I didn't complete my first answer so here is a second:
This is a long post. Eventually it will come to the part where there is a problem that can be solved several ways. One of the possible solutions is using the impulse reactor to power the warp drive. Anyone who rejects the other solutions is forced to accept that the impulse reactor was used to power the warp drive at least once.
The data about interstellar travel from the earliest episodes of Star Trek is rather hard to interpret.
Part 1 Time Warps.
In "The Cage"/"Menagerie" Captain Pike and Doctor Boyce discuss:
BOYCE: Sometimes a man'll tell his bartender things he'll never tell his doctor. What's been on your mind, Chris, the fight on Rigel Seven?
PIKE: Shouldn't it be? My only yeoman and two others dead, seven injured.
Later Pike relives the fight on Rigel Seven:
PIKE: It's starting just as it happened two weeks ago. Except for you.
In the first scene Pike decides to continue to the Vega Colony:
SPOCK: We aren't going to go, to be certain?
PIKE: Not without any indication of survivors, no. Continue to the Vega Colony and take care of our own sick and injured first. You have the helm. Maintain present course.
These bits of dialog indicate that the Enterprise was travelling from the star Rigel (Beta Orionis), then believed to be about 500 to 1,000 light years from Earth, to the star Vega (Alpha Lyrae) which is about 25 light years From Earth, and the trip already lasted about two weeks.
If "two weeks" equals 11 to 17 days, and the total distance is about 500 to 1,000 light years or 182,625 to 365,250 light days, then speeds of about 10,742.6 to 33,204.5 times the speed of light would be needed. But since the trip was not yet completed after 11 to 17 days, the average speed of the Enterprise during the voyage must have been less than 10,742.6 to 33,204.5 times the speed of light.
Pike does decide to make a detour to Talos IV 18 light years away to search for survivors of the Columbia. With speeds an unknown amount less than 10,742.6 to 33,204.5 times the speed of light, the trip to Talos IV will take an unknown about of time more than 0.198 to 0.612 days.
At Talos IV Pike Tells the Talosians:
PIKE: Can you hear me? My name is Christopher Pike, commander of the space vehicle Enterprise from a stellar group at the other end of this galaxy. Our intentions are peaceful. Can you understand me?
If they have traveled to the "other end of the galaxy", a distance of at least 20,000 light years (or 7,305,000 light days) and possibly several times as far, in less than one week, the average speed in the voyage to Talos IV must be at least 1,043,571.4 times the speed of light.
But Pike must be lying to the Talosians, since a star group 18 light years from the line from Rigel to Vega must be less than 1,000 light years from Earth.
Pike orders the 18 light year trip to Talos IV to be made at time warp factor 7, which may be faster or slower than the speed for the rest of the time from Rigel to Vega.
And Pike mentions "time warp" in both "The cage" and "The Menagerie Part 1":
PIKE: Address intercraft.
TYLER: System open.
PIKE: This is the captain. Our destination is the Talos star group.
PIKE [on screen]: Our time warp, factor seven.
The "time warp" of Pike's Enterprise is totally canonical in TOS.
In "The Cage" and also "The Menagerie Part 1" they talk to survivors on Talos IV:
SURVIVOR: Is Earth all right?
PIKE: The same old Earth, and you'll see it very soon.
TYLER: And you won't believe how fast you can get back. Well the time barrier's been broken. Our new ships can
This shows that the "time barrier" - whatever that is - has been broken comparatively recently and new ships, presumably including Pike's Enterprise, can make interstellar voyages in much less time than before. This may mean that the new ships actually travel much faster, or that they use some sort of "time warp" to make time pass much slower aboard them and thus make the voyages seem shorter, or something else I can't think of.
But use of time warps in Pike's era is clearly established by the first pilot episode in dialog that was quoted in "The Menagerie Part 1" and aired back in 1966.
"The Alternative Factor" has some interesting dialog:
BARSTOW [on viewscreen]: You may not be aware of its scope. It occurred in every quadrant of the galaxy and far beyond. Complete disruption of normal magnetic and gravimetric fields, timewarp distortion, possible radiation variations. And all of them centring on the general area which you are now patrolling.
