|Warp Speed in TOS
In TOS, maximum warp appears consistently interpreted as between 200,000c and 1,000,000c for trips of up to 2000 light years. Occasional cruising speeds in the range of 1000c are witnessed, and can apparently be maintained indefinitely. An overall rate of 12,000 light years per season seems a reasonable estimate for total distance covered.
Although many would place warp 1 as being lightspeed, we see warp 1 ordered frequently in the first season - nearly the entire first half of it, in fact. We also hear occasionally of impulse vessels going in excess of the speed of light. During this season, the Enterprise seems to have traveled a geographic minimum of 6,000 light years under its own power.
"Where No Man Has Gone Before" suggests the Enterprise can go at least 12,000c on a month long trip back to Earth. In "The Naked Time," we learn that warp engines are not intended to be started cold, with a half hour warmup recommended. A hot start risks explosion, implosion, or - apparently - accidental time travel. In "Balance of Terror," we learn that warp engines can engage in a hurry normally. In "Arena" we see the Enterprise in a hurry for the first time; the speeds used seem to fall in the likely range of 40,000-100,000c.
During the first half of this season, most episodes occurred at known star systems fairly close to Earth; during the latter half, mostly untraceable generic names were used. The Enterprise covered 9,000-12,000 light years of easily traced milage in this season.
"Bread and Circuses" shows a very brief speed of ~200,000c for the Enterprise, with a wide margin of error. "Obsession" gives a longer haul speed of 300,000-550,000c."A Piece of the Action" suggests an unhurried cruise averaging ~1200c. From "By Any Other Name" we come to 1,000-10,000c for a very long range trip.
By the third season, hard and fast distances and real stars were almost completely avoided. The distance travelled in this season seems similar to that of prior seasons, but with few explicit distance figures and no familiar stars visited, it is difficult to be certain.
"That Which Survives" gives us a previously unprecedented speed of 766,000c for 990.7 light years in 11.337 hours at warp 8.4 - although the time elapsed between when Spock gives the distance of 990.7 and the ETA of 11.337 hours allows us to reduce this somewhat.
"Where No Man Has Gone Before"starts us off with the Enterprise investigating the voyages of an impulse vessel, which was lost two hundred years ago after it "Swept past this point, about 1/2 light-year out of the galaxy," They encounter the same problem as the long lost S. S. Valiant, which is to say esper-sensitives turning into godlike creatures; they make an emergency stop rather than head straight for earth, as by then, it would be too late. "In a month he'll have as much in common with us as we'd have with a ship full of white mice." We also learn Kirk has been to Deneb, 1600-3200 light years away. Traveling straight out of the galaxy either upwards or downwards in a month is about 1,000 light years in a month, or roughly 12,000c. Alternatively, if outside of the galaxy means beyond the rim of the galaxy, this figure is amplified about 25 times, to 250,000-375,000c. This would also increase the definite milage traveled under the Enterprise's own power by 40,000-50,000 light years.
"Arena" has the Enterprise, in a very brief pursuit of an enemy vessel, pass 22.3 parsecs outside of the charted limits; this suggests that, in a matter of no more than a couple hours (several minutes of onscreen time involving various weapons fire attempts) the Enterprise traveled at least 70 light years; this suggests a speed somewhere on the order of 100,000c (and possibly up to 1,000,000c, if we took onscreen time as elapsed time) as Cestus III is definitely on the charts. The return trip takes, as far as we can tell, somewhere in the neighborhood of several days to several weeks; 4-14 days gives 40,000-150,000c.
"Obsession" begins with the Enterprise expecting to meet up with another ship in 8 hours; after failing this deadline, with the other ship stationary, the Enterprise makes what is either a 1.7 day or 48 hour trip to the original location of the vampire-cloud, stated earlier as 1,000 light years from the place the Enterprise started. Given somewhere around 33-57 hours to travel 2,000 light years from the two locations it attacked in, this gives a broad range of speeds in the neighborhood of 300,000-500,000c.
"By Any Other Name" is an episode that asks a great deal of attention. The Kelvans, very advanced, have traveled from the Andromeda galaxy in less than 300 years. They modify the Enterprise to do likewise, but while in the galaxy, the speed reached is easily comparable with the above, bringing the Enterprise to the edge of the galaxy before the end of the episode. What is particularly interesting is that this episode places "Where No Man Has Gone Before" at the rim of the galaxy, which would mean beyond the arms - in other words, with a sustained return cruise in the range of 300,000c.
