The dynamical analysis of rotation curves of galaxies, binary galaxies, clusters of galaxies and large scales structures show significant discrepancies between the observed behaviour and the one expected from the application of General Relativity and its Newtonian limit to the visible mass. This disagreement has led many astrophysicists to believe in the existence of a large amount of non-visible matter and is, thus, commonly known as the "dark matter problem".
In spite of this, there is no direct evidence for the validity of either Einstein's General Relativity or Newtonian gravity at scales much larger than those of the Solar system (See, e.g., Will 1993). There is, therefore, no experimental or observational reason to ascertain that unmodified General Relativity holds at larger distances. This leads us to think that we should be open to the possibility that it had to be revised (perhaps in the same spirit as Newton's law had to be modified for strong fields and large velocities).
In this paper we consider the possibility that Newton's law of gravity is just a good approximation at short distances of a more general expression for the force. It is interesting to identify which, if any, extensions of the usual inverse square law are compatible with the dynamics observed at large scales.
Work along these lines has already been done (Tolhine 1983, Kuhn & Kruglyak 1987, Mannheim & Kazanas 1989) assuming a specific functional form for the force, and then evaluating the field generated by a mass distribution (for instance, a galaxy) by performing the corresponding three-dimensional integrals. This approach has helped to find very interesting results, as the scale (5 - 10 kpc) where the breakdown of Newtonian gravity takes place when the correction to the force is assumed to be proportional to (Kuhn & Kruglyak 1987), but its success obviously depends on the election of the initial form for the force. We present and work out a method that allows us to follow the inverse methodology, that is, to infer, directly from observations, the phenomenological law of gravity that is able to generate a given macroscopic gravitational field. We do it for the cases of a spherical mass distribution and a thin disk with exponential density.
In Sect. 2 we give the general definitions that will be used later in Sect. 3 for the case of spherical symmetry, and in Sect. 4 for a thin disk (in both cases assuming an exponential density). In Sect. 5 we study the possibility of using the results obtained in the previous sections to study the problem of rotation curves of spiral galaxies under a non-Newtonian point of view, and in Sect. 6 an example is shown on how to use these results for a real galaxy. Finally, some conclusions are offered. In Appendix A we show the mathematical basis underlying the results presented in Sects. 3 and 4.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998