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5Mon. Not. R. Astron. Soc.000, 1–17 (2005) Printed 12 March 2018 (MN LATEX style file v2.2)

Kinemetry: a generalisation of photometry to the higher moments ofthe line-of-sight velocity distribution

Davor Krajnovic,1,2⋆ Michele Cappellari,1 P. Tim de Zeeuw,1 Yannick Copin31Sterrewacht Leiden, Postbus 9513, 2300 RA Leiden, The Netherlands2Denys Wilkinson Building, University of Oxford, Keble Road, OX1 3RH, UK3Institut de Physique Nucleaire de Lyon, 69622, Villeurbanne, France

12 March 2018

ABSTRACTWe present a generalisation of surface photometry to the higher-order moments of the line-of-sight velocity distribution of galaxies observed with integral-field spectrographs. The gener-alisation follows the approach of surface photometry by determining the best fitting ellipsesalong which the profiles of the moments can be extracted and analysed by means of harmonicexpansion. The assumption for the odd moments (e.g. mean velocity) is that the profile alongan ellipse satisfies a simple cosine law. The assumption for the even moments (e.g velocity dis-persion) is that the profile is constant, as it is used in surface photometry. We test the methodon a number of model maps and discuss the meaning of the resulting harmonic terms. Weapply the method to the kinematic moments of an axisymmetricmodel elliptical galaxy andprobe the influence of noise on the harmonic terms. We also apply the method toSAURONobservations of NGC 2549, NGC 2974, NGC 4459 and NGC 4473 where we detect multipleco- and counter-rotating (NGC 2549 and NGC 4473 respectively) components. We find thatvelocity profiles extracted along ellipses of early-type galaxies are well represented by the sim-ple cosine law (with 2% accuracy), while possible deviations are carried in the fifth harmonicterm which is sensitive to the existence of multiple kinematic components, and has some anal-ogy to the shape parameter of photometry. We compare the properties of the kinematic andphotometric ellipses and find that they are often very similar, but a study on a larger sampleis necessary. Finally, we offer a characterisation of the main velocity structures based only onthe kinemetric parameters which can be used to quantify the features in velocity maps.

Key words: methods: data analysis – techniques: photometric –techniques: spectroscopic –galaxies: kinematic and dynamics –galaxies: photometry

1 INTRODUCTION

Over the last three decades broad band observations of early-type galaxies were successfully analysed by a method commonlycalled surface photometry or, simply, photometry. This method isbased on the analysis of isophotal shapes of the projected sur-face brightness. The development of the method was stimulatedby the empirical discovery that the isophotes of early-typegalax-ies are reproduced by ellipses to better than 1 per cent (Kent1984;Lauer 1985b; Davis et al. 1985; Jedrzejewski 1987; Peletieret al.1990). Although the isophotes are elliptical in shape to high ac-curacy, under careful examination, many isophotes of early-typegalaxies do show differences from pure ellipses at a level of≈0.5% (e.g. Bender et al. 1988; Peletier et al. 1990). The true suc-cess of photometry was the ability to measure these deviationsand classify early-type galaxies accordingly into disky and boxyobjects (Lauer 1985a; Jedrzejewski 1987; Bender & Moellenhoff

⋆ E-mail: dxk@astro.ox.ac.uk

1987). When combined with information on total luminosity andspatially resolved spectroscopy, it followed that the duality ofphotometric properties of early-type galaxies is reflectedin aduality of kinematic properties, where faint disky objectswerefound to rotate faster than luminous boxy objects (Davies etal.1983; Davies & Illingworth 1983; Bender 1988; Nieto et al. 1988;Wagner et al. 1988; Bender et al. 1989; Nieto & Bender 1989;Busarello et al. 1992; Bender et al. 1994). High resolution imag-ing studies with the Hubble Space Telescope (Jaffe et al. 1994;Lauer et al. 1995; Rest et al. 2001), again based on photometricanalysis, revealed new properties of early-type galaxies that deep-ened the division of the galaxies in two groups. These remark-able set of discoveries resulted in a revised classificationschemeof galaxies (Kormendy & Bender 1996).

It is fair to say that our increased knowledge of early-typegalaxies (partially) comes from photometry and the abilityof themethod to harvest and describe in a compact way the informa-tion from two-dimensional images. However, the integratedlightremains a limited source of information about the internal struc-

c© 2005 RAS

2 Davor Krajnovic et al.

ture and thus the true nature of galaxies. Kinematic informationis also essential to fathom the complexity of galaxies and, es-pecially, two-dimensional kinematic information is required (e.g.Franx et al. 1991; Statler 1991; Statler & Fry 1994; Statler 1994a,b;Arnold et al. 1994).

Two-dimensional velocity maps were, until recently, onlypossible for objects with clear emission lines. Such maps werethe result of studies of, e.g., HI with radio interferometers,(e.g. Swaters et al. 2002), CO with millimetre interferometers(e.g. Helfer et al. 2003) and Hα with Fabry-Perot spectrographs(e.g. Hernandez et al. 2005). The advent of integral-field spectro-graphs (e.g.TIGER, OASIS, SAURON, PMAS, GMOS, SINFONI,OSIRIS), however, has brought two-dimensional kinematic mea-surements to classical optical and near-infrared wavelengths. Hereit is possible to probe both the stellar absorption- and gas emission-lines, which may co-exist in the same potential with very dif-ferent spatial distributions and dynamical structures. The wealthof features seen in stellar kinematic maps of early-type galaxies(Emsellem et al. 2004) confirms the usefulness of two-dimensionaldata, but also poses a problem to efficiently harvest and interpretthe important features from the maps.

An approach using harmonic expansion was developed forthe analysis of two-dimensional velocity maps of disk galaxies.This method divides a velocity map into individual rings (the so-called tilted-ring method, Begeman 1987) and performs a har-monic expansion along these rings (e.g. Binney 1978; Teuben1991; Franx et al. 1994; Schoenmakers et al. 1997; Wong et al.2004). This analysis, however, is based on the assumption that theemission-line emitting material is confined to a thin disk struc-ture. Clearly, a spheroidal distribution of stars typical of early-typegalaxies does not have the same dynamical properties as a gasdisk.To explore those intrinsic properties, one needs a more generalmethod which will not be based on assumptions about the natureof the observed system (e.g. a disk), but will rely solely on theproperties of the investigated observables (e.g. surface brightnessor velocity).

Here we explore such a general method. Photometry earnedits spurs in intensive applications over the last three decades andoffers a natural starting point for the analysis of two-dimensionalkinematic maps, although one can, in principle, invent a numberof basis functions which can describe a two-dimensional distribu-tion. In this paper, we present a generalisation of photometry tothe higher-order moments of the line-of-sight velocity distribution(LOSVD). This generalisation is based on the theoretical fact thatsurface brightness is itself a moment of the LOSVD, and on ourem-pirical discovery that the velocity maps of many early-typegalax-ies can be well reproduced by a simple cosine law along samplingellipses. We call our methodkinemetry1, which reduces to photom-etry for the investigation of surface brightness distributions.

In Section 2 we present the theoretical background and moti-vation for the new method. Section 3 presents the technical aspectsof the method. The meaning of the kinemetric coefficients andtheirdiagnostic merits for different model maps are presented inSec-tion 4. In Section 5 we apply the method on kinematic maps of anaxisymmetric model galaxy, present the application of kinemetry toactual observations and characterise typical structures on velocitymaps. We summarise our conclusions in Section 6.

1 This name was introduced by Copin et al. (2001) who presenteda prelim-inary discussion on this topic.

2 THEORETICAL BACKGROUND AND MOTIVATION

The dynamics of a collisionless stellar system is fully specified byits phase-space density or distribution functionf = f(~x,~v, t) (e.g.Binney & Tremaine 1987), but this quantity is not measurabledi-rectly. When observing external galaxies, we measure propertiesthat are integrated along the line-of-sight (LOS). An additionalcomplication is that the galaxies are viewed from a certain direc-tion, and we actually observe only those projected properties of theintegrated distribution function.

The observables, which reveal only averages over a largenumber of unresolved stars, are the surface brightness and theLOSVD, sometimes also called the velocity profile2 (for a reviewsee de Zeeuw 1994). The surface brightness is given by

µ(x, y) =

∫

LOS

dz

∫ ∫ ∫

d~v f(~r,~v). (1)

The rest of the observable information is carried by the LOSVD,which relates to the distribution function via

L(v;x, y) =∫

LOS

dz

∫ ∫

dvxdvy f(~r,~v), (2)

where (x,y,z) are the three spatial coordinates, oriented such that theLOS is along thez-axis. From these equations it is trivial to showthat surface brightness is the zero-th order moment of the LOSVD.Higher-order moments, such as〈V 〉, or 〈V 2〉, can easily be relatedto the observables of the LOSVD: mean velocityV , velocity dis-persionσ and higher-order moments, commonly parametrised byGauss-Hermite coefficients (van der Marel & Franx 1993; Gerhard1993),h3 andh4 being the most used.