So what were the "timewarp distortions"? Were they weird phenomena that distorted and warped time, or were they weird phenomena that distorted the effects of Federation time warp technology and made it warp time in unintended ways instead of desired ways?
In "The Naked Time" Spock invents a new formula to rapidly start the warp engines of the Enterprise with an "implosion" to escape a peril.
SULU: Captain, my velocity gauge is off the scale.
SPOCK: Engine power went off the scale as well. We're now travelling faster than is possible for normal space.
KIRK: Checked elapsed time, Mister Sulu.
SULU: My chronometer's running backwards, sir.
KIRK: Time warp. We're going backward in time. Helm, begin reversing power. Slowly.
Obviously that particular time warp that sent the Enterprise back in time was something new. But if the warp drive normally warps time in some way that would make its ability to travel back in time in unusual circumstances much more plausible.
It certainly seems possible that the Federation uses time warps in Kirk's era.
In the era of TNG, Worf's son Alexander, Molly O'Brien, and Naomi Wildman seemed to grow older super fast in their early years and then grow older at more normal rates later. And anyone trying to make TNG era programs seem as plausible as they can must be very irritated by what seems to be examples of SORAS (Soap Opera Rapid Aging Syndrome) in their science fiction shows. Warning! TV Tropes link:
But the aging of those characters doesn't have to be examples of SORAS. Soap Opera Rapid Aging Syndrome is never commented on by the characters or explained in any way, as the writers like to pretend nothing happened. But in some stories kids rapidly age with some sort of magical or scientific explanation or are actually seen being aged by some process. This is called Plot-Relevant Age-Up.
Warning!: TV Tropes link:
Fans of TNG era shows should prefer Plot-Relevant Age-Up to Soap Opera Rapid Aging Syndrome. Bu there is no magic in Star Trek. If only TNG-era Star Trek programs happened in a society with technology advanced centuries beyond our own to make Plot-Relevant Age-Up seem plausible. Wait, they do happen in a society with science and technology centuries beyond our own. Star Trek fans are thus free to come up with technological explanations that make Alexander, Molly, and Naomi examples of Plot-Relevant Age-Up instead of Soap Opera Rapid Aging Syndrome.
My theory is that many Federation citizens and especially Starfleet members take time off from their jobs when their children are born. They stay inside special quarters with accelerated time rates for months or years of time inside until their babies grow into toddlers that don't need so much parental attention. Then they turn off the time warps and return to their jobs after perhaps a few days have passed outside.
That might also explain why Amanda seemed a bit young to have a son as old as Spock. This has led me to imagine 23rd century tabloid headlines saying: "Alien Ambassador Elopes With Teenage Earth Girl". Perhaps Spock (and Amanda) aged super fast in a time warp to grow him from a baby to a toddler, and after Spock later left home for Starfleet Academy Amanda used time warps to freeze time for herself every other day, so that she would age not much faster than Sarek instead of dying of a old age when he was still young.
So IMHO Rick Sternbach and Michael Okuda should have included time warps among Federation technologies in the Star Trek: The Next Generation Technical Manual in 1991. And if they didn't want to use time warps as part of the warp drive in the TNG era, fine. But they still had to explain how time warps were part of the warp drive in the era of Pike's Enterprise and explain why time warps were no longer part of the TNG era warp drive.
Part 2: "Where No Man Has Gone Before".
The second pilot episode begins with:
Captain's log, Star date 1312.4. The impossible has happened. From directly ahead, we're picking up a recorded distress signal, the call letters of a vessel which has been missing for over two centuries. Did another Earth ship once probe out of the galaxy as we intend to do? What happened to it out there? Is this some warning they've left behind?
And a few minutes later on the bridge:
KELSO: Screen on, sir. Approaching galaxy edge, sir.
KIRK: Neutralise warp, Mister Mitchell. Hold this position.
MITCHELL: Neutralise warp, sir.
KIRK: Address intercraft.
MITCHELL: Intercraft open.
KIRK: This is the Captain speaking. The object we encountered is a ship's disaster recorder, apparently ejected from the S.S. Valiant two hundred years ago.
