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This FAQ does not discuss subspace or the mechanics of warp travel. See the Warp and Subspace FAQ for discussions of the how warp drive works, and what subspace actually is.
1. TOS-era: The Original Series
In all of the formulas in this FAQ, the following hold:
v = velocity
"Who the heck is Mike?"
Mike is Michael Okuda, a member of the Star Trek television production crew as a scenic artist and technical advisor. He designs the computer displays and alien writing seen in TNG, DS9 and VOY. He is also co-author of the TNG Technical Manual, which mentions that the the TNG Warp formula exists in a Excel spreadsheet on his Macintosh. Thus, when it comes to warp calculations, Mike is god. At the very least, a patron deity. He is also co-author of the Encyclopedia and Chronology.
"So who's Rick?"
Rick is Rick Sternbach, the "other" author of the TNG Technical Manual, and the main writer of the DS9 Technical Manual. He is also a member of the Star Trek television production crew, designing most of the ships and props seen in TNG and VOY.
1. TOS-era: The Original Series
The original series warp equation is generally accepted to be: v = (W ^ 3) * c
But this has never appeared in any episode. However, it has such wide acceptance that it has pretty much stuck. It's even in the Encyclopedia.
Boris S. writes:
The equation is almost certainly the work of TOS producers. Stephen Whitfield extensively researched the show in the period of 1967-68 and published the information in his book The Making of Star Trek (1968). He states that Warp 1 is the speed of light, Warp 3, 24 times the speed of light, Warp 6, 216c and Warp 8, 512c. With the exception of the value for Warp 3 (which should be 27c), the W^3*c formula holds. Whitfield almost certainly obtained the numbers from the TOS writer's guide which contained a small technical manual. This chart compares TOS-era Warp speeds with the speed of light:
It is also generally accepted that the TOS scale was also used for the first few movies. Since speeds are rarely quoted in the movies, however, that's only speculation.
Joe Chiasson, describing Star Trek Maps, a map and manual combination by Bantam Books from 1980, offers:
The booklet contains quite a lot of written information on the development of warp drive systems and how warp travel is affected by matter density in a given area of space. The above formula was written as v = Wf^3 * c. This was further modified to include the Greek letter chi (X), which was a variable denoting the local density of matter, which changed depending on where you happened to be. So the proper formula for TOS level warp drive is v = c * Wf^3 * X
where Wf was the warp factor, and c was the speed of light. Included was a table of corrected warp speed for a given average value of X.
This correction factor does make a lot of sense, given that v = W ^ 3 by itself is almost ludicrously slow given the speeds quoted by TOS. Joe also suggests that by the time of TNG warp fields have been refined to the point that the chi factor is dropped from the formula. I think that the numbers are a little too high, however, when compared to TNG speeds.
As a side note, wf(n) = n * c appears in James Blish's TOS script adaptations, which have been widely read, so you may see that formula cropping up from time to time as well. Those speeds would be ridiculously slow, so that formula isn't really worth considering. (Thanks to Taki Kogoma for pointing that out.)
John "Eljay" Love-Jensen points out that in "By Any Other Name" [TOS] the Kelvans were using there technology to propel the Enterprise "to Andromeda, 300 years of travel". Andromeda is 2.3 million light-years away. For 300 years of travel, that translates into Warp 19.7! They probably intended to accelerate to that speed once they made it outside the galaxy, and the Enterprise didn't end up going that fast during the episode, but it means they thought it was possible for the ship to make it.
2. TNG-era: The Next Generation, Deep Space Nine, Voyager
By the time of Star Trek: The Next Generation, the warp scale has changed. Warp 1-9 are roughly the same, but Warp 10 is infinite speed. Going Warp 10 or faster is hogwash on the TNG scale. It isn't a speed barrier that can be or needs to be broken, but an energy barrier.
At least, that's what the Tech Manual says. Many fans disagree, saying that this has been contradicted on air, most clearly by the episode "Where No One Has Gone Before" [TNG] where someone says "We are passing Warp 10." See the Warp and Subspace FAQ for more discussion of this.
