Distances and Speeds in TOS

[Falconer]

Let’s take a deep dive into distances and speeds in TOS.

I considered posting this in the TOS forum, but I figured I would get a lot of, “LOL who cares speed of plot,” kind of answers. So I posted it here in Trek Tech. However, this IS related solely to TOS and TOS-based official technical reference works, not really the later shows or their respective manuals at all.

Now, speed of plot is not a bad concept, and it is a simple fact that the writers were never really rigorously adhering to a particular warp formula, let alone a map. So, for all the pretentiousness of this thread, it’s basically just an exercise in identifying distances and speeds that more-or-less “feel right” — and ones that really don’t.

Let’s begin.

The Star Trek Writer’s Guide (3rd rev., 1967) has a few general remarks on distances:

For long measurements, such as distance between stars, we use light year measurements. For example, the closest star to Earth is Proxima Centauri, which is 4.2 light years away. Other stars in our galaxy are hundreds or thousands of light years away.…

The term PARSEC is also used in measuring vast distances -- Parsec is 3.26 light years…

There are seventeen Starfleet Command Centers strategically located throughout our galaxy.… The STAR TREK FORMAT is to use Starbases with Starbase Commanders only when vital to a story, preferring to keep Kirk and the Enterprise far away and out of touch.… When necessary, we can establish our distance from a Starfleet Base is such that it takes hours or even many days for subspace radio messages to be exchanged.…

Our galaxy… is approximately 100,000 light years in diameter and 12,000 light years in depth at the center. Thus, to patrol only a small part of this gigantic cluster of matter, our starship must be capable of traveling hundreds of times the speed of light. Our galaxy has not yet been fully explored by the Federation's starships there are still vast unknown areas even in the sector assigned to the U.S.S. Enterprise.

NOTE: Our starship will never leave our galaxy -- by conservative scientific estimate, its uncounted millions of suns and planets include at least several billion planets quite like Earth -- more than enough adventures for even an unusually long television run.

For speeds, it is rather more specific:

The Enterprise engines (the two outboard nacelles) use matter and anti-matter for propulsion, the annihilation of dual matter creating the fantastic power required to warp space and exceed the speed of light.…

Hyper-light speeds or space warp speeds (the latter is the terminology we prefer) are measured in WARP FACTORS. Warp factor one is the speed of light -- 186,000 miles per second (or somewhat over six hundred million miles per hour.) Note: warp factors two, three and four and so on are based upon a geometrical formula of light velocity. Warp factor two is actually eight times the speed of light; warp factor three is twenty-four times the speed of light; warp factor four is sixty-four times the speed of light, and so on.

Maximum safe speed is warp six. At warp eight the vessel begins to show considerable strain. We have established in preceding episodes that warp seven or eight are used only in emergencies, in hot pursuit and so on, and can be highly dangerous.

In summary:

Like many things in the Writer’s Guide, this was and is taken as gospel by some fans, and disputed by other fans on the basis that the on-screen evidence pointed to something else. Let us summarize some of the stronger evidence from the series:

“Rigel…is…approximately 860 light-years (260 pc) from Earth.” (Wikipedia) (This real star system was visited by the Enterprise in The Cage, Mudd’s Women, and The Doomsday Machine, and was mentioned in many other episodes.)

“We’re hundreds of light years from Earth, Mister Spock. No colonies or vessels out this far.” (Miri)

“900 light years from Earth.” (The Squire of Gothos)

“All of a sudden, we’re clear across the galaxy, 500 parsecs from where we are, I mean, were.” “Take us back to where we’re supposed to be, Mister Sulu. Warp Factor 1.” (Arena)

“Only ¹⁄₁₆ₜₕ parsec away, Captain. We should be there in seconds.” (Bread and Circuses)

“A planet over 1,000 light years from here.” “Go to Warp 6.” “Round-trip time?” “1.7 days.”(Obsession)

“Not even a Vulcan can know the unknown, Captain. We are hundreds of light years past where any Earth ship has ever explored.” (Return to Tomorrow)

“990.7 light years away.” “Warp 8.4… our estimated time of arrival is…” “11.337 hours.” (That Which Survives)

Now, before we ourselves grapple with what the speeds and distances “should” be, let us summarize the TOS-based works which have done so before us.

