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Line-of-sight velocity dispersions and a mass distribution model of the Sa galaxy NGC 4594

Mon. Not. R. Astron. Soc. 000, 1–12 (2006)

Printed 5 February 2008

A (MN L TEX style ?le v2.2)

Line-of-sight velocity dispersions and a mass distribution model of the Sa galaxy NGC 4594
E. Tempel1,2? and P. Tenjes1,2 1

Institute of Theoretical Physics, Tartu University, T¨ ahe 4, 51010 Tartu, Estonia Tartu Astrophysical Observatory, 61602 T? oravere, Estonia

Accepted 2006 June 27. Received 2006 June 21; in original form 2006 April 25.

arXiv:astro-ph/0606680v1 28 Jun 2006


In the present paper we develop an algorithm allowing to calculate line-of-sight velocity dispersions in an axisymmetric galaxy outside of the galactic plane. When constructing a self-consistent model, we take into account the galactic surface brightness distribution, stellar rotation curve and velocity dispersions. We assume that the velocity dispersion ellipsoid is triaxial and lies under a certain angle with respect to the galactic plane. This algorithm is applied to a Sa galaxy NGC 4594 = M 104, for which there exist velocity dispersion measurements outside of the galactic major axis. The mass distribution model is constructed in two stages. In the ?rst stage we construct a luminosity distribution model, where only galactic surface brightness distribution is taken into account. Here we assume the galaxy to consist of the nucleus, the bulge, the disc and the stellar metal-poor halo and determine structure parameters of these components. Thereafter, in the second stage we develop on the basis of the Jeans equations a detailed mass distribution model and calculate line-of-sight velocity dispersions and the stellar rotation curve. Here a dark matter halo is added to visible components. Calculated dispersions are compared with observations along di?erent slit positions perpendicular and parallel to the galactic major axis. In the best-?tting model velocity dispersion ellipsoids are radially elongated with σθ /σR ? 0.9 ? 0.4, σz /σR ? 0.7 ? 0.4, and lie under the angles 30? with respect to the galactic equatorial plane. Outside the galactic plane velocity dispersion behaviour is more sensitive to the dark matter density distribution and allows to estimate dark halo parameters. For visible matter the total M/LB = 4.5 ± 1.2, M/LR = 3.1 ± 0.7. The central density of the dark matter halo is ρDM (0) = 0.033 M⊙ pc?3 . Key words: galaxies: individual: NGC 4594 – galaxies: kinematics and dynamics – galaxies: spiral – galaxies: structure – cosmology: dark matter.



The study of the dark matter (DM) halo density distribution allows us to constrain possible galaxy formation models and large scale structure formation scenarios (Navarro & Steinmetz 2000; Khairul Alam, Bullock & Weinberg 2002; Gentile et al. 2004). For this kind of analysis, it is necessary to know both the distribution of visible and dark matter. Without additional assumptions rotation curve data alone are not su?cient to discriminate between these two kinds of matter (Dutton et al. 2005). It does not su?ce either to use additionally velocity dispersions along the major axis. Realistic mass and light distribution models must be consistent, i.e. the same model must describe the luminos?

ity distribution and kinematics. Three main classes of selfconsistent mass distribution models can be discriminated: the Jeans equations based models, the speci?c phase space density distribution models and the Schwarzschild orbit superposition based models. Mass distribution models based on solving the Jeans equations have an advantage that the equations contain explicitly observed functions – velocity dispersions. On the other hand, there are three equations, but at least ?ve unknown functions (three dispersion components, centroid velocity and the velocity dispersion ellipsoid orientation parameter) and thus the system of equations is not closed. In addition, the use of the Jeans equations neglects possible deviations of velocity distributions from Gaussians and does not garantee that the derived dynamical model has non-negative phase density distribution everywhere. However, within certain approximations the Jeans equations

