Warp factor
Warp factor was the primary means of measuring speeds attained using warp drive. The term was often shortened to warp when followed by its value, so that saying "warp six" was the same as saying "warp factor six." An alternative term time-warp factor was also used. () Light speed travel began at warp one, whereas lower fractional values sometimes measured sublight speeds or sublight factors. (; ;  display graphic) Spacecraft ordinarily traveled at a higher integer warp factor.  (Strictly speaking, warp factor only applies when a warp drive is in use; therefore traveling at sublight speed using only impulse thrusters would not be measured as a warp factor.) 

By the mid-24th century, warp ten became infinite velocity and thus unattainable by conventional means. () Because of this, extremely high warp speeds mapped to decimal values between nine and ten, such as warp 9.975. (; )

Warp factor was one of the vocabulary words listed on the chart "A Tunnel in the Sky". This chart was seen in the schoolroom aboard Deep Space 9 in 2369. ()

Warp factor vs. average speed

Warp ten and above

23rd century

In the 23rd century, warp factors of 10 and higher were known as generally unsafe velocities. () Speeds on the order of warp 15 were called multiwarp speeds. () 

 Warp factor 10. In 2267, the  engaged an Orion scout ship capable of warp 10, if not higher speeds, since crew safety was of no concern to them, prompting Spock to remark that it was "interesting." The ship was too fast for the Enterprise to hit it with phasers or photon torpedoes. () In 2268, Bele, upon commandeering the engine and directional controls of the Enterprise, propelled the starship faster than warp 10 towards Cheron. ()
 Warp factor 11. In 2267, the Nomad probe improved efficiency in the antimatter input valve and energy release controls on the Enterprise, allowing the ship to achieve at least warp 11. When this happened, Montgomery Scott was in disbelief. Captain James T. Kirk ordered Nomad to reverse the modifications though, as the structure of the Enterprise was not designed to handle the stress of that much power output. () In 2268, the Kelvans who commandeered the ship made similar modification. At that time the ship could maintain warp 11 without danger. ()

 Warp factor 14. In 2268, the Enterprise achieved a speed of warp 14.1 when the engine of the ship was sabotaged to overload by a Kalandan planetary defense system. At that velocity the ship came within moments of destroying itself. ()
 Warp factor 15. In 2267, the Nomad probe was armed with a weapon system capable of firing energy bolts that traveled at the speed of warp 15. ()

 Warp factor 22. In 2270,  the Enterprise was accelerated to speeds in the excess of warp 22, while being linked to Karla Five's vessel with a tractor beam. ()
 Warp factor 36. In 2270, the Enterprise encountered Karla Five's vessel, that was about to enter the Beta Niobe nova. At maximum speed, the ship was traveling at approximately warp 36. ()

Infinite velocity

In 24th century warp theory, warp factor 10 had been redesignated to correspond with infinite velocity. A vessel traveling at warp 10 occupied all points in the universe simultaneously. Warp 10 was also known as the transwarp threshold. () Warp 10 had also become a slang term referring to anything extremely fast. Kathryn Janeway made the observation in 2376 that rumors traveled fast on the . Chakotay agreed with Janeway, quipping at "warp 10." ()

 Warp factor 10-. In 2364, the Traveler used the energy of his thoughts to move the  through space at a speed that registered on instruments as exceeding warp factor 10 and going off the warp scale. ()

 Warp factor 10. Although considered a theoretical impossibility at the time, Tom Paris of the  reached the warp 10 threshold in 2372, using shuttlecraft , which was equipped with a transwarp drive and an extraordinarily rare form of dilithium discovered earlier that year. After it was discovered that such travel induced hyper-evolution, this technology was discontinued after the initial test. ()
 Warp factor 10+. It was possible to travel backwards in time by surpassing warp 10. In 2365, Captain Jean-Luc Picard and Commander William T. Riker speculated on what phenomenon could have thrown the El-Baz back through time. While Riker knew the shuttlepod didn't have warp drive, he still suggested the theory that it could have somehow accelerated beyond warp 10. Picard suggested this could have been achieved by a warp-powered slingshot using the gravitational pull of a star. ()

Alternate timelines

In the original future, which was changed by Jean-Luc Picard, around the turning point of the 24th century, warp factor values beyond warp 10 were again used to describe extremely fast speeds. ()

 Warp factor 13. The Medical Spaceship Pasteur under Captain Beverly Crusher's command traveled at warp 13 in the incident concerning the anti-time eruption in the Devron system. ()

Appendices

Related topics 
 Bessel function
 
 Cochrane equation
 Cruising speed
 Emergency warp
 Maximum warp
 Time warp
Slingshot effect
 Warp equation
 Warp barrier
Time barrier
Transwarp threshold
Warp 2 barrier
 Warp theory

Background information

Variations in relative speed 
Although formulas to calculate a relative speed from a warp factor have existed in the writer's guides, these were rarely used for reference in the episodes and films. To explain the apparent discontinuities of relative speed equivalents for warp factor speeds, reference sources have given several explanations: 

Star Trek: The Next Generation Technical Manual (p. 55) states the actual speed values of a warp factor are dependent upon interstellar conditions, for example gas density, electric and magnetic fields in different regions of the galaxy, and fluctuations of the subspace domain. Also quantum drag forces and motive power oscillation cause energy penalties to a ship using warp drive. 

Star Trek Maps (p. 6) introduced a similar concept as the Cochrane factor, that influences the actual speed by multiplying it. It can be as high as a multiplication of 1,500 to the relative speed within the curvature of space caused by the interstellar dust and gas of a galaxy, and as little as 1 in the empty intergalactic void. In the vicinity of massive objects it is so high that disproportionately high speeds are created when approaching them, and they tend to result in the slingshot effect. Between the galaxies there is only the empty void, so the speed follows only the basic cubic formula. (see below) Within the interstellar medium of Federation space the average value for the Cochrane factor has been calculated to be 1292.7238. This value explains, for example, the ball park of the fast relative speed equivalent for warp factor 8.4 from : 8.43 × 1,292.7238 = 766,202.57 times the speed of light.

The slowing down effect of moving away from a source of gravity to the relative speed has been well established in canon. For example in , the Enterprise is at maximum warp but is not moving in space at all, due to the gravity of a black hole behind it. Similarly, in , the  engages warp speed while in the atmosphere of Earth, and it takes over two minutes in the film for the ship to achieve and break out of Earth's orbit, but reaches the Sun's event horizon only a few minutes later. In , a  starship moving away from a star system at a speed better than warp 6 is moving slower than the speed of light. In some areas of space with unstable or disrupted subspace, it is impossible to use warp drive at all, as was established in such episodes as  and .

Many examples of more subtle variations exist. For example, in "By Any Other Name", the Kelvans modified the  to accelerate to a speed of warp 11 in order to safely cross the galactic barrier. If this was also meant to represent the velocity of travel to the Andromeda Galaxy, a travel time of three hundred years would indicate a far greater speed than can be derived from the basic cubic scale from the writer's guide. Warp 8.4 was stated to be much faster in "That Which Survives" than warp 9.9 in "The 37's". In , warp 7 was stated to be about 55 times faster than warp factor 2, again confirming that fluctuations in the relative speeds exist that are not covered by the basic formula.

Star Trek: The Original Series 
In his initial draft proposal, Star Trek is..., Gene Roddenberry established the maximum velocity of the starship as ".73 of one light year per hour". This would translate to a top speed of approximately 6,400 (equivalent to TOS warp 18.57, or somewhere between TNG warp 9.9 and 9.99).

The original warp scale was described in the writer's guide, The Star Trek Guide, (third revision, p. 8) as a set of warp factors and multiples of light speed that can be obtained by raising a warp factor to the third power.  This information appeared in widespread print in The Making of Star Trek (1968, p. 191). The book also states a shift in relative time occurs while traveling at warp, an hour might equal to three hours experienced outside the ship. (p. 198) In 1975, the warp scale given a more technical gloss in Franz Joseph's Star Fleet Technical Manual, now extended to include warp factors below 1. In 1977 Roddenberry again adopted the scale for the aborted Star Trek: Phase II series, but abandoned it for . It was not until the  episode  of , that the warp factor scale made an official on screen debut. Warp factors from 1 to 5 were depicted with their corresponding relative speed values on a large computer graphic. 

The scale used by Starfleet in the 22nd and 23rd century is based on a geometric progression, where the speed of a vessel (measured in multiples of , the speed of light) is equal to the cube of the given warp factor. The warp factor was calculated as follows:

with
 being the speed of the signal or starship
 being the speed of light (3.0 × 108 m/s) and
 being the resulting warp factor

Or, to calculate speed () in terms of , the formula would be:

At warp 1, a starship would reach ; at warp 6, it would reach 216 . This is a much slower velocity than initially proposed by Roddenberry.

