4 NGC 3198

4.1 r-band photometry

This gas-rich spiral galaxy has a generally symmetric H  I distribution, and there are no large scale significant warps or distortions of the velocity field. The rotation curve extends to roughly 10 radial scale lengths and is, to first order, flat and featureless (Begeman 1987). For these reasons it has become the classic case of a spiral galaxy evidencing a large mass discrepancy in its outer regions (van Albada et al. 1985). If any theory, such as MOND, fails to predict the rotation curve of this galaxy, then it would be problematic for that theory.

Figure 2: MOND rotation curves for NGC 3198 assumed to be at various distances. The Cepheid-based distance is 13.8 Mpc. The dotted and dashed lines are the Newtonian rotation curves of the stellar and gas discs respectively.

In Fig. 2 we show the MOND rotation curves of NGC 3198 when the galaxy is assumed to be at distances of 10 Mpc, 12.5 Mpc and 13.8 Mpc. Again, the MOND acceleration parameter is assumed to be the BBS value rescaled to the new distance scale, i.e., $0.9 \times 10^{-8}$ cm s^-2.

The closest assumed distance, 10 Mpc, is roughly the Hubble law distance given the radial velocity of NGC 3198 with respect to the local group; it is also the least-square-fit distance if distance is left as a free parameter in the context of MOND. The distance of 13.8 Mpc is the final Cepheid-based distance given by Freedman et al. (2001); and 12.5 Mpc corresponds to the Cepheid distance less 10%. The disc M/L values in the B-band corresponding to the MOND fits at these various distances are also given in Fig. 2.

Here we see that the MOND rotation curve for a distance of 10 Mpc is essentially a perfect fit to the observed curve. At the distance of 12.5 Mpc, the MOND curve is less than a perfect match, but, nowhere that the rotation curve is well measured, does the predicted rotation curve deviate by more than 5 km s^-1 from the curve derived from the observed velocity field. This is typically within the difference in the rotation curves derived from the two sides of the galaxies considered separately- a sensible estimate of the uncertainties (the error bars are formal errors determined from the tilted ring fitting procedure).

At the Cepheid distance of 13.8 Mpc, the MOND rotation curve deviates in the same sense but by now up to -10 km s^-1 in the inner regions (8-14 kpc) and by +10 km s^-1 in the outer regions (30-40 kpc). The reason for the deteriorating fit with increasing assumed distance is the relatively larger contribution of the gaseous component to the rotational velocity. The rotation curve of NGC 3198 in the outer regions (r>20 kpc) is constant at about 150 km s^-1. This would imply, in the context of MOND, that essentially the entire mass of the galaxy is enclosed within about 20 kpc, but this is obviously not the case given the significant surface density of neutral gas in the outer regions - contributing more than 50 km s^-1 to the Newtonian rotation curve at the last measured point.

At a distance of 12.5 Mpc, the MOND rotation curve appears to be consistent with the observed curve (within the likely errors of the method for estimating rotation curves from 21 cm line data). Although this distance is formally 2$\sigma$below the Cepheid-based distance, it is unclear if all systematic effects connected with this method are well-understood. It has been noted, for example, that for the galaxy NGC 4258 the kinematic water-maser-based distance is also about 10% less than the Cepheid-based distance (Maoz et al. 1999). The error budget of the Cepheid method is probably on the order of 10%.

Figure 3: The I-band Tully-Fisher relation for local calibrator galaxies with Cepheid-based distances. The positions of NGC 3198 (diamonds) and NGC 2841 (crosses), assumed to be at various distances, are indicated.

Sakai et al. (1999) have calibrated the T-F relation using 21 spiral galaxies with known Cepheid distances in five color bands: B, V, R, I, and H. If one places NGC 3198 on the mean B-band relation its distance should be 12.2 Mpc, while for the I-band this distance is 13.3 Mpc. Thus the Tully-Fisher distance is essentially consistent with the maximum MOND distance. Although the MOND rotation curve fit clearly prefers a somewhat smaller distance than the Cepheid-based distance, the idea is in no sense falsified by this well-determined rotation curve.

The I-band Tully-Fisher relation from Sakai et al. is shown in Fig. 3. The open points show the position of NGC 3198 when at a distance of 10.0 Mpc, 12.5 Mpc, and 13.8 Mpc. It is evident that, given the scatter in the observed relation, it is impossible to distinguish between these possibilities although distances of 12.5 to 13.8 Mpc are clearly preferred.

4.2 K$^{\prime }$ band photometry

One possible reason for the small deviation of the MOND curve from the observed curve at the Cepheid-based distance is that the r-band photometry is not a precise tracer of the stellar light distribution due to possible contamination by newly-formed stars and dust absorption. For this reason we have also considered recent near-infrared photometry of this galaxy.

