Two-parametric families of orbits created by three-dimensional galactic-type potentials

Abstract

One of the main problems of astrophysics is to determine the mean field potential of galaxies. The potentials describe the motion in the central region of the galaxy. We can obtain families of star orbits in a galaxy from the astronomical observations. In the present paper, using the tools of the 3D Inverse problem of Dynamics, we study three-dimensional potentials of the form $V=\mathcal{A}(x^{2}+py^{2}+qz^{2})$, ($\mathcal{A}$ is an arbitrary function and $p, \; q=const.$), which are compatible with a preassigned two-parametric family of spatial regular curves $f(x,y,z)=c_{1}$, $g(x,y,z)=c_{2}$ ($c_{1}, c_{2}=const$). We establish three differential conditions which are fulfilled by the “slope functions” $\alpha (x,y,z)$, $\beta (x,y,z)$ corresponding to the above two-parametric family of orbits. If these conditions are satisfied, then we can find such a potential by quadratures. We offer pertinent examples of potentials which are mainly used in Galactic Dynamics, e.g. axisymmetric potentials, the potential of a homogeneous sphere, potentials coming from power law density profile and potentials which represent the central region of a perturbed triaxial galaxy. Special cases are also examined. Finally, potentials producing families of straight lines are taken into account. Two-dimensional potentials is a special category and it is studied separately.