Lecture Thirteen:

Elliptical Galaxies

Sparke & Gallagher, chapter 628th May 2014

http://www.astro.rug.nl/~etolstoy/pog14

1

Galaxy mass functions 949

Figure 4. Bivariate distribution for SB versus mass. The contours representthe volume-corrected number densities from the sample: logarithmicallyspaced with four contours per factor of 10. The lowest dashed contourcorresponds to 10!5 Mpc!3 per 0.5 " 0.5 bin, while the lowest solid contourcorresponds to 5.6 " 10!5 Mpc!3. The grey dashed-line regions representareas of low completeness (70 per cent or lower as estimated by Blantonet al. 2005a). The diamonds with error bars represent the median and 1!

ranges over certain masses with a straight-line fit shown by the middle dottedline. The outer dotted lines represent ±1 ! .

survey (Cross & Driver 2002). From the tests of Blanton et al.(2005a), as shown in fig. 3 of that paper, the SDSS main galaxysample has 70 per cent or greater completeness in the range 18–23 mag arcsec!2 for the effective SB µR50,r .

In order to identify at what point the GSMF becomes incompletebecause of the SB limit, we computed the bivariate distribution inSB versus stellar mass. Fig. 4 shows this distribution represented bysolid and dashed contours (1/Vmax and 1/n LSS corrected). There isa relationship between peak SB and logM, which is approximatelylinear in the range 108.5 to 1011 M#. At lower masses, the distri-bution is clearly affected by the low-SB incompleteness at µR50,r

> 23 mag arcsec!2. Therefore, any GSMF values for lower massesshould be regarded as lower limits if there are no corrections for SBcompleteness.

The other important consideration is the fact that the r-bandselection is not identical to the mass selection required for theGSMF. This is nominally corrected for by 1/Vmax but it should benoted that galaxies with high M/L at a given mass are viewed oversignificantly smaller volumes than those with low M/L. Fig. 5 showsthe bivariate distribution in M/L versus mass. The limits at variousredshifts for the SDSS main galaxy sample are also identified. Forexample, galaxies with M < 108 M# and log (M/Lr ) > 0.1 areonly in the sample at z < 0.008. At these low redshifts, the stellarmass and Vmax depend significantly on the Hubble-flow corrections.However, it does appear that the SB limit affects the completenessof GSMF values at higher masses than the M/L limits. At M <

108.5 M#, the SB limit becomes significant, while atM< 108 M#,the GSMF is affected both by the constrained volume for high M/Lgalaxies and, more severely, by the SB incompleteness.

3.2 Corrected GSMF with lower limits at the faint end

Fig. 6 shows the results of the GSMF determination. The binnedGSMF is represented by points with Poisson error bars, with lower

Figure 5. Bivariate distribution for M/L versus mass. The contours rep-resent the volume-corrected number densities: logarithmically spaced withfour contours per factor of 10. The lowest dashed contour corresponds to10!4 Mpc!3 per 0.5 " 0.2 bin. The dotted lines represent the observablelimits for an r < 17.8 mag limit and different redshift limits (ignoringk-corrections). The grey dashed line region represents galaxies that can onlybe observed at z < 0.008 where Hubble-flow corrections are significant(cz < 2400 km s!1).

Figure 6. GSMF extending down to 107 M# determined from the NYU-VAGC. The points represent the non-parametric GSMF with Poisson errorbars; at M < 108.5 M# the data are shown as lower limits because of theSB incompleteness (Fig. 4). The dashed line represents a double-Schechterfunction extrapolated from a fit to the M> 108 M# data points. The dottedline shows the same type of function with a faint-end slope of "2 = !1.8(fitted to M > 108.5 M# data). The dash–dotted line represents a power-law slope of !2.0. The shaded region shows the range in the GSMF fromvarying the stellar mass used and changing the redshift range.

limits represented by arrows. The GSMF has been corrected forvolume (1/Vmax) and LSS (1/n) effects.9 The masses used were

9 We compared the GSMF computed using 1/n(z) correction for LSS vari-ations with the stepwise maximum-likelihood method (Efstathiou et al.1988). There was good agreement between the two methods after matching

C$ 2008 The Authors. Journal compilation C$ 2008 RAS, MNRAS 388, 945–959

ellipticalsspirals

dwarfs

most of themass is here

most of thegalaxies are

here

2

Elliptical galaxies

Elliptical galaxies look very simple...

