134 TNG-era: The Next Generation, Deep Space Nine, Voyager (Warp Velocities - Star Trek)
Description
This article is from the Star Trek Tech FAQ, by Joshua Bell inexorabletash@hotmail.com with numerous contributions by others.
134 TNG-era: The Next Generation, Deep Space Nine, Voyager (Warp Velocities - Star Trek)
By the time of Star Trek: The Next Generation, the warp scale has
changed. Warp 1-9 are roughly the same, but Warp 10 is infinite speed.
Going Warp 10 or faster is hogwash on the TNG scale. It isn't a speed
barrier that can be or needs to be broken, but an energy barrier.
At least, that's what the Tech Manual says. Many fans disagree, saying
that this has been contradicted on air, most clearly by the episode
"Where No One Has Gone Before" [TNG] where someone says "We are
passing Warp 10." See the Warp and Subspace FAQ for more discussion of
this.
Graph
Here's the graph of warp vs. speed and warp vs. power consumption from
the Technical Manual:
Chart
This chart was compiled with data from episodes, the Encyclopedia, and
the TNG Technical Manual:
Warp Factor Velocity Source Comment
1 1 TM Speed of Light
2 10 TM
3 39 TM
4 102 TM
5 214 TM Federation speed limit (2370)*
6 392 TM
7 656 TM
8 1024 TM
9 1516 TM Defiant maximum speed**
9.2 1649 Encyc
9.6 1909 Encyc
9.9 3053 Encyc NCC-1701-D maximum speed
9.975 ? Episode Voyager "cruise velocity"***
9.99 7912 Encyc
9.9997 ~198696 TM (derived) Subspace radio speed
9.9999 199516 Encyc Maximum boosted subspace radio speed
* * "Force of Nature" [TNG]
* ** "The Sound of Her Voice" [DS9]
* *** "Caretaker" [VOY]
Sources
The Tech Manual (on page 111) says that a subspace radio signal
travels at Warp 9.9997, and takes 45 minutes to reach 17 light years,
which works out to 198696c.
As an interesting anomaly, Pete Carr also points out the following
tidbit from the Tech Manual:
______________________________________________________________
... the TM goes on to say that TNG Warp 9.7 is about 14.1 on the
TOS scale. So [TNG Warp] 9.7 ~= 14.1^3 [c] and 14.1^3 [c] = 2803
[c]. I went and graphed the new value with our current values.
Unfortunately the new value doesn't fit into the exponential curve
... it should be lower.
______________________________________________________________
I suspect Mike made a rounding error; TOS Warp 14.1 is much closer to
TNG Warp 9.8 by all of the accurate formulas that have been found or
sheer reckoning off the graph.
Formulae
Unlike TOS (where we have a formula but no scale), for TNG we have a
scale but no formula! The reason for this is that the graph was drawn
by Mike Okuda rather than calculated, as is related in the following:
On June 22, 1995, Jeff Reinecke forwarded the following letter from
Michael Okuda to rec.arts.startrek.tech:
______________________________________________________________
Date: Fri, Jan 27, 1995 02:09 AM EST
From: MOkuda
Subj: Re: Star Trek Warp
To: Yar of Spit
The warp factors we've used in ST:TNG were computed in an arbitrary
way to fit some specific characteristics we needed.
First, the speed for any given warp factor had to be greater than
it was in the original Star Trek series. This was primarily to
satisfy fan expectations.
Second, the new warp speeds couldn't be TOO much faster, or it
would be possible for the ship to cross the galaxy in a fairly
brief time. (In a way, maintaining this restriction made Voyager's
story situation possible. If we hadn't done this, Voyager could
have gotten home too quickly.)
We used an exponent of (I think) 3.33 or 3.33333... for warp
factors less than 9.
Between 9 and 10, I gradually increased the exponent so that it
approached infinity as the warp factor approached 10. Lacking
knowledge of calculus, I just drew what looked to me to be a
credible curve on graph paper, then pulled the points from there. I
think I re-created the curve fairly accurately in the Star Trek:
The Next Generation Technical Manual.
Hope this helps.
-Mike
______________________________________________________________
So it looks like there isn't a grand formula to end all formulas after
all!
