Probing a regular orbit with spectral dynamics

We have extended the spectral dynamics formalism introduced by Binney & Spergel, and have implemented a semi‐analytic method to represent regular orbits in any potential, making full use of their regularity. We use the spectral analysis code of Carpintero & Aguilar to determine the nature of an orbit (irregular, regular, resonant, periodic) from a short‐time numerical integration. If the orbit is regular, we approximate it by a truncated Fourier time series of a few tens of terms per coordinate. Switching to a description in action‐angle variables, this corresponds to a reconstruction of the underlying invariant torus. We then relate the uniform distribution of a regular orbit on its torus to the non‐uniform distribution in the space of observables by a simple Jacobian transformation between the two sets of coordinates. This allows us to compute, in a cell‐independent way, all the physical quantities needed in the study of the orbit, including the density and the line‐of‐sight velocity distribution, with much increased accuracy. The resulting flexibility in the determination of the orbital properties, and the drastic reduction of storage space for the orbit library, provide a significant improvement in the practical application of Schwarzschild's orbit superposition method for constructing galaxy models. We test and apply our method to two‐dimensional orbits in elongated discs, and to the meridional motion in axisymmetric potentials, and show that for a given accuracy, the spectral dynamics formalism requires an order of magnitude fewer computations than the more traditional approaches.