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

## 2. General definitions

Let us assume that the gravitational potential generated by a point-like mass does not correspond to the usual Newtonian form but can be written in terms of a function that describes the deviation from the Newtonian law, that is,

where is the gravitational potential experienced by two point-like particles of masses and separated by a distance r and is the Newton's constant. Of course, the Newtonian limit, , must be recovered as .

This modification could, for example, be due to the many body nature of the mass distribution making up the galaxy, a relativistic theory different from General Relativity... This is irrelevant in what follows.

The force per unit mass is, by definition, the gradient of the potential,

where we have introduced

with as , as required by the Newtonian limit.

In this way, to find the total potential or the total force generated by a mass distribution with density , one must integrate over the volume spanned by to get:

for the potential experienced by a point mass at a distance R from the centre of , and

for the force.

In the case that the gravitational potential is only a function of the distance to the centre of the distribution, it is convenient to introduce two new functions and such that:

and the rotation velocity of a test particle in a circular orbit bound to the distribution will be:

where the auxiliary functions and satisfy the following functional relationship:

Our goal is to design a procedure where, assuming that is known (say from observation of the rotation velocity) for all values of R, we obtain a that generates the given rotation velocity. Or, in other words, given the potential as inferred from observations we want to find which could have generated it. Actually, what we will find are and as functions of and respectively. This will be done, in the following sections, for the cases of spherical symmetry and a thin disk, in both cases assuming an exponential density.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998