be-tween 110 and 200 days. The shortest and longest period belong to the halo orbits which are the closest ones to the L2 and the Earth, respectively. The family of orbits shown in Figure 3.6are classified as southern orbits, since the maximum out-of-plane components of all orbits are under the x-y plane. These orbits concurrent with their corresponding symmetrical northern orbits are shown in Figure 3.7. The northern orbits are computed using only the symmetric property of the three-body problem.
0.998 1 1.002 1.004 1.006 1.008 1.01 1.012 1.014
−0.015
−0.01
−0.005 0 0.005 0.01 0.015
L2
x (AU) Earth
z (AU)
Figure 3.7: A family of halo orbits aroundL2 point in the Sun-Earth system. The period of this family varies between 100.05 days till 182.17 days.
46 Chapter 3. The Three-Body Problem
as the three-body system as
˙
x=f(x). (3.24)
The vector of statesxis real and defined in an-dimensional space,x∈Rn. The reference solution ¯x is supposed to be known. Ifx = ¯x+y which y is the variations of the states, and ingnor the higher order terms, the linear differential equation governingy is
˙
y=A(t)y, (3.25)
where A(t) is an n×n matrix which generally can be time varying. However, for the special case when ¯x is an equilibrium point, it is constant. The solution of (3.25) for the constant matrixAcan be written as
y(t) =P eΛtP−1y(0), (3.26)
where the matrixP transformsAto its Jordan normal form, and the matrix Λ is a block diagonal matrix which has the same eigenvalues as A. Without loss of generality, suppose that the eigenvalues ofA are distinct real scalarsλi fori= 1, . . . , n. Thus,
y(t) =
n
X
i=1
cieλitvi, (3.27)
wherevi is thei-th eigenvector ofAcorresponding to the engenvector λi, and the scalars ci are the integration constants which can be determined from the initial conditions. The solution of (3.25) in the form (3.27) makes it easy to address the concept of stability. To this end, suppose that λ1, . . . , λs are s eigenvalues which their real parts are negative, λs+1, . . . , λs+u are u eigenvalues with positive real parts, andλs+u+1, . . . , λs+u+c are the pure imaginary eigenvalues. Then the stable, unstable and center subspaces of Rn are [78] defined as
Es=span{v1, . . . , vs}, Eu =span{vs+1, . . . , vs+u+1},
Ec =span{vs+u+1, . . . , vs+u+c}.
(3.28)
If the initial point lies on each of these stable, unstabl or center subspaces, then the solution will stay in the same subspace for all time. This demonstrates the concept of invariance. Indeed, the subspace E ⊆ Rn is invariant, if for any set of initial conditions inE the solution belongs toE for all time. The mentioned subspaces are related to the
invariant manifolds which is explained in the following theorem [37].
Theorem 3.2.1. Suppose that x¯ is a hyperbolic equilibrium point of the system (3.24).
Then there exists local stable and unstabole manifolds Wlocs (¯x) and Wlocu (¯x) of the same dimensions as stable and unstable subspaces at x. Furthermore, The stable and unstable¯ manifolds are tangent to the stable and unstable subspaces, respectively.
The manifolds are also invariant. The manifolds referred in this dissertation are global manifolds which can be obtain by numerical integration through the vector field over a larger time interval. Figure3.8shows the first equilibrium point of the three-body system and its stable and unstable subspaces and manifolds. Indeed, the manifols shown in this figure as Ws and Wu are global manifolds. Note that the manifolds are tangent to the subspaces and beyond the vicinity of equilibrium point, they branch out into subspaces and manifolds. The instability of the collinear libration points of the three-body problem
0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89
−0.08
−0.06
−0.04
−0.02 0 0.02 0.04 0.06 0.08 0.1
L1
x (AU)
y (AU)
Eu Eu
Es
Es
Ws Wu
Wu Ws
To Moon
Figure 3.8: The stable and unstable subspaces and manifolds corresponding to the equilibrium point L1 in the Earth-Moon three-body system
is a strong motivation to look for another types of solutions which are stable. Among the solutions of the three-body problem, periodic orbits are frequently used as a reference solution for investigating the nonlinear system. Consideration of a periodic orbit as a reference solution for the linearization makes the matrix A in (3.25) to be time-varying.
48 Chapter 3. The Three-Body Problem
To use the priodicity of a periodic orbit in stability analysis, most of the works use the monodromy matrix. The monodromy matrix is the same state transition matrix associated with the periodic orbit after one period of the motion.The eigenvalues of the monodromy matrix can be used to estimate the local geometry in the vicinity of the periodic orbit. Due to the Floquet theory [95], the state transition matrix Φ(t, t0) can be rewritten as
Φ(t, t0) =F(t)eJtF−1(t0), (3.29) whereJ is a block diagonal matrix which its diagonal elements are the Floquet multipliers.
SinceF(t) is a periodic matrix, i.e. F(T) =F(t0), therefore
Φ(T, t0) =F(t0)eJTF−1(t0), (3.30) thus
eJT =F−1(t0)Φ(T, t0)F(t0), (3.31) whereT is the period of the orbit. The eigenvectors of the monodromy matrix are equal for all the fixed points on the periodic orbit, but the associated eigenvectors vary in direction at each fixed point [95]. From (3.31), we have
λi=e̟iT, (3.32)
̟i= 1
T ln(λi), (3.33)
where̟iare the Poincar´e exponents. Due to the Poincar´e exponents’ stability properties, we have
• |λi|<1 results in stability.
• |λi|>1 results in instability.
• |λi|= 1 gives no information about the stability.
In [96], it is explained that the monodromy matrix derived from a periodic orbit has at least one eigenvalue with a modulus of one. On the other hand, based on the next theorem, one can imply existance of two eigenvalues with modulus of one.
Theorem 3.2.2. If λ is an eigenvalue of the monodromy matrix of a time invariant system, thenλ−1 is also an eigenvalue.
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.