A New Torus Generator for AGAMA

TL;DR

The paper introduces AGAMA's new torus generator, significantly broadening the range of orbital tori that can be constructed for axisymmetric galactic potentials, including highly eccentric orbits. It couples flexible Hamilton-Jacobi maps (isochrone and harmonic-oscillator) with advanced point transformations and a generating-function framework to produce accurate angle-action coordinates and to interpolate between tori, enabling efficient Schwarzschild-like modelling and rapid stream generation. A dedicated action finder yields true angle-action coordinates by anchoring to a torus through a given phase-space point, improving upon the Stäckel Fudge in accuracy and reliability. The approach is demonstrated through applications to tidal streams, notably the GD1 stream, showing how torus-based models can reproduce observed stream morphologies and constrain the Galactic potential, with Python wrappers and accessible code now available for broad use. Overall, AGAMA provides a fast, flexible, and scalable toolkit for action-based galaxy modelling, including resonant dynamics via eTorus and practical data-model comparisons for Gaia-era stellar streams.

Abstract

Code is presented that computes and exploits orbital tori for any axisymmetric gravitational potential. The code is a development of the AGAMA software package for action-based galaxy modelling and can be downloaded as the AGAMAb code library. Although coded in C++, most of its functions can be accessed from Python. We add to the package functions that facilitate confronting models with data, which involve sky coordinates, lines of sight, distances, extinction, etc. The new torus generator can produce tori for both highly eccentric and nearly circular orbits that lie beyond the range of the earlier torus-mapping code. Tori can be created by interpolation between tori at very low cost. Tori are fundamentally devices for computing ordinary phase-space coordinates from angle-action coordinates, but AGAMAb includes an action finder that returns angle-action coordinates from any given phase-space location. This action finder yields the torus through the given point, so it includes the functionality of an orbit integrator. The action finder is more accurate and reliable but computationally more costly than the widely used Staeckel Fudge. We show how AGAMAb can be used to generate sophisticated but cheap models of tidal streams and use it to analyse data for the GD1 stream. With the most recently published distances to the stream, energy and angular momentum imply that the end that must be leading is trailing, but extremely small changes to the distances rectify the problem.

Figures (16)

• Figure 1: In black an orbit produced by the orbit method of a torus and in red the result of directly integrating the equations of motion. The orange cross at upper right shows the velocities with which the torus will eventually visit the marked point, according to the method containsPoint.
• Figure 2: Plots of the logarithm of the density contributed by stars on a torus. The upper plot was made using the method density while the lower plot was made by uniformly sampling angle space at four times as many points as there are pixels in each panel.
• Figure 3: $(R,p_R)$ (left) and $(\vartheta,p_\vartheta)$ surfaces of section for the orbit plotted in Fig. \ref{['fig:t-seqs']}. The green curves are provided by a toy map that comprises the isochrone orbit of the given actions mapped into the cylindrical system by a point transformation defined by a choice of $\Delta$ and the functional relation $v(\vartheta)$ shown in Fig. \ref{['fig:thetaPsi']}
• Figure 4: A surface of section at energy $E=-0.2878$ constructed by using the method Torus::zSoS on the sequence of tori returned by the method TorusGenerator::constE. The black curves were produced by tori in that sequence with values of $J_r$ that decrease from $0.125$ to $0.001$; the broken red curve was produced by a torus interpolated between the first and third tori in the sequence for the radial action of the second torus in the sequence. The red circles are consequents produced by Runge-Kutta integration of the equations of motion from an initial condition provided by the torus with third smallest value of $J_r$. The broken green curves were produced by the toy maps of alternate tori acting alone.
• Figure 5: Testing the new action finder. In both panels an eccentric orbit in an NFW potential is plotted in red and then over-plotted in black by a time sequence obtained from a torus, with the location at $t=0$ taken to be the orbit's initial condition. The tori for the upper and lower panels were obtained by applying to the orbit's initial condition the new action finder and the standard Stäckel-Fudge action finder, respectively.