models of elliptical galaxies: ngc 3 3 7 9,4 ... - oxford academic

Mon. Not. R. astr. Soc. (1990) 245,582-596

Models of elliptical galaxies: NGC 3 3 7 9,4 2 6 1 ,4 2 7 8 and 4 4 7 2

Roeland P. van der Mare1 Sterrewacht Leiden, Postbus 9513, 2300 RA Leiden, The Netherlands

James Binney Department of Theoretical Physics, Keble Road, Oxford OX1 3NP

Roger L. Davies Department of Astrophysics, Keble Road, Oxford OX1 3RH

Accepted 1990 February 6. Revised 1990 January 9

SUMMARY We investigate whether the observed kinematics and surface photometry for the elliptical E1/E2 galaxies NGC 3379, 4261, 4278 and 4472 can be modelled under the assumptions of (i) axisyrnmetry, (ii) a distribution function of the form f = f (E, L,) and (iii) constant mass-to-light ratio. The methods used are an extension of the work of Binney, Davies & Illingworth (Paper I).

Models satisfying the above assumptions fit the observations of all four galaxies remarkably well. For all galaxies, the rotation curves rule out isotropic velocity dispersion tensors. However, an excellent fit to the data can usually be obtained by including a measure velocity of anisotropy. Only in the case of NGC 4472 do the velocity dispersion profiles suggest that the radial and vertical dispersions are decoupled and thus the distribution function depends upon a third integral. In NGC 4278 the rotation velocity changes sign at R = 30 arcsec. Such behaviour is allowed by our models, but since it is associated with a large isophote twist, it is likely that it indicates a triaxial figure. We have developed a prolate model for NGC 4261 which accurately predicts the kinematics, although the comparison is limited to the rms line-of-sight velocities.

We investigate how the predicted kinematics vary as a galaxy's isophotes are varied from 'boxy' to 'discy' in shape merely by deforming the bulge rather than introducing a thin disc. A discy model rotates faster on the major axis than the equivalent boxy system, but the effect is too small to account for Bender's suggested correlation of (v/a)* with disciness (which exhibits a large scatter). In boxy galaxies the rotation velocity falls more slowly away from the equatorial plane than in the equivalent discy galaxies.

1 INTRODUCTION

The structure and dynamics of a collisionless stellar system such as an elliptical galaxy are completely determined by the phase-space distribution function f (r, v), which gives the distribution of the stars in the system over position r and velocity v. Plenty of photometric and kinematic data are now available for elliptical galaxies, but the information we have is restricted to projected quantities and the kinematic data are usually confined to a few position angles. Recovering f from the existing data is thus an impossible task. An approach often used is to make certain assumptions concerning the

distribution function f, and to find out if the data on a certain elliptical galaxy are consistent with these assumptions.

A number of observational lines of evidence support the hypothesis that most elliptical galaxies are triaxial (Schechter 1987; Franx 1988). Unfortunately, we cannot at present build consistent dynamical models of triaxial systems, but a useful first step is to ask whether a given galaxy can or cannot be consistently modelled under the assumption of axisymmetry. Such a model not only provides a convenient benchmark against which to set future triaxial models, but should be a reliable guide to the dynamics of a mildly triaxial system. In particular, it enables us to seek evidence for

Dow

nloaded from https://academ

ic.oup.com/m

nras/article/245/4/582/6525373 by guest on 23 October 2022

variation within an individual galaxy of the mass-to-lightratio. Such variations could be interpreted as evidence foreither a dark halo (Dressier 1979; Efstathiou, Ellis & Carter1982) or a central black hole (Sargent etal. 1978).

Jeans' theorem states that the distribution function, /, of asteady-state galaxy may be assumed to depend on the phase-space coordinates (r,v) only through the isolating integrals ofmotion in the galaxy's potential. An axisymmetric potentialalways admits at least two exact isolating integrals, the energyE and the component of angular momentum parallel to thesymmetry axis, Lz. It has been argued that the distributionfunctions of elliptical galaxies, like that of the Milky Way (e.g.Binney & Tremaine 1988), depend on a third integral I3. Formost galaxies, however, it is still unclear whether the presentdata exclude axisymmetric models with distribution func-tions of the form/( E, Lz).

This paper extends the work of Binney, Davies & Illing-worth (1990; hereafter Paper I), in which the E3/E4 galaxiesNGC 720,1052 and 4697 were modelled under the assump-tions of

(i) constant mass-to-light ratio;(ii) axisymmetry;(iii) / = / ( £ , Lz).

Paper I introduced a technique, based on Lucy's (1974) iter-ative scheme, for fitting three-dimensional models to surfacephotometry. The main advantage of this technique is that itdoes not involve models based on a few parameters, butallows the density in every point in the meridional (R,z)plane to be independently varied. Once the density is known,the Jeans' equations are used with some assumed mass-to-light ratio to predict velocity dispersions and streamingvelocities. These can then be compared with known kine-matical data. It is thus possible to state if the data on a galaxydo or do not exclude models that conform to the aboveassumptions. Such a statement is much stronger than those towhich the methods of Wilson (1975) or Satoh (1980) lead;these authors built models that depend on only a handful ofparameters and can therefore explore only a small subspaceof the space of all possible models that conform to assump-tions (i)-(iii) above.

In the present paper we extend the work of Paper I to fourE1/E2 galaxies: NGC 3379, 4261, 4278 and 4472. Thesegalaxies were chosen because good kinematical data areavailable for them (Davies & Birkinshaw 1988; DB here-after). NGC 4261, 4278 and 4472 are known to have somepeculiar kinematic properties. It is interesting to see if it ispossible to relate these properties to the particular failure ofone or more of our assumptions.

No new observations are presented. The surface photo-metry used is that of Peletier et al. (1990; PDIDC hereafter),but is reduced differently.

In recent years relations have been claimed between thedeviations of isophotes from perfect ellipticity [in particularthe cos(40) Fourier component] and other properties ofelliptical galaxies such as the degree of rotational support,radio power and X-ray power (Bender et al. 1989). In view ofthis we also used our technique to study the kinematics ofmodel galaxies that differ only in the magnitude of the devia-tions of their isophotes from ellipses.

The paper is organized as follows. Section 2 describes thegalaxies, the data and the way the data were reduced. Section

Models of four elliptical galaxies 583

3 outlines the modelling technique introduced in Paper I.Section 4 compares different numerical implementations ofthe technique and estimates the errors in the kinematicalpredictions to which it leads. Section 5 compares the trueand the predicted kinematical quantities for each galaxy.Section 6 studies the effect of cos(40) deviations from per-fect ellipticity on the kinematics of elliptical galaxies, andSection 7 sums up.

2 THE SAMPLE, SURFACE PHOTOMETRYAND KINEMATICAL DATA

2.1 The sample

The galaxies studied in this paper, NGC 3379, 4261, 4278and 4472, are those galaxies in DB with measured velocitydispersion profiles and significant rotation velocities, whichwere not modelled in Paper I. They are all El/2 galaxies,their basic properties are listed in Table 1. NGC 4261, 4278and 4472 each have their own peculiar kinematic properties.

NGC 3379 is the 'standard' elliptical galaxy (de Vau-couleurs & Capaccioli 1979), and has been modelled before,as a spherical system by Miller & Prendergast (1962), and asan oblate system with known f(E,Lz) by Prendergast &Tomer (1970) and by Wilson (1975).

Davies & Birkinshaw (1986) showed that NGC4261rotates around its apparent major axis, as do a few otherelliptical galaxies (Bender et al. 1989; Franx, Illingworth &Heckman 1989a). In view of this we attempt to fit to thisgalaxy a prolate model conforming to the assumptions (i)-(iii)of Section 1. PDIDC found NGC 4261 to have the mostboxy isophotes of the 39 ellipticals they studied; it is also apowerful radio source, 3C270.

NGC 4278 is a little fainter than NGC 3379 and containsdust (Ebneter & Balick 1985) and neutral hydrogen(Raimond et al. 1981). The galaxy's rotation curve does notresemble that of a typical elliptical; the rotation velocity

Table 1. Properties of the galaxies.

Galaxy Classification BT re c ugrp D MB

RC2 RSA " Mpc

(1) (2) (3) (4) (5) (6) (7) (8) (9)

NGC 3379 El /El EO

NGC 4261 E2/E2 E3

NGC 4278 El /El El

10.33 37.5 0.10 667 13

11.38 42.5 0.20 2087 42

11.13 35 0.12 754 15

NGC 4472 E2/E4 E1/S0] 9.32 114 0.17 1074 21

-20.20

-21.78

-19.87

-22.34

Notes: Columns (2) and (3) give the galaxy classification from theSecond Reference Catalogue (de Vaucouleurs, de Vaucouleurs &Corwin 1976; hereafter RC2) and the Revised Shapley-Ames Cata-logue (Sandage & Tammann 1981; hereafter RSA). Column (4)gives the total apparent magnitude BT and column (5) lists the effec-tive radius re in arcsec, both from Burstein et al. (1987). Column (6)gives the mean ellipticity from PDIDC. Column (7) gives the groupvelocity from Davies et al. (1987) in km s~', corrected for motionwith respect to the centroid of the local group according to theprecepts of RC2. The distance derived from these group velocitieswith Ho = 50 km s"' Mpc"' is given in column (8). Column (9) givesthe absolute magnitude MB for distance D including reddening andK-corrections.

Dow


ic.oup.com/m


584 R. P. van der Marel, J. Binney and R. L. Davies

increases to R«10 arcsec from the nucleus and thendeclines at larger distances, reaching zero at R ~ 30 arcsec(see Fig. 5). NGC 4278 has a large isophote twist ( ~ 20° overhalf a decade in radius), and it is likely that the turndown inthe rotation curve is related to this isophote twist as thesephenomena occur at about the same distance from the core.Isophote twists are conventionally attributed to changes inthe true axis ratios of a triaxial galaxy. In this picture a changeof axis ratio may be expected to be accompanied by a changein the magnitude of the mean streaming velocity.

NGC 4472 is the brightest elliptical galaxy in the Virgocluster. It rotates slowly. In the core, at i?S 5 arcsec, there isno rotation, suggesting that the core is decoupled from therest of the galaxy (DB; Franx et al. 1989a). NGC 4472 wasthe only galaxy in the DB sample for which the rotation axiswas not consistent with its lying parallel to either of the pro-jected axes of the galaxy's figure - it makes an angle of ~ 13°with the minor axis. Franx et al. (1989a) observed NGC 4472but the small minor-axis rotation detected by DB would notbe apparent in their data, given their error estimates.

2.2 Surface photometry

In this paper we use the R-band CCD frames from PDIDCtaken on the # 1 0.9-m telescope on Kitt Peak (see PDIDC).We used frames that had been trimmed to 320 x 512 format,bias-subtracted and flat-fielded as described by PDIDC.These frames were then reduced using software written byRobert Jedrzejewski, and described in Jedrzejewski (1987;hereafter RJ).

All bad pixels, rows and columns were identified and setto a 'don't know' value of zero. Cosmic rays and interferingstars were identified interactively and circular areas aroundthem also set to zero. After this, typically 10 per cent of allthe pixels on a frame had been set to zero.

The principal difference from the PDIDC reductioninvolved the removal of a sloping background in the data thatwas identified by a systematic shift in the coordinates of thecentre of the outer isophotes. All frames were corrected forthis effect by fitting the frame with a galaxy symmetricalabout its (assumed) centre, and a background linear in x andy and with zero mean. The fitted backgrounds (up to 3 percent of sky across the frame) were then subtracted from theframes.

Having reduced the data in this way, we ran the ellipse-fitting program described by RJ. Our results for the lumi-nosity, ellipticity, and position angle profiles agreed well withthe results of PDIDC. Throughout this paper the followingmajor-axis position angles were adopted: 70° (NGC 3379),158° (NGC 4261), 22° (NGC 4278) and 162° (NGC 4472).

The next step in the reduction procedure was to deter-mine the sky brightness on all the frames. Since offset frameswere not available, the sky brightnesses were determinedthrough the 'boxes' procedure (Davis et al. 1985; PDIDC), inwhich the sky brightness is estimated by taking the mean ofthe pixel values in square regions on the CCD that seem tobe free of galaxy light. The boxes were typically 60 pixels ona side.

For NGC 3379 five frames were available; two longexposures (300 s) and three short (100 s). For NGC 4261,4278 and 4472 only two frames were available; one longexposure (600 s for NGC 4261 and 4278; 300 s for

NGC 4472) and one short (100 s). Comparison of differentframes of the same galaxy was used to assess the accuracy ofthe sky-brightness estimates. The error in the adopted valuesof the sky is ~ 0.5 per cent for NGC 3379 and ~ 1 per centfor the other galaxies. We use our surface photometry out tothe radius at which the surface brightness reaches 10 percent of the sky brightness (see Section 4). The systematicerror that may be introduced into the surface photometry bythe uncertainty in the sky determination could thus be 0.1mag for NGC 4261, 4278 and 4472, and 0.05 mag forNGC 3379. The correction made for the gradients in thebackground of the frames means that the sky estimates madeare more accurate and are typically 1-2 per cent higher thanthose found by PDIDC.

In this work we use the same photometric constants asPDIDC (including corrections for galactic extinction and/^-corrections) although strictly speaking a correction isnecessitated by our change in the assumed background. Thiscorrection would be S0.02 mag s~' and would affect onlythe derived mass-to-light ratios, Y, not our conclusions con-cerning the viability of models based on our fundamentalassumptions (i)-(iii) of Section 1. Uncertainties in the dis-tance estimates give rise to very much larger uncertainties inY.

