Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 3, pp. 551-564, Warsaw 2008

LOW-THRUST CHAOTIC BASED TRANSFER FROM

THE EARTH TO A HALO ORBIT

Elbert E.N. Macau

Laboratório Associado de Computaçǎo e Matemática Aplicada – LAC, Instituto Nacional

de Pesquisas Espaciais – INPE, São José dos Campos, SP, Brazil

e-mail: elbert@lac.inpe.br

Halo orbits around the Lagrange points serve as excellent platforms for
scientificmissions involving the Sun aswell as planetary explorations.The
satellites ISEE-3,GGS WIND,SOHOandACEhaveexploitedtheseorbits
to accomplish their missions. The trajectory design in support of such
missions is increasingly challenging asmore complexmissions, like the case
of theNextGeneration SpaceTelescope, are envisioned in the near future.
The purpose of this paper is to introduce a new chaotic-based transfer
strategy that can be used to transfer a spacecraft from the vicinity of
the Earth to a halo orbit around the equilibrium point L1 of the Earth-
Sun system. The strategy exploits the inherent exponential sensitivity of
chaotic time evolutions to perturbations to guide a spacecraft from its
starting position to an invariant stable manifold of the chosen halo orbit.
As just judiciously chosen perturbations are used, the transfer operation
requires very small amounts of fuel.As an example,we applied themethod
in context of the restricted three-body problem,which gives a good insight
into the strategies to be used in a real situation.

Key words: chaos, control of chaos, orbital transfer

1. Introduction

In the restricted three-body problem, halo orbits are spatial periodic solu-
tions that are present around the collinear libration points. They have been
calculated both numerically (Breakwell and Brown, 1979; Howell, 1984; Ho-
well and Pernicka, 1988) from the equation of motion and analytically (Gou-
das, 1963; Zagouras and Kazantzis, 1979; Richardson, 1980a,b) by means of
the Lindstedt-Poincaré procedure. The analytical approaches show that the
linearized motion about the collinear libration points encompass a periodic



552 E.E.N. Macau

orbit in the plane of the primary motion and an uncoupled periodic out-of-
plane motion. Increasing the amplitude of these orbits, certain combinations
of in-plane and out-of-plane amplitudes exist, such that the corresponding fre-
quencies are equal and a perfectly periodic three-dimensional motion occurs.
Robert Farquhar coined the term ”halo” for these orbits in his PhD thesis
(Farquhar, 1968b). Later on, he proposed placing a communication satellite
in such an orbit about the libration point L2 in the Earth-Moon system (Fa-
rquhar, 1968a). This satellite was envisioned in the context of the Apollo 18
mission and would allow a continuous contact with both the far side of the
Moon and the Earth, as it would never be blocked from view by the Moon.
As the Apollo 18 mission was canceled, his idea was archived.

Farquhar’s idea of exploiting halo orbits was resumed, but this time asso-
ciatedwith the collinear libration points in the Sun-Earth system. Spacecrafts
in halo orbits near these libration points offer valuable opportunities for scien-
tific investigations concerning solar exploration, cosmic background radiation,
and solar and heliospheric effects on planetary environments. Thus, in 1978,
the International Sun-Earth Explorer-3 (ISEE-3) spacecraft was the first one
that was successfully deployed into a halo orbit, specifically about the interior
collinear libration point L1 (Farquhar et al., 1977; Richardson, 1979). A num-
ber of other missions succeeded this pioneer mission: WIND, SOHO, ACE,
Genesis and others are currently in development.

