Perturbations in Cislunar Space; Perturbation Analysis & Applications (Part II)

Analyzing perturbations in cislunar space involves studying and quantifying the impact of the perturbative forces on the spacecraft and adjusting the spacecraft's trajectory and the control commands according to the resultant behavior of the spacecraft in response to these perturbations. In other words, it means analyzing deviations from the ideal two-body motion conditions of the spacecraft due to external forces in the Earth-Moon system. Several methods are always employed in analyzing perturbations.

The mathematical approach precisely groups perturbative forces and adds them as corrections to the equations of the spacecraft motion, such that the total force acting on a spacecraft also includes the sum of the gravity force, Third-body effects, Solar Radiation Pressure (SRP), perturbation due to Earth's oblateness, drag force, relativistically induced forces, tidal torques, and other perturbations, depending on the spacecraft's orbit and mission duration, which also determines the degree of impact of the perturbation on the spacecraft. Choosing reference frames also plays a crucial role in this analysis. The Earth-Centered Inertial (ECI) Frame is often used for initial trajectory design while Moon-Centered Inertial (MCI) Frames are useful for lunar orbits and Moon-centric mission phases. On the other hand, the Earth-Moon Rotating Frame is mostly suitable for studying orbits around Lagrange points.

Mathematical models applied also depend on the region of interest in cislunar space. Patched-Conic Approximation is useful for initial trajectory design but does not account for perturbations. The Three-Body Problem includes third-body effects but neglects high-order perturbative forces. Most of the time, High-Fidelity N-body simulations cater for more perturbative forces and are therefore used for precise orbit determination. The gravitational potentials of the Earth and Moon are expanded using spherical harmonics to account for their non-spherical shapes or oblateness. Most High-fidelity simulations use numerical integrators like the Runge-Kutta or Gauss-Jackson methods to solve the equations of spacecraft motion.

Furthermore, Perturbation theories do provide analytical and numerical techniques for the study of how small forces can cause deviations from an ideal Keplerian orbit. These theories are essential for trajectory design, long-term orbit prediction, and mission planning in cislunar space, where multiple perturbations interact over different periods. General Perturbation (GPT) methods, like the Lagrange Planetary Equations, KAM (Kolmogorov-Arnold-Moser) theorems, and Hamiltonian Perturbation Theory, often focus on the long-term behavior by averaging or simplifying the equations of motion, while Special Perturbation (SPT) methods, like the Cowell’s Method, Encke’s Method, and Gauss’ Variational Equations integrate perturbations directly into the equations of motion and other numerical simulations.

One of the major conditions for analyzing perturbative forces is to check for stability and resonance. Oftentimes, Cislunar missions target stable or quasi-stable orbits like those around Lagrange points or halo orbits, thus, stability analysis in these regions usually accounts for periodic perturbations. At the same time, gravitational resonances, like mean-motion resonances, can amplify perturbative effects. This analysis prepares the need for control strategies for orbit maintenance of the spacecraft. Through station-keeping maneuvers, perturbations are counteracted to maintain desired orbits (e.g. Near Rectilinear Halo Orbits for Gateway Space Station). Also, continuous low-thrust engines like ion thrusters mitigate perturbative drift efficiently.

Analyzing perturbations in cislunar space plays so many roles in current-day and future cislunar missions. In planning for the Lunar Gateway space station, the understanding of solar radiation pressure, third-body effects, and station-keeping in a Near Rectilinear Halo Orbit is required for the Gateway's stability in the Earth-Moon system. Additionally, precise trajectory correction maneuvers are required for Artemis missions so as to counter perturbations during trans-lunar injection and lunar orbit insertion of Orion spacecraft. Furthermore, for secondary payloads deployed in cislunar space, perturbation modeling ensures mission success given their limited propulsion capabilities. Perturbation analysis also helps account for gravitational anomalies caused by mascons so as to optimize lunar landing trajectories.

Perturbation analysis in cislunar space is crucial for trajectory design, station-keeping, and long-term orbit stability. Various forces influence spacecraft motion over different times, and the combination of both analytical and numerical methods is employed in modeling and mitigating these effects. Technological advances in AI-driven orbit prediction and quantum-enhanced models will further improve precision for future lunar and deep-space missions. Understanding and controlling perturbations ensures efficient and sustainable spaceflight and exploration within the Earth-Moon system.