In order to keep the numbers somewhat tangible and relatable, I calculate each speed first in absolute terms of parsecs traveled per day (I could as easily have chosen light years, but all the mapmakers preferred parsecs), and second in relative terms of the time needed to cross a percentage of the map, i.e., a sector. Each mapmaker defined a sector differently, and mapped on a completely different scale, so, in order to compare apples to apples, for the purposes of this discussion I am defining a sector as ¹⁄₁₉ₜₕ of the diameter of the Federation’s sphere of exploration/influence. It’s a little bit arbitrary, but for now it’s good enough to help us “eyeball it.”

The Star Fleet Technical Manual (1975) shows us a Federation Exploration Area with a 9,500-parsec diameter, which is huge compared to later maps, but small compared to the Guide’s suggestion that Starfleet has the run of the galaxy. But the great Franz Joseph also stuck with the Guide’s WF scale. Thus giving us the very longest travel times that we will see:

Next up we have Star Trek Maps (1980), which reduced the diameter of the Federation to 480 pc (smaller than a sector in SFTM terms). Furthermore, STM used a much faster WF scale, retaining the basic formula from the Guide, but “correcting” it by multiplying it by “Cochrane’s factor” — or at least the “average value” thereof. Thus giving us the very shortest travel times that we will see:

The Star Trek: The Role Playing Game by FASA (1983-97) returned to the Guide speeds, but reduced the Federation diameter yet again so that it was now 180 pc:

Meanwhile, the Star Fleet Universe (1979-), a family of tabletop games based on the SFTM — and therefore retaining its 9,500-pc map — invented different speed scales for the needs of different games, but what matters for this discussion is the fact that eventually, in GURPS Prime Directive (2002), the settled on non-combat speeds equal to 64 times the Guide speeds. Coincidentally or non-coincidentally, this gives us relative travel times very similar to FASA’s.

Analysis forthcoming…

 

[Nerroth]

The warp factors you list here for the SFU don't quite line up with the ones I've been quoting elsewhere.

The "tactical" speeds used in Star Fleet Battles, as listed in Captain's Log #16, follow the "cube root of c" formula; Warp 1 is SFB Speed 1 (or 1 c); Warp 2 is SFB Speed 8; Warp 3 is SFB Speed 27, etc. Note that the tactical "speed limit" for a Constitution-class heavy cruiser is Warp 3.14 (SFB Speed 31).

The "operational" speeds in GURPS Prime Directive 4th Edition, as listed here, are closer to what you have listed: A Federation CA can "cruise" at Warp 7, which GPD4e lists at 18.7 parsecs a day.

But the "strategic" (or "dash") speeds are significantly faster: a Fed CA can "dash" at Warp 9.25, or 436.27 parsecs per day.

Also, the SFU uses "sectors" to refer to its 24 sub-divisions of the galactic map. On the "hex map" used in Federation and Empire, each map hex is 500 parsecs across. (As noted in GURPS Federation, the UFP refers to each of its claimed map hexes as a "district"; these are grouped into "provinces", which in turn are grouped into larger "regions".)

 

[Falconer]

Thanks, Nerroth! My SFU calculations were off because I forgot the SFU uses 360-day years. Compensated. (You may have to reload the image.)

I mentioned above the fact that the SFU has multiple different speed scales, and so I picked the one that seemed most germane to the comparison—the main Warp Factor scale they use for travel. Dash is a whole other animal.

And I did mention that everyone has a different definition of sector, so I was inventing a uniform one for the purposes of comparison. But maybe I should have picked a different term so as not to sow unnecessary confusion. Sorry about that.

 

[Falconer]

Preliminary remarks about scale

Right from the get-go I must rule out FASA’s 180-pc diameter, since it’s established that the Enterprise travels as far as Rigel (265 pc away from Earth). Not to mention Gothos being 276 pc away from Earth, the 304-pc journey in “That Which Survives,” the 307-pc journey in “Obsession,” or the 500-pc journey in “Arena.” And all these trips are undertaken without batting an eye.

Indeed, more than one quote suggest that the limit of human (and Vulcan) knowledge extends in a radius only “hundreds of light years” from Earth—apparently no greater than 1,000 ly (307 pc). That would make the diameter 613 pc or less. Considering Sulu considers 500 pc to be “clear across the galaxy,” I’d say we’re in the right ballpark.

 

[blssdwlf]

I'll toss in my old space terrain chart as another idea for TOS speeds...

Basically: slow in-system/near stars and planets; fast between systems; slow outside of galaxy

 

[Henoch]

I'll toss in my old space terrain chart as another idea for TOS speeds...