The model-building technique of Paper I assumes that thegalaxy is four-fold symmetric. This assumption is an over-simplification: some ellipticals have big isophote twists andseveral ellipticals have non-zero 3 6 Fourier components. ForNGC 3379, 4261 and 4472 the 30 terms are negligible andthe isophote twists are smaller than ~ 3° over more than adecade in radius. By contrast, NGC 4278 has both appreci-able 3d terms (~ 1 per cent for radii ^ 30 arcsec) as well asan isophote twist of ~ 20° over half a decade in radius. Thisisophote twist will undoubtedly introduce systematic errorsinto our models of this galaxy. We address this in Section 5.3.The 30 terms on the other hand are presumably caused bydust absorption (PDIDC), and not by the distribution of thestars, which is what we are interested in here.

The models require surface brightnesses on a grid that islinear in azimuth and logarithmic in radius. Such a grid hasthe virtue that at large radii equal numbers of photons from aHubble law galaxy would fall in every cell. In Paper I thegalaxy images were divided into wedges of opening angle90°/(N-l) with N=l, all the wedges having the centre ofthe galaxy as a common point, and the major-axis being thebisector of one of the wedges. Then for every wedge theluminosity profile along the bisector of the wedge was foundby azimuthally averaging over the wedge. Finally the profilesof wedges that should be related by four-fold symmetry wereaveraged together to give N cuts. Let us call this method ofgenerating cuts 'method A'. For the work reported here weused an alternative method ('method B'). In this method theluminosity along each ray is generated by superposing thePDIDC 4 0 corrections on the ellipses that were fitted usingRJ's program. The luminosity profile along each of TV rays isthen determined from the intersections of the ray with theseisophotes. The cuts are extrapolated to values below x2 timesthe sky brightness by fitting an i?1/4 profile to each ray overthe radius range from xx to x2 times the sky brightness. Pos-sible choices for *, and x2 will be discussed in Section 4. Theinterpolation of the data from a set of cuts to a regular polargrid proceeded as in Paper I.

Dow


ic.oup.com/m


2.3 The kinematical data

We compare the velocities predicted by our models with datafrom DB which are plotted in Figs 4-7 (triangles andsquares). We have added the data of Franx et al. (1989a) forNGC 4472 to Fig. 6 (pentagons and hexagons).

Squares and hexagons correspond to radii to the east ofthe centre of the galaxy, triangles and pentagons to points tothe west. Velocities greater than zero correspond to redshiftshigher than the redshift of the nucleus. Open symbols markvelocities that were multiplied by — 1 before being plotted.

In Paper I spectra were presented along slits that did notpass through the galaxy's nucleus but were offset parallel andperpendicular to the major axis. Unfortunately, such spectraare not available for the galaxies studied here.

3 THE MODELS

Our modelling procedure is described in detail in Paper I. Inoutline, one of the galaxy's principal axes on the sky ischosen as the projected symmetry axis and an inclinationangle i (the angle between the symmetry axis and the line-of-sight) is assumed. Then

(i) A least-squares fit to the surface brightnesses of either aflattened Jaffe (1983) model or a flattened modified Hubblemodel (Rood et al. 1972) is carried out.

(ii) In the next stage Lucy's (1974) iterative algorithm isused to generate from the result of step (i) a series of modelsin which the luminosity density j varies freely at each point ofa polar grid of (R, z) values.

(iii) The Jeans' equations and some assumed constantmass-to-light ratio, Y, are then used to calculate for thisluminosity model the radiaj_yelocity dispersion a and mean-square azimuthal velocity (v^ implied by p = Yj(R, z) and theassumption f(E, Lz).

(iv) Finally, the line-of-sight velocities are projected backon to the plane of the sky and compared with observation.

If the adopted symmetry axis projects to the apparentminor axis, the Lucy iterations generate an oblate body. Thepredicted velocities then satisfy {v\-a2)>0_and there is anatural way to parameterize the division of i/J into contribu-tions from random and mean azimuthal streaming: we write

%-v\-v\= k2o2 - k2)v2, (1)

where k is the free parameter introduced by Satoh (1980)and employed in Paper I. When k < 1 the azimuthal velocitydispersion exceeds the radial dispersion a rather than beingsmaller than o as in the solar neighbourhood. When k = 1 thevelocity dispersion tensor is isotropic and the excess azi-muthal motion is associated with azimuthal streaming.

If, by contrast, the assumed symmetry axis projects to theapparent major axis, the Lucy iterations generate a prolatebody in which v J < o2. Equation (1) now gives rise to imagin-ary ijp In fact, since the model now has less motion in theazimuthal direction than perpendicular to it, the velocitydispersion tensor cannot be isotropic and there is no excessazimuthal motion to associate with azimuthal streaming. Inthe absence of a natural way of breaking v^ into contribu-tions d\ and v^ from random and mean motions in the azi-muthal direction, the correct procedure is to project themean-square line-of-sight velocities on to the plane of the sky


and compare the resulting velocities with the observed mean-square line-of-sight velocity o2 + v2.

4 IMPLEMENTATION AND ACCURACY OFTHE MODELS

One has certain choices to make when inferring a three-dimensional luminosity distribution from surface photo-metry.

(i) Does one start the Lucy iterations from a Jaffe or amodified Hubble model?

(ii) Should the initial surface brightness profiles (cuts)through the galaxy be generated using method A or methodB of Section 2?

(iii) How many (L) Lucy iterations should one perform?(iv) The TV cuts are extrapolated to values below x2 times

the sky brightness by fitting an RllA profile to each ray overthe radius range from x{ to x2 times the sky brightness (seeSection 2). What values should one use for xl and x{l

(v) How many (TV) rays should there be in the polar gridson which are represented the surface brightnesses and lumi-nosity densities?

One can feel confidence in the models only if the velocitiesthey predict are substantially independent of the choices onemakes. We have conducted an extensive series of tests tocheck that this is so. Based on our tests we estimate the errorin our predicted velocity dispersions to be ~ 1 per cent. Thepredicted streaming velocities, which depend on gradients ofthe dispersions, are slightly more sensitive, the error being~ 3 per cent.

Which model to start iterations from? The left panels of Fig.1 show the difference between oblate models of NGC 4261obtained by starting from (a) a modified Hubble model or (b)a Jaffe model. The right panels of Fig. l(a) and (b) show theresidual surface brightnesses along four of seven rays after1-6 Lucy iterations. At /?S2 arcsec these are 5 0.02 magarcsec"2 for both final models (full curves). At smaller radiithe profiles derived from a Jaffe model predict significantlylarger surface brightnesses than naive extrapolation of thedata would suggest. However, it is important to bear in mindthat in projecting the models we do not take seeing intoaccount, so the projected model should be brighter than thedata at R 5 2 arcsec. Hence the galaxy could have a centralcusp like that shown in the left panel of Fig. l(b). At R> 2arcsec the three-dimensional density profiles shown in theleft panels of Fig. 1 agree well, indicating that beyond thezone materially affected by seeing, Lucy iteration recoversprofiles that are independent of the model from which theiterations start.

An unattractive feature of profiles derived from a Jaffemodel is that they tend to cross where seeing becomesimportant. In the left panel of Fig. l(b) this occurs at r- 300pc. This effect occurs rather generally when Lucy iterationsare started from a Jaffe model and is an artefact. In view ofthis and the fact that we can obtain satisfactory fits to ourdata starting from modified Hubble models, we decided tomodel all our galaxies in that way even though it is, in prin-ciple, preferable to start from a model which in projection isbrighter at the centre than the observations.

