star trek space normal speed

Screen Rant

Star trek: how fast warp speed is (& how it compares to hyperspace).

Warp speed is the key to making Star Trek work - but just how does this fictional form of travel work, and how does it compare to Star Wars?

Here's how fast  Star Trek 's warp speed actually is, and how it compares to other forms of travel in science-fiction, such as  Star Wars ' Hyperspace. When Gene Roddenberry began to work on  Star Trek , he was faced with a thorny problem: he wanted the show to have some semblance of scientific accuracy, but at the same time, he also needed the Starship Enterprise to be able to move at incredible speeds.

The solution was the warp drive (or, as it was actually called in  Star Trek 's pilot episode, the hyperdrive). This brings matter and antimatter into collision in order to generate a phenomenal amount of power, and the reaction is regulated by a rare mineral called dilithium . The matter/antimatter reaction allows the starship to travel faster than light - several multiples faster.

Related:  Star Trek: How Stardates Are Calculated (& What They Really Mean)

But just how fast is warp speed? The original  Star Trek series was unclear, but over time the franchise has established clear rules. Let's explore them.

Warp Speed In The Original Series

The original  Star Trek series was wildly inconsistent in its portrayal of warp speed, with the concept basically used as a plot device to get from A to B at whatever speed an episode's writer wanted. Warp speeds of 10 or higher were treated as unsafe, at least until the Enterprise was refitted in  Star Trek: The Motion Picture , when it gained the ability to safely travel up to warp 12. The fastest warp speed shown on-screen was an alien vessel that could actually reach warp 35. All these inconsistencies clearly bugged Gene Roddenberry, and as a result warp speed was recalibrated for  Star Trek: The Next Generation in order to create something a little more well-structured.

Warp Speed In The Next Generation Era

According to the Star Trek Encyclopedia , in simple terms, the new warp speed factor 1 is the exact speed of light, 299,792,458 m/s. Each factor above is a multiple of that warp speed, although what those values are vary depending on the show in question.

• Warp Factor 1 - 1x lightspeed
• Warp Factor 2 - 10x lightspeed
• Warp Factor 3 - 39x lightspeed
• Warp Factor 4 - 102x lightspeed
• Warp Factor 5 - 214x lightspeed
• Warp Factor 6 - 392x lightspeed
• Warp Factor 7 - 656x lightspeed
• Warp Factor 8 - 1,024x lightspeed
• Warp Factor 9 - 1,516x lightspeed
• Warp Factor 9.99 - 7,912x lightspeed
• Warp Factor 10 - Infinity

In theory, a ship traveling at warp 10 is moving so fast that it essentially exists in all places and all times simultaneously. It's only known to have been achieved once in the entire history of  Star Trek , in the  Star Trek: Voyager episode "Threshold". There, Tom Paris successfully traveled at warp 10, but the experience proved to have a transformative, evolutionary effect upon any organic matter. The crew of Voyager wisely ditched the idea, concerned about the consequences. It's worth noting that warp speeds seem to have been re-calibrated again in the future timeline glimpsed in "All Good Things," where two vessels are said to travel at warp 13. Presumably as technology became more efficient, Starfleet saw the value in creating a new scale with a wider range. In a technical note, science adviser Andre Bormanis suggested that in this future timeline, warp 15 would be the Threshold limit.

In truth, warp travel is usually little more than a plot device to help the Federation's starships jump across the galaxy at speed. Almost every sci-fi series has its own equivalent, simply because the alternative is to slow your story down for quite a while as the crew jumps from one star-system to another.

How Warp Speed Compares To Hyperspace

In  Star Wars , for example, ships that break the speed of light are able to access another dimension called hyperspace . This is essentially another plane of reality, one where the laws of physics operate differently. By accessing hyperspace, a spaceship can move at a phenomenal speed; there are different classes of hyperdrive, which allow different speeds.

It's important to note that a journey through hyperspace can be disrupted; if a vessel encounters the gravity shadow of a stellar mass, such as a planet, star, or black hole, they are ripped out of hyperspace. That's why the trip from Tatooine to Alderaan in  Star Wars took 16 hours, which is actually pretty slow for a 10,000 light year journey in a ship with an illegal class 0.5 hyperdrive, and is roughly equivalent to Star Trek 's warp 7. The dream scenario would be to acquire a hyperdrive of class 0.0, which would be analogous to Star Trek 's warp 10.

More:  Picard Isn’t The First Star Trek Captain To Say “Engage”

Log in or Sign up

You are using an out of date browser. It may not display this or other websites correctly. You should upgrade or use an alternative browser .

"Ahead, Space Normal Speed"

Discussion in ' Star Trek - The Original & Animated Series ' started by NuFan , Mar 31, 2014 .

NuFan Lieutenant Commander Red Shirt

What is Space Normal Speed?  

F. King Daniel Fleet Admiral Admiral

Not space-warp speed. Impulse power. Sub-light.  

mach7 Commander Red Shirt

Yup, What he said  

USS Triumphant Vice Admiral Admiral

King Daniel Into Darkness said: ↑ Not space-warp speed. Impulse power. Sub-light. Click to expand...

Melakon Admiral In Memoriam

Slow enough with scanners aft to see if Spock does anything out of desperation.  

BigJake Vice Admiral Admiral

USS Triumphant said: ↑ King Daniel Into Darkness said: ↑ Not space-warp speed. Impulse power. Sub-light. Click to expand...

J.T.B. Rear Admiral Rear Admiral

USS Triumphant said: ↑ Not very bloody specific, though, is it? I mean, he's basically leaving it at Sulu's discretion to go full impulse, or the slowest possible speed the engines can idle at (2 miles an hour, so everybody sees you ), or anywhere in between. Click to expand...

Metryq Fleet Captain Fleet Captain

It's about the speed you moved as a kid when being sent to bed for the night. Ahead full RCS. "Give me a ping."  

Shawnster Rear Admiral Rear Admiral

I always thought it was more like a drifting speed. Like when you turn the motor off on a boat and let the current carry you. Of course, there are no currents in space. Still, that's how I thought of it as a kid. Slow. The speed of a comet or an asteroid that's just floating around out there without any propulsion.  

ZapBrannigan Rear Admiral Rear Admiral

J.T.B. said: ↑ I figure "space normal speed" means some specific speed; captains and helmsmen know what that speed is but we don't. Click to expand...

scotpens Professional Geek Premium Member

It means the speed of plot.  

JarodRussell Vice Admiral Admiral

60s scifi writing. Put random words together and put "Space" in it.  

Ssosmcin Rear Admiral Rear Admiral

I'm more curious as to why Kirk only orders warp one after getting the shuttle crew back, since he dilly dallied his ass up to that point.  

ClayinCA Commodore Commodore

ssosmcin said: ↑ I'm more curious as to why Kirk only orders warp one after getting the shuttle crew back, since he dilly dallied his ass up to that point. Click to expand...

Kevman7987 Fleet Captain Fleet Captain

ClayinCA said: ↑ ssosmcin said: ↑ I'm more curious as to why Kirk only orders warp one after getting the shuttle crew back, since he dilly dallied his ass up to that point. Click to expand...

FredH Fleet Captain Fleet Captain

Well, if the point is "Not warp speed," then it doesn't matter what slower-than-light speed they're actually at. Kirk is telling Sulu to head away from the planet without going to warp, basically "staying in place while moving" since at either .9c or 3mph, you're not reaching the next star system for a long, long time. Kirk doesn't care what slow speed Sulu actually sets, since the point is to stick around and not get anywhere while technically obeying orders to leave.  

Kevman7987 said: ↑ ClayinCA said: ↑ ssosmcin said: ↑ I'm more curious as to why Kirk only orders warp one after getting the shuttle crew back, since he dilly dallied his ass up to that point. Click to expand...

Irwin Allen apparently loved John Crawford, using him four times in The Time Tunnel, once on Lost in Space, Land of the Giants and Voyage to the Bottom of the Sea. Not to mention The Poseidon Adventure and The Towering Inferno.  

Timo Fleet Admiral Admiral

I'd favor the "it's any speed below warp" interpretation, for the reason FredH gives: it's a unique order to refrain from leaving the star system too soon. If it were some specific, agreed-upon speed every helmsman knows about, one would assume the order to be given in every third episode at least - not to be applied here and here only. Why go to the trouble of agreeing upon something if you never actually put it to use? That we hear the order only once in the history of Star Trek is consistent with it being an order of low applicability. More specific things such as "half impulse" or "forward thrusters" have more utility value and thus are heard more often. On the other hand, while warp one seems to be a perfectly useful interstellar speed in many episodes, it's odd for Kirk to use it when in a hurry. If warp one will do at the conclusion of the adventure, supposedly the ship could have waited for a couple of days and then sailed out at warp three, or taken a two-month leave of absence and then shot out at warp seven. I guess we have to assume that it's not wise to apply high-cochrane warp fields close to mini-quasars... Timo Saloniemi  

LMFAOschwarz Fleet Captain Fleet Captain

This makes me think also of ST:TMP , where Kirk orders "Ahead warp...point five!". Wouldn't a sub-warp velocity be in the dominion of an impulse speed? As far as 'space normal', my take is that it depends where you are. It might be the speed of a given object (artificial or natural), and the velocity required to maintain its position against gravitational pull from planets, stars, etc....kind of like an orbital velocity, really: too slow and you begin to arc toward the gravity source. Space normal speed when ten million miles from a star will be faster than space normal speed when a hundred million miles from the same star. I hope I explained that well. If I'm not going to make any sense, I at least want to say it right.  

• Log in with Facebook
• No, create an account now.
• Yes, my password is:
• Forgot your password?
• Search titles only

Separate names with a comma.

• Search this thread only
• Display results as threads

Useful Searches

• Recent Posts

How Star Trek's Warp Speed Works

Explaining the seemingly wonky science behind The Starship Enterprise.

The following is an excerpt of The Science of Superheroes and Space Warriors published by HowStuffWorks.

When the writers of Star Trek sat down to plan the series, they found themselves confronted with a few problems. They were essentially creating a space opera, a subgenre of science fiction that takes place in space and covers the span of several galaxies and millions of light-years. As the “opera” part of the name suggests, a show like Star Trek isn’t meant to be slow or ordinary. When people think of the series, they probably think of melodramatic plots involving aliens, space travel, and action-packed laser fights.

So the creator of the series, Gene Roddenberry, and the other writers had to find a way to move the show’s characters around the universe in a timely, dramatic fashion. At the same time, they wanted to do their best to stick to the laws of physics. The biggest problem was that even if a starship could travel at the speed of light, the time to go from one galaxy to another could still take hundreds, maybe thousands of years. A journey from Earth to the center of our galaxy, for example, would take about twenty-five thousand years if you were to travel just under the speed of light. This, of course, wouldn’t make very exciting television.

The invention of warp speed solved the opera part of the problem, since it allowed the Enterprise to go much faster than the speed of light. But what was the explanation? How could they explain an object traveling faster than the speed of light, something Einstein proved impossible in his special theory of relativity?

The USS Enterprise

The first obstacle the writers had to confront is much simpler than you’d think. It comes down to Newton’s third law of motion: how for every action, there is an equal and opposite reaction.

What does this have to do with Star Trek and the Enterprise? Even if it were possible to accelerate to something like half the speed of light, such intense acceleration would kill a person by smashing him against his seat. Even though he’d be pushing back with an equal and opposite force, his mass compared to the starship is just too small. The same kind of thing happens when a mosquito hits your windshield and splatters. So how can the Enterprise possibly go faster than the speed of light without killing the members on board?

To sidestep the issue of Newton’s third law of motion and the impossibility of matter traveling faster than the speed of light, we can look to Einstein and the relationship between space and time. Taken together, space (consisting of three dimensions: up-down, left-right, and forward-backward) and time are part of what’s called the space-time continuum.

In his special theory of relativity, Einstein states two postulates:

• The speed of light (about 300 million meters per second) is the same for all observers, whether or not they’re moving.
• Anyone moving at a constant speed should observe the same physical laws.

Putting these two ideas together, Einstein realized that space and time are relative—an object in motion actually experiences time at a slower rate than one at rest. Although this may seem absurd to us, we travel incredibly slowly when compared to the speed of light, so we don’t notice the hands on our watches ticking slower when we’re running or traveling on an airplane.

USS Enterprise at warp speed in Into The Darkness.

What does this mean for the Captain Kirk and his team? As an object gets closer and closer to the speed of light, that object actually experiences time at a significantly slower rate. If the Enterprise were traveling safely at close to the speed of light to the center of our galaxy from Earth, it would take twenty-five thousand years of Earth time. For the crew, however, the trip would probably only take ten years.