The kinematic moments of stationary triaxial systems showa high degree of symmetry which can be expressed through theirparity. The mean velocity is an odd moment, while the velocitydispersion is an even moment. In practice, this means that a two-dimensional map of a given moment shows corresponding symme-try. Maps of even moments arepoint-symmetric, while maps of oddmoments arepoint-anti-symmetric. In polar coordinates on the skyplane this gives:

µe(r, θ + π) = µe(r, θ),

µo(r, θ + π) = −µo(r, θ), (3)

where µe and µo are arbitrary even and odd moments of theLOSVD, respectively. Furthermore, if the observed system is ax-isymmetric, the even moment of the LOSVD will also bemirror-symmetricor, correspondingly, an odd moment will bemirror-anti-symmetric:

µe(r, π − θ) = µe(r, θ),

µo(r, π − θ) = −µo(r, θ). (4)

A kinematic moment that satisfy both symmetries (eqs. 3 and 4) issaid to be bi-(anti)-symmetric. The existence of these symmetriescan be used to simplify the harmonic analysis. Point-symmetricmoments give rise only to even harmonic terms, while point-anti-symmetric moments will require only odd harmonic terms. Mirror-(anti)-symmetry additionally requires that there is no change in themoment’s position angle. For a more detailed discussion on the in-fluence of these equations on the terms of the harmonic expansionsee Appendix A.

2 In this work, however, the term ‘velocity profile’ (or generally, kinematicprofile) will be used to denote a set of velocity (kinematic) measurementsextracted from a map along a certain curve.

c© 2005 RAS, MNRAS000, 1–17

Kinemetry 3

Following these conventions, surface brightness, as the zerothorder moment, is an even moment. As the surface brightness and thekinematic moments of the LOSVD are moments of the same distri-bution function, it is natural to analyse them similarly, and we canask in which way one can generalise the method of photometry towork on the higher moments of the LOSVD. Considering symme-try, the even moments do not require much change in the method,but the odd moments need a new working assumption. Also, whilethe isophotes of integrated light are elliptical to a high accuracy,the choice of sampling curves is not as obvious for odd kinematicmoments.

Since the light distribution of early-type galaxies is ellipticalit is reasonable to assume that an expansion along ellipses wouldalso be suitable for the extraction of profiles from kinematic mo-ment maps giving some insight in their structure and properties.The choice of the actual ellipticity, however, is somewhat arbitrary.For example, it is possible to expand along the galaxy isophotesor along ellipses that correspond to deprojected circles. In the firstcase, one follows the photometry and probes regions of the sameprojected surface brightness. In the second case, one samples lo-cations with equal intrinsic radii, but it is necessary to assume aninclination for the galaxy, which is usually difficult to estimate. Itmay also not be physically justified to use a constant ellipticity forthe whole velocity map, as the ellipticity of the light distributionmay change with radius.

As a special case, it is possible to expand a kinematicmap along circles. This approach was investigated by Copin et al.(2001), Copin (2002) and Krajnovic (2004), and although itisstraight-forward and requires no a priori assumption, a large num-ber of harmonic terms are often necessary, preventing a simpledescription of the maps and complicating the analysis (see Ap-pendix A for a discussion). Additionally, an expansion along el-lipses can reduce to expansion along circles in limiting cases (seesection 4.3). Franx et al. (1989) also note the advantages ofextrac-tion along circles, even for photometry, but conclude that an ap-proach based on ellipses is superior.

Consider the first moment of the LOSVD, the mean velocity.In specific cases, the geometry of the system offers a choice forcurves along which velocity profiles can be extracted. If a velocitymap traces material in a thin disk, such as is often the case inHI,CO and Hα gas emission maps, velocity can be well represented bycircular motion. If the inclination of the disk is known, a circularorbit in the plane of the galaxy can be projected as an ellipseon thesky. A velocity profile extracted along this ellipse is then of simplecosine form, which can be expressed by

V (R,ψ) = V0 + VC(R) sin i cosψ, (5)

whereR is the radius of a circular ring in the plane of the galaxy(or the semi-major axis length of the ellipse on the sky),V0 is thesystemic velocity,VC is the ring circular velocity,i is the ring in-clination (i = 0 for a ring seen face-on) andψ is the azimuthalangle measured from the projected major axis in the plane of thegalaxy. The inclinationi is related to the axial ratio or flattening,q, of the ring’s major and minor axes. For a given flattening, thering’s velocity profile is most similar to the velocity of a circularorbit with radiusR and inclinationcos i = q. In other words, theflattening defines an ellipse on the sky, with ellipticityǫ = 1 − q,which corresponds to a circle describing the orbits in the plane ofthe galaxy.

If one assumes that the velocity profiles extracted from veloc-ity maps (either stellar or gaseous) of early-type galaxiescan bedescribed by the simple cosine law of eq. (5) to a high accuracy,

then the first step in the analysis of velocity maps is to determinethose ellipses along which the velocity profile is best described byeq. (5). In analogy with photometry, the next step would be tode-termine the deviations of the velocity profile from the cosine law,which can be achieved by the higher-order Fourier analysis.

We discovered that the velocity profiles extracted from themaps of theSAURON sample can indeed be well described by thecosine law of eq. (5). Note that this observation suggests that thevelocity maps of early type galaxies closely resemble the observedkinematics of inclined circular disks. One has, however, totake astep back and recognise that spheroids are not disks, and thetwocases should be clearly differentiated during the interpretation ofthe results. The fact that the velocity maps of galaxies are well ap-proximated by the thin disk model allows us to develop a simplegeneralisation of photometry based on eq. (5), but also reflects aninteresting property of the internal structure of galaxies, which isworth exploring further (see section 5.2 for a few examples whilethe results for the rest of theSAURON sample will be reported in afuture paper).

The method presented in this paper is applicable to all mo-ments of the LOSVD. The quality of kinematic data, however, usu-ally decreases rapidly with increasing order of the kinematic mo-ment: the signal is generally simply too weak for a good enoughextraction of the higher-order moments from raw data. Perhaps it iseven fair to say that only the mean velocity maps reach the qualityof photometric observations from thirty or so years ago, when sur-face photometry was introduced. For these reasons, as well as forpresentation purposes, we restrict our analysis to the meanvelocitymaps. The method, however, can be straight-forwardly applied tothe higher order moments (see Section 5.1 for an example).

3 THE METHOD

In this section we present the details of the new algorithm extendingsurface photometry to the maps of odd moments of the LOSVD.

3.1 Harmonic expansion

Fourier analysis is the most straight-forward approach to charac-terise any periodic phenomenon. A velocity mapK(a, ψ) can bedivided into a number of elliptical rings yielding velocityprofileswhich can be described by a finite (and small) number (N + 1) ofharmonic terms (frequencies):

K(a, ψ) = A0(a) +N∑

n=1

An(a) sin(nψ) +Bn(a) cos(nψ), (6)

whereψ is the eccentric anomaly, that in case of disks correspondsto the azimuthal angle of eq. (5), anda is the length of the semi-major axis of the elliptical ring.

As presented by Carter (1978), Kent (1983) and Jedrzejewski(1987), to find the best sampling ellipse along which a profileµe

should be extracted, one minimises the harmonic terms sensitive tothe parameters of the ellipse (centre, position angle and ellipticityfor a given semi-major axis length). In the case of an even moment(e.g. surface brightness) the best fitting ellipse parameters are givenby minimisingA1,B1,A2 andB2 coefficients of the harmonic ex-pansion. This means that a photometric ellipse is determined whena sufficiently good fit toµe(a,ψ) = A0(a) is obtained. In the caseof an odd moment of the LOSVD (e.g. mean velocity), given ourempirical finding, the sampling ellipse parameters are determined

c© 2005 RAS, MNRAS000, 1–17

4 Davor Krajnovic et al.

by requiring that the profile along the ellipse is well described byeq. (5), or, generally,µo(a,ψ) = B1(a) cosψ.

Ellipse parameters (position angle, centre and flattening)canbe determined by a minimising a small number of harmonic terms.In Appendix B we investigate which terms are sensitive to changesof the ellipse parameters. We find that:

- an incorrect kinematic centre produces non-zeroA0, A2, B2,and to a lower levelA1,A3 andB3 coefficients,

- an incorrect flattening (inclination) produces a non-zeroB3

coefficient,- an incorrect position angle produces non-zeroA1,A3 andB3

coefficients.

In the study of disk velocity maps, Schoenmakers et al. (1997)in their Appendix A2, discuss the same sensitivity, but for smalldeviations. In this limit, our results reduce to the analytically de-rived conclusions of Schoenmakers et al. (1997). It is, however, im-portant to note, that whatever the combination of incorrectcentre,flattening or position angle, only then ≤ 3 harmonic terms areaffected.

