
           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 This essay may be periodically superceded by revised versions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 way 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 the5.84o 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 d sues 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-1inguchurg 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 SF: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 = 0.71793









                         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

