6. An example of how to work with real data
In this section we just give a simple example of how to use the results previously obtained to study the problem of the rotation curves of spiral galaxies under a non-Newtonian point of view. We will do it only for one galaxy, just to show how the method can be used. In a forthcoming paper (Rodrigo-Blanco & Pérez-Mercader 1997) a similar and more detailed study of a sample of nine galaxies will be done.
We can only apply Eq. (22) to a real galaxy if it can be well described, at least as a first approximation, as a thin disk with exponential density. We have chosen NGC 6503, a galaxy with a luminosity profile that can be well fitted using a thin disk model with exponential density (with a scale length kpc), with no bulge, and without a very large amount of neutral gas (Begeman 1987). Assuming a constant value of for the disk, the luminous mass density of this galaxy can be reasonably described by Eq. (17).
Once we have a galaxy that can be described in the manner described above, the next step is to fit its observed rotation velocity by some mathematical function, so that we can take its derivatives in Eq. (22). It is important to note that there is no physical reason for choosing one function or another to fit the observed data. Thus, we have arbitrarily chosen a functional form for the observed velocity given by:
where is a third degree polynomial and some real number. For twelve values of , going from 1.0 to 2.2, the coefficients of the polynomial have been fitted to the observed rotation curve (all the fittings are shown in Fig. 4, upper graph). Once the rotation curve is fitted by a function, Eq. (7) can be used to obtain , where is the total mass of the galaxy. In this way, after putting it in Eq. (22), we obtain a form for for each functional fitting. In Fig. 4, lower graph, we have plotted the 's corresponding to the fits shown in the upper graph, multiplied by a common factor: . This 's can be introduced into Eq. (2) to obtain an elemental law of gravity (between any two point-like masses) that would generate the observed rotation curve of NGC 6503 without the need of dark matter. The total mass of the galaxy, a priori, can be found requiring that . But, actually, Fig. 4 also tells us that the exact form of the elemental force is not constrained by the observed rotation curve at small distances. Thus, since we cannot constrain the exact form of for intermediate distances, we cannot either fix the exact value of the mass of this galaxy.
It must be kept in mind that we are looking for a universal gravitational law, that is, one that is valid for any two point-like particles, with no dependence on where they are located. Thus, the law obtained for one galaxy must be also at work in any other mass distribution. If the laws necessary to explain the rotation curve of other spiral galaxies without dark matter were shown not to be compatible with the one obtained here, that would be a proof against the existence of a law like the one written in Eq. (2).
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998