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

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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 [FORMULA] that describes the deviation from the Newtonian law, that is,

[EQUATION]

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

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,

[EQUATION]

where we have introduced

[EQUATION]

with [FORMULA] as [FORMULA], as required by the Newtonian limit.

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

[EQUATION]

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

[EQUATION]

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 [FORMULA] and [FORMULA] such that:

[EQUATION]

[EQUATION]

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

[EQUATION]

where the auxiliary functions [FORMULA] and [FORMULA] satisfy the following functional relationship:

[EQUATION]

Our goal is to design a procedure where, assuming that [FORMULA] is known (say from observation of the rotation velocity) for all values of R, we obtain a [FORMULA] that generates the given rotation velocity. Or, in other words, given the potential as inferred from observations we want to find which [FORMULA] could have generated it. Actually, what we will find are [FORMULA] and [FORMULA] as functions of [FORMULA] and [FORMULA] 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.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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