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Orbit classification in the Hill problem: I. The classical case

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Abstract

The case of the classical Hill problem is numerically investigated by performing a thorough and systematic classification of the initial conditions of the orbits. More precisely, the initial conditions of the orbits are classified into four categories: (i) non-escaping regular orbits; (ii) trapped chaotic orbits; (iii) escaping orbits; and (iv) collision orbits. In order to obtain a more general and complete view of the orbital structure of the dynamical system, our exploration takes place in both planar (2D) and the spatial (3D) version of the Hill problem. For the 2D system, we numerically integrate large sets of initial conditions in several types of planes, while for the system with three degrees of freedom, three-dimensional distributions of initial conditions of orbits are examined. For distinguishing between ordered and chaotic bounded motion, the Smaller Alignment Index method is used. We managed to locate the several bounded basins, as well as the basins of escape and collision and also to relate them to the corresponding escape and collision time of the orbits. Our numerical calculations indicate that the overall orbital dynamics of the Hamiltonian system is a complicated but highly interested problem. We hope our contribution to be useful for a further understanding of the orbital properties of the classical Hill problem.

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Notes

  1. Sticky orbits need an extremely long time interval in order to move away from the invariant sticky tori [36]. Therefore, sticky orbits behave, for long time interval, as regular ones before revealing their true chaotic nature.

  2. When we state that an area is fractal we simply suggest that it has a fractal-like geometry without conducting any specific calculations regarding the fractal dimensions as in [3].

  3. The two saddle points \(L_1\) and \(L_2\) lie along the x axis and the zero velocity surfaces are symmetrical with respect to the three axes. It is easy to prove that if (x(t), y(t), z(t)) is a solution of the Hill problem, \((-x(t), -y(t), -z(t))\) is also a solution, of course with appropriately chosen momenta (because of the presence of the Coriolis force). Therefore, for every orbit escaping through \(L_1\) there is a symmetry-related orbit escaping through \(L_2\), because the dynamics of the system must be symmetrical. On this basis, the observed preference for escape through exit channel 2 is due to the particular choice of the initial conditions. Similar choices of initial conditions of orbits can be found in several earlier related papers (e.g., [13, 33, 34, 67,68,69]).

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Acknowledgements

The author would like to express his warmest thanks to the three anonymous referees for the careful reading of the original manuscript and for all the apt suggestions and comments which allowed us to improve both the quality as well as the clarity of the paper.

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  1. Department of Physics, School of Science, Aristotle University of Thessaloniki, 541 24, Thessaloniki, Greece

    Euaggelos E. Zotos

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Appendix: Derivation of the potential function of the classical Hill problem

Appendix: Derivation of the potential function of the classical Hill problem

The first step is to perform the coordinate transformation given in Eq. (4) along with \(\mu = M^3\). Then Eq. (1) becomes a polynomial function \(\varOmega = \varOmega (x,y,z,M)\).

It is very easy to prove that \(\varOmega _{\mathrm{lim}} = \displaystyle {\lim _{M \rightarrow 0} \varOmega } = 3/2\).

Next we expand \(\varOmega \) into a power series around \(M = 0\) (Taylor series), keeping terms up to second order, thus having

$$\begin{aligned} W_0(x,y,z,M) = \frac{3}{2} + M^2 \left( \frac{3x^2}{2} - \frac{z^2}{2} + \frac{1}{r}\right) , \end{aligned}$$
(13)

where \(r = \sqrt{x^2 + y^2 + z^2}\).

Now for obtaining the potential function W(x, y, z) of the classical Hill problem all we have to do is to eliminate the parameter M for Eq. (13). This can be achieved as

$$\begin{aligned} W(x,y,z) = \frac{W_0 - \varOmega _{\mathrm{lim}}}{M^2}, \end{aligned}$$
(14)

thus deriving the final form of Eq. (6)

$$\begin{aligned} W(x,y,z) = \frac{3x^2}{2} - \frac{z^2}{2} + \frac{1}{r}. \end{aligned}$$
(15)

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Zotos, E.E. Orbit classification in the Hill problem: I. The classical case. Nonlinear Dyn 89, 901–923 (2017). https://doi.org/10.1007/s11071-017-3491-4

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