3.2 The algorithm

Given our discovery that the cosine law along sampling ellipses de-scribes well velocity maps of galaxies, a small number of harmonicterms is needed to determine the best fitting ellipse. In thiscaseeq. (6) reduces to

K(ψ) = A0 +A1 sin(ψ) +B1 cos(ψ)

+A2 sin(2ψ) +B2 cos(2ψ)

+A3 sin(3ψ) +B3 cos(3ψ) (7)

whereψ is the eccentric anomaly. In practice, as it is done in pho-tometry, we sample the points uniformly inψ and the ellipse coor-dinates are then given byxe = a cosψ andye = q a sinψ. Thisimplies that, if the ellipse is projected onto a circle, the samplingpoints are equidistant. The selection ofK(ψ) values is obtained bya bilinear interpolation of the observed map, also in the same wayit is usually done in photometry (Jedrzejewski 1987).

The parameters of the best sampling ellipse for an odd kine-matic moment are obtained by minimising:

χ2 = A2

1 + A2

2 +B2

2 +A2

3 +B2

3 (8)

from eq. (7). This non-linear fit is performed in two stages. Thefirst stage consists of minimisingχ2 on an input grid of flatteningsq and position anglesΓ. The second stage repeats the fit using anon-linear least-square curve fitting algorithm, but now also fittingfor the coordinates of the centre, taking as initial valuesq = qmin

andΓ = Γmin from the first stage and an estimate of the centre(x0, y0). In this way, the trueχ2 minimum can be robustly deter-mined and, hence, the parameters of the best sampling ellipse: cen-tre (xc, yc), flattening (ellipticity) and position angle. We use theMINPACK implementation (More et al. 1980) of the Levenberg-Marquardt method3. This two stage approach was implemented toimprove the robustness of the method, preventing the minimisationprogram from choosing possible secondary minima.

The choice for the length of semi-major axis of consecutiveellipses is set by the decrease of surface brightness of the observed

3 We used an IDL version of the code written by Craig B. Markwardt andavailable from: http://astrog.physics.wisc.edu/∼craigm/idl

object. The kinematic maps are necessarily spatially binned (e.g.using adaptive Voronoi binning (Cappellari & Copin 2003), as inthe examples of Section 5), and as the bins typically increase insize with radius, the distance between the rings has to be increasedto avoid correlation between the ring parameters. The correlationshould also be avoided in the centre where the bins are usually iden-tical to the original pixels. For these reasons, we adopted acombina-tion of linear and logarithmic increase of the semi-major axis lengthgiven by this expression:a = n+ 1.1n, wheren = 1, 2, 3....

The final ellipse parameters obtained by minimisation are thenused to describe an elliptical ring from which a kinematic profile isextracted and expanded onto the harmonic series of eq. (6), wherethe coefficients(An, Bn) are determined by a least-squares fit witha basis{1, cosψ, sinψ, ..., cosNψ, sinNψ}.

The harmonic series can be presented in a more compact way,

K(a, ψ) = A0(a) +N∑

n=1

kn(a) cos[n(ψ − φn(a))]. (9)

where the amplitude and phase coefficients (kn, φn) are easily cal-culated from theAn, Bn coefficients:

kn =√

A2n +B2

n and φn = arctan

(

An

Bn

)

. (10)

The sin andcos terms have different properties and, in prin-ciple, describe different characteristics of the maps. Theprop-erties of the corresponding coefficients have been studied anddescribed for the case of thin gaseous disks (Franx et al. 1994;Schoenmakers et al. 1997; Wong et al. 2004). In a general casehowever, namely for the analysis of kinematic maps of triaxial (stel-lar) systems, those properties do not apply (e.g. theb1 term rep-resents circular rotation only in the thin disk approximation). Forthese reasons, we find it more instructive in this paper to combinethe coefficients of the same order and characterise the global prop-erties of kinematic maps. In what follows, although the implemen-tation of the method determines individual coefficients (and can bestraight-forwardly used for the special cases of disks), they will bepresented in the spirit of eq. (9).

3.3 Internal error estimates

The uncertainties of the resulting parameters can be estimated fromthe measurement errors of the input kinematic data. The ellipseparameters determined by the Levenberg-Marquardt least-squaresminimisation code have formal one-sigma uncertainties computedfrom the covariance matrix. If an input parameter is held fixed, orif it touches a boundary, then the error is reported as zero. Simi-larly, the harmonic terms obtained in the linear least-squares fit inthe second part of the procedure have their formal one-sigmaerrorsestimated from the diagonal elements of the corresponding covari-ance matrix, as described in Section 15.6 of Press et al. (1992).

We performed Monte Carlo simulations to estimate the ro-bustness of the obtained internal errors. We used theSAURON

(Bacon et al. 2001) velocity map of NGC 2974 presented inEmsellem et al. (2004) and its measurement errors to construct onehundred realisations of the velocity map. The velocity in each binwas taken from a Gaussian distribution of mean equal to the ob-served velocity and standard deviation given by the correspondingmeasurement error. Typical errors on the velocities of theSAURON

observations are about5km s−1. All realisations provide a distribu-tion of values from which we estimate one-sigma errors. The MonteCarlo uncertainties are in a good agreement with the formal errors.

c© 2005 RAS, MNRAS000, 1–17

Kinemetry 5

Figure 1. Four model maps representing different but typical velocity maps of triaxial early-type galaxies. Velocity maps were constructed of two components(see text for details). The position angle of the first (nuclear) component changes from left to right: 0, 45, 90 and 180◦. The size of bins increases towards theedge of the maps, as expected form observations, to keep constant signal-to-noise ratio. Note also the different opening angles of the two components on themaps.

Note, however, that these errors are statistical errors only and do notinclude systematic effects. We investigate the resulting accuracy ofthe kinemetric coefficients in Section 5.

4 KINEMATIC PARAMETERS AND THEIR MEANING

The crucial aspect of any analysis method is the interpretation ofits results. In this section, we describe in more detail the way themethod works and demonstrate the meaning of the resulting kine-metric parameters.

4.1 Model velocity maps

We constructed a set of model velocity maps combining two com-ponents using the circular velocity from the Hernquist (1990) po-tential, weighted with two different surface brightness distributions.The Hernquist potential was chosen because it approximatesrea-sonably well the density of real early-type galaxies. Its circular ve-locity is given byVc =

√GMr/(r + r0), whereG is the grav-

itational constant,M is total mass of the system, andr0 is scalelength of the potential. The

√GM factors for the two components

were 850 and 1500 (in units ofkm s−1 arcsec1/2), respectively,while r0 scale lengths were set to 5′′and 15′′, defining a central andlarge scale component, respectively.

The light distribution of the first component was described byan exponential law typical of disks, and that of the second com-ponent by anr1/4 law typical of spheroids. They were both nor-malised to a maximum value of unity. The light distribution scalelengths,re, of the two components were set to 7′′and 15′′, re-spectively. The components were inclined (60◦and 40◦, respec-tively) and combined with 4:1 relative weights, respectively, ro-tating the central component for different position anglesΓ =(0◦, 45◦, 90◦, 180◦) in order to construct different features on ve-locity maps possible to occur in triaxial early-type galaxies.

The final result are four model velocity maps, which satisfythe symmetry relations of eq. (3), presented in Fig. 1. Each modelhas a distinct nuclear component with a somewhat higher maxi-mum velocity and a smaller opening angle than the large scale,spheroidal component. We define the opening angle loosely asaquantity that describes the tightness of the iso-velocity contours. Itis discussed in more detail below. The main difference between themodels is in the orientation of the nuclear component. All modelmaps were constructed to simulate realistic observations with theSAURON integral-field spectrograph, where the size of the spatial

bins increases from the centre outwards to maintain constant signal-to-noise ratio.

4.2 Kinemetric analysis of model maps

The parity of an odd kinematic moment guarantees that the eventerms in the expansion vanish in the absence of measurement errorsand assuming that the centre is given correctly. These conditions aresatisfied for the model velocity maps and, while applying thealgo-rithm of section 3.2, we simplified eqs. (7) and (8) by not minimis-ing the contribution ofA2 andB2. However, during the harmonicanalysis we expanded the extracted velocity profiles including twoadditional odd harmonic terms,A5 andB5, which describe the de-viations from circular rotation (see eq. 5).

A special case is that of the zeroth-order term given by theA0 coefficient. It is not shown in the subsequent analysis becausethe absolute level of the maps was set to zero by construction. Forvelocity maps,A0 gives the systemic velocity of each ring. In thecase of even kinematic moments, the meaning ofA0 is different: itis the dominant term describing the amplitude of the moment (e.g.intensity for surface brightness, amplitude for velocity dispersionof velocity dispersion maps, see section 5.1).

