Actions of highly eccentric orbits

TL;DR

This work tackles the challenge of calculating actions for highly eccentric, axisymmetric orbits by analyzing how centrifugal barriers and a critical action $J_{z,\mathrm{crit}}(E)$ separate box and loop orbits, even when $J_\phi$ is nonzero. It develops an algorithm to determine $J_{z,\mathrm{crit}}(E)$ and $I_{3,\mathrm{crit}}(E)$, and applies these concepts to improve the Stäckel Fudge via a dynamic focal distance $\Delta(I_3)$ to better place barriers near the transition. The study reveals a notable failure mode of the Stäckel Fudge near $J_{z,\mathrm{crit}}$, proposes a practical workaround, and analyzes the resulting impact on both actions and orbital frequencies, finding partial improvement for $J_r$ and limited gains for frequencies. The findings enhance torus mapping and the construction of distribution functions in flattened galactic potentials, with the AGAMA implementation providing a robust tool for action-angle analysis in realistic systems.

Abstract

The challenge presented by computing actions for eccentric orbits in axisymmetric potentials is discussed. In the limit of vanishing angular momentum about the potential's symmetry axis, there is a clean distinction between box and loop orbits. We show that this distinction persists into the regime of non-zero angular momentum. In the case of a Staeckel potential, there is a critical value I_{3crit}(E) of the third integral I_3 below which I_3 does not contribute to the centrifugal barrier. An orbit is of box or loop type according as its value of I_3 is smaller or greater than I_{3crit}. We give algorithms for determining I_{3crit}(E) and the critical action Jzcrit below which orbits in any given potential are boxes. It is hard to compute the actions and especially the frequencies of orbits that have Jz ~ Jzcrit using the Staeckel Fudge. A modification of the Fudge that alleviates the problem is described.

Figures (14)

• Figure 1: Surface of section at $E=\Phi(0)/2$ and $J_\phi=0$ in the potential of a perfect oblate ellipsoid with scale lengths $a=1$ and $c=0.6$. At this energy the circular angular momentum is $L_{\rm circ}=0.663$.
• Figure 2: The ratio of vertical to horizontal frequencies as a function of circularity $J_z/(J_r+J_z)$ for the orbits with the energy of Fig. \ref{['fig:StackSoS']}. The four curves are for $J_\phi=0$ (black), $J_\phi=0.02$ (red), $J_\phi=0.1$ (green) and $J_\phi=0.2$ (blue). We plot $\Omega_z/2\Omega_x$ for box orbits and $\Omega_z/\Omega_R$ for loop orbits.
• Figure 3: Surface of section at $J_\phi=0.02$ for the orbits with the energy of Fig. \ref{['fig:StackSoS']}.
• Figure 4: The analogues of Figs. \ref{['fig:StackSoS']} and \ref{['fig:StackSoSFreq']} at a higher energy, namely $E=\Phi(0)/4$.
• Figure 5: Frequency ratios for orbits of low $J_\phi$ in a flattened perfect ellipsoid. All orbits pass through $(R,z)=(3a,0)$, where $a$ is the scale radius of the ellipsoid. In the case of the top full curve in each panel the velocity there is $(v_R,v_z,v_\phi)=(V_m\sin\theta,V_m\cos\theta,0.01V_c)$ where $V_c$ is the circular speed at $R$ and $V_m$ yields the energy of the circular orbit at $R$. The $n^{\rm th}$ broken curves below the top curve on the left corresponds to $V_\phi=(0.01+n/10)V_c$.