Basically: slow in-system/near stars and planets; fast between systems; slow outside of galaxy

Yes, an excellent chart to explain the warp speed vs. distances given in canon. We know from WNMHGB and The Galileo Seven, strong magnetic potentials/fields affect a ship's speed. Observations from your space terrain chart:

• Whether a star system's magnetic field, gravity, mass, and heliosphere affects the warp speed is unknown but probably does based on the idea of space terrain.
• Some theorize that outside the galaxy (intergalactic space) contains antimatter and/or dark matter, which can be space terrain and affect warp speed.
• Within the galactic boundary (interstellar space), the antimatter and/or dark matter have been reacted out of existence or blown out of the galaxy hence having less space terrain effects within the galaxy.

I would love to see a similar chart, perhaps rescaled with uniform log scales for distance (in light-minutes) and speed (in c) superimposed with warp speed lines of warp one through eight, displayed in the background or on a navigation/helm display as a ship approaches a star system on SNW ...

 

[blssdwlf]

Yes, an excellent chart to explain the warp speed vs. distances given in canon. We know from WNMHGB and The Galileo Seven, strong magnetic potentials/fields affect a ship's speed. Observations from your space terrain chart:

• Whether a star system's magnetic field, gravity, mass, and heliosphere affects the warp speed is unknown but probably does based on the idea of space terrain.
• Some theorize that outside the galaxy (intergalactic space) contains antimatter and/or dark matter, which can be space terrain and affect warp speed.
• Within the galactic boundary (interstellar space), the antimatter and/or dark matter have been reacted out of existence or blown out of the galaxy hence having less space terrain effects within the galaxy.

I would love to see a similar chart, perhaps rescaled with uniform log scales for distance (in light-minutes) and speed (in c) superimposed with warp speed lines of warp one through eight, displayed in the background or on a navigation/helm display as a ship approaches a star system on SNW ...

If I were to redraw the chart today I would make Warp 1 between systems behave faster to account for all the times Warp 1 is called for when leaving or heading back to pick people up. Warp 1 could be used by the crew to figure out how fast the space is for their calculations.

I don't know if this chart would work with SNW or any of the modern Treks though. IIRC, Discovery warp takes on the appearance and behaves more like Star Wars' jumps to hyperspace. Were there any warp speed runs next to a planet/star or in the atmosphere of one?

 

[Henoch]

If I were to redraw the chart today I would make Warp 1 between systems behave faster to account for all the times Warp 1 is called for when leaving or heading back to pick people up. Warp 1 could be used by the crew to figure out how fast the space is for their calculations.

The Enterprise leaves orbits <edit. star systems> mostly at warp 1 in season one, then mostly leaves orbits at warp 2 in seasons two and three. In-universe, maybe this has something to do with the Engineering upgrade between seasons one and two.

Personally, I find exactly warp one equaling c is okay (including whether the speed of light is affected by local space terrain). I like that space terrain affects the speed of ships in warp, even ships barely exceeding warp 1.000 such as warp 1.2 where the effect on speed might be huge.

 

[DEWLine]

FASA's map was never meant to operate as a 3-D map either, so... *shrugs*

 

[blssdwlf]

The Enterprise leaves orbits mostly at warp 1 in season one, then mostly leaves orbits at warp 2 in seasons two and three. In-universe, maybe this has something to do with the Engineering upgrade between seasons one and two.

Personally, I find exactly warp one equaling c is okay (including whether the speed of light is affected by local space terrain). I like that space terrain affects the speed of ships in warp, even ships barely exceeding warp 1.000 such as warp 1.2 where the effect on speed might be huge.

Yeah. I personally look at Warp One or Two as a way to measure the local space to determine how fast space warp can be before the ship accelerates to a target speed. Kind of like how the speed of sound varies so does the speed of space warp.

 

[publiusr]

If I were to redraw the chart today I would make Warp 1 between systems behave faster to account for all the times Warp 1 is called for when leaving or heading back to pick people up. Warp 1

Warp 1 might be what you need to “engage”space lanes with.

The Star Fleet Universe speeds are when a ship also accelerated while inside a space lane, along a Slaver positioned cosmic string. Earlier scales are free space…first or second gen warp drive. TNG pushes things up again maybe?

 

[Falconer]

Obsessed with Obsession

SCOTT: Captain, we can't maintain Warp 8 speed much longer. Pressures are approaching the critical point.
…
KIRK: Do what you can to increase our speed, Scott.
SCOTT: Aye, sir.
…
SCOTT: Captain, we can't do it. If we keep this speed, we'll blow up any minute now.
KIRK: Go to Warp 6.

