Torus-fitting method for obtaining action variables in two-dimensional
  Galactic potentials
  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.