Figure 2 shows the kinemetric parameters which describe themodel velocity maps. All parameters are plotted versus the semi-major axis length of the ellipses. The first and second row show theorientation (Γ) and flattening (q) of the ellipse along which the ve-locity profile was extracted. These parameters uniquely specify thebest fitting ellipse, along which the contribution ofA1, A3 andB3

terms is minimal. The following rows show plots of the harmoniccoefficients given in the compact form of eq. (10):k1 andk5/k1.

The meaning of these parameters can be associated with thevisible structures on the maps as follows:

(i) Γ is the kinematic position angle. Like the photometric po-sition angle, it describes the orientation of the velocity map, andit is related to the projection of the angular momentum vector (seeAppendix C).Γ traces the position of the maximum velocity on themap.

(ii) q is the axial ratio or flattening of the ellipse and it describesthe flattening of the map. It can be interpreted as a measure oftheopening angle of the iso-velocity contours. Maps that have largeopening angles appearround (large values ofq), while maps withsmall opening angle appearflat (small values ofq). In the specialcase of a disk, where the motion is confined to a plane (spiral galax-ies, gaseous disks, etc), the flattening is directly relatedto the incli-

c© 2005 RAS, MNRAS000, 1–17

6 Davor Krajnovic et al.

Figure 2. Kinemetric coefficients of the model maps. From left to rightthe plots show the results formodel 0, model 45, model 90andmodel 180. From topto bottom: kinematic position angleΓ, flatteningq, coefficients of harmonic expansionk1 andk5/k1. Note that the sudden rise ofk5/k1 in model 180at 9′′ isdue to the nearly zero value of the rotation velocityk1 at that radius.

nation of the disk,q = cos(i), where the projected ellipses corre-spond to intrinsically circular orbits.

(iii) k1 is the dominant coefficient on maps of odd kinematicmoments. For velocity maps, it describes the amplitude of bulk mo-tions (rotation curve). It contains contributions from both cosψ andsinψ terms, and in the special case of a thin disk geometry, theA1

andB1 terms should be considered separately. In the case of perfectcircular motions (A1 and other higher terms are zero),k1 describesthe full motion on the map. In other words, if higher order coeffi-cients are zero (or negligible), then the velocity map is identical tothe velocity map of a disk viewed at inclinationi.

(iv) k3 is the combination of harmonic coefficients that are min-imised in the fitting process. This coefficient is sensitive to the, cen-tre, flattening and position angle of the ellipse and in special cases itwill be larger then zero indicating a velocity profile with a complexstructure (multiple extremes) originating from the overlap of mul-tiple kinematic components with different orientations (see Fig. 3).This coefficient, however, does not bring any additional informa-tion to q andΓ and we do not show it. On the other hand, ifq andΓ are fixed and not fitted, as is often the case in studies of disk ve-

locity maps, then this term brings the first correction to thesimplerotational motion.

(v) k5 is the first higher-order term which does not define theparameters of the best fitting ellipse. It represents the higher-orderdeviations from simple rotation and points to complex structures onthe maps. This term is thus sensitive to the existence of separatekinematic components. It is a kinemetric analogous of the photo-metric term that describes the deviation of the isophote shape froman ellipse (diskiness and boxiness)

Comparison of the structures on the model maps (Fig. 1) andthe radial dependence of the kinemetric coefficients (Fig. 2) illus-trates the descriptive power of the method. Our model maps werebuilt of two components which had different flattening and kine-matic position angle. It is the role of the kinemetric coefficients torecognise these characteristics and describe them (Fig. 2).

The components ofmodel 0were aligned, but the nuclear com-ponent iso-velocities had a small opening angle, while those of thelarge-scale component had a large opening angle. This is reflectedin the different flatteningsq. In the inner 10′′, the disky componentdominates and the map is ratherflat, while beyond this region the

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Figure 3. Velocity profile taken from themodel 90map along a circle(q = 1) at r=9′′ , in the transition region between the two components. Over-plotted with diamonds is the circular velocity profile at thesame position.The observed profile exhibits a complex structure which can not be repre-sented by requiring that the third-order terms in the harmonic expansion bezero. The local extremes atψ ≈ 100◦ and280◦ originate in the nuclearcomponent, while the global maximum and minimum are given bythe largescale component.

spheroidal component dominates and the map looks moreround.In the regions where only one component dominates, the recoveredaxial ratio corresponds correctly to the inclination of that compo-nent. While thek1 terms show the amplitude of the rotation, thek3 andk5 coefficients are close to zero. This is expected becausethe models were built of two maps of essentially circular velocity.A small k5 terms exists in thetransition regionbetween the com-ponents. This suggests that the combination of aligned but separatevelocity components will give rise to non-zerok5.

A confirmation of this effect is given bymodel 180, where thetwo components are aligned but counter-rotating. In this configu-ration the components are clearly visible on theq andk1 panelsandk5 rises strongly in the region coinciding with the change ofrotational sense (change in the kinematic position angle by180◦).

Differences in kinematic position angle between the compo-nents, as in the maps ofmodel 45and model 90, generate morepower in higher order coefficients. The position angle is recoveredcorrectly in the regions dominated by one component, while in theregion between the components it traces the position of (combined)maximum velocity. It is the same with the flattening, which followsthe change of the map topology, reaching extreme round valuesin the transition region where both components equally dominate.The reason for this is the complex shape of the velocity profile, forwhich it becomes impossible to make theA3 andB3 terms alongany ellipse equal to zero (Fig. 3). In such a case, the fitting routinechooses the equal ratio of axes as this is the most general case, con-firming the robustness of the approach. The two stage fit, direct gridfitting followed by non-linear minimisation, is necessary to providethe required robustness in such regions.

The extent of the transition region depends on the relative posi-

tion angle, larger angles resulting in a larger transition region. Thisis reflected in the behaviour of the higher-order terms, which be-come more important and influence a larger area with increasingmisalignment of the components. The increasing misalignment iseffectively separating the components, creating a complex, multi-peaked velocity profile such as in Fig. 3. Note that strong contribu-tion of higher-order terms in this examples is not necessarily repre-sentative of real galaxies, as the features on the velocity maps werenot modelled to represent specific observed cases.

Figure 4 shows the two-dimensional representation of the anal-ysis. For each model, we reconstructed a map using only theb1 termwhich is not minimised, and we named those new maps ’circularvelocity’. We also reconstructed a map using all 6 terms of the har-monic expansion, which we named ’kinemetric velocity’. Thedif-ference between the two velocity maps shows the contribution ofthe higher order terms (k3 andk5). These residual maps ofmodel 0andmodel 180show distinct five-fold symmetries coming from thehighk5 term. Residuals of other models show a combination of thek3 andk5 terms in the transition region where velocity profiles aresimilar to the one in Fig. 3

The two examples with five-fold symmetries are interestingwhen compared with the corresponding model maps. Both resid-ual maps have positive amplitudes on the negative side of thema-jor axis. This is, however, not the case for the model maps. Themodel 0map has two components which both rotate in the samedirection, while formodel 180the inner component counter-rotateswith respect to the outer component. Both models have a strong k5term, and hence five-fold residual structure in the transition region.Interestingly, the amplitude of the residuals ofmodel 180at e.g.r = 7′′ along the major axis is negative while the mean rotation atthe same radius is positive, belonging to the region dominated bythe counter-rotating component. It is thus possible to infer that theresiduals actually trace the contribution of the main body rotation inthis region, although this is not visible on the velocity map. A simi-lar situation is visible inmodel 0, but the two components co-rotateand the residuals show the same sense of rotation.

We conclude this exercise stating thatk5 is sensitive to theexistence of different components, where the increase ofk5 in thepresented models is correlated to larger misalignments between thecomponents.

4.3 Special cases

There are two possible cases of velocity maps for which the de-termination of kinemetric coefficients will be degenerate.The firstcase arises from the effect of solid body rotation. As shown inSchoenmakers et al. (1997), while it is possible to measure the po-sition angleΓ(a), it is impossible to uniquely determine the flatten-ing q(a), the centre and the systemic velocity. This degeneracy ispresent because all iso-velocity contours are parallel, and the open-ing angle is180◦.

Another extreme case occurs when rotation is very slow. Suchvelocity maps are often found in giant ellipticals which arethoughtto be triaxial bodies. Here the velocity profiles are roughlyconstantand do not show the parity expected from an odd moment of theLOSVD. There is no gradient in velocity and as a consequence theflattening and position angle cannot be determined.

In these two cases the new method cannot be applied blindlybecause it is not possible to interpret harmonic terms in thewaywe showed up to now. There are two possible ways to proceed. Oneway is to make an assumption about the flattening and the kinematicposition angle (e.g. use photometric flattening), and continue with

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Figure 4. Model velocity maps and the results of the kinemetric analysis. From top to bottom: maps corresponding tomodel 0, model 45, model 90andmodel180 (as noted at the end of each row). From left to right: model maps, maps of circular velocity, reconstructed model maps using 6 harmonic terms, residualbetween circular and reconstructed velocity maps. Overplotted on the model maps are the best fitting ellipses along which the velocity profiles were extractedand analysed. Note the change in flattening and position angle of the ellipses.

the harmonic analysis of the velocity profiles extracted along thoseellipses.