This increased interest in libration point missions has motivated the rese-
arch involving a new trajectory design process in the scope of the three-body
problem (Gómez et al., 1993; Howell et al., 1994, 1997). In fact, traditional
approaches based on conic approximations are not adequate, while standard
targeting andoptimization strategies that usevariationalmethodsdonotwork
properlybecause of the extremenonlinearities andhigh sensitivities in thepro-
blemof finding a proper transfer trajectory from theEarth to the desiredHalo
orbit. Despite the success in guiding the previously cited missions, more ef-
ficient and flexible techniques are still needed, especially such which would
exploit the richness of behaviour present in a nonlinear system, and also in-
corporate ideas insights from the theoretical understanding of this type of the
multi-body problem. A remarkable step in this direction was introduced by
Gómez et al. (1993). Their strategy involves computation of a stable mani-
fold of a halo orbit and determination of a transfer trajectory that allows the
injection of the spacecraft onto this manifold. In the transfer from a parking
orbit about the Earth, their strategy can be easily achieved when the orbits
of the stable manifold come close to the parking orbit. However, this happens
in general when the halo orbit is large enough. If it is not the case, a strategy



Low-thrust chaotic based transfer... 553

of transfer between halo orbits can be applied to guide the spacecraft from a
large halo orbit to the desired small one.

In this paper, we consider another approach to the problem of finding a
transfer trajectory from the neighborhood of the Earth to a halo orbit. This
approach, named the targeting type of the control problem for a chaotic sys-
tem, was introduced in the scope of the Theory of Chaotic System to drive
trajectories in chaotic systems (Shinbrot et al., 1990). Its main characteristic
is to take advantage of the richness of behaviour present in chaotic dynamics
to accomplish the goal of finding a proper trajectory from a source point to a
target point. In fact, the hallmark of chaotic behaviour is the extremely high
sensitive dependence on initial conditions, which prevents long-term predic-
tion of the state of the system from measured data. On the other hand, this
inherent exponential sensitivity of chaotic time evolution to perturbations can
be exploited in driving trajectories to some desired final state by the use of a
carefully chosen sequence of small perturbations to some control parameters.
The key point is that in being so small, the perturbations does not change
the system dynamics significantly, but make it work properly to accomplish
the transfer from the initial state to the desired final state (Macau and Gre-
bogi, 2001). In Celestial Mechanics, this approach has been used before to
accomplish both theoretical (Bollt andMeiss, 1995a,b; Macau, 2000) and real
mission (Belbruno, 1994). Its main appeal is that it requires small amounts of
fuel to attain the desired goal. However, usually a transfer takes a significantly
long time if compared with traditional techniques.

The problem of transfer from the Earth to a halo orbit by using the tar-
geting control of chaos presents a high level of complexity if compared with
the previously cited problems. For these ones, the problemswere solved in the
environment of the planar, circular, restricted three-body problem.Because of
the presence of a constant ofmotion, i.e., the Jacobi integral, theHamiltonian
flow is confined to a three-dimensional submanifold. The introduction of a Po-
incaré section allow the problem tobe solved in the scope of a two-dimensional
area-preservingmap.This is a specific situation inwhich chaotic targetingme-
thodsarewell studied andhave evenbeenappliedwith success in experimental
situations. On the other hand, the problem that we consider here takes place
in the environment of the three-dimensional, circular, restricted three-body
problem, which means that the state space is of dimensional six. The Jacobi
integral confines the flow to a five-dimensional submanifold. For this case, the
problem can not be easily solved with the introduction of a Poincaré section,
because even with it we are in the scope of a four-dimensional area preserving
map. It means that the proper perturbations to be applied must be determi-



554 E.E.N. Macau

ned in stable and unstable manifolds of high dimension, which, in general, is
a hard task. For this specific situation, Kostelich et al. (1993) introduced a
method on which we based our work. This method must also be adapted to
the Celestial Mechanics situation where perturbations can just be applied to
the velocity, as a perturbation in a position has no physical meaning.

We proceed as follows. In the next section, we describe the high-dimension
chaotic targeting method in detail. Then, we present halo orbits in context
of the three-dimensional, circular, restricted three-body problem. In the sub-
sequent section, we show how the presented high-dimension chaotic targeting
method can be applied to find a transfer trajectory from the neighbourhood of
theEarth to ahalo orbit. A general discussion about this approach is provided
in the last section.