• roughly round

• smooth light distribution

• no obvious patches of star formation

• no obvious strong dust lanes (except sometimes...)

the brightest and the faintest galaxies in the Universe...

...but they’re really not! Detailed studies show significant complexity:

• large range of shapes: oblate to triaxial

• large range of luminosities and concentrations

• rotation vs. pressure support

• cuspy vs. cored centers

• complex star formation histories

3

Generically, elliptical galaxies are classified by their luminosities:

Giant L > L*

Intermediate ~3x109 L⊙ < L < L*

Dwarf L < ~3x109 L⊙ or MB > –18 mag

Elliptical galaxies

Unlike for disc galaxies, where Hubble types can cover a large range in luminosity, these classes have physical relevance

Ellipticals (and most S0 galaxies) are predominantly a one-parameter family: an elliptical galaxy’s properties depend primarily on its mass

4

Elliptical galaxies have very smooth profiles over 2 orders of magnitude in radius, usually falling off as R1/4

Light in ellipticals is highly concentrated

Surface brightness profilesSurface brightness profiles

Surface brightness profile � for giant elliptical galaxy: as f(R) and f(R1/4).

Very good fit over 2 decades in radius

The surface brightness falls 9 magnitudes from centre to outskirts: 109 fall-off in projected luminosity!

The light in elliptical galaxies is quite centrally concentrated

NGC1700

5

Sources of ErrorSky Background Subtraction –if the sky is over or under subtracted this can have a dramatic effect on the shape of the profile.

Sources include: light pollution from nearby cities, photochemical reactions in the Earth’s upper atmosphere, the zodiacal light, unresolved stars in the Milky Way, and unresolved galaxies.

Seeing effects – unresolved points are spread out due to effects of our atmosphere, quantified by the Point Spread Function (PSF) FWHM (σ) on the images

-makes central part of profile flatter-makes isophote rounder

6

de Vaucouleurs Profile

In specifying the radius of a galaxy it is necessary to define the surface brightness of the isophote being used to determine that radius.

Effective radius, re, is the projected radius within which one-half of the galaxies light is emitted. This means that the surface brightness level at re (µe) depends on the distribution of surface brightness with radius.For large ellipticals, the surface brightness distribution typically follows an r1/4 law (de Vaucouleurs profile):

mag arcsec-2

L! pc-2

Or, in physical units:

7

Some properties of the de Vaucouleurs law

The central surface brightness is I0 = 103.33Ie � 2000Ie

...and the definition that half of the light is emitted within Re we can deduce the following:

7.22�R2eIeThe total light of a galaxy that follows a de Vaucouleur profile is

Ie is surface brightness at Re

I(R) = Ie � 10�3.33[(R/Re)1/4�1]Starting from

�I�e = 3.61Ieand the mean surface brightness contained within Re is

8

Sérsic ProfileGeneralised version of r1/4 law is frequently used in which 1/4 is replaced by 1/n. Often called Sérsic (1968) models have been shown (Caon et al 1993) to be an even better fit to E’s, though it increases the number of free parameters:

Where µe re and n are all free parameters used to obtain the best possible fit to the actual surface brightness profile.

mag arcsec-2

9

This formalism can be used to discriminate between disk-dominated and bulge-dominated galaxies. Especially suited to automated analyses of large samples.

Distribution of Sersic-n for 10 095 galaxies selected from the Millennium Galaxy Catalogue (Driver et al. 2006, MNRAS, 368, 414)

ellipticals and bulges of spirals

disc galaxies

Sérsic Profile

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Elliptical galaxies in the centers of clusters with extended, “too-bright” profiles are called cD galaxies

cDs are amongst the brightest of all galaxies

1986ApJS...60..603S Outskirts

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CentresObservations from space suffer no seeing because there is no atmosphere!

Don’t think that space observations have a δ-function PSF, though — the optics themselves impose a PSF...

Ellipticals have either a

core: giant ellipticals

cusp: intermediate ellipticals (note that there is still a break radius)

core

cusp

12

"R1/4 and Sersic fits tend to fail in the inner regions of Elliptical "Regions of special interest because they host supermassive black holes"Need HST since largest E’s lie far away and seeing effects degrade profile centers

"Core-less also reveal “extra light” which may be result of nuclear starburst resulting from “wet mergers” (with gas)

Centres of Elliptical Galaxies

"More luminous E’s (Mv<-21.7) tend to have cores, where the SB profile flattens towards center

"Midsize E’s (-21.5<Mv<-15.5 with L<2x1010L!) are typically core-less systems where the SB profile steeply rises to centre

"Cores could be the result of mergers so central nucleus is more diffuse – caused by binary BHs scouring out centers in “dry mergers” (no gas)

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Core

Coreless

Brig

hter

cen

tral s

urfa

ce b

right

ness

#

$ Brighter total galaxy light

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Isophotal analysis of elliptical galaxiesElliptical galaxies have (obviously) elliptical isophotes with ellipticities

where a,b are the semi-major and -minor axes of the ellipse

In the traditional Hubble sequence, ellipticals are classified by the apparent ellipticity: En, where

e = 1� (b/a)

n = 10[1� (b/a)] = 10e

Isophotes are rarely perfect ellipses

Excess of light in the “corners” of the ellipse: boxy

Excess of light along the principal axes: disky

disky

elliptical

twisted

boxy

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Shapes of elliptical galaxies

Long axis

Short axis

Intermediate axis

xy

z

The contours of constant density are ellipsoids with m2=constant

α≠β≠γ: triaxial

α=γ<β: prolate (cigar-shaped)

α=γ>β: oblate (pancake-shaped)

What, can we learn about the intrinsic shapes of elliptical galaxies from the observed distribution of their axis ratios?

In the most general case, the luminosity density ρ(x) can be expressed as ρ(m2), where

m2 =x2

�2+

y2

⇤2+

z2

⇥2

16

Family of ellipsoids

Oblate Prolate Triaxial

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Deviation from elliptical isophotes: a4

The diskiness/boxiness of an isophote is measured by the difference between the real isophote and the best-fit ellipse:

If the isophotes have 4-fold symmetry (typical), then terms with n<4 and all bn should be small, and a4 gives the shape:

a4<0: boxy

a4>0: disky

Deviations from ellipsesIsophotes are not perfect ellipses:

4Excess of light on the major axis: disky4Excess on the <corners= of the ellipse: boxy.

The diskiness/boxiness of an isophote is measured by the difference between the real isophote and the best-fit ellipse:

�(�) = <� > + � an cos n� + � bn sin n�

4If isophotes have 4-fold symmetry: terms with n < 4 and all bn should be small. The value of a4 tells us the shape:

4a4 > 0 : disky E 4a4 < 0 : boxy E

Deviations from ellipsesIsophotes are not perfect ellipses:

4Excess of light on the major axis: disky4Excess on the <corners= of the ellipse: boxy.

The diskiness/boxiness of an isophote is measured by the difference between the real isophote and the best-fit ellipse:

�(�) = <� > + � an cos n� + � bn sin n�

4If isophotes have 4-fold symmetry: terms with n < 4 and all bn should be small. The value of a4 tells us the shape:

4a4 > 0 : disky E 4a4 < 0 : boxy E

disky

boxy

�(⇥) = ��� +�

n

(an cos n⇥ + bn sinn⇥)

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Isophote twistingIf the intrinsic shape of a galaxy is triaxial — that is, all three principal axes have different lengths — then the orientations of the projected ellipses depend on the inclination of the galaxy to the line of sight and the galaxy’s true axis ratios

Isophote twistingIf intrinsic shape of a galaxy is triaxial, the orientation of the projected ellipses depends on:8 the inclination of the body8 the body:s true axis ratio.

