A hydrodynamic analogy of the solution to double-averaged Hill problem

Abstract

PRELIMINARY REMARKSA geometric pattern of integral curves describing the evolution of the elements of elliptic orbits under the action of gravitational perturbations from external bodies was studied in great detail in a series of papers by V.I. Prokhorenko (see [4][5][6]). It corresponded to the solution of doubly averaged system of evolution equations [1] (in the following, "the solution of averaged Hill problem"). Comparing this pattern with the pattern of stream lines of two-dimensional eddy motion of a fluid in a circular domain [7,8] we can see a close resemblance. More detailed analysis allows one to establish not only geometric but also kinematic similarity of these patterns. This suggests that there exists an analogy between two mentioned physical phenomena, different in their nature. A proof of the existence of such an analogy and an analysis of some of its significant consequences would allow one to supplement and widen the known celestial-mechanical facet of the problem.The geometric studies are based on the analytical solutions to an approximate system of equations describing the motion of point P of zero mass in the field of gravitation of central point S with mass M under the action of gravitation perturbations of the third point J with mass M 1 . Let us present some fragments of geometric representation of solutions [1] proposed in [4][5][6]. We introduce a spherical frame of reference O ε i ω , where the radius is ε = 1 -e 2 (0 ≤ ε ≤ 1) , where e is eccentricity, i (colatitude) is the inclination of the orbit (0° ≤ i ≤ 180°), and ω (longitude) is the argument of the pericenter ( 0° ≤ ω ≤ 360° ). Angular elements are measured with respect to the plane of the orbit of a perturbing point.In the corresponding Cartesian coordinate system OXYZ the plane OXY coincides with the plane i = 90° , plane OXZ corresponds to the values ω = 0°, ω = 180° , and plane OYZ corresponds to values ω = 90°, ω = 270° , and the OX axis lies in the meridian plane ω = 0° . The range of possible values of parameters ε , i , and ω in this frame of reference is limited by the spherical surface of unit radius. The region of evolution of orbital elements determines the following first integrals obtained in [1]: c 1 = ε cos 2 i ; c 2 = (1 -ε )(2/5 -sin 2 ω sin 2 i ). Let us use as parameters the integral constants c 1 (0 ≤ c 1 ≤ 1) and c 2 (-0.6 ≤ c 2 ≤ 0.4) determined by initial values of the orbital moments ε 0 , i 0 , and ω 0 : c 1 = ε 0 cos 2 i 0 ; c 2 = (1 -ε 0 )(2/5 -sin 2 ω 0 sin 2 i 0 ). An example of integral curves corresponding to the solution [1] to averaged Hill problem at c 1 = 0.4 is presented in the projection onto the OXY plane in Fig. 1. When c 2 > 0, the orbit evolution has a rotational character (it is determined by closed curves going around the center of the circle). When c 2 < 0, the evolution has libration character (determined by closed curves located eccentrically).Above we wrote about the similarity of the integral curves of a solution of the Hill problem and the stream lines for some class of motions of...

Hydrodynamic Analogy for the Solution to the Doubly Averaged Hill Problem Disturbed by Precession of the Orbit Plane of a Disturbing Body

“…Summarizing the results presented above, it is possible to state that the introduced hydrodynamic analogy allows us to expand the boundaries of applicability of the analogy [1]. …”

“…In [1], a hydrodynamic analogy of the analytical solution of a satellite version of the doubly averaged three-body problem obtained by M.L. Lidov [2] (hereafter, for brevity, the Lidov's solution) is described.…”

“…We take as the basis the model introduced in [1] corresponding to fictitious homogeneous fluid. We subject this model to a certain modification permitting to reproduce better the general picture of orbit evolution.…”

“…-vanishing of the radial velocity V ε on the outer boundary C 0 of the area, i.e., at ε = 1; RABINOVICH, PROKHORENKO -vanishing of the function itself on the boundary C 0 ; -equality of the tangential velocity V θ and velocity V θ (5)* from [1] on the boundary C 0 at point θ = 0 at ε = 1 (here and below, formulas from [1] are designated by index * ); -vanishing of tangential velocity V θ on the boundary C 0 at the same conjugate points θ = , as V θ (5)*; θ 1 2 , * -vanishing of the function at point with coordinate -ε ° close to -; -the existence of a logarithmic singularity of the function at the point with coordinate -ε * close to -ε L .…”

“…The solution of the formulated boundary value problem can be presented, as in [1], in the form of a flow function (imaginary part of the complex potential, whose real part is the potential of velocities). In the case under consideration, these functions have the following form:…”