The angles result from the derivatives of the generating function W(x,J) w.r.t. the actions (A.1)(Binney & Tremaine 2008, Eq. (3.204)). The complete generating function for a spherical system is given by (modified from Eq. (3.220), Binney & Tremaine 2008) (A.2)where we integrate over the particles trajectory in phase space, which means for example that during a whole radial period, the particle transverses twice the branch from pericentre to apocentre, but once in reversed direction. The latitudinal momentum pϑ is (A.3)where cosi = Lz/L. The radial momentum pr is (A.4)where L = Jθ + | Jφ | and Jφ ≥ 0 is assumed8.
We find the radial angle as (A.5)where we note that the angles are always defined modulo 2π because there is no information on how many loops a particle made around the Galactic centre. The derivative of the generating function is given by (A.6)where the conditions on pr are necessary to take the right branch of f1. After one full period, we find θr = 2π as expected.
The function Wϑ,Jφ is oscillatory in nature (w.r.t. ϑ), while the combination Wr,Jφ + ΩφWr,H is also oscillatory (w.r.t. r): after one radial period it evaluates to −2Lf2(rapo) + 2Ωφf1(rapo) = 0.
The latitudinal angle is given by (A.11)Because we assume Jφ ≥ 0, we find Wr,Jϑ = Wr,Jφ and Ωϑ = Ωφ. The remaining derivative of the generating function is (A.12)where the term Wr,Jϑ + ΩϑWr,H vanishes after one radial period, while the term Wϑ,Jϑ contains the dependence on ϑ and increases by 2π after one period in ϑ.
The linearised transformation between action-angle coordinates and Cartesian coordinates is (B.1)For simplicity, we provide here the matrix for the 2D case (i.e. when the orbit is in the plane). In that case, Jθ = 0 and Jφ = L = Lz, so that we find (B.2)where the above terms are given by (B.7)The functions κ and η are and the WJi,Jj are found by differentiating the generating function
© ESO, 2015