Resonant Stellar Orbits in Spiral Galaxies

This paper describes an epicyclic theory of the orbits of stars near the Lindblad resonances of a spiral galaxy. The spiral component of the gravitational field is assumed to be stationary, in a uniformly rotating frame of reference, and tightly wound. The equations of motion are solved by the method of harmonic balance of Bogoliubov and Mitropolsky. The solution is obtained explicitly to the lowest order in the epicyclic amplitude, the spacing of the spiral arms, and the amplitude of the spiral field. The solution represents families of periodic orbits, tube orbits, and nonresonant orbits in excellent qualitative agreement with the results of numerical investigations by Contopoulos. Integrals of the motion are obtained in a representation suitable for the extension of the theory of density waves into the regions of the Lindblad resonances. Subject headings: galactic structure - galaxies - stellar dynamics