tokens: in the bottom panel we show distributions as a function of R and φ . Figure 2. The – plane coloured by mean φ for the simulation described in Section 2.2.1 . The bins have size Δ = 0.05 kpc, and Δ = 3 km s −1 . The bins with fewer than 10 particles are ignored. Fig. 2 shows the – plane coloured by mean φ . Stars at large radii in the – plane, i.e. stars with high E preferrentially come from stars in population 1, which have lower φ , as seen in the bottom panel of Fig. 1 . The initial DF shown in Fig. 2 is perturbed by displacing it by in . The particles are then evolved using a standard leap-frog algorithm for and the results are shown in Fig. 3 . The number density shows the spiral pattern typical of phase wrapping. Stars orbit the – plane in a clockwise sense and since those at large E have a lower vertical epicycle frequency, the DF quickly develops a trailing spiral. Also shown is the – plane coloured by mean R and φ . Here, the pattern is essentially featureless, apart from the imprint of the number density distribution: with our separable potential the in-plane and vertical epicyclic motions completely decouple. Figure 3. Number density and mean radial and azimuthal velocities in the – plane for the toy model considered in Section 2.2.1 (toy model with additively separable potential). Top panel shows the number density. The bins in the middle and lower panels are coloured by R and φ , respectively. The bin sizes and particle number threshold are the same as in Fig. 2 . 2.2.2 Kinematic spirals in a coupled potential Next we consider the addition of a R – term to the toy model potential R s is a constant radius characteristic of the distribution [if a Taylor expansion of the effective potential is performed for each L population, R s is replaced by R c ( L )]. For illustrative purposes, we choose and . These values are consistent with the cubic terms that arise in realistic Milky Way potentials. The main effect of the new term is to introduce a linear dependence in R to the vertical epicyclic frequency. In particular the difference between the vertical frequency for a population 1 star with and a population 2 star with is . As before we sample an initial distribution (equation 3 ) with 500k particles, perturb it and integrate the perturbed distribution for . The results are shown in Fig. 4 . Not surprisingly, the spirals that appear in the – number density distribution are virtually identical to the ones that appear in our separable model. However, the R and φ spirals are qualitatively different. In particular, there are gradients across the spiral arms and along their inner and outer edges. The addition of the ( R − R s ) 2 term implies a coupling between the in-plane and vertical epicyclic motions. Figure 4. Same as Fig. 3 but for the model considered in Section 2.2.2 (toy model with R – coupling included in the potential). 2.3 Realistic Milky Way potential We conclude this section on test-particle simulations by considering a realistic
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