What Math Can Tell Us about Star Trek and Its "Warp Speeds"

 “Warp factor six, engage.”

“Scotty, I need warp power in three minutes or we’re all dead!”

“Set a course for Earth, maximum warp.”

 If you’ve ever watched Star Trek, you’ve heard variants of these lines numerous times. But what do they mean? Obviously, warp power and warp factors are inventions by Gene Roddenberry and the writers in order to place the characters in new locations to initiate stories and move them forward. But these terms actually have meaning, and connections to real physics. Let’s use PTC Mathcad to make sense of warp travel in the Star Trek galaxy.

 Stock image of USS Enterprise

The warp drive of the USS Enterprise uses a matter-antimatter reaction to provide the necessary power to generate a warp field that distorts the space-time continuum, allowing starships to travel faster than the speed of light – several times faster, as a matter of fact.

The Speed of Warp

First, I’ll create a new length unit ly for light years, using the built-in constant c for the speed of light:

Specifying LY (light years) as a new unit in Mathcad

Let’s look at the distances that make warp speed necessary:

Distance to select stars and planets

Mind blowing aside: Wolf 359, the site of the devastating defeat of Starfleet by the Borg in “Best of Both Worlds Part II,” is a real star! And Vulcan is also located in a real (and relatively close) star system, 40 Eridani A!

Starfleet Changes the Standards

Traveling at light speed (warp factor 1) would still require a journey of years to relatively “close” stars. Luckily, warp factors define an exponential, not linear, scale. Let’s write some functions that convert warp factor to velocity:

Velocity of warp factors in two seasons of Star Trek

The warp factor scale was redefined between The Original Series (TOS) and The Next Generation (TNG). Here we can see the difference in warp factors in the Original Cochrane Units (OCU) of TOS vs. the Modified Cochrane Units (MCU) of TNG:

Warp factors graphed

Warp 5 on the new scale is just about warp 6 on the old scale, and warp 8 on the new scale is twice as fast as the old warp 8!

Let’s see how long it would take to cruise to some places in the original Enterprise versus the Enterprise-D:

Time needed to travel select distances at Star Trek Warp Speeds

Even at 200 – 400 times the speed of light, these trips are long. In Kirk’s time, it would take the Enterprise a month to travel from Earth to Vulcan, and Picard could halve that time. And forget about Rigel; they must have been talking about a different Rigel system. The trip to Rigel and back would have been longer than the original five-year mission.

Are We There Yet?

This also gives us a sense of how big the Milky Way is. Traveling from one side to the other at 216 c would take almost half a millennium; Picard’s Enterprise would need a quarter of a millennium. From this, we can gather that Starfleet’s range is a very small portion of the Milky Way galaxy.

Let’s look at a couple emergency situations. When Spock was experiencing the pon farr (a Vulcan mating ritual), he had 8 days to get back to Vulcan or he would die. We don’t know where the Enterprise was at the time, but let’s say it was traveling from Earth:

How long it would take to save Spock at warp 9.1 speeds.

You’d have to travel at warp 9.1; traveling at that speed for 8 days would mean a lot of work for Scotty afterward. (I don’t believe the original Enterprise could sustain warp 9 for 8 days.)

What about the Borg threat?

How long would it take to intercept the Borg from earth travelling at Warp 9?

This Warp Goes to 11!

Now let’s address warp factors above 9. In TOS, warp factors above 10 were seen as unsafe. On a few occasions, encounters with aliens drove the Enterprise to warp factors of 10, 11, and 14. For TNG, the writers decided to make warp factor 10 the top “speed limit.” Warp factor 10 was infinite velocity, at which an object occupied all points in the universe. The exponential scale applied up to warp 9; we have the following values from the writers for warps above 9:

Finding warp speeds for values over 9 in Star Trek

The first column of the matrix shows the warp factor, and the second is the multiplier of c. Even at 200,000 times the speed of light, it would still take half a year to travel across the Milky Way!

Let’s see if we can find a formula that explains this data, using PTC Mathcad’s curve fitting functions:

Using curve fitting to estimate speeds between warp 9 and 9.999 in PTC Mathcad

Nope. I actually tried a few different fits, and exponential came the closest. But warps below almost 9.9 are negative. The writers’ warp scales above 9 appear to be as real as corbomite. Oh well.

Defying SciFi with PTC Mathcad

Somewhere, William Shatner is yelling at me to “Get a life!” Personally, I expected (and found) a lack of internal consistency when it comes to the “science” of warp travel in Star Trek. But this fun exercise with PTC Mathcad gave me a new appreciation of the vastness of our galaxy.

You can take a closer look at the science in your favorite science fiction, too. Download PTC Mathcad Express, your free-for-life copy of PTC’s engineering math software.

Download PTC Mathcad Express free