From: ph111-bx@violet.berkeley.edu (;;;;7202)
Newsgroups: rec.arts.startrek.tech
Subject: Leon Myerson's Star Trek technology article
Date: 7 Oct 1992 04:55:04 GMT
Organization: University of California, Berkeley
NNTP-Posting-Host: violet.berkeley.edu

Well no one else would do it, so I borrowed a friend's CompuServe account
and downloaded the entire Leon Myerson article (all previous posts here
were missing the last part of it).  A couple of RLE graphics files came
with it and the entire set may be found as warpte.zip on pdq.coe.montana.edu
in the /tmp directory, available through anonymous ftp.  So now, once again
here's Leon Myerson's [complete!] treatise on trektech.  Enjoy!
_______________

                S T A R   T R E K   T E C H N O L O G Y

                          by Leon Myerson

COPYRIGHT 1988 by Leon Myerson - permission to download and reprint this
essay for free distribution within the ranks of Star Trek fandom is hereby
granted provided the author's name and this copyright notice are retained.
This essay may be periodically superceded by revised versions uploaded to
Data Library 2 of CompuServe's Science Fiction Forum.

    DISCLAIMER SECTION:  None of the ideas expressed in this essay are
"official".  All concepts put forth are solely my own opinions and
speculations, and as such, might be completely contradicted by "official"
Star Trek material issued in the future.  I have drawn as much as possible
upon the filmed Star Trek episodes and features, and refer to such
"references" as the Franz Joseph blueprints and Technical Manual, and to the
"Spaceflight Chronology" book, when I have found it useful to do so.  At
other times, I made it up.  This material and any companion essays I may
upload, are for the sole purpose of having fan-fun with the Star Trek
universe.  I have no connection whatsoever with ST:TNG or with Paramount, I
just like to speculate regarding futuristic science. -Leon.



    Warp numbers do not directly refer to speed, but to power.  Warp 1 is the
power level required to enter the warp continuum, and is known as Threshold
power.  Warp 2 is twice that power level, etc.  Fractional warp is simply
less than Threshold power while the ship, though traveling via warp field
effect, is still "in" the Einstein space-time continuum at sub-light speed.
The unit of power between whole warp factors for a given vessel is one
"Impulse", as in the ST:TNG episode "Conspiracy" when Geordi answers Riker's
command to increase from Warp 5 to Warp 6 by acknowledging the addition of
yet another full impulse to the power already coursing thru the warp
nacelles.
    The formula relating the Warp number W to velocity in terms of C is not
the hopelessly inadequate V = W^3.  In Trek Classic's very first episode the
Enterprise was seen at the edge of our galaxy.  Even assuming this to be the
near edge reached by going perpendicular to the galactic plane, it is still
at least 1500 light years from Earth.  At a cruising speed of Warp 6 = 216 C,
the ship would have spent at least 7 years getting out there, then 7 more
back.
    Nor would that formula fit the size of the United Federation of Planets'
Treaty Exploration Zone mapped in the "StarFleet Technical Manual".  This zone
was pictured as being approximately 12,000 light years in radius, with both
the Klingon and Romulan empires located at the rim some 60 degrees apart.
Clearly, Enterprise did not require an excess of 50 years to reach the neutral
zone.
    In the ST:TNG episode "Conspiracy", Picard and Riker confront the parasite
mother creature in the guise of Lt.  Cmdr.  Remmick as he/it sends a beacon to
the parasite beings' homeworld via StarFleet's own CommNet.  The 3-D map of
that network on the wall behind him fits almost perfectly the Treaty
Exploration Zone of the Trek Classic Era.

    Instead of V = W^3, velocity is defined by the sum of an infinite series
known as the 3rd-order Cochrane function, which is applicable to Tertiary
warp effect fields such as are utilized by major Federation vessels from the
Tritium class onward.  The first term of this series is the familiar W^3, the
second term is the integral of the first term, W^4/4, the third is the
integral of the second, W^5/20, and so on, ad infinitum.  Each term is the
integral of the preceding term.  Thus the common mistake so often made is to
quote only the first term of the series as if it were the entire function.
    The behavior of this series is such that the terms rise in value at
first, then become increasingly smaller so as to converge on a definite
value.  This may be calculated by the equivalent formula:

                   V =  6 * { e^W - [ (W^2)/2 + W + 1] }

where V = velocity, W = Warp factor, and e = base for natural logs 2.71828..

    When calculated in this manner, this function gives the following values:

                    Generated              Uncorrected
                       Power               Warp Speed x C

                           1                      1.31
                           2                     14.33
                           3                     69.51
                           4                    249.59
                           5                    779.48
                           6                   2270.57
                           7                   6384.80
                           8                  17639.75
                           9                  48315.50
                          10                 131792.79
                          11                 358809.85
                          12                 976018.75
                          13                2653889.35
                          14                7214947.68
                          15               19613332.78

    For starship designers, these numbers seemed too good to be true, and
indeed they were.  From the earliest days of starship operations, warp
engines had always registered a small power loss as they were fed more than
Warp 1 power.  Defined as the difference between Generated Power and
Delivered Power, this drain was ascribed to the faintly conceived notion of
"continuum drag".  It was Delivered Power that determined actual velocity
according to the 3rd-order Cochrane function.  As the phenomena was still too
poorly understood for mathematical description, progressive increases in
power generation capability had to be matched empirically with increases in
Delivered Power via actual flight testing, and the term Warp Factor continued
to refer to Generated Power.
    The Dilithium breakthru made it possible to generate unprecedented
multiples of threshold power, and led to the Federation's investment in the
Constitution class vessels.  Able to safely generate and sustain Warp 8
power, these ships found the drag/drain worsening rapidly at the higher
levels.
    It was the USS Enterprise, under Christopher Pike, that first challenged
the "Warp Barrier".  After three month's total overhaul at the Terran Orbital
Shipyards personally supervised at every stage by Montgomery Scott, the ship
went on speed runs pushing her anti-matter reactors as high as Warp 13 for a
few seconds at a time.  The resulting measurements at last permitted Scott to
define the continuum drag equation:

                                   tan(A)
          CDF = G - ------------------------------------ + 10
                    (G-S) + (tan^2(A)+((G-S)^2)-1)^(1/2)

and thus

                                 D = G - CDF

    where D = Delivered Power; G = Generated Power; CDF = Continuum Drag
Factor; A = 5.1050881 radians; and S = 9.8658770244 (Scott's constant).  The
corrected table of Warp speeds is therefore:

                  Generated         Delivered      Warp Speed
                    Power             Power           x C

                      1              1.00000            1.31
                      2              1.98354           13.91
                      3              2.96260           65.98
                      4              3.93509          230.94
                      5              4.89755          696.42
                      6              5.84370         1926.80
                      7              6.76140         4999.38
                      8              7.62571        12075.26
                      9              8.38615        26048.20
                     10              8.96633        46707.91
                     11              9.33067        67348.90
                     12              9.53548        82717.85
                     13              9.65322        93087.64
                     14              9.72615       100151.85
                     15              9.77477       105155.01

                  Old Warp          New Warp

    A graph of Scott's equation plotting Generated Power as X against
Delivered Power as Y, shows that at threshold power (Scott's equation and the
3rd-order Cochrane's function are not applicable below this point) X = Y = 1,
and the graph line proceeds at an almost 45 degree angle assuming equal
scales.  (This graph is available as WARP10.RLE in DL2 for those with IBM PCs
or compatibles.)
    But as Generated Power exceeds 8 times threshold level, Delivered Power
deviates ever more significantly and the graph curves sharply to the right.
The curve is half of a hyperbola, rotated by angle A, with the significant
asymptote line represented by the equation Y = 10, so that while the
Generated Power may go arbitrarily high, the Delivered Power will only
approach ever more closely but never equal 10.  The speed value for Warp 10
from the uncorrected chart, 131792.39 times the speed of light, is the
theoretical limit of the Tertiary warp effect, and can only be approached,
never equaled or exceeded.  This is also the velocity of such warp continuum
energy transmission phenomena as sub-space radio and the standard phaser
effect.  (The complete hyperbola is graphed in WARP_X.RLE, also in DL2.)
    By the time of ST:TNG, it had become standard practice to quote Warp
factors in terms of Delivered, rather than Generated, power.  This explains
the apparent discrepancy between the eras.  Overall Generated Power
capabilities are still crucial to military vessels, as even a few dozen extra
C's may mean the difference between success and failure when outrunning or
persuing an opponent.  Here then is the standard warp factor scale used in the
24th century:


                 Delivered      Generated         Tertiary
                   Power          Power               Warp

                     1         1.0000000000           1.31
                     2         2.0167653720          14.33
                     3         3.0383208502          69.51
                     4         4.0670614879         249.59
                     5         5.1072983806         779.48
                     6         6.1676537197        2270.57
                     7         7.2682459514        6384.80
                   7.5         7.8487197368       10628.50
                     8         8.4694304149       17639.75
                   8.2         8.7364919027       21588.78
                   8.4         9.0203187626       26414.32
                   8.6         9.3280961537       32310.48
                   8.8         9.6717993420       39514.34
                     9        10.0729838055       48315.50
                   9.1        10.3071067812       53422.73
                   9.2        10.5747605008       59067.65
                   9.3        10.8903152831       65306.85
                   9.4        11.2777216596       72202.80
                   9.5        11.7800905867       79824.61
                   9.6        12.4836439773       88248.61
                   9.7        13.5895662949       97559.17
                   9.8        15.7014109302      107849.55
                   9.9        21.8369448362      119222.79
                    10          INFINITE         131792.79

                 New Warp       Old Warp        Velocity x C

    To calculate the Generated Power corresponding to a given Delivered
Power level, use the formula:

                        ((D-10)^2*(tan(A)^2-1)-tan(A)^2
               G = S - ---------------------------------
                               2*(D-10)*tan(A))


    An interesting aspect of the 3rd-order Cochrane function is that Warp 1
is not C but 1.31 x C.  Taking the reciprocal of this number, 0.763, gives
what is known as threshold velocity.  Under fractional warp power, a starship
"accelerates" as the power is steadily increased.  At Warp .99999 etc., the
ship is traveling at 0.763 x C.  Transition occurs, an explosive event
accompanied by the hauntingly beautiful phenomena known as the Chromatic
Detonation, the optical analog of a sonic boom.  In the next micro-instant,
the ship is on "the other side", traveling through the warp continuum at 1.31
x C.  The ship is never observed at speeds 0.763 < V < 1.31 under normal
conditions.
    It should be noted however, that the boundary layer of the warp field
effect creates an envelope of 4 dimensional Einsteinian space-time within
which the ship travels.  Therefore, all the familiar physical laws of the
"ordinary" continuum still apply within the envelope.  From the outside, it
appears as though a space-time anomaly were manifested sequentially along a
linear path.  Fleeting, multiple images of the vessel in the center of the
anomaly are created at widely spaced intervals which grow more distant at
higher warp factors.  Light coming from within the envelope gathers at the
boundary layer until it reaches optical crossover threshold, at which point
it "pulses" through, thus re-entering normal space-time to project the image
of the ship.  This effect was nicely filmed for the climactic scene in
ST:TSFS when we see the Enterprise fleeing the detonation of the Genesis
Device.
    External light enters the envelope via complex optical interaction with
the warp field boundary layer.  The micro-instant lost for photons in front
of the ship's path to cross the boundary layer causes them to appear to
originate from directions shifted away from the line of motion in favor of
apparent origins perpendicular to the direction of travel.  While an optical
blind spot exists directly behind the ship along the direction of motion, due
to the superluminal velocity involved, the tear-drop shape of the overall
warp field minimizes the area so blanked out to a vanishingly small region.
    The overall effect is curiously symmetrical to that observed by vessels
approaching light speed in normal space-time.  Such a vessel would see its
3-dimensional field of view collapsed into twin circles of light in front of
and behind the ship, with a band of darkness around its mid-section.  A
vessel in the warp field traveling at superluminal velocities experiences a
tunnel-like effect in which the dark region consists of circles in front of
and behind the vessel, and its view of the universe is projected onto a
cylindrical tube which the ship appears to travel through.
    Of course, the ship's computers correct for this effect to present an
intuitively "normal" view upon the bridge and other viewscreens.  Windows
facing port or starboard reveal a relatively normal view without
sophisticated correction, others have internal holographic layers which serve
as the functional equivalents of corrective lenses to keep the view at least
intelligible, if not exactly accurate.
    Sometimes a foreign body, such as small pieces of asteroidal rock or
chunks of cometary ice are pulled into the forming continuum envelope as a
starship achieves transition.  Usually this is a harmless occurence, unless
the "dragger" is massive enough to damage the hull if it should collide with
the vessel.  If so, the ship will usually power down below threshold to
release the object, otherwise it can remain within the influence of the warp
field effect and go along for the ride to the starship's scheduled
destination.  An unusually extreme instance of this effect occurs in ST:TMP
when the old Enterprise, bucking wildly from her imbalanced engines, pulled a
whole asteroid into the warp envelope formed around herself, and was forced
to pulverize it with a photon torpedo.
    Old space junk from various inhabited systems often gets distributed
about the galaxy in this fashion, centuries in orbit about their star of
origin affording plenty of time for a chance encounter with a transitioning
starship.  Some of places identifiable objects ultimately turn up can be
downright humorous.  Items too small to possibly damage a vessel thru its
deflector shield are usually ignored, especially when they have no possible
salvage value.
    An example would be the cryonics satellite found just prior to the
NCC-1701-D's recent visit to the Neutral Zone which originally WAS orbiting
Sol, minding its own business for centuries.  People in the future tend to
leave space junk that old alone, the objects most popular as tourist sights
actually being protected with "landmark" status.  A sleeper ship such as
Khan's would certainly have been detected, but the cryonauts registered NO
life signs at all, so no one ever knew what was in this craft.  Eventually, a
starship pulled it into its envelope and carried it thousands of light years
out to the vicinity of the starbase Enterprise was visiting for Captain
Picard's conference with StarFleet authorities regarding the apparent loss of
stations near the Neutral Zone.
    This is also now considered the most probable explanation for the early
1990's Voyager 6 probe having reached a black hole capable of sending it to
the "machine" planet, as various research ships have made many voyages
directly from the Sol system to known black holes since warp drive was first
employed.  Its return to the Sol system as "V'ger" prompted some talk
off a system wide clean up of old hardware, but nothing ever came of it.

