Periodic solutions in the spatial elliptic restricted three-body problem
Introduction
The study of periodic orbits of a non-integrable dynamical system is a very useful tool to obtain information on the topology of phase space. In the vicinity of a periodic orbit, the study of the phase space can be reduced to the study of the invariant curves of a Poincaré map on a surface of section and then the fixed points, together with their stability character, determine critically the topology of the problem. For this reason, the computation of periodic orbits plays an important role in the study of dynamical systems.
In the circular restricted three-body problem, in spite of the large amount of numerical work available, there are relatively few analytical results on the whole subject of periodic solutions. The analytic continuation method has been used by several authors to show the existence of such orbits. A classic result is that of Arenstorf [1] showing the existence of second kind periodic solutions (i.e. which come from analytic continuation of unperturbed elliptic Keplerian orbits, with arbitrary eccentricity) in the planar circular restricted three-body problem in a neighbourhood of one of the primaries, irrespective of the mass ratio. In the three-dimensional circular restricted problem, Jefferys [2] showed there exist doubly symmetric, almost circular periodic solutions if one of the primaries is sufficiently small. He showed as well [3] the existence of families of elliptic orbits for any value of the eccentricity and a critical inclination. In the same problem and with the same symmetries, Howison and Meyer [4] showed the existence of doubly symmetric periodic orbits, with no restriction on the mass ratio, when the infinitesimal body is either very far from both primaries or very close to one of them. There are even fewer results in the case of the general (not restricted) problem (see [5]).
As far as we know, the restricted elliptic three-body problem has not yet been fully explored, and the three-dimensional problem even less so, although a number of papers have been devoted to it: Katsiaris [6] computes some periodic orbits when the eccentricity e of the primaries lies in the range [0, 0.4]; Olle and Pacha [7] compute families of periodic solutions, very far from the primaries, which are assumed to be of the same mass and in a rectilinear collision orbit.
Analytical results for the existence of periodic solutions are even more scarce. Serysels-Lamy [8] shows the existence of second kind periodic solutions in a neighbourhood of the primary of small mass in the planar case. Another result in the planar case is that of Llibre and Saari [9], connecting the circular and the elliptic problem. To our knowledge, no results have been published in the three-dimensional elliptic restricted three-body problem.
In this paper, we show the existence of some symmetric periodic solutions of the three-dimensional elliptic restricted three-body problem when the primaries are of equal mass, no restriction being imposed on the eccentricity. These solutions are slightly perturbed circular Keplerian orbits lying in a plane perpendicular to that of the primaries, with very large semiaxes, and the small parameter used in the continuation is the inverse of the semiaxis itself. They can be looked at as orbits of radius unity after a change of scale which brings the primaries very close to the origin. The problem can thus be seen as a perturbed Kepler problem, where the small parameter is just the semimajor axis of the primaries orbits. Then a new difficulty shows up due to the fast motion of the primaries, the equations are no longer analytic when the parameter equals zero, which precludes the use of the standard implicit function theorem. This difficulty can be easily overcome by resorting to Arenstorf’s version of that theorem (where weaker assumptions on differentiability are needed) which turns out to be fundamental in the proof of the result. Fig. 1 shows one of the orbits predicted by the theorem, numerically computed with initial values q1=0, q2=−0.042819579, q3=0.999600822, , for the case ε=6−2/3, η=0.99. The equations defining the initial conditions for periodic solution were solved with Newton’s method, starting at the point given by the non-perturbed Keplerian circular orbit of radius 1. The equations of motion and the first variationals were numerically integrated with a Runge–Kutta 7–8 routine. In the picture, the primaries lie on the q1q2-plane and the infinitesimal body starts at a point on the q2q3-plane, near the q3-axis. Short lines parallel to the coordinate axes are shown for the sake of greater clarity both at the starting point and the intersection with the q1-axis.
Section snippets
Equations of motion and symmetries
We consider two primaries of mass moving on elliptic orbits around their centre of mass, with eccentricity η∈[0,1) and semimajor axis ε. The distance between both is given bywhere ϕ is the true anomaly of m1 (see [10]), i.e. its angular position measured from the pericentre. The angular motion of the primaries is governed by the equationwhere t is the time. If β is defined to bethen ϕ(t) is given by the following expansion (see
Approximate solutions
In this section, we will show how those solutions of the three-dimensional elliptic restricted three-body problem in which the infinitesimal body keeps far away from the primaries can be approximated through successive corrections to the Keplerian motion. As the system (5) is not defined when ε=0, the analysis is somewhat more delicate because the standard techniques, such as expansion in power series, are no longer available. In the next section, we will use those approximations to continue
Continuation of doubly symmetric periodic solutions
In this section, we use the results of the latter to show the existence of doubly symmetric periodic solution of the spatial elliptic restricted three-body problem. The periodicity conditions in Poincaré variables state that at time t=0, we must haveand at time ,The condition P2=0 implies either or L=G, i.e. m3 is either on an elliptic orbit with its pericentre on the q2q3-plane or on a circular orbit, respectively. In a similar way
Concluding remarks
In the periodic orbits found, the primaries are at the pericentre of their orbit at both t=0 and t=π/2. There exist periodic solutions when the primaries are at the apocentre, or for the combination apocentre–pericentre and pericentre–apocentre as well.
It could be shown, likewise, the existence of such solutions in the general case when the primaries have undergone an exact number of semiorbits in the interval of time between t=0 and t=π/2+kπ, k a positive integer (then the infinitesimal body
References (13)
Continuation of periodic solutions in three dimensions
Physica D
(1998)A new method of perturbation theory and its application to the satellite problem of celestial mechanics
J. Die Reine Angew. Math.
(1966)Doubly symmetric periodic orbits in the three-dimensional restricted problem
Astronom. J.
(1965)A new class of periodic solutions of the three-dimensional restricted problem
Astronom. J.
(1966)- R.C. Howison, K.R. Meyer, Doubly symmetric periodic solutions of the spatial restricted three-body problem, J. Diff....
- G. Katsiaris, The three-dimensional elliptic problem, in: Recent Advances in Dynamical Astronomy, Proc. of NATO...
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2011, Scientia IranicaCitation Excerpt :In this problem, with the same symmetries, Howison and Meyer [4] showed that double symmetry exists if the third mass is so far or so near. Cors et al. [5] studied a general case of TBP, wherein two primaries have equal mass, but work on general cases is scarce. In a numerical scope, many studies in different ranges have been undertaken.
Searching for periodic orbits of the spatial elliptic restricted three-body problem by double averaging
2006, Physica D: Nonlinear PhenomenaCitation Excerpt :In [4] the reader can find a good summary of the main contributions existing in the literature on this subject. In the same way as done in [4] we formulate the problem as a perturbed Keplerian system. The small parameter is of the order of the inverse of the semimajor axis of the orbit of the third body or, equivalently, the distance between the two primaries.
Analytic continuation in the case of non-regular dependency on a small parameter with an application to celestial mechanics
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