On May 29th, 1996, Dominic Berry wrote:
______________________________________________________________
Since Mike calculated the speeds for the various warp factors up to
9 simply using the exponent 10/3, it is more sensible to use a
piecewise function for the speeds that gives an exponent of 10/3
for W<9 and gives higher exponents for higher warp factors. My
suggestion is
n
( 10/3 + u(W-9) * A * (-ln(10-W)) )
v = W
where u is the step function, i.e. u(x)=0 for x<0 and u(x)=1 for
x>0. Note that the term multiplying the step function is zero at
W=9, so the step function introduces no discontinuity in the
formula. If the value of n used is greater than 1, then both the
function and its derivative are continuous at W=9. (In order to
have continuous higher order derivatives a function like c(w) could
be used.)
[Martin Shields amends that with:
There is a better alternative to the step function as follows:
/ 0 ; x <= 0
|
u(x) = <
|
\ e ^ ( -1 / (bx^2)) ; x > 0
Where b is a constant whose value remains to be determined. This
function is "infinitely smooth" (that is, no matter how many times
you differentiate it, the value of the differential is 0 at x=0).
As b approaches infinity, the function approaches the pure step
function.
]
I take A and n as
A = 0.03684678
n = 1.791275
I then get the correct warp factors for W<9, and for the warp
factors above 9 I get
Warp | Actual Formula Modified*
-------------------------------------
9.2000 | 1649 1641 1640
9.6000 | 1909 2019 2017
9.9000 | 3053 3035 3029
9.9900 | 7912 7939 7912
9.9997 | 198696 79352 79240
9.9999 | 199516 199415 199516
[* Martin Shields offers A = 0.036528749373 and n = 1.79522947028
which are slightly better for some values.]
My formula agrees with the values for warp factors of 9.2, 9.9,
9.99 and 9.9999 to within 0.6%, though it is about 6% out at 9.6
and it is way out at 9.9997. If you calculate the exponents for the
data points at 9.9997 and 9.9999, however, you get 5.29826 and
5.30000, suggesting that the exponent 5.3 was used to calculate the
speed at both of these warp factors. Since the exponent should be
increasing with the warp factor, one of these data points should be
ignored. Ignoring the data point for W=9.9997, my formula is
perfect for W<9, and is slightly better than that of Tahk for W>9.
Now the exponent corresponding to the speed given for W=9.2 was
about 3.33810. If we linearly extrapolate this to W=9.6 then the
exponent should be about 3.34763. The exponent corresponding to the
speed for W=9.6 is 3.34002, which is slightly less! This means that
a formula for the exponent that gives values similar to the given
values for warp factors of 9, 9.2 and 9.6 must have a derivative
that decreases. (This means that the function for the exponent
would have to curve downwards between 9.2 and 9.6.) Since this is
not a desirable property if we want an exponent that gradually
increases, I also left out the data point for W=9.6 in fitting the
curve.
I mentioned before that the speeds for W=9.9997 and W=9.9999 seemed
to have both been calculated using an exponent of 5.3. If you
calculate the exponents corresponding to the other warp factors
above 9, you get:
Warp | Exponent
-----------------
9.2000 | 3.33810
9.6000 | 3.34002
9.9000 | 3.50000
9.9900 | 3.89998
The speeds for warp factors of 9.6, 9.9 and 9.99 were obviously
calculated using exponents of 3.34, 3.5 and 3.9 respectively, and
the speed for a warp factor of 9.2 was probably calculated using an
exponent of 3.338. Therefore it is not reasonable to ascribe any
greater accuracy to the warp factors given than is implied by the
number of significant digits in the exponents used to calculate
them. By this criterion my formula gives speeds well within the
uncertainty for warp factors of 9.9, 9.99 and 9.9999, although it
gives an exponent of about 3.336 for W=9.2, which is a little low.
Since the points were originally taken off a hand drawn curve, this
is still reasonable accuracy.
______________________________________________________________
I used to have a bunch of formulae in here from various posters who
made some pretty good attempts at finding the Holy Grail of an
accurate formula. However, due to length considerations I'm only going
to keep the current best. Older formulae (basically an excised chunk
of this FAQ) can be found at
http://www.calormen.net/Star_Trek/FAQs/warp_formulae.html, but that
page will probably never look too pretty.
