THE SLOPE OF THE BLACK HOLE MASS VERSUS VELOCITY DISPERSION CORRELATION
Scott Tremaine,
1
Karl Gebhardt,
2
Ralf Bender,
3
Gary Bower,
4
Alan Dressler,
5
S. M. Faber,
6
Alexei V. Filippenko,
7
Richard Green,
8
Carl Grillmair,
9
Luis C. Ho,
5
John Kormendy,
2
Tod R. Lauer,
8
John Magorrian,
10
Jason Pinkney,
11
and Douglas Richstone
11
Received 2002 January 17; accepted 2002 April 3
ABSTRACT
Observations of nearby galaxies reveal a strong correlation between the mass of the central dark object
M
BH
and the velocity dispersion of the host galaxy, of the form logðM
BH
=M
Þ¼ þ logð=
0
Þ; how-
ever, published estimates of the slope span a wide range (3.75–5.3). Merritt & Ferrarese have argued that
low slopes (d4) 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 because of sys-
tematic 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 r
e
=8(r
e
is the effective radius) using an
empirical formula. Another component may arise from dispersion-dependent systematic errors in the mea-
surements. A new determination of the slope using 31 galaxies yields ¼ 4:02 0:32, ¼ 8:13 0:06 for
0
¼ 200 km s
1
. The M
BH
- relation has an intrinsic disper sion in log M
BH
that is no larger than 0.25–0.3
dex and may be smaller if observational errors have been underestimated. In an appendix, we present a sim ple
kinematic model for the velocity-dispersion profile of the Galactic bulge.
Subject headings: black hole physics — galaxies: bulges — galaxies: fundamental parameters —
galaxies: nuclei — Galaxy: bulg e — Galaxy: kinematics and dynamics
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
(Sootan 1982; Fabian & Iwasawa 1999; Yu & Tremaine
2002). There appears to be a strong correlation between the
mass M
BH
of the black hole and the velocity dispersion of
the host galaxy, of the form
12
logðM
BH
=M
Þ¼ þ logð=
0
Þ ; ð1Þ
where
0
is some reference value (here chosen to be
0
¼ 200 km s
1
). The first published estimates of the slope
,5:27 0:40 (L. Ferrarese & D. Merritt 2000, astro-ph/
0006053 v1) and 3:75 0:3 (Gebhardt et al. 2000a), differed
by 3 standard deviations. Subsequently, Ferrarese & Merritt
(hereafter FM) revised their slope downward, to 4:8 0:5
(Ferrarese & Merritt 2000), 4 :72 0 :36 (Merritt & Ferrar-
ese 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 monotoni-
cally with time and is now only 1.4 standard deviations, it is
still worthwhile to understand the reasons behind it. In par-
ticular, the slope is the most important point of comparison
to theoretical models that attempt to explain the M
BH
-
relation (Adams, Graff, & Richstone 2001; Burkert & Silk
2001; Haehnelt & Kauffmann 2000; Ostriker 2000).
This paper has three main goals. (1) In xx 2–4 we explore
the reasons for the wide range in estimated slopes of the
M
BH
- relation. In x 2 we focus on the statistical techniques
used to estimate slopes by the two groups; we 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 x 3 we describe the
data sets used by the two groups. In x 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 dif-
ferences in sample selection, and show that none of these is
viable. We argue instead that the slope range reflects system-
atic differences in the velocity dispersion measurements used
by the two groups. (2) In x 5 we present a new analysis of the
M
BH
- relation using recent data. (3) Finally, in the Appen-
dix we model the velocity-dispersion profile of the Milky
Way bulge, which helps to fix the low-mass end of the
M
BH
- relation.
1
Princeton University Observatory, Peyton Hall, Princeton, NJ 08544;
tremaine@astro.princeton.edu.
2
Department of Astronomy, University of Texas, RLM 15.308, Austin,
TX 78712; gebhardt@astro.as.utexas.edu, kormendy@astro.as.utexas.edu.
3
Universita
¨
ts-Sternwarte, Scheinerstrasse 1, Munich 81679, Germany;
bender@usm.uni-muenchen.de.
4
Computer Sciences Corporation, Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21218; bower@stsci.edu.
5
Observatories of the Carnegie Institution of Washington, 813 Santa
Barbara Street, Pasadena, CA 91101; dressler@ociw.edu, lho@ociw.edu.
6
UCO/Lick Observatories, University of California, Santa Cruz,
CA 95064; faber@ucolick.org.
