Citation: Carletta, S.; Pontani, M.;
Teofilatto, P. Characterization of
Low-Energy Quasiperiodic Orbits in
the Elliptic Restricted 4-Body
Problem with Orbital Resonance.
Aerospace 2022, 9, 175. https://
doi.org/10.3390/aerospace9040175
Academic Editors: Mikhail
Ovchinnikov and Dmitry Roldugin
Received: 22 February 2022
Accepted: 15 March 2022
Published: 22 March 2022
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Article
Characterization of Low-Energy Quasiperiodic Orbits in the
Elliptic Restricted 4-Body Problem with Orbital Resonance
Stefano Carletta
1,
*
,†
, Mauro Pontani
2,†
and Paolo Teofilatto
1,†
1
School of Aerospace Engineering, Sapienza University of Rome, 00185 Roma, Italy;
paolo.teofilatto@uniroma1.it
2
Department of Astronautical, Electrical and Energy Engineering, Sapienza University of Rome,
00185 Roma, Italy; mauro.pontani@uniroma1.it
* Correspondence: stefano.carletta@uniroma1.it
† These authors contributed equally to this work.
Abstract:
In this work, we investigate the behavior of low-energy trajectories in the dynamical frame-
work of the spatial elliptic restricted 4-body problem, developed using the Hamiltonian formalism.
Introducing canonical transformations, the Hamiltonian function in the neighborhood of the collinear
libration point
L
1
(or
L
2
), can be expressed as a sum of three second order local integrals of motion,
which provide a compact topological description of low-energy transits, captures and quasiperiodic
libration point orbits, plus higher order terms that represent perturbations. The problem of small
denominators is then applied to the order three of the transformed Hamiltonian function, to identify
the effects of orbital resonance of the primaries onto quasiperiodic orbits. Stationary solutions for
these resonant terms are determined, corresponding to quasiperiodic orbits existing in the presence
of orbital resonance. The proposed model is applied to the Jupiter-Europa-Io system, determining
quasiperiodic orbits in the surrounding of Jupiter-Europa
L
1
considering the 2:1 orbital resonance
between Europa and Io.
Keywords:
ER4BP; libration point; quasiperiodic orbits; orbital resonance; Jupiter–Europa–Io; Hamil-
tonian; normal forms
1. Introduction
Low-energy trajectories have long been used in space exploration, starting from mis-
sions to the libration points of the Sun–Earth system [
1
–
5
] and including lunar transfers
passing close to the intermediate Earth–Moon libration point
L
1
[
6
–
8
], named internal trans-
fers, or extending beyond the orbit of the Moon to take advantage of the Sun gravitational
attraction [9,10], named external transfers.
The spatial Circular Restricted 3-Body Problem (CR3BP) [
11
] represents an effective
model for the preliminary analysis of both missions to the libration points [
12
–
15
] and
low-energy transfers [
16
–
18
]. As proved by Conley [
19
], in this dynamical framework the
ultimate behavior of low-energy trajectories can be determined based on their phase space
description in the neighborhood of the collinear libration points [
20
], a property which
has been extensively used in mission oriented works, since it allows a rapid design of
low-energy missions [21–25].
More accurate solutions can be generated based on more sophisticated models, such
as the Elliptic Restricted 3-Body Problem (ER3BP) [
26
], in which the primaries move
along elliptic orbits, or the Circular Restricted 4-Body Problem (CR4BP), in which the
spacecraft dynamics evolves under the gravitational attraction of four primaries (i.e., the
Sun–Earth–Moon system) [27]. Unfortunately, because no integrals of motion, such as the
Jacobi constant for the CR3BP, are known for neither the ER3BP nor the CR4BP, a general
characterization of low-energy trajectories in these dynamical frameworks has not been
derived yet.
Aerospace 2022, 9, 175. https://doi.org/10.3390/aerospace9040175 https://www.mdpi.com/journal/aerospace