Here's the graph of warp vs. speed and warp vs. power consumption from the Technical Manual:
Warp speed/power graph
| :| .- 10^10 | : | := | + : | :- | ::: ' :- 10^9 10^4 -| + : | := =| + : : | :- =| + : ::::: ,/ :- 10^8 -| + : : :: __*-' := 1000 -| : : :::: __*--' | ???:- =| + : :: __*--' | | :- 10^7 =| : :: __*--': | | | := -| + : __*--': |::: | | | :- 100 -| : _*--': |:: | | | | :- 10^6 =| : _-':| : | | | | | := =| + _*' : |:: | | | | | :- -| : _-':| : | | | | | | :- 10^5 10 -| *' :| : | | | | | | := =| + ,'| : |:: | | | | | | :- =| : ,' :| : | | | | | | | :- 10^4 -| :/ :| : | | | | | | | := 1 -| :* : |:: | | | | | | | :- =| :/| : | | | | | | | | :- 10^3 =| ::/ | : | | | | | | | | := -|:_:-' |:: | | | | | | | | :- -| ,-----|-----|-----|-----|-----|-----|-----|-----|-----|-----. :- 10^2 ------' | 1| 2| 3| 4| 5| 6| 7| 8| 9| | `:::::::: | Warp Factor | `--- Velocity in Power usage in ---' multiples of c Megajoules/cochrane __.---' Velocity * Integral Warp Values ::::::: Power + Power Thresholds
This chart was compiled with data from episodes, the Encyclopedia, and the TNG Technical Manual:
The Tech Manual (on page 111) says that a subspace radio signal travels at Warp 9.9997, and takes 45 minutes to reach 17 light years, which works out to 198696c.
As an interesting anomaly, Pete Carr also points out the following tidbit from the Tech Manual:
...the TM goes on to say that TNG Warp 9.7 is about 14.1 on the TOS scale. So [TNG Warp] 9.7 ~= 14.1^3 [c] and 14.1^3 [c] = 2803 [c]. I went and graphed the new value with our current values. Unfortunately the new value doesn't fit into the exponential curve... it should be lower.
I suspect Mike made a rounding error; TOS Warp 14.1 is much closer to TNG Warp 9.8 by all of the accurate formulas that have been found or sheer reckoning off the graph.
Unlike TOS (where we have a formula but no scale), for TNG we have a scale but no formula! The reason for this is that the graph was drawn by Mike Okuda rather than calculated, as is related in the following:
On June 22, 1995, Jeff Reinecke forwarded the following letter from Michael Okuda to rec.arts.startrek.tech:
Date: Fri, Jan 27, 1995 02:09 AM EST From: MOkuda Subj: Re: Star Trek Warp To: Yar of Spit The warp factors we've used in ST:TNG were computed in an arbitrary way to fit some specific characteristics we needed. First, the speed for any given warp factor had to be greater than it was in the original Star Trek series. This was primarily to satisfy fan expectations. Second, the new warp speeds couldn't be TOO much faster, or it would be possible for the ship to cross the galaxy in a fairly brief time. (In a way, maintaining this restriction made Voyager's story situation possible. If we hadn't done this, Voyager could have gotten home too quickly.) We used an exponent of (I think) 3.33 or 3.33333... for warp factors less than 9. Between 9 and 10, I gradually increased the exponent so that it approached infinity as the warp factor approached 10. Lacking knowledge of calculus, I just drew what looked to me to be a credible curve on graph paper, then pulled the points from there. I think I re-created the curve fairly accurately in the Star Trek: The Next Generation Technical Manual. Hope this helps. -Mike
So it looks like there isn't a grand formula to end all formulas after all!
On May 29th, 1996, Dominic Berry wrote:
Since Mike calculated the speeds for the various warp factors up to 9 simply using the exponent 10/3, it is more sensible to use a piecewise function for the speeds that gives an exponent of 10/3 for W<9 and gives higher exponents for higher warp factors. My suggestion is n ( 10/3 + u(W-9) * A * (-ln(10-W)) ) v = W
I used to have a bunch of formulae in here from various posters who made some pretty good attempts at finding the Holy Grail of an accurate formula. However, due to length considerations I'm only going to keep the current best. Older formulae (basically an excised chunk of this FAQ) can be found at http://www.calormen.net/Star_Trek/FAQs/warp_formulae.html, but that page will probably never look too pretty.
Martin Shields has a warp speed calculator up on his web site: http://www-personal.usyd.edu.au/~mshields/TV/ along with discussion of the formula used.
