Basis function expansions for galactic dynamics: Spherical versus cylindrical coordinates

Abstract

Aims. The orbital structure of galaxies is strongly influenced by the accuracy of the force calculation during orbit integration. We explore the accuracy of force calculations for two expansion methods and determine which one is preferable for orbit integration. Methods. We specifically compare two methods, one was introduced by Hernquist & Ostriker (HO), which uses a spherical coordinate system and was built specifically for the Hernquist model, and the other by Vasiliev & Athanassoula (CylSP) has a cylindrical coordinate system. Our comparisons include the Dehnen profile, its triaxial extension (of which the Hernquist profile is a special case) and a multicomponent system including a bar and disk density distributions for both analytical models and N-body realizations. Results. For the generalized Dehnen density, the CylSP method is more accurate than the HO method for nearly all inner power-law indices and shapes at all radii. For N-body realizations of the Dehnen model, or snapshots of an N-body simulation, the CylSP method is more accurate than the HO method in the central region for the oblate, prolate, and triaxial Hernquist profiles if the particle number is more than 5 × 105. For snapshots of the Hernquist models with spherical shape, the HO method is preferred. For the Ferrers bar model, the force from the CylSP method is more accurate than the HO method. The CPU time required for the initialization of the HO method is significantly shorter than that for the CylSP method, while the HO method costs subsequently much more CPU time than the CylSP method if the input corresponds to particle positions. From surface of section analyses, we find that the HO method creates more chaotic orbits than the CylSP method in the bar model. This could be understood to be due to a spurious peak in the central region when the force is calculated with the HO expansion. Conclusions. For an analytical model, the CylSP method with an inner cutoff radius of interpolation Rmin as calculated by the AGAMA software, is preferred due to its accuracy. For snapshots or N-body realizations not including a disk or a bar component, a detailed comparison between these two methods is needed if a density model other than the Dehnen model is used. For multicomponent systems, including a disk and a bar, the CylSP method is preferable.