How can Earth-Sun Lagrange points L1 & L2 even be semi stable considering the moon?

Question

I know that the Earth-Sun Lagrange L1, L2, and L3 points are not considered stable over longer periods, especially when compared to L4 and L5... But, with the moon orbiting the Earth in the general path of the ecliptic, it would seem that the first time the moon got in between the Earth and the object at the L1 or L2 points, it would be enough to perturb the object a good bit, and the next lunar month would be even worse, and so on.

It seems that if the moon was taken out of the equation, the L1 and L2 points would be much more stable. And, as a result of this line of thinking, I would think that the L3 point would be much more stable than L1 and L2, since the moon would be so much farther away and unimportant, so its effect could be ignored.

Am I wrong in thinking this? Is the moon's influence not large enough to be a major factor? And, just to clarify, I am talking specifically about L1 and L2 with respect to a large 3rd body, like the moon, instead of some idealized 2 body only system.

Answer 1

The L1, L2, L3 points are not stable. (full stop) Small deviations grow exponentially, even in the perfectly circular restricted three-body problem. In reality, we have a non-circular many-body problem, when (1) these points are not strictly well-defined and (2) deviations from the simple case have to be accounted for to obtain the exact trajectories (and to keep any artificial satellites near these points).

The L1 & L2 points are attractive for space missions exactly because they are unstable, since this prevents the accumulation of debris (natural and artificial in origin) and hence reduces the probability of collisions.

Answer 2

Problem solved (well, partially). A project at colorado.edu goes into horrific detail, and end with the conclusion,

Lagrange points are a feature of the three-body problem being the equilibrium points from the CRTBP equations of motion. The collinear points are considered unstable and any perturbations for the exact location will cause exponentially departure. As such, fourth-body perturbations should be considered. We should have particular care because the Sun-Earth-Moon system is one such case. An initial look into the effect the Moon has on the Earth-Sun L1 and L2 points shows little concern. The mass ratio changes as little as 1% and even in the original case the Sun completely out-masses the other bodies. The positions of L1 and L2 change by a few thousand kilometers which is small in comparison to being approximately 1.5 million km away from Earth. Looking at the individual components of acceleration did not help. The Moon is about 2% of the Earth’s acceleration at its best (closest) and 3/4% at its worst. Compiling all the results I conclude that the Moon has about 1% of the effect that the Earth does on these Lagrange points. This seems small however over time this can add up in station keeping costs. Additionally, sensitive equipment such as the James Webb Space Telescope require steady orbits and knowing the perturbation will help keep it steady. In summary, the Moon does not have a large effect but a time aspect must be considered. Further work should include the four-body problem to completely understand changes to these equilibrium points.