SPOCK: The tapes are burnt out. Trying the memory banks.
KIRK: We hope to learn from the recorder what the Valiant was doing here and what destroyed the vessel. We'll move out into our probe as soon as we have those answers. All decks, stand by.
So the Enterprise has reached the edge of the galaxy and found that the S.S. Valiant was there 200 years earlier. The outermost part of our galaxy is a very thin spherical halo about 200,000 light years in diameter and the edge of the halo should be about 75,000 to 125,000 light years from Earth. but the main part of the galaxy is the galactic disc about 100,000 to 120,000 light years in diameter and quite thin. If the edge of the galactic disc is meant then I will arbitrarily say the Enterprise and the Valiant should have reached it at a spot between about 500 and 85,000 lightyears from Earth.
On the bridge the department heads are introduced:
PIPER: Life sciences ready, sir. This is Doctor Dehner, who joined the ship at the Aldebaran colony.
DEHNER: Psychiatry, Captain. My assignment is to study crew reaction in emergency conditions.
Aldebaran is about 65 light years from Earth, or just about next door when traveling to the edge of the galaxy. Apparently Kirk was never introduced to a beautiful female officer on the entire trip from Aldebaran to the edge of the galaxy.
If the Enterprise traveled 500 to 85,000 light years, or 182,625 to 31,046,250 light days, in a time so short that Kirk never met Dr. Dehner, then it must have been going very fast. If we arbitrarily assume it took 0.5 to 7.0 days to go from Aldebaran to the edge of the galaxy, 182,625 to 31,046,250 light days away, the average speed of the Enterprise should have been between 26,089.285 and 62,092,500 times the speed of light.
What about the "time warp" in The Cage"? Suppose that the Enterprise used time warps to slow down the passage of time aboard ten times, so that 5 to 70 days passed in the outside universe while 0.5 to 7.0 days passed on the Enterprise in the voyage from Aldebaran to the edge of the galaxy. Then the average speed on the voyage would be between 2,608.9285 and 6,209,250 times the speed of light.
Suppose they used time warps to slow down time 100 times, so that 50 to 700 days passed in the outside universe while 0.5 to 7.0 days passed on Enterprise in the voyage from Aldebaran to the edge of the galaxy. Then the average speed on the voyage would be between 260.89285 and 620,9250 times the speed of light.
Thus, if they travel to a spot on the edge of the galaxy very close to Earth, it is possible for the speeds of the Enterprise in "The Cage" and "Where No Man Has Gone Before" to be consistent.
Soon after Dehner is introduced:
SPOCK: Decoding memory banks. I'll try to interpolate. The Valiant had encountered a magnetic space storm and was being swept in this direction.
KIRK: The old impulse engines weren't strong enough.
Kirk's statement has several possible explanations:
1) Early Earth ships used impulse drive to travel faster than light. The warp drive was invented after the time of the Valiant.
2) Early Earth ships used impulse engines to travel slower than light and used the power from the impulse engines to power the warp drive when going faster than light.
3) An early form of warp drive was called impulse warp and Kirk was referring to impulse warp engines.
4) Kirk believed the warp engines would go offline in such a situation and the Valiant had to use the impulse engines.
5) Kirk meant to say "warp engines" but said "impulse engines" by mistake.
Soon afterwards the Enterprise encounters the force field at the edge of the galaxy and is damaged.
SPOCK: Main engines are out, sir. We're on emergency power cells. Casualties, nine dead.
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?
Let us study this statement in detail.
Ship's condition, heading back on impulse power only.
This can mean that the Enterprise is only using its impulse engines, and thus is travelling slower than light, unless impulse engines can be used to travel faster than light. But since Kirk said "impulse power" this could also mean that energy from the impulse reactor is being used to power the warp engines - or what is left working of them - instead of energy from the warp drive reactor that would normally be used but is off line.
There is another mention of "impulse power", in "Balance of Terror".
KIRK: Yes, well gentlemen, the question still remains. Can we engage them with a reasonable possibility of victory?
SCOTT: No question. Their power is simple impulse.