The difference in central surface brightness betweenmodels obtained by performing four Lucy iterations starting

Dow


ic.oup.com/m



r/kpc10

R/arcsec100

Figure 1. Fitting oblate models to NGC 4261 starting from (a) a modified Hubble model and (b) a Jaffe model. The sets of curves in the right-hand panels show the residual surface brightnesses fimoM - ,uobs after each Lucy iteration, the sixth iteration leading to the full curves. Residualsare shown for just four rays running from the major axis at top to the minor axis at the bottom. Exact agreement corresponds to the curvesrunning along the dotted lines. Where a curve is below its dotted line, the model is brighter than the galaxy. The left panels show for four raysthe final density profiles in units of 10'" Lo kpc~3.

from either a Hubble or Jaffe model is about 0.2 magarcsec"2 (right panels of Fig. 1), which is comparable in sizeto the effect seeing can have in the centre (PDIDC). Fig. 2shows, however, that the corresponding predicted rotationvelocities differ by more than a few per cent only in the innerfew arcsec, where the rotation velocity is rising steeply; theJaffe systems, being more centrally concentrated, tend tohave the larger v^ at a given radius. Consequently, we con-clude that the effect of seeing on our predicted velocities isvery small and completely insignificant away from the core.

How to make cuts? We find that cuts are best generated bymethod B (in which the cuts are generated from smoothisophotes) since the cuts are then much smoother and thereis less noise for the Lucy iterations to amplify - see Fig. 3.However, the final velocities derived for the same galaxy bythe two methods differ by S 1 per cent. This demonstrateshow well Lucy's scheme works on noisy data.

How many Lucy iterations? Very similar results are obtainedfor LS 4 Lucy iterations. Hence the results shown here wereobtained with L= 4 as in Paper I.

Choice of extrapolation parameters. The final velocitiesbarely depend on the extrapolation parameters x{. Weadopted x, = l, x2 = 0.1 in preference to the values (xl = 2,x2 = 0.2) employed in Paper I as giving slightly better fits tothe brightness profiles at large R.

Number of rays and choice of interpolation scheme. In PaperI the streaming velocities predicted for slits that do not passthrough the galaxy's centre undulated and it was suggestedthat this phenomenon was an artefact of the interpolationscheme employed. We have therefore tried (i) varying thenumber N of rays employed in the interpolation scheme, and(ii) changing the interpolation in angle from the cosine seriesemployed in Paper I to one based on cubic splines in d.

Dow


ic.oup.com/m



200 —

> 100 -

50 60

Figure 2. Major-axis rotation curves predicted by oblate models of NGC 4261 that have been constructed by iterating from either a modifiedHubble profile performing four (solid curve) or six (dotted curve) Lucy iterations, or a Jaffe model performing four (short dashed curve) or six(long dashed curve) Lucy iterations.

NGC 4261 (method A . method B)

100

Radius (Arcsec)

150

Figure 3. Photometric cuts for NGC 4261, extracted from the CCD frames by method A (at left) and method B (at right).

We found that increasing the number of rays from N= 7 toN=10 or N=13 simply shortened the wavelength of theunwanted undulations without diminishing their amplitude.Thus the undulations are definitely unphysical.

Fitting a cosine series to a function /(6) specified by itsvalue at N points 0, is equivalent to finding a polynomial inc = cos 0 that passes through the given points. If N is large,the order of the polynomial will be large and there is adanger that the polynomial will undulate between the 0t.Hence a safer, if computationally more expensive, strategy isto find a cubic spline in 0 that passes through the/( 0,-). Whenwe implemented this strategy the undulations in the pre-dicted velocities were entirely eliminated and all velocitiesbecame essentially independent of N for N>1. All themodels presented in Section 5 are based on A?= 7.

Using the new interpolation scheme we tested our pro-grams (as in Paper I) by having them recover some of Satoh's(1980) analytic results. The errors were typically reduced toabout 75 per cent of the values quoted in Paper I. Furtherimprovement can probably only be achieved by taking a finerpolar grid (increasing N).

5 RESULTS FOR INDIVIDUAL GALAXIES

5.1 Oblate models

Figs 4-6 show results for NGC 3379, 4278 and 4472. Ineach plot the full curves show the velocities expected in theisotropic case k = 1.

As in Paper I, the mass-to-light ratio, Y, has been chosento optimize the fit between the predicted and observedminor-axis velocity dispersions. Table 2 lists these valuesof Y, which are based on the distances given in Table 1.The last column of Table 2 shows the corresponding resultsLauer (1985) obtained using core fitting. The mass-to-light ratios were calculated using MR(O) = 4.31 (Allen1973; Bessell 1979), (B-R)3{Q) = 1.17 (Allen 1973) and(B -/?)j(galaxy)« 1.85 (PDIDC).

In addition to k and Y, we are at liberty to optimize theagreement between theory and observation by adjusting eachgalaxy's assumed inclination i.

The predicted major-axis rotation speed is affected bychanges of i in two ways: in three dimensions, v^ increases asi is decreased (since the model becomes intrinsically flatter),

Dow


ic.oup.com/m


00

n

i i

r300

200

100 -

0

20

0

> 1

00

3379

H

ray

0 —

_l I I I L

_Ld

~i—

I—i—

i—i—

i—I—

i—i—

i—i—

[—I—

i—i—

r

tj

i ii

i i

i i

i i

300

200

100

3379

H

ray

90

—

bj—

I 1

I |

I I

I 1

I I

0

102

030 R

40

506

0

300

200

10

0 ~—

0 '—

i i

i i

I i

i i

\ i

\I

I I I I I I I I I I I I I I

I -

20

0

>

100

~i—

i—I—

i—r

i-1

i i

i i

-i

i i

Ii

J i

i —

i

a s

300

200

b

100 0

200

> 100

3379

H

ray

-30

—

X^j_^_L

I I I I I I L

50

60

Figu

re 4

. Pr

edic

ted

and

obse

rved

kin

emat

ics

of N

GC

337

9. T

he l

egen

ds a

t th

e to

p ri

ght o

f ea

ch p

anel

indi

cate

the

rel

evan

t sl

it po

sitio

n w

ith r

espe

ct t

o th

e m

ajor

axi

s. T

he 'H

' ind

icat

es th

at t

heL

ucy

itera

tion

star

ted

from

a m

odif

ied

Hub

ble

mod

el. T

he f

ull

curv

es a

re o

btai

ned

with

inc

linat

ion

i = 9

0° a

nd a

niso

trop

y pa

ram

eter

k=

1.0.

T

he d

otte

d cu

rves

are

for

i =

90°

and

/c =

0.5

. T

heda

shed

cur

ves

are

for

/ = 6

0° a

nd k

= 0

.5.

Dow


ic.oup.com/m


Table 2. Parameters of the best-fitting models.

