An exact analytical method for inferring the law of gravity from the macroscopic dynamics: Spherical mass distribution with exponential density

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We consider the gravitational potential and the gravitational rotation field generated by an spherical mass distribution with exponential density, when the force between any two mass elements is not the usual Newtonian one, but some general central force. We invert the usual integral relations and obtain the elemental interaction (between two point-like masses) as a function of the macroscopic gravitational field (the one generated by the distribution). Thus, we have a direct way for testing the possibility of finding a correction to the Newtonian law of gravity that can explain the observed dynamics at large scales without the need of dark matter. We show that this method can be used even in the case of spiral galaxies with a good level of confidence.