[/Begin Thread Tangent/Derailment]
I've owned a Franz-Joseph TOS Tech Manual since 1975 - and even when I saw it I always thought (WF^3)*c was TOO SLOW!
Before all the 'official' things trying to explain/codify various aspects of Trek tech - I always assumed WF = LY travelled per hour of "real time".
^^^
That was the only way one could rationalize Kirk often ordering low warp speeds when leaving planets at the end of or during episodes). Most times he'd order Warp 1 or Warp 2 and occasionally a blistering Warp 3. <--- But that was the only way to even rationalize being able to get to another star system in a few hours/days.
I have to assume the writers felt something similar because in TOS S2 "Friday's Child"; when the 1701 is out looking for a Frieghter (it was a faked distress call done by Klingons of course) - Sulu makes the comment:
Sulu: "At best a freighter might manage Warp 2..."
to which Mr. Scott replies:
Scotty: "I'm WELL AWARE of a freighter's maximum speed..."
My point being: baring all the stuff that's come out AFTER TOS - during the series run, in the 23rd century world of that era, "Warp 2" would have to be a VERY FAST FTL speed; hence my old (outdated/debunked by ancillary 'official' Tech Manuals) fan idea that
WF = LY per hour
was the only thing that made such interstellar travel even remotely possible as depicted in TOS. (WF^3)*c is just way too slow.
[/End Thread Tangent/Derailment]
So according the formula that warp factors equal the number of light years (LY) travelled per hour:
Warp factor 1 is 1 ly per hour, 24 ly per days, 168 ly per week, 8,766 ly per year, 87,660 ly per decade, and 876,60 ly per century.
Warp factor 2 is 2 ly per hour, 48 ly per day, 336 ly per week, 17,532 ly per year, 175,320 ly per decades, and 1,753,200 ly per century.
Warp factor 3 is 3 ly per hour, 72 ly per day, 504 ly per week, 26,298 ly per year, 262,980 ly per decade, and 2,929,800 ly per century.
And so on.
Warp factor 6, the fastest safe speed, is 6 ly per hour, 144 ly per day, 1,008 ly per week, 52,596 ly per year, 525,960 ly per decade, and 5,259,600 ly per century.
Warp factor 6, the fastest emergency speed safe for only a short period of time, is 8 ly per hour, 192 ly per day, ly per week, 70,128 ly per year, 701,280 ly per decade, and 7,012,800ly per century.
In Obssesson":
KIRK: And what if it is the same creature that attacked eleven years ago from a planet over a thousand light years from here?
CHEKOV: Computed and on the board.
KIRK: Ahead full.
CHEKOV: Ahead full, sir.
KIRK: Lieutenant Uhura, contact Starfleet and the USS Yorktown.
UHURA: Frequency open and clear, sir.
KIRK: Inform them that we are pursuing the creature to planet four of that system. That's the location of its attack on the USS Farragut eleven years ago.
KIRK: Yes, I think I do. I don't know how I know, but home is where it fought a starship once before. Ito Uhura) Inform them of our tactical situation and inform them I'm committing this vessel to the destruction of the creature. We will rendezvous. Round-trip time, Mister Chekov.
CHEKOV: One point seven days, sir.
So a round trip of over 2,000 light years would take 1.7 days, at a speed of over 49.09196 light years per hour. But the Enterprise would be travelling at warp factor 8, not warp factor 50.
In "That Which survives":
RAHDA: No debris of any kind, sir. I've made two full scans. If the planet had broken up, there'd be some sign. But what bothers me is the stars, Mister Spock.
SPOCK: The stars.
RAHDA: Yes, sir. They're wrong.
SPOCK: Wrong?
RAHDA: Yes, Mister Spock. Look. Now here's a replay of the star pattern just before the explosion.
SPOCK: A positional change.
RAHDA: It doesn't make any sense. But somehow I'd say that in a flash we've been knocked one thousand light years away from where we were.
SPOCK: Nine hundred and ninety point seven light years to be exact, Lieutenant.
Later the Enterprise heads back to the planet:
SPOCK: Good. Then prepare to come to warp eight.
Later:
RAHDA: We're holding warp eight point four, sir. If we can maintain it, our estimated time of arrival is eleven and one half solar hours.
SPOCK: Eleven point three three seven hours, Lieutenant. I wish you would be more precise.
Travelling 990.7 light years, or about 8,684,476.2 light hours, in 11.337 hours requires a speed of about 766,029.4787 times the speed of light, many times warp factor 8.4, according to you, whch would be only 73,634.4 times the speed of light.
So in some episodes the warp factor formula you suggest would be much too slow for the distances and times in voyages.
Bu the warp factor formula you suggest would also be too fast for some episodes.
In "This Side of Paradise", set on Omicron Ceti III, Kirk says:
KIRK: Another dream that failed. There's nothing sadder. It took these people a year to make the trip from Earth. They came all that way and died.
Even at warp factor 1 according to your proposed scale, a ship could travel 8,766 light years in one year. And of course every good science fiction fan should know that Mira, or Omicron Ceti, is only a small fraction of that distance from Earth.
In "By Any Other Name":
ROJAN: Your ship, Captain Kirk. It will serve us well in the long voyage that is to come.
KIRK: Voyage? Where?
ROJAN: To your neighbouring galaxy, which you call Andromeda.
And:
KIRK: What's the point of capturing my ship? Even at maximum warp, the Enterprise couldn't get to Andromeda galaxy for thousands of years.
ROJAN: Captain, we will modify its engines, in order to produce velocities far beyond the reach of your science. The journey between galaxies will take less than three hundred of your years.
Spock Fascinating. Intergalactic travel requiring only three hundred years. That is a leap far beyond anything man has yet accomplished.
The distance to the Andromeda Galaxy is estimated to be approximatley 2,540,000 light years. If it takes about 250 to 300 years to travel that distance, the speeds will be approximately 8,466.66666 to 10,160 times the speed of light. And your suggested warp factor 1 is about 8,766 times the speed of light, within that speed range.
And Kirk said the Enterprise couldn't make the journey in thousands of years, even at maximum warp. If that is interpreted as meaning the trip will take 2,000 to 10,000 years, whcih may not be correct, kirk would imagine the speed of the Enterprise to be on the order of 254 to 1,270 times the speed of light, which is slower than your proposed warp factor scale.
Thus it is seen that your proposed warp factor scale is to slow for some episodes and too fast for others.