Galaxy Shape A^rms k i ^O 'TR/TQ / ^ T B / T Q h

(NGC) This paper This paper Lauer (1985)

3379 oblate 0.020 0.5 60° 4.4 7.8 8.5

4261 prolate 0.016 - 30° 6.0 10.7 10.3

4278 oblate 0.016 0.5 90° 6.6 11.7

4472 oblate 0.026 0.4 90° 4.7 8.5 10.7

Note: A/urms is the rms difference in the model and true surfacebrightnesses expressed in magnitudes, k is the parameter of equa-tion (1).

but the portion of v+ that projects along the line-of-sightdiminishes with /. In Paper I, in which the galaxies wereflatter E3/E4, the first effect was dominant and the major-axis rotation velocities increased significantly when i waslowered. In our sample the galaxies are rounder and the twoeffects are more nearly in balance - in the cases of NGC4278 and 4472 the projected rotation velocity rises by :£ 10per cent as i is lowered from 90°-60°. Since our predictionsfor these galaxies are so insensitive to i, we show here onlyresults for ( = 90°. We were able to obtain a significantlybetter fit to the data for NGC 3379 with i = 60°, so we alsoshow this smaller inclination in this case.

Figs 4-6 were all obtained after four Lucy iterations froma modified Hubble profile. Table 2 shows the rms residualsbetween the projected models and the initial photometriccuts. (The latter were obtained by method B of Section 2.)

5.1.1 NGC3379

Our model of NGC 3379 is based upon an average of fiveCCD frames, each frame being weighted by 1004xPC, wherePC is the frame's photometric constant.

Fig. 4 shows that with k=l and i = 90° the predicted rota-tion velocities along the 30° and — 30° slits are too high. Thedotted curves show the result of lowering k to 0.5. The pre-dicted rotation speeds far down the 30° and — 30° slits nowagree well with the observations but the speeds predicted inthe inner parts are too low, as is the rotation velocity all alongthe major axis.

The dashed curves in Fig. 4(a)-(d) correspond to the pre-dictions for A: =0.5 and j = 60°. With these parameters thepredicted rotation velocities along all slits are consistent withthe data. In DB there is an indication of rotation along theminor axis at radii beyond ~ 40 arcsec, but this may not besignificant. The only possible conflict between theory andobservation is that the predicted velocity dispersions consist-ently fall with increasing R, where the data points suggestthat a is independent of R.

Franx (1988) presents principal-axis velocities forNGC 3379. His rotation velocities agree well with the data ofDB but his dispersions are systematically 15 per cent lowerthan those of DB. Franx's data were obtained at higher reso-lution than those of DB, and in view of the relatively lowdispersion in this galaxy, Franx's data may be more reliable.The only modification they require is setting Y and k toY^ = 3.1 and A: = 0.6.

We conclude that the kinematics of NGC 3379 are con-sistent with a distribution function of the form f{E, Lz).


5.1.2 NGC 4278

Our models of NGC 4278 are based on a single long-exposure (600 s) frame.

NGC 4278 has a very large isophote twist. The positionangle of the isophotes swings from ~ 17° for R<20 arcsec to~ 37° for R > 50 arcsec (PDIDC). Consequently, the valuewe adopt for the major axis position angle is rather arbitrary- we chose 22°. With this position angle the kinematic datado not correspond to the principal axes. In fact two of the slitpositions are inclined at ~ 23° to the minor axis.

Isophote twists are generally interpreted as indications oftriaxiality. Are axisymmetric models such as ours of a triaxialgalaxy of any value? If the galaxy is only mildly triaxial theanswer must be yes, and it is perfectly possible for an appar-ently rather round galaxy such as NGC4278 (£ = 0.10, seePDIDC) to be nearly axisymmetric and yet show a substan-tial isophote twist.

Fig. 5(a)-(d) compares the data with the predictions of twomodels: k=l (full curves) and 0.5 (dotted curves), both atinclination / = 90°. Again the isotropic model (k = 1) predictstoo much rotation along every slit. Dropping k to k = 0.5eliminates this discrepancy at R<20 arcsec, but further outthe model still fails to reproduce the observed decline of v tozero and perhaps even below.

The fact that the abrupt fall in v occurs where the iso-photes are twisting most rapidly suggests that this fall shouldbe interpreted in terms of triaxiality rather than, for example,radial variation of k. However, formally it would be straight-forward to construct a model that reproduced the observedrotation curve, including a reversal of v at R ~ 30 arcsec: theJeans' equations have nothing to say about the sign of v, so amodel can rotate one way near the centre, and in the oppo-site sense far out. Since the rms line-of-sight velocity is welldetermined, independently of v, in such a model the velocitydispersion would peak where v passed through zero. Themajor-axis dispersion data of Fig. 5 cannot rule out such apeak.

As in the case of NGC 3379, there is a slight tendency forthe predicted velocity dispersions to fall towards large Rmore than do the data.

5.7.5 NGC4472

Our models of NGC 4472 are based on a single long-exposure frame. Fig. 6(a)-(d) compares the data with the pre-dictions of two models: k=l (full curves) and k = 0.4 (dottedcurves), both at inclination i = 90°. Again the isotropic model(k=\) predicts too much rotation along every slit. The/c=0.4 model accounts well for the rotation velocities atR S; 15 arcsec, but at small R still predicts more rotation thanis observed - NGC 4472 does not rotate in its inner 5 arcsec,and has a kinematically decoupled core (Davies 1989; Franxet al. 1989a).

Away from the minor axis the k = 0A model predictsvelocity dispersions that are slightly too high. This cannot beinterpreted' as a radial variation of mass-to-light ratio, sincethis would affect the velocities on all slits in the same sense.Since we see predominantly o^ on the major axis and pre-dominantly CTR on the minor axis, the observed discrepancywould be explained by the presence of a third integral, sincethe galaxy could then be flattened by a large CTR instead of bya large o^ as is the case for our models. A similar, but much

Dow


ic.oup.com/m


b > b >

30

0

20

0

10

0 0

20

0

10

0 n

30

0

20

0

10

0

20

0

10

0 o

=1—

1—— - — z -• 5

1

- -i n—

i—

— i i 1 - — : 5

i4

~l

— - -

! 1

1 1

—i—

I—r

x-— %

i i

i

—i—

i—r

1 !

!

—i—

i—r

i i

[i

I i

t *

*——

s_

_j— i

i i

—1—

1—

r-

.—-

I f,

f

_j

,i

i i

—i—

i—r

L—

-J

I

i 1

i i

i

~i—

|—i—

i—r

• ,1

i i

i 1

i i

i

i •

1

i

n—

i

—i i

~~

i—

i

—r~

—

i

*

i r

i

s

i

i i

i

-i

i i

~i—

i—i—

•

i i

~i

i—i—

i i

~i—

i—i—

_J

1 L

_

1

{

1 1 1—

I ]

1 1—

i1

I I

I I

i

i—i—

I*

\i

T—

r~

i—-—

i

i—r-

J L_

1 |

1

1 1

4278H

ra

y

—•

i i—i

i i

—I—

I—I—

i—

i

1—'—

'—'—

4278H

ra

y

i Ii

i i

—I—

I—I—

i i

i

Ii

i-

-2

2

—-

ih.

