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Does there exist two orbits synchronized so that a telescope in one would line up with a star shade in the other on a line pointing to a certain star? And this during a period of time relevant for great astronomical imaging, such as hours or days. And in the Solar system.
Is there any deeper meaning to why it is so hard or impossible to put three points on a straight line in this spacetime?
Well, in theory you could launch a telescope to some orbit and a starshade to the orbit obtained by reflection in the plane that passes through the central body perpendicularly to the direction to the object you want to observe:
Then, assuming perfectly Keplerian orbits, if you synchronize the telescope and the starshade to move perfectly symmetrically, then the line which passes through them will always point to the observed object. (Of course, if they move perfectly symmetrically, then they will collide in the points where their orbits inntersect, so there's that.) Part of the time, the telescope will be in front of the starshade, though, but you can make this fraction small by making orbits highly elliptical.
These two orbits don't have to lie in the same plane, they just have to be symmetric to each other w.r.t. the plane perpendicular to the direction to the star. If you make the orbits non-coplanar, then the central body won't block the view when the telescope and the starshade are in apogee.
I doubt that this would work in practice, though. Influences of other celestial bodies and irregularities in the gravity of the central body would affect the orbits, and the direction of the line connecting the telescope and the starshade will change. But I don't know how quickly.
As far as I can tell, this is the only solution under the assumption of perfectly Keplerian orbits. If the direction of the line that passes through two orbiting objects stays constant over some period of time, then the components of their accelerations perpendicular to that direction must be the same. This is possible only if they are at the same distance from the central body, i.e., they are symmetric to each other w.r.t. to the plane passing through the central body and perpendicular to the line connecting the two objects.
I do not think the problem is the orbit here. What you really need is the capacity for both satellites (starshade and telescope) to fly in formation (at least until they are properly positioned) in order to align themselves over the telescope-star axis, and point in the right direction.
An example would be the Proba3 mission of the ESA. Proba3 is a formation flying mission (to be launched) comprised of two satellites, an occulter and a coronograph.
As the names indicate the occulter will occult the sun and the coronograph will be used to study the solar corona. However, because it is mainly a demonstration mission (i.e. meant to test new technologies), the orbit of the system formed by the two satellites will be HEO, not an orbit around the Lagrangian L1 as many other sun observation missions).