See figure below:

Since the ellipticity changes with radius, even if the major axis of all the ellipses have the same orientation, they appear as if they were rotated in the projected image.

This is called isophote twisting.

It is not possible, from an observation of a twisted set of isophotes to conclude whether there is a real twist, or whether the object is triaxial.

Observer

True

Because the ellipticity changes with radius, even if the major axis of all the ellipses have the same true orientation, they will appear as if they were rotated in the projected image

This is isophote twisting

Isophote twisting is generally taking to imply triaxiality, but it is impossible to distinguish a real twist from true triaxiality from images alone...

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NGC 205, one of the dwarf elliptical companions of M31, has strongly twisted isophotes

compare upper shallow image to lower deep image

likely due to tidal effects in this case!

Twisted isophotes in a satellite galaxy of Andromeda (M31).

The shallower exposure shows the brightest part of the galaxy.

The deeper exposure shows the weaker more extended emission.

A twist between both images of the same galaxy are apparent (the orientation in the sky is the same in both figures).

Isophote twisting

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In general,

Boxy galaxies are

• more luminous

• more likely to show isophote twists

• probably triaxial

Disky galaxies are

• intermediate ellipticals

• often oblate

• faster rotators

Boxy vs. Disky

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Shells - seen at faint levels around most E’s- Origin could be merger remnants or captured satellites- prominent shells goes with evidence for some young stars in the galaxy

Dust - visible dust clouds seen in many nearby E’s (maybe 50% of E’s have some - but not much – dust)

Shells in Cen A

…and dust

Are Ellipticals really so smooth?

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~10-20% of ellipticals show distinct “edges” in their surface brightness profiles, known as shells

Probably the result of the accretion (or merger) of a small galaxy that was originally on a nearly radial orbit

Fine structure

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Kinematics of elliptical galaxiesHow do we measure velocities in elliptical galaxies?

NGC 4873: data

Template

Smoothed, shifted template

Use the absorption lines in the spectrum, which is the composite spectrum of all of the stars in the galaxy

Each star emits a spectrum which is then Doppler-shifted in wavelength according to its motion, which widens the absorption lines.

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Rotation of Ellipticals

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Rotation of Ellipticals

Low luminosity Ellipticals and bulges rotate rapidly and have nearly isotropic velocity dispersions and are flattened by rotation

Bright Ellipticals rotate slowly and are pressure supported and owe their shapes to velocity anisotropy.

This could be explained by proto-ellipticals acquiring angular momentum through tidal torques and then if mergers produce brighter ellipticals the rotation gets scrambled in the process.

Solid lines: amount of rotation necessary to account for observed ellipticity of galaxy relative to σ of stars.

Ratio of rotation to dispersion plotted against projected ellipticity, for various galaxies. Note that many Ellipticals have very little rotation, even though they are quite flattened.

Most Ellipticals rotate too slowly for centrifugal forces to be the causes of their observed flattening.

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Because the relaxation times of elliptical galaxies are much longer than a Hubble time, these galaxies “remember” their formation.

Scaling relations tell us something about the formation process(es)

We’ll start with scaling relations between structural parameters, like luminosity, velocity dispersion, and surface brightness...

Scaling relations of elliptical galaxies

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One of the earliest known structural scaling relations was the Faber-Jackson relation:

1976ApJ...204..668F

L � �4

=log

σ

Scaling relations of elliptical galaxies

strong correlation between luminosity (L) and central velocity dispersion (σ)

A natural consequence of the virial theorem.

Plenty of scatter, and the slope of the relation is different than the virial theorem.

This was assumed to indicate a missing parameter… and there was originally a lot of debate about what was the missing parameter..

