Astron. Astrophys. 321, 444-451 (1997)

## 7. Summary and conclusions

We have found the solution to the problem of inverting the integral relation between the elemental law of gravity and the overall gravitational field generated by two interesting mass distributions: (i) a sphere with exponential density (where the solution is exact), and (ii) a thin-disk with exponential density. The problem in the case of the thin disk has been solved in an approximation that has been called Gaussian as it is equivalent to use the Gauss' law for calculating the gravitational field generated by the distribution (that is, to assume that the gravitational force at a distance R to the centre of the disk is proportional to the mass inside that radius). Although this is not exact, we have shown that it is a very good approximation, and it gets much better when grows with r, which is the expected behaviour if the observations must be explained without the need of dark matter. Actually we have also shown that this Gaussian approximation is always better than using the results for the case of the sphere as an approximation to the disk case.

In summary, we now have a method for inferring given the observed rotation velocity. It can be said, in some sense, that we have a way to travel from the world of macroscopic interactions to the world of elemental interactions (where, by elemental interaction we mean that between two point-like particles). It can be sketched as follows: Given the observed for a given galaxy, use Eq. (8) to obtain , fit it by a mathematical function and then use Eq. (22) to get the that describes the elemental gravitational force (through Eq. (2)) that can explain the observed rotation curve.

We are now ready for applying the differential expression that we have found to the observed rotation curves of spiral galaxies, and find the required for explaining those rotation curves. We have given an example of how it can be done for a particular galaxy. By repeating the exercise for a sample of several galaxies, we will see whether it is possible or not to find a universal law of gravity that can explain all the rotation curves requiring only the observed luminous matter. This will be done in a separate publication (Rodrigo-Blanco & Pérez-Mercader 1997).

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998