A more general way, for both cases, is to extract the velocityprofiles along concentric circles and perform the harmonic analy-sis on those. In this way one does not bring any assumption in theanalysis, but it is still possible to determine the rotationvelocity andkinematic position angle (if present) and detect trends in higher-order harmonic terms. A detailed discussion of extraction along cir-cles and the meaning of the coefficients is given in Appendix A. Anautomatic transition towards circles can be implemented byaddinga penalty term to theχ2, to bias the solution towards round shapes,when the gradient is small (or iso-velocities are parallel). The good-ness of fit is then determined byχ2

p = χ2 + αP (q). We alsonote that an expansion on circles might be good for the analysis ofhigher-order moments of the LOSVD which normally have ratherpoor signal-to-noise ratio.

5 APPLICATION

In this section we apply our kinemetry formalism to three differentexamples. Our wish is to justify the choice of expansion on ellipsesalong which the profiles satisfy the simple cosine law. We first applythe method to the odd and even kinematic moments of an axisym-metric model of a galaxy. Then we apply the method to a few ob-served galaxies. Finally, we offer a characterisation of structures onthe velocity maps which can be recognised through the behaviourof the kinemetric parameters.

5.1 Axisymmetric two-integral model

The working assumption of the method introduced in Section 2is that there exists an ellipse along which the profile of a kine-matic moment can be described by a simple cosine law (eq. 5).This profile is easily associated with the circular motions in a thindisk (e.g. Schoenmakers et al. 1997), which show regular velocity

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Figure 5. Maps of mean velocity (right), velocity dispersion (middle) σ, and Gauss-Hermite momenth3 (left) for the model galaxy with added intrinsic scattercorresponding to a realistic measurement withSAURON. The overplotted ellipses on the velocity map are the surface brightness contours. The plots in eachright and left column represent the kinemetric parameters (Γ, q, k1, k5/k1) describing the maps. The plots in the middle column are kinemetric analogs ofthe standard photometric parameters: position angle, flattening, intensity (A0 term) and the shape parameter (B4). Horizontal line in the last plot of the middlecolumn marks the division between boxiness and diskiness ofiso-σ contours. Blue lines in the middle and right column plots correspond to the noiseless mapsof σ andh3, whose contours are overplotted on theσ andh3 map.

maps. More general stellar systems with axial symmetry, however,also show very regular velocity maps. Here we demonstrate thatour working assumption of eq. (5) is satisfied in realistic models foraxisymmetric galaxies.

We constructed an axisymmetric model galaxy with a distri-bution function which depends only on the energy and one com-ponent of the angular momentum. The model was constructed us-ing the Hunter & Qian (1993) contour integration as in methodEmsellem et al. (1999) to resemble theSAURON observations ofNGC 2974 (Emsellem et al. 2004; Krajnovic et al. 2005). The dis-tribution function of the model yields the full LOSVD from whichthe observable kinematics can be extracted. This was done onalarge100′′ × 100′′ field by fitting a Gauss-Hermite series (V , σ,h3 andh4) to spatially binned data resembling typicalSAURON ob-servations. In order to mimic real observations, we also assigned

typicalSAURON observational errors to each model observable (therelevant errors here are on velocity, velocity dispersion andh3, re-spectively5 kms−1, 7 km s−1 and 0.03). These errors were usedto add intrinsic scatter to the noiseless model data by meansof aMonte Carlo realisation. For further details of the model, we referthe reader to Section 5.1 of Krajnovic et al. (2005). Model maps ofvelocity, velocity dispersion andh3 are presented in Fig. 5.

We performed a kinemetric analysis of two odd and one evenmoment of the model galaxy. The left column of Fig. 5 presentstheresults for the velocity map. The flattening is constant overmostof the map with a small but continuous change in the centre. Thekinematic position angle is very well determined and remains con-stant. The first harmonic term,k1, describes the rotational motionin the model galaxy and peaks around 15′′. The higher order term

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k5, mostly consistent with zero, shows some deviations on a 0.5-1%level in the centre and slightly more at larger radii.

The middle column of Fig. 5 presents the analysis of the firsteven moment of the LOSVD: the velocity dispersion map. As men-tioned earlier, because even moments have the same parity asthesurface brightness (zeroth moment), the kinemetric analysis of aneven moment, such as velocity dispersion, is analog to the photo-metric analysis of the surface brightness. We demonstrate this byfitting the velocity dispersion map using our code, but with the re-quirement thatµe(a,ψ) = A0(a) along a trial ellipse. This yieldsthe usual photometeric parameters: position angle, ellipticity, inten-sity and the shape parameters describing the deviations from the el-lipse. In our case of an even kinematic moment we recognise themas the: kinematic position angleΓ, flattening (q = 1 − ǫ), A0 andB4 harmonic terms, while other harmonic terms in eq. (6) are zero.The zeroth harmonic term,A0, is the dominant term in the expan-sion and it represents velocity dispersion profile with radius. Thefourth harmonic term,B4, can be used to describe the deviations ofthe iso-σ contours from an ellipse, where, as in photometry, nega-tive values are indication of boxiness, while positive values of disk-iness. The given velocity dispersion map is in general boxy,but be-tween 10′′and 20′′it becomes disky. The same region is marked bya change in flattening (elipticity) and orientation (position angle).

Observed velocity dispersion maps are usually noisier thanthevelocity maps. We analysed a noiseless model velocity dispersionmap (without intrinsic scatter) to determine with what confidenceone could analyse an observed velocity dispersion map. The resultsare also plotted in the middle column of Fig. 5, and they show thatthe main features of kinemetric coefficients are well recovered, sug-gesting that the method could be used to analyse velocity dispersionmaps obtained from data with high signal-to-noise ratio.

The right column of Fig. 5 presents the results of a similar ex-ercise, but this time on the observed third moment of the LOSVD(given by Gauss-Hermite parameterh3). The noise on this map con-siderably distorts the structure of the map. The kinematic positionangle is again very well determined, being, as expected, offset 180◦

from the kinematic angle of the velocity map, and thek1 term traceswell the amplitude of theh3 with radius. The flattening changeswith a jump at≈ 25′′. The higher-order coefficients are howeversubstantially influenced by the noise. Thek5 term is often morethan 10 % ofk1. We repeated the analysis on the noiseless (no in-trinsic scatter) map ofh3 moment. The new values ofq, PA andk1, also shown in Fig. 5, are in a good agreement with the valuesextracted from the map with the intrinsic scatter. This is, however,not the case for the last term,k5, where the noiseless model termis small (consistent with zero) everywhere except in the region ofthe change of the flatteningq. This feature is completely absentfrom the noise dominatedk5 term of the intrinsic scatter map. Weconclude from this exercise that, although the method can beap-plied straight-forwardly to the higher moments of the LOSVD, thesignal-to-noise ratio in current state of the art measurements is gen-erally not high enough to reliably recover kinemetric parameters ofthe kinematic moments beyond velocity dispersion.

The negligible higher-order harmonic terms for both odd mo-ments confirm that the assumption of a simple cosine law for thekinematic moments is justified in the case of an axisymmetricgalaxy. In other words, the method assumes that to zeroth ordergalaxies are axisymmetric rotators. The non-zero higher-order har-monic terms suggest the presence of multiple components (k5 6= 0).

A change inq andΓ, however, hints to departures from axisymme-try4.

5.2 SAURON velocity maps

We now apply the method to a few galaxies observed in theSAURON survey (de Zeeuw et al. 2002). We selected the velocitymaps of NGC 2549, NGC 2974, NGC 4459 and NGC 4473 becausethey show diverse velocity structures. The maps were presented inEmsellem et al. (2004) and are reproduced in the first row of Fig. 6.Inspection of the two-dimensional maps shows that the iso-velocitycontours of NGC 2549 behave differently in the central and outerregions, suggesting the existence of two velocity components. Thevelocity map of NGC 2974 is remarkably regular, with a constantopening angle of the iso-velocities. NGC 4459 has a round velocitymap with a large opening angle, and the velocity map of NGC 4473shows an unusual decrease of the velocity along the major axis.