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Multi-parameter correlations: the Fundamental Plane

A break through in our understanding of scaling laws came from large homogeneous data sets (from CCDs & long-slit spectroscopy), and the application of statistical tools.

plane - means correlation in a 3D space and we see this space with three parameters that “see” the correlation from different angles.

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Projections of Fundamental Plane of Ellipticals & Bulges

Rad

ius

Mean Surface Brightness

Lum

inos

ity

Velocity Dispersion

Faber-Jackson

Mea

n S

urfa

ce B

right

ness

Velocity Dispersion

plane face-on

Observed “Cooling Diagram” from galaxy formation theory (virial temprature vs. density)

Rad

ius

Combined Velocity Dispersion & surface brightness

Plane edge-on

Djorgovski & Davis 1987

Radius derived from surface brightness profiles, e.g., core or effective radius.

Tilt to plane leads to scatter.

physical meaning of these measurements links to formation theory!

position of galaxy in this diagram relates to amount of dissipation during its formation.

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Ellipticals: Fundamental Plane

Radius mean Surface BrightnessVelocity Dispersion

R / �1.24hIi�0.82e

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Where does the Fundamental Plane of elliptical galaxies come from?

The Fundamental Plane

M = V 2R/GAssume the Virial Theorem holds, then the mass is

I = �

�M

L

��1

The surface brightness is this divided by the mass-to-light ratio:

� =M

⇥R2� V 2

RDivide this by the area to get the mass surface density:

combining this: I � V 2

R(M/L)Rewrite this in terms of the radius Re=R and identifying the velocity V as the velocity dispersion σ, we have

Re ��

M

L

��1

�2�I��1e

But note the observed coefficients aren't 2 & -1 (they are 1.25 and -0.8) and so M

L� L1/4

R / �1.24hIi�0.82e

observed relation:

32

That is, the mass-to-light ratios of elliptical increase as they become more luminous (or more massive)

...if (and only if) the structure of ellipticals doesn’t change as a function of luminosity (or mass)

Why is this?

Possibility 1: more dark matter in more massive galaxies

Possibility 2: bigger galaxies are older

Possibility 3: structure changes with size

The Fundamental Plane

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Stellar populations of elliptical galaxies

The spectra of elliptical galaxies look, to first order, like G or K stars (with a few features of M stars)

This implies that they must be, on average, older than a few Gyr in order not to have light from hot stars

34

Some Elliptical galaxies contain HI, but nearly all contain a significant amount of hot X-ray gas

~10–20% of the baryonic mass is in the form of gas at 106–7 K in a halo ≥30 kpc in radius

Gas in elliptical galaxies

35

Dark matter in elliptical galaxies

X-rays — hydrostatic equilibrium of hot gas gives a mass if the temperature and density structure are known: M/L~100 M⊙/L⊙ for r~100 kpc

Careful modeling of kinematical data shows that many ellipticals have flat rotation curves

0.5 1 1.5 2 2.5

400

500

600

N 315

N 4472N 4486

N 1399

N 4374N 7626

N 7507

300

400

N 5846

N 4278

N 4589

N 4636 N 2434

N 3193

N 3640

0.5 1 1.5 2 2.5

200

300N 4168

N 3379

N 6703N 7145

N 7192

N 4486B

N 4494

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 1 1.5 2 2.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

DYNAMICS OF GIANT ELLIPTICAL GALAXIES 1937

pro!les begin to rise at around and are consistent0.5È2Rewith X-ray and other data where available, although from

the kinematic data alone constant M/L models can only beruled out at 95% con!dence in a few galaxies.

This sample provides a new and much improved basis forinvestigating the dynamical family properties of ellipticalgalaxies, which is the subject of the present study. In ° 2, weanalyze the unexpectedly uniform dynamical structure ofthese elliptical galaxies. In ° 3, we investigate the depen-dence on luminosity, discussing the Faber-Jackson, Tully-Fisher, and fundamental plane relations. In ° 4, we relate thedynamical mass-to-light ratios to the stellar populationproperties. In ° 5 we discuss the structure of the dark halosof these ellipticals. Our conclusions are summarized in ° 6.

