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

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Appendix A: mathematical development

The line for the proof of Eqs. (13, (14), (15), and (16), for the spherical mass distribution, is very similar to the one for the proof of Eqs. (20, (21), (22), and (23), for the thin disk. Thus, the calculations are first shown in some detail for the sphere (where the solution is exact) and, then, the main differences for the case of the disk are explained.

A.1. Sphere with exponential density

First, we show the solution to what we call the potential problem, that is, how to go from Eq. (11) to Eqs. (13) and (14). Later we will use these results to solve the force problem, i.e, how to go from Eq. (12) to Eqs. (15) and (16).

In order to go from equation Eq. (11) to Eqs. (13) and (14), it is convenient to work with the integration variables in a way that is as independent as possible of the form of the function g. In order to do this, it is useful to use the Fourier sine transform:

[EQUATION]

as this allow us to use the addition theorem for Bessel functions (see Gradshteyn 1980) for decoupling the integration variables in the integrals:

[EQUATION]

Making use of all these equations, Eq. (11) can now be written as:

[EQUATION]

And, then, the orthogonality of Legendre polynomials,

[EQUATION]

leads to:

[EQUATION]

Then, we use the inverse Fourier transform,

[EQUATION]

to obtain, after some straightforward calculations, the following more useful form:

[EQUATION]

In order to simplify this expression, we introduce an auxiliary function [FORMULA] that makes the integrals exact:

[EQUATION]

We can insert Eq (A8) into Eq. (A7) and, upon integration by parts, we get:

[EQUATION]

where [FORMULA] is a solution to the ordinary differential Eq. (A8) that satisfies the conditions of being an analytic function at [FORMULA], and

[EQUATION]

where [FORMULA] stands for [FORMULA] and its first three derivatives.

These conditions are easily fulfilled in all the cases of interest. Actually, the analyticity is satisfied in the Newtonian limit, that is the behaviour that we expect to recover at [FORMULA]. Although it is possible to artificially build a pathological [FORMULA] such that it can represent a physical system without satisfying Eq. (A10), it can be seen that almost every function [FORMULA] that does not satisfy it corresponds to a rotation velocity that grows almost exponentially with the distance, which clearly seems to contradict the observations.

Actually, it is straightforward to see that, provided [FORMULA] is a solution to Eq. (A8), then [FORMULA] is also a solution to the same equation. Moreover, the terms proportional to [FORMULA] and [FORMULA] in Eq. (A9) assure that [FORMULA] and its second derivative are both zero at the origin. Taking all this into consideration, we finally obtain that:

[EQUATION]

[EQUATION]

Once [FORMULA] is known we can calculate [FORMULA] using Eq. (9). Equivalently, once [FORMULA] is known, [FORMULA] can be obtained through Eq. (3). Using these two equations together with Eq. (A11), and after some straightforward calculations, we can find a direct relation between [FORMULA] and [FORMULA]:

[EQUATION]

From the behaviour of [FORMULA] at the origin (Eq. (A12), and Eq. (9)), it is easy to see that, at the origin, [FORMULA] will satisfy:

[EQUATION]

(Eqs. (A11), (A12), (A13) and (A14) already appear in Sect.  3, but we prefer to rewrite them again here to keep the flow of the paper).

A.2. Thin-disk with exponential density

The line of reasoning for the solution of the problems outlined in Eqs. (18) and (19) is very similar to the one used in the case of the sphere. The use of the Fourier sine transform (Eq. (A1)) and the addition theorem for Bessel functions (Eq. (A2)) allows us to decouple the integration variables and arrive to an expression for [FORMULA] that is slightly different from the one found in the spherical case:

[EQUATION]

In this case, the absence of the [FORMULA] term in the volume element (that appears in the spherical case, but not for a thin disk) does not allow us to use the orthogonality of Legendre polynomials but the following expression:

[EQUATION]

and then Eq. (A15) becomes:

[EQUATION]

We apply our approximation here. We only consider the first term in the series of Bessel functions, that is, we drop all the terms in the series but the one with [FORMULA]. We have shown in Sect.  5 that in the Newtonian limit (i.e, when [FORMULA] = [FORMULA] = 1) this approximation corresponds to applying the Gauss' law to the mass distribution, i.e, to saying that the gravitational force at a distance R is proportional to the total mass contained in the sphere of radius R. This is the reason why we have called this method the "Gaussian approximation ". As it is quite a good approximation in the Newtonian case, and we also have seen that it is exact for some forms of [FORMULA] and a very good approximation for the most interesting forms of [FORMULA], we restrict ourselves to consider only the first term in the series.

Thus, we write:

[EQUATION]

Now, using the functional form of [FORMULA],

[EQUATION]

and applying (A6) to invert the Fourier transformation, and after some straightforward calculations, we get:

[EQUATION]

At this point, it is useful to introduce an auxiliary function [FORMULA] that makes the integrals exact:

[EQUATION]

Eq (A21) can be inserted into Eq. (A20) and, upon integration by parts, we get:

[EQUATION]

where [FORMULA] is a solution of the ordinary differential Eq. (A21) that satisfies the conditions of being an analytical function at [FORMULA], and

[EQUATION]

These conditions are easily satisfied in all the astrophysical systems. The analyticity is satisfied in the Newtonian limit, which is the behaviour we expect to recover at [FORMULA] and it can be seen that if Eq. (A23) were not satisfied, the rotation velocity would grow almost exponentially with the distance, which clearly seems to contradict the observations.

It is straightforward to see that, provided [FORMULA] is a solution of Eq. (A21), then [FORMULA] is also a solution of the same equation. Moreover, the term proportional to [FORMULA] in Eq. (A22) implies that [FORMULA] is zero at the origin. Taking all that into account, we finally obtain:

[EQUATION]

and

[EQUATION]

where we have omitted the subscript 0 everywhere as we will in what follows.

Once [FORMULA] is known, [FORMULA] can be calculated using Eq. (9). Equivalently, once [FORMULA] is known, [FORMULA] can be obtained through Eq. (3). Using these two equations together with Eq. (A24), and after some straightforward calculations, a direct relation between [FORMULA] and [FORMULA] can be obtained:

[EQUATION]

From Eq. (9) and the behaviour of [FORMULA] at the origin, Eq. (A25), it is easy to see that, at the origin, [FORMULA] will satisfy:

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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