Maximum Impulse Speed, Minimum Warp Speed, and the Transition

[Donovan Loucks]

As the thread title indicates, I'm wondering about the maximum speed of impulse engines, the minimum speed of warp engines, and the nature of the transition between the two. Further, I'm wondering specifically about the TOS Enterprise and her refit as depicted in the films, though examples from other ships and other shows -- and various technical documents, canonical or not -- might be enlightening.

Unsurprisingly, different sources cite different maximum impulse speeds. I've found numbers anywhere from 25% up to 99% of the speed of light, and none seem any more conclusive than any other. Searching through the forums, this has been discussed before, and it appears there's no clear-cut answer. If anyone has any ideas, I'd love to hear them, but I suspect the answer to this is wide open, so I'll move on to my next question.

Can the warp engines be used to travel at sublight speeds? Looking at the old Franz Joseph Star Fleet Technical Manual, there's some implication of this in the "Velocity/Time Relationship" chart (TO:02:06:20). At the upper right of the diagram, there's a section that deals with "Real Time (Terrestrial)" and "Sub-light Velocity". This may just be for completeness and not bear on the functioning of the warp drive, though the chart seems to have been made to expressly address warp speeds. The Star Trek: The Next Generation Technical Manual also implies warp travel at sublight speeds when it states, "Fields below Warp 1 are measured in millicochranes."

In addition, there's an implication that warp engines can be used to travel lower than the speed of light in the very fact that the secondary hull of the Enterprise can operate separately from the primary hull. Since the secondary hull doesn't have impulse engines, it would have to travel on its warp engines alone. That is, unless separation occurred at a speed above warp and the secondary hull continued at warp, though that raises the question of how the primary hull gets clear...

Finally, what about the transition between impulse speed and warp speed? Is there a reason to get near light speed before engaging the warp engines or is it usually done just because a ship first needs to get away from an astronomical body's gravitational effects? Another reason might be to evade enemies until the warp engines can be engaged. For example, in the Federation Reference Series, Volume 3, there's a reference to "Onset Critical Momentum", which may imply that once the warp engines are "engaged", they take some time to run up. Thus, first accelerating up to near light speed may be standard procedure, but not always necessary.

So, here's what I currently think about these three questions. First, there's no clear maximum impulse speed, so it's pretty much arbitrary (as, technically, all of this is). Second, the warp engines can be used at speeds less than that of light. Third, because of the second point, a stationary ship can go directly to warp. Thus, the only reason for any "transition" between impulse and warp is either because of gravitational reasons or "run up" time. Your thoughts?

 

[Timo]

Unsurprisingly, different sources cite different maximum impulse speeds.

However, remarkably, canon sources never cite maximum impulse speeds.

Theoretically, lightspeed should be the tangential limit. In practice, various kinds of interstellar drag may result in some sort of a limiting value. But a limit never comes up in any episode or movie, and I don't see any reason for a limit (other than lightspeed, and assuming the subspace magic apparently also involved in impulse flight doesn't remove this limit as well) to exist based on current onscreen evidence.

an the warp engines be used to travel at sublight speeds?

There are two TOS episodes where the ship moves effortlessly enough, even though in one, the impulse engines of the Enterprise are down for repairs ("Obsession"), and in another, the impulse engines of the Lexington are knocked out completely ("Ultimate Computer").

Neither episode involves explicit movement at less than lightspeed - but both episodes assume the ship deprived of impulse would still be capable of meaningful combat maneuvering. But it does appear as if the wounded Lexington closes in on Kirk's ship at less than warp speed at the end of the episode, suggesting either the ability to use warp engines for sublight flight - or perhaps the repairing of the impulse engines some time prior.

the secondary hull of the Enterprise can operate separately from the primary hull.

This is never indicated in any episode or movie - and I don't think it is indicated in too many noncanon works, either. Rather, the secondary hull becomes so much useless junk when separated from the primary hull.

That is, unless we're talking about the E-D, which can explicitly operate in two pieces. But the secondary hull of that ship does have a very prominent impulse engine "nozzle", glowing red all the time.

Is there a reason to get near light speed before engaging the warp engines or is it usually done just because a ship first needs to get away from an astronomical body's gravitational effects?

