- Email: [email protected]
A note on libration point orbits, temporary capture and low-energy transfers
Acta Astronautica 67 (2010) 1038–1052
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
A note on libration point orbits, temporary capture and low-energy transfers$ E. Fantino a,b,, G. Go´mez a,c, J.J. Masdemont a,b, Y. Ren a,b a
Institut d’Estudis Espacials de Catalunya, Gran Capita 2-4, 08034 Barcelona, Spain Universitat Polite cnica de Catalunya, Dpt. Matematica Aplicada I, Diagonal 647, 08028 Barcelona, Spain c Universitat de Barcelona, Dpt. Matematica Aplicada i Ana lisi, Gran Via 585, 08007 Barcelona, Spain b
a r t i c l e i n f o
abstract
Article history: Received 28 January 2010 Received in revised form 6 June 2010 Accepted 20 June 2010
In the circular restricted three-body problem (CR3BP) the weak stability boundary (WSB) is defined as a boundary set in the phase space between stable and unstable motion relative to the second primary. At a given energy level, the boundaries of such region are provided by the stable manifolds of the central objects of the L1 and L2 libration points, i.e., the two planar Lyapunov orbits. Besides, the unstable manifolds of libration point orbits (LPOs) around L1 and L2 have been identified as responsible for the weak or temporary capture around the second primary of the system. These two issues suggest the existence of natural dynamical channels between the Earth’s vicinity and the Sun–Earth libration points L1 and L2. Furthermore, it has been shown that the Sun– Earth L2 central unstable manifolds can be linked, through an heteroclinic connection, to the central stable manifolds of the L2 point in the Earth–Moon three-body problem. This concept has been applied to the design of low energy transfers (LETs) from the Earth to the Moon. In this contribution we consider all the above three issues, i.e., weak stability boundaries, temporary capture and low energy transfers, and we discuss the role played by the invariant manifolds of LPOs in each of them. The study is made in the planar approximation. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Circular restricted three-body problem Libration points Periodic orbits Invariant manifolds Two-body problem Weak stability boundary
1. Introduction In the CR3BP, the WSB is defined as a boundary set in the phase space between stable and unstable motion relative to the second primary: Keplerian orbits about this body, perturbed by the first primary, are said to be stable if, after a prescribed number of revolutions, preserve the character of bounded motion, unstable otherwise. This definition of WSB is due to [3]. The concept and its application have later been analysed by several authors, e.g., [4,12], also in connection with the construction of
$
This paper was presented during the 60th IAC in Daejeon.
Corresponding author at: Institut d’Estudis Espacials de Catalunya,
Gran Capita 2-4, 08034 Barcelona, Spain. E-mail addresses: [email protected] (E. Fantino), [email protected] (G. Go´mez), [email protected] (J.J. Masdemont), [email protected] (Y. Ren). 0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.06.037
ballistic capture trajectories around the second primary [11]. This contribution takes the steps from [8] and, instead of the eccentricity, we parametrize the regions of weak stability by the Jacobi constant. We aim at clarifying some structural aspects of the WSB, particularly in relation with the dynamics of the invariant manifolds of LPOs and the temporary capture around them as a function of the energy of the third body. Then, it is thanks to the work of [5,10] that some wellknown phenomena of temporary capture around the second primary of a CR3BP (i.e., the resonance transitions of the comets Oterma and Gehrels 3 as a result of temporary captures on behalf of Jupiter in the Sun–Jupiter system) have been understood as controlled by the invariant manifolds of Jupiter’s LPOs. Here, the characteristics of this effect are investigated in terms of the dynamics of the unstable manifolds of LPOs with respect to the second primary.
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
Maneuver (ΔV) at Patch Point
Sun-earth-S/C system
Earth-Moon-S/C system
Moon
Sun-Earth L2 portion using ‘twisting’
Lunar capture portion Sun
Moon’s orbit
Earth x
L2
y
L2
y
Sun
Maneuver (ΔV) at Patch Point
1039
Earth L2 orbit x
Fig. 1. A low-energy transfer from the Earth to the Moon through the L2 libration points of the Sun–Earth and Earth–Moon systems as seen in the Sun–Earth synodical barycentric reference frame (a), and its construction based on patching invariant manifolds of the Sun–Earth and Earth–Moon CR3BPs (b) [9].
The LETs from the Earth to the Moon consist in connecting trajectories guided by the invariant manifolds of the central objects around the L2 collinear libration points of the Sun–Earth and Earth–Moon CR3BPs in such a way that the spacecraft can reach the Moon’s vicinity from a low-Earth orbit (LEO) practically at the only cost associated to leaving the LEO. The mechanism for the LETs was first presented in [9]. See also [6,7]. Fig. 1 illustrates the idea for the planar case in the Sun–Earth barycentric synodical reference frame: a branch of a stable manifold of a planar Lyapunov orbit around LSE 2 (i.e., the Sun–Earth L2 point) drives the spacecraft away from the Earth, then the unstable manifold of the same orbit redirects it to a region where also the stable manifold of a planar Lyapunov orbit around LEM (i.e., the Earth–Moon L2 point) flows. 2 There, the intersection in configuration space (through a conveniently set Poincare section) is sought. The difference in velocity between the two intersecting trajectories (equal to zero if there is a heteroclinic connection) represents the cost of the transfer. The dynamical characteristics of the two CR3BPs and their kinematical relationship are such that low cost (and even zero cost) connections of the EM type LSE can easily be found. We have investigated 2 L2 the existence and the characteristics (particularly the cost) of transfers between any combination of the two EM collinear libration points, the so-called LSE (i,j=1,2) i Lj connections. Section 2 illustrates the main findings concerning the EM exploration of the LSE (i,j = 1,2) connections. Section 3 i Lj contains an analysis of the WSB regions of the Sun–Earth system and discusses the role of the invariant manifolds of LPOs therein. The temporary capture of WSB unstable trajectories in the vicinity of the collinear libration points of the system is examined in Section 4. Finally, Section 5 discusses the issue of temporary capture around the Earth of trajectories belonging to the unstable manifold of LPOs in the Sun–Earth CR3BP. 2. LSE LEM connections i j In the coupled CR3BP, the Sun–Earth and the Earth– Moon three-body problems are kinematically linked in EM inertial space through their orbital phases aSE at 0 and a0 t = 0. This allows to transform at any time the invariant
manifold trajectories of the Earth–Moon problem into the Sun–Earth barycentric synodical reference frame and to look for intersections with the manifold trajectories of the Sun–Earth problem at a given Poincare section. More specifically, the transfer sought is one between the unstable manifold of a LPO in the Sun–Earth model and the stable manifold of a LPO in the Earth–Moon system.1 There are four types of connections because there are four libration point pairs of the type LSE (i= 1,2) LEM (j= 1,2). i j We have explored each type by varying the progenitor Lyapunov orbits in the two CR3BPs over a wide range of energies. Then, for each pair of LPOs, the initial orbital SE phase a0 ¼ aEM 0 a0 of the Earth–Moon problem in the Sun–Earth synodical barycentric reference frame has been varied at increments of 51 between 0 and 360. The Poincare section is the yy_ plane orthogonal to the x-axis of the Sun–Earth synodical barycentric reference frame at some x-coordinate xP. The value of xP is varied discretely between the location of the given Sun–Earth Lyapunov orbit and the Earth. This analysis provides information concerning the feasibility, the cost and the time of flight of each type of connection. Concerning the cost, it has to be noted that the maneuver at the yy_ Poincare section is obtained as the difference between the x velocity components of two intersecting orbits belonging to the two connecting manifolds. This method does not consider transit orbits, i.e., orbits which flow inside the manifold tubes. Besides, the maneuvers considered are entirely in the x direction. EM For each LSE combination, the variation of the i Lj above listed parameters, i.e., the relative orbital phase a0 at t =0 of the two models and the x coordinate of the Poincare section, allow to construct maps of the minimum (i.e., within the present method) cost (i.e., maneuver size) between the connecting trajectories as a function of the Jacobi constants JSE and JEM of the LPOs of the two CR3BPs. Figs. 2–5 show that low and zero cost connections exist EM EM for LSE and LSE over a wide range of energy 1 L2 2 L2 EM EM levels. Connections of the type LSE and LSE 1 L1 2 L1 , instead, are always more expensive, regardless of the energy of the departure and destination orbits. A cheaper
1 The first portion of the LET trajectory, i.e., that which departs from a low-Earth orbit and reaches the Sun–Earth LPO is not considered here.