    The relativistic time dilation experienced at Tertiary threshold
velocity is such that time passes at 64.6% per cent, or roughly 2/3's,
the "normal" rate of objects "at rest".  This time dilation factor goes
along with the ship as the warp effect envelope separates from normal
space/time in crossing over the threshold, and remains stable
thereafter, so that all the time spent under way at superluminal
velocities is discounted by 1/3 for those on the vessel vs.  those
staying behind.  The effect is rather conveinient for starship crews, as
it effectively cuts by 1/3 the travel time between stop-overs, and since
all Tertiary warp vessels experience it, there is no disadvantage in
reaction time against opponents.
    There are social aspects to the cumulative effect of a lifetime career
devoted to star travel, in that one's age starts falling behind that of
friends, family, and above all spouse's left behind.  In the 2nd, 3rd, and
4th feature films, we see James Kirk wearing four bars and three dots on his
sleeve, indicating 23 years service in StarFleet.  Yet his birthday
depression in ST:TWOK and the presence of the fully grown David Marcus all
point towards a 50th birthday.  Assuming Kirk graduated the Academy at the
normal age of 22, adding 23 years leaves a 5 year gap.  The gap is simply the
cumulative effect of the time he's spent cruising at warp speed.  For married
personnel, this "age gaping" becomes a serious problem over a lifetime, and
was a major factor in StarFleet's decision to allow families to go along on
its latest vessels of the ST:TNG era.
    A very important aspect of this effect derives from the behavior of the
threshold cross-over phenomena in the presence of intense gravitational
fields, such as would be found near stellar bodies.  The intense warping of
space/time already imposed upon the region of the continuum nearest the star
causes it to become more tolerant of extreme profile skewing than normal
space.  As a nearby ship accelerates, the threshold velocity is reached, but
cross-over does not occur, one has to increase the degree of skew with still
more power.  This means going nearer to lightspeed while still in the normal
continuum, thus the time dilation factor increases.  Since the time dilation
at cross-over remains in effect throughout the period spent in the warp
continuum's sub-space, it is possible to retard one's own rate of time
passage to an arbitrarily high degree to assist in making extremely long
voyages.
    Some of the early Federation exploration ships, such as the famous USS
Horizon, used this sort of maneuver on occasion, but more often avoided it
due to the detrimental effect upon shipboard reaction time it causes.
Merchant vessels sometimes tried it, but the extreme danger of maneuvering so
close to a star led first to uninsurability and finally to outright
regulatory prohibitions against the practice.  Ships full of colonists almost
always housed them in sleeper chambers, an old and proven technology dating
as far back as the late 20th century, leaving only the crew awake.
    One of the greatest scientific discoveries made by the original
NCC-1701 Enterprise was that if a ship went EXTREMELY close to an object
of stellar mass while in the normal continuum, then poured on maximum
power to force its way to threshold before putting significant distance
between itself and the gravity field of the celestial body in question,
then the effective threshold velocity could actually be slightly above
lightspeed, and the associated time dilation not only extremely large
but NEGATIVE.  This is the essence of time travel under what has become
known as the breakaway maneuver.
    The class of phenomena known as "time travel" are extremely complex
and remain poorly understood.  Most recorded incidents have involved
multiple effects which, in the absence of a fully developed theory of
time, are often difficult to untangle for separate description and
analysis.  The Enterprise's unintentional journey to the Terra of the
late 1960's began with an accidental encounter with an uncharted black
hole.  The unusual properties of this particular hole had attracted
their attention, resulting in the Enterprise making a low warp speed
sensor pass.  The anomalous readings prevented them from realizing the
nature of this object until it was too late.  The hole's intense
distortion of the continuum pulled the Enterprise out of warp, where the
ship was in iminent danger of being sucked into the hole itself.
    On Kirk's orders, Sulu applied full emergency power in a desperate
attempt to fight their way back to threshold so as to to re-enter the
warp continuum, but even as the mighty starship trembled under the
effort, the threshold power level was moving higher and higher as they
neared the event horizon.  With seconds left before the end, Mr.  Scott
in engineering surmised the nature of their situation.  Knowing the ship
could never make the rising tertiary warp threshold in time, he engaged
the emergency circuit breakers to take the tertiary booster coils
offline, and diverted 100% of the reactor output into what was now a
lower threshold secondary warp field system.  The collapse of the
tertiary field into a secondary one "collided" with the rapidly growing
overall power level, kicking the ship into the warp continuum with such
explosive force that she briefly left sub-space itself on a kind of
"ballistic arc" OVER rather than thru the warp-space she would normally
traverse.
    It would take Spock many weeks of theoretical study and analysis
before he would devise a tentative explanation for their seemingly
miraculous appearance within the Terran atmosphere.  Ultimately, his
explanation for their movement thru space as well as time rested upon
two major points.
    First, time travel does not permit violation of the conservation of
mass law.  One cannot simply send 200,000 metric tons of starship back
in time to coexist with an "earlier" copy of the same 200,000 tons of
matter without in some way compensating for the effect such functional
duplication of mass will have on the overall gravitational process of
the cosmos.
    Second, in this particular incident the mode of compensation took
the form of an exchange or displacement of the 20th century matter that
would one day be the Enterprise and her crew, this material swapping out
of the normal plane of existance to reside in the hyper-continuum the
ship had traversed to reach its destination.  Therefore, in a manner
related to the phenomena of "symmetry breaking", the cosmos "selected"
as the ship's re-entry point a location determined by the whereabouts at
that time of the raw materials which would one day be the Enterprise and
her crew.
    As most of this material would be found on Terra in the 1960's, that
is where the ship materialized.  Fortunately, not quite all of the
material constituting the Enterprise was of Terrestrial origin, or the
ship would appeared at the center of the Earth instead of 5 miles above
its surface.  That it wasn't 5 miles below the surface instead was
simply good luck as to the total net effect of the mass-origin location
factors.  When the Enterprise returned to its proper place in time, the
older version of her material constituents resumed their proper place in
the continuum as well.
    Later studies of the "breakaway maneuver" and its associated
parameters revealed that had this early incident not involved such
extreme conditions, the time traveling starship would have remained
"linked" to the net gravitational influence of the star used as the
initiator mass.  This would have caused the celestial body itself to
assume the role of adjusting its own impact on the expansion of the
universe to compensate for sending a vessel back in time, and would
permit such voyages thru time while retaining the ability to target
spatial destinations as well.  This type of controled temporal
translation was successfully demonstrated by the Enterprise via Sol
during the mission Kirk's log describes as "Assignment:  Earth", and
was later employed from a captured Klingon cruiser to solve the
"Whalesinger" crisis.
    Given the operational parameters of starship reactor systems, the
time it takes to build up power applied to generating the warp field
effect normally requires an initiator mass the size of a star or greater
to perform the breakaway maneuver.  A planetary mass is just too small
under most circumstances as the vessel will have already moved too far
from the center of its gravitational field before attaining threshold
power where the time dilation effects are manifested.  This does not
mean it isn't possible to use a planetary mass as the initiator, only
that the ship in question would have to bring up its power output in an
incredibly rapid surge to do so.  The only known means of doing this is
the all but suicidal technique of deliberate implosion to "cold-start"
completely shut down power systems.  Only one ship, NCC-1701, is known
to have ever survived this procedure.  Historians remained baffled as to
why the crew dubbed the gambit an "Irishman's Chance".
    Were you to travel back in time without triggering some form of
gravitational impact compensation for your mass, the continuum would
soon destroy you via an effect strikingly similar to the manner in which
a living creature's immune system destroys that which does not belong.
The unfortunate time traveler would experience progressive
disintegration as the particles of his/her body are randomly pushed back
to their own correct time.
    An advanced form of such compensation was an integral part of the
Atavachron, which functioned by actually forcing open "portals" between
times.  As Kirk, Spock, and McCoy went through the portal but bypassed
the compensation stage, they were in grave danger and had but little
time to return.  Sarabeth could not return with them unless they could
have learned to use the machine to compensate for her entry into their
era, but alas there was no time for that before the star in that system
went nova.