E-mail: elmo@aai.ee; peeter.tenjes@ut.ee

c 2006 RAS


E. Tempel and P. Tenjes
In the present paper, we construct a more sophisticated self-consistent mass and light distribution model. We decided to base it on the Jeans equations. For all visible components, both rotation and velocity dispersions are taken into account. The velocity dispersion ellipsoid is assumed to be triaxial and line-of-sight velocity dispersions are calculated. Mass distribution of a galaxy is axisymmetric and inclination of the galactic plane with respect to the plane of the sky is arbitrary. In order to discriminate between DM and visible matter, it is most complicated to determine the contribution of the stellar disc to the galactic mass distribution. Quite often the maximum disc approximation is used. In the present paper, we attempt to decrease degeneracy, comparing calculated models with the observed stellar rotation curve, velocity dispersions along the major axis and in addition, along several cuts parallel to the major and minor axis. In the case of two-integral models for edge-on galaxies this allowed to constrain possible dynamical models (Merri?eld 1991). First measurements of velocity dispersions along several slit positions were made by Kormendy & Illingworth (1982) and Illiingworth & Schechter (1982). Later, similar measurements were performed by Binney et al. (1990), Fisher, Illingworth & Franx (1994), Statler & Smecker-Hane (1999), Cappellari et al. (2002). In recent years, with the help of integral ?eld spectroscopy, complete 3D velocity and dispersion ?elds have been measured already for several tens of galaxies. We apply our model for the nearby spiral Sa galaxy M 104 (NGC 4594, the Sombrero galaxy). This galaxy is suitable for model testing, being a disc galaxy with a signi?cant spheroidal component. The galaxy has a detailed surface brightness distribution and a well-determined stellar rotation curve. M 104 has a signi?cant globular cluster (GC) subsystem. And as most important in our case, the line-of-sight velocity dispersion has been measured along the slit at di?erent positions parallel and perpendicular to the projected major axis. We construct the model in two stages. First, a surface brightness distribution model is calculated. Here we distinguish stellar populations and calculate their structure parameters with the exception of masses. In the second stage, we calculate line-of-sight velocity dispersions and the stellar rotational curve and derive a mass distribution model. Sections 2 and 3 describe the observational data used in the modelling process and construction of the preliminary model. In Section 4 we present the line-of-sight dispersion modelling process. Section 5 is devoted to the ?nal M 104 modelling process. In Section 6 the discussion of the model is presented. Throughout this paper all luminosities and colour indices have been corrected for absorption in our Galaxy according to Schlegel, Finkbeier & Davis (1998). The distance to M 104 has been taken 9.1 Mpc, corresponding to the scale 1 arcsec = 0.044 kpc (Ford et al. 1996; Larsen, Forbes & Brodie 2001; Tonry et al. 2001). The angle of inclination has been taken 84? .
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are widely used for the construction of mass distribution models. In the case of spherical systems with biaxial velocity dispersion ellipsoids, such models have been constructed, for example, by Binney & Mamon (1982), Merritt (1985), Gerhard (1991), Tremaine et al. (1994). In the case of ?attened systems with biaxial velocity dispersion ellipsoids, a general algorithm for the solution of the Jeans equations was developed by Binney, Davies & Illingworth (1990), Cinzano & van der Marel (1994). Another algorithm in the context of the multi-Gaussian expansion (MGE) formalism was developed by Emsellem et al. (1994). An approximation for cool stellar discs (random motions are small when compared with rotation) has been developed by Amendt & Cudderford (1991). Dynamical models with a speci?c phase density distribution have the advantage that velocity dispersion anisotropy can be calculated directly. On the other hand, due to rather complicated analytical calculations, only rather limited classes of distribution functions can be studied. Spherical models of this kind have been constructed by Carollo, de Zeeuw & van der Marel (1995), Bertin et al. (1997). In the case of an axisymmetric density distribution, velocity dispersion pro?les have been calculated for certain speci?c mass and phase density distribution forms by van der Marel, Binney & Davies (1990), Evans (1993), Dehnen (1995), de Bruijne, van der Marel & de Zeeuw (1996), de Zeeuw, Evans & Schwarzschild (1996), Merritt (1996), An & Evans (2006) and others. A special case is an analytical solution with three integrals of motion for some speci?c potentials: an axisymmetric model with a potential in the St¨ ackel form (Dejonghe & de Zeeuw 1988), isochrone potential (Dehnen & Gerhard 1993). Probably the most complete class of dynamical models has been developed on the basis of the Schwarzschild linear programming method (Schwarzschild 1979). Thus, it is not surprising that just for this method most signi?cant developments occured in last decade. Rix et al. (1997) and Cretton et al. (1999) have developed this method in order to calculate line-of-sight velocity pro?les. Thereafter, Cappellari et al. (2002) and Verolme et al. (2002) generalized it for an arbitrary density distribution linking it with MGE method. A modi?cation of the least-square algorithm was done by Krajnovi? c et al. (2005). Interesting comparisons of the results of the Schwarzschild method with phase density calculations within a two-integral approximation have been made by van der Marel et al. (1998) and Krajnovi? c et al. (2005). At present, nearly all dynamical models have been applied for one-component systems. However, the structure of real galaxies is rather complicated – galaxies consist of several stellar populations with di?erent density distribution and di?erent ellipticities. In addition, in di?erent components velocity dispersions or rotation may dominate. In our earlier multi-componental models (see Tenjes, Haud & Einasto 1994, 1998; Einasto & Tenjes 1999) we approximated ?at components with pure rotation models and spheroidal components with dispersion dominating kinematics. For spheroidal components mean velocity dispersions were calculated only on the basis of virial theorem for multi-componental systems. These models ?t central velocity dispersions, gas rotation velocities and light distribution with self-consistent models.

Mass distribution of NGC 4594
Table 1. Photometrical data
mB (mag arcsec ) 16




Faintest isophote

Colour system R B B B B V BV RI R V B R V

18 20 22 24 26 28 0.01 14 n


mR (mag arcsec )

Spinrad et al. (1978) Burkhead (1979) Boroson (1981) Hamabe & Okamura (1982) Beck et al. (1984) Jarvis & Freeman (1985) Burkhead (1986) Kent (1988) Kormendy (1988) Crane et al. (1993) Emsellem et al. (1994) Emsellem et al. (1996)


28.2 24.7 22.7 27.3 25.0 23.1 23.3 19.2 17.4 18.5 17.8






16 d 18 20 22 24 26 0.01 0.1 1 b h



By now a surface photometry of M 104 is available in U BV RIJHK colours. In the present study, we do not use the U -pro?le, as this pro?le has a rather limited spatial extent and is probably most signi?cantly distorted by absorption. In certain regions also the B -pro?le is probably in?uenced by absorption, but the B -pro?le has the largest spatial extent and we decided to use it with some caution outside prominent dust lane absorption. The JHK pro?les have a rather limited spatial extent and resolution and we decided not to use them in here. In this way the surface brightness pro?les in BV RI colours were compiled. Di?erent colour pro?les help to distinguish stellar populations and allow to calculate corresponding M/L ratios, and thereafter, colour indices of the components. Table 1 presents references, the faintest observed isophotes (mag arcsec?2 ), and the colour system used. Di?erent R colour system pro?les are transferred into the Cousins system, using the calibration by Frei & Gunn (1994). The observations by Spinrad et al. (1978) were made without absolute calibration. They were calibrated with the help of other R colour observations. Hes & Peletier (1993) observed M 104 in BV RI colours but in their paper only colour indices are given and we cannot use them in here. The composite surface brightness pro?les in the BV RI colours along the major and/or the minor axes were derived by averaging the results of di?erent authors. Due to dust absorption lane, surface brightnesses only on one side along the minor axis have been taken into account. All the surface brightness pro?les obtained in this way belong to the initial data set of our model construction. To spare space, we present here the surface brightness distributions in B and R only (Fig. 1 upper panels), and the axial ratios (the ratio of the minor axis to the major axis of an isophote) (Fig. 1 lower panel) as functions of the galactocentric distance. The observed surface density distribution of GC candidates was derived by Bridges & Hanes (1992), Larsen et al. (2001) and Rhode & Zepf (2004). The derived distributions were averaged, taking into account di?erent background levels. The resulting surface density distribution of GC candidates is given in Fig. 2 by ?lled circles and was used to constrain stellar metal-poor halo parameters. Line-of-sight
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axial ratio

0.8 0.6 0.4 0.2 0.0 0.01 0.1 1 R (kpc) 10 100

Figure 1. Upper panels: the averaged surface brightness pro?les of M 104 in the B and R colours. Filled circles – observations, solid line – model, dashed lines – models for components (n – the nucleus, b – the bulge, h – the metal-poor halo, d – the disc). Lower panel: the axial ratios of M 104 isophotes as a function of the galactocentric distance.

velocities of GCs were measured by Bridges et al. (1997) and the calculated mean velocity dispersion of GC subsystem σGC = 255 km s?1 was derived. Rotation velocities of stars and line-of-sight velocity dispersion pro?le along the major axis in very good seeing conditions (0.2–0.4 arcsec) for the central regions was obtained by Kormendy et al. (1996) with HST and CFHT . In addition, the central regions were measured by Carter & Jenkins (1993), Emsellem et al. (1996). In the central and intermediate distance interval, dispersions and stellar rotation have been measured by Kormendy & Illingworth (1982), Hes & Peletier (1993) and van der Marel et al. (1994). We averaged the stellar rotation velocities at various distance intervals with weights depending on seeing conditions and velocity resolution, and derived the stellar rotation curve presented by ?lled circles in Fig. 3. Averaged in the same way line-of-sight velocity dispersions along the major axis are presented by ?lled circles in Fig. 4. In addition, Kormendy & Illingworth (1982) derived dispersion pro?les along several slit positions (at 0, 30, 40, 50 arcsec parallel and at 0, 50 arcsec perpendicular to the major axis) in the bulge component. We use them in mass distribution model calculations (?lled circles in Figs. 10, 11). Ionized gas radial velocities were obtained and the


E. Tempel and P. Tenjes

200 σobs (km s )

1 log S





-1 0.1 1 R (kpc) 10 100
0 0.01 0.1 R (kpc) 1 10

Figure 2. The surface density distribution of GCs in M 104. The observations are presented by ?lled circles. The continuous line gives our best-?tting model distribution for the halo.

Figure 4. Observed line-of-sight velocity dispersions of M 104 along the major axis. Filled circles – observations, solid line – model.

450 400

300 250 VΘ (km s-1) 200 150 b


Vcirc (km s )



300 250 200 150 100 dm h b d

100 h 50 0 0 1 2 3 R (kpc) 4 5 6

50 0 0 1 2 3 4 5 6 7

R (kpc)



Figure 3. The rotation curve of M 104 on the basis of stellar rotation velocities. Filled circles – observations, solid line – model, dashed lines – models for components.

Figure 5. Calculated circular velocity for the best-?tting model of M 104 (solid line). Dashed lines give circular velocities for components (dm – dark matter). Observed gas velocities are given by ?lled circles. Only outermost points are given where stellar motions are not known.

3 rotation curve was constructed by Schweizer (1978) and Rubin et al. (1985). HI velocities at 11.4 arcsec (0.5 kpc) resolution were obtained by Bajaja et al. (1984). Unfortunately, we have no detailed information about the gas velocity dispersions. When using gas rotation velocities, often an assumption is made that gas dispersions are small when compared with rotation velocities, and in this way, rotation velocities are taken to be circular velocities. However, in the case of M 104, up to distances ? 3 kpc, rotation velocities of stars and gas are comparable and thus we may expect also dispersions to be comparable, and therefore, gas dispersions cannot be neglected. For this reason, we cannot use gas rotation velocities directly in ?tting the model. We use gas rotation only to have an approximate mass distribution estimate at large galactocentric distances where stellar rotation and dispersion data do not extend. In these outer parts, the velocities from di?erent studies were averaged and the resulting gas rotation velocities are given by ?lled circles in Fig. 5.