If there existed a reason of any sort, Kirk would probably not be so willing to order warp straight from orbit, as he does several times in TOS. He does that on low Earth orbit in ST:TMP and ST4, too (in the latter, once with suspect Klingon hardware, the other time with a somewhat suspect Starfleet vessel). Even more to the point, Archer would not risk going to immediate warp from Earth orbit with his rather experimental ship in the pilot "Broken Bow" already.

It is somewhat odd that so many of the older literary works assume a "critical velocity" is needed when there is zero canonical call for it, then - the first canon suggestion of sublight acceleration to warp comes from ST:FC where the Phoenix seems to absolutely need such a thing before engaging warp.

Although actually, Cochrane first starts the warp engine and then begins the acceleration, making it look very much as if the acceleration is performed by the warp engine. It just takes the better part of a minute there, rather than a split second like in most later occurrences. But if we accept that the warp engine did the work there (rather than just being "engaged" and ready but not actually propelling yet, while the big rocket-like thing at the tail of the Phoenix accelerated the test rig towards a "threshold"), we can canonically argue that warp engines do offer sublight mobility.

So, here's what I currently think about these three questions. First, there's no clear maximum impulse speed, so it's pretty much arbitrary (as, technically, all of this is). Second, the warp engines can be used at speeds less than that of light. Third, because of the second point, a stationary ship can go directly to warp. Thus, the only reason for any "transition" between impulse and warp is either because of gravitational reasons or "run up" time. Your thoughts?

I'd tend to agree on all that. We have seen direct jumps from standstill to warp often enough, and never heard of an upper limit for impulse speeds, so those points seem to be close to proven. Various fictional reasons might exist for prolonged transition, though. Gravity doesn't appear to be a factor, as many a warp flight has started from the depths of a gravity well (sometimes even from inside a black hole!), but other sorts of obstacles might have to be cleared; tactical reasons might exist; and it may take time to get the warp drive working, time well spent already getting the hell out of Dodge on impulse.

Timo Saloniemi

 

[blssdwlf]

As the thread title indicates, I'm wondering about the maximum speed of impulse engines, the minimum speed of warp engines, and the nature of the transition between the two. Further, I'm wondering specifically about the TOS Enterprise and her refit as depicted in the films,
...
So, here's what I currently think about these three questions. First, there's no clear maximum impulse speed, so it's pretty much arbitrary (as, technically, all of this is). Second, the warp engines can be used at speeds less than that of light. Third, because of the second point, a stationary ship can go directly to warp.

That's along the same line of thinking that I have. For "no clear max impulse speed", I'd even argue that the TOS impulse engines were capable of low FTL speeds as well, based on "Where No Man Has Gone Before".

As Timo points out about warp engines possibly in use at sublight from "The Ultimate Computer" and "Obsession", there is also a bit from "The Mark of Gideon":

KIRK: Took the ship out of warp speed.
ODONA: Out of what?
KIRK: Space terminology. We're no longer moving faster than the speed of light. I've trimmed down to sublight speed until we find out where we are.
​

Kirk doesn't say he switched to impulse engines or disengaged the warp engines but instead reduced (presumably power setting) to slow to sublight speed.

Which leads to your 3rd point which we've seen many of times where a ship jumps to warp speed.

Thus, the only reason for any "transition" between impulse and warp is either because of gravitational reasons or "run up" time. Your thoughts?

I phrase it a little differently. More like "factors that affect acceleration between sublight and FTL" can be from the environment, tactical reasons, and also power available to the warp drive, IMHO.

 

[Ronald Held]

I see no technical reason why the warp engines cannot run at any sublight speed. Going to warp from a standstill(compared to c) has been done. Short of shielding at impulse speeds, those engines should be able to go close to c.

 

[SWHouston]

The only "limitation" that I'm aware of is elective, that being limiting the Impulse speed to .25 C.

 

[zephramc]

I agree that warp engines should be able to move the ship at any speed (sublight or FTL), however they may be more power hungry than impulse. Maybe at sublight speeds, the impulse engines can do the same work at 1/3 the energy (or whatever amount).

 

[Timo]

...Also, there are two curious cases where warp drive appears to be the wrong choice for hurrying towards the destination in an extreme crisis, and impulse is preferred.

In "Best of Both Worlds", both the Borg and the E-D drop out of warp long before reaching Earth. The Borg probably do it because they want to defeat Jupiter, Mars and other UFP installations of note before tackling Earth. Riker on the E-D makes no attempt to defend Jupiter or Mars, though, and is headed straight for Earth, yet feels that impulse is the faster way to go (even though previously this vessel has warped right down to a low orbit of a comparable planet deep within its own star system).

In "By Inferno's Light", going to warp towards Bajor's central star is considered risky in a situation where any time wasted would mean said star blowing up. Moreover, if the star did blow up, it would supposedly hurt a fleet of warp-capable starships some 2 AU away from it - that is, the wave of destruction would reach said fleet no sooner than in about fifteen-twenty minutes and still the fleet wouldn't get out of the way in time!

This appears to indicate that warp travel is sometimes very slow, perhaps owing to the local properties of subspace ("subspace weather") or somesuch. Bajor is notorious for its space weather anyway, as indicated in "Invasive Procedures" and "Things Past"; Sol might be another such place. Impulse would be preferable during bad weather, then.

Further tying into this might be the eagerness of our heroes to use shuttlecraft (seemingly at impulse) to access star systems even when they could sail all the way in using their starship (say, "The Neutral Zone", "Samaritan Snare" etc.). Perhaps the weather was bad on those particular days, and rather than have the starship struggle inwards and then outwards at warp, it was better to have a shuttle do the struggling and leave the ship free to maneuver? This would also explain many cases of shuttlecraft use in TOS.

In a careful analysis of the timing cues in the episode, "Paradise Syndrome" is another example of high warp being suspiciously slow, indeed warp nine being slower than impulse; it is also an example of this warp travel taking place deep within a star system. For whatever reason, Spock decided that high warp was a better way to move STL than impulse was...

Timo Saloniemi

 

[Ronald Held]

If one uses the Cochrane factor as a coefficient for warp speed calculations. that number may be small when the density of matter is high(in the plane of a planetary system), versus interstellar space where the density is significantly lower.

 

[Timo]

...But only at times. Most of the time, going straight to warp and achieving high speeds thereby is practicable in the Sol and Bajor systems, too.

Perhaps the local variation in subspace is more due to the energies expelled by the local star than to something as permanent as mass? As far as we know today, Sol is a fairly stable star - but perhaps things are different in the Trek universe. Or perhaps some agent, natural or not, malevolent or not, has affected Sol somehow and made it "subspatially" unstable a bit before Earth science learned to measure such things?

Timo Saloniemi

 

[Crazy Eddie]

There's something to be said for stylistic differences. TMP was the last time there was any direct implication that warp speed and impulse speeds would in any way overlap (where the very slow velocities later described as "impulse power" were produced by maneuvering thrusters). The first canon dialog for "one quarter impulse power" occurs in Wrath of Khan and is probably a meyerism that has since been grandfathered in to all subsequent trek productions.

So if we're to take this from TWOK on, I think we ought to buckle down and accept three bad-tasting truths:
1) 90% of the time, the impulse engines are used to produce very slow accelerations (a few hundred m/s) and are thus the "orbital engines" of a starship, with maneuvering thrusters used only for attitude control.
2) Warp drives spool up to a sort of "critical mass" at which it is possible to move the ship at any arbitrary speed, but it is difficult to control that exact speed except by huge factors of the speed of light. Doing so is like trying to dial the yield of a hydrogen bomb.
3) There are situations where impulse power can be used for interstellar travel (we're not sure what these are).

I'm thinking we just go the Mass Effect route and assume that the real trick to warp drive is manipulating the inertial mass of the ship. In this case, impulse engines can produce FTL velocities if you reduce the mass of the ship to a tiny fraction of its true value (although this is only ever done if your warp engines aren't available for some reason). Warp drive occurs when the engines push the ship to the point of achieving zero mass, at which point the ship couldn't move slower than light if it wanted to. Any factor above warp one has the ship achieving negative mass, probably with an inverse relationship with special relativity so that you require the same amount of energy increase to achieve higher and higher increases in speed.

 

[blssdwlf]

The first canon dialog (from TOS) for "x impulse power" comes from "The Doomsday Machine":

SCOTT: ...The best I can give you on impulse is one third power..
​

As to Warp Speed and Impulse Speed overlap, there are a couple instances post-TWOK:

• "The Voyage Home" - Klingon BOP takes a couple of minutes to leave orbit of Earth at warp speed.
• "The Undiscovered Country" - Excelsior is going home on impulse power.
• "The Undiscovered Country" - Spock's countdown rate to transporter range stays consistent between warp and impulse speeds.