    Just as the 3rd-order Cochrane function is known as Tertiary Warp, the
1st and 2nd orders represent Primary, and Secondary Warp.  Primary Warp is
the function consisting of the sum of the infinite series begining with X
plus (X^2)/2 plus (X^3)/6 etc.  As with the 3rd-order series, it may be
calculated with the equivalent formula (e^W)-1.  This was the first type of
warp field effect propulsion system developed, and it is still in use on
later vessels as the Impulse Drive sub-system.
    When Secondary Warp drive systems were developed, governed by the
2nd-order Cochrane function consisting of the sum of the infinite series
begining with X^2 + (X^3)/3 +(X^4)/12 etc., equivalent formula:
2*((e^W)-(W+1)), it was learned that they, and all higher order warp fields,
were dangerously unstable at low fractions of threshold power.  This forced
the retention of some form of Primary warp drive, though it need not handle
enough power to go superluminal.
    All warp field effects are created via the use of superconducting
Cochrane coils, which are wound according to the complex topological patterns
defined by Impulsor Calculus, the branch of mathematics developed by Zephram
Cochrane to express the new kinematics and mechanics resulting from his
successful unification of gravity with the electro-strong-weak force of
quantum physics.  As this essay is intended for a 20th century audience,
ethical constraints place severe limits on the range of comments that can be
made on this subject, but the inference should be obvious that if theoretical
physics has mastered the unification of the primal forces of nature, it
becomes possible to use a force easily generated and controled, such as
electromagnetism, to manipulate phenomena normally governed by another force,
such as gravity.
    Cochrane's mechanics superceded Einstein's, as his in its time
supplanted Newton's.  Each is "true" or at least acceptably valid,
within its range, and may be thought of as a special case approximation
of its successor, which is itself regarded as a superset of its
predecessor.  The following clues to Cochrane's accomplishment,
paraphrased from the preface to his own textbook, are deemed safe for
20th century humans.
    The first is that while current attempts to build ever larger particle
accelerators will lead to the unification of the strong nuclear force with
the electro-weak force, this approach will not be successful with gravity.
The reason is that while accelerators of sufficient power approximate the
fantastic extremes of temperature and pressure found during the era
immediately following the Big Bang, it was not these aspects of the early
universe but rather the extreme curvature of space-time then in force which
wedded gravity to the other forces.  As space-time expanded, or flattened,
gravity was the first force to de-couple from the others.
    The second clue is that while Newton's mechanics were based upon the
Euclidean model of geometry, and Einstein's was grounded in 19th century
alternatives such as that of Riemann, Cochrane found the mathematical tools
he needed to join the probability functions of quantum physics to the
structures defined by distortions of space-time in the "strange attractors"
of Fractal Geometry's framework for the study of "chaos".
    The warp effect itself derives from Cochrane's advanced concepts of
gravitation under which the interaction between the mass of a physical body
and the surrounding space/time matrix defines a complex mathematical field
known as a continuum profile.  On a purely theoretical level, Cochrane was
able to establish a new understanding of the term velocity by demonstrating
an intriguing difference in the continuum profiles of moving objects versus
those stationary relative to the observer.  All objects having mass distort
the space/time continuum around them, but when an object is in motion
relative to the observer, the pattern of the this distortion, known as the
continuum profile, becomes skewed along the direction of travel.
    Space/time is not infinitely malleable, it takes a minute but finite
interval for gravitational distortions to be fully manifested upon newly
encountered regions.  Because of this propagation-time factor, the region of
space/time in front of a moving object at a given instant is not as distorted
as it would be had the object in question been excerting its gravitational
influence on it for an arbitrarily long period, and the region behind the
traveling body shows excess distortion because of the time it takes to flatten
back to its undisturbed state.  The concept of relative motion remains in
force, for the skewing of the continuum profiles of all objects in the
universe is measured from the vantage point of the observer's own comparably
skewed line of travel.  In measuring the velocity relative to himself, the
observer is actually noting the degree of continuum profile skewing relative
to his own, and an inertial frame of reference becomes one with a constant
degree of skew.
    In astrophysics, this effect is largely muted by the ability of
space/time to "remember" repeated transits, so that all cyclic motions, such
as the orbits of planets, literally "groove" their paths into the very fabric
of the continuum, diminishing the skewing effect to almost vanishing levels.
Also, such circular motions involve the interaction of mutually influencing
bodies, so that each experiences far more change in the direction of its
skewing factor than in its absolute magnitude.
    But for non-cyclic motions, such as that of spacecraft executing huge
linear translations thru the continuum, the effect is sufficiently pronounced
to impact observations made from onboard instruments.  Generations after
Cochrane, the ability of the space/time continuum to store such
information-laden "memories" would be used by Dr.  Carol Marcus to establish
the theoretical basis for the long suspected existence of morphogenetic
fields, and would lead her to attempt the exploitation of this phenomena via
the "Genesis" technology.
    In creating his unified field theory, Cochrane opened the door to
full-scale interaction/exchange between the primal forces of nature.  Using
the analytic tools of his carefully derived Impulsor Calculus, he has able to
map out complex yet stable forms of interwoven electro-magnetic fields which
would cross "the line" by manifesting part of their effect in the form of
gravitational phenomena.  He was then able to follow the conceptual trail
back to the actual design and contruction of field generating coils that
could transform his theories into useful technology.  In his first great
practicle success, he proved that if his coil systems were used to
reconfigure the continuum profile of a "stationary" object so that it
acquired the relative "skew" of a moving one, it moved accordingly.
    This led first to the developement of the long wished-for "jetless" space
drive, ultimately called "Impulse drive", in which designers no longer needed
to bother about reaction mass carried onboard only to serve as kinetic
exhaust.  Later studies revealed that the application of sufficient power to
the skewing field would produce a degree of skew effect so highly pronounced
as to be insupportable by the familiar Einsteinian continuum.  Attaining this
"threshold" level would so stress the ordinary continuum that a vessel and its
surrounding field envelope would literally be ejected into a higher order
continuum in which the speed of light was no longer relevant as a limiting
factor.  Cochrane himself visualized our familiar continuum as "floating"
above the larger realm, and so described the transition process as "dropping
into sub-space" rather than apply an upward linguistic bias and the overused
"hyperspace".
    A gentle, private, and in some respects almost old-fashioned man,
Cochrane lived far enough into his twilight years to see his work send
humanity to the stars, before he mysteriously vanished.  Some say that the
warp-driven space yacht presented to him by the greatful governments of
several worlds disappeared at the same time, fueling speculation that he
headed into unknown space on some final adventure.  While historians argue
over his ultimate fate, none dispute the enormity of his contributions,
without which the very founding of the Federation could never have occurred.