To construct a model of the M 104 galaxy, we limit the main stellar components to the central nucleus, the bulge, the disc and the metal-poor halo. To construct a dynamical model in the following sections, a DM component – the dark halo – must be added to visible components. To construct the light distribution model, the surface luminosity distribution of components is usually approximated by the S? ersic formula (S? ersic 1968). If, in addition to the photometrical data, kinematic data are also used, the corresponding dynamical model must be consistent with the photometry, i.e. the same density distribution law must be used for rotation curve modelling (and for the velocity dispersion curve, if possible). For spherical systems, an expression for circular velocity with an integer S? ersic index can be derived (Mazure & Capelato 2002). For a non-integer index and ellipsoidal surface density distribution, a consistent solution for rotation curve calculations is not known. In the present paper, the density distribution parameters are determined by the least squares method and may
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Mass distribution of NGC 4594
have any value. In addition, our intention is to use the model also for velocity dispersion calculations. For the reasons given above, we decided to construct models starting from a spatial density distribution law for individual components, which allows an easier ?tting simultaneously for light distribution and kinematics. In such models, the visible part of a galaxy is given as a superposition of the nucleus, the bulge, the disc and the metal-poor halo. The spatial density distribution of each visible component is approximated by an inhomogeneous ellipsoid of rotational symmetry with the constant axial ratio q and the density distribution law l(a) = l(0) exp[?(a/(ka0 ))1/N ], hL/(4πqa3 0) (1)


where l(0) = is the central density and L is the component luminosity; a = R2 + z 2 /q 2 , where R and z are two cylindrical coordinates, a0 is the harmonic mean radius which characterizes rather well the real extent of the component, independently of the parameter N . Coe?cients h and k are normalizing parameters, depending on N , which allows the density behavior to vary with a. The de?nition of the normalizing parameters h and k and their calculation is described in appendix B of Tenjes et al. (1994). Equation (1) allows a su?ciently precise numerical integration and has a minimum number of free parameters. The density distributions for the visible components were projected along the line-of-sight, and their sum gives us the surface brightness distribution of the model
4 ∞

the nucleus are more uncertain because no su?ciently highresolution central luminosity distribution observations are available for us. On the other hand our aim is to study general mass distribution in M 104 where nuclear contribution in small. For this reason convolution and deconvolution processes were not used in luminosity distribution model and in subsequent mass distribution model. The ?nal parameters of the model (the axial ratio q , the harmonic mean radius a0 , the structural parameters N , the dimensionless normalizing constants h and k, BV RI luminosities) are given in Table 2. The model is represented by solid lines in Figs. 1, 2. The mean deviation of the model from the observations of surface brightnesses is ?obs ? ?model = 0.16 mag.

4 4.1


Knowing spatial luminosity densities of the components li (a) and ascribing a mass-to-light ratios to each component fi (i indexes the nucleus, the bulge, the disc and the stellar metal-poor halo), we have spatial mass density distribution of a galaxy

ρ (a ) =

fi li (a) + ρDM (a)


L(A) = 2

qi Qi

li ( a ) a d a , (a2 ? A2 )1/2


where A is the major semiaxis of the equidensity ellipse of the projected light distribution and Qi are their apparent axial ratios Q2 = cos2 δ + q 2 sin2 δ . The angle between the plane of the galaxy and the plane of the sky is denoted by δ . The summation index i designates four visible components. For the nucleus and the stellar metal-poor halo, parameters q , a0 , N were determined independently of other subsystems. For the nucleus, these parameters were determined on the basis of the central light distribution; for the metalpoor halo, these parameters were determined on the basis of the GC distribution. In subsequent ?tting processes, these parameters were kept ?xed. This step allows to reduce the number of free parameters in the approximation process. The model parameters q , a0 , L, and N for the bulge and the disc, and L for the nucleus and the halo were determined by a subsequent least-squares approximation process. First, we made a crude estimation of the population parameters. The purpose of this step is to avoid obviously non-physical parameters – relation (2) is non-linear and ?tting of the model to observations is not a straightforward procedure. Next, a mathematically correct solution was found. Details of the least squares approximation and the general modelling procedure were described by Einasto & Haud (1989), Tenjes et al. (1994, 1998). Total number of free parameters (degrees of freedom) in least-squares approximation was 18, the number of observational points was 231. Transition from the bulge to the disc and from the disc to the metal-poor halo is rather welldetermined by comparing the light pro?les along the major and the minor axis (see Fig. 1 lower panel). Parameters of
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(ρDM (a) is the DM density). On the basis of spatial mass density distributions, derivatives of the gravitational poten?Φ Φ tial ?R and ? can be calculated (see Binney & Tremaine ?z 1987). In stationary collisionless stellar systems with axial symmetry the Jeans equations in cylindrical coordinates are
2 ? (ρvR ) ? (ρvR vz ) + +ρ ?R ?z 2 2 vR ? vθ ?Φ + R ?R

= 0,

(4) (5) (6)

? (ρvR vθ ) ? (ρvθ vz ) 2ρ vθ vR = 0, + + ?R ?z R
2 ? (ρvR vz ) ? (ρvz ) ρvR vz ?Φ + + +ρ = 0, ?R ?z R ?z

where vR , vz , vθ are velocity components. 2 The velocity dispersion tensor σij = vi vj ? v i v j in the diagonal form for the axisymmetric case can be described by four variables: dispersions along the coordinate axis (σR , σz and σθ ) and an orientation angle α in Rz -plane (see Fig. 6). Mixed components of the tensor are
2 2 2 σRz = γ ( σR ? σz ), 2 2 σRθ = σzθ = 0,


where γ= 1 tan 2α. 2 (8)

As a result of axial symmetry the second Jeans equation (5) vanishes. Introducing the dispersion ratios kz ≡
2 σz , 2 σR

kθ ≡

2 σθ , 2 σR



E. Tempel and P. Tenjes
The third coordinate x3 = θ. Foci of ellipses and hyperbolae are determined by ±z0 . The relations between elliptic and cylindrical coordinates are as follows: 1 2 1/2 ? + (?2 ? 4z 2 z0 ) , 2 1 2 1/2 x2 ? ? (?2 ? 4z 2 z0 ) , 2 = 2 where x2 1 =
2 ? ≡ R 2 + z0 + z2 .

(16) (17)


Later we also need the relation z 2 ? x2 2 . (19) Rz In this case, the parameter γ related to the angle between the ellipsoid major axis and the galactic disc is tan α = γ= Rz . 2 R 2 + z0 ? z2 (20)

Figure 6. Elliptical coordinates (x1 , x2 ) and their relations with cylindrical coordinates (R, z ) in galactic meridional plane.

the remaining Jeans equations can be written in a more convenient form for us
2 ?ρσR + ?R 2 1 ? kθ ?κ ?ρσR 2 + ρσR +κ = R ?z ?z V2 ?Φ ? θ , = ?ρ ?R R 2 ξ ?ξ ?Φ ?ρσz 2 + = ?ρ , ρσz +ξ R ?R ?R ?z


2 ?ρσz + ?z


The position of foci z0 is at present a free parameter, which must be determined within the modelling process. For z0 = const the orientation of the velocity ellipsoid would be along the elliptic coordinates. Velocity dispersions along elliptical coordinates (x1 , x2 , x3 ) are denoted as (σ1 , σ2 , σ3 ). In the case of a triaxial velocity ellipsoid, the phase density of a stellar system is a function of three integrals of motion. For an axisymmetrical system, in addition to energy and angular momentum integrals, a third non-classical integral is needed. As it was stressed by Kuzmin (1953), this third integral should be quadratic with respect to velocities (in this case minimum number of constraints result for gravitational potential). On the basis of this assumption, Kuzmin (1953) derived a corresponding form of the third integral. Starting from the form of Kuzmin’s third integral, Einasto (1970) derived that dispersion ratios can be written in the form k12 ≡ k13 ≡
2 2 σ2 a1 z0 + a2 x2 2 = , 2 2 σ1 a1 z0 + a2 x 2 1 2 2 σ3 a1 z0 + a2 x2 2 = , 2 2 σ1 a 1 z 0 + a 2 z 2 + b2 R 2

where Vθ is rotational velocity, κ ≡ γ (1 ? kz ), κ ξ≡ . kz (12)

(21) (22)

For each component the rotation velocity have been taken Vθ = βVc , where Vc is circular velocity and β is a constant speci?c for each subsystems. Taking into account the de?nition of the circular velocity we can substitute in Eq. (10) Vθ2 ?Φ = β2 . R ?R (13)

where a1 , a2 and b2 are unknown parameters. As a simplifying assumption these three parameters were taken in Einasto (1970) a1 = a2 = b2 . In the present paper we determine these parameters by demanding that a1 , a2 and b2 must satisfy the relation (Kuzmin 1961) 1 1 =1+ . k12 k13 (23)

The Jeans equations (10) and (11) include unknown values kz , kθ and γ . Spatial density, gravitational potential and rotational velocity can be determined on the basis of the galactic surface brightness distribution (Eq. 2), the Poisson equation and the observed stellar rotation curve. Dispersions 2 2 σR and σz must be calculated from the Jeans equations. Following the notation of Landau & Livshits (1976), we de?ne confocal elliptic coordinates (x1 , x2 ) as the roots of R2 z2 + 2 = 1, 2 2 x ? z0 x where x2 = x2 1 x2 2
2 z0 2 . z0

This relation was derived by Kuzmin in the case of disclike systems and we must keep in mind that therefore our results may not be a good approximation far from the galactic plane. Using the relations between cylindric and elliptic coordinates we derive kz = kθ = sin2 α + k12 cos2 α tan2 α + k12 , = 2 2 1 + k12 tan2 α cos α + k12 sin α cos2 k13 (1 + tan2 α) k13 = . 2 1 + k12 tan2 α α + k12 sin α (24) (25)



The quantity z0 determines the orientation of the velocity ellipsoid. For speci?c density distribution (gravitational
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Mass distribution of NGC 4594
2 2 2 σl = σ? cos2 Θ + σz sin2 Θ.


To compare the calculated dispersions with the measured data, we must calculate averaged along the line-ofsight dispersion. Integrating dispersions along the line-ofsight we may write
∞ 2 σint (X, Y

1 )= L(X, Y )

2 l(R, z )σl (R, z )dl,


Figure 7. Dispersion projection in a plane parallel to galactic disc.

where l(R, z ) denotes galactic spatial luminosity density, and L(X, Y ) is the surface luminosity density pro?le (please note that integration dl means integration along the line-ofsight). Changing variables in the integral to have integration along the radius, we obtain
∞ 2 σint (X, Y ) =

1 L(X, Y )


R √ dR, cos Θ R2 ? X 2


2 2 Ψ ≡ l(R, z1 )σl (R, z1 ) + l(R, z2 )σl (R, z2 ),

(32) (33)

z1,2 =

Figure 8. Projection of dispersions to the line-of-sight.

Y ± sin Θ

R2 ? X 2 tan Θ.

Equation (31) gives the line-of-sight dispersion for one galactic component. As in our model we have several components, we must sum over all components considering the luminosity distribution pro?le

potential) forms within the theory of the third integral of motion z0 = const. In the case of general density distributions z0 = f (R, z ). For example, it was demonstrated by Kuzmin (1962) that in a galactic disc R ?γ ?z
z =0

σobs (X, Y ) =

? ? ?

