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Register a new userAngle-action variables for orbits trapped at a Lindblad resonance
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Authors
James Binney
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The conventional approach to orbit trapping at Lindblad resonances via a pendulum equation fails when the parent of the trapped orbits is too circular. The problem is explained and resolved in the context of the Torus Mapper and a realistic Galaxy model. Tori are computed for orbits trapped at both the inner and outer Lindblad resonances of our Galaxy. At the outer Lindblad resonance, orbits are quasiperiodic and can be accurately fitted by torus mapping. At the inner Lindblad resonance, orbits are significantly chaotic although far from ergodic, and each orbit explores a small range of tori obtained by torus mapping.
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In this paper we develop a general conceptual approach to the problem of existence of action-angle variables for dynamical systems, which establishes and uses the fundamental conservation property of associated torus actions: anything which is preserved by the system is also preserved by the associated torus actions. This approach allows us to obtain, among other things: a) the shortest and most conceptual easy to understand proof of the classical Arnold--Liouville--Mineur theorem; b) basically all known results in the literature about the existence of action-angle variables in various contexts can be recovered in a unifying way, with simple proofs, using our approach; c) new results on action-angle variables in many different contexts, including systems on contact manifolds, systems on presymplectic and Dirac manifolds, action-angle variables near singularities, stochastic systems, and so on. Even when there are no natural action variables, our approach still leads to useful normal forms for dynamical systems, which are not necessarily integrable.
Torus mapping yields constants of motion for stars trapped at a resonance. Each such constant of motion yields a system of contours in velocity space at the Sun and neighbouring points. If Jeans theorem applied to resonantly trapped orbits, the density of stars in velocity space would be equal at all intersections of any two contours. A quantitative measure of the violation of this principal is defined and used to assess various pattern speeds for a model of the bar recently fitted to observations of interstellar gas. Trapping at corotation of a bar with pattern speed in the range 33-36 /Gyr is favoured and trapping at the outer Lindblad resonance is disfavoured. As one moves around the Sun the structure of velocity space varies quite rapidly, both as regards the observed star density and the zones of trapped orbits. The data seem consistent with trapping at corotation.
Galaxy modelling is greatly simplified by assuming the existence of a global system of angle-action coordinates. Unfortunately, global angle-action coordinates do not exist because some orbits become trapped by resonances, especially where the radial and vertical frequencies coincide. We show that in a realistic Galactic potential such trapping occurs only on thick-disc and halo orbits (speed relative to the guiding centre >~80 km/s). We explain how the Torus Mapper code (TM) behaves in regions of phase space in which orbits are resonantly trapped, and we extend TM so trapped orbits can be manipulated as easily as untrapped ones. The impact that the resonance has on the structure of velocity space depends on the weights assigned to trapped orbits. The impact is everywhere small if each trapped orbit is assigned the phase space density equal to the time average along the orbit of the DF for untrapped orbits. The impact could be significant with a different assignment of weights to trapped orbits.
Starting from the action-angle variables and using a standard asymptotic expansion, here we present a new derivation of the Wave Kinetic Equation for resonant process of the type $2leftrightarrow 2$. Despite not offering new physical results and despite not being more rigorous than others, our procedure has the merit of being straightforward; it allows for a direct control of the random phase and random amplitude hypothesis of the initial wave field. We show that the Wave Kinetic Equation can be derived assuming only initial random phases. The random amplitude approximation has to be taken only at the end, after taking the weak nonlinearity and large box limits. This is because the $delta$-function over frequencies contains the amplitude-dependent nonlinear correction which should be dropped before the random amplitude approximation applies. If $epsilon$ is the small parameter in front of the anharmonic part of the Hamiltonian, the time scale associated with the Wave Kinetic equation is shown to be $1/epsilon^2$. We give evidence that random phase and amplitude hypotheses persist up to a time of the order $1/epsilon$.
A phase-space distribution function of the steady state in galaxy models that admits regular orbits overall in the phase-space can be represented by a function of three action variables. This type of distribution function in Galactic models is often constructed theoretically for comparison of the Galactic models with observational data as a test of the models. On the other hand, observations give Cartesian phase-space coordinates of stars. Therefore it is necessary to relate action variables and Cartesian coordinates in investigating whether the distribution function constructed in galaxy models can explain observational data. Generating functions are very useful in practice for this purpose, because calculations of relations between action variables and Cartesian coordinates by generating functions do not require a lot of computational time or computer memory in comparison with direct numerical integration calculations of stellar orbits. Here, we propose a new method called a torus-fitting method, by which a generating function is derived numerically for models of the Galactic potential in which almost all orbits are regular. We confirmed the torus-fitting method can be applied to major orbit families (box and loop orbits) in some two-dimensional potentials. Furthermore, the torus-fitting method is still applicable to resonant orbit families, besides major orbit families. Hence the torus-fitting method is useful for analyzing real Galactic systems in which a lot of resonant orbit families might exist.
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