    Just as the 20th century's mastery of undreamed of natural forces such as
electricity produces technological wonders inconceivable to 17th century
minds, so did Cochrane's breakthru set the stage for a vast family of related
discoveries and devices that seem almost magical to residents of our time.
In the decades following the construction of the first "impulsor drives",
further experimentation and theoretical studies led to totally different,
often unexpected, applications of the basic Cochrane coil system.  The coil
itself would become as basic a concept to an entire branch of technology as
the "circuit" is to the field of electrical engineering.
    Physicist Alicia Chalmbers interwove two coils, one wound clockwise, the
other anti-clockwise, and sent twin currents thru them in opposite
directions.  The "Chalmbers" coil did not move, as its external effect upon
the continuum was balanced between opposite and equal influences, but within
the dual-coil itself a profound disruption of space/time took place.  Wave
like patterns of variation in the "topological gradient" or distortability of
space/time, went out equally in all directions.  A second Chalmbers coil,
though unenergized, reacted to the distortion pattern by converting part of
its energy content back into electricity.
    Of course, modulations in the current flow to the first Chalmbers coil
were echoed analog fashion in the current output of the second "receiving"
coil, giving birth to sub-space radio.  The effect propagates at the
theoretical limit of the warp effect, Warp 10, the actual speed depending on
whether the Chalmbers coils are of the Primary, Secondary, or Tertiary
variety.  All StarFleet, and virtually all modern civilian vessels, use
Tertiary Chalmbers systems, allowing communications at 131792.79 x C.
Passive listening for natural occurences of this phenomena, and the active
use of a form of sub-space radio in "radar" mode, constitute much of the
sensor technology of Starships.
    Another variation of the basic Cochrane device bends the coil away from
its "barber-pole" configuration, to double back on itself full circle, in
effect coiling the coil in a single loop.  The result is an artificial
gravity field projected perpendicular to the plane of the loop, in either
pull or push mode depending on the orientation of the windings and/or the
direction of current flow.  Within its housing, the loop coil is physically
anchored or it would simply spin in a warp driven circle rather than impart
its effect to the gravity field.  Such units are always paired so that the
torque from each cancels the other rather than be imparted to the vessel via
the structural elements holding them in place.
    Other variants of the Cochrane coil take the form of conical shaped pairs
of coils nested within each other facing in opposite directions.  The conical
shape causes the warp field's skewing effect to be projected away from the
coil system rather than centered upon it.  By using the two coils in tandem,
one can induce any desired combination of push or pull force up to the
system's operational limits on a distant object, moving it arbitrarily close
to the starship's hull and holding it there.  Known as a tractor beam, this
piece of equipment is indispensable for modern spacecraft operations, without
it sleek warp-driven starships would be reduced to reliance upon primitive
manipulator arms such as the one found on the 1980's space shuttle.  When
holding a derelict vessel via tractor beam, it is possible to apply the the
repulsive force against selected portions of the outer hull, concentrating
the attractive force thru the vehicles' center, so as to not only retrieve
and stabilize it, but provide artificial gravity as well for the comfort of
boarding parties.
    In man's first experience with interstellar combat, the technological
level of the participants had the vessels of both sides drop into sub-light
speeds to maneuver against each other in a tight volume of laser crossfired
space.  These primitive battles were analogous to the way in which late 20th
century fighter planes would reach a combat zone via supersonic travel, then
go subsonic for the actual dogfight.  The advantages of a weapon that could
unleash its effect at warp speed were so obvious that an all out technology
race to build such a device began even before the first Romulan War was over.
    Ultra high velocity missiles carrying powerful matter/anti-matter
warheads were already in use.  As the M/A anhilation produces a shower of
photons in the extremely high energy gamma ray portion of the spectrum, these
missiles were dubbed photon mines.  Though their highly developed fusion
thrusters could accelerate them at hundreds of G's, they were still so slow
compared to even the sublight capabilties of impulse driven starships that
one had to use them in the manner of depth charges, simply deploying them in
the expected path of the enemy ship and hoping for the best.  Attempts to
replace the fusion thruster with a warp engine enjoyed some success against
vehicles moving at sublight speeds, but against vessels traveling at warp
speeds what was needed was a weapon that could travel substantially faster
than any ship.
    The answer was ultimately inspired by the ancient submarine torpedo,
which used steam power pumped into the torpedo by the submarine rather
generated onboard the weapon itself.  The modern analog of the torpedo tube
emerged as an inside out warp engine coil which generated its field within
its own interior and imparted an enormous skewing effect on any object placed
inside.  The specially designed warhead pod would zip out of the tube at
extremely high warp speeds, having an unprecendented degree of skew, but free
of the mass of any onboard warp field generating equipment.  Though the
warhead pod is designed to retain its imparted skew as long as possible, it
does begin to decay immediately after leaving the tube.  As this takes at
least several minutes, the effective range is quite adequate for the tactical
role these weapons play.  Note that these devices have almost no steering,
only a slight course correction capability, and so must be carefully aimed.
The parallel to ancient submarine weapons was so close that the term "photon
torpedo" became permanently attached to these deadly implements of celestial
combat, which in the ST:TNG era are capable of as much as 10 to 15 minutes
travel at speeds approaching warp 9.9.

