Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.stsci.edu/~marel/abstracts/psdir/mnras245.ps Äàòà èçìåíåíèÿ: Sat Apr 26 22:15:11 1997 Äàòà èíäåêñèðîâàíèÿ: Sat Dec 22 11:30:27 2007 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ngc 4261

Mon. Not. R. Astr. Soc. (1990) 245, 582
Models of elliptical galaxies: NGC 3379, NGC
4261, NGC 4278 and NGC 4472
Roeland van der Marel
Sterrewacht Leiden, Postbus 9513, 2300 RA Leiden, The Netherlands
James Binney
Department of Theoretical Physics, Keble Road, Oxford, OX1 3NP,
Great­Brittain
Roger L. Davies
Department of Astrophysics, Keble Road, Oxford, OX1 3RH,Great­
Brittain
Summary. We investigate whether the observed kinematics and surface pho­
tometry for the elliptical E1/E2 galaxies NGC 3379, NGC 4261, NGC 4278 and
NGC 4472 can be modelled under the assumptions of (i) axisymmetry, (ii) a dis­
tribution function of the form f = f(E; L z ) and (iii) constant mass­to­light ratio.
The methods used are an extension of the work of Binney, Davies & Illingworth
(1990; 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 00 . Such
behaviour is allowed by our models but since it is associated with a large isophote
twist, it likely 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 line­of­sight velocities. We investigate how the predicted kine­
matics vary as a galaxy's isophotes are varied from ``boxy'' to ``disky'' in shape
merely by deforming the bulge rather than introducing a thin disk. A disky model
rotates faster on the major axis than the equivalent boxy system, but the effect is
too small to account for Bender's (1988) suggested correlation of (v=oe) \Lambda with disk­
iness (which exhibits a large scatter). In boxy galaxies the rotation velocity falls
more slowly away from the equatorial plane than in the equivalent disky galaxies.

2 Models of elliptical galaxies
1. Introduction
The structure and dynamics of a collisionless stellar system such as an ellipti­
cal galaxy are completely determined by the phase­space 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 el­
liptical 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 assump­
tion 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 variation within
an individual galaxy of the mass­to­light ratio. Such variations could be interpreted as
evidence for either a dark halo (Dressler 1979; Efstathiou et al. 1982) or a central black
hole (Sargent et al. 1978).
Jeans' theorem states that the distribution function f of a steady­state galaxy
may be assumed to depend on the phase­space coordinates (r; v) only through the
isolating integrals of motion in the galaxy's potential. An axisymmetric potential always
admits at least two exact isolating integrals, the energy E and the component of angular
momentum parallel to the symmetry axis, L z . It has been argued that the distribution
functions of elliptical galaxies, like that of the Milky Way (e.g. Binney & Tremaine
1987), depend on a third integral I 3 . For most galaxies however it is still unclear
whether the present data excludes axisymmetric models with distribution functions of
the form f(E; L z ).
This paper extends the work of Binney, Davies & Illingworth (1990; hereafter Paper
I), in which the E3/E4 galaxies NGC 720, NGC 1052 and NGC 4697 were modelled
under the assumptions of:
(i) constant mass­to­light ratio;
(ii) axisymmetry;
(iii) f = f(E; L z ).
Paper I introduced a technique, based on Lucy's (1974) iterative scheme, for fitting
three­dimensional models to surface photometry. The main advantage of this technique
is that it does not involve models based on a few parameters, but allows the density in
every point in the meridional (R; z) plane to be independently varied. Once the density

R. van der Marel, J. Binney and R. L. Davies 3
is known the Jeans equations are used with some assumed mass­to­light ratio to predict
velocity dispersions and streaming velocities. These can then be compared with known
kinematical data. It is thus possible to state if the data on a galaxy do or do not exclude
models that conform to the above assumptions. Such a statement is much stronger than
those to which the methods of Wilson (1975) or Satoh (1980) lead; these authors built
models that depend on only a handful of parameters and can therefore explore only a
small subspace of the space of all possible models that conform to assumptions (i) --
(iii) above.
In the present paper we extend the work of Paper I to four E1/E2 galaxies: NGC
3379, NGC 4261, NGC 4278 and NGC 4472. These galaxies were chosen because good
kinematical data is available for them (Davies and Birkinshaw 1988; DB hereafter).
NGC 4261, NGC 4278 and NGC 4472 are known to have some peculiar kinematic
properties. It is interesting to see if it is possible to relate these properties to the
particular failure of one or more of our assumptions.
No new observations are presented. The surface photometry used is that of Peletier
et al. (1990; PDIDC hereafter), but is reduced differently.
In recent years relations have been claimed between the deviations of isophotes from
perfect ellipticity (in particular the cos(4`) Fourier component) and other properties of
elliptical galaxies such as the degree of rotational support, radio power and X­ray power
(Bender et al. 1989). In view of this we also used our technique to study the kinematics
of model galaxies that differ only in the magnitude of the deviations of their isophotes
from ellipses.
The paper is organized as follows. Section 2 describes the galaxies, the data, and
the way the data were reduced. Section 3 outlines the modelling technique introduced
in Paper I. Section 4 compares different numerical implementations of the technique
and estimates the errors in the kinematical predictions to which it leads. Section 5
compares the true and the predicted kinematical quantities for each galaxy. Section
6 studies the effect of cos(4`) deviations from perfect ellipticity on the kinematics of
elliptical galaxies, and Section 7 sums up.
2. The sample, surface photometry and kinematical data
2.1 The sample
The galaxies studied in this paper, NGC 3379, NGC 4261, NGC 4278 and NGC
4472 are those galaxies in DB with measured velocity dispersion profiles and significant
rotation velocities, which were not modelled in paper I. They are all E1/2 galaxies, their
basic properties are listed in Table 1. NGC 4261, NGC 4278 and NGC 4472 each have
their own peculiar kinematic properties.