Martin Shields has a warp speed calculator up on his web site:
http://www-personal.usyd.edu.au/~mshields/TV/ along with discussion of
the formula used.
Evidence
Do any of these values actually match up with what we've seen on the
show? There are often claims that these speeds are much to slow to
allow the kind of adventuring that the Star Trek series portrays. But
amazingly enough, when they do quote numbers and we can time things
without cuts (wherein we may miss hours of ship-time), the numbers do
match up:
* "The Most Toys" [TNG]:
Ges Seger offers:
The numbers I remember were about how far a ship doing warp 3 for
23 hours would travel, and the answer they came up with was 0.102
light-years. I worked the math just now and got 0.1022 light-years.
* "Bloodlines" [TNG]:
Riker calculated in his head the time required for the Enterprise
to travel 300 billion kilometers at Warp 9, and gets 20 minutes:
Warp 9 = (300e12 m) / (20 min * 60s/min) ~= 2.5e11 m/s
From the chart: Warp 9 = 1516c ~= 4.548e11 m/s
Discrepancy? Riker did the calculations in his head in about 5
seconds given arbitrary numbers. He's within a factor of two, so I
won't complain. Bok's ship was "holding position", so it was a
simple flight path.
* "Emergence" [TNG]:
The Enterprise jumped to Warp 7.3, and traveled 30 billion
kilometers in a couple of minutes.
All of the formulas we have for warp speeds predict Warp 7.3 to be
approximately 746c. Using c = 3e8 m/s, we get v = 2.24e11 m/s. 30
billion km = 3e13m. So t = 134s, or just over two minutes.
* "Allegiance" [TNG]
c/o Boris S.:
Wesley gives the ETA of the Enterprise to Lonka Pulsar as 34
minutes at Warp 7. When Picard orders Warp 2 instead, he comments
that at that speed it would take 31 hours to get there. Using the
first two data points, 34 minutes at Warp 7, I calculated a
distance of 4.012e14 m. At Warp 2, it would take the Enterprise 37
hours to travel that distance. This clearly shows that the TNG
production staff used the established warp scale when they
calculated the travel time, and the 6-hour discrepancy can be
explained by the use of a less accurate value for the speed of
light.
* "Clues" [TNG]
c/o Boris S.:
the Enterprise is transported 0.54 parsecs by the Paxans. Riker
says something like "nearly a day's travel in 30 seconds" (I cannot
give you the exact quote since I am watching TNG on German TV). At
Warp 6 (Enterprise cruising speed), the Enterprise would need 1.6
days to travel that distance. Given that Riker calculated the
travel time without a computer in a couple of seconds, you can
allow for the deviation. On the other hand, if you calculate the
travel time at Warp 7, you get 23.5 hours, which fits the quote.
* "Caretaker" [VOY]
The basic numbers involved in Voyager's journey home support the
TNG formula. Voyager is transported 70,000 LY from home, and
expects to take 70 years to make the journey. This speed of 1000c
corresponds closely to Warp 8, a high but conceivable average
speed for a long journey for an Intrepid-class ship.
* "The 37s" [VOY]
Paris states that Warp 9.9 is equal to 4 billion miles per second.
Unfortunately, that turns out to be over 20,000c, which doesn't
fit in at all. Bummer. But then, Paris is an idiot.
* "Maneuvers" [VOY]
Kim states the ships speed as 2 billion km/s, which is 2*10^12
m/s, which is roughly 6667c. This is in the same ballpark as what
Warp 9.975 (Voyager's top cruise speed), it turns out.
Boris S. speculates that if Okuda picked an exponent of 3.83 (a
nice roundish number off the graph) for Warp 9.975, you get 6696c.
Pretty close to the value above.
* "Threshold" [VOY]
Commentary aboard the ship confirms that Warp 10 is indeed
infinite speed.
* "Dreadnought" [VOY]
c/o Boris S.:
B'Elanna gives the distance to Rakosan system as more than 10
light-years. A day or so later Chakotay states that the vessel has
resumed its journey at Warp 9 and will reach Rakosan V in 51 hours,
which works out to a distance of 8.8 light-years.
