An approximate analytical method for inferring the law of gravity from the macroscopic dynamics: Thin-disk mass distribution with exponential density

Abstract

The gravitational potential and the gravitational rotation field generated by a thin-disk mass distribution with exponential density are considered in the case when the force between any two mass elements is not the usual Newtonian one, but some general central force. We use an approximation such that in the Newtonian case the gravitational field generated by the disk reduces to the familiar expression that results from applying the Gauss' law. In this approximation, we invert the usual integral relations in such a way that the elemental interaction (between two point-like masses) is obtained as a function of the overall gravitational field (the one generated by the distribution). Thus, we have a direct way for testing whether it is possible or not to find a correction to the Newtonian law of gravity that can explain the observed dynamics in spiral galaxies without dark matter.