Fitting the light distribution
The B,I broad band brightness profiles along the major axis of the lenticular, and along the polar ring are available in the literature (WMS, S94); the K band profile comes from this work. The lenticular light profile is exponential in the three bands, with a characteristic radial exponential scale of rd R: 4.5”. S94 have shown that the profiles could be well reproduced by both a thin and a thick exponential disk with respectively 4.7” and 4.4” radial scale (0.9” and 1.8” exponential scale height), provided that the inclination is then adjusted to 68° for the thin disk, and 78° for the thick disk. However, we stress that the observed axis ratio, 0.42, is still compatible with the typical thickness of a lenticular galaxy seen edge-on, so the inclination and the thickness parameters are not uniquely determined. We choose to model the lenticular with a thick disk (4.4” radial scale and 1.8” z-scale), which better corresponds to the galaxy morphological type (see Fig. 2). The fits to the luminosity profiles in the two cases are not very different, as shown by S94, but a thick disk gives a slightly better fit. We also add a small bulge, with a luminosity of 4.3 - 109 LG (SS), and a scale rb z 1". This bulge is detected only with good seeing conditions, and its low luminosity does not correspond to what can be expected for a typical bulge in a SO. It could correspond to a region of recent star formation in the nucleus, triggered by the gas accretion event following the polar ring formation, as suggested by S94. Moreover, S94 fits to the observed radial velocities and the velocity dispersion indicate a very low ratio (0.5) for this bulge. In any case, this component is not adding uncertainty to the parameters, since it has negligible effect on the various fits.
The fit to the PR luminosity profile gives more problems, because its inclination and position angles are slightly varying as function of radius. WMS pointed out that it is not possible to have the whole polar ring in the same slit, due to its twisting and S-shape bending. The underexposed blue image from WMS, with a seeing of‘ 1.6”, reveals clearly that the polar ring is not edge-on in the central part until roughly 30", where it becomes edge-on. This behaviour confirms that the polar ring in NGC 4650a is indeed a ring whose surface density is maximum at 1' 30". The I-band image from S94 (seeing 2.25”), and our K-band image (1 ) do not allow us to distinguish clearly the central hole of the polar ring, possibly because the dust lane is less prominent at larger wavelength, and there is a strong effect caused by the seeing conditions. The inclination of the central region of the polar ring is not far from edge-on, of the order of 85°. The velocities will not be affected, as far as their projections are concerned; but the luminosity profile, and the apparent velocity profile are significantly modified even in the case of a 5° departure from edge-on, due to the integration along the line of sight, as will be shown later on. Because of all these uncertainties, a fit with varying inclinations and position angles is not worthwhile. We choose a constant inclination of 85°, as suggested by the blue image, and an exponential thickness of 300 pc (corresponding to an axis ratio of w 0.05) to reproduce
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
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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
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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.
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.
IIILIIIII