4 Models of elliptical galaxies
TABLE 1
Properties of the Galaxies
Galaxy Classification BT r e ffl v grp D MB
RC2 RSA '' Mpc
(1) (2) (3) (4) (5) (6) (7) (8) (9)
NGC 3379 E1/E1 E0 10.33 37.5 0.10 667 13 ­20.20
NGC 4261 E2/E2 E3 11.38 42.5 0.20 2087 42 ­21.78
NGC 4278 E1/E1 E1 11.13 35 0.12 754 15 ­19.87
NGC 4472 E2/E4 E1/S0 1 9.32 114 0.17 1074 21 ­22.34
Notes : Columns (2) and (3) give the galaxy classification from the RC2 and the RSA.
Column (4) gives the total apparent magnitude BT and column (5) lists the effective
radius r e in arcsec, both from Burstein et al. (1987). Column (6) gives the mean
ellipticity from PDIDC. Column (7) gives the group velocity from Davies et al. (1987)
in km/s, corrected for the motion with respect to the centroid of the local group, using
the approach in RC2. The distance derived from these group velocities with H 0 = 50
km s \Gamma1 Mpc \Gamma1 is given in column (8). The absolute magnitude MB is calculated from
BT and the distance D, including reddening and K­corrections.
NGC 3379 is the ``standard'' elliptical galaxy (de Vaucouleurs & Capaccioli 1979),
and has been modelled before, as a spherical system by Miller and Prendergast (1962),
and as an oblate system with known f(E; L z ) by Prendergast and Tomer (1970) and
by Wilson (1975).
Davies & Birkinshaw (1986) showed that NGC 4261 rotates around its apparent
major axis. [As do a few other elliptical galaxies (Bender et al. , 1989; Franx et al. ,
1989a)]. In view of this we attempt to fit to this galaxy a prolate model conforming to
the assumptions (i) -- (iii) of Section 1. PDIDC found NGC 4261 to have the most boxy
isophotes of the 39 ellipticals they studied, it is also a powerful radio source, 3C270.
NGC 4278 is a little fainter than NGC 3379 and contains dust (Ebneter & Balick
1985) and neutral hydrogen (Raimond et al. 1981). The galaxy's rotation curve does
not resemble that of a typical elliptical; the rotation velocity increases to R ú 10 00 from
the nucleus and then declines at larger distances, reaching zero at R ú 30 00 (see Fig.
5). NGC 4278 has a large isophote twist (¸ 20 ffi over half a decade in radius), and
it is likely that the turndown in the rotation curve is related to this isophote twist as
these phenomena occur at about the same distance from the core. Isophote twists are
conventionally attributed to changes in the true axis ratios of a triaxial galaxy. In this
picture a change of axis ratio may be expected to be accompanied by a change in the
magnitude of the mean streaming velocity.

R. van der Marel, J. Binney and R. L. Davies 5
NGC 4472 is the brightest elliptical galaxy in the Virgo cluster. It rotates slowly.
In the core, at R ¸ ! 5 00 , there is no rotation, suggesting that the core is decoupled from
the rest of the galaxy (DB, Franx et al. 1989a). NGC 4472 was the only galaxy in
the DB sample for which the rotation axis was not consistent with its lying parallel to
either of the projected axes of the galaxy's figure -- it makes an angle of ¸ 13 ffi with the
minor axis. Franx et al. (1989a) observed NGC 4472 but the small minor­axis rotation
detected by DB would not be apparent in their data given their error estimates.
2.2 Surface photometry
In this paper we use the R­band CCD frames from PDIDC taken on the #1 0:9 m
telescope on Kitt­Peak (see PDIDC). We used frames that had been trimmed to 320\Theta512
format, bias­subtracted and flat­fielded as described by PDIDC. These frames were then
reduced using software written by Robert Jedrzejewski, and described in Jedrzejewski
(1987; hereafter RJ).
All bad pixels, rows and columns were identified and set to a ``don't know'' value of
zero. Cosmic rays and interfering stars were identified interactively and circular areas
around them also set to zero. After this typically 10% of all the pixels on a frame had
been set to zero.
The principal difference from the PDIDC reduction involved the removal of a sloping
background in the data that was identified by a systematic shift in the co­ordinates of
the centre of the outer isophotes. All frames were corrected for this effect by fitting the
frame with a galaxy symmetrical about its (assumed) centre, and a background linear
in x and y and with zero mean. The fitted backgrounds (up to 3% of sky across the
frame) were then subtracted from the frames.
Having reduced the data in this way, we ran the ellipse­fitting program described
in RJ. Our results for the luminosity, ellipticity, and position angle profiles agreed well
with the results of PDIDC. Throughout this paper the following major­axis position
angles were adopted: 70 ffi (NGC 3379), 158 ffi (NGC 4261), 22 ffi (NGC 4278) and 162 ffi
(NGC 4472).
The next step in the reduction procedure was to determine the sky brightness on all
the frames. Since offset frames were not available, the sky brightnesses were determined
through the ``boxes'' procedure (Davis et al. 1985; PDIDC), in which the sky brightness
is estimated by taking the mean of the pixel values in square regions on the CCD that
seem to be free of galaxy light. The boxes were typically 60 pixels on a side.
For NGC 3379 five frames were available, two long exposures (300 s), and three
short (100 s). For NGC 4261, NGC 4278 and NGC 4472 only two frames were available,
one long exposure (600 s for NGC 4261 and NGC 4278; 300 s for NGC 4472) and one
short (100 s). Comparison of different frames of the same galaxy was used to assess the
accuracy of the sky brightness estimates. The error in the adopted values of the sky is