7
Department of Astronomy, University of California, Berkeley,
CA 94720-3411; alex@astro.berkeley.edu.
8
National Optical Astronomy Observatory, P.O. Box 26732, Tucson,
AZ 85726; green@noao.edu, lauer@noao.edu.
9
SIRTF Science Center, MS 220-6, 1200 East California
Boulevard, Pasadena, CA 91125; carl@ipac.caltech.edu.
10
Department of Physics, University of Durham, Rochester Building,
Science Laboratories, South Road, Durham DH1 3LE, UK;
john.magorrian@durham.ac.uk.
11
Department of Astronomy, Dennison Building, University of
Michigan, Ann Arbor, MI 48109; jpinkney@astro.lsa.umich.edu,
dor@astro.lsa.umich.edu.
12
All logarithms in this paper are base 10.
The Astrophysical Journal, 574:740–753, 2002 August 1
# 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.
740
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 ¼ þ x ; ð2Þ
where y ¼ logðM
BH
=M
Þ, x ¼ logð=
0
Þ. We assume that
the measurement errors are symmetric in x and y with rms
values
xi
and
yi
for galaxy i. The goal is to estimate the
best-fit values of and 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 disadvan-
tages. In subsequent sections we usually give results for both
estimators; we find that the differences are significant but
not large enough to explain the slope range.
The Nukers’ estimate is based on minimizing
2
X
N
i¼1
ðy
i
x
i
Þ
2
2
yi
þ
2
2
xi
ð3Þ
(e.g., Press et al. 1992, whose procedures we use). The ‘‘ 1 ’’
uncertainties in an d are given by the maximum range of
and for which
2
2
min
1. An attractive feature of
this approach is that the variables x and y are treated sym-
metrically; in other words, if we set
~
¼ 1=,
~
¼=,
equation (3) can be rewritten in the form
2
X
N
i¼1
ðx
i
~
~
y
i
Þ
2
2
xi
þ
~
2
2
yi
; ð4Þ
which has the same form as equation (3) if x $ y, $
~
,
and $
~
. This symmetry 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 mech-
anism that links black hole mass to dispersion. We call esti-
mators of this kind ‘‘
2
estimators ’’ and denote them by
,
.
One limitation is that this approach does not account for
any intrinsic dispersion in the M
BH
- relation (i.e., disper-
sion due to the galaxies themselves rather than to measure-
ment errors). Thus, for example, one or two very precise
measurements with small values of
xi
and
yi
can dominate
2
, even though the large weight given to these observations
is unrealisti c if the intrinsic dispersion is larger than the
measurement errors. There are two heuristic approaches
that address this concern. (1) Simply set
yi
y
¼ constant,
corresponding to the same fractional uncertainty in all the
black hole mass estimates. The value of
y
is adjusted so that
the value of
2
per degree of freedom is equal to its expecta-
tion value of unity. This approach was adopted by Gebhardt
et al. (2000a). (2) Replace
yi
by ð
2
yi
þ
2
0
Þ
1=2
, where the
unknown constant
0
, which represents the intrinsic dis-
persion, is adjusted so that the value of
2
per degree
of freedom is unity. The second procedure is preferable
if and only if the individual error estimates
yi
are reli-
able. We use both approaches in x 5.2.
FM use the estimator
AB
¼
P
N
i¼1
ðy
i
hyiÞðx
i
hxiÞ
P
N
i¼1
ðx
i
hxiÞ
2
P
N
i¼1
2
xi
;
AB
¼hyi
AB
hxi ;
ð5Þ
here hxiN
1
P
N
i¼1
x
i
and hyiN
1
P
N
i¼1
y
i
are the sam-
ple means of the two variables. This estimator is described
by Akritas & Bershady (1996), who also provide formulae
for the uncertainties in and . The Akritas-Bershady
(hereafter AB) estimator accounts for measurement uncer-
tainties in both variables and is asymptotically normal and
consistent. When
xi
¼ 0 and
yi
y
¼ constant, the AB
and
2
estimators give the same estimates for and (but
not their uncertainties).
Despite its merits, the AB estimator has several unsettling
properties. (1) The measuremen t errors in velocity disper-
sion,
xi
, enter equation (5) only through the sum
P
2
xi
.
Thus, for example, a single low-precision measurement can
dominate both
P
2
xi
and
P
N
i¼1
ðx
i
hxiÞ
2
, rendering the
estimator useless, no matter how many high-precision mea-
surements are in the sample. (2) The errors in the black hole
mass determinations
yi
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. (3) We have argued
above that the variables x and y should be treated symmetri-
cally, but this is not the case in equation (5). (4) Even if the
variables x
i
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
AB
will
be very large. These occasional large excursions are frequent
enough that the variance of
AB
in a population of galaxy
samples is infinite, no matter how large the number N of
data points may be. (5) Figure 1 shows the distribution of
estimates of
(solid line)and
AB
(dashed line) obtained
from 100,000 Monte Carlo trials drawn from a population
that has ¼ 4:5 and other parameters similar to the sample
FM1, defined below (for details see figure legend). The dis-
tribution of
is substantially narrower than
AB
(note that
values of either estimator outside the range of the histogram
are plotted in the outermost bins). The estimator
has a
mean of 4.52 and a standard deviation of 0.36. The distribu-
tion of
AB
has a mean of 4.69, and as stated above, the
standard deviation of this mean is infinite. Thus, in this
example at least,
AB
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 devi-
ations
a
and
b
, respectively, and the major axis having
slope tan .If
b
were zero, all of the points would lie
exactly on a line of slope ; thus
b
characterizes the intrin-
sic dispersion in the correlation between x and y. Let us also
assume that the measurement errors are Gaussian, with
standard deviations
x
and
y
that are the same for all gal-
axies. The observed distribution of x and y, which is
obtained by convolving the intrinsic Gaussian with the mea-
surement errors, is still Gaussian. The shape of this Gauss-
ian is fully described by the three independent components
of the symmetric 2 2 dispersion tensor
ij
hðx
i
hx
i
iÞðx
j
hx
j
iÞi ; i ¼ 1; 2; j ¼ 1; 2 ;
ð6Þ
where ðx
1
; x
2
Þðx; yÞ and hi denotes a sample average.
In this idealized but plausible model, at most three of the
five parameters
x
,
y
,
a
,
b
, and can be determined from
the data, no matter how many galaxies we observe. For
BLACK HOLE MASS–DISPERSION CORRELATION 741
example, if
x
and
y
are known, the other parameters can be
estimated using the formulae
tan 2 ¼
2
xy
xx
yy
þ
2
y
2
x
;
2
b
¼
xx
2
x
xy
cot ¼
yy
2
y
xy
tan ;
2
a
¼
xx
2
x
þ
xy
tan ¼
yy
2
y
þ
xy
cot ; ð7Þ
there are two solutions for h differing by =2, and we choose
the solution for which
a
>
b
> 0. These equations, which
we call Gaussian estimators, are related to the AB estimator
(eq. [5]), which in this notation is simply
tan ¼
xy
=ð
xx
2
x
Þ. However, the Gaussian estimators
have the advantages that (1) they are symmetric in x and y
and (2) they account naturally for the possibility that there
is an intrinsic dispersion
b
in the M
BH
- correlation. The
Gaussian estimators can easily be extended to include mea-
surement errors that differ from galaxy to galaxy and to pro-
vide uncertainties in the estimators (e.g., Gull 1989; see
Feigelson & Babu 1992 for a general review of linear regres-
sion procedures), and with these extensions they are likely
to provide a more reliable slope estimator than either the
2
or AB estimators.
We close this section with a general comment on fitting
linear relations such as equation (1). The choice of the refer-
ence value
0
affects the uncertainty in and the covariance
between the estimated values of and . A rough rule of
thumb is that
0
should be chosen near the middle of the
range of values of in the galaxy sample to minimize the
uncertainty in and the correlation between and .Asan
example, Ferrarese & Merritt (2000) use
0
¼ 1kms
1
and
find an uncertainty in of 1.3. However, most of this
uncertainty arises because errors in and are strongly
correlated at this value of
0
(correlation coefficient
r ¼0:998). Simply by choosing
0
¼ 200 km s
1
, the
uncertainty in is reduced by a factor of more than 10, to
0.09.
3. THE DATA
The M
BH
- relation has been explored in the literature
using a number of distinct data sets:
1. Sample FM1.—Much of FM’s analysis is ba sed on a
set of 12 galaxies with ‘‘ secure ’’ black hole mass estimat es
(sample A, Table 1 of Ferrarese & Merritt 200 0). However,
their definition of ‘‘ secure ’’ is not itself secure: in x 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 6. 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 kine-
matics. Unless otherwise indicated, when discussing this
sample we use the upper and lower limits to the dispersion
and black hole mass given by Ferrarese & Merritt (2000).
