## Appendix A: the saddle-point evaluation of the density integralThe following saddle-point analysis justifies our choice of the normalization function as given by Eqs. (4) and (9). We start from the definition of the density: where the integral in the radial velocity is: here and . Along the
positive real The argument of the exponential in Eq. (A1) vanishes at and can be expanded to give: Here the function is defined as: Note that the function is well behaved. Let us now introduce the scaled velocity variable and further expand the argument of the integral: with , , and , and we recall that . This leads to the final expression: The two limits, and 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 . The limitation of this saddle-point method is recognized in the fact that the assumption of slow variations of the quantity is violated in the region where , because the two integrals and 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 integrationIf 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
, 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
where the quantities and are defined in Sect. 2.4. We multiply both sides by and integrate them with respect to the "radial" variable along the imaginary axis at constant . By exchanging the order of integration on the right hand side, this procedure leads to follows by noting that, to leading order in , This result can be derived by expanding for , provided the ratios 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 The resulting ovaloid shape of the integration path in the complex
where the residue at is best computed numerically by integrating Eq. (B4) along a path enclosing and on the positive real axis such that . This completes the derivation of Eq. (18).
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |