Triaxial orbit based galaxy models with an application to the (apparent) decoupled core galaxy NGC 4365

Abstract

We present a flexible and efficient method to construct triaxial dynamical models of galaxies with a central black hole, using Schwarzschild's orbital superposition approach. Our method is general and can deal with realistic luminosity distributions, which project to surface brightness distributions that may show position angle twists and ellipticity variations. The models are fit to measurements of the full line-of-sight velocity distribution (wherever available). We verify that our method is able to reproduce theoretical predictions of a three-integral triaxial Abel model. In a companion paper (van de Ven, de Zeeuw & van den Bosch), we demonstrate that the method recovers the phase-space distribution function. We apply our method to two-dimensional observations of the E3 galaxy NGC 4365, obtained with the integral-field spectrograph SAURON, and study its internal structure, showing that the observed kinematically decoupled core is not physically distinct from the main body and the inner region is close to oblate axisymmetric. Comment: 21 Pages, 14 (Colour) Figures, Companion paper is arXiv:0712.0309 Accepted to MNRAS. Full resolution version at http://www.strw.leidenuniv.nl/~bosch/papers/RvdBosch_triaxmethod.pdf

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Mon. Not. R. Astron. Soc. 385, 647–666 (2008)doi:10.1111/j.1365-2966.2008.12874.x
Triaxial orbit based galaxy models with an application to the (apparent)
decoupled core galaxy NGC 4365
R. C. E. van den Bosch,1?G. van de Ven,1,2,3† E. K. Verolme,1,4M. Cappellari1,5
and P. T. de Zeeuw1,6
1Sterrewacht Leiden, Universiteit Leiden, Postbus 9513, 2300 RA Leiden, the Netherlands
2Department of Astrophysical Sciences, Peyton Hall, Princeton, NJ 08544, USA
3Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
4TNO Defense, Security and Safety, Lange Kleiweg 137, 2280 AA, Rijswijk, the Netherlands
5Sub-Department of Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH
6European Southern Observatory, D-85748 Garching bei M¨ unchen, Germany
Accepted 2007 December 14. Received 2007 December 3; in original form 2006 November 29
ABSTRACT
We present a flexible and efficient method to construct triaxial dynamical models of galaxies
withacentralblackhole,usingSchwarzschild’sorbitalsuperpositionapproach.Ourmethodis
generalandcandealwithrealisticluminositydistributions,whichprojecttosurfacebrightness
distributionsthatmayshowpositionangletwistsandellipticityvariations.Themodelsarefitto
measurementsofthefullline-of-sightvelocitydistribution(whereveravailable).Weverifythat
our method is able to reproduce theoretical predictions of a three-integral triaxial Abel model.
In a companion paper by Ven, de Zeeuw & van den Bosch, we demonstrate that the method
recovers the phase-space distribution function. We apply our method to two-dimensional ob-
servationsoftheE3galaxyNGC4365,obtainedwiththeintegral-fieldspectrographSAURON,
and study its internal structure, showing that the observed kinematically decoupled core is not
physically distinct from the main body and the inner region is close to oblate axisymmetric.
Key words: galaxies: elliptical and lenticular, cD – galaxies: kinematics and dynamics –
galaxies: structure.
1 INTRODUCTION
Binney (1976, 1978) argued convincingly that elliptical galaxies
may well have triaxial intrinsic shapes, based on the observed
slow rotation of the stars (Bertola & Capaccioli 1975; Illingworth
1977),thepresenceofisophotetwistsinthesurfacebrightness(SB)
distribution (e.g. Williams & Schwarzschild 1979), the presence
of velocity gradients along the apparent minor axis (‘minor-axis
rotation’,Schechter&Gunn1978),andevidencefromN-bodysim-
ulations (Aarseth & Binney 1978). Schwarzschild’s (1979, 1982)
numerical models demonstrated that such systems can be in dy-
namical equilibrium, and suggested that their observed kinematics
can be rich (see also e.g. Statler 1991). This is supported by the dis-
covery of kinematically decoupled cores (KDCs) in the late 1980s
(Bender 1988; Franx & Illingworth 1988) and, more recently, by
observations with integral-field spectrographs such as SAURON,
which reveal that some ellipticals have point symmetric rather than
bisymmetric velocity fields, and often contain kinematically decou-
pled components (e.g. Emsellem et al. 2004). This means that these
galaxies are not axisymmetric.
?E-mail: bosch@strw.leidenuniv.nl
†Hubble Fellow.
Subsequent work on triaxial dynamical models focused mostly
on models with a cusp in the central density profile, on the effect of
a central black hole, and on the range of shapes for which triaxial
models could be in dynamical equilibrium (e.g. Gerhard & Binney
1985; Levison & Richstone 1987; Statler 1987; Hunter & de Zeeuw
1992; Schwarzschild 1993; Merritt & Fridman 1996; Siopis 1998;
Terzi´ c 2002). With the exception of studies of the Galactic bulge
(Zhao 1996; H¨ afner et al. 2000), most of this work was restricted to
finding (numerical) distribution functions (DFs) consistent with a
given triaxial density. This showed that many different DFs may re-
produce the same triaxial density, and that these dynamical models
allhavedifferentobservablekinematicproperties,butdetailedcom-
parison to observations received little attention (Arnold, de Zeeuw
& Hunter 1994; Mathieu & Dejonghe 1999). Ad hoc kinematic
models were used to constrain the distribution of intrinsic shapes
(Binney 1985; Franx, Illingworth & de Zeeuw 1991) or of individ-
ual objects (Tenjes et al. 1993; Statler 1994b; Statler, Dejonghe &
Smecker-Hane 1999; Statler 2001; Statler et al. 2004).
The possibility to measure accurate line-of-sight velocity distri-
butions (LOSVDs) in elliptical galaxies from observations of the
stellar absorption lines (e.g. Bender 1990; van der Marel & Franx
1993), and the realization that these are the only way to distinguish
radial variations in mass-to-light ratioM/L from radial variations in
the anisotropy of the orbital structure (Dejonghe 1987; Gerhard
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R. C. E. van den Bosch et al.
1993), led to the development of detailed spherical, and subse-
quently axisymmetric, numerical dynamical models aimed to fit
all these kinematic measurements. These are generally constructed
withavariantofSchwarzschild’s(1979)orbitsuperpositionmethod,
in which occupation numbers are found for a representative library
of orbits calculated in the gravitational potential of the galaxy. The
aim is to measure the mass of the central black holes (van der Marel
etal.1998;Gebhardtetal.2003;Valluri,Merritt&Emsellem2004;
Valluri et al. 2005; Shapiro et al. 2006; van den Bosch et al. 2006),
to deduce the properties of dark haloes (e.g. Rix et al. 1997; Cretton
etal.1999;Gerhardetal.2001;Thomasetal.2005),andtoderivethe
internal orbital structure and intrinsic shape (Cappellari et al. 2002;
Verolme et al. 2002; Krajnovi´ c et al. 2005; Cappellari et al. 2006;
vandeVenetal.2006).Somegalaxiesdisplaysignificantsignatures
of non-axisymmetry, suggesting they are intrinsically triaxial.
The logical next step is to construct realistic triaxial mod-
els, which fit the details of the observed SB, including isophote
twists, nuclear stellar discs and a central cusp, as well as the two-
dimensional kinematic measurements. This is a non-trivial under-
taking, as the parameter range to be explored for a given model is
significantly larger than in axisymmetric geometry, and the internal
dynamical structure is more complicated, as it includes four major
orbit families, a host of minor families and chaotic orbits. However,
the ability to construct such models will make it possible to derive
reliable intrinsic parameters for giant elliptical galaxies, and opens
thewayforasystematicexplorationoftheirproperties.Inthispaper,
we describe a practical method for doing this, and report an appli-
cation which accurately reproduces the two-dimensional kinematic
measurements of the triaxial E3 galaxy NGC 4365, obtained with
SAURON. In the companion paper (van de Ven, de Zeeuw & van
den Bosch 2008, hereafter vdV08) we apply the method to analyti-
cal triaxial three-integral models and show that it reliably recovers
the input three-integral DF.
We start with a short section on Schwarzschild’s method (Sec-
tion 2), which includes a brief overview of our implementation. We
then give a step-by-step description of the main properties of our
formalism (Sections 3–5). In Section 6 we test the method, includ-
ing the ability to recover the global input parameters. We construct
a triaxial model for NGC 4365 in Section 7, and we summarize our
conclusions in Section 8.
2 SCHWARZSCHILD’S METHOD
2.1 Brief historical overview
Schwarzschild’s (1979) orbit superposition method is a flexible
method to build dynamical models of early-type galaxies. The
original implementation was aimed at reproducing a given triax-
ial density distribution. Subsequently, it was applied to a large
variety of density distributions, from spherically and axially sym-
metric (Richstone 1980, 1982, 1984; Richstone & Tremaine 1984;
Levison & Richstone 1985; Valluri et al. 2004) to triaxial shapes
(e.g. Schwarzschild 1982; Vietri 1986; Levison & Richstone 1987;
Statler 1987; Schwarzschild 1993; Merritt & Fridman 1996; H¨ afner
et al. 2000; Siopis & Kandrup 2000).
Pfenniger (1984) showed that it is possible to include measure-
ments of the mean line-of-sight velocity and the second velocity
moment,providedthatthetruesecondvelocitymoment?v2?isused
andnotthevelocitydispersionσ2=?v2?−?v?2.Thereasonforthis
requirement is that the dispersion depends quadratically on the first
velocity moment and can therefore not be included in a linear orbit
superposition method (but see Dejonghe 1989). Zhao (1996) used
this principle to build triaxial models of the Galactic bulge. At the
same time, theoretical (Dejonghe 1987) and observational (Franx
& Illingworth 1988) investigations showed that LOSVDs are gen-
erally not Gaussian-shaped and higher order velocity moments are
required to describe the true profile. This stimulated the use of the
so-called Gauss–Hermite (GH) moments (Gerhard 1993; van der
Marel & Franx 1993).
The first implementations of Schwarzschild’s method that used
additional kinematic information were designed for the modelling
of spherical galaxies (Richstone & Tremaine 1984; Rix et al. 1997).