Do any of these values actually match up with what we've seen on the show? There are often claims that these speeds are much to slow to allow the kind of adventuring that the Star Trek series portrays. But amazingly enough, when they do quote numbers and we can time things without cuts (wherein we may miss hours of ship-time), the numbers do match up:
There have been several times where the warp velocities proposed don't match what we see on-screen. The most blatant example of this kind is a call by the captain to head somewhere at Warp 1, or some other ridiculously slow speed. This happened several times in TOS, but does crop up from time to time. Here are some examples:
A bigger problem which crops up on rec.arts.startrek.tech is the size of the Federation. Sizes of up to 10,000 LY across have been quoted as diameters, and this corresponds to the occasional graphic displayed on screen showing the Federation's size and position within the galaxy. Other evidence points to a somewhat smaller size, but such questions as the distance from Earth to Bajor appear to present a paradox: some routes between Federation locations which are known to be far apart are traveled much more quickly than the TNG formula appears to allow.
The leading speculation on the newsgroup is a concept called "Warp Highways". Distinct from wormholes, these "highways" represent either natural (pre-existing) or artificial (thanks to heavy traffic) pathways where warp travel is much faster than the TNG formula, which represents a baseline.
The highways do not require additional technology beyond warp drive. Highways are not easily detectable in unknown space. This means that an exploration ship, such as the Enterprise, or a ship in unknown territory, such as Voyager, will travel between two arbitrary points at the nominal velocities presented in the TNG formula. A well-known region of space - such as the route from Bajor to Earth - would probably contain several well-known warp highways and allow less powerful ships to make the route in weeks rather than years, and top-of-the-line Starfleet ships to make the trip in mere days. Contact with local civilizations would allow Voyager to take shortcuts through the Delta Quadrant - which they frequently seem to.
Perhaps the Hekaras Corridor in "Force of Nature" [TNG] is one such route, explaining the frequent travel in that area. The whole notion of starship travel affecting local subspace properties in a permanent way supports the notion that at least some warp highways are created by frequent warp travel - that is, as a route is used it becomes more efficient. Other speculation includes the notion that gaseous anomalies are indicators of the presence of warp travel. Why else would Excelsior - one of Starfleet's latest ships - be engaged in charting such anomalies in Star Trek IV?
This is strongly reminiscent of the X (chi) factor first presented in Star Trek Maps, where the warp equation varies with local spatial conditions.
(Other speculation or comments?)
Why did it change?
3. AGT-era: "All Good Things..." (TNG final episode)
Quoted in the final episode of Star Trek: The Next Generation, in a possible or imaginary future, is the speed Warp 13. Both Admiral Riker and Captain Beverly Picard call for this speed, and at another point in the episode, Admiral Riker calls for "maximum warp", which is either Warp 13 or greater.
While we haven't a clue how fast this is, they're presumably faster than Warp 9 on the TNG scale, and necessarily slower than Warp 10 on the TNG scale (since TNG Warp 10 is infinite speed). A few possibilities present themselves:
The last one is demonstrably incorrect (see below). Among the others, there's no way to tell which is correct. Sharp-eyed Boris S. found the following explanation by Andre Bormanis, Star Trek's science advisor:
I raised that question in a TECH note. Basically, the idea there was that they recalibrated the warp scale. I don't think that ended up in the final draft teleplay, but the idea there was that if you've got ships that can routinely travel at speeds in excess of Warp 9, then maybe it makes sense to recalibrate your speed scale so that Warp 10 is no longer infinite velocity. Maybe Warp 15 will be the ultimate speed limit, and Warp 13 in that scale will be the equivalent of warp 9.95 or something like that.
Tom Bagwell writes:
Martin Shields updates that with:
4. VOY-era: Transwarp Frogs In Spaaaaaace!
As you may have guessed, the .tech community was less than impressed with "Threshold" [VOY] in which Voyager - a ship running low on supplies, with half its crew dead, stranded away from repair or research facilities, on the other side of the Galaxy from the Federation - manages to upgrade one of its never-ending supply of shuttles to make a Transwarp flight, something that has defied the best minds in the Federation for a century. Things go higglety-pigglety after that, and many people (including some of the production crew) just pretend it didn't happen.
Forgiving that, however, the episode's technobabble isn't too bad. The Voyager crew mention how Warp 10 = Infinite Speed = being everywhere at once. An interesting tidbit is that once Transwarp drive is active, the shuttle's speed registers at Warp 10.
