According to this Memory Alpha article
, the Enterprise orbited Excalbia at an altitude of almost 643.5 miles. If we assume that this orbit was a "powered" one, and that it would be typical of "powered" orbits, would it require huge amounts of energy to sustain? I could see this notion of "powered" orbits being used, but I would think it would have to be arranged to avoid excessive power consumption.
Momentum does part of the work of keeping the ship from falling out of orbit. The higher the orbit, the less power is required. How much power must be applied depends on the planets's rotation you are matching. Faster rotation means less power needed. You could think of it as flying backward within your orbit in relation to the advance of the area you want to scan, since an unpowered orbit would take you around too fast in most cases. And you orbit above the planet's equator, maintaining the desired distance from the spot of interest, albeit at an angle. If you don't need to stay in sight of a certain spot, you can move to an orbit 600 miles above the surface and coast until the plot thickens.
The Space Shuttle orbiter flies forward within its initial orbit to catch up with the ISS but uses little power to do that and usually takes many hours (many trips around the world)
to match the orbit of the ISS.