Integrals of Motion in Kuzmin-Like Potentials

Details Integrals of Motion in Kuzmin-Like Potentials Integrals of Motion in Kuzmin-Like Potentials

Jul 2001

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2001apj...555..524t&link_type=abstract

The Astrophysical Journal, Volume 555, Issue 1, pp. 524-531.

Astronomy and Astrophysics

Astronomy

1

Celestial Mechanics, Galaxies: Kinematics And Dynamics, Galaxies: Structure, Stellar Dynamics

Scientific paper

We have identified an orthogonal curvilinear coordinate system (ξ1,ξ2,ξ3) for which surfaces of constant ξ1 exactly coincide with equipotential surfaces in the Kuzmin potential. The three principal coordinates are ξ1≡[x2+y2+(a+|z|)2]1/2, ξ2≡arctan[(|z|/z)(x2+y2)1/ 2/(a+|z|)], and ξ3≡arctan[(y/x)]. When the equation of motion is written in terms of this ``Kuzmin'' coordinate system, the integrals of motion for particles moving in Kuzmin-like potentials are easily recognized. Specifically, any time-independent potential Φ that is a function only of ξ1 will exhibit the same number of integrals of motion as does a spherically symmetric potential. In such Kuzmin-like potentials, the vector J≡e1ξ1Xv is the analog of the specific angular momentum vector in spherically symmetric potentials. Furthermore, in potentials of the form Φ~1/ξ1, all three Cartesian components of the vector L≡[vXJ+e1(ξ1Φ)] (an analog of the Laplace-Runge-Lenz vector) also prove to be integrals of the motion. For two specific potentials, the Kuzmin potential and a logarithmic Kuzmin-like potential, we have derived analytical expressions defining the surfaces of section for particles in polar orbits, and in both cases we have identified the domain of occupancy of box orbits, loop orbits, and periodic orbits.

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