Stability of extended rigid bodies around Lagrange-points $L_1$, $L_2$ and $L_3$

Question

$L_1$, $L_2$ and $L_3$ are known unstable for point-like bodies, cf. e.g. this Phys.SE post. They assume for point-like objects, which might not cover all practical scenarios. For example, there are sci-fi like ideas about huge solar sails protecting a planet from its star.

However, a non-point-like body has also much more degrees of freedom (3 positional and 3 orientational, also their time-derivatives, sum is 12), but also a literal infinity of possible forms.

Particularly $L_1$ and $L_2$ looks quite interesting.

The idea is, that while a point-like body is known unstable in $L_{1-3}$...

...maybe a non-point like body has a different potential well:

Is there some result known about it? How could such an object look, what might be its size limits?

Answer 1

This is a very interesting question and it seems that the answer is quite complicated. There is a ~400 page report from the ESA on this topic, Dynamics and Stability of Tethered Satellites at Lagrangian Points (PDF link). It covers both inert and electrodynamic tethers alongside their control mechanisms, stability conditions, exploratory applications, calculation of forces and geometries, etc.

Briefly, the answer is yes — an extended body can have a stable orbit near a colinear Lagrange point. From the attached report's executive summary:

[An] extended body like a tethered satellite has a specific behavior at a Lagrangian point... For non-negligible values of the ratio $L/R$ (where $L$ is the tether length and $R$ the distance of the spacecraft from the center of mass of the two primaries) the effect of the higher-order terms of the mutual gravity field play a role in the dynamics and stability of the satellite near a Lagrangian point. These terms can be exploited (by changing the tether length) to stabilize the orbit of a tethered satellite in the proximity of a collinear Lagrangian point that would be otherwise unstable for a compact-body satellite.