Torus construction, dynamical tomography and galactic modelling
Dynamical tomography reconstructs the potential and orbit distribution
in phase space of a large dynamical system from kinematic sample data.
For near-integrable Hamiltonian systems, an efficient tool for this
is torus construction, with which we create an integrable Hamiltonian
closely approximating any given Hamiltonian.
We use this approach in galactic modelling, where we construct the
complete dynamical model
of our Galaxy, including the distribution of dark matter,
using the data from various current
and future sky surveys.
Galaxies are, as a first approximation, near-integrable collisionless
Hamiltonian systems.
Understanding the formation, evolution and structure of galaxies is one
of the major quests of contemporary astrophysics, largely comparable
to the similar scientific quest with respect to stars in the last century.
To understand galaxies, the imminent aim is to construct comprehensive
models at several physical levels. From the dynamical point of view,
a galaxy is a collection of orbits rather than
stars. Mapping the Galaxy dynamically is
comparable to the main goal and great accomplishment of celestial mechanics,
that of understanding the dynamics of the solar system. In the same way
as the two-body Kepler orbits form the integrable ``ground state'' of
the solar system upon which the huge success of Hamiltonian perturbation
theory has been built, our goal is to define such an integrable state
for the Galaxy, and to provide the tools for efficient application of
perturbation theory.
Some references:
M. Kaasalainen (2008):
Dynamical tomography of gravitationally bound systems. Inverse Problems and Imaging, 2, 527.
M. Kaasalainen and J. Binney (1994):
Construction of invariant tori and integrable Hamiltonians.
Phys. Rev. Lett. 73, 2377.
M. Kaasalainen and J. Binney (1994):
Torus construction in potentials supporting different orbit families. Mon. Not. Roy. Ast. Soc. 268, 1033.
M. Kaasalainen (1994):
Hamiltonian perturbation theory for numerically constructed phase-space tori.
Mon. Not. Roy. Ast. Soc. 268, 1041.
M. Kaasalainen (1995):
Construction of invariant tori around closed orbits. Mon. Not. Roy. Ast. Soc. 275, 162.
M. Kaasalainen (1995):
Construction of invariant tori in chaotic regions. Phys. Rev. E 52, 1193.
M. Kaasalainen and J. Binney (1996):
Integrable Hamiltonians from close approximations
to invariant tori. Comm. Fields Inst. 10, 93.