Isolating integrals of the motion for stellar orbits in a rotating galactic bar

Abstract

The study of the equilibrium of a rotating galactic bar requires an enumeration of the isolating integrals of the motion of a star in the prevailing gravitational field. In general, Jacobi's integral is the only exact isolating integral known. This paper describes a search for an additional isolating integral for orbits confined to a plane perpendicular to the axis of the bar's rotation. It is shown that, in general, the equations of motion admit an additional integral exactly which is a nonhomogeneous quadratic form in the momenta of the star only if (1) the gravitational potential is axisymmetric, (2) the gravitational potential is harmonic, or (3) the bar does not rotate and the gravitational potential is separable in elliptic coordinates. A formal integral of the motion is constructed for orbits in a slightly anharmonic potential. Numerical solutions of the equations of motion for orbits in a slightly anharmonic potential behave as if there were indeed an additional isolating integral, and that behavior is represented very well in terms of the formal integral. If the rotation of the bar is rapid and/or the nonaxisymmetry of the bar is weak, then the additional integral restricts the motion of a star in muchmore » the same way that the angular momentum restricts motion in an axisymmetric potential. Conversely, if the rotation of the bar is slow and/or the nonaxisymmetry of the bar is strong, then the additional integral restricts the motion in much the same way that the difference of the separable energies would if the motion were separable in Cartesian coordinates.« less