0 Radius (arcsec)

Fig. 2. Surface brightness profiles along the major axis of the S0 disk of NGC 4650a (Bzsquares, Kzdots) and our corresponding fit (solid line), with the parameters listed in Table 1

the polar ring luminosity profile. We examine the influence of different inclination and thickness in the Appendix.

As in S94, we model the polar ring using the difference of two Toomre disks, with the same characteristic scales (see Table 1). Although we adopt the same constant inclination as S94, our best fit to the surface brightness profile along the PR major axis requires a total luminosity for the stellar component 3 8% larger than that adopted by S94, an effect which we attribute to our finite ring height. To model the visible mass potential, the H2 mass is added to the ring, with the same radial distribution as the blue light (cf. Young & Scoville 1991). The H2 mass is 1.2- 10° M9 (Watson et al. 1994), i.e. 15% of the total mass in stars, which is a typical value for late-type spiral disks.

The surface brightness profile along the major axis of the polar ring has two marked maxima in the B-band, at 1' 30" on each side of the centre. This is not well reproduced in models with constant inclination, but it could be caused by a warp and the sudden edge-on inclination occurring at this radius (S94). Even when the warping is taken into account, the true distribution remains uncertain, since the B, I, and K-band profiles differ significantly. Since the two 30"-maxima are almost absent in the K-band, they could be a dust-induced effect. The worse seeing in the K-band photometry can also contribute (see Fig. 3).

The HI surface density was also reproduced, to fit the observations by van Gorkom et al. (1987). The difference between two Toomre disks used by S94 is a reasonable fit to the observed HI column densities, with a total HI mass of 6.4 - 109 M9 inside 140" = 24 kpc, although the polar ring is quite asymmetric (see Fig. 4). This value of the atomic component mass corresponds to a distance of 35 Mpc (H0 75 km s" Mpc“), and has been corrected for helium abundance by a factor 1.4, as in SS and S94. The HI distribution is modelled with an exponential thickness of 300 pc, and constant inclination 2' 85°.

3.2. The mass model

Our multi-component mass model is quite similar to the previous ones by SS and S94, we just generalised the Toomre disks

© European Southern Observatory ' Provided by the NASA Astrophysics Data System

NGC 4650A minor axis

-40 -20 O 20 40 60

Radius (arcsec)

Fig. 3. Surface brightness profiles along the major axis of the polar ring of NGC 4650a (Bzsquares, Ktdots) and our corresponding fit (solid line), with the parameters listed in Table l

Rm! Radius (arcsec)

Fig.4. HI column density along the major axis of the polar ring of NGC 4650a (squares, from van Gorkom et al. 1987) and our corresponding fit (solid line: direct; dashed line: smoothed to 20" resolution), with the parameters listed in Table 1

into Miyamoto~Nagai potential-density pairs (Miyamoto & Nagai 1975), to take into account the finite thickness of the polar ring. The bulge is represented by a Plummer component, of characteristic size T1, and total luminosity LB (mass M B). It has a very low and is not dynamically important. The main lenticular disk is represented by a double-exponential disk (rd, Ld, Md) with scale height hd. The stellar and gaseous polar rings are represented by differences of Miyamoto-Nagai disks, which are simple 3D density-potential analytical pairs.

A dark halo is added to the luminous mass distribution. Its mass density is given by a pseudo-isothermal ellipsoid (SS, S94):

R2 2 2 ph(R, Z) = Th

where p0 is the central density, rh the core radius, and q is the axial ratio of the isodensity curves, which vary from spherical to flattened ellipsoids.

F. Combes & M. Arnaboldir The dark halo of polar-ring galaxy NGC 4650a

When each luminous component present in the optical im~ ages of NGC 4650a has been modelled using the observed luminosity profiles, and every characteristic scale has been fixed, the only free parameters are the mass-to-light ratios, and the parameters for the dark halo. All these parameters are interdependent, and the solution will not be unique. In this work, we will choose to maximise the M/LB of the visible components,

within the allowed values derived from the observed colours and

stellar populations, and within the constraints of the observed kinematics. We will try to explore extreme flattenings for the dark matter halo, to determine the maximum range compatible with the observations, including oblate distributions perpendicular to the S0 disk.

Given the large number of solutions to explore, a first guess of the parameters is obtained in the circular orbit approximation. The model is then refined, taking into account the ellipticity of the orbits in triaxial potentials. Because we will choose to consider two extreme models, for which the dark matter distribution has the largest possible flattening 1) along the S0, and 2) along the polar ring, the correction due to elliptical orbits will be minimized.

