Donovan:
Good points about both the TOS "no impulse, warp okay" episodes. Arguably, impulse engines might still be necessary for maneuvering at warp.
Although Kirk may "order warp straight from orbit", that doesn’t necessarily indicate that the Enterprise immediately goes to warp. Instead, perhaps the helmsman’s standard procedure is to accelerate under impulse power until the Enterprise leaves the gravitational effects of the system (or the warp engines are ready), at which point they’re engaged.
Agreed; this probably applies to most of the closing scenes of TOS.
I checked out Star Trek: The Motion Picture, and didn’t see the Enterprise leaving Earth orbit under warp
This happens at the very end of the movie ("Thataway").
Perhaps the Enterprise needed to leave the system under impulse and that took much of that time.
But early on, Spock had made it clear that the only way to intercept the asteroid was to "warp out of orbit". And since he also specifies this has to happen within thirty minutes, he's clearly assuming maximum possible speed of warping out or else there would be variables allowing for a longer stay. So the question goes, do the heroes understand without saying that warping out of orbit at maximum possible warp in fact means warping out at less than warp nine, for technical reason X? This is somewhat unlikely in light of Spock's subsequent log about how it is "imperative" to maintain maximum warp against all recommendations.
The return trip on impulse alone
...But that's the interesting thing: Spock doesn't return at impulse. He coasts all the way, the ship's bow remaining pointed more or less at the asteroid and the distance remaining constant.
Spock says he wants to "resume heading", but in the same phrase he also emphasizes the asteroid will remain four hours behind the ship "all the way", indicating either that the ship will not be accelerating at all, or that the asteroid will be accelerating (beyond the freefall accelerations affecting both objects, that is). He further establishes that he considers a rescue time window of four hours to be "another calculated risk", thus probably a choice.
Perhaps there exists an absolute speed limit there for some reason? Say, dense interplanetary medium that can only be pushed aside with the raw power of a warp field? Full impulse thrust against the medium would thus result in zero speed gain, so it's best to remain in the asteroid's protective shadow (which for some reason is in front of it!) and coast...
That sounds contrived, though. It already appears that Spock doesn't want to get to the planet quickly - there's no point until and unless he has solved the riddle of the obelisk, and he calculates this will take him two months anyway. So, coasting is a likely explanation, likelier than propulsive shortcomings.
As for the speeds and distances involved, supposedly the asteroid moves at a respectable fraction of lightspeed or else deflection at two-month distance should be a breeze, even for an object almost the size of the Moon. But not at a high fraction, because the asteroid is said to block out the sun just before it hits. Incidentally, this may mean the asteroid is going to go rather straight past the sun, which (if we assume the sun is creating the warping difficulty) must pose maximal propulsive problems for our starship, both out and in.
...head in the opposite direction of the asteroid to leave the system, and then skirt the system on warp drive...
Makes good sense.
apparent time has slowed for the crew so it seems like it will take minutes
...While for the chronometers, the time has slowed even more and is going backwards. I really wonder how the chronometers get their time cues! Supposedly, they're in synch with "outside time" somehow, and ignore the passage of time aboard the ship - another nifty built-in capability that might suggest relativity is a regular concern for starships.
Timo Saloniemi