2 Li (X, Y ) σint (X, Y ) i


Li (X, Y )

?1/2 ? ?



where i denotes the subsystem and the summation is taken over all subsystems.


1 ? ln ρt , 4 ? ln R


where ρt is total galactic spatial mass density. From Eq. (26) 2 we can determine z0
2 z0 (R, 0) = ?R 4ρt (R, 0)

?ρt (R, 0) ?R


+R .


The dependence of z0 on z is derived to have the best-?tting with measured dispersions. 4.2 Line-of-sight dispersions

Calculated on the basis of hydrodynamic models, dispersions 2 2 2 σR , σz and σθ cannot be compared directly with measurements. We project the velocity dispersions in two steps (see Figs. 7, 8). First, we make a projection in a plane parallel 2 2 to the galactic disc. For this we must project σR and σθ to the disc, going along the line-of-sight and being parallel to the galactic disc. Projected dispersions are
2 2 σ? = σθ

X2 2 + σR R2


X2 R2



In the present section we apply the above constructed model to a concrete galaxy. We selected the Sa galaxy NGC 4594 having enough observational data to construct a detailed mass distribution model. The velocity dispersions for NGC 4594 have been measured also along several slit positions outside of the galactic disc. To avoid calculation errors, we ?rst made several tests: we calculated dispersions for several simple density distribution pro?les, varied the viewing angle between the disc and the line-of-sight, varied density distribution parameters. All test results were in accordance with our physical expectations. In velocity dispersion calculations all the luminosity distribution model parameters derived in Section 3 will be handled as ?xed. The visible part of the galaxy is given as a superposition of the nucleus, the bulge, the disc and the metal-poor halo. The spatial luminosity and mass density distributions of each visible component are consistent, i.e. their mass density distribution is given by ρ(a) = ρ(0) exp[?(a/(ka0 ))1/N ], where ρ(0) = l(0)M/L = hM/(4πqa3 0) (35) is the central mass

2 2 Second, we must project dispersions σz and σ? to the line-ofsight. Designating Θ as the angle between the line-of-sight 2 and the galactic disc, the line-of-sight dispersion σl is

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E. Tempel and P. Tenjes
300 250 σobs (km s-1) 200 150 100 50 0 300 250 σobs (km s-1) 00’’ || major

30’’ || major

Figure 9. Orientation of the calculated velocity dispersion ellipsoids in galactic meridional plane.

200 150 100 50 0 300 250

density and M is the component mass. For designations see Eq. (1). In the case of mass distribution models, a DM component must be added to visible components. The DM distribution is represented by a spherical isothermal law ρ (a ) =

40’’ || major

ρ(0){[1 + 0

a 2 ?1 (a ) ] c

? [1 +

0 2 ?1 (a ) ] } ac

σobs (km s-1)

a a (36) a > a0 .


200 150 100 50 0 300 250

Here a is the outer cuto? radius of the isothermal sphere, ac = ka0 . Our model includes an additional unknown value – velocity ellipsoid orientation. We have to ?nd the best solution to z0 , when ?tting the model to the measured dispersions. Figure 9 gives the shape and orientation of the velocity dispersion ellipsoid in the galactic meridional (R, z ) plane. In Figure 10 calculated line-of-sight dispersions along and parallel to galactic major axis are given. In Figure 11 calculated line-of-sight dispersions parallel to the minor axis are given. It is seen that moving further o? from the galactic disc, the results become a little di?erent from the data observed. One reason may be that we could not ?nd an appropriate solution for z0 . It is possible to ?t the data far from the galactic plane with appropriate selection of z0 but in this case the ?t with dispersions along the major axis is not so good. As a result, the ?gures present the best compromise solution we could ?nd. In addition we must take into account the observed average velocity dispersion of GCs σGC = 255 km s?1 . This value is clearly higher than the above mentioned last points in stellar dispersion curve ? 160 ? 180 km s?1 (Fig. 11). All these dispersions correspond to a region where DM takes effect. For this reason we think that the mean velocity dispersion of GCs and and stellar velocity dispersions far outside of galactic plane can be ?tted consistently by introducing a ?attened DM halo density distribution. This kind of models have not yet been constructed by us within the present algorithm. Figure 12 gives the calculated velocity dispersion in the Rz -plane illustrating the behavior of dispersions.

50’’ || major

σobs (km s-1)

200 150 100 50 0 0 1 2 3 R (kpc) 4 5

Figure 10. Line-of-sight velocity dispersions of NGC 4594 along and parallel to major axis. Solid line – calculated model dispersions, ?lled circles – observations.



In the present paper we developed an algorithm, allowing to construct a self-consistent mass and light distribution model

and to calculate projected line-of-sight velocity dispersions outside galactic plane. We assume velocity dispersion ellipsoids to be triaxial and thus the phase density is a function of three integrals of motion. The galactic plane may have an arbitrary angle with respect to the plane of the sky. The developed algorithm is applied to construct a mass and light distribution model of the Sa galaxy M 104. In the ?rst stage a luminosity distribution model was constructed on the basis of the surface brightness distribution. The inclination angle of the galaxy is known and the spatial luminosity distribution can be calculated directly with deprojection. Using the surface brightness distribution in BV RI colours and along the major and minor axis, we assume that our components represent real stellar populations and determine their main structure parameters. In the second stage, the Jeans equations are solved and the line-of-sight velocity dispersions and the stellar rotation curve are calculated. Observations of velocity dispersions outside the apparent galactic major
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Mass distribution of NGC 4594
Table 2. Calculated model parameters. Popul. Nucleusa Bulge Halo Disc Dark matter M 0.001 3.4 7.4 12.0 180 a0 0.0015 0.28 11.0 3.4 40.0 q 0.99 0.54 0.75 0.25 1.0 N 3.0 2.1 4.0 0.78 k 0.00297 0.03539 1.465E-4 0.7429 0.1512 h 314.3 44.51 2807. 2.607 14.82 β LB 3.9E-4 0.34 3.2 1.6 LV LR LI


0.7 0.3 0.88

0.48 2.3

0.56 4.4 2.45

0.92 5.0

Masses and luminosities are in units of 1010 M⊙ and 1010 L⊙ ; components radii are in kpc.

