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Copyright © 1996, Joshua Sean Bell. Not in the public domain. Permission to distribute this document, unedited and including this copyright notice is granted, provided no fees are charged for access beyond charges for downloading or connection time from a commercial information service. Publication of this document in a magazine or journal (in any media format) must be approved by the author.
Star Trek ®, Star Trek: The Next Generation ® and Star Trek: Deep Space Nine ® are trademarks of Paramount Pictures registered in the United States Patent and Trademark Office. Star Trek: Voyager is a trademark of Paramount Pictures.
v = velocity c = speed of light in vacuum W = Warp factor ~ = approximate ^ = exponential operator ("to the power of") log = log base 10 ln = log base e
"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 another member of the Star Trek television production crew.
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.
Taki Kogoma writes:
I'm reasonably confident that this first appeared in the old Star Fleet Technical Manual by Franz Joseph and was quickly adopted by fandom. (Note that in some of James Blish's episode transcriptions, he had WF n = n*c.)
This chart compares TOS-era Warp speeds with the speed of light:
Warp Factor Velocity Comment 1 1 Speed of Light 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729 10 1000 11 1331 12 1728 13 2197 14.1 2803.221 "That Which Survives" [TOS]
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 isX * Wf^3 * c = v
where Wf was the warp factor, and c was the speed of light. Inclued was a table of corrected warp speed for a given average value of X.Wf Wf^3 X * Wf^3 Time per parsec hrs min sec 1 1 1,292.7238 22 05 29 2 8 10,341.7904 02 45 41 3 27 34,903.5426 00 49 05 4 64 82,734.3232 00 20 43 5 125 161,590.4750 00 10 36 6 216 279,228.3407 00 06 08 7 343 443,404.2634 00 03 52 8 512 661,874.5856 00 02 35 9 729 942,395.6502 00 01 49 10 1000 1,292,723.8 00 01 19
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.
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 Mini-FAQ for more discussion of this.
Here's the graph of warp vs. speed and warp vs. power consumption from the Technical Manual:
This chart was compiled with data from episodes, the Encyclopedia, and the TNG Technical Manual:
Warp Factor Velocity Source Comment 1 1 TM Speed of Light 2 10 TM 3 39 TM 4 102 TM 5 214 TM Federation speed limit 6 392 TM 7 656 TM 8 1024 TM 9 1516 TM 9.2 1649 Encyc 9.6 1909 Encyc 9.9 3053 Encyc Enterprise-D maximum speed. 9.975 ? Episode Voyager maximum speed ("Caretaker" [VOY]) 9.99 7912 Encyc 9.9997 ~198696 TM (derived) Subspace radio speed 9.9999 199516 Encyc Maximum boosted subspace radio speed
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
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.
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 isn ( 10/3 + u(W-9) * A * (-ln(10-W)) ) v = W
where u is the step function, i.e. u(x)=0 for x<0 and u(x)=1 for x>0. Note that the term multiplying the step function is zero at W=9, so the step function introduces no discontinuity in the formula. If the value of n used is greater than 1, then both the function and its derivative are continuous at W=9. (In order to have continuous higher order derivatives a function like c(w) could be used.) I take A and n asA = 0.03684678 n = 1.791275
I then get the correct warp factors for W<9, and for the warp factors above 9 I getWarp | Actual Formula --------------------------- 9.2000 | 1649 1641 9.6000 | 1909 2019 9.9000 | 3053 3035 9.9900 | 7912 7939 9.9997 | 198696 79352 9.9999 | 199516 199415
My formula agrees with the values for warp factors of 9.2, 9.9, 9.99 and 9.9999 to within 0.6%, though it is about 6% out at 9.6 and it is way out at 9.9997. If you calculate the exponents for the data points at 9.9997 and 9.9999, however, you get 5.29826 and 5.30000, suggesting that the exponent 5.3 was used to calculate the speed at both of these warp factors. Since the exponent should be increasing with the warp factor, one of these data points should be ignored. Ignoring the data point for W=9.9997, my formula is perfect for W<9, and is slightly better than that of Tahk for W>9.
Now the exponent corresponding to the speed given for W=9.2 was about 3.33810. If we linearly extrapolate this to W=9.6 then the exponent should be about 3.34763. The exponent corresponding to the speed for W=9.6 is 3.34002, which is slightly less! This means that a formula for the exponent that gives values similar to the given values for warp factors of 9, 9.2 and 9.6 must have a derivative that decreases. (This means that the function for the exponent would have to curve downwards between 9.2 and 9.6.) Since this is not a desirable property if we want an exponent that gradually increases, I also left out the data point for W=9.6 in fitting the curve.
