10 80 Radius along lhlur Am (nmao)

Fig. 5a and b. Comparison between the observed SO velocities (a) and the velocity dispersion (b) with those predicted by the luminous mass model. The kinematical data are from S94 (open symbols= approaching east side, full symbols= receding west side). a Rotation curve - the best-fit model is represented by lines: long-dash: circular velocities; short-dash: azimuthal velocity taking into account the asymmetric drift; smooth solid line: final predicted velocities (analytical) afterintegrating along the line-of-sight; solid ragged line: results of simulations taking into account the ellipticity of the orbits. b Velocity dispersion - longdash: empirical law for the dispersion profile of the disk (see text); smooth solid line: predicted dispersion, after line-of-sight integration, and inclusion of the bulge dispersion; solid ragged line: as in a

4.2. Elliptical orbits

After an initial guess for the values of the parameters is derived in the circular orbits approximation, we explored the effects of triaxiality of the potential for the closed orbits in the plane of the S0 disk, and in the polar ring. We have computed these closed orbits using the shooting method (cf. Katz & Richstone 1984): in both planes, the 2D potential was tabulated, and particles were launched on the long-axis, with a tangential velocity slightly smaller than the circular one; a few trials are repeated until a closed orbit is found.

Fig. 7 displays the shape of the potential for the luminous mass model in the equatorial plane of the lenticular galaxy, and in the polar ring. The isopotentials are quite round in the central SO disk, since the polar ring mass is essentially at large distance. In the polar ring plane, the isopotentials are round in the outer parts, since the potential is then dominated by the polar ring disk itself, but they are quite elongated at short radii, within the S0 optical radius. This is where the corrections for non-circularity are crucial.

Pi g. 8 displays the tangential velocity as a function of radius on both the symmetry axes of the closed orbits. Because we will always measure only the tangential velocity projected along the long axis, the effect of orbit ellipticity will always decrease the

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

NGC 4850A polar ring

150

100

HI Velocities (km/I)

50 100 Ring Radius (crease)

Fig. 6a and b. Comparison between the observed polar ring velocities and the prediction from the luminous mass model. a The polar Ha velocities are from WMS (open symbols: approaching south side, full symbols receding north side). Dash-line: circular velocities, smooth solid-line: predictions after line-of-sight integration; solid ragged line: results of simulations taking into account the ellipticity of the orbits. b The HI velocities are from van Gorkom et al. (1987): the symbols indicate the maxima of the PV diagram along the major axis (full symbols= receding north side), the error bars indicates the first contour extension (open symbols correspond to dashed error bars, and to approaching south side). Long-dash, solid and broken line, as in a). Small-dash: effect of convolution by the 20" beam. Two models have been computed, corresponding to a constant inclination of 85° (higher curves) and 90° (lower curves) to show the effect of the variable orientation in the polar

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observed velocities with respect to the ones expected from the underlying potential.

To account for the line of sight integration in the non circular orbit case, we have computed a 3D simulation of the galaxy and polar ring with 104 particles. The total potential (due to all visible mass components from the lenticular galaxy and the polar ring) was tabulated on a grid of cells 1" = 170 pc (actually only one eighth of this space is necessary by symmetry). Particles were launched according to their density laws on the symmetry axes, with initial velocities derived from the shooting algorithm, as shown in Fi g. 8, and assuming cylindrical rotation for the two flattened systems. The pressure support (asymmetric drift) was subtracted from the square of the velocities, and the velocity dispersion was added statistically according to gaussian laws. The particles were then advanced in the potential with a random number of time-steps, allowing for few rotations. We launched about 50% particles on the long axis, and 50% on the short axis of their orbits, and changing this fraction did not affect the re

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

50

I

Fig. 7a. Isopotentials of the luminous mass plane. Contours are linear

model in the equatorial

-100

— 50 0 50

Fig. 7b. Same as Fig. 7a, for the polar ring plane

sults. The derived spatial distributions was then compared with the light profiles.

To derive the velocity distribution in the lenticular disk, we have subtracted the asymmetric drift from the circular velocity, according to Eq. (1) and assuming an isotropic dispersion, then we add a random Maxwellian component, following the observed dispersion as functions of radius (Fig. 5): the orbits are then integrated in the tabulated potential. The final system is displayed in Fig. 9.

We “observe” the simulated galaxy along the major axes of the S0 and of the polar ring, with a slit of 3", and convolving with a seeing of 2.25” . The final result is plotted and compared to the observed profiles in Figs. 5-6.

Elliptical orbits strongly effect the velocity profile of the lenticular disk, while their effect is almost negligeable for the

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

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