Orbits in elementary, power-law galaxy bars: 1. Occurence and role of
  single loops
  Orbits in galaxy bars are generally complex, but simple closed loop orbits
play an important role in our conceptual understanding of bars. Such orbits are
found in some well-studied potentials, provide a simple model of the bar in
themselves, and may generate complex orbit families. The precessing, power
ellipse (p-ellipse) orbit approximation provides accurate analytic orbit orbits
in symmetric galaxy potentials. It remains useful for finding and fitting
simple loop orbits in the frame of a rotating bar with bar-like and symmetric
power-law potentials. Second order perturbation theory yields two or fewer
simple loop solutions in these potentials. Numerical integrations in the
parameter space neighborhood of perturbation solutions reveal zero or one ac-
tual loops in a range of such potentials with rising rotation curves. These
loops are embedded in a small parameter region of similar, but librating
orbits, which have a subharmonic frequency superimposed on the basic loop.
These loops and their librat- ing companions support annular bars. Solid bars
can be produced in more complex potentials, as shown by an example with
power-law indices varying with radius. The power-law potentials can be viewed
as the elementary constituents of more complex potentials. Numerical
integrations also reveal interesting classes of orbits with multiple loops. In
two-dimensional, self-gravitating bars, with power-law potentials, single loop
orbits are very rare. This result suggests that gas bars or oval distortions
are unlikely to be long-lived, and that complex orbits or three-dimensional
structure must support self-gravitating stellar bars.
