How do the Moon and other planets affect Earth-Sun Lagrange points' locations?

Question

We can calculate Earth-Sun Lagrange points based on Sun & earth mass/gravity. However moon and other planets must be affecting the location of these points. How this effect is analyzed and exact Lagrange points location is calculated considering moon & other planets gravity?

Answer 1

Lagrangian points are mathematical entities, and unless a specific list of criteria are met, they simply don't exist. The constraints are those of the circularly restricted three body problem (CR3BP), and in real-life situations there are imperfections.

In face of those imperfections, the theoretical stability properties of some specific points in space no longer hold. Sure, something located in that approximate region (both spatially and with the correct velocity) may hang around for a long time, but the difference between such a location and say a kilometre farther away is merely one of duration, not a yes/no for "is this a Lagrangian point?".

The effects of other planets are usually modelled as perturbations. Such models work well when the perturbing force is much smaller than the main forces affecting an orbiting object.

In fact, the gravity of the Sun is usually so much stronger than everything else that the orbit can be treated as a simple solar orbit (that is, no 3-body mechanics), with the gravity of some planet just being another perturbation. (This especially for L3, L4 and L5, where the secondary mass is very large)

Planetary orbits diverging from a perfect circle can also be modelled as perturbations. Readers that have been around for a couple of millennia may be familiar with this in the form of deferents and epicycles...

For the specific example you mention of the Moon affecting Earth-Sun L-points, keep in mind that the distance between the Earth and the Moon is almost three orders of magnitude smaller than the distance between the Earth and the Sun. For all practical purposes, treating the Earth-Moon system as a single point mass is sufficient (I even did some simulations for this once. The "wobbliness" the Moon adds is usually pretty balanced).

The Moon is actually about as bad as it gets when it comes to orbiters being significant in size compared to their primaries (and at the same time distant). The perturbation effects from moon systems will be much smaller in almost any other scenario you may encounter.

Modelling trajectories and orbits around imperfect Lagrangian "points" is a rich field, with various numerical methods often being the tool of choice (not trying to discredit the analytical approaches).