Publication Abstracts

Contopoulos and Moutsoulas 1965

Contopoulos, G., and M. Moutsoulas, 1965: Resonance cases and small divisors in a third integral of motion. II. Astron. J., 70, 817-835, doi:10.1086/109822.

This paper contains a complete description of the resonance case A=B, i.e., when the unperturbed frequencies in two perpendicular directions are equal. The form of the third integral is different from that of the nonresonance case. The secular terms are eliminated, step by step, and higher-order terms are calculated by means of a computer. The third integral is better conserved in actual orbits when more higher-order terms are included in it. The invariant curves give the main characteristics of the orbits. The theoretical invariant curves represent sufficiently well the empirically found invariant curves (by means of orbital calculations) when terms up to the fourth degree in the perturbation parameter E are included. A complete classification of the orbits can be achieved even by using the zero-order terms of the third integral. For accurate numerical results, however, we need to include terms up to the second or even to the fourth order, especially in the case that E approaches the value for which the curve of zero velocity opens and the moving point may go to infinity.

There are three main types of orbits: the A-, B-, and C-type orbits. Their boundaries are calculated numerically and some characteristic points are found by means of the third integral numerically or by series expansions. A detailed comparison between theory and numerical experiments gives always good agreement when sufficient terms of the third integral are included. Five periodic orbits have been found, three stable and two unstable. The transition types have also been discussed in detail.

These calculations are applied to the galactic orbits on the plane of symmetry of a distorted (nonaxisymmetric) galaxy. If the distortion is of the order of 20%, we find that the circular orbits become almost rectilinear through the central region and then reverse the sense of rotation in a few billion years.

A second application refers to the energy exchange between two coupled oscillators. The third integral predicts the correct amount of energy exchange. In one case the energy of one of the oscillators varies between 0 and 2/3 of the total energy.

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BibTeX Citation

@article{co05200y,
  author={Contopoulos, G. and Moutsoulas, M.},
  title={Resonance cases and small divisors in a third integral of motion. II},
  year={1965},
  journal={Astronomical Journal},
  volume={70},
  pages={817--835},
  doi={10.1086/109822},
}

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RIS Citation

TY  - JOUR
ID  - co05200y
AU  - Contopoulos, G.
AU  - Moutsoulas, M.
PY  - 1965
TI  - Resonance cases and small divisors in a third integral of motion. II
JA  - Astron. J.
JO  - Astronomical Journal
VL  - 70
SP  - 817
EP  - 835
DO  - 10.1086/109822
ER  -

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