Although that time frame might be possible for the individuals on board, we’re presented with yet another problem—a Federation attempting to run an intergalactic civilization would run into some problems if it took fifty thousand years for a starship to hit the center of our galaxy and come back.

So the Enterprise has to avoid the speed of light to keep the passengers onboard in sync with Federation time. At the same time, the Enterprise must reach speeds faster than that of light to move around the universe in an efficient manner. Unfortunately, as Einstein states in his special theory of relativity, nothing is faster than the speed of light. Space travel therefore would be impossible if we’re looking at the special relativity.

That’s why we need to look at Einstein’s later theory, the general theory of relativity, which describes how gravity affects the shape of space and flow of time. Imagine a stretched-out sheet. If you place a bowling ball in the middle of the sheet, the sheet will warp as the weight of the ball pushes down on it. If you place a baseball on the same sheet, it will roll toward the bowling ball. This is a simple design, and space doesn’t act like a two-dimensional bed sheet, but it can be applied to something like our solar system. More massive objects like our sun can warp space and affect the orbits of the surrounding planets. The planets don’t fall into the sun, of course, because of the high speeds at which they travel.

The ability to manipulate space is the most important concept in regard to warp speed. If the Enterprise could warp the space-time continuum by expanding the area behind it and contracting the area in front, the crew could avoid going the speed of light. As long as it creates its own gravitational field, the starship could travel locally at very slow velocities, therefore avoiding the pitfalls of Newton’s third law of motion and keeping clocks in sync with its launch site and destination. The ship isn’t really traveling at a “speed,” per se—it’s more like it’s pulling its destination toward it while pushing its starting point back.

Don’t Miss: The Scientist Who Keeps Science Fiction Real

• Science Fiction

Sublight propulsion

• Edit source
• View history

Sublight (or space-normal speed) is a term that refers to anything below the speed of light, approximately 3 × 10E8 meters/second.

Sublight speed is usually used to describe non- warp velocities -- typically when either an impulse drive or thrusters would be used as propulsion for a vessel -- though an impulse-powered flight of half lightspeed may be described as "Warp point 5". ( Star Trek: The Motion Picture )

As far as it is known, Earth spacecraft had no means of propelling ships to the speed of light or greater until Zefram Cochrane 's Phoenix test-flight in 2063 . ( Star Trek: First Contact ) Impulse drive, developed c. 2018 ( TOS : " Space Seed "), initially drove spacecraft to 0.25 c. ( TOS novel: The Rings of Time ) This made exploration and colonization within the Sol system feasible, less so interstellar exploration and colonization -- though many generation ships of the 21st and even early 22nd centuries undertook decades-long journeys to do so. ( TAS : " The Terratin Incident ", Star Trek Spaceflight Chronology , Star Trek: Shadowstar Station : "Last Night I Dreamed Somebody Loved Me", "Esteban")

All sublight vessels were subject to time-dilatation effects, the more so as they approached light speed.

In addition to impulse (essentially a nuclear-fusion drive), thruster and ion drives, other means of sublight propulsion included the use of solar sails and even the use of nuclear-fission bombs against a ship's pusher plate. Many starfaring civilizations experimented with this latter method; on Earth, it was widely known as Project Orion . Another system, using nuclear fission and chemical propellants, was known as NERVA , for N uclear E ngine for R ocket V ehicle A pplication.

Bajoran lightships traveled at sublight speeds. ( DS9 : " Explorers ")

• 1 Dominion War
• 2 USS Enterprise (NCC-1701-G) (Excalibur class)
• 3 USS Enterprise (NCC-1701-H) (Endurance class)

Warp Speed Calculator

Table of contents

Omni Calculator's warp speed calculator is a tool that allows you to determine the speed of your favorite Star Trek ship . You can also compute the distance covered and the time of travel .

In the text below, we will explain the fundamental aspects of the warp propulsion system, warp factors, and how we can compute the speed of our starship. The starship speed can be calculated using three different equations: TOS (The Original Series) and two versions of TNG (The New Generation) .

Read on if you want to learn about the warp factor, the Cochrane scale , and how Star Trek ships can overcome the speed of light. Let us move to our calculator and live long and prosper🖖!

⚠️ Disclaimer: We try our best to make our Omni Calculators as precise and reliable as possible. However, this tool can never replace the professional advice of a starship's Chief Engineer.

Warp propulsion and warp factor

As it is known by any Star Trek fan or by anyone who eventually watched the USS Enterprise exploring the frontiers of space, the warp propulsion system (WPS) is the tool that allowed Captain James T. Kirk and his crew to access deep space. The WPS is a fictional technology that makes a starship move in space with superluminal speeds (speeds faster than the speed of light). The system uses a warp drive device, distorting spacetime and enabling a starship to cross several light-years in seconds!

We know it seems non-practical or strange that the spacetime can be distorted. However, this is the mechanism behind the propagation of the gravitational waves . These waves are generated by cataclysmic events in the Universe, such as the collision between supermassive black holes or neutron stars. Before their final collision, these objects are orbiting each other, and this spiral movement distorts the spacetime among them. This distortion is propagated in spacetime in the form of waves. The phenomenon was predicted by Einstein's general relativity in 1916 and observed for the first time in 2015 by LIGO detectors — Laser Interferometer Gravitational-Wave Observatory .

Back to Star Trek, the advent of the WPS is credited to a scientist called Zefram Cochrane , whose surname is used as the scale to measure the speed of a starship in multiples of the speed of light . This means that the light in a vacuum moves at one cochrane . Meanwhile, a star cruiser may reach 2450 cochranes . These superluminal speeds depend on the value of the so-called warp factor. The warp factor is a parameter that scales the starship's velocity due to the warp drive.

The warp factor formula has two main definitions: The Original Series (TOS) and The New Generation (TNG) equations. These equations are going to be discussed deeply in the following sections.

🙋 If you wish to know more about the physical limits of the speed and energy of an object, access Omni Calculator's relativistic kinetic energy calculator .

The original series equation — Cochrane Scale

The original series equation is presented in Star Trek: Star Fleet Technical Manual written by Franz Joseph, and its form is:

• v v v — Warp speed;
• w w w — Warp factor; and
• c c c — Speed of light in vacuum ( c = 299 , 792 , 458   m / s c = 299{,}792{,}458\rm \ m/s c = 299 , 792 , 458   m/s ).

The TOS equation is also known as the Cochrane scale among the community of Star Trek fans. From the previous formula, we can observe that for w = 2 w = 2 w = 2 , our ship is moving with v = 8c or 8 cochranes .

The Next Generation equations for superluminal speed

The Next Generation equations are a set of theoretical formulae based on the values for the warp factor presented by Rick Sternbach and Michael Okuda in the Star Trek: The Next Generation Technical Manual . There, the authors introduced the following table:

The data shown in the table above were used to define both TNG-1 and TNG-2 equations for w ≤ 9 w \leq 9 w ≤ 9 , whose formula is:

You can compare the speeds for the TOS with the TNG-1 and TNG-2 equations and see how the new generation starships are faster than the original series ones.

Would you like to know why we use two names for the new generation warp speed equations? Keep reading, and you will find the answer.

The technical manual for the new generation starships also talks about the physical limit for the warp factor, known as Eugene limit . The Eugene limit establishes that the maximum warp factor is w = 10 w = 10 w = 10 and that even if a civilization could reach the energy boundary of such a warp factor, an object traveling at warp factor 10 would occupy all the points of the Universe simultaneously .

You can find below a table containing the warp factor predictions for w > 9 w > 9 w > 9 :

The conjectured values are part of a warp factor chart and were approximately derived by two different equations built by fans .

This is why we use two different equations for the Next Generation starships. The TNG-1 formula for w > 9 w > 9 w > 9 is:

And the TNG-2 equation for w > 9 w > 9 w > 9 has the following form:

The numerical parameters shown in the last equations were carefully chosen to fit the warp factor chart prediction in the superluminal regime. Besides, both TNG-1 and TNG-2 go to infinity at warp factor 10, recovering the Eugene limit for superluminal velocities.

We still do not have a final answer for a warp speed equation that recovers the Eugene limit, so you can choose your favorite proposal and explore the behavior of traveling close to infinity .

Now, it is time to play with our calculator and find the speed of your favorite starship. Enjoy exploring the power of your warp drive!

How to use the warp speed calculator

Using our warp speed calculator is simple and intuitive . Just type the warp factor of your starship and then click on the name of the equation you wish to use. You will see the warp factor converted to the superluminal velocity as a multiple of the speed of light (or in cochranes).

You can also include the distance you would like to travel, and you will obtain the time necessary to complete your journey instantly. Remember that our Omni calculators work in the inverse direction, so you may enter the time interval of your space trip and derive the traveled distance in the blink of an eye.

Do you know what value to choose for the warp factor? You do not need to worry; we provide a table with the maximum warp factor for some Star Trek ships.

Searching for a new planet — an example

The Universe is so vast and full of interesting places that it is difficult to decide a destination for our starship. A potential target recently discovered by the James Webb Space Telescope is a planet known as LHS 475b .

This Earth-size planet orbits the star LHS 475, localized in the constellation of Octans. You can find more information about other fascinating new worlds in our exoplanet discovery calculator .

So, let us consider the USS ENTERPRISE D as our starship to explore this new world. Then, by taking warp factor 9.3 and 41 light years as the distance between Earth and LHS 475b in our warp speed calculator, the TOS, TNG-1, and TNG-2 equations give us:

v = 804   c v = 804\,c v = 804 c , t = 447   hours t = 447 \, \text{hours} t = 447 hours ;

v = 1713   c v = 1713\,c v = 1713 c , t = 210   hours t = 210 \, \text{hours} t = 210 hours ; and

v = 1693   c v = 1693\,c v = 1693 c , t = 212   hours t = 212 \, \text{hours} t = 212 hours , respectively.

🔎 Now, you can compare these results with those derived in our exoplanet travel planner calculator and check the power of the warp propulsion system!

How fast is Star Trek warp speed?

In Star Trek, the warp speed is a velocity scaled by the warp factor . The warp factor formula is inconsistent throughout the series . In the original series, for instance, warp factors were described using the formula v = w³⋅c , where c is the speed of light in a vacuum. Therefore, a Star Trek ship may travel faster than light , allowing the civilizations to explore the Universe.

How do I calculate the warp speed of a starship?

You can calculate the warp speed in Star Trek using two formulae: The Original Series (TOS) and The Next Generation (TNG) equations. In both cases, the speeds are given as multiples of the speed of light c . To do that, let us suppose that your starship has a warp factor w = 5 , then follow the steps below:

Input the warp factor in the TOS equation v = w 3 ⋅c .

Input the warp factor in the TNG equation v = w 10/3 ⋅c .

That is it! You will find that the warp speeds are v = 125⋅c and v = 214⋅c , for the TOS and TNG equations, respectively.

How long it takes for a starship to leave the solar system?

The most distant object made by humanity is Voyager 1 , which will take more than 30,000 years to reach the outer edge of the Oort Cloud , whose distance is estimated at 100,000 AU from the sun. The USS Enterprise D, flying in cruise mode ( w = 6 ), could reach this distance in 35 hours . To find this result:

Input the warp factor in the formula v = w 10/3 ⋅c .

Compute the warp velocity v = 392⋅c .

Isolate the time in the velocity formula t = d/v .

Input the distance d = 100,00 AU and c = 299,792,458 m/s in the last formula.

Convert the units properly.

That is it! You found that t = 35 hours .

Is a warp drive possible?

With the present science and technology, the answer is no . However, some theoretical models allow the creation of a warp bubble in flat spacetime, which can move faster than the speed of light. The most famous theoretical proposal in this direction was introduced by Miguel Alcubierre in 1994 , also known as the Alcubierre metric or Alcubierre drive .