Orbits in these models obey four integrals of motion: the energy E
and all three components of the angular momentum L = (Lx, Ly,
Lz). While useful, this software was still of limited applicability, as
most galaxies are not round, but axisymmetric or triaxial. Orbits
in oblate axisymmetric galaxies conserve at least the two classical
integrals E and Lz(which is the component of the angular momen-
tum along the short axis), while it has been known for a long time
that most orbits in our Galaxy conserve an additional non-classical
third integral of motion (e.g. Contopoulos 1960; Ollongren 1962).
A more general version of the Schwarzschild software was there-
fore developed to model axisymmetric galaxies with three-integral
DFs (van der Marel et al. 1998; Cretton et al. 1999; Thomas et al.
2005). Results that were obtained with the extended Schwarzschild
method indeed showed that the third integral is an essential ingre-
dient of realistic axisymmetric galaxy models (van der Marel et al.
1998; Verolme et al. 2002), that we can derive information on the
phase-space structure of galaxies (Cappellari et al. 2002; Krajnovi´ c
et al. 2005), that we can use the method to measure the mass of
the central black hole in galaxies (Gebhardt et al. 2003) and that
propermotionkinematicobservationscanbeused(vandeVenetal.
2006), provided that the models have sufficient internal freedom,
e.g. the total number of orbits is large enough (Cretton & Emsellem
2004; Thomas et al. 2004; Richstone et al. 2004; Valluri et al. 2004;
Magorrian 2006).
2.2 Generalization to triaxial geometry
The method described here uses many of the ideas and algorithms
described in Rix et al. (1997), van der Marel et al. (1998), Cretton
et al. (1999), Verolme et al. (2002) and Cappellari et al. (2006). The
computer program for triaxial geometry was written largely from
scratch.
The standard implementation of the extended Schwarzschild
method starts from a SB distribution, which we parametrize with
a sum of Gaussians (Section 3.1). The intrinsic mass distribution
and potential are then obtained by deprojecting the surface density,
whichrequiresachoicefortheviewingangle(s)alongwhichtheob-
ject is observed (Section 3.3). The potential calculation is outlined
in Section 3.8.
In the potential, the initial conditions for a representative orbit
libraryarefound(Section4).Theseorbitalcomponentsmustinclude
all types of orbits that the potential supports, to avoid any bias
(e.g. Thomas et al. 2004).
Schwarzschild’s method tries to find a steady-state model of a
galaxy, requiring orbital building blocks to be time independent.
We integrate the orbits for a fixed time of 200 times the period of a
closed elliptical orbit with the same energy.
During orbit integration, the intrinsic and projected properties
are stored on grids, in order to allow for comparison with the data
(Section 4.5). The quantities that will be compared to observations
are spatially convolved with the same point spread function (PSF)
as the observations.
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After orbit integration, the superposition of orbits whose proper-
ties best match the observational data is determined. The superpo-
sition can be constructed by using linear or quadratic programming
(Schwarzschild 1979, 1982; Vandervoort 1984; Dejonghe 1989;
Schwarzschild 1993), maximum entropy methods (Richstone &
Tremaine 1988; Gebhardt et al. 2003; Thomas et al. 2004) or with a
least-squares solver as was used in many of the axisymmetric three-
integralimplementations(Rixetal.1997;vanderMareletal.1998;
Cappellarietal.2006).Hereweuseaquadraticprogrammingsolver
as it finds the best-fitting superposition in a least-squares sense,
while allowing for additional constraints (Section 5.1).
3 MASS PARAMETRIZATION, POTENTIAL
AND ACCELERATIONS
In this section, we describe the method that we use to obtain a
triaxialmassmodelfromtheobservedSB.Wedescribeaconvenient
mass parametrization and derive the corresponding potential and
accelerations. A summary of symbols introduced in this section is
given in Table 1.
3.1 The MGE parametrization
In order to derive the intrinsic luminosity density from the observed
galaxy SB, a deprojection is required. For a spherical galaxy, this
leadstoauniquesolution(Binney&Tremaine1987).Thisisnotthe
case for an axisymmetric object, unless it is seen edge-on (Rybicki
1987). This non-uniqueness is even stronger for triaxial shapes,
where the deprojection is not unique from any viewing direction
(e.g. Gerhard 1996). For this reason, the assumption that an object
istriaxialisnotsufficienttouniquelyrecovertheintrinsicluminosity
density from an observed image, and additional assumptions have
to be made.
The simplest option is to assume that the intrinsic density is strat-
ified on similar triaxial ellipsoids. The isophotes that are produced
by such a mass model are similar coaxial ellipses (Contopoulos
1956; Stark 1977), which is approximately consistent with obser-
vations of some galaxies. However, many objects display position
angle twists and ellipticity variations, which cannot be reproduced
by these simple models. More flexible mass models are therefore
required to reproduce these observed features.
A general approach to the triaxial deprojection problem would
be to use fully non-parametric methods (e.g. Scott 1992). This has
already been done in the axisymmetric case by Romanowsky &
Kochanek (1997) and in the triaxial case by Bissantz & Gerhard
(2002). Unfortunately, these methods are complicated, require a
Table 1. Summary symbols introduced in Section 3.
Symbol Definition
(x, y, z)
(x?, y?, z?)
(ϑ, ϕ, ψ)
(p, q, u)
q?
Lj, σ?
Intrinsic coordinate system
Projected coordinate system
Viewing angles
Intrinsic shape parameters
Averaged projected flattening
Projected luminosity, dispersion, flattening
of individual Gaussians
Shape parameters of individual Gaussians
Intrinsic dispersion of individual Gaussians
Misalignment angle of individual Gaussians
Isophotal twist of individual Gaussians
j, q?
j
(pj, qj, uj)
σj
ψ?
j
?ψ?
j
significant amount of time before convergence is reached, and do
not always provide a global solution.
We therefore decided to parametrize the mass distribution
by using a multi-Gaussian expansion (MGE; Monnet, Bacon &
Emsellem 1992; Emsellem, Monnet & Bacon 1994; Cappellari
2002). We assume that the intrinsic density can be described as
a sum of coaxial triaxial Gaussian distributions. The Gaussians do
not constitute a complete basis of functions and therefore cannot re-
produce any arbitrary positive density distribution. However, MGE
models can reproduce a large variety of densities, which appears re-
alistic when projected along any viewing direction, including mass
models with radially varying triaxiality, multiple photometric com-
ponents and discs.
Accordingly, we write the triaxial MGE luminosity density as
ρ(x, y,z) =
N
?
j=1
(M/L)
Lj
(σj
√2π)3pjqj
?
×exp
?
−
1
2σ2
j
x2+y2
p2
j
+z2
q2
j
??
,
(1)
where N is the number of required Gaussian components, Ljis the
luminosity of the jth Gaussian, pjand qjare the axial ratios andσjis
thecorrespondingdispersionalongthex-axis.Moreover,M/Listhe
mass-to-lightratio,and(x,y,z)isasystemofcoordinatescentredon
the common origin of the Gaussians and aligned with the common
principal axes of the Gaussians.
3.2 Transformation from intrinsic to projected coordinates
To be able to compute the projection of the density in equation (1)
on the sky plane, we introduce a new coordinate system, (x?, y?, z?)
as defined in Binney (1985). Here, z?is located along the line of
sight and x?is in the (x, y) plane.
To go between these coordinate systems two transformations are
needed. First, a projection to the sky plane given by a projection
matrix
P =



−sinϕ
−cosϑ cosϕ
sinϑ cosϕ
cosϕ
0
−cosϑ sinϕ
sinϑ sinϕ
sinϑ
cosϑ

,

(2)
where the two usual spherical coordinates (ϑ, ϕ) define the orien-
tation of the line of sight with respect to the principal axes of the
object. For example, (90◦, 0◦), (90◦, 90◦), (0◦, 0◦, ... , 90◦) are
the views down the long, intermediate and short axis, respectively.
Secondly, a rotation on the sky plane is given by the matrix
R =



sinψ
−cosψ
sinψ
0
cosψ
0
001

.

(3)
The angle ψ is required to specify the rotation of the object around
the line of sight. The rotation ψ is chosen to align the major axis of
the projected ellipse (of the innermost MGE component, see equa-
tion 6 below) with the x?-axis. For an oblate axisymmetric intrinsic
shape ψ equals 90◦.
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R. C. E. van den Bosch et al.
3.3 The observed surface brightness of an MGE
TheprojectedSBthatcorrespondstothedensityofequation(1)can
be written as a sum of two-dimensional Gaussians of the form
?
with
SB(R?,θ?) =
N
j=1
Lj
2πσ?2
jq?
j
exp
?
−
1
2σ?2
j
?
x?2
j+
y?2
q?2
j
j
??
,
(4)
x?
j= R?sin(θ?− ψ?
where (R?, θ?) are polar coordinates on the sky plane. The Gaussian
components have axial ratio 0 ? q?
majoraxis,andpositionangleψ?
the y?-axis to the major axis of each Gaussian. The misalignment
angle ψ?
y?-axis is not observable. We define
j) and
y?
j= R?cos(θ?− ψ?
j),
(5)
j? 1, dispersion σ?
j,measuredcounterclockwisefrom
jalong the
jcannot be measured directly as the position of the intrinsic
ψ?
j= ψ + ?ψ?
where ?ψ?
measured directly.
j
with
?ψ?
1≡ 0,
(6)
jis the isophotal twist of each Gaussian, which can be
3.4 From projected to intrinsic shape
To determine the parameters of the Gaussians in equation (1), we
fit the two-dimensional MGE model of equation (4) to the observed
SB. After assuming the space orientation (ϑ, ϕ, ψ) of the galaxy,
the relations between the observed quantities (σ?
intrinsic ones (σj, pj, qj) are given by Cappellari et al. (2002; for a
different formalism see Monnet et al. 1992):
?2cos2ψ?
?2cos2ψ?
?
where δ?
j
pression factor, which together with the dimensionaless parameters
pjandqjdefinetheintrinsicshape.Themathematicalconstraintqj>
0 and pj> 0 (or the stronger and more physical constraint qj> 0.2
and pj> 0.4, which gives the range of reasonable axis ratios for an
elliptical galaxy, Binney & de Vaucouleurs 1981) implies that each
Gaussian can be deprojected only for a limited range of orientations
(see also Monnet et al. 1992). The orientations for which the whole
MGEmodelcanbedeprojectedarelocatedintheintersectionofthe
regions that are allowed by the individual Gaussian components.
j, q?
j, ψ?
j) and the
1−q2
j=
δ?
jj+sin2ψ?
jcosψ?
j(secϑ cotϕ−cosϑ tanϕ)?
j(cosϑ cotϕ−secϑ tanϕ)?
2sin2ϑ?δ?
δ?
j
2sin2ϑ?δ?
1
q?
j
j(cosψ?
j+cotϕ secϑ sinψ?
j)−1?, (7)
j)−1?, (8)
p2
j−q2
j=
j+sin2ψ?
jcosψ?
j(cosψ?
j+cotϕ secϑ sinψ?
u2
j=
p2
jcos2ϑ+q2
jsin2ϑ(p2
jcos2ϕ + sin2ϕ),
(9)
j= 1 − q?
2, and uj≡ σ?
j/σj, the scalelength projection com-
3.5 Constructing a realistic triaxial MGE
The individual Gaussian components have no direct physical sig-
nificance, but their parameters provide constraints on other, more
important quantities. We must therefore be careful that the MGE
model does not result in spurious conditions on the physical prop-
erties of the galaxy. The allowed intrinsic orientation of the galaxy
depends on the axis ratios of the Gaussians in the superposition. It
can be easily verified numerically that the region in the space of the
rotationangles(ϑ,ϕ,ψ?
flattening q?
(q?
j)forwhichaGaussianwithagivenobserved
jcan be deprojected increases with q?
j= 1) can be deprojected for any assumed intrinsic orientation,
j: a round Gaussian
Figure 1. Contours of the deprojectable volume of a hypothetical MGE
as a function of the observed isophotal twist and flattening. The horizontal
axis shows the minimum projected flattening and the vertical axis the maxi-
mum isophotal twist of all the Gaussians in the MGE. The labels denote the
percentage of Euler angle space that can be deprojected.
while an extremely flat one (q?
the object is observed along one of its principal planes. Moreover,
when a Gaussian has an photometric twist ?ψ?
other Gaussians in the MGE, than the allowed deprojection region
(ϑ, ϕ, ψ?
The Gaussians in a given MGE superposition generally have dif-
ferent values of q?
wholecanonlybedeprojectedforanglesthatappearintheintersec-
tion of the allowed individual regions (ϑ, ϕ, ψ?
of the individual Gaussians. The largest deprojectable volume is
obtained by maximizing min {q?
while still fitting the photometry within a certain accuracy (see also
Cappellari2002).ThisisverifiednumericallyinFig.1whichshows
contoursoftheallowedvolumeavailablefordeprojection,forgiven
minimum flattening and isophotal twist of an MGE model.
TheMGEmodelsthatareobtainedinthiswayhave,byconstruc-
tion, the largest set of orientations for which a triaxial deprojection
is possible. For any given orientation, the roundest projection on
the sky corresponds to the roundest triaxial intrinsic density. This
means that this MGE model will be the roundest one that fits the
observations for any given intrinsic orientation of the galaxy. Very
boxy or very discy models are therefore excluded.
j? 1) can only be deprojected when
jwith respect to the
j) becomes even smaller.
jand ψ?
j. This means that the MGE model as a
j) of the deprojection
j} and minimizing max{|?ψ?
j|},
3.6 Light to mass
At this stage it is possible to add the contribution of invisible mass
to the gravitational potential of the model. This can be done by
creating two MGE models for the galaxy: one for the gravitating
matter and the other for the visible light. The matter MGE is then
usedforthecalculationofthepotentialandthelightMGEisusedto
reproducetheintrinsicandobservedlightdistribution.Tosimulatea
radial M/L profile one can construct a matter MGE by multiplying
the luminosity of each Gaussian of the light distribution with the
desired (M/L)jat that radius, see e.g. van den Bosch et al. (2006).
In this way, it is possible to construct a large range of potentials or
SB distributions, as long as the matter and light distributions can be
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represented by Gaussians. Alternatively one could make (M/L) in
equation (1) a function of (x, y, z).
3.7 Deprojection
The parameter range that has to be explored when fitting general
triaxial models to observations of elliptical galaxies is large. Two
axis ratios and three angles are needed to specify the intrinsic shape
and orientation of a triaxial ellipsoid, while only the projected flat-
tening q?
axis can be deduced from photometric observations. As there is a
relation between q?
rameters of the galaxy, the allowed range of intrinsic shapes can be
constrained to some degree, but a large freedom remains. In some
cases, additional information, such as the value of the kinematic
offset angle or the relative position of a gas disc or dust lane, can
provide further constraints (e.g. Bertola et al. 1991). However, un-
less two perpendicular gas discs are observed, no unique intrinsic
shapecanbededuceddirectlyfromtheobservations.Themethodto
obtain triaxial MGE mass models from SB data that were described
in the above does not solve these problems. However, it produces a
range of regular, searchable and well-behaved triaxial density dis-
tributionsthatareconsistentwiththeobservedSB,whilebeingeasy
to handle computationally.
The shape of the reconstructed potential used in our models is di-
rectly related to the viewing angles by equations (7)–(9). By chang-
ing the viewing angles the potential of the model changes with it.
However, the intrinsic shape parameters are much more natural pa-
rameters than the viewing angles (ϑ, ϕ, ψ), as they influence the
appearance of the orbits, and thus the kinematics they represent,
much more directly. For example, for an axisymmetric deprojection
of the SB the angle ϕ has no meaning, as rotating along the symme-
try axis does not change anything. Therefore we choose to study the
effectsofthedeprojectionintermsoftheintrinsicshapeparameters
(p, q, u), which can be computed from the viewing angles (ϑ, ϕ, ψ)
given the averaged flattening q?of the galaxy. The conversion from
the intrinsic shape parameters to viewing angles (which is the input
for the models) is given by1
jand the relative position angle ?ψ?
jof the projected major
j, the viewing angles and the intrinsic shape pa-
cos2ϑ =(u2− q2)(q?2u2− q2)
(1 − q2)(p2− q2)
tan2ϕ =(u2− p2)(p2− q?2u2)(1 − q2)
tan2ψ =(1 − q?2u2)(p2− q?2u2)(u2− q2)
(1 − u2)(u2− p2)(q?2u2− q2)
valid for q ? p ? 1,q ? q?and max (q/q?, p) ? u ? min (p/q?, 1).
Four of the eight possible solutions are unphysical or have q > p.
The valid solutions are
,
(1 − u2)(1 − q?2u2)(p2− q2),
,
(10)
{
{ π − ϑ , ϕ , ψ },
{
{ π − ϑ , −ϕ , −ψ }.
They represent the same intrinsic shape only seen from the oppo-
site side and mirror images. They are thus identical and need not
be modelled separately. However, for Gaussians in the MGE with
ϑ, −ϕ , ψ },
ϑ, ϕ , −ψ },
(11)
1The quantities u2and u2q?2are recognized as the conical coordinates µ
andν,withwhichtheprojectedpropertiesofatriaxialellipsoidofaxisratios
p and q can be evaluated in an elegant manner (e.g. Franx 1988).
isophotal twist (|?ψ?
viewing angle ψ and −ψ are not the same, since the ?ψ?
deprojects (equations 7–9) them to a different (pj, qj, uj), and thus
a different intrinsic shape. Hence, in the case of isophotal twist we
have to consider one solution from the first two lines in (11), and
one from the last two lines.
To convert from (p, q, u) to (pj, qj, uj) one uses equation (10) and
the averaged flattening q?to go to (ϑ, ϕ, ψ), and then equation (6)
(and the observed isophotal twists) to go to ψ?
q?
To find the best-fitting intrinsic shape and corresponding viewing
anglesforanobservedgalaxytheparameterspacehastobesearched
effectively. Since the models are computationally expensive the
numberofmodelscannotbetoolarge.TheMGEparametrizationof
the SB already excludes some viewing angles since their deprojec-
tionisunphysical(p<0.4orq<0.4).Especiallyanisophotaltwist
reducestheallowedviewingangles.Butalsothesamplinginthein-
trinsic shape, instead of viewing angles, helps reducing the number
of models required, as this will avoid having models with (nearly)
the same intrinsic shape. Overall, for galaxies which are mildly flat-
tenedapproximately100distinctmodelsareneededwhensampling
(p, q, u) in steps of 0.05.
j| > 0) the intrinsic shape of the models with
joffset
j. From there one uses
jand equations (7)–(9) to go to (pj, qj, uj).
3.8 Potential and accelerations
The next step is to calculate the potential that corresponds to the
mass distribution of equation (1). This is done by using the classical
Chandrasekhar (1969) formula for the potential that corresponds to
a density stratified on similar concentric ellipsoids. This results in
(Emsellem et al. 1994)
?1
with
?
and
?
?(1 − δjτ2)(1 − ?jτ2)
where
V(x, y,z) = −
N
?
j=1
V0,j
0
dτF(x, y,z,τ),
(12)
V0,j= (M/L)
2
π
GLj
σj
(13)
F(x, y,z,τ) =
exp
−τ2
2σ2
j
?
x2+
y2
1 − δjτ2+
z2
1 − ?jτ2
??
,
(14)
δj= 1 − p2
Here, G is the gravitational constant and M/L is the mass-to-light
ratio. Equation (12) has no simple analytic expression and must
be evaluated numerically. The integrand is badly behaved in the
centralandoutermostregions.Itisthereforemoreefficienttoreplace
equation (12) by analytical approximations in those regions.
The central density of each Gaussian can be expanded as
j
and
?j= 1 − q2
j.
(15)
ρj(x, y,z) = ρ0,j
∞
?
n=0
αnm2n,
(16)
with m2= x2+ y2/p2+ z2/q2and
1
n!
2σ2
j
αn=
?
−
1
?n
.
(17)
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R. C. E. van den Bosch et al.
This expansion generates a potential (e.g. equation 29 of de Zeeuw
& Lynden-Bell 1985)
?
1
8σ4
j
Vj(x, y,z) = −V0,j
√?j
Fj−
1
2σ2
j
?A1,jx2+ A2,jy2+ A3,jz2?
?A11,jx4+ A22,jy4+ A33,jz4
+2A12,jx2y2+ 2A13,jx2z2+ 2A23,jy2z2?
+
+ ···
?
(18)
The index symbols Aiand Ailare given in Chandrasekhar (1969).
Foramoderatelytriaxialmodel,theexpression(18)differslessthan
10−4from the exact potential for r < 0.1σj, with r2≡ x2+ y2+
z2. A higher order Taylor expansion does not extend this limiting
radius significantly.
Thepotentialoutsider>45σjcanbeapproximatedtowithin10−4
by the monopole term in a multipole expansion, which corresponds
to the potential of a central point mass with mass equal to that of
the Gaussian
G Lj
?
Higher order multipole terms hardly extend the range of applicabil-
ity. Using equations (18) and (19), numerical integrations only have
to be performed over the range 0.1σj< r < 45σj, which speeds up
the orbit integration significantly.
The contribution of a central supermassive black hole is repre-
sented by a Plummer potential
Vj(x, y,z) = −(M/L)
x2+ y2+ z2.
(19)
V•(x, y,z) = −
G M•
?
r2
s+ x2+ y2+ z2,
(20)
inwhichM•isthemassoftheblackholeandrsisasofteninglength,
which can be set to a non-zero value to prevent the central poten-
tial to be infinite. In most applications, this smoothing is used, and
rsis chosen to be significantly smaller than the smallest kinematic
aperture. The black hole potential is added to V(x, y, z) from equa-
tion (12) to obtain the total galaxy potential. A separate dark halo
potential can also be added at this stage, using either the MGE (see
Section 3.6) or another, specific, expression.
The orbit integration is performed in Cartesian coordinates. The
stellar accelerations are given by the derivatives of equation (12)
with respect to x, y and z. Similar to what is done for the poten-
tial, the numerical calculation of the accelerations in the central and
outerregionsofthemodelarereplacedby,respectively,aTaylorex-
pansion and the dipole approximation. If we differentiate the terms
in equation (18), we obtain as first-order approximations
?
yV0,j
σ2
2σ2
j
?
ax,j=
xV0,j
σ2
j√?j
A1,j−
1
2σ2
j
?A11,jx2+A12,jy2+A13,jz2??
?A21,jx2+A22,jy2+A23,jz2??
?A31,jx2+A32,jy2+A33,jz2??
,
ay,j=
j√?j
?
A2,j−
1
,
az,j =
zV0,j
σ2
j√?j
A3,j−
1
2σ2
j
,
(21)
where we have suppressed the dependence of the left-hand side of
the equations on (x, y, z). These expressions differ less than a factor
of 10−4from the exact accelerations inside r < 0.1σj. As before,
outside r > 45σjthe accelerations can be approximated to within
10−4via the monopole term
aj,ξ= (M/L)
ξ G Lj
(x2+ y2+ z2)3,
?
ξ G M•
s+ x2+ y2+ z2)3,
ξ = x, y,z.
(22)
Similarly, the accelerations due to the black hole are given by
a•,ξ=
?
(r2
ξ = x, y,z.
(23)
To make accurate and fast orbit integration possible, we interpolate
the total accelerations (ax, ay, az) on to a three-dimensional polar
grid linearly in [log(r), θ, φ]. For each grid point (r, θ, φ) we store
[log(−ax/x), log(−ay/y), log(−az/z)]. We can then compute the ac-
celerations (ax, ay, az) at point (r, θ, φ) with trilinear interpolation.
After the interpolation grid has been computed we ensure that the
minimum relative accuracy is better than 10−4.
4 ORBITS
Schwarzschild’s method tries to find a numerical representation of
the DF of a galaxy by assigning weights to a set of orbits. To avoid
any bias and to allow for the maximum degree of freedom, the
sample of orbits that the fitting routine can choose from must be
as general as possible and ‘representative’ of the potential. In this
section,wedescribehowthisisachieved.Wefirstintroduceatriaxial
Abel model from the companion paper (vdV08) that we use to test
our method. We then discuss the orbit structure in separable and
more general triaxial potentials. We continue with a description of
the orbital initial conditions, orbit integration and storage grids that
are used in our method.
4.1 Separable test models
The Abel models with a separable potential from the companion
paper are a generalization of the spherical Osipkov–Merritt models,
introducedbyDejonghe&Laurent(1991)andextendedbyMathieu
& Dejonghe (1999). These models have a DF that depends on three
integrals of motions, contain a central core, and allow for a large
range of (triaxial) shapes. The observables of these models, includ-
ing the LOSVD, can be calculated efficiently and they can be used
to generate test models that simulate realistic wide-field imaging
and integral-field spectrograph kinematics of galaxies. These mock
observations serve as input for the triaxial Schwarzschild method
presented in this paper.
We use the triaxial test model from section 4.3 of vdV08, which
has an isochrone St¨ ackel potential. This model resembles a triaxial
1011M?galaxy at 20 Mpc with a kinematically decoupled com-
ponent. We infer the potential from the MGE fit to the projected
(total) surface mass density. To obtain the luminous mass density,
weuseaseparateMGEthatfitstheSB[assumingaconstant(stellar)
mass-to-light ratio of M/L = 4M?/L?]. The kinematics are con-
structed in such a way that they resemble SAURON observations
(Bacon et al. 2001).
We will use this test model to demonstrate our method. More
details and tests of the recovery of global parameters are given in
Section 6 of this paper, whereas tests of the recovery of the internal
structure and the DF can be found in the companion paper.
4.2 Orbit structure
In a separable triaxial potential, all orbits are regular and conserve
threeintegralsofmotionE,I2andI3,whichcanbecalculatedanalyt-
ically. Four different orbit families exist: three types of tube orbits,
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Figure 2. The (x, z) plane of a triaxial galaxy with a separable potential, for
avalueoftheenergyEthatislargeenoughthatallorbitfamiliesappear.The
figure shows the equipotential that corresponds to E, the focal hyperbola,
the curve at which I2= 0, and the location of the thin orbits. The regions
where the different orbit families cross the (x, z) plane perpendicularly are
indicated: ‘B’ denotes box orbits, ‘S’ corresponds to short-axis tubes and
‘I’ and ‘O’ label inner and outer long-axis tubes. It can be seen that all
tube orbits cross the (x, z) plane perpendicularly in two points: once in the
regionoutsidethethinorbitcurveandonceinside.Thismeansthatthe(grey)
region between the thin orbit curves comprises all orbits just once, which is
important for the orbital sampling (Schwarzschild 1993).
which avoid the centre and are therefore sometimes referred to as
‘centrophobic’, and a set of orbits that can cross the centre, usually
referred to as boxes or ‘centrophilic’ orbits (e.g. Kuzmin 1973; de
Zeeuw 1985; Statler 1987). These different orbit families conserve
uniquecombinationsoftheseintegralsandcanthereforebelinkedto
distinct volumes in phase space. May be even more remarkably, all
four orbit families in a separable potential cross the (x, z) plane per-
pendicularly in well-defined regions (Fig. 2; Schwarzschild 1993).
Similar to axisymmetry, all tubes except the so-called thin orbits (in
which the inner and outer radial turning points coincide) cross the
(x,z)planeperpendicularlytwice.Atagivenenergy,thesepointsare
located in two distinct areas, separated by the line that connects the
points of the thin orbits. This line can be parametrized analytically
in a separable potential.
These properties are summarized in Fig. 2, where we have used
the isochrone separable potential of the triaxial Abel model. The
figure shows the (x, z) plane for a value of the energy that is large
enough that all orbit families are populated. The thick outermost
curveistheequipotentialatthisenergy,theinnermostandoutermost
decreasing curves inside the equipotential connect the points where
thethinorbitscrossthe(x,z)planeperpendicularly,theintermediate
decreasing curve corresponds to I2= 0, and the rising curve is the
focal hyperbola. The four areas corresponding to the different orbit
families are also indicated (see section 5.4 of vdV08 for further
details).
This orbital structure depends crucially on the presence of a cen-
tral core and is (partially) destroyed by the addition of a super-
massive black hole and/or a central cusp (Gerhard & Binney 1985).
Someorbitsinthesenon-separablepotentialsdonotconserveglobal
integrals of motion other than the energy E and may not all cross
the (x, z) plane perpendicularly. The three types of tube orbits, in-
cluding the thin tubes, are still supported (cf. Schwarzschild 1993).
Most box orbits are transformed into boxlets (Miralda-Escud´ e &
Schwarzschild 1989) and orbits that occupy certain parts of phase
space become chaotic. The amount of chaotic motion and the radial
range inside which it is present depends on the central cusp slope
(see Section 4.6).
4.3 Initial conditions
The orbits in our models are more complicated than those in a sep-
arable potential, as we use a more realistic MGE potential with a
supermassive black hole. Still, we use the properties of separable
models in our sampling of initial conditions. We sample the orbital
energy implicitly through a logarithmic grid in radius. When the
model has to reproduce observational data, it is important to sample
the orbital energy on a grid with a minimum radius that is at least an
order of magnitude smaller than the pixel size of the observations.
In the case of Hubble Space Telescope (HST) data, this typically
corresponds to ∼10−2arcsec. The outer grid radius is determined
by our constraint that the grid must include ?99.9 per cent of the
mass.
Each of the grid radii riis linked to an energy by calculating
the potential at (x, y, z) = (ri, 0, 0). The orbital initial conditions
are then sampled from a dense grid in the (x, z) plane. Since most
orbits cross the (x, z) plane perpendicularly twice above z > 0 it is
not necessary to sample the whole plane. The double countings are
avoided by finding the location of the thin orbit curves iteratively:
we launch orbits at different radii [keeping θ = arctan(x/z) fixed]
until the width of the orbit is minimal. This is similar to what was
done in the axisymmetric three-integral models, where all orbits are
short-axis tubes.
