Corrected entry: Whenever the Romulan mining ship lowers its laser drill into the atmosphere of a planet, the drill and chain hang straight down. Regardless of whether the ship is station-keeping over the same spot on the rotating planet or has found the proper geosynchronous orbital altitude to stay there, the lowered chain would experience orbital tidal forces sideways along its length that would not allow it to stay "straight up and down", even if one believes that it is strong enough to endure the tension involved.
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Correction: Force Fields, Shields, and Inertial Dampers all keep the Drill steady. All are regularly used technology in the Star Trek Universe.
dablues7