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Asymptotic orbits in the restricted four-body problem
ARTICLE IN PRESS
Planetary and Space Science 55 (2007) 1368–1379 www.elsevier.com/locate/pss
Asymptotic orbits in the restricted four-body problem K.E. Papadakis Department of Engineering Sciences, Division of Applied Mathematics and Mechanics, University of Patras, GR-26504 Patras, Greece Received 3 October 2006; received in revised form 31 January 2007; accepted 11 February 2007 Available online 27 February 2007
Abstract This paper studies the asymptotic solutions of the restricted planar problem of four bodies, three of which are finite, moving in circular orbits around their center of masses, while the fourth is infinitesimal. Two of the primaries have equal mass and the most-massive primary is located at the origin of the system. We found the invariant unstable and stable manifolds around the hyperbolic Lyapunov periodic orbits which emanate from the collinear equilibrium points Li ; i ¼ 1; . . . ; 4, as well as the invariant manifolds from the Lagrangian critical points L5 and L6 . We construct numerically, applying forward and backward integration from the intersection points of the appropriate Poincare´ cuts, homo- and hetero-clinic, symmetric and non-symmetric asymptotic orbits. We present the characteristic curves of the 24 families which consist of symmetric simple-periodic orbits of the problem for a fixed value of the mass parameter b. The stability of the families is computed and also presented. Sixteen families contain as terminal points asymptotic periodic orbits which intersect the x-axis perpendicularly and tend asymptotically to L5 for t ! þ1 and to L6 for t ! 1, spiralling into (and out of) these points. The corresponding 16 terminating heteroclinic asymptotic orbits, for b ¼ 2, are illustrated. r 2007 Elsevier Ltd. All rights reserved. Keywords: Restricted four-body problem; Asymptotic orbit; Homoclinic orbit; Heteroclinic orbit; Cut of Poincare´ surface of section; Periodic orbit; Lyapunov orbit
1. Introduction and equations of motion There are many reasons for studying the four-body problem besides the historical ones (Lagrange, Hill, Darwin; Multon, 1900), since it is known that approximately two-thirds of the stars in our Galaxy exist as part of multistellar systems. Around one-fifth of these are part of triple systems, while a rough estimate suggests that a further one-fifth of these triples belong to quadruple or higher systems, which can be modeled by the four-body problem. The four-body problem is also increasingly being used to study and explain many of the complex dynamical phenomena found in the Solar system and exoplanetary systems. Studying the stability, the periodic solutions, the asymptotic ones, and the regions of motion in four-body systems is therefore fundamental to understanding the evolution of quadruple stellar and exoplanetary systems (Sze´ll et al., 2004).
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E-mail address: [email protected]. 0032-0633/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2007.02.005
The importance of the asymptotic solutions of a dynamical system is well known. From the time of Stro¨mgren (1935) we know that in a restricted three-body problem there are heteroclinic asymptotic orbits connecting the two triangular critical points. Koon et al. (2000), in the same problem, presented asymptotic orbits which connect pairs of hyperbolic Lyapunov periodic orbits around the collinear equilibrium points. In this work we have done a systematic numerical computation of the invariant stable and unstable manifolds and the corresponding homoclinic and heteroclinic orbits associated with all the equilibrium configurations of the restricted four-body problem. An alternative approach, using analytical procedure based on high-order normal forms computations around the equilibria, could be used in order to obtain invariant manifolds and asymptotic homoclinic and heteroclinic orbits (Deprit and Henrard, 1965; Llibre et al., 1985; Markellos et al., 1995, etc.). We consider m0 , m1 and m2 the masses of three primary bodies where m0 is located at the center of mass (the origin of the coordinate system) and the other two bodies with equal masses are moving in circular orbits around the center
ARTICLE IN PRESS K.E. Papadakis / Planetary and Space Science 55 (2007) 1368–1379
of mass of the system. This model can be considered as a particular case of the restricted regular polygon problem of N þ 1 bodies when the number of the peripheral primaries equals 2 and the mass parameter of the central mass to a peripheral one is different from zero (Kalvouridis, 1999) and it has been used, among others, by Desiniotis and Kazantzis (1993) in their three-magnetic dipoles problem, where they studied the motion of a charged particle in the electromagnetic field produced by the dipoles, Markellos et al. (1995) studied zero velocity curves, periodic orbits, etc. of the specific four-body problem Maran˜hao (1995), Maran˜hao and Llibre (1999) investigated ejection–collision orbits and invariant punctured tori of the same problem, Cors et al. (2004) studied the central configurations of the problem, and in this journal Kalvouridis et al. (2006) explored the dynamical properties of the photogravitational version of the problem. The equations of motion of the infinitesimal mass, m3 , of the restricted four-body problem, in the usual dimensionless rectangular rotating coordinate system are written as (Kalvouridis et al., 2006) qO 1 bx x 12 x þ 12 ¼x x€ 2y_ ¼ þ þ , qx D r30 r31 r32 qO 1 b 1 1 ¼y 1 þ þ , ð1Þ y€ þ 2x_ ¼ qy D r30 r31 r32
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of length is chosen in such a way that the distance between the primaries m1 and m2 is equal to 1. So, the peripheral primaries m1 , m2 are placed at the position ð12; 0Þ and ð12; 0Þ, correspondingly (Fig. 1, left). We remark that for b ¼ 0, the Copenhagen case of the restricted three-body problem is produced. The energy (Jacobi) integral of this problem is given by the expression x_ 2 þ y_ 2 ¼ 2O C,
(4)
(3)
where C is the Jacobi constant. The equations of motion of the restricted four-body problem, as in the classical three-body one, have the property that if x ¼ xðtÞ, y ¼ yðtÞ is a solution, then x ¼ xðtÞ, y ¼ yðtÞ is also a solution. This symmetry follows from inspection of the terms occurring in the differential equations (1), where if we substitute x ! x, _ y_ ! y, _ x€ ! x€ and y€ ! y, € the y ! y, x_ ! x, equations remain unchangeable, a fact that verifies the preceding statement. This symmetry will be useful in the computations in the fourth section of the present work. The restricted four-body problem has six equilibrium points (in contrast to the classical restricted three-body problem which has five), four collinear L1;2;3;4 , and two ‘‘triangular’’ L5;6 (Kalvouridis et al., 2006). The first four are on the primaries line: L1 on the positive axis between m0 and m1 , L2 on the positive axis outside m1 and L4 and L3 are symmetric to L1 and L2 , respectively, with respect to the origin. From the differential equations of motion (1) we obtain the coordinates x0 , y0 , of the Lagrangian equilibrium points as solutions of the equations: qO 1 b 1 1 1 1 1 þ þ ¼ x0 1 þ ¼ 0, qx D r300 r310 r320 2D r310 r320 qO 1 b 1 1 ¼ y0 1 þ þ ¼ 0, ð5Þ qy D r300 r310 r320
Here, b ¼ m0 =m is the ratio of the central mass m0 to one of the other two equal primaries m ¼ m1 ¼ m2 and the unit
where r00 , r10 and r20 are the distances given by (3) at the equilibrium. For the Lagrangian points L5;6 , ðy0 a0Þ we
where dots denote time derivatives while the gravitational potential O in synodic coordinates is defined as 1 1 b 1 1 þ þ O ¼ ðx2 þ y2 Þ þ , (2) 2 D r0 r1 r2 where D ¼ 2ð1 þ 4bÞ;
r20 ¼ x2 þ y2 ,
2 r21 ¼ x 12 þ y2 ;
2 r22 ¼ x þ 12 þ y2 .
y
0.5
1.0
0.5
μ∗
r0
r2
0.4
m3 (x,y)
L5
r1
0.3
L5
X -1.0
L3
-0.5 L4 0 0 L1 m0 m2 -0.5
L6
0.5 m1
L2
1.0
0.2 L1
0.1
L2 distance
-1.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fig. 1. Left: The restricted four-body problem. Right: the exact positions, obtained numerically, of the equilibrium points on the x-axis (for L1 and L2 ) and on the y-axis (for L5 ), as functions of the mass parameter m .
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obtain r10 ¼ r20 which means that x0 ¼ 0 and therefore L5;6 are located symmetrically with respect to the origin, on the y-axis and their positions are given by the following equation (L5 on the positive axis), 1 b 2 1 þ ¼ 0, (6) D r300 r310
equilibrium points are unstable for every value of the mass ratio m, and now in the present problem the collinear equilibria are unstable for every value of the mass parameter b. The corresponding eigenvectors can be normalized to 1 (because of the type of the matrix A), u1 ¼ ð1; s; l; lsÞT ;
w1 ¼ ð1; it; in; ntÞT ,
where,
u2 ¼ ð1; s; l; lsÞT ;
w2 ¼ ð1; it; in; ntÞT ,
r00 ¼ jy0 j and
r10
qffiffiffiffiffiffiffiffiffiffiffiffi ¼ y20 þ 14.
(7)
In Fig. 1 (right) the exact positions of the equilibrium points on the x-axis (for L1 and L2 ) and on the y-axis (for L5 ), obtained numerically, are plotted as functions of the mass parameter m ¼ 1=ð2 þ bÞ ¼ m=ðm0 þ 2mÞ corresponds to the classical mass parameter m of the restricted three-body problem. 2. Linearization around the equilibrium configurations 2.1. Collinear equilibrium points Li ; i ¼ 1; . . . ; 4 The linearized equations for infinitesimal motions near the collinear equilibrium points are x_ ¼ Ax;
_ yÞ _T x ¼ ðx; y; x;
(8)
where x is the state vector of the fourth particle with respect to the equilibrium points and the time-independent coefficient matrix A is 0 1 0 0 1 0 B 0 0 0 1C B C A¼B (9) C, 0 0 2A @ 1 þ 2A0 0 1 A0 2 0 where
! 1 b 1 1 A0 ¼ þ þ , D jx0 j3 jx0 12j3 jx0 þ 12j3
(10)
and x0 is the abscissa of the collinear equilibrium point as solution of the equation, 1 bx0 x0 12 x0 þ 12 þ 3 þ 3 x0 ¼ 0, (11) D r30 r1 r2 where r0 ¼ jx0 j, r1 ¼ jx0 12j and r2 ¼ jx0 þ 12j are the distances at the equilibrium. The characteristic equation of the linear system (8) is l4 þ ð2 A0 Þl2 þ ð2A20 þ A0 þ 1Þ ¼ 0
(12)
and it has one real and one imaginary pair of eigenvalues which have the form l and in where l and n are the positive constants, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ 12 ½A0 2 þ A0 ð9A0 8Þ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ 12 ½2 A0 þ A0 ð9A0 8Þ. ð13Þ As in the restricted three-body problem, where the collinear
where s and t are the constants, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ 3A0 A0 ð9A0 8Þ s ¼ pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 2 A0 2 þ A0 ð9A0 8Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ 3A0 þ A0 ð9A0 8Þ t ¼ pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 2 A0 2 A0 ð9A0 8Þ
ð14Þ
ð15Þ
It is known, in the restricted three-body problem, that from collinear equilibrium points Li ; i ¼ 1; 2; 3, emanate the families a (from L2 ), b (from L3 ) and c (from L1 ) (according to the classical nomenclature). For Jacobi constant C just below C Li hyperbolic periodic orbits around each Li exist, called Lyapunov orbits. Similarly, in the four-body problem, emanates a Lyapunov family from each collinear equilibrium point Li ; i ¼ 1; 2; 3; 4 as we will see in Fig. 3 of Section 3 (family f 1 from L1 , f 7 from L2 , f 13 from L3 and family f 19 from L4 ). 2.2. Lagrangian equilibrium points L5ð6Þ To study analytically the solutions in the neighborhood of the equilibrium configuration L5ð6Þ , we transfer the origin at L5ð6Þ , we linearize the equations of motion and we receive for coplanar motion, the expression 0 1 0 0 1 0 B 0 0 0 1C B C (16) x_ ¼ Bx with B ¼ B C, 0 0 2A @ c31 0 3 c31 2 0 where we have abbreviated c31 ¼
48 Dð1 þ 4y20 Þ5=2
,
(17)
and y0 is given by (6). The corresponding characteristic equation of B is " # 144 D 16 4 2 l þ l þ d ¼ 0; d ¼ 2 D ð1 þ 4y20 Þ5=2 ð1 þ 4y20 Þ5 (18) and its eigenvalues are: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l1;2;3;4 ¼ pffiffiffi 1 1 4d ¼ p iq 2
(19)
with p and q being the reals and strictly positive quantities, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi p ¼ 12 2 d 1 and q ¼ 12 2 d þ 1. (20)
ARTICLE IN PRESS K.E. Papadakis / Planetary and Space Science 55 (2007) 1368–1379
The Lagrangian equilibrium points L5ð6Þ are linearly stable if these four roots are purely imaginary i.e. if, do14
(21)
which happens when b4bcrit ffi 11:72034889, as we see in Fig. 2 in which the diagram of the mass parameter b versus the quantities d and 1 4d is presented. Note that d is always positive for any value of the mass parameter b (Fig. 2, left frame). The Lagrangian equilibrium points L5ð6Þ are linearly unstable if bobcrit . In the first case (b4bcrit ) we have the imaginary roots, l1;2 ¼ io1 and l3;4 ¼ io2 . In the right frame of Fig. 2 we present the variation of the eigenfrequencies or mean motions o1 , o2 when the mass parameter b varies. In that case we have solution with ‘‘short’’ and ‘‘long’’ period terms corresponding to large and small values of the eigenfrequencies (for details, see Papadakis, 2006). In the second case, i.e. for bobcrit , the solution of the linear part of (1) is, ¯ ¼ ept ða1 cos qt þ a2 sin qtÞ þ ept ða3 cos qt þ a4 sin qtÞ, xðtÞ Z¯ ðtÞ ¼ ept ðb1 cos qt þ b2 sin qtÞ þ ept ðb3 cos qt þ b4 sin qtÞ, where the coefficients ai and bi are not independent but are connected by the following set of equations: b1 ¼ c 1 a1 þ c 2 a2 ;
b2 ¼ c2 a1 þ c1 a2 , b4 ¼ c2 a3 c1 a4 ,
manifold W sL5;6 in the vicinity of libration point. Similarly, if a3 ¼ a4 ¼ 0 we have an approximation of the unstable invariant manifold W uL5;6 . For a numerical globalization of the invariant unstable manifold W sL5 , we adopt an appropriate set of initial conditions ¯ xð0Þ ¼ a1 ; Z¯ ð0Þ ¼ c1 a1 þ c2 a2 , ¯_ xð0Þ ¼ pa1 þ qa2 ; Z_¯ ð0Þ ¼ ðpc1 qc2 Þa1 þ ðpc2 þ qc1 Þa2 , ð25Þ where a1 , a2 are chosen (see Go´mez et al., 1988) a1 ¼ r cos y
ð23Þ
where pffiffiffi 3 þ 2ð3 c31 Þc31 8 d þ 2d p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi and pffiffiffi pffiffiffi 4ðc31 þ d 3Þ 2 d 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi ð2c31 3Þ 2 d þ 1 pffiffiffi . ð24Þ c2 ¼ 4ðc31 þ d 3Þ The four initial conditions x¯ 0 , Z¯ 0 , x_¯ 0 and Z_¯ 0 are linear functions of the four constants a1 ; . . . ; a4 and selecting a1 ¼ a2 ¼ 0 (then b1 ¼ b2 ¼ 0 from (23)), the solution approaches the equilibrium point asymptotically and so the solution (22) give an approximation of the stable invariant c1 ¼
b
r sin y c1 r cos y c2
(26)
with r40, y 2 ½0; 2p, which means that when t ¼ 0 then ¯ ðxð0Þ; Z¯ ð0ÞÞ describes a small circle ðr cos y; r sin yÞ around the libration point L5 on the linear part of the manifold.
In this section we present all the families of the planar symmetric simple-periodic orbits of the restricted fourbody problem, for b ¼ 2 (see also Kalvouridis et al., 2006), as well as the asymptotic orbits which associated to these. There are 24 families, named f i ; i ¼ 1; . . . ; 24, which consist of simple symmetric periodic orbits (i.e. having only two both perpendicular, intersections with the x-axis). We can separate them in four groups one in each collinear equilibrium point area. So we have 6 families in L1 area, 6 in L2 , etc. The heavy red lines, in Fig. 3, indicate the horizontal stable arcs of the simple families. The small families f 9 and f 22 are shown in two inside windows of Fig. 3. The positions of the bodies m0 , m1 and m2 are denoted by three vertical lines, while the shaded areas represent the regions of forbidden motion in the ðx0 ; CÞ plane. We note that we have considered the periodic orbits (symmetric with respect to the x-axis) as represented by their initial conditions x0 , y0 ¼ 0, x_ 0 ¼ 0 and y_ 0 40 (‘‘positive’’ perpendicular intersection of the x-axis). 1.0
12
a2 ¼
and
3. Simple-families and asymptotic orbits
ð22Þ
b3 ¼ c1 a3 þ c2 a4 ;
1371
bcrit
ωi
0.9
10
0.8
8
0.7
ω2
ω1 = ω2 =1/21/2
0.6 6 0.5 4
ω1
0.4 δ
1-4δ
0.3
2
bcrit =11.7203
0.2
b
0 -6
-5
-4
-3
-2
-1
0
1
2
5
10
15
20
25
30
35
40
45
50
Fig. 2. Left: the evolution of the quantities d and 1 4d as the mass parameter b varies. Right: the eigenfrequencies at the libration points L5ð6Þ .
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Sixteen families contain as terminal points asymptotic periodic orbits which intersect the x-axis perpendicularly and tend asymptotically to L5 for t ! þ1 and to L6 for t ! 1, spiralling into (and out of) these points. Actually these orbits are a combination of two heteroclinic asymptotic orbits running from L5 to L6 and vice versa. All these asymptotic periodic orbits have C ¼ C L5ð6Þ and the characteristics of the respective families spiral around points in the ðx0 ; CÞ plane with C ¼ C L5ð6Þ . These are the natural termination points I–IV which are presented in Fig. 3. In each point terminate four families. Details of the areas of these points are illustrated in four frames of Fig. 4. All the 16 terminating heteroclinic asymptotic orbits, for b ¼ 2, are presented in Fig. 5. As in the restricted three-body problem, and in the present problem we have Lyapunov families which emanate from the collinear equilibrium points Li ; i ¼ 1; . . . ; 4. From L1 emanates the family f 1 , from L2 the family f 7 , from L3 the family f 13 and from L4 the 2 C f20
L3
f17
f13
1
f3
L f8
f6 f21
f18 CL(5,6)
L4
f5
f19
f22
f4 IV
III
L2
f9 f f11 f1
I
10
f
f7
II f2
f14
1.44
f23
1.41 f15
f9
1.38
0 f24
0.500
0.505
1.14 1.11 f16
f22
1.08
-1 -0.500 -0.495 m0
m2
-2
-1
m1
X
0
1
Fig. 3. The family networks of the symmetric simple-periodic orbits of the four-body problem for b ¼ 2.
1.05
C
family f 19 . In contrast of the three-body problem, all the Lyapunov families terminate with asymptotic heteroclinic orbits at the points I–IV (Figs. 3 and 4). 4. Asymptotic orbits around Lyapunov periodic orbits In the first subsection we are going to present invariant stable W sLi p:o: and unstable W uLi p:o: manifolds, which correspond to the Lyapunov periodic orbits around the equilibrium points Li ; i ¼ 1; . . . ; 4 as well as homoclinic and heteroclinic orbits associated with the Lyapunov orbits for two values of energy C and for b ¼ 2. In the second subsection we study numerically the asymptotic homoclinic and heteroclinic orbits associated with the Lagrangian equilibrium points L5 and L6 . 4.1. Asymptotic orbits ‘‘around’’ collinear equilibrium points As we have already mentioned, for fixed values of the Jacobi constant C just smaller than the values of C at the collinear equilibrium points Li ; i ¼ 1; . . . ; 4 we have unstable periodic orbits around Li , called Lyapunov orbits. The Hill’s region (keeping the same terminology as in the restricted three-body problem) corresponding to such values of the Jacobi constant contains ‘‘necks’’ about the equilibrium points as we see in Figs. 6 and 8. We have chosen two values of C so that two cases are created. First case (Fig. 6) where we have two ‘‘necks’’ close to L1 and L4 , respectively, and a second one (Fig. 8) where four ‘‘necks’’ close to each of the four equilibrium collinear points exist. The aim here is to describe the invariant unstable and stable manifolds of the Lyapunov periodic orbits around the collinear equilibrium points and to compute numerically the corresponding homoclinic and heteroclinic orbits which exist in these regions. The existence and the determination of such orbits are described later. We note that if we know the unstable manifold of a Lyapunov orbit, then the corresponding stable manifold is obtained using the symmetric property of the equations of the problem as we have mentioned in the Section 1. This observation will be used to find the transversal homoclinic or heteroclinic orbits. By transversal homoclinic (heteroclinic) orbit, we mean that appropriate unstable and stable manifolds intersect transversally. We recall that a homoclinic orbit 1.2
C
C
f18 CL(5,6)
CL(5,6) 1.0
1.00
I
f2
f1
f3
1.0
II
f21
1.1
CL(5,6) 1.0
III
CL(5,6) IV
f12
f4 0.9
C
1.1
1.1
f7 X
0.95 0.00 0.01 0.02 0.03 0.04 0.05 0.500
0.9 f10
f11 0.505
f15
f14
0.9
f13 X
X 0.510
-1.4
-1.3
-1.2
-1.1
f24
0.8 -1.0 -0.50
Fig. 4. Details of the map of the family characteristics around areas I–IV, respectively.
f23 -0.45
f19 X -0.40
-0.35
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1.0
1373
1.0 1.0
f1 0.5
L5
0.0
m2
0.5
m0
0.0
m1
-0.5
L6
-0.5
1.0 f4
f3
f2 0.5
0.5
0.0
0.0
-0.5
-0.5
-1.0 -1.0
-1.0 -0.5
0.0
0.5
-0.5
0.0
0.5
1.0
1.5
1.0
0.0
0.5
0.0 -0.5
0.0
0.5
1.0 f11
f10
0.5
-0.5
1.0
f7
1.0
-1.0 -0.5
f12
0.5
0.5
0.5
0.0
0.0
0.0
-0.5
-0.5
-0.5
-1.0 -1.0 -0.5
0.0
0.5
1.0
1.0
1.5 f13
-1.0 -0.5
0.0
0.5
1.0 f14
1.0
-1.0 -0.5
0.0
0.5
-0.5
f15
1.0
0.5
0.5
0.5
0.0
0.0
0.0
0.0
-0.5
-0.5
-0.5
-0.5
-1.0
-1.0
-1.0
-1.0
-1.0
-0.5
0.0
0.5
-1.5 1.0
-1.0
-0.5
0.0
0.5
-1.5
-1.0
-0.5
0.0
0.5
0.5 f18
1.0
0.5
-1.5 1.0
0.0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
1.0 f21
f19
f24
1.0
f23
0.5
0.5
0.5
0.5
0.0
0.0
0.0
0.0
-0.5
-0.5
-0.5
-1.0
-1.0
-1.0 -1.0
-0.5 -1.0
-0.5
0.0
0.5
-0.5
0.0
0.5
-0.5
0.0
0.5
1.0
-0.5
0.0
0.5
1.0
1.5
Fig. 5. The 16 asymptotic orbits which are the termination of the corresponding families as marked on the frames, respectively.
related to an equilibrium point L or to a periodic orbit P is an orbit that tends to L (or to P) for t ! 1. Using the analysis of Deprit and Henrard (1969), or the analysis of Llibre et al. (1985), or the purely numerical method presented by Simo´ and Stuchi (2000), we can determine the invariant unstable manifold W uLip:o: emerging from each unstable Lyapunov orbit close to any of the collinear equilibrium points L1 , L2 , L3 and L4 . It is known (Conley, 1968) that unstable (and stable W sp:o: ) manifolds of these Lyapunov orbits are two dimensional, locally diffeomorphic to cylinders (e.g. Fig. 6), while the invariant unstable W uLi and stable W sLi manifolds associated to
Li ; i ¼ 1; 2; 3; 4, are one dimensional (case which is not examined in the present paper). In this way we compute the surface of section of the invariant manifold with the plane y ¼ 0 (or other appropriate plane) one or two etc. times, and obtain a closed curve which is diffeomorphic to a circle (e.g. first frame of Fig. 7). We call this intersection the (first or second, etc) ‘‘cut’’ of W uLip:o: with the appropriate plane. In Fig. 6 we present the unstable manifolds W uL4 p:o: which correspond to the Lyapunov periodic orbit (white dashed line-left) around the collinear equilibrium point L4 when the middle primary is twice larger than m1 , m2 (i.e. b ¼ 2) and for C ¼ 1:46. Similarly the stable invariant
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y
this homoclinic point A and the value of the Jacobi constant C (i.e. x0 ¼ ðx0 ; 0; 0; y_ 0 ðCÞÞ), we produce the homoclinic trajectory which is presented in the second frame of the first row of Fig. 7. In the second row-left the first three unstable cuts of L1 and the second unstable cut of L4 are presented. The stable third cut of L1 is indicated by the dashed line. The intersection of the second unstable cut of L4 and the third stable cut of L1 defines the heteroclinic intersection point B. In the next frame (same row) the corresponding heteroclinic orbit which connect the unstable Lyapunov periodic orbit around L1 and the Lyapunov one around L4 is illustrated. We note here that only the third cut of W sL1 p:o: and the second cut of W uL4 p:o: are used to define point B. In the last frame of Fig. 7 we present the first 10 cuts of the unstable manifolds W uLð1;4Þ p:o: and the stable W sLð1;4Þ p:o:
C=1.46
0.2 W uL
4p.o.
W uL4p.o.
W sL1p.o.
m0
0.0 L3
m1
m2
L2
W sL
1p.o.
-0.2 b=2 X -1.0
-0.5
-0.0
-0.5
-1.0
Fig. 6. Unstable manifolds W uL4 p:o: of the Lyapunov periodic orbit (white dashed line-left) near L4 (yellow and green tubes). Stable manifolds W sL1 p:o: of the Lyapunov periodic orbit (wh dashed line-right) near L1 (red and blue tubes).
manifolds W sL1 p:o: of the Lyapunov periodic orbit around L1 (white dashed line-right) are also illustrated for the same values of b and C. So in this figure we illustrate the position space projection of the unstable manifold W uL4 p:o: (yellowgreen tube) and the stable one of L1 (red-blue tube) until the first intersection with the Poincare´ section at y ¼ 0. These (and the next) intersections as well as their _ symmetric, with respect to the x-axis, we are going to present in Fig. 7. In the next figures homoclinic orbits and heteroclinic ones which connect the Lyapunov periodic orbits are presented. These asymptotic orbits constructed numerically by finding an intersection of the manifolds W uLi p:o: and W sLi p:o: ; i ¼ 1; 4, on the appropriate chosen Poincare´ section at y ¼ 0 (the line passing through the three primary bodies in the rotating frame). Due to reversibility of the restricted four-body problem, if we have an intersection point A of the unstable invariant manifold with the x-axis, then the stable one will also intersect the x-axis at the same point. The common point A corresponds to symmetric orbit (with respect to the x-axis) which is homoclinic to the Lyapunov orbit around the equilibrium point Li . In the first row-left frame of Fig. 7 we present the first Poincare´ cut of the unstable invariant manifold W uL1 p:o: with the plane y ¼ 0 of the Lyapunov periodic orbit around the equilibrium point L1 for b ¼ 2 and C ¼ 1:46. Letter A stands for the intersection point of the first cut of W uL1 p:o: with x-axis. As we have already mentioned, the corresponding cut of the stable invariant manifold W sL1 p:o: will intersect the x-axis at the same point A. With forward and backward integration using the initial conditions of
for b ¼ 2 and C ¼ 1:46, which define the permissible region _ plane. of motion in the ðx; xÞ So far we study the asymptotic orbits in the case where the energy constant has the value C ¼ 1:46 and the Hill region is closed and two of the four collinear equilibrium points are out of it. Now we decrease the energy and we take C ¼ 1:36 (a value just below the energy of L2;3 ) where the closed region opens and four ‘‘necks’’ close to equilibrium points exist. In that case and for b ¼ 2 the unstable invariant manifolds of the Lyapunov periodic orbits around the four collinear equilibrium points in the interior and the exterior Hill’s region are presented (Fig. 8). Like earlier, we compute the appropriate cuts with the plane y ¼ 0 in order to find the intersections with x-axis (homoclinic intersections) or between cuts (heteroclinic intersections). In Fig. 9 (left frame) the first Poincare´ cut of the unstable manifold of the Lyapunov periodic orbit around L1 i.e. W uL1 p:o: and the second cut of the stable one of W sL2 p:o: for b ¼ 2 and C ¼ 1:36 are presented. In the middle frame we present an heteroclinic point which is the intersection of the first Poincare´ cut of the stable manifold W sL3 p:o: and the second cut of the unstable one of W uL4 p:o: . In the last frame the homoclinic points F and G which are the intersections of the first Poincare´ cut of the stable manifold W sL4 p:o: and the second cut of the unstable one of W uL1 p:o: with x-axis correspondingly are presented. From these intersection points we calculate and present in Fig. 10 the four corresponding orbits, two homoclinic and two heteroclinic, around the Lyapunov periodic orbits which emanate from the collinear equilibrium points. 4.2. Asymptotic orbits ‘‘around’’ Lagrangian equilibrium points In this subsection we study the unstable and stable manifolds as well as the asymptotic homoclinic and heteroclinic orbits associated with the Lagrangian equilibrium points L5 and L6 of the restricted four-body problem. Using Eqs. (25) and (26) we compute and present in Fig. 11 the unstable manifold of the equilibrium
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0.0
A
x
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y 0.2
-0.5 A
cut1 uL1
-1.0
0.0 L3
m2 L4
m0
L1 m1
L2
-1.5 -0.2 -2.0
x -0.08
2
-0.06
-0.04
-0.02
-0.00
-1.0
xx
-0.5
0.0
0.5
1.0
y 0.2
cut3 sL1 1 cut2 uL4
B B
0
-1
0.0
L3
m2
L4
m0
L1 m1
L2
cut2 uL1
cut3 uL1
-0.2 cut1 uL1
-2 -0.4
-0.2
x
x
0.0
0.2
-1.0
0.4
-0.5
0.0
0.5
1.0
2 x
1
0
-1
x -2 -0.4
-0.2
0.0
0.2
0.4
W uL1 p:o:
Fig. 7. First row-left: the first Poincare´ cut of the unstable manifolds with the plane y ¼ 0, for b ¼ 2 and C ¼ 1:46. Right: the corresponding homoclinic orbit of the Lyapunov periodic orbit around L1 . Second row-left: the first three unstable cuts of L1 and the second unstable cut of L4 . The stable third cut of L1 is indicated by the dashed line. Right: the corresponding heteroclinic orbit of the Lyapunov periodic orbits around L1 and L4 . Last frame: the first 10 cuts of the unstable manifolds W uLð1;4Þ p:o: and the stable W sLð1;4Þ p:o: for the same values of b and C as above.
point L5 . In order to avoid congestion of the picture we present in the left frame the part of the unstable manifold for y 2 ½0; p while in the right frame we give the part of the unstable manifold for y 2 ðp; 2p. As we have already mentioned in the Introduction, the equations of motion of the problem have the property that if x ¼ xðtÞ, y ¼ yðtÞ is a solution, then x ¼ xðtÞ,
y ¼ yðtÞ is also a solution. The stable manifold of L5 is the symmetric picture of Fig. 11 (both frames as one figure) with respect to y-axis due to the above symmetry property of the problem. The stable and unstable manifolds of the equilibrium point L6 are symmetric with respect to the x-axis and are also not presented here.
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Fig. 8. Left: all the unstable manifolds W uLi p:o: of the Lyapunov periodic orbits near the collinear Li ; i ¼ 1; . . . ; 4, for b ¼ 2 and C ¼ 1:36, just below the energy of L2;3 . Right: details of the interior region of the equilibrium points.
Fig. 9. Left: the first Poincare´ cut of the unstable manifold W uL1 p:o: and the second cut of the stable one of W sL2 p:o: , for b ¼ 2 and C ¼ 1:36. Middle: the first Poincare´ cut of the stable manifold W sL3 p:o: and the second cut of the unstable one of W uL4 p:o: . Right: the first Poincare´ cut of the stable manifold W sL4 p:o: and the second cut of the unstable one of W uL1 p:o: .
Fig. 10. The corresponding homoclinic and heteroclinic orbits of the points D–G of Fig. 9.
In the first row-left of Fig. 12 the first cut of the unstable invariant manifold of L5 , which we denote it by cut1uL5 , and the stable one cut1sL5 from the same Lagrangian point are
presented. The four intersection points A, B, D and E on the x-axis are heteroclinic points from which we can calculate symmetric, with respect to the x-axis (as x_ 0 ¼ 0), heteroclinic orbits running from L5 to L6 . These four heteroclinic orbits are computed and we found that they are the termination asymptotic orbits of the families f 13 (the heteroclinic which corresponds to point A), f 3 (point B), f 15 (point D) and f 18 (point E) correspondingly. The characteristic curves of these families are presented in Fig. 3 and the four heteroclinic orbits keeping the same notation ðf 13 ; f 3 ; f 15 ; f 18 Þ are illustrated in Fig. 5. In the right frame of the first row of Fig. 12 we present the second cut of the stable and unstable invariant manifold of L5 . We observe now that we have many intersection points on the x-axis but we have also many intersection points between the unstable and the stable cuts off the x-axis. This means that except of symmetric, nonsymmetric heteroclinic orbits from L5 to L6 will exist. Indeed we calculate, for example, two points on the x-axis (points F and G) and two off the x-axis (points H and K)
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Fig. 11. The unstable invariant manifold of L5 for y 2 ½0; p (left) and for y 2 ðp; 2p (right).
Fig. 12. First row: the first Poincare´ cut (left) and the second one (right) of the stable and unstable invariant manifold of L5 for b ¼ 2. Second row: two symmetric and two non-symmetric heteroclinic orbits correspond to the intersection points F, G, H and K.
and we compute the four corresponding heteroclinic orbits. In the second row of Fig. 12 two symmetric with respect to x-axis heteroclinic orbits and two non-symmetric ones are presented. Similar to above, we can find the invariant manifolds and cuts as well as the corresponding heteroclinic orbits of the symmetric Lagrangian point L6 . We found, so far, homoclinic or heteroclinic points using the x–x_ cuts of the unstable (stable) manifolds. We can construct asymptotic orbits numerically, by finding inter-
sections of the unstable W u5ð6Þ and stable W s5ð6Þ , manifolds on an other appropriate Poincare´ section map. So, if we choose the Poincare´ section at the plane x ¼ 0, then the corresponding y–y_ cuts of the unstable and stable manifolds of the L5 ð6Þ are shown in Fig. 13. In particular, we present the fifth (first row-left) and the combination of the sixth and seventh (right) Poincare´ cut of the stable and unstable invariant manifolds of L5ð6Þ with the plane x ¼ 0, for b ¼ 2. Some of the existing intersection points which produce homo-heteroclinic orbits are denoted in these
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Fig. 13. First row: the fifth Poincare´ cut of the stable and unstable invariant manifold of L5 (left) and the sixth cut of unstable manifold of L6 with the seventh cut of stable manifold of L5 (right) for b ¼ 2. Second row: two symmetric, with respect to y-axis, and one non-symmetric homoclinic orbits correspond to the intersection points M, N, and O. Last frame: a non-symmetric heteroclinic orbit from L6 to L5 corresponds to the intersection point P.
figures. We found a lot of such orbits using the intersection points on and off the y-axis in order to construct numerically, applying forward and backward integration, homo- and hetero-clinic, symmetric and non-symmetric asymptotic orbits. Four of them, as example, which correspond to the points M, N, O (left frame) and P (right frame of the first row of Fig. 13) are presented in the second row of the same figure. 5. Conclusions In the present work we extended previous studies (Go´mez et al., 1988; Koon et al., 2000) about asymptotic solutions of the restricted three-body problem to the restricted four-body problem. We studied numerically the behavior of the orbits of the equations of motion of the problem when the Jacobi constant is just below that of the collinear critical points Li ; i ¼ 1; . . . ; 4. The asymptotic solutions associated with the Lagrangian equilibrium points L5ð6Þ were also calculated. The geometrical behavior of the stable–unstable manifolds of the six libration points was studied. The network of the 24 existing families of the symmetric simple periodic orbits of the problem as well as the asymptotic termination orbits of 16 of them were found and presented. From the results we conclude that there is an infinite number of asymptotic orbits (simple and higher-multiplicity) since the stable and unstable invariant manifolds produce an infinite number of homoclinic and heteroclinic points. Compared with the Copenhagen restricted three-body problem (He´non, 1965) the simple symmetric
heteroclinic orbits (i.e. having only two both perpendicular intersections with the x-axis) of the problem are now four more, namely 16 instead of 12. All these asymptotic orbits have C ¼ C L5ð6Þ and the characteristics of the respective families spiral around points in the (x0 ; C) plane which have ordinate C ¼ C L5ð6Þ (Figs. 3 and 4). The network of the simple families of the present problem consists of 24 members instead of the 22 families of the Copenhagen problem. The Lyapunov families are now four because of the existing of the fourth collinear libration point. Symmetric and non-symmetric homo- and hetero-clinic asymptotic orbits associated with the six critical points of the restricted four-body problem were calculated and presented. The corresponding Poincare´ cuts of the unstable and stable invariant manifolds, on appropriate planes, were also shown. We found that these orbits are not changed, qualitatively, to the corresponding asymptotic orbits associated with the equilibrium points of the restricted three-body problem. References Conley, C., 1968. Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math. 16, 732–746. Cors, J., Llibre, J., Olle, M., 2004. Central configurations of the planar coorbital satellite problem. Cel. Mech. 89, 319–342. Deprit, A., Henrard, J., 1965. Symmetric doubly asymptotic orbits in the restricted three-body problem. Astron. J. 70, 271–274. Deprit, A., Henrard, J., 1969. Construction of orbits asymptotic to a periodic orbit. Astron. J. 74, 308–316. Desiniotis, C., Kazantzis, P., 1993. The equilibrium configurations of the three-dipole problem. Astrophys. Space Sci. 202, 89–112.
ARTICLE IN PRESS K.E. Papadakis / Planetary and Space Science 55 (2007) 1368–1379 Go´mez, G., Llibre, J., Masdemont, J., 1988. Homoclinic and heteroclinic solutions in the restricted three-body problem. Cel. Mech. 44, 239–259. He´non, M., 1965. Exploration nume´rique du proble´me restreint. Ann. Astrop. 28, 499–511. Kalvouridis, T.J., 1999. A planar case of the n þ 1 body problem: ‘ring’ problem. Astrophys. Space Sci. 260, 309–325. Kalvouridis, T.J., Arribas, M., Elipe, A., 2006. Parametric evolution of periodic orbits in the restricted four-body problem with radiation pressure. Planet. Space Sci. 55, 475–493. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D., 2000. Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10, 427–469. Llibre, J., Martı´ nez, R., Simo´, C., 1985. Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem. J. Differential Equations 58, 104–156. Maran˜hao, D.L., 1995. Estudi del flux d’un problema restringit de quatre cossos. Ph.D. Thesis, Universitat Autonoma de Barcelona, Barcelona.
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Maran˜hao, D.L., Llibre, J., 1999. Ejection–collision orbits and invariant punctured tori in a restricted four-body problem. Cel. Mech. Dyn. Astron. 71, 1–14. Markellos, V.V., Perdios, E.A., Papadakis, K.E., 1995. In: Roy, A., Stevens, B. (Eds.), From Newton to Chaos. Plenum Press, New York. Multon, F., 1900. On a class of particular solutions of the problem of four bodies. Am. J. Math. 1, 17–29. Papadakis, K.E., 2006. Asymmetric periodic orbits from the triangular equilibrium points of the restricted four-body problem. In: Second International Conference ‘‘From Scientific Computing to Computational Engineering’’, Athens, Greece, in press. Simo´, C., Stuchi, T.J., 2000. Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem. Physica D 140, 1–32. Sze´ll, A., E´rdi, B., Sa´ndor, Zs., Steves, B., 2004. Chaotic and stable behaviour in the Caledonian symmetric four-body problem. Mon. Not. R. Astron. Soc. 347, 380–388. Stro¨mgren, E., 1935. Connaissance actuelle des orbites dans le proble´me des trois corps., Publ. Copenhague Obs., No. 100.
Planetary and Space Science 55 (2007) 1368–1379 www.elsevier.com/locate/pss
Asymptotic orbits in the restricted four-body problem K.E. Papadakis Department of Engineering Sciences, Division of Applied Mathematics and Mechanics, University of Patras, GR-26504 Patras, Greece Received 3 October 2006; received in revised form 31 January 2007; accepted 11 February 2007 Available online 27 February 2007
Abstract This paper studies the asymptotic solutions of the restricted planar problem of four bodies, three of which are finite, moving in circular orbits around their center of masses, while the fourth is infinitesimal. Two of the primaries have equal mass and the most-massive primary is located at the origin of the system. We found the invariant unstable and stable manifolds around the hyperbolic Lyapunov periodic orbits which emanate from the collinear equilibrium points Li ; i ¼ 1; . . . ; 4, as well as the invariant manifolds from the Lagrangian critical points L5 and L6 . We construct numerically, applying forward and backward integration from the intersection points of the appropriate Poincare´ cuts, homo- and hetero-clinic, symmetric and non-symmetric asymptotic orbits. We present the characteristic curves of the 24 families which consist of symmetric simple-periodic orbits of the problem for a fixed value of the mass parameter b. The stability of the families is computed and also presented. Sixteen families contain as terminal points asymptotic periodic orbits which intersect the x-axis perpendicularly and tend asymptotically to L5 for t ! þ1 and to L6 for t ! 1, spiralling into (and out of) these points. The corresponding 16 terminating heteroclinic asymptotic orbits, for b ¼ 2, are illustrated. r 2007 Elsevier Ltd. All rights reserved. Keywords: Restricted four-body problem; Asymptotic orbit; Homoclinic orbit; Heteroclinic orbit; Cut of Poincare´ surface of section; Periodic orbit; Lyapunov orbit
1. Introduction and equations of motion There are many reasons for studying the four-body problem besides the historical ones (Lagrange, Hill, Darwin; Multon, 1900), since it is known that approximately two-thirds of the stars in our Galaxy exist as part of multistellar systems. Around one-fifth of these are part of triple systems, while a rough estimate suggests that a further one-fifth of these triples belong to quadruple or higher systems, which can be modeled by the four-body problem. The four-body problem is also increasingly being used to study and explain many of the complex dynamical phenomena found in the Solar system and exoplanetary systems. Studying the stability, the periodic solutions, the asymptotic ones, and the regions of motion in four-body systems is therefore fundamental to understanding the evolution of quadruple stellar and exoplanetary systems (Sze´ll et al., 2004).
Tel.: +30 2610 997394; fax: +30 2610 997394.
E-mail address: [email protected]. 0032-0633/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2007.02.005
The importance of the asymptotic solutions of a dynamical system is well known. From the time of Stro¨mgren (1935) we know that in a restricted three-body problem there are heteroclinic asymptotic orbits connecting the two triangular critical points. Koon et al. (2000), in the same problem, presented asymptotic orbits which connect pairs of hyperbolic Lyapunov periodic orbits around the collinear equilibrium points. In this work we have done a systematic numerical computation of the invariant stable and unstable manifolds and the corresponding homoclinic and heteroclinic orbits associated with all the equilibrium configurations of the restricted four-body problem. An alternative approach, using analytical procedure based on high-order normal forms computations around the equilibria, could be used in order to obtain invariant manifolds and asymptotic homoclinic and heteroclinic orbits (Deprit and Henrard, 1965; Llibre et al., 1985; Markellos et al., 1995, etc.). We consider m0 , m1 and m2 the masses of three primary bodies where m0 is located at the center of mass (the origin of the coordinate system) and the other two bodies with equal masses are moving in circular orbits around the center
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of mass of the system. This model can be considered as a particular case of the restricted regular polygon problem of N þ 1 bodies when the number of the peripheral primaries equals 2 and the mass parameter of the central mass to a peripheral one is different from zero (Kalvouridis, 1999) and it has been used, among others, by Desiniotis and Kazantzis (1993) in their three-magnetic dipoles problem, where they studied the motion of a charged particle in the electromagnetic field produced by the dipoles, Markellos et al. (1995) studied zero velocity curves, periodic orbits, etc. of the specific four-body problem Maran˜hao (1995), Maran˜hao and Llibre (1999) investigated ejection–collision orbits and invariant punctured tori of the same problem, Cors et al. (2004) studied the central configurations of the problem, and in this journal Kalvouridis et al. (2006) explored the dynamical properties of the photogravitational version of the problem. The equations of motion of the infinitesimal mass, m3 , of the restricted four-body problem, in the usual dimensionless rectangular rotating coordinate system are written as (Kalvouridis et al., 2006) qO 1 bx x 12 x þ 12 ¼x x€ 2y_ ¼ þ þ , qx D r30 r31 r32 qO 1 b 1 1 ¼y 1 þ þ , ð1Þ y€ þ 2x_ ¼ qy D r30 r31 r32
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of length is chosen in such a way that the distance between the primaries m1 and m2 is equal to 1. So, the peripheral primaries m1 , m2 are placed at the position ð12; 0Þ and ð12; 0Þ, correspondingly (Fig. 1, left). We remark that for b ¼ 0, the Copenhagen case of the restricted three-body problem is produced. The energy (Jacobi) integral of this problem is given by the expression x_ 2 þ y_ 2 ¼ 2O C,
(4)
(3)
where C is the Jacobi constant. The equations of motion of the restricted four-body problem, as in the classical three-body one, have the property that if x ¼ xðtÞ, y ¼ yðtÞ is a solution, then x ¼ xðtÞ, y ¼ yðtÞ is also a solution. This symmetry follows from inspection of the terms occurring in the differential equations (1), where if we substitute x ! x, _ y_ ! y, _ x€ ! x€ and y€ ! y, € the y ! y, x_ ! x, equations remain unchangeable, a fact that verifies the preceding statement. This symmetry will be useful in the computations in the fourth section of the present work. The restricted four-body problem has six equilibrium points (in contrast to the classical restricted three-body problem which has five), four collinear L1;2;3;4 , and two ‘‘triangular’’ L5;6 (Kalvouridis et al., 2006). The first four are on the primaries line: L1 on the positive axis between m0 and m1 , L2 on the positive axis outside m1 and L4 and L3 are symmetric to L1 and L2 , respectively, with respect to the origin. From the differential equations of motion (1) we obtain the coordinates x0 , y0 , of the Lagrangian equilibrium points as solutions of the equations: qO 1 b 1 1 1 1 1 þ þ ¼ x0 1 þ ¼ 0, qx D r300 r310 r320 2D r310 r320 qO 1 b 1 1 ¼ y0 1 þ þ ¼ 0, ð5Þ qy D r300 r310 r320
Here, b ¼ m0 =m is the ratio of the central mass m0 to one of the other two equal primaries m ¼ m1 ¼ m2 and the unit
where r00 , r10 and r20 are the distances given by (3) at the equilibrium. For the Lagrangian points L5;6 , ðy0 a0Þ we
where dots denote time derivatives while the gravitational potential O in synodic coordinates is defined as 1 1 b 1 1 þ þ O ¼ ðx2 þ y2 Þ þ , (2) 2 D r0 r1 r2 where D ¼ 2ð1 þ 4bÞ;
r20 ¼ x2 þ y2 ,
2 r21 ¼ x 12 þ y2 ;
2 r22 ¼ x þ 12 þ y2 .
y
0.5
1.0
0.5
μ∗
r0
r2
0.4
m3 (x,y)
L5
r1
0.3
L5
X -1.0
L3
-0.5 L4 0 0 L1 m0 m2 -0.5
L6
0.5 m1
L2
1.0
0.2 L1
0.1
L2 distance
-1.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fig. 1. Left: The restricted four-body problem. Right: the exact positions, obtained numerically, of the equilibrium points on the x-axis (for L1 and L2 ) and on the y-axis (for L5 ), as functions of the mass parameter m .
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obtain r10 ¼ r20 which means that x0 ¼ 0 and therefore L5;6 are located symmetrically with respect to the origin, on the y-axis and their positions are given by the following equation (L5 on the positive axis), 1 b 2 1 þ ¼ 0, (6) D r300 r310
equilibrium points are unstable for every value of the mass ratio m, and now in the present problem the collinear equilibria are unstable for every value of the mass parameter b. The corresponding eigenvectors can be normalized to 1 (because of the type of the matrix A), u1 ¼ ð1; s; l; lsÞT ;
w1 ¼ ð1; it; in; ntÞT ,
where,
u2 ¼ ð1; s; l; lsÞT ;
w2 ¼ ð1; it; in; ntÞT ,
r00 ¼ jy0 j and
r10
qffiffiffiffiffiffiffiffiffiffiffiffi ¼ y20 þ 14.
(7)
In Fig. 1 (right) the exact positions of the equilibrium points on the x-axis (for L1 and L2 ) and on the y-axis (for L5 ), obtained numerically, are plotted as functions of the mass parameter m ¼ 1=ð2 þ bÞ ¼ m=ðm0 þ 2mÞ corresponds to the classical mass parameter m of the restricted three-body problem. 2. Linearization around the equilibrium configurations 2.1. Collinear equilibrium points Li ; i ¼ 1; . . . ; 4 The linearized equations for infinitesimal motions near the collinear equilibrium points are x_ ¼ Ax;
_ yÞ _T x ¼ ðx; y; x;
(8)
where x is the state vector of the fourth particle with respect to the equilibrium points and the time-independent coefficient matrix A is 0 1 0 0 1 0 B 0 0 0 1C B C A¼B (9) C, 0 0 2A @ 1 þ 2A0 0 1 A0 2 0 where
! 1 b 1 1 A0 ¼ þ þ , D jx0 j3 jx0 12j3 jx0 þ 12j3
(10)
and x0 is the abscissa of the collinear equilibrium point as solution of the equation, 1 bx0 x0 12 x0 þ 12 þ 3 þ 3 x0 ¼ 0, (11) D r30 r1 r2 where r0 ¼ jx0 j, r1 ¼ jx0 12j and r2 ¼ jx0 þ 12j are the distances at the equilibrium. The characteristic equation of the linear system (8) is l4 þ ð2 A0 Þl2 þ ð2A20 þ A0 þ 1Þ ¼ 0
(12)
and it has one real and one imaginary pair of eigenvalues which have the form l and in where l and n are the positive constants, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ 12 ½A0 2 þ A0 ð9A0 8Þ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ 12 ½2 A0 þ A0 ð9A0 8Þ. ð13Þ As in the restricted three-body problem, where the collinear
where s and t are the constants, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ 3A0 A0 ð9A0 8Þ s ¼ pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 2 A0 2 þ A0 ð9A0 8Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ 3A0 þ A0 ð9A0 8Þ t ¼ pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 2 A0 2 A0 ð9A0 8Þ
ð14Þ
ð15Þ
It is known, in the restricted three-body problem, that from collinear equilibrium points Li ; i ¼ 1; 2; 3, emanate the families a (from L2 ), b (from L3 ) and c (from L1 ) (according to the classical nomenclature). For Jacobi constant C just below C Li hyperbolic periodic orbits around each Li exist, called Lyapunov orbits. Similarly, in the four-body problem, emanates a Lyapunov family from each collinear equilibrium point Li ; i ¼ 1; 2; 3; 4 as we will see in Fig. 3 of Section 3 (family f 1 from L1 , f 7 from L2 , f 13 from L3 and family f 19 from L4 ). 2.2. Lagrangian equilibrium points L5ð6Þ To study analytically the solutions in the neighborhood of the equilibrium configuration L5ð6Þ , we transfer the origin at L5ð6Þ , we linearize the equations of motion and we receive for coplanar motion, the expression 0 1 0 0 1 0 B 0 0 0 1C B C (16) x_ ¼ Bx with B ¼ B C, 0 0 2A @ c31 0 3 c31 2 0 where we have abbreviated c31 ¼
48 Dð1 þ 4y20 Þ5=2
,
(17)
and y0 is given by (6). The corresponding characteristic equation of B is " # 144 D 16 4 2 l þ l þ d ¼ 0; d ¼ 2 D ð1 þ 4y20 Þ5=2 ð1 þ 4y20 Þ5 (18) and its eigenvalues are: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l1;2;3;4 ¼ pffiffiffi 1 1 4d ¼ p iq 2
(19)
with p and q being the reals and strictly positive quantities, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi p ¼ 12 2 d 1 and q ¼ 12 2 d þ 1. (20)
ARTICLE IN PRESS K.E. Papadakis / Planetary and Space Science 55 (2007) 1368–1379
The Lagrangian equilibrium points L5ð6Þ are linearly stable if these four roots are purely imaginary i.e. if, do14
(21)
which happens when b4bcrit ffi 11:72034889, as we see in Fig. 2 in which the diagram of the mass parameter b versus the quantities d and 1 4d is presented. Note that d is always positive for any value of the mass parameter b (Fig. 2, left frame). The Lagrangian equilibrium points L5ð6Þ are linearly unstable if bobcrit . In the first case (b4bcrit ) we have the imaginary roots, l1;2 ¼ io1 and l3;4 ¼ io2 . In the right frame of Fig. 2 we present the variation of the eigenfrequencies or mean motions o1 , o2 when the mass parameter b varies. In that case we have solution with ‘‘short’’ and ‘‘long’’ period terms corresponding to large and small values of the eigenfrequencies (for details, see Papadakis, 2006). In the second case, i.e. for bobcrit , the solution of the linear part of (1) is, ¯ ¼ ept ða1 cos qt þ a2 sin qtÞ þ ept ða3 cos qt þ a4 sin qtÞ, xðtÞ Z¯ ðtÞ ¼ ept ðb1 cos qt þ b2 sin qtÞ þ ept ðb3 cos qt þ b4 sin qtÞ, where the coefficients ai and bi are not independent but are connected by the following set of equations: b1 ¼ c 1 a1 þ c 2 a2 ;
b2 ¼ c2 a1 þ c1 a2 , b4 ¼ c2 a3 c1 a4 ,
manifold W sL5;6 in the vicinity of libration point. Similarly, if a3 ¼ a4 ¼ 0 we have an approximation of the unstable invariant manifold W uL5;6 . For a numerical globalization of the invariant unstable manifold W sL5 , we adopt an appropriate set of initial conditions ¯ xð0Þ ¼ a1 ; Z¯ ð0Þ ¼ c1 a1 þ c2 a2 , ¯_ xð0Þ ¼ pa1 þ qa2 ; Z_¯ ð0Þ ¼ ðpc1 qc2 Þa1 þ ðpc2 þ qc1 Þa2 , ð25Þ where a1 , a2 are chosen (see Go´mez et al., 1988) a1 ¼ r cos y
ð23Þ
where pffiffiffi 3 þ 2ð3 c31 Þc31 8 d þ 2d p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi and pffiffiffi pffiffiffi 4ðc31 þ d 3Þ 2 d 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi ð2c31 3Þ 2 d þ 1 pffiffiffi . ð24Þ c2 ¼ 4ðc31 þ d 3Þ The four initial conditions x¯ 0 , Z¯ 0 , x_¯ 0 and Z_¯ 0 are linear functions of the four constants a1 ; . . . ; a4 and selecting a1 ¼ a2 ¼ 0 (then b1 ¼ b2 ¼ 0 from (23)), the solution approaches the equilibrium point asymptotically and so the solution (22) give an approximation of the stable invariant c1 ¼
b
r sin y c1 r cos y c2
(26)
with r40, y 2 ½0; 2p, which means that when t ¼ 0 then ¯ ðxð0Þ; Z¯ ð0ÞÞ describes a small circle ðr cos y; r sin yÞ around the libration point L5 on the linear part of the manifold.
In this section we present all the families of the planar symmetric simple-periodic orbits of the restricted fourbody problem, for b ¼ 2 (see also Kalvouridis et al., 2006), as well as the asymptotic orbits which associated to these. There are 24 families, named f i ; i ¼ 1; . . . ; 24, which consist of simple symmetric periodic orbits (i.e. having only two both perpendicular, intersections with the x-axis). We can separate them in four groups one in each collinear equilibrium point area. So we have 6 families in L1 area, 6 in L2 , etc. The heavy red lines, in Fig. 3, indicate the horizontal stable arcs of the simple families. The small families f 9 and f 22 are shown in two inside windows of Fig. 3. The positions of the bodies m0 , m1 and m2 are denoted by three vertical lines, while the shaded areas represent the regions of forbidden motion in the ðx0 ; CÞ plane. We note that we have considered the periodic orbits (symmetric with respect to the x-axis) as represented by their initial conditions x0 , y0 ¼ 0, x_ 0 ¼ 0 and y_ 0 40 (‘‘positive’’ perpendicular intersection of the x-axis). 1.0
12
a2 ¼
and
3. Simple-families and asymptotic orbits
ð22Þ
b3 ¼ c1 a3 þ c2 a4 ;
1371
bcrit
ωi
0.9
10
0.8
8
0.7
ω2
ω1 = ω2 =1/21/2
0.6 6 0.5 4
ω1
0.4 δ
1-4δ
0.3
2
bcrit =11.7203
0.2
b
0 -6
-5
-4
-3
-2
-1
0
1
2
5
10
15
20
25
30
35
40
45
50
Fig. 2. Left: the evolution of the quantities d and 1 4d as the mass parameter b varies. Right: the eigenfrequencies at the libration points L5ð6Þ .
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Sixteen families contain as terminal points asymptotic periodic orbits which intersect the x-axis perpendicularly and tend asymptotically to L5 for t ! þ1 and to L6 for t ! 1, spiralling into (and out of) these points. Actually these orbits are a combination of two heteroclinic asymptotic orbits running from L5 to L6 and vice versa. All these asymptotic periodic orbits have C ¼ C L5ð6Þ and the characteristics of the respective families spiral around points in the ðx0 ; CÞ plane with C ¼ C L5ð6Þ . These are the natural termination points I–IV which are presented in Fig. 3. In each point terminate four families. Details of the areas of these points are illustrated in four frames of Fig. 4. All the 16 terminating heteroclinic asymptotic orbits, for b ¼ 2, are presented in Fig. 5. As in the restricted three-body problem, and in the present problem we have Lyapunov families which emanate from the collinear equilibrium points Li ; i ¼ 1; . . . ; 4. From L1 emanates the family f 1 , from L2 the family f 7 , from L3 the family f 13 and from L4 the 2 C f20
L3
f17
f13
1
f3
L f8
f6 f21
f18 CL(5,6)
L4
f5
f19
f22
f4 IV
III
L2
f9 f f11 f1
I
10
f
f7
II f2
f14
1.44
f23
1.41 f15
f9
1.38
0 f24
0.500
0.505
1.14 1.11 f16
f22
1.08
-1 -0.500 -0.495 m0
m2
-2
-1
m1
X
0
1
Fig. 3. The family networks of the symmetric simple-periodic orbits of the four-body problem for b ¼ 2.
1.05
C
family f 19 . In contrast of the three-body problem, all the Lyapunov families terminate with asymptotic heteroclinic orbits at the points I–IV (Figs. 3 and 4). 4. Asymptotic orbits around Lyapunov periodic orbits In the first subsection we are going to present invariant stable W sLi p:o: and unstable W uLi p:o: manifolds, which correspond to the Lyapunov periodic orbits around the equilibrium points Li ; i ¼ 1; . . . ; 4 as well as homoclinic and heteroclinic orbits associated with the Lyapunov orbits for two values of energy C and for b ¼ 2. In the second subsection we study numerically the asymptotic homoclinic and heteroclinic orbits associated with the Lagrangian equilibrium points L5 and L6 . 4.1. Asymptotic orbits ‘‘around’’ collinear equilibrium points As we have already mentioned, for fixed values of the Jacobi constant C just smaller than the values of C at the collinear equilibrium points Li ; i ¼ 1; . . . ; 4 we have unstable periodic orbits around Li , called Lyapunov orbits. The Hill’s region (keeping the same terminology as in the restricted three-body problem) corresponding to such values of the Jacobi constant contains ‘‘necks’’ about the equilibrium points as we see in Figs. 6 and 8. We have chosen two values of C so that two cases are created. First case (Fig. 6) where we have two ‘‘necks’’ close to L1 and L4 , respectively, and a second one (Fig. 8) where four ‘‘necks’’ close to each of the four equilibrium collinear points exist. The aim here is to describe the invariant unstable and stable manifolds of the Lyapunov periodic orbits around the collinear equilibrium points and to compute numerically the corresponding homoclinic and heteroclinic orbits which exist in these regions. The existence and the determination of such orbits are described later. We note that if we know the unstable manifold of a Lyapunov orbit, then the corresponding stable manifold is obtained using the symmetric property of the equations of the problem as we have mentioned in the Section 1. This observation will be used to find the transversal homoclinic or heteroclinic orbits. By transversal homoclinic (heteroclinic) orbit, we mean that appropriate unstable and stable manifolds intersect transversally. We recall that a homoclinic orbit 1.2
C
C
f18 CL(5,6)
CL(5,6) 1.0
1.00
I
f2
f1
f3
1.0
II
f21
1.1
CL(5,6) 1.0
III
CL(5,6) IV
f12
f4 0.9
C
1.1
1.1
f7 X
0.95 0.00 0.01 0.02 0.03 0.04 0.05 0.500
0.9 f10
f11 0.505
f15
f14
0.9
f13 X
X 0.510
-1.4
-1.3
-1.2
-1.1
f24
0.8 -1.0 -0.50
Fig. 4. Details of the map of the family characteristics around areas I–IV, respectively.
f23 -0.45
f19 X -0.40
-0.35
ARTICLE IN PRESS K.E. Papadakis / Planetary and Space Science 55 (2007) 1368–1379
1.0
1373
1.0 1.0
f1 0.5
L5
0.0
m2
0.5
m0
0.0
m1
-0.5
L6
-0.5
1.0 f4
f3
f2 0.5
0.5
0.0
0.0
-0.5
-0.5
-1.0 -1.0
-1.0 -0.5
0.0
0.5
-0.5
0.0
0.5
1.0
1.5
1.0
0.0
0.5
0.0 -0.5
0.0
0.5
1.0 f11
f10
0.5
-0.5
1.0
f7
1.0
-1.0 -0.5
f12
0.5
0.5
0.5
0.0
0.0
0.0
-0.5
-0.5
-0.5
-1.0 -1.0 -0.5
0.0
0.5
1.0
1.0
1.5 f13
-1.0 -0.5
0.0
0.5
1.0 f14
1.0
-1.0 -0.5
0.0
0.5
-0.5
f15
1.0
0.5
0.5
0.5
0.0
0.0
0.0
0.0
-0.5
-0.5
-0.5
-0.5
-1.0
-1.0
-1.0
-1.0
-1.0
-0.5
0.0
0.5
-1.5 1.0
-1.0
-0.5
0.0
0.5
-1.5
-1.0
-0.5
0.0
0.5
0.5 f18
1.0
0.5
-1.5 1.0
0.0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
1.0 f21
f19
f24
1.0
f23
0.5
0.5
0.5
0.5
0.0
0.0
0.0
0.0
-0.5
-0.5
-0.5
-1.0
-1.0
-1.0 -1.0
-0.5 -1.0
-0.5
0.0
0.5
-0.5
0.0
0.5
-0.5
0.0
0.5
1.0
-0.5
0.0
0.5
1.0
1.5
Fig. 5. The 16 asymptotic orbits which are the termination of the corresponding families as marked on the frames, respectively.
related to an equilibrium point L or to a periodic orbit P is an orbit that tends to L (or to P) for t ! 1. Using the analysis of Deprit and Henrard (1969), or the analysis of Llibre et al. (1985), or the purely numerical method presented by Simo´ and Stuchi (2000), we can determine the invariant unstable manifold W uLip:o: emerging from each unstable Lyapunov orbit close to any of the collinear equilibrium points L1 , L2 , L3 and L4 . It is known (Conley, 1968) that unstable (and stable W sp:o: ) manifolds of these Lyapunov orbits are two dimensional, locally diffeomorphic to cylinders (e.g. Fig. 6), while the invariant unstable W uLi and stable W sLi manifolds associated to
Li ; i ¼ 1; 2; 3; 4, are one dimensional (case which is not examined in the present paper). In this way we compute the surface of section of the invariant manifold with the plane y ¼ 0 (or other appropriate plane) one or two etc. times, and obtain a closed curve which is diffeomorphic to a circle (e.g. first frame of Fig. 7). We call this intersection the (first or second, etc) ‘‘cut’’ of W uLip:o: with the appropriate plane. In Fig. 6 we present the unstable manifolds W uL4 p:o: which correspond to the Lyapunov periodic orbit (white dashed line-left) around the collinear equilibrium point L4 when the middle primary is twice larger than m1 , m2 (i.e. b ¼ 2) and for C ¼ 1:46. Similarly the stable invariant
ARTICLE IN PRESS K.E. Papadakis / Planetary and Space Science 55 (2007) 1368–1379
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y
this homoclinic point A and the value of the Jacobi constant C (i.e. x0 ¼ ðx0 ; 0; 0; y_ 0 ðCÞÞ), we produce the homoclinic trajectory which is presented in the second frame of the first row of Fig. 7. In the second row-left the first three unstable cuts of L1 and the second unstable cut of L4 are presented. The stable third cut of L1 is indicated by the dashed line. The intersection of the second unstable cut of L4 and the third stable cut of L1 defines the heteroclinic intersection point B. In the next frame (same row) the corresponding heteroclinic orbit which connect the unstable Lyapunov periodic orbit around L1 and the Lyapunov one around L4 is illustrated. We note here that only the third cut of W sL1 p:o: and the second cut of W uL4 p:o: are used to define point B. In the last frame of Fig. 7 we present the first 10 cuts of the unstable manifolds W uLð1;4Þ p:o: and the stable W sLð1;4Þ p:o:
C=1.46
0.2 W uL
4p.o.
W uL4p.o.
W sL1p.o.
m0
0.0 L3
m1
m2
L2
W sL
1p.o.
-0.2 b=2 X -1.0
-0.5
-0.0
-0.5
-1.0
Fig. 6. Unstable manifolds W uL4 p:o: of the Lyapunov periodic orbit (white dashed line-left) near L4 (yellow and green tubes). Stable manifolds W sL1 p:o: of the Lyapunov periodic orbit (wh dashed line-right) near L1 (red and blue tubes).
manifolds W sL1 p:o: of the Lyapunov periodic orbit around L1 (white dashed line-right) are also illustrated for the same values of b and C. So in this figure we illustrate the position space projection of the unstable manifold W uL4 p:o: (yellowgreen tube) and the stable one of L1 (red-blue tube) until the first intersection with the Poincare´ section at y ¼ 0. These (and the next) intersections as well as their _ symmetric, with respect to the x-axis, we are going to present in Fig. 7. In the next figures homoclinic orbits and heteroclinic ones which connect the Lyapunov periodic orbits are presented. These asymptotic orbits constructed numerically by finding an intersection of the manifolds W uLi p:o: and W sLi p:o: ; i ¼ 1; 4, on the appropriate chosen Poincare´ section at y ¼ 0 (the line passing through the three primary bodies in the rotating frame). Due to reversibility of the restricted four-body problem, if we have an intersection point A of the unstable invariant manifold with the x-axis, then the stable one will also intersect the x-axis at the same point. The common point A corresponds to symmetric orbit (with respect to the x-axis) which is homoclinic to the Lyapunov orbit around the equilibrium point Li . In the first row-left frame of Fig. 7 we present the first Poincare´ cut of the unstable invariant manifold W uL1 p:o: with the plane y ¼ 0 of the Lyapunov periodic orbit around the equilibrium point L1 for b ¼ 2 and C ¼ 1:46. Letter A stands for the intersection point of the first cut of W uL1 p:o: with x-axis. As we have already mentioned, the corresponding cut of the stable invariant manifold W sL1 p:o: will intersect the x-axis at the same point A. With forward and backward integration using the initial conditions of
for b ¼ 2 and C ¼ 1:46, which define the permissible region _ plane. of motion in the ðx; xÞ So far we study the asymptotic orbits in the case where the energy constant has the value C ¼ 1:46 and the Hill region is closed and two of the four collinear equilibrium points are out of it. Now we decrease the energy and we take C ¼ 1:36 (a value just below the energy of L2;3 ) where the closed region opens and four ‘‘necks’’ close to equilibrium points exist. In that case and for b ¼ 2 the unstable invariant manifolds of the Lyapunov periodic orbits around the four collinear equilibrium points in the interior and the exterior Hill’s region are presented (Fig. 8). Like earlier, we compute the appropriate cuts with the plane y ¼ 0 in order to find the intersections with x-axis (homoclinic intersections) or between cuts (heteroclinic intersections). In Fig. 9 (left frame) the first Poincare´ cut of the unstable manifold of the Lyapunov periodic orbit around L1 i.e. W uL1 p:o: and the second cut of the stable one of W sL2 p:o: for b ¼ 2 and C ¼ 1:36 are presented. In the middle frame we present an heteroclinic point which is the intersection of the first Poincare´ cut of the stable manifold W sL3 p:o: and the second cut of the unstable one of W uL4 p:o: . In the last frame the homoclinic points F and G which are the intersections of the first Poincare´ cut of the stable manifold W sL4 p:o: and the second cut of the unstable one of W uL1 p:o: with x-axis correspondingly are presented. From these intersection points we calculate and present in Fig. 10 the four corresponding orbits, two homoclinic and two heteroclinic, around the Lyapunov periodic orbits which emanate from the collinear equilibrium points. 4.2. Asymptotic orbits ‘‘around’’ Lagrangian equilibrium points In this subsection we study the unstable and stable manifolds as well as the asymptotic homoclinic and heteroclinic orbits associated with the Lagrangian equilibrium points L5 and L6 of the restricted four-body problem. Using Eqs. (25) and (26) we compute and present in Fig. 11 the unstable manifold of the equilibrium
ARTICLE IN PRESS K.E. Papadakis / Planetary and Space Science 55 (2007) 1368–1379
0.0
A
x
1375
y 0.2
-0.5 A
cut1 uL1
-1.0
0.0 L3
m2 L4
m0
L1 m1
L2
-1.5 -0.2 -2.0
x -0.08
2
-0.06
-0.04
-0.02
-0.00
-1.0
xx
-0.5
0.0
0.5
1.0
y 0.2
cut3 sL1 1 cut2 uL4
B B
0
-1
0.0
L3
m2
L4
m0
L1 m1
L2
cut2 uL1
cut3 uL1
-0.2 cut1 uL1
-2 -0.4
-0.2
x
x
0.0
0.2
-1.0
0.4
-0.5
0.0
0.5
1.0
2 x
1
0
-1
x -2 -0.4
-0.2
0.0
0.2
0.4
W uL1 p:o:
Fig. 7. First row-left: the first Poincare´ cut of the unstable manifolds with the plane y ¼ 0, for b ¼ 2 and C ¼ 1:46. Right: the corresponding homoclinic orbit of the Lyapunov periodic orbit around L1 . Second row-left: the first three unstable cuts of L1 and the second unstable cut of L4 . The stable third cut of L1 is indicated by the dashed line. Right: the corresponding heteroclinic orbit of the Lyapunov periodic orbits around L1 and L4 . Last frame: the first 10 cuts of the unstable manifolds W uLð1;4Þ p:o: and the stable W sLð1;4Þ p:o: for the same values of b and C as above.
point L5 . In order to avoid congestion of the picture we present in the left frame the part of the unstable manifold for y 2 ½0; p while in the right frame we give the part of the unstable manifold for y 2 ðp; 2p. As we have already mentioned in the Introduction, the equations of motion of the problem have the property that if x ¼ xðtÞ, y ¼ yðtÞ is a solution, then x ¼ xðtÞ,
y ¼ yðtÞ is also a solution. The stable manifold of L5 is the symmetric picture of Fig. 11 (both frames as one figure) with respect to y-axis due to the above symmetry property of the problem. The stable and unstable manifolds of the equilibrium point L6 are symmetric with respect to the x-axis and are also not presented here.
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Fig. 8. Left: all the unstable manifolds W uLi p:o: of the Lyapunov periodic orbits near the collinear Li ; i ¼ 1; . . . ; 4, for b ¼ 2 and C ¼ 1:36, just below the energy of L2;3 . Right: details of the interior region of the equilibrium points.
Fig. 9. Left: the first Poincare´ cut of the unstable manifold W uL1 p:o: and the second cut of the stable one of W sL2 p:o: , for b ¼ 2 and C ¼ 1:36. Middle: the first Poincare´ cut of the stable manifold W sL3 p:o: and the second cut of the unstable one of W uL4 p:o: . Right: the first Poincare´ cut of the stable manifold W sL4 p:o: and the second cut of the unstable one of W uL1 p:o: .
Fig. 10. The corresponding homoclinic and heteroclinic orbits of the points D–G of Fig. 9.
In the first row-left of Fig. 12 the first cut of the unstable invariant manifold of L5 , which we denote it by cut1uL5 , and the stable one cut1sL5 from the same Lagrangian point are
presented. The four intersection points A, B, D and E on the x-axis are heteroclinic points from which we can calculate symmetric, with respect to the x-axis (as x_ 0 ¼ 0), heteroclinic orbits running from L5 to L6 . These four heteroclinic orbits are computed and we found that they are the termination asymptotic orbits of the families f 13 (the heteroclinic which corresponds to point A), f 3 (point B), f 15 (point D) and f 18 (point E) correspondingly. The characteristic curves of these families are presented in Fig. 3 and the four heteroclinic orbits keeping the same notation ðf 13 ; f 3 ; f 15 ; f 18 Þ are illustrated in Fig. 5. In the right frame of the first row of Fig. 12 we present the second cut of the stable and unstable invariant manifold of L5 . We observe now that we have many intersection points on the x-axis but we have also many intersection points between the unstable and the stable cuts off the x-axis. This means that except of symmetric, nonsymmetric heteroclinic orbits from L5 to L6 will exist. Indeed we calculate, for example, two points on the x-axis (points F and G) and two off the x-axis (points H and K)
ARTICLE IN PRESS K.E. Papadakis / Planetary and Space Science 55 (2007) 1368–1379
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Fig. 11. The unstable invariant manifold of L5 for y 2 ½0; p (left) and for y 2 ðp; 2p (right).
Fig. 12. First row: the first Poincare´ cut (left) and the second one (right) of the stable and unstable invariant manifold of L5 for b ¼ 2. Second row: two symmetric and two non-symmetric heteroclinic orbits correspond to the intersection points F, G, H and K.
and we compute the four corresponding heteroclinic orbits. In the second row of Fig. 12 two symmetric with respect to x-axis heteroclinic orbits and two non-symmetric ones are presented. Similar to above, we can find the invariant manifolds and cuts as well as the corresponding heteroclinic orbits of the symmetric Lagrangian point L6 . We found, so far, homoclinic or heteroclinic points using the x–x_ cuts of the unstable (stable) manifolds. We can construct asymptotic orbits numerically, by finding inter-
sections of the unstable W u5ð6Þ and stable W s5ð6Þ , manifolds on an other appropriate Poincare´ section map. So, if we choose the Poincare´ section at the plane x ¼ 0, then the corresponding y–y_ cuts of the unstable and stable manifolds of the L5 ð6Þ are shown in Fig. 13. In particular, we present the fifth (first row-left) and the combination of the sixth and seventh (right) Poincare´ cut of the stable and unstable invariant manifolds of L5ð6Þ with the plane x ¼ 0, for b ¼ 2. Some of the existing intersection points which produce homo-heteroclinic orbits are denoted in these
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Fig. 13. First row: the fifth Poincare´ cut of the stable and unstable invariant manifold of L5 (left) and the sixth cut of unstable manifold of L6 with the seventh cut of stable manifold of L5 (right) for b ¼ 2. Second row: two symmetric, with respect to y-axis, and one non-symmetric homoclinic orbits correspond to the intersection points M, N, and O. Last frame: a non-symmetric heteroclinic orbit from L6 to L5 corresponds to the intersection point P.
figures. We found a lot of such orbits using the intersection points on and off the y-axis in order to construct numerically, applying forward and backward integration, homo- and hetero-clinic, symmetric and non-symmetric asymptotic orbits. Four of them, as example, which correspond to the points M, N, O (left frame) and P (right frame of the first row of Fig. 13) are presented in the second row of the same figure. 5. Conclusions In the present work we extended previous studies (Go´mez et al., 1988; Koon et al., 2000) about asymptotic solutions of the restricted three-body problem to the restricted four-body problem. We studied numerically the behavior of the orbits of the equations of motion of the problem when the Jacobi constant is just below that of the collinear critical points Li ; i ¼ 1; . . . ; 4. The asymptotic solutions associated with the Lagrangian equilibrium points L5ð6Þ were also calculated. The geometrical behavior of the stable–unstable manifolds of the six libration points was studied. The network of the 24 existing families of the symmetric simple periodic orbits of the problem as well as the asymptotic termination orbits of 16 of them were found and presented. From the results we conclude that there is an infinite number of asymptotic orbits (simple and higher-multiplicity) since the stable and unstable invariant manifolds produce an infinite number of homoclinic and heteroclinic points. Compared with the Copenhagen restricted three-body problem (He´non, 1965) the simple symmetric
heteroclinic orbits (i.e. having only two both perpendicular intersections with the x-axis) of the problem are now four more, namely 16 instead of 12. All these asymptotic orbits have C ¼ C L5ð6Þ and the characteristics of the respective families spiral around points in the (x0 ; C) plane which have ordinate C ¼ C L5ð6Þ (Figs. 3 and 4). The network of the simple families of the present problem consists of 24 members instead of the 22 families of the Copenhagen problem. The Lyapunov families are now four because of the existing of the fourth collinear libration point. Symmetric and non-symmetric homo- and hetero-clinic asymptotic orbits associated with the six critical points of the restricted four-body problem were calculated and presented. The corresponding Poincare´ cuts of the unstable and stable invariant manifolds, on appropriate planes, were also shown. We found that these orbits are not changed, qualitatively, to the corresponding asymptotic orbits associated with the equilibrium points of the restricted three-body problem. References Conley, C., 1968. Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math. 16, 732–746. Cors, J., Llibre, J., Olle, M., 2004. Central configurations of the planar coorbital satellite problem. Cel. Mech. 89, 319–342. Deprit, A., Henrard, J., 1965. Symmetric doubly asymptotic orbits in the restricted three-body problem. Astron. J. 70, 271–274. Deprit, A., Henrard, J., 1969. Construction of orbits asymptotic to a periodic orbit. Astron. J. 74, 308–316. Desiniotis, C., Kazantzis, P., 1993. The equilibrium configurations of the three-dipole problem. Astrophys. Space Sci. 202, 89–112.
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