Sevenster et al. (1997) ” ” Terndrup, Sadler, & Rich (1995) Beaulieu, Dopita, & Freeman (1999) ” ” Sevenster et al. (1997) Beaulieu, Dopita, & Freeman (1999) ” ” Sevenster et al. (1997)

– 36 – 4. Blum et al. (1994, 1995) have measured the dispersion of samples of M giants in four ﬁelds between 160 and 300 pc from the Galactic Center. Terndrup, Sadler, & Rich (1995) have measured the dispersion of K giants in Baade’s window (0.56 kpc from the Galactic Center). We include only stars with V > 16.0, which they believe restricts the sample to bulge stars and eliminates the foreground disk. The data at r > 0.1 pc shown in Figure 9 and Table 2 have been ﬁt to the functional form (r/r0 )α 2 1/2 + d(r/r0)−1/2 . (A3) hvlos i =c 1 + b(r/r0 )β For r0 = 500 pc the best-ﬁt values are c = 633 km s−1 , α = 0.67, β = 1.14, b = 4.64, d = 2.52. The general features of this curve—a minimum in the dispersion near 5 pc and a maximum of ∼ 130 km s−1 at a few hundred pc—are not new (Kent 1992). A possible concern is that the bulge is ﬂattened, with an axis ratio of about 0.5, so the dispersion at a given radius may depend on the angle between the radius vector and the Galactic plane. To address this concern, we have divided the data points from outside 4 pc from the Galactic Center into those biased towards the minor axis, plotted with ﬁlled symbols (the criterion is h|ℓ|i > h|b|i, where ℓ and b are the Galactic longitude and latitude; these are objects such as planetary nebulae and late-type giants that are found optically), and those biased towards the major axis (the OH/IR stars, found in surveys along the Galactic plane), which are plotted with open symbols. There is no obvious systematic diﬀerence between the dispersion curves deﬁned by the ﬁlled and open symbols. We employ the ﬁt (A3) at the end of §4.3 to estimate the appropriate Milky Way dispersion to use in the M• –σ relation.

– 37 –

Fig. 9.— The rms line-of-sight velocity in the bulge of the Milky Way, as a function of radius. PN = planetary nebulae (Beaulieu, Dopita, & Freeman 1999); OH/IR = OH/IR stars (Lindqvist et al. 1992a; Lindqvist, Habing, & Winnberg 1992b; Sevenster et al. 1997); BW = giant stars in Baade’s window (Terndrup, Sadler, & Rich 1995); K,M = giant stars (Blum et al. 1994, 1995); GC = stars near the Galactic Center (Genzel et al. 2000). Filled symbols denote observations biased toward the Galactic plane, and open symbols denote observations biased away from the plane. The curve is the ﬁtting function (A3).

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