•—

—

i 1

i i-

i——

i—c — —

-

1 ,

^i

1

i i-

23

—

- 2 -

1 j

1 U —

-

i1

1-

R—

i—I—

i i—

r

1020

30

R4

05

0

300

200

b

100 0

200

>

100 0

300

200

b

100 0

200

>

100

1 1

1

1

F

b

i_

P—

r~

4278

H ra

y

68

—

S

I I

I I

I I

I I

I 1

1

I

I H

^

Q S'

3

I I

I

II

~1

1

1

1

1

1

1

1 F

4278

H ra

y -6

7

~

i i

i i

i i

I i

ii

i i

i I

\ \

\

a

i—I—

I—i—

I—i—

i—I—

I—I—

I—r

1020

30

R4

05

0

Figu

re 5

. T

he s

ame

as F

ig. 4

but

for

NG

C 4

278.

The

dot

ted

curv

e no

w c

orre

spon

ds t

o i =

90°

and

k =

0.4

.

Dow


ic.oup.com/m


>

100

—

4472

H

ray

-2

1 I

I I

I I

I I

I 1

I I

I—I—

I—I—

I—I—

I—I—

I—I—

I—I—

I—I—

I

i 44

72H

ra

y 88

10

20

30

40

50

60

70

> b >

300

200

100

200

100

300

200

100 0

200

100 0

R" 1

1 1 1 1

1

— ':

-

i i

-

i i

'-

— -1

i

_ -

1 i

i

i

i i

iii

i

i

i i

_-—

4*--

*--

i

i i i i i i

... !

4 ^-~—

i

1 I f 1 »• 1

1 t i 1 1 1 1

1 1

1 1 1

1 1

- 1 i

1 ID 1 1 1 1

1 1

I 1 t 1 \

11

{—i—

i

nI

I1

l1

1 I '

11

I 1

i i

1 1

-f

1 1

| 1

1 1 [ 1 l

1 1

1 ii—

i i i

i

il

l.

i i

i i i

i i i

4472H

•

ii i

D (i

i i

1

4472H

4- i 1 i i

i

i 1 i i i

I!

•

t 1 i

1 '

1(

1 ra1

V i

i 1 1

i

1 1 1 [-

2R

— 1 1 1 II I ! 1

—

ll

i

I-

1 1 1 |_ _

1 1 1

-32 -- ~ — -

-r-r

-r-p

I I I R

0 10

20

30

40

50

60

Figu

re 6

. T

he s

ame

as F

ig. 4

, but

for

NG

C 4

472.

The

dot

ted

curv

e no

w c

orre

spon

ds t

o / =

90°

and

k =

0.5

.

VO

Dow


ic.oup.com/m


592 R.P. van der Mar el, J. Binney and R. L. Davies

more pronounced phenomenon was encountered in Paper Iin the case of NGC 720.

In summary, our model with k = 0.4 fits the kinematics ofthe main body of NGC 4472 adequately, but there is someevidence that the distribution function of NGC 4472involves a third integral.

5.2 Prolate models

5.2.1 NGC4261

NGC 4261 rotates at up to 100 km s~' around its apparentmajor axis. Obviously such a system cannot be successfullymodelled as an oblate body. Hence in Fig. 7 we showpredicted and observed rms line-of-sight velocities for twoprolate models of NGC 4261. The full curves are for the case/ = 90° in which we see the galaxy broadside-on, while thedotted lines are for /=30°. Only the second, nearly end-on

model correctly predicts that the rms velocities along the twoprincipal axes are approximately equal. It is instructive tounderstand how this result comes about.

In any two-integral model, be it oblate or prolate, theminor-axis velocities tend to be smaller than those along themajor axis since, in the oblate case, the enhanced azimuthalvelocities are seen along the major axis, while in the prolatecase one sees the depressed azimuthal velocities along theminor axis. If the inclination / of a prolate model is reduced,the system needs to be more elongated to project to thegalaxy's observed shape (e.g. in three dimensions our i = 30°model of NGC 4261 has axis ratio a/b—1.1 compared witha/b — 1.2 in the case ( = 90°), thereby increasing the discrep-ancy between o^ and oR = ov However, as / is decreased, theazimuthal velocities contribute less to the line-of-sight,thereby making the rms velocities along all slits more nearlyequal.

400

300

b 200

100

0

- 1 i | l [ i i i

E- !

E-— j= I I i i i i I I

"T~ 1 1 1 1 1 1 1 1 1

i

I I i i i i i i i i

I I I I I I I I 1 i i -

4261H ray 3 ~

~E

11111

-i i i i 1 i i i i I I -

400

300

b 200

100

0

400

300

b 200

100

0

400

300

b 200

100

0

=T—1—

E-

II

II

—

-1

-1

E-

11

M

z_

- 1

—

E-E-

E--

—I—1—1—1—

I i r

I I I

1 1 1

i i i i i i i i i i i i i i

~ m^m

i i i i

i i i

I I

— i — i — i — i —

i i i i

— i — i — i — i —

i i

—i—i—

I

1 1 1 I I I I i i i i i i

i

426i i

H ray -I i -

i -Z

II 1 I

I

-i

i—i—

426

i i

—i—i—i—i—

H ray C

i i -

to II 1

II

i ~z

i

I 1 i i i i i -

I I I

i i i

I I I I 1 1 1 I I i i i 1 i i

" T~

i i i i i i i i 1 i i i i 1 i i

426i i i

H Jay -i 1 -

27 ~

I i

MI

-i i i i i i -

10 20 30R

40 50 60

Figure 7. Predicted and observed kinematics of NGC 4261. The data points show the square root oiU^=a^ + vl, where op and vp are themeasured velocity dispersion and streaming velocity, respectively. The full curves show the predictions for ,/tr of a prolate model seenbroadside on (/ = 90°). The dotted curves show the predictions of a prolate model inclined at / = 30°. The dashed curve shows the predictions ofan edge-on oblate model whose mass-to-light ratio is larger by a factor of 1.1. All models are based on a single 600 s frame.

Dow


ic.oup.com/m


The introduction of a third integral makes it possible todecouple the dispersion parallel to the symmetry axis fromthe radial dispersion parallel to the equatorial plane. Hencein a three-integral model it should be possible to obtaincomparable dispersions along both apparent principal axeseven at inclination ;' = 90°.

Prolate models can reproduce given velocities with smallermass-to-light ratios Y than oblate models. For example, thedashed curves in Fig. 7 show the rms velocities predictedby an oblate model with Y^ = 6.6 compared with YR = 6.0 inthe prolate case.

6 THE EFFECT OF NON-ZERO 4 0 TERMSON THE KINEMATICS OF A GALAXY

We have used our technique to study the effect of 46 devia-tions from perfect ellipticity on the kinematics of a galaxy. To


this end we created toy surface photometry frames, contain-ing 'galaxies' whose isophotes were concentric, coaxial andbuilt up by superposing a fixed 46 Fourier component on anested sequence of similar ellipses. Three-dimensionalmodels were then constructed from these frames as from realCCD frames.