The starting points (x, z) are selected from a linear open polar
grid (R, θ) in between the thin orbit and the equipotential (the grey
area in Fig. 2). The initial velocity in the y direction is determined
from v2
dimensionalset(E,R,θ)ofstartingconditionsiscommonlyreferred
to as the ‘(x, z) start space’ (Schwarzschild 1993). It is sufficient to
launch orbits in only one direction when the density (or another
quantity that is even in the velocity, such as the second moment)
has to be reproduced. When the velocity (and odd higher order
velocity moments of the DF) is fitted in the model, the direction
of the orbital motion is also important. This information could be
taken into account directly by launching orbits in both the positive
and negative y direction. However, the trajectories of the prograde
and retrograde orbits are identical, which means it is much more
efficient to include the counter-rotating orbits only at the fitting
stage by reversing the velocity sign appropriately. This is only valid
when figure rotation is ignored (cf. Schwarzschild 1982).
Since boxes are essential for the support of the triaxial shape
(Schwarzschild 1979; Hunter & de Zeeuw 1992), a library with
relatively few of them cannot be expected to reproduce a triaxial
mass model. The (x, z) start space has few box orbits, especially
at large radii (see Fig. 3). To make sure that the orbit library pro-
vides enough freedom in the outer parts of the model, we add ad-
ditional box orbits, like Schwarzschild (1993). Box orbits always
touch the equipotential (Schwarzschild 1979). We therefore sample
start points on (successive) equipotential curves, using linear steps
in the two spherical angles θ and φ. At each combination of (θ, φ)
y,0= 2[V(x0, 0, z0) − Ei] and (vx, vz) = (0, 0). This three-
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Figure 3. Representation of the (x, z) and the stationary start space and their symmetries for the triaxial Abel model from vdV08. The panels show the orbital
startingpointsforincreasingenergies(denotedatthetop),fromaninnershellofthemodel(topleft-handdiagram)toanoutershell(bottomright-handdiagram).
The symbols represent the position of the orbits in the start spaces. The orbits in the inner right-hand quarter are in the (x, z) start space and the orbits placed in
the outer right-hand quarter are in the stationary start space (Section 4.3). The thick black line represents the equipotential (cf. Fig. 2). The orbits in the inner
left-hand quarter are the orbits from the (x, z) start space with reversed angular momentum and the orbits placed in the outer left-hand quarter is identical to the
outer right-hand quarter and are only drawn to make the panels symmetric. The symbols show the result of the orbit classification (based on angular momentum
conservation, Section 4.5): the crosses are box orbits, the stars correspond to short-axis tubes and the diamonds correspond to (both types of) long-axis tubes.
We have also overplotted the analytical curves that separate the different type of orbits (see also Fig. 2 and vdV08). The solid rising curve is the focal hyperbola,
with above it the long-axis tubes and below it the short-axis tubes and boxes. The crossing solid declining curve separates, respectively, between the inner and
outer long-axis tubes, and between the short-axis tubes and boxes. The thin curves indicate the location of the corresponding thin tube orbits.
and E, we use bisection to find the corresponding value of r0that is
located on the equipotential. This three-dimensional set (E, θ, φ) of
start conditions, the ‘stationary start space’ (Schwarzschild 1993),
results in box orbits or boxlets only. Tube orbits always conserve
the sign of at least one component of the angular momentum and
therefore never reach zero velocity. Since the direction of the orbits
in this start space is not important it is not necessary to add velocity
mirrored copies of them during the fit.
By design the set of energies E and angles in θ in both start
spaces are identical, so that the orbits on the equipotential bound-
ary of the (x, z) start space have obvious neighbours in stationary
start space. While not necessary, the size of the third dimension
of the start spaces is chosen to be the same for consistency. Both
sets of orbits can be represented in a single figure, by graphically
connectingthecorrespondingstartingspacesattheequipotential,as
shown in Fig. 3, where selected energies of the triaxial Abel model
(Section 4.1) are shown. In this figure we have overplotted the same
lines from Fig. 2, which shows that our numerical scheme to locate
the thin orbits indeed results in an orbit sampling from the correct
region. The stationary start space intersects with the xz start space
at the equipotential. In the figure all the orbits in the stationary start
spacethatarenearesttotheequipotentialareplottedjustoutsidethe
equipotential.Subsequentrowsinthestationarystartspaceareplot-
ted radially outwards. A mirror image of the stationary start space
is also plotted for symmetry.
4.4 Dithered orbit integration
The orbits in the start space are integrated numerically and their
properties stored. The integration is done in Cartesian coordinates,
using an explicit Runga–Kutta method of order 5(4) (DOPRI5 rou-
tine by Hairer, Norsett & Wanner 1993). With this method, the
majority of the orbits can be integrated with energy accuracies of
better than one part in 105. This routine is capable of dense output,
which enables you to get an interpolated position and velocity at
any time in current time-step during the integration.
To ensure numerical precision the Runga–Kutta integrator uses
more steps where the orbital trajectory changes direction quickly.
Since this usually happens when the ‘star’ is travelling with a
high velocity, the integrated time-steps do not represent the time-
averaged path of the orbit. To make sure this is not a problem we
use the dense output of the integrator, to sample the orbit on equal
time intervals, ensuring that the orbits are properly time weighted.
Single orbits correspond to delta-functions in integral space,
whiletheDFofa(partially)phase-mixedgalaxyisexpectedtovary
smoothly (Tremaine, Henon & Lynden-Bell 1986). This limitation
maybereducedbycombiningnearbyorbits(Richstone&Tremaine
1988; Rix et al. 1997). Here we extend this method by ‘dithering’
orbits in all three dimensions in the initial starting space. We do
this by taking a bundle of 53orbits with different, but adjacent, ini-
tial conditions and sum their observables. This method is also used
in the construction of axisymmetric models (see Cappellari et al.
2006).
Whencalculatingthestartingspacesfortheorbitswecreatemore
starting points for the dithering. We enlarge the sampling three-
dimensional (E, θ, φ) start spaces five times in each direction. This
leads to 125 orbits per bundle. The odd number five was chosen
so that each bundle has a clearly defined central orbit (see fig. 6 in
Cappellarietal.2006).Theorbitalpropertiesofeachoftheorbitsin
each bundle are simply co-added. As an alternative, it is be possible
to apply some form of (Gaussian) weighting. In this way the orbit
bundles could be made to overlap, but the effects of this have not
been studied.
Effectively, the model thus contains 125 times more orbits. The
ditheringcausestheorbitalbuildingblockstobesmoother,eliminat-
ing aliasing effects, especially when modelling spatially extended
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kinematic data. We found that this dithering is essential to obtain
smooth orbital observables and remove numerical noise, using a
limited amount of orbits.
4.5 Storage grids and symmetries
For spherical galaxies, it is in principle sufficient to store the orbital
properties in one dimension, along a line. The three-dimensional
model can be reconstructed afterwards by deprojecting the radial
profile back on to the sphere. Similarly, axial symmetry allows one
to carry out the calculations in the meridional plane only. Revolu-
tion of the model around the intrinsic short axis returns the three-
dimensional intrinsic properties. As we restrict ourselves to station-
ary, non-rotating galaxies that are symmetric in the three principal
planes,allorbitalpropertieshavetobecalculatedinonlyoneoctant.
The properties in the other octants follow by symmetry.
Thedensityofeveryorbitinthelibraryisstoredonasphericalgrid
in (rg, θg, φg) [θg= 0 corresponds to the short axis and (θg= π/2,
φg= 0) to the long axis]. The radial sampling is logarithmic with
the inner and outer boundary set to zero and infinity. The angular
grids θgand φgare sampled linearly between 0 and π/2. The grid
has Nrg= 15, Nθg= 4 and Nφg= 5. This leads to 20 bins per
radius and 300 bins in total, which is enough to ensure that the mass
isreproducedwellandthemodelisself-consistent.Whenfittingthe
model, the intrinsic mass grid is used as a constraint and is fitted
everywhere with an accuracy of 2 per cent (see Section 5).
Similar to the intrinsic symmetries, the projected properties of
spherical galaxies are one-dimensional and those of axisymmetric
galaxies are symmetric in the projected axes. It is therefore suffi-
cient to store the projected properties of spherical galaxies in one
dimension and those of axisymmetric objects in one quadrant of the
sky. The projected properties of triaxial galaxies are at most point
symmetric, with respect to the projected centre, which implies that
the model-data comparison must be done in one half (or more) of
the sky plane.
To convert the intrinsic coordinates (x, y, z) to the projected co-
ordinates (x?, y?, z?) we use equations (2) and (3). After this step the
PSF is included by randomly perturbing the projected coordinates
(x?, y?) with a probability described by the MGE PSF, before being
included into the observational apertures (identical to Cappellari
et al. 2006). We use a three-dimensional rectangular storage grid in
the projected Cartesian coordinates x?and y?and the line-of-sight
velocity v in the sky plane. The resolution and rotation of this grid
is adapted to the kinematical data that have to be reproduced. Op-
tionally, the observational apertures can be binned as a final step to
match any observational binning.
Only orbits with the correct degree of symmetry can be used to
reproduce the density and potential. All orbits in a separable poten-
tial are indeed eightfold symmetric, but this need not be the case
for resonant and irregular orbits in more general potentials. These
orbits can therefore not be used directly in the reconstruction of the
potential and density that we are interested in. This does not mean
that they are useless, as we can enforce the required symmetries by
apply a folding scheme to these orbits. Again, this scheme is sim-
ilar to what is done for axisymmetric potentials, except that only
orbits that are not symmetric with respect to the z = 0 plane have
to be corrected in that case [e.g. 1:1 (R, z) resonances, Richstone
1982].
The folding scheme is based on the fact that a given asymmetric
orbit has up to seven mirror images that are obtained by reflection
in the principal planes (see the first column of Table 2). These mir-
ror images are also supported by the potential, but do not appear in
Table2. Therecipethatisusedtomirrororbitsinthethreeprincipalplanes.
Long-axis tubes are abbreviated by L-tube and short-axis tubes by S-tube.