3.3. Asymmetric drift in the S0 disk

One of the biggest uncertainties in the modelling resides in the velocity distribution of the lenticular galaxy, because the velocity dispersion measured by S94 is apparently quite high, even at large distances from the centre. This suggests a high asymmetric drift (difference between circular and azimuthal velocities) and requires a precise determination. Contradictory results on the dark halo shape in polar ring galaxies (WMS, SS) came in mainly because of different modelling of this drift. The large velocity dispersion may also be caused by the elliptic orbits present because of the polar ring potential.

For an axisymmetric exponential disk of characteristic scale rd in steady-state, the Jeans equations give the relation between the azimuthal velocity 11¢, and the circular velocity 11¢:

where 0, and are the velocity dispersions in the radial and azimuthal directions respectively (cf. Binney & Tremaine 1987, S94). Several different models relate 0, and 04>, ranging from the epicyclic theory to the isotropic 04, or model. In the case of a spiral disk, dominated by rotation, the epicyclic approximation holds in particular when the velocity dispersion is small. The ratio of velocity dispersion is then:

In NGC 4650a, the velocity dispersion orthogonal to the SO plane is significantly higher than what is expected when the scale-height h is independent of radius, and in the isothermal approximation. For an exponential disk of scale rd (cf. Bottema

© European Southern Observatory ' Provided by the NASA Astrophysics Data System

F. Combes & M. Arnaboldi: The dark halo of polar-ling galaxy NGC 4650a

Table 1. K band luminosity along the SO major axis (P.A. 61°) and the polar ring major axis (P.A. 161°)

R [arcsec] px R [arcsec] pK

P.A. 61° SW P.A. 161° NW

-61 20.89

-60.39 21.44 -59.78 22.45 -59.17 22.79

-58.56 22.62

-57.95 22.8

-57.34 24.71 -55.51 23.22

-54.9 24.22 -54.29 26.08

-53.68 22.99 -53.07 22.59 -52.46 23.06

-51.24 24.31 -50.02 23.34 -49.41 24.12

-47.58 24.49

-46.97 23.19 -46.36 23.73 -45.75 24.65

-45.14 23.02

-44.53 22.81

-43.92 22.9

-43.31 22.05

-42.7 22.4

-42.09 22.79 -42.09 22.78

-41.48 23.31 -41.48 22.31 -40.87 22.78 -40.87 21.78 -40.26 25.41 -40.26 21.63 -39.65 -39.65 21.53 -39.04 23.73 -39.04 21.69 -38.43 23.39 -38.43 22.36

-37.82 23.4 -37.82 22.07 -37.21 -37.21 21.79 -36.6 -36.6 21.67 -35.99 -35.99 21.44 -35.38 22.8 -35.38 21.21 -34.77 24.66 -34.77 21.35 -34.16 -34.16 21.3 -33.55 24.32 -33.55 21.35 -32.94 24.16 -32.94 21.25

-32.33 25.31 -32.33 21.04 -31.72 23.54 -31.72 21.12 -31.11 22.04 -31.11 21.33

-30.5 23.37 -30.5 21.11 -29.89 -29.89 20.79 -29.28 23.6 -29.28 20.97 -28.67 -28.67 20.92 -28.06 23.71 -28.06 21.07

-27.45 -27 .45 20.87 -26.84 -26.84 20.73 -26.23 -26.23 20.71 -25 .62 -25.62 20.66 -25.01 25.27 -25.01 20.66

1993), the velocity dispersion orthogonal to the SO plane is given by:

and it is adopted in our modelling. In the S0 plane, the velocity dispersion profile is modelled by the empirical law:

as is indicated by a long-dash-line in Fig. 5, where the kinematics of the SO disk is compared with observations. Better fits of the data were obtained when we assumed the velocity dispersion isotropic in the plane, i.e. 11¢ UT.

One of the main differences between our models and those of S94 is precisely our fit to the velocity dispersion profile. In the outer parts, the errors in the measurements allow both low (10 km/s, our model) as well as high (40km/s, S94) values for the radial velocity dispersion. The use of the high value of the radial velocity dispersion at 1" = 20" reduces the expected rotational velocities in the SO disk for a given mass in the disk, and it implies adding more mass inside 3.4 kpc. This explains why S94 adopted a large dark mass (7.6 109 inside 20" for their E7 halo model, to be added to the SO luminous disk mass of 5.25 - 109 MG), while we have no dark matter inside the same radius (see Sect. 4).

Because of the large uncertainties in the radial velocity measurements at large radii, the fit of our model with a radial velocity dispersion at 1' 30" of 40 km/s is still in agreement with the observations.

The observed dispersion profile appears rather peculiar with respect to what we know from normal spiral galaxies. But the S0 disk in NGC 4650a must have been strongly perturbed during the accretion/merging event which had likely caused the formation of the polar ring, and because there is no gas in the S0 to dissipate energy, the disk has remained hot. We just model empirically the velocity dispersion, and build a gravitationally coherent model through the Jeans equations.