And although I don't consider TOS-TFS in the same tech continuity as TNG+, TNG has made a couple of slips

• "The Best of Both Worlds" - "impulse" travel time from Saturn to Earth for both Borg and Enterprise-D calculates out to 2-3c.
• "Conspiracy" -
  RIKER: Increase to warp six.
  LAFORGE: Aye sir. Full impulse.

Precision controlling a ship at warp in TOS-TFS doesn't appear to be a problem. The E-D might have difficulty with it from the TNG episodes although several times we've seen her be able to move somewhat precisely around other ships/objects at warp although perhaps not as easily as other smaller ships. Voyager showed many aliens as well Voyager herself being able to have fine control while at warp...

Well, IMHO, of course.

 

[Timo]

"The Best of Both Worlds" - "impulse" travel time from Saturn to Earth for both Borg and Enterprise-D calculates out to 2-3c.

...Before one factors in relativistic concerns, that is. Our heroes and villains are only concerned with subjective travel time, after all.

Timo Saloniemi

 

[blssdwlf]

"The Best of Both Worlds" - "impulse" travel time from Saturn to Earth for both Borg and Enterprise-D calculates out to 2-3c.

...Before one factors in relativistic concerns, that is. Our heroes and villains are only concerned with subjective travel time, after all.

Actually, there would be no relativistic concerns involved from the E-D's POV at the time of the query.

The Borg are not at warp and their time to Earth from Jupiter is 27 minutes (that's between 1.2c-2c). The crew of the E-D are still at warp when the ETA is generated.

SHELBY: The Borg have dropped out of warp, sir. Jupiter outpost nine two reported visual contact at twelve hundred hours, thirteen minutes.
RIKER: Planetary defenses?
SHELBY: Responding. No reports on effectiveness but I can't believe that against the Borg
RIKER: Ensign Crusher, at their current speed, when will they reach Earth?
WESLEY: Twenty seven minutes.
​

 

[Crazy Eddie]

The first canon dialog (from TOS) for "x impulse power" comes from "The Doomsday Machine":

SCOTT: ...The best I can give you on impulse is one third power..
​

Not as a discrete throttle setting called for intentionally. Only one third of that power is available at all, but it's again unclear what this implies for the movement of the ship or the engine's propulsive capabilities.

As to Warp Speed and Impulse Speed overlap, there are a couple instances post-TWOK:

• "The Voyage Home" - Klingon BOP takes a couple of minutes to leave orbit of Earth at warp speed.
• "The Undiscovered Country" - Excelsior is going home on impulse power.
• "The Undiscovered Country" - Spock's countdown rate to transporter range stays consistent between warp and impulse speeds.

Those are instances where an overlap is inferred as opposed to being explicit. For example, in Elaan of Troyus Kirk orders a velocity "sublight factor point zero three five." This is probably meant to be read as "three point five percent of the speed of light" and directly suggests a range of speeds available to the impulse engines (short of FTL velocity, of course). TMP, likewise, has the impulse engines firing at "warp point five," which is clearly some sort of very high sublight velocity.

Never again, however, is there any overt reference to impulse speeds being anything more than standard orbital velocities. It all becomes either highly ambiguous (as in TUC where it's not totally clear what Spock is counting down or why) or directly suggestive of those same low velocities (e.g. every time someone calls for impulse power and the ship visibly moves at about 20m/s).

Precision controlling a ship at warp in TOS-TFS doesn't appear to be a problem.

That's because it was never necessary above warp one. The first time we ever hear references to fractional warp speeds is in The Voyage Home, where the Klingon bird of prey is struggling for every last bit of speed it can get for the breakaway timewarp maneuver. This doesn't appear to be an example of precise control so much as the Bird of Prey not really having enough power to make it to the next largest warp factor and maxing out at a sub-factor somewhere (hence in the second case, the bird of prey gets warp 7.9 when Scotty says "Mister Sulu, that's all I can give you!").

