Orbital Tori Research Articles

Orbital tori for non-axisymmetric galaxies

Our Galaxy's bar makes the Galaxy's potential distinctly non-axisymmetric. All orbits are affected by non-axisymmetry, and significant numbers are qualitatively changed by being trapped at a resonance with the bar. Orbital tori are used to compute these effects. Thick-disc orbits are no less likely to be trapped by corotation or a Lindblad resonance than thin-disc orbits. Perturbation theory is used to create non-axisymmetric orbital tori from standard axisymmetric tori, and both trapped and untrapped orbits are recovered to surprising accuracy. Code is added to the TorusModeller library that makes it as easy to manipulate non-axisymmetric tori as axisymmetric ones. The augmented TorusModeller is used to compute the velocity structure of the solar neighbourhood for bars of different pattern speeds and a simple action-based distribution function. The technique developed here can be applied to any non-axisymmetric potential that is stationary in a rotating from - hence also to classical spiral structure.

Distribution functions for Galactic disc stellar populations in the presence of non-axisymmetric perturbations

AbstractThe present-day response of a Galactic disc stellar population to a non-axisymmetric perturbation of the potential, in the form of a bar or spiral arms, can be treated, away from the main resonances, through perturbation theory within the action-angle coordinates of the unperturbed axisymmetric system. The first order moments of such a perturbed distribution function (DF) in the presence of spiral arms give rise to non-zero radial and vertical mean stellar velocities, called breathing modes. Such an Eulerian linearized treatment however diverges at resonances. The Lagrangian approach to the impact of non-axisymmetries at resonances avoids this problem. It is based on the construction of new orbital tori in the resonant trapping region, which come complete with a new system of angle-action variables. These new tori can be populated by phase-averaging the unperturbed DF over the new tori. This boils down to phase-mixing the DF in terms of the new angles, such that the DF for trapped orbits only depends on the new set of actions. This opens the way to quantitatively fitting the effects of the bar and spirals to Gaia data with an action-based DF.

Choice of high-apogee AES orbits on the basis of the qualitative methods of the theory of perturbations and situational analysis. Part I. Situational studies based on orbital tori

The paper discusses the problems of the choice of high-apogee orbits of artificial Earth satellites (AES), proceeding from the tasks of space experiments aimed at studying near-earth space and taking into account the features of the orbital evolution and ballistic lifetime. The suggested methods of the choice of orbits consist of two components. The first is based on the use of mathematical models of studied regions of near-earth space and various techniques of situation analysis, among which the annual and daily orbital tori developed by the author about 35 years ago are key. The second component is based on qualitative methods of the theory of perturbations of high-apogee AES orbits developed by M.L. Lidov more than 50 years ago.

Torus mapper: a code for dynamical models of galaxies

We present a freely downloadable software package for modelling the dynamics of galaxies, which we call the Torus Mapper (TM). The package is based around `torus mapping', which is a non-perturbative technique for creating orbital tori for specified values of the action integrals. Given an orbital torus and a star's position at a reference time, one can compute its position at any other time, no matter how remote. One can also compute the velocities with which the star will pass through any given point and the contribution it will make to the time-averaged density there. A system of angle-action coordinates for the given potential can be created by foliating phase space with orbital tori. Such a foliation is facilitated by the ability of TM to create tori by interpolating on a grid of tori. We summarise the advantages of using TM rather than a standard time-stepper to create orbits, and give segments of code that illustrate applications of TM in several contexts, including setting up initial conditions for an N-body simulation. We examine the precision of the orbital tori created by TM and the behaviour of the code when orbits become trapped by a resonance.

Analysing surveys of our Galaxy - I. Basic astrometric data

We consider what is the best way to extract science from large surveys of the Milky Way galaxy. The diversity of data gathered in these surveys, together with our position within the Galaxy, imply that science must be extracted by fitting dynamical models to the data in the space of the observables. Models based on orbital tori promise to be superior for this task than traditional types of models, such as N-body models and Schwarzschild models. A formalism that allows such models to be fitted to data is developed and tested on pseudodata of varying richness.

Models of our Galaxy - II

Stars near the Sun oscillate both horizontally and vertically. In Paper I the coupling between these motions was modelled by determining the horizontal motion without reference to the vertical motion, and recovering the coupling by assuming that the vertical action is adiabatically conserved as the star oscillates horizontally. Here we show that, although the assumption of adiabatic invariance works well, more accurate results can be obtained by taking the vertical action into account when calculating the horizontal motion. We use orbital tori to present a simple but fairly realistic model of the Galaxy's discs in which the motion of stars is handled rigorously, without decomposing it into horizontal and vertical components. We examine the ability of the adiabatic approximation to calculate the model's observables, and find that it performs perfectly in the plane, but errs slightly away from the plane. When the new correction to the adiabatic approximation is used, the density, streaming velocity and velocity dispersions are in error by less than 10 per cent for distances up to $2.5\kpc$ from the Sun. The torus-based model reveals that at locations above the plane the long axis of the velocity ellipsoid points almost to the Galactic centre, even though the model potential is significantly flattened. This result contradicts the widespread belief that the shape of the Galaxy's potential can be strongly constrained by the orientation of velocity ellipsoid near the Sun. An analysis of orbits reveals that in a general potential the orientation of the velocity ellipsoid depends on the structure of the model's distribution function as much as on its gravitational potential, contrary to what is the case for Staeckel potentials. We argue that the adiabatic approximation will provide a valuable complement to torus-based models in the interpretation of current surveys of the Galaxy.

Spectral Decomposition of Orbital Tori

The Kolmogorov―Arnold―Moser theorem states that lightly perturbed integrable Hamiltonian systems maintain their multiply periodic, toroidal motion in the phase space. The assertion that Earth-orbiting satellites under the influence of the geopotential mimic this behavior is the underlying premise of this work. This paper focuses on applying trajectory-following spectral methods on selected orbits to decompose them into multiperiodic Fourier series, effectively compressing ephemerides for long-term use. The proposed approach focuses on fitting local spectral structures, denoted as frequency clusters, within the sampled orbital data to the analytical form of the windowed, truncated, continuous Fourier transform. This approach is significantly more numerically efficient than fitting every coefficient within the N-tuple Fourier series simultaneously. Numerical simulations using integrated data yield root-mean-square error in orbital tori fits at 10 m or less per coordinate axis over a one-year period for most low-inclination, low-eccentricity orbits with altitudes lower than 1900 km.

Modelling the Galaxy with orbital tori

AbstractThe principles and advantages of torus modelling are explained.

Angle variables for numerically fitted orbital tori

Angle variables are constructed for orbital tori least-squares fitted to general potentials by the method of McGill & Binney . These angle variables enable one to determine the densities ⍴J(x) associated with the orbit that has given actions J. They also make it possible to treat any non-integrable potential as a perturbation on a nearby integrable one. As an illustration of this approach, Hamiltonian perturbation theory is used to derive the width of the 1:1 resonant-orbit family in a realistic model of the potential of a disk galaxy.

Situation studies of aes orbits for solar-Earth space experiments

The questions on the orbit selection for studying various areas of the Earth magnetosphere are discussed. The method of the orbit selection is based on the use of orbital tori and on the construction of areas of orbit set intersection with model surfaces, forming specific areas in the near Earth space: bow shock (BSh), magnetopause (MP) neutral sheet in the magnetosphere tail, cusp etc. Intersections of the “Prognoz” 1–8 orbit set with model surfaces BSh and MP are shown.