1040
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0
50
100
150
200
250
300
350
400
450
500
500
Δ V (m/s)
400 300 200 100 0 3.00091 3.00086 3.00081 3.00076 3.00072 3.00067 JSE 3.00063 3.00058
3.18 3.17 3.16 3.15 JEM
3.14 3.13
EM Fig. 2. 3D color maps of minimum DV (m/s) for connections of type LSE as a function of the Jacobi constants of the LPOs in the two systems. 1 L2
0
50
100
150
200
250
300
350
400
450
500
500
ΔV (m/s)
400 300 200 100 0 3.18 3.17 3.16 3.15 JEM
3.14 3.13
3.00091 3.00086 3.00081 3.00076 3.00072 3.00067 JSE 3.00063 3.00058
EM Fig. 3. 3D color maps of minimum DV (m/s) for connections of type LSE as a function of the Jacobi constants of the LPOs in the two systems. 2 L2
and more appropriate way of reaching a periodic orbit around LEM is by a heteroclinic connection with an orbit 1 around LEM 2 . There are typically one or two Poincare section positions (i.e., xP values) associated to the largest majority of the DV minima. The time of flight is the sum of the time intervals spent on each manifold trajectory. Theoretically, each such interval corresponds to the time of flight between the Lyapunov orbit and the Poincare section, but a large fraction of it is spent in the vicinity of the Lyapunov orbit (winding on and off) and for practical applications it can be eliminated by application of a small maneuver. In this way it is possible to define an effective time of flight, measured between two given distances from the departure and arrival Lyapunov orbits. Example solutions of the four types are illustrated in Figs. 6–9: the invariant manifolds as well as the connecting trajectories are represented in the Sun–Earth
synodical barycentric reference frame. The Jacobi constants of the two periodic orbits, the value of a0 , the cost of the connection at the Poincare section, the (full) time of flight and the effective time of flight (neglecting the winding on and off the periodic orbit up to distances of 0.001 in adimensional CR3BP units) are also provided. The coupled CR3BP is an approximate model, in that it neglects, among other contributions, the effect of the Sun’s gravity over the Earth–Moon portion of the trajectory. These are known to be quite relevant, especially around the exterior equilibrium point LEM (see [2]). 2 Hence, to check the reliability of the above trajectories in a more accurate model is mandatory. Fig. 10 illustrates the result of the refinement of the case shown in Fig. 7 EM (i.e., an LSE connection) in a complete Solar System 2 L2 force-field (using the JPL files for the determination of the ephemeris of the planets) [1]. Note that the refined
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
250
300
350
400
1041
450
500
500
ΔV (m/s)
450 400 350 300 250 3.20
3.19
3.18
3.17
3.16
3.15 3.14 JEM 3.13
3.12
3.11
3.10
3.00091 3.00086 3.00081 3.00076 3.00072 3.00067 JSE 3.00063 3.00058
EM Fig. 4. 3D color maps of minimum DV (m/s) for connections of type LSE as a function of the Jacobi constants of the LPOs in the two systems. 1 L1
280
300
320
340
360
380
400
420
440
460
480
500
500
ΔV (m/s)
450 400 350 300 3.20
3.19
3.18
3.17
3.16
3.15 JEM 3.14 3.13
3.12
3.11
3.10
3.00091 3.00086 3.00081 3.00076 3.00072 3.00067 JSE 3.00063 3.00058
EM Fig. 5. 3D color maps of minimum DV (m/s) for connections of type LSE as a function of the Jacobi constants of the LPOs in the two systems. 2 L1
trajectory is a natural, ballistic connection, thus requiring no maneuvers. The most evident changes with respect to the initial velocity (i.e., the coupled CR3BP solution, shown in red in the upper plot of Fig. 10) is the 3D development of the refined orbit (which reflects the different orbital planes of the relevant bodies), together with some differences at the beginning of the Sun–Earth portion and throughout the Earth–Moon portion.
3. WSB points and invariant manifolds of LPOs According to the algorithmic definition given in [3], in a CR3BP the motion of the third body P3 around the second primary P2 is stable if, after leaving a given initial absidal line lðyÞ (Fig. 11), the former makes one full
revolution about the latter without having turned around the first primary P1, and its orbit remains endowed with negative Keplerian energy relative to P2 on return to lðyÞ. Whenever such requirement is violated, the corresponding initial condition is said to be unstable. This includes trajectories which escape or collide with either primary. In this study the stability criterion has been applied with a number nR of revolutions different from one and also non-integer. nR can be viewed as a parameter of the problem: its value determines the instantaneous border of the stable region, i.e., the set of initial conditions in the phase space which satisfy the stability definition. The WSB regions can be defined for a fixed energy level J. Under planar approximation, the Jacobi constant J in barycentric synodical coordinates can be expressed as a function of the following quantities: the
1042
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.015
0.01
Earth−Moon RTBP:
Sun−Earth RTBP:
J EM =3.14112
J
Stable manifold of L2
Unstable manifold of L1
SE
=3.00063
α 0 =87 deg
SE
0.005
L
y
Earth 0
−−> Sun
1
−0.005
−0.01
Δ V at connection =23 m/s Full time of flight =229 days Effective time of flight =30 days −1.004
−1.002
−1
−0.998
SE winding−off time =183 days EM winding−on time =16 days Full time to connection =210 days Effective time to connection =26 days −0.996 −0.994 x SE
−0.992
−0.99
−0.988
−0.986
EM Fig. 6. Example of a low-cost connection of the type LSE 1 L2 : view of the LPOs, their manifolds and the connecting trajectory in configuration space (Sun–Earth synodical barycentric coordinates).
Earth−Moon RTBP: JEM =3.1807
Sun−Earth RTBP: JSE =3.00072
0.008
Stable manifold of L2
Unstable manifold of L2
0.006
α0 =307 deg
0.01
0.004
SE
0.002 L2
y
0
Earth
−−> Sun
−0.002 −0.004 −0.006 −0.008 −0.01
Δ V at connection =9 m/s Full time of flight =225 days Effective time of flight =37 days
SE winding−off time =174 days EM winding−on time =14 days Full time to connection =203 days Effective time to connection =29 days
−1.014 −1.012 −1.01 −1.008 −1.006 −1.004 −1.002 xSE
−1
−0.998 −0.996
EM Fig. 7. Example of a low-cost connection of the type LSE 2 L2 : view of the LPOs, their manifolds and the connecting trajectory in configuration space (Sun–Earth synodical barycentric coordinates).
pericenter distance r2 at t =0, the angle y formed by the absidal line with the positive x-axis of the synodical barycentric reference frame, and the eccentricity e of the osculating Keplerian orbit: J ¼ 2cosyðm1Þr2 þ
mð1eÞ r2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ1m 7 2 mð1 þeÞr2 :
2ð1mÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r22 2cosyr2 þ 1 ð1Þ
Here the + and signs refer to prograde and retrograde initial orbits, respectively, and m is the mass ratio of the
two primaries. Varying e and y over the intervals [0.0,1.0] and ½0:0,2p, respectively, and solving Eq. (1) for r2, provides the initial conditions that correspond to osculating Keplerian orbits around P2 at t= 0 (see Fig. 12 for a plot of the initial conditions in configuration space obtained in this way for the Sun–Earth CR3BP). These are integrated in the force field due to P2 and P1. Here we investigate the relationship between the set of unstable points of the WSB in the Sun–Earth CR3BP and the stable invariant manifolds of planar Lyapunov orbits around L1 and L2 with the same Jacobi constant. We also
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
1043
−3
x 10 8
Earth−Moon RTBP: J =3.14112
Sun−Earth RTBP: JSE =3.0008
Stable manifold of L1
Unstable manifold of L
EM
6
1
α =355 deg 0
4
y
SE
2 L
Earth
0
−−> Sun
1
−2 −4 −6
Δ V at connection =293 m/s Full time of flight =224 days Effective time of flight =57 days
−8 −1.004 −1.002
−1
SE winding−off time =155 days EM winding−on time =12 days Full time to connection =198 days Effective time to connection =43 days
−0.998 −0.996 −0.994 −0.992 x
−0.99
−0.988
SE
EM Fig. 8. Example of a low-cost connection of the type LSE 1 L1 : view of the LPOs, their manifolds and the connecting trajectory in configuration space (Sun–Earth synodical barycentric coordinates).
0.015
Earth−Moon RTBP:
Sun−Earth RTBP:
J
J
EM
=3.1824
SE
Stable manifold of L1 0.01
=3.00058
Unstable manifold of L 2
α =315 deg 0
y
SE
0.005
L 0
Earth
2
−−> Sun
−0.005
−0.01
−0.015
SE winding−off time =185 days EM winding−on time =11 days Full time to connection =225 days Effective time to connection =40 days
Δ V at connection =603 m/s Full time of flight =245 days Effective time of flight =50 days −1.014
−1.012
−1.01
−1.008
−1.006 x
−1.004
−1.002
−1
−0.998
−0.996
SE
EM Fig. 9. Example of a low-cost connection of the type LSE 2 L1 : view of the LPOs, their manifolds and the connecting trajectory in configuration space (Sun–Earth synodical barycentric coordinates).
examine the causes of the unstable behaviour and relate them to the fine geometrical structure of the WSB. The WSB regions are computed for values of nR equal to 0.5, 1.0 and 1.5. Figs. 13 and 14 show the superposition between the unstable points of the WSB with nR = 1.5 and SE Jacobi constant J ¼ 3:000583 and the points of the stable
manifolds of the two Lyapunov orbits of the same energy for which the radius vector is orthogonal to the velocity vector (the two are meant relative to the second primary). The unstable points originate from both initial prograde and retrograde motion. The stable manifolds have been computed by means of a semi-analytic expansion of
1044
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.01 0.008 0.006 0.004 0.002 SE
L
y
Earth
2
0
−0.002 −0.004 −0.006 −0.008 −0.01 −1.012
−1.01
−1.008 −1.006 −1.004 −1.002 xSE
−1
−0.998
−4
x 10 6 4
L2 2
Earth
0
0.01 0.005
−2 0 −4 −1.012 −1.01 −1.008
−0.005 −1.006 −1.004 −1.002 xSE
−1 −0.998
y
SE
−0.01
EM Fig. 10. Refinement of the LSE 2 L2 connection in a complete Solar System force-field, shown in Sun–Earth synodic barycentric coordinates: xy-plane projection (top) and 3D view (bottom). In black the refined solution, in red the original two coupled CR3BP connection. Note that the refined trajectory is a natural connection, i.e., requiring no maneuvers. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
order 17 and propagated up to 100 normalized time units. The agreement2 between the two sets is a clear indication that the stable manifolds confine the WSB and play a role in driving the unstable points (see also [4]). Furthermore, Fig. 15 shows that the initial conditions which remain stable at nR = 1.5 effectively form the boundary of the corresponding unstable zone.