KIRK: Meaning we can outrun them?
There seems to be three possible interpretations of This dialog:
a) That the Romulan ship only has impulse engines and travels slower than light and the Romulans can't have a large interstellar empire.
b) That the Romulan ship only has impulse engines and can travel many times faster than light, thus enabling the Romulans to have a large interstellar empire.
c) That the Romulan ship has both impulse engines and warp drive engines, but uses an impulse reactor to power both sets of engines, since they haven't invented warp drive reactors yet. So the Romulans ships travel many times the speed of light and they have a large interstellar empire but their ships are much slower than Federation ships.
Interpretation c) would be consistent with the Enterprise being able to power its warp engines with the impulse reactor.
Main engines burned out.
If the main engines are the warp drive engines, and are burned out, it seems like they can't be used at all, even if impulse power is diverted to them. But possibly the warp engines include the main space warp engines and the lesser time warp engines and only the "main [space warp] engines" are burnt out and the lesser time warp engines can still be used with power from the impulse engines. Or maybe it means that the power supply for the main engines is burned out, not the main engines themselves.
The ship's space warp ability gone.
Since "time warp" is mentioned in "The Cage" and "The Menagerie Part 1", the "space warp" in this episode could be another function of the warp drive. Thus the space warp capability of the warp engines could be out but they might still be able to generation time warps to make the real or perceived duration of the voyage shorter.
Earth bases which were only days away are now years in the distance.
Clearly Enterprise is much slower now. If "days" are 1.0 days to 7.0 days (up to one week) and "years" are 1.0 to 10.0 years (up to one decade) or 365.25 to 3,652.5 days, the normal speed of the Enterprise should be about 52.178 to 3,652.5 times faster than its speed after the accident.
If the Enterprise after the accident can travel at one tenth the speed of light, the normal speed of the Enterprise should be 5.2178 to 365.25 times the speed of light.
If the Enterprise after the accident can travel at exactly the speed of light, the normal speed of the Enterprise should be 52.178 to 3,652.5 times the speed of light.
If the Enterprise after the accident can travel ten times speed of light, the normal speed of the Enterprise should be 521.78 to 36,525 times the speed of light.
Kelso visits Mitchell in sickbay:
MITCHELL: So, er, so, how go the repairs?
KELSO: Well, the main engines are gone, unless we can find some way to re-energise them.
MITCHELL: You'd better check the starboard impulse packs. Those points have about decayed to lead.
KELSO: Oh, yeah, sure, Mitch.
MITCHELL: I'm not joking, Lee! You activate those packs, and you'll blow the whole impulse deck.
Later, in the briefing room:
KELSO: Well, it didn't make any sense that he'd know, but naturally, I checked out the circuit anyway. I don't know how, but he was right. This point is burned out exactly the way he described it.
That leaves two possibilities:
a) The ship has been using the impulse engines, but the impulse packs are used only rarely when using the impulse engines, and so had not been used since the accident at the galactic barrier.
b) The impulse packs are activated whenever the impulse engines are used, but the ship had not used the impulse engines since the accident at the galactic barrier. Thus the ship should be using "impulse power" but not the impulse engines, and seems to be using the impulse reactor to power the warp drive engines.
Spock offers a course of action:
SPOCK: Recommendation one. There's a planet a few light days away from here. Delta Vega. It has a lithium cracking station. We may be able to adapt some of its power packs to our engines.
KIRK: And if we can't? We'll be trapped in orbit there. We haven't enough power to blast back out.
This implies that the problem with the warp engines is not how they work but lack of energy, and thus that the warp reactor is offline. Thus the ship might be be using the impulse reactor to power the warp drive engines.
When they approach Delta Vega, Kirk's log entry says:
...Kelso's task, transport down with a repair party, try to regenerate the main engines, save the ship...
The fuel bins in the lithium cracking station apparently contain a lot of energy:
KIRK: Can you do it, Lee?
KELSO: Maybe, if we can bypass the fuel bins without blowing ourselves up.
KIRK: The fuel bins, Lee. Could they be detonated from here?
KELSO: A destruct switch? I guess I could wire one up right there.
