5 NGC 2841

In Fig. 9 we show the MOND rotation curve for NGC 2841 compared to the observed curve at various assumed distances: 15.6 Mpc, which is the 1$\sigma$ upper limit on the Cepheid-based distance, 17 Mpc which is 20% larger than the Cepheid-based distance, and 23 Mpc which is the MOND-preferred distance. As in Fig. 2 the M/L values for the disc and bulge are also given in the figure.

Figure 5: The orientation and inclination of ellipses fitted to the $K^{\prime }$ band image of NGC 3198. The same fixed central positions have been adopted for all ellipses. The distinct signature of the central bulge (bar) is apparent.

Figure 6: Radial light profile in the $K^{\prime }$ band compared with that in the r-band (Kent 1987). The photometry of the disc is similar for both bands, however, in the near infrared the bulge is considerably brighter than in the optical.

Figure 7: Radial light profile in the central regions and a possible decomposition into a bulge (dotted line) and a disc (dashed line).

Figure 8: The MOND rotation curve of NGC 3198 where the surface density distribution of the stellar component is taken to be traced by the $K^{\prime }$-band photometry using the decomposition shown in Fig. 7. The distance is Cepheid-based distance of 13.8 Mpc. The filled squares show the rotation curves derived for the two sides of the galaxy independently; this gives a better estimate of the errors.

Figure 9: MOND rotation curves for NGC 2841 at various distances ranging from the one-sigma upper limit on the Cepheid-based distance (15.6 Mpc) to the Tully-Fisher and SNIa distance (23 Mpc). The  $M/L_{\rm B}$ values for the disc and bulge respectively are given in parenthesis.

The rotation curve, as a tracer of the radial force distribution in this galaxy, is actually not as well-determined as that of NGC 3198. There is a significant warp in the outer regions which must be modelled by the tilted-ring technique, and this adds uncertainty to the derived rotation curve (see comments by Bosma 2002). None-the-less, it is clear that, while the Cepheid distance goes in the right direction (it is significantly larger than the Hubble based distance), it is not enough to bring the MOND-predicted rotation curve into agreement with the observed curve. Moreover, not only does the form of the predicted curve differ systematically from that observed, but it is clear that the M/L value for the disc is un-naturally large (6.8  $M_\odot/L_\odot$) - larger than that required for the bulge (2.2).

Both of these problems are relieved somewhat if the distance is taken to be 20% larger than the Cepheid-based determination - at 17 Mpc. There are still systematic deviations in the form of the rotation curve, but these become large (in the outer regions), only where the gas layer of the galaxy is observed to be significantly warped. We may take this as a lower limit on the distance which would be compatible with MOND, although the disc M/L does remain uncomfortably large (5.9  $M_\odot/L_\odot$) and that of the bulge rather small (2.7).

Leaving distance as a free parameter in the fit yields a MOND-preferred distance of 23 Mpc, and here we see that the rotation curve fit is perfect with very reasonable implied M/L values for the disc and bulge. This is entirely consistent with the distance implied by the Cepheid-calibrated Tully-Fisher relation as is also shown in Fig. 3. Taking the galaxy to be at the Cepheid distance of 14.1 Mpc, we see that the galaxy lies about one entire magnitude below the mean line of the TF relation. The distance implied by I-band TF relation is 24 Mpc.

There has been a recent supernova in NGC 2841 (SN 1999by), which is type Ia, i.e., the fundamental extragalactic "standard candle''. However, if the galaxy is at the Cepheid distance of 14.1 Mpc, SN 1999by is one of the least luminous supernovae Ia ever observed, with a peak absolute magnitude of M_B = -17.15 $\pm$ 0.23. Based upon an estimate of the decline-rate parameter ($\Delta$ $m_{15}\approx 1.9$) Garnivich et al. (2001) argue that this supernova is a peculiar low luminosity event, and they use this event and several others to recalibrate the Phillips relation (Phillips et al. 1999) between decline rate and peak luminosity. However, if we take the Phillips relation at face value then the peak luminosity of this object would be M_B = -18.3, which would imply that the distance to the galaxy would be 23.5 Mpc. It is interesting that an earlier SN event in NGC 2841, SN 1957A, would be, if the galaxy is at the Cepheid distance, the faintest supernova type Ia ever observed (M_B = -16.4). It is curious that this galaxy only seems to provide sub-luminous supernovae.

The deviation of the galaxy from the TF relation and the abnormally low peak powers of supernovae, suggest that the Cepheid distance to this object may be substantially too low. It has been argued that the Cepheid method may be adversely affected by blending: the true apparent brightness of Cepheids is enhanced by blending with the light of nearby stars. This would lead to an underestimate of distances based upon the period-luminosity relation, and would affect, in particular, the more distant objects (see Paczynski & Pindor 2001 for a discussion of these points). All we can conclude, at the moment, is that the MOND-preferred distance to NGC 2841 remains significantly larger than the present Cepheid-based distance.