I mentioned before that the speeds for W=9.9997 and W=9.9999 seemed to have both been calculated using an exponent of 5.3. If you calculate the exponents corresponding to the other warp factors above 9, you get:Warp | Exponent ----------------- 9.2000 | 3.33810 9.6000 | 3.34002 9.9000 | 3.50000 9.9900 | 3.89998
The speeds for warp factors of 9.6, 9.9 and 9.99 were obviously calculated using exponents of 3.34, 3.5 and 3.9 respectively, and the speed for a warp factor of 9.2 was probably calculated using an exponent of 3.338. Therefore it is not reasonable to ascribe any greater accuracy to the warp factors given than is implied by the number of significant digits in the exponents used to calculate them. By this criterion my formula gives speeds well within the uncertainty for warp factors of 9.9, 9.99 and 9.9999, although it gives an exponent of about 3.336 for W=9.2, which is a little low. Since the points were originally taken off a hand drawn curve, this is still reasonable accuracy.
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.dimensionx.com/jsbell/Star_Trek/FAQs/warp_formulae.html, but that page will probably never look too pretty.
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:
Ges Seger offers:
The numbers I remember were about how far a ship doing warp 3 for 23 hours would travel, and the answer they came up with was 0.102 light-years. I worked the math just now and got 0.1022 light-years.
Riker calculated in his head the time required for the Enterprise to travel 300 billion kilometers at Warp 9, and gets 20 minutes:
Warp 9 = (300e12 m) / (20 min * 60s/min) ~= 2.5e11 m/s
From the chart: Warp 9 = 1516c ~= 4.548e11 m/s
Discrepancy? Riker did the calculations in his head in about 5 seconds given arbitrary numbers. He's within a factor of two, so I won't complain. Bok's ship was "holding position", so it was a simple flight path.
The Enterprise jumped to Warp 7.3, and travelled 30 billion kilometers in a couple of minutes.
All of the formulas we have for warp speeds predict Warp 7.3 to be approximately 746c. Using c = 3e8 m/s, we get v = 2.24e11 m/s. 30 billion km = 3e13m. So t = 134s, or just over two minutes.
c/o Boris Skrbic:
Wesley gives the ETA of the Enterprise to Lonka Pulsar as 34 minutes at Warp 7. When Picard orders Warp 2 instead, he comments that at that speed it would take 31 hours to get there. Using the first two datapoints, 34 minutes at Warp 7, I calculated a distance of 4.012e14 m. At Warp 2, it would take the Enterprise 37 hours to travel that distance. This clearly shows that the TNG production staff used the established warp scale when they calculated the travel time, and the 6-hour discrepancy can be explained by the use of a less accurate value for the speed of light.
c/o Boris Skrbic:
the Enterprise is transported 0.54 parsecs by the Paxans. Riker says something like "nearly a day's travel in 30 seconds" (I cannot give you the exact quote since I am watching TNG on German TV). At Warp 6 (Enterprise cruising speed), the Enterprise would need 1.6 days to travel that distance. Given that Riker calculated the travel time without a computer in a couple of seconds, you can allow for the deviation. On the other hand, if you calculate the travel time at Warp 7, you get 23.5 hours, which fits the quote.
Paris states that Warp 9.9 is equal to 4 billion miles per second. Unfortunately, that turns out to be over 20,000c, which doesn't fit in at all. Bummer.
Kim states the ships speed as 2 billion km/s, which is 2*10^12 m/s, which is roughly 6667c. This is in the same ballpark as what Warp 9.975 (Voyager's top cruise speed), it turns out.
Commentary aboard the ship confirms that Warp 10 is indeed infinite speed.
Looking at the graph, you can see that the energy costs for cruising at integral Warp values are much lower than for non-integral Warp factors. The first explorers to travel past Warp 1 must have realized this. Since for Warp values in the 1-3 range follow the v = (W ^ 3) * c formula, it makes sense that a scale based on the formula would come into use.
When ships started cruising at Warp values larger than 5, the difference between what v = (W ^ 3) * c predicted to be the most energy efficient speeds and what actually were must have become noticeable. It may have taken a long time for a new, accurate scale based on new observations came into use. (Look at the USA and SI, for an example of a large sociopolitical body taking a long time to adopt a more useful, universally used scale.)
Sulu's readings of Warp velocity in Star Trek IV seem to hint that the Klingons had moved to an accurate scale by the 2280s, but the Federation didn't catch up until much later, even though it must have been painfully obvious that the old scale was next to useless. Fortunately, some time before TNG, the new accurate scale was adopted by Starfleet.
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.