KIRK: Do it.
KELSO: Direct to the power bins. From here you could blow up this whole valley.
Since the fuel bins are not emptied, the "power packs" taken from the station to the ship may be reactors or generators that use fuel to generate power.
Consider the stardate in "Where No Man Has Gone Before".
In the opening teaser:
Star date 1312.4.
After encountering the Galactic barrier:
Star date 1312.9.
Hours or days later, in the briefing room, Kirk and Spock discuss options and Kirk decides to head for Delta Vega.
The next scene is another log entry:
Star date 1313.1. We're now approaching Delta Vega. Course set for a standard orbit.
Soon after they put Mitchell in a cell in the station on Delta Vega.
The next stardate is when they are about to leave Delta Vega.
Star date 1313.3.
In the next line Dehner says:
DEHNER: He's been like that for hours now.
If 1.0 to 24.0 (up to one day) hours have passed in 0.15 to 0.25 stardate units between 1313.1 and 1313.3, there should be about 4 to 159.99 hours per stardate unit.
Right after that Kirk is Knocked unconscious. He is revived by Dr. Piper after minutes or hours. Kirk gives Piper an order before chasing Mitchell and Dehner:
KIRK: If you have not received a signal from me within twelve hours, you'll proceed at maximum warp to the nearest Earth base with my recommendation that this entire planet be subjected to a lethal concentration of neutron radiation. No protest on this, Mark. That's an order.
When Kirk confronts Michell, Mitchell creates a tombstone indicating that Kirk is about to die on stardate 1313.7. After the confrontation, Kirk calls the Enterprise which is still in orbit. A few minutes or hours after being beamed up, Kirk makes the final log entry:
KIRK: Captain's log, Star date 1313.8. Add to official losses, Doctor Elizabeth Dehner. Be it noted she gave her life in performance of her duty. Lieutenant Commander Gary Mitchell, same notation. I want his service record to end that way. He didn't ask for what happened to him.
I guess that about 1.00 to 24.00 hours pass between stardates 1313.3 and 1313.8, or in about 0.45 to 0.55 stardate units. Thus a stardate unit should be about 1.81 to 53.33 hours long. If there are 4.0 to 159.99, and also 1.81 to 53.33, hours in a stardate unit, a "Where No Man Has Gone Before" era stardate unit should be 4.0 to 53.33 hours long.
Stardates can either measure time in the outside universe which may pass at a different rate than on a starship, or they can measure time aboard a starship which can pass at a different rate than time in the outside universe.
Sometime after stardate 1312.9 Kirk decides to go to Delta Vega. On Stardate 1313.1 they are approaching Delta Vega. Thus the trip to Delta Vega takes sometime less than the entire 0.15 to 0.25 stardate units between stardate 1312.9 and stardate 1313.1. If a stardate unit is 4.0 to 53.33 hours long, the trip to Delta Vega takes less than 0.6 to 13.325 hours.
Spock says that Delta Vega is:
a few light days away from here.
A light day is the distance light travels in a day. If Delta Vega was more than 7 light days away Spock would probably call it a light week away. Thus The trip to Delta Vega should have been about 1.0 to 7.0 light days, or about 24.0 to 168.0 light hours.
If stardates measure the passage of time in the outside universe, traveling a distance of 24.0 to 168.0 light hours in 0.6 to 13.325 hours would require speeds of 1.801 to 2,800 times the speed of light. And if the normal speed of the Enterprise is about 52.178 to 3,652.5 times faster than that, the normal speed of the Enterprise should be about 93.97 to 10,227,000 times the speed of light.
Of course the trip to Delta Vega lasted an unknown amount of time less than 0.6 to 13.225 hours, and thus the speed of the Enterprise should be an unknown amount faster than calculated.
So if the Enterprise traveled faster than light to reach delta Vega, there seem like two possible methods.
1) The Enterprise used its impulse engines which could make travel faster than light. Probably when the Enterprise was repaired and updated between "where No Man Has Gone before" and the first season of TOS it got upgrade warp engines and the impulse engines were downgraded to merely slower than light speeds.