Position Box L-tube S-tube
(x, y, z)
(−x, y, z)
(x, −y, z)
(x, y, −z)
(−x, −y, z)
(−x, y, −z)
(x, −y, −z)
(−x, −y, −z)
(vx, vy, vz)
(−vx, vy, vz)
(vx, −vy, vz)
(vx, vy, −vz)
(−vx, −vy, vz)
(−vx, vy, −vz)
(vx, −vy, −vz)
(−vx, −vy, −vz)
(vx, vy, vz)
(−vx, vy, vz)
(vx, vy, −vz)
(vx, −vy, vz)
(−vx, vy, −vz)
(−vx, −vy, vz)
(vx, −vy, −vz)
(−vx, −vy, −vz)
(vx, vy, vz)
(vx, −vy, vz)
(−vx, vy, vz)
(vx, vy, −vz)
(−vx, −vy, vz)
(vx, −vy, −vz)
(−vx, vy, −vz)
(−vx, −vy, −vz)
the library because we sample orbital initial conditions only from
one octant. All eight mirror orbits have identical properties and are
equally useful for the model. We may therefore add these eight or-
bits to obtain an orbit that has three planes of symmetry. In practice,
this is done as follows. During orbit integration, the orbital weight
that corresponds to a given point (x, y, z) is equally distributed over
the eight mirror points. The contributions to both the intrinsic and
projected density of all eight points are added up into one orbital
building block. In this way, orbits that are asymmetric are included
correctly, while orbits that already are eightfold symmetric are sim-
ply sampled more densely.
More attention is required when calculating the kinematical ob-
servablesoftheorbitalbuildingblocks.Ifwereflectthevelocitiesin
the same way as the coordinates, the total orbital building block has
no net angular momentum. This is only correct for box orbits, while
tubeorbits,whichareessentialwhenfittingtotheobservedvelocity
field, conserve the sign of at least one component of the angular
momentum vector. Therefore, the sign of the angular momentum of
theseorbitsmustbepreservedalsointhetotalorbitalbuildingblock.
Thisisensuredbyclassifyingtheorbitsonthebasisoftheirangular
momentumproperties.Boxorbitsoscillateinallthreedirections,so
that no components of the angular momentum are conserved, while
long-axis and short-axis tubes conserve the sign of the angular mo-
mentumparalleltothelongandshortaxis,respectively.Thisallows
us to distinguish orbits by checking for which angular momentum
component(s), if any, the sign is conserved during orbit integration.
In doing this, inner and outer long-axis tubes cannot be recognized
separately. This is, however, not a problem for the present applica-
tion (cf. Schwarzschild 1979). As can be seen from Fig. 3, where
we have plotted the different orbital types with different symbols,
the numerical classification agrees with the analytical calculations.
Wethenapplythefollowingschemeforreflectionsintheprincipal
planes (see Table 2). Box orbits: the average angular momentum is
zero, which allows us to reflect the velocity components in exactly
the same manner as the coordinates. Long-axis tube orbits: the sign
of the angular momentum around the long axis, Lx= yvz− zvy,
is conserved, which means that Lxmust be the same for all eight
mirror points. Short-axis tube orbits: the sign of Lz= xvy− yvxis
conserved.
4.6 The influence of a central mass concentration
Central mass concentrations have considerable influence on the or-
bital structure of the galaxy as a whole and may induce chaotic
behaviour. In an axisymmetric potential, the non-integrable regions
of phase space (usually referred to as the Arnold web) are not con-
nected. This means that the diffusion of chaotic orbits is limited and
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their influence on the model is probably not significant, which jus-
tifies the fact that chaotic orbits are not treated in a special manner
in axisymmetric models. Realistic triaxial potentials can support a
muchlargerfractionofchaoticorbits.Theoverallamountofchaotic
motion depends on the cusp slope and on the mass of the central
mass concentration (Gerhard & Binney 1985; Merritt & Fridman
1996; Valluri & Merritt 1998), but the different orbit families ex-
perience fundamentally different effects, due to the central mass
concentration.
The box orbits that are started from the inner most equipotentials
are very difficult to integrate numerically. The central accelerations
arelargeandthetime-stepsthatarerequiredtoconservetheintegra-
tion accuracy are correspondingly small, resulting in prohibitively
long orbital integration times. This effect can be reduced by using
a non-zero value for the softening length that was introduced in
Section 3.8. The DOPRI5 routine that we use to integrate the orbits
varies the time-steps to match the desired accuracy. We found that
evenorbitsthatpassclosetotheblackholecanbeintegratedwithan
accuracy of 10−5in a reasonable time. The softening length that we
usedistypicallytwoordersofmagnitudessmallerthantheradiusof
the sphere of influence of the black hole. This sphere of influence is
defined as R•= GM/σ2, which is the radius inside which the black
hole potential dominates.
Test particles that are dropped from equipotentials at somewhat
larger distances from the black hole are scattered off their orig-
inal (box) orbits and may become chaotic (Gerhard & Binney
1985). The trajectories of such particles can be described by a
series of regular segments and, given enough time, will fill most
of the equipotential that corresponds to the orbital energy. Since
equipotentials are rounder than equidensity curves and box or-
bits are the backbone of the triaxial shape, this process may de-
stroy the triaxial shape from the inside out (e.g. Poon & Merritt
2002).
Chaotic orbits are not time independent, since their orbital den-
sities do not average out on physical time-scales. In principle, this
meansthataSchwarzschildmodelwithanorbitlibrarythatincludes
irregular orbits is not stationary. However, evolutionary studies of
models that include chaotic orbits (Schwarzschild 1993) and N-
body simulations (Merritt & Fridman 1996; Holley-Bockelmann
et al. 2002) display no dramatic shape changes, even after very
long times. This means that a model with chaotic orbits may be
stationary for as long as a Hubble time and Schwarzschild so-
lutions can be constructed also for models that contain chaotic
orbits.
The use of dithered orbital components (Section 2.2) is critical to
create nearly time-independent models. In fact orbits started from
similar initial conditions can follow very different trajectories. Be-
cause the dithering single ‘sticky’ chaotic orbits do not play a major
role, since orbits are always bundled with nearby orbits.
The influence of a central mass concentration on tube orbits is
radically different. Low-energy tubes, which orbit at large radii,
never approach the central mass concentration close enough to be
significantly disturbed. Tubes that are launched from the principal
planes close to the radius of influence of the black hole turn into
precessing ellipses. Depending on the shape of the volume that the
ellipseeventuallyfills,theymaybelabelledaspyramidorbits(Poon
& Merritt 2002) or lens orbits (Sridhar & Touma 1999; Sambhus &
Sridhar2000).Theprecessionrateoftheellipseisdeterminedbythe
ratio of the stellar mass that is enclosed by the orbit and the central
black hole mass. The integration time that is required for conver-
gence of the orbital properties is therefore inversely proportional to
the orbital radius.
4.7 Number of orbits
To summarize, we use two start spaces with three dimensions (E, θ,
R) and (E, θ, φ), which are connected at the equipotential boundary
of every energy. The total number of orbits in the fit, excluding
dithering, is three times the number of orbits in the one start space,
as the (x, z) start space is used twice and the stationary start space
once. The number of orbits is denoted as 3 × NE × Nθ × NR.
Because of the dithering each effective orbit consists of 125 orbits,
significantly smoothing the orbital component. The total number of
orbits necessary to make a model is dependent on several factors:
the number and spatial distribution of the observed kinematics, the
size of the galaxy model and the shape of the potential.
The effect of the number of orbits can be studied by determin-
ing the quality-of-fit χ2of the model as a function of the number
of orbits. With increasing orbit numbers the χ2decreases. When
enough orbits are present, the model does not improve anymore and
the χ2does not decrease anymore. In our test cases we find that the
modeldoesnotimproveconsiderablywhentheorbitlibraryconsists
of 2000 orbits or more. Self-consistent models with smaller orbit
libraries have significantly larger χ2, due to the fact that there are
not enough orbits to reproduce the mass, especially at larger radii.
We therefore decided to use an orbital sampling of 21 equipotential
shells with 8 × 7(θ, R) starting points each, totalling 3528 (×125)
orbits.
5 SUPERPOSITION AND REGULARIZATION
To make a model with the computed orbits we need to combine the
orbits in such a way that they fit the observations, while reproduc-
ing the (intrinsic) mass distribution for self-consistency. Here we
describe the construction of the orbital superposition and a way to
ensure that our numerical solution is realistic.
5.1 Finding the orbital weights
The model has two components that need to be fitted: the kinematic
observationsandthe(intrinsicandprojected)massdistribution.The
kinematics are fitted using linearly superposed mass-weighted GH
moments (Rix et al. 1997). The fit is done by combining the orbits
linearly by assigning each orbit an orbital weight (γi). These orbital
weights directly represent the mass in each orbit and must therefore
be positive (γi? 0).
The intrinsic mass grid (Section 4.5) and the aperture masses
(the total amount of mass in each observed aperture: the zeroth
GH moment) must be added to the fit to ensure that the model is
self-consistent with the density in which the orbits were calculated.
Often this is done by including them in the fit as an ‘observable’
(e.g. van der Marel et al. 1998; Valluri et al. 2004). However, they
are not actually observed and therefore it is difficult to assign an
error. To include them into the fit they are usually assigned a hand-
tuned fractional error so that the mass is reproduced well without
influencing the fit of the kinematics. Here we use a different ap-
proach by including them as ‘constraints’ with bounds in the fit
(similartoRichstone&Tremaine1988).Thismeansthattheorbital
superpositionreproducestheintrinsicandaperturemassestowithin
2 per cent at all times, while finding the best-fitting kinematics. The
total normalized mass of all the orbital weights is fixed using an
equality constraint. The reason for including constraints is that the
mass can almost always be reproduced up to numerical precision
(van der Marel et al. 1998; Poon & Merritt 2002) and is thus not
relevant for finding the best-fitting solution. We only want the mass
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to be reproduced to within 2 per cent, because this is the typical
accuracyof(theMGEof)theobservedSB.Withintheseboundsthe
solver allows the mass to vary to find the best-fitting kinematics in
the least-squares sense.
We use the sparse quadratic programming solver QPB from the
GALAHAD library (Gould, Orban & Toint 2003) to make the su-
perposition, as this algorithm is capable of fitting the kinematics
in the least-squares sense while satisfying mass constraints. This
algorithm optimizes the orbital weight γ in the least-squares sense:
min
γ∈Rn?Aγ − b?2
subject to the positivity constraint
γ ? 0,
and the linear constraints,
0.98p ? Mγ ? 1.02p.