2 The small discrepancies observed on the right side of Fig. 13 are essentially due to the growing numerical errors in the invariant manifolds computation as time increases. Furthermore, the few, scattered WSB points in the lower right part of the plot are initial conditions of trajectories which have become unstable very early in the given interval, i.e., 1:0o nR r 1:5; therefore their distribution in configuration space recalls that of the unstable points pertaining to the ‘‘previous interval’’ in nR (i.e., 0:5 o nR r 1:0).
Fig. 11. A drawing showing how a stable (solid curve) and an unstable (dashed curve) trajectory look like, according to the WSB stability criterion stated in the text.
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.02
J
SE
1045
=3.00058
prograde
0.015 0.01 0.005 y
L2
L1
0 −0.005 −0.01 −0.015 −0.02 −0.02 −0.015 −0.01 −0.005
0 0.005 x −μ + 1
0.02
0.01
J
SE
0.015
0.02
=3.00058
retrograde
0.015 0.01 0.005 y
L2
L1
0 −0.005 −0.01 −0.015 −0.02 −0.02 −0.015 −0.01 −0.005
0 0.005 x −μ + 1
0.01
0.015
0.02
Fig. 12. The Sun–Earth CR3BP in synodical coordinates centered on the Earth: location in configuration space of the initial conditions giving rise to osculating Keplerian orbits around the Earth at t= 0 (filled areas: prograde orbits on top, retrograde orbits at the bottom) for a given Jacobi constant value. The two planar Lyapunov orbits with the same Jacobi constant value are also plotted.
For each of the three available sets of unstable points, i.e., with nR = 0.0, 0.5, and 1.5, we have identified the cause of instability:
completion of at least one revolution around P1 before the return to the initial absidal line;
return to the initial absidal line with positive Keplerian energy relative to P2.
escape (i.e., reaching the end of the integration before
the return to the initial absidal line relative to P2); collision with P2 (when the distance of P3 from the Earth is smaller than its equatorial radius); collision with P1 (when the heliocentric distance of P3 is smaller than the Sun’s radius);
SE
Initial conditions corresponding to J ¼ 3:000583 and prograde motion, produce unstable points characterized by the various unstable behaviours in the proportions shown in Figs. 16–18, respectively, for the three selected values of nR. This information allows to understand
1046
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
−3
8
x 10
WSB unstable for 1.0 < nR < 1.5 stable man. of PLOs of L1 and L2
6 4
y
2 0 −2 −4 −6 −8 −8
−6
−4
−2
0
2
4
6
8 −3
x−μ+1
x 10
−3
2
x 10
WSB unstable for 1.0 < nR < 1.5 stable man. of PLOs of L and L
1.5
1
2
1
y
0.5 0 −0.5 −1 −1.5 −2 −2
−1.5
−1
−0.5
0 x−μ+1
0.5
1
1.5
2 −3
x 10
SE
Fig. 13. Sun–Earth system: initial conditions with Jacobi constant J ¼ 3:000583 producing unstable trajectories at nR = 1.5 (crosses) and points of the stable manifolds of the two Lyapunov orbits around L1 and L2 with the same energy for which the radius vector is orthogonal to the velocity vector (dots). The reference frame is the synodical one, centered on the second primary. Both prograde and retrograde motions are represented. The bottom plot provides an enlarged view of the top plot, showing details of the superposition between the two sets of points close to the Earth.
the nature of the empty areas inside the unstable zones of Fig. 13: they are occupied by trajectories which turn into unstable at either of the two previous steps, i.e., nR = 0.5 or 1.0. Besides, as Fig. 19 shows, such
trajectories either escape from P2 or collide with it. Close inspection of their location in configuration space (Fig. 20 shows two details of Fig. 19 separately obtained for initial prograde and retrograde motions)
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
1047
0.15
0.1
vy
0.05
0
−0.05
−0.1
−0.1
−0.05
0 vx
0.05
0.1
0.15
Fig. 14. Same as the previous figure but illustrating the x_ and y_ components of the unstable initial conditions for nR = 1.5 (crosses) and the points of the stable manifolds of the two Lyapunov orbits around L1 and L2 with the same energy for which the radius vector is orthogonal to the velocity vector (dots).
0.01 0.008
nu1.0−1.5 revs :2073 stable at1.5 revs:312
0.006 0.004
y
0.002 0 −0.002 −0.004 −0.006 −0.008 −0.01 −0.01
−0.005
0 x−μ+1
0.005
0.01 SE
Fig. 15. Initial conditions producing unstable (crosses) and stable (squares) trajectories for nR = 1.5 and J ¼ 3:000583. Synodical reference frame centered on the Earth. The legend provides the number nu of unstable points and the number ns of stable ones.
suggests that there is no finer structure, and by small variations in the initial conditions one behaviour, i.e., escape, changes into the other, i.e., collision, and vice versa (Figs. 21–23).
4. Capture around the collinear libration points Among the unstable trajectories, some approach either L1 or L2. It is possible to identify those which perform a
1048
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.01
0.005
escape (t>t ) :1891 f 1 rev. P :5911 1 coll. P :0
0.01
E > 0 :1 k coll. P :469
nu1.0−1.5 revs :2073
0.008
escape :2622 coll P2 :520
2
0.006
1
0.004
0
0
y
y
0.002
−0.002 −0.004
−0.005
−0.006 −0.008
−0.01 −0.01
−0.005
0 x−μ+1
0.005
Fig. 16. Initial conditions corresponding to prograde motions and producing unstable trajectories for nR =0.5. The legend indicates the type of instability and the number of points that fall into each category.
0.01
0.005
−0.01 −0.01
0.01
−0.005
0 x−μ+1
0.005
0.01
Fig. 19. Unstable points for nR = 1.5 (crosses) and a subset of the unstable points for nR = 0.5 and 1.0 which either collide with P2 (dots) or escape from it (circles).
−3
escape (t>tf) :731 1 rev. P :260 1 coll. P :0
−1
Ek > 0 :14325 coll. P :51
x 10
2
−1.5
1
0 y
y
−2
−2.5
−3
−0.005
−3.5
−0.01 −0.01
−0.005
0 x−μ+1
0.005
0.01
Fig. 17. Initial conditions corresponding to prograde motions and producing unstable trajectories at nR =1.0. The legend indicates the type of instability and the number of points that fall into each category.
0.01
y
0.005
escape (t>tf) :437 1 rev. P :101 1 coll. P :0
−6.8
−6.6
−6.4
−6.2
x−μ+1
−6 −3
x 10
Fig. 20. A higher-resolution slice of Fig. 19 corresponding to prograde motion.
temporary capture in either of the two appropriately defined regions around the two libration points. The captured trajectories appear to follow for some time the trajectories of the stable manifolds of the two planar Lyapunov orbits of the same energy, thus suggesting that the stable manifolds play a role also in this kind of temporary capture. Fig. 24 shows for initial prograde motion, the position of the unstable points that for each of the three values of nR perform temporary capture, or just enter either or both the capture regions, or are rejected by the capture criterion.
Ek > 0 :1454 coll. P :81 2
1
0
−0.005
−0.01 −0.01
−4 −7
5. Temporary capture around the second primary −0.005
0 x−μ+1
0.005
0.01
Fig. 18. Initial conditions corresponding to prograde motions and producing unstable trajectories at nR = 1.5 revolutions. The legend indicates the type of instability and the number of points that fall into each category.
That the invariant manifolds of LPOs play a role in the temporary capture around the second primary of a CR3BP is a known fact (see [5]). In this contribution we examine the dynamics of the captured trajectories relative to the second primary and we relate the resulting orbital
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
−4
5
0.01
x 10
0.008
4
1049
captured at L1 :375
0.006
through L1 only :0
0.004
through L2 only :0
3
nR =0.5 revs.
captured at L2 :197
2 0.002
y
1
0
0
−0.002
−1
−0.004
−2 −3
−0.006
stable points :312 L1 first :0
−0.008
L2 first :0 rejected :7700
−4
−0.01 −0.01
−5 −1
−0.5
0
0.5
−0.005
1 −3
x 10
x−μ+1
Fig. 21. A higher-resolution slice of Fig. 19 corresponding to retrograde motion.
0.015
0.01 0.008
0 x −μ + 1
captured at L1 :71
0.005
0.01
nR =1.0 revs.
captured at L2 :334
0.006
through L1 only :0
0.004
through L2 only :0
0.002 0
0.01
−0.002 −0.004
0.005
y
L2
L1
0
−0.006
stable points :312 L first :0
−0.008
L2 first :1
1
rejected :14961 −0.01 −0.01
−0.005
0.01
−0.01
0.008
−0.005
0 x −μ + 1
captured at L1 :385
0.005
0.01
nR =1.5 revs.
captured at L :316 2
−0.015 −0.015
−0.01
−0.005
0 x −μ + 1
0.005
0.01
0.015
0.006 0.004
Fig. 22. Examples of collision trajectories with initial conditions in the subset of Fig. 20.
through L only :0 1
through L only :0 2
0.002 0 −0.002 −0.004
0.015
0.01
−0.006
stable points :312 L1 first :10
−0.008
L2 first :13 rejected :1349
−0.01 −0.01
0.005 L
2
y
−0.005
0 x −μ + 1
0.005
0.01
L
1
0
Fig. 24. Sun–Earth system: position of the unstable points for nR = 0.5 (top), 1.0 (middle), 1.5 (bottom). Different colors and symbols mark the several behaviours with respect to the capture criterion, as specified in the legend. The location of the stable points is also shown (crosses).