.css-slt4t3.css-slt4t3{color:#2B3148;background-color:transparent;font-family:"Roboto","Helvetica","Arial",sans-serif;font-size:20px;line-height:24px;overflow:visible;padding-top:0px;position:relative;}.css-slt4t3.css-slt4t3:after{content:'';-webkit-transform:scale(0);-moz-transform:scale(0);-ms-transform:scale(0);transform:scale(0);position:absolute;border:2px solid #EA9430;border-radius:2px;inset:-8px;z-index:1;}.css-slt4t3 .js-external-link-button.link-like,.css-slt4t3 .js-external-link-anchor{color:inherit;border-radius:1px;-webkit-text-decoration:underline;text-decoration:underline;}.css-slt4t3 .js-external-link-button.link-like:hover,.css-slt4t3 .js-external-link-anchor:hover,.css-slt4t3 .js-external-link-button.link-like:active,.css-slt4t3 .js-external-link-anchor:active{text-decoration-thickness:2px;text-shadow:1px 0 0;}.css-slt4t3 .js-external-link-button.link-like:focus-visible,.css-slt4t3 .js-external-link-anchor:focus-visible{outline:transparent 2px dotted;box-shadow:0 0 0 2px #6314E6;}.css-slt4t3 p,.css-slt4t3 div{margin:0px;display:block;}.css-slt4t3 pre{margin:0px;display:block;}.css-slt4t3 pre code{display:block;width:-webkit-fit-content;width:-moz-fit-content;width:fit-content;}.css-slt4t3 pre:not(:first-child){padding-top:8px;}.css-slt4t3 ul,.css-slt4t3 ol{display:block margin:0px;padding-left:20px;}.css-slt4t3 ul li,.css-slt4t3 ol li{padding-top:8px;}.css-slt4t3 ul ul,.css-slt4t3 ol ul,.css-slt4t3 ul ol,.css-slt4t3 ol ol{padding-top:0px;}.css-slt4t3 ul:not(:first-child),.css-slt4t3 ol:not(:first-child){padding-top:4px;} .css-4okk7a{margin:auto;background-color:white;overflow:auto;overflow-wrap:break-word;word-break:break-word;}.css-4okk7a code,.css-4okk7a kbd,.css-4okk7a pre,.css-4okk7a samp{font-family:monospace;}.css-4okk7a code{padding:2px 4px;color:#444;background:#ddd;border-radius:4px;}.css-4okk7a figcaption,.css-4okk7a caption{text-align:center;}.css-4okk7a figcaption{font-size:12px;font-style:italic;overflow:hidden;}.css-4okk7a h3{font-size:1.75rem;}.css-4okk7a h4{font-size:1.5rem;}.css-4okk7a .mathBlock{font-size:24px;-webkit-padding-start:4px;padding-inline-start:4px;}.css-4okk7a .mathBlock .katex{font-size:24px;text-align:left;}.css-4okk7a .math-inline{background-color:#f0f0f0;display:inline-block;font-size:inherit;padding:0 3px;}.css-4okk7a .videoBlock,.css-4okk7a .imageBlock{margin-bottom:16px;}.css-4okk7a .imageBlock__image-align--left,.css-4okk7a .videoBlock__video-align--left{float:left;}.css-4okk7a .imageBlock__image-align--right,.css-4okk7a .videoBlock__video-align--right{float:right;}.css-4okk7a .imageBlock__image-align--center,.css-4okk7a .videoBlock__video-align--center{display:block;margin-left:auto;margin-right:auto;clear:both;}.css-4okk7a .imageBlock__image-align--none,.css-4okk7a .videoBlock__video-align--none{clear:both;margin-left:0;margin-right:0;}.css-4okk7a .videoBlock__video--wrapper{position:relative;padding-bottom:56.25%;height:0;}.css-4okk7a .videoBlock__video--wrapper iframe{position:absolute;top:0;left:0;width:100%;height:100%;}.css-4okk7a .videoBlock__caption{text-align:left;}@font-face{font-family:'KaTeX_AMS';src:url(/katex-fonts/KaTeX_AMS-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_AMS-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_AMS-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Caligraphic';src:url(/katex-fonts/KaTeX_Caligraphic-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Caligraphic-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Caligraphic-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Caligraphic';src:url(/katex-fonts/KaTeX_Caligraphic-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Caligraphic-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Caligraphic-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Fraktur';src:url(/katex-fonts/KaTeX_Fraktur-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Fraktur-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Fraktur-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Fraktur';src:url(/katex-fonts/KaTeX_Fraktur-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Fraktur-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Fraktur-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-BoldItalic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-BoldItalic.woff) format('woff'),url(/katex-fonts/KaTeX_Main-BoldItalic.ttf) format('truetype');font-weight:bold;font-style:italic;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Math';src:url(/katex-fonts/KaTeX_Math-BoldItalic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Math-BoldItalic.woff) format('woff'),url(/katex-fonts/KaTeX_Math-BoldItalic.ttf) format('truetype');font-weight:bold;font-style:italic;}@font-face{font-family:'KaTeX_Math';src:url(/katex-fonts/KaTeX_Math-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Math-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_Math-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Script';src:url(/katex-fonts/KaTeX_Script-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Script-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Script-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size1';src:url(/katex-fonts/KaTeX_Size1-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size1-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size1-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size2';src:url(/katex-fonts/KaTeX_Size2-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size2-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size2-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size3';src:url(/katex-fonts/KaTeX_Size3-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size3-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size3-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size4';src:url(/katex-fonts/KaTeX_Size4-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size4-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size4-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Typewriter';src:url(/katex-fonts/KaTeX_Typewriter-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Typewriter-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Typewriter-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}.css-4okk7a .katex{font:normal 1.21em KaTeX_Main,Times New Roman,serif;line-height:1.2;text-indent:0;text-rendering:auto;}.css-4okk7a .katex *{-ms-high-contrast-adjust:none!important;border-color:currentColor;}.css-4okk7a .katex .katex-version::after{content:'0.13.13';}.css-4okk7a .katex .katex-mathml{position:absolute;clip:rect(1px, 1px, 1px, 1px);padding:0;border:0;height:1px;width:1px;overflow:hidden;}.css-4okk7a .katex .katex-html>.newline{display:block;}.css-4okk7a .katex .base{position:relative;display:inline-block;white-space:nowrap;width:-webkit-min-content;width:-moz-min-content;width:-webkit-min-content;width:-moz-min-content;width:min-content;}.css-4okk7a .katex .strut{display:inline-block;}.css-4okk7a .katex .textbf{font-weight:bold;}.css-4okk7a .katex .textit{font-style:italic;}.css-4okk7a .katex .textrm{font-family:KaTeX_Main;}.css-4okk7a .katex .textsf{font-family:KaTeX_SansSerif;}.css-4okk7a .katex .texttt{font-family:KaTeX_Typewriter;}.css-4okk7a .katex .mathnormal{font-family:KaTeX_Math;font-style:italic;}.css-4okk7a .katex .mathit{font-family:KaTeX_Main;font-style:italic;}.css-4okk7a .katex .mathrm{font-style:normal;}.css-4okk7a .katex .mathbf{font-family:KaTeX_Main;font-weight:bold;}.css-4okk7a .katex .boldsymbol{font-family:KaTeX_Math;font-weight:bold;font-style:italic;}.css-4okk7a .katex .amsrm{font-family:KaTeX_AMS;}.css-4okk7a .katex .mathbb,.css-4okk7a .katex .textbb{font-family:KaTeX_AMS;}.css-4okk7a .katex .mathcal{font-family:KaTeX_Caligraphic;}.css-4okk7a .katex .mathfrak,.css-4okk7a .katex .textfrak{font-family:KaTeX_Fraktur;}.css-4okk7a .katex .mathtt{font-family:KaTeX_Typewriter;}.css-4okk7a .katex .mathscr,.css-4okk7a .katex .textscr{font-family:KaTeX_Script;}.css-4okk7a .katex .mathsf,.css-4okk7a .katex .textsf{font-family:KaTeX_SansSerif;}.css-4okk7a .katex .mathboldsf,.css-4okk7a .katex .textboldsf{font-family:KaTeX_SansSerif;font-weight:bold;}.css-4okk7a .katex .mathitsf,.css-4okk7a .katex .textitsf{font-family:KaTeX_SansSerif;font-style:italic;}.css-4okk7a .katex .mainrm{font-family:KaTeX_Main;font-style:normal;}.css-4okk7a .katex .vlist-t{display:inline-table;table-layout:fixed;border-collapse:collapse;}.css-4okk7a .katex .vlist-r{display:table-row;}.css-4okk7a .katex .vlist{display:table-cell;vertical-align:bottom;position:relative;}.css-4okk7a .katex .vlist>span{display:block;height:0;position:relative;}.css-4okk7a .katex .vlist>span>span{display:inline-block;}.css-4okk7a .katex .vlist>span>.pstrut{overflow:hidden;width:0;}.css-4okk7a .katex .vlist-t2{margin-right:-2px;}.css-4okk7a .katex .vlist-s{display:table-cell;vertical-align:bottom;font-size:1px;width:2px;min-width:2px;}.css-4okk7a .katex .vbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;-webkit-align-items:baseline;-webkit-box-align:baseline;-ms-flex-align:baseline;align-items:baseline;}.css-4okk7a .katex .hbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;width:100%;}.css-4okk7a .katex .thinbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;width:0;max-width:0;}.css-4okk7a .katex .msupsub{text-align:left;}.css-4okk7a .katex .mfrac>span>span{text-align:center;}.css-4okk7a .katex .mfrac .frac-line{display:inline-block;width:100%;border-bottom-style:solid;}.css-4okk7a .katex .mfrac .frac-line,.css-4okk7a .katex .overline .overline-line,.css-4okk7a .katex .underline .underline-line,.css-4okk7a .katex .hline,.css-4okk7a .katex .hdashline,.css-4okk7a .katex .rule{min-height:1px;}.css-4okk7a .katex .mspace{display:inline-block;}.css-4okk7a .katex .llap,.css-4okk7a .katex .rlap,.css-4okk7a .katex .clap{width:0;position:relative;}.css-4okk7a .katex .llap>.inner,.css-4okk7a .katex .rlap>.inner,.css-4okk7a .katex .clap>.inner{position:absolute;}.css-4okk7a .katex .llap>.fix,.css-4okk7a .katex .rlap>.fix,.css-4okk7a .katex .clap>.fix{display:inline-block;}.css-4okk7a .katex .llap>.inner{right:0;}.css-4okk7a .katex .rlap>.inner,.css-4okk7a .katex .clap>.inner{left:0;}.css-4okk7a .katex .clap>.inner>span{margin-left:-50%;margin-right:50%;}.css-4okk7a .katex .rule{display:inline-block;border:solid 0;position:relative;}.css-4okk7a .katex .overline .overline-line,.css-4okk7a .katex .underline .underline-line,.css-4okk7a .katex .hline{display:inline-block;width:100%;border-bottom-style:solid;}.css-4okk7a .katex .hdashline{display:inline-block;width:100%;border-bottom-style:dashed;}.css-4okk7a .katex .sqrt>.root{margin-left:0.27777778em;margin-right:-0.55555556em;}.css-4okk7a .katex .sizing.reset-size1.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size1{font-size:1em;}.css-4okk7a .katex .sizing.reset-size1.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size2{font-size:1.2em;}.css-4okk7a .katex .sizing.reset-size1.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size3{font-size:1.4em;}.css-4okk7a .katex .sizing.reset-size1.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size4{font-size:1.6em;}.css-4okk7a .katex .sizing.reset-size1.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size5{font-size:1.8em;}.css-4okk7a .katex .sizing.reset-size1.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size6{font-size:2em;}.css-4okk7a .katex .sizing.reset-size1.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size7{font-size:2.4em;}.css-4okk7a .katex .sizing.reset-size1.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size8{font-size:2.88em;}.css-4okk7a .katex .sizing.reset-size1.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size9{font-size:3.456em;}.css-4okk7a .katex .sizing.reset-size1.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size10{font-size:4.148em;}.css-4okk7a .katex .sizing.reset-size1.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size11{font-size:4.976em;}.css-4okk7a .katex .sizing.reset-size2.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size1{font-size:0.83333333em;}.css-4okk7a .katex .sizing.reset-size2.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size2{font-size:1em;}.css-4okk7a .katex .sizing.reset-size2.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size3{font-size:1.16666667em;}.css-4okk7a .katex .sizing.reset-size2.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size4{font-size:1.33333333em;}.css-4okk7a .katex .sizing.reset-size2.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size5{font-size:1.5em;}.css-4okk7a .katex .sizing.reset-size2.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size6{font-size:1.66666667em;}.css-4okk7a .katex .sizing.reset-size2.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size7{font-size:2em;}.css-4okk7a .katex .sizing.reset-size2.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size8{font-size:2.4em;}.css-4okk7a .katex .sizing.reset-size2.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size9{font-size:2.88em;}.css-4okk7a .katex .sizing.reset-size2.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size10{font-size:3.45666667em;}.css-4okk7a .katex .sizing.reset-size2.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size11{font-size:4.14666667em;}.css-4okk7a .katex .sizing.reset-size3.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size1{font-size:0.71428571em;}.css-4okk7a .katex .sizing.reset-size3.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size2{font-size:0.85714286em;}.css-4okk7a .katex .sizing.reset-size3.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size3{font-size:1em;}.css-4okk7a .katex .sizing.reset-size3.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size4{font-size:1.14285714em;}.css-4okk7a .katex .sizing.reset-size3.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size5{font-size:1.28571429em;}.css-4okk7a .katex .sizing.reset-size3.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size6{font-size:1.42857143em;}.css-4okk7a .katex .sizing.reset-size3.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size7{font-size:1.71428571em;}.css-4okk7a .katex .sizing.reset-size3.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size8{font-size:2.05714286em;}.css-4okk7a .katex .sizing.reset-size3.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size9{font-size:2.46857143em;}.css-4okk7a .katex .sizing.reset-size3.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size10{font-size:2.96285714em;}.css-4okk7a .katex .sizing.reset-size3.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size11{font-size:3.55428571em;}.css-4okk7a .katex .sizing.reset-size4.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size1{font-size:0.625em;}.css-4okk7a .katex .sizing.reset-size4.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size2{font-size:0.75em;}.css-4okk7a .katex .sizing.reset-size4.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size3{font-size:0.875em;}.css-4okk7a .katex .sizing.reset-size4.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size4{font-size:1em;}.css-4okk7a .katex .sizing.reset-size4.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size5{font-size:1.125em;}.css-4okk7a .katex .sizing.reset-size4.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size6{font-size:1.25em;}.css-4okk7a .katex .sizing.reset-size4.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size7{font-size:1.5em;}.css-4okk7a .katex .sizing.reset-size4.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size8{font-size:1.8em;}.css-4okk7a .katex .sizing.reset-size4.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size9{font-size:2.16em;}.css-4okk7a .katex .sizing.reset-size4.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size10{font-size:2.5925em;}.css-4okk7a .katex .sizing.reset-size4.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size11{font-size:3.11em;}.css-4okk7a .katex .sizing.reset-size5.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size1{font-size:0.55555556em;}.css-4okk7a .katex .sizing.reset-size5.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size2{font-size:0.66666667em;}.css-4okk7a .katex .sizing.reset-size5.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size3{font-size:0.77777778em;}.css-4okk7a .katex .sizing.reset-size5.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size4{font-size:0.88888889em;}.css-4okk7a .katex .sizing.reset-size5.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size5{font-size:1em;}.css-4okk7a .katex .sizing.reset-size5.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size6{font-size:1.11111111em;}.css-4okk7a .katex .sizing.reset-size5.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size7{font-size:1.33333333em;}.css-4okk7a .katex .sizing.reset-size5.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size8{font-size:1.6em;}.css-4okk7a .katex .sizing.reset-size5.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size9{font-size:1.92em;}.css-4okk7a .katex .sizing.reset-size5.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size10{font-size:2.30444444em;}.css-4okk7a .katex .sizing.reset-size5.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size11{font-size:2.76444444em;}.css-4okk7a .katex .sizing.reset-size6.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size1{font-size:0.5em;}.css-4okk7a .katex .sizing.reset-size6.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size2{font-size:0.6em;}.css-4okk7a .katex .sizing.reset-size6.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size3{font-size:0.7em;}.css-4okk7a .katex .sizing.reset-size6.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size4{font-size:0.8em;}.css-4okk7a .katex .sizing.reset-size6.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size5{font-size:0.9em;}.css-4okk7a .katex .sizing.reset-size6.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size6{font-size:1em;}.css-4okk7a .katex .sizing.reset-size6.