Counter Evidence
There have been several times where the warp velocities proposed don't
match what we see on-screen. The most blatant example of this kind is
a call by the captain to head somewhere at Warp 1, or some other
ridiculously slow speed. This happened several times in TOS, but does
crop up from time to time. Here are some examples:
* "Where Silence Has Lease" [TNG]:
Roger M. Wilcox offers:
The Enterprise-D gets sucked into a black nebulous void. Before
Nagilum announces his/her/its presence to our intrepid crew, they
find an opening in the void "1.3 parsecs away". (1.3 parsecs would
be 4.243 light-years.) Picard orders the crew to head for the
opening at Warp 2.
It may be best to just pretend that these didn't happen, or
rationalize them on a case by case basis (going Warp 1 until outside
of the solar system, then switching to a higher speed "off camera").
...
A bigger problem which crops up on rec.arts.startrek.tech is the size
of the Federation. Sizes of up to 10,000 LY across have been quoted as
diameters, and this corresponds to the occasional graphic displayed on
screen showing the Federation's size and position within the galaxy.
Other evidence points to a somewhat smaller size, but such questions
as the distance from Earth to Bajor appear to present a paradox: some
routes between Federation locations which are known to be far apart
are traveled much more quickly than the TNG formula appears to allow.
The leading speculation on the newsgroup is a concept called "Warp
Highways". Distinct from wormholes, these "highways" represent either
natural (pre-existing) or artificial (thanks to heavy traffic)
pathways where warp travel is much faster than the TNG formula, which
represents a baseline.
The highways do not require additional technology beyond warp drive.
Highways are not easily detectable in unknown space. This means that
an exploration ship, such as the Enterprise, or a ship in unknown
territory, such as Voyager, will travel between two arbitrary points
at the nominal velocities presented in the TNG formula. A well-known
region of space - such as the route from Bajor to Earth - would
probably contain several well-known warp highways and allow less
powerful ships to make the route in weeks rather than years, and
top-of-the-line Starfleet ships to make the trip in mere days. Contact
with local civilizations would allow Voyager to take shortcuts through
the Delta Quadrant - which they frequently seem to.
Perhaps the Hekaras Corridor in "Force of Nature" [TNG] is one such
route, explaining the frequent travel in that area. The whole notion
of starship travel affecting local subspace properties in a permanent
way supports the notion that at least some warp highways are created
by frequent warp travel - that is, as a route is used it becomes more
efficient. Other speculation includes the notion that gaseous
anomalies are indicators of the presence of warp travel. Why else
would Excelsior - one of Starfleet's latest ships - be engaged in
charting such anomalies in Star Trek IV?
This is strongly reminiscent of the X (chi) factor first presented in
Star Trek Maps, where the warp equation varies with local spatial
conditions.
(Other speculation or comments?)
Why did it change?
* In terms of a real-world "Star Trek is just a TV show" reason,
Gene Roddenberry himself put Warp 10 at infinite speed, according
to the TNG Tech Manual. To keep the scale fluid, Mike and Rick
made it asymptotic at Warp 10, while starting off similar to the
TOS scale.
* From the characters' perspective, the best explanation is that the
TOS scale was established before warp was fully understood.
Looking at the graph, you can see that the energy costs for
cruising at integral Warp values are much lower than for
non-integral Warp factors. The first explorers to travel past Warp
1 must have realized this. Since for Warp values in the 1-3 range
follow the v = (W ^ 3) * c formula, it makes sense that a scale
based on the formula would come into use.
When ships started cruising at Warp values larger than 5, the
difference between what v = (W ^ 3) * c predicted to be the most
energy efficient speeds and what actually were must have become
noticeable. It may have taken a long time for a new, accurate
scale based on new observations came into use. (Look at the USA
and SI, for an example of a large sociopolitical body taking a
long time to adopt a more useful, universally used scale.)
Sulu's readings of Warp velocity in Star Trek IV seem to hint that
the Klingons had moved to an accurate scale by the 2280s, but the
Federation didn't catch up until much later, even though it must
have been painfully obvious that the old scale was next to
useless. Fortunately, some time before TNG, the new accurate scale
was adopted by Starfleet.
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