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Astron. Astrophys. 321, 703-712 (1997)

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Appendix A: the saddle-point evaluation of the density integral

The following saddle-point analysis justifies our choice of the normalization function [FORMULA] as given by Eqs. (4) and (9). We start from the definition of the density:

[EQUATION]

where the integral in the radial velocity [FORMULA] is:

[EQUATION]

here [FORMULA] and [FORMULA]. Along the positive real J -axis, the phase of the exponential function in Eq. (A1) is stationary at [FORMULA] defined by the relation [FORMULA]. Since, for [FORMULA] the quantity [FORMULA] goes to infinity uniformly in r, we can effectively set the upper integration limits in Eqs.(A1) and (A2) to infinity. Thus Eq. (14) follows.

The argument of the exponential in Eq. (A1) vanishes at [FORMULA] and can be expanded to give:

[EQUATION]

Here the function [FORMULA] is defined as:

[EQUATION]

Note that the function [FORMULA] is well behaved. Let us now introduce the scaled velocity variable [FORMULA] and further expand the argument of the integral:

[EQUATION]

with [FORMULA], [FORMULA], and [FORMULA], and we recall that [FORMULA]. This leads to the final expression:

[EQUATION]

where:

[EQUATION]

and

[EQUATION]

The two limits, [FORMULA] and [FORMULA] yield the well known normalization of an isotropic Maxwellian and the epicyclic normalization of Eq. (4), respectively. These limits justify our choice (10) for the function [FORMULA]. The limitation of this saddle-point method is recognized in the fact that the assumption of slow variations of the quantity [FORMULA] is violated in the region where [FORMULA], because the two integrals [FORMULA] and [FORMULA] are found to present large derivatives there. One possible way out that has been attempted has been to introduce an appropriate "fudge factor" to compensate for the undesired fast variations; in practice, the problem is best resolved by a contour integration, as illustrated in the following Appendix B.

Appendix B: contour integration

If we want to generate models that reproduce the chosen density distribution more accurately at small radii, we can proceed by an approximate inversion of the relationship between the density and the distribution function in the following way (as suggested to us by J. Schmidt). The method is still based on the smallness of the parameter [FORMULA], but, by means of a suitable transformation of the relevant integrals, the approximations are all carried out in one integral that does not depend on the unknown normalization factor P.

We start from Eq. (15)

[EQUATION]

where the quantities [FORMULA] and [FORMULA] are defined in Sect. 2.4. We multiply both sides by [FORMULA] and integrate them with respect to the "radial" variable

[EQUATION]

along the imaginary axis [FORMULA] at constant [FORMULA]. By exchanging the order of integration on the right hand side, this procedure leads to

[EQUATION]

[EQUATION]

The inversion formula

[EQUATION]

follows by noting that, to leading order in [FORMULA],

[EQUATION]

This result can be derived by expanding [FORMULA] for [FORMULA], provided the ratios [FORMULA] between the following coefficients and the first in the Taylor expansion be bounded.

In order to evaluate the inverse integral transform (B4) explicitly, it is more convenient to bring the integral over z along the imaginary axis into an integral over [FORMULA]. The detailed form of the integration path in the complex u -plane depends on the explicit form of the potential [FORMULA], but its main features are quite general and are governed by the contribution arising from the centrifugal potential, which dominates at [FORMULA], i.e., [FORMULA], and by the crossing of the real u -axis at [FORMULA] for [FORMULA].

The resulting ovaloid shape of the integration path in the complex u -plane is shown for the case of the isochrone potential in Fig. 5. The integrand in Eq. (B4) has neither poles nor branch-points inside this ovaloid, but has an essential singularity at [FORMULA]. In the half-plane to the left (negative real-part of u) of the essential singularity, the integrand in Eq. (B4) is exponentially small. Then, the integration path can be deformed so as to fully encircle the singularity at [FORMULA] and the inversion formula (B4) becomes

[EQUATION]

where the residue [FORMULA] at [FORMULA] is best computed numerically by integrating Eq. (B4) along a path enclosing [FORMULA] and [FORMULA] on the positive real axis such that [FORMULA]. This completes the derivation of Eq. (18).

[FIGURE] Fig. 5. Integration path in the complex u -plane, illustrated for the isochrone potential at [FORMULA]. This corresponds to the integration in the z -variable along the imaginary axis
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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