Since the monodromy matrix of a periodic orbit as a reference solution of the three-body problem is a 6×6 matrix, and according to the Theorem3.2.2, the eigenvalues are related in the following way [17]
λ1 = 1 λ2
, λ3= 1 λ4
, λ5 =λ6. (3.34)
Therefore, there are exactly two eigenvalues with modulus less than one. Furthermore, there are also exatly two eigenvalues with modulus greater than one, and two eigenvalues with modulus one. Bray and Goudas in [15] presented a fast and simple way to compute the eigenvalues of the monodromy matrix for a periodic orbit in the three-body problem.
Parker also summerized this way as following. Suppose
α= 2−trace(Φ(T,0)), (3.35)
β = α2−trace(Φ2(T,0))
2 + 1, (3.36)
p= α+p
α2−4β+ 8
2 , (3.37)
q = α−p
α2−4β+ 8
2 , (3.38)
then, we have
λ1 = 1 λ2
= −p+p p2−4
2 , (3.39)
λ3 = 1
λ4 = −q+p q2−4
2 , (3.40)
λ5 =λ6 = 1. (3.41)
These eigenvalues can be used to calculate the corresponding eigenvectors. In the next section, the stable and unstable eigenvalues will be used to construct numerically the invariant stable and unstable manifolds.
50 Chapter 3. The Three-Body Problem
periodic orbit’s picture of the stablity and instabilty. To this end, suppose that the stable and unstable eigenvectors of the monodromy matrix arevsandvu, respectively. Then one selects a fixed point xi on the periodic orbit corresponding to the time ti. Suppose that the corresponding stable and unstable vectors at the pointxi arevisand vui, respectively, which are indeed
vis= Φ(ti, t0)vs, (3.42)
vui = Φ(ti, t0)vu. (3.43)
To compute the stable manifold, the considered fixed point on the halo orbit should be perturbed by a small perturbationǫin the direction of the stable vector, i.e.
xWi s =xi+ǫ vsi
kvsik. (3.44)
The corresponding state for unstable manifold as (3.44) is xWi u =xi+ǫ vui
kvuik. (3.45)
Note that the normalization of the stable and unstable vectors is necessary, since the states resulted in Equation 3.45 grows exponentially. To compute the global manifold, the perturbed states should be integrated along the system. In both Equations3.44 and 3.45, there is an initial displacement from the periodic orbit denoted by ǫ. The small perturbation cuases a better approximation of the manifold, but it tends to diverge from the orbit much slower than a larger perturbation. On the other hand, this perturbation cannot be chosen arbitrarily large, since the linear approximation must remain whitin a valid range. In the Sun-Earth system,ǫis usually varies between 100 and 200 kilometers.
But in this dissertation, whole manifolds are computed withǫ= 7.0E−8 which is almost 10 kilometers. If more than one fixed point xi be considered on the halo orbit, then the manifolds form a tube made of trajectories corresponding to each point along the halo orbit. Figure 3.9shows the instant points on a halo orbit which are indeed twenty points considered equidistantly along the halo orbit. The trajectories of stable manifolds corresponding to each of these points shape a tube which can be seen in three-dimensional.
Furtheremore, two-dimensional views of the stable tube projected onx-y and x-zplanes are also shown in this figure. To see the rest of the stable manifold shown in Figure 3.9, Figure 3.10shows the enterior and exterior part of the stable manifold associated to the same halo orbit aroundL2 of the Sun-Earth system in three different views. Due to the concept of assymptotic stability of the stable manifolds, a spacecraft following one of
1.0071.0075 1.008 1.0085 1.009 1.0095 1.01 1.0105 1.011
−5 0 5
x 10−3
−5
−4
−3
−2
−1 0 1 2 3 4
x 10−3
L2
x (AU) Halo Orbit
y (AU)
z (AU)
1 1.002 1.004 1.006 1.008 1.01
−6
−4
−2 0 2 4 6
x 10−3
L2
Halo Orbit
x (AU) Stable Manifolds Earth
y (AU)
1 1.002 1.004 1.006 1.008 1.01 −5
0 5
x 10−3
−5
−4
−3
−2
−1 0 1 2 3 4
x 10−3
y (AU) L2 Halo Orbit
x (AU) Stable Manifolds Earth
z (AU)
1 1.002 1.004 1.006 1.008 1.01
−5
−4
−3
−2
−1 0 1 2 3 4x 10−3
L2
Halo Orbit
x (AU) Stable Manifolds Earth
z (AU)
Figure 3.9: Different plots of stable manifolds associated with a halo orbit around L2 in the Sun-Earth system. This tube of stable manifolds is constructed using 20 fixed points which are shown along the halo orbit (up left).
stable trajectories will asymptotically reach the halo orbit. That means the spacecraft’s arrival at the halo orbit will happen in infinity. This fact does not make any problem for mission design, since it can be solved by an extra Halo-Insert Maneuver (HIM). To see the behaviour of the stable trajectories approaching the halo orbit, the propagated stable manifolds in different times are sketched in Figure3.11. The time intervals for propagation of the stable manifols are 179.23, 217.07, 250.92, 286.76, 322.61, 358.45, 393.77 and 429.57 days. In this figure, one can see that the particle spends a long period of time orbiting quite close to the halo orbit before departing from it. In the second figure, after 217.07 days, the spacecraft is still orbiting the halo orbit. In the last figures embedded in Figure 3.11, the stable manifolds pass the way toward the Earth and cross the vicinity of the Earth. The minimum distance of these close approaches of each trajectory to the Earth are calculated and Figures 3.12and 3.13summerize the results. In these figures, six halo orbits of a family around theL2libration point in the Sun-Earth system are considered on which fifty instant points as initial conditions to construct the stable trajectories. Each color is relevent to the information of a specific halo orbit. The minimum distance of the first perigee of stable manifolds among these six halo orbits is 2167.26 km from the Earth’s surface. This minimum approach takes place in the first perigee (197.38 days) of
52 Chapter 3. The Three-Body Problem
−1
−0.5 0
0.5
1 −1
−0.5 0
0.5 1
−0.01
−0.005 0 0.005 0.01
y (AU) Stable Manifold Sun
x (AU) Solar Orbit
z (AU)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Earth
Solar Orbit
x (AU) Sun
Stable Manifold
y (AU)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−0.01
−0.008
−0.006
−0.004
−0.002 0 0.002 0.004 0.006 0.008 0.01
Solar Orbit Sun
x (AU) Stable Manifold
z (AU)
Figure 3.10: Different plots of a set of interior and exterior stable manifolds associated with a halo orbit aroundL2 in the Sun-Earth system.
the stable trajectory departing from the fifth instant point on the halo orbit with Jacobi constant 3.0007 and 179 days period.
0.998 1 1.002 1.004 1.006 1.008 1.01 1.012 1.014
−6
−4
−2 0 2 4 6x 10−3
Halo Orbit
L2
x (AU) Stable Manifold Earth
y (AU)
0.998 1 1.002 1.004 1.006 1.008 1.01 1.012 1.014
−6
−4
−2 0 2 4 6x 10−3
Halo Orbit
L2
x (AU) Stable Manifold Earth
y (AU)
1 1.002 1.004 1.006 1.008 1.01 1.012
−6
−4
−2 0 2 4 6x 10−3
Halo Orbit
L2
x (AU) Stable Manifold Earth
y (AU)
1 1.002 1.004 1.006 1.008 1.01 1.012
−6
−4
−2 0 2 4
x 10−3
Halo Orbit
L2
x (AU) Stable Manifold Earth
y (AU)
0.995 1 1.005 1.01
−6
−4
−2 0 2 4 6
x 10−3
Halo Orbit
L2
x (AU) Earth
Stable Manifold
y (AU)
0.99 0.995 1 1.005 1.01
−6
−4
−2 0 2 4 6
x 10−3
Halo Orbit
L2
x (AU)
Earth
Stable Manifold
y (AU)
0.98 0.985 0.99 0.995 1 1.005 1.01 1.015
−8
−6
−4
−2 0 2 4 6 8 10 12
x 10−3
Halo Orbit L2 Earth
x (AU) Stable Manifold
y (AU)
0.99 0.995 1 1.005 1.01 1.015
−8
−6
−4
−2 0 2 4 6 8 10x 10−3
Halo Orbit
L2
x (AU) Earth
Stable Manifold
y (AU)
Figure 3.11: The propagation of the stable manifold associated to a halo orbit around L2 in the Sun-Earth system. The time durations of propagation are t = 179.23, 217.07, 250.92, 286.76, 322.61, 358.45, 393.77, and 429.57 days.
54 Chapter 3. The Three-Body Problem
0 20 40 60 80 100 120 140 150
1 2 3 4 5 6 7 8 9 10x 105
Point on the halo orbit
Distance to the Earth’s surface (km)
Figure 3.12: Distance analysis of 50 trajectories making stable manifold tubes for thirteen halo orbits aroundL2 in the Sun-Earth system. The y-axis is scaled linearly.
0 20 40 60 80 100 120 140 150
103 104 105 106
Point on the halo orbit
Distance to the Earth’s surface (km)
Figure 3.13: Distance analysis of 50 trajectories making stable manifold tube for thirteen halo orbits aroundL2 in the Sun-Earth system. The y-axis is scaled logarithmically.
Optimization
As mentioned before, the approach of this dissertation uses the optimization theory to model and solve the transfer problem. Therefore, the concept of optimization is important in understanding the method used here, and shall be explained in more details. An optimization problem generally deals with a set of independent variables or parameters which can vary to achieve the optimal value of an objective function. These parameters may be restricted by some conditions which are calledconstraints. Optimization problems can be categorized into two different categories;Static OptimizationandOptimal Control.
Static optimization problems are an optimization problem which time is not a parameter in it. Lack of continuous parameters in this kind of optimization problems is the reason that it is also called Discrete Optimization. On the other hand, the optimal control deals with the optimization problems which usually have time as independent variable and elements vary continuously with respect to the time. Hence, it is sometimes referred as Continuous Optimization.