In that case, it's like trying to get a few extra megatons out of a strategic nuke by enriching the primary just a little bit more. You probably can't dial it for a specific addition (so no saying "Let's try to boost the yield by two point five six megatons") but you can get a hard-to-calculate fraction of an increase with a little more work.

The E-D might have difficulty with it from the TNG episodes although several times we've seen her be able to move somewhat precisely around other ships/objects at warp although perhaps not as easily as other smaller ships.

Instances which I tend to attribute to thruster action while at warp; I find it a bit far fetched that an engine that can propel a starship at over 500C and maintain that velocity is all that well suited for fine attitude control and relative motion on the order of a few dozen meters. That'd be like the space shuttle using its SSMEs to control a docking maneuver with the ISS.

Voyager showed many aliens as well Voyager herself being able to have fine control while at warp...

Voyager showed MANY things I prefer not to think about.

 

[blssdwlf]

The first canon dialog (from TOS) for "x impulse power" comes from "The Doomsday Machine":

SCOTT: ...The best I can give you on impulse is one third power..
​

Not as a discrete throttle setting called for intentionally. Only one third of that power is available at all, but it's again unclear what this implies for the movement of the ship or the engine's propulsive capabilities.

Sorry then, I was only replying to the "first canon dialog" in your reply. I do subscribe to the "X impulse power" being independent of actual speed.

As to Warp Speed and Impulse Speed overlap, there are a couple instances post-TWOK:

• "The Voyage Home" - Klingon BOP takes a couple of minutes to leave orbit of Earth at warp speed.
• "The Undiscovered Country" - Excelsior is going home on impulse power.
• "The Undiscovered Country" - Spock's countdown rate to transporter range stays consistent between warp and impulse speeds.

Those are instances where an overlap is inferred as opposed to being explicit. For example, in Elaan of Troyus Kirk orders a velocity "sublight factor point zero three five." This is probably meant to be read as "three point five percent of the speed of light" and directly suggests a range of speeds available to the impulse engines (short of FTL velocity, of course).

That's very likely although the use of "sublight factor" would suggest that it is not something obvious as a direct translation to x speed of light, IMO.

Never again, however, is there any overt reference to impulse speeds being anything more than standard orbital velocities.

During TOS, there were never any overt reference to impulse speeds to actual speeds, either. Even the reference to "sublight factor" doesn't directly equate to an actual speed. There are only two episodes where we hear of ships' actual speeds at sublight:

The Tholian ship on final approach at 0.51c in "The Tholian Web" and being towed by Balok's ship at 0.64c in "The Corbomite Maneuver".

It all becomes either highly ambiguous (as in TUC where it's not totally clear what Spock is counting down or why)

Spock's countdown is specifically time to transport range.

SPOCK: Coming up on transporter range in fifty-seven seconds. Transporter room. Stand by to beam down, ...fifty-three
​

or directly suggestive of those same low velocities (e.g. every time someone calls for impulse power and the ship visibly moves at about 20m/s).

Unless Kirk's called for "maximum acceleration" ala "The Squire of Gothos" then I've not seen any visible movement of the ship at impulse in TOS or even the TFS to appear to be super fast relative to a "fixed" object like a planet or base.

That's because it was never necessary above warp one. The first time we ever hear references to fractional warp speeds is in The Voyage Home, where the Klingon bird of prey is struggling for every last bit of speed it can get for the breakaway timewarp maneuver.

If it was never necessary above warp one, how does the Enterprise maneuver within hundreds of meters around V'ger in TMP? Or how does Spock use the acceleration and braking thrusters to maneuver around the sun in TVH?

 

[Timo]

RIKER: Ensign Crusher, at their current speed, when will they reach Earth?
WESLEY: Twenty seven minutes.

Good point. Had Riker simply asked "When will they reach Earth?", then the answer could indicate ETA that factors in all the flight modes that will be involved: the Borg will be at Earth at a time that will feel like 27 minutes for the E-D heroes as they pursue the Borg there. That is all that matters tactically, after all.

But Riker specifies the "at their current speed" thing, which appears to call for "objectivity" of sorts.

Spock's countdown is specifically time to transport range.

It is specifically time to the moment when transporting can be started, yes. But he isn't counting kilometers. For all we know, he knows exactly the speed profile the ship will follow when approaching transporter range, and his countdown thus is accurate even though the ship at first travels faster than light, subsequently slower, and possibly keeps on slowing down all the time. It thus doesn't follow from a steady countdown that the speed across warp and impulse flight modes would also remain steady.