Our purpose here is not to undertake a detailed study of theproperties of these galaxies, but to illustrate quantitatively the appli-cation of the new method. Kinemetric profiles for the velocity mapsof the four galaxies are shown in Fig. 7. We also show the pho-tometric position angle and flattening (whereqphot = 1 − ǫphot)for a tentative comparison between the surface brightness (as aneven moment of the LOSVD) and velocity (an odd moment of theLOSVD). A detailed study of the photometric properties of theSAURON galaxies will be presented in a paper in preparation bytheSAURON team. We summarise the main results as follows:NGC 2974 is largely consistent with axisymmetry (but seeEmsellem et al. (2003) and Krajnovic et al. (2005) for bar signa-tures), and the simplicity of the velocity map is reflected inthekinemetric coefficients: thek5 term is below 1-2%, while the flat-tening and position angle are both constant over the map except inthe central few arcsecs where the departures from the axisymmetryare the strongest. The photometric and kinematic position anglesand, especially, the flattenings agree very well.NGC 4459 is a kinematically round map with large opening anglesreflected in the high values ofq. The position angle of the veloc-ity map changes in the central region but remains constant beyond7′′. In the same region, the velocity reaches a maximum and thendecreases, with a hint of a rise towards the edge of the map. Sim-ilarly to NGC 2974, thek5 coefficient is small and below 2% inthe investigated region. It is possible that there is a separate centralvelocity component, indicated by the change inΓ andk1. The sim-ilarity with NGC 2974 is also reflected in the comparison withthephotometric parameters: the position angles agree very well beyond5′′ while flattenings agree over most of the map.NGC 2549 has a multi-component velocity map. Its central region(r < 5′′) is described by a linear rise in the velocity and a lowflattening, rising up to7′′ − 8′′ where the velocity reaches its max-imum. Further away the velocity drops and beyond10′′ so doesthe flattening (changing again atr > 17′′). These features are alsorecognisable on the velocity map. The behaviour of thek5 term is,however, not obvious from the map. It is on average larger thanin the two previous examples over the whole map, and it shows aclear rise between 10 and 15′′, marking the change between the twocomponents. The kinematic and photometric position anglesdifferinside 7′′ and outside 15′′, while flattening of the contours is equal

4 Strictly speaking, in axisymmetric galaxiesΓ has to be constant and equalto the photometric position angle.

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Figure 6. Velocity maps, reconstructed kinemetric velocity maps andthe corresponding residual map (constructed as in Fig. 4) ofNGC 2974, NGC 4459,NGC 2549 and NGC 4473 (from left to right). The best fitting ellipses are overplotted on the velocity maps. Maps were oriented such that the major axis ishorizontal, and the north-east orientation of the maps is shown with the arrows. Residuals of NGC 2974 and NGC 4459 are smaller then the five-fold residuals ofNGC 2549 and NGC 4473. Along the major axis, the residuals of NGC 2549 have the same sense of rotation as the velocity map, while residuals of NGC 4473show the opposite sense of rotation.

only beyond 15′′, but a general similarity between the two flatten-ings is noticeable.

NGC 4473 is our final example. The decrease in the velocity alongthe major axis coincides with the rise of the flattening andk5. Thekinematic flattening does not match well the flattening of thephoto-metric ellipses, although the position angles agree very well in theregion of the velocity decrease.

Fig. 6 shows the reconstructed (kinemetric velocity) and resid-ual (constructed as in Fig. 4) maps in the second and third row,respectively. All velocity maps are well reconstructed with a smallnumber of terms. NGC 2974 and NGC 4459 have small residu-als, and the features on their maps are well described with only k1terms. Both NGC 2549 and NGC 4473, however, have a strongk5term. In section 4.2, using model maps, we showed that largek5 islikely to indicate the existence of two kinematic components. Therelative orientation of the components can be any, but we draw theattention of the reader to the co- and counter-rotating cases pre-sented bymodel 0andmodel 180. Precisely the same behaviour isvisible in Figs. 6 and 7 for NGC 2549 and NGC 4473, respectively.

The two co-rotating velocity components are clearly visible inthe map of NGC 2549: a central fast rotating component with highflattening and an outer slower rotating component with somewhatsmaller flattening. On Fig. 6 it can be seen that along the major axisthe contribution of the residuals is in the same direction asthe bulkrotation, confirming the co-rotation of the components.

NGC 4473 is not as clear a case. The velocity map (Fig. 6)does not show any clear signature of the two kinematic compo-nents which are inferred from the higher-order kinemetric terms.The residual map (Fig. 6), however, shows that the residualsare op-posite to the bulk rotation, suggesting that the two components are

actually counter-rotating. This configuration can naturally explainthe observed decrease of rotation with radius. It is thus likely thatNGC 4473 is made up of two components: a dominant one respon-sible for the bulk rotation of the stars, and a lighter counter-rotatingone (perhaps a disk because of its confined contribution along themajor axis) which brings more weight to large radii. A confirmationof this result is given by Cappellari & McDermid (2005), who con-structed Schwarzschild dynamical models of this galaxy andrecov-ered two counter-rotating components in the distribution function.

The velocity maps presented in this section are by no meansmeant to be representative of all possible velocity structures. It is,however, striking to see how the assumption of the method (eq. 5)is satisfied to a level of 2% or better for a number of real galax-ies. On the other hand, some galaxies do show strong deviations inthe higher-order terms, corresponding to substructure andmultiplekinematic components. The correspondence between photometricand kinemetric parameters is also interesting. While it is expectedthat the two position angles should agree if a galaxy is axisymmet-ric, it is not expected that the flattening of the isophotes and thekinematic ellipses will be the same. This is satisfied in the case ofthin disks, whereqphot = qkin = cos(i), but this relation does notnecessarily apply to spheroidal systems. The correspondence be-tweenqphot andqkin for spheroidal galaxies is likely related to thedegree of anisotropy of the galaxies. It may also indicate that thesegalaxies contain stellar disks.

5.3 Characterisation of features on stellar velocity maps

It is possible to describe the variety of structures presentin veloc-ity maps of galaxies, as seen in Emsellem et al. (2004), or observed

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Figure 7. Application of the method toSAURON observations of NGC 2974, NGC 4459, NGC 2549 and NGC 4473. Each panel shows kinemetric coefficientsfor the velocity maps: the kinematic position angle, axial ratio of the best fitting ellipses, and harmonic terms:k1 andk5. Dashed horizontal lines are globalkinematic position angle determined as the median ofΓ (blue line) and with the method outlined in Appendix C. They are almost always indistinguishable. Forcomparison, the photometric values for the position angle and the flattening of the best fitting photometric ellipses areoverplotted (diamonds).

with other integral-field units, using the results of our analysis. Thefeatures can be present uniformly over a whole map or can be dif-ferent between the inner and outer parts of the map. Here we offer acharacterisation of features on the velocity maps (or the maps them-selves) based only on the kinemetric parameters. Building on theresults from the previous sections, velocity features can be sorted inat least the following four descriptive groups:Disk-like Rotation (DR). This group consists of the simplest andfeatureless velocity maps. A DR map shows clear rotation wherethe amplitude ofk1 is substantial, but not necessary constant or ris-ing. Thek5 coefficient is consistent with zero, while the kinematicposition angleΓ and the flatteningq remain constant.Multiple Components (MC). Similar to ourmodel 0, MC mapsexhibit features which can be related to kinematically different, butaligned components. The components are kinematically separate ifthe flattening changes in the transition region, or if it is different foreach component. Additionally, MC are detected by a rise of thek5coefficients in the transition region and possibly but not necessarilyby a drop ofk1.Kinematic Twists (KT). Twists in the iso-velocity contours arecharacterised by a change in the kinematic position angleΓ. Ad-ditionally, several characteristics of the MC group can be presentas well, e.g.k5 > 0 or drop ink1, indicating different kinematiccomponents.

Low-level Rotation (LR). A rather special feature on the veloc-ity maps occurs when there is no detectable rotation. This effectis dependent on the instrumental resolution (rotation is too low tobe measured), but its origin is in the formation scenario andin-trinsic distribution of orbits. This group can be characterised by arequirement thatk1 remains below a certain limiting value ofk′1.Maps belonging to this class are characterised by high amplitude ofk5, often bigger thenk1. Due to the indeterminacy ofq andΓ forthe LR type, the analysis should be performed along circles,whichchanges the meaning of the harmonic terms.

It is important to note that the exact details, such as the limitingvalue ofk′1, or the tolerated change inΓ, which serve to characterisethe structures, depend on the instrumental settings and thequality ofthe observed data. However, it is clear that the kinemetric parame-ters could be used to describe and classify the properties ofvelocitymaps.

6 SUMMARY

We have presented a generalisation of surface photometry tothehigher-order moments of the LOSVD, observed with integral-fieldspectrograph. We call our methodkinemetry. For even moments ofthe LOSVD, kinemetry reduces to photometry. For odd moments,

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kinemetry is based on the assumption that it is possible to define anellipse such that the kinematic profile extracted along the ellipse canbe well described by a simple cosine law. This assumption is satis-fied in the case of simple axisymmetric rotators. Kinematic profilesthat deviate from axisymmetry or contain multiple kinematic com-ponents will be more complex, and the residuals can be measuredthrough a harmonic analysis of the profiles. We also find that theassumption is well satisfied for a number of observed galaxies.