2. DYNAMICAL STRUCTURE

The elliptical galaxies analyzed by K]2000 divide in twosubsamples, one with new extended kinematic data, reach-ing typically to (““ EK sample,ÏÏ the data are fromD2R

eKronawitter et al. and from several other sources referencedthere), and one with older and less extended kinematic mea-surements (““ BSG sample ÏÏ ; this is a subsample fromBender, Saglia & Gerhard 1994). Based on these data andmostly published photometry, K]2000 constructed non-parametric spherical models from which circular velocitycurves, radial pro!les of mass-to-light ratio, and anisotropypro!les for these galaxies were derived, including con!denceranges.

The galaxies were selected to rotate slowly if at all and tobe as round as possible on the sky. They are luminouselliptical galaxies The expected mean(M

B^ [21 ^ 2).1

intrinsic short-to-long axis ratio for such a sample of lumi-nous ellipticals is Sc/aT \ 0.79. The mean systematic e†ectsarising from the use of spherical models and the possiblepresence of small embedded face-on disks are small for thesample as a whole, but may be non-negligible in individualcases (see K]2000, ° 5.1).

2.1. Circular Velocity CurvesCircular velocity curves (CVCs) for all galaxies in the

sample are shown in Figure 1, in three bins roughly orderedby luminosity. CVCs normalized by the respectivemaximum circular velocity are shown in Figure 2 separatelyfor the two subsamples. The plotted curves correspond tothe ““ best ÏÏ models of K]2000, which are taken from thecentral region of their 95% con!dence interval for eachgalaxy, respectively. Based on dynamical models near theboundaries of the con!dence interval, the typical uncer-tainty in the outermost circular velocity is ^10%È15%.The expected mean systematic error from Ñattening alongthe line of sight is smaller ; cf. ° 5.1 of K]2000.

The most striking result from these diagrams is that atthe ^10% level all CVCs are Ñat outside ThisR/R

e^ 0.2.

result is most signi!cant for the galaxies with the extendeddata, while for many galaxies from the BSG sample theradial extent of the data is insufficient to show clear trends.However, in cases where X-ray data are available (NGC4472, 4486, 4636) the mass pro!les of the ““ best ÏÏ modelsapproximately match those from the X-ray analysis even forthose galaxies (see K]2000).

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ1 Throughout this paper we use a Hubble constant km s~1H0 \ 65

Mpc~1 unless explicitly noted otherwise.

FIG. 1.ÈThe ““ best model ÏÏ circular velocity curves of all galaxies fromthe K]2000 sample plotted as a function of radius scaled by the e†ectiveradius The panels are roughly ordered by luminosity.R

e.

This result is illustrated further by Figure 3, which showsthe derived ratio for all galaxies of the EK-v

c(Rmax)/vc

maxsample. Here is the circular velocity at the radius ofv

c(Rmax)the last kinematic data point, and is the maximumv

cmax

circular velocity in the respective ““ best ÏÏ model. For NGC315 was used instead of The error barsv

c(0.6R

e) v

cmax.

plotted correspond to the 95% con!dence range forcompared to which the uncertainty in can bev

c(Rmax), v

cmax

FIG. 2.ÈSame circular velocity curves, normalized by the maximumcircular velocity. The upper panel now shows the galaxies from the EKsubsample of K]2000, the lower panel those from the BSG subsample.The extended curve in the lower panel is for the compact elliptical NGC4486B.

36

If a super-massive black hole (SMBH) lives at the center of a galaxy, we should be able to detect this by looking at the speeds of stars that pass near to the black hole

So we need to find stars that have speeds of

Black holes in elliptical galaxies

2002MNRAS.335..517V

V 2(r) � GMBH

r� �2

c

This means we need to look within a radius

rBH � 45 pc�MBH

108 M�

� � �c

100 km s�1

��2

In M32, 2x106 M⊙ are required inside the central parsec!