A point mass 109 M⊙ have been added to the center of the galaxy.

300 250 σobs (km s-1) 200 150 100 50 0 300 250 σobs (km s-1) 200 150 100 50 0 0 1 2 z (kpc) 3

00’’ || minor

50’’ || minor

Figure 12. Projected line-of-sight dispersions (in km s?1 ) in galactic meridional plane.


Figure 11. Line-of-sight velocity dispersions of NGC 4594 parallel to the minor axis. Solid line – calculated model dispersions, ?lled circles – observations.

axis allow to determine the velocity ellipsoid orientation, anisotropy and to constrain DM halo parameters. The total luminosity of the galaxy M 104 resulting from the best-?tting model is LB = (5.1 ± 0.6) · 1010 L⊙ , LR = (7.4 ± 0.7) · 1010 L⊙ . The total mass of the visible matter is Mvis = (22.9 ± 3.2) · 1010 M⊙ , giving the mean mass-to1 light ratio of the visible matter M/LB = 4.5 ± 1.2 M⊙ L? ⊙, 1 M/LR = 3.1 ± 0.7 M⊙ L? ⊙ . The surface brightness distributions in V and I have not su?cient extent to determine the luminosities of the stellar halo and we do not either give galactic total luminosities in these colours and corresponding M/L ratios. Calculated from the model, the LB coincides well with the total absolute magnitude MB = ?21.3 (= 5.2 · 1010 L⊙ ) obtained by Ford et al. (1996). In our model, the mass of the disc is Mdisc = 12 · 1010 M⊙ . This coincides rather well with the disc mass 11.4 · 1010 M⊙ calculated with the help of Toomre’s stability criterion by van den Burg & Shane (1986) and with the mass 9.6 · 1010 M⊙ derived by Emsellem et al. (1994). On the other hand, Emsellem et al. (1994) derived for the bulge mass > 5 · 1011 M⊙ , giving Mdisc /Mbulge = 0.2. This is similar to the value 0.25 derived by Jarvis & Freeman (1985),
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but this is much less than Mdisc /Mspher = 1.1 resulting from our model. An explanation may be that in the models by Emsellem et al. (1994) and Jarvis & Freeman (1985) no DM halo was included and hence the extended bulge mass is higher. In our model, the disc is rather thick (q = 0.25). However, disc thickness can be easily reduced to q = 0.15 ? 0.2 when taking the galactic inclination angle instead of δ = 84? to be 83–82? . Other parameters remain nearly unchanged. At present this was not done. Derived in the present model bulge parameters can be used to compare them with the results of chemical evolu1 tion models. Our model gives M/LV = 7.1 ± 1.4 M⊙ L? ⊙ and (B ? V ) = 1.06 for the bulge. Comparing spectral line intensities with chemical evolution models, Vazdekis et al. (1997) obtained for the bulge region the metallicity Z = 0.03 and the age 11 Gyrs. According to Bruzual & Charlot 1 (2003), these parameters give M/LV = 7 ? 8 M⊙ L? ⊙ and (B ? V ) = 1.06 ? 1.08 for simple stellar population (SSP) models. Bulge parameters from our dynamical model agree well with these values and suggest that our model is realistic. In our calculations, we corrected luminosities from the absorption in the Milky Way only and did not take into account the inner absorption in M 104. According to Emsellem et al. (1996), absorption in the centre may be at least AV ? 0.13 mag and thus M/LV = 6.3 for the bulge. This is slightly too small when compared with the Bruzual & Charlot (2003) SSP models. However, decreasing the bulge age to 10.5 Gyrs allows to ?t the results. Rather sophisticated models of M 104 have been


