Leveraging the Interplanetary Superhighway for Propellant–Optimal Orbit Insertion into Saturn–Titan System

Abstract

This paper presents an innovative approach using Dynamical Systems Theory (DST) for interplanetary orbit insertion into Saturn−Titan three−body orbits. By leveraging DST, this study identifies invariant manifolds guiding a spacecraft into Titan−centered Distant Retrograde Orbits (DROs), strategically selected for their scientific significance. Subsequently, Particle Swarm Optimization (PSO) is employed to fine−tune the insertion parameters, thereby minimizing ΔV. The results demonstrate that the proposed method allows for a reduction in ΔV of over 70% compared to conventional approaches like patched conics−based flybys (2.68 km/s vs. 9.23 km/s), albeit with an extended time of flight, which remains notably faster than weak stability boundary transfers. This paper serves as an interplanetary mission planning methodology to optimize spacecraft trajectories for the exploration of the Saturn−Titan system.

1. Introduction

Titan is Saturn’s largest moon and the second−largest moon in the Solar System. Following the Cassini−Huygens mission, which acquired over 600 GB of data [1], scientists and researchers have shown particular interest in this moon due to its unique characteristics. Titan, in fact, is the only moon known to have a dense atmosphere extending over 900 km, primarily composed of nitrogen, methane, and hydrocarbons. Additionally, it hosts various hydrocarbon lakes [2]. These features make the moon a candidate for prebiotic chemistry and potentially sustaining environments. Consequently, Titan is a primary target for astrobiology and origin−of−life research due to its complex carbon−rich chemistry [3].

The Dragonfly mission, scheduled to launch in 2027, is designed to explore the equatorial region of Titan, enabling the study of its surface composition. The mission will utilize a rotorcraft, specifically designed for multiple−site exploration [4]. However, the localized nature of this mission imposes certain limitations. There are no plans for a comprehensive study of Titan’s surface, thus restricting the examination of significant formations such as those found at the poles or high latitudes. Despite this constraint, the mission represents a significant advancement in Titan exploration, providing a more in−depth understanding of this moon.

To broaden the scope of exploration to a global scale, an orbiter mission around Titan would be necessary. Such a mission could acquire extensive data by studying regions far from Dragonfly’s landing site. For example, it could conduct more comprehensive studies of hydrocarbon lakes or cryovolcanoes erupting icy materials instead of lava [5], thereby enhancing our comprehension of ongoing geological processes and Titan’s evolutionary history.

The goal of this paper is to analyze the interplanetary orbit insertion maneuvers that would enable reaching a candidate three−body orbit for further and more comprehensive exploration and study of Titan while minimizing the required $\Delta V$. Specifically, the focus initially is on analyzing various families of three−body orbits, assessing their propellant consumption, orbital stability, and scientific advantages.

In this context, we consider the gravitational interactions among Titan, Saturn, and the spacecraft, which create a complex dynamical environment. These three−body interactions significantly influence the orbital behavior and stability of potential trajectories. We used Dynamical System Theory (DST) to develop accurate and precise mathematical models of these gravitational interactions, allowing us to account for the combined effects of all three bodies in our analysis.

In the next section, we will introduce Titan, Saturn’s largest moon and the primary target of our mission. We will explore the major missions that have focused on Titan and discuss why it is a focal point of this research.

Following the introduction of Titan, we will delve into a section dedicated to background theory. This will allow us to address the Circular Restricted Three−Body Problem (CR3BP), the principles of Dynamical Systems Theory (DST), and the Particle Swarm Optimization (PSO) algorithm, including its application in analyzing interplanetary trajectories.

The Section 4 will focus on the various types of three−body orbits. Here, we will assess the stability of these orbits and their capacity for monitoring Titan in alignment with our mission objectives.

Once we have defined the family of three−body orbits, we will turn our attention to optimizing the orbital insertion trajectory. This will involve leveraging invariant manifolds derived from DST alongside the PSO method. This approach will enable us to evaluate the advantages of using such optimization techniques for trajectory planning.

2. Titan

Titan, characterized by a dense atmosphere and evidence of liquids on its surface, is the only celestial body besides Earth to exhibit such features. Its atmosphere is primarily composed of nitrogen, with traces of hydrocarbons such as methane and ethane, which are crucial for atmospheric processes and cloud formation. Additionally, the orange hue of Titan’s surface is attributed to the presence of complex compounds [6]. Titan orbits Saturn every 15 days and 22 h, and like many moons, including Earth’s Moon, it is tidally locked in synchronous rotation with its planet, always presenting the same face to Saturn.

The Cassini mission confirmed the existence of hydrocarbon lakes on Titan [7], with impact craters filled by methane rain. Despite having lakes covering only a small portion of its surface, Titan is drier than Earth. Radar observations by Cassini revealed long−lasting hydrocarbon lakes in equatorial desert areas, including one near the Huygens landing site in the Shangri−La region [8]. Most lakes are situated around the poles, where sunlight scarcity inhibits evaporation.

Launched in 1973, Pioneer 11 provided the first close−up images of Titan in 1979, revealing its opaque and hazy surface [9]. Voyager 1, in 1980, conducted a flyby suggesting the presence of liquid on Titan [10]. In 1997, the Cassini−Huygens mission, a collaboration between NASA, ESA, and ASI, was launched and arrived at Saturn in 2004. Cassini captured high−resolution images of Titan’s surface, discovering liquid in the north polar regions. The Huygens probe, part of the mission, landed on Titan in 2005, providing valuable data on its atmosphere and surface composition. Subsequent Cassini flybys confirmed the existence of permanent liquid hydrocarbon lakes in Titan’s polar regions [11].

NASA’s proposed Dragonfly mission, announced in April 2017, is set to send a unique robotic rotorcraft to Titan. The mission’s goal is to explore primordial chemistry and extraterrestrial habitability, with a landing site chosen within the Selk impact structure. The rotorcraft, designed to function like a drone, will cover extensive distances and navigate diverse terrains on Titan. Launch is scheduled for June 2027, with Dragonfly expected to reach Titan in 2034, providing crucial insights into Titan’s formation, evolution, and potential habitable environments [12].

The Dragonfly mission promises to yield invaluable insights into Titan; however, its localized focus will restrict our ability to obtain a comprehensive understanding of the moon’s broader environmental context and its intricate geological and atmospheric phenomena. While Dragonfly can conduct detailed analyses of specific surface areas and investigate organic chemistry, it may not adequately capture the global climate patterns or geological processes occurring in regions distant from its landing sites.

In the upcoming section, we will introduce several fundamental theoretical concepts essential to our research, beginning with the Circular Restricted Three−Body Problem (CR3BP). This framework is crucial for evaluating the gravitational interactions among the spacecraft, Titan, and Saturn. By accurately modeling these interactions, we can enhance our understanding of the underlying dynamics and optimize the spacecraft’s trajectory in relation to both Titan and its host planet.

3. Background Theory

In this chapter, we introduce fundamental theoretical concepts essential for determining the arrival orbit around Titan and optimizing the trajectory to reach such an orbit. We delve into the circular restricted three−body problem, investigating various orbit types. Next, we delve into dynamical systems theory to analyze different possible trajectories. Additionally, we employ the particle swarm optimization algorithm to identify the most optimal trajectories. These frameworks allow us to refine the trajectory for achieving orbital insertion around Titan, thereby facilitating the examination of its surface.

3.1. Circular Restricted Three−Body Problem

The Circular Restricted Three−Body Problem (CR3BP) is a mathematical model used to analyze the motion of three celestial bodies. The model is based on three key hypotheses:

• The CR3BP considers three bodies, m₁, m₂ and m₃, such that the third mass is much smaller than the two main ones, termed primaries:

  $$m_1>m_2>>m_3$$

  (1)
• The third body m₃ has no gravitational influence on the motion of m₁ and m₂;
• The two primaries m₁ and m₂ orbit in circular orbits relative to their center of mass (CM).

Figure 1 shows the three masses in an inertial reference frame IRF ($\widehat{I},\widehat{J},\widehat{K}$) centered at the barycenter of masses m₁ and m₂ and in the rotational reference frame RFF ($\widehat{i},\widehat{j},\widehat{k}$) with the versor $\widehat{i}$ connecting the barycenters and pointing toward the largest mass. The versor $\widehat{j}$ are perpendicular to the plane of motion, and the versor $\widehat{k}$ is positioned accordingly (right hand rule).

The equations of motion are derived in a RFF, and the position, the velocity, and the acceleration of $m_3$ are expressed in this frame.

The mass ratio $\mu$ is defined as follows:

$$\mu ={\displaystyle \frac{m_2}{m_1+m_2}}$$

(2)

Having $m_1>m_2$ (from assumption (1)), we determine that $\mu \in (0,0.5]$, even though within the Solar System, $\mu =0.5$ is never reached.

The non−dimensional equations of motion (EOMs) of $m_3$ in the CR3BP are as follows:

$$\left\{\begin{array}{c}\widehat{i}:\ddot{x}-2\dot{y}-x=-{\displaystyle \frac{(1-\mu )(x-\mu )}{r_1^3}}-{\displaystyle \frac{\mu (x+1-\mu )}{r_2^3}}\hfill \\ \widehat{j}:\ddot{y}-2\dot{x}-y=-{\displaystyle \frac{(1-\mu )y}{r_1^3}}-{\displaystyle \frac{\mu y}{r_2^3}}\hfill \\ \widehat{k}:\ddot{z}=-{\displaystyle \frac{(1-\mu )z}{r_1^3}}-{\displaystyle \frac{\mu z}{r_2^3}}\hfill \end{array}\right.$$

(3)

The CR3BP EOMs are 2nd order non−linear coupled ordinary differential equations. In 1836, Jacobi calculated one integral of motion (Jacobi integral [13]), while Poincarè, in 1892, proved that there are no other independent integrals of motion besides the Jacobi integral [14]. This means that Equation (3) generally has no analytical solution.

3.2. Lagrange Points

The Lagrange points are the equilibrium points of the CR3BP, where the gravitational forces of two large bodies create regions of enhanced attraction and repulsion, enabling the stable placement of satellites or spacecrafts. These points can be obtained starting from Equation (3) and setting all velocities and accelerations to zero: $\ddot{x}=\ddot{y}=\ddot{z}=\dot{x}=\dot{y}=\dot{z}=0$. An object placed at one of these points would experience no acceleration and have zero velocity, thus remaining there indefinitely.

Thus, Equation (3) become a set of coupled algebraic equations in x, y, and z:

$$\left\{\begin{array}{c}x={\displaystyle \frac{(1-\mu )(x-\mu )}{r_1^3}}+{\displaystyle \frac{\mu (x+1-\mu )}{r_2^3}}\hfill \\ y={\displaystyle \frac{(1-\mu )y}{r_1^3}}+{\displaystyle \frac{\mu y}{r_2^3}}\hfill \\ 0=-{\displaystyle \frac{(1-\mu )z}{r_1^3}}-{\displaystyle \frac{\mu z}{r_2^3}}\hfill \end{array}\right.$$

(4)

It can be seen that the $x-$ and $y-$equations are coupled, while the $z-$equation is completely decoupled from the other two. The only solution for z is $z=0$, and consequently, all equilibrium points must lie in the xy plane; the plane of motion of the primaries.

• Equilateral points. Setting $r_1=r_2$, we can find two of the five solutions of Equation (4):

  $$\sqrt{{(x-\mu )}^2+y^2+z^2}=\sqrt{{(x+1-\mu )}^2+y^2+z^2}$$

  (5)

  Solving for x and using the non−dimensionalized equation $r_1=r_2=a_1+a_2=1$ to substitute in y, the equilateral points can be found:

  $$(x,y)=\left(\mu -{\displaystyle \frac{1}{2}},\pm {\displaystyle \frac{\sqrt{3}}{2}}\right)$$

  (6)

  These two points are called $L_4$ and $L_5$.

These points are numbered based on the overall dynamics of the three−body system, particularly in relation to the Jacobi constant, which is a conserved quantity that describes the energy and motion of the bodies involved.

• Collinear points. Noting that $y=0$ is a solution of the second equation of Equation (3), one can compute the $x-$values from the first equation. In particular, one can use the fact that $r_1=x-\mu$ and $r_2=x+1-\mu$, both of which must be positive quantities, since they represent physical distances, and substitute them into the $x-$equation. This yields the following:

  $$x={\displaystyle \frac{(1-\mu )(x-\mu )}{{|x-\mu |}^3}}+{\displaystyle \frac{\mu (x+1-\mu )}{{|x+1-\mu |}^3}}$$

  (7)

  This algebraic equation has three real roots for x that correspond to the $x-$coordinates of the collinear points $L_1$, $L_2$, and $L_3$, which are all unstable for any value of $\mu$. On the other hand, $L_4$ and $L_5$ are stable for all Sun−planet and planet−moon combinations, except for the Pluto−Charon system ($\mu =0.1085112>{\mu }^{*}$).

For the Saturn−Titan system, the mass ratio is $\mu =2.3663931583\times {10}^{-4}$, which is the second largest mass ratio between a planet and a moon in the Solar System. For reference, the only larger value is the Earth−Moon’s mass ratio, $\mu =1.2150585609\times {10}^{-2}$. The Lagrange points are represented in Figure 2. The non−dimensionalized and dimensional coordinates of these points for the Saturn−Titan system are summarized in

Table 1. Lagrange point coordinates of the Saturn−Titan system.

Lagrange Point	(x, y) [ND]	(x, y) [km]
$L_1$	(0.9574961733, 0)	(1,144,856.15, 0)
$L_2$	(1.0432564213, 0)	(1,247,397.71, 0)
$L_3$	(−1.0000985997, 0)	(−1,195,794.89, 0)
$L_4$	(0.4997633607, 0.8660254038)	(597,555.56, 1,035,486.66)
$L_5$	(0.4997633607, −0.8660254038)	(597,555.56, −1,035,486.66)

ND stands for non−dimensional.

.

The upcoming section will delve into dynamical systems theory, a crucial framework for analyzing the Circular Restricted Three−Body Problem (CR3BP), which presents a complex and intricate system of gravitational interactions. This theory provides the mathematical tools necessary to study the stability and behavior of orbits within this system.

A key aspect of our analysis will involve the concept of invariant manifolds. These manifolds represent the trajectories in phase space that connect different dynamical states, offering a pathway for the spacecraft to transition from one orbit to another. By identifying these invariant manifolds, we can uncover optimal trajectories that allow us to reach Titan efficiently while minimizing the required $\Delta V$.

3.3. Dynamical Systems Theory

Dynamical Systems Theory (DST), commonly referred to as chaos theory, provides a mathematical framework for analyzing complex, evolving systems by examining how they respond to specific initial conditions and external influences. Within astrodynamics, the N−body problem serves as a prime example of the intricacy and sensitivity present in chaotic systems. This problem involves the gravitational interaction among multiple celestial bodies, resulting in chaotic behavior, where minor variations in initial conditions lead to significantly different orbital trajectories. For instance, even slight adjustments in the initial positions or velocities of objects within a planetary system can cause orbits to diverge drastically over time. DST plays a critical role in comprehending and modeling such phenomena within interplanetary trajectories, underscoring the inherent unpredictability and complexity inherent in chaotic systems.

DST has played a significant role in notable achievements in trajectory design for libration point missions [15]. Examples include the Genesis mission, which collected and analyzed solar particles in a halo orbit around the Sun−Earth L1 point [16], and the James Webb Space Telescope mission, positioned at the L2 point of the Sun−Earth system, facilitating astronomical discoveries across various wavelengths [17].

• Invariant manifolds are mathematical structures that play a crucial role in trajectory optimization for interplanetary missions. Invariant manifolds help define regions in space that remain unchanged under the influence of gravitational forces, providing stable paths for spacecrafts. Understanding and utilizing these manifolds contribute to the optimization of trajectories, allowing for more efficient and fuel−effective interplanetary orbit insertion maneuvers.

By utilizing the monodromy matrix, which is derived from the State Transition Matrix (STM) and describes the evolution of a dynamical system over one complete period, we can compute eigenvalues and eigenvectors to understand a dynamical system’s evolution and long−term trajectory behavior. The eigenvalues indicate perturbation growth or decay, while the eigenvectors show perturbation directions. With eigenvalues (${\lambda }_i$) and eigenvectors (${\mathbf{w}}_i$) from the monodromy matrix, we can perturb an initial state vector ${\mathbf{X}}_0$ of a periodic orbit by a small amount $ϵ$ along stable or unstable eigenvectors.

$${\mathbf{X}}_{perturbed}={\mathbf{X}}_0\pm ϵ\mathbf{w}$$

(8)

Integrating the new state vector ${\mathbf{X}}_{perturbed}$ forward or backward in time yields trajectories that leave from or arrive at the periodic orbit whose initial condition ${\mathbf{X}}_0$ was used [18].

By repeating this process for various values of ${\mathbf{X}}_0$ within the orbit, a family of approximated asymptotic solution trajectories is obtained that form a so−called invariant manifold tube in the phase space. Some examples of stable and unstable invariant manifolds can be observed in Figure 3.

Even distant from a periodic orbit, a spacecraft approaching a stable manifold will be drawn closer to the orbit and can be inserted into it with minimum $\Delta V$. Any spacecraft with arbitrary initial conditions cannot cross the surface of the manifold tubes without executing an additional propulsive maneuver. In practical computations, numerical integrations are halted when the spacecraft reaches a predetermined Poincaré section.

The use of these structures has been well documented by Kelly et al. [19], who explore the application of stretching directions in the orthogonal approximation of invariant manifolds within the circular restricted three−body problem. Their study focuses on stable or nearly stable orbits, such as nearly rectilinear halo orbits (NRHOs) and distant retrograde orbits (DROs), which are essential for long−term exploration missions. By analyzing state variations along the maximum stretching direction, they demonstrate how efficient trajectories can be designed for both departures from and arrivals at periodic orbits, addressing the challenges of timely insertion and departure.

Additionally, Vivek Muralidharan and Kathleen C. Howell [20] leveraged invariant manifolds for lunar explorations, extending the concept to cislunar space. Their paper discusses how stable or nearly stable orbits often lack distinct manifold structures that facilitate trajectory design for departures and arrivals at periodic orbits. They highlight that while nearly stable orbits reduce the risk of rapid departure, their linear stability poses challenges for timely insertion and exit.

The authors emphasize the potential of stable or nearly stable near rectilinear halo orbits (NRHOs), distant retrograde orbits (DROs), and lunar orbits for long−horizon missions that demand efficient operations. By utilizing stretching directions, they propose a methodology for effectively designing trajectories.

From Figure 3, it is possible to observe that there are hundreds of possible stable and unstable trajectories. In the next section, we introduce particle swarm optimization, which is a heuristic algorithm that will allow us to assess a variety of orbital transfers that make use of invariant manifolds in order to obtain the most $\Delta V$−efficient trajectory.

3.4. Particle Swarm Optimization

Particle Swarm Optimization (PSO) is a computational technique inspired by the collective behavior observed in natural swarms, such as bird flocks. In PSO, each potential solution is represented as a particle, and these particles collectively explore the solution space by adjusting their positions based on individual experiences and shared knowledge within the swarm. Unlike many other optimization methods, PSO does not necessitate a precise initial guess, making it advantageous for scenarios like interplanetary trajectory design, especially within the context of the Circular Restricted Three−Body Problem (CR3BP) model. The particle swarm in PSO collaborates to explore and exploit the solution space, enabling it to adapt and discover optimal solutions even in situations where initial estimations are unknown.

PSO has proven to be a powerful and efficient technique in interplanetary trajectory design. Its computational approach facilitates the exploration of extensive solution spaces and optimization of trajectories, as evidenced by studies conducted by Pontari and Conway [21] and Baraldi and Conte [22]. Utilizing PSO enables mission planners to identify optimal paths and maneuver sequences for interplanetary orbit insertion into Saturn−Titan three−body orbits by iteratively adjusting candidate solutions based on their fitness.

The PSO algorithm has been successfully integrated with other optimization methods, such as Differential Evolution (DE) or Genetic Algorithms (GAs), to enhance overall optimization performance [23,24,25]. Combining PSO with DE or GA aims to create hybrid algorithms that capitalize on the strengths of both techniques. While PSO is known for efficiently exploring solution spaces and rapidly converging towards promising solutions, DE and GA bring attributes like global search capability and robustness in handling complex and multi−modal optimization problems.

• PSO algorithm: The idea of applying PSO is to minimize a certain objective function related to a dynamic system during its temporal evolution.

Let us consider the objective function J that must be minimized. This function depends on the variables of the dynamic system, denoted as $x_1,\dots x_n$, which are constrained within their respective ranges:

$$a_k\le x_k\le b_k(k=1,\dots ,n)$$

(9)

In this context, the goal is to find values of $x_1,\dots x_n$ that minimize the objective function J while adhering to the constraints imposed by the system’s dynamics.

PSO is a population−based method, with the population being a swarm comprising $N_{par}$ particles. Each particle, denoted as i ($i=1,\dots ,N_{par}$), consists of $k=1,\dots ,n$ components represented by ${\chi }_k$. Each component is associated with a position $x_k\left(i\right)$ and a velocity $w_k\left(i\right)$. Note that within this context, the terms “position” and “velocity” do not carry physical meanings; rather, they are abstract representations that govern the movement and interaction of particles.

The values of the problem’s n variables are contained in the position vector:

$$x\left(i\right)\triangleq {[x_1\left(i\right),\dots ,x_n\left(i\right)]}^T$$

(10)

whereas the position update is determined by the velocity vector. Given that the position is bounded, the corresponding velocity must also be restricted within suitable ranges:

$$-(b_k-a_k)\le w_k\le (b_k-a_k)\Rightarrow -d_k\le w_k\le d_k(k=1,\dots ,n)$$

(11)

In the PSO algorithm, the position vector encodes the variable values, making each particle’s position vector a representation of a potential solution. The velocity vector controls the rate and direction of change for these values.

To initialize the PSO algorithm, an initial population is randomly generated. $N_{par}$ particles are introduced, and their positions and velocities are distributed randomly within the search space. Through iterative updates of positions and velocities, the particles collectively seek optimal solutions by interacting and sharing knowledge. Figure 4 provides a schematic representation of the PSO algorithm structure.

For a generic j iteration:

• For i = 1, …, N_par

  (a)

  Evaluate the objective function for the i−th particle: $J^j\left(i\right)$

  (b)

  Define the best position visited by the i−th particle up to the current j−th iteration:

  $${\Psi }^j\left(i\right)=[x_1,\dots ,x_k,\dots ,x_n]{|}_{min\left\{J^p\left(i\right)\right\}}(p=1,\dots ,j)$$

  (12)

• Define the global best position ever visited by the entire swarm up to the current j−th iteration:

  $${\mathbf{Y}}^j=[x_1,\dots ,x_k,\dots ,x_n]{|}_{min\left\{J_m^p\right\}}(m=1,\dots ,N_{par})$$

  (13)
• Update the velocity w for each particle i and each component ${\chi }_k$:

  $$w_k^{j+1}\left(i\right)=C_I{w}_k^j\left(i\right)+C_C[{\Psi }_k^j\left(i\right)-x_k^j\left(i\right)]+C_S[Y_k^j-x_k^j\left(i\right)]$$

  (14)

  The inertial, cognitive, and social weights have the following expressions [21]:

  $$C_I=(1+r_1)/2C_C=1.49445r_2{C}_S=1.49445r_3$$

  (15)

  where $r_1,r_2$, and $r_3$ represent three independent random numbers between 0 and 1.

  (a)

  If $w_k^{j+1}\left(i\right)<a_k-b_k\Rightarrow w_k^{j+1}\left(i\right)=a_k-b_k$

  (b)

  If $w_k^{j+1}\left(i\right)>b_k-a_k\Rightarrow w_k^{j+1}\left(i\right)=b_k-a_k$

• Update the position x for each particle i and each component ${\chi }_k$:

  $$x_k^{j+1}\left(i\right)=x_k^j\left(i\right)+w_k^j\left(i\right)$$

  (16)

  (a)

  If $x_k^{j+1}\left(i\right)<a_k\Rightarrow x_k^{j+1}\left(i\right)=a_k$ and $w_k^{j+1}\left(i\right)=0$

  (b)

  If $x_k^{j+1}\left(i\right)>b_k\Rightarrow x_k^{j+1}\left(i\right)=b_k$ and $w_k^{j+1}\left(i\right)=0$

The algorithm terminates when the objective function J reaches a certain tolerance or when the maximum number of iterations, denoted as $N_{\mathrm{iter}}$, is reached. At the end, the vector ${\mathbf{Y}}^{N_{\mathrm{iter}}}$ will contain the optimal values of the unknown parameters that correspond to the minimum of the objective function J, denoted as $J_{\mathrm{opt}}^{N_{\mathrm{iter}}}$.

Whenever at least one inequality constraint is violated, the objective function of the i−th particle at the j−th iteration is set to ∞, and the velocity of the i−th particle for the (j+1)−th iteration is set to 0.

The distinctive feature of the algorithm can be found in the formula to update the velocity, as shown in Equation (14), which incorporates three stochastic terms:

• The first term, $C_I$, is referred to as the inertial term, and for each particle, it is proportional to the particle’s velocity in the previous iteration.
• The second term, $C_C$, represents the cognitive term and is proportionate to the distance between the current position of the particle and the best position the particle has achieved up to the j−th iteration.
• The third term, $C_S$, is the social term, and in this case, it is proportional to the difference between the current position of the particle and the best position encountered among all particles up to the j−th iteration.

In the next section, we will analyze various types of three−body orbits to identify the optimal family of orbits that maximize visibility of Titan’s surface while ensuring orbital stability. Subsequently, we will integrate the Particle Swarm Optimization (PSO) algorithm with the applications of Dynamical Systems Theory (DST) discussed earlier. This combination will allow us to numerically compute the most $\Delta V$−efficient trajectory for Saturn−Titan orbit insertion maneuvers from interplanetary space. By leveraging both DST and PSO, we aim to identify trajectories that minimize fuel consumption while achieving our mission objectives.

4. A Trade−Off Analysis of CR3BP Orbits

Having introduced the fundamental theory for the development of this paper, we can begin evaluating the various families of CR3BP orbits to select which family allows for a better study of Titan’s surface.

Within the framework of the CR3BP, various orbit families arise due to the consideration of different initial conditions and subsequent integration of the equations of motion (Equation (3)). This section embarks on an exploration of these orbit families, examining their distinct characteristics and advantages. Through a thorough assessment of these orbital configurations, our objective is to determine which family aligns most suitably with our mission objectives and scientific goals in the study of Titan’s surface.

This comprehensive evaluation forms the foundation for making well−informed decisions, thereby maximizing the likelihood of success for the mission. By delving into the intricacies of each orbit family, we aim to pinpoint the orbital parameters that best fulfill the mission requirements, ensuring the effectiveness and efficiency of our trajectory design for studying Titan’s surface.

Taking into account the accessibility of Saturn (see Figure 5 as an example of a HALO orbit) and the capability to observe the surface of Titan (see Figure 6 as an example of a Butterfly orbit),

Table 2. Orbit candidate CR3BP.

Orbit	Period [Days]	Saturn Occultated [%]	Saturn Occultated [Days]	Titan’s Surface Visibility [%]
Lyapunov ($L_1$)	8.377	0	0	76.20
Assiale ($L_2$)	10.259	9.417	0.9661	91.90
Verticale ($L_1$)	14.556	8.651	1.2591	88.27
Halo ($L_1$)	7.3308	0	0	64.79
Halo ($L_2$)	7.7367	0	0	64.35
NRHO ($L_1$)	4.6321	0	0	99.68
Butterfly	9.4883	0	0	100.00
Dragonfly	15.314	0	0	99.87
DRO (≈9 days)	9.2473	10.072	0.9314	87.10
DRO (≈5 days)	4.6768	13.889	0.6496	80.65

is obtained.

Furthermore, it is imperative to consider the stability of these orbits. While we derived them by accounting for the CR3BP, which incorporates the gravitational influences of Titan and Saturn while excluding external perturbations, in reality, these orbits are influenced by other celestial bodies such as the Sun, Jupiter, and all other moons of Saturn.

Assessing orbit stability is crucial to understand an orbit’s ability to withstand perturbations. It is unrealistic to expect that a spacecraft following a CR3BP orbit will maintain the same trajectory indefinitely. Over time, external perturbations will gradually induce variations, necessitating spacecraft maneuvers and consequently fuel consumption.

For assessing orbital stability, we introduce the Stability Index [26]. This index ranges from less than 1, indicating a stable orbit, equal to 1, indicating a marginally stable orbit, and greater than 1, indicating an unstable orbit that requires frequent corrections.

Essentially, while these orbits offer valuable insights into visibility and coverage, their long−term viability depends on their stability against external perturbations. This aspect is crucial for spacecraft mission planning, as it determines the practicality of utilizing these orbits over extended durations.

Table 3. CR3BP candidate orbits: stability index.

Orbit	Stability Index2
Lyapunov ($L_1$)	$3.89469275627888\times {10}^2$
Axial ($L_2$)	$1.58028851350330\times {10}^2$
Vertical ($L_1$)	$2.61939577137195\times {10}^2$
Halo ($L_1$)	$6.39280628625028\times {10}^2$
Halo ($L_2$)	$5.39208380823829\times {10}^2$
NRHO ($L_1$)	$1.93244112461674$
Butterfly	$5.28330760821393$
Dragonfly	$3.70087836653840\times {10}^2$
DRO (≈9 days)	$1.00000000071544$
DRO (≈5 days)	$1.00000000000414$

provides the stability index for each orbit we considered:

Orbit Selection

The Dragonfly mission, scheduled for the mid−2030s to explore Titan’s equatorial region, aims to deepen our understanding of the moon’s surface composition around specific areas like the Selk crater [27]. However, its localized focus comes with inherent limitations, excluding comprehensive global coverage and exploration of polar regions and high−altitude locations.

Despite these constraints, the data collected by Dragonfly will represent a significant advancement in our understanding of Titan. Yet, the chosen landing site, the Selk crater, aligns with specific scientific objectives while leaving much of Titan unexplored.

To overcome these limitations, considering the progress of the Dragonfly mission, there is a growing need for future exploration missions to cover a broader geographical range on Titan. Analyzing various parameters, a Distant Retrograde Orbit (DRO) is proposed as a preferable orbital choice due to its stability and extensive surface coverage. This decision involves a trade−off, sacrificing access to polar regions for enhanced orbit stability and reduced fuel consumption during station−keeping maneuvers.

A mission in a DRO around Titan would complement Dragonfly’s regional exploration, offering a global−scale perspective. It provides an opportunity to investigate phenomena such as cryovolcanoes [5], which can enhance our understanding of ongoing geological processes on Titan.

In conclusion, a DRO mission around Titan strategically complements the efforts of Dragonfly, enabling a comprehensive global−scale investigation. By studying features beyond Dragonfly’s reach, we can gain essential context for understanding Titan’s surface composition, geological dynamics, and evolution. The potential of DROs is underscored by recent research by Li et al. [28], which highlights the increasing popularity of DROs due to their conspicuous stability. However, this same stability presents challenges in designing transfer orbits into and out of these orbits. By leveraging the insights from this study, we can optimize mission designs to Titan, ensuring a holistic understanding of its complex environment and geological history.

At this point, having selected the family of orbits around Titan and introduced the principles of DST and PSO, we can now evaluate the invariant manifolds associated with our chosen orbits. Our goal will be to optimize the trajectory to achieve orbit around Titan while minimizing propellant consumption. This approach will effectively demonstrate the advantages of combining these theories, highlighting their synergy in enhancing trajectory planning and mission efficiency.

5. Trajectory Optimization

In this chapter, we delve into the optimization of trajectories with the goal of achieving insertion into the previously chosen DRO discussed in Section 4. The primary focus is on minimizing the required $\Delta V$, leveraging the foundational concepts of dynamical systems theory and the particle swarm optimization algorithm introduced in earlier chapters. As elucidated in Section 3.3, the multitude of periodic orbits, both stable and unstable, necessitates a meticulous optimization process. The variability in $\Delta V$ requirements for transitioning into these trajectories underscores the importance of fine−tuning trajectory optimization methods. The objective is to identify the most efficient path by considering diverse parameters and characteristics of the system, ultimately determining the optimal trajectory and entry point into the specified DRO.

First, we will analyze a direct arrival to the chosen DRO. Subsequently, we will compare these results with those obtained through trajectory optimization using invariant manifolds. This comparison will highlight the benefits of leveraging the natural dynamics of the system via invariant manifolds to potentially reduce fuel consumption and improve the overall efficiency of the transfer to the DRO, as opposed to a direct approach.

Before proceeding with calculations and comparisons, it is important to make a consideration: DROs, being marginally stable, do not allow for the computation of invariant manifolds, which, as described, are structures derived precisely from orbit perturbations. For this reason, in both cases (direct arrival or use of invariant manifolds), we will employ a Lyapunov orbit that is tangent to the selected DRO as a support. This double maneuver will allow us, later on, to exploit the invariant manifolds of the Lyapunov orbit in order to then maneuver the spacecraft into the selected DRO.

Table 4. Chosen DRO and Lyapunov orbits.

Orbit	Period [Days]	Jacobi Constant $[{\mathit{LU}}^2/{\mathit{TU}}^2]$	Stability Index
DRO	9.5786	2.9922448517210601	1.0000000023234800
Lyapunov	10.2010	3.0034876561564201	$1.6615852828089501\times {10}^2$

presents some characteristics of the selected DRO and Lyapunov orbits.

5.1. Direct Arrival to Titan

Arriving directly onto Titan through a hyperbolic trajectory involves complex calculations and considerations due to the dynamics of the Saturn−Titan system. This approach necessitates a spacecraft to enter Titan’s orbit with a velocity exceeding its escape velocity, utilizing the moon’s gravitational pull to decelerate effectively.

Synchronizing the spacecraft’s trajectory with Titan’s orbit is crucial for a successful encounter, considering Saturn’s gravitational effects, Titan’s elliptical orbit, and changes in its orbital speed.

While challenging, the calculations and insights gained from planning such a mission will serve as valuable benchmarks for evaluating and contrasting alternative approaches for reaching Titan, providing invaluable data for future deep space missions.

The plan involves leveraging the hyperbolic excess of velocity provided by [29] to set up our calculations. The concept is to construct the arrival hyperbola so that it has its periapsis at a point along Titan’s trajectory. This would enable the spacecraft to reach Titan’s sphere of influence.

Once the spacecraft reaches Titan’s SOI, the goal is to circularize the orbit. By slowing down the spacecraft, it becomes possible to establish a circular orbit around Titan. It is essential to note that by transitioning from the three−body problem to this approach, we are essentially simplifying it into a two−body problem. After reaching Titan’s sphere of influence, we no longer consider gravitational attraction and perturbations caused by Saturn in the system.

The circular orbit is designed to intersect the Lyapunov orbit, which in turn intersects the selected DRO. The scenario of the direct arrival at Titan is shown in Figure 7. In this figure, within the Inertial Reference Frame (IRF), the orbits in the Rotational Reference Frame (RRF) are also plotted.

The use of the RRF simplifies the visualization, providing a clearer understanding of the dynamics involved in these orbits in relation to Titan. Additionally, the distinct methodologies used in calculating these orbits−one involving the complex gravitational interactions in the CR3BP and the other simplifying it to the 2BP−highlight different modeling approaches for orbital analysis in celestial mechanics.

At the hyperbolic periapsis, a $\Delta V_1=9.2296$ km/s is needed to circularize the orbit at an altitude of h = 13,217 km from Titan’s surface. Once in the circular orbit, only half of the orbital path is traversed before reaching the intersection with the Lyapunov orbit. At this intersection, an additional $\Delta V_2=0.2557$ km/s maneuver facilitates the transition to the Lyapunov orbit. The spacecraft does not spend much time in the circular orbit, as it serves as a stepping stone for the subsequent intersection with the Lyapunov orbit. This scenario proves useful when we seek comparative terms for the following scenarios.

Consequently, a total of $\Delta V_{TOT}=\Delta V_1+\Delta V_2=9.4853$ km/s would be needed for direct arrival at Titan.

5.2. Hyperbolic Arrival in Invariant Manifolds

This paper focuses on optimizing insertion trajectories around Titan, using Saturn’s sphere of influence as the initial reference for the optimization process. A comprehensive study by Hajdik and Ramsey et al. [29] employed advanced software tools and custom scripts to incorporate an aerogravity assist at Titan, facilitating the strategic exploration of Saturn’s moon Enceladus from a carefully chosen orbit. Gajeri [30] made significant contributions by analyzing trajectories for a robotic mission to Titan. Additionally, the Dragonfly mission, designed by McQuaide et al. [31], aims to conduct a thorough scientific survey of Titan using a relocatable lander. Together, these research efforts provide valuable insights and methodologies for trajectory optimization, forming a critical foundation for missions targeting Titan while minimizing $\Delta V$ requirements.

These studies highlight Titan’s importance as an exploration target and advocate for the adoption of innovative methodologies in mission design. They lay the groundwork for this research by providing essential insights and data that inform our approach.

In this context, this paper introduces a novel strategy aimed at reducing $\Delta V$, which can be effectively applied to the mission design of various multi−body systems. By leveraging invariant manifolds, this method takes advantage of the complex dynamics and gravitational interactions between celestial bodies.

This technique not only enhances trajectory optimization around Titan but also has broader implications for other celestial systems where similar gravitational interactions can be utilized. By offering a flexible framework that can adapt to different mission profiles, this research significantly contributes to the fields of astrodynamics and mission planning, opening up new avenues for exploration in intricate gravitational environments.

Upon entering Saturn’s sphere of influence (SOI) with a hyperbolic excess velocity of $v_{\infty }=5.9177$ km/s, obtained from [29], the trajectory optimization process is initiated. The objective is to select invariant manifold paths, starting from the hyperbolic arrival trajectory, to guide the spacecraft towards the previously chosen DRO while minimizing $\Delta V$. This trajectory optimization begins following the spacecraft’s arrival on 2 July 2043, after a launch planned for 11 December 2036.

The optimization strategy involves transitioning from the hyperbolic arrival trajectory to the stable invariant manifold using a $\Delta V$. This manifold then asymptotically directs the spacecraft towards the Lyapunov orbit, which, in turn, guides the spacecraft to the DRO. The incorporation of the Lyapunov orbit is crucial, since DROs, being marginally stable, lack useful invariant manifolds. This comprehensive approach ensures an efficient trajectory optimization process, considering the specific dynamics of Saturn’s system.

For a more precise estimation of the required $\Delta V$ for this maneuver, the relative positions of Titan and Saturn were determined using the JPL Horizons System3, a JPL−provided software for celestial bodies’ ephemerides. This became feasible thanks to the work by Hajdik et al. [29], which also provides the dates of a simulated mission.

At this point, constructing the arrival hyperbola involves considering Saturn as the focus and utilizing the known $v_{\infty }$ to derive specific orbital parameters of the hyperbola.

The major semi–axis a and the eccentricity e can be defined as follows:

$$a=-{\displaystyle \frac{{\mu }_{\hslash }}{v_{\infty }^2}}e=1+{\displaystyle \frac{r_p{v}_{\infty }^2}{{\mu }_{\hslash }}}$$

(17)

where $r_p$ is the distance between Saturn and Titan, and ${\mu }_{\hslash }$ = 37,940,584.842 ${\mathrm{km}}^3/{\mathrm{s}}^2$ represents Saturn’s gravitational parameter4.

Having established the arrival hyperbola in the Saturn−Titan system, the PSO analysis comes into play. As the spacecraft approaches Saturn along the hyperbolic trajectory, a $\Delta V_1$ will be applied to deviate it from the hyperbolic path, followed by a subsequent $\Delta V_2$ for insertion into the invariant manifold. This two−step process, facilitated by PSO, aims to optimize the spacecraft’s trajectory for a successful transition from the hyperbolic arrival to the stable invariant manifold, contributing to the overall efficiency of the mission.

5.2.1. PSO Structure Applied to Trajectory Optimization

The application of the PSO algorithm aims to evaluate the combination of maneuvers that minimize $\Delta V_{TOT}=\Delta V_1+\Delta V_2$.

Three specific parameters are used to compute the propellant−optimal trajectory (see Figure 8):

• The i−th trajectory of the invariant manifold; hence, the velocity required for the spacecraft to enter that trajectory. Indirectly, this parameter provides $\Delta V_2$.
• The $\Delta V_1$, indicating the retrograde change in velocity required.
• The position on the arrival hyperbolic trajectory where the first maneuver described by $\Delta V_1$ will occur. This parameter will be evaluated based on the angle $\delta$ between Saturn and the considered point of the hyperbola.

Figure 8. Variables of the PSO algorithm.

Figure 8. Variables of the PSO algorithm.

By combining these three parameters with the PSO framework, the algorithm aims to converge towards the solution that minimizes $\Delta V_{TOT}$. The algorithm involves creating $N_{par}=15$ particles, each containing a randomly generated value for the point where the first maneuver, described by $\Delta V_1$, takes place, the magnitude of $\Delta V_1$, and a generic i−th trajectory of the invariant manifolds. Specific limits are defined for generating these values.

The maximum number of iterations, $N_{iter}$, is set at 300 iterations, as the algorithm typically converges within this limit.

The random generated initial population can indeed affect both the accuracy of the results and the computational time required for convergence. Since each simulation uses different random particles, the convergence rate can vary significantly. Some initial populations may lead to faster convergence towards a solution, while others might take longer or get stuck in local minima. In our experiments, the PSO algorithm was applied multiple times—dozens of iterations were performed to ensure that we capture a global minimum rather than settling for a local one. We observed that, after a certain number of iterations, despite different initial conditions, the particles converged towards similar positions, indicating that the algorithm is indeed capable of finding the absolute minimum.

One of our simulations along with its outcomes is available at this link5.

The algorithm converges after 27 iterations. At the 27th iteration, all particles become identical, halting further progression of the algorithm. The provided solution is characterized by the following:

$$\left\{\begin{array}{c}\Delta V_1=2.456530\mathrm{km}/\mathrm{s}\hfill \\ \delta ={85.91}^{\circ }\Rightarrow R_p=1.7065\times {10}^7\mathrm{km}\mathrm{from}\mathrm{Saturn}\hfill \\ \mathrm{i}-\mathrm{th}\mathrm{trajectory}=97\hfill \end{array}\right.$$

(18)

While the first two parameters are readily understandable, the third represents the i−th trajectory considered. As we have seen, invariant manifolds consist of hundreds of trajectories, and the algorithm has enabled us to evaluate various combinations of parameters to achieve the minimum $\Delta V_{TOT}$, which amounts to the following:

$$\Delta V_{TOT}=\Delta V_1+\Delta V_2=2.681346\mathrm{km}/\mathrm{s}$$

(19)

The $\Delta V_2$ is smaller than $\Delta V_1$: the more significant maneuver involves diverting the spacecraft from the hyperbolic arrival trajectory, while the subsequent maneuver for entering the invariant manifold trajectory is less demanding. Both maneuvers utilize a retrograde $\Delta V$, causing the spacecraft to decelerate. In comparison to a direct arrival on Titan, we achieved a reduction of over 70.9% in terms of $\Delta V$.

This significant decrease in propellant requirements translates directly to lower operational costs, making missions more financially feasible. By applying the Tsiolkovski rocket equation, which relates the spacecraft’s velocity change to the ratio of initial to final mass, we can see that the reduced $\Delta V$ leads to an even more dramatic reduction in the amount of fuel needed. In fact, the reduction in $\Delta V$ by 70.9% results in an 89.2% reduction in the mass of fuel required. This exponential relationship underscores the importance of minimizing $\Delta V$ for mission planning.

5.2.2. Benefits of Reduced $\Delta V$

First, a lower $\Delta V$ value means that significantly less fuel is needed for the spacecraft. This not only the reduces costs associated with fuel procurement but also allows for more efficient use of the launch vehicle’s capacity. Consequently, missions can allocate the savings in mass and budget to other crucial areas, such as adding more advanced scientific instruments, extending the mission duration, or including additional payloads. For example, reducing the required fuel mass by 89.2% could allow for the addition of specialized landers or atmospheric probes to further scientific goals.

Second, with less fuel required, the spacecraft’s total mass at launch can be reduced. This reduction can enable the use of smaller and less powerful launch vehicles, which are often significantly cheaper. A spacecraft designed with a reduced $\Delta V$ profile could potentially switch from a heavy−lift rocket to a medium−class rocket, providing further cost savings. For instance, a mission that initially required a Delta IV Heavy could, thanks to the mass reduction, use a Falcon 9, resulting in savings of several hundreds of millions of dollars.

Finally, the success of this approach sets a strong precedent for future space missions. By demonstrating that significant $\Delta V$ reductions can be achieved through optimized trajectory planning, space agencies may be more inclined to adopt similar strategies. The large−scale reduction in fuel requirements opens up new possibilities for mission design, allowing for more frequent launches or even the inclusion of multiple missions within the same budget. This can lead to a new era of space exploration, where missions can reach distant and challenging targets like Titan or Europa with enhanced efficiency and effectiveness. Such trajectory optimizations pave the way for more ambitious scientific endeavors, potentially enabling exploration of further−off moons or planets that were previously considered to be too costly or complex to reach.

6. Trajectory Overview and Final Results

This section summarizes the key findings, tracing the complete trajectory of the spacecraft from Saturn’s SOI to the DRO orbit, chosen after a meticulous analysis discussed in Section 4.

The spacecraft’s journey starts as it enters Saturn’s SOI with a hyperbolic excess velocity of $v_{\infty }=5.9177\mathrm{km}/\mathrm{s}$ (Figure 9−left). At a distance of approximately $1.7065\times {10}^7$ km from Saturn, the first critical maneuver occurs, characterized by $\Delta V_1=2.4565$ km/s, facilitating a deviation from the initial hyperbolic trajectory and enabling a transition to a lower−velocity one (Figure 9−right).

Continuing along this new, lower−velocity hyperbolic trajectory, the spacecraft approaches the invariant manifold, prompting the second maneuver $\Delta V_2=0.2248$ km/s. This maneuver optimizes the spacecraft’s velocity by further slowing it down, aiding its precise alignment with the invariant manifold and ensuring a smoother intersection (Figure 10−left).

The spacecraft is propelled through the invariant manifold, asymptotically reaching the Lyapunov orbit. Hence, a negligible $\Delta V$ adjustment on the order of a few m/s is necessary for precise alignment within the orbit (Figure 10−right).

Upon attaining the Lyapunov orbit, transitioning to the DRO mandates $\Delta V_3=0.1697$ km/s. In this case, the maneuver accelerates the spacecraft, facilitating its transfer to the DRO orbit, which is more stable than the Lyapunov orbit.

Ultimately, as the spacecraft enters Saturn’s SOI, the cumulative $\Delta V_{TOT}$ required is as follows:

$$\Delta V_{TOT}=\Delta V_1+\Delta V_2+\Delta V_3=2.851\mathrm{km}/\mathrm{s}$$

(20)

7. Conclusions and Future Work

This paper effectively combines DST and PSO to optimize spacecraft trajectories for interplanetary orbit insertion into Saturn−Titan orbits. It emphasizes Titan’s scientific significance, including its unique atmosphere and potential for prebiotic chemistry.

After the initial analysis, this study explores the CR3BP, pinpointing DROs as optimal trajectories for comprehensive Titan observations, considering orbital stability.

The obtained results show promising outcomes, indicating a $\Delta V$ requirement of 2.6813 km/s. This represents a noteworthy reduction of approximately 70.9% compared to a direct insertion onto Titan.

This achievement underscores the importance of advanced computational techniques in optimizing interplanetary trajectories within the Saturn−Titan system, paving the way for accurate and efficient future explorations.

Moving forward, several avenues for future research and development emerge from this paper:

• Refinement of optimization techniques: Continual refinement of optimization techniques could involve exploring machine learning or evolutionary algorithms to supplement or replace PSO. This exploration has the potential to yield even more optimized trajectories.
• Enhanced mission flexibility: Trajectories that enable exploration across various regions of Titan, particularly the polar areas, should be investigated while ensuring orbital stability.
• Multi−objective optimization: Optimization strategies should be extended to consider multiple objectives, such as reducing time of flight, maximizing scientific data collection, and minimizing propellant consumption simultaneously.
• Optimization of alternative trajectory selection: Alternative Lyapunov or other family orbits should be explored for insertion into DROs to potentially enhance efficiency and meet specific mission requirements while ensuring stability.

In conclusion, the successful fusion of advanced computational techniques with dynamical systems theory in optimizing interplanetary trajectories for Saturn−Titan missions sets the stage for continued innovation and refinement in space exploration, promising more efficient and accurate mission planning in the future.