Now there could be a loop hole. If the Enterprise's impulse engines can generate a subspace field and allow them to get a bit faster than Warp 1 then they can make it to Delta Vega in a few days and be spared any time dilation effects. The one wrinkle to that is Warp 1 can't be = to the speed of light, something I've long suspected in TOS.
I'm pretty sure also that Warp 1 is not = to the speed of light. It probably is dependent on what is nearby. A planet or a star would drop Warp 1 to well below light speed. Out between star systems, Warp 1 could be many, many times the speed of light.
Looking back at the episode "WNMHGB"...
- There is a 0.2 stardate difference from when they head back to an Earth base and when they arrive at Delta Vega. They've traveled a "few light days from here".
- There is also a 0.2 stardate difference from when they arrived on Delta Vega to when the ship's engines are repaired AND Mitchell has been "like that for hours now".
- If the Enterprise was a "few light days from" Delta Vega then at sublight speeds (without time dilation) they'd have to take more than a few days to get there.
But a 0.2 stardate difference appears to be only a few hours or less than a day when they are planetside. That would suggest that it only took a few hours or less than a day on the Enterprise to arrive at Delta Vega and that would be FTL speeds (if no time dilation was involved).
If we were to throw time dilation in, then we can use guess times based on the dialogue. We could say the observed flight time was 72 hours and the shipboard flight time only 6 hours. That works out to be around 0.986 speed of light for the trip to Delta Vega. Certainly doable with their technology speed-wise but again, their is a dearth of examples of time dilation used for flight and it seems to run counter to the stardate comparisons between stationary and traveling times, IMHO.
Also, another problem with time dilation at that speed, a trip to the nearest Earth base is still likely 19,000 light years away and even at 0.9999c, thousands of years would've passed for the folks at the Earth base. It might not even be there when they arrive!
The only way "WNMHGB"'s scenario would work out is if she had impulse engines capable of eventually getting up to FTL speeds, IMO.
Captain's log, Star date 1312.9. Ship's condition, heading back on impulse power only. Main engines burned out. The ship's space warp ability gone. Earth bases which were only days away are now years in the distance. Our overriding question now is what destroyed the Valiant? They lived through the barrier, just as we have. What happened to them after that?
KIRK: Set course for Delta Vega.
Star date 1313.1. We're now approaching Delta Vega. Course set for a standard orbit. This planet, completely uninhabited, is slightly smaller than Earth. Desolate, but rich in crystal and minerals. Kelso's task, transport down with a repair party, try to regenerate the main engines, save the ship. Our task, transport down a man I've known for fifteen years, and if we're successful, maroon him there.
Captain's log, Star date 1313.3. Note commendations on Lieutenant Kelso and the engineering staff. In orbit above us, the engines of the Enterprise are almost fully regenerated. Balance of the landing party is being transported back up. Mitchell, whatever he's become, keeps changing, growing stronger by the minute.
[Delta Vega Brig]
DEHNER: He's been like that for hours now.