3. Spherical mass distribution with exponential density
In this section, we study a spherically symmetric distribution with an exponentially decaying density:
Our ultimate goal is to find a method to study the discrepancies between the observed rotation curves of spiral galaxies and the curves predicted by using Newton's law of gravity. The luminosity profile of many spiral galaxies can be well fitted assuming that the density of luminous matter decreases exponentially with distance from the centre of the galaxy (Kent 1987). This is the reason why we are interested in studying such a density function, even though spiral galaxies are not, obviously, spherical.
Using Eqs. (6), (7) and (10), in Eqs. (4) and (5), and considering spherical symmetry, the two problems sketched in Sect. 2 can be conveniently recast in the form of two integral equations:
(i) Given , find a function that satisfies the equation:
(ii) Given , find a function such that:
The solution to these integral equations will be described in detail in Appendix A.1 The results can be summarised as:
(i) Potential problem (viz. Eqs. (6) and (11))
where the function has the following behaviour at the origin:
(ii) Force and velocity problem (viz. Eqs. (7), (8) and (12)).
Here, the exact solution is given by the following expression:
The behaviour of at the origin is as follows:
The behaviours at the origin just tell us that for , and thus, for , which is in fact in good agreement with the observations (as the observed rotation curves are usually well fitted in the inner regions by a straight line) and, from a non-Newtonian gravity point of view, it is also in agreement with the fundamental experimental constrain that for short distances the gravitational interaction must be well described by a Newtonian limit.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998