4. Trials and results

We first try to obtain the best fit to the observed velocities and velocity dispersion profile with the luminous mass distributions only, assuming constant mass-to-light ratios. We describe the effects of spatial convolution, line-of-sight integration, and triaxiality of the potential, and discuss the uncertainties at every step of the modelling in the following sections.

4.1. Seeing and line-of-sight integration

The effects of convolution with a 2 — 3" spatial resolution for the optical kinematical data (both due to the seeing and slit width) were tested and they were found negligible for most of the disk (except for the very centre, but here the mass model is also not well determined). This is due to spatial gradients corresponding to characteristic scales larger than 3" = 0.5 kpc: the bulge has a smaller scale, but this component is not dynamically

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

Table 1. (continued) Table 1. (continued)

R [arcsec] ,uK R [arcseC] MK R [arcsec] pg R [arcsec]

-24.4 22.74 -24.4 20.5 12.81 19.2 12.81 20.21 -23.79 23.77 -23 .79 20.53 13.42 19.42 13.42 20.22 -23.18 21.97 -23.18 20.69 14.03 19.57 14.03 20.17 -22.57 22.28 -22.57 20.51 14.64 19.76 14.64 20.27 -21.96 23.14 -21.96 20.62 15.25 19.95 15.25 20.29 -21.35 21.67 -21.35 20.59 15.86 20.11 15.86 20.31 -20.74 21.89 -20.74 20.67 16.47 20.17 16.47 20.39 -20. 13 21.88 -20.13 20.62 17.08 20.22 17.08 20.45 -19.52 21.19 -19.52 20.53 17.69 20.27 17.69 20.6

-18.91 21.31 -18.91 20.44 18.3 20.03 18.3 20.75 -18.3 21.39 -18.3 20.33 18.91 20.45 18.91 20.63 -17.69 20.73 -17.69 20.28 19.52 20.85 19.52 20.68 -17.08 20.94 -17.08 20.39 20.13 20.92 20.13 20.61 -16.47 20.61 -16.47 20.43 20.74 21.07 20.74 20.59 -15.86 20.15 -15.86 20.33 21.35 21.56 21.35 20.52 -15.25 20.02 -15.25 20.25 21.96 22.14 21.96 20.45 -14.64 19.89 -14.64 20.19 22.57 22.68 22.57 20.46 -14.03 19.6 -14.03 20.2 23.18 23.81 23.18 20.68 -13.42 19.45 -13.42 20.23 23.79 23.79 20.63 -12.81 19.34 -12.81 20.12 24.4 24.4 20.49 -12.2 19.17 -12.2 20.13 25.01 23.91 20.42 -11.59 18.99 -11.59 20.03 25.62 22.96 25.62 20.45 -10.98 18.94 -10.98 20.13 26.23 23.12 26.23 20.38 -10.37 18.75 -10.37 20.21 26.84 24.38 26.84 20.62 -9.76 18.56 -9.76 19.99 27.45 23.59 27.45 20.8

-9.15 18.47 -9.15 19.82 28.06 22.71 28.06 20.75 -8.54 18.38 -8.54 19.71 828.67 28.67 20.89 -7.93 18.25 -7.93 19.68 29.28 29.28 20.87 -7.32 18.07 -7.32 19.52 29.89 26.88 29.28 20.87 -6.71 17.9 -6.71 19.34 30.5 25.77 29.89 20.89 -6.1 17.82 -6.1 19.15 31.11 30.5 20.97 -5.49 17.68 -5.49 18.9 31.72 31.11 20.92 -4.88 17.63 -4.88 18.64 32.33 24.25 31.72 21

-4.27 17.4 -4.27 18.25 32.94 32.33 20.95 -3.66 17.22 -3.66 17.9 33.55 23.03 32.94 21.12 -3.05 16.95 -3.05 17.46 34.16 22.29 33.55 21.19 -2.44 16.68 -2.44 16.97 34.77 22.57 34.16 21.26 -1.83 16.35 -1.83 16.4 35.38 23.62 34.77 21.21 -1.22 15.91 -1.22 15.82 35.99 35.38 21.29 -0.61 15.42 -0.61 15.28 36.6 35.99 21.46 7.629e-06 15.11 7.629e-06 15.11 37.21 36.6 21.51 0.61 15.58 0.61 15.4 37.82 37.21 21.62 1.22 16.23 1.22 15.8 38.43 37.82 21.73 1.83 16.74 1.83 16.25 39.04 23.9 38.43 22.06 2.44 16.56 2.44 17.22 39.65 23.72 39.04 21.77 3.05 16.81 3.05 17.61 40.26 24.34 39.65 21.73 3.66 17.05 3.66 17.98 40.87 40.26 22.34 4.27 17.28 4.27 18.33 41.48 40.87 22.17 4.88 17.44 4.88 18.65 41.48 21.89 5.49 17.65 5.49 18.9 42.09 21.81 6.1 17.86 5.49 18.89 42.7 22.28 6.71 18.01 6.1 19.08 43.92 22.93 7.32 18.13 6.71 19.28 44.53 23.95 7.93 18.24 7.32 19.51 45.14 24.05 7.93 18.26 7.93 19.71 45.75 23.35 8.54 18.4 8.54 19.79 46.36 23.02 9.15 18.52 9.15 19.87 46.97 23.8

9.76 18.6 9.76 19.97 49.41 27.08 10.37 18.82 10.37 20.05 50.02 23.28 11.59 19.05 11.59 19.93 50.63 24.76 12.2 19.12 12.2 20.05 51.24 22.79

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

Table 1. (continued)

Table 2. Luminosity and mass model (best-fit) of NGC 4650a and its Polar Ring

Masses have been derived assuming = 4 for the S0 disk, 5 for the stellar polar ring, and 0.5 for the bulge. and rp; are the characteristic radii of the two Toomre disks for the stellar PR and Toomre disks for the HI PR (see text). Masses are in units of 109 blue luminosities in 109 L9, and sizes in kpc.

significant, and the effect of the disk smoothed out by the line-of-sight integration, which produces averages over regions of the galaxy at different radii. We have taken into account the spatial convolution with the HI beam of 20" = 3.4 kpc, but even there the effects are quite small (see Fig. 6), for the same reasons.

On the other hand, the line-of-sight integration has a strong effect on the predicted velocity profiles, as it can be seen in Figs. 5 and 6. This is due to the almost edge-on orientation of both the S0 disk and the polar ring, because the line-of-sight samples very different and distant regions in the two components. It is the main limitation of the modelling, since spatial distributions and kinematics cannot be determined independently. The effect of the integration along the line-of-sight on the predicted velocities is maximum in the central region, since the projected zero velocity contribution from the outer material decreases significantly the average.

150

100

IIILIIIII

100