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size7{font-size:1.2em;}.css-4okk7a .katex .sizing.reset-size6.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size8{font-size:1.44em;}.css-4okk7a .katex .sizing.reset-size6.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size9{font-size:1.728em;}.css-4okk7a .katex .sizing.reset-size6.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size10{font-size:2.074em;}.css-4okk7a .katex .sizing.reset-size6.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size11{font-size:2.488em;}.css-4okk7a .katex .sizing.reset-size7.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size1{font-size:0.41666667em;}.css-4okk7a .katex .sizing.reset-size7.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size2{font-size:0.5em;}.css-4okk7a .katex .sizing.reset-size7.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size3{font-size:0.58333333em;}.css-4okk7a .katex .sizing.reset-size7.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size4{font-size:0.66666667em;}.css-4okk7a .katex .sizing.reset-size7.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size5{font-size:0.75em;}.css-4okk7a .katex .sizing.reset-size7.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size6{font-size:0.83333333em;}.css-4okk7a .katex .sizing.reset-size7.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size7{font-size:1em;}.css-4okk7a .katex .sizing.reset-size7.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size8{font-size:1.2em;}.css-4okk7a .katex .sizing.reset-size7.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size9{font-size:1.44em;}.css-4okk7a .katex .sizing.reset-size7.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size10{font-size:1.72833333em;}.css-4okk7a .katex .sizing.reset-size7.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size11{font-size:2.07333333em;}.css-4okk7a .katex .sizing.reset-size8.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size1{font-size:0.34722222em;}.css-4okk7a .katex .sizing.reset-size8.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size2{font-size:0.41666667em;}.css-4okk7a .katex .sizing.reset-size8.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size3{font-size:0.48611111em;}.css-4okk7a .katex .sizing.reset-size8.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size4{font-size:0.55555556em;}.css-4okk7a .katex .sizing.reset-size8.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size5{font-size:0.625em;}.css-4okk7a .katex .sizing.reset-size8.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size6{font-size:0.69444444em;}.css-4okk7a .katex .sizing.reset-size8.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size7{font-size:0.83333333em;}.css-4okk7a .katex .sizing.reset-size8.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size8{font-size:1em;}.css-4okk7a .katex .sizing.reset-size8.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size9{font-size:1.2em;}.css-4okk7a .katex .sizing.reset-size8.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size10{font-size:1.44027778em;}.css-4okk7a .katex .sizing.reset-size8.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size11{font-size:1.72777778em;}.css-4okk7a .katex .sizing.reset-size9.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size1{font-size:0.28935185em;}.css-4okk7a .katex .sizing.reset-size9.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size2{font-size:0.34722222em;}.css-4okk7a .katex .sizing.reset-size9.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size3{font-size:0.40509259em;}.css-4okk7a .katex .sizing.reset-size9.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size4{font-size:0.46296296em;}.css-4okk7a .katex .sizing.reset-size9.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size5{font-size:0.52083333em;}.css-4okk7a .katex .sizing.reset-size9.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size6{font-size:0.5787037em;}.css-4okk7a .katex .sizing.reset-size9.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size7{font-size:0.69444444em;}.css-4okk7a .katex .sizing.reset-size9.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size8{font-size:0.83333333em;}.css-4okk7a .katex .sizing.reset-size9.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size9{font-size:1em;}.css-4okk7a .katex .sizing.reset-size9.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size10{font-size:1.20023148em;}.css-4okk7a .katex .sizing.reset-size9.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size11{font-size:1.43981481em;}.css-4okk7a .katex .sizing.reset-size10.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size1{font-size:0.24108004em;}.css-4okk7a .katex .sizing.reset-size10.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size2{font-size:0.28929605em;}.css-4okk7a .katex .sizing.reset-size10.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size3{font-size:0.33751205em;}.css-4okk7a .katex .sizing.reset-size10.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size4{font-size:0.38572806em;}.css-4okk7a .katex .sizing.reset-size10.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size5{font-size:0.43394407em;}.css-4okk7a .katex .sizing.reset-size10.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size6{font-size:0.48216008em;}.css-4okk7a .katex .sizing.reset-size10.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size7{font-size:0.57859209em;}.css-4okk7a .katex .sizing.reset-size10.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size8{font-size:0.69431051em;}.css-4okk7a .katex .sizing.reset-size10.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size9{font-size:0.83317261em;}.css-4okk7a .katex .sizing.reset-size10.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size10{font-size:1em;}.css-4okk7a .katex .sizing.reset-size10.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size11{font-size:1.19961427em;}.css-4okk7a .katex .sizing.reset-size11.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size1{font-size:0.20096463em;}.css-4okk7a .katex .sizing.reset-size11.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size2{font-size:0.24115756em;}.css-4okk7a .katex .sizing.reset-size11.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size3{font-size:0.28135048em;}.css-4okk7a .katex .sizing.reset-size11.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size4{font-size:0.32154341em;}.css-4okk7a .katex .sizing.reset-size11.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size5{font-size:0.36173633em;}.css-4okk7a .katex .sizing.reset-size11.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size6{font-size:0.40192926em;}.css-4okk7a .katex .sizing.reset-size11.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size7{font-size:0.48231511em;}.css-4okk7a .katex .sizing.reset-size11.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size8{font-size:0.57877814em;}.css-4okk7a .katex .sizing.reset-size11.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size9{font-size:0.69453376em;}.css-4okk7a .katex .sizing.reset-size11.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size10{font-size:0.83360129em;}.css-4okk7a .katex .sizing.reset-size11.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size11{font-size:1em;}.css-4okk7a .katex .delimsizing.size1{font-family:KaTeX_Size1;}.css-4okk7a .katex .delimsizing.size2{font-family:KaTeX_Size2;}.css-4okk7a .katex .delimsizing.size3{font-family:KaTeX_Size3;}.css-4okk7a .katex .delimsizing.size4{font-family:KaTeX_Size4;}.css-4okk7a .katex .delimsizing.mult .delim-size1>span{font-family:KaTeX_Size1;}.css-4okk7a .katex .delimsizing.mult .delim-size4>span{font-family:KaTeX_Size4;}.css-4okk7a .katex .nulldelimiter{display:inline-block;width:0.12em;}.css-4okk7a .katex .delimcenter{position:relative;}.css-4okk7a .katex .op-symbol{position:relative;}.css-4okk7a .katex .op-symbol.small-op{font-family:KaTeX_Size1;}.css-4okk7a .katex .op-symbol.large-op{font-family:KaTeX_Size2;}.css-4okk7a .katex .op-limits>.vlist-t{text-align:center;}.css-4okk7a .katex .accent>.vlist-t{text-align:center;}.css-4okk7a .katex .accent .accent-body{position:relative;}.css-4okk7a .katex .accent .accent-body:not(.accent-full){width:0;}.css-4okk7a .katex .overlay{display:block;}.css-4okk7a .katex .mtable .vertical-separator{display:inline-block;min-width:1px;}.css-4okk7a .katex .mtable .arraycolsep{display:inline-block;}.css-4okk7a .katex .mtable .col-align-c>.vlist-t{text-align:center;}.css-4okk7a .katex .mtable .col-align-l>.vlist-t{text-align:left;}.css-4okk7a .katex .mtable .col-align-r>.vlist-t{text-align:right;}.css-4okk7a .katex .svg-align{text-align:left;}.css-4okk7a .katex svg{display:block;position:absolute;width:100%;height:inherit;fill:currentColor;stroke:currentColor;fill-rule:nonzero;fill-opacity:1;stroke-width:1;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;}.css-4okk7a .katex svg path{stroke:none;}.css-4okk7a .katex img{border-style:none;min-width:0;min-height:0;max-width:none;max-height:none;}.css-4okk7a .katex .stretchy{width:100%;display:block;position:relative;overflow:hidden;}.css-4okk7a .katex .stretchy::before,.css-4okk7a .katex .stretchy::after{content:'';}.css-4okk7a .katex .hide-tail{width:100%;position:relative;overflow:hidden;}.css-4okk7a .katex .halfarrow-left{position:absolute;left:0;width:50.2%;overflow:hidden;}.css-4okk7a .katex .halfarrow-right{position:absolute;right:0;width:50.2%;overflow:hidden;}.css-4okk7a .katex .brace-left{position:absolute;left:0;width:25.1%;overflow:hidden;}.css-4okk7a .katex .brace-center{position:absolute;left:25%;width:50%;overflow:hidden;}.css-4okk7a .katex .brace-right{position:absolute;right:0;width:25.1%;overflow:hidden;}.css-4okk7a .katex .x-arrow-pad{padding:0 0.5em;}.css-4okk7a .katex .cd-arrow-pad{padding:0 0.55556em 0 0.27778em;}.css-4okk7a .katex .x-arrow,.css-4okk7a .katex .mover,.css-4okk7a .katex .munder{text-align:center;}.css-4okk7a .katex .boxpad{padding:0 0.3em 0 0.3em;}.css-4okk7a .katex .fbox,.css-4okk7a .katex .fcolorbox{box-sizing:border-box;border:0.04em solid;}.css-4okk7a .katex .cancel-pad{padding:0 0.2em 0 0.2em;}.css-4okk7a .katex .cancel-lap{margin-left:-0.2em;margin-right:-0.2em;}.css-4okk7a .katex .sout{border-bottom-style:solid;border-bottom-width:0.08em;}.css-4okk7a .katex .angl{box-sizing:border-box;border-top:0.049em solid;border-right:0.049em solid;margin-right:0.03889em;}.css-4okk7a .katex .anglpad{padding:0 0.03889em 0 0.03889em;}.css-4okk7a .katex .eqn-num::before{counter-increment:katexEqnNo;content:'(' counter(katexEqnNo) ')';}.css-4okk7a .katex .mml-eqn-num::before{counter-increment:mmlEqnNo;content:'(' counter(mmlEqnNo) ')';}.css-4okk7a .katex .mtr-glue{width:50%;}.css-4okk7a .katex .cd-vert-arrow{display:inline-block;position:relative;}.css-4okk7a .katex .cd-label-left{display:inline-block;position:absolute;right:calc(50% + 0.3em);text-align:left;}.css-4okk7a .katex .cd-label-right{display:inline-block;position:absolute;left:calc(50% + 0.3em);text-align:right;}.css-4okk7a .katex-display{display:block;margin:1em 0;text-align:center;}.css-4okk7a .katex-display>.katex{display:block;white-space:nowrap;}.css-4okk7a .katex-display>.katex>.katex-html{display:block;position:relative;}.css-4okk7a .katex-display>.katex>.katex-html>.tag{position:absolute;right:0;}.css-4okk7a .katex-display.leqno>.katex>.katex-html>.tag{left:0;right:auto;}.css-4okk7a .katex-display.fleqn>.katex{text-align:left;padding-left:2em;}.css-4okk7a body{counter-reset:katexEqnNo mmlEqnNo;}.css-4okk7a table{width:-webkit-max-content;width:-moz-max-content;width:max-content;}.css-4okk7a .tableBlock{max-width:100%;margin-bottom:1rem;overflow-y:scroll;}.css-4okk7a .tableBlock thead,.css-4okk7a .tableBlock thead th{border-bottom:1px solid #333!important;}.css-4okk7a .tableBlock th,.css-4okk7a .tableBlock td{padding:10px;text-align:left;}.css-4okk7a .tableBlock th{font-weight:bold!important;}.css-4okk7a .tableBlock caption{caption-side:bottom;color:#555;font-size:12px;font-style:italic;text-align:center;}.css-4okk7a .tableBlock caption>p{margin:0;}.css-4okk7a .tableBlock th>p,.css-4okk7a .tableBlock td>p{margin:0;}.css-4okk7a .tableBlock [data-background-color='aliceblue']{background-color:#f0f8ff;color:#000;}.css-4okk7a .tableBlock [data-background-color='black']{background-color:#000;color:#fff;}.css-4okk7a .tableBlock [data-background-color='chocolate']{background-color:#d2691e;color:#fff;}.css-4okk7a .tableBlock [data-background-color='cornflowerblue']{background-color:#6495ed;color:#fff;}.css-4okk7a .tableBlock [data-background-color='crimson']{background-color:#dc143c;color:#fff;}.css-4okk7a .tableBlock [data-background-color='darkblue']{background-color:#00008b;color:#fff;}.css-4okk7a .tableBlock [data-background-color='darkseagreen']{background-color:#8fbc8f;color:#000;}.css-4okk7a .tableBlock [data-background-color='deepskyblue']{background-color:#00bfff;color:#000;}.css-4okk7a .tableBlock [data-background-color='gainsboro']{background-color:#dcdcdc;color:#000;}.css-4okk7a .tableBlock [data-background-color='grey']{background-color:#808080;color:#fff;}.css-4okk7a .tableBlock [data-background-color='lemonchiffon']{background-color:#fffacd;color:#000;}.css-4okk7a .tableBlock [data-background-color='lightpink']{background-color:#ffb6c1;color:#000;}.css-4okk7a .tableBlock [data-background-color='lightsalmon']{background-color:#ffa07a;color:#000;}.css-4okk7a .tableBlock [data-background-color='lightskyblue']{background-color:#87cefa;color:#000;}.css-4okk7a .tableBlock [data-background-color='mediumblue']{background-color:#0000cd;color:#fff;}.css-4okk7a .tableBlock [data-background-color='omnigrey']{background-color:#f0f0f0;color:#000;}.css-4okk7a .tableBlock [data-background-color='white']{background-color:#fff;color:#000;}.css-4okk7a .tableBlock [data-text-align='center']{text-align:center;}.css-4okk7a .tableBlock [data-text-align='left']{text-align:left;}.css-4okk7a .tableBlock [data-text-align='right']{text-align:right;}.css-4okk7a .tableBlock [data-vertical-align='bottom']{vertical-align:bottom;}.css-4okk7a .tableBlock [data-vertical-align='middle']{vertical-align:middle;}.css-4okk7a .tableBlock [data-vertical-align='top']{vertical-align:top;}.css-4okk7a .tableBlock__font-size--xxsmall{font-size:10px;}.css-4okk7a .tableBlock__font-size--xsmall{font-size:12px;}.css-4okk7a .tableBlock__font-size--small{font-size:14px;}.css-4okk7a .tableBlock__font-size--large{font-size:18px;}.css-4okk7a .tableBlock__border--some tbody tr:not(:last-child){border-bottom:1px solid #e2e5e7;}.css-4okk7a .tableBlock__border--bordered td,.css-4okk7a .tableBlock__border--bordered th{border:1px solid #e2e5e7;}.css-4okk7a .tableBlock__border--borderless tbody+tbody,.css-4okk7a .tableBlock__border--borderless td,.css-4okk7a .tableBlock__border--borderless th,.css-4okk7a .tableBlock__border--borderless tr,.css-4okk7a .tableBlock__border--borderless thead,.css-4okk7a .tableBlock__border--borderless thead th{border:0!important;}.css-4okk7a .tableBlock:not(.tableBlock__table-striped) tbody tr{background-color:unset!important;}.css-4okk7a .tableBlock__table-striped tbody tr:nth-of-type(odd){background-color:#f9fafc!important;}.css-4okk7a .tableBlock__table-compactl th,.css-4okk7a .tableBlock__table-compact td{padding:3px!important;}.css-4okk7a .tableBlock__full-size{width:100%;}.css-4okk7a .textBlock{margin-bottom:16px;}.css-4okk7a .textBlock__text-formatting--finePrint{font-size:12px;}.css-4okk7a .textBlock__text-infoBox{padding:0.75rem 1.25rem;margin-bottom:1rem;border:1px solid transparent;border-radius:0.25rem;}.css-4okk7a .textBlock__text-infoBox p{margin:0;}.css-4okk7a .textBlock__text-infoBox--primary{background-color:#cce5ff;border-color:#b8daff;color:#004085;}.css-4okk7a .textBlock__text-infoBox--secondary{background-color:#e2e3e5;border-color:#d6d8db;color:#383d41;}.css-4okk7a .textBlock__text-infoBox--success{background-color:#d4edda;border-color:#c3e6cb;color:#155724;}.css-4okk7a .textBlock__text-infoBox--danger{background-color:#f8d7da;border-color:#f5c6cb;color:#721c24;}.css-4okk7a .textBlock__text-infoBox--warning{background-color:#fff3cd;border-color:#ffeeba;color:#856404;}.css-4okk7a .textBlock__text-infoBox--info{background-color:#d1ecf1;border-color:#bee5eb;color:#0c5460;}.css-4okk7a .textBlock__text-infoBox--dark{background-color:#d6d8d9;border-color:#c6c8ca;color:#1b1e21;}.css-4okk7a .text-overline{-webkit-text-decoration:overline;text-decoration:overline;}.css-4okk7a.css-4okk7a{color:#2B3148;background-color:transparent;font-family:"Roboto","Helvetica","Arial",sans-serif;font-size:20px;line-height:24px;overflow:visible;padding-top:0px;position:relative;}.css-4okk7a.css-4okk7a:after{content:'';-webkit-transform:scale(0);-moz-transform:scale(0);-ms-transform:scale(0);transform:scale(0);position:absolute;border:2px solid #EA9430;border-radius:2px;inset:-8px;z-index:1;}.css-4okk7a .js-external-link-button.link-like,.css-4okk7a .js-external-link-anchor{color:inherit;border-radius:1px;-webkit-text-decoration:underline;text-decoration:underline;}.css-4okk7a .js-external-link-button.link-like:hover,.css-4okk7a .js-external-link-anchor:hover,.css-4okk7a .js-external-link-button.link-like:active,.css-4okk7a .js-external-link-anchor:active{text-decoration-thickness:2px;text-shadow:1px 0 0;}.css-4okk7a .js-external-link-button.link-like:focus-visible,.css-4okk7a .js-external-link-anchor:focus-visible{outline:transparent 2px dotted;box-shadow:0 0 0 2px #6314E6;}.css-4okk7a p,.css-4okk7a div{margin:0px;display:block;}.css-4okk7a pre{margin:0px;display:block;}.css-4okk7a pre code{display:block;width:-webkit-fit-content;width:-moz-fit-content;width:fit-content;}.css-4okk7a pre:not(:first-child){padding-top:8px;}.css-4okk7a ul,.css-4okk7a ol{display:block margin:0px;padding-left:20px;}.css-4okk7a ul li,.css-4okk7a ol li{padding-top:8px;}.css-4okk7a ul ul,.css-4okk7a ol ul,.css-4okk7a ul ol,.css-4okk7a ol ol{padding-top:0px;}.css-4okk7a ul:not(:first-child),.css-4okk7a ol:not(:first-child){padding-top:4px;} Warp Factor