Using this scale:

Star Trek: The Next Generation 

A document dated May 14, 1986 and attributed to Gene Roddenberry places warp 10 at the top of the scale: "Beyond that time-space continuity is disoperative." The corresponding velocity is given as "the speed of light multiplied by the speed of light ten times", whereas warp 2 is now "the speed of light squared", implying a general rule of the speed of light to the power of the warp factor. Aside from warp 1 mapping to the speed of light, it is unclear how this was to be applied in practice, since a unit of speed to the second or higher power is no longer a unit of speed. There is, however, a clue in the statement that only 34% of the galaxy has been explored as opposed to 18% in TOS, suggesting improvements without major breakthroughs. 

The Writers'/Directors' Guide revision of March 23, 1987 confirms that warp 1 remains the speed of light and accepts warp 10 as "the physical limit of the universe – beyond that normal time-space relationships do not exist and a ship at that velocity may simply cease to exist." As in the classic series, warp 6 is the highest cruising speed, though the stated equivalent of a light year per hour is more in keeping with .73 in the format of 1964 than the 41-hour light year by the cubed scale, or the 22 hours it would take to traverse the distance in the final revision. At this early version of warp 6, however, the Enterprise would need 308 years to travel the 2,700,000 light years it covered in , consistent with Geordi La Forge's "over three hundred years" in the episode.

By the creation of Star Trek: The Next Generation Technical Manual, the warp factor scale used by Starfleet in the 24th century was based on a recalibration of the scale used in the Original Series. Rather than a simple geometric progression based on relative speed, warp factors were established to be based upon the amount of power required to transition from one warp plateau to another. For example, the power to initially get to warp factor 1 was much more than the power required to maintain it; likewise warp 2, 3, 4, and so on. Those transitional power points rather than observed speed were then assigned the integer warp factors. These transitional points were established to apply to the original warp scale as well in the canonical warp chart presented in "First Flight".

According to an article in  by André Bormanis, this scale change occurred in 2312. A term was added to the above equation that caused the speed to rise slightly at lower warp factor, but to become infinite at warp 10. The ratio / at a given warp factor is equal to the corresponding  value that describes the subspace distortion.

Gene Roddenberry stated that he wanted to avoid the ever-increasing warp factors used in the original series to force added tension to the story, and so imposed the limit of warp 10 as infinite speed.

For warp factors up to 9, the revised formula became:

with
 being the speed of the signal or starship
 being the speed of light (3.0 × 108 m/s) and
 being the resulting warp factor

Or, to calculate speed in terms of  (up to warp 9), the formula would be:

In this case, warp 1 is equivalent to  (as it was in the 23rd century scale), but above warp 9 the speed increases exponentially, approaching infinity as the warp factor approaches 10. 

Using this scale:

Star Trek: Deep Space Nine 
According to  and , the Bajoran Wormhole connected the Bajoran system to a region in the Gamma Quadrant 70,000 light years away. In "Battle Lines", Sisko stated that it would take Starfleet's fastest ship over sixty-seven years to cross the distance, suggesting the fastest ship in 2369 could travel at approximately 1,044 times the speed of light on a flight of that duration. The figure was further explained in the series bible, that it is more specifically a sixty-year journey at warp 9,  suggesting warp 9 would be about 1,167 times the speed of light.

In , the , traveling at warp 9, is three days away from a planet. Increasing speed to warp 9.5 took almost a full day away from the travel time. This indicates that warp 9.5 is almost 50% faster than warp 9.

Star Trek: Voyager 
In Star Trek: Voyager Technical Manual (pp. 12 & 13) several other speed equivalents are established: Warp 9.6 is 1,909 times the speed of light. Warp 9.99 is 7,912 times the speed of light, which in turn is nearly three times the speed of warp 9.9. Subspace communication signals travel at warp 9.9999, a hundred times faster than warp 9.6, 199,516 times the speed of light.

In the pilot episode of the series, , it is established that "at maximum speeds" it would take seventy-five years for Voyager to reach Earth, which was at that time approximated to be 75,000 light years away. This would mean that the maximum speeds of the Voyager were around approximately 933-1,000 times the speed of light. According to the Star Trek: Voyager Technical Manual, this calculation was meant to be based on an unrealistic non-stop direct journey at the speed of warp 9.6 (p. 14) or at warp 9.99 (p. 36). A realistic estimate, according to the manual, was that the journey would last somewhere between two and four hundred years when taking into account the required engine cooling time needed on such an extended journey.

According to Star Trek: Voyager Technical Manual (pp. 4 & 27), a sector (about twenty light years) took four days to cross at warp 9.6, five days at warp 9 and about nineteen days at warp 6. However, in , the ship cuts three years off its journey by crossing thirty sectors, implying that they expected to travel more than a month (or approximately 36.5 days) to cross a sector.

In the episode , Captain Kathryn Janeway stated that the current Starfleet starships in 2373 were twice as fast to what the  and the  were in the 2290s. According to Star Trek: Voyager Technical Manual, (p. 13) the maximum rated speed of the ship was warp 9.975 or 3,053 times the speed of light. According to the Star Trek Spaceflight Chronology (p. 180) and Mr. Scott's Guide to the Enterprise (p. 14), the maximum speed of ships like the Enterprise-A was warp 12 in the old scale, or 1,728 times the speed of light.

Star Trek: Enterprise 
Officially and according to the large warp chart featured in , the warp drive of  used the TOS scale. Speeds in , which were mentioned as traveling at 30,000,000 kilometers per second and going to "Neptune and back in six minutes", fit well into the ballpark of cubic warp factors between 4 and 5. In , Trip Tucker states that warp 4.8 (approximately 111 times the speed of light in the TOS scale) is double the speed of warp 3.9 (approximately 59), which is also a close enough margin of error considering it is an offhand comment made without navigational implications. 

In the episode , a location was given to Jonathan Archer as to where to look for the Xindi inside the Delphic Expanse. The location was stated to be a three-month trip away from Earth at warp 5. In the next episode, , when Enterprise had arrived to look for the Xindi in that region, it was said they were fifty light years away from Earth. This indicates warp 5 would equal to a speed of approximately two hundred times the speed of light. This would fall closer to the TNG scale figure for warp 5 instead of the TOS scale figure of 125 times the speed of light estimated in the canonical chart.

There are, however, instances in "Broken Bow" that do not appear to be compatible with any of the basic scales. Zefram Cochrane notes in his recorded speech that the warp five engine would allow a ship to travel a hundred times faster than what they could in 2119. Warp 2 was later on established to be the maximum warp ships in the early 22nd century had achieved in  and . Warp 5, however, was only sixteen or twenty-one times faster than warp 2 in the scales. The journey from Earth to Qo'noS in four days was another instance. In either scale, Enterprise wouldn't even reach the closest star to Earth in four days.

In , it is stated that a warp three engine would allow a ship to travel ten times faster than warp factor 1.8. This doesn't work out in either of the basic formulas, unless we interpret the statement to indicate that a warp three engine would allow a speed of warp factor 3.9 in the TOS scale or 3.6 in the TNG scale. Warp factor 3 would be only around five times faster in either scale.

Alternate reality 

In the alternate reality seen in , the  traveled from Earth to  at maximum warp. According to , Vulcan is located slightly over sixteen light years away from Earth. According to background sources maximum warp of the ship was Warp factor 8.  Directly after the ship had accelerated to and attained maximum warp, Captain  ordered  to give an announcement of the mission to the crew. At the end of the broadcast, Chekov stated that the ship would arrive within three minutes. 

However, there was an unknown amount of time the ship spent accelerating to maximum velocity, so there is no accurate way to ascertain the total travel time of the Enterprise from Earth to Vulcan beyond the obvious implication that it was not an especially lengthy trip. By comparison to the prime reality, in , when the crew was returning to Earth from  on board the , Sulu reported they would arrive in 1.6 hours.

In , the Enterprise and the three-times-faster  were capable of traveling from the Sol system through the Neutral Zone to the edge of Klingon space and back in less than a day. Co-writer Roberto Orci acknowledged 's line about his time away from the Enterprise should have been something like "one week" rather than "one day".  As a comparison, in , the internal clock of the  read 08:27 as the ship left Earth Spacedock and 16:12 when it arrived to the edge of Klingon space to meet up with Kronos One, for a trip of a little under eight hours.

Apocrypha 
In the 25th-century timeline of the video game Star Trek Online, the warp speed scale appears to have been re-calibrated yet again to allow for the spread of new technologies such as a transwarp conduit network and quantum slipstream drive systems. Warp factors higher than 10 appear in the game, but only when a ship is using a quantum slipstream drive or exotic equipment such as Borg-enhanced "Assimilated Subtranswarp Engines". Speeds higher than warp 10 are classified as "transwarp factors", with higher numbers equating to faster speed. Borg subtranswarp engines allow ships to travel at an average speed of "warp 15", while activating quantum slipstream gives a temporary speed boost of up to warp 35. 

The relation between warp factor and speed is s(F) = (F/20) light-years per second, F being the warp factor.

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Measurements
Warp