−0.005
−0.01
−0.015 −0.015
−0.01
−0.005
0 x −μ + 1
0.005
0.01
0.015
Fig. 23. Examples of escape trajectories with initial conditions in the subset of Fig. 20.
parameters to the dynamical features of the invariant manifold to which they belong. Let us consider the unstable invariant manifolds of planar Lyapunov orbits around L1 and L2 and, in particular, the branches that develop in the region around P2, confined by the zero
1050
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.015
400 = peri
= apo
= revs. P
2
=E >0 K
= collision with P (:90) 2
350
0.01
300 0.005
y
orbit
250 200
0
150
−0.005
100
−0.01
50 −0.015 −0.015
0
5
10
15 time
20
25
−0.01
−0.005
0 x−μ+1
0.005
0.01
0.015
−0.01
−0.005
0 x−μ+1
0.005
0.01
0.015
30 0.015
400 = peri
= apo
= revs. P
2
=E >0 K
= collision with P (:0) 2
0.01
orbit
350
0.005
250
0
y
300
200 −0.005
150 −0.01
100 −0.015 −0.015
50
0
5
10
15 time
20
25
30
Fig. 25. Development with time of some dynamical parameters related to the Keplerian orbit around the second primary, obtained from the unstable manifolds of two planar Lyapunov orbits around L1 in the Sun–Earth system, one with J= 3.00059 (top), the other with J = 3.000888 (bottom): the occurrence of perigees (crosses), apogees (dots), collisions with P2 (squares), the completion of successive loops around P2 (asterisks) and finally the sign of the Keplerian energy Ek (white background for negative, yellow for positive, respectively). The legend also reports the number of collisions (in parentheses). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
velocity curves and the openings of the bottleneck at L1 and L2. Fig. 25 shows the behaviour with time (up to 30 normalized time units) of a number of parameters related to the dynamics relative to P2 of 360 orbits in which the unstable manifolds of two Lyapunov orbits (with J= 3.00059 and 3.000888, respectively, in the two plots) around the L1 point in the Sun–Earth system have been divided. These parameters are:
the occurrence of perigees and apogees (crosses and dots, respectively);
Fig. 26. Some trajectories belonging to the invariant manifolds of Fig. 25: J = 3.00059 (top) and J = 3.000888 (bottom).
the occurrence of collisions with P2 (squares, which
also constitute the end-point of the corresponding trajectory); the sign of the Keplerian energy Ek (white background for negative, yellow for positive).
Note that positive energy means escape, i.e., the trajectory leaves the lobe around P2. Therefore, the necessary condition for (temporary) capture3 to occur is that Ek be negative. Comparison between the two plots of Fig. 25 tells that invariant manifolds of Lyapunov orbits of smaller size (i.e., of larger J) experience fewer collisions and perform more revolutions around P2 while in the capture region. Fig. 26 shows some example trajectories from the two manifolds of Fig. 25. Fig. 27 illustrates one trajectory for each of the two manifold: that corresponding to the smaller Lyapunov orbit remains captured through all the integration time considered, whereas the trajectory associated to the larger Lyapunov orbit completes one loop around P2 and then collides with it.
the completion of full revolutions around P2 (asterisks), counted starting from a given line orthogonal to the x-axis and located between the progenitor orbit and the second primary;
3 A trajectory is said to be in a temporary capture if it enters the lobe and completes a minimum number of revolutions around P2 before escaping.
0.015
0.03
0.01
0.02
0.005
0.01
semi−major axis
y
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0
−0.005
−0.01
−0.015 −0.015
0
−0.01
360 300 240 180 120 60
−0.02
−0.01
−0.005
0 x−μ+1
0.005
0.01
1051
−0.03
0.015
0
5
10
15 time
20
25
30
5
10
15 time
20
25
30
0.015
0.03 0.01
0.02
semi−major axis
y
0.005
0
−0.005
0.01
0
−0.01
360 300 240 180 120 60
−0.01
−0.02 −0.015 −0.015
−0.01
−0.005
0 x−μ+1
0.005
0.01
0.015
Fig. 27. One trajectory from each of the samples of Fig. 26: the crossing of the vertical lines indicates the entrance into the region around P2 and the starting point for loop counting.
Fig. 28 shows for the orbits of Fig. 26, the behaviour with time of the semi-major axis of the Keplerian orbit relative to P2: all the trajectories are initially endowed with positive semi-major axis; then, those which escape exhibit a sudden growth (to infinity), followed by a change into a constant, negative value (hyperbolic orbit relative to P2). The unstable invariant manifolds of planar Lyapunov orbits around L2 behave similarly. Fig. 29 provides global information concerning the temporary capture as a function of the energy, over two families of 70 planar Lyapunov orbits each, around L1 and L2, respectively. The two plots give the percentage of trajectories that complete at least 5 (respectively, 10) loops around P2 before the energy turns to positive or the integration ends. The horizontal range of the plot is limited to the first 40 orbits of each family, the remaining ones having zero percentage of captured orbits of either type.
6. Conclusions In this contribution we considered four transfer mechanisms in systems composed of three (CR3BP) or four (two coupled CR3BPs) bodies, with special interest in the Sun–Earth(–Moon) system: the low energy transfers
−0.03
0
Fig. 28. Semi-major axis as a function of time for the trajectories of Fig. 26.
from the Earth to the Moon, the weak stability boundaries, the temporary capture around the libration points on behalf of trajectories in the WSB, and the temporary capture of invariant manifold trajectories around the second primary of a CR3BP. The understanding of these four issues is related to the dynamics of the invariant manifolds of periodic orbits around the collinear libration points L1 and L2 of the system. Concerning the LETs, of the four combinations of libration EM point connections examined, only two, i.e., LSE and 1 L2 SE EM L2 L2 , can be obtained at low or zero cost, and this capability depends on the energy (and hence the size) of the LPOs to be connected, being higher for lower energies. The relationship between the WSB unstable points and the stable invariant manifolds of LPOs of the same energy has been clarified: the points of the invariant manifold trajectories that are characterized by orthogonality between the radius and the velocity vector relative to P2 effectively confine the unstable points of the WSB region. The destination of the unstable WSB points has also been understood, and different behaviours, in particular collisions and escape, have been found to be closely related. The stable invariant manifolds of LPOs also play a role in the temporary capture of the unstable WSB points on behalf of L1 and L2. Finally, the temporary capture around the second primary, a well known phenomenon induced by the unstable invariant manifolds of LPOs,
1052
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
100
1
90 % of orbits with capture >5 revs.
all the above issues will appear in a subsequent, more extended publication.
=L
= L2
80 70
Acknowledgements
60 50 40 30 20 10 0 5
10
15 20 25 Lyapunov orbit index
30
100
40
=L
1
90 % of orbits with capture >10 revs.
35
A special thanks is expressed to Elisa Maria Alessi for the valuable assistance in the trajectory refinement based on the JPL ephemeris model. E. Fantino and Y. Ren have been supported by the Marie Curie Actions Research and Training Network AstroNet MCRTN-CT-2006-035151. G. Go´mez and J.J. Masdemont have been partially supported by the Grants MTM2006-05849/Consolider, MTM2009-06973 and 2009SGR859. The authors also acknowledge the use of EIXAM, the UPC Applied Math cluster system for research computing (see http://www. ma1.upc.edu/eixam/), and in particular Pau Roldan for providing technical support in the use of the cluster.
=L
2
80
References
70 60 50 40 30 20 10 0
5
10
15 20 25 Lyapunov orbit index
30
35
40
Fig. 29. Capability of being captured around P2 within the unstable invariant manifolds of two families of planar Lyapunov orbits around L1 (circles) and L2 (asterisks), expressed as the percentage of trajectories that perform a minimum number of loops (i.e., 5 (top) and 10 (bottom)) around P2 before the Keplerian energy turns to positive or the integration ends.
has been put in relation with the dynamical characteristics (size, energy, semi-major axis, perigees and apogees) of the invariant manifold trajectories relative to the second primary: temporary capture is more efficient when the Jacobi constant of the invariant manifold is larger and the size of the progenitor LPO is smaller. Further analysis and discussion of
[1] E.M. Alessi, Study of some dynamical problems in the Earth–Moon system, Ph.D. Dissertation, Universitat de Barcelona, 2010. [2] M.A. Andreu, The quasi-bicircular problem, Ph.D. Dissertation, Universitat de Barcelona, 1999. [3] E.A. Belbruno, Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers, Princeton University Press, 2004. [4] E.A. Belbruno, M. Gidea, F. Topputo, Weak stability boundary and invariant manifolds, preprint, 2009. [5] E. Belbruno, B.G. Marsden, Resonance hopping in comets, Astronomical Journal 97 (1997) 1433–1444. [6] C. Circi, P. Teofilatto, On the dynamics of weak stability boundary lunar transfers, Celestial Mechanics and Dynamical Astronomy 79 (2001) 41–72. [7] C. Circi, P. Teofilatto, Weak stability boundary trajectories for the deployment of lunar spacecraft constellations, Celestial Mechanics and Dynamical Astronomy 95 (2006) 371–390. [8] F. Garcı´a, G. Go´mez, A note on weak stability boundaries, Celestial Mechanics and Dynamical Astronomy 97 (2007) 87–100. [9] W.S. Koon, M. Lo, J.E. Marsden, S.D. Ross, Low energy transfers to the Moon, Celestial Mechanics and Dynamical Astronomy 81 (2001) 63–73. [10] M. Lo, S. Ross, SURFing the solar system: invariant manifolds and the dynamics of the solar system, JPL IOM 312-97, 1997, pp. 2–4. [11] J.K. Miller, E.A. Belbruno, A method for the construction of a lunar transfer trajectory using ballistic capture, in: Proceedings of the 1st AAS/AIAA Spaceflight Mechanics Meeting, Paper No. AAS 97-109, 1991. [12] D. Romagnoli, C. Circi, Earth–Moon weak stability boundaries in the restricted three and four body problem, Celestial Mechanics and Dynamical Astronomy 103 (2009) 79–103.