Input the warp factor and open sections of the calculator to compute the speed of your starship using different equations.

Warp factor

TOS: The Original Series

TOS is the first equation to compute the speed of a starship scaled by the warp factor.

TNG-1: The Next Generation version 1

TNG-1 combines the new speed scaled by the warp factor with the first proposal to map the Eugene Limit.

TNG-2: The Next Generation version 2

TNG-2 combines the new speed scaled by the warp factor with the second proposal to map the Eugene Limit.

Warp speeds in 'Star Trek' are achingly slow, and a simple animation by a former NASA scientist proves it

• In the sci-fi series "Star Trek," spaceships rapidly travel from one part of the universe to another using warp drives.
• The speed of light in a vacuum is about 186,282 miles per second (299,792 kilometers per second). In "Star Trek," a warp factor of 1 is light speed, and a warp factor of 9.9 is more than 2,000 times greater than light speed.
• But James O'Donoghue, a former NASA scientist who's now at JAXA, animated warp-speed velocities in the solar system using the Enterprise spaceship. He posted his video on Twitter on Monday.
• O'Donoghue's animation shows it'd take nearly 10 seconds to reach Pluto , and 18 hours to reach the nearest star, at close to maximum warp speed.
• The planetary scientist said he feels "despair" about the distances and speeds involved in space travel and, in part, made the animation "to make everyone else feel as bad as me."
• Visit Business Insider's homepage for more stories .

In the sci-fi universe of "Star Trek," spaceships with warp drives can zoom past the normally impenetrable limit of light speed , or about 186,282 miles per second (299,792 kilometers per second) in a vacuum.

This trouncing of theoretical physics makes reaching alien-rich planets across the galaxy seem like just a convenient TV-commercial-break-length trip away.

But a new animation by the planetary and space scientist James O'Donoghue , who used to work at NASA and is now employed by JAXA (Japan's national space agency), grounds the warp drives of those fictional spaceships in reality. He says the work gives him "a sense of despair" about traveling through space, even at superluminal speeds.

O'Donoghue previously animated the speed of light within the solar system, and the results were depressing . After receiving widespread attention for those animations, he began wondering what going faster might look like in reality.

So O'Donoghue took the Federation starship USS Enterprise, commanded by Captain Jean-Luc Picard (played by Patrick Stewart) in "Star Trek: The Next Generation," and sent it flying from the sun to Pluto at varying warp-speed velocities.

The animated video above, which O'Donoghue posted on Twitter on Monday, is almost as deflating as the scientist's first set of popular animations .

"I have genuinely felt a sense of despair at the distances involved in our solar system and beyond," O'Donoghue told Business Insider, adding: "It's been one of my aims to make everyone else feel as bad as me."

Incidentally, a follow-on series titled "Star Trek: Picard" is scheduled to premiere on CBS All Access on January 23 and on Amazon Prime the following day, according to CNN .

What the new 'Star Trek' warp-speed animation shows — and why it's depressing

There's no set-in-stone scale of "warp-factor" speeds in the "Star Trek" universe. Over the more than 50 years of productions, different series and episodes and movies throw out conflicting numbers.

However, Rick Sternbach and Michael Okuda — two technical advisers to "The Next Generation" series — published a technical manual in 1991 that includes some solid figures, and it's those numbers (vis-a-vis a Wikipedia page ) that O'Donoghue said he leaned on for his animation.

Related stories

That scale suggests a warp factor of 1 is light speed (shown below between Earth and the moon) and the typical upper limit warp of 9.99 is more than 2,140 times light speed.