6 Models of elliptical galaxies
¸0.5% for NGC 3379 and ¸1% for the other galaxies. We use our surface photometry
out to the radius at which the surface brightness reaches 10% of the sky brightness (see
Section 4). The systematic error that may be introduced into the surface photometry
by the uncertainty in the sky determination could thus be 0:1 mag for NGC 4261, NGC
4278 and NGC 4472 and 0:05 mag for NGC 3379. The correction made for the gradients
in the background of the frames means that the sky estimates made are more accurate
and are typically 1--2% higher than those found by PDIDC.
In this work we use the same photometric constants as PDIDC (including correc­
tions for galactic extinction and K­corrections) although strictly speaking a correction
is necessitated by our change in the assumed background. This correction would be
¸ ! 0:02 mag s \Gamma1 and would affect only the derived mass­to­light ratios, \Upsilon, not our con­
clusions concerning the viability of models based on our fundamental assumptions (i)--
(iii) of Section 1. Uncertainties in the distance estimates give rise to very much larger
uncertainties in \Upsilon.
The model­building technique of Paper I assumes that the galaxy is four­fold sym­
metric. This assumption is an over simplification: some ellipticals have big isophote
twists and several ellipticals have non­zero 3` Fourier­components. For NGC 3379, NGC
4261 and NGC 4472 the 3` terms are negligible and the isophote twists are smaller than
¸ 3 ffi over more than a decade in radius. By contrast, NGC 4278 has both appreciable
3` terms (¸1% for radii ¸ ! 30'') as well as an isophote twist of ¸ 20 ffi over half a decade
in radius. This isophote twist will undoubtedly introduce systematic errors into our
models of this galaxy. We address this in Section 5.3. The 3` terms on the other hand
are presumably caused by dust absorption (PDIDC), and not by the distribution of the
stars, which is what we are interested in here.
The models require surface brightnesses on a grid that is linear in azimuth and
logarithmic in radius. Such a grid has the virtue that at large radii equal numbers of
photons from a Hubble law galaxy would fall in every cell. In paper I the galaxy images
were divided into wedges of opening angle 90 ffi =(N \Gamma 1) with N = 7, all the wedges having
the centre of the galaxy as a common point, and the major­axis being the bisector of
one of the wedges. Then for every wedge the luminosity profile along the bisector of the
wedge was found by azimuthally averaging over the wedge. Finally the profiles of wedges
that should be related by four­fold symmetry were averaged together to give N cuts.
Let us call this method of generating cuts ``method A''. For the work reported here we
used an alternative method (``method B''). In this method the luminosity along each ray
is generated by superposing the PDIDC 4` corrections on the ellipses that were fitted
using RJ's program. The luminosity profile along each of N rays is then determined
from the intersections of the ray with these isophotes. The cuts are extrapolated to
values below x 2 times the sky brightness by fitting an R 1=4 profile to each ray over the
radius range from x 1 to x 2 times the sky brightness. Possible choices for x 1 and x 2 will
be discussed in Section 4. The interpolation of the data from a set of cuts to a regular

R. van der Marel, J. Binney and R. L. Davies 7
polar grid proceeded as in Paper I.
2.3 The kinematical data
We compare the velocities predicted by our models with data from DB which are
plotted in Figures 4--7 (triangles and squares). We have added the data of Franx et al.
(1989a) for NGC 4472 to Figure 6 (pentagons and hexagons).
Squares and hexagons correspond to radii to the East of the centre of the galaxy,
triangles and pentagons to points to the West. Velocities greater than zero correspond
to redshifts higher than the redshift of the nucleus. Open symbols mark velocities that
were multiplied by \Gamma1 before being plotted.
In paper I spectra were presented along slits that did not pass through the galaxy's
nucleus but were offset parallel and perpendicular to the major axis. Unfortunately
such spectra are not available for the galaxies studied here.
3 The Models
Our modelling procedure is described in detail in Paper I. In outline, one of the
galaxy's principal axes on the sky is chosen as the projected symmetry axis and an in­
clination angle i (the angle between the symmetry axis and the line of sight) is assumed.
Then
1) a least­squares fit to the surface brightnesses of either a flattened Jaffe (1983) model
or a flattened modified Hubble model (Rood et al. 1972) is carried out.
2) In the next stage Lucy's (1974) iterative algorithm is used to generate from the
result of step (1) a series of models in which the luminosity density j varies freely
at each point of a polar grid of (R; z) values.
3) The Jeans equations and some assumed constant mass­to­light ratio, \Upsilon, are then
used to calculate for this luminosity model the radial velocity dispersion oe and
mean­square azimuthal velocity v 2
OE implied by ae = \Upsilonj (R; z) and the assumption
f(E; L z ).
4) Finally, the line­of­sight velocities are projected back onto the plane of the sky and
compared with observation.
If the adopted symmetry axis projects to the apparent minor axis, the Lucy itera­
tions generate an oblate body. The predicted velocities then satisfy (v 2
OE \Gamma oe 2 ) ? 0 and
there is a natural way to parameterize the division of v 2
OE into contributions from random
and mean azimuthal streaming: we write
v OE = k
q
v 2
OE \Gamma oe 2 ) oe 2
OE j v 2
OE \Gamma v 2
OE = k 2 oe 2 + (1 \Gamma k 2 )v 2
OE ; (1)