Some authors (e.g. Lauer 1985; PDIDC) measure devia-tions from perfect ellipticity in terms of the brightness varia-tions around a perfect ellipse, while others (e.g. Bender &Mollenhoff 1987) work in terms of the variation of the differ-ence dR between the distances from the centre of a point onan ellipse and the point with the same azimuth on the corre-sponding isophote. We have adopted the latter approach,writing

dR(6)

J(acos0)2+(bsin6f=AA COS46. (2)

(a) 300

200

b

100

0

200

> 100

0

(b) 300

200

b

100

0

200

> 100

n

- 1 1

—

L

- 1 i

- i i

r——- 1 1

=n—i

^_

-_

—

- i

- i i

-

- i i

i i i i 1 1 1 1

—i—i—i—i—

i 1 1

—!—1—1—1

/

1 1 1

i—i—i—i—

1 1 1

—1—1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

, 1 1 1 1 1

I I I 1 1 ! 1 I I I I i l l I I -

9999H ray 0 —

=^^—^=-=-=^^^-^___ —

-

i l l

—i—i—i—i—

I I I I I I I I

1 1 [ "1 1 1 1 1 1

I I I I

i , i i

i i -

i -

-

—

i i i i

—i i—i—i—

1 1 I I

—1 1 1—1—

I I I

—1 1 1 1

999$

I I I I

— i — i — i — i —

m ray ^

i -

—i r=

-5 —_

— — — _ _ _— — - -

—

_

I I I I

I I I I

1 , 1

I I I

, i ,

i l l

i I i i 1 , i -

i l l —'—'"-I

—

—

-

I I I I I I I I I I I , , , i 1 i i -

0 10 20 30R

40 50 60

Figure 8. Predicted kinematics along (a) the major axis and (b) a slit that makes a 45° angle with the major axis, for three model galaxieswithout isophote twists or centre shifts, with fixed ellipticity (1 — b/a) = 0.2, the same major axis luminosity profile as NGC4261, and withrespectively no 40 Fourier component (solid curves), fixed 40 Fourier component A4= -0.02 (dotted curves; slightly 'boxy' isophotes) andfixed 40 Fourier component A4 = 0.02 (dashed curves; slightly 'pointed' isophotes).

Dow


ic.oup.com/m



Here 9 is the true polar angle rather than the elliptical angleemployed by Bender & Mollenhoff (1987), and on the right-hand side we have dropped higher order terms and termsodd in 9. Thus defined, our parameter A4 is for b/a—lidentical with the ratio a4/a of Bender & Mollenhoff. WhenA4 > 0 the galaxy has pointed isophotes (is discy), while whenA4 < 0 the isophotes are boxy.

Fig. 8(a) and (b) show the predicted kinematics of a modelwhose ellipticity is £ = 0.2 and whose major-axis luminosityprofile is the same as that of NGC4261. Three differentcases were studied: (i) no 49 Fourier component (A4 = 0) -solid curves); (ii) slightly boxy isophotes ( J 4 4 = - 0 . 0 2 -dotted curves); (iii) slightly discy isophotes (A4 = 0.02 -dashed curves). Note that A4 = 0.02 is about the highestvalue that is encountered in real galaxies (Bender et al.1989). Fig. 8(a) is for a slit along the major axis while Fig.8(b) is for a slit inclined at 45° to the major axis, both underthe assumptions i = 90°, k = 1 and YR = 5.1 Y0.

Fig. 8(a) and (b) show that changing A4 by 2 per centchanges the predicted velocity dispersions by ~ 2 per centalso. By contrast the rotation velocities change by up to 10per cent. On the major axis the discy model rotates faster

than the boxy one, the situation being reversed along the 45°slit since the discy model's rotation is more strongly confinedto the equatorial regions. These conclusions were found tobe insensitive to the ellipticity, radial density profile etc. ofthe model studied excepting that a boxy system rotates fasterthan its discy equivalent along slits that make an angle > /?with the major axis, where 20° 5 /? 5 40° depending on thesystem's ellipticity, luminosity profile, etc.

Can we observe any of these effects in real galaxies?Bender et al. (1989) argue from a plot similar to Fig. 9 thatdiscy galaxies have higher rotation on the major axis thanboxy galaxies. From this figure it is apparent that any corre-lation between (v/a)* and A4 is weak, but if present the lineof regression in Fig. 9 would be inclined at near 45°. Thedashed lines in Fig. 9 show that the effect seen in our models,while qualitatively the same as that suggested by Bender etal., gives rise to a very different slope in a plot of (v/a)*versus A4. Thus we cannot explain why so few discy galaxieswith low rotation have been observed yet.

A satisfactory test of the prediction that in ellipticalgalaxies the rotation velocity will prove to fall more slowlyaway from the major axis in boxy galaxies than in discy ones,

1.5o

1

5

i i i

__ — — ~~ ~~

~~ A

_ _ _

A

1 1 1

- — ~~

A

- - A -

D

•

1

1

- — —

a

D•

•o

D C

1

•

-

ft

1

D

—

a

1

_ A - —

—

D

A

A

D

a

D

A

1

V

-

-

_

1

1

a

" a

D

—

A

1

_ - -

A

D

_•

a

1

—

D

1

a _

" " " • " '

1

1

-

_ - -

_

—

-

-

i

- .02 -.01 0a4/a

.01 .02

Figure 9. Plot of aA/a - A4 versus (vjaf. A4 measures the amplitude of cos 40 deviations of the isophotes from ellipses (equation 2). (vja)*measures the amount of rotational support of an elliptical galaxy (Davies et al. 1983). Open squares denote galaxies in the sample of Bender(1988). Solid triangles denote those galaxies in the sample of PDIDC that are not part of the sample of Bender (1988). The C4 parameters ofPDIDC were transformed to a4/a parameters by using the galaxies that both samples have in common as calibrators. The figure shows aweaker correlation than the plot of Bender et al. largely because we have plotted {vjaf rather than its logarithm. The dashed lines show themovement predicted by our models: a change of 2 per cent in A4 changes the major axis rotation by 10 per cent.

Dow


ic.oup.com/m


must await better kinematical data than are presently avail-able. However, Kormendy & Illingworth (1982) observedsuch an effect in the extremely boxy bulge of NGC4565.Binney & Petrou (1985) suggested that boxiness and cylin-drical rotation are both consequences of over-populatingorbits inclined at a narrow range of angles with respect to theequatorial plane, and argued that mergers might lead to suchover-population.

7 CONCLUSIONS

We have used the technique of Paper I to predict kinematicalquantities from CCD surface photometry under the assump-tions of (i) axisymmetry, (ii) f=f(E,Lz) and (iii) constantmass-to-light ratio. Four galaxies were studied: NGC 3379,4278 and 4472 (to which oblate models were fitted), andNGC 4261 (to which a prolate model was fitted, since itrotates around its apparent major axis). For oblate models weemployed Satoh's (1980) parameter k to split the predictedv^ into streaming and random motion, whereas for prolatemodels we merely calculate the predicted rms line-of-sightvelocities. The kinematical predictions have been comparedwith the observations of DB and Franx et al. (1989a).