The countdown isn't affected by any unforeseen speed changes, mind you. Once Chang intercepts, the countdown ceases, apparently because it becomes irrelevant.

If it was never necessary above warp one, how does the Enterprise maneuver within hundreds of meters around V'ger in TMP?

We don't know how nested warp fields behave, but merging them seems to be a difficult thing in ENT at least. It sounds quite likely that Kirk was no longer at warp once immersed in V'Ger's own warp fields. Various other means of precision propulsion would be at use, then.

Or how does Spock use the acceleration and braking thrusters to maneuver around the sun in TVH?

Good question. The movement there seems consistent with the insystem warping in "Paradise Syndrome" or "Tomorrow is Yesterday" - that is, application of high warp factors results in sublight movement when near a star. Perhaps the important thing in the time travel sense is to maintain a high warp factor field; in all other senses, the ship is just moving fairly slowly through space, and can be maneuvered using systems designed for those slow speeds?

One could argue that warp drive allows for precision maneuvering in most circumstances, but not when it is being degraded by whatever degrades warp propulsion near stars. In these exceptional cases, other means of steering have to be brought in.

Timo Saloniemi

 

[Saturn0660]

Heck i'd say you could run your warp engines at any speed. Even as slow as 0. In TNG the warp engines are always on. As in we see the glow all the time. I can't think it's a very fuel efficient way to travel. But can be done nonetheless. I guess I always thought of running the warp engine below FTL speeds is much like running your car in 1st or 2nd gear. A lot of "power" but slow and inefficient.

 

[Timo]

Even as slow as 0.

We have indeed quite explicitly seen warp engines revved up at zero speed: they fried the cheese out of the Voyager computers that way in "Learning Curve".

In TNG the warp engines are always on. As in we see the glow all the time.

...Except in "Skin of Evil" where they temporarily yank out the dilithium for a bit of refurbishment.

I can't think it's a very fuel efficient way to travel.

...Although it probably is, considering how they keep the engines on all the time!

It doesn't sound bad at all that warp fields would be used to lower the inertial mass of the ship during impulse operations, making it possible to move swiftly at STL. After all, we know for sure that subspace fields can reduce inertia, and we know warp nacelles are typically kept glowing even when warp drive as such is offline.

I guess I always thought of running the warp engine below FTL speeds is much like running your car in 1st or 2nd gear.

Perhaps it's indeed a bit like that - and sailing deep inside a star system is like driving up a really steep slope, so you either use 1st and put the pedal to the metal, or you get out and walk. Walking might in fact be faster.

Timo Saloniemi

 

[blssdwlf]

Spock's countdown is specifically time to transport range.

It is specifically time to the moment when transporting can be started, yes. But he isn't counting kilometers. For all we know, he knows exactly the speed profile the ship will follow when approaching transporter range, and his countdown thus is accurate even though the ship at first travels faster than light, subsequently slower, and possibly keeps on slowing down all the time. It thus doesn't follow from a steady countdown that the speed across warp and impulse flight modes would also remain steady.

After re-watching that part again, I'm somewhat in agreement now.

I think what threw me off is that Kirk seems to have randomly picked a time to ask Spock if they are close enough to beam down (while at warp) at about 2 minutes ETA. Then he immediately says "Go to impulse power for Khitomer orbit" which if done immediately would not have been something Spock could have pre-calculated into his countdown.

KIRK: Close enough to beam down?
SPOCK: Not yet, Captain. In two minutes, ...one fifty-eight..
KIRK: Go to impulse power for Khitomer orbit.
​

But, if I read it literally, then they could have stayed with warp drive all the way till the point right before orbit and when the BOP fires on them. Then the warp drive could've been running at sublight speed based on the FX...

 

[Donovan Loucks]

Timo,

First, thanks as always for your very thorough and thoughtful response! I only joined The Trek BBS five months ago and you’re one of the many experts I already respect greatly for your knowledge. Now, I’m by no means as intimately familiar with the episodes as you are. But I’ve gone through the transcripts and watched the pertinent points in the shows and have several comments on your points.

There are two TOS episodes where the ship moves effortlessly enough, even though in one, the impulse engines of the Enterprise are down for repairs ("Obsession")...