2. Higher-dimension chaotic targeting

Let us consider a discrete in time dynamical system

Xn+1 =F(Xn) (2.1)

where Xi ∈ℜ
n, and F is a smooth function.Let us suppose that our goal is to

target the point Q from the point P, both located inside the chaotic invariant
set. Our aim is to find ”small” perturbations δ so that if those numbers are
adequately applied to the original trajectory that passes through P we have
a perturbed trajectory that eventually hits the target point Q. Let us now
suppose that in the neighbourhood of P there is a point X0 that belongs to
the trajectory {Xi}, and in the neighbourhood of Q there is another point YN
that belongs to the trajectory {Yi} so that those trajectories come close to
each other at the point Xj of the trajectory {Xi}, where j > 0, and at the po-
int Yk of the trajectory {Yi}, where i <N. Note that the restriction over the
indices of the trajectory points indicates that the trajectory {Xi} passes first
through point X0 before getting to the point Xj, while the trajectory {Yi}
first passes through the point Yk, before getting to YN. It means that if we
could follow the trajectory {Xi} until the point Xj, and then from this point
move to point Yk of the trajectory {Yi}, we would have a transfer trajectory
from the neighbourhood of the point P to the neighbourhood of the point Q.
The goal of the chaotic targetingmethod is to findproper perturbations to ac-
complish it. This can be done in the followingway: For a hyperbolic situation,
associated to each point on the invariant set, there are stable and unstable



Low-thrust chaotic based transfer... 555

manifolds. The stable manifold of Z is a set of points W with the property
‖Fk(Z)−Fk(W)‖→ 0 as k→∞, while the unstablemanifold of Z is the set
of points U with the property ‖F−k(Z)−F−k(U)‖→ 0 as k→∞. According
toMacau andGrebogi (2001), in a hyperbolic situaition, if the distance betwe-
en Xj and Yk is sufficiently small, then the unstable manifold of Xj and the
stablemanifold of Yj intersect each other at a point q. This fact can be explo-
ited to accomplish our goal if a proper perturbation is applied to the sequence
of points of the orbit that passes through Xj. In fact, if the point q belongs
to the intersection of the unstable manifold of Xj with the stable manifold
of Yj, then forward iterations of q converge to forward iterations of Yk and
backward iterations of q converge to backward iterations of Xj. It is known
that in each point p of the trajectory, the local stable and unstable mani-
folds are respectively tangent to the eigenspaces Esp and E

u
p of the linearized

system, where Esp span the subspaces whose eigenvalues have modulus < 1,
while Eup span the subspaces whose eigenvalues have modulus > 1. Thus, if
the proper perturbation αXj−m is applied in the direction of the E

u
Xj−m
, it

produces a perturbed orbit that passes through q, and converges to the tra-
jectory {Yi} after Yk. Consequently, this procedure generates the desired path
that allows us to smothly pass fromthe trajectory {Xi} to the trajectory {Yi}
and so be transfered to the neighbourhood of the point Q. In addition, that
argument indicates that the perturbationαXj−m can be calculated by solving
the following equation

Fm+t(Xj−m+αXj−mE
u
Xj−m
)=Yk+t+βk+tE

s
Yk+t

(2.2)

In this equation, the values of m and t can be adequately adjusted for each
system by an empirical procedure. In applying this procedure, another impor-
tant simplification can be introduced. If we consider any orbit {Zk}

n
k=1 that

contains Zi, almost any variation near Zi−m will expand along the unstable
manifold of Zi if the value of m chosen is large enough. A similar statement
can be made regarding the stable manifold of Zi for variations near Zi+m
iterated in the backward direction (Macau and Grebogi, 2001). Thus, to cal-
culate the perturbation, we introduce a small perturbation β̂eβ where eβ is
a unit vector in the direction of the perturbation at the position Yk+t and
iterate this perturbed point t times backward in time. This will typically ge-
nerate a nearby trajectory that will deviate progressively from the original
trajectory at each backward iteration, expanding away from {Yi} along the
direction of the stable manifold at the points on the orbit {Yi} (Macau and
Grebogi, 2001) (we assume that the direction of the small perturbation β̂eβ is
not precisely such that it has no component in the stable direction). We also