37

It is likely that every elliptical galaxy — in fact, every spheroidal system, including bulges — has a super massive black hole (SMBH)

Moreover, there is a reasonable correlation between the mass of the SMBH and the velocity dispersion of the spheroid:

204 GULTEKIN ET AL. Vol. 698

Figure 1. M–! relation for galaxies with dynamical measurements. The symbol indicates the method of BH mass measurement: stellar dynamical (pentagrams), gasdynamical (circles), masers (asterisks). Arrows indicate 3!68 upper limits to BH mass. If the 3!68 limit is not available, we plot it at three times the 1!68 or at 1.5 timesthe 2!68 limits. For clarity, we only plot error boxes for upper limits that are close to or below the best-fit relation. The color of the error ellipse indicates the Hubbletype of the host galaxy: elliptical (red), S0 (green), and spiral (blue). The saturation of the colors in the error ellipses or boxes is inversely proportional to the area ofthe ellipse or box. Squares are galaxies that we do not include in our fit. The line is the best fit relation to the full sample: MBH = 108.12 M!(!/200 km s"1)4.24. Themass uncertainty for NGC 4258 has been plotted much larger than its actual value so that it will show on this plot. For clarity, we omit labels of some galaxies incrowded regions.

relation from sample S. The distribution of the residuals appearsconsistent with a normal or Gaussian distribution in logarithmicmass, although the distribution is noisy because of the smallnumbers. For a more direct test of normality we look at log(MBH)in galaxies with !e between 165 and 235 km s"1, correspondingto a range in log(!e/200 km s"1) from approximately "0.075to 0.075. The predicted masses for the 19 galaxies in thisnarrow range differ by at most a factor of 4.3, given ourbest-fit relation. The power of having a large number ofgalaxies in a narrow range in velocity dispersion is evidenthere, as there is no need to assume a value for the slope of

M–! or even that a power-law form is the right model. Theonly assumption required is that the ridge line of any M–!relation that may exist does not change substantially acrossthe range of velocity dispersion. The mean of the logarithmicmass in solar units is 8.16, and the standard deviation is0.45. The expected standard deviation in mass is 0.19, basedon the rms dispersion of log(!e/200 km s"1) (0.046) in thisrange times the M–! slope "; thus the variation in the ridge lineof the M–! relation in this sample is negligible compared tothe intrinsic scatter. We perform an Anderson–Darling test fornormality with unknown center and variance on this sample of

log(MBH/M�) = 4.24 log(�/200 km s�1) + 8.12

Black holes in elliptical galaxies

38

MBH

Mbulge= 2.2+1.6

�0.9 � 10�3

No. 1, 2003 MARCONI & HUNT L23

Fig. 1.—Left: vs. for the galaxies of group 1. The solid lines are obtained with the bisector linear regression algorithm of Akritas & Bershady (1996),M LBH K, bulwhile the dashed lines are ordinary least-squares fits. Middle: vs. with the same notation as in the previous panel. Right: Residuals of vs. , inM M M -j RBH bul BH e e

which we use the regression of T02.M -jBH e

TABLE 2Fit Results ( )log M p a! bXBH

X

Group 1 Galaxies All Galaxies

a b rms a b rms

. . . . . .log L " 10.0B, bul 8.18! 0.08 1.19! 0.12 0.32 8.07! 0.09 1.26! 0.13 0.48

. . . . . .log L " 10.7J, bul 8.26! 0.07 1.14! 0.12 0.33 8.10! 0.10 1.24! 0.15 0.53

. . . . . .log L " 10.8H, bul 8.19! 0.07 1.16! 0.12 0.33 8.04! 0.10 1.25! 0.15 0.52

. . . . . .log L " 10.9K, bul 8.21! 0.07 1.13! 0.12 0.31 8.08! 0.10 1.21! 0.13 0.51. . . . . . .logM " 10.9bul 8.28! 0.06 0.96! 0.07 0.25 8.12! 0.09 1.06! 0.10 0.49