The impulse engines start out being down for repairs, but Scotty says the Enterprise will "be ready to leave orbit in under half an hour." There doesn’t seem to be any indication that they’re still out when Kirk later says, "Prepare to leave orbit." The fact that the vent is open doesn’t seem to imply that this makes the impulse engines non-functional. When we come back from commercial, the Enterprise is at warp 8, so we don’t know what sort of transition took place. In fact, Scotty’s earlier statement makes the strong implication that the impulse engines are necessary to leave orbit.

...and in another, the impulse engines of the Lexington are knocked out completely ("Ultimate Computer").

Spock never says that the impulse engines of the Lexington are "knocked out completely". Instead, he says there is "possible damage to her impulse engines." He does go on to say that "She’s still maneuverable on warp drive". Since the Lexington may be at warp to begin with it doesn’t seem to matter at the moment even if the impulse engines are knocked out.

If there existed a reason of any sort, Kirk would probably not be so willing to order warp straight from orbit, as he does several times in TOS. He does that on low Earth orbit in ST:TMP and ST4, too (in the latter, once with suspect Klingon hardware, the other time with a somewhat suspect Starfleet vessel).

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.

I checked out Star Trek: The Motion Picture, and didn’t see the Enterprise leaving Earth orbit under warp. Kirk does order "warp point five", but at that point they’re under impulse. Soon after this, Kirk states in his log, "In order to intercept the intruder at the earliest possible time, we must now risk engaging warp drive while still within the solar system." Given the urgency of reaching V’ger, it seems reasonable to assume that "warp point five" was maximum impulse speed.

Note, however, that as the Enterprise goes to warp, Sulu then calls out the speed: "Accelerating to warp one, sir. Warp point seven. Point eight. Warp one, sir." If maximum impulse speed was "warp point five", then under what motive force is the Enterprise accelerating? The obvious conclusion is that it’s the warp drive, echoing your belief that sublight speeds are possible using warp drive. (However, note that it’s possible that "warp point five" is simply the maximum speed within a system and the above acceleration is under impulse power…)

In Star Trek IV: The Voyage Home, the Klingon Bird-of-Prey obviously goes to warp within the solar system. After all, using warp speed within the system to time travel is the whole point of the movie. But this is the Bird-of-Prey, not the Enterprise. There may be something unique about this vessel, perhaps even related to its diminutive size, that permits it to go to warp from within orbit.

In a careful analysis of the timing cues in the episode, "Paradise Syndrome" is another example of high warp being suspiciously slow, indeed warp nine being slower than impulse; it is also an example of this warp travel taking place deep within a star system.

You’re correct that in this episode high warp appears to be too slow, or that full impulse is too fast. If you assume that the Enterprise has been going at warp 9 for "several hours" (to quote Spock), then things just don’t sync up. However, no one ever actually says the Enterprise has been going that speed the whole time. Perhaps the Enterprise needed to leave the system under impulse and that took much of that time. For example, the distance from the sun to Pluto would take over five hours at the speed of light.

Let’s assume, based on my comments above regarding Star Trek: The Motion Picture, that full impulse speed is one-half light speed. If the Enterprise traveled five hours at that speed to leave the system and a mere one hour more at warp 9, it would travel about 30.5 light-days in a total of six hours. (This, of course, assumes that warp 9 really is 729 times the speed of light.) The return trip on impulse alone would therefore take 61 days -- close to the 59.223 days that Spock cited.

All of this also assumes that both trips are in straight lines and, for the same reason that the Enterprise has to leave a solar system to engage warp drive, it might mean that the first route wasn’t direct but the return route was. The fastest way to get to the asteroid in "The Paradise Syndrome" with both impulse and warp drives functional might be to head in the opposite direction of the asteroid to leave the system, and then skirt the system on warp drive. Since the warp engines are then damaged when attempting to divert the asteroid, the return trip is in a straight line under impulse power.

In short, I agree with you that there doesn’t seem to be a maximum impulse speed (other than light speed) and that the Enterprise can move at sublight speeds using warp. However, in the cases you cite, the Enterprise doesn’t go to warp within the solar system or even directly from orbit (though the Bird-of-Prey does). Of course, there may be other examples that demonstrate otherwise.