Tom Bagwell writes:
I timed the interval in AGT between when Data reported the second Klingon ship to be disengaging and when Riker's helmsman reported it to be a "half a light year away" at about 22 seconds, so I calculated the speed assuming 20 seconds to reach 1/2 a light year and assuming 30 seconds to reach 1/2 a light year.
At 30 seconds, the velocity would be approximately 525,960c which equates to roughly Warp 9.97244 on the TNG scale and approx. Warp 81 on the TOS scale.
At 20 seconds, the velocity would be approximately 788,940c which equates to roughly Warp 9.97535 on the TNG scale and approx. Warp 92.4 on the TOS scale.
Martin Shields updates that with:
Assuming 30 seconds to travel 1/2 a light year, v = 525,960c which he estimates is Warp 9.97244 on the TNG scale. However, the Tech Manual and Encyclopedia tell us that Warp 9.9999 (a higher warp factor) is set at 199,516c (less than half the speed calculated). This figure comes from the M = -11/3 equation. My equation gives the TNG warp factor of 9.999974 (approx.) which better fits the known data.
If a damaged AGT-era Klingon ship can limp home at TOS Warp 81, while a Federation ship trying to be sneaky can only manage TOS Warp 13, the Klingons have nothing to worry about. I consider this adequate evidence that the TOS scale was not returned to use in the AGT future.
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. And then Paris (killed by the trip) turns into a frog, mates with Janeway, and alters the future of humanity.
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.
"Whats 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 barier 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 Traveller's tweaking of the warp drive and the Enterprise's speedometer don't get along well.
"This new Warp 5 speed limit - whats 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 Blackwell(?) 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] ?"
It's safe to say that the USS Pasteur and USS Enterprise, cruising at Warp 13, were able to ignore the Warp 5 limitation enforced by Starfleet. There are a couple of 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 happy Star Trek Universe, however.
The second is that changes to warp technology allow warp travel without the nasty side effects. The Pasteur had very different nacelle designs, the Enterprise had "fins" on the nacelle pylons which would affect the shape of the warp fields, and it even had a third nacelle which might be used to eliminate the nasty effects of warp drive.
The third possibility is that these new integral speeds above Warp 9 just don't hurt subspace the same way that other speeds do. Personally, I go for the second option.
"And how about Star Trek: Voyager?"
The ship is quoted in "Caretaker" [VOY] as having a "maximum sustainable cruise velocity of Warp 9.975". That seems to be in violation of the new speed limit. But much of the behind-the-scenes type information about Voyager indicates that it has "green" drives; a redesigned warp core and newer nacelles that do not cause subspace damage. Or perhaps it has something to do with the tilting nacelles.
Another point - that can't be a speed at which Voyager can cruise for a long period of time, despite the quote; at that speed, Voyager could cross 70,000 light years in a little under 10 years, making their journey much shorter.
"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]?"
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 intertially while the warp drive provides propulsion, etc.
"Whoah! Hold on! They must be moving faster - look at the stars that shoot past while they're in warp!"
Joseph Haller offers:
The most extreme ship induced speed discussed ... is W(ST:TNG) = 9.97535, or 788,940c.
This would give a characteristic angular speed for nearby stars of 1578 arcseconds per second or 1 degree every 2.3 seconds. This is indeed verified in the simulations. Travel at high warp speeds, on the ST:TNG warp scale, does not match very well the appearance of the bridge view screen on a typical episode. Indeed, most visible stars are not nearby but are further away with correspondingly lower angular speeds. I offer no solutions to this discrepancy other than the dramatic necessity that stars go wooshing by at high warp speed.
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. The predominant theories are:
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!) blueish 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.
Those are stars, but we're not seeing them where they really are; the warp field distorts our view so that, say, the 0.01 arcseconds of the ship's view in a belt around the center of the warp field is stretched to surround the ship as a whole and the rest of the view is compressed ahead and behind into smaller areas. This also accounts for the rainbow-banding sometimes seen, as the field also produces a prism effect.
As a side note, in "The Cage" [TOS], the moving particles seen through the forward viewscreen are explicitly identified as meteoroids.
See the Reading List Mini-FAQ for full details on the volumes mentioned above and below.
More recently presented information is considered to supercede old information, unless the weight of the evidence supports the original data.
Greatest faith is placed on aired live-action material (canon) and documents produced by or quoting the production crews for Star Trek (quasi-canon), most notably the technical advisors to TNG, DS9 and VOY: Michael Okuda and Rick Sternbach.
Other materials are not considered reliable sources of information, and anything gleaned from these is of questionable relevance.
Questionable (but useful) materials:
Material that is ignored (other than where it reproduces material from the above, e.g. photographs, descriptions, etc.):
Joshua Sean Bell <firstname.lastname@example.org>