"Mudd's Women" has the Enterprise visiting Rigel. In "Miri" we are "hundreds of light years from Earth." In "Court Martial" the Enterprise visits Starbase 11, and subsequently revisits it and then Talos IV in "Menagerie." In "The Squire of Gothos," the Enterprise starts 900 light years from Earth; Beta 6 is eight days away at warp 3. In "Arena," while visiting Cestus III 3045.6, the Enterprise ends up 500 parsecs away from Cestus III, after chasing a Gorn vessel 22.3 parsecs outside of the charts 3046.2, and then sets course back to Cestus III. In the next episode, "The Alternative Factor," the Enterprise is within Starfleet patrolled space again and on course for Starbase 200 3087.6. "Tommorrow is Yesterday" sees the Enterprise return to Earth 3113.2 after a slingshot around a block hole on the way to Starbase 9. By "Space Seed" 3141.9, the Enterprise is again in an area where Earth ships hadn't been for years, then sets course for Starbase 12, then drops Khan & co. off at Ceti Alpha V 3143.3 (Alpha Ceti is ~150 light years from Earth); they then progress further afield in that direction in "This Side of Paradise" 3417.3 with Omicron Ceti, 400 light years from Earth and a failed colony of Earth. That episode ends with the Enterprise on course to Starbase 27. At the end of "Operation: Annihilate!" the Enterprise sets course for Starbase 10, ending Season 1.
"Friday's Child" 3497.2 involves a visit to Capella, a bright star 42.2 light years from Earth. "Who Mourns for Adonais" 3468.1 takes place in orbit of Pollux, 33.7 light years from Earth in a slightly different direction and in comm range of Starbase 12. In "Amok Time" 3372.7, Spock asks to divert to Vulcan on the way to Altair, which is 17 light years from Earth. The "Doomsday Machine"'s next stop is the Rigel colony - Rigel being 700-900 light years. "Wolf in the Fold" 3614.9 takes place on Argelius; allegedly, Rigel and Deneb lie on a straight line between Argelius and Earth; unfortunately, Rigel, Deneb, and Earth don't make a very straight line, but this firmly places Argelius as being further than either Deneb or Rigel. "The Apple" 3715.0 shows the U.S.S Enterprise taking a minute to explore Gamma Trianguli VI - Gamma Trianguli is only 118 light years from Earth. It is also possible that this is southerly star Gamma Trianguli Australis, 180 light years away. "The Deadly Years" 3478.2 pass by on Gamma Hydra IV; if this is indeed Gamma Hydrae, that is 132 light years away and apparently in comm range of Starbase 10. In "The Gamesters of Triskelion" 3211.7, a short detour adding a couple dozen light years is added in a hurry. "Obsession" 3619.2 places a cloud that Kirk encountered a thousand light years and 11 years ago under scrutiny; a round trip to and from the cloud's origins is made. The next stop is Starbase 6 in "The Immunity Syndrome" 4307.1. "A Piece of the Action" 4598.0 then happens "near the edge of the galaxy," although how near is not given; the Enterprise acted on a 100 year old radio signal in a month's time. In "By Any Other Name" 4658.9, the Enterprise is modified and hijacked to the edge of the galaxy by Kelvans, who then relent; as they continue, they are still "hundreds of light years past" where any ship has explored in "Return to Tommorrow" 4768.3., but only around ~313 light years away at Alpha Carinae in "The Ultimate Computer" 4729.4. The season closes at Earth in "Assignment: Earth."
"The Enterprise Incident" 5027.3 brings us back to the Romulan Neutral Zone. "And the Children Shall Lead" 5029.5 ends with the Enterprise setting course for Starbase 4. "Is There In Truth No Beauty?"5630.7 finds the Enterprise reaching warp 9.5 and getting thrown clear out of the galaxy. "The Tholian Web" 5693.2 is spun in "unsurveyed territory." Again in "Day of the Dove," the Enterprise goes to warp 9 on a course out of the galaxy under control of an alien hijacker. "That Which Survives" involves a 990.7 light year transport; after this, the Enterprise returns to the close neighborhood of Starbase 4 in "Let That Be Your Last Battlefield." The Enterprise is again "hundreds of light years" from Earth in "The Savage Curtain" 5906.4.