E. Tempel and P. Tenjes
1.0 0.8 z= 0 z= 1 z= 3 z=10

constructed by Emsellem et al. (1994, 1996) and Emsellem & Ferruit (2000). Due to our di?erent approaches, it is di?cult to compare our components and their parameters with those of Emsellem et al. (1994, 1996). On the basis of the data used by us, we had no reason to add an additional inner disc or a bar to the bulge region. However, we did not analyse I and H colours and ionized gas kinematics in inner regions as it was done by Emsellem & Ferruit (2000). Modelling of gas kinematics in central regions is beyond the scope of the present paper as gas is not collision-free. On the basis of velocity dispersion observations only along the major axis it is di?cult to decide about the presence of the DM even when dispersions extend up to 2–3 Re (Samurovi? c & Danziger 2005). In the case of M 104, additional dispersion measurements can be used. Velocity dispersions in the case of the slit positioned parallel and perpendicular to the galactic major axis, have been measured by Kormendy & Illingworth (1982). The calculated mass distribution model describes rather well the observed stellar rotation curve and line-of-sight velocity dispersions. Only the two last measured points at a cut 50 arcsec perpendicular to the major axis deviate rather signi?cantly when compared to the model. On the other hand, in addition to stellar velocity dispersion measurements, the mean line-of-sight velocity dispersion of the GC subsystem σ = 255 km s?1 was measured by Bridges et al. (1997). This corresponds to GCs at average distances 5–10 kpc from the galactic center and is in rather good agreement with the dispersions calculated from the model. In the best-?tting model the DM halo harmonic mean radius a0 = 40 kpc and M = 1.8 · 1012 M⊙ giving slightly falling rotation curve in outer parts of the galaxy (Fig. 5). The central density of the DM halo in our model is ρ(0) = 0.033 M⊙ pc?3 , being also slightly less than it was derived for distant (z ? 0.9) galaxies (ρ(0) = 0.012?0.028 M⊙ pc?3 , Tamm & Tenjes 2005). On the other hand, the result ?ts with the limits derived by Boriello & Salucci (2001) for local galaxies ρ(0) = 0.015 ? 0.050 M⊙ pc?3 . An essential parameter in mass distribution determination is the inclination of the velocity dispersion ellipsoid with respect to the galactic plane (see e.g. Kuijken & Gilmore 1989; Merri?eld 1991). Velocity dispersion ellipsoid inclinations calculated in the present paper are moderate, being 30? . In a sense, our approach to the third integral of stellar motion is similar to that by Kent & de Zeeuw (1991) – the local St¨ ackel ?t. In their modelling of the local Milky Way structure, they derived that at 0 < z < 600 pc and 6.8 < R < 8.8 kpc, the inclination of the velocity dispersion ellipsoid is less than z/R, and they studied the corresponding correction in detail. In our model, in the same distance regions (although the Milky Way and M 104 are not very similar objects), inclination correction values are slightly smaller. Variations of the corrections with R and z are qualitatively similar. The largest di?erence is the variation of the correction value with z , for which Kent & de Zeeuw (1991) obtained an increase by 0.1, when moving from z = 0 to z = 0.6 kpc, but in our model, corresponding increase was only by 0.01. In our model, a signi?cant increase of the ellipsoid inclination angle begins at larger z , which can be explained by higher thickness of the disc component of M 104 (q = 0.25). In a following paper we intend to construct sim-


0.6 0.4 z= 0 z= 1 z= 3 z=10



0.6 0.4 0 2 4 6 R (kpc) 8 10

Figure 13. Dispersion ratios in galactic meridional plane. Coordinate z is in kpc.

ilar models for other galaxies with velocity dispersion measurements outside the galactic major axis. This may lead to more ?rm conclusions about the inclination of velocity dispersion ellipsoids outside the galactic plane. Ratios of the line-of-sight velocity dispersions are given in Fig. 13. It is seen that velocity dispersion ellipsoids are quite elongated – anisotropies in the symmetry plane at outer parts of the galaxy are < 0.5. Modelling the disc Sb galaxy NGC 288 within a constant velocity ellipsoid inclination approximation, Gerssen, Kuijken & Merri?eld (1997) estimated that the dispersion ratio σz /σR = 0.70. Within epicycle approximation Westfall et al. (2005) derived for Sb galaxy NGC 3982 the dispersion anisotropy σz /σR = 0.73. These two galaxies are morphologically close to the Sa galaxy modelled in the present paper, and it is seen that dispersion ratios are more anisotropic in our case. In general, a radially elongated dispersion ellipsoid is rather common (Shapiro, Gerssen & van der Marel 2003). On the other hand, there are exceptions – galaxy NGC 3949 (Westfall et al. (2005) has the dispersion ratio σz /σR = 1.18). By using the quadratic programming method (Dejonghe 1989), the distribution function within the three-integral approximation has been numerically calculated for the S0 galaxy NGC 3115 by Emsellem, Dejonghe & Bacon (1999). Assuming some similarity between S0 and Sa galaxies, it is interesting to compare the derived velocity dispersion behaviour outside the galactic plane. Although detailed comparison is di?cult a similar structure of isocurves is seen. But the results disagree in values of σz /σR . Our ratios are radially elongated, the ratios by Emsellem et al. (1999) are vertically elongated. Qualitatively this is in agreement with the general trend that galaxies of earlier morphological type have larger σz /σR ratio (Shapiro et al. 2003). At larger R, the dispersion ratio σz /σR decreases. This is in agreement with the decrease of dispersion ratios due to the decrease of the role of interactions with molecular clouds at greater galactocentric distances (see Jenkins & Binney 1990). The use of this explanation in the case of gas-poor S0 galaxies is not clear. At greater distances z both our and
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Mass distribution of NGC 4594
Emsellem et al. (1999) dispersion ellipsoids become more spherical. By using the Schwarzschild method, dispersion ratios for E5–6 galaxy NGC 3377 have been calculated by Copin, Cretton & Emsellem (2004). Taking into account relation between spherical and cylindrical coordinates, the behaviour of the dispersion ratios as a function of R and z near the galactic plane is in approximate accordance (dispersion ratios by Copin et al. (2004) are slightly more spherical). Rotational properties of elliptical and Sa galaxies are too di?erent to compare the ratios σθ /σR . Unfortunately, it is not possible to compare also orientations of velocity dispersion ellipsoids.


ACKNOWLEDGEMENTS We thank Dr. U. Haud for making available his programs for the light distribution model calculations. We would like to thank the anonymous referee for useful comments and suggestions helping to improve the paper. We acknowledge the ?nancial support from the Estonian Science Foundation (grants 4702 and 6106). This research has made use of the NASA/IPAC extragalactic database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the NASA.

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