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
A note on libration point orbits, temporary capture and low-energy transfers$ E. Fantino a,b,, G. Go´mez a,c, J.J. Masdemont a,b, Y. Ren a,b a
Institut d’Estudis Espacials de Catalunya, Gran Capita 2-4, 08034 Barcelona, Spain Universitat Polite cnica de Catalunya, Dpt. Matematica Aplicada I, Diagonal 647, 08028 Barcelona, Spain c Universitat de Barcelona, Dpt. Matematica Aplicada i Ana lisi, Gran Via 585, 08007 Barcelona, Spain b
a r t i c l e i n f o
abstract
Article history: Received 28 January 2010 Received in revised form 6 June 2010 Accepted 20 June 2010
In the circular restricted three-body problem (CR3BP) the weak stability boundary (WSB) is defined as a boundary set in the phase space between stable and unstable motion relative to the second primary. At a given energy level, the boundaries of such region are provided by the stable manifolds of the central objects of the L1 and L2 libration points, i.e., the two planar Lyapunov orbits. Besides, the unstable manifolds of libration point orbits (LPOs) around L1 and L2 have been identified as responsible for the weak or temporary capture around the second primary of the system. These two issues suggest the existence of natural dynamical channels between the Earth’s vicinity and the Sun–Earth libration points L1 and L2. Furthermore, it has been shown that the Sun– Earth L2 central unstable manifolds can be linked, through an heteroclinic connection, to the central stable manifolds of the L2 point in the Earth–Moon three-body problem. This concept has been applied to the design of low energy transfers (LETs) from the Earth to the Moon. In this contribution we consider all the above three issues, i.e., weak stability boundaries, temporary capture and low energy transfers, and we discuss the role played by the invariant manifolds of LPOs in each of them. The study is made in the planar approximation. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Circular restricted three-body problem Libration points Periodic orbits Invariant manifolds Two-body problem Weak stability boundary
1. Introduction In the CR3BP, the WSB is defined as a boundary set in the phase space between stable and unstable motion relative to the second primary: Keplerian orbits about this body, perturbed by the first primary, are said to be stable if, after a prescribed number of revolutions, preserve the character of bounded motion, unstable otherwise. This definition of WSB is due to [3]. The concept and its application have later been analysed by several authors, e.g., [4,12], also in connection with the construction of
$
This paper was presented during the 60th IAC in Daejeon.
Corresponding author at: Institut d’Estudis Espacials de Catalunya,
Gran Capita 2-4, 08034 Barcelona, Spain. E-mail addresses: [email protected] (E. Fantino), [email protected] (G. Go´mez), [email protected] (J.J. Masdemont), [email protected] (Y. Ren). 0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.06.037
ballistic capture trajectories around the second primary [11]. This contribution takes the steps from [8] and, instead of the eccentricity, we parametrize the regions of weak stability by the Jacobi constant. We aim at clarifying some structural aspects of the WSB, particularly in relation with the dynamics of the invariant manifolds of LPOs and the temporary capture around them as a function of the energy of the third body. Then, it is thanks to the work of [5,10] that some wellknown phenomena of temporary capture around the second primary of a CR3BP (i.e., the resonance transitions of the comets Oterma and Gehrels 3 as a result of temporary captures on behalf of Jupiter in the Sun–Jupiter system) have been understood as controlled by the invariant manifolds of Jupiter’s LPOs. Here, the characteristics of this effect are investigated in terms of the dynamics of the unstable manifolds of LPOs with respect to the second primary.
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
Maneuver (ΔV) at Patch Point
Sun-earth-S/C system
Earth-Moon-S/C system
Moon
Sun-Earth L2 portion using ‘twisting’
Lunar capture portion Sun
Moon’s orbit
Earth x
L2
y
L2
y
Sun
Maneuver (ΔV) at Patch Point
1039
Earth L2 orbit x
Fig. 1. A low-energy transfer from the Earth to the Moon through the L2 libration points of the Sun–Earth and Earth–Moon systems as seen in the Sun–Earth synodical barycentric reference frame (a), and its construction based on patching invariant manifolds of the Sun–Earth and Earth–Moon CR3BPs (b) [9].
The LETs from the Earth to the Moon consist in connecting trajectories guided by the invariant manifolds of the central objects around the L2 collinear libration points of the Sun–Earth and Earth–Moon CR3BPs in such a way that the spacecraft can reach the Moon’s vicinity from a low-Earth orbit (LEO) practically at the only cost associated to leaving the LEO. The mechanism for the LETs was first presented in [9]. See also [6,7]. Fig. 1 illustrates the idea for the planar case in the Sun–Earth barycentric synodical reference frame: a branch of a stable manifold of a planar Lyapunov orbit around LSE 2 (i.e., the Sun–Earth L2 point) drives the spacecraft away from the Earth, then the unstable manifold of the same orbit redirects it to a region where also the stable manifold of a planar Lyapunov orbit around LEM (i.e., the Earth–Moon L2 point) flows. 2 There, the intersection in configuration space (through a conveniently set Poincare section) is sought. The difference in velocity between the two intersecting trajectories (equal to zero if there is a heteroclinic connection) represents the cost of the transfer. The dynamical characteristics of the two CR3BPs and their kinematical relationship are such that low cost (and even zero cost) connections of the EM type LSE can easily be found. We have investigated 2 L2 the existence and the characteristics (particularly the cost) of transfers between any combination of the two EM collinear libration points, the so-called LSE (i,j=1,2) i Lj connections. Section 2 illustrates the main findings concerning the EM exploration of the LSE (i,j = 1,2) connections. Section 3 i Lj contains an analysis of the WSB regions of the Sun–Earth system and discusses the role of the invariant manifolds of LPOs therein. The temporary capture of WSB unstable trajectories in the vicinity of the collinear libration points of the system is examined in Section 4. Finally, Section 5 discusses the issue of temporary capture around the Earth of trajectories belonging to the unstable manifold of LPOs in the Sun–Earth CR3BP. 2. LSE LEM connections i j In the coupled CR3BP, the Sun–Earth and the Earth– Moon three-body problems are kinematically linked in EM inertial space through their orbital phases aSE at 0 and a0 t = 0. This allows to transform at any time the invariant
manifold trajectories of the Earth–Moon problem into the Sun–Earth barycentric synodical reference frame and to look for intersections with the manifold trajectories of the Sun–Earth problem at a given Poincare section. More specifically, the transfer sought is one between the unstable manifold of a LPO in the Sun–Earth model and the stable manifold of a LPO in the Earth–Moon system.1 There are four types of connections because there are four libration point pairs of the type LSE (i= 1,2) LEM (j= 1,2). i j We have explored each type by varying the progenitor Lyapunov orbits in the two CR3BPs over a wide range of energies. Then, for each pair of LPOs, the initial orbital SE phase a0 ¼ aEM 0 a0 of the Earth–Moon problem in the Sun–Earth synodical barycentric reference frame has been varied at increments of 51 between 0 and 360. The Poincare section is the yy_ plane orthogonal to the x-axis of the Sun–Earth synodical barycentric reference frame at some x-coordinate xP. The value of xP is varied discretely between the location of the given Sun–Earth Lyapunov orbit and the Earth. This analysis provides information concerning the feasibility, the cost and the time of flight of each type of connection. Concerning the cost, it has to be noted that the maneuver at the yy_ Poincare section is obtained as the difference between the x velocity components of two intersecting orbits belonging to the two connecting manifolds. This method does not consider transit orbits, i.e., orbits which flow inside the manifold tubes. Besides, the maneuvers considered are entirely in the x direction. EM For each LSE combination, the variation of the i Lj above listed parameters, i.e., the relative orbital phase a0 at t =0 of the two models and the x coordinate of the Poincare section, allow to construct maps of the minimum (i.e., within the present method) cost (i.e., maneuver size) between the connecting trajectories as a function of the Jacobi constants JSE and JEM of the LPOs of the two CR3BPs. Figs. 2–5 show that low and zero cost connections exist EM EM for LSE and LSE over a wide range of energy 1 L2 2 L2 EM EM levels. Connections of the type LSE and LSE 1 L1 2 L1 , instead, are always more expensive, regardless of the energy of the departure and destination orbits. A cheaper
1 The first portion of the LET trajectory, i.e., that which departs from a low-Earth orbit and reaches the Sun–Earth LPO is not considered here.