    Early experiments with Dilithium crystals found that two such crystals, a
mirror, a semi-reflector, and a light source made a marvelously efficient
laser, as Spock once demonstrated in escaping from Gestapo headquarters on
the planet Ekos.  When Science Officer Bruno Wilhelm placed a dilithium laser
setup inside a Chalmbers coil, the crystals synchronized so as to overlap the
coinciding lightwaves exactly out of "phase" making the light energy
effectively vanish from our continuum, only to reimerge as a uni-directional
highly intense disruption of the space/time continuum now known as the
"phaser effect".  Such synchronization of the crystals required a
super-luminal transfer of coordinating influences, and so was only possible
in the context of a coil-induced sub-space environment.  Within the coil, one
can reasonably construe the laser as being "in" sub-space.
    When fully powered, the phaser effect travels at the Warp 10 limit for
the type of Chalmbers coil used, be it Primary, Secondary, or Tertiary.
Naturally, StarFleet vessels are armed only with Tertiary phasers, anything
less would produce a "beam" literally too slow to catch a Tertiary warp
starship with Dilithium focused anti-matter reactors.
    However, hand phasers don't have access to quite enough power to energize
the coil component to its equivalent threshold power level.  The result is
that the phaser beam produced travels at a speed dependent upon the power
level applied to the coil.  Whereas a beam emitted from a coil at threshold
power would always move at Warp 10, with additional coil power just boosting
the intensity or striking power of the beam, at just below threshold power
the beam's speed is the reciprocal of Warp 10.  This is a mere 7.58766 x
10^(-6) x C, or approximately 7300 feet per second from a Tertiary coil,
therefore hand phasers use Primary coils so that the phaser effect
propagatation velocity is proportional to the reciprocal of the Primary warp
field's Warp 10 limit of 22025.47 x C.  The reciprocal value is therefore
approxiamately 8.45 miles per second.  At still less coil power, the speed
diminishes in direct proportion to the fraction of threshold power applied to
the coil.  Operational maximums for ST:TNG hand phasers take their coils to
about 1/3 of threshold velocity, so that the weapons full power effect moves
at roughly 2.82 miles per second.
    One can vary the proportion of coil vs.  initiating light energy levels
only so far without overloading the hand phaser, causing burnout or even
detonation.  Thus to moderate the phaser effect down to stun levels, the beam
in some models of hand phaser may travel as slowly as 200 or 300 feet per
second.  We've seen this effect quite clearly when Kirk once fired his phaser
set for stun at the metabolically accelerated Deela of Scalos, who simply
stepped out of the way.  Hand phaser on stun is definitely a close quarters
only weapon, where speed is not significant.
    Unlike sub-space "radio", which simply attenuates under an inverse square
law, phaser beams have a much shorter range due their propensity to "decay"
by losing their energy to the creation of visible spectrum photons all along
their path of travel.  This is what the observer sees, not the phaser beam
itself.  The actual phaser effect is one of micro-range random space/time
fluctuations in the topological gradient of the space encountered, imparting
warp impulses to the atoms encountered.  The effect tends to spread and
propagate thru solid matter, so that material objects are likely to
distribute the effect throughout their particularly shaped volumes.
    At high power, the effect is so severe that all molecular bonds are
snapped, and all of the particles are "impulsed" in random directions.  What
had been a solid object becomes an expanding cloud of particles moving fast
enough to penetrate other solid matter to an enormous extent.  A body so
"disintegrated" on a ship would pass right thru the hull like a burst of
gamma rays, but because the particles are traveling via impulse rather than
momentum, their behavior apes that of neutrinos in that they do almost no
damage to the matter they pass thru.
    Lower power simply streches the molecular bonds without breaking them,
their rebounding motions translating into simple heat.  In this manner, a
hand phaser may be used to heat rocks for warmth, cook food, or even act as a
very precise cutting torch.  At the lowest useful power, the jolting of
molecules is too slight to really impact inanimate matter, but does tend to
produce neurological shock as large numbers of synapses have their firing
threshold randomly raised or lowered.  The vast number of additional versus
inhibited synaptic firings causes a biological equivalent of "systems crash"
leading to unconsciousness, as the nervous system becomes hopelessly confused
and overloaded by spurious signals.  As no actual tissue damage is sustained,
the nervous system "reboots" itself eventually.  Somewhat higher power can do
permanent, even lethal damage to the nervous system however, and can cause a
seizure-like muscular convulsion.  This minimally lethal effect is not unlike
electric shock.