13
The slope estimators then yield
¼ 4:47 0:44;
AB
¼ 4:81 0:55 : ð8Þ
The minimum
2
per degree of freedom is 0.69, which indi-
cates an acceptable fit; thus, there is no evidence for any
intrinsic dispersion in this sample.
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 mod-
els 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
adopt
y
¼ 0:30, which yields a minimum
2
per degree of
freedom equal to unity. Gebhardt et al. (2000a) take
x
¼ 0,
corresponding to negligible uncertainties in the dispersions;
this approximation is discussed in x 4.1. The slope estima-
tors then yield
¼ 3:74 0:30 ;
AB
¼ 3:74 0:23 : ð9Þ
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) supple-
ment sample FM1 with 10 additional galaxies, mostly taken
from Kormendy & Gebhardt (2001), for a total of 22 gal-
axies. The stated rms fractional uncertainty in the black hole
masses is 0.24 dex. The slope estimators yield
¼ 4:78 0:43 ;
AB
¼ 4:65 0:49 : ð10Þ
The minimum
2
per de gree 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.
13
The error bars in x and y are given by ðlog
upper
log
lower
Þ=2 and
ðlog M
BH; upper
log M
BH; lower
Þ=2, respectively.
Fig. 1.—Distribution of the estimators
(eq. [3]; solid line) and
AB
(eq.
[5]; dashed line) for 100,000 Monte Carlo simulations of a sample with
¼ 4:5 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
x
¼ 0:06,
y
¼ 0:18). Values greater than
5.5 or less than 3.3 are plotted in the outermost bins of the histogram. The
sample means are marked by arrows.
742 TREMAINE ET AL. Vol. 574
By comparing samples FM2 and G2, we can isolate the
effects of different treatments of the same galaxies. W e
assume 20% uncertainty in the dispersion of the Milky Way
and 5% uncertainty in the velocity dispersions of external
galaxies (see xx 4.1 and 4.3). Using G2’s stated uncertainties
in the black hole masses, the slope estimators yield
¼ 3:70 0:20,
AB
¼ 3:61 0:31. The minimum
2
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
2
per degree of freedom to unity and reduces the best-fit slope
to
¼ 3:61 0:29 ;
AB
¼ 3:61 0:31 : ð11Þ
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.
4. WHY ARE THE SLOPES DIFFERENT?
Our goal is to determine why different invest igations yield
such a wide range of slopes. In particular, the two samples
from FM give slopes e4.5 (‘‘ high ’’ slopes) with both the
2
and AB estimators, while the two samples from the Nukers
give slopes d4.0 (‘‘ low ’’ slopes) with both estimators. In
xx 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 x 4.5 we suggest that system-
atic 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 measure-
ment errors in the velocity dispersion can have a significant
effect on the slope of the M
BH
- regression. In particular,
they claim that the Nukers’ assumption of zero measure-
ment error in leads them to underestimate the slope. To
test this claim, we plot in Figure 2 the slope derived from
the G1 sample using both the AB and
2
estimators, as a
function of the assumed rms error
x
in the log of the veloc-
ity dispersion.
For nearly all of the galaxies in sample G1, the data typi-
cally have signal-to-noise ratios around 100, and the formal
uncertainties in the dispersions are around 2%–3%
(
x
¼ 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 disper sion. To
account crudely for these systematic errors, we double the
uncertainties in the dispersions, to 5% (
x
¼ 0:021). The
uncertainty is larger in the Milky Way (see x 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 cor-
rect for the sample FM1, where the rms fractional error in
the dispersions is 14% (
x
¼ 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 dis-
persions is negligible: at the 5% level, the change in for
sample G1 is only 0.03 and 0.04 for the
2
and AB estima-
tors, respectively, and even at the 10% level the correspond-
ing changes are only 0.12 and 0.16.
4.2. Measurement Errors in Black Hole Mass
We next ask whether the combined effects of varying
assumptions about measurement errors in both velocity dis-
persion and black hole mass can explain the discrepancy
between the high and low slopes. As usual, we parameterize
these uncertainties by
x
and
y
, 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 can then be explored using
the Gaussian estimators (eq. [7]). These estimat ors have tw o
advantages over the
2
or AB estimators for this purpose:
(1) the slope estimator depends only on the difference
2
y
2
x
and hence is a function of only one variable and (2)
the condition that the derived intrinsic dispersion
2
b
be posi-
tive-definite provides an upper limit to the allowable errors.