2MASS images are photometrically calibrated with a typicalaccuracy of a few percent. More details can be found in L. K.Hunt & A. Marconi (2003, in preparation, hereafter Paper II).We performed a two-dimensional bulge/disk decomposition

of the images using the program GALFIT (Peng et al. 2002),which is made publicly available by the authors. This codeallows the fitting of several components with different func-tional shapes (e.g., generalized exponential [Sersic] and simpleexponential laws); the best-fit parameters are determined byminimizing . More details on GALFIT can be found in Peng2xet al. (2002). We fitted separately the J, H, and K images. Eachfit was started by fitting a single Sersic component and constantbackground. When necessary (e.g., for spiral galaxies), an ad-ditional component (usually an exponential disk) was added.In many cases, these initial fits left large residuals, and we thusincreased the number of components (see also Peng et al. 2002).The fits are described in detail in Paper II. In Table 1, wepresent the J, H, and K bulge magnitudes, effective bulge radiiin the J band, and their uncertainties. The J, H, and KRe

magnitudes were corrected for Galactic extinction using thedata of Schlegel, Finkbeiner, & Davis (1998). We used the Jband to determine because the images tend to be flatter, andRethus the background is better determined.

4. RESULTS AND DISCUSSION

In Figure 1, we plot, from left to right, versus ,M LBH K, bulversus , and the residuals of versus (basedM M M -j RBH bul BH e e

on the fit from T02). Only group 1 galaxies are shown. Mbulis the virial bulge mass given by ; if bulges behave as2kR j /Ge e

isothermal spheres, . However, comparing our virialk p 8/3estimates of with those of , obtained from dynamicalM Mbul dynmodeling (Magorrian et al. 1998; Gebhardt et al. 2003), showsthat and are well correlated ( ); settingM M r p 0.88bul dyn

(rather than 8/3) gives an average ratio of unity. There-k p 3fore, we have used in the above formula. Consideringk p 3the uncertainties of both mass estimates, the scatter of the ratio

is 0.21 dex. We fitted the data with the bisector linearM /Mbul dynregression from Akritas & Bershady (1996) that allows foruncertainties on both variables and intrinsic dispersion. TheFITEXY routine (Press et al. 1992) used by T02 gives con-sistent results (see Fig. 1). Fit results of versus the galaxyMBHproperties for group 1 and the combined samples are sum-marized in Table 2. The intrinsic dispersion of the residuals(rms) has been estimated with a maximum likelihood methodassuming normally distributed values. Inspection of Figure 1and Table 2 shows that and correlate well with theL MK, bul bulBH mass. The correlation between and is equivalentM MBH bulto that between the radius of the BH sphere of influence RBH(p ) and .2GM /j RBH e e

4.1. Intrinsic Dispersion of the Correlations

To compare the scatter of for different wave bands,M -LBH bulwe have also analyzed the B-band bulge luminosities for oursample. The upper limit of the intrinsic dispersion of the

correlations goes from !0.5 dex in whenM -L logMBH bul BHconsidering all galaxies to !0.3 dex when considering onlythose of group 1. Hence, for galaxies with reliable andMBH

, the scatter of correlations is !0.3 dex, indepen-L M -Lbul BH buldently of the spectral band used (B or JHK), comparable tothat of . This scatter would be smaller if the measurementM -jBH e

errors are underestimated. McLure & Dunlop (2002) and Erwinet al. (2003) reached a similar conclusion using R-band ,L bulbut on smaller samples. The correlation between the R-bandbulge light concentration and has a comparable scatterMBH(Graham et al. 2001).Since and have comparable disper-M -L M -LBH B, bul BH NIR, bul

Black holes in elliptical galaxies

In other words, the black hole at the center of a galaxy is ≈0.2% of the mass of its bulge!

Because

this implies a MBH–Mbulge relation

Mbulge � �2r

39