    To residents of the 20th century, the transporter is perhaps a more
incredible application of Cochrane's Unified Field Theory than superluminal
travel, since the later affords no real Terrestrial gauge for appreciating
the effect, whereas the wonder of instantaneously materializing elsewhere has
been part and parcel of Earth's mythology/magic belief systems for millenia.
    Building on the ability of the "looped coil" to project gravitational
fields, experimenters eventually learned to handle gravity waves in ways that
parallel optical technology's capabilities with light waves.  Ultimately,
command of these techniques was sufficient to produce a gravitational wave
"hologram" in which the system literally captured the continuum profile of an
object down to the minutest detail of atomic constituents and molecular
bondings in the intersection between its stationary "reference beam" and the
rotating "scanning beam".  Sophisticated split beam techiniques permitted the
"projection" of a second "continuum profile image", which, depending on the
operational limits of the equipment, could be located at an arbitrarily
large distance and direction from the source.  These experiments were
originally conceived in pursuit of improved medical technology following the
progression of X-rays, ultrasound, nuclear magnetic resonance, and positron
emmision tomography, with the result enabling Dr. Crusher to obtain a clear
view of the parasite creature embedded in Admiral Quinn during the
"Conspiracy" period.
    The transporter breakthru grew out of experiments attempting to
manipulate matter via alterations of the continuum profile associated
with an object.  If a continuum profile projection were maintained long
enough, it began to fill itself in with atoms picked up from the
environment.  Eventually, it would recreate the original, though in the
meantime, if sufficient power was used to intensify the projection, this
profile construct could behave like the original, even appearing to be
solid matter, as long it remained within range of the projection radius.
At the same time, it was shown that changes in the profile of the
original were reflected in the original object as well in the
projection, establishing the real-time linkage between the two.  Early
attempts at matter manipulation were usually destructive, not until the
early 24th century would the raw computer power be available for such
things as the holodeck, where the projection could be based on computer
simulations rather than real life / real time models, but in these
pioneering efforts, the ability to project a profile back on its own
source object, while maintaining an independent second projection
elsewhere, was developed.
    Dr.  Janet Hester of the Deneva Research Station first conceived the idea
that if one reversed the "topological polarity" of the image projected back
upon the source, in effect FLATTENING the impression it made in space/time,
while simultaneously boosting the gravitational intensity, and thus the DEPTH
of the spatially projected image, one could create a situation in which the
probability of finding any given constituent of the source object at the
original location could be reduced to zero, even as the probability of
finding it at the projection's location went up to unity.  Every component of
an object, its atoms, the chemical bonds between them, even the ongoing
molecular processes, would cascade back and forth between the twin loci of
probable locations, finally coming to rest at the one brought to unity.  Of
all the marvels that have sprung from Zephram Cochrane's insights, none more
clearly demonstrate his success at unifying gravitational space/time
continuum phenomena with quantum mechanical probability functions.
    It would take another four decades of dedicated experiment and study
before Science Officer Winston of the USS Moscow became the first human to
transport across to the USS Tehran.  Still more work was required before the
ability of the tranporter to project a "virtual" yet functional copy of the
active components of the scanning and projection processes to envelope the
retrieval site would eliminate the need for physical hardware at both ends of
the transport linkage, and theb to learn to bend the projection around the
surfaces of planets using the natural gravitational field so that transport
could be free of line-of-sight restraints.  The depth of dense planetary
matter the transporter can penetrate is still limited, but the often
life-saving speed and conveinience of transport in general has proved well
worth the time, cost, and often sacrifice it took to perfect.

    The Secondary Warp field effect was originally achieved by winding a
second-stage "booster" coil around a specially designed Primary coil.  The
early versions of this system would energize the Primary coil first to
navigate at low percentages of threshold power.  Once clear of stellar and
planetary gravitational fields, they would engage the booster coil
reconfiguring their warp field into the 2nd order type.  When this was
accomplished, power would be steadily increased until the threshold level was
attained and transition to the warp continuum occured.  The Primary and the
booster together constitute the Secondary coil.  Should the booster fail
under operational stress, a fairly common occurence in the early days, the
Primary alone could be used and could operate above its threshold level to
take the ship to superluminal velocities.
    While later vessels retained the above system layout, experience proved
it far more efficient to energize the whole Secondary coil system as a single
circuit, and to navigate at very low power and speeds with an independent
miniature Primary system.  This became known as the Impulse Drive.  As it was
intended only for low speed operations, this system would not normally be
capable of handling the power load it would require to bring the vessel past
the threshold point.  However, engineers took advantage of this dual
propulsion system to split the vessel itself, letting each major sub-division
of the hull house one of the systems.  It became customary to place the major
living quarters in the hull with the smaller Impulse Drive, both to better
shield the crew from the higher radiation levels the more powerful Secondary
system created, and also with the idea of better accomodating the entire crew
should "coil burnout" force the abandonment of the other hull.
    The terminology of vessel design adopted the convention of referring to
the hull housing the Secondary coil system as the Secondary Hull, and
the other housing the Primary coil only Impulse Drive as the Primary Hull.
Tertiary drive systems simply wound yet another type of booster coil around
the Primary and Secondary stages nested inside it, but as there were still
only two drive systems and two main hull sections, the one with the large
engine system continued to be called the Secondary Hull.
    In the event of separation, the Primary Hull's Impulse Drive, freed of
the weight of the entire Secondary Hull and the even more massive main drive
engine nacelles, is usually large enough for superluminal propulsion.  This
has been shown quite clearly in ST:TNG during the initial encounter with Q,
when the Primary Hull found its way to Farpoint after the entire ship spent
some 10 minutes pushing itself to its operational limits while going in
exactly the opposite direction.  It is equally well implied by Geordi's
instructions to Engineer Logan to take the Primary Hull to a Starbase if
unable to re-establish contact with him after performing the saucer-sep
manuevar in the "Arsenal of Freedom" incident.
    The first three orders of warp field phenomena correspond to the first
three "generations" of warp drive technology in the "Spaceflight Chronology".
Logically, a "Fourth generation" designation should have waited for the
developement of Quarternary warp, the sum of X^4 + (X^5)/5 + (X^6)/30 etc.,
equivalent formula 24*((e^W)-((W^3)/6 + (W^2)/12 + W + 1)), but the impact of
Dilithium on power generation, and thus overall performance, was so great
that the "Fourth generation" label took hold for the Constitution class.  All
orders of warp field phenomena remain subject to the Warp 10 limit on
Delivered Power, but higher order warps produce greater velocity for the same
Delivered Power than lower orders.  (See Appendix for tables of Primary,
Secondary, and Quartenary Warp Factor Equivalent Velocities).
    The term "Fifth generation" is usually applied to the abortive attempt to
harness "Trans-Warp", a misbegotten application of the Interphase phenomena
first observed by the Enterprise NCC-1701 during the "Tholian Web" incident.
The abandonment of this dangerous system was made doubly disappointing by the
continued failure of Federation science to perfect a workable Quartenary warp
drive.  The seemingly insurmountable difficulties encountered in the early
attempts at Quarternary drive design were the prime reason for the costly
"Trans-Warp" interlude.
    However, in the intervening decades advanced theoretical studies have led
to vastly simpler, more reliable Tertiary drives which can be pushed, and
above all held, far closer to the Warp 10 limit of Delivered Power than the
original design multi-stage units.  These single stage "integrated" units
were first used in ship's of the NCC-1701-C's Ambassador class, and marked
the arrival of warp technology's "Sixth generation".  A highly refined and
advanced version of this type of drive serves as the main propulsion for
"Galaxy" class starships such as Enterprise NCC-1701-D.  Gone are the
inefficientcies of the nested, three coil approach, advances in Impulsor
Calculus theory and supercomputer simulation techniques having found a single
coil equivalent.
    As the early efforts at Quarternary warp floundered on the complexities
of a four level multi-stage approach, the success of the single stage
"integrated" approach for Tertiary warp has scientists of ST:TNG's era once
more confident of eventual success, and aggressively paced research programs
are again under way in the race for the Quartenary drive.  It should be noted
that the extra heavy warp nacelle mountings and overall structural strength
rating of the Galaxy class design should easily permit retrofitting of
Quarternary Warp engines when they become available.
    Montgomery Scott correctly predicted the crippling deficientcies of the
Trans-Warp system, but was unable to dissuade StarFleet from investing in it.
Rightly convinced that Quartenary warp would have to await improvements in
warp theory permitting "integrated" designs, he attempted to convince
StarFleet to allow him to challenge the Warp 10 Barrier itself.  Alas, Scott
was never able to secure StarFleet backing for his proposal, and only a
handful of ST:TNG era technical persons who've studied his original notes
even know what he had in mind.
    Realizing that the "SuperWarp" scheme was far too radical for his era,
Scott dedicated his leisure time engineering studies to the design of the
ship he felt StarFleet should build in place of more "Excelsior" class
vessels.  Yet this project also offered too many radical advances, as Scott
was allowing for upgrades to integrated Tertiary or even Quartenary main
drives in his huge dreamship.  But while the Galaxy class would ultimately be
larger and incorporate advances beyond his wildest imaginings, even a cursory
glance at Scott's old plans and drawings reveals the striking similarities
that mark the true lineage of these greatest of all StarShips.  NCC-1701-D's
operational status is the way Scott would most have wanted StarFleet
Engineering to acknowledge its continuing debt to its greatest practitioner.
    As for the mechanics of SuperWarp, the mathematically inclined are invited
to contemplate the significance of the other half of the hyperbola relating
Generated to Delivered power, which most Federation scientists dismiss as a
mere geometric curiousity.  Of course, scientists once thought that C itself
represented an impassable barrier, yet as Spock would say, "There are always
possibilities".
    Without giving too much away, I can offer the following clue, that the
Constitution class USS Enterprise NCC-1701 under James Kirk, once broke
through the Warp Barrier by accident, the result of her Captain's famous
propensity for taking desperate gambles in otherwise hopeless situations.
Students of warp physics correctly identifying the occassion are eligible to
win a scholarship to StarFleet academy, which, alas, may not be used until the
23rd century.