O'Donoghue chose to depict the Enterprise flying away from the sun and across the solar system toward a finish line at Pluto . The spaceship starts out at warp 1 and eventually accelerates to warp 9.9, or about 2,083 times light speed.

• Warp 1 , or light speed, makes the Enterprise look like it's at a standstill over the sun. At this light-speed rate, the ship would take 5 hours and 28 minutes just to reach Pluto, which is about 3.67 billion miles (5.9 billion kilometers) away from the sun. Meanwhile, Proxima Centauri — the nearest star to our own — is a dismal four years and three months away.
• Warp 5 is about 213 times faster, making a sun-Pluto journey just 1 minute and 30 seconds long. Proxima Centauri is still a weeklong voyage.
• Warp 9.9 makes Pluto less that a 10-second trip away, and Proxima Centauri an 18-hour cruise.

This last rate of travel is thousands of times faster than the physics of our universe may ever permit.

However, traveling at a warp factor of 9.9 from one end of the Milky Way galaxy — a body of hundreds of billions of stars that may stretch 150,000 to 200,000 light-years wide, according to a recent study — to the other could take 96 years. That's almost a decade longer than an average human life span today.

Even considering the fastest "transwarp" (or "beyond warp") speed achieved by the Enterprise, which is about 8,323 times light speed, according to " Star Trek: The Next Generation — Technical Manual ," a transgalactic voyage would take 24 years. A transwarp voyage to Andromeda, which is the nearest galaxy to ours at about 2.5 million light-years away , would last about 300 years.

A quarter of a century is a grueling amount of time that no holodeck , artificially intelligent companions, and extremely well-stocked spaceship bar may stack up against, let alone three centuries.

"It is difficult to feel sorry for them, however," O'Donoghue said of the people in "Star Trek," since "we ourselves haven't achieved even a 10th of 1% of the speed of light in our fastest-ever spacecraft."

Humanity moves really, really slowly through space 

The fastest any human-built object has ever gone relative to the sun is about 119 miles per second (192 kilometers per second), or 430,000 mph (692,000 kilometers per hour). NASA's Parker Solar Probe briefly achieves this speed when it careens around the sun, and flying to Pluto from the sun at that rate would take nearly a year.

Engineers with the Breakthrough Starshot project are working toward achieving partial light-speed travel with tiny "nanocraft." The idea is to rapidly accelerate them by shooting reflective light sails with powerful laser beams, ultimately flying them past nearby stars like Proxima Centauri (a red dwarf that just might host habitable planets).

Read more : A startup is developing a 100-gigawatt laser to propel a probe to another star system. That may be powerful enough to 'ignite an entire city.'

Yet even at a planned cruise velocity of 20% of light speed, it could take more than 21 years for the probes to fly past and photograph the nearby star system. On top of that, it'd take another 4.24 years for their radio signals (traveling at light-speed) carrying image data to reach antennas on Earth.

If we really want to feel lonely, though, O'Donoghue thinks we need look no farther than science fiction itself.

"I think it adds a layer of isolation and grittiness to know that even our nearest star takes dozens of hours at a rarely used warp 9.9 to get to," O'Donoghue said. "Perhaps this gives viewers a better sense of the magnitude of space and the frontier nature of the Federation ships and crew."

Correction : James O'Donoghue recently left NASA to join JAXA, which is Japan's national space agency. We also misstated how long it'd take radio signals from Proxima Centauri to reach Earth; it's about 4.24 years, not 21 years.

Watch: Albert Einstein has a famous loophole in the special theory of relativity that could predict faster-than-light travel

• Main content

Maximum warp

• View history

Maximum warp (shortened from maximum warp speed and also known as top warp speed , maximum speed , high warp speed , emergency speed ) was the maximum warp factor velocity a starship was designed for and a term whereby a bridge officer could ask the flight control officer to accelerate the ship to its highest possible velocity.

The maximum warp velocity and the time it could be maintained by a starship varied between classes. This was due to energy output limitations of the warp core and limitations of other systems, such as the warp coils , the structural integrity field , and the amount of energy the propulsion system could handle without overloading . ( ENT : " In a Mirror, Darkly "; DS9 : " The Sound of Her Voice "; TOS : " The Changeling ")

In the early days of warp drive development, the top speed capabilities took massive leaps in the mid- 22nd century after the warp 2 barrier was broken. ( ENT : " First Flight ") The invention of the warp three engine allowed early Earth starships to travel ten times faster, ( ENT : " Fortunate Son ") while ships installed with the warp five engine could travel a hundred times faster. ( ENT : " Broken Bow ") The increase in capabilities slowed down later on, as the Starfleet ships of the 2370s were only twice as fast as the ships used in the 2290s . ( VOY : " Flashback ")

Though originally rated for warp 5, by 2154 the installation of new warp injectors allowed the Enterprise NX-01 to push to warp 5.06. ( ENT : " Babel One ")

In 2256 of an alternate timeline , Captain Michael Burnham from the future had the USS Discovery taken to maximum warp and then had the warp bubble broken. This was a part of Commander Paul Stamets successful attempt to use relativity to break through a temporal shield and remove a time bug . Such a move had the potential to tear the ship apart, but Stamets was able to calibrate the inertial dampeners to compensate, resulting in a jolt, but no other damage to the ship and crew. ( DIS : " Face the Strange ")

In 2268 on stardate 4843.6, it was imperative for the USS Enterprise to proceed at maximum warp speed for a period which exceeded the recommended safety margin in order to intercept an asteroid headed on a collision course with the planet Amerind . Chief Engineer Montgomery Scott noted that they couldn't maintain warp 9 for much longer as the engines began to show signs of stress . ( TOS : " The Paradise Syndrome ")

Galaxy -class ships, such as the USS Enterprise -D , could theoretically achieve warp 9.8, but the maximum warp speeds above 9.6 could be maintained for a few hours only. ( TNG : " Encounter at Farpoint ", " The Best of Both Worlds ") Intrepid -class ships such the USS Voyager , on the other hand, could maintain it from three days up to two months. ( VOY : " Workforce ", " Workforce, Part II ", " Friendship One ")

In 2370 , the Federation imposed a maximum speed limit of Warp 5 on all vessels (except for emergencies) due to concerns over high warp factors causing damage to subspace. ( TNG : " Force of Nature ", " The Pegasus ", " Eye of the Beholder ") This limit would appear to have been temporary.

The Klaestron starship that visited Deep Space 9 in 2369 was said to have a high warp capability, and was able to achieve warp 5 or better. ( DS9 : " Dax ")

The USS Carolina traveled 0.7 light years in two hours when at maximum warp. ( VOY : " Inside Man ")

Even though the Nova -class was designed to travel at a maximum speed of warp factor 8, when the USS Equinox powered its warp drive with energy derived from the corpses of nucleogenic lifeforms , an increase of 0.03% was added to the warp factor, allowing the velocity of traveling ten thousand light years in less than two weeks. ( VOY : " Equinox ")

During its time in the Delta Quadrant , the USS Voyager encountered technologies that enabled it to travel great distances at massively greater speeds than its rated maximum warp velocity, including the quantum slipstream drive , transwarp coil and Tash's catapult . The vessel had arrived in the Delta Quadrant from the Alpha Quadrant , approximately 75,000 light years, via the Caretaker's array . ( VOY : " Hope and Fear ", " Dark Frontier ", " The Voyager Conspiracy ", " Caretaker ")

• 1 Maximum warp limitations by class
• 2.1 Maximum warp requests
• 2.2 Background information
• 4 External link

Maximum warp limitations by class [ ]

• Warp 1.4 † – Arctic One -type
• Warp 1.8 † – Y-class
• Warp 2 † – Neptune -class , J-class
• Warp 3.2 – Defiant -class , USS Valiant had suffered combat damage
• Warp 4 – Whorfin -class † , type 9 shuttlecraft
• Warp 5 – Danube -class
• Warp 5.2 † – NX-class
• Warp 6 † – D5-class , Class III neutronic fuel carrier
• Warp 6 – D'deridex -class , maximum speed that can be used where the use of the Cloaking device is not detected
• Warp 6.5 † – Suurok -class ( Ti'Mur )
• Warp 7 † – Suurok -class ( Sh'Raan ) , Coridan ship
• Warp 8 † – Constitution -class
• Warp 8 – Nova -class
• Warp 8.7 – Baran's mercenary ship
• Warp 9 – Defiant -class ††
• Warp 9.3 – Nebula -class (2368)
• Warp 9.5 – Nebula -class (2370, could be pushed to warp 9.6)
• Warp 9.6 – D'deridex -class , can be exceeded in exchange for engine damage
• Warp 9.8 – Galaxy -class , Xepolite Free Trader
• Warp 9.9 – Prometheus -class
• Warp 9.97 – Protostar -class (with conventional warp drive)
• Warp 9.975 – Intrepid -class
• Warp 9.99 – Constitution III -class
• Warp 36 † – Karla Five's vessel

Appendices [ ]

Maximum warp requests [ ].

• " Where No Man Has Gone Before " (first mentioned)
• " The Corbomite Maneuver "
• " Balance of Terror "
• " The Squire of Gothos "
• " Spock's Brain "
• " That Which Survives "
• " Unnatural Selection "
• " Time Squared "
• " Redemption "
• " The Chase "
• " Descent, Part II "
• " Parallels "
• " Homeward "
• " Lower Decks "
• " Bloodlines "
• " All Good Things... "
• " Whispers "
• " Defiant "
• " The Die is Cast "
• " The Adversary "
• " The Way of the Warrior "
• " Return to Grace "
• " For the Cause "
• " For the Uniform "
• " By Inferno's Light "
• " Soldiers of the Empire "
• " A Time to Stand "
• " Sacrifice of Angels "
• " The Sound of Her Voice "
• " Parallax "
• " Cold Fire "
• " Threshold "
• " Flashback "
• " Macrocosm "
• " Scorpion "
• " Scorpion, Part II "
• " The Omega Directive "
• " Counterpoint "
• " Dark Frontier "
• " Think Tank "
• " Equinox "
• " Equinox, Part II "
• " Tinker Tenor Doctor Spy "
• " Flesh and Blood "
• " Body and Soul "
• " The Aenar "
• " In a Mirror, Darkly "
• Star Trek Generations
• Star Trek: First Contact

Background information [ ]

As noted above, the tolerance of a ship's systems for the stresses of warp travel are a primary limiting factor, so while exceeding the ship's maximum rated speed can be technically possible, it carries the risk of severe damage to the warp drive or to the ship's structural integrity.

When a ship is ordered to accelerate to maximum warp, it does not always mean the absolute highest reachable warp factor. For example, in ENT : " Affliction ", Enterprise NX-01 already travels at maximum warp, but then accelerates to warp 5.2. In both TNG : " Time Squared " and TNG : " Bloodlines ", maximum warp denoted warp 9, although the Enterprise -D was established to be capable of faster speeds in other episodes. A similar situation occurred in DS9 : " The Sound of Her Voice ", when the USS Defiant , traveling at maximum warp, accelerates to warp 9.5.

According to Star Trek Encyclopedia  (3rd ed., p. 556) updated with information up to 2375 and Star Trek: Insurrection , warp factor 9.2 was the normal maximum warp speed of Federation starships.

TNG : " Force of Nature " established Warp 5 as the maximum allowable Federation-vessel speed (except in emergencies), due to environmental concerns. Following this episode, the use of the term "maximum warp" became ubiquitous in lieu of a specific warp factor being stated. Even though later episodes and series strongly indicated that the limit was no longer in place (in particular throughout Star Trek: Voyager ), the use of the phrase "maximum warp" has continued, even in series set well before the events of "Force of Nature".

See also [ ]

• Cruising speed
• Emergency warp

External link [ ]

• Warp factor at Memory Beta , the wiki for licensed Star Trek works
• 2 ISS Enterprise (NCC-1701)

• Skip to main content
• Keyboard shortcuts for audio player

Pop Culture Happy Hour

• LISTEN & FOLLOW
• Apple Podcasts
• Google Podcasts
• Amazon Music

Your support helps make our show possible and unlocks access to our sponsor-free feed.

Does 'Star Trek: Strange New Worlds' attain warp speed?

Glen Weldon

Chris Klimek

Laura Sirikul

Mike Katzif

Jessica Reedy

Anson Mount and Ethan Peck star on the Paramount+ series Star Trek: Strange New Worlds . Marni Grossman/Paramount+ hide caption

Anson Mount and Ethan Peck star on the Paramount+ series Star Trek: Strange New Worlds .

In Star Trek: Strange New Worlds , USS Enterprise Captain Christopher Pike (Anson Mount), Spock (Ethan Peck) and their diverse crew boldly go exploring the galaxy. The Paramount+ series takes place a few years before the original series from the '60s, and new show's tone and narrative certainly feel like a return to old-school Star Trek .

