2.1. The Circular Restricted 3-Body Problem
The CR3BP represents the most common dynamic framework to study the motion of a spacecraft in the Earth–Moon system. This model is based on the hypothesis that the mass
m of the spacecraft is negligible compared to that of the primaries, the Earth (
) and the Moon (
), and that in the inertial space, the relative trajectory of
about
is circular [
36].
The dynamic equations of motion for the CR3BP are expressed in a reference frame
, centered at the center of mass of the system
O, with
pointing from
to
,
orthogonal the orbital plane of the primaries and
completing the rectangular reference frame [
37]
with
where
indicates the distance between the spacecraft and the
i-th primary. It is worth recalling that Equation (
1) are expressed in terms of units of distance
and time
, where
and
are, respectively, the mean distance and the orbital angular speed of the Moon with respect to the Earth.
The CR3BP admits five equilibrium points
, also named the Lagrange or libration points [
38], and one constant integral of motion, named the Jacobi constant
In fact, the Jacobi constant is proportional to the opposite of the total energy of the spacecraft; therefore, C decreases as increases.
Trajectories that can evolve in the neighborhood of both the primaries are characterized by
, where
is the Jacobi constant calculated in
, the libration point lying on
between
and
[
13,
39]. The deployment strategy proposed in this work is based on the use of those low-energy trajectories, which verify the condition
, where
is an arbitrarily small constant named the energy level. The characterization of these solutions is discussed in the following section.
2.2. Characterization of Low-Energy Trajectories Crossing the Equilibrium Region
It is known that the ultimate behavior of low-energy trajectories characterized by a low energy level
h can be predicted based on their state representation inside the equilibrium region, i.e., the phase space surrounding the libration point
[
34,
40]. The rigorous definition of the equilibrium region and the representation are developed hereafter using the Hamiltonian formalism and the results are later applied to design trajectories that, after crossing the equilibrium region, reach the Moon with desired semimajor axis, eccentricity, inclination, and RAAN.
The Hamilton’s equations are given by
where
and
are the position and conjugate momenta. The Hamiltonian function for the CR3BP is by definition
. Aiming at investigating the dynamics in the neighborhood of
, the following expression for
H can be conveniently derived from quadratic expansion of Equation (
4) about
[
35]
with
.
The linear system associated with Equations (
5) and (
6) is characterized by a saddle–center–center type of equilibrium, having one couple of real eigenvalues (
) and two couples of complex conjugate ones (
,
). Therefore, according to Morse’s lemma, the Hamiltonian function can be represented as the sum of three local integrals of motion, each dependent on a different pair of state variables and associated with one of the three above-mentioned eigenspaces. This new form of
H can be conveniently derived by applying a canonical transformation
, originally proposed by Siegel and Moser [
41], producing
with
,
and
is the arbitrarily small energy level. Equation (
7) is the mathematical representation of the equilibrium region, the phase space in the neighborhood of
.
Based on a theorem by Moser [
42], Conley proved that if a low-energy trajectory is inside the equilibrium region at a given time
, then its long-term behavior is fully characterized by its topological location in the equilibrium region and provided the following classification [
34]
, transit trajectories which evolve alternately around one of the two primaries, crossing the equilibrium region multiple times
, bouncing trajectories which never cross the equilibrium region, thus evolve only around one of the two primaries
or , capture trajectories that cross only once the equilibrium region and then evolve around one of the primaries indefinitely in time
, quasi-periodic orbits which never depart from the equilibrium region.
Within the scope of this research, trajectories transiting from the Earth to the Moon (i.e., transits or captures) are of interest. In particular, low-energy captures are attractive because they do not require any further maneuvering to keep the satellites in the neighborhood of the Moon, thus ensuring relaxed operation times to perform any eventual corrective maneuver.
For this class of trajectories, the osculating orbital elements at capture can be characterized, as for the long-term behavior, by their state representation at time
when the trajectory is inside the equilibrium region [
40]. Equation (
7) can be rearranged as follows [
43]
with
arbitrarily small. The two constant terms of Equation (
8) are named energy fractions and indicated as
and
. The value of the energy fractions can be related to those of some osculating orbital elements. For the sake of clarity, the following expressions for the properties of the canonical transformation
shall be recalled:
From the conservation of the angular momentum, the following expression for the inclination
i can be derived [
44]
Therefore, a target inclination at capture can be fixed by selecting any couple (
) verifying Equation (
10). By virtue of Equation (
9), Equation (
10) represents a constraint for the six state variables in the position space. Moreover, if
h is not fixed (i.e., it can be varied within an admissible range) for a given set of in-plane variables
, the target inclination can be fixed by selecting only the out-of-plane variables
.
Another relationship between the energy fractions can be derived based on Tisserand’s parameter
where
a and
e are the semimajor axes and the eccentricity of the capture orbit. Introducing Equation (
10) into (
11) leads to
Equations (
10) and (
12), which were verified using numerical analysis in a previous study by the author [
44], indicate that fixing the value of the inclination and a combination of
a and
e (or equivalently, the desired pericenter distance
at capture) the energy fractions
and
-and therefore the values of
and
-are determined.
In
Section 3, the above-mentioned constraints are rearranged in terms of position and velocity in
and implemented to design the deployment strategy.
2.3. Extension to the Sun–Earth–Moon System
The design of low-energy captures by the Moon, as well as the deployment strategy derived from it, cannot ignore two conditions that are overlooked in the CR3BP model: neither the eccentricity of the Earth and Moon orbits nor the gravitational field of the Sun are negligible in the real environment [
45,
46].
The topological characterization presented in
Section 2.2 is here extended to the more accurate dynamic framework of the elliptic restricted four-body problem (ER4BP) that is used to model the Sun–Earth–Moon system. The dynamic equations of motion are developed in a reference frame
centered in the center of mass of the system, with
pointing from the Sun (
) to the center of mass of the Earth–Moon system,
orthogonal to the ecliptic plane and
completing the rectangular frame [
47]. Denoting by
and
the distance and the true anomaly of the Earth–Moon center of mass
O with respect to the Sun, the equations can be expressed as follows [
48]
where
,
,
is the mass of the Sun,
, and the prime
indicates the derivative with respect to
. Equation (
13) are expressed in terms of the units of distance
and time
where
G is the gravitational constant.
Because of its dependence on time, the ER4BP does not admit equilibrium points. Nevertheless, their instantaneous dynamic substitutes can be computed at any given time considering the corresponding geometric configuration of the primaries [
48]. Operating as in
Section 2.2, the set of Equation (
13) can be linearized about the instantaneous libration point
of the Earth–Moon system and expressed using the Hamiltonian formalism using the following Hamiltonian function [
49]
where
denotes the mean distance between the Sun and the center of mass of the Earth–Moon system and the expressions for the coefficients
are reported in
Appendix A. It can be observed that the expressions of
H and
are equivalent, except for the last two terms of Equation (
14), which represent the effects of the eccentric motion of the primaries (
) and the solar gravitational perturbation (
). Examining in detail the two coefficients, it is easy to verify that for the Earth–Moon system
and in the neighborhood of
also
, so they can be regarded as small perturbations acting onto the CR3BP.
Conley and Easton proved that the basic topological properties of the CR3BP in the neighborhood of
are preserved in the presence of small perturbations [
50]. In fact, a canonical transformation
, developed in
Appendix A, can be introduced to rearrange Equation (
14) to a form equivalent to
H plus negligible higher order terms in the perturbations [
49]
Because
is equivalent to
, all the results derived from the application of the canonical transformation
, and in particular the characterization of low-energy captures, are still valid [
51]. Capture orbits with the desired orbital elements can be designed in the
coordinates and then converted to position and velocity coordinates in
, applying in order the inverse of the two canonical transformations
and
.