International Journal of Aerospace Engineering

Volume 2018, Article ID 6890173, 12 pages

https://doi.org/10.1155/2018/6890173

## Preliminary Capture Trajectory Design for Europa Tomography Probe

Department of Mechanical and Aerospace Engineering, Sapienza-University of Rome, Via Eudossiana 18, Rome, Italy

Correspondence should be addressed to Alessandro Zavoli; ti.1amorinu@ilovaz.ordnassela

Received 31 March 2018; Accepted 9 July 2018; Published 9 August 2018

Academic Editor: Linda L. Vahala

Copyright © 2018 Lorenzo Federici et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The objective of this work is the preliminary design of a low- transfer from an initial elliptical orbit around Jupiter into a final circular orbit around the moon Europa. This type of trajectory represents an excellent opportunity for a low-cost mission to Europa, accomplished through a small orbiter, as in the proposed Europa Tomography Probe mission, a European contribution to NASA’s Europa Multiple-Flyby Mission (or Europa Clipper). The mission strategy is based on the leveraging concept, and the use of resonant orbits to exploit multiple gravity-assist from the moon. Possible sequences of resonant orbits are selected with the help of the Tisserand graph. Suitable trajectories are provided by an optimization code based on the parallel running of several differential evolution algorithms. Different solutions are finally compared in terms of propellant consumption and flight time.

#### 1. Introduction

The Jovian moon Europa is a celestial body of primary interest for astrophysicists. The likely existence of a global subsurface ocean, proved by measurements carried out during Galileo mission, makes Europa one of the most promising environments in the Solar System to sustain human habitability. The presence of an ocean may also imply that Europa hosts (or, at least, hosted) life [1]. The importance of the determination of the ice-water layer characteristics is clearly stated in NASA’s 2013–2022 Decadal Survey [2].

Europa Clipper is the next mission planned by NASA with the aim of exploring Europa. Because of the extremely harsh Jovian environment in the proximity of Europa, the initial concept of an orbiter was abandoned in favour of a multi-flyby strategy, the same considered for Galileo mission. The present mission profile, with more than 40 flybys of Europa, allows for a paramount investigation of Europa surface and subsurface properties, but is not very favourable to the investigation of Europa’s deep interior structure.

A scientific enhancement to Europa Clipper mission was investigated in [3]. There, a small probe deployed on a polar orbit around Europa, hosting just one scientific instrument (a magnetometer) and a transponder required for the Intersatellite Link (ISL) with the mother spacecraft, is proved to be capable of providing crucial information on the interior structure of the moon, such as depth, thickness, and conductivity of the subsurface ocean. Also, ISL could support the reconstruction of the mother spacecraft orbit, hence significantly improving the accuracy of the topographic reconstruction of Europa’s surface.

Standing on these arguments, a scientific and engineering team at Sapienza University of Rome, in collaboration with the Imperial College of London, carried out a feasibility study for a probe that could be hosted by the main spacecraft during the interplanetary cruise and released in the Jovian system with the aim at entering into a low-altitude circular quasi-polar orbit around Europa [4]. The result is a small spacecraft named Europa Tomography Probe (or ETP), which could fit the provisional 250 kg allowance that NASA has assigned to a secondary flight element hosted by the main spacecraft.

The feasibility study was carried out under the design philosophy of determining the minimum total mass and volume that allows for the scientific measurements considered in [3]. All subsystems have been described with some details, with the relevant exception of only two elements: (a) the transponder and (b) the trajectory which moves ETP into a polar orbit around Europa, which should interfere as less as possible with the mother spacecraft mission plan. This paper investigates the latter point, that is, the capture strategy for an Europa orbiter at the level of preliminary mission analysis.

The problem of optimizing the capture trajectory of an orbiter (or a lander) directed towards a moon of an outer planet, such as Europa [5], Enceladus [6], or Titan [7], has been the subject of many investigations. A two-body patched-conic approximation is usually assumed for interplanetary missions and transfers in a multibody planetary system [8]. This dynamical model retains the most prominent features of the real system, while keeping the numerical difficulties low. Three-dimensionality and eccentricity of the planetary bodies can be easily taken into account, and several flybys of different moons can be dealt with.

The same kind of missions have been also studied by using dynamical system techniques, which rely on circular restricted three-body problem (CRTBP). Low-energy trajectories are searched for, attempting the construction of “transfer tubes,” whose boundaries are typically given by invariant manifolds originating from invariant sets (such as and Lyapunov orbits). As an example, dynamical chains formed by linking heteroclinic connections and homoclinic orbits [9] are proposed for the analysis of fast resonance transitions between exterior and interior resonant orbits (in the Sun-Jupiter system) [10] or “loose” capture trajectories [11]. Similar concepts are exploited for Halo-to-Halo [12] or libration-to-libration [13] transfers between planetary moons in the Jovian system, adopting a “patched” CRTBP model.

In the present problem, the probe comes from a high-energy condition and approaches the moon with a high hyperbolic excess velocity. The latter techniques are thus not efficient for attaining a solution, while a patched-conic approximation can be profitably adopted.

Delta velocity gravity assist (V-GA) or leveraging [14] has proved a powerful concept to improve the design of capture (or escape) trajectories. Large changes of the hyperbolic excess velocity at the encounter () are obtained by using small deep-space maneuvers. When this strategy is used in conjunction with a series of resonant gravity assists, a significant reduction of propellant requirement can be achieved, with respect to a direct insertion maneuver [15].

In the present paper, the design of ETP capture trajectory using leveraging is pursued by blending the patched-conic model and a modern global optimization procedure based on a differential evolution algorithm. The trajectory is modeled as a sequence of legs between two moon encounters; only one deep-space maneuver is permitted in each leg. This approach, proposed in [16] and hereafter referred as “MGA-1DSM,” permits a quite general parameterization of the whole trajectory, which is not limited to V-GA maneuvers. A preliminary solution (the sequence of resonant orbits and intercepted bodies) is defined by using two simple tools: the suboptimal solution of the leveraging problem proposed by Sims and Longuski [17] and the Tisserand graph [18]. The former permits an easy design of a mission based only on leveraging maneuvers, by suggesting a viable sequence of resonant orbits. The latter is a powerful graphical aid for the design of the same class of missions, when multiple bodies are intercepted, and some deep-space maneuvers are conveniently replaced by gravity assists of other moons in the planetary system.

The paper organization is here outlined. In Section 2, the physical problem of interest is described, and the adopted dynamical model and relevant assumptions are stated. A mathematical formulation (MGA-1DSM), which parameterizes a generic interplanetary trajectory as a sequence of legs containing a gravity assist and one deep-space maneuver, is outlined, leading to the purposeful definition of a global optimization problem. Section 3 presents the multipopulation differential evolution algorithm that has been used to solve the optimization problem. Fundamental tools for preliminary mission design, that is, leveraging and Tisserand graph, are discussed in Section 4. A tentative solution is devised and used to prune global optimization search. Numerical results of this investigation are presented in Section 5. A conclusion section ends the paper.

#### 2. Problem Statement and Mathematical Modeling

##### 2.1. Problem Overview

According to the ongoing proposal [4], the probe is assumed to be released by the main spacecraft after a few Europa flybys have been completed. In particular, ETP starts its own transfer at the apocenter of a Jovian orbit of period four times the period of Europa (), pericenter equal to the Europa semi-major axis, and coplanar with the Europa orbit; the orientation of the major axis is left free. A target circular quasi-polar orbit around Europa is desired, of assigned altitude km over Europa’s surface. Four R-6D bipropellant thrusters form the primary propulsion system of the probe, which allows for a total thrust of 88 N with specific impulse s. This propulsion system will be used for deep-space maneuvers, Europa Orbit Insertion (EOI) maneuver, and orbit maintenance during the scientific part of the mission.

A probe “net” mass kg, which does not account for the propellant and tank masses, was estimated in [4]. Assuming a structural coefficient , which is a reasonable value for liquid propellant systems, a maximum value of velocity increment m/s can be obtained if the spacecraft wet mass is constrained at 250 kg. This value of must cover the orbit maintenance (about 43.2 m/s for a 6-month mission) and capture cost. The goal is to reduce the required for the capture, so that a convenient safety margin is left.

##### 2.2. Dynamical Model

A patched-conic model is assumed for the present analysis. Flybys are modeled as instantaneous changes in velocity. Subscripts “−” and “+” are used to distinguish between values immediately before or after the discontinuity, respectively. The radius of the sphere of influence (SOI) of the secondary bodies and the travel time inside these regions are assumed to be negligible. Powered flybys are neglected, as considered useless to reduce propellant consumption [19]. An impulsive-thrust model is adopted. This assumption well suits deep space maneuvers (DSMs), which are performed at a large distance from the main body and require usually a quite short time if compared to the orbital period, as chemical engines are here considered. This assumption is also used for the EOI maneuver, even though finite-thrust losses might be considered. Only one DSM is permitted between a flyby and the other. As a further assumption, Jovian moons move on Keplerian orbits (even though the proposed procedure is soon applicable to the general case that uses planetary ephemeris).

Despite its simplicity, this model allows to capture the most prominent features of the mission, while keeping the analysis simple enough. In fact, under the hypothesis of impulsive thrust, the trajectory can be computed analytically, without involving the numerical integration of the complete equations of motion.

##### 2.3. Trajectory Parameterization

Let us assume that a sequence of body encounters, where identifies the encountered body (1 = Io, 2 = Eu, 3 = Ga, and 4 = Ca), has been established. Europa is the first and last body in the series. The spacecraft trajectory can be modeled according to the multiple gravity assist-one deep space maneuver (MGA-1DSM) formulation [16]. The trajectory is broken down into a series of body-to-body legs. Each leg starts with a flyby and is made up of two ballistic arcs, joined by an impulsive maneuver.

This general formulation for a multigravity assist trajectory can be adapted to the problem at hand by adding an initial leg, which moves the probe from the assigned initial conditions to the first encounter with Europa. The mission ends with a last approach to Europa’s surface, where an impulsive maneuver inserts the probe into the assigned polar orbit.

###### 2.3.1. Departure Leg

The departure leg, which connects ETP release position to the first encounter with Europa, is modeled as a Lambert arc, where release epoch , flight angle , and flight time are design parameters to optimize.

Let be a Jovicentric radial-traversal-normal reference frame connected to a Europa position at epoch , that is,
where *r* and *v* indicates position and velocity vectors of Europa () at time , respectively, which are provided by the ephemeris.

The probe departure point is located on a circle of radius which lies on the plane ; hence, it can be expressed as while the velocity immediately before the release maneuver is where the values and are, respectively, the radius and velocity magnitude at the apocenter of a Jovian orbit with pericenter equal to the Europa semi-major axis, and period . In this respect, the problem solution will eventually define the optimal orientation of the line of apses of the initial orbit with respect to Europa’s orbit.

The velocity vectors after the release maneuver *v* and immediately before the first flyby *v* can now be evaluated by solving the associated Lambert problem:

The propulsive cost of the release maneuver is evaluated as

###### 2.3.2. Intermediate Legs

The leg can be parametrized by using four parameters , , , and , which represent, respectively, the flyby radius, the flyby plane orientation, the (overall) leg flight time, and the fraction of the leg flight time at which DSM occurs. The trajectory associated to a generic intermediate leg is presented in Figure 1.