• Insider Reviews
• Tech Buying Guides
• Personal Finance
• Insider Explainers
• Sustainability
• United States
• International
• Deutschland & Österreich
• South Africa

• Home ›
• Science ›
• Space ›
• news »

Warp speeds in 'Star Trek' are achingly slow, and a simple animation of the starship Enterprise by a NASA scientist proves it

Paramount Pictures

Patrick Stewart plays Jean-Luc Picard in the sci-fi space series "Star Trek."

• In the sci-fi series "Star Trek," spaceships rapidly travel from one part of the universe to another using warp drive.
• The speed of light in a vacuum is about 186,282 miles per second (299,792 kilometers per second). In "Star Trek," a warp factor of 1 is light-speed, and a warp factor of 9.9 is more than 2,000 times greater.
• But James O'Donoghue, a scientist at NASA, animated warp-speed velocities in the solar system using the "Enterprise" spaceship and posted the video to Twitter on Monday.
• O'Donoghue's animation shows it'd take nearly 10 seconds to reach Pluto near maximum warp and 18 hours just to reach the nearest star.
• The NASA scientist says he feels "despair" about the distances and speeds involved in space travel , and in part made the animation "to make everyone else feel as bad as me."
• Visit Business Insider's homepage for more stories .

In the sci-fi universe of "Star Trek," spaceships equipped with warp drives can zoom past the normally impenetrable limit of light-speed , or about 186,282 miles per second (299,792 kilometers per second) in a vacuum.

This trouncing of physics makes reaching alien-rich planets across the galaxy seem like just a convenient TV-commercial-break-length trip away.

But a new animation by NASA scientist James O'Donoghue grounds the warp drives of those fictional spaceships in reality, and he says the work gives him "a sense of despair" about even super-luminal (or beyond-light-speed) travel.

O'Donoghue previously animated the speed of light within the solar system, and the results were depressing . After receiving widespread attention for those animations, though, he began wondering what going faster might look like in real life.

So O'Donoghue took the Federation starship "Enterprise," commanded by Captain Jean-Luc Picard (played by Patrick Stewart) in "Star Trek: The Next Generation," and sent it flying from the sun to Pluto at varying warp-speed velocities.

The animated video above, which O'Donoghue posted to Twitter on Monday, is almost as deflating as the scientist's first set of popular animations.

"I have genuinely felt a sense of despair at the distances involved in our solar system and beyond," O'Donoghue told Business Insider, adding: "[I]t's been one of my aims to make everyone else feel as bad as me!"

Incidentally, a new version of Star Trek, titled "Star Trek: Picard," returns to CBS in January 2020.

What the 'Star Trek' warp-speed animation shows - and why it's sort of depressing

The "Enterprise" ship from "Star Trek: The Next Generation" moving through space.

There's no set-in-stone scale of "warp factor" speeds in the "Star Trek" universe. Over more than 50 years' worth of productions, different series and episodes and movies throw out conflicting numbers.

However, Rick Sternbach and Michael Okuda - two technical advisors to "The Next Generation" series - first published a technical manual in 1991 that includes some solid figures, and it's those numbers (vis-a-vis a Wikipedia page ) that O'Donoghue says he leaned on for his animation.

That scale suggests a warp factor of 1 is light-speed (shown below between Earth and the moon) and the typical upper limit warp of 9.99 is more than 2,140 times light-speed.

O'Donoghue chose to depict the "Enterprise" flying away from the sun and across the solar system toward a finish line at Pluto . The spaceship starts out at warp 1 and eventually accelerates to warp 9.9, or about 2,083 times light-speed.

• Warp 1 , or light-speed, makes the "Enterprise" look like it's at a standstill over the sun. At this light-speed rate, the ship would take 5 hours and 28 minutes just to reach Pluto, which is about 3.67 billion miles (5.9 billion kilometers) away from the sun. Meanwhile, Proxima Centauri - the nearest star to our own - is a dismal 4 years and 3 months away.
• Warp 5 is about 213 times faster, making a sun-Pluto journey just 1 minute and 30 seconds long. Proxima Centauri is still a weeklong voyage.
• Warp 9.9 makes Pluto less that a 10-second trip away, and Proxima Centauri an 18-hour cruise.

This last number is thousands of times faster than the physics of our universe may ever permit.

However, traveling at a warp factor of 9.9 from one end of the Milky Way galaxy - a body of hundred of billions of stars that may stretch 150,000 to 200,000 light-years wide, according to a recent study - to the other could take 96 years. That's a decade longer than a generous human lifespan.

Even considering the fastest "transwarp" (or "beyond warp") speed achieved by the "Enterprise," which is about 8,323 times light-speed, according to " 'Star Trek': The Next Generation - Technical Manual ," a transgalactic voyage would take 24 years.

A quarter of a century is a grueling amount of time that no Holodeck , artificially intelligent companions, or extremely well-stocked spaceship bar may stack up against.

"It is difficult to feel sorry for them, however," O'Donoghue said, "as we ourselves haven't achieved even a tenth of 1% of the speed of light in our fastest-ever spacecraft."

The fastest any human-built object has ever gone relative to the sun, meanwhile - NASA's Parker Solar Probe - is about 119 miles per second (192 kilometers per second). If flying toward Pluto from the sun at that speed, it'd take nearly a year for the probe to arrive.

Breakthrough Foundation

An illustration of a Breakthrough Starshot "nanocraft" being propelled toward the Alpha Centauri star system with a powerful laser beam.

Read more : A startup is developing a 100-gigawatt laser to propel a probe to another star system. That may be powerful enough to 'ignite an entire city.'

Even their planned cruise velocity of 20% of the speed of light, though, it could take more than 21 years for the probes to fly past the star system, then another 21 years for those signals to reach Earth.

If we really want to feel lonely, though, O'Donoghue thinks we need look no farther than science-fiction itself.

"I think it adds a layer of isolation and grittiness to know that even our nearest star takes dozens of hours at a rarely-used warp 9.9 to get to," O'Donoghue said. "Perhaps this gives viewers a better sense of the magnitude of space and the frontier-nature of the Federation ships and crew."

NOW WATCH: Albert Einstein has a famous loophole in the special theory of relativity that could predict faster-than-light travel

• Single Use Plastics
• Southwest Monsoon
• Avoid food during Diarrhea
• Home Remedies for Wasp Sting
• Overdose side effects
• ABHA Health Card
• Lemon Water
• Solar Eclipse vs Lunar Eclipse
• 10000 Steps
• Best printers for Home
• Best Mixer Grinder
• Best wired Earphones
• Best 43 Inch TV in India
• Best Wi Fi Routers
• Best Vacuum Cleaner
• Best Home Theatre in India
• Smart Watch under 5000
• Best Laptops for Education
• Best Laptop for Students

• Advertising
• Write for Us
• Privacy Policy
• Policy News
• Personal Finance News
• Mobile News
• Business News
• Ecommerce News
• Startups News
• Stock Market News
• Finance News
• Entertainment News
• Economy News
• Careers News
• International News
• Politics News
• Education News
• Advertising News
• Health News
• Science News
• Retail News
• Sports News
• Personalities News
• Corporates News
• Environment News
• Nothing Phone (2a) India-exclusive edition launched
• JNK India IPO allotment
• JioCinema New Subscription Plans
• Realme 70X 5G Launched
• Apple Let Loose Launch event
• Top 10 Richest people
• Cibil Score vs Cibil Report
• Top 10 Largest Economies
• Lucky Color for 2023
• How to check pan and Aadhaar
• Deleted Whatsapp Messages
• How to restore deleted messages
• 10 types of Drinks
• Instagram Sad Face Filter
• Unlimited Wifi Plans
• Recover Whatsapp Messages
• Google Meet
• Check Balance in SBI
• How to check Vodafone Balance
• Transfer Whatsapp Message

Copyright © 2024 . Times Internet Limited. All rights reserved.For reprint rights. Times Syndication Service.

Normal Space

Normal Space , or N-Space , is the universe in which we exist. It follows the fundamental rules of physics and must adhere to the rules of conservation of energy. It is the universe in which most known life forms exist and is the universe in which the majority of beings inhabit throughout their lives.

There exist at least three other major layers of Space, and an almost infinite number of sub-levels including fluidic space. Each major layer of Space possess different quantum resonance signatures. It is there fore possible to detect the origin of an object by scanning it's quantum resonance frequency. The chart at the left shows a very basic concept of various quantum resonance states as defined by their place of origin.

Travel between or through the D-Branes that separate the various layers of space is possible. One means of doing so is through the use of an Interphasic Rift , a Wormholes , using a Phase Variant Modulator , or through Subspace Rifts .

The principles of the Warp Drive operates by breaking down the D-Brane between N-Space and S-Space. The result of this is a relative reduction in mass, so much so, that the ship is capable of being propelled beyond the the speed of light, making interstellar travel possible.

The other major known layers of space are:

Subspace : subspace, or S-Space , is a layer of space immediately beneath Normal Space . It is characterized by the bending of several fundamental rules of conventional physics, including the absolute limits of velocity. It is Subspace that allows the utilization of Warp Drive .

Subspace is an integral part of the space-time continuum, distinct, yet coexistent with normal space. Subspace and normal space are confluenced together. However, there are in some regions an interfold layer forms between the two realms. (VOY: "Real Life") Subspace has an infinite number of domains. Geordi La Forge compared it with "...a huge honeycomb with an infinite number of cells". (TNG: "Schisms")

Tetryon particles are a form of subatomic particle which can only exist naturally in subspace. If they are released into normal space, they exhibit erratic properties. This can only occur when the fabric of subspace comes into contact with normal space in some manner.

Fractal Space : Fractal space, or F-Space , was considered to be nothing more than a mathematical theory until 2408. The conceptualization of Fractal Space, however dates back to 1890, and was first put forward by Guiseppe Peano. The exact properties of F-Space are unmeasured, although reinforced wave motion has been observed in the Tch'Kairn fractal spacial communications signal.

Phase Space : Phase space, or P-Space , is comprised of an infinite number of dimensional membranes which are only separated from the other layers of space due to quantum resonance. it is possible to shift one's quantum resonance into another layer of P-Space, through the use of a Phase variant modulator.

Navigation menu

Personal tools.

• Not logged in
• Contributions
• View history
• Recent changes
• Random Image
• Random Article

Popular Wiki Pages

Retired or destroyed.

• USS Templar
• USS Paladin
• USS Rosenante
• USS Boudicca
• USS Cochrane
• USS Firebrande
• USS Dennison
• USS Champlain
• USS Mithrandir
• USS Mystique
• USS Starfire
• USS Spectre

STF History

• What links here
• Related changes
• Special pages
• Printable version
• Permanent link
• Page information
• Cite this page
• This page was last edited on 4 March 2010, at 20:57.
• Privacy policy
• About Star Trek : Freedom's Wiki
• Disclaimers

Star Trek N00b, Episode 17: The Galileo Seven (Logic v. Emotion Cage Match)

• Click to share on Facebook (Opens in new window)
• Click to share on Twitter (Opens in new window)
• Click to share on Tumblr (Opens in new window)
• Click to share on Pinterest (Opens in new window)
• Click to email a link to a friend (Opens in new window)
• Click to print (Opens in new window)

N00bs, get your philosophy caps on! Galileo Seven is an episode that really critiques Spock’s logical methods, which often lack emotion, during a fight for survival. It’s the whole head vs. heart conflict, which you can watch with me here .

Why these two make such an adorable and effective couple…of leaders.

Star Trek, Remastered

So many of you have noted in the comments that the TOS I’ve been reviewing is the remastered version, due to the updated CGI graphics added in 2007. This has never been more apparent during this episode, which relies on some pretty lengthy CGI shots. Just to give you an idea of the old vs the new:

…and new.

Fancy! Old TOS picture credit: Memory Alpha

So what’s this giant green thing they’re flying into? Well, it seems to be a quasar known as Murasaki 312. Kirk really wants to explore this thing, but there’s a Federation head honcho by the name of Commissioner Ferris who wants the Enterprise to deliver some medical supplies directly to a planet, Makus III. To Ferris’ displeasure, Kirk decides to go adventuring first!

Spock , Bones , Scotty and some new characters — Yeoman Mears , Lt. Latimer , Lt. Gaetano , and Lt. Boma — clamber into the shuttlecraft Galileo to go flying through the green cloud.

Any Mass Effect gamers in the house? The Galileo slightly reminded me of the Mako. Driving that shit around was hard. In fact, there are comics and memes dedicated to it!

