## 4. Thin-disk mass distribution with exponential densityThe luminosity profile of many spiral galaxies can be well fitted assuming that the luminous matter is placed along a thin disk with a density that decreases exponentially with the distance to the centre of the galaxy (Kent 1987). being a normalisation constant with units of a two-dimensional density, related to the total mass of the galaxy by . Considering a thin-disk distribution and using Eqs. (6), (7) and (17), in Eqs. (4) and (5), the two problems outlined in Sect. 2 can be recast as two integral equations: (i) Given , defined by (6), find a function such that: (ii) Given , defined by (7), find a function such that: In this case the problem cannot be solved exactly. We use an
approximation that we call The calculations are described in detail in Appendix A.2. The
solution to the two problems outlined above, in the (i) Potential problem: ( In this case, the approximate solution to the problem is where the function has the following behaviour at the origin: (ii) Force and velocity problem: ( Here, the approximate solution is given by the following expression: and the behaviour of at the origin is as follows: © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |