Gravity in Cislunar Space (Part I); Analysis of Cislunar Gravity and Gravity Transitions

The cislunar space, which is the region beyond Earth's geosynchronous orbit (GEO) governed by the gravity influence of both the Earth and the moon, encompasses a complex gravity-induced environment. Proper understanding of gravity and gravity maneuvers in this domain is highly crucial for mission planning, development and design of mission spacecraft and the spacecraft's navigation technology.

Cislunar Gravity; Gravity interplay between the Earth and the Moon

A major characteristic feature of the cislunar space is that gravity is influenced by both the earth and the moon. The influence of Earth's gravity is dominant in near-Earth regions, while that of the moon becomes dominant on approaching lunar orbit. However, gravity dominance in the cislunar space depends largely and significantly on the mass of the body, largely following, Newton's Gravity equation;

F=GMm/r^2 where:

• G: Gravitational constant (6.674×10^?11?Nm^2/kg^2)
• M: Mass of the celestial body
• m: Mass of the spacecraft
• r: Distance from the center of the celestial body

Earth's mass is 81 times the mass of the Moon. This significantly implies that a spacecraft equidistant from the earth and the moon will experience gravity effect much more from the earth's gravity than from the moon's gravity. Proximity to Earth's barycenter additionally contributes to Earth's domination of cislunar gravity. Spacecrafts closer to Earth experience higher gravity pulls, requiring more energy to escape orbits. Additionally, mission planning relies on an intuitive understanding of Earth's gravity dominance to aid maneuvers from low-Earth orbits (LEO) to high-Earth orbits (HEO). On approaching lunar orbit, the Moon's gravity pull becomes dominant at about 60,000km radius from the moon (this radius is termed the moon's sphere of influence, SOI). Spacecraft would need to perform trajectory adjustments for soft landing. Also, astronauts on board may feel the moon's gravity dominance in its SOI as counterintuitive, given Earth's dominance of the cislunar space.

This gravity tug-of-war experienced by spacecraft in the cislunar space requires accurate and necessary transitional dynamics. With the prediction of spacecraft paths in real-time becoming quite complex and human intuition often insufficient, autonomous systems are often preferred for navigation, as the use of machine learning and AI algorithms continually achieve success by continuous trajectory plotting and recalculation based on gravity models.

Cislunar Gravity balance; Lagrange Points to the rescue

For a spacecraft in the earth-moon system, the equations of motion are given as;

x¨?2y˙=?U/?x, y¨+2x˙=?U/?y with x and y being spacecraft positions in the rotating frame.

The Lagrange points in the earth-moon system serve as unstable and stable regions of equilibrium where earth's gravity and moon's gravity technically cancel out in the rotating reference frame. These Lagrange points are derived when;

?U/?x=0, and ?U/?y=0

Astronauts do not experience any form of intuition at these points, but adequate training is often required for station-keeping or utility of these points for orbital transfers. Observatories and satellites stationed at Lagrange points usually require a high level of precision modeling, as the smallest or most minute perturbation could influence observatory results.

Five of these points exist in the Cislunar space;

• L1 (Earth-Moon inner point), lying at about 58,000 km from the moon towards Earth, a semi-unstable equilibrium point useful for communication relay for lunar missions, as well as for earth orbit to lunar orbit transitioning.
• L2 (Earth-Moon outer point), lying on the line connecting the Earth and moon at roughly 64,500 km from the moon beyond the moon, also semi-unstable but offers an uninterrupted view of the far side of the moon. It offers a staging point for exploration of the far-side of the moon and for deep space observations.
• L3 (Earth-Moon opposite point), lying at 384,000 km from the moon on the far side of Earth, beyond Earth's orbit, but on the line connecting the Earth and the moon. This point is influenced by the gravity pull of the Sun, hence, it is highly unstable, giving limited practicability in cislunar exploration, but hypothetically useful for studying Sun-Earth interaction.
• L4 (Earth-Moon leading point), equilateral with the Earth and the moon by staying ahead of the moon's orbit by 60 degrees. It is a stable equilibrium point, as gravity perturbations draws the spacecraft or object back to it (L4). Useful for long-term observation, resources storage and staging for lunar missions. It is often characterized as a Trojan region.
• L5 (Earth-Moon Trailing point), equilateral with the Earth and the moon but stays behind the moon's orbit by 60 degrees. Its stability is similar to that of L4, and it is largely useful for the stationing of spacecraft.

The stability of the Lagrange points are influenced by the interplay of gravity forces and centrifugal forces in the rotating reference frame. L4 and L5 are stable due to the balance between the aforementioned forces, while L1, L2 and L3 are unstable, requiring active station keeping as a counter-action against perturbation-induced drifts. Halo orbits are useful for spacecraft in these unstable Lagrange points for position maintenance at minimal fuel usage.

In Cislunar exploration, Lunar Gateway orbits work suitably at L1 for lunar mission hubs. L2 is suitable for deploying communication relay observations from the far side of the moon. L4 and L5 serve significantly as the potential point for long-term infrastructure to aid lunar missions.

To be continued...