The left and right panels of Figure 3 show the slope an d
the maximum allowed value of
x
for each of the galaxy
samples in x 3. For each sample there is a minimum slope
and a maximum value of
2
y
2
x
, beyond which the intrinsic
dispersion
2
b
is negative. In particular, for sample FM1 the
minimum allowable slope is ¼ 4:39; thus, there are no
assumptions about the measurement errors that can lead to
a slope in the low range. The slope versus 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.
Fig. 2.—Dependence of the slope on the assumed measurement
uncertainty in velocity dispersion for the sample G1. The abscissa is the rms
measurement error in log . Solid and dashed lines show the slopes derived
from the AB and
2
estimators, respectively. The error bars show the com-
puted uncertainty in the slope at zero error. The formal uncertainty in the
dispersion measurements of G1 is
x
’ 0:01; allowing for possible system-
atic errors in the stellar template and continuum subtraction increases
x
to
0.02 (5%).
No. 2, 2002 BLACK HOLE MASS–DISPERSION CORRELATION 743
Consistent slopes would require that ð
2
y
2
x
Þ
Nuker
’ð
2
y
2
x
Þ
FM
0:07. This relation, combined with the con-
straint
2
b
> 0, cannot be satisfied with any plausible combi-
nation of measurement errors—note in particular that
y
should be similar for the two groups since they rely on many
of the same black hole mass determinations, and
x
should
be smaller for the Nuker samples than the FM samples,
since the Nukers employ high signal-to-noise ratio slit spec-
tra 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
¼ 75 km s
1
should be increased to ¼ 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 is quite insensitive to the Milky Way dis-
persion used in the G1 sample: increasing the dispersion
from 75 to 100 km s
1
as suggested by Merritt & Ferrarese
(2001a) increases only by 0.13. The corresponding slope
change is substantially larger for sample FM1—0.27 for the
2
estimator and 0.44 for the AB estimator—but this strong
sensitivity reflects the small size of that sample and is not rel-
evant 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
BH
- 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 equa-
tion (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 for-
mal 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 aper-
ture of radius r
e
=8, where r
e
is the effective radius. For
r
e
¼ 0:7 kpc as derived in the Appendix, we find ¼ 95 km
s
1
. Because the bulge is triaxial, we correct the dispersion
that we measur e 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
0
¼ 20
from
the Sun-center line. If the density is stratified on similar
ellipsoids, the ratio r
2
2
ð
0
¼ 20
Þ=h
2
ð
0
Þi depends
only on the axis ratios (Roberts 1962). For the axis ratios
given by Binney et al., r ¼ 1:07. Thus, our best estimate for
the dispersion within r
e
=8is
FM
¼ 90 18 km s
1
; if we use
this instead of FM’s estimate of ¼ 100 20 km s
1
, the
slope derived from sample FM1 is reduced from
AB
¼ 4:81 0:55 to
AB
¼ 4:66 0:42 and for sample
FM2 from
AB
¼ 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
r
e
. This dispersion depends weakly on the slit width, which
we take to be 70 pc (corresponding to 1
00
at Virgo). In this
case we find ¼ 110 km s
1
; reducing this by a factor r to
account for triaxiality, we have ¼ 103 20 km s
1
, close
Fig. 3.—Dependence of the slope on the assumed rms errors in black hole mass and velocity dispersion. The rms errors in log M
BH
and log are
y
and
x
,
respectively (assumed the same for all galaxies). The left panel shows the slope derived from the Gaussian estimator (eq. [7]) for samples FM1 (solid line), FM2
(short-dashed line), G1 (dotted line), and G2 (long-dashed line). The lines stop where the intrinsic dispersion
2
b
< 0. The right panel shows the maximum
allowed value of
x
; for larger values the intrinsic dispersion is negative. The filled circles denote the locations corresponding to the estimated values of
x
and
y
in each survey; in the right panel these are connected by vertical lines to the curves for the corresponding survey.
744 TREMAINE ET AL. Vol. 574
![Fig. 8.—Residuals between the black hole masses and dispersions for the galaxies in Table 1 and the best-fit correlation described by eq. (1) with ¼ 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.](/figures/fig-8-residuals-between-the-black-hole-masses-and-184atjv3.webp)
![Fig. 1.—Distribution of the estimators (eq. [3]; solid line) and AB (eq. [5]; dashed line) for 100,000 Monte Carlo simulations of a sample with ¼ 4:5 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 x ¼ 0:06, y ¼ 0:18). Values greater than 5.5 or less than 3.3 are plotted in the outermost bins of the histogram. The sample means are marked by arrows.](/figures/fig-1-distribution-of-the-estimators-eq-3-solid-line-and-ab-3r7ils4e.webp)