The Sun is BIG
Quick question, which exerts a greater force on the Moon, The Sun, or the Earth?
Newton’s law of universal gravitation provides us with the following relation for calculating the force on an object
where G is the universal gravitational constant, m1/m2 are the masses of the two objects, and r12 is the distance between the two objects. While the Sun is more massive than the Earth, its distance to the Moon is much greater (and force drops off with distance squared) so it isn’t obvious. Using NASA’s DE430 dataset, we find out that the Sun exerts approximately 2.2 times the Earth’s force on the Moon. If we look at it over a five-year span, the relative positions of the 3 bodies cause it to fluctuate, between slightly higher than 2.5 down to 1.9 times. Because the Sun exerts so much force on the moon, it will also exert a great force on spacecraft trajectories in and around Lunar orbit. This is especially important as NASA looks to create a semi-permanent cis-Lunar presence as part of the Artemis missions, so in this post, we’ll look at a way to include the effects of the Sun in a reduced-order dynamical system, the Bicircualr Restricted 4 Body Problem (BR4BP).
What is the BR4BP?
In the past, we’ve used the Circular Restricted 3 Body Problem (CR3BP) to uncover structures in 3 body problem. There, 2 large bodies (like a planet and moon), circularly orbit about their combined barycenter unaffected by the third body, which is much smaller than the first two (like a spacecraft or comet). In the Bicircualr Restricted 4 Body Problem (BR4BP) we introduce a fourth body that orbits ina circular orbit about the initial barycenter.
This equations of motion for this system has been written about before, although I enjoy this masters thesis (Starting on page 22)
What Changes
Now, thanks to the fourth body’s positions, the system is no longer autonomous. additionally, instead of being points fixed in space, the Lagrange points are now closed orbits. Note how the L1 Lagrange point has a winding number of -2 (two clockwise turns to return to close)
Additionally, the formally closed orbits in the CR3BP like the Lyapunov orbits are no longer closed
While correcting for the solar perturbation would be trivial, any correction would only close a single orbit. The period of the Lyapunov orbit and the solar orbit are not multiples of each other so the sun would be in a different location and the trajectory would diverge on the second orbit. Starting from the same initial conditions, but with the sun at different locations produces the following plot.
If we then plot the non-dimensional error for position and velocity at the end, it is clear that velocity dominates. Also, note the close, but not quite symmetrical behavior. This likely comes from the fictitious force generated by being in the synodic frame imparting a bias towards counterclockwise rotations.
Stable Orbits
Now, just because our formerly closed orbits, are now open, this doesn’t mean there aren’t stable orbits. For example Take the following DRO, it is stable for approximately 100 years in the Sun-Earth-Moon BR4BP, making it ideal for long-term storage of a captured asteroid.
Another relatively stable BR4BP orbit is the NRHO’s. These orbits, while less stable than the DROs, can be maintained for small amounts of fuel. Additionally, because the L2 Southern NRHO spends most of its time above the Lunar south pole, NASA is interested in placing a space station in one of these orbits to support the Artemis missions. (There are also other benefits to placing gateway in an L2 Southern NRHO orbit like favorable geometry to avoid most eclipses)
Now, plotted together
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Alex Hoffman