## 5. Real thin disks: comparison between the spherical and the Gaussian approximationsIn this section we consider the possibility of using the results obtained in the previous sections to study the rotation curves of spiral galaxies (provided they can be described as thin disks with exponential density). At first sight, we would think that we cannot apply the results of
Sect. 3 to spiral galaxies, because they have been obtained
assuming spherical symmetry, and spiral galaxies are disk-shaped, not
spherical. In spite of this, it seems logical that the difference
between the gravitational field generated by a sphere and the one
generated by a disk becomes negligible at very large distances (here
"large distances" means large compared with some typical length scale
for the distribution; it could be, for instance,
, i.e, the exponential length scale of the mass
distribution). Furthermore, when we consider the effect of an
increasing , it is evident that the meaning of
what is a "large distance" changes as we change the functional form of
. That is, the faster
grows as a function of Concerning the solution summarised in Sect. 4 for a thin disk, it is not an exact solution to the problem. In Appendix A.2 the details are shown, but the main point is that in the step from Eq. (A17) to Eq. (A18) we made the approximation of considering only the first term of an infinite series. The terms dropped depend also on , which is not known, so it is not trivial to evaluate the goodness of the approximation. In this section, we choose some functions
and compare the By Concerning the and thus, the rotation velocity is where is the disk mass inside the sphere of
radius Next, both approximations must be checked for other different forms of . For doing that we choose a parametric family of 's given by where We have found the For the other values of In the Gaussian approximation, it can be seen that, for the solution is (28), that is, the approximation is exact. In Fig. 1 we have plotted the exact rotation velocities
compared with those corresponding to both approximations for some
values of
It can be seen in Fig. 1 that the where the sub-scripts So defined, is a measure of the mean square
error that we make in the rotation velocity if the approximation is
used instead of the numerical integrals, for each value of
In Fig. 2, we plot the value of versus
There remains the problem that must be
Newtonian for small distances and thus, the fact that the
where stands for the exact disk solution and
is the velocity in the
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |