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Resonant motion, ballistic escape, and their applications in astrodynamics
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Advances in Space Research 42 (2008) 1318–1329 www.elsevier.com/locate/asr
Resonant motion, ballistic escape, and their applications in astrodynamics Francesco Topputo a,*, Edward Belbruno b, Marian Gidea c a Aerospace Engineering Department, Politecnico di Milano, Via La Masa, 34, 20156 Milan, Italy Department of Astrophysical Sciences, Princeton University, Peyton Hall, Ivy Lane, Princeton, NJ 08544, USA c Department of Mathematics, Northeastern Illinois University, 5500 N. St. Louis Avenue, Chicago, IL 60625, USA b
Received 30 September 2007; received in revised form 3 January 2008; accepted 3 January 2008
Abstract A special set of solutions governing the motion of a particle, subject to the gravitational attractions of the Earth, the Moon, and, eventually, the Sun, is discussed in this paper. These solutions, called resonant orbits, correspond to a special motion where the particle is in resonance with the Moon. For a restricted set of initial conditions the particle performs a resonance transition in the vicinity of the Moon. In this paper, the nature of the resonance transition is investigated under the perspective of the dynamical system theory and the energy approach. In particular, using a new definition of weak stability boundary, we show that the resonance transition mechanism is strictly related to the concept of weak capture. This is shown through a carefully computed set of Poincare´ surfaces, at different energy levels, on which both the weak stability boundary and the resonant orbits are represented. It is numerically demonstrated that resonance transitioning orbits pass through the weak stability boundaries. In the second part of the paper the solar perturbation is taken into account, and the motion of the resonant orbits is studied within a four-body dynamics. We show that, for a wide class of initial conditions, the particle escapes from the Earth–Moon system and targets an heliocentric orbit. This is a free ejection called a ballistic escape. Astrodynamical applications are discussed. Ó 2008 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Restricted three/four-body problem; Resonance transition; Weak capture; Weak stability boundary; Low energy transfers; Ballistic escape
1. Introduction In recent years, the study of the restricted n-body problem has gained a considerable interest. (In these problem, the motion of a small particle, subject to n 1 gravitational attractions, is analyzed.) Compared to the classic Kepler dynamics, the n-body problem gives rise to a wider class of solutions defined in a more complicated vector field. The n-body problem is not, in general, integrable by quadratures when n P 3; its solutions are not analytic. Hence, in the step from the two-body to the n-body problem, useful tools, such as the orbital elements, become unavailable. Moreover, even though governed by the Newtonian *
Corresponding author. E-mail addresses: [email protected] (F. Topputo), belbruno@ princeton.edu (E. Belbruno), [email protected] (M. Gidea).
dynamics, the motion of a small particle subject to n 1 gravitational attractions, n P 3, turns out to be in general chaotic. This avoids long-term predictions on the particle’s motion as small changes in the initial conditions produce large trajectory deviations. Nevertheless, these highly nonlinear orbits can be exploited to design transfer trajectories requiring less propellant than the standard patched-conics orbits: this is the case of the low energy transfers. With a low energy transfer, a spacecraft may achieve a ballistic capture in an automatic fashion, therefore no rocket engine would be required in the capture process. Conley (1968) conjectured that a low energy transfer between the Earth and Moon, leading to a ballistic capture, might exist. A low energy transfer was used in 1991 to resurrect a failed Japanese lunar mission and get its spacecraft Hiten successfully to the Moon, thereby demonstrating the existence of weak capture (Belbruno and Miller, 1993; Belbruno,
0273-1177/$34.00 Ó 2008 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2008.01.017
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
2004, 2007). This lunar transfer and demonstration satisfactorily answer Conley’s conjecture. Another transfer, based on this methodology, was used in 2004 for SMART-1 of the European Space Agency (Belbruno, 1987; Schoenmaekers et al., 2001; Racca, 2003). The Earth-to-Moon low energy transfers have been analyzed under the perspective of the dynamical system theory (Belbruno, 1994; Koon et al., 2001a). This approach has led to the extension of low energy transfers to the planetto-planet case (Topputo et al., 2005); lastly, the ESA’s BepiColombo spacecraft is being designed to perform a gravitational capture at Mercury (Jehn et al., 2004). The Earth-to-Moon low energy transfers are based on the concept of weak capture at the Moon. This is a capture where the Kepler energy with respect to the Moon is nonpositive and the motion of the particle with respect to the Moon is unstable. Such captures are generally temporary. Weak capture occurs in a special region in phase space about the Moon called the weak stability boundary, WSB, rigorously defined by Belbruno (2004); it has been proven that the motion of the particle in the WSB is chaotic due to the existence of a hyperbolic invariant set. This boundary can generally be viewed as a location where a particle lies between capture and escape with respect to the Moon, or any other secondary body such as Jupiter, in the case of the Sun–Jupiter system. The link between the weak capture and the resonance transition is described for the first time by Belbruno (1990): when the particle is initially in weak capture at the Moon, then it transitions onto a resonant orbit about the Earth, in resonance with respect to the Moon, in both forward and backward time. Therefore, this yields resonance transitions which occur due to passing through weak capture at the Moon, or equivalently, when the particle’s orbit passes through the weak stability boundary. An analogous motion has been observed for a special set of short period comets that moves about the Sun in resonance with Jupiter. When these comets approach Jupiter, their orbit abruptly transitions into another orbit about the Sun, also in resonance with respect to Jupiter, but generally of a different type (Belbruno and Marsden, 1997; Koon et al., 2001b). The first of these was observed for the comet Lexell which transitioned from a 5:4 resonance into a 2:1 resonance, denoted by 5:4 ? 2:1. (In general, an m:n resonant orbit means that m times the comet’s period is approximately n times the period of Jupiter.) In this paper we study the motion of a particle of negligible mass, a rock or a spacecraft, P 3 , that moves in the same plane as two nonzero bodies, P 1 ; P 2 , of masses m1 ; m2 , respectively, where we assume m1 m2 , and where P 1 ; P 2 move about their common center of mass in uniform circular orbits. This defines the planar circular restricted three-body problem. The motion of P 3 is studied in the Earth–Moon system, thus P 1 and P 2 are the Earth and the Moon, respectively. If the total energy of P 3 , called the Jacobi energy, C, is suitably restricted, then in a coordinate system rotating with P 1 ; P 2 , retrograde unstable
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periodic orbits, called Lyapunov orbits, exist on both sides of P 2 on the P 1 –P 2 line. They exist within the allowable region of motion of P 3 , called the Hill’s region. From the work of Conley (1968), local invariant hyperbolic manifolds emanate from the Lyapunov orbits, which are topological cylinders. Global extension of these manifolds about P 2 and showing that they transversally intersect would imply that a complicated hyperbolic network exists about P 2 (Llibre et al., 1985). Such a network would give rise to a chaotic dynamics about P 2 , that could possibly be associated to the resonance transitions. This network was numerically shown to exist by Koon et al. (2000, 2001b), in the case where P 1 ; P 2 are Sun, Jupiter, respectively, and where the value of C was suitably restricted. Although this demonstration partially clarifies the structure of the phase space in the proximity of P 2 , obtaining a satisfactory understanding the WSB for a large range of C and mass ratios m2 =m1 has been elusive. The main goal of this paper is to provide a way to numerically explore the resonance structure of the WSB and nearby dynamics. We give a way to do this by formulating a novel definition of WSB, giving rise to the extended WSB, B, where P 3 is no longer constrained to lie at the periapsis/apoapsis of an osculating orbit around the Moon. The dynamics is then analyzed on a suitable set of Poincare´ sections under the perspective of the energy method described below. In particular, by projecting B on this surface of section, we numerically demonstrate that the resonance transition occurs when P 3 passes through the extended WSB. Finally, we analyze the motion of P 3 in the more realistic model that takes into account the gravitational attraction of the Sun. We find that, when the initial resonant motion of P 3 is integrated in the restricted four-body problem, the resonance transition mechanism breaks and P 3 leaves the Earth–Moon system. This is a ballistic escape of which we provide some preliminary examples. This interesting dynamics has potential applications in the frame of low energy interplanetary transfers and in designing transfers to the libration point orbits of the Sun–Earth system. The remainder of the paper is organized as follows. The dynamics of the restricted three- and four-body problems is described in the next section. The energy method is introduced in Section 3 together with three different definitions of WSB. In Section 4 we show the connections between the WSB and the resonance transitions by means of a set of Poincare´ sections showing the global orbit structure. In Section 5 examples of ballistic escaping orbits are given in the frame of the restricted four-body problem. Final remarks and future works are pointed out in Section 6. 2. Dynamics In this section, we define the modeling used to describe the motion of P 3 in both the restricted three- and four-body problems. A detailed derivation of the equations of motion can be found in Szebehely (1967) and Belbruno (2004), for
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the restricted three-body problem, and in Simo´ et al. (1995), for the restricted four-body problem. 2.1. Restricted three-body problem The motion of P 3 , a rock or a spacecraft, is studied in the gravitational field generated by the mutual circular motion of two particles, P 1 ; P 2 of masses m1 ; m2 , respectively, about their common center of mass. It is assumed P 1 ; P 2 represent the Earth and the Moon, respectively, and that P 3 moves in the same plane of P 1 ; P 2 . This is the planar circular restricted three-body problem, or simply RTBP. The equations of motion are written in a barycentric rotating coordinate frame with dimensionless units: the angular frequency of the circular motion of P 1 ; P 2 , around their center of mass, is normalized to 1; the distance between P 1 and P 2 is normalized to 1; the sum of their masses is normalized to 1, m1 þ m2 ¼ 1. The equations of motion can be written in terms of l, the mass parameter, l ¼ m2 =ðm1 þ m2 Þ. P 1 , of mass 1 l, is located at ðl; 0Þ; P 2 , of mass l, is located at ð1 l; 0Þ. The dynamical system describing the motion of P 3 is oX ; ox oX €y þ 2_x ¼ ; oy
€x 2_y ¼
ð1Þ
where 1 1l l 1 Xðx; yÞ ¼ ðx2 þ y 2 Þ þ þ þ lð1 lÞ; 2 r1 r2 2
ð2Þ
and ðx; yÞ is the position of P 3 . The two distances in Eq. (1) are
retrograde Lyapunov orbits and two-dimensional stable and unstable manifolds emanating from them. The value of the Jacobi energy evaluated at Lk , where x_ ¼ y_ ¼ 0, is denoted by C k . For the two collinear points L1 and L2 one obtains C 1 3:20034 and C 2 3:18416. For a given value of C, the Jacobi energy can be used to establish some allowed and forbidden regions for the motion of P 3 . These regions are bounded by the zero-velocity curves, ZðCÞ, defined as ZðCÞ ¼ fðx; yÞ 2 R2 j2Xðx; yÞ C ¼ 0g:
ð6Þ
The set Z is not defined for C 6 3; in these regimes P 3 is allowed to move in the entire configuration space. 2.2. Restricted four-body problem In the last part of the paper, the perturbation produced by the Sun on the motion of P 3 is taken into account. In this case, the resulting four-body problem, RFBP, is no longer autonomous as the Sun is not fixed in the Earth– Moon rotating frame. Moreover, the key properties of the RTBP, as the Lagrangian points, the Jacobi energy, and the Hill’s curves, are no longer defined in a four-body scenario. In this case the equations of motions are oX0 ; ox oX0 €y þ 2_x ¼ ; oy
€x 2_y ¼
ð7Þ
and have the form of Eq. (1) but this time the potential is ms ms X0 ðx; yÞ ¼ Xðx; yÞ þ 2 ðx cos h þ y sin hÞ: ð8Þ rs q
The mass parameter used in this study is l ¼ 0:01215. The dynamical system (1) admits an integral of motion, the Jacobi energy,
The physical constants introduced to describe the motion of the Sun are assumed to be in agreement with the rotating and normalized system introduced previously. Thus, the distance between the Sun and the Earth–Moon barycenter corresponds to q ¼ 3:8881 102 , the mass of the Sun is ms ¼ 3:2890 105 , and its angular velocity with respect to the rotating frame is x ¼ 0:9251. The phase of the Sun, h, is given by
J ðx; y; x_ ; y_ Þ ¼ 2Xðx; yÞ x_ 2 þ y_ 2 ;
h ¼ h0 þ xt;
2
r21 ¼ ðx þ lÞ þ y 2 ; 2
r22 ¼ ðx þ l 1Þ þ y 2 :
ð3Þ
ð4Þ
that, for a given constant C, defines a three-dimensional manifold _ 2 R4 jJ ðx; y; x_ ; y_ Þ C ¼ 0g: J ðCÞ ¼ fðx; y; x_ ; yÞ
ð5Þ
The vector field defined by (1) has five well known equilibrium points, known as the Lagrange points, labeled Lk ; k ¼ 1; . . . ; 5. The collinear points (L1 ; L2 ; L3 ) lie along the x-axis, while the triangular points (L4 ; L5 ) are located at the vertices of two equilateral triangles having P 1 ; P 2 at the other two vertices. The two collinear points, L1 and L2 , lie at approximate x position 0:83691 and 1.15568, respectively. In a linear analysis, collinear points behave like the product saddle center. Thus, in a small neighborhood of the collinear points there exists a family of small
ð9Þ
where h0 is the initial angle. With these constants, the instantaneous position of the Sun is ðq cos h; q sin hÞ, and therefore the P 3 –Sun distance is r2s ¼ ðx q cos hÞ2 þ ðy q sin hÞ2 :
ð10Þ
It is worth noting that the bicircular model (Eqs. (7)– (10)) is not coherent because all the three primaries are assumed to move in circular orbits, and therefore they violate the Newton’s equations (Simo´ et al., 1995). Nevertheless, this model is expected to depict a good approximation of the real four-body (the eccentricities of the Earth’s and Moon’s orbits are 0.016 and 0.054, respectively, and the Moon’s orbit is inclined on the ecliptic by 5 deg).
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
3. Energy method and WSB definitions The aim of this paper is to explore the connections between the resonance transition and the weak capture. In order to define the weak capture condition, it is convenient to analyze the two-body energy of P 3 with respect to the primaries. This is the essence of the energy approach. Analyzing the Kepler energies, and defining the weak capture, allows to locate the weak stability boundaries and to study their structure on appropriate surfaces of section. We denote by H 1 ; H 2 , and H s the Kepler energy of P 3 relative to the Earth, Moon, and Sun, respectively. Letting r2 ; v2 be the distance and the speed, respectively, of P 3 expressed in an inertial Moon-centered frame, then 1 l H 2 ¼ v22 : 2 r2 Analogously, the Kepler energy H 1 reads
ð11Þ
1 1l ; ð12Þ H 1 ¼ v21 2 r1 where r1 ; v1 are the distance and speed of P 3 relative to an inertial Earth-centered frame. In the case of the restricted four-body problem, H s is given by 1 ms ð13Þ H s ¼ v2s 2 rs where rs ; vs are expressed in a Sun-centered inertial frame. Let the flow of either (1) or (7) be represented by uðtÞ ¼ ðxðtÞ; yðtÞ; x_ ðtÞ; y_ ðtÞÞ; t 2 R1 ; we define some special captures of P 3 with respect to P 2 (the Moon in our case). Definition 1 (Weak capture). P 3 is ballistically captured by P 2 at time t, if H 2 ðuðtÞÞ 6 0. P 3 is temporary ballistically captured by P 2 , if H 2 ðuðtÞÞ 6 0 for t1 6 t 6 t2 H 2 ðuðtÞÞ > 0
for t < t1
and
and t > t2 ;
for finite times t1 ; t2 ; t1 < t2 . Temporary ballistic capture is also referred to as weak capture. Definition 2 (Ballistic ejection). P 3 is ballistically ejected (or ballistically escapes) from P 2 along the solution uðtÞ at a time t1 if H 2 ðuðtÞÞ < 0
for t < t1
and
H 2 ðuðtÞÞ P 0
for t P t1 :
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Analogous definitions are made for capture or ejection with respect to the Earth, P 1 , by substituting P 1 ; H 1 , in place of P 2 ; H 2 , respectively. Weak capture and ballistic ejection are conceptually shown in Fig. 1. We now define the resonant orbits. Definition 3 (Resonant orbit). An m:n resonant orbit about the Earth, in resonance with the Moon, is an orbit whose period T is related to the Moon’s period T 2 by mT ¼ nT 2 ; where m; n are positive integers. It can be easily shown (see Belbruno et al., 2008) that Definition 3 yields the following condition on the semimajor axis: n 2=3 1=3 a¼ ð1 lÞ : ð14Þ m The energy method, and in particular the condition on the Keplerian energy in order to get weak capture at P 2 , is illustrated through an example. The energy method does not require restricting C, but rather only the semimajor axis a has to be determined from condition (14) once the two resonance numbers, m and n, are specified. In Fig. 2 we have integrated in forward time a 2:1 resonant orbit with the Moon having the initial eccentricity e ¼ 0:316. It can be seen that, when performing the second revolution around the Earth, P 3 gets weakly captured by the Moon (Fig. 2(a)) where H 2 K 0 (Fig. 2(c)). P 3 performs one revolution around the Moon in an unstable ellipse (Fig. 2(c)). After this temporary capture, P 3 lies again in an approximate elliptic state relative to the Earth but this time it lies on a 1:3 resonant orbit with the Moon. Hence, after a time interval equal to three Moon periods, P 3 approaches the Moon. After this second interaction its orbit is in 1:2 resonance with the Moon and so on. It is noted, in Fig. 2(c), that the condition H 2 K 0 is satisfied along the weak capture as required by the energy method. 3.1. WSB definitions The focus of the paper is to study resonance hops due to P 3 passing near P 2 in the RTBP. In addition, we are interested in studying this process when the Sun perturbation is
Fig. 1. Temporary ballistic capture and ballistic ejection. (The subscript of H is not shown to indicate either the Earth, P 1 , or Moon, P 2 .)
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x 10
x 10
4
18
6
16
1:3
14
4
2:1
0
y (km)
y (km)
12
1:2
2
10 8 6
–2
4
–4
Moon
2
–6
0
–2 –10
–8
–6
–4
–2
0
2
x (km)
0
4
5
10
x (km)
5
x 10
4
x 10
3 140
2.5
H (adim.)
1.5
H2
120
hop #1
hop #2
hop #3
100
T (days)
2
1 0.5
1:3 80 60
1:2
0 40
–0.5
H
1
–1 –1.5
0
20
40
20
60
80
100
120
140
160
t (days)
0
2:1 20
40
60
80
100
120
140
160
t (days)
Fig. 2. Example of resonance transition and weak capture at the Moon. The resonances sequence is 2:1 ? 1:3 ? 1:2. The initial orbit has the semimajor axis given by Eq. (14) with m ¼ 2; n ¼ 1. The temporary capture in Fig. 2(b) refers to the first Moon approach. Figs. 2(c) and 2(d) show the two-body energies, relative to the Earth (H 1 ) and to the Moon (H 2 ), and the orbit’s period, respectively. It is noted that, as P 3 moves through the different resonances, H 1 varies (increases or decreases), while H 2 K 0. The orbits are in approximate resonance with the Moon since the period is not exactly equal to those associated to a perfect resonance.
taken into account, that is when P 3 moves in the RFBP. As mentioned previously, it has been observed that numerical integration from weak capture near P 2 results in P 3 transitioning into resonant orbits with respect to P 2 (Belbruno, 1990). Thus, in order to study resonance hops, it is necessary to study weak capture at P 2 . The region where weak capture occurs can be used to define the WSB. It is defined by Belbruno and Miller (1993), Belbruno (2004), and further refined by Garcı´a and Go´mez (2007). The first algorithmic definition of WSB is recalled below. We consider trajectories starting on a radial line lðhÞ departing from P 2 and making an angle of h with the x-axis (Fig. 3). The trajectories are assumed to start at the periapsis of an osculating ellipse around P 2 , whose semimajor axis a lies on lðhÞ and whose eccentricity e is fixed; this makes r2 ¼ að1 eÞ. The initial velocity of the trajectory is taken perpendicular to lðhÞ, and the Keplerian energy of P 3 relative to P 2 is H 2 < 0. We want to emphasize that
Fig. 3. The stability criterion used to numerically define W, the algorithmic WSB.
in this definition r_ 2 ¼ 0. The motion, for fixed values of h and e, depends only on the distance r2 . The motion is called stable if, after leaving lðhÞ; P 3 makes a full turn about P 2 , without going around P 1 , and returns to lðhÞ at a point with H 2 < 0 (Fig. 3). The motion is otherwise called unstable.
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
Definition 4 (Algorithmic WSB). The algorithmic weak stability boundary is the locus of all points r ðh; eÞ along the radial line lðhÞ for which there is a change of stability of the initial trajectory, that is, r ðh; eÞ is one of the endpoints of an interval ½r1 ; r2 characterized by the fact that for all r 2 ½r1 ; r2 the motion is stable, and there exist r0 62 ½r1 ; r2 , arbitrarily close to either r1 or r2 for which the motion is unstable. Thus W ¼ fr ðh; eÞjh 2 ½0; 2pÞ; e 2 ½0; 1Þg: It has been numerically demonstrated that the weak stability boundary W is contained in the set given by the union of the stable manifolds, having zero radial velocity, associated to the L1 and L2 Lyapunov orbits (Garcı´a and Go´mez, 2007). Although Definition 4 yields the location of the set W in the configuration space, that is ideal to construct low energy spacecraft transfers from the Earth to the Moon, it does not give any insight into the capture dynamics and the phase portrait characterizing the WSB. We are interested in studying the dynamical properties of the WSB on a suitable surface of section. In order to do this, we give two analytic definitions of the WSB based on the consideration made in Belbruno (2004) and Belbruno et al. (submitted for publication). In these definitions, the WSB depends on the value of the Jacobi energy C. Definition 5 (Classic WSB). Let the two sets r and R be r ¼ fðx; y; x_ ; y_ Þ 2 R4 j_r2 ðx; y; x_ ; y_ Þ ¼ 0g; R ¼ fðx; y; x_ ; y_ Þ 2 R4 jH 2 ðx; y; x_ ; y_ Þ 6 0g; then, for C 2 ½C ; C 1 Þ, the analytic WSB, W, is defined as
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where C 2 ½C ; C 1 Þ and C is estimated such that B is nonempty. Definition 6 corresponds to the definition of W in Definition 5 with the condition r_ 2 ¼ 0 removed. This means that the condition that P 3 is at the periapsis of its orbit around the Moon in Definition 4 is removed. Through B, the capture may occur when P 3 is at any point of its osculating orbit around the Moon. We have estimated numerically that the greatest lower bound C of Jacobi constants such that both W and B are nonempty, with the introduced value of l, is approximately C ’ 2:8868 (Belbruno et al., 2008). In the next section, we represent B on a surface of section and numerically show that resonant orbits pass through B when they perform a resonance hop. 4. The role of the WSB in resonance transitions In this section, we relate the weak stability boundary to the resonance transition. In order to do this, both the resonant orbits and the WSB are visualized on an appropriate subsets of the phase space represented by a Poincare´ section. On this section, not only the resonant orbits but the chaotic dynamics and the global orbit structure of the RTBP, including the WSB, can be studied. When represented on this surface of section, the set B lets us unlock the role of the WSB in resonance hops. For the sake of brevity, we show both the Poincare´ section and the WSB corresponding to just one energy level. A detailed analysis, including a number of sections at different energy levels, can be found in Belbruno et al. (2008).
W ¼ J ðCÞ \ R \ r; where J ðCÞ is the three-dimensional manifold (5) and C is estimated so that W exists. A value of C ¼ 2:2 is obtained in Belbruno (2004), which is a crude estimate since the approximation is not dynamically based. Definition 5 implies that W has a dimension of 2, and is generally an annular region in phase space. The condition of C < C 1 is assumed in order that the inner and outer Hill’s regions are connected. The dependence on l is suppressed since it is fixed. We refer to W as the classic WSB. It is worth pointing out that, with Definition 5, the set W can be represented on a suitable surface of section. W is a more realistic and precise representation of WSB than W, and, in general, W is smaller than W (Belbruno et al., 2008). Analogously to W, the set W implies that P 3 is on the apsidal line (i.e. r_ 2 ¼ 0) of the osculating orbit around the Moon when it lies on the WSB. We remove this condition in the extended WSB. Definition 6 (Extended WSB). Let J ðCÞ be the threedimensional manifold (5) and r be the set in Definition 5, then the extended WSB, B, is B ¼ J ðCÞ \ R
4.1. Poincare´ section and global orbit structure We consider the flow of (1) on the three-dimensional energy surface J ðCÞ, defined by Eq. (5), for a given value of C. On this manifold we define a two-dimensional surface of section by the set _ 2 J ðCÞjy ¼ 0; y_ > 0g: SðCÞ ¼ fðx; y; x_ ; yÞ A Poincare´ section consists of the set of points given by the intersection of the flow of (1) with the surface S. They are obtained by computing the intersection between hundreds of flows, given by hundreds of initial conditions, and S. In the following, we discuss a Poincare´ map obtained with C ¼ 3:18176, that is the Jacobi energy of a ‘‘normalized” 2:1 orbit with zero eccentricity (see Belbruno et al., 2008). It is convenient to show sections in both the (x; x_ ) and (a; h) planes, where a and h are the semimajor axis and the true anomaly, respectively, of the osculating twobody orbits around the Earth. Fig. 4 shows the Poincare´ map in the (x; x_ )-plane. The shape of the region is due to the projection of J ðCÞ onto this plane. It is noted that the flow is organized on invariant tori in the regions close to the Earth (x 0) and to the Moon (x 1), while a ‘‘chaotic sea” of points characterizes
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Fig. 4. Poincare´ section of the flow of the RTBP for C ¼ 3:1817683176.
the region between them (the orbits are not organized in any discernable pattern). There are invariant curves in Fig. 4(a) surrounding 4:1, 3:1, 8:3, 7:3, and 2:1 resonant orbits. The invariant tori on the right of Fig. 4(a) are associated to stable orbits around the Moon. In Fig. 4(b), the island of invariant closed curves surrounding the 2:1 resonant point is magnified. These invariant curves are due to two-dimensional invariant tori in the phase space, which are indicative of KAM tori, projected onto the section. An exactly resonant, stable, orbit exists at the center of each island. The term island indicates the region on the section where there is a set of invariant curves. If they surround a resonant periodic point, we will refer to them as resonance islands. Orbits will not exhibit a resonance hop within an island since they would be trapped within the tori themselves. The chaotic sea surrounding the islands represents the region of the phase space where the resonance transitions take place, where P 3 moves between resonance islands. This is discussed further below. Fig. 5 represents the section of Fig. 4 viewed in the coordinates (a; h). It is worth noting that, in this set of coordinates, standard two-body orbits around the Earth generate vertical straight lines. In Fig. 5(a) the global resonant structure is much more evident, where invariant tori around the 2:1, 3:1, 4:1, 7:3, and 8:3 orbits can be distinguished. For a given m:n resonance, through the condition (14), we are able to locate and label the semimajor axis corresponding to the resonant orbit (vertical dashed lines in Fig. 5(a)). The resonant islands are centered on these lines, therefore their nature depends on which line they are centered. The size of each island is indicative of the importance of the corresponding resonance. Once P 3 is initialized on one of these tori, its motion will take place on the same torus for all future time. In particular, as mentioned above, an exact, stable, resonant orbit is defined at the center of each resonance island. Correspondingly, an exact, unstable, resonant orbit is defined between the stable islands along a strip of constant a (Koon et al., 2000). If P 3 is moving in the chaotic sea outside these resonance tori islands, then,
with the exception of the 4:1 island orbits, the other resonance islands are connected with each other. P 3 could pass between island boundary regions by executing a substantial variation in the semimajor axis. In this way, resonance hops can occur since the chaotic sea in this plane spans a wide range of semimajor axes. Fig. 5(a) magnifies the tori around the 2:1 orbits. The effect of the temporary capture by the Moon can be again viewed in the top- and bottom-right side of the picture. Here, the gravitational attraction of the Moon acts to increase the semimajor axis of the P 3 ’s orbit and breaks the invariant structure. This mechanism occurs at h ffi p meaning that P 3 gets captured when it is approximately at the apoapsis of its orbit about the Earth. We recall that the present value of the Jacobi integral is less than C 1 , so the size of P 3 ’s orbit can be raised up to reach the outer Hill’s region beyond the Moon’s orbit. 4.2. Representation of the extended WSB In this section, we visualize the WSB on SðCÞ. We have defined the algorithmic, W, the classic, W, and the extended, B, weak stability boundaries (Definitions 4–6). It is convenient to view W and B on SðCÞ in ðx; x_ Þ coordinates, for the fixed value of C (we recall that SðCÞ is the set on J ðCÞ where y ¼ 0; y_ > 0). It turns out that, on SðCÞ; W corresponds to a subset of the x-axis. This follows from the fact that the condition: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r_ 2 ¼ ½ðx þ l 1Þ_x þ y y_ = ðx þ l 1Þ2 þ y 2 ¼ 0; ð15Þ combined with y ¼ 0 implies that x_ ¼ 0 (we assume r2 < þ1). Thus, SðCÞ \ W is a one-dimensional set. In Fig. 6 we show a portion of SðCÞ; C ¼ 3:18176, where both W and B are represented. The projection of J ðCÞ is also shown; its boundaries are the curves with y_ ¼ 0. The location of the Moon is seen at x 1. The points of the iterates of the Poincare´ map are contained within the projection of the Jacobi energy surface as expected. The 2:1
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Fig. 5. The points of Fig. 4 mapped into the (a; h) set.
roles in hop dynamics. The set B is an interesting region; we describe some of its structure that was observed in the following two examples.
Fig. 6. A portion of the phase space containing the classic and the extended WSB, W and B, respectively. The boundaries of both J ðCÞ (the line y_ ¼ 0) and R (the line H 2 ¼ 0) are also shown.
resonance island is also labeled, as well as a family of invariant curves about the Moon. The region H 2 6 0 is shown, with a dashed curve for the boundary, H 2 ¼ 0. The set B is represented by bold points; note that some of these points in B are about the Moon location. In this plot, the set of invariant curves about the Moon is excluded since its points do not give rise to the unstable behavior characterizing the temporary capture required in the WSB. The set of invariant curves about the Earth is also excluded since they do not yield the unstable behavior of temporary required for the WSB. The subset of B which lies on the x-axis, as indicated, form the set W. That is W ¼ B \ f_x ¼ 0g. 4.3. Resonance transitions through B It is noted that B largely overlaps with the chaotic sea on S. Also, within this section, there are a lot of resonance islands in which P 3 can travel from the boundaries to B via the chaotic sea of points. In this sense B, and its subset W, are associated to resonance motion, and play important
4.3.1. Example 1 In Fig. 7 we have reported the Poincare´ section SðCÞ; C ¼ 3:18176, illustrated in Fig. 5(a). As already observed, the section, when viewed in the ða; hÞ plane, is made up by resonant islands surrounded by a chaotic sea of irregular orbits. On this plane, the points belonging to B (those points with H 2 6 0) are shown bold. A sample orbit having the points inside both B and the chaotic sea has been selected and a portion of this orbit is plotted in the inertial frame in Fig. 7(b). As it can be seen, the initial P 3 ’s orbit is in an 8:3 resonance with the Moon; it is then captured by the Moon and ejected into a 5:2 resonant orbit. This is an 8:3 ? 5:2 resonance transition. There is evidence that this mechanism occurs when the orbit is temporary captured by the Moon and hence when it lies in the extended WSB set B. 4.3.2. Example 2 This example is defined with the energy level C ¼ 3:15109. In Fig. 8(a) we have reported the Poincare´ section of the flow of the RTBP at this energy level. Two intersections of the orbit with the plane of section, contained in B, are marked as #1 and #2. These are not two consecutive intersections between the orbit of P 3 and the surface of section, but they rather correspond to two consecutive weak captures where P 3 orbits the Moon. Thus, by the given definition of B, both the two states #1 and #2 lie on B. We study the piece of the orbit between the two weak captures and in particular in Fig. 8(b) we show the P 3 ’s osculating orbital period. Before the first capture, the orbit of P 3 is in 8:3 resonance with the Moon; after orbiting the Moon for a while, P 3 is placed into a 2:1 resonant orbit with the Moon and performs the following sequence of resonance transitions 2:1 ? 5:2 ? 2:1 ? 2:1 ? 7:3 ? 2:1. Then the particle is again captured by the Moon and is
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x 10
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5:2
y (km)
1 0 –1
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–2 –3 –3
–2
–1
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x (km)
a
3
x 105
Fig. 7. Example 1 – The extended WSB set B viewed in the (a; h) plane (a). B corresponds to the region (top and bottom) where the 2:1 resonant island is distorted due to the Moon’s gravitational attraction. A portion of a sample orbit intersecting B exhibits an 8:3 ? 5:2 resonance transition (b).
periods 30
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T (days)
25
20
weak 1:1 capture #1
15
10
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x
8:3 5:2 250
300
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2:1 400
450
500
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Fig. 8. Example 2 – Two states defined inside B corresponding to two consecutive weak captures where P 3 orbits the Moon. Between the two weak captures the motion of P 3 goes through resonant states and performs a number of resonance transitions.
ejected into another 2:1 orbit. It can be noted that the resonances are not exact. This series of resonance transitions is generated between two states defined within B.
5. Comments on ballistic escape and applications The perturbation of the Sun is now taken into account. In particular, we study how the initial orbits, in resonance with the Moon, behave in the four-body problem described by Eqs. (7)–(10). This dynamics is even more complicated than of the RTBP since the potential (8) contains two nonautonomous terms describing the gravitational attraction of the Sun on P 3 . These terms are particularly relevant when P 3 moves in the region outside the Moon’s orbit. In the outer region, indeed, the resonance motion, characteristic of the RTBP, is broken. Moreover, as already observed by Belbruno (1990), the particle P 3 leaves the Earth–Moon
system an targets a heliocentric orbit: this is a ballistic escape. Fig. 9 reports the orbit, obtained with the same initial condition that generates the orbit of Fig. 2, integrated under the four-body dynamics (the time span is equal to 10 Moon’s periods). In the RTBP, the initial 2:1 resonant orbit transitions into a 1:3 and then into a 1:2 orbit (P 3 performs a 2:1 ? 1:3 ? 1:2 resonance hop). When the same initial conditions is flowed in the RFBP, P 3 orbits the Earth one time and then gets temporary captured by the Moon up to reach the exterior region (Fig. 9(a)). Here, the 3:1 orbit is missed and P 3 is simply driven away from the Earth–Moon system by the gravitational attraction of the Sun. The particle targets a heliocentric orbit and falls in a temporary state where its motion is mainly governed by the Sun (Fig. 9(b)). It is worth pointing out that this kind of escape takes place without carrying out any maneuver; it just exploits the
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
1
x 10
6
1 0.5
y (AU)
y (km)
0
–0.5
0
–1
–1.5 –2 –1
–1 –0.5
0
x (km)
0.5
1 6
x 10
–1
–0.5
0
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1
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Fig. 9. The orbit of Fig. 2(a) integrated under the four-body dynamics (7). The particle P 3 is initially in a 2:1 resonant orbit and gets weakly captured by the Moon as expected. Once in the exterior region, P 3 does not hop into another resonant orbit (like in Fig. 2(a)) as the resonance transition mechanism is broken by the Sun perturbation. P 3 escapes from the Earth– Moon system in a totally ballistic fashion and targets a heliocentric orbit. This ballistic escape occurs in a relatively short time.
intrinsic dynamics of the four-body problem. In this example h0 ¼ 0 in Eq. (9). Another interesting example is presented in Fig. 10. In this case an initial 2:1 resonant orbit with the Moon, e ¼ 0:333, is integrated for 13 Moon’s periods. After the weak capture at the Moon, P 3 is ejected into a high elliptical orbit about the Earth having the apogee of approximately 106 km. The perturbation of the Sun lowers the perigee of this orbit and so P 3 flies-by the Earth and performs a ballistic escape. The shape of the trajectory – from the weak capture at the Moon up to the Earth approach – is very similar to the exterior WSB transfers (Belbruno and Miller, 1993; Belbruno, 1994, 2004; Belbruno, 2007). The point is that the exterior WSB transfers were designed to go from the Earth to the Moon; it has been demonstrated by Belbruno (2004) that these transfers are regulated by a chaotic dynamics. In this study the behavior of the resonant orbits, in the four-body dynamics, was investigated and we came up with that there is a sort of link between the resonant motion and the low energy transfers. The key is that both processes take place in the weak stability boundaries of the Moon and the Earth. In Fig. 10(b) the heliocentric orbit is shown. Adjusting the phase of the Sun results in a heliocentric orbit inside or outside the Earth’s orbit. In this case we have assumed h0 ¼ p in Eq. (9). Through numerical experiments it has been found that this kind if escape can bring P 3 up to 0.15 AU far from the Earth’s orbit. We recall once again that this mechanism exploits the simultaneous gravitational attractions of the Sun, Earth, and Moon and hence the escape occurs at zero cost. This is ideal for spacecraft applications, for instance in the frame of low energy interplanetary transfers aimed at reaching Mars and Venus with the lowest propellant mass (Topputo et al., 2005). This process, indeed, can be optimized and there is evidence that a small
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maneuver performed at the closest point to the Earth (see Fig. 10(a)) may bring the spacecraft very far from the Earth’s orbit. The Keplerian energies, H 1 and H 2 , are reported in Fig. 10(c). The condition H 2 K 0 is again verified when P 3 gets weakly captured by the Moon. After this weak capture, the function H 1 monotonically increases and changes the sign from negative to positive. This is a ballistic escape in the sense of the Definition 2. H 1 is zero approximately 120 days after the Moon encounter. Fig. 10(d) shows the trend of the distance from P 3 to the Sun. Through numerical experiments we have found that, after the ballistic escape, the Keplerian energy of P 3 relative to the Sun, H s (Eq. (13)), exhibits small steps like the function H 1 does in Fig. 2(c) where P 3 is in resonance with the Moon. Thus, we have assessed that the heliocentric orbits resulting from the ballistic escape are in resonance with the Earth. There are in fact two integers, m0 ; n0 , that verify m0 T ¼ n0 T 1 , where T 1 is the Earth’s period and T is the period of the osculating orbit of P 3 around the Sun. We have found that the two resonance numbers, m0 and n0 , in this case are typically greater than 10. We conjecture that the geocentric resonant orbits with the Moon are connected to the heliocentric resonant orbits with the Earth. In this paper, we have numerically demonstrated the role of the WSB in the resonance transition. There is evidence (Koon et al., 2000) that the WSB of the Moon and that of the Earth are defined at almost the same energy level. 6. Conclusions In this paper, we have investigated the nature of the resonance motion affecting a particle that moves in the vector field generated by the gravitational attractions of the Earth and Moon. The resonance transition has been approached with the energy method: from this point of view, the Keplerian energy of the particle relative to the Moon is key in revealing the nature of the motion. In particular, with suitable restrictions on this energy, three different definitions of weak stability boundaries have been give. With the aid of suitable Poincare´ sections, that reveal the global orbit structure of the RTBP, and by representing the extended WSB on these surfaces, it has been numerically shown that resonance transitioning orbits pass through the WSB. In this sense, the WSB is a ‘‘resonance hub”. In the last part of the paper, the resonant motion under the perturbation of the Sun has been studied. In this case the resonance transition mechanism is broken and the particle leaves the Earth–Moon system, at zero cost, by means of a ballistic escape. This kind of escape occurs in reasonable times and so it seems to be particularly appealing for potential spacecraft applications. In particular, the ballistic escaping trajectory can be optimized and integrated with a suitable maneuver to yield a low energy interplanetary transfer.
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15
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5
1
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y (km)
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8
ballistic escape (from P )
weak capture (at P2)
1.04
1
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1
x (AU)
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10
6
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6
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100
150
200
250
300
350
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0.99 0
50
100
150
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250
300
350
400
t (days)
Fig. 10. An initial 2:1 resonant orbit with the Moon, e ¼ 0:333, integrated under the four-body dynamics. (a) The trajectory of P 3 and the Moon’s orbit (dashed) in a Earth-centered frame. After getting weakly captured by the Moon, P 3 is placed in an ‘‘exterior-like” WSB transfer trajectory (Belbruno and Miller, 1993) that approaches the Earth. The particle P 3 flies-by the Earth and escapes the Earth–Moon system in a hyperbolic state relative to the Earth. In (b) we have reported the trajectory of P 3 and the Earth’s orbit (dashed) in a Sun-centered frame. (c) The Keplerian energies of P 3 relative to the Earth, H 1 , and the Moon, H 2 . According to the Definition 2, P 3 performs a ballistic escape as the function H 1 becomes positive. (d) The distance from P 3 to the Sun along the trajectory. Numerical experiments shows that this distance may grow up to 1.15 AU. The cost for this escape is zero. It is ideal for spacecraft applications.
Acknowledgment The research of M.G. is partially supported by NSF Grant DMS 0601016. References Belbruno, E. Lunar capture orbits, a method of constructing Earth–Moon trajectories and the lunar GAS mission, AIAA Paper No. 97-1054, in: Proc. AIAA/DGLR/JSASS Inter. Elec. Propl. Conf., 1987. Belbruno, E. Examples of nonlinear dynamics of ballistic capture and escape in the Earth–Moon system, AIAA Paper No. 90-2896, in: Proc. Annual Astrodyn. Conf., 1990. Belbruno, E., Miller, J. Sun-perturbated Earth-to-Moon transfers with ballistic capture. J. Guid. Control Dyn. 16, 770–775, 1993. Belbruno, E. The Dynamical Mechanism of Ballistic Lunar Capture Transfers in the Four-Body Problem from the Perspective of Invariant Manifolds and Hill’s Regions, CRM Research Report 270, Centre de Recerca Matematica, Institute d’Estudis Catalans, Barcelona, 1994.
Belbruno, E., Marsden, B. Resonance hopping in comets. Astron. J. 113, 1433–1444, 1997. Belbruno, E. Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press, Princeton, 2004. Belbruno, E. Fly Me to the Moon: An Insiders Guide to the New Science of Space Travel. Princeton University Press, Princeton, 2007. Belbruno, E., Topputo, F., Gidea, M. Resonance transitions associated to weak capture in the restricted three-body problem. J. Adv. Space Res. 42, 1330–1351, 2008. Conley, C. Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math. 16, 732–746, 1968. Garcı´a, F., Go´mez, G. A note on weak stability boundaries. Celest. Mech. Dyn. Astron. 97, 87–100, 2007. Jehn, R., Campagnola, S., Garcı´a, D., Kemble, S. Low-thrust approach and gravitational capture at Mercury, in: Proc. 18th Int. Symp. Space Flight Dynamics, pp. 487–492, 2004. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D. Heteroclinic connections between periodic orbits and resonance transition in celestial mechanics. Chaos 10, 427–469, 2000.
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329 Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D. Low energy transfer to the Moon. Celest. Mech. Dyn. Astron. 81, 63–73, 2001a. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D. Resonance and capture of jupiter comets. Celest. Mech. Dyn. Astron. 81, 27–38, 2001b. Llibre, J., Martinez, R., Simo´, C. Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem. J. Differ. Eqns. 58, 104–156, 1985. Racca, G. New challenges to trajectory design by the use of electric propulsion and other means of wandering in the solar system. Celest. Mech. Dyn. Astron. 85, 1–24, 2003.
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Schoenmaekers, J., Horas, D., Pulido, J.A. SMART-1: with solar electric propulsion to the Moon, in: 16th International Symposium on Space Flight Dynamics, Pasadena, California, 3–7 December, 2001. ´ ., Masdemont, J. The Bicircular Model Simo´, C., Go´mez, G., Jorba, A Near the Triangular Libration Points of the RTBP, From Newton to Chaos. Plenum Press, New York, 1995. Szebehely, V. Theory of Orbits: The Restricted Problem of Three Bodies. Academic, New York, 1967. Topputo, F., Vasile, M., Bernelli-Zazzera, F. Low energy interplanetary transfers exploiting invariant manifolds of the restricted three-body problem. J. Astronaut. Sci. 53, 353–372, 2005.
Advances in Space Research 42 (2008) 1318–1329 www.elsevier.com/locate/asr
Resonant motion, ballistic escape, and their applications in astrodynamics Francesco Topputo a,*, Edward Belbruno b, Marian Gidea c a Aerospace Engineering Department, Politecnico di Milano, Via La Masa, 34, 20156 Milan, Italy Department of Astrophysical Sciences, Princeton University, Peyton Hall, Ivy Lane, Princeton, NJ 08544, USA c Department of Mathematics, Northeastern Illinois University, 5500 N. St. Louis Avenue, Chicago, IL 60625, USA b
Received 30 September 2007; received in revised form 3 January 2008; accepted 3 January 2008
Abstract A special set of solutions governing the motion of a particle, subject to the gravitational attractions of the Earth, the Moon, and, eventually, the Sun, is discussed in this paper. These solutions, called resonant orbits, correspond to a special motion where the particle is in resonance with the Moon. For a restricted set of initial conditions the particle performs a resonance transition in the vicinity of the Moon. In this paper, the nature of the resonance transition is investigated under the perspective of the dynamical system theory and the energy approach. In particular, using a new definition of weak stability boundary, we show that the resonance transition mechanism is strictly related to the concept of weak capture. This is shown through a carefully computed set of Poincare´ surfaces, at different energy levels, on which both the weak stability boundary and the resonant orbits are represented. It is numerically demonstrated that resonance transitioning orbits pass through the weak stability boundaries. In the second part of the paper the solar perturbation is taken into account, and the motion of the resonant orbits is studied within a four-body dynamics. We show that, for a wide class of initial conditions, the particle escapes from the Earth–Moon system and targets an heliocentric orbit. This is a free ejection called a ballistic escape. Astrodynamical applications are discussed. Ó 2008 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Restricted three/four-body problem; Resonance transition; Weak capture; Weak stability boundary; Low energy transfers; Ballistic escape
1. Introduction In recent years, the study of the restricted n-body problem has gained a considerable interest. (In these problem, the motion of a small particle, subject to n 1 gravitational attractions, is analyzed.) Compared to the classic Kepler dynamics, the n-body problem gives rise to a wider class of solutions defined in a more complicated vector field. The n-body problem is not, in general, integrable by quadratures when n P 3; its solutions are not analytic. Hence, in the step from the two-body to the n-body problem, useful tools, such as the orbital elements, become unavailable. Moreover, even though governed by the Newtonian *
Corresponding author. E-mail addresses: [email protected] (F. Topputo), belbruno@ princeton.edu (E. Belbruno), [email protected] (M. Gidea).
dynamics, the motion of a small particle subject to n 1 gravitational attractions, n P 3, turns out to be in general chaotic. This avoids long-term predictions on the particle’s motion as small changes in the initial conditions produce large trajectory deviations. Nevertheless, these highly nonlinear orbits can be exploited to design transfer trajectories requiring less propellant than the standard patched-conics orbits: this is the case of the low energy transfers. With a low energy transfer, a spacecraft may achieve a ballistic capture in an automatic fashion, therefore no rocket engine would be required in the capture process. Conley (1968) conjectured that a low energy transfer between the Earth and Moon, leading to a ballistic capture, might exist. A low energy transfer was used in 1991 to resurrect a failed Japanese lunar mission and get its spacecraft Hiten successfully to the Moon, thereby demonstrating the existence of weak capture (Belbruno and Miller, 1993; Belbruno,
0273-1177/$34.00 Ó 2008 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2008.01.017
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
2004, 2007). This lunar transfer and demonstration satisfactorily answer Conley’s conjecture. Another transfer, based on this methodology, was used in 2004 for SMART-1 of the European Space Agency (Belbruno, 1987; Schoenmaekers et al., 2001; Racca, 2003). The Earth-to-Moon low energy transfers have been analyzed under the perspective of the dynamical system theory (Belbruno, 1994; Koon et al., 2001a). This approach has led to the extension of low energy transfers to the planetto-planet case (Topputo et al., 2005); lastly, the ESA’s BepiColombo spacecraft is being designed to perform a gravitational capture at Mercury (Jehn et al., 2004). The Earth-to-Moon low energy transfers are based on the concept of weak capture at the Moon. This is a capture where the Kepler energy with respect to the Moon is nonpositive and the motion of the particle with respect to the Moon is unstable. Such captures are generally temporary. Weak capture occurs in a special region in phase space about the Moon called the weak stability boundary, WSB, rigorously defined by Belbruno (2004); it has been proven that the motion of the particle in the WSB is chaotic due to the existence of a hyperbolic invariant set. This boundary can generally be viewed as a location where a particle lies between capture and escape with respect to the Moon, or any other secondary body such as Jupiter, in the case of the Sun–Jupiter system. The link between the weak capture and the resonance transition is described for the first time by Belbruno (1990): when the particle is initially in weak capture at the Moon, then it transitions onto a resonant orbit about the Earth, in resonance with respect to the Moon, in both forward and backward time. Therefore, this yields resonance transitions which occur due to passing through weak capture at the Moon, or equivalently, when the particle’s orbit passes through the weak stability boundary. An analogous motion has been observed for a special set of short period comets that moves about the Sun in resonance with Jupiter. When these comets approach Jupiter, their orbit abruptly transitions into another orbit about the Sun, also in resonance with respect to Jupiter, but generally of a different type (Belbruno and Marsden, 1997; Koon et al., 2001b). The first of these was observed for the comet Lexell which transitioned from a 5:4 resonance into a 2:1 resonance, denoted by 5:4 ? 2:1. (In general, an m:n resonant orbit means that m times the comet’s period is approximately n times the period of Jupiter.) In this paper we study the motion of a particle of negligible mass, a rock or a spacecraft, P 3 , that moves in the same plane as two nonzero bodies, P 1 ; P 2 , of masses m1 ; m2 , respectively, where we assume m1 m2 , and where P 1 ; P 2 move about their common center of mass in uniform circular orbits. This defines the planar circular restricted three-body problem. The motion of P 3 is studied in the Earth–Moon system, thus P 1 and P 2 are the Earth and the Moon, respectively. If the total energy of P 3 , called the Jacobi energy, C, is suitably restricted, then in a coordinate system rotating with P 1 ; P 2 , retrograde unstable
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periodic orbits, called Lyapunov orbits, exist on both sides of P 2 on the P 1 –P 2 line. They exist within the allowable region of motion of P 3 , called the Hill’s region. From the work of Conley (1968), local invariant hyperbolic manifolds emanate from the Lyapunov orbits, which are topological cylinders. Global extension of these manifolds about P 2 and showing that they transversally intersect would imply that a complicated hyperbolic network exists about P 2 (Llibre et al., 1985). Such a network would give rise to a chaotic dynamics about P 2 , that could possibly be associated to the resonance transitions. This network was numerically shown to exist by Koon et al. (2000, 2001b), in the case where P 1 ; P 2 are Sun, Jupiter, respectively, and where the value of C was suitably restricted. Although this demonstration partially clarifies the structure of the phase space in the proximity of P 2 , obtaining a satisfactory understanding the WSB for a large range of C and mass ratios m2 =m1 has been elusive. The main goal of this paper is to provide a way to numerically explore the resonance structure of the WSB and nearby dynamics. We give a way to do this by formulating a novel definition of WSB, giving rise to the extended WSB, B, where P 3 is no longer constrained to lie at the periapsis/apoapsis of an osculating orbit around the Moon. The dynamics is then analyzed on a suitable set of Poincare´ sections under the perspective of the energy method described below. In particular, by projecting B on this surface of section, we numerically demonstrate that the resonance transition occurs when P 3 passes through the extended WSB. Finally, we analyze the motion of P 3 in the more realistic model that takes into account the gravitational attraction of the Sun. We find that, when the initial resonant motion of P 3 is integrated in the restricted four-body problem, the resonance transition mechanism breaks and P 3 leaves the Earth–Moon system. This is a ballistic escape of which we provide some preliminary examples. This interesting dynamics has potential applications in the frame of low energy interplanetary transfers and in designing transfers to the libration point orbits of the Sun–Earth system. The remainder of the paper is organized as follows. The dynamics of the restricted three- and four-body problems is described in the next section. The energy method is introduced in Section 3 together with three different definitions of WSB. In Section 4 we show the connections between the WSB and the resonance transitions by means of a set of Poincare´ sections showing the global orbit structure. In Section 5 examples of ballistic escaping orbits are given in the frame of the restricted four-body problem. Final remarks and future works are pointed out in Section 6. 2. Dynamics In this section, we define the modeling used to describe the motion of P 3 in both the restricted three- and four-body problems. A detailed derivation of the equations of motion can be found in Szebehely (1967) and Belbruno (2004), for
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the restricted three-body problem, and in Simo´ et al. (1995), for the restricted four-body problem. 2.1. Restricted three-body problem The motion of P 3 , a rock or a spacecraft, is studied in the gravitational field generated by the mutual circular motion of two particles, P 1 ; P 2 of masses m1 ; m2 , respectively, about their common center of mass. It is assumed P 1 ; P 2 represent the Earth and the Moon, respectively, and that P 3 moves in the same plane of P 1 ; P 2 . This is the planar circular restricted three-body problem, or simply RTBP. The equations of motion are written in a barycentric rotating coordinate frame with dimensionless units: the angular frequency of the circular motion of P 1 ; P 2 , around their center of mass, is normalized to 1; the distance between P 1 and P 2 is normalized to 1; the sum of their masses is normalized to 1, m1 þ m2 ¼ 1. The equations of motion can be written in terms of l, the mass parameter, l ¼ m2 =ðm1 þ m2 Þ. P 1 , of mass 1 l, is located at ðl; 0Þ; P 2 , of mass l, is located at ð1 l; 0Þ. The dynamical system describing the motion of P 3 is oX ; ox oX €y þ 2_x ¼ ; oy
€x 2_y ¼
ð1Þ
where 1 1l l 1 Xðx; yÞ ¼ ðx2 þ y 2 Þ þ þ þ lð1 lÞ; 2 r1 r2 2
ð2Þ
and ðx; yÞ is the position of P 3 . The two distances in Eq. (1) are
retrograde Lyapunov orbits and two-dimensional stable and unstable manifolds emanating from them. The value of the Jacobi energy evaluated at Lk , where x_ ¼ y_ ¼ 0, is denoted by C k . For the two collinear points L1 and L2 one obtains C 1 3:20034 and C 2 3:18416. For a given value of C, the Jacobi energy can be used to establish some allowed and forbidden regions for the motion of P 3 . These regions are bounded by the zero-velocity curves, ZðCÞ, defined as ZðCÞ ¼ fðx; yÞ 2 R2 j2Xðx; yÞ C ¼ 0g:
ð6Þ
The set Z is not defined for C 6 3; in these regimes P 3 is allowed to move in the entire configuration space. 2.2. Restricted four-body problem In the last part of the paper, the perturbation produced by the Sun on the motion of P 3 is taken into account. In this case, the resulting four-body problem, RFBP, is no longer autonomous as the Sun is not fixed in the Earth– Moon rotating frame. Moreover, the key properties of the RTBP, as the Lagrangian points, the Jacobi energy, and the Hill’s curves, are no longer defined in a four-body scenario. In this case the equations of motions are oX0 ; ox oX0 €y þ 2_x ¼ ; oy
€x 2_y ¼
ð7Þ
and have the form of Eq. (1) but this time the potential is ms ms X0 ðx; yÞ ¼ Xðx; yÞ þ 2 ðx cos h þ y sin hÞ: ð8Þ rs q
The mass parameter used in this study is l ¼ 0:01215. The dynamical system (1) admits an integral of motion, the Jacobi energy,
The physical constants introduced to describe the motion of the Sun are assumed to be in agreement with the rotating and normalized system introduced previously. Thus, the distance between the Sun and the Earth–Moon barycenter corresponds to q ¼ 3:8881 102 , the mass of the Sun is ms ¼ 3:2890 105 , and its angular velocity with respect to the rotating frame is x ¼ 0:9251. The phase of the Sun, h, is given by
J ðx; y; x_ ; y_ Þ ¼ 2Xðx; yÞ x_ 2 þ y_ 2 ;
h ¼ h0 þ xt;
2
r21 ¼ ðx þ lÞ þ y 2 ; 2
r22 ¼ ðx þ l 1Þ þ y 2 :
ð3Þ
ð4Þ
that, for a given constant C, defines a three-dimensional manifold _ 2 R4 jJ ðx; y; x_ ; y_ Þ C ¼ 0g: J ðCÞ ¼ fðx; y; x_ ; yÞ
ð5Þ
The vector field defined by (1) has five well known equilibrium points, known as the Lagrange points, labeled Lk ; k ¼ 1; . . . ; 5. The collinear points (L1 ; L2 ; L3 ) lie along the x-axis, while the triangular points (L4 ; L5 ) are located at the vertices of two equilateral triangles having P 1 ; P 2 at the other two vertices. The two collinear points, L1 and L2 , lie at approximate x position 0:83691 and 1.15568, respectively. In a linear analysis, collinear points behave like the product saddle center. Thus, in a small neighborhood of the collinear points there exists a family of small
ð9Þ
where h0 is the initial angle. With these constants, the instantaneous position of the Sun is ðq cos h; q sin hÞ, and therefore the P 3 –Sun distance is r2s ¼ ðx q cos hÞ2 þ ðy q sin hÞ2 :
ð10Þ
It is worth noting that the bicircular model (Eqs. (7)– (10)) is not coherent because all the three primaries are assumed to move in circular orbits, and therefore they violate the Newton’s equations (Simo´ et al., 1995). Nevertheless, this model is expected to depict a good approximation of the real four-body (the eccentricities of the Earth’s and Moon’s orbits are 0.016 and 0.054, respectively, and the Moon’s orbit is inclined on the ecliptic by 5 deg).
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
3. Energy method and WSB definitions The aim of this paper is to explore the connections between the resonance transition and the weak capture. In order to define the weak capture condition, it is convenient to analyze the two-body energy of P 3 with respect to the primaries. This is the essence of the energy approach. Analyzing the Kepler energies, and defining the weak capture, allows to locate the weak stability boundaries and to study their structure on appropriate surfaces of section. We denote by H 1 ; H 2 , and H s the Kepler energy of P 3 relative to the Earth, Moon, and Sun, respectively. Letting r2 ; v2 be the distance and the speed, respectively, of P 3 expressed in an inertial Moon-centered frame, then 1 l H 2 ¼ v22 : 2 r2 Analogously, the Kepler energy H 1 reads
ð11Þ
1 1l ; ð12Þ H 1 ¼ v21 2 r1 where r1 ; v1 are the distance and speed of P 3 relative to an inertial Earth-centered frame. In the case of the restricted four-body problem, H s is given by 1 ms ð13Þ H s ¼ v2s 2 rs where rs ; vs are expressed in a Sun-centered inertial frame. Let the flow of either (1) or (7) be represented by uðtÞ ¼ ðxðtÞ; yðtÞ; x_ ðtÞ; y_ ðtÞÞ; t 2 R1 ; we define some special captures of P 3 with respect to P 2 (the Moon in our case). Definition 1 (Weak capture). P 3 is ballistically captured by P 2 at time t, if H 2 ðuðtÞÞ 6 0. P 3 is temporary ballistically captured by P 2 , if H 2 ðuðtÞÞ 6 0 for t1 6 t 6 t2 H 2 ðuðtÞÞ > 0
for t < t1
and
and t > t2 ;
for finite times t1 ; t2 ; t1 < t2 . Temporary ballistic capture is also referred to as weak capture. Definition 2 (Ballistic ejection). P 3 is ballistically ejected (or ballistically escapes) from P 2 along the solution uðtÞ at a time t1 if H 2 ðuðtÞÞ < 0
for t < t1
and
H 2 ðuðtÞÞ P 0
for t P t1 :
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Analogous definitions are made for capture or ejection with respect to the Earth, P 1 , by substituting P 1 ; H 1 , in place of P 2 ; H 2 , respectively. Weak capture and ballistic ejection are conceptually shown in Fig. 1. We now define the resonant orbits. Definition 3 (Resonant orbit). An m:n resonant orbit about the Earth, in resonance with the Moon, is an orbit whose period T is related to the Moon’s period T 2 by mT ¼ nT 2 ; where m; n are positive integers. It can be easily shown (see Belbruno et al., 2008) that Definition 3 yields the following condition on the semimajor axis: n 2=3 1=3 a¼ ð1 lÞ : ð14Þ m The energy method, and in particular the condition on the Keplerian energy in order to get weak capture at P 2 , is illustrated through an example. The energy method does not require restricting C, but rather only the semimajor axis a has to be determined from condition (14) once the two resonance numbers, m and n, are specified. In Fig. 2 we have integrated in forward time a 2:1 resonant orbit with the Moon having the initial eccentricity e ¼ 0:316. It can be seen that, when performing the second revolution around the Earth, P 3 gets weakly captured by the Moon (Fig. 2(a)) where H 2 K 0 (Fig. 2(c)). P 3 performs one revolution around the Moon in an unstable ellipse (Fig. 2(c)). After this temporary capture, P 3 lies again in an approximate elliptic state relative to the Earth but this time it lies on a 1:3 resonant orbit with the Moon. Hence, after a time interval equal to three Moon periods, P 3 approaches the Moon. After this second interaction its orbit is in 1:2 resonance with the Moon and so on. It is noted, in Fig. 2(c), that the condition H 2 K 0 is satisfied along the weak capture as required by the energy method. 3.1. WSB definitions The focus of the paper is to study resonance hops due to P 3 passing near P 2 in the RTBP. In addition, we are interested in studying this process when the Sun perturbation is
Fig. 1. Temporary ballistic capture and ballistic ejection. (The subscript of H is not shown to indicate either the Earth, P 1 , or Moon, P 2 .)
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F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329 5
x 10
x 10
4
18
6
16
1:3
14
4
2:1
0
y (km)
y (km)
12
1:2
2
10 8 6
–2
4
–4
Moon
2
–6
0
–2 –10
–8
–6
–4
–2
0
2
x (km)
0
4
5
10
x (km)
5
x 10
4
x 10
3 140
2.5
H (adim.)
1.5
H2
120
hop #1
hop #2
hop #3
100
T (days)
2
1 0.5
1:3 80 60
1:2
0 40
–0.5
H
1
–1 –1.5
0
20
40
20
60
80
100
120
140
160
t (days)
0
2:1 20
40
60
80
100
120
140
160
t (days)
Fig. 2. Example of resonance transition and weak capture at the Moon. The resonances sequence is 2:1 ? 1:3 ? 1:2. The initial orbit has the semimajor axis given by Eq. (14) with m ¼ 2; n ¼ 1. The temporary capture in Fig. 2(b) refers to the first Moon approach. Figs. 2(c) and 2(d) show the two-body energies, relative to the Earth (H 1 ) and to the Moon (H 2 ), and the orbit’s period, respectively. It is noted that, as P 3 moves through the different resonances, H 1 varies (increases or decreases), while H 2 K 0. The orbits are in approximate resonance with the Moon since the period is not exactly equal to those associated to a perfect resonance.
taken into account, that is when P 3 moves in the RFBP. As mentioned previously, it has been observed that numerical integration from weak capture near P 2 results in P 3 transitioning into resonant orbits with respect to P 2 (Belbruno, 1990). Thus, in order to study resonance hops, it is necessary to study weak capture at P 2 . The region where weak capture occurs can be used to define the WSB. It is defined by Belbruno and Miller (1993), Belbruno (2004), and further refined by Garcı´a and Go´mez (2007). The first algorithmic definition of WSB is recalled below. We consider trajectories starting on a radial line lðhÞ departing from P 2 and making an angle of h with the x-axis (Fig. 3). The trajectories are assumed to start at the periapsis of an osculating ellipse around P 2 , whose semimajor axis a lies on lðhÞ and whose eccentricity e is fixed; this makes r2 ¼ að1 eÞ. The initial velocity of the trajectory is taken perpendicular to lðhÞ, and the Keplerian energy of P 3 relative to P 2 is H 2 < 0. We want to emphasize that
Fig. 3. The stability criterion used to numerically define W, the algorithmic WSB.
in this definition r_ 2 ¼ 0. The motion, for fixed values of h and e, depends only on the distance r2 . The motion is called stable if, after leaving lðhÞ; P 3 makes a full turn about P 2 , without going around P 1 , and returns to lðhÞ at a point with H 2 < 0 (Fig. 3). The motion is otherwise called unstable.
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
Definition 4 (Algorithmic WSB). The algorithmic weak stability boundary is the locus of all points r ðh; eÞ along the radial line lðhÞ for which there is a change of stability of the initial trajectory, that is, r ðh; eÞ is one of the endpoints of an interval ½r1 ; r2 characterized by the fact that for all r 2 ½r1 ; r2 the motion is stable, and there exist r0 62 ½r1 ; r2 , arbitrarily close to either r1 or r2 for which the motion is unstable. Thus W ¼ fr ðh; eÞjh 2 ½0; 2pÞ; e 2 ½0; 1Þg: It has been numerically demonstrated that the weak stability boundary W is contained in the set given by the union of the stable manifolds, having zero radial velocity, associated to the L1 and L2 Lyapunov orbits (Garcı´a and Go´mez, 2007). Although Definition 4 yields the location of the set W in the configuration space, that is ideal to construct low energy spacecraft transfers from the Earth to the Moon, it does not give any insight into the capture dynamics and the phase portrait characterizing the WSB. We are interested in studying the dynamical properties of the WSB on a suitable surface of section. In order to do this, we give two analytic definitions of the WSB based on the consideration made in Belbruno (2004) and Belbruno et al. (submitted for publication). In these definitions, the WSB depends on the value of the Jacobi energy C. Definition 5 (Classic WSB). Let the two sets r and R be r ¼ fðx; y; x_ ; y_ Þ 2 R4 j_r2 ðx; y; x_ ; y_ Þ ¼ 0g; R ¼ fðx; y; x_ ; y_ Þ 2 R4 jH 2 ðx; y; x_ ; y_ Þ 6 0g; then, for C 2 ½C ; C 1 Þ, the analytic WSB, W, is defined as
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where C 2 ½C ; C 1 Þ and C is estimated such that B is nonempty. Definition 6 corresponds to the definition of W in Definition 5 with the condition r_ 2 ¼ 0 removed. This means that the condition that P 3 is at the periapsis of its orbit around the Moon in Definition 4 is removed. Through B, the capture may occur when P 3 is at any point of its osculating orbit around the Moon. We have estimated numerically that the greatest lower bound C of Jacobi constants such that both W and B are nonempty, with the introduced value of l, is approximately C ’ 2:8868 (Belbruno et al., 2008). In the next section, we represent B on a surface of section and numerically show that resonant orbits pass through B when they perform a resonance hop. 4. The role of the WSB in resonance transitions In this section, we relate the weak stability boundary to the resonance transition. In order to do this, both the resonant orbits and the WSB are visualized on an appropriate subsets of the phase space represented by a Poincare´ section. On this section, not only the resonant orbits but the chaotic dynamics and the global orbit structure of the RTBP, including the WSB, can be studied. When represented on this surface of section, the set B lets us unlock the role of the WSB in resonance hops. For the sake of brevity, we show both the Poincare´ section and the WSB corresponding to just one energy level. A detailed analysis, including a number of sections at different energy levels, can be found in Belbruno et al. (2008).
W ¼ J ðCÞ \ R \ r; where J ðCÞ is the three-dimensional manifold (5) and C is estimated so that W exists. A value of C ¼ 2:2 is obtained in Belbruno (2004), which is a crude estimate since the approximation is not dynamically based. Definition 5 implies that W has a dimension of 2, and is generally an annular region in phase space. The condition of C < C 1 is assumed in order that the inner and outer Hill’s regions are connected. The dependence on l is suppressed since it is fixed. We refer to W as the classic WSB. It is worth pointing out that, with Definition 5, the set W can be represented on a suitable surface of section. W is a more realistic and precise representation of WSB than W, and, in general, W is smaller than W (Belbruno et al., 2008). Analogously to W, the set W implies that P 3 is on the apsidal line (i.e. r_ 2 ¼ 0) of the osculating orbit around the Moon when it lies on the WSB. We remove this condition in the extended WSB. Definition 6 (Extended WSB). Let J ðCÞ be the threedimensional manifold (5) and r be the set in Definition 5, then the extended WSB, B, is B ¼ J ðCÞ \ R
4.1. Poincare´ section and global orbit structure We consider the flow of (1) on the three-dimensional energy surface J ðCÞ, defined by Eq. (5), for a given value of C. On this manifold we define a two-dimensional surface of section by the set _ 2 J ðCÞjy ¼ 0; y_ > 0g: SðCÞ ¼ fðx; y; x_ ; yÞ A Poincare´ section consists of the set of points given by the intersection of the flow of (1) with the surface S. They are obtained by computing the intersection between hundreds of flows, given by hundreds of initial conditions, and S. In the following, we discuss a Poincare´ map obtained with C ¼ 3:18176, that is the Jacobi energy of a ‘‘normalized” 2:1 orbit with zero eccentricity (see Belbruno et al., 2008). It is convenient to show sections in both the (x; x_ ) and (a; h) planes, where a and h are the semimajor axis and the true anomaly, respectively, of the osculating twobody orbits around the Earth. Fig. 4 shows the Poincare´ map in the (x; x_ )-plane. The shape of the region is due to the projection of J ðCÞ onto this plane. It is noted that the flow is organized on invariant tori in the regions close to the Earth (x 0) and to the Moon (x 1), while a ‘‘chaotic sea” of points characterizes
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F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
Fig. 4. Poincare´ section of the flow of the RTBP for C ¼ 3:1817683176.
the region between them (the orbits are not organized in any discernable pattern). There are invariant curves in Fig. 4(a) surrounding 4:1, 3:1, 8:3, 7:3, and 2:1 resonant orbits. The invariant tori on the right of Fig. 4(a) are associated to stable orbits around the Moon. In Fig. 4(b), the island of invariant closed curves surrounding the 2:1 resonant point is magnified. These invariant curves are due to two-dimensional invariant tori in the phase space, which are indicative of KAM tori, projected onto the section. An exactly resonant, stable, orbit exists at the center of each island. The term island indicates the region on the section where there is a set of invariant curves. If they surround a resonant periodic point, we will refer to them as resonance islands. Orbits will not exhibit a resonance hop within an island since they would be trapped within the tori themselves. The chaotic sea surrounding the islands represents the region of the phase space where the resonance transitions take place, where P 3 moves between resonance islands. This is discussed further below. Fig. 5 represents the section of Fig. 4 viewed in the coordinates (a; h). It is worth noting that, in this set of coordinates, standard two-body orbits around the Earth generate vertical straight lines. In Fig. 5(a) the global resonant structure is much more evident, where invariant tori around the 2:1, 3:1, 4:1, 7:3, and 8:3 orbits can be distinguished. For a given m:n resonance, through the condition (14), we are able to locate and label the semimajor axis corresponding to the resonant orbit (vertical dashed lines in Fig. 5(a)). The resonant islands are centered on these lines, therefore their nature depends on which line they are centered. The size of each island is indicative of the importance of the corresponding resonance. Once P 3 is initialized on one of these tori, its motion will take place on the same torus for all future time. In particular, as mentioned above, an exact, stable, resonant orbit is defined at the center of each resonance island. Correspondingly, an exact, unstable, resonant orbit is defined between the stable islands along a strip of constant a (Koon et al., 2000). If P 3 is moving in the chaotic sea outside these resonance tori islands, then,
with the exception of the 4:1 island orbits, the other resonance islands are connected with each other. P 3 could pass between island boundary regions by executing a substantial variation in the semimajor axis. In this way, resonance hops can occur since the chaotic sea in this plane spans a wide range of semimajor axes. Fig. 5(a) magnifies the tori around the 2:1 orbits. The effect of the temporary capture by the Moon can be again viewed in the top- and bottom-right side of the picture. Here, the gravitational attraction of the Moon acts to increase the semimajor axis of the P 3 ’s orbit and breaks the invariant structure. This mechanism occurs at h ffi p meaning that P 3 gets captured when it is approximately at the apoapsis of its orbit about the Earth. We recall that the present value of the Jacobi integral is less than C 1 , so the size of P 3 ’s orbit can be raised up to reach the outer Hill’s region beyond the Moon’s orbit. 4.2. Representation of the extended WSB In this section, we visualize the WSB on SðCÞ. We have defined the algorithmic, W, the classic, W, and the extended, B, weak stability boundaries (Definitions 4–6). It is convenient to view W and B on SðCÞ in ðx; x_ Þ coordinates, for the fixed value of C (we recall that SðCÞ is the set on J ðCÞ where y ¼ 0; y_ > 0). It turns out that, on SðCÞ; W corresponds to a subset of the x-axis. This follows from the fact that the condition: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r_ 2 ¼ ½ðx þ l 1Þ_x þ y y_ = ðx þ l 1Þ2 þ y 2 ¼ 0; ð15Þ combined with y ¼ 0 implies that x_ ¼ 0 (we assume r2 < þ1). Thus, SðCÞ \ W is a one-dimensional set. In Fig. 6 we show a portion of SðCÞ; C ¼ 3:18176, where both W and B are represented. The projection of J ðCÞ is also shown; its boundaries are the curves with y_ ¼ 0. The location of the Moon is seen at x 1. The points of the iterates of the Poincare´ map are contained within the projection of the Jacobi energy surface as expected. The 2:1
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
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Fig. 5. The points of Fig. 4 mapped into the (a; h) set.
roles in hop dynamics. The set B is an interesting region; we describe some of its structure that was observed in the following two examples.
Fig. 6. A portion of the phase space containing the classic and the extended WSB, W and B, respectively. The boundaries of both J ðCÞ (the line y_ ¼ 0) and R (the line H 2 ¼ 0) are also shown.
resonance island is also labeled, as well as a family of invariant curves about the Moon. The region H 2 6 0 is shown, with a dashed curve for the boundary, H 2 ¼ 0. The set B is represented by bold points; note that some of these points in B are about the Moon location. In this plot, the set of invariant curves about the Moon is excluded since its points do not give rise to the unstable behavior characterizing the temporary capture required in the WSB. The set of invariant curves about the Earth is also excluded since they do not yield the unstable behavior of temporary required for the WSB. The subset of B which lies on the x-axis, as indicated, form the set W. That is W ¼ B \ f_x ¼ 0g. 4.3. Resonance transitions through B It is noted that B largely overlaps with the chaotic sea on S. Also, within this section, there are a lot of resonance islands in which P 3 can travel from the boundaries to B via the chaotic sea of points. In this sense B, and its subset W, are associated to resonance motion, and play important
4.3.1. Example 1 In Fig. 7 we have reported the Poincare´ section SðCÞ; C ¼ 3:18176, illustrated in Fig. 5(a). As already observed, the section, when viewed in the ða; hÞ plane, is made up by resonant islands surrounded by a chaotic sea of irregular orbits. On this plane, the points belonging to B (those points with H 2 6 0) are shown bold. A sample orbit having the points inside both B and the chaotic sea has been selected and a portion of this orbit is plotted in the inertial frame in Fig. 7(b). As it can be seen, the initial P 3 ’s orbit is in an 8:3 resonance with the Moon; it is then captured by the Moon and ejected into a 5:2 resonant orbit. This is an 8:3 ? 5:2 resonance transition. There is evidence that this mechanism occurs when the orbit is temporary captured by the Moon and hence when it lies in the extended WSB set B. 4.3.2. Example 2 This example is defined with the energy level C ¼ 3:15109. In Fig. 8(a) we have reported the Poincare´ section of the flow of the RTBP at this energy level. Two intersections of the orbit with the plane of section, contained in B, are marked as #1 and #2. These are not two consecutive intersections between the orbit of P 3 and the surface of section, but they rather correspond to two consecutive weak captures where P 3 orbits the Moon. Thus, by the given definition of B, both the two states #1 and #2 lie on B. We study the piece of the orbit between the two weak captures and in particular in Fig. 8(b) we show the P 3 ’s osculating orbital period. Before the first capture, the orbit of P 3 is in 8:3 resonance with the Moon; after orbiting the Moon for a while, P 3 is placed into a 2:1 resonant orbit with the Moon and performs the following sequence of resonance transitions 2:1 ? 5:2 ? 2:1 ? 2:1 ? 7:3 ? 2:1. Then the particle is again captured by the Moon and is
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F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329 5
x 10
3 2
5:2
y (km)
1 0 –1
8:3
–2 –3 –3
–2
–1
0
1
2
x (km)
a
3
x 105
Fig. 7. Example 1 – The extended WSB set B viewed in the (a; h) plane (a). B corresponds to the region (top and bottom) where the 2:1 resonant island is distorted due to the Moon’s gravitational attraction. A portion of a sample orbit intersecting B exhibits an 8:3 ? 5:2 resonance transition (b).
periods 30
weak capture #2 3:2
T (days)
25
20
weak 1:1 capture #1
15
10
5
x
8:3 5:2 250
300
7:3 350
2:1 400
450
500
t (days)
Fig. 8. Example 2 – Two states defined inside B corresponding to two consecutive weak captures where P 3 orbits the Moon. Between the two weak captures the motion of P 3 goes through resonant states and performs a number of resonance transitions.
ejected into another 2:1 orbit. It can be noted that the resonances are not exact. This series of resonance transitions is generated between two states defined within B.
5. Comments on ballistic escape and applications The perturbation of the Sun is now taken into account. In particular, we study how the initial orbits, in resonance with the Moon, behave in the four-body problem described by Eqs. (7)–(10). This dynamics is even more complicated than of the RTBP since the potential (8) contains two nonautonomous terms describing the gravitational attraction of the Sun on P 3 . These terms are particularly relevant when P 3 moves in the region outside the Moon’s orbit. In the outer region, indeed, the resonance motion, characteristic of the RTBP, is broken. Moreover, as already observed by Belbruno (1990), the particle P 3 leaves the Earth–Moon
system an targets a heliocentric orbit: this is a ballistic escape. Fig. 9 reports the orbit, obtained with the same initial condition that generates the orbit of Fig. 2, integrated under the four-body dynamics (the time span is equal to 10 Moon’s periods). In the RTBP, the initial 2:1 resonant orbit transitions into a 1:3 and then into a 1:2 orbit (P 3 performs a 2:1 ? 1:3 ? 1:2 resonance hop). When the same initial conditions is flowed in the RFBP, P 3 orbits the Earth one time and then gets temporary captured by the Moon up to reach the exterior region (Fig. 9(a)). Here, the 3:1 orbit is missed and P 3 is simply driven away from the Earth–Moon system by the gravitational attraction of the Sun. The particle targets a heliocentric orbit and falls in a temporary state where its motion is mainly governed by the Sun (Fig. 9(b)). It is worth pointing out that this kind of escape takes place without carrying out any maneuver; it just exploits the
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329
1
x 10
6
1 0.5
y (AU)
y (km)
0
–0.5
0
–1
–1.5 –2 –1
–1 –0.5
0
x (km)
0.5
1 6
x 10
–1
–0.5
0
0.5
1
x (AU)
Fig. 9. The orbit of Fig. 2(a) integrated under the four-body dynamics (7). The particle P 3 is initially in a 2:1 resonant orbit and gets weakly captured by the Moon as expected. Once in the exterior region, P 3 does not hop into another resonant orbit (like in Fig. 2(a)) as the resonance transition mechanism is broken by the Sun perturbation. P 3 escapes from the Earth– Moon system in a totally ballistic fashion and targets a heliocentric orbit. This ballistic escape occurs in a relatively short time.
intrinsic dynamics of the four-body problem. In this example h0 ¼ 0 in Eq. (9). Another interesting example is presented in Fig. 10. In this case an initial 2:1 resonant orbit with the Moon, e ¼ 0:333, is integrated for 13 Moon’s periods. After the weak capture at the Moon, P 3 is ejected into a high elliptical orbit about the Earth having the apogee of approximately 106 km. The perturbation of the Sun lowers the perigee of this orbit and so P 3 flies-by the Earth and performs a ballistic escape. The shape of the trajectory – from the weak capture at the Moon up to the Earth approach – is very similar to the exterior WSB transfers (Belbruno and Miller, 1993; Belbruno, 1994, 2004; Belbruno, 2007). The point is that the exterior WSB transfers were designed to go from the Earth to the Moon; it has been demonstrated by Belbruno (2004) that these transfers are regulated by a chaotic dynamics. In this study the behavior of the resonant orbits, in the four-body dynamics, was investigated and we came up with that there is a sort of link between the resonant motion and the low energy transfers. The key is that both processes take place in the weak stability boundaries of the Moon and the Earth. In Fig. 10(b) the heliocentric orbit is shown. Adjusting the phase of the Sun results in a heliocentric orbit inside or outside the Earth’s orbit. In this case we have assumed h0 ¼ p in Eq. (9). Through numerical experiments it has been found that this kind if escape can bring P 3 up to 0.15 AU far from the Earth’s orbit. We recall once again that this mechanism exploits the simultaneous gravitational attractions of the Sun, Earth, and Moon and hence the escape occurs at zero cost. This is ideal for spacecraft applications, for instance in the frame of low energy interplanetary transfers aimed at reaching Mars and Venus with the lowest propellant mass (Topputo et al., 2005). This process, indeed, can be optimized and there is evidence that a small
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maneuver performed at the closest point to the Earth (see Fig. 10(a)) may bring the spacecraft very far from the Earth’s orbit. The Keplerian energies, H 1 and H 2 , are reported in Fig. 10(c). The condition H 2 K 0 is again verified when P 3 gets weakly captured by the Moon. After this weak capture, the function H 1 monotonically increases and changes the sign from negative to positive. This is a ballistic escape in the sense of the Definition 2. H 1 is zero approximately 120 days after the Moon encounter. Fig. 10(d) shows the trend of the distance from P 3 to the Sun. Through numerical experiments we have found that, after the ballistic escape, the Keplerian energy of P 3 relative to the Sun, H s (Eq. (13)), exhibits small steps like the function H 1 does in Fig. 2(c) where P 3 is in resonance with the Moon. Thus, we have assessed that the heliocentric orbits resulting from the ballistic escape are in resonance with the Earth. There are in fact two integers, m0 ; n0 , that verify m0 T ¼ n0 T 1 , where T 1 is the Earth’s period and T is the period of the osculating orbit of P 3 around the Sun. We have found that the two resonance numbers, m0 and n0 , in this case are typically greater than 10. We conjecture that the geocentric resonant orbits with the Moon are connected to the heliocentric resonant orbits with the Earth. In this paper, we have numerically demonstrated the role of the WSB in the resonance transition. There is evidence (Koon et al., 2000) that the WSB of the Moon and that of the Earth are defined at almost the same energy level. 6. Conclusions In this paper, we have investigated the nature of the resonance motion affecting a particle that moves in the vector field generated by the gravitational attractions of the Earth and Moon. The resonance transition has been approached with the energy method: from this point of view, the Keplerian energy of the particle relative to the Moon is key in revealing the nature of the motion. In particular, with suitable restrictions on this energy, three different definitions of weak stability boundaries have been give. With the aid of suitable Poincare´ sections, that reveal the global orbit structure of the RTBP, and by representing the extended WSB on these surfaces, it has been numerically shown that resonance transitioning orbits pass through the WSB. In this sense, the WSB is a ‘‘resonance hub”. In the last part of the paper, the resonant motion under the perturbation of the Sun has been studied. In this case the resonance transition mechanism is broken and the particle leaves the Earth–Moon system, at zero cost, by means of a ballistic escape. This kind of escape occurs in reasonable times and so it seems to be particularly appealing for potential spacecraft applications. In particular, the ballistic escaping trajectory can be optimized and integrated with a suitable maneuver to yield a low energy interplanetary transfer.
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Fig. 10. An initial 2:1 resonant orbit with the Moon, e ¼ 0:333, integrated under the four-body dynamics. (a) The trajectory of P 3 and the Moon’s orbit (dashed) in a Earth-centered frame. After getting weakly captured by the Moon, P 3 is placed in an ‘‘exterior-like” WSB transfer trajectory (Belbruno and Miller, 1993) that approaches the Earth. The particle P 3 flies-by the Earth and escapes the Earth–Moon system in a hyperbolic state relative to the Earth. In (b) we have reported the trajectory of P 3 and the Earth’s orbit (dashed) in a Sun-centered frame. (c) The Keplerian energies of P 3 relative to the Earth, H 1 , and the Moon, H 2 . According to the Definition 2, P 3 performs a ballistic escape as the function H 1 becomes positive. (d) The distance from P 3 to the Sun along the trajectory. Numerical experiments shows that this distance may grow up to 1.15 AU. The cost for this escape is zero. It is ideal for spacecraft applications.
Acknowledgment The research of M.G. is partially supported by NSF Grant DMS 0601016. References Belbruno, E. Lunar capture orbits, a method of constructing Earth–Moon trajectories and the lunar GAS mission, AIAA Paper No. 97-1054, in: Proc. AIAA/DGLR/JSASS Inter. Elec. Propl. Conf., 1987. Belbruno, E. Examples of nonlinear dynamics of ballistic capture and escape in the Earth–Moon system, AIAA Paper No. 90-2896, in: Proc. Annual Astrodyn. Conf., 1990. Belbruno, E., Miller, J. Sun-perturbated Earth-to-Moon transfers with ballistic capture. J. Guid. Control Dyn. 16, 770–775, 1993. Belbruno, E. The Dynamical Mechanism of Ballistic Lunar Capture Transfers in the Four-Body Problem from the Perspective of Invariant Manifolds and Hill’s Regions, CRM Research Report 270, Centre de Recerca Matematica, Institute d’Estudis Catalans, Barcelona, 1994.
Belbruno, E., Marsden, B. Resonance hopping in comets. Astron. J. 113, 1433–1444, 1997. Belbruno, E. Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press, Princeton, 2004. Belbruno, E. Fly Me to the Moon: An Insiders Guide to the New Science of Space Travel. Princeton University Press, Princeton, 2007. Belbruno, E., Topputo, F., Gidea, M. Resonance transitions associated to weak capture in the restricted three-body problem. J. Adv. Space Res. 42, 1330–1351, 2008. Conley, C. Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math. 16, 732–746, 1968. Garcı´a, F., Go´mez, G. A note on weak stability boundaries. Celest. Mech. Dyn. Astron. 97, 87–100, 2007. Jehn, R., Campagnola, S., Garcı´a, D., Kemble, S. Low-thrust approach and gravitational capture at Mercury, in: Proc. 18th Int. Symp. Space Flight Dynamics, pp. 487–492, 2004. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D. Heteroclinic connections between periodic orbits and resonance transition in celestial mechanics. Chaos 10, 427–469, 2000.
F. Topputo et al. / Advances in Space Research 42 (2008) 1318–1329 Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D. Low energy transfer to the Moon. Celest. Mech. Dyn. Astron. 81, 63–73, 2001a. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D. Resonance and capture of jupiter comets. Celest. Mech. Dyn. Astron. 81, 27–38, 2001b. Llibre, J., Martinez, R., Simo´, C. Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem. J. Differ. Eqns. 58, 104–156, 1985. Racca, G. New challenges to trajectory design by the use of electric propulsion and other means of wandering in the solar system. Celest. Mech. Dyn. Astron. 85, 1–24, 2003.
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Schoenmaekers, J., Horas, D., Pulido, J.A. SMART-1: with solar electric propulsion to the Moon, in: 16th International Symposium on Space Flight Dynamics, Pasadena, California, 3–7 December, 2001. ´ ., Masdemont, J. The Bicircular Model Simo´, C., Go´mez, G., Jorba, A Near the Triangular Libration Points of the RTBP, From Newton to Chaos. Plenum Press, New York, 1995. Szebehely, V. Theory of Orbits: The Restricted Problem of Three Bodies. Academic, New York, 1967. Topputo, F., Vasile, M., Bernelli-Zazzera, F. Low energy interplanetary transfers exploiting invariant manifolds of the restricted three-body problem. J. Astronaut. Sci. 53, 353–372, 2005.