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Astron. Astrophys. 321, 444-451 (1997)

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5. Real thin disks: comparison between the spherical and the Gaussian approximations

In 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, [FORMULA], i.e, the exponential length scale of the mass distribution). Furthermore, when we consider the effect of an increasing [FORMULA], it is evident that the meaning of what is a "large distance" changes as we change the functional form of [FORMULA]. That is, the faster [FORMULA] grows as a function of r, the smaller the distance at which a sphere is indistinguishable from a disk becomes, when considered from the gravitational point of view.

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 [FORMULA], which is not known, so it is not trivial to evaluate the goodness of the approximation.

In this section, we choose some functions [FORMULA] and compare the exact rotation velocity with the two approximate ones: (i) approximating a disk with a sphere, and (ii) using the Gaussian approximation for the thin disk. By exact we mean the rotation velocity generated by the considered [FORMULA] through Eqs. (19) and (8). Nevertheless, for most functional forms of [FORMULA] an analytical solution cannot be found and it is necessary to perform numerical integrations for finding that exact rotation velocity. Whenever that happens, we have decided to perform the integrals assuming that the disk has some non-zero thickness. Actually, this is a more realistic model for a spiral galaxy, the astrophysical system to which our approximate method is applied.

By approximate rotation velocity we mean the one such that the corresponding [FORMULA] satisfies the adequate equations: Eqs. (15) and (16) (for the spherical approximation); and Eqs. (22) and (23) for the Gaussian one.

Concerning the Gaussian approximation, as a test of consistency, we first consider the Newtonian case, i.e., [FORMULA]. When Eqs. (22) and (23) are used we obtain:

[EQUATION]

and thus, the rotation velocity is

[EQUATION]

where [FORMULA] is the disk mass inside the sphere of radius R. Of course, this is not the exact result, but it is what we find if we apply the Gauss' law as an approximation for evaluating the gravitational field. That is why the approximation is called Gaussian.

Next, both approximations must be checked for other different forms of [FORMULA]. For doing that we choose a parametric family of [FORMULA] 's given by

[EQUATION]

where µ parametrises how fast [FORMULA] grows.

We have found the exact analytical solution (for the thin-disk case) for [FORMULA] and [FORMULA]:

[EQUATION]

[EQUATION]

For the other values of µ the integrals are performed numerically for a disk with a small thickness [FORMULA]

In the Gaussian approximation, it can be seen that, for [FORMULA] 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 µ. For each case, the solutions are normalised dividing by a convenient constant [FORMULA] defined as:

[EQUATION]

[FIGURE] Fig. 1. Exact rotation curves (dots), Gaussian approximation (solid line) and spherical approximation (dashed line) when [FORMULA] for some values of µ ([FORMULA] is the Newtonian case). In every case, for the sake of clarity, the velocities are normalised by dividing by an appropriate constant [FORMULA].

It can be seen in Fig. 1 that the Gaussian approximation (that is always better than the spherical one) is quite good in every case, and is better when [FORMULA] is a growing function of r. However, it would be interesting to have a more quantitative way of describing the difference in the rotation velocity obtained in the exact case and in both approximations as a function of µ. In order to do that, we define the quantity [FORMULA] as follows:

[EQUATION]

where the sub-scripts D and A stand for the disk (exact) and the approximation (either the spherical or the Gaussian one) respectively. We sum over [FORMULA], which are the points where the integrals are calculated. The total number of points for each value of µ is [FORMULA].

So defined, [FORMULA] 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 µ.

In Fig. 2, we plot the value of [FORMULA] versus µ. As a result of what is seen in Figs. 1 and 2 two conclusions can be extracted: (i) the Gaussian approximation is a much better tool for studying the gravitational field in thin disk galaxies, and (ii) the faster [FORMULA] grows with r, the better is the approximation.

[FIGURE] Fig. 2. Mean square error in the velocity when the Gaussian (full points) and the spherical approximation (open points) are used instead of the exact solution for the disk, as a function of µ, and using [FORMULA]. The exact meaning of [FORMULA] is explained in the text.

There remains the problem that [FORMULA] must be Newtonian for small distances and thus, the fact that the Gaussian approximation is not very good for the Newtonian case could introduce large errors in the final results. We have calculated the exact curve and the one corresponding to the Gaussian approximation in the simple, but interesting, case when [FORMULA], for a galaxy with a length scale [FORMULA] and we have taken [FORMULA] which is consistent with the [FORMULA] that is obtained in Rodrigo-Blanco & Pérez-Mercader (1997) when working with real data. In Fig. 3 we have plotted the rotation curve for this [FORMULA] compared with the Newtonian one and the one for [FORMULA]. We also have plotted the value of [FORMULA] defined as:

[EQUATION]

where [FORMULA] stands for the exact disk solution and [FORMULA] is the velocity in the Gaussian approximation. It can be seen that, although the error in the Newtonian case can reach a 15%, the final error for [FORMULA] is more than three times smaller. This is due to the fact that for distances of the order of 5 to 10 kpc, [FORMULA] begins to be dominated by the non-Newtonian contribution and thus the approximation becomes more accurate.

[FIGURE] Fig. 3. Comparison between the exact rotation curve and the one in Gaussian approximation, for a galaxy with length scale [FORMULA], for three cases of interest: [FORMULA], [FORMULA], [FORMULA], being [FORMULA]. All the curves are normalised dividing by [FORMULA] The lower graph shows the percentage of difference between both curves in the three cases.
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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