Mean Warp Speed Project

[Jedi Master Spock]

Objective:
To determine the typical order of magnitude of speed assigned to a vessel moving at warp, regardless of whether it's in a hurry or not.

Methodology:
Due to the sensitive (and highly variant) range of speeds, and the degree to which high examples bias an arithmatic mean, we will take the mean value of the order of magnitude of every perceivable incident.

Each incident shall be multiplied by a weight. Per normal statistical methods, this weight shall be proportionate to the inverse square of the fractional margin of error. Per typical treatment of accuracy and significant figures, the margin of error will be estimated as +/- 0.5 of the last significant figure given, or +/- 2.5 if the last digit ends in a 5 in a singles digit (assumed rounding in such cases), or 20% if numbers are given but the above would be too large (i.e., "a thousand" goes to 1000+/-200 rather than 1000+/-500). (Seems reasonable? I refuse, due to the weighting involved, to treat any figures as exact.)

Both time and distance will be evaluated based on the mean, and error propagated normally in determining appropriate MOEs for velocity (i.e., root of the sum of the squares.)

"Incident" shall include each and all of the following:

• Direct references in which a distance is traveled in a length of time or a velocity is given.
• Indirect references in which a distance is said to have been traveled.
• Estimates based on current value.
• "In excess of" chase figures, which shall be treated as no more than double the closing speed.
• Dependent references based on any two or more episodes in combination, including each minimum combination for which a time figure can be derived.
• References which are otherwise identical to those seen in another episode, or reference another episode, or of a family with those in other episodes (i.e., including all the Voyager-going-home countdowns.)
• Radically different, but equally plausible, interpretations of the same event.

Does this project sound worthwhile?

[Jedi Master Spock]

There is one episode in particular which throws the weighting off, namely, "The Most Toys," in which we know a very slow trip to very fantastic precision.

[2046]

Mind-reading goob. ;)

I've had one of my many articles-in-progress (with the working title "Warp Speeds Suck") wherein I've been collecting all the examples of warp velocity I can.

My underlying strategy was to attempt to create an approximate graphical curve which adequately described the most reasonable range of values for each given warp factor. Some preliminary charts were already complete, though it's been well over a year since I touched them.

This included an analysis of fractional 9.x speeds, for which we got a certain level of comparative data in "Legacy"[TNG4]. I still have the envelope I did the calculations on circa 2002 . . . somewhere. (Yes, there were literal back-of-the-envelope calculations. And yes, I am a packrat.)

In some cases information is limited . . . e.g. if memory serves, I don't think we have much info on the speed of warp two, or four even. Hence the best-fit range, which would at least pretend to fill in the gaps.

And it would also give me something to point to other than that "warp highways" stuff I dislike.

[SSFPhoenix]

This project sounds worthwhile. :)

[Jedi Master Spock]

OK... so I've tabulated the 88 incidents I could come up with off the top of my head. I should go back through them tomorrow, since I'm not sure all the errors I ascribed were entirely appropriate, especially given the quirky results, and I've probably missed some. I included all the Voyager progress references where they were calculable.

In fact, for certain levels of accuracy, I can probably add in many more combination pieces. (2046, I'll e-mail you a copy of the spreadsheet soon - send me back any incident or combination I've missed that you notice.)

Some interesting things. Current (all in factors of c):

Weighted means:
Geometric Mean: 741
Arithmatic Mean: 2,046,956
Geometric Arithmatic
TOS 17,372 112,629.
TNG 258 122,850
DS9 4,993 13,424
ENT 53 2,969
VOY 964 8,053

Unweighted means:
Geometric Mean: 7,380
Arithmatic Mean: 629,700
Geometric Arithmatic
TOS 42,315 442,807
TNG 13,224 628,086
DS9 6,333 25,442
ENT 2,343 22,175
VOY 2,929 30,030

Now, my favorite - throw out STV and weight by series - i.e., give each series equal say in determining the average:
Geometric: 7537
Arithmatic: 229,709

Universally, the weighted means are lower, thanks to the phenomena of there being always one very well known and very slow incident in just about every series, and the fact that the fastest cases are often remarkably hard to pin down. These are all incidents using normal warp cores of either Federation or lower level technology (Hirogen, for example, who are slower than Voyager in calculable incidents, and a wide assortment of civilian freighters.)

The consistently enormous difference between the geometric and arithmatic means (typically a factor of ten or more) highlights what we've always known about warp drive:

Fast ships in a hurry move much, much faster than ships at cruising speed and slow ships.

[Jedi Master Spock]

Oh... and the median is 4100. If anyone wants to be e-mailed the working spreadsheet in its current form, speak up; I'll eventually post it on the website somewhere.

[Mike DiCenso]

There is one episode in particular which throws the weighting off, namely, "The Most Toys," in which we know a very slow trip to very fantastic precision.

In the other extreme, in "That Which Survives" [TOS3], we also have extremely well-given precision regarding the return ETA (11.337 hours at warp 8.4) for the E-1701 across some 990.7 light years.
-Mike

[Jedi Master Spock]

We actually don't know that with absolute certainty and precision - how long is it until Scotty babies the engines up to warp 8.4? (The transcript says 11.337 hours - is that wrong?)

[Mike DiCenso]

Uh, Scotty said no such thing, JMS. Here's the ETA:


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.


If you mean the time it would take from when they set course initally at warp 8 to when they got to warp 8.4, it cannot have been very long at all since the jump cuts between the Enterprise and Kirk and the landing party on the planet indicate at most a few hours went by. Also bear in mind that the landing party had no food or water, and we only see one time where the crew spends the night, which is well after Spock and Rahda's time and speed estimate. So we have it nailed fairly well down here.
-Mike

[Jedi Master Spock]

Uh, Scotty said no such thing, JMS. Here's the ETA:

I mean 11.337 re: the figure, not what Scotty said. Sorry. I'm not clear sometimes.

Even then, our MOE has to be related to that "less than several hours" time rather than the significant figures of 11.337 hours.

I probably can report greater precision than 11.337-24 hours, though.

[Mike DiCenso]

While TWS may not be quite as precise as TMT, it still is overall a remarkably detailed ETA, especially when compared to other speed estimates given throughout Trek, and it can help serve as a bracket on the opposite end of the speed spectrum from TMT.
-Mike

[Jedi Master Spock]

Fair enough. I'll put out another surge of work on it probably within the next week. Combing through piles of slightly different warp speed incidents and resolving their context more clearly is a bit headache-inducing.

I actually should upgrade to include MOEs on the various means, and stick in warp numbers when given. Some of the usual statistical techniques for data simply don't work very well when we have data that's basically contradictory at its core, as warp speed is.

[Jedi Master Spock]

Some added methods that I will be factoring in soon:


To take the distance to a random location within the Federation, consider the Federation is some 8,000 light years across. Assuming that the Federation is slightly squashed and rather disc-like (being within a largely flat galaxy), this means the average distance between two randomly selected points is roughly 4,000 light years, and the average distance to the far end of the Federation is about 6,500 light years - give or take.

Assume 1 stardate = 8.8 hours. (1000 per year.)

Assume that total milage is 1.5 +/- 0.5 of the direct distance traveled for journeys not stated to be direct, i.e., elapsed-episode measures, per Voyager's known meanders, unless a more direct factor is called for.

Between these two, we should have more cruising speeds to examine.

Assume a distance or time stated in a ballpark with a plural ("thousands," "tens of thousands," etc) is 5.5+/-3.5 of those units.

Fresh conjecture: The total distance a Trek ship can travel at V times the speed of light on one load of fuel is generally proportionate to 1/V. Thus, if a ship travels X distance in T time, X~T/X or X^2~T.

Problem: Even accounting for differences between kinds of ships, the extragalactic ranges won't fit properly.

Technique for correlating:

Measure distance squared and correlate with time. Calculate variance of X^2/T overall, and for each ship. To compare with the idea of a constant warp speed method, compare this variance with the variance present in V.

[Jedi Master Spock]

Assuming that the Federation is spread over a region roughly 8000 LY across (as with the above assumptions, rather squashed to only ~2000 light years thick) and a largely squashed disc, what would the average distance to a starbase be at an arbitrary point within the Federation?

Well, we can calculate this very simply if we assume that the Federation uses optimal "stacking" of its starbases within this volume. Assuming the space the Federation is spread out to be a squashed ellipse, and has 1,000 starbases, we can easily see that each starbase is responsible for the volume equivalent to a ~250 LY radius sphere, or ia ~400 LY grid, which gives an average distance of ~200 LY from any arbitrary point to the starbase - assuming that we are on the interior of Federation space.

Accordingly, taking a week to get to the nearest starbase while within the Federation assumes a speed of ~10,000 c.

If we are on or near the edge of space settled by Federation members (although not necessarily within Federation borders proper), the number of directions we can go to get to a starbase is approximately halved - i.e., the starbases per unit volume. Accordingly, the distance increases per cube root of two, or ~250 light years.

If we are working within the confines of "explored" space, we are instead dealing with (in the TNG era) the roughly 10-20% of the galaxy charted by the Federation, which is a ~16,000-22,000 light year radius. Overwhelmingly, outside the space the Federation is sprawled in, the distance is largely the average distance to the border in most cases; distance to nearest starbase ~ sqrt(250^2+distance squared); however, it is difficult to estimate what the typical distance from the border of an average incident outside Federation space is.