5. Speed limits
"What's this about a Warp 10 barrier?"
In the TNG scale, Warp 10 is infinite speed. As you approach a position on the graph corresponding to Warp 10, your power requirements increase astronomically compared to your increase in speed. But you can keep speeding up forever, unlike the light barrier, which keeps you from getting to the speed of light.
In other words, keep piling on the 9s. Warp 9.99 is a lot faster than Warp 9.9, while Warp 3.99 is only marginally faster than Warp 3.9. The barrier is only one of energy, not velocity.
Once again, in case you missed it, TNG Warp 10 is not a speed barrier; it cannot be broken like the sound barrier. Any warp factor greater than 10 must be on a different scale than the TNG scale (either TOS or AGT or something else), since a speed faster than infinite speed is nonsensical.
"But in "Is There in Truth no Beauty?" [TOS] and "That Which Survives" [TOS], the old Enterprise went over Warp 14!"
Yes, but that's on the old scale. By the new scale, that translates to about Warp 9.7 (TM), which the Enterprise-D can do for brief periods. The original Enterprise was being shaken apart. Voyager can cruise at that speed without blinking.
"But in "Where No One Has Gone Before" [TNG] they went past Warp 10!"
Chalk this one up to instrument failure. While Geordi did say they'd passed Warp 10, later in the episode they were booting along at some outrageously huge speed, while the instruments only read Warp 1.5. So there's canonical evidence that the Traveler's tweaking of the warp drive and the Enterprise's speedometer don't get along well.
Daryle Walker points out that the real-world explanation for this is probably that the Warp 10 rule hadn't been established yet - this was an early first-season episode.
"This new Warp 5 speed limit - what's up with that?"
In "Force of Nature" [TNG] it is discovered that in the Hekaras Corridor, a region of space where warp travel is hindered except for a narrow path the intense use of warp drives in an already sensitive area can (over time) cause subspace rifts to form, where subspace manifests itself in real space on a macroscopic scale. This is not a good thing.
"Does this take effect everywhere?"
Yes. In "The Pegasus" [TNG] an Admiral Pressman gives Picard permission to travel faster than Warp 5 for the duration of the mission. Ditto in "Eye of the Beholder" [TNG], when Picard is given permission to exceed the speed limit to delivery needed medical supplies. The Encyclopedia concurs as well, naming Warp 5 as the new cruising speed for starships. Overkill? Probably. Typical bureaucratic overcompensation? Yep.
"So what about in "All Good Things..." [TNG] and post-TNG shows?"
It's safe to say that the U.S.S. Pasteur and U.S.S. Enterprise, cruising at Warp 13, were able to ignore the Warp 5 limitation enforced by Starfleet. While the limitation was mentioned in a few later TNG episodes, it was ignored in DS9 and VOY episodes set only a few years later. There are a few possible explanations. The first is that Starfleet simply repealed the ruling, and is allowing ships to muck up subspace. That isn't what we'd expect in the eco-friendly Star Trek Universe, however.
Another is that changes to warp technology allow warp travel without the nasty side effects. Rumors abound that Voyager's folding nacelles and/or warp core design mitigate the effect, although Rick Sternbach (the designer of Voyager) isn't so sure. The most probable explanation is that internal technological changes allow warp drive without damaging subspace.
Franz Joseph's "Field Restoration" nacelle end-cap, anyone? (Star Fleet Technical Manual)
6. Q & A
"What causes fractional warp speeds?"
As you can see from the above chart, travelling at integral Warp factors is much more energy efficient. But there are times when a fractional value must be used - for example, staying a certain distance from another ship, or keeping pace with some phenomenon. Also, beyond Warp 9, only fractional speeds are possible. (Modulo "All Good Things..." [TNG], of course.)
"Why not use impulse drive within the warp field to create a higher velocity?"
There's no reason to think that a Newtonian drive (Impulse) would augment a non-Newtonian drive (warp). Also, consider that the maximum velocity attainable with a Newtonian drive is c. At Warp 2, which is ~= 10c, this gives you a whole 11c at maximum (overloading, fuel wasting) impulse. Warp 2.1 is about 12c anyway, so overloading the impulse drive doesn't get you much.
"What about "The Corbomite Maneuver" [TOS] or The Voyage Home?"
