# Dynamical Systems Applications for Missions to Detect Life in Ocean Worlds

Water is a requirement for all life as we know it, so the search for extraterrestrial life generally begins there. Luckily, Earth is not the only place in our solar system to have oceans with liquid water. Several moons of the outer planets show signs of salty oceans beneath their icy exteriors; in fact, measurements of Europa, Enceladus, Titan, Ganymede, and Callisto directly indicate the presence of subsurface oceans [8]. Europa (a moon of Jupiter) and Enceladus (a moon of Saturn) are two of the most interesting ocean worlds, as they both exhibit evidence of interactions between their oceans and the surface in the form of geysers or plumes [7].

**Figure 1.**The circular restricted three-body problem in the rotating \((x, y)\) frame with normalized coordinates. Figure courtesy of Jared Blanchard.

Dynamical systems theory provides a mathematical framework to study the behavior and evolution of complex systems over time. Applications to astrodynamics and celestial mechanics return this area of study to its origins — Henri Poincaré invented dynamical systems theory while investigating the three-body problem. When applied to ocean world missions, dynamical systems theory helps design trajectories that optimize fuel consumption and minimize travel time while navigating the gravitational interactions between celestial bodies.

### The Circular Restricted Three-body Problem

The circular restricted three-body problem (CR3BP) is a useful model for determining the motion of a spacecraft under the influence of two massive bodies (e.g., Earth-Moon, Jupiter-Europa, Saturn-Enceladus). The key simplifying assumptions are that (i) the massive bodies move in circular orbits about their mutual barycenter, and (ii) the mass of the spacecraft is small enough that it has a negligible effect on the massive bodies [9, 13].

Normalizing the units of the CR3BP—such that the distances between the primary and secondary bodies, total mass, and mean motion are all unity—is certainly convenient. Doing so greatly simplifies the system and shows that it is dependent on only a single parameter: the mass parameter \(\mu\), which is defined by the masses of the massive bodies \(m_1\) and \(m_2\) such that

\[\mu = \frac{m_2}{m_1+m_2}. \tag1\]

Normally \(m_1 \ge m_2\); therefore \(0 < \mu \le 0.5\).

**Figure 2.**A stable family of inclined 5:6 resonant orbits in the Jupiter-Europa system. Figure courtesy of Jared Blanchard.

To remove any dependence on time and obtain an autonomous system, we define a rotating frame that is aligned with the line between the primary and secondary bodies. As such, these bodies remain fixed on the \(x\)-axis at \(-\mu\) and \(1-\mu\), respectively. Figure 1 depicts the setup for the CR3BP in the inertial and rotating frames. The vector \(\pmb{r} = [x,y,z]^\intercal\) is the spacecraft’s position with respect to the barycenter, expressed in the rotating frame coordinate system. We can write the vectors from the primary and secondary bodies to the spacecraft as \(\pmb{r}_1 = [(x+\mu), \; y, \; z]^\intercal\) and \(\pmb{r}_2 = [(x-1+\mu), \; y, \; z]^\intercal\). In the rotating frame, the potential function \(\Omega\) is

\[\Omega(\pmb{r}) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}, \quad r_i = \|\pmb{r}_i\|. \tag2\]

The Jacobi constant is the only conserved quantity in the CR3BP and is defined as

\[\begin{eqnarray} &C&(\pmb{r},\dot{\pmb{r}}) = 2\Omega(\pmb{r}) - \dot{\pmb{r}}^\intercal\dot{\pmb{r}} \tag3 \\ &C& = (x^2 + y^2) + 2\frac{1-\mu}{r_1} + 2\frac{\mu}{r_2} - \left(\dot{x}^2+\dot{y}^2+\dot{z}^2\right). \tag4 \end{eqnarray}\]

The full equations of motion are

\[\begin{eqnarray} \ddot{x} &=& -\left(\frac{1-\mu}{r_1^{3}}\left(x+\mu\right) \right.&+&\left. \frac{\mu}{r_2^{3}}\left(x-1+\mu\right) \right) &+& x + 2\dot{y} \tag5 \\ \ddot{y} &=& -\left(\frac{1-\mu}{r_1^{3}}y \right.&+&\left.\frac{\mu}{r_2^{3}}y \right) &+& y - 2\dot{x} \tag6 \\ \ddot{z} &=& -\left(\frac{1-\mu}{r_1^{3}}z \right.&+&\left. \frac{\mu}{r_2^{3}}z \right). \tag7 \end{eqnarray}\]

**Figure 3.**A family of \(\textrm{L}_2\) Lyapunov orbits in the Earth-Moon system. Figure courtesy of Jared Blanchard.

### Periodic Orbits

In general, analytical solutions to the CR3BP’s nonlinear equations of motion do not exist. However, the periodic orbits—repeating paths that spacecraft can follow within the gravitational field of the two primary bodies—can help us understand the dynamics at a global level. Poincaré said that periodic orbits are “the only opening through which we can try to penetrate in a place which, up to now, was supposed to be inaccessible” [12]. These orbits provide a convenient way to characterize motion through the complex gravitational dynamics of the ocean world systems. They also possess unique properties and can serve as staging points for complicated chains of trajectories. The many types of periodic orbits in the CR3BP are organized into families based on whether the orbit is centered around the primary body, secondary body, or one of the Lagrange points, as well as other geometric properties.

### Resonant Orbits

Resonant orbits are often a key component of mission planning because of their utility for gravity-assist maneuvers. A resonant orbit is typically centered around the primary body and only interacts with the secondary body during brief flybys that occur at regular intervals. The period of a resonant orbit is commensurate with the period of the secondary body. For example, 5:6 resonant orbits—such as those in Figure 2—complete five orbits around Jupiter in the same amount of time that it takes Europa to complete six orbits, meaning that a flyby takes place once roughly every 21 days. The upcoming Europa Clipper mission will take advantage of resonant orbits to complete flybys for several of Jupiter’s moons and perform gravity assist maneuvers [5].

### Lyapunov and Halo Orbits

**Figure 4.**A portion of the \(\textrm{L}_2\) halo orbit family around Enceladus (one of Saturn’s moons). These orbits reach low periapsis over the south pole of Enceladus, which makes them attractive candidates for missions to observe the plumes. Figure courtesy of Jared Blanchard.

### Quasi-periodic Orbits

Quasi-periodic orbits (QPOs) oscillate around a central periodic orbit. Unlike ordinary periodic orbits that exist on a one-dimensional curve, QPOs exist on a two-dimensional torus. This geometry provides an extra degree of freedom that is apparent in Figure 5, which depicts a QPO that is centered around one of the halo orbits from Figure 4. The QPO’s width means that its coverage of the south pole is greater than that of a single periodic orbit.

**Figure 5.**A quasi-periodic orbit (QPO) that is centered around a near-rectilinear halo orbit of Enceladus (one of Saturn’s moons). The QPO spans a wide region of the south pole and passes over a set of long cracks known as the tiger stripes (blue-green regions). Scientists could potentially use this orbit to sample plumes from a larger region than would be possible with a single periodic orbit. Figure courtesy of Jared Blanchard.

### Invariant Funnels

Normally, identifying trajectories to and from periodic orbits involves the computation of stable/unstable invariant manifolds. The stable manifold of a periodic orbit is the set of trajectories that converges asymptotically to the periodic orbit over time, while the unstable manifold is the set that diverges. Though these trajectories provide the most efficient transfers onto and away from the periodic orbit, they may sometimes be too slow, as in the case of stable orbits. And if the desired target state is not on a periodic orbit at all, there is no clear way to target it using the natural dynamics. We have therefore developed a method to compute a closed set of trajectories that converge to an arbitrary state near the secondary body [2]. This set of trajectories is called an* invariant funnel*. Figure 6 displays funnels that we computed in the Jupiter-Europa, Saturn-Enceladus, and Earth-Moon systems. By using the invariant funnel of a trajectory as terminal sets in a model predictive control scheme, we can significantly reduce the amount of control effort that is required to reach a given state within some tolerance [1]. We have demonstrated this technique to target landing orbits on Europa [1] as well as the NRHO periapsis for Enceladus.

**Figure 6.**Invariant funnels in the Jupiter-Europa, Saturn-Enceladus, and Earth-Moon systems. Significant contraction of trajectories is evident near \(\textrm{L}_2\), meaning that a distant spacecraft could target a very large area and follow the natural dynamics to a point that is quite close to the desired state. Figure courtesy of Jared Blanchard.

### Mission Phases

A potential mission to sample the plumes of Enceladus would involve several phases. First, in the *interplanetary phase*, the spacecraft achieves escape velocity from the Earth and travels to Saturn. This phase would last the longest and could take up to 10 years. Next, the *capture phase* requires the spacecraft to slow down so that it does not simply fly past Saturn. It then begins the *pump-down phase*, employing resonant orbits to string together gravity assist flybys of Saturn’s other moons and slow enough to enter an NRHO around Enceladus. At this point, the* science phase* begins. The spacecraft would be moving slowly enough at periapsis to collect plume material emitted from the tiger stripes—a set of long cracks near the south pole of Enceladus (see Figure 5)—without destroying any organic matter. Sophisticated onboard sensors could analyze this collected material for signs of life. The spacecraft could transition to the nearby QPO to maximize the sampled area over the tiger stripes and identify areas with more interesting material.

In conclusion, dynamical systems theory is a powerful tool for understanding the periodic orbits of the CR3BP and taking advantage of the coupled gravitational pull from multiple bodies to devise new, efficient strategies to explore the ocean worlds. These special orbits greatly increase the variety of prospective missions, and many past and future missions have used three-body orbits to meet their requirements. Our novel contribution of invariant funnels that target arbitrary states near the secondary body—including states on periodic orbits—expands the available options for mission designers who are seeking to make the next generation of ocean world missions a reality.

*Jared Blanchard delivered a contributed presentation on this research at the 2023 SIAM Conference on Applications of Dynamical Systems, which took place in Portland, Ore., last year.*

**Acknowledgments: **A portion of the work was funded by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). The author offers a special thanks to Martin Lo, Damon Landau, Brian Anderson, Ricardo Restrepo, and Ryan Burns from NASA Jet Propulsion Laboratory for their collaboration on this work, as well as Cassandra Webster, Dave Folta, and Ariadna Farres from NASA Goddard Space Flight Center. This work was supported by a NASA Space Technology Graduate Research Opportunity and the JPL Visiting Student Research Program.

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