556 E.E.N. Macau

introduce a small perturbation δ̂eδ to the orbit {Xi} at the iteration j−m
where eδ is a unit vector in the direction of the perturbation, and iterate this
perturbed point forward in time m iterates. This will typically generate a ne-
arby trajectory that will deviate progressively from the original trajectory at
each forward iteration, expanding away from {Xi} along the direction of the
unstablemanifold at the points on the orbit {Xi} (Macau andGrebogi, 2001).
Thus, the perturbations can be calculated by solving the following equation

Fm(Xj−m+ δ̂eδ)=F
−t(Yk+t+ β̂eβ) (2.3)

which can be calcuated by using the Broyden method (Press et al., 1996).

The use of the previous equation makes the problem more easily to be
solved, in particular because it is not necessary to determine local approxi-
mations of the stable and unstable manifolds at each step of the trajectories.
However, to be effective, it requires a closer approximation of the trajectories
than applying Eq. (2.2).

Note that because of the approximations and numerical roundoff erros,
this described method must be repeated from time to time in order to keep
the new trajectory close to the path leading to the target point Q.

Whether using Eqs. (2.2) or (2.3), this approach can just be applied if
the trajectories are sufficiently close to each other. Thus, before using it, the
points X0 in the neighbourhood of P and YN in the neighbourhood of Q
must be found so that foward iterations of X0 and backward iterations of YN
imply in trajectories that come close enough to each other.As a chaotic system
has the property of being topologically transitivite, a trajectory starting from
almost any point of the chaotic invariant set is dense in this set. It means
that starting from almost any point P and Q of the chaotic attractor, we can
rely on the ergodic wander of the forward orbit from P and backward orbit
from Q to bring these orbits sufficiently close to each other so that equations
(2.2) or (2.3) can be applied with success. However, for such a process, in
especially in the case of high-dimensional system, the transport time can be
extremely long. This situation is worst in the case of Hamiltonian systems,
because in this case the phase space is divided into layered components which
are separated from each other by Cantory (Mackay et al., 1984). Typically, a
trajectory initialized in one layer of the chaotic region wanders in that layer
for a long time before it crosses the Cantory and wanders in the next region.
Thus, if P and Q are located in different layers, the time that is required
before the trajectory come close to each other can be prohibitively long.

In particular, in high-dimensional systems and in situations in which the
existence of layer is not critical, Kostelich et al. (1993) proposed to face this



Low-thrust chaotic based transfer... 557

problem by building a hierarchy or a ”tree” of trajectories. Thus, we start a
trajectory from P, say {X0,X1, . . .}.To eachpoint of this trajectory,weapply
a small perturbationandwe start another trajectory fromtheperturbedpoint.
So, if δ is a perturbation and Z0O

i = Xi + δ, we have a tree of trajectories
{Zi0,Z

i
1, . . .}, for i=0,1, . . . associated with the trajectory from P. The same

procedure is applied to the backward trajectory from Q. By doing so, we
increase the probability of finding iterated points from the two trees close to
each other and, consequently, the time (number of iterations) needed to find
those points is substantially decreased.