8 Models of elliptical galaxies
where k is the free parameter introduced by Satoh (1980) and employed in Paper I. When
k ! 1 the azimuthal velocity dispersion exceeds the radial dispersion oe rather than being
smaller than oe as in the solar neighbourhood. When k = 1 the velocity dispersion tensor
is isotropic and the excess azimuthal motion is associated with azimuthal streaming.
If, by contrast, the assumed symmetry axis projects to the apparent major axis,
the Lucy iterations generate a prolate body in which v 2
OE ! oe 2 . Equation (1) now gives
rise to imaginary v OE . In fact, since the model now has less motion in the azimuthal
direction than perpendicular to it, the velocity dispersion tensor cannot be isotropic
and there is no excess azimuthal motion to associate with azimuthal streaming. In the
absence of a natural way of breaking v 2
OE into contributions oe 2
OE and v 2
OE from random
and mean motions in the azimuthal direction, the correct procedure is to project the
mean­square line­of­sight velocities onto the plane of the sky and compare the resulting
velocities with the observed mean­square line­of­sight velocity oe 2
p + v 2
p .
4 Implementation and Accuracy of the Models
One has certain choices to make when inferring a three­dimensional luminosity
distribution from surface photometry:
­ Does one start the Lucy iterations from a Jaffe or a modified Hubble model?
­ Should the initial surface brightness profiles (cuts) through the galaxy be generated
using method A or method B of Section 2?
­ How many (L) Lucy iterations should one perform?
­ The N cuts are extrapolated to values below x 2 times the sky brightness by fitting
an r 1=4 profile to each ray over the radius range from x 1 to x 2 times the sky
brightness (see Section 2). What values should one use for x 1 and x 2 ?
­ How many (N) rays should there be in the polar grids on which are represented
the surface brightnesses and luminosity densities?
One can feel confidence in the models only if the velocities they predict are substan­
tially independent of the choices one makes. We have conducted an extensive series of
tests to check that this is so. Based on our tests we estimate the error in our predicted
velocity dispersions to be ¸1%. The predicted streaming velocities, which depend on
gradients of the dispersions, are slightly more sensitive, the error being ¸3%.
Which model to start iterations from? The left panels of Figure 1 show the
difference between oblate models of NGC 4261 obtained by starting from (a) a modified
Hubble model or (b) a Jaffe model. The right panels of Fig 1a & b show the residual
surface brightnesses along four of seven rays after 1--6 Lucy iterations. At R ¸ ? 2 arcsec

R. van der Marel, J. Binney and R. L. Davies 9
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 right­hand panels show the residual surface brightnesses
¯ model \Gamma ¯ obs after each Lucy iteration, the sixth iteration leading to the full curves. Residuals are
shown for just four rays running from the major axis at top to the minor axis at the bottom. Exact
agreement corresponds to the curves running 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 rays the final density profiles
in units of 10 10 L fi kpc \Gamma3 .

10 Models of elliptical galaxies
Figure 2. Major­axis rotation curves predicted by oblate models of NGC 4261 that have been con­
structed by iterating from either a modified Hubble 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.
these are ¸ ! 0:02 mag/arcsec 2 for both final models (full curves). At smaller radii the
profiles derived from a Jaffe model predict significantly larger surface brightnesses than
na¨ive extrapolation of the data would suggest. However, it is important to bear in
mind that in projecting the models we do not take seeing into account, so the projected
model should be brighter than the data at R ¸ ! 2 arcsec. Hence the galaxy could have
a central cusp like that shown in the left panel of Figure 1b. At R ? 2 arcsec the three­
dimensional density profiles shown in the left panels of Figure 1agree well, indicating
that beyond the zone materially affected by seeing, Lucy iteration recovers profiles that
are independent of the model from which the iterations start.
An unattractive feature of profiles derived from a Jaffe model is that they tend
to cross where seeing becomes important. In the left panel of Figure 1b this occurs
at r ' 300 pc. This effect occurs rather generally when Lucy iterations are started
from a Jaffe model and is an artifact. In view of this and the fact that we can obtain
satisfactory fits to our data starting from modified Hubble models, we decided to model
all our galaxies in that way even though it is in principle preferable to start from a
model which in projection is brighter at the centre than the observations.
The difference in central surface brightness between models obtained by perform­
ing four Lucy iterations starting from either a Hubble or a Jaffe model is about 0:2
mag/arcsec 2 (right panels of Figure 1), which is comparable in size to the effect see­
ing can have in the center (PDIDC). Figure 2 shows however, that the corresponding
predicted rotation velocities differ by more than a few percent only in the inner few
arcseconds, where the rotation velocity is rising steeply; the Jaffe systems, being more
centrally concentrated, tend to have the larger v OE at a given radius. Consequently, we
conclude that the effect of seeing on our predicted velocities is very small and completely
insignificant away from the core.

