4. Thin-disk mass distribution with exponential density
The 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 Gaussian as, in the Newtonian case, it is equivalent to use the Gauss' law for calculating the gravitational field.
The calculations are described in detail in Appendix A.2. The solution to the two problems outlined above, in the Gaussian approximation can be summarised as:
(i) Potential problem: (viz. Eqs. (6) and (18))
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: (viz. Eqs. (7), (8) and (19)).
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