1040
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0
50
100
150
200
250
300
350
400
450
500
500
Δ V (m/s)
400 300 200 100 0 3.00091 3.00086 3.00081 3.00076 3.00072 3.00067 JSE 3.00063 3.00058
3.18 3.17 3.16 3.15 JEM
3.14 3.13
EM Fig. 2. 3D color maps of minimum DV (m/s) for connections of type LSE as a function of the Jacobi constants of the LPOs in the two systems. 1 L2
0
50
100
150
200
250
300
350
400
450
500
500
ΔV (m/s)
400 300 200 100 0 3.18 3.17 3.16 3.15 JEM
3.14 3.13
3.00091 3.00086 3.00081 3.00076 3.00072 3.00067 JSE 3.00063 3.00058
EM Fig. 3. 3D color maps of minimum DV (m/s) for connections of type LSE as a function of the Jacobi constants of the LPOs in the two systems. 2 L2
and more appropriate way of reaching a periodic orbit around LEM is by a heteroclinic connection with an orbit 1 around LEM 2 . There are typically one or two Poincare section positions (i.e., xP values) associated to the largest majority of the DV minima. The time of flight is the sum of the time intervals spent on each manifold trajectory. Theoretically, each such interval corresponds to the time of flight between the Lyapunov orbit and the Poincare section, but a large fraction of it is spent in the vicinity of the Lyapunov orbit (winding on and off) and for practical applications it can be eliminated by application of a small maneuver. In this way it is possible to define an effective time of flight, measured between two given distances from the departure and arrival Lyapunov orbits. Example solutions of the four types are illustrated in Figs. 6–9: the invariant manifolds as well as the connecting trajectories are represented in the Sun–Earth
synodical barycentric reference frame. The Jacobi constants of the two periodic orbits, the value of a0 , the cost of the connection at the Poincare section, the (full) time of flight and the effective time of flight (neglecting the winding on and off the periodic orbit up to distances of 0.001 in adimensional CR3BP units) are also provided. The coupled CR3BP is an approximate model, in that it neglects, among other contributions, the effect of the Sun’s gravity over the Earth–Moon portion of the trajectory. These are known to be quite relevant, especially around the exterior equilibrium point LEM (see [2]). 2 Hence, to check the reliability of the above trajectories in a more accurate model is mandatory. Fig. 10 illustrates the result of the refinement of the case shown in Fig. 7 EM (i.e., an LSE connection) in a complete Solar System 2 L2 force-field (using the JPL files for the determination of the ephemeris of the planets) [1]. Note that the refined
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
250
300
350
400
1041
450
500
500
ΔV (m/s)
450 400 350 300 250 3.20
3.19
3.18
3.17
3.16
3.15 3.14 JEM 3.13
3.12
3.11
3.10
3.00091 3.00086 3.00081 3.00076 3.00072 3.00067 JSE 3.00063 3.00058
EM Fig. 4. 3D color maps of minimum DV (m/s) for connections of type LSE as a function of the Jacobi constants of the LPOs in the two systems. 1 L1
280
300
320
340
360
380
400
420
440
460
480
500
500
ΔV (m/s)
450 400 350 300 3.20
3.19
3.18
3.17
3.16
3.15 JEM 3.14 3.13
3.12
3.11
3.10
3.00091 3.00086 3.00081 3.00076 3.00072 3.00067 JSE 3.00063 3.00058
EM Fig. 5. 3D color maps of minimum DV (m/s) for connections of type LSE as a function of the Jacobi constants of the LPOs in the two systems. 2 L1
trajectory is a natural, ballistic connection, thus requiring no maneuvers. The most evident changes with respect to the initial velocity (i.e., the coupled CR3BP solution, shown in red in the upper plot of Fig. 10) is the 3D development of the refined orbit (which reflects the different orbital planes of the relevant bodies), together with some differences at the beginning of the Sun–Earth portion and throughout the Earth–Moon portion.
3. WSB points and invariant manifolds of LPOs According to the algorithmic definition given in [3], in a CR3BP the motion of the third body P3 around the second primary P2 is stable if, after leaving a given initial absidal line lðyÞ (Fig. 11), the former makes one full
revolution about the latter without having turned around the first primary P1, and its orbit remains endowed with negative Keplerian energy relative to P2 on return to lðyÞ. Whenever such requirement is violated, the corresponding initial condition is said to be unstable. This includes trajectories which escape or collide with either primary. In this study the stability criterion has been applied with a number nR of revolutions different from one and also non-integer. nR can be viewed as a parameter of the problem: its value determines the instantaneous border of the stable region, i.e., the set of initial conditions in the phase space which satisfy the stability definition. The WSB regions can be defined for a fixed energy level J. Under planar approximation, the Jacobi constant J in barycentric synodical coordinates can be expressed as a function of the following quantities: the
1042
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.015
0.01
Earth−Moon RTBP:
Sun−Earth RTBP:
J EM =3.14112
J
Stable manifold of L2
Unstable manifold of L1
SE
=3.00063
α 0 =87 deg
SE
0.005
L
y
Earth 0
−−> Sun
1
−0.005
−0.01
Δ V at connection =23 m/s Full time of flight =229 days Effective time of flight =30 days −1.004
−1.002
−1
−0.998
SE winding−off time =183 days EM winding−on time =16 days Full time to connection =210 days Effective time to connection =26 days −0.996 −0.994 x SE
−0.992
−0.99
−0.988
−0.986
EM Fig. 6. Example of a low-cost connection of the type LSE 1 L2 : view of the LPOs, their manifolds and the connecting trajectory in configuration space (Sun–Earth synodical barycentric coordinates).
Earth−Moon RTBP: JEM =3.1807
Sun−Earth RTBP: JSE =3.00072
0.008
Stable manifold of L2
Unstable manifold of L2
0.006
α0 =307 deg
0.01
0.004
SE
0.002 L2
y
0
Earth
−−> Sun
−0.002 −0.004 −0.006 −0.008 −0.01
Δ V at connection =9 m/s Full time of flight =225 days Effective time of flight =37 days
SE winding−off time =174 days EM winding−on time =14 days Full time to connection =203 days Effective time to connection =29 days
−1.014 −1.012 −1.01 −1.008 −1.006 −1.004 −1.002 xSE
−1
−0.998 −0.996
EM Fig. 7. Example of a low-cost connection of the type LSE 2 L2 : view of the LPOs, their manifolds and the connecting trajectory in configuration space (Sun–Earth synodical barycentric coordinates).
pericenter distance r2 at t =0, the angle y formed by the absidal line with the positive x-axis of the synodical barycentric reference frame, and the eccentricity e of the osculating Keplerian orbit: J ¼ 2cosyðm1Þr2 þ
mð1eÞ r2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ1m 7 2 mð1 þeÞr2 :
2ð1mÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r22 2cosyr2 þ 1 ð1Þ
Here the + and signs refer to prograde and retrograde initial orbits, respectively, and m is the mass ratio of the
two primaries. Varying e and y over the intervals [0.0,1.0] and ½0:0,2p, respectively, and solving Eq. (1) for r2, provides the initial conditions that correspond to osculating Keplerian orbits around P2 at t= 0 (see Fig. 12 for a plot of the initial conditions in configuration space obtained in this way for the Sun–Earth CR3BP). These are integrated in the force field due to P2 and P1. Here we investigate the relationship between the set of unstable points of the WSB in the Sun–Earth CR3BP and the stable invariant manifolds of planar Lyapunov orbits around L1 and L2 with the same Jacobi constant. We also
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
1043
−3
x 10 8
Earth−Moon RTBP: J =3.14112
Sun−Earth RTBP: JSE =3.0008
Stable manifold of L1
Unstable manifold of L
EM
6
1
α =355 deg 0
4
y
SE
2 L
Earth
0
−−> Sun
1
−2 −4 −6
Δ V at connection =293 m/s Full time of flight =224 days Effective time of flight =57 days
−8 −1.004 −1.002
−1
SE winding−off time =155 days EM winding−on time =12 days Full time to connection =198 days Effective time to connection =43 days
−0.998 −0.996 −0.994 −0.992 x
−0.99
−0.988
SE
EM Fig. 8. Example of a low-cost connection of the type LSE 1 L1 : view of the LPOs, their manifolds and the connecting trajectory in configuration space (Sun–Earth synodical barycentric coordinates).
0.015
Earth−Moon RTBP:
Sun−Earth RTBP:
J
J
EM
=3.1824
SE
Stable manifold of L1 0.01
=3.00058
Unstable manifold of L 2
α =315 deg 0
y
SE
0.005
L 0
Earth
2
−−> Sun
−0.005
−0.01
−0.015
SE winding−off time =185 days EM winding−on time =11 days Full time to connection =225 days Effective time to connection =40 days
Δ V at connection =603 m/s Full time of flight =245 days Effective time of flight =50 days −1.014
−1.012
−1.01
−1.008
−1.006 x
−1.004
−1.002
−1
−0.998
−0.996
SE
EM Fig. 9. Example of a low-cost connection of the type LSE 2 L1 : view of the LPOs, their manifolds and the connecting trajectory in configuration space (Sun–Earth synodical barycentric coordinates).
examine the causes of the unstable behaviour and relate them to the fine geometrical structure of the WSB. The WSB regions are computed for values of nR equal to 0.5, 1.0 and 1.5. Figs. 13 and 14 show the superposition between the unstable points of the WSB with nR = 1.5 and SE Jacobi constant J ¼ 3:000583 and the points of the stable
manifolds of the two Lyapunov orbits of the same energy for which the radius vector is orthogonal to the velocity vector (the two are meant relative to the second primary). The unstable points originate from both initial prograde and retrograde motion. The stable manifolds have been computed by means of a semi-analytic expansion of
1044
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.01 0.008 0.006 0.004 0.002 SE
L
y
Earth
2
0
−0.002 −0.004 −0.006 −0.008 −0.01 −1.012
−1.01
−1.008 −1.006 −1.004 −1.002 xSE
−1
−0.998
−4
x 10 6 4
L2 2
Earth
0
0.01 0.005
−2 0 −4 −1.012 −1.01 −1.008
−0.005 −1.006 −1.004 −1.002 xSE
−1 −0.998
y
SE
−0.01
EM Fig. 10. Refinement of the LSE 2 L2 connection in a complete Solar System force-field, shown in Sun–Earth synodic barycentric coordinates: xy-plane projection (top) and 3D view (bottom). In black the refined solution, in red the original two coupled CR3BP connection. Note that the refined trajectory is a natural connection, i.e., requiring no maneuvers. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
order 17 and propagated up to 100 normalized time units. The agreement2 between the two sets is a clear indication that the stable manifolds confine the WSB and play a role in driving the unstable points (see also [4]). Furthermore, Fig. 15 shows that the initial conditions which remain stable at nR = 1.5 effectively form the boundary of the corresponding unstable zone.