Figure 4: The $K^{\prime }$ image of the central regions of NGC 3198. Brightness is represented by a linear gray-scale until 16.94 mag. arcsec-2 and black beyond this level. The size of the image is 5.6 by 5.6 arcmin. North is at the top, east to the left. NGC 3198 has a clear bulge in the near-infrared surrounded by what seems to be a small light depression and then a ring of spiral arm features.

An image of NGC 3198 in the $K^{\prime }$ band has been obtained by Rothberg et al. (2000) in order to calibrate the near infrared Tully-Fisher relation. The observations and initial stages of the data reduction, like sky-subtraction and flat-fielding are described in that paper. The detector was 1024 $\times$ 1024 square pixels of size 1 $.\!\!^{\prime\prime}$68 $\times$ 1 $.\!\!^{\prime\prime}$68. Consequently the total image measures 28.7 arcmin along the sides and NGC 3198 which has a scale-length of approximately 1 arcmin fits completely within the image leaving ample margins of pure sky around the galaxy. Rothberg et al. (2000) derived a total brightness of 7.79 $K^{\prime }$ magnitudes which translates to 3.4 $\times$ 10¹⁰  $L_{\odot}^{K^{\prime}}$ for a distance of 13.8 Mpc.

In Fig. 4 the image of the central regions of NGC 3198 is reproduced. Clearly discernible is a prominent bulge which is much less obvious in images at bluer wavelengths. As a consequence this central bulge region must be enshrouded in an appreciable amount of dust, which explains the reddening going inward. Surrounding the bulge appears to be a ring of spiral arm features with a light depression between the bulge and this ring.

To determine the radial luminosity profile, ellipses have been fitted to the image which provided the position and orientation of the galaxy (Fig. 5). As a next step the intensities have been averaged over elliptic annuli. In the inner regions the orientations of the annuli were equal to those determined by the ellipse fit, while for the intermediate and outer regions a constant position angle and ellipticity was adopted. The error of each radial intensity value was calculated by quadratically adding the error generated by sky-level variations and the noise appropriate for each annulus. The result is shown in Fig. 6. The radial profile in the r-band is also plotted in that figure, and one may notice that the photometry of the disc is of similar shape for the r and  $K^{\prime }$ bands.

It is without doubt that NGC 3198 has a bulge or central light concentration. A possible bulge/disc light decomposition is shown in Fig. 7. Here, it is assumed that the mass surface density is exactly proportional to the observed intensity level, and that the light and mass distribution are axisymmetric. For that case the bulge/disc decomposition illustrated in Fig. 7 is essentially a decomposition by eye. Here, it is further assumed that the stellar disc has a central hole with a radius corresponding to that of the light depression and that the bulge extends slightly beyond this radius.

The light depression might well be caused by the presence of a central bar. The influence of a bar on the radial velocity field of the gas is suggested in high-resolution H$\alpha$ images of the galaxy, where characteristic distortions from circular motion are evident (Corradi et al. 1991). The light depression would then be due to a real deficiency of matter near the L4 and L5 Lagrangian points along the minor axis of the bar (Bosma 1978). In that case, the bar would be oriented nearly parallel to the line-of-sight and would not be photometrically conspicuous. Moreover, the bar would affect the derived rotation curve in the inner regions, or, at least, the interpretation of the rotation curve as a tracer of the radial force distribution. A bar aligned with the minor axis of the galaxy image would have the effect of increasing the apparent rotation velocities in the inner region (Teuben & Sanders 1985); however, this would be significant only within the inner 30 arcsec ($\approx$2 kpc) and would have little influence upon the overall shape of the derived 21 cm line rotation curve.

Keeping this caveat in mind, we proceed using the decomposition depicted in Fig. 7: assuming a spherical bulge and disc with observed ellipticity, the total $K^{\prime }$ luminosity of 3.4 $\times$ 10¹⁰  $L_{\odot}^{K^{\prime}}$, is divided into 8.23 $\times$ 10⁹ and 29.09 $\times$ 10⁹  $L_{\odot}^{K^{\prime}}$for the bulge and disc respectively.

Because the scale-length of the disc in $K^{\prime }$ is nearly equal to the disc scale-length in the optical, it is not to be expected that the MOND fit will be much different from that for the r-band photometry. This is the case, as can be seen in Fig. 8 where the MOND rotation curve again has been determined at the Cepheid-based distance of 13.8 Mpc. Here, except for a spike in the central regions which is due to the bulge, the predicted rotation curve is essentially the same as derived from the r-band photometry; that is to say, the conclusions are unchanged by the near-infrared results.