Since the profiles are extracted along the best fitting ellipse, thenumber of harmonic parameters necessary to describe the profile isusually small. A total of six harmonic terms is generally necessaryto determine the best fitting ellipse (A0, A1, A2, B2, A3 andB3).The deviations from the simple cosine law are carried in higher har-monic terms. It is not expected that many more higher-order termswill be necessary in the general case, and we stop the analysis at theA5 andB5 harmonics.

The most important parameters of the analysis are the kine-matic position angleΓ, flatteningq of the ellipses, and the harmonicterms:k1 andk5, where we combined the terms of the same order.We briefly summarise their properties:

- the kinematic position angle traces the maximum velocity anddescribes the orientation of the map.

- the flattening is related to the opening angle of the iso-velocities and describes the different components of the maps,which can appearflat or round. In the special case of a thin diskit gives the inclination of the galaxy;

- k1 is the dominant harmonic coefficient and describes the am-plitude of bulk motions. It is related to the circular velocity. Higherorder coefficients, which we normalise tok1, describe the devia-tions from simple circular motion;

- k5 is the first higher-order term which is not fitted and quan-tifies higher-order deviations from rotational motions. Itindicatescomplexity on the maps and is sensitive to the existence of separatekinematic components. A photometric analog of this term is the co-efficient that describes the diskiness/boxiness of the isophotes.

This method can be straight-forwardly applied to all momentsof the LOSVD observed from different objects (e.g. early- and late-type galaxies). However, the analysis and the intepretation of theharmonic terms has to be related to the physical nature of theob-served objects. An example are velocity maps of thin disks. In thiscase, the kinemetric flattening and position angle are directly re-lated to the inclination and orientation of the disk, and theresultsof kinemetry are equal to results of the tilted-ring method.If qandΓ are kept fixed, kinemetry reduces to the harmonic analysisof Schoenmakers et al. (1997) and Wong et al. (2004). In this case,since the velocity profiles are not extracted along the best fitting el-lipses, all harmonic terms will carry information about departuresfrom the simple circular motion.

We applied the method on one model and four observed galax-ies. The model galaxy was a two-integral model of an observedaxisymetric galaxy. We constructed maps of observable odd kine-matic moments of the LOSVD: mean velocity, velocity dispersionand Gauss-Hermite coefficienth3. We analysed both maps to showthat the method works well on an axisymmetric galaxy (all higher-order terms were small or consistent with zero). We also introducedan intrinsic scatter and currently realistic measurement uncertain-ties to show the difficulties expected analysing the higher-order oddmoments (e.g.h3) obtained from state-of-the-art integral-field ob-servations. The position angle, flattening and amplitude ofthe mo-ment can be realistically recovered, but the signal-to-noise ratio istoo low for expansion to higher-order harmonics.

The four galaxies were taken from theSAURON sample. Weused them to show the descriptive power of the new method: de-tection of co- or counter-rotating subcomponents well hidden in thekinematics, which could be detected previously only through de-tailed modelling. We also showed that there are early-type galaxieswith rotation velocity described by a simple cosine law. Thekine-metric position angle and flattening of these galaxies are ina goodagreement with the photometric position angle and flattening.

Based on the kinemetric parameters we characterise severalfeatures on the velocity maps: Disk-like Rotation, Multiple Compo-nents, Kinematic Twists and Low-level Rotation. Other, more spe-cific and instrument dependant features on the maps can be con-structed from these basic groups.

We wish to stress that the method works well on velocity mapsbecause many of them satisfy the basic assumption of the method tothe level of a few per cent. This empirical fact is related to the inter-nal structure of the galaxies. We developed the method of kinemetryto harvest the information from the observed maps of momentsofthe LOSVD probing the nature of galaxies.

An IDL implementation of the described algorithm is avail-able on the web address: http://www-astro.physics.ox.ac.uk/∼dxk/.

Acknowledgements We thank Eric Emsellem for providing amodel axisymmetric galaxy and Maaike Damen for providingthe photometric data used in this work. DK thanks Martin Bu-reau and Eric Emsellem for a careful reading of the manuscriptand Marc Sarzi, Glenn van den Ven and Richard McDermid forfruitful discussions. This research was supported by NOVA,theNetherlands Research School for Astronomy and by PPARC grantPPA/G/O/2003/00020 ’Observational Astrophysics in Oxford’. MCacknowledges support from a VENI grant 639.041.203 awardedbythe Netherlands Organization for Scientific Research (NWO).

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APPENDIX A: EXTRACTION ALONG CIRCLES

The simplest case of kinemetric expansion is by extracting velocityprofiles along concentric circles. This approach has several advan-tages over the more general case of ’ellipse fitting’. They are:

- there is no a priori assumption about the structure of the map- extraction of harmonic coefficients is a purely linear problem

(no degeneracy between parameters)- reconstruction of maps is straight-forward even in the most

complicated cases

These points, however, are weighted against the fact that thenumber of the extracted harmonic terms which describe a morecomplicated map is large, and often the coefficients cannot beclearly associated with the physical properties of the objects. In thisAppendix, for completeness, we briefly summarise the analysis ofCopin et al. (2001) and Krajnovic (2004).

Requiring that axial ratioq = 1, kinemetry reduces to Fourierexpansion along concentric circles via eq. (6), where the coeffi-cientsAn, Bn are determined by a least-squares fit with a basis{1, cos θ, sin θ, ..., cosNθ, sinNθ}, whereθ is now the polar an-gle, to which the eccentric anomaly,ψ, reduces in this case. De-pending whether the fit is to an odd or an even kinematic moment,the terms in the expansion can be selected such thatn is odd oreven, respectively. In a more compact way, the harmonic series ispresented by the eq. (9), where the amplitude and phase coefficients(kn, φn) are easily calculated from theAn, Bn coefficients follow-ing eq. (10).

The following description of the harmonic coefficients is sim-ilar to the description of the coefficients obtained by ellipse fitting,but differs in some crucial details.

The zeroth-order term,A0, measures the mean level of themap. For the first kinematic moment, this is equivalent to thesys-temic velocity of the galaxy. Forh3 maps, which have mean levelequal to zero by construction,A0 will, like other even terms, beconsistent with zero.

The coefficientk1 gives the general shape and amplitude ofthe odd moment map and is the dominant term. The first-order cor-rection is given by the next odd term,k3. This term can be namedthemorphologyterm because it describes most of the additional ge-ometry of the map. Often, the first two odd terms are enough todescribe the velocity map, although the kinemetric expansion mayrequire higher terms as well,k5 and evenk7.

Connected to the amplitude terms are the corresponding phaseterms,φn. They determine the orientation of the map, but their con-tribution depends on the relative strength of the amplitudeterms.The first phase coefficient,φ1, gives the mean position angle of thevelocity map (measured from the horizontal axis,θ = 0). The an-gle thatφ1 measures is the position angle of thek1 term. This is,in general, slightly different from the positions of the maxima onthe map, which are also influenced by the contribution from higher-order terms. However, it does give the global orientation ofthe map.This phase corresponds to previously defined kinematic position an-gleΓ.

The angleφ3 is the phase angle of the third harmonic: the nextsignificant term. For a small amplitude ofk3, its contribution tothe overall orientation will be small, and the position of the maxi-mum velocity will be given accurately byθmax = φ1. The phasesof higher-order terms are interesting because, for an axisymmetricgalaxy, they satisfy the relation:

φ1 − φi =nπ

i(A1)

c© 2005 RAS, MNRAS000, 1–17

Kinemetry 15

wheren ∈ Z andφi is thei-th order term in the expansion.This condition can be easily derived considering that for an

axisymmetric velocity map the position of the zero velocitycurve,the curve along which the velocity is zero on the map, is orthogonalto the kinematic angle given byφ1. This means that K(a, θ) = 0,for θ = φ1 + π/2. Neglecting the higher order terms, (ki > k3),and substitutingθ = φ1 + π/2 into eq. (9) one obtains the result ofeq. (A1).

Alternatively, deviations from axisymmetry can be quantifiedfollowing eqs. (9) and (A1). If K(a, θ = φ1 + π/2) = ∆V 6= 0,then one finds:

∆V

k1=k3k1

sin 3(φ1 − φ3), (A2)

where we express the relation as a ratio of the dominant term inthe expansion. This relation quantifies the contribution ofthe k3term due to departures from axisymmetry and can be generalisedto other higher terms. In cases of large misalignments between thekinematic and photometric position angles,φ1 should be replacedby the photometric PA in the above equation.

In many ways, analysis of the even moments is analogous tothat for the odd moments, taking the proper parity into account.

The dominant term of the even maps isA0, and it describes theabsolute level of the map as a function of radius. The next importantterm in the expansion isk2, and it is themorphologyterm of theeven moment maps, which describes features such as elongationalong the minor axis or “bow-tie” shapes, often seen in observedvelocity dispersion maps (Emsellem et al. 2004). The orientation ofthe morphologically distinct features are determined by the phaseanglesφ2 of thek2 term.