Kirk and Sulu use a combination of warp drive and Impulse to break free of the First Federation pilot craft. The combination of a tractor beam, impulse drive, and warp drive would be very strange, and many explanations come to mind, such as the warp field causing the tractor effect to "slip" away, while the impulse provides propulsion, or the impulse fighting the tractor beam inertially while the warp drive provides propulsion, etc.
In The Voyage Home, for the trip back to the future, thrusters are used by Spock to get the last burst of speed just before entering time warp. Also, during both trips, the ship is brought out of time warp by braking thrusters. The H.M.S. Bounty is visibly moving slower than the speed of light toward the sun and certainly slower than the Warp 8 quoted by Sulu, so the time warp slingshot (in an intense gravity well) may be one case where Impulse drives are useful to augment warp drives.
"Whoah! Hold on! They must be moving faster - look at the stars that shoot past while they're in warp!"
Joseph Haller offers:
Or should we give up so easily?
There's a lot of support on rec.arts.startrek.tech for the notion that those things aren't really stars. For one, as the Enterprise drops out of warp (with the camera tagging along for the ride) some of the "stars" do some pretty strange things, such as suddenly angling off in various directions, disappearing, etc.
Also, in Star Trek: First Contact, the Phoenix barely breaks Warp 1 and stays relatively close to Earth, but we still see the streaks. Definitely not stars.
The predominant theory is that what we're seeing are free particles in space interacting with the expanding boundaries of the warp field. As they cross the warp field, they are repeatedly accelerated to FTL velocities and then slowed to STL speeds, and start spewing out something like Cerenkov radiation, a (real!) bluish light emitted when particles moving faster than the local speed of light (in a dense medium) are forced to slow down. If not exactly Cerenkov radiation, then something similar.
Jon Mitchell tells me that in the TNG video game for the Sega Genesis console platform states the streaks are part of the visual manifestation of Einsteinian space in subspace. So people other than us .techers have noticed this problem too.
As a side note, in "The Cage" [TOS], the moving particles seen through the forward view-screen are explicitly identified as meteoroids.
John "Eljay" Love-Jensen, Daryle Walker, Roger M. Wilcox, Jon Mitchell, Jonah Rapp, Boris S., Dominic Berry, Alex Tahk, Jeff Reinecke, Martin Shields, Joe Chiasson, Jason Hinson, Greg Berigan, A.J. Madison, Axis, Sharon Collicutt, Ges Seger, Michael M. WelchTom Bagwell, Joseph Haller, Chris Franklin, Pete Carr, Taki Kogoma
See the Reading List FAQ for more details on the reference volumes mentioned above and below.
The question of "what is canon" has been argued for years in the Star Trek newsgroup hierarchy. In the realm of technical discussions, this can be refined to the question of "what evidence is factual, and what is apocryphal". These FAQs follow the currently dominant notion that "canon" is aired live-action material and nothing more, with the caveat that materials produced off-camera by the production crew are often (but not always) reliable predictors of the direction future canonical material will follow, and are therefore granted a special "quasi-canonical" status. Any other material falls into the realm of speculation - it may be perfectly well grounded speculation useful for building up technical arguments, or wild flights of fancy that have no rational basis.
In addition, more recently presented information is considered to supercede old information, unless the weight of the evidence supports the original data. While this may seem highly biased and may be eyed with some skepticism as a form of Orwellian "newthink", it is a more useful predictor of what those directly responsible for the creation of the series are likely to include as canonical material in the future.
For example, the excellent and groundbreaking Star Fleet Technical Manual, by Franz Joseph created in the 1970's was a very well thought out look at the technical world of Starfleet just slightly beyond what was seen in the original series. Unfortunately, and perhaps for purely arbitrary reasons, the future development of "canon" Star Trek diverged from this speculation. This in no way implies that there was anything wrong with that volume or any others, merely that due to later "evidence", it can no longer be regarded as an authoritative overview of Trek technology. On the other hand, the author performed a lot of research to create it, and therefore its speculation should not be dismissed out of hand.
That said, we are dealing with a universe in the process of being created by scores of (usually) non-technical people, aiming to provide weekly entertainment for a mass audience. There are many inconsistencies even amid the canonical material, and often times the wildest speculation on the newsgroup makes more sense than what we see in the episodes.
Highly regarded, but non-canonical material:
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