3. Equations of motion

The three-dimensional, circular, restricted three-body problem involves two
finite masses m1 and m2, assumed to be point masses, moving around the-
ir common mass center under gravitational influence of each other. We use a
rotating coordinate systemwith the origin at the barycenter and angular velo-
city normalized to the unity, as shown inFig.1, andwe also normalize the sum
of themasses to one, i.e., m1+m2 =1.Wedefine the characteristic parameter
µ as the mass ratio m2 to the sum m1 +m2. The position of the primaries
in the rotating frame are fixed at x1 = (−µ,0,0) and x2 = (1−µ,0,0). The
xy-plane is the plane of motion of m1 and m2. The third body, m3, is as-
sumed massless but may travel in all of the three dimensions of the space.
In this frame, the equations of motion for m3, which we associated with the
spacecraft, are the following

ẍ−2ẏ=
∂U

∂x
ÿ+2ẋ=

∂U

∂y
z̈=
∂U

∂z
(3.1)

where

U =
1

2
(x2+y2)+

1−µ

d1
+
µ

d2
d1 =

√
(x+µ)2+y2+z2

(3.2)

d2 =
√
(x−1+µ)2+y2+z2

This systemdoes admit a constant of integration, the Jacobi constant C, such
that

C =2U− (ẋ2+ ẏ2+ ż2) (3.3)



558 E.E.N. Macau

Fig. 1. Libration points in the three-body problem

In this system, there are five equilibirum points, or libration points, whe-
re the gravitational and centrifugal forces balance each other. In Fig.1, we
represent these points considering that m2 <m1. As the primaries travel aro-
und the barycenter, all five points remain in the same position relative to the
masses for a given µ.

As stated before, halo orbits are spatial periodic solutions that are present
around the collinear libration points. Those orbits can be calculated both
numerically and analytically. For this work, using µ=0.04 and C =2.9307,
wedetermined about the libration point L1 a halo orbit that appears inFig.2.
In the next section, we explain our method of transfer from an orbit in the
neighbourhood of the first primary to this halo orbit.

Fig. 2. Halo orbit to be targeted



Low-thrust chaotic based transfer... 559

4. Chaotic targeting to a halo orbit

Let us consider a spacecraft at a point P in the neighbourhood of the prima-
ry m1, but located inside the chaotic region for C = 2.9307. Figure 3 shows
together the trajectory of this spacecraft from the point P and the halo orbit
for which we aim to find a transfer trajectory from P. Note that, usually, we
start the design of a mission considering that the spacecraft is located in a
parking orbit around the primary. If this is the case, an initial ∆v is required
to inject the spacecraft into the chaotic region.

Fig. 3. ”Parking” and halo orbits

Thefirst step is to determine two trajectories that come close to each other
so that one passes in the neighbourhood of P before it gets the position of the
closest proximity,while the other, hereafter called the reference trajectory, first
passes through this position before attains the halo orbit. Following Section 2,
we identify the first trajectory by {Xi}, and the second one by {Yi} aswell as
thepointof closest approach ineach trajectoryby, respectivelly, Xj and Y0.To
make the taskof findingthese trajectories easy,we introduceaPoincaré section
at z = 0, mapping the xy-plane to itself whenever the trajectory traverses
this plane with ż > 0. We start it by selecting a large number of points
{Y 1,Y 2, . . . ,YM} uniformly distributed on the halo orbit.We choose a small
integer number k. To each of those points, we apply a small perturbation δv
on thevelocities andobtain thepoints {Y 1k ,Y

2
k , . . . ,Y

M
k } so that Y

i
k =Y

i+δv,
for i=1, . . . ,M. We call the set of these numbers by Hk. Having each of the
points Y ik as the initial condition, we integrate system (3.1) backward in time
to obtain a set of points Hk−1 = {Y

1
k−1,Y

2
k−1, . . .Y

M
k−1} inwich each trajectory

crosses the Poincaré section for the first time. We continue this procedure to
get a set of points Hk−2 = {Y

1
k−2,Y

2
k−2, . . . ,Y

M
k−2} of the second crossing of

the Poincaré section, and so one until we obtain a set of points H0. In our
exemple, we selected k=3. For the halo orbit, this is a ”tree” of trajectories
mentioned in Section 2.