(later)
KIRK: Mister Scott, I'll need all the speed you can deliver. Keep at it until we begin to shake apart.
…
Ahead full.
…
Round-trip time, Mister Chekov.
CHEKOV: 1.7 days, sir.
KIRK: We will rendezvous with the U.S.S. Yorktown in 48 hours.

(later)
KIRK: Have Mister Chekov lay in a course for a rendezvous with the Yorktown, Maximum Warp.

So, questions for you guys. Obviously, Kirk will take WF 8 or greater if he can get it. Even though the groundwork has been laid to establish that the ship could blow up from the pressure, Kirk IS ordering WF 8, right? Otherwise he wouldn’t have said, “until we begin to shake apart.” Here’s what I wonder, though:

1. Does “Ahead full” always mean WF 8?
2. Does “Maximum Warp” always mean WF 8?
3. Does Chekov does he take into account the unmaintainability of WF 8, and compute a general decline and/or alternating pattern of unsafe and safe speeds (as we saw earlier)?
4. Or does he assume WF 8 for the whole trip in his calculation, and the audience can go ahead and assume that the ship may, in fact, blow up at any moment?

Next question. So the Enterprise takes a detour away from a scheduled rendezvous with the Yorktown to pick up emergency medical supplies for Theta VII. This is a major source of dramatic tension in the episode. The Enterprise detours to Tycho IV, which is “in another part of the galaxy,” “over 1,000 light years” away.

Just within this particular episode script, do you assume the Enterprise is the only vessel in range of…?

1. Theta
2. the Yorktown rendezvous point
3. Tycho

In other words, why is the Yorktown “waiting … at the rendezvous point” when they could be doing something like bloody well delivering the vaccines to Theta, or meeting Enterprise part-way? Why are there no other ships which can take over this delivery? or no other ships nearer to Tycho? Is it because the Enterprise is the hero ship* and everyone else is lame, or is it because all the other ships have very important duties of their own?

More specifically, are the other ships far away by distances far greater than 1,000 ly? Is the Yorktown coming from 10,000 ly away, and has 20,000 ly to go? Does “another part of the galaxy” just mean “the next sector over”?

Or is WF 8 just an insanely fast speed that nobody but Kirk in their right minds would dare? And the only reason Kirk is taking matters into his own hands is that he doesn’t trust he’ll be able to convince any ship to go to Tycho and handle the situation properly? Some combo of all the above factors?

* - I mean, I know it is, and that’s just the format of the show, but it should be the only ship in range because it’s really the only ship in range, right?

 

[blssdwlf]

So, questions for you guys. Obviously, Kirk will take WF 8 or greater if he can get it. Even though the groundwork has been laid to establish that the ship could blow up from the pressure, Kirk IS ordering WF 8, right? Otherwise he wouldn’t have said, “until we begin to shake apart.” Here’s what I wonder, though:

1. Does “Ahead full” always mean WF 8?
2. Does “Maximum Warp” always mean WF 8?

Maybe? In "Obsession" it's a warp chase but in "Mudd's Women" it isn't specific but we do assume Kirk means to warp out of system. In "The Corbomite Maneuver" "Full speed" comes before "Warp speed". In "Plato's Stepchildren" Kirk calls for "engines at full speed" to get out of orbit and into space and we assume it is also warp engines. As to the speed, the Enterprise's engines improve as the series progresses so in S1 it might be WF6 while in S3 it is WF 9...

3. Does Chekov does he take into account the unmaintainability of WF 8, and compute a general decline and/or alternating pattern of unsafe and safe speeds (as we saw earlier)?
4. Or does he assume WF 8 for the whole trip in his calculation, and the audience can go ahead and assume that the ship may, in fact, blow up at any moment?

I think if there is no specific warp factor specified then Chekov is calculating it based on a sustainable speed that won't damage the engines. Only when the commanding officer orders a high warp factor that could hurt the engines is when we get additional dialogue about it...

Next question. So the Enterprise takes a detour away from a scheduled rendezvous with the Yorktown to pick up emergency medical supplies for Theta VII. This is a major source of dramatic tension in the episode. The Enterprise detours to Tycho IV, which is “in another part of the galaxy,” “over 1,000 light years” away.

Just within this particular episode script, do you assume the Enterprise is the only vessel in range of…?