Here A is the m × n projection matrix whose n columns give the
model contribution of every orbit to the m kinematical observables
b.ThematrixMisaprojectionmatrixgivingthemodelcontribution
of the orbits to the mass in the various apertures and p is the mass
derived by integrating the MGE model over the projected apertures
and intrinsic mass grid. The total number of constraints in the fit
(p) is 300 (intrinsic mass grid) + 1 (total mass) + the number of
apertures (aperture masses).
The quality of the model is determined by measuring the discrep-
ancy between the model and the observations for different values of
the input parameters. This is done by calculating the χ2, defined as
?D∗
in which Ndis the number of observables (the number of apertures
times the GH moments), Diis the observation for the ith observ-
able, D∗
associated with this value (the observational error).
(24)
(25)
(26)
χ2=
Nd
?
i=1
i− Di
?Di
?2
,
(27)
iis the model prediction and ? Diis the uncertainty that is
5.2 Regularization scheme
The quadratic programming problem to be solved is ill conditioned
in most applications, due to (close to) degenerate orbits. As a con-
sequence, the orbital weight distribution for the solution with the
smallest χ2may be a rapidly varying function, which is not likely
to be realistic (Merritt 1993; Verolme & de Zeeuw 2002). This ef-
fect can be reduced by adding linear regularization equations to the
problem (e.g. Zhao 1996; Rix et al. 1997), which is also known as
‘damping’ in the field of linear programming. Such regularization
terms can be added to force the orbital weights towards a smoother
function by minimizing their higher order derivatives.
Our two start spaces are sampled in three dimensions (E, R, θ)
and (E, θ, φ) and they are connected at the equipotential boundary.
The three dimensions of the (x, z) start space roughly correspond to
the three integrals of motion (in a separable potential). We can thus
enforce smoothness of our solution by adding regularization terms
to our minimization routine in each of the three directions (k, l, m)
of our start space. We adopt the second-order finite differencing
(Press et al. 1992) regularization from Cretton et al. (1999), so for
each orbit we add three equations to the array A given above, and
minimizes them in a least-squares sense:
λ(2ξkγ(k,l,m)− ξk−1γ(k−1,l,m)− ξk+1γ(k+1,l,m)) = 0,
λ(2ξkγ(k,l,m)− ξkγ(k,l−1,m)− ξkγ(k,l+1,m))
λ(2ξkγ(k,l,m)− ξkγ(k,l,m−1)− ξkγ(k,l,m+1))
= 0,
= 0,
(28)
where γ(k,l,m) represents the orbital weight at position (k, l, m) in
the start space grid. The ξkweights are added to include the radial
energy dependence of the model. It is estimated, a priori, as the
normalized mass enclosed by each radial shell in the start space:
1
ξk
=
1
No
???ρ dx dy dz
? ?
R(k−1)
R(k+1) ?
ρ(x, y,z)dx dy dz,
(29)
where Nois the number of orbits. The regularization error λ de-
termines how much smoothing is performed. Increasing λ increases
theamountofsmoothing.Theoptimalvalueofthisλisusuallytime
consuming to determine (see e.g. Cretton et al. 1999). However, it
has been shown elsewhere that a theoretical axisymmetric galaxy
with a two-integral DF can be accurately reproduced by using this
approach (Verolme et al. 2002; Cretton & Emsellem 2004). Appli-
cations of the axisymmetric Schwarzschild method have often used
a value of 1/? ≡ λ = 0.25 as regularization (Cappellari et al. 2002;
Krajnovi´ c et al. 2005). As we will show in the next section it is also
acceptable for the triaxial method.
5.3 Testing the regularization
InthecompanionpapertheDFofthetriaxialtestmodeliscompared
directly in terms of the integrals of motion (section 5.4.3 in vdV08)
andwefindthattherecoveryoftheDFisconsistentwiththequality
of the input kinematics (figs 12 and 13 in vdV08). The triaxial test
model is ideally suited to test regularization. To do this we compare
the orbital mass weights directly. The top row in Fig. 4 displays
the computed orbital weights and kinematics for the triaxial Abel
model from vdV08. The other rows show the orbital weights and
kinematicsforthebest-fittingSchwarzschildmodelwithdecreasing
regularization from top to bottom. The orbits at the radius of 2 and
33 arcsec are outside the range where they are constrained by the
kinematics, and as such the orbital weights for the Schwarzschild
models are not expected to compare well with those of the Abel
model.
The distribution of (analytical) orbital weights for the Abel
model is smooth with some sharp peaks. The reconstructed orbital
weights from the Schwarzschild agree well with the analytical or-
bital weights, except for high values of λ, which corresponds to
strong regularization. The orbital weights of strongly regularized
models are distributed more smoothly; adjacent orbits receive sim-
ilar weight and the kinematics start to change. From Fig. 4, we see
that the kinematics is affected by the regularization at λ ? 0.2, and
thus the optimal regularization in this case has to be chosen to be
λ ? 0.2. This will give satisfactory orbital weights, and kinematics
that are consistent with the observations. The comparison is best for
a λ ∼ 0.1.
There are many reasons why the reconstructed orbital weights
do not exactly match the analytical orbital weights. Most of them
are minor numerical and discretization effects, as discussed in the
following. (i) The Schwarzschild method samples the galaxy with
discrete orbits (computed in the reconstructed potential), which are
related to the DF via their phase-space volume (Vandervoort 1984).
Theresultingorbitalmassweightsareevaluatedinanapproximated
and numerical way (section 5.4 in vdV08). (ii) Some symmetric
orbits appear twice (or more often) in the orbit library. Without
regularization the (quadratic) solver ignores the second identical
copy of this orbit, as they do not improve the fit. However, these
orbits do get assigned weight in the test case. A good example of
thisaretheboxorbitsinthe(x,z)startspace,astheyareaddedtwice
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Figure 4. The effect of regularization on the models. The top row shows the triaxial Abel from van de vdV08, while the other rows show the best-fitting
Schwarzschild models with decreasing regularization, from top to bottom. (For a detailed comparison of the DF see vdV08.) The left-hand columns show the
orbital weights of the models in the same configuration as Fig. 3, whereas the two rightmost columns show the velocity and velocity dispersion fields.
to the fit [like all (x, z) orbits]. These differences are only visible in
this direct comparison of the orbital weights.
To estimate the effect of regularization on the kinematical fit
more quantitatively, we investigated the χ2difference between the
models with different values of the regularization λ. For the best-
fittingmodelwith2370kinematicalobservablesthetotalχ2is2588.
Adding regularization does increase the χ2of the model. With λ =
0.01 (little regularization) it does not affect the fit to the kinematics
(?χ2∼ 1). When increasing λ further to 0.2 or even 4 (very strong
regularization), the ?χ2changes to ∼200 and ∼1000, respectively.
These numbers reflect what one sees in Fig. 4: for λ ? 0.2 the kine-
matical fit does not change visibly, whereas for higher λ (stronger
regularization) the kinematical fit becomes rapidly worse.
One other important question is whether the regularization
changes the recovered input parameters, including the viewing an-
gles, mass-to-light ratio, anisotropy and black hole mass. This is
nearly impossible to test with real galaxies, as their properties are
unknown. The Abel model has known parameters and was used to
test the recovery. We found that there is no difference in the best-
fitting parameters when a regularization of λ = 0.2 was chosen. The
confidence intervals of the parameters do become smaller by using
regularization. This is expected, as the added regularization terms
decrease the freedom of the model and therefore increase the χ2.
A notable exception, that we do not test here, is the recovery of
the black hole mass. There are often few observables in the models
nearthesphereofinfluence.Thenumberofmassbinsneartheblack
holeisextremelylimitedandthekinematicalobservationsinsidethe
sphereofinfluenceoftheblackholeisverylimited,usuallylessthan
10 observables. In this scenario it is conceivable that regularization
is needed, as the model might otherwise adapt the orbital structure
to be able to accommodate the black hole (see also e.g. Magorrian
2006). Recovery of the black hole mass using regularization will be
presented elsewhere.
6 TESTS ON THE TRIAXIAL ABEL MODEL
We test our method on the triaxial Abel model from the compan-
ion paper vdV08, introduced in Section 4.1 and already used in
Section 5.3 above. Here, we outline further tests done on the Abel
model.
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Figure 5. Schwarzschild models with different intrinsic shapes fitted to the observables of the triaxial Abel model from vdV08, reveal that the kinematics vary
significantly by changing the axis ratios p and q. The panels on the left-hand side show the velocity field of the Schwarzschild model for each value for p and
q, while the input velocity field from the Abel model is shown in the top left-hand corner. The panels on the right-hand side show the same for the velocity
dispersion. The input model has axis ratios (p, q) = (0.82, 0.67).
6.1 Internal orbital structure and DF recovery
In vdV08 the best-fitting triaxial Schwarzschild model to the input
triaxial Abel is presented. The Schwarzschild method only uses
the information that can be observed in real galaxies, i.e. the two-
dimensional SB and the two dimensional stellar kinematics. The
resulting best-fitting model is an excellent fit and has a (reduced)
χ2per degree of freedom (d.o.f.) of ∼1.1.
Since Schwarzschild models only fit the projected observables it
is not obvious that these models can recover the three-integral DF
and the internal structure of the test model. By comparing the mass
weights of the Schwarzschild model to the DF of the test model,
vdV08 demonstrate that both the internal orbital structure and DF
are recovered with an accuracy similar to the typical (simulated)
errors on the kinematics.
6.2 Recovering the global parameters
When constructing Schwarzschild models of the Abel model, we
expect the kinematics of the Schwarzschild model to vary when
changing intrinsic shape parameters (p, q, u), and thus the viewing
angles (ϑ, ϕ, ψ). The most obvious are the change of the zero-
velocity curve as the projected axes change on the sky. Also, the
characteristics of the orbits in the model are dependent on the shape
of the potential. These effects are shown in Fig. 5, by showing
models of the analytic test data in linear steps of 0.05 in p or q. The
models become significantly worse when changing the parameters
away from the correct values. This shows that different intrinsic
shapes support different orbits and that one cannot expect a model
with the wrong potential to be able to fit the kinematics in all cases.
