Is it possible to have a satellite (natural or artificial) in orbit around a planer which is tidally locked to its sun?
1$\begingroup$ TL;DR no, that orbit would be outside the planet's Hill sphere thus unstable, and the satellite will eventually start orbiting the sun instead of the planet. See here, regarding orbital period vs year, for example, and referenced links. $\endgroup$– VesperFeb 8 at 13:35
Yes, kind of.
"Geostationary orbit" in such a case would be an orbit around the Sun at the Lagrange points 1 or 2.
(distances in the image are not in the right scale, the points L1 and L2 are much closer to the planet)
Such a satellite would always be above the same spot of the planet. Its orbit around the Sun would last exactly as long as the orbit of the planet. Its orbit around the planet would be "infinitely long", so the satellite would not move in the planet's sky. Therefore it would work just as a satellite in geostationary orbit. As for Earth the points L1 and L2 are about 1.5 million km away.
EDIT: As others have pointed out, orbits at L1 and L2 are not completely stable, so orbital adjustments would have to be made. The James Webb telescope is at the L2 point A space elevator between the hypothetical planet and the L1/2 points would be possible, and be just as good as a space elevator in geostationary orbit (if you can make a long enough tether).
$\begingroup$ If L1 and the planet complete an orbit in the same time, something is wrong: the square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit. $\endgroup$– L.Dutch ♦Feb 8 at 13:55
5$\begingroup$ "The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit." - That's not taking into account the gravity of a third body, which the Lagrange points specifically do. But this answer does neglect to mention the instability of the L1 and L2 points...I'm not sure how long a satellite would last in those "orbits" without station-keeping. $\endgroup$– QamiFeb 8 at 13:59
2$\begingroup$ You may prefer to place it at Lagrange points 4 or 5, around which stable orbits are possible. Spacecraft at L1 or L2 need station-keeping engines. See for example how the James Webb Space Telescope does it around Earth's L2. (But that's not a show stopper, because if you want the geostationary satellite to be truely stationary you need station keeping engines anyway.) $\endgroup$– AlexPFeb 8 at 14:04
1$\begingroup$ The JWST is at Earth's L2 with respect to the Sun; the entire point is to make it stay in Earth's shadow, because it works in infrared and needs to be as cold as possible. $\endgroup$– AlexPFeb 8 at 18:25