Time to ragequit! Via Mass Effect Fuck Yeah

Eaaaasy now. Kirk’ll have a fit if you ding the Enterprise!

Things go wrong almost immediately. An ion storm causes the Galileo’s controls to go out of whack and they crash land on a planet smack in the center of the green cloud — Taurus II.

Lost in Space

Everyone on the Enterprise is horrified by the loss of the Galileo, which they cannot communicate with. The only way to find them in the storm is visually. Sensors and transporters won’t work in this electrical storm. According to Kirk, “Finding a needle in a haystack would be child’s play.”

See that meanie standing behind Kirk? Well that’s Ferris, and he doesn’t really give a damn about the crew. All he wants is to fly off and complete their medical equipment mission. He gives Kirk 48 hours to find the crew or abandon them. Ugh!

Conveniently, the Galileo has landed on a planet with breathable air and a survivable atmosphere. Also great news: Scotty, the Chief Engineer is there to make repairs and save the day!

Spock: Okay, now use your MacGyver powers. Here’s a paperclip.

Spock in Command

Bones notes that this is Spock’s opportunity for leadership, since he’s the highest ranking officer present. Characteristically, Spock replies, “I can neither enjoy the idea of command, nor am I frightened of it. It simply exists.”

But some bad news! They can’t fly away unless they drop at least 500 lbs (226.8 kg for everyone else in the world and space). So they might need to leave someone behind…

All that leadership! So much responsibility! Spock’s officially up there with Heidi Klum, RuPaul and Jeff Probst. He might need to vote someone off the island soon.

We’re Not Alone

Spock asks Latimer and Gaetano to explore their environment for any information that might help. This is when we learn that Makus III is inhabited by giant, furry beings (they’re never named in this episode) that throw GIGANTIC spears. Latimer gets killed, and I’m surprised since they were wearing yellow.

Credit: An Introspective World . Just in case you haven’t caught onto this trend with the redshirts.

Spock compares the weapon to an early human tool known as the Folsom Point . Except these spears are like 20 feet long.

Tensions rise as Spock continues to think about the situation logically, while the rest of the crew wants to give Latimer a proper burial. “We may all die here. At least let us die like men, not machines!” exclaims Bones.

Spock decides that they should attempt to scare the furry peeps away by demonstrating the superiority of their phasers. LOGICALLY this should work. EMOTIONALLY, it just pisses the furry peeps off even more.

The Problem with Reason

At this point, two crewmen are dead and everyone is mad at Spock. Boma and Bones insist on a proper burial for the dead. Spock argues that it’s irrational and too dangerous — he’s just trying to keep everyone alive.

Here are a couple of issues we’re seeing under Spock’s leadership:

• He made the deadly mistake of assuming that their enemies would also react rationally and retreat, instead of getting super angry.
• His lack of emotional response is upsetting the team unity. Refusing to mourn the crewmen’s deaths causes group turmoil, which is slowing them down.

Poor Spock! This episode did give me some clarity on why Kirk is such a great leader — Kirk represents a blend of calculated thinking and human emotion.

Spock can’t figure out where his logic went wrong.

Luckily, Scotty’s thinking clearly and SAVES THE DAY! He tells the crew that he can convert their phasers into a power source, which will get them into orbit around the planet. Hopefully, the Enterprise will be able to see them, and rescue the Galileo crew.

The times has come for “less analysis, more action!” according to Bones. The furry peeps are back, and they’re SUPER ANGRY now! They’re trying to break open the Galileo with a good old-fashioned rock smashing.

Spock decides to help the crew with a rush burial to appease them. Of course, the furry peeps attack, and Spock is pinned under a giant rock! He screams, “No, go back! Lift off!” Of course, since everyone else has FEELINGS, they rush and help Spock. Faced with the possible death of my fav character, I was screaming at my computer at this point.

The most intense team building exercise that HR could think of.

Meanwhile, on the bridge, Ferris is counting down the seconds, telling Kirk that he needs to abandon the rescue mission. This guy is such a freaking asshole. He isn’t doing ANYTHING to help out, leaving Uhura , Sulu , and Kirk to desperately seek contact with the Galileo.

Their time has run out. The Enterprise turns to leave.

It seems like Spock and Friends are doomed.

Everyone rushes back into the Galileo (which meets its weight requirement since two people died). Scotty activates their hacked energy source, and they blast off into orbit. But Spock’s worried that the Enterprise might have left already.

He and Bones have a catty exchange.

Spock: Gentlemen, by coming after me, you may well have destroyed what slim chance you had for survival. The logical thing for you to have done was to have left me behind. Bones: Mister Spock, remind me to tell you that I’m sick and tired of your logic.

I’m with Bones on this one.

Kirk’s got a trick up his sleeve — he’s asked Sulu to fly away at “Space Normal” speed to their next destination. It’s a really time-consuming way to travel, especially if you’ve got warp speed. His Space Normal trick made me think of these kittens flying in slo-mo:

Spock Surprises Everyone

The Galileo is orbiting Makus III, but no one can see them. In a very illogical move, Spock presses a button to dump ALL their fuel out.

Everyone’s about to panic, but then Scotty realizes what’s going on: The ignited fuel will give a giant visual signal to the Enterprise – kinda like an enormous flare.

Unfortunately, their survival time has now dropped to SIX MINUTES, since their orbit will degrade and the Galileo will burn up in the planet’s atmosphere. Like this:

Time to get out your prayer beads and booze. This ship is going down!

Luckily, Sulu saw the giant fuel signal and the transporter room has locked onto the crew! They’re beamed aboard just before the Galileo turns into a pile of ash.

Talk about a last-minute!

Let’s all laugh at Spock. Again!

Remember how last episode, how everyone laughed at Spock ? Well, they’re doing it again!

Kirk is thrilled to be reunited with his one true love. He insists that Spock performed an “act of desperation,” which is totally human. When Spock denies it, Kirk laughs, “You’re not going to admit that for the first time in your life, you committed a purely human emotional act?”

Uhura and Mears are gonna write fanfic about this later.

Spock refuses to admit that he’s got an emotional and human side. Cue Enterprise giggles.

Can I just say again, I love episodes with Spock character development! He is the best ever.

Also fun fact : The shuttle prop for this episode fell apart and got all rusty in storage, but it was saved by the amazing Galileo Restoration team! They’re planning to display it in a museum once it’s totally finished. Check out their fantastic progress pics !.

Growing up, I had never watched a Star Trek episode all the way through. My family members weren’t huge sci-fi fans, and they’d flip the channel whenever Star Trek came on. Now I am embarking on an epic nerd rite of passage, chronicling my reactions to every episode of Star Trek: The Original Series (TOS).

Header by  Rory Midhani

Queer hapa writer inspired by gadgets. Attending the Ada Developers Academy in the third cohort. Uninterested in quitting her coffee habit. Reads and writes sci-fi and horror. Find her at lorainekv.com or on Twitter .

Loraine has written 33 articles for us.

You May Also Like...

that walking dead pic got me a few odd looks at the coffee shop. I guess my loling was a disturbance.

Hehe I’ve written enough of these to know I shouldn’t do it in public. And if I do, I must be prepared for awkward stares. XD

Holy shit! I had no idea this existed and I am so happy. o w o Autostraddle and Star Trek are my two favorite things!

Excitement aside, the first time I saw this episode I cried. To me I saw everyone getting really mad at Spock ( god. poor baby. wait are we talking about him or me? ) because while he is half human, his logical Vulcan side put him at odds with the rest of the very human crew who were all reacting in a very human way. All of them under intense pressure to survive. You see it being where you’d think Spock would absolutely THRIVE but you realize that even his logical behavior is a weakness (I guess that’s why we have Spock and Bones to point out that Kirk is that perfect middle ground). : c I probably wouldn’t survive the Galileo crash in my red shirt tbh.

I felt real bad for Spock too! But I also get how everyone would get frustrated with him. If I had to work with Spock, I’d probably be like Bones and Boma- just like have a feeling for 2 seconds, you know it all, sheeeeeeeeesh.

I wouldn’t survive the galileo crash either. Yay red shirts!

Wheeeeee indeed! I liked this episode. Good old fashioned space shuttle adventures! I like that you can see bits of the set chip off whenever some one throws a spear. Yup- definitely seems dangerous!

Also I was not aware that CGI graphics had been added… wow! Now I’m kind of curious to see how all the episodes originally looked.

This is one of my very favorite episodes, which may not mean that much, since that is an extremely difficult thing to decide! This summary was lovely and amazing and made me very happy. I have to say, since this column was started, Autostraddle feels even more like my cyber home-away-from-home. Star Trek is so near and dear to my nerdy heart, and to see it so well represented and appreciated on one of my favorite websites is a true joy. Thank you, Loraine. :)

You don’t even know all of the feels this comment gave me, Samantha. I’m pretty sure that validation of my life and nerdiness is in there somewhere.

Thank you so much for reading!

Yay! I thought maybe you’d left us, Loraine. This is, as you say, a great episode for illustrating what makes a Captain. We talked about this in Conscience of the King, too, that Kirk “feels” his way through these tricky situations, relying heavily on instinct and other things difficult to put into words or quantify. Long past the series and into the movies, we see Spock able to do the same to some limited degree. He learns, slowly, that logic isn’t always the best way. He says in this episode something to the effect of “Everything I’ve done has been perfectly correct.” Clearly there’s a big difference between “correct” and “right.”

Regarding the special effects being “improved.” Most of us classicists hate the new spangly effects. The old ones were so bad they were fabulous–you could see the mattes moving through space, for example. It was part of the show’s charm that they were so clunky.

A wonderful recap, as usual, Loraine. Thanks for the Flying Kiddens! That must the only kidden video I’d never seen before.

Comments are closed.

• Cast & crew
• User reviews
• Episode aired Feb 16, 1967

While on patrol in deep space, Captain Kirk and his crew find and revive a genetically-engineered world conqueror and his compatriots from Earth's Twentieth Century. While on patrol in deep space, Captain Kirk and his crew find and revive a genetically-engineered world conqueror and his compatriots from Earth's Twentieth Century. While on patrol in deep space, Captain Kirk and his crew find and revive a genetically-engineered world conqueror and his compatriots from Earth's Twentieth Century.

• Marc Daniels
• Gene L. Coon
• Carey Wilber
• Gene Roddenberry
• William Shatner
• Leonard Nimoy
• Ricardo Montalban
• 35 User reviews
• 13 Critic reviews

• Captain James Tiberius 'Jim' Kirk

• Mister Spock

• (as Blaisdell Makee)

• Transporter Technician
• Ingenieur Fields
• (uncredited)

• Lieutenant Hadley

• Female Guard
• All cast & crew
• Production, box office & more at IMDbPro

Did you know

• Trivia Gene Roddenberry questioned Carey Wilber 's notion of wasting a high-tech spaceship and expensive resources on criminals - just like Kirk and Spock came up with the same question in the story itself - and came up with the concept of "a bunch of Napoleons" self-exiling in space.
• Goofs Marla, when examining a hibernating Khan prior to his revival, speculates that he is "probably a Sikh". There is nothing about Khan's appearance that would suggest that he was a Sikh. Being clean-shaven and bareheaded, he lacked the prerequisite beard and turban. Sikh is not an ethnicity but a cultural subset of India.

Dr. McCoy : [lying in sickbay, Khan surprises McCoy and holds a knife to his throat, then pauses] Well, either choke me or cut my throat. Make up your mind.

Khan Noonien Singh : English... I thought I dreamed hearing it. Where am I?

Dr. McCoy : You're in...

[Khan tightens his grip. McCoy continues in a lower voice]

Dr. McCoy : You're in bed, holding a knife at your doctor's throat.

Khan Noonien Singh : Answer my question.

Dr. McCoy : It would be most effective if you would cut the carotid artery, just under the left ear.

Khan Noonien Singh : [impressed, releases McCoy] I like a brave man.

Dr. McCoy : [takes the knife from Khan] I was simply trying to avoid an argument.

• Alternate versions Special Enhanced version Digitally Remastered with new exterior shots and remade opening theme song. In the new version, Khan's ship from the 20th century, Botany Bay, is much more aged and weathered than in the original version.
• Connections Featured in Leonard Nimoy: Star Trek Memories (1984)
• Soundtracks Theme From Star Trek Written by and credited to Alexander Courage

User reviews 35

• Apr 18, 2007
• February 16, 1967 (United States)
• United States
• Official Facebook
• Desilu Studios - 9336 W. Washington Blvd., Culver City, California, USA
• Desilu Productions
• Norway Corporation
• See more company credits at IMDbPro

Technical specs

• Runtime 50 minutes

Related news

Contribute to this page.

• See more gaps
• Learn more about contributing

More to explore

Recently viewed