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Introduction

One of the more well-known sci-fi terms specific to Star Trek is the warp factor. Star Trek spaceships travel at many times the speed of light, and the warp factor is a convenient way to measure how fast a ship is traveling. Instead of referring to speeds in somewhat awkward terms, like "five hundred times the speed of light" or "three point nine times ten to the eleventh meters per second," a starship captain can simply say, "Warp Factor Seven," or just "Warp Seven."

Over the years two different methods were used to calculate the speed of each warp factor. The first warp function was used during the original Star Trek television series and the early feature films. This function was a simple exponential function. The second function was introduced for Star Trek: The Next Generation, and it has been used on all Star Trek productions set during the era of that series. This function was made about by a Gene Roddenberry edict that Warp 10 was to become the new maximum speed of the universe -- infinite speed. The Star Trek: The Next Generation warp scale thus became something like a Richter scale for faster-than-light speeds. I thought it might be fun to use a different set of restrictions to come up with a different warp function that would incorporate a Richter scale-like construct, yet could be applied retroactively to the speeds mentioned in the original series. The original series occasionally mentioned speeds in excess of Warp 10 -- values disallowed by Gene Roddenberry's later restriction.

Background

Star Trek (The Original Series)

Stephen E. Whitfield explained the rationale behind the warp factor on page 191 of The Making of Star Trek (ISBN 0-345-34019-1):

The speed of light... is in itself a speed with which much of the audience has difficulty relating. Even greater problems result when it becomes necessary to express a speed many times faster than the speed of light. STAR TREK dialogue solves the problem by measuring all faster-than-light speeds in terms of "Warp Factors."

Star Trek's television audience had no idea how fast a given warp factor was, but The Making of Star Trek provided some hints on page 191.

Warp Factor One is the speed of light. Warp Factor Three is 24 times the speed of light. Maximum safe cruising speed of the Enterprise is Warp Factor Six, or 216 times the speed of light. At Warp Factor Eight (512 times the speed of light) the ship's structure begins to show considerable strain...

By way of contrast, particularly fast objects, or the Enterprise when under the influence of a highly advanced technology, could travel at speeds such as Warp 10 or Warp 14. The highest quoted speed in the original Star Trek television series was Warp 15 in "The Changeling."

Over the years, Star Trek fans adopted a "warp-factor-cubed" equation to determine how fast a given warp factor was in terms of multiples of the speed of light. The equation is as follows:

where
s is the speed as a multiple of the speed of light (c)
w is the warp factor
(c is the traditional symbol for the speed of light)

This formula yields the following speeds for each warp factor:

Note that the description of Warp Factor Three in the table disagrees with the value given in The Making of Star Trek; this is presumably a mathematical error.

In practice, the Star Trek writers made little to no effort to remain consistent with speed references.

Star Trek: The Next Generation

During the development of Star Trek: The Next Generation, Gene Roddenberry decided for reasons unknown to "recalibrate" the warp factor scale. He reset Warp Factor Ten to be an unattainable maximum speed -- infinite speed. The references in the original Star Trek to speeds equal or greater than Warp Factor Ten thus suggested that the warp scale in use during the fictional era of Star Trek: The Next Generation -- the mid-24th century -- was a different scale than the one used during the fictional era of the original series -- the mid-23rd century.

As technical consultants for Star Trek: The Next Generation, Michael Okuda and Rick Sternbach were tasked with detailing the recalibration of the warp scale, given Roddenberry's new restriction that Warp Factor Ten was infinite speed. Okuda and Sternbach describe the machinations of the new warp scale on page 55 of the Star Trek: The Next Generation Technical Manual (Pocket Books, ISBN 0-671-70427-3):

Figuring out how "fast" various warp speeds are was pretty complicated, but not just from a "scientific" viewpoint. First, we had to satisfy the general fan expectation that the new ship was significantly faster than the original. Second, we had to work with Gene's recalibration, which put Warp 10 at the absolute top of the scale. These first two constraints are fairly simple, but we quickly discovered that it was easy to make warp speeds TOO fast.... Finally, we had to provide some loophole for various powerful aliens like Q, who have a knack for tossing the ship millions of light years in the time of a commercial break. Our solution was to redraw the warp curve so that the exponent of the warp factor increases gradually, then sharply as you approach Warp 10. At Warp 10, the exponent (and the speed) would be infinite, so you could never reach this value.

The warp factor equation used on Star Trek: The Next Generation was never published in a licensed Star Trek publication, but Angel Swan's World of Star Trek web site quotes a message attributed to Mike Okuda that explains the formula.

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.... 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.

There would thus seem to be a warp scale equation for only warp factors 0 through 9. That formula is:

where
s is the speed as a multiple of the speed of light (c)
w is the warp factor

Observations

Mike Okuda's warp equation works well for Star Trek: The Next Generation and other Star Trek productions set in the same era. On the other hand, in the twenty-one television seasons and four feature films set in the era of Star Trek: The Next Generation era, it is likely that the writers weren't always consistent with Okuda's warp equation, just as the writers of the original series weren't always consistent with that series' warp equation.

A major drawback to the warp scale of Star Trek: The Next Generation is that it's confusing to the average viewer. Consider a chase scene, in which the pursued ship accelerates in an attempt to avoid capture, and the pursuing ship accelerates in an attempt to overtake the pursued ship. For Warp Factors from 0 through 9, speeds can be incremented by integer warp factors. For example, a ship can increase in speed from Warp 2 to Warp 3. Once a ship reaches Warp 9, though, its speed will seem to increment at a slower rate. It can't increase by a full warp factor, because Warp 10 is infinite speed. An average television viewer doesn't know this, and even if Lieutenant Commander Data were to explain this situation to the viewing audience, the average viewer probably still wouldn't understand.

This drawback is not the fault of the average television viewer. A television viewer shouldn't have to comprehend infinite speed and asymptotic warp curves in order to enjoy a Star Trek story. The drawback also isn't the fault of Mike Okuda, since he was forced to recalibrate the warp curve in this way in order to follow Roddenberry's edict that Warp 10 was at the absolute top of the scale.

One could argue that Roddenberry shouldn't be faulted for the complexity of the Star Trek: The Next Generation warp scale, since he seemed to be applying a Richter scale-like approach to the recalibrated warp curve. Unfortunately this approach made the warp curve much more difficult to understand for the casual viewer.

Another drawback is that the change in the warp scale between the original Star Trek series and Star Trek: The Next Generation was never mentioned in any Star Trek episode or feature film. If a casual viewer were to watch the Star Trek: Voyager episode "Threshold," in which Warp 10 is stated to be infinite speed, and the same viewer also watches the original series episode "By Any Other Name," in which the Enterprise travels at Warp 14.1, the seeming contradiction is confusing.

Mental Exercise

I thought it would be fun to do my own "recalibration" of the warp curve that was originally defined by the "warp-factor-cubed" equation during the original Star Trek television series. Yeah, it's geeky, but so is reading this web site!

In redefining the warp scale, I imagined myself to be in the shoes of Mike Okuda and Rick Sternbach in 1987, during the preproduction phase of Star Trek: The Next Generation. Unlike Rick and Mike, however, I decided to see what might develop under a different set of restrictions than the "Warp Ten equals infinite speed" restriction imposed by Gene Roddenberry. My self-imposed restrictions are:

1. To prevent warp factors from becoming ridiculously high for very fast ships and technology. For example, there would be no "Warp 78".
2. To maintain the validity of all warp factor references in the original Star Trek television series. The highest warp factor mentioned in the series was Warp 15 in "The Changeling."
3. To represent an absolute maximum (infinite) speed with a finite warp factor value.
4. To maintain Warp Factor 0 as being a speed of zero, or "absolute rest"¹.
5. To keep the speeds of warp factors reasonably close to those defined by the old "warp-factor-cubed" formula

I wanted the warp factors mentioned in the original Star Trek series to remain valid in the new scale, so that the casual viewer of the original show and later shows wouldn't be confused by seemingly invalid warp factors in the original series. The idea was that the "new" warp scale was "really" always in effect, even during the original series, superseding the "warp-factor-cubed" equation. This seemed like a reasonable approach, since the writers of the original series were rarely consistent with that show's warp scale anyway.

Once I defined the restrictions on the "new" warp scale, I was off to the races.

One of the simplest asymptotic functions in mathematics is the tangent function. The standard tangent function is:

The value of tan(0) is 0, which is consistent with restriction #4. The tangent function thus seemed to be a logical starting point for the new warp curve.

Unfortunately the standard tangent function approaches infinite value as the independent variable approaches pi/2. That is, tan(pi/2) = infinity. If the standard tangent function were used as the new warp function, then any warp factor greater than or equal to Warp Factor Pi/2 (approximately Warp 1.57) would be invalid. This violates restriction #2.

The next step was to define the Warp Factor at which the speed would be infinite. To be consistent with restriction #2, the Warp Factor would have to be greater than 15. I imagined that Gene's use of Warp 10 as infinite speed was motivated because 10 is a "nice, round number." I decided to also choose a "nice, round number." To avoid very high warp factors, violating restriction #1, I chose a "nice, round number" greater than but fairly close to Warp Factor 15. I decided on Warp Factor 20.

The warp function thus began as a modified tangent function:

where
s is the speed as a multiple of the speed of light (c)
w is the warp factor

This warp function has the following values at integer warp factors:

At this point it became necessary to scale the warp formula to make the values more closely resemble those from the original series, satisfying restriction #5. The old warp curve defined Warp 1 to be equal to the speed of light, so the first step in scaling the new warp formula was to "normalize" it so that Warp 1 would equal the speed of light. The new warp function then became:

where
s is the speed as a multiple of the speed of light (c)
w is the warp factor

This warp function has the following values at integer warp factors:

Interestingly lower warp factors have speeds remarkably close to the warp factors themselves. Warp Three, for example, is close to three times the speed of light. It then follows that cubing the warp function would yield results close to the old "warp-factor-cubed" warp function at low warp factors, satisfying restriction #5.

where
s is the speed as a multiple of the speed of light (c)
w is the warp factor

The table below shows the speeds at various warp factors using the warp function for the original Star Trek series, as well as the speeds using my "recalibrated" warp function.

Note that, for most warp factors, the two functions match within an order of magnitude. The functions match quite closely at warp factors 1 through 8, the typical speeds attained by the Enterprise in the original series.

Images 1, 2 Comparison of TOS and Proposed TNG Warp Scales

Conclusion

If I were in charge of recalibrating the warp scale for Star Trek: The Next Generation, and I had the set of restrictions defined above, then my recalibrated warp function, which would also apply retroactively to the speeds of the original Star Trek television series, would be:

where
s is the speed as a multiple of the speed of light (c)
w is the warp factor

On page 372 of the first edition of the Star Trek Encyclopedia (Pocket Books, ISBN 0-671-86905-1), the Enterprise-D's top speed is listed as Warp 9.2, or 1649c by that series' warp function. This speed is roughly equivalent to Warp 9.54 using my recalibrated warp function.

In conclusion, my warp function and twenty-five cents are worth one quarter.

Footnotes

¹ The concept of "absolute rest" is scientifically inaccurate, but technically so is warp propulsion, by our current understanding of physics!