560 E.E.N. Macau

From thepoint P, we proceedbygenerating its ”tree” trajectory. Todo so,
we associate this position with X00, and having this value as the initial condi-
tion, we integrate system (3.1) forward in time until the trajectory crosses the
Poincaré section for thefirst time.We call this point X01. In the orbit that goes
from X00 to X

0
1,we select a largenumberof points {X

1,X2, . . . ,XL}uniform-
lydistributed.Toeach of thosepoints,weapplya small perturbation δṽ on the
velocities and obtain a set of points {X10,X

2
0 , . . . ,X

L
0 } so that X

i
0 =X

i+δṽ,
for i = 1, . . . ,L. We call this set of points {X00,X

1
0,X

2
0 , . . . ,X

L
0 } P0. We

choose a small integer number j. Having each of the points Xi0 as the initial
condition, we integrate system (3.1) forward in time to obtain a set of po-
ints P1 = {X

0
1 ,X

1
1,X

2
1, . . . ,X

L
1 } inwhich each trajectory crosses thePoincaré

section for the first time. As before, we continue this procedure to get a set
of points P2 = {X

0
2,X

1
2,X

2
2, . . . ,X

L
2 } of the second crossing of the Poincaré

section, and so one until we obtain a set of points Pj. In our example, we
selected j=3.

Now,we plot the set of points H0 and Pj in thePoincaré section as shown
in Fig.4. Using the scenario displayed in the Poincaré section, we select pairs
of points (αi,βi),αi ∈Pj and βi ∈H0, within at least adistance dmx between
one another. Doing so, we obtain the set PH = {(αi,βi), i= 1,2, . . .}. This
set belong to the trajectories from P and to the halo orbit that come spatially
close to each other. From this set, we select the pair that comes closest to each
other in the full state space, i.e., not just in relation to its position values, but
also in relation to the velocity. Let us say that these point are Xαj and Y

β
0 .

Note that the Poincaré section is used as a kind of reference to allow both an
easy identification of the desired trajectories that come close to each other and
the building of the ”tree” of trajectories. Thus, we have the trajectory {Xαi }

that comes from the point P, and the trajectory {Y
β
i } that goes to the halo

orbit, and these two trajectories come closest to each other in their points Xαj
and Y

β
0 , respectively.

As these trajectories are found, we can proceed to the next step, which is
to findproper perturbations to ”path” the trajectory {Xαi } to {Y

β
i }. In order

to accomplish that, we use the method presented in Section 2, which implies
solving equation (2.3) for those trajectories about the points Xαj and Y

β
0 .

However, this methodwas originaly developed for dynamical systems discrete
in time, while in our situation we deal with a continuous dynamical system.
For this specific situation, we can associate a system continuous in time to a
discrete system by considering the solution of the former system at discrete
time intervals, i.e., at t = 0,1τ,2τ, . . .. As we use a numerical integrator to
solve the system of differential equations, we employ amultiple of the integra-



Low-thrust chaotic based transfer... 561

Fig. 4. Reference Poincaré section used to find convenient trajectories

tion step as the discretization interval, i.e., τ =nh, where h is the integration
step used. By doing so, we have a six dimensional discrete in time dynamical
system for which themehtod is applicable.
Another consideration is related to the perturbation. For celestial mecha-

nics problems, just ∆v makes sense. Thus, the perturbations to be found
are suitable ∆v that allow a smoth ”path” from the trajectory {Xαi } to the

trajectory {Y
β
i }.

Inour example,weuseequation (2.3) to thepoints Xαj and Y
β
0 .Usinga try

and error process, the parameters t and m were set to 2 and 8, respectively,
which imply the smallest value of ∆v. Thus, we have found the following
values: ∆ẋ=7.4611 ·10−3; ∆ẏ=1.0848 ·10−2; ∆ż=1.6492 ·10−2.
The final transfer trajectory that we have obtained by using our method

is shown in Fig.5. To get this result, besides the ∆v that was used to ”path”
the trajectories {Xαi } and {Y

β
i }, another ∆v was required, associated to the

”tree” of trajectories from P, in order todirect the trajectory to thepoint Xαj .
That ∆v was equal to 2.3 ·10−3.

