Planetary Orbital Equations in Externally-Perturbed Systems: Position
  and Velocity-Dependent Forces
  The increasing number and variety of extrasolar planets illustrates the
importance of characterizing planetary perturbations. Planetary orbits are
typically described by physically intuitive orbital elements. Here, we
explicitly express the equations of motion of the unaveraged perturbed two-body
problem in terms of planetary orbital elements by using a generalized form of
Gauss' equations. We consider a varied set of position and velocity-dependent
perturbations, and also derive relevant specific cases of the equations: when
they are averaged over fast variables (the "adiabatic" approximation), and in
the prograde and retrograde planar cases. In each instance, we delineate the
properties of the equations. As brief demonstrations of potential applications,
we consider the effect of Galactic tides. We measure the effect on the
widest-known exoplanet orbit, Sedna-like objects, and distant scattered disk
objects, particularly with regard to where the adiabatic approximation breaks
down. The Mathematica code which can help derive the equations is freely
available upon request.
