1 Introduction

It is well established that, in the context of Newtonian dynamics, the observable mass in spiral galaxies cannot account for the observed flat rotation curves in the outer regions of galaxies (Bosma 1978; Begeman 1987; van Albada et al. 1985). The standard explanation for this discrepancy is the proposal that galaxies are embedded in an extended dark halo which dominates the gravitational field in the outer regions (Trimble 1987).

An alternative explanation for the discrepancy is the possibility that dynamics becomes non-Newtonian in the limit of low accelerations. The most successful such proposal is Milgrom's (1983) modified Newtonian dynamics or MOND. Here the idea is that below a certain acceleration threshold ($a_{\rm o}$) the effective gravitational acceleration approaches $\sqrt{a_{\rm o}g_{\rm n}}$ where $g_{\rm n}$ is the usual Newtonian acceleration. This modification yields asymptotically flat rotation curves of spiral galaxies and a luminosity - rotation velocity relationship of the observed form, $L \propto v^4$, the Tully-Fisher relation (Tully & Fisher 1977). But apart from these general aspects the prescription also successfully predicts the observed form of galaxy rotation curves from the observed distribution of stars and gas with reasonable values for the mass-to-light ratio of the stellar component (Begeman et al. 1991; Sanders 1996; Sanders & Verheijen 1998; McGaugh & de Blok 1998). A crucial element of a very specific prescription like MOND is that the precise form of the rotation curve is predicted by the observed mass distribution given the value of a single universal parameter; in this case, the critical acceleration a₀. Consequently MOND can in principle be falsified as soon as there is one galaxy for which the predicted rotation curve disagrees significantly with the observed curve; although, in practice, the usual uncertainties inherent in astronomical data render a definitive falsification problematic in any individual case.

In Begeman et al. (1991, hereafter BBS) MOND is applied to a sample of galaxies for which high quality H  I rotation curves are available. For a value of a₀ equal to 1.21 $\times$ 10^-8 cm s^-2the rotation curves of the sample could be reasonably reproduced, the free parameter in each case being the mass-to-light ratio of the visible disc. Because MOND is an acceleration dependent modification, this derived value of $a_{\rm o}$depends upon assumed distance scale ( $H_{\rm o} = 75$ km s^-1 Mpc^-1 in this case). Moreover, the quality of an individual fit depends upon the adopted distance to the galaxy, and, since the relative distances to these nearby galaxies have not been known to within an accuracy, typically, of 25%, this has provided some freedom to adjust the distance in order to improve the MOND fit; i.e., distance, within certain limits, can be considered as an additional second parameter in the fitting procedure. For most of the galaxies in the sample of BBS, the distance did not have to be adjusted significantly (<$10\%$) to improve the MOND fits, and the improvement was not significant. However, one object, NGC 2841, required a large readjustment: the Hubble law distance to this galaxy is about 9 Mpc, but MOND clearly prefers a distance which is twice as large.

Using ground-based and Hubble Space Telescope observations Cepheid distances to 21 inclined galaxies have now been determined as part of the HST key program on the extragalactic distance scale (e.g. Sakai et al. 2000). Three of the galaxies in this Cepheid sample are also in the sample with high quality rotation curves considered by BBS. These are NGC 2403 (Freedman & Madore 1988), NGC 3198 (Kelson et al. 1999) and NGC 7331 (Hughes et al. 1998). For these three galaxies the MOND prescription can now be considered in the context of the Cepheid distance that is generally considered to be the most precise indicator.

NGC 2841 has been discussed as a critical case for MOND by Sanders (1996). For this galaxy, there is also a large discrepancy between the Tully-Fisher distance and the Hubble law distance (for plausible values of the Hubble constant). Moreover, the galaxy was the site of a recent SNIa (1999by). For these reasons this galaxy has been included, subsequently, in the HST program (Macri et al. 2001).

Here we demonstrate that for two galaxies in the BBS sample the rotation curve predicted by MOND is consistent with the observed curve when the galaxies are placed at the Cepheid distance. However, for NGC 3198 at the Cepheid distance of 13.8 $\pm$ 0.6 Mpc, the shape of the rotation curve predicted by MOND systematically deviates (by up to 10 km s^-1) from the observed curve, both in the inner and outer regions. The largest distance which can be compatible with MOND is about 10% lower than the Cepheid-based distance. This is not particularly problematic because of likely uncertainties in the Cepheid method and in the determination of a rotation curve from the observed two-dimensional velocity field. NGC 2841, however, remains a difficult case for MOND. The minimum distance which is consistent with MOND is about 17 Mpc whereas the Cepheid-based distance is 14.1 $\pm$ 1.5 Mpc. We discuss the implications and seriousness of this discrepancy for MOND, or, alternatively, for the Cepheid method.