Star Trek Warp Factor Equations

[Oriet]

So I was poking around things the other day and I decided it could be fun to run a Star Trek game at some point. I don't yet know which system I'll use for it (possibly the one I've been working on creating), but before I get any game rule specifics I want to make sure I have a decent set-up on how to handle things within the Trek universe itself.

This, of course, leads to the old nemesis of TNG Warp Factors.

Below or at Warp 9.0 the official calculation (speed in multiples of C = warp factor ^(10/3)) work excellently, so remains untouched. After that it gets really messy, and with so many ships being able to travel above Warp 9.0 it is needed to know just how fast in comparison they are, or how long a journey will take. I had used some online Warp Factor calculators in the past, but the main one I used seems to have vanished, so instead of trying to find another one I decided to take it upon myself to find or create an equation for it.

Looking around the 'net (though only for a short while) didn't yield me anything useful, other than re-confirmation that the TNG Warp scale past Warp 9.0 was a hand drawn line that's supposed to reach infinity at Warp 10. In a way this didn't help, but in another it meant that I was free to create my own calculation for it as long as it both continued to increase past Warp 9.0 (I've seen some things listing Warp 9.6 as being slower than Warp 8) and that it approaches infinity as it approaches Warp 10, and does so at an ever increasing rate.

Surprisingly it didn't take all that long to whip up something functional for this. In it there are two variables: 'speed', which is the speed in multiples of C (lightspeed), and 'decimal', which is the trailing numbers after the decimal for Warp Factors above 9.0.

speed = 9^(10/3) / (1 - decimal)

I know it might look strange, but let me demonstrate. Say you are travelling at Warp Factor 9.85, and so want to know how fast you are going. 'decimal' is then 0.85, so we plug that in to get:
speed = 9^(10/3) / (1 - 0.85)
speed = 9^(10/3) / 0.15
speed = 1,516.38 / 0.15
speed = 10,109.2
This means with this formula Warp 9.85 means you're travelling at 10,109.2 times C (or 3.03x10^12 m/s, for those who want an inane measurement for it).

"That's great!" one might say, but unfortunately it by itself is not enough, at least for those who are not mathematically inclined. As such I have devised (with Vene's help; damn my mind wanting to do this stuff while being tired!) the reverse equation, for when you know the speed but want to find the Warp Factor.

decimal = (9^(10/3) / -speed) + 1

First you should see if the speed is actually at or below Warp 9.0, but using: WF = speed^(3.10). If that yields more than 9.0 then you know to plug it into the above equation. For this, let's give an example. Say you know the speed (whether it's of a vessel they're chasing or the speed they need to use to get somewhere in time for something) is 2,000 times C. Plugging it in we get:
decimal = (9^(10/3) / -2,000) + 1
decimal = (1,516.38 / -2000) + 1
decimal = -0.758 + 1
decimal = 0.242
This then gives us a Warp Factor of 9.242.

These equations don't factor in any other variables, such as subspace disturbances, though they do give a way to convert to and from other FTL drive speeds, such as TOS Warp Factors (which uses a simple equation: speed = WF^3).

[Napoleon the Clown]

The reason things get so silly with Warp Factors is because TNG and after uses a different equation than TOS. TNG set it as being related to the energy needed to attain the speed or something like that whereas TOS had a simple, straightforward equation. Which is also why Warp 9 can be slower than Warp 8. TOS era Warp 9 is slower than TNG era Warp 8.

The reason for this is to prevent the ever-increasing Warp factors seen in TOS.

[Oriet]

Oh, I know of the differences between TOS and TNG Warp speed scales, hence why I said "the old nemesis of TNG Warp Factors." I just find their rationales a little silly, especially when they had already introduced an alternative with transwarp drives in Star Trek III: The Search for Spock on the Excelsior. I also find they just made the problem worse by having what would be interpreted as a very minute change (say, anything past Warp 9.9) being an incredibly large difference that would be better shown by continuing the count. A new scale would make perfect sense for a transwarp drive as well, since it is using different mechanisms to achieve FTL travel.

I also find them having it based on power usage to be laughable and utterly useless for anything other than figuring out how long you can maintain a specific speed. I mean, really, it'd be like measuring how fast your car is going with the engine's RPM; while it correlates with different speeds it also is then correlating with multiple speeds at different outputs, and will be an entirely different speed in a different craft.

Because of this I prefer to stick with the TOS for traditional Warp travel. These equations, though, will allow me to freely convert speeds back and forth between TOS and TNG scales, which is actually the entire reason I poked at making it to begin with.

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