What rotations occur due to gravitation for a body placed/anchored at Lagrange point $L_1$ in the Sun-Earth system, which is an unstable equilibrium point?
Is the rotation axis vector normal to the solar planetary system ecliptic plane?
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L1 is a saddle point in the potential:
(From here)
As such, a body at L1 feels tidal forces.
An extended body will tend to align due to these tidal forces.
This is similar to behavior near just one planet, where extended bodies align up and down. In close orbit, like ISS, the stable orientation is to rotate once per orbit, so that “up and down” are always toward the planet.
The same will be true for L1: tidal forces will tend to align an extended body so that it’s turning once per orbit.
Bob Jacobsen
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I have not checked the magnitudes numerically, but I do know that solar radiation pressure causes measurable torques on spinning spacecraft buses orbiting L1 (e.g., nutation of ecliptic-pointing spin axis with ~yearly period of oscillation). Corrections for tidal forces I have not seen but this may be part of the inherent instability of the orbit (in addition to its saddle-point potential geometry). L1 is far enough away that I am not even sure if the variation of the Earth-moon system over 28 days noticeably affects spacecraft orbits. I suppose one could play with Mathematica to find out. – honeste_vivere Jan 20 '20 at 19:45