Synchronous orbit
Synchronous orbit was a spatial relation in which an object orbits a massive body (usually a planet) in the same period that the body rotates, and does so in the same direction, often at an equatoral incline. To an observer an object so orbiting would generally appear to oscillate north and south (in a figure eight, teardrop, or elliptical pattern) before returning to its original position at the completion of one day. The term geosynchronous orbit generally refers to such an orbit around Earth or Earth-type planets. A (geo)synchronous orbit positioned in an equatorial orbit was known as a (geo)stationary orbit.

The  maintained a "synchronous orbit over the capital city of " in 2268. () 

The Vaadwaur placed a sentry satellite in orbit directly above their capital city. ()

A space elevator or orbital tether was a particular type of stationary space station, connected to the equatorial surface by an extremely long cable. The Nezu of the Delta Quadrant used such structures in their colonizations. () 

USS Voyager parked in a high synchronous orbit over Kelemane's planet in 2375, at an altitude of 57,000 kilometers above the planet's equator. ()

Appendices

Background information

Satellites, stations and ships orbiting in lockstep with Earth rotation are said to be "geosynchronous"; those about Mars "areosynchronous". 

These terms can however be used loosely, to describe orbits about other planets. In the  novel Captain's Glory by William Shatner and Garfield and Judith Reeves-Stevens, James T. Kirk found himself in geostationary orbit above . The novel Spock's World used "hephaistosynchronous". 

According one draft of the  for , the harmonic resonance Wesley Crusher and Salia heard while observing Rosseau V was caused when "the orbits of the moons [became] synchronous," rather than to say that "the harmonic resonance from the neutrino clouds will become synchronous"

Script orbital references 
Several script references were made to vessels being in (geo)synchronous of various planets, that were not stated as being as such in the following episodes' dialogues:

 In the  for , the , the , and the space vessel lifeform were described as being in a geosynchronous orbit above Cygnus IV. It is said in the script that, with the Enterprise-D at this orbit, the ship required minimal monitoring.

 In the  for , the Enterprise-D moved into a synchronous orbit with the  over Braslota II.

 In the  for , the Enterprise-D moved into a synchronous orbit over Alpha Onias III.

 In the  for , the Enterprise-D was in synchronous orbit over Tarchannen III.

 In the  for , the Enterprise-D moved into a synchronous orbit around Kaelon II.

 In the  for , the  was described as being in geo-synchronous orbit of Gaia.

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Astronomy