For all the oblate galaxies the data are incompatible withan isotropic model. Models in which the tangential velocitydispersion exceeds those in the radial and vertical directionsare remarkably successful. Such a model provides an accept-able fit to the observations for NGC 3379. One of ourmodels can also fit the observations for the main body ofNGC 4472 but fails to account for NGC 4472's kinemati-cally decoupled core. There is a slight indication of thepresence of a third integral in NGC 4472. NGC 4278 has arotation curve that appears to change sign soon after peakingat the radius at which the photometry reveals a marked iso-phote twist. It would be straightforward to reproduce thesuggested counter-rotation of the outer portion of this galaxyif one were to introduce an additional parameter into themodels. However, axisymmetric models such as ours cannotmodel the observed isophote twist, and it seems probablethat both the kinematic and the photometric anomalies arecaused by triaxiality.

The data on NGC 4261 can be fitted with a nearly end-onprolate model, with axis ratio ~ 1.7. Three-integral modelsof NGC 4261 would probably not be confined to near end-on orientations.

An extensive series of tests of our technique leads to thefollowing conclusions:

(i) changing the projected shape of a perfectly ellipticalmodel galaxy, by including a 40 Fourier component with anamplitude of 2 per cent, changes the velocity dispersionsonly slightly but can change the rotation velocities by - 10per cent;

(ii) a model galaxy whose projected shape is discy, rotatesfaster on the major axis than the equivalent boxy model. Thiseffect is consistent with known data on elliptical galaxysamples, but does not explain the absence in the observationsof slowly rotating, discy galaxies;

(iii) the rotation velocity in a discy model falls off muchmore steeply away from the major axis than the rotation


velocity of the equivalent boxy model. To observe this effectin elliptical galaxies, better kinematical data are needed.

ACKNOWLEDGMENTS

The authors want to thank Reynier Peletier for providing theCCD surface photometry frames from PDIDC, and RobertJedrzejewski for providing his galaxy surface photometryFORTRAN programs. RPvdM wishes to thank the SterrewachtLeiden, and the Departments of Theoretical Physics andAstrophysics of Oxford University for grants to work on thisproject in Oxford. Furthermore, he wishes to thank MertonCollege, Oxford for its hospitality during his stays.

REFERENCES

Allen, C. W., 1973. Astrophysical Quantities, 3rd edn, AthlonePress, London.

Bender, R., 1988. Astr. Astrophys., 193, L7.Bender, R. & Mollenhoff, C, 1987. Astr. Astrophys., 177, 71.Bender, R., Surma, P., Dobereiner, S., Mollenhoff, C. & Madejsky,

R., 1989. Astr. Astrophys., 217, 35.Bessell, M. S., 1979. Publs astr. Soc. Pacif., 91, 589.Binney, J. J. & Petrou, M., 1985. Mon. Not. R. astr. Soc, 214, 449.Binney, J. J. & Tremaine, S. D., 1988. Galactic Dynamics, Princeton

University Press, Princeton.Binney, J. J., Davies, R. L. & Illingworth, G. D., 1990. Astrophys. J.,

in press (Paper I).Burstein, D., Davies, R L., Dressier, A., Faber, S. M, Stone, R. P. S.,

Lynden-Bell, D., Terlevich, R. & Wegner, G., 1987. Astrophys. J.Suppl, 64, 601.

Davies, R. L., 1989. In: The World of Galaxies, p. 312, eds Corwin,H. G. & Bottinelli, L., Springer-Verlag, New York.

Davies, R. L. & Birkinshaw, M., 1986. Astrophys. J., 303, L45.Davies, R. L. & Birkinshaw, M., 1988. Astrophys. J., 68, 409 (DB).Davies, R. L., Efstathiou, G., Fall, S. M., Illingworth, G. & Schechter,

P., 1983. Astrophys. J., 266, 41.Davies, R. L., Burstein, D., Dressier, A., Faber, S. M., Lynden-Bell,

D., Terlevich, R. & Wegner, G., 1987. Astrophys. J. Suppl., 64,581.

Davis, L. E., Cawson, M, Davies, R. L. & Illingworth, G. D., 1985.Astr. J., 90, 169.

de Vaucouleurs, G. & Capaccioli, M., 1979. Astrophys. J. Suppl, 40,699.

de Vaucouleurs, G., de Vaucouleurs, A. & Corwin, H. G., 1976.Second Reference Catalogue of Bright Galaxies, University ofTexas Monographs in Astronomy No. 2, Austin (RC2).

Dressier, A., 1979. Astrophys. /., 231, 659.Ebneter, K. & Balick, B., 1985. Astr. J., 90, 183.Efstathiou, G., Ellis, R. S. & Carter, D., 1982. Mon. Not. R. astr.

Soc, 201,975.Franx, M., 1988. PhD thesis, Leiden University.Franx, M., Illingworth, G. D. & Heckman, T., 1989a. Astrophys. J.,

344,613.Franx, M., Dlingworth, G. D. & Heckman, T., 1989b. Astr. J., 98,

538.Jaffe, W., 1983. Mon. Not. R. astr. Soc, 202, 995.Jedrzejewski, R. I., 1987. Mon. Not. R. astr. Soc, 226, 747 (RJ).Kormendy, J. & Illingworth, G., 1982. Astrophys. J., 256, 460.Lauer, T. R., 1985. Astrophys. J., 292, 104.Lucy, L. B., 1974. Astr. J., 79, 745.Miller, R. H. & Prendergast, K. H., 1962. Astrophys. J., 136, 713.

Dow


ic.oup.com/m



Peletier, R. E, Davies, R. L., Illingworth, G. D., Davis, L. E. & ton, Washington, D.C. (RSA).Cawson,M, 1990. As/r. /., in press (PDIDC). Sargent, W. L. W., Young, P. J., Boksenberg, A., Shortridge, K.,

Prendergast, K. H. & Tomer, E., 1970. Astr. /., 75, 674. Lynds, C. R. & Hartwick, F. D. A., 1978. Astrophys. J., 221,Raimond, E., Faber, S. M., Gallagher, J. S. & Knapp, G. R., 1981. 731.

Astrophys. J., 246, 708. Satoh, C, 1980. Publs astr. Soc. Japan, 32,41.Rood, H. J., Page, T. L., Kilmer, E. C. & King, I. R., 1972. Atrophys. Schechter, P. L., 1987. Structure and Dynamics of Elliptical

J., 175,627. Galaxies, IAU Symp. No. 127, p. 217, ed. de Zeeuw, P. T.,Sandage, A. & Tammann, G. A., 1981. A Revised Shapley-Ames Reidel, Dordrecht.

Catalogue of Bright Galaxies, Carnegie Institution of Washing- Wilson, C. P., 1975. Astr. J., 80, 175.

Dow


ic.oup.com/m


models of elliptical galaxies: ngc 3 3 7 9,4 ... - oxford academic

Documents