                                        -Leon Myerson; 72157,3432; 6/23/88


                         APPENDIX 1 - PRIMARY WARP


                    Generated     Delivered         Primary
                    Power             Power         Warp x C

                           1        1.00000            1.72
                           2        1.98354            6.27
                           3        2.96260           18.35
                           4        3.93509           50.17
                           5        4.89755          132.96
                           6        5.84370          344.05
                           7        6.76140          862.85
                           8        7.62571         2049.24
                           9        8.38615         4384.92
                          10        8.96633         7833.82


                    Theoretical Limit = 22025.47 x C

                    Threshold Velocity = 0.5814 x C

                    Time Dilation at threshold = 0.813205




                         APPENDIX 2 - SECONDARY WARP


                    Generated     Delivered       Secondary
                    Power             Power       Warp x C

                           1        1.00000            1.44
                           2        1.98354            8.57
                           3        2.96260           30.77
                           4        3.93509           92.46
                           5        4.89755          256.13
                           6        5.84370          676.42
                           7        6.76140         1712.18
                           8        7.62571         4083.24
                           9        8.38615         8753.06
                          10        8.96633        15649.70


                    Theoretical Limit = 44030.93 x C

                    Threshold Velocity = 0.6944 x C

                    Time Dilation at threshold = 0.717939




                         APPENDIX 3 - QUARTERNARY WARP



                    Delivered      Generated      Quarternary
                      Power          Power               Warp

                              1   1.0000000000           1.24
                              2   2.0167653720          25.34
                              3   3.0383208502         170.05
                              4   4.0670614879         742.36
                              5   5.1072983806        2617.92
                              6   6.1676537197        8218.29
                              7   7.2682459514       24167.20
                            7.5   7.8487197368       40826.52
                              8   8.4694304149       68510.99
                            8.2   8.7364919027       84149.66
                            8.4   9.0203187626      103286.47
                            8.6   9.3280961537      126697.69
                            8.8   9.6717993420      155331.49
                              9  10.0729838055      190346.01
                            9.1  10.3071067812      210676.62
                            9.2  10.5747605008      233155.87
                            9.3  10.8903152831      258009.95
                            9.4  11.2777216596      285488.88
                            9.5  11.7800905867      315868.94
                            9.6  12.4836439773      349455.49
                            9.7  13.5895662949      386586.00
                            9.8  15.7014109302      427633.43
                            9.9  21.8369448362      473009.97
                             10    INFINITE         523171.18



                    Theoretical Limit = 523171.18 x C

                    Threshold Velocity = 0.8065 x C

                    Time Dilation at threshold = 0.590200





    For comparison, here is a chart of Quarternary Warp Factor equivalent
velocities keyed on the older "Generated Power" scale.


                    Generated     Delivered     Quarternary
                    Power             Power      Warp x C

                           1        1.00000            1.24
                           2        1.98354           24.41
                           3        2.96260          159.92
                           4        3.93509          680.00
                           5        4.89755         2315.80
                           6        5.84370         6908.99
                           7        6.76140        18761.08
                           8        7.62571        46527.25
                           9        8.38615       101833.70
                          10        8.96633       183948.24
                          11        9.33067       266146.24
                          12        9.53548       327403.32
                          13        9.65322       368752.42
                          14        9.72615       396927.10
                          15        9.77477       416884.29
                          16        9.80915       431599.84
                          17        9.83463       442835.76
                          18        9.85421       451667.92
                          19        9.86971       458779.77
                          20        9.88225       464622.34
                          21        9.89262       469503.75
                          22        9.94445       494688.02