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1. Introduction

There is a wide consensus that the rotation curves of spiral galaxies constitute an observational proof - perhaps the best proof - for the existence of dark matter in the Universe. Dark matter is of evident interest in Cosmology, hence the interest of a review on the topic, in this case from a post-graduate didactic point of view.

Dark matter is not an exotic or sophisticated hypothesis. Neutrinos, brown dwarfs and black holes are all candidates to be identified with dark matter and are, nonetheless, classical concepts in Physics, introduced by fully established theories. High Energy Physics actually predicts a number of particles that do not interact with photons. We cannot claim that all the existing matter emits or absorbs photons.

However, although the necessity of dark matter was proposed more than 60 years ago (Zwicky, 1937), this hypothesis is still not whole-heartedly accepted by some workers. This scepticism is never explicitly expressed, but is subjacent, implicitly revealed in sentences such as "a galaxy has a halo". We should rather say "a galaxy is a halo" as a galaxy's mass may be at least 10 times its visible mass, the visible element then being a mere minor component, only important for us because it is what we see.

(We must, however, accept that when we state that if a galaxy is a halo, the discovery of just one exception, i.e. of a visible galaxy with no halo, would represent a serious problem of interpretation, a problem as fundamental as finding the light of a firefly not associated with a firefly.)

Once the existence of dark matter is recognized as a conservative possibility, let us establish a difference between the problem of dark matter in the Universe and the problem of dark matter in galaxies. We implicitly assume throughout the paper that $ \Omega$ $ \sim$ 1, and that visible matter only contributes with $ \Omega_{V}^{}$ $ \sim$ 0.003. The total matter contributes either with $ \Omega_{M}^{}$ $ \sim$ 1, as classically assumed, or with $ \Omega_{M}^{}$ $ \sim$ 0.3, coherent with the more recently assumed scenario deduced from the observations of early supernovae, leading to the re-accelerating Universe, the non-vanishing cosmological constant or other identifications of dark energy (e.g. Turner, 1999). The assumption of a plane Universe, $ \Omega$ $ \sim$ 1, is based on theoretical ideas about inflation, observations of large-scale dynamics, the interpretation of the CMB and even on some philosophical arguments. Our position on dark matter is based on the difference between $ \Omega_{M}^{}$ $ \sim$ 0.3 and $ \Omega_{V}^{}$ $ \sim$ 0.003, as observed. We only see, in the best of cases, 1% of the matter. This is the basic fact, rather than the existence and possible observation of dark matter in galaxies, that demonstrates that dark matter exists.

Rotation curves and other dynamical effects in galaxies suggest that total galactic mass is about 10 times larger than that observed. Even in this case, we should still find about 90% of dark matter elsewhere in the Universe, which means, given the relative uncertainty in all these figures, that practically all the required dark matter lies outside galaxies. The possible galactic contribution to dark matter is negligible to close the Universe. Therefore, the existence of galactic dark matter is clearly very important for our knowledge of galaxies and their dynamics, but not so decisive for the cosmological problem of identifying its nature and amount.

This notion allows us to make a more objective analysis of the topic of dark matter in galaxies, once it has been partially disconnected from the cosmological one. Rotation curves of spirals, and many other observations, are currently interpreted as evidence of the existence of massive large dark halos, but a critical analysis of the observations, and the theoretical interpretations involved, is permanently necessary. Even if dark matter is a major ingredient in the most widely accepted and orthodox picture of a galaxy, most authors only differing about its mass and size, we will see that the hypothesis of a total absence of galactic dark matter cannot be completely ruled out.

The problems involved in determining galactic dark matter may be summarized as follows: the internal regions of galaxies require little or no dark matter and we must examine the external ones, where there are few stars and we must observe gas to determine the gravitational potential. However gas, in these regions where the gravitational force is low, may be influenced by magnetic fields. Thus, it is convenient to look for stellar systems lying far from the galaxy, and particularly satellite or companion galaxies. Then, however, it is very difficult to distinguish between galaxies with a halo and halo with galaxies, i.e. the hypothesis of a large common halo is difficult to reject.

Though the rotation curves of spirals is the main topic to be discussed, we must pay attention to other methods of determining DM, in particular those based on globular clusters, satellite galaxies, binary systems, polar-ring galaxies and so on. Lensing by spiral galaxies could become an important tool in the near future (Maller et al. 1999). We should also review the methods of DM estimation in other types of galaxies. With respect to the DM problem, spirals do not seem to be exceptional.

Other models, which do not require DM to explain rotation curves, have been reported in the literature and also require our attention in this study. One such model is MOND (Modified Newtonian Dynamics) and another is the magnetic model. The authors have contributed to the development of this latter model, which is commented in some detail at the end, together with its cosmological implications. Despite the inclusion of MOND and the magnetic approach as interesting possibilities, through out this paper we adhere to the most conservative point of view based on DM. Nevertheless, we emphasize the difficulties inherent to most methods and models.

The purpose of this review is mainly didactic, as a bibliographic source for postgraduate courses, but also critical. This critical approach might be considered unnecessary, in view of the wide acceptance of the dark matter hypothesis by the scientific community; but apart from its didactic interest, it is always pertinent to reconsider apparently solid beliefs. In this respect, it is convenient to remind the reader of four historical aspects related to the early adoption of the DM halo hypothesis, which were based on arguments that eventually became open to discussion or were made obsolete:

a) Kahn and Woltjer (1959) considered the dynamics of the double system formed by M31 and the Milky Way and concluded that their motion would require a binary system mass of $ \ge$1.8 × 1012M$\scriptstyle \odot$, much greater than the visible matter in the two galaxies. Kahn and Woltjer formed the opinion that there existed an as yet unobserved mass in some invisible form. Although they identified this invisible mass with hot gas rather than dark matter in its present sense (unobserved rather than unobservable) this work gave a first proof of the missing mass. It should be remembered, however, that this mass was considered to lie either in M31 and the Milky Way or in the intergalactic space (either two halos or a common halo). It is still not clear whether this second possibility can be completely ruled out. Therefore, this paper established the existence of dark matter, but not necessarily within galaxies. The doubt remained: either in or in between.

b) Oort (1960, 1965) found evidence for the presence of dark matter in the disk of our galaxy, although van den Bergh (1999) wrote: "However, late in his life, Jan Oort told me that the existence of missing mass in the galactic plane was never one of his most firmly held scientific beliefs". After a long debate since then (see Binney and Tremaine, 1987; Ashman 1992) the discussion finally seems to be closed. Using HIPPARCOS data, Crezé et al. (1998) have found that there is no evidence for dark matter in the disk: gas and stars perfectly account for the gravitational potential.

c) Ostriker and Peebles (1973) suggested that spherical dark matter halos around the visible component of the spiral galaxies were necessary to suppress bar instabilities. However, their arguments did not convince Kalnajs (1983) and Sellwood (1985), who showed that a central bulge was equally efficient to stabilize disks.

d) Babcock (1939) observed that the stars in M31 were rotating at an unexpectedly high velocity, indicating a high outer mass-to-light ratio, although he also considered other possibilities: either strong dust absorption or, as he stated, "new dynamical considerations are required, which will permit of a smaller relative mass in the outer parts". This sentence, quoted by van den Bergh (1999), is interesting, bearing in mind that gravitation is the only force considered at present to explain rotation curves, while other "dynamical considerations" are ignored. Optical rotation curves of other galaxies were obtained, until those published by Rubin et al. (1980), that were considered to be clear evidence of dark matter in galaxies. Today, however, many authors consider that optical rotation curves can be explained without dark matter (even without rejecting its contribution). For instance, Broeils and Courteau (1997) by means of r-band photometry and H$\scriptstyle \alpha$ rotation curves for a sample of 290 spirals concluded that "no dark halo is needed".

Therefore, the basic initial arguments leading to the belief of dark matter halos around spiral galaxies failed or were not conclusive. Only the interpretation of the HI rotation curves by Bosma (1978) subsists unmodified. For early thoughts on dark matter, the paper by van den Bergh (1999) and the Ph.D. Thesis of Broeils (1992) are essential reading.

In brief, there is a standard interpretation of the rotation curve of spiral galaxies that is implicitly adopted throughout this paper, but we should not ignore other possibilities.

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