The parity of kinematic moments, expressed by eqs. (3) and(4) in section 2, has the following consequences on the harmonicexpansion along circles:

- point-symmetry requires only even terms in the harmonic ex-pansion

- point-anti-symmetry requires only odd terms in the harmonicexpansion

- additional mirror-(anti)-symmetry requires only cosinetermsin the harmonic expansion

It follows that kinemetry can easily be used for constructingtwo-dimensional kinematic maps of a given symmetry. This isdoneby fixing certain harmonic terms (odd, even or sine terms) to zero;e.g. a bi-anti-symmetric map is created by keeping only odd cosineterms in the harmonic expansion. Additionally, if the number ofharmonic terms in the expansion is small, the reconstructedmapwill be smoothed. In this way, kinemetry can be used for removingthe higher-order harmonics from the data and effectively filteringthe map.

APPENDIX B: INFLUENCE OF INCORRECT CENTRE,FLATTENING AND ORIENTATION

Here we test the sensitivity of harmonic expansion of a velocityprofile given by eq. (7) with respect to a chosen incorrect flattening(inclination in a thin disk case), position angle and the centre of theellipse along which the profile was extracted. We constructed a sim-ple velocity map using the circular velocity form the Hernquist po-tential and an exponential light profile corresponding to the central(disk) component of themodel 0. The velocity map had a constantflattening ofq0 = 0.5 and kinematic position angle ofΓ0 = 0◦.

Figure B1. The effect of an incorrect flattening on the coefficients of har-monic expansion of a velocity profile. Upper panels shows themodel ve-locity map having constant orientation (kinematic position angle) and flat-tening. The solid ellipse has correct flattening and orientation, while thedashed ellipse has incorrect axial ratio and correct orientation. The lowerpanel shows residuals between the extracted velocity profile and the cir-cular velocityB1 cos(ψ) (solid symbols). Overplotted are contributions ofA1 sin(ψ) (red line) andB3 cos(3ψ) (blue curve).

Figure B2. Same as Fig. B1 where the dashed ellipse has correct flatten-ing but incorrect orientation. Next to the residuals on the lower panel over-plotted are curves indicating the contribution of different terms:A1 sin(ψ)(red curve),A3 sin(3ψ) + B3 cos(3ψ) (blue curve), and a combinationA1 sin(ψ) + A3 sin(3ψ) +B3 cos(3ψ) (green curve).

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16 Davor Krajnovic et al.

Figure B3. Same as Fig. B1 where the dashed ellipse of correct flatteningand position angle was shifted by 1′′horizontally and 1′′vertically from thecentre. The lower panel presents the residual and contributions from differ-ent terms:A0 (black line),A1 sinψ (red line),A2 sin(2ψ) +B2 cos(2ψ)(orange curve) andA3 sin(3ψ) +B3 cos(3ψ) (blue curve). The combinedcontribution of all these terms is overplotted with the green curve.

Systemic velocity was set to zero. We first extracted a velocity pro-file from this map ata = 10′′ along a pre-designed ellipse. Theellipse was first chosen to have an incorrect flatteningq′ = 0.6,but correct kinematic position angleΓ0 (Fig. B1). We repeated theexercise selecting the ellipse with incorrectΓ′ = 10◦ and correctq (Fig. B2). Each of the profiles were then harmonically expandedwith eq. (7). Finally, we shifted an ellipse, defined by correct valuesof q andΓ, horizontally and vertically for 1′′(Fig. B3). The corre-sponding velocity profile was expanded using firstN = 6 (evenand odd) terms in eq. (6).

Harmonic expansion of a profile extracted along an ellipse de-fined by the correct centre, flattening and orientation would(byconstruction) yields all harmonic coefficients zero expecttheB1,which gives the circular velocity of the map. Hence, on the lowerpanels of Figs. B1, B2 and B3 we plot the difference (residuals) ofthe extracted velocity profile and theB1 cos(ψ) term.

On Fig. B1 overplotted are the contributions of terms:A1 sin(ψ) andB3 cos(3ψ). TheA3 sin(3ψ) is nearly zero, like thea1, and is not shown. The contribution of only theB3 cos(3ψ) termcan explain the distribution of residuals, confirming that only B3

coefficient is affected when choosing an incorrect flattening.On Fig. B2 overplotted are the following terms:A1 sin(ψ),

A3 sin(3ψ)+B3 cos(3ψ),B3 cos(3ψ), as well as the combinationA1 sin(ψ) + A3 sin(3ψ) + B3 cos(3ψ). All of these terms con-tribute significantly to describe the distribution of residuals. Thedominant coefficient isA1, whileA3 andB3 contribute in this ex-ample with about 45% and 15% in amplitude, respectively. In thelimit of a small incorrect position angle, Schoenmakers et al. (1997)showed that onlyA1 andA3 are affected, but our example showsthat for larger incorrect position angles theB3 term has to be con-sidered as well.

Fig. B3 presents contributions forA0, A1, A2, B2, A3 and

B3 harmonic terms. To the first order, onlyA0, A2 andB2 termsare sensitive to incorrect centring. The same result was also analyt-ically derived by Schoenmakers et al. (1997), in the limit ofsmallmiscentering. However, for large deviation from the true centre co-ordinates, contribution from theA3 andB3 terms becomes impor-tant, whileA1 remains much smaller.

We conclude this exercise by stating that the ellipse parametersfor an odd kinematic moment map can determined by harmonicexpansion of theA0,A1,A2,B2,A3 andB3 terms only.

APPENDIX C: DETERMINATION OF GLOBALKINEMATIC POSITION ANGLE

The apparent angular momentumLp of a galaxy, as a projection ofthe intrinsic angular momentum to the plane of sky, is definedas(Franx 1988):

Lp =

∫

P

rp ×Σvrdrp (C1)

whererp is a projection to the plane of sky of the vectorr at whichthe mean velocity of stars,v, projects to the observed radial velocityvectorvr. Σ is surface density and the integration is over the wholeprojection plane. As noted by Franx et al. (1991), it followsfromeq. (C1) that the apparent angular momentum is fully specified bythe observed surface brightness and the observed two dimensionalvelocity map. In addition to that, if figure rotation is absent, the ap-parent angular momentum is parallel to the projection of theintrin-sic angular momentum. This means that the projection of the fullthree-dimensional velocity space is reduced to the projection of theangular momentum. Reversing the argument, measuring the (lumi-nosity weighted) kinematic position angle of the two-dimensionalprojection of velocities, it is possible to determine the position ofthe apparent angular momentum.

The position angle of the apparent angular momentumΓkin

can be determined from eq. (C1) using integral-field observationsof galaxies. This is the most physical way of determination of Γkin,but it is somewhat dependant on the geometry of the maps5, like therelative orientation of the map with respect to the orientation of theapparent angular momentum.

A more robust way of determiningΓkin is based on a direct de-termination of the kinematic position angleΓ. Kinemetry measuresΓ as a function of radius, but we are interested in a global (indepen-dent of radius) value of the kinematic position angle,ΓG, becauseof a direct correspondence betweenΓG andΓkin. ΓG can be deter-mined in a robust way using the information in the whole velocitymap. For any chosenΓG, we construct a bi-anti-symmetric velocitymapV ′(x, y) with thex-axis along the givenΓG. This is done byreplacing the measured mean velocityV (x, y) inside each Voronoibin, of centroidal coordinates(x, y), with the weighted average ofthe corresponding velocity in the four quadrants, using linear inter-polation when needed to estimate the velocity:

V ′(x, y) =V (x, y) + V (x,−y)− V (−x, y)− V (−x,−y)

4(C2)

5 We do not discuss here the influence of the size of the maps. Clearly, onlyΓkin of the observed part of the galaxy can be determined, becauseeq. (C1)is defined over the whole projection plane

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Figure C1. Comparison of observed velocity map (left) and bi-anti-symmetric map (right) at the best fittingΓG using the method outlined inthe text. We usedSAURON observations of NGC 4473. North is up and eastto the left.

When any of the four symmetric coordinates was outside the con-vex hull defined by the bins centroids, we used only the remainingvalues to determinedV ′. Fig. C1 shows an example of a bi-anti-symmetric map. The trueΓG was defined as the angle which min-imises

χ2 =N∑

n=1

(

V ′(x, y)− V (x, y)

∆V (x, y)

)2

(C3)

In this way it is possible to assign simple error estimates totheobtainedΓG as the range of angles for which∆χ2 < 9, whichcorresponds to the3σ confidence level for one parameter.

We used velocity maps from section 5.2 to compute theglobal kinematic position angle. Obtained values for NGC 2974,NGC 4459, NGC 2549 and NGC 4473 are 43.2, 280.6, 1.0, 92.5,respectively. The results are overplotted on Fig. 7 where they canbe compared with the kinemetric position angle,Γ, which changeswith radius describing specific features on the maps. They are ina very good agreement as can be seen comparing with the medianvalues of the kinemetric position angle.

c© 2005 RAS, MNRAS000, 1–17