Celest Mech Dyn Astr DOI 10.1007/s10569-014-9594-z ORIGINAL ARTICLE
Transport orbits in an equilateral restricted four-body problem M. Alvarez-Ramírez · E. Barrabés
Received: 8 June 2014 / Revised: 26 September 2014 / Accepted: 20 October 2014 © Springer Science+Business Media Dordrecht 2014
Abstract In this paper we consider a restricted equilateral four-body problem where a particle of negligible mass is moving under the Newtonian gravitational attraction of three masses (called primaries) which move on circular orbits around their center of masses such that their configuration is always an equilateral triangle (Lagrangian configuration). We consider the case of two bodies of equal masses, which in adimensional units is the parameter of the problem. We study numerically the existence of families of unstable periodic orbits, whose invariant stable and unstable manifolds are responsible for the existence of homoclinic and heteroclinic connections, as well as of transit orbits traveling from and to different regions. We explore, for three different values of the mass parameter, what kind of transits and energy levels exist for which there are orbits with prescribed itineraries visiting the neighborhood of different primaries. Keywords Lagrangian configuration · Four-body problem · Invariant manifolds · Transfer orbits · Homoclinic and heteroclinic connections
1 Introduction One of the most extensively studied problem in Celestial Mechanics is the restricted three body problem, where one of the bodies is considered massless, so it does not affect the motion of the other two that move in a Keplerian orbit (circular or elliptic trajectory solution of a two-body problem). The restricted four body problem is similar in the sense that the problem
M. Alvarez-Ramírez (B) Departamento de Matemáticas, UAM–Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, 09340 Mexico, D.F., Mexico e-mail:
[email protected] E. Barrabés Departament d’Informàtica, Matemàtica Aplicada i Estadística Campus Montilivi, Universitat de Girona, 17071 Girona, Spain e-mail:
[email protected]
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deals with the motion of an infinitesimal particle under the attraction of three masses, whose trajectories are solution of the three Newtonian body problem. There are many four-body systems in the Solar System and in the Galaxy that can be approximated, in a first order, by a four body problem (restricted or not). An example is the Sun–Jupiter–Saturn-X model where the two bigger planets of the Solar System are considered and X can be a planet, an asteroid or a satellite of Jupiter or Saturn. Another example is the Sun–Venus–Earth–Jupiter system. In our galaxy of 1011 stars an estimate of the number of four-star systems is of order 109 , see Roy (2004). In February 1906, the astronomer Max Wolf discovered (photographically) an asteroid named 588 Achilles. It was the first of the Trojan asteroids to be discovered, orbiting around the L 4 Lagrangian point in the Sun–Jupiter system. The neighborhood of L 4 is known as the Greek camp, while around L 5 is set the Trojan camp, and in both of them there exists a cluster of asteroids orbiting in tadpole orbits around the equilibrium points. The L 4 point (as well as the L 5 ) is at one vertex of an equilateral triangle with the other vertices at the Sun and Jupiter, respectively. The Sun–Jupiter–Trojan asteroid—infinitesimal particle (even an spacecraft) can be viewed as a restricted four body problem, where the primaries are in the particular configuration of an equilateral triangle, (see, for example Marzari et al. 2002; Mazari 2006; Steves et al. 1998a, or Steves et al. 1998b). In Ceccaroni and Biggs (2012) the authors consider a perturbation of a Sun–Jupiter–asteroid–spacecraft model by the inclusion of low-thrust propulsion. In this paper we study a restricted planar four-body problem in which the three primaries are in the Lagrange equilateral triangle configuration, like the Sun–Jupiter–Trojan asteroid, which we denote by Equilateral Restricted Four Body Problem (ERFBP). We consider that the motion of the massless particle is contained in the orbital plane of the motion of the three primaries (planar ERFBP). There does exist a number of studies on the restricted four-body problem in such configuration. Alvarez-Ramírez and Vidal (2009) study the ERFBP when the three primaries have equal masses. Baltagiannis and Papadakis (2011) determine the number of the equilibrium points of the ERFBP for any value of the masses, and study numerically their linear stability varying the values of the masses. They also present a description and compute numerically some families of symmetric periodic orbits. In Burgos-García and Delgado (2012) the authors obtain similar results to those given in Baltagiannis and Papadakis (2013) for the case of two equal masses approximately at Routh’s critical value, and in BurgosGarcía and Delgado (2013) the authors show the existence of the blue sky catastrophe around an specific collinear equilibrium point (the so called blue sky catastrophe phenomenon, named after Devaney, see Devaney (1977), refers to the existence of an infinite set of periodic orbits accumulating to a given homoclinic orbit). Recently, Baltagiannis and Papadakis (2013) found a large number of families of non-symmetric orbits, in particular two families of stable (horizontally and vertically) retrograde non-symmetric periodic orbits around Jupiter and the Trojan asteroids. Also, the linear stability of each periodic solution is studied. Furthermore, Alvarez-Ramírez et al. (2014) proved that any double collision can be regularized by using a Birkhoff-type transformation. These papers indicate the intensity of research in the ERFBP in the recent years. The main objective of this paper is to illustrate transport from one primary to another one. By transport, we mean trajectories that depart from the neighborhood of one of the primaries and arrive to the neighborhood of another one, so that “matter” from one body can end up in a different one following such paths. In order to do so, we will study the dynamics of the problem, without trying to be exhaustive. We will restrict our attention to the ERFBP with two equal masses. One example of this model problem is the Saturn–Tethys–Telesto (or Calypso)spacecraft system where Tethys is the large Saturn’s moon, and two small co-orbital satellites, Telesto and Calypso lie in L 4 and L 5 Lagrange points—60◦ along the orbit, in front of and
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behind Tethys. Is it possible to have trajectories moving from the neighborhood of one of the moons to the other? We will explore the behavior of certain hyperbolic periodic orbits that born from unstable equilibrium points to determine the existence of homoclinic/heteroclinic orbits and transit orbits from one region to another. The invariant manifold structures of the collinear and non collinear equilibrium points for the restricted equilateral four-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated to some periodic orbits are the phase space conduits for transporting material between the neighborhood of different primary bodies. The paper is organized as follows. In Sect. 2 we give a brief introduction to the principal aspects of the equilateral four-body problem ERFBP, and in Sect. 3 we present the results obtained with respect to transit orbits and orbits with prescribed trajectories for three different values of the mass parameter. The main features of the problem can be found, for example, in Baltagiannis and Papadakis (2011); Alvarez-Ramírez and Vidal (2009) or Ceccaroni and Biggs (2012) (and the references therein). We include here the description of the problem and some of their properties for completeness and clarity of the exposition of the results in Sect. 3.
2 Main features We consider three masses m i , i = 1, 2, 3, in a Lagrangian (or triangular) configuration, in which the three bodies move in circular orbits in the same plane around their common center of masses, and are located at all time at the vertices of an equilateral triangle (see Wintner 1941). The Equilateral Restricted Four Body Problem (ERFBP) describes the motion of a massless particle under the gravitational attraction of the three masses in the triangular configuration. We consider the case in which two of the masses are equal. Considering m 1 + m 2 + m 3 = 1, the masses will be m 1 = 1 − 2μ and m 2 = m 3 = μ, where μ ∈ (0, 1/2) is called the mass parameter of the problem. We also consider a rotating coordinate system (x, y) that rotates with the uniform angular velocity of primaries, so that they are fixed in the (x, y) plane. Without loss of generality, we take the primary of mass m 1 located on the positive x-axis at (x1 , 0), and then the other two primaries m 2 and m 3 are at x2 , 21 and x2 , − 21 , where √ √ x1 = 3 μ and x2 = − 23 (1 − 2μ). The equations of motion of the infinitesimal mass in the rotating coordinate system are (see, for example, Baltagiannis and Papadakis 2011) x¨ − 2 y˙ = Ωx , y¨ + 2 x˙ = Ω y ,
(1)
where
1 − 2μ μ μ 1 2 + + , (x + y 2 ) + 2 r1 r2 r3 with r1 = (x − x1 )2 + y 2 , r2 = (x − x2 )2 + (y − 21 )2 and r3 = (x − x2 )2 + (y + 21 )2 . Notice that the Eq. (1) are invariant under the following symmetry: Ω = Ω(x, y) =
(t, x, y, x, ˙ y˙ )
−→
(−t, x, −y, −x, ˙ y˙ )
(2)
Thus, an orbit with one perpendicular cross to the axis y = 0 is symmetric with respect to the same axis.
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The system (1) has a first integral given by x˙ 2 + y˙ 2 = 2Ω(x, y) − C J .
(3)
This expression is the same than the first integral of the circular restricted three-body problem, differing only in the expression of the function Ω. In the circular restricted three-body problem C J is known as the Jacobi constant, named after Carl Gustav Jacob Jacobi. If the problem is written in canonical coordinates, then the first integral corresponds to the value of the Hamiltonian, and it is known as the energy. The existence of the first integral, allows to study the problem fixing the energy level, of the value of the Jacobi constant. In our model, we will follow the same strategy, and due to the similarity, we will use here the same name, and we will call C J the Jacobi constant. For each value of the Jacobi constant, the zero velocity curves given by C J = 2Ω(x, y).
(4)
limit, in the configuration space, the Hill’s region where the motion is allowed. We will start presenting some known results with respect to the equilibrium points of the ERFBP and their stability, and the geometry of the zero velocity curves. Some of the equilibrium points are of type center × saddle, so that there exists families of unstable periodic orbits. Their invariant manifolds are the main responsible of the existence of transit between neighborhoods of different primaries within the Hill’s region. We are interested in those families of periodic orbits and in the behavior of their invariant manifolds. We will focus on them in Sect. 3. In order to present the results, we have chosen different values of the mass parameter, each one corresponding to a different scenario with respect to the number of equilibrium points and the evolution of the shape of the zero velocity curves. Qualitatively, there are three different scenarios (we explain that with more detail later) and the values of μ chosen have been 0.1, 0.3, and 0.46. In particular, these values are not far from μ = 0, 1/2, and 1/3, which correspond to the well-known cases of the central force problem, the restricted three-body problem, and the ERFBP symmetric problem with three-equal-masses, respectively. 2.1 Equilibrium points and their stability We include here a description of the the number of equilibrium points and the bifurcations depending on the mass parameter μ, as well as their linear stability. An analytical study of the stability of the equilibrium points can be found in Budzko and Prokopenya (2012), and numerically in Baltagiannis and Papadakis (2011). In Leandro (2006), the author proved that the existence and the number of equilibrium points depends on the mass parameter. In Tables 1 and 2 we show the number of equilibria and the approximate values of μ at which bifurcations occur. For a fixed value of μ, one convention to label the equilibrium points is to denote them as L i , i = 1, 2, . . ., where the ordering is given by the value of the Jacobi constant C J at them (at a decreasing or increasing order). When the mass parameter varies, that order changes (see Fig. 2), so it is not possible to label them following that rule. We opted to name them in the same way as in the works of Baltagiannis and Papadakis (2011, 2013) and their location is (x L i , y L i ), i = 1, 2, . . .. With respect to the collinear equilibrium points (along the y = 0 axis), for μ ∈ (0, μ∗ ), where μ∗ 0.2882761912, there are only two collinear equilibrium points L 2 and L 3 . At the bifurcation value μ = μ∗ two new equilibria, L 1 and L 4 , appear, and for μ ∈ (μ∗ , 1/2) there are four collinear equilibrium points. Their coordinates are y L i = 0, i = 1, . . . , 4,
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Transport orbits in an equilateral restricted four-body problem Table 1 Collinear equilibrium points of the ERFBP depending on the mass parameter μ
Number of equilibria
Collinear points
(0, μ∗ )
2
μ∗
3
(μ∗ , 0.5)
L3, L2 L3, L4 = L1, L2
4
L3, L4, L1, L2
The bifurcation value is approximately μ∗ = 0.28827619129. They are sorted in increasing order with respect to their x coordinate. See Fig. 1
Table 2 Non-collinear equilibrium points of the ERFBP depending the mass parameter μ
Number of equilibria
Non-collinear points
(0, μ∗∗ )
6
μ∗∗
L 5 , L 6 , L 7 , L 8 , L 9 , L 10
5
(μ∗∗ , 0.5)
L 5 , L 6 , L 7 , L 8 , L 9 = L 10
4
L5, L6, L7, L8
The bifurcation value is approximately μ∗∗ = 0.44020149999
0
-1
0.5
L9 L3
L2 L10
-0.5 L8 -1
0
0
x
0.5
1
-1
-1
-0.5
L2
0
x
0
L3
L1
L4
-0.5 L6
L8
L5
0.5
L1 L4 L10
L3
L7
1
L5 L9
-0.5 L6
-0.5
L7
y
y
0.5
1
L5
L7
y
1
0.5
L6 L8
-1 1
L2
-1
-0.5
0
x
0.5
1
Fig. 1 Location of the equilibrium points and masses for μ = 0.1 (left), μ = 0.3 (middle) and μ = 0.46 (right)
x L 3 < 0 < x L 2 for all μ > 0 and x L 3 < x L 4 < x L 1 < x L 2 , for μ > μ∗ . See Table 1 and Fig. 1, where we illustrate the location of the equilibrium points for μ = 0.1, 0.3, 0.46. With respect to the non-collinear equilibrium points, for μ ∈ (0, μ∗∗ ), where μ∗∗ 0.44020149999, there are six non-collinear equilibrium points L i , i = 5, . . . , 10, and at μ∗∗ , two of them, L 9 and L 10 merge with the collinear point L 1 , so only four non-collinear points remain. Due to the symmetry with respect to x-axis, for each non-collinear equilibrium point L i at (x L i , y L i ) with y L i > 0, there exists another equilibrium point at (x L i , −y L i ). The equilibrium points on the upper half plane are L 5 , L 7 and L 9 , and the symmetric ones are, L 6 , L 8 and L 10 respectively. See Table 2 and Fig. 1. We are interested in the equilibrium points that are center × saddle, so that the Lyapunov’s center theorem applies and there exist a one-parameter family of unstable periodic orbits emanating from them. At each equilibrium point, we compute the eigenvalues associated to the differential matrix of the field at the point. In Table 3, the number of real (nr ), complex (n c ) and pure imaginary (n i ) eigenvalues is summarized. We see that the equilibrium points that can be of our interest are L 2 , L 4 , L 7,8 and L 9,10 for all possible values of μ, and L 1 for a small range of values of μ.
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M. Alvarez-Ramírez, E. Barrabés Table 3 Number of real (nr ), complex (n c ) and pure imaginary (n i ) eigenvalues associated to each equilibrium point varying μ L1 L2 L3 L4
μ ∈ (μ∗ , 0.428337819) μ ∈ (0.428337819, μ∗∗ ) μ ∈ (μ∗∗ , 0.5) μ ∈ (0, 0.5) μ ∈ (0, 0.0027096302) μ ∈ (0.0027096302, 0.5) μ ∈ (0, 0.5)
nc = 4 nr = 4 nr = n i = 2 nr = n i = 2 ni = 4 nc = 4 nr = n i = 2
L 5,6
μ ∈ (0, 0.018858526) μ ∈ (0.018858526, 0.5)
ni = 4 nc = 4
L 7,8 L 9,10
μ ∈ (0, 0.5) μ ∈ (0, μ∗∗ )
nr = n i = 2 nr = n i = 2
4
Fig. 2 Variation of the Jacobi constant C J of each equilibrium point as a function of the mass parameter. The vertical lines correspond to the values μ = μ∗ and μ = μ∗∗
3.8 L4
L9
3.6
L1
L7
CJ
3.4
L2
3.2
L3
3
L5
2.8 2.6
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
µ
2.2 Hill’s region In this section we describe the geometry of the Hill’s region of the ERFBP, that is, the regions in the configuration space where the motion of the particle takes place for a fixed value of the Jacobi constant. From (3), the Hill’s regions are limited by the zero velocity curves (4) (see, for example, Szebehely 1967 for the circular restricted three-body problem, or Baltagiannis and Papadakis 2011 for more details in the ERFBP). For enough big values of C J , the Hill’s region has four disjoints components: three around each one of the primaries, and one contained in {(x, y); x 2 + y 2 > R} for a suitable R. As the Jacobi constant decreases, the components merge and channels connecting different regions open. That happens exactly when the equilibrium points appear. Thus, in the description of the evolution of the shape of the Hill’s region is important the order in which the equilibrium points are born as C J decreases. In Fig. 2, we plot the variation of the value of the Jacobi constant at each equilibrium point C J (L i ). Due to the symmetry (2), C J (L i ) = C J (L i+1 ), i = 5, 7, 9. Figures 3, 4 and 5 show the Hill’s regions for μ = 0.1, μ = 0.3 and μ = 0.46, respectively, and some vaules of C J in a decreasing order (the forbidden region in white). The values of the Jacobi constant chosen are close to the value of C J at different equilibrium points. It can be seen how the channels connecting the different allowed regions appear around the corresponding equilibrium points (you can see the location of the equilibrium points in Fig. 1). Thus, trajectories with a Jacobi constant just below that of the equilibrium points are energetically permitted to make a transit through the bottleneck connecting the corresponding two regions.
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Transport orbits in an equilateral restricted four-body problem 2
2
2
2
1
1
1
1
0
0
0
0
1
1
1
1
2
2 2
1
0
1
2
2
0
1
1
2
2
2
0
1
1
2
2
2
1
0
1
2
Fig. 3 Zero velocity curves and Hill’s region (shown by shading) in the configuration space for μ = 0.1 and fixed values of the Jacobi constant. The white areas correspond to the forbidden regions of motion. The values of the Jacobi constant are, from left to right, C J = 3.4876, C J = 3.368, C J = 3.147 and C J = 3.0379. All the frames correspond to [−2, 2] × [−2, 2] in the configuration plane (x, y) 2
2
2
2
1
1
1
1
0
0
0
0
1
1
1
1
2
2 2
1
0
1
2
2 2
0
1
1
2
2 2
0
1
1
2
2
1
0
1
2
Fig. 4 Zero velocity curves and Hill’s region (shown by shading) in the configuration space for μ = 0.3 and fixed values of the Jacobi constant. The white areas correspond to the forbidden regions of motion. The values of the Jacobi constant are, from left to right, C J = 3.55, C J = 3.45865, C J = 3.36 and C J = 3.35. All the frames correspond to [−2, 2] × [−2, 2] in the configuration plane (x, y) 2
2
2
1
1
1
0
0
0
1
1
1
2
2 2
1
0
1
2
2 2
1
0
1
2
2
1
0
1
2
Fig. 5 Zero velocity curves and Hill’s region (shown by shading) in the configuration space for μ = 0.46 and fixed values of the Jacobi constant. The white areas correspond to the forbidden regions of motion. The values of the Jacobi constant are, from left to right, C J = 3.869 C J = 3.41 and C J = 3.21. All the frames correspond to [−2, 2] × [−2, 2] in the configuration plane (x, y)
Notice the main differences between the three values of μ: for μ = 0.1 and 0.3, the firsts channels to open are those corresponding to L 9 and L 10 , whereas for μ = 0.46, the first channel to open is the one corresponding to L 4 . Furthermore, in the first two cases the neighborhoods of the three masses are connected before there can exist trajectories escaping to infinity, whereas in the last one orbits departing from m 2 or m 3 can (theoretically) scape before the channel connecting to m 1 is opened.
3 Transit orbits and trajectories with prescribed itineraries We consider an equilibrium point L i such that it is of type center × saddle, so that nr = n i = 2 (see Table 3). Applying the Lyapunov center theorem (see, for example, Meyer and Hall 1992), there exists a one-parameter family of periodic orbits emanating from the equilibrium
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W ku
s
Wk
u
Wj s
Wk
Wj s
Wj u
W ks
Wj u
Fig. 6 Sketch in configuration space of the transit (left) and non-transit (right) orbits. The solid curves represent a Lyapunov periodic orbit O L i and the zero-velocity curves. The dotted lines represent the manifold tubes. The arrows indicate direction of motion of trajectories following the manifold branches
point. The orbits are called Lyapunov orbits, and an orbit of the family will be denoted by O L i , where i means the index of the equilibrium point from which the family is born. For each family, the Lyapunov orbits exist for a certain range of values of the Jacobi constant C J < C J (L i ), and at least for values of C J close to the value at the equilibrium point, the periodic orbits inherit its hyperbolicity. Thus, there exist 2-dimensional unstable and stable invariant manifolds associated to the periodic orbits. These manifolds, which are locally diffeomorphic to cylinders R × S 1 , can be viewed geometrically as two-dimensional tubes approaching (forwards and backwards in time) the periodic orbit (see Conley 1969). As we have seen, for big values of C J , the Hill’s region has four different connected components: three around each one of the primaries, and the fourth one around all of them (see Figs. 3, 4, 5). We denote by R1 , R2 and R3 the neighborhood, in the (x, y) plane, around each one of the bodies of mass m 1 , m 2 and m 3 , respectively, and by X the exterior region around all the bodies where the orbits can escape to infinity. Decreasing the Jacobi constant, the equilibrium points appear and the regions can be connected through bottlenecks that grow in size as C J decreases. The Lyapunov periodic orbits “live” in that bottlenecks, the transitions between the regions R1 , R2 , R3 and X are governed by the invariant manifolds structures associated with that periodic orbits, and in particular, transit and non-transit orbits exist (see Conley 1969). A transit orbit can be defined as a trajectory that approach the Lyapunov periodic orbit from one region and traverses the bottleneck to a different region. A non-transit orbit, on the contrary, bounces back to the region it comes from. See Fig. 6. It is known that transit orbits lie in the interior of the invariant manifold tubes of the Lyapunov orbits, whereas the non-transit ones lie in the exterior of the tubes, so that the invariant manifolds separate the two types of motion (see Conley 1969; McGehee 1969). For each invariant manifold W u/s associated to an O L i , there exist two branches, each one entering u/s into one region. In order to identify them, we denote by Wk the branch that enters into the region Rk . We will focus on the transits between the regions Ri , i = 1, 2, 3. For a given value of the Jacobi constant, in order to have a transit from Rk to R j , k = j, a trajectory must be inside the tube of the branch Wks (O L i ), where O L i is the periodic orbit at the bottleneck connecting Rk and R j for that C J . Such an orbit will enter the region R j inside the tube W uj (O L i ) (see Fig. 6 left). But then, for the same orbit to have another transit to Rl (l = j, but maybe l = k) it must also be inside the branch W sj (O L m ), where O L m is the periodic orbit at the bottleneck connecting R j and Rl for the same C J (if l = k, then m = i). Therefore, to complete the sequence of visits Rk − R j − Rl the set W uj (O L i ) ∩ W sj (O L m ) = ∅, so there must exist homoclinic (if i = m) or heteroclinic (if i = m) connections. Then, the intersections of the tubes of the stable and unstable manifolds define sets of initial conditions of orbits that visit different regions. Given a sequence of regions to visit,
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Transport orbits in an equilateral restricted four-body problem
to see if there exists a trajectory with that prescribed itinerary it is necessary to look at the corresponding sequence of intersections between different invariant manifolds. A rigorous study using symbolic dynamics would be necessary to show the existence of orbits with prescribed itineraries. Our aim is to show that there exist orbits visiting two different regions for a wide range of values of the Jacobi constant, so we will explore the existence of trajectories following sequences of visits of type Rk − R j − Rl , k, l = j. We will describe the evolution of the dynamics and the existence of transit orbits as the value of C J decreases in three cases, μ = 0.1, 0.3, 0.46. In each case we consider a range of values for which the zero velocity curves exist, so there is a region of forbidden motion, and the regions Ri , i = 1, . . . , 3 can be recognized. For values of the Jacobi constant below that values, and specially once the zero velocity curves disappear, it makes no sense to talk about transfers from one region to another. In all the cases, we start computing the families of Lyapunov periodic orbits that are born from the equilibrium points, and looking at the linear stability of the orbits in order to know up to which value of C J the orbits are unstable. For that, we compute the stability parameter s = λ + 1/λ, where λ and 1/λ are the eigenvalues associated to the linear stability obtained from the monodromy matrix. Then, in order to explore the existence of homo/heteroclinic connections, we compute the branches of the corresponding invariant manifolds and we follow them up to a fixed section Σ, typically x = c or y = c, for a given constant c. As C J and the value of one coordinate are fixed, we can visualize the intersection of the tubes in a plane, (typically we plot the intersection in the (x, x ) or (y, y ) planes). We denote by u n or sn the intersection of the unstable or stable branches with the section Σ at the n-th crossing of the manifolds with the section. We are interested in the study of the intersection of the curves u n and sk , as that intersection correspond to homo/heteroclinic connections, and determines the region corresponding to transit orbits. In general, the intersection of each tube with the section is a closed curve until it breaks due to the presence of a homoclinic or heteroclinic connection, or a collision orbit with one of the primaries. For the numerical integrations, we have used a variable step Runge–Kutta–Felbergh method of orders 7 and 8. For the computation of the families of periodic orbits we have used a standard predictor–corrector method. See, for example, Gómez and Mondelo (2001), Sect. 2.1 where the methodology is explained in detail. For the computation of the invariant manifolds, we have considered the linear approximation given by ψ u/s (θ, ξ ) = φ
θ 2π
u/s u/s −θ/(2π ) ) Dφ θ T (z 0 )v0 , T (z 0 ) + ξ(
2π
where φt (z) is the time–t flow of the RTBP, z 0 = (x0 , y0 , x˙0 , y˙0 ) is an initial condition of u/s the periodic orbit, T is its period, v0 is an eigenvector of the monodromy matrix DφT (z 0 ) corresponding to the unstable/stable manifold, and u/s is the corresponding eigenvalue. Since all the computations have been done with double precision, we have taken ξ of the order of 10−6 . See, for example, Barrabés et al. (2009) for more details. Finally the intersections of the invariant manifolds with a given section have been obtained taking a set of points on the (linear approximation of the) invariant manifolds {ψ u/s (θ j , ξ )} j=1,...,N , and integrating them until the section (defined implicitly g(z) = 0) is reached with a precision of 10−12 . In each case the section used is specified, and will be taken at most convenience depending on the geometry of the invariant manifolds.
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Stability parameter
Fig. 7 Families of periodic orbits that are born from the equilibrium points L 7 , and L 9 for μ = 0.1. For C J > C J (L 2 ) (vertical dotted line) all the orbits are unstable
OL9
3000 2500 2000 1500 1000
OL7
500 0 -500 2.8 2.9
3
3.1 3.2 3.3 3.4 3.5
CJ
3.1 Case μ = 0.1 We study the behavior of the invariant manifolds for the values of the Jacobi constant in (C J (L 2 ), C J (L 9 )), see Figs. 2 and 3. The values of C J at the equilibrium points in this case are, approximately, C J (L 2 ) = 3.147231, C J (L 7,8 ) = 3.368690, C J (L 9,10 ) = 3.48768130. For values of C J > C J (L 9 ), the regions R j , j = 1, 2, 3 and X are disconnected, until the apparition of L 9 and L 10 . Then, as C J decreases, the region R1 connects with R2 and R3 through the bottlenecks opened around these equilibrium points. Next, the regions R2 and R3 connect with X through the channels opened by L 7 and L 8 , so that the Hill’s region has only one component. Finally, after L 2 appears, the region R1 connects with the region X directly through the channel opened on the right. See Fig. 3. The equilibrium points L i , i = 7, . . . , 10 are center × saddle (see Table 3). We compute the families of periodic orbits O L 7 and O L 9 (using the symmetry of the equations we also have the families of O L 8 and O L 10 ) and the value of the stability parameter s at each orbit. In Fig. 7 we plot the characteristic curves of such families in the (C J , s) plane. The families are followed until the orbits are close to collision with one of the primaries. For the values of C J > C J (L 2 ), the orbits are unstable, so there exist stable/unstable invariant manifolds associated to each one of them. We focus only in transitions between the regions Ri , i = 1, 2, 3, so in this case we study, for a fixed value of C J , the behavior of the invariant manifolds of W u/s (O L 9 ) and W u/s (O L 10 ), and specifically their intersections, where O L 9 and O L 10 are the Lyapunov periodic orbits of that energy level (see Fig. 8). We use that, from the symmetry given by (2), W u (O L 9 ) (W s (O L 9 )) is symmetric to W s (O L 10 ) (respectively W u (O L 9 )). An orbit inside any of the tubes will have a transit from one region to another. First, we want to see if there are orbits that, after one transit, have a second transit bouncing back to the initial region. That is, if there exist trajectories following the sequence of visits R2 –R1 –R2 or R1 –R2 –R1 (using the symmetry of the problem, the same results can be obtained replacing R2 by R3 ). Next, we also explore the transport from R2 and R3 , or the other way around, going through R1 , so that the sequence of visits are R2 –R1 –R3 or R3 –R1 –R2 . We start exploring the transits R1 –R2 –R1 . In this case, the trajectory must be inside the tubes W2u (O L 9 ) and W2s (O L 9 ) (branches in R2 , see Fig. 8 right). In Fig. 9 we plot the intersection of these tubes with the section Σ = {x = −0.7} after a fixed number of crossings (curves u k and s j ). We see that the curves intersect, so there exists homoclinic connections (corresponding to the common points to both curves) and transit orbits following
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Transport orbits in an equilateral restricted four-body problem 0.75
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s W1
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0
0.25 0.5
Wu2 s
0.5
W2
0.2 -1
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3
x
Fig. 8 Branches of the invariant manifolds associated to a Lyapunov periodic orbit O L 9 entering R1 (left) and R2 (right) up to their first (W u ) and second (W s ) intersection with a fixed section (the periodic orbit in black, the unstable branches in red, the stable branches in blue). The dotted lines show the zero velocity curve and the section used in each case. R1 , x = −0.7 in R2 ). The small circles mark the position of the masses. The value of the Jacobi constant is C J = 3.4700192109 CJ=3.4667254661
0.5
’
y
’
-1.5 u1
-2.5 -3.5
CJ=3.3686768326
1
s10
-0.5
y
Fig. 9 Intersection of W2u (O L 9 ) (in red) with W2s (O L 9 ) (in blue) at their intersection with the section x = −0.7 in the the plane (y, y ). The region interior to both curves correspond to transit orbits from R1 –R2 –R1
0
u1
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0.5
0.54
0.58
0.62
-5
0.66
0.5 0.55 0.6 0.65 0.7 0.75
y
2 1.5
CJ=3.4098466645
2.5 1.5
s3
0
’
0.5
x
’
CJ=3.3686768326
2
u2
1
x
Fig. 10 Intersection of the invariant manifolds W1u (O L 9 ) (in red) and W1s (O L 9 ) (in blue) with the section y = 0 in the (x, x ) plane for the values of the Jacobi constant shown
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that sequence of visits for C J ≤ 3.4667254661. As the value of C J decreases, the region that correspond to transit orbits grows. Next, we explore the existence of transits R2 –R1 –R2 . A trajectory with that sequence of visits must be inside the tubes W1u (O L 9 ) and W1s (O L 9 ). We follow the branches W1u (O L 9 ) and W1s (O L 9 ) until their intersection with the section Σ = {y = 0} (see Fig. 8, left). The first value for which we find that they intersect is C J = 3.4098466645, and the intersection takes place at their 2nd and 3rd cross with the section respectively. That is, the curves u 2 and s3 intersect (see Fig. 10). On one hand, when the value of C J decreases, the region enclosed by both tubes grows, so that, the set of transit orbits that leave R2 and returns back after some revolutions around m 1 increases. On the other hand, for bigger values of C J we have followed the invariant manifolds up to their 5th and 6th crossing with the section, without finding transit orbits. Notice in Fig. 10 that the curve s3 , intersection between W1s (O L 9 ) with the section at its 3rd crossing, is not closed (in fact, we have not plotted the whole set). This is because for
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M. Alvarez-Ramírez, E. Barrabés CJ=3.4876011612
CJ=3.4700192109
0.08
1 u6
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Fig. 11 Intersection of the invariant manifolds W1s (O L 9 ) (in blue) and W1u (O L 10 ) (in red) at their 6th intersection with the section y = 0 in the plane (x, x ) for the values of the Jacobi constant shown. The common points, at x = 0 correspond to symmetric heteroclinic orbits. The region interior to both curves correspond to transit orbits from R3 –R1 –R2
CJ=3.4700192109
0.8
u3
0.4
x’
Fig. 12 Intersection of W1u (O L 9 ) (in red) with W1s (O L 10 ) (in blue) at their 3rd intersection with the section y = 0 in the the plane (x, x ). The region interior to both curves correspond to transit orbits from R2 –R1 –R3
0 -0.4 s3 -0.8
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that values of the Jacobi constant there already exist heteroclinic connections from O L 10 to O L 9 , and orbits on the W1s (O L 9 ) manifold that go, forwards in time, to the R3 region. That means that there exist transit orbits following the sequence of visits R3 –R1 –R2 . We will see this in more detail next. Finally, we explore the transits from R2 to R3 through R1 , or the other way around. In these cases we need to explore the intersections W1u (O L 9 ) ∩ W1s (O L 10 ) and W1s (O L 9 ) ∩ W1u (O L 10 ). The invariant manifolds W s (O L 9 ) and W u (O L 10 ) are symmetric with respect to y = 0, so are the curves sk and u k , for any k number of crossings with the section Σ = {y = 0}. In fact, the points where the curve sk crosses (transversally) the axis x = 0 correspond to heteroclinic connections from O L 10 to O L 9 . We find that for values of C J close to the value at the equilibrium point, the invariant manifold W s (O L 9 ) already intersects W u (O L 10 ), see Fig. 11. Notice that the first plot correspond to a Jacobi constant very close to C J (L 9 ), and when C J decreases the region of transit orbits grows (compare the range of the variables in both plots). Therefore, as soon as the bottlenecks between R3 or R2 and R1 open, there exists heteroclinic connections and are transit orbits that follow the sequence R3 –R1 –R2 . Nevertheless, the same is not true for the transits from R2 to R3 . There exist such transit orbits, but it is necessary for the Jacobi constant to decrease until C J 3.47 to find them. In Fig. 12 we show the region intersection between W u (O L 9 ) and W s (O L 10 ), at their 3rd crossing with the section, for that value of the Jacobi constant. For bigger values of C J , we have not found any intersection between them (at least, up to eight crossings with the section).
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Transport orbits in an equilateral restricted four-body problem 1.5
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Fig. 13 Heteroclinic connections from O L 10 to O L 9 (left) and from O L 9 to O L 10 (right) for C J = 3.4700192109
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In Fig. 13 we show two heteroclinic connections between O L 9 and O L 10 with different number of revolutions around m 1 . We want to emphasize the following facts: – The methodology we use to find homoclinics (and, therefore, transit orbits) depends on the number of intersections of the invariant manifolds with the specific section. Looking up to three or four crossings with the section, we find transit orbits following the sequence R1 –R2 –R1 for values of C J < 3.466725, whereas we have to decrease to C J 3.409 to find the transit R2 –R1 –R2 . For bigger values of C J , we would have to take a higher number of crossings with the section in order to see if there exist homoclinic orbits. Due to the fact that there are orbits on the stable manifold that after six revolutions go to the region R3 , the tube of the stable manifold breaks and its intersection with the section becomes more complicated as C J decreases. Thus, it becomes more difficult to find intersections between the curves sk an u j . For example, for the Jacobi constant 3.48, we have followed W1s (O L 9 ) for more than 20 intersections with the section and we have not found any orbit entering R2 . – Similarly, there exist orbits visiting R3 –R1 –R2 almost since the channels open, but it is necessary to decrease to C J 3.47 to obtain orbits that follow the sequence of visits R2 –R1 –R3 . 3.2 Case μ = 0.3 As in the previous case, we consider values of C J in (C J (L 2 ), C J (L 9 )), see Figs. 4 and 2. The values corresponding to the equilibrium points are, approximately, C J (L 2 ) = 3.3504838, C J (L 7,8 ) = 3.36094049, C J (L 4 ) = 3.4586913, C J (L 9,10 ) = 3.55775701. The evolution of the Hill’s region in this case is similar to the previous one, with the difference of the existence of the equilibrium points L 1 and L 4 . First the equilibrium points L 9,10 appear connecting the regions R2 and R3 with R1 , then a small channel connecting R2 and R3 opens with the apparition of L 4 . For the range of values C J ∈ (C J (L 1 ), C J (L 4 )), with C J (L 1 ) = 3.4508245244, there exists a tiny island of forbidden motion (see Fig. 4, third plot), and in principle direct transits between regions R2 and R3 can exists. Then, for C J ≤ C J (L 1 ) transit between them must go through R1 . Finally, the channels that connect those regions with the exterior one, L 7,8 and L 2 open. In this case, we focus in the differences with the previous one, and explore the behavior and the intersections of the invariant manifolds associated to the Lyapunov orbits O L 4 , that can allow direct transit between R2 and R3 without going through R1 . The orbits O L 4
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M. Alvarez-Ramírez, E. Barrabés
Stability parameter x 103
Fig. 14 Families of periodic orbits that are born from the equilibrium point L 4 , for μ = 0.3
40 20
OL4
0 -20 -40 -60 -80 -100 -120 3.45
1
1
0.6
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CJ
3.456
3.459
0.2
y
Wu2
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0.2 Ws2
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Fig. 15 Left: Branches of the invariant manifolds associated to a periodic orbit O L 4 (the unstable branches in red, the stable branches in blue) up to their intersection with a given section. The dotted lines show the zero velocity curve and the section. Right: transit orbit visiting R2 –R1 –R3 –R1 –R2 . The value of the Jacobi constant is C J = 3.4585985029
are unstable until approximately C J 3.4506774081, see Fig. 14, where we show the characteristic curve of stability parameter of the family. First, we explore the existence of transits visiting R3 − R2 − R3 . We consider the branches W2u (O L 4 ) and W2s (O L 4 ), that enter the R2 region, and their intersections. For values of C J less than, but close to C J (L 4 ), the whole branches of the invariant manifolds have a transit from the region R2 to R1 , see Fig. 15 left, where the invariant manifolds of a periodic orbit O L 4 for C J = 3.4585985029, only 10−4 less that C J (L 4 ), are plotted. That means that there is no direct connection between R2 and R3 , but there are orbits that can leave and return to R2 after visiting both others regions. For example, the trajectory shown in Fig. 15 right correspond to an orbit belonging to W2s (O L 4 ) that follows the sequence of visits R2 –R1 – R3 –R1 –R2 . Therefore, there must exists an heteroclinic connection between O L 4 and O L 9 , O L 9 and O L 10 , and a homoclinic connection to O L 10 . Knowing that the branches of W u/s (O L 4 ) enter the region R1 , we explore the behavior of W2u (O L 4 ) (branch entering R2 ) and W3s (O L 4 ) (branch entering R3 ) and their possible intersections (in fact that branches are symmetric with respect to y = 0). We find that there also exist symmetric homoclinic connections to O L 4 for the same energy level, see Fig. 16. The region in between the curves u 1 (first intersection of the W u brach with a fixed section) and s2 (second intersection of the W s brach with a fixed section) correspond to transit orbits R3 –R2 –R1 –R3 –R2 . As the Jacobi constant decreases down to the value at which the family becomes stable C J 3.4506774081 (see Fig. 14), the periodic orbit O L 4 gets wider, but still the invariant manifolds have a transit towards the region R1 . At each level there exist homoclinic connec-
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Transport orbits in an equilateral restricted four-body problem 1
-0.36
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Fig. 16 Left: Branches W2u (O L 4 ) and W3s (O L 4 ) of the invariant manifolds associated to a periodic orbit O L 4 up to their intersection with a given section. The dotted lines show the zero velocity curve and the section. Right: intersection curves of the invariant manifolds with the section x = 0.2. The value of the Jacobi constant is C J = 3.4585985029 CJ=3.4585985029
1
CJ=3.4506774082
1 0.6
0.2
0.2
y
0.6
y
Fig. 17 Homoclinic connections from/to O L 4 for different values of the Jacobi constant. Notice that in the first case (left) the equilibrium point L 1 has not appeared, so still a small forbidden region persist close to the periodic orbit O L 4 . In the right, the periodic orbit O L 4 is shown in black
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tions to the periodic orbit, which go through the region R1 , see Fig. 17, right. In particular, there also exist transit orbits following the sequence R2 –R3 –R1 –R2 –R3 . 3.3 Case μ = 0.46 We will study the behavior of the invariant manifolds for the values of the Jacobi constant in (C J (L 2 ), C J (L 4 )), see Figs. 2 and 5. The values corresponding to the equilibrium points are, approximately, C J (L 2 ) = 3.1716076, C J (L 1 ) = 3.21248121. C J (L 7,8 ) = 3.41110339, C J (L 4 ) = 3.86978078. In this case, as C J decreases, first R2 and R3 connect through the bottleneck opened by L 4 , and then these two regions connect with the exterior one, X , through the passages opened by L 7 and L 8 , so there can be orbits visiting R2 and R3 coming from or escaping to the exterior region without the possibility of visiting R1 . Next, after the apparition of L 2 , R1 is connected with R2 and R3 , and finally after the apparition of L 1 , R1 is connected directly with the exterior region. We focus on the families of Lyapunov orbits O L 4 and O L 1 which are the ones that live in the channels that connect the regions Ri , i = 1, 2, 3. We compute the families for wide range of values of the Jacobi constant. In Fig. 18, we plot the value of the linear stability parameter of both families (due to the difference of ranges, each curve is plotted in a different y axis). We can see that for values of C J ≥ C J (L 2 ), the orbits of both families are hyperbolic. In particular, the orbits O L 1 are quite unstable, with values of the stability parameter bigger than 2 × 104 for that range of values of C J .
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M. Alvarez-Ramírez, E. Barrabés 4000 3000 2000 1000 0 -1000 -2000 -3000 -4000 -5000 -6000 2.6 2.8
1.4e+05 OL4
1.2e+05 1.0e+05 8.0e+04 6.0e+04 4.0e+04
OL1 3
3.2 3.4 3.6 3.8
CJ
2.0e+04 4
Stab. parameter OL1
Stab. parameter OL4
Fig. 18 Families of periodic orbits that are born from the equilibrium points L 4 (in red, left vertical axis), and L 1 (in blue, right vertical axis) for μ = 0.46. For C J > C J (L 2 ) (vertical dotted line) all the orbits are unstable
0.0e+00
1 Ws2
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Fig. 19 Branches of the invariant manifolds associated to a Lyapunov periodic orbit around O L 4 (the periodic orbit in black, the unstable branches in red, the stable branches in blue) that enter into the region R2 up to their first (W u ) and second (W s ) intersection with the section {x = −0.3}. The dotted lines show the zero velocity curve and the section used. The small circles mark the position of the masses. The value of the Jacobi constant is C J = 3.8659068300504340
First, we investigate the transit between R2 and R3 by considering successive intersections of the invariant manifolds W u/s (O L 4 ) with a section Σ = {x = c}, for a suitable constant c. Due to the symmetry of the problem, the branches of the invariant manifolds that enter into the region R3 are symmetric to the branches that go towards R2 , so we will consider only the intersections of W2u (O L 4 ) and W2s (O L 4 ) with Σ. See Fig. 19. For values of the Jacobi constant close to the value at L 4 , we already find that W2u (O L 4 ) and W2s (O L 4 ) intersect, see Fig. 20. For C J = 3.8659068301, we show the first intersection of the unstable branch with Σ = {x = −0.3}, and the consecutive intersections of the stable branch with Σ up to its 8th crossing. That means that, as soon as the bottleneck between the regions R2 and R3 opens, there exists orbits transiting from one region to another and bouncing back to the initial region. For smaller values of the Jacobi constant, the tubes grow in width (their projection in the configuration space become wider) and it is difficult to obtain a “clear” intersection with a section due, on one hand, to the loops of the orbits around the primaries, and on the other hand to the presence of orbits with passages close to the primary, and even collision orbits. Next we consider the invariant manifolds associated to the periodic orbits around L 1 that exist in the bottleneck that connects the region R1 with R2 and R3 . In fact, when L 1 appears, the regions R2 and R3 cannot be anymore well identified, so we will consider the region R23 as the neighborhood around masses m 2 and m 3 in between the two forbidden regions of motion in configuration space (see Fig. 5). In this case, we will study the intersection of the
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Transport orbits in an equilateral restricted four-body problem -0.7
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Fig. 20 Intersection of the invariant manifolds W1u (O L 4 ) (in red) and W s (O L 4 ) (in blue) with the section x = −0.3 in the (y, y ) plane for C J = 3.8659068301. Left: the 1st intersection u 1 of W1u (O L 4 ) and the 2nd, 4th and 8th intersection of W s (O L 4 ). Right: detail around u 1 and s8 2
1
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Fig. 21 Branches of the invariant manifolds associated to a periodic orbit O L 1 (the periodic orbit in black, the unstable branches in red, the stable branches in blue) up to their intersection with a given section. The dotted lines show the zero velocity curve and the section. The small circles mark the position of the masses. The value of the Jacobi constant is C J = 3.2119780763
u (O L ) and W s (O L ) in the branches W1u (O L 1 ) and W1s (O L 1 ) in the region R1 and W23 1 1 23 region R23 , see Fig. 21. As soon as the channel around L 1 opens, there exists homoclinic connections and therefore transit orbits that coming from R23 spend some time around m 1 and returns back to the original region, so they follow the sequence R23 –R1 –R23 . In Fig. 22 we show the intersections of the branches W1u (O L 1 ) and W1s (O L 1 ) with the section Σ = {x = 0.8}. The unstable manifold is followed until the first crossing with Σ (curve u 1 ), whereas the stable branch is followed up to different crossings (curves s j in the plots). Each two crossings correspond to one loop around m 1 . As C J decreases, the number of loops around m 1 needed before the stable and the unstable manifold match decreases. As in previous cases, the smaller the value of C J , the bigger the tubes of the manifolds, and the lest the time that the orbits spend inside R1 before having a transit back to R23 . For C J = 3.2119780763088257, the homoclinics are found following W2s (O L 1 ) up to its 10th intersection with Σ, so the transit orbits perform five loops around m 1 . For C J = 3.2043958090 the stable tube matches the unstable one at its 6th intersection. Also, the smaller the value of C J , the more complicated the curves W2s (O L 1 ), due both to the presence of orbits close to collision and by the geometry of the tubes. With respect to the intersections of the invariant manifold in the R23 region, we find u (O L ) and W s (O L ) intersect, that for the value C J = 3.2119848042 the branches W23 1 1 23 which is close to the value of the Jacobi constant at L 1 . Thus, there exists transit following the sequence R1 –R23 –R1 . In this case, the intersection takes place at the 3rd and 4th crossing of W u and W s with the section, respectively, see Fig. 23, left. As the Jacobi constant decreases, the tubes become more complicated and, very soon, there appear collision
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M. Alvarez-Ramírez, E. Barrabés CJ=3.2119780763
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Fig. 23 Projection in the (y, y ) plane of the curves intersection between the invariant manifolds u (O L ) (in red) and W23 1 s (O L ) (in blue) with the W23 1 section x = 1 in the region R23
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Fig. 22 Projection in the (y, y ) plane of the curves intersection between the invariant manifolds W1u (O L 1 ) (in red) and W1s (O L 1 ) (in blue) with the section x = 0.8 in the region R1
7
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orbits with the primaries. That means that the invariant tubes break and their intersection with the section are not any more closed curves. In Fig. 23, center and right, we can see s (O L ) with the section Σ = {x = 1} becomes more complihow the intersection of W23 1 cated in a small range of values of C J . In particular, the section s4 shown in the plot on the right is not a closed curve as the invariant manifold has collision orbits with the primary m 2 . In any case, we observe that the region that correspond to transit orbits is small, although we can expect as in the other cases, that it becomes bigger as the Jacobi constant decreases.
4 Conclusions We have shown transit orbits visiting neighborhoods of different primaries in the restricted planar equilateral four-body problem by using numerical methods. Those transit orbits can be seen as possible trajectories for particles abandoning one mass and traveling to a different one, so we can view them as ways for natural transport of matter. In order to find the transit orbits, we have studied the behavior of the invariant manifolds associated to the equilibrium points located at the neck regions in between the neighborhoods around each mass. The existence of transversal intersections of that invariant manifolds give rise to homoclinic and heteroclinic connections, as well as transit orbits. The exploration has been done for three Table 4 Summary of the transit orbits sequences and the values of the Jacobi constant for which they are found (μ = 0.1) Jacobi constant
Connection and manifolds
Transit orbits sequences
C J ≤ 3.4667254661
Homoclinic to O L 9 in R2 W2u (O L 9 ) ∩ W2s (O L 9 )
R1 –R2 –R1
C J ≤ 3.409
Homoclinic to O L 9 in R1 W1u (O L 9 ) ∩ W1s (O L 9 )
R2 –R1 –R2
C J ≤ 3.48768130
Heteroclinic O L 10 −−O L 9 W1s (O L 9 ) ∩ W1u (O L 10 )
R3 –R1 –R2
C J ≤ 3.4700192109
Heteroclinic O L 9 −−O L 10 W1u (O L 9 ) ∩ W1s (O L 10 )
R2 –R1 –R3
The first value of the Jacobi constant for which they can exist is C J (L 9,10 ) = 3.48768130
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Transport orbits in an equilateral restricted four-body problem Table 5 Summary of the transit orbits sequences and the values of the Jacobi constant for which they are found (μ = 0.3) Jacobi constant C J ≤ 3.4586913 C J ≤ 3.4585985029 C J ≤ 3.4506774081
Connection and manifolds W2u (O L 4 ), W2s (O L 4 ) entering R1 Homoclinic to O L 4 W2u (O L 4 ) ∩ W3s (O L 4 ) Homoclinic to O L 4 W2s (O L 4 ) ∩ W3u (O L 4 )
Transit orbits sequences R2 –R1 –R3 –R1 –R2 R3 –R2 –R1 –R3 –R2 R2 –R3 –R1 –R2 –R3
The first value of the Jacobi constant for which they can exist is C J (L 9,10 ) = 3.55775701 Table 6 Summary of the transit orbits sequences and the values of the Jacobi constant for which they are found (μ = 0.46) Jacobi constant
Connection and manifolds
Transit orbits sequences
C J ≤ 3.8659068301
Homoclinic to O L 4 W2u (O L 4 ) ∩ W2s (O L 4 )
R3 − R2 − R3
C J ≤ 3.8659068301
Homoclinic to O L 4 W3u (O L 4 ) ∩ W3s (O L 4 )
R2 − R3 − R2
C J ≤ 3.2119780763
Homoclinic to O L 1 in R1 W1u (O L 1 ) ∩ W1s (O L 1 )
R23 –R1 –R23
C J ≤ 3.2119848042
Homoclinic to O L 1 in R23 u (O L ) ∩ W s (O L ) W23 1 1 23
R1 –R23 –R1
The first value of the Jacobi constant for which they can exist is C J (L 4 ) = 3.86978078
representative values of the mass parameter μ = 0.1, 0.3 and 0.45, and in all the cases we show that the transit orbits appear soon after the channels through the bottlenecks open, corresponding to values of the Jacobi constant close to the value at the equilibrium points considered. Tables 4, 5, and 6 show a summary of the transit orbits sequences found in the three cases. Acknowledgments M. Alvarez-Ramírez is supported by the project grant Red de cuerpos académicos Ecuaciones Diferenciales. Proyecto sistemas dinámicos y estabilización. PROMEP 2011–SEP Mexico. E. Barrabés is supported by the Grants MTM2010-16425, MTM2013-41168, and 2014 SGR 1145.
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