R. van der Marel, J. Binney and R. L. Davies 11
Figure 3. Photometric cuts for NGC 4261, extracted from the CCD frames by method A (at left) and
method B (at right).
How to make cuts? We find that cuts are best generated by method B (in which
the cuts are generated from smooth isophotes) since the cuts are then much smoother
and there is less noise for the Lucy iterations to amplify -- see Figure 3. However, the
final velocities derived for the same galaxy by the two methods differ by ¸ ! 1%. This
demonstrates how well Lucy's scheme works on noisy data.
How many Lucy iterations? Very similar results are obtained for L ¸ ? 4 Lucy
iterations. Hence the results shown here were obtained with L = 4 as in Paper I.
Choice of extrapolation parameters: The final velocities barely depend on
the extrapolation parameters x i . We adopted x 1 = 1, x 2 = 0:1 in preference to the
values (x 1 = 2, x 2 = 0:2) employed in Paper I as giving slightly better fits to the
brightness profiles at large R.

12 Models of elliptical galaxies
Table 2 Parameters of the best­fitting models
Galaxy Shape \Delta¯ rms k i h \Gamma1
50 \Upsilon R=\Upsilon fi h \Gamma1
50 \Upsilon B =\Upsilon fi h \Gamma1
50 \Upsilon B =\Upsilon fi
(NGC) This paper This paper Lauer (1985)
3379 oblate 0:020 0:5 60 ffi 4:4 7:8 8:5
4261 prolate 0:016 \Gamma 30 ffi 6:0 10:7 10:3
4278 oblate 0:016 0:5 90 ffi 6:6 11:7 \Gamma
4472 oblate 0:026 0:4 90 ffi 4:7 8:5 10:7
Number of rays and choice of interpolation scheme: In Paper I the stream­
ing velocities predicted for slits that do not pass through the galaxy's centre undulated
and it was suggested that this phenomenon was an artifact of the interpolation scheme
employed. We have therefore tried (a) varying the number N of rays employed in the
interpolation scheme, and (b) changing the interpolation in angle from the cosine series
employed in Paper I to one based on cubic splines in `.
We found that increasing the number of rays from N = 7 to N = 10 or N = 13
simply shortened the wavelength of the unwanted undulations without diminishing their
amplitude. Thus the undulations are definitely unphysical.
Fitting a cosine series to a function f(`) specified by its value at N points ` i is
equivalent to finding a polynomial in c j cos ` that passes through the given points.
If N is large, the order of the polynomial will be large and there is a danger that
the polynomial will wiggle between the ` i . Hence a safer, if computationally more
expensive, strategy is to find a cubic spline in ` that passes through the f(` i ). When
we implemented this strategy the undulations in the predicted velocities were entirely
eliminated and all velocities became essentially independent of N for N – 7. All the
models presented in Section 5 are based on N = 7.
Using the new interpolation scheme we tested our programs (as in paper I) by
having them recover some of Satoh's (1980) analytic results. The errors were typi­
cally reduced to about 75% of the values quoted in Paper I. Further improvement can
probably only be achieved by taking a finer polar grid (increasing N ).
5 Results for individual galaxies
5.1 Oblate models
Figures 4--6 show results for NGC 3379, NGC 4278 and NGC 4472. In each plot
the full curves show the velocities expected in the isotropic case k = 1.
As in Paper I the mass­to­light ratio \Upsilon has been chosen to optimize the fit between
the predicted and observed minor­axis velocity dispersions. Table 2 lists these values of

R. van der Marel, J. Binney and R. L. Davies 13
\Upsilon, which are based on the distances given in Table 1. The last column of Table 2 shows
the corresponding results Lauer (1985) obtained using core fitting. The mass­to­light
ratios were calculated using MR (fi) = 4:31 (Allen, 1973; Bessell, 1979), (B \Gamma R) J (fi) =
1:17 (Allen 1973), and (B \Gamma R) J (galaxy) ú 1:85 (PDIDC).
In addition to k and \Upsilon, we are at liberty to optimize the agreement between theory
and observation by adjusting each galaxy's assumed inclination i.
The predicted major­axis rotation speed is affected by changes of i in two ways: in
three dimensions v OE increases as i is decreased (since the model becomes intrinsically
flatter), but the portion of v OE that projects along the line of sight diminishes with i. In
Paper I, in which the galaxies were flatter E3/E4, the first effect was dominant and the
major­axis rotation velocities increased significantly when i was lowered. In our sample
the galaxies are rounder and the two effects are more nearly in balance---in the cases
of NGCs 4278 and 4472 the projected rotation velocity rises by ¸ ! 10% as i is lowered
from 90 ffi \Gamma 60 ffi . Since our predictions for these galaxies are so insensitive to i, we show
here only results for i = 90 ffi . We were able to obtain a significantly better fit to the
data for NGC 3379 with i = 60 ffi , so we also show this smaller inclination in this case.
Figures 4 -- 6 were all obtained after four Lucy iterations from a modified Hubble
profile. Table 2 shows the rms residuals between the projected models and the initial
photometric cuts. (The latter being obtained by Method B of Section 2.)
(i) NGC 3379 Our model of NGC 3379 is based upon an average of five CCD
frames, each frame being weighted by 10 0:4\ThetaP C where PC is the frame's photometric
constant.
Figure 4 shows that with k = 1 and i = 90 ffi the predicted rotation velocities along
the 30 ffi and \Gamma30 ffi slits are too high. The dotted curves show the result of lowering k
to 0:5. The predicted rotation speeds far down the 30 ffi and \Gamma30 ffi slits now agree well
with the observations but the speeds predicted in the inner parts are too low, as is the
rotation velocity all along the major axis.
The dashed curves in Figures 4a--d correspond to the predictions for k = 0:5 and
i = 60 ffi . With these parameters the predicted rotation velocities along all slits are
consistent with the data. In DB there is an indication of rotation along the minor axis
at radii beyond ¸ 40 00 , but this is may not be significant. The only possible conflict
between theory and observation is that the predicted velocity dispersions consistently
fall with increasing R where the data points suggest that oe is independent of R.
Franx (1988) presents principal­axis velocities for NGC 3379. His rotation velocities
agree well with the data of DB but his dispersions are systematically 15% lower than
those of DB. Franx's data were obtained at higher resolution than those of DB, and in
view of the relatively low dispersion in this galaxy, Franx's data may be more reliable.
The only modification they require is setting \Upsilon and k to \Upsilon R = 3:1 and k = 0:6.