2) The Enterprise used its impulse reactor to power the warp engines which retained full or partial functionality but couldn't travel as fast as before due to the lower energy levels provided.
But what if stardates measure time aboard starships, and not the outside universe?
If so, there are three possibilities:
1) Time aboard the Enterprise passed at the same rate as in the outside universe. Thus the Enterprise traveled faster than light on the voyage to Delta Vega and the only possible methods are the same as for stardates measuring time in the outside universe.
2) Time aboard the Enterprise passed much faster than in the outside universe. Thus the Enterprise traveled faster than light on the voyage to Delta Vega and the only possible methods are the same as for stardates measuring time in the outside universe.
3) Time aboard the Enterprise passed much slower than in the outside universe. Thus the voyage to Delta Vega took much more time in the outside universe than aboard the Enterprise. That means that the Enterprise traveled much slower than it would if stardates measure time in the outside universe.
This gives two possibilities:
3a) Time traveled slower on the Enterprise, the voyage lasted longer, and the Enterprise traveled slower, but still much faster than light. And the only possible methods are the same as for stardates measuring time in the outside universe.
3b) Time traveled slower on the Enterprise, the voyage lasted longer, and the Enterprise traveled slower, at or slower than the speed of light. And presumably the Enterprise used the impulse engines to travel at or slower than the speed of light.
If the Enterprise had an average speed of one percent of the speed of light on the Voyage to Delta Vega it should take about 2,400.0 to 16,800.0 hours - 100 to 700 days. If 0.6 to 13.325 hours passed on the Enterprise the time warps would slow down time 180.112 to 28,000 times.
If the Enterprise had an average speed of ten percent of the speed of light on the Voyage to Delta Vega it should take about 240.0 to 1,680.0 hours - 10 to 70 days. If 0.6 to 13.325 hours passed on the Enterprise the time warps would slow down time 18.011 to 2,800 times.
As far as I know I am the original creator of the theory that the Enterprise has any time warp capability. Naturally I am in favor the theory that the Enterprise could have and might have traveled to Delta Vega slower than light and using time warps to slow down time aboard the ship. Other Star Trek fans might not like it that much.
What other way would be there be for the Enterprise to travel to Delta Vega slower than light with time slowed down? If the Enterprise traveled almost as fast as light time dilation could slow down time aboard the Enterprise enough.
I don't know if impulse engines can reach speeds fast enough for significant time dilation. In Star Trek: The Motion Picture the possibly highly upgraded impulse engines of the the Enterprise reached warp 0.5:
KIRK: Impulse power, Mister Sulu. Ahead, warp point five. ...Departure angle on viewer.
Warp 0.5 should be 0.125 times the speed of light, too slow for significant time dilation.
If impulse engines can reach speeds fast enough for significant time dilation, then leaving orbit around any plausible habitable planet in any plausible solar system should be incredibly easy for them.
When Spock recommends repairing the engines with power packs from the station on Delta Vega, Kirk says:
KIRK: And if we can't? We'll be trapped in orbit there. We haven't enough power to blast back out.
How could the impulse engines have enough power to accelerated to relativistic speeds with time dilation and then decelerate and yet not have enough power to reach escape velocity from Delta Vega and its sun(s)?
Maybe Delta Vega and its sun(s) orbit close to the event horizon of a black hole. But the gravity of ultra dense objects like black holes falls away rapidly with distance. I find it hard to picture Delta Vega and its suns(s) orbiting far enough from the event horizon to avoid tidal disruption and yet having such a strong pull from the black hole's gravity that the escape velocity is too high to achieve with impulse engines capable of relativistic speeds.
So the possibilities for the voyage to Delta Vega include:
1) Faster than light impulse engines.
2) Impulse reactor powering warp engines.
3) Slower than light impulse engines with time warps slowing time.
4) Slower than light impulse travel at relativistic velocities and significant time dilation.
Different persons will prefer different solutions. If 1), 3), and 4) seem impossible to someone, 2) will be the only solution left.
Thus some Star Trek fans will believe that the most likely method of traveling to Delta Vega would be to use the impulse reactor to power the warp engines.