To search the global parameters we sample the parameter space,
by making linear steps of 0.1M?/L?in M/L, and 0.05 in p, q
and u (resulting in 100 different intrinsic shapes). For each corre-
sponding Schwarzschild model, the changes are quantified by the
goodness-of-fit parameter ?χ2. To visualize this four-dimensional
parameter space, we calculate for a pair of parameters, say p and q,
the minimum ?χ2as a function of the remaining parameters, u and
M/L in this case. The contour plots of the resulting marginalized
?χ2foralldifferentparametersfortheSchwarzschildmodelsfitted
to the observables of the Abel model are shown in Fig. 6. Since we
sampled in intrinsic shape and not in viewing direction, the viewing
angle sampling is not uniform. In particular the very round models,
whichareindependentofφ arenotrepresentedproperly.Tothisend,
wecreateadensegridin(p,q,u)andinterpolatetheχ2linearlyover
this dense grid, resulting in the contour plots of (ϑ, ϕ, ψ) in Fig. 6.
The input parameters for which the simulated observables of the
Abel model were obtained are M/L = 4M?/L?and (ϑ, ϕ, ψ) =
(70◦,30◦,101◦).AsoutlinedinSection3.7,thelatterviewingangles
converttotheintrinsicshapeparameters(p,q,u)=(0.82,0.67,0.88),
given the average projected flattening q?= 0.76 of (the MGE model
of) the SB. These input parameters are denoted by a red diamond in
the contour plots of Fig. 6. We find that the input M/L and (p, q, u)
(andhencealsotheviewingangles)oftheAbelmodelareaccurately
recovered, with a typical uncertainty of 10 per cent or less.
Krajnovi´ c et al. (2005) suggested that the recovery of the inclina-
tionforaxisymmetricmodelsisdegenerate,whichseemsinconflict
with our recovery of the intrinsic shape. However, the kinematics
of the Abel model have a significant feature, namely a orthogonal
decoupled core, and this makes it plausible that the viewing angles
are constrained quite strongly. We verified that for galaxies with no
such distinguished kinematic feature, e.g. in the case of a (nearly)
zero mean velocity map like for M87, the intrinsic shape is not well
constrained.
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Figure 6. Marginalized contours maps of the Schwarzschild models fitted
to the observables of the triaxial Abel model for different intrinsic shapes.
The contours denote 2, 4 (thick line) 8 and 32σ confidence levels. Areas for
which the MGE cannot be deprojected are left blank. The six upper panels
show the intrinsic shape parameters (p, q, u) and mass-to-light ratio M/L;
the three lower panels show the viewing angles (ϑ, ϕ, ψ). The combination
ofϑ andϕ isshowninaLambertequal-areaprojection,seendownthenorth
pole (z-axis). The x,y and z symbols give the location of views down those
axis. The red diamond in each panel indicates the input parameters from the
Abel model.
7 APPLICATION TO NGC 4365
We now apply our method to NGC 4365, one of the prototypical
galaxies with a KDC. It is a giant E3 elliptical and it was one of the
first objects in which minor axis rotation was discovered (Wagner,
Bender & Moellenhoff 1988; Bender, Saglia & Gerhard 1994). The
Figure 7. SAURON observations of NGC 4365. Top panels: from left- to right-hand side: the mean velocity, velocity dispersion and GH moments h3and h4
of NGC 4365, as observed with the integral-field spectrograph SAURON. The pixel scale of the observations is 0.8 arcsec. Middle panels: point-symmetrized
kinematics with respect to the galaxy centre. Non-symmetric deviations cannot be reproduced by a triaxial model anyway and the symmetrization guides the
eye. Bottom panels: kinematic maps of the best-fitting Schwarzschild model, obtained by adding the weighted contributions of the best-fitting set of orbits. The
same colour levels are used for both data and model.
peculiarvelocitystructureofthisgalaxywaspartiallyunravelledby
multiple long-slit observations (Surma & Bender 1995), but the full
two-dimensional kinematical structure was only revealed with the
integral-field spectrograph SAURON (Davies et al. 2001).
KDCs can be the result of a merger event, but can also occur
when the galaxy is triaxial and supports different orbital types in
the core and main body (Statler 1991). Davies et al. (2001) stud-
ied the first option and investigated the link between the kinematics
and the line-strength distribution of NGC 4365. They found that
the core and the main body are of similar age and that any mergers
that led to the formation of the KDC must have occurred at least
12 Gyr ago, as otherwise younger stellar populations would have
been detected. The orbital structure that supports the KDC and the
main body cannot be observed directly and must be inferred from
dynamical models. Statler et al. (2004) studied the viewing angles
and triaxiality of the system using an approach developed by Statler
(1994a), which uses Bayesian analysis to fit analytic solutions of
the continuity equation to an observed velocity field. They found
NGC 4365 to be strongly triaxial and seen almost along the long
axis. The triaxial Schwarzschild method that was presented in the
previoussectionsallowsustobuildcomprehensivedynamicalmod-
els of this galaxy and investigate its intrinsic structure.
7.1 Observations
NGC 4365 was observed with SAURON on the nights of 2000
March 29 and 30 for two different pointings, with an overlap in the
central region. The exposures were combined and rebinned into a
map with a slightly better spatial sampling (0.8 arcsec, compared
to 0.94 arcsec for the individual lenslets) and a coverage of 33 ×
63 arcsec2. Davies et al. (2001) give a full description of the obser-
vations.
To increase the signal-to-noise ratio (S/N) to sufficient levels for
accuratedeterminationofthekinematics,thedatacubewasspatially
binned into 964 non-overlapping bins using the two-dimensional
Voronoi binning of Cappellari & Copin (2003). A minimum S/N
of 100 per spectral element was imposed. However, many of the
spectra have a much higher S/N value (up to ∼300), and more than
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Table 3. The parameters of the 11 Gaussians in the MGE fit to the com-
bined HST/WFPC2/F814W and the ground-based image of NGC 4365.
The columns give for each Gaussian, respectively, its number j, amplitude
SB0=L/(2πσ?2q?),dispersionσ?,projectedflatteningq?andpositionangle
offset ?ψ?, as defined in equation (4).
j
logSB0(L?,Ipc−2)
3.424
3.319
3.238
3.435
3.820
3.740
3.576
3.106
2.874
2.400
2.122
1.329
logσ?(arcsec)
q?
?ψ?(◦)
1
2
3
4
5
6
7
8
9
10
11
12
−1.024
−0.727
−0.320
−0.027
0.138
0.402
0.648
0.955
1.224
1.499
1.833
2.362
0.800
0.800
0.800
0.670
0.709
0.698
0.798
0.737
0.739
0.741
0.775
0.670
0.0
0.0
0.0
0.0
0.5
0.8
0.0
0.0
0.0
3.5
3.6
4.5
one quarter of the spectral elements remain unbinned. The stellar
kinematicswhereextractedusingthepenalizedpixel-fittingmethod
(pPXF) of Cappellari & Emsellem (2004). For every Voronoi bin
we extracted the velocity V, velocity dispersion σ and the higher
order GH moments h3and h4of the stellar LOSVD.
TheSAURONspectrahaveveryhighS/NsothattheLOSVDcan
be reliably extracted from the data; however, care has to be taken
to minimize the effect of template mismatch, which dominates the
errorbudgetinthebright,high-S/Ncentralregionsofthegalaxy.To
this end an accurate template was determined during the pPXF fit
usingthe∼1000starsoftheMILESlibrary(S´ anchez-Bl´ azquezetal.
2006), which span a large range of stellar atmospheric parameters.
Out of the MILES stars, only 14 are selected by pPXF to provide
anaccuratematchtotheobservedaveragegalaxyspectrum,withan
rms scatter in the residuals of only 0.17 per cent. From the observed
residuals and fig. B3 of Emsellem et al. (2004), we infer an upper
limit of ?0.02 on the systematic error of the GH moments, due to
any remaining template mismatch.
The rereduced kinematics, shown in Fig. 7, are a significant im-
provement over the kinematics shown in Davies et al. (2001). They
show a core in the inner ∼6 arcsec that rotates around the minor
axis. At larger radii the stars rotate around an axis offset by 82◦,
which is evidence that the system is intrinsically triaxial. The peak
mean streaming velocities are 55 kms−1. The dispersion peaks at a
value of ∼260 kms−1.
7.2 Mass model
We used an HST/WFPC2/F814W image and a ground-based image
of NGC 4365 obtained with the 1.3-m McGraw-Hill at the MDM
observatory (from Falc´ on-Barroso et al., in preparation) to make
an MGE (mass) model, using the software by Cappellari (2002).
We ensure that the model is the roundest that is consistent with the
observations. This is done by setting a lower limit to the allowed
projectedflatteningof0.67andanupperlimitof45onthedifference
in position angle between the individual Gaussians. The modest
difference between the rms error of the free model (0.99 per cent)
and the constrained MGE model (1.02 per cent) suggests that these
constraints do not lead to systematic errors in the mass model. The
parameters of the MGE model are given in Table 3, the SB map and
the MGE model fit are shown in Fig. 8.
Figure 8. Top: the contours of HST/WFPC2/F814W image of NGC 4365,
overplotted with (smooth) contours of the best-fitting MGE. Bottom: the
contours of the ground-based image of NGC 4365 obtained with the 1.3-m
McGraw–Hill telescope and the best-fitting MGE.
7.3 Dynamical models
We calculate triaxial Schwarzschild models using orbit libraries of
3 × 1176 orbits, 2352 of which are started in the (x, z) plane, the
remainingaredroppedfromtheequipotential.Weassumeadistance
of 23 Mpc for NGC 4365 (Mei et al. 2005). The assumed distance
does not influence our conclusions about the internal structure of
the galaxy, but lengths and masses scale linearly with the distance,
while mass-to-light ratios are inversely proportional to the distance.
A given triaxial model is determined by the mass-to-light ratio
M/L, the shape parameters (p, q, u) – or equivalently, the viewing
angles (ϑ, ϕ, ψ) – and the mass M•of the central black hole. We fix
the latter to M•= 3.6 × 108M?, consistent with the black hole–σ
C ?2008 The Authors. Journal compilation C ?2008 RAS, MNRAS 385, 647–666