14 Models of elliptical galaxies
Figure 4. Predicted and observed kinematics of NGC 3379. The legends at the top right of each panel
indicate the relevant slit position with respect to the major axis. The ``H'' indicates that the Lucy
iteration started from a modified Hubble model. The full curves are obtained with inclination i = 90 ffi
and anisotropy parameter k = 1:0. The dotted curves are for i = 90 ffi and k = 0:5. The dashed curves
are for i = 60 ffi and k = 0:5.

R. van der Marel, J. Binney and R. L. Davies 15
We conclude that the kinematics of NGC 3379 are consistent with a distribution
function of the form f(E; L z ).
(ii) NGC 4278 Our models of NGC 4278, are based on a single long­exposure
(600 s) frame.
NGC 4278 has a very large isophote twist. The position angle of the isophotes
swings from ¸ 17 ffi for R ! 20 00 to ¸ 37 ffi for R ? 50 00 (PDIDC). Consequently the value
we adopt for the major axis position angle is rather arbitrary---we chose 22 ffi . With this
position angle the kinematic data do not correspond to the principal axes. In fact two
of the slit positions are inclined at ¸ 23 ffi to the minor axis.
Isophote twists are generally interpreted as indications of triaxiality. Are axisym­
metric models such as ours of a triaxial galaxy of any value? If the galaxy is only mildly
triaxial the answer must be yes, and it is perfectly possible for an apparently rather
round galaxy such as NGC 4278 (ffl ú 0:10, see PDIDC) to be nearly axisymmetric and
yet show a substantial isophote twist.
Figures 5a--d compare the data with the predictions of two models: k = 1 (full
curves) and 0:5 (dotted curves), both at inclination i = 90 ffi . Again the isotropic model
(k = 1) predicts too much rotation along every slit. Dropping k to k = 0:5 eliminates
this discrepancy at R Ÿ 20 00 , but further out the model still fails to reproduce the
observed decline of v to zero and perhaps even below.
The fact that the abrupt fall in v occurs where the isophotes are twisting most
rapidly suggests that this fall should be interpreted in terms of triaxiality rather than,
for example, radial variation of k. However, formally it would be straightforward to
construct a model that reproduced the observed rotation curve, including a reversal of
v at R ¸ 30 00 : the Jeans equations have nothing to say about the sign of v, so a model
can rotate one way near the centre, and in the opposite sense far out. Since the rms
line­of­sight velocity is well determined, independently of v, in such a model the velocity
dispersion would peak where v passed through zero. The major­axis dispersion data of
Figure 5cannot rule out such a peak.
As in the case of NGC 3379, there is a slight tendency for the predicted velocity
dispersions to fall towards large R more than do the data.
(iii) NGC 4472 Our models of NGC 4472 are based on a single long­exposure
frame. Figures 6a--d compare the data with the predictions of two models: k = 1 (full
curves) and k = 0:4 (dotted curves), both at inclination i = 90 ffi . Again the isotropic
model (k = 1) predicts too much rotation along every slit. The k = 0:4 model accounts
well for the rotation velocities at R ¸ ? 15 00 , but at small R still predicts more rotation

16 Models of elliptical galaxies
Figure 5. The same as Figure 4 but for NGC 4278. The dotted curve now corresponds to i = 90 ffi
and k = 0:4.

R. van der Marel, J. Binney and R. L. Davies 17
Figure 6. The same as Figure 4, but for NGC 4472. The dotted curve now corresponds to i = 90 ffi
and k = 0:5.

18 Models of elliptical galaxies
than is observed---NGC 4472 does not rotate in its inner 5 00 , and has a kinematically
decoupled core (Davies, 1989; Franx et al. 1989a).
Away from the minor axis the k = 0:4 model predicts velocity dispersions that
are slightly too high. This cannot be interpreted as a radial variation of mass­to­light
ratio, since this would effect the velocities on all slits in the same sense. Since we
see predominantly oe OE on the major axis and predominantly oe R on the minor axis, the
observed discrepancy would be explained by the presence of a third integral, since the
galaxy could then be flattened by a large oe R instead of by a large oe OE as is the case for
our models. A similar, but much more pronounced phenomenon was encountered in
Paper I in the case of NGC 720.
In summary, our model with k = 0:4 fits the kinematics of the main body of NGC
4472 adequately, but there is some evidence that the distribution function of NGC 4472
involves a third integral.
5.2 Prolate models
NGC 4261 NGC 4261 rotates at up to 100 km s \Gamma1 around its apparent major axis.
Obviously such a system cannot be successfully modelled as an oblate body. Hence in
Figure 7 we show predicted and observed rms line­of­sight velocities for two prolate
models of NGC 4261. The full curves are for the case i = 90 ffi in which we see the
galaxy broadside­on, while the dotted lines are for i = 30 ffi . Only the second, nearly
end­on model correctly predicts that the rms velocities along the two principal axes are
approximately equal. It is instructive to understand how this result comes about.
In any two­integral model, be it oblate or prolate, the minor­axis velocities tend
to be smaller than those along the major axis since in the oblate case the enhanced
azimuthal velocities are seen along the major axis, while in the prolate case one sees
the depressed azimuthal velocities along the minor axis. If the inclination i of a prolate
model is reduced, the system needs to be more elongated to project to the galaxy's
observed shape (e.g. in three dimensions our i = 30 ffi model of NGC 4261 has axis
ratio a=b ' 1:7 compared with a=b ' 1:2 in the case i = 90 ffi ), thereby increasing
the discrepancy between oe OE and oe R = oe z . However, as i is decreased, the azimuthal
velocities contribute less to the line of sight, thereby making the rms velocities along
all slits more nearly equal.
The introduction of a third integral makes it possible to decouple the dispersion
parallel to the symmetry axis from the radial dispersion parallel to the equatorial plane.
Hence in a three­integral model it should be possible to obtain comparable dispersions
along both apparent principal axes even at inclination i = 90 ffi .

R. van der Marel, J. Binney and R. L. Davies 19
Figure 7. Predicted and observed kinematics of NGC 4261. The data points show the square root of
v 2
p j oe 2 p +v 2 p , where oe p and vp are the measured velocity dispersion and streaming velocity, respectively.
The full curves show the predictions for
q
v 2
p of a prolate model seen broadside on (i = 90 ffi ). The
dotted curves show the predictions of a prolate model inclined at i = 30 ffi . The dashed curve shows the
predictions of an edge--on oblate model whose mass­to­light ratio is larger by a factor 1:1 . All models
are based on a single 600 s frame.

20 Models of elliptical galaxies
Prolate models can reproduce given velocities with smaller mass­to­light ratios \Upsilon
than oblate models. For example, the dashed curves in Figure 7show the rms velocities
predicted by an oblate model with \Upsilon R = 6:6 compared with \Upsilon R = 6:0 in the prolate
case.
6 The effect of non­zero 4` terms on the kinematics of a galaxy
We have used our technique to study the effect of 4` deviations from perfect el­
lipticity on the kinematics of a galaxy. To this end we created toy surface photometry
frames, containing ``galaxies'' whose isophotes were concentric, coaxial and built up
by superposing a fixed 4` Fourier component on a nested sequence of similar ellipses.
Three­dimensional models were then constructed from these frames as from real CCD
frames.
Some authors (e.g. Lauer, 1985; PDIDC) measure deviations from perfect ellipticity
in terms of the brightness variations around a perfect ellipse, while others (e.g. Bender
& M¨ollenhoff, 1987) work in terms of the variation of the difference ffiR between the
distances from the centre of a point on an ellipse and the point with the same azimuth
on the corresponding isophote. We have adopted the latter approach, writing
ffiR(`)
p
(a cos `) 2 + (b sin `) 2
= A 4 cos 4`: (2)
Here ` is the true polar angle rather than the elliptical angle employed by Bender &
M¨ollenhoff (1987), and on the right hand side we have dropped higher­order terms and
terms odd in `. Thus defined our parameter A 4 is for b=a ' 1 identical with the ratio
a 4 =a of Bender & M¨ollenhoff. When A 4 ? 0 the galaxy has pointed isophotes (is disky),
while when A 4 ! 0 the isophotes are boxy.
Figures 8a & b show the predicted kinematics of a model whose ellipticity is ffl = 0:2
and whose major­axis luminosity profile is the same as that of NGC 4261. Three different
cases were studied: (i) no 4` Fourier component (A 4 = 0) -- solid curves); (ii) slightly
boxy isophotes (A 4 = \Gamma0:02 -- dotted curves); (iii) slightly disky isophotes (A 4 = 0:02
-- dashed curves). Note that A 4 = 0:02 is about the highest value that is encountered
in real galaxies (Bender et al. 1989). Figure 8a is for a slit along the major axis while
Figure 8b is for a slit inclined at 45 ffi to the major axis, both under the assumptions
i = 90 ffi , k = 1 and \Upsilon R = 5:1\Upsilon fi .
Figures 8a & b show that changing A 4 by 2% changes the predicted velocity dis­
persions by ¸ 2% also. By contrast the rotation velocities change by up to 10%. On
the major axis the disky model rotates faster than the boxy one, the situation being
reversed along the 45 ffi slit since the disky model's rotation is more strongly confined to
the equatorial regions. These conclusions were found to be insensitive to the ellipticity,

R. van der Marel, J. Binney and R. L. Davies 21
Figure 8. Predicted kinematics along (a) the major axis and (b) a slit that makes a 45 ffi angle with
the major axis, for three model galaxies without isophote twists or centre shifts, with fixed ellipticity
(1 \Gamma b=a) = 0:2, the same major axis luminosity profile as NGC 4261, and with respectively no 4` Fourier
component (solid curves), fixed 4` Fourier component A 4 = \Gamma0:02 (dotted curves; slightly ``boxy''
isophotes) and fixed 4` Fourier component A 4 = 0:02 (dashed curves; slightly ``pointed'' isophotes).

22 Models of elliptical galaxies
radial density profile etc. of the model studied excepting that a boxy system rotates
faster than its disky equivalent along slits that make an angle ? fi with the major axis,
where 20 ffi
¸ ! fi ¸ ! 40 ffi depending on the system'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 Figure 9 that disky galaxies have higher rotation on the major
axis than boxy galaxies. From this figure it is apparent that any correlation between
(v=oe) \Lambda and A 4 is weak, but if present the line of regression in Figure 9would be inclined
at near 45 ffi . The dashed lines in Figure 9show that the effect seen in our models, while
qualitatively the same as that suggested by Bender et al. , gives rise to a very different
slope in a plot of (v=oe) \Lambda versus A 4 . Thus we cannot explain why so few disky galaxies
with low rotation have been observed yet.
A satisfactory test of the prediction that in elliptical galaxies the rotation velocity
will prove to fall more slowly away from the major axis in boxy galaxies than in disky
ones, must await better kinematical data than are presently available. However, Kor­
mendy & Illingworth (1982) observed such an effect in the extremely boxy bulge of NGC
4565. Binney & Petrou (1985) suggested that boxiness and cylindrical rotation are both
consequences of over­populating orbits inclined at a narrow range of angles with respect
to the equatorial plane, and argued that mergers might lead to such over­population.
7 Conclusions
We have used the technique of Paper I to predict kinematical quantities from CCD
surface photometry under the assumptions of (i) axisymmetry, (ii) f = f(E; L z ) and
(iii) constant mass­to­light ratio. Four galaxies were studied: NGC 3379, NGC 4278 and
NGC 4472 (to which oblate models were fitted) and NGC 4261 (to which a prolate model
was fitted, since it rotates around its apparent major axis). For oblate models we em­
ployed Satoh's (1980) parameter k to split the predicted v 2
OE into streaming and random
motion, whereas for prolate models we merely calculate the predicted rms line­of­sight
velocities. The kinematical predictions have been compared with the observations of
DB and Franx et al. (1989a).
For all the oblate galaxies the data are incompatible with an isotropic model. Mod­
els in which the tangential velocity dispersion exceeds those in the radial and vertical
directions are remarkably successful. Such a model provides an acceptable fit to the ob­
servations for NGC 3379. One of our models can also fit the observations for the main
body of NGC 4472 but fails to account for NGC 4472's kinematically decoupled core.
There is a slight indication for the presence of a third integral in NGC 4472. NGC 4278
has a rotation curve that appears to change sign soon after peaking at the radius at
which the photometry reveals a marked isophote twist. It would be straightforward to
reproduce the suggested counter­rotation of the outer portion of this galaxy if one were
to introduce an additional parameter into the models. However, axisymmetric models

R. van der Marel, J. Binney and R. L. Davies 23
Figure 9. Plot of a 4 =a ' A 4 versus (vm =oe) \Lambda . A 4 measures the amplitude of cos 4` deviations of
the isophotes from ellipses [equation (2)]. (vm =oe) \Lambda 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 C 4 parameters of PDIDC were transformed to a 4 =a parameters by using the galaxies that
both samples have in common as calibration. This figure shows a weaker correlation than the plot of
Bender et al. largely because we have plotted (v=oe) \Lambda rather than its logarithm. The dashed lines show
the movement predicted by our models: a change of 2% in A 4 changes the major axis rotation by 10%.
such as ours cannot model the observed isophote twist, and it seems probable that both
the kinematic and the photometric anomalies are caused by triaxiality.
The data on NGC 4261 can be fit with a nearly end­on prolate model, with axis
ratio ' 1:7. Three­integral models of NGC 4261 would probably not be confined to
near end­on orientations.
An extensive series of tests of our technique leads to the following conclusions:
(i) changing the projected shape of a perfectly elliptical model galaxy, by including a
4` Fourier component with an amplitude of 2%, changes the velocity dispersions
only slightly but can change the rotation velocities by ¸ 10%;
(ii) A model galaxy whose projected shape is disky, rotates faster on the major axis
than the equivalent boxy model. This effect is consistent with known data on
elliptical galaxy samples, but does not explain the absence in the observations of

24 Models of elliptical galaxies
slowly rotating, disky galaxies;
(iii) The rotation velocity in a disky model falls off much more steeply away from the
major axis than the rotation velocity of the equivalent boxy model. To observe this
effect in elliptical galaxies, better kinematical data are needed.
Acknowledgements
The authors want to thank Reynier Peletier for providing the CCD surface pho­
tometry frames from PDIDC, and Robert Jedrzejewski for providing his galaxy surface
photometry fortran programs. RvdM wishes to thank the Sterrewacht Leiden and the
Departments of Theoretical Physics and Astrophysics of Oxford University for grants
to work on this project in Oxford. Furthermore he wishes to thank Merton College,
Oxford for its hospitality during his stays.
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