5. Conclusions

The method introduced here uses a chaotic-based strategy to transfer a spa-
cecraft from a point located in the neighbourhood of the first primary, but
inside the chaotic region, to a halo orbit. This method requires very a small
amount of fuel, because it exploits the extremely high sensitive dependence of
the chaotic evolution to a perturbation to accomplish the goal. Although in



562 E.E.N. Macau

Fig. 5. Transfer trajectory to the halo orbit

this work we have applied the method to transfer a spacecraft to a halo orbit
around the equilibriumpoint L1, it can be applied to transfer into halo orbits
around other equilibrium points as well.

An important characteristic of this approach is that the issue if the stable
manifold of the desired halo orbit comes close to the parking orbit or not
is irrelevant. This happens because the search for the position of minimum
proximity between the trajectory that comes from the parking orbit to the
one that goes to the halo orbit is conducted inside the chaotic region. Because
of the topologically transitiviness of a chaotic evolution, these trajectories will
eventually come sufficiently close to each other to allow the determination of a
path trajectory between them by themethod explained in the text. However,
as in Hamiltonian systems the phase space is divided into layered components
which are separated from each other by Cantory (Mackay et al., 1984), we
may expect situations in which it is necessary to wait for a long time until
the trajectories come close to each other. Despite we have not found any
situation like that in the cases that we had considered, the determination of
the circumstances in which this situation might happen and how to overcome
it are issues that deserve amore thorough better investigation.

In the context of theory of chaos, chaotic targeting methods are regarded
as procedures to find the fastest trajectory that allows the transportation
between two points in the phase space. Although in this work our goal is to
finda low-thrust transfer orbit,we consider that exploitation of other concepts
from that theorymay conceive amethod to be used in Celestial Mechanics to
allows not only low-thrust, but also fast low-thrust transfer trajectories.

Acknowledgments

This work was supported in part by the Conselho Nacional de Desenvolvimento

Cient́ıfico e Tecnológico – CNPq, of the Brazilian Scientific andTechnologyMinistry.



Low-thrust chaotic based transfer... 563

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Nisko-energetyczny, chaotyczny transfer sztucznego satelity z Ziemi na

orbitę typu halo

Streszczenie

Orbity typu halowokół punktówLagrange’a są doskonałymiplatformami dla sta-
cji kosmicznychwypełniającychmisje naukowewpobliżuSłońca i planet.Przykładem
są satelity ISEE-3, GGS-WIND, SOHO i ACE. Projektowanie orbit dla takich misji
stanowi ogromne wyzwanie, zwłaszcza że one same stają się coraz bardziej wyszuka-
nymi i złożonymi zadaniami, tak jak np. w przypadku teleskopu kosmicznego następ-
nej generacji. Konieczność ich eksploracji łatwo przewidzieć w przyszłych badaniach.
Wtej pracy zajęto się zagadnieniemwynoszenia statku kosmicznego z sąsiedztwaZie-
mi na orbitę halowokół punktuLagrange’aL1 wukładzie Ziemia-Słońce, opartymna
nowej strategii zawierającej ruch chaotyczny statku. Strategia bazuje na wykładni-
czejwrażliwości trajektorii chaotycznejna zakłóceniawarunkówpoczątkowych, coma
bezpośrednieprzełożeniena sposóbprzeniesienia statkukosmicznego z orbitywyjścio-
wej na stabilną rozmaitość orbity halo. Przy odpowiednio dobranych perturbacjach,
transfer statkuwymagaminimalnego zapotrzebowanianapaliwo.Wpracy rozważono
taki transfer w odniesieniu do zagadnienia trzech ciał, co pozwoliło uzyskać bardzo
dobry wgląd do problemu projektowania zmiany orbity w realnych warunkach.

Manuscript received February 22, 2008; accepted for print March 6, 2008