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Warp Velocity
Posted Stardate: 2007.1228

Cited From: Wikipedia

Warp travel velocity in Star Trek is generally described in "warp factor" units, which - according to the Star Trek Technical Manuals - correspond to the strength of the warp field. Achieving warp factor 1 is equivalent to breaking the light-speed barrier, while the actual speed of higher factors is determined according to an ambiguous "warp formula". Several episodes of the original series placed the Enterprise in peril by having it travel at high warp factors; in "That Which Survives", this factor was as high as 14.1. However, the actual speed of any given warp factor is rarely explicitly stated on screen, and travel times for specific interstellar distances are not consistent through the various series.

According to the Star Trek episode writer's guide for The Original Series, warp factors are supposedly converted to multiples of light speed with the cubic function s(w) = w3c. Accordingly, "warp 1" is equivalent to the speed of light, "warp 2" is eight times the speed of light, "warp 3" is 27 times the speed of light, and so on. However, this conflicts with the on-screen application of the technology, as it would make the Enterprise far too slow for the voyages depicted in the television series. These speeds do not even correlate with details presented in some of the episodes. For example, in "That Which Survives" (1969), the Enterprise travels at warp 8.4 for 11.33 hours and traverses 990.7 light years (as indicated in Spock's dialog), which makes the speed more than 600,000 times the speed of light. The Enterprise has also easily traveled to and from the edge of the Milky Way galaxy ("Is There in Truth No Beauty" and "By Any Other Name" (1968)), a journey which should take years at "warp 8" if the actual speed is merely a cube of the warp factor.

For Star Trek: The Next Generation and the subsequent series, Star Trek artist Michael Okuda devised a formula based on the original one but with important differences. For warp 1–9, if w is the warp factor, s(w) is the speed in km per second, and c is the speed of light, then s(w) = w^{10 \over 3}c. In the half-open interval from warp 9 to warp 10, the exponent of w increases toward infinity. Thus, in the Okuda scale, warp speeds approach warp 10 asymptotically. There is no exact formula for this interval because the quoted speeds are based on a hand-drawn curve.

It should be noted that warp speeds tend to warp 10 asymptotically, and at speeds greater than warp 9 the form of the warp function changes because of an increase in the exponent of the warp factor, w. Due to the resultant increase in the derivative, a small change in the warp factor corresponds to a large increase in speed.

The later series were better at keeping to calculated velocities than the original; however, they were still far from perfect. Later episodes of Star Trek: The Next Generation (such as "Descent" (1993)) contradicted these speeds and Star Trek: Deep Space Nine depicted Federation Starfleet strategic operations (fleet movements) which would have been impossible under the Okuda scale. Star Trek: Voyager, though its premise was generally based on the Okuda scale, had several notable instances, such as in the episode "Parallax" or "The '37s" (1995), where the stated warp velocities varied wildly from the Okuda standard.

In general, the farther away a Star Trek show is in production date from the publish date of the Star Trek Technical Manual, the more likely a ship would be to travel at the "speed of plot". For example, in the Star Trek: Enterprise pilot episode they give a time and speed to Neptune that accords with the original series' formula, but then they estimate a trip to the Klingon Homeworld of Qo'noS at warp 5 as a four-day journey, placing it just one light-year away from Earth — far closer than the nearest stellar system, Alpha Centauri.

     
   

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