## Appendix A: axial ratios for surfaces of equal density and triaxiality parameterAs mentioned in Sect. 3, the smooth density function of the logarithmic potential has the following form: where , , , are functions of and : It can be directly perceived from the form of Eq. (A1) that the
surfaces of equal density are not ellipsoidal. We may thus calculate
approximate axial ratios through the parameters
, and
defined as the points where the considered
isodensity cuts the corresponding with and the convention that . By resolving three quadratic equations with respect to and excluding the negative solutions, we have: Thus, for each value of the density , we are able to determine the density axial ratios. Now, we may notice that the value of the density on each axis is minimised when the corresponding spatial variable takes its maximum value. However, the values of the density for maximum values of the variables corresponding to the middle and short axes are always smaller than the value of the last isodensity cutting the long axis. Hence, the smaller possible value for an isodensity surface is given by the maximum value of the variable labelling the long axis. More specifically, the smaller possible value of the isodensity surfaces is given by the equation: and or 2 or 3 depending on which of the
spatial variables where the axial ratios are denoted as and , with , labelling the short, middle and long axes, respectively, with the previously employed convention for the correspondence between the numbers and the spatial variables. Finally, through these axial ratios we may determine a triaxiality parameter (Eq. 28): © European Southern Observatory (ESO) 1998 Online publication: December 8, 1997 |