1. Theta
2. the Yorktown rendezvous point
3. Tycho

In other words, why is the Yorktown “waiting … at the rendezvous point” when they could be doing something like bloody well delivering the vaccines to Theta, or meeting Enterprise part-way? Why are there no other ships which can take over this delivery? or no other ships nearer to Tycho? Is it because the Enterprise is the hero ship* and everyone else is lame, or is it because all the other ships have very important duties of their own?

More specifically, are the other ships far away by distances far greater than 1,000 ly? Is the Yorktown coming from 10,000 ly away, and has 20,000 ly to go? Does “another part of the galaxy” just mean “the next sector over”?

Or is WF 8 just an insanely fast speed that nobody but Kirk in their right minds would dare? And the only reason Kirk is taking matters into his own hands is that he doesn’t trust he’ll be able to convince any ship to go to Tycho and handle the situation properly? Some combo of all the above factors?

In TOS, I got the sense that the Enterprise could traverse the entire galaxy but they have their assigned patrol zone just like other ships in the fleet. Yorktown and Enterprise probably had an agreed upon transfer point between their respective patrols.

 

[publiusr]

Well, We don't know the state the Enterprise is in from episode to episode.

Warp 6 is the usual Maximum safe cruising speed but an accident may have happened off screen that could have impinged on things off to the side.

 

[Falconer]

Greetings, again!

Based on a multitude of quotes about “hundreds of light years,” I thought it would be eminently practical and idiomatic if a sector were 100 ly across.

In an earlier post, I said the diameter of the Federation seemed to be between 500 pc and 2,000 ly (613 pc). Since, for the purposes of this thread, I have defined the length of a sector as equal to 1/19 of the diameter of the Federation, I am now working under the hypothesis that the diameter is 1,900 ly.

Now, I devised some experimental WF scales playing with the assumption that Chekov’s calculation in “Obsession” is meant to represent WF 8. (I’m not entirely convinced it is, but let’s check it out.) Given the WF 8.4 calculation from “That Which Survives,” I still needed a third point in order to create a formula, so I decided to give WF 1 three different values:

 

[Jedi Marso]

I love this kind of analysis, but the sad truth is that there is only one speed in Star Trek: The Speed of Plot.

 

[Mytran]

I love this kind of analysis, but the sad truth is that there is only one speed in Star Trek: The Speed of Plot.

The prophecy has been fulfilled!

I considered posting this in the TOS forum, but I figured I would get a lot of, “LOL who cares speed of plot,” kind of answers. So I posted it here in Trek Tech. However, this IS related solely to TOS and TOS-based official technical reference works, not really the later shows or their respective manuals at all.

Now, speed of plot is not a bad concept, and it is a simple fact that the writers were never really rigorously adhering to a particular warp formula, let alone a map. So, for all the pretentiousness of this thread, it’s basically just an exercise in identifying distances and speeds that more-or-less “feel right” — and ones that really don’t.

For my part, I love trying to make sense of the data presented in the episodes, even if it's not what the writers originally had in mind.
@Falconer if you don't mind me asking, what's the formula for your TOS v.3 chart? I realise you've already mentioned the other two upthread

 

[Falconer]

Actually all three formulae are new:

TOS v. 1: 0.0032982943802368 * (4.2633759365 ^ WF) - 0.0132224390521947 pc/day

TOS v. 2: 0.00329347612940653 * (4.264075596 ^ WF) + 0.040463296249298 pc/day

TOS v. 3: 0.00327092282521483 * (4.2673658538 ^ WF) + 0.292643524481639 pc/day

Ugleeee! Yet actual mathematical formulae which provide the desired results.

 

[blssdwlf]

Another data point - in "Balance of Terror" the Enterprise at season 1 maximum warp takes 8 minutes to go from roughly bottom left corner to Outpost 4 (center) on the Sector map. It can be estimated that it takes a minimum of 16-20 minutes to cross the entire diagonal of a sector (we don't know the vertical position the Enterprise or other objects have on the map) at season 1 maximum warp.

This episode appears before "Arena" so the Enterprise's max warp would've been Warp 7 or 8.

 

[Jedi Marso]

This was on of the things I enjoyed back in the day of playing the FASA Trek RPG. We had these big maps with the distances laid out, and we knew how fast the ships were. There was never any 'crossing the Federation in two days' kind of stuff. When a ship left spacedock at Earth, it could take weeks or months to reach its destination. We didn't roleplay all of that time, but we kept track of it in-campaign. A couple of crews actually completed 5-year missions where much of it was transit-time, with plenty of adventures in between. Good stuff.