2 The small discrepancies observed on the right side of Fig. 13 are essentially due to the growing numerical errors in the invariant manifolds computation as time increases. Furthermore, the few, scattered WSB points in the lower right part of the plot are initial conditions of trajectories which have become unstable very early in the given interval, i.e., 1:0o nR r 1:5; therefore their distribution in configuration space recalls that of the unstable points pertaining to the ‘‘previous interval’’ in nR (i.e., 0:5 o nR r 1:0).
Fig. 11. A drawing showing how a stable (solid curve) and an unstable (dashed curve) trajectory look like, according to the WSB stability criterion stated in the text.
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.02
J
SE
1045
=3.00058
prograde
0.015 0.01 0.005 y
L2
L1
0 −0.005 −0.01 −0.015 −0.02 −0.02 −0.015 −0.01 −0.005
0 0.005 x −μ + 1
0.02
0.01
J
SE
0.015
0.02
=3.00058
retrograde
0.015 0.01 0.005 y
L2
L1
0 −0.005 −0.01 −0.015 −0.02 −0.02 −0.015 −0.01 −0.005
0 0.005 x −μ + 1
0.01
0.015
0.02
Fig. 12. The Sun–Earth CR3BP in synodical coordinates centered on the Earth: location in configuration space of the initial conditions giving rise to osculating Keplerian orbits around the Earth at t= 0 (filled areas: prograde orbits on top, retrograde orbits at the bottom) for a given Jacobi constant value. The two planar Lyapunov orbits with the same Jacobi constant value are also plotted.
For each of the three available sets of unstable points, i.e., with nR = 0.0, 0.5, and 1.5, we have identified the cause of instability:
completion of at least one revolution around P1 before the return to the initial absidal line;
return to the initial absidal line with positive Keplerian energy relative to P2.
escape (i.e., reaching the end of the integration before
the return to the initial absidal line relative to P2); collision with P2 (when the distance of P3 from the Earth is smaller than its equatorial radius); collision with P1 (when the heliocentric distance of P3 is smaller than the Sun’s radius);
SE
Initial conditions corresponding to J ¼ 3:000583 and prograde motion, produce unstable points characterized by the various unstable behaviours in the proportions shown in Figs. 16–18, respectively, for the three selected values of nR. This information allows to understand
1046
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
−3
8
x 10
WSB unstable for 1.0 < nR < 1.5 stable man. of PLOs of L1 and L2
6 4
y
2 0 −2 −4 −6 −8 −8
−6
−4
−2
0
2
4
6
8 −3
x−μ+1
x 10
−3
2
x 10
WSB unstable for 1.0 < nR < 1.5 stable man. of PLOs of L and L
1.5
1
2
1
y
0.5 0 −0.5 −1 −1.5 −2 −2
−1.5
−1
−0.5
0 x−μ+1
0.5
1
1.5
2 −3
x 10
SE
Fig. 13. Sun–Earth system: initial conditions with Jacobi constant J ¼ 3:000583 producing unstable trajectories at nR = 1.5 (crosses) and points of the stable manifolds of the two Lyapunov orbits around L1 and L2 with the same energy for which the radius vector is orthogonal to the velocity vector (dots). The reference frame is the synodical one, centered on the second primary. Both prograde and retrograde motions are represented. The bottom plot provides an enlarged view of the top plot, showing details of the superposition between the two sets of points close to the Earth.
the nature of the empty areas inside the unstable zones of Fig. 13: they are occupied by trajectories which turn into unstable at either of the two previous steps, i.e., nR = 0.5 or 1.0. Besides, as Fig. 19 shows, such
trajectories either escape from P2 or collide with it. Close inspection of their location in configuration space (Fig. 20 shows two details of Fig. 19 separately obtained for initial prograde and retrograde motions)
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
1047
0.15
0.1
vy
0.05
0
−0.05
−0.1
−0.1
−0.05
0 vx
0.05
0.1
0.15
Fig. 14. Same as the previous figure but illustrating the x_ and y_ components of the unstable initial conditions for nR = 1.5 (crosses) and the points of the stable manifolds of the two Lyapunov orbits around L1 and L2 with the same energy for which the radius vector is orthogonal to the velocity vector (dots).
0.01 0.008
nu1.0−1.5 revs :2073 stable at1.5 revs:312
0.006 0.004
y
0.002 0 −0.002 −0.004 −0.006 −0.008 −0.01 −0.01
−0.005
0 x−μ+1
0.005
0.01 SE
Fig. 15. Initial conditions producing unstable (crosses) and stable (squares) trajectories for nR = 1.5 and J ¼ 3:000583. Synodical reference frame centered on the Earth. The legend provides the number nu of unstable points and the number ns of stable ones.
suggests that there is no finer structure, and by small variations in the initial conditions one behaviour, i.e., escape, changes into the other, i.e., collision, and vice versa (Figs. 21–23).
4. Capture around the collinear libration points Among the unstable trajectories, some approach either L1 or L2. It is possible to identify those which perform a
1048
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.01
0.005
escape (t>t ) :1891 f 1 rev. P :5911 1 coll. P :0
0.01
E > 0 :1 k coll. P :469
nu1.0−1.5 revs :2073
0.008
escape :2622 coll P2 :520
2
0.006
1
0.004
0
0
y
y
0.002
−0.002 −0.004
−0.005
−0.006 −0.008
−0.01 −0.01
−0.005
0 x−μ+1
0.005
Fig. 16. Initial conditions corresponding to prograde motions and producing unstable trajectories for nR =0.5. The legend indicates the type of instability and the number of points that fall into each category.
0.01
0.005
−0.01 −0.01
0.01
−0.005
0 x−μ+1
0.005
0.01
Fig. 19. Unstable points for nR = 1.5 (crosses) and a subset of the unstable points for nR = 0.5 and 1.0 which either collide with P2 (dots) or escape from it (circles).
−3
escape (t>tf) :731 1 rev. P :260 1 coll. P :0
−1
Ek > 0 :14325 coll. P :51
x 10
2
−1.5
1
0 y
y
−2
−2.5
−3
−0.005
−3.5
−0.01 −0.01
−0.005
0 x−μ+1
0.005
0.01
Fig. 17. Initial conditions corresponding to prograde motions and producing unstable trajectories at nR =1.0. The legend indicates the type of instability and the number of points that fall into each category.
0.01
y
0.005
escape (t>tf) :437 1 rev. P :101 1 coll. P :0
−6.8
−6.6
−6.4
−6.2
x−μ+1
−6 −3
x 10
Fig. 20. A higher-resolution slice of Fig. 19 corresponding to prograde motion.
temporary capture in either of the two appropriately defined regions around the two libration points. The captured trajectories appear to follow for some time the trajectories of the stable manifolds of the two planar Lyapunov orbits of the same energy, thus suggesting that the stable manifolds play a role also in this kind of temporary capture. Fig. 24 shows for initial prograde motion, the position of the unstable points that for each of the three values of nR perform temporary capture, or just enter either or both the capture regions, or are rejected by the capture criterion.
Ek > 0 :1454 coll. P :81 2
1
0
−0.005
−0.01 −0.01
−4 −7
5. Temporary capture around the second primary −0.005
0 x−μ+1
0.005
0.01
Fig. 18. Initial conditions corresponding to prograde motions and producing unstable trajectories at nR = 1.5 revolutions. The legend indicates the type of instability and the number of points that fall into each category.
That the invariant manifolds of LPOs play a role in the temporary capture around the second primary of a CR3BP is a known fact (see [5]). In this contribution we examine the dynamics of the captured trajectories relative to the second primary and we relate the resulting orbital
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
−4
5
0.01
x 10
0.008
4
1049
captured at L1 :375
0.006
through L1 only :0
0.004
through L2 only :0
3
nR =0.5 revs.
captured at L2 :197
2 0.002
y
1
0
0
−0.002
−1
−0.004
−2 −3
−0.006
stable points :312 L1 first :0
−0.008
L2 first :0 rejected :7700
−4
−0.01 −0.01
−5 −1
−0.5
0
0.5
−0.005
1 −3
x 10
x−μ+1
Fig. 21. A higher-resolution slice of Fig. 19 corresponding to retrograde motion.
0.015
0.01 0.008
0 x −μ + 1
captured at L1 :71
0.005
0.01
nR =1.0 revs.
captured at L2 :334
0.006
through L1 only :0
0.004
through L2 only :0
0.002 0
0.01
−0.002 −0.004
0.005
y
L2
L1
0
−0.006
stable points :312 L first :0
−0.008
L2 first :1
1
rejected :14961 −0.01 −0.01
−0.005
0.01
−0.01
0.008
−0.005
0 x −μ + 1
captured at L1 :385
0.005
0.01
nR =1.5 revs.
captured at L :316 2
−0.015 −0.015
−0.01
−0.005
0 x −μ + 1
0.005
0.01
0.015
0.006 0.004
Fig. 22. Examples of collision trajectories with initial conditions in the subset of Fig. 20.
through L only :0 1
through L only :0 2
0.002 0 −0.002 −0.004
0.015
0.01
−0.006
stable points :312 L1 first :10
−0.008
L2 first :13 rejected :1349
−0.01 −0.01
0.005 L
2
y
−0.005
0 x −μ + 1
0.005
0.01
L
1
0
Fig. 24. Sun–Earth system: position of the unstable points for nR = 0.5 (top), 1.0 (middle), 1.5 (bottom). Different colors and symbols mark the several behaviours with respect to the capture criterion, as specified in the legend. The location of the stable points is also shown (crosses).
