Astron. Astrophys. 329, 451-481 (1998)

Appendix A: axial ratios for surfaces of equal density and triaxiality parameter

As 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 x, y and z axis, respectively. Unfortunately, these parameters are not unique, i.e. they are parameterized by the value of the considered isodensity. Indeed, by nullifying successively two out of three values of the spatial variables we have the following equations for the density function:

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 x, y or z labels the long axis of the system, following the values of and . In that way better estimates of the isodensity axial ratios can be determined:

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
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