−0.005
−0.01
−0.015 −0.015
−0.01
−0.005
0 x −μ + 1
0.005
0.01
0.015
Fig. 23. Examples of escape trajectories with initial conditions in the subset of Fig. 20.
parameters to the dynamical features of the invariant manifold to which they belong. Let us consider the unstable invariant manifolds of planar Lyapunov orbits around L1 and L2 and, in particular, the branches that develop in the region around P2, confined by the zero
1050
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0.015
400 = peri
= apo
= revs. P
2
=E >0 K
= collision with P (:90) 2
350
0.01
300 0.005
y
orbit
250 200
0
150
−0.005
100
−0.01
50 −0.015 −0.015
0
5
10
15 time
20
25
−0.01
−0.005
0 x−μ+1
0.005
0.01
0.015
−0.01
−0.005
0 x−μ+1
0.005
0.01
0.015
30 0.015
400 = peri
= apo
= revs. P
2
=E >0 K
= collision with P (:0) 2
0.01
orbit
350
0.005
250
0
y
300
200 −0.005
150 −0.01
100 −0.015 −0.015
50
0
5
10
15 time
20
25
30
Fig. 25. Development with time of some dynamical parameters related to the Keplerian orbit around the second primary, obtained from the unstable manifolds of two planar Lyapunov orbits around L1 in the Sun–Earth system, one with J= 3.00059 (top), the other with J = 3.000888 (bottom): the occurrence of perigees (crosses), apogees (dots), collisions with P2 (squares), the completion of successive loops around P2 (asterisks) and finally the sign of the Keplerian energy Ek (white background for negative, yellow for positive, respectively). The legend also reports the number of collisions (in parentheses). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
velocity curves and the openings of the bottleneck at L1 and L2. Fig. 25 shows the behaviour with time (up to 30 normalized time units) of a number of parameters related to the dynamics relative to P2 of 360 orbits in which the unstable manifolds of two Lyapunov orbits (with J= 3.00059 and 3.000888, respectively, in the two plots) around the L1 point in the Sun–Earth system have been divided. These parameters are:
the occurrence of perigees and apogees (crosses and dots, respectively);
Fig. 26. Some trajectories belonging to the invariant manifolds of Fig. 25: J = 3.00059 (top) and J = 3.000888 (bottom).
the occurrence of collisions with P2 (squares, which
also constitute the end-point of the corresponding trajectory); the sign of the Keplerian energy Ek (white background for negative, yellow for positive).
Note that positive energy means escape, i.e., the trajectory leaves the lobe around P2. Therefore, the necessary condition for (temporary) capture3 to occur is that Ek be negative. Comparison between the two plots of Fig. 25 tells that invariant manifolds of Lyapunov orbits of smaller size (i.e., of larger J) experience fewer collisions and perform more revolutions around P2 while in the capture region. Fig. 26 shows some example trajectories from the two manifolds of Fig. 25. Fig. 27 illustrates one trajectory for each of the two manifold: that corresponding to the smaller Lyapunov orbit remains captured through all the integration time considered, whereas the trajectory associated to the larger Lyapunov orbit completes one loop around P2 and then collides with it.
the completion of full revolutions around P2 (asterisks), counted starting from a given line orthogonal to the x-axis and located between the progenitor orbit and the second primary;
3 A trajectory is said to be in a temporary capture if it enters the lobe and completes a minimum number of revolutions around P2 before escaping.
0.015
0.03
0.01
0.02
0.005
0.01
semi−major axis
y
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
0
−0.005
−0.01
−0.015 −0.015
0
−0.01
360 300 240 180 120 60
−0.02
−0.01
−0.005
0 x−μ+1
0.005
0.01
1051
−0.03
0.015
0
5
10
15 time
20
25
30
5
10
15 time
20
25
30
0.015
0.03 0.01
0.02
semi−major axis
y
0.005
0
−0.005
0.01
0
−0.01
360 300 240 180 120 60
−0.01
−0.02 −0.015 −0.015
−0.01
−0.005
0 x−μ+1
0.005
0.01
0.015
Fig. 27. One trajectory from each of the samples of Fig. 26: the crossing of the vertical lines indicates the entrance into the region around P2 and the starting point for loop counting.
Fig. 28 shows for the orbits of Fig. 26, the behaviour with time of the semi-major axis of the Keplerian orbit relative to P2: all the trajectories are initially endowed with positive semi-major axis; then, those which escape exhibit a sudden growth (to infinity), followed by a change into a constant, negative value (hyperbolic orbit relative to P2). The unstable invariant manifolds of planar Lyapunov orbits around L2 behave similarly. Fig. 29 provides global information concerning the temporary capture as a function of the energy, over two families of 70 planar Lyapunov orbits each, around L1 and L2, respectively. The two plots give the percentage of trajectories that complete at least 5 (respectively, 10) loops around P2 before the energy turns to positive or the integration ends. The horizontal range of the plot is limited to the first 40 orbits of each family, the remaining ones having zero percentage of captured orbits of either type.
6. Conclusions In this contribution we considered four transfer mechanisms in systems composed of three (CR3BP) or four (two coupled CR3BPs) bodies, with special interest in the Sun–Earth(–Moon) system: the low energy transfers
−0.03
0
Fig. 28. Semi-major axis as a function of time for the trajectories of Fig. 26.
from the Earth to the Moon, the weak stability boundaries, the temporary capture around the libration points on behalf of trajectories in the WSB, and the temporary capture of invariant manifold trajectories around the second primary of a CR3BP. The understanding of these four issues is related to the dynamics of the invariant manifolds of periodic orbits around the collinear libration points L1 and L2 of the system. Concerning the LETs, of the four combinations of libration EM point connections examined, only two, i.e., LSE and 1 L2 SE EM L2 L2 , can be obtained at low or zero cost, and this capability depends on the energy (and hence the size) of the LPOs to be connected, being higher for lower energies. The relationship between the WSB unstable points and the stable invariant manifolds of LPOs of the same energy has been clarified: the points of the invariant manifold trajectories that are characterized by orthogonality between the radius and the velocity vector relative to P2 effectively confine the unstable points of the WSB region. The destination of the unstable WSB points has also been understood, and different behaviours, in particular collisions and escape, have been found to be closely related. The stable invariant manifolds of LPOs also play a role in the temporary capture of the unstable WSB points on behalf of L1 and L2. Finally, the temporary capture around the second primary, a well known phenomenon induced by the unstable invariant manifolds of LPOs,
1052
E. Fantino et al. / Acta Astronautica 67 (2010) 1038–1052
100
1
90 % of orbits with capture >5 revs.
all the above issues will appear in a subsequent, more extended publication.
=L
= L2
80 70
Acknowledgements
60 50 40 30 20 10 0 5
10
15 20 25 Lyapunov orbit index
30
100
40
=L
1
90 % of orbits with capture >10 revs.
35
A special thanks is expressed to Elisa Maria Alessi for the valuable assistance in the trajectory refinement based on the JPL ephemeris model. E. Fantino and Y. Ren have been supported by the Marie Curie Actions Research and Training Network AstroNet MCRTN-CT-2006-035151. G. Go´mez and J.J. Masdemont have been partially supported by the Grants MTM2006-05849/Consolider, MTM2009-06973 and 2009SGR859. The authors also acknowledge the use of EIXAM, the UPC Applied Math cluster system for research computing (see http://www. ma1.upc.edu/eixam/), and in particular Pau Roldan for providing technical support in the use of the cluster.
=L
2
80
References
70 60 50 40 30 20 10 0
5
10
15 20 25 Lyapunov orbit index
30
35
40
Fig. 29. Capability of being captured around P2 within the unstable invariant manifolds of two families of planar Lyapunov orbits around L1 (circles) and L2 (asterisks), expressed as the percentage of trajectories that perform a minimum number of loops (i.e., 5 (top) and 10 (bottom)) around P2 before the Keplerian energy turns to positive or the integration ends.
has been put in relation with the dynamical characteristics (size, energy, semi-major axis, perigees and apogees) of the invariant manifold trajectories relative to the second primary: temporary capture is more efficient when the Jacobi constant of the invariant manifold is larger and the size of the progenitor LPO is smaller. Further analysis and discussion of
[1] E.M. Alessi, Study of some dynamical problems in the Earth–Moon system, Ph.D. Dissertation, Universitat de Barcelona, 2010. [2] M.A. Andreu, The quasi-bicircular problem, Ph.D. Dissertation, Universitat de Barcelona, 1999. [3] E.A. Belbruno, Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers, Princeton University Press, 2004. [4] E.A. Belbruno, M. Gidea, F. Topputo, Weak stability boundary and invariant manifolds, preprint, 2009. [5] E. Belbruno, B.G. Marsden, Resonance hopping in comets, Astronomical Journal 97 (1997) 1433–1444. [6] C. Circi, P. Teofilatto, On the dynamics of weak stability boundary lunar transfers, Celestial Mechanics and Dynamical Astronomy 79 (2001) 41–72. [7] C. Circi, P. Teofilatto, Weak stability boundary trajectories for the deployment of lunar spacecraft constellations, Celestial Mechanics and Dynamical Astronomy 95 (2006) 371–390. [8] F. Garcı´a, G. Go´mez, A note on weak stability boundaries, Celestial Mechanics and Dynamical Astronomy 97 (2007) 87–100. [9] W.S. Koon, M. Lo, J.E. Marsden, S.D. Ross, Low energy transfers to the Moon, Celestial Mechanics and Dynamical Astronomy 81 (2001) 63–73. [10] M. Lo, S. Ross, SURFing the solar system: invariant manifolds and the dynamics of the solar system, JPL IOM 312-97, 1997, pp. 2–4. [11] J.K. Miller, E.A. Belbruno, A method for the construction of a lunar transfer trajectory using ballistic capture, in: Proceedings of the 1st AAS/AIAA Spaceflight Mechanics Meeting, Paper No. AAS 97-109, 1991. [12] D. Romagnoli, C. Circi, Earth–Moon weak stability boundaries in the restricted three and four body problem, Celestial Mechanics and Dynamical Astronomy 103 (2009) 79–103.