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 formula relating the Warp number W to velocity in terms of C is not the ridiculous 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. 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 Speeds 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. 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. 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 catching an opponent. 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 drive system developed, and it is still in use on later vessels as the Impulse Drive 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. These 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. 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 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 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. 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. 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. The mathematically inclined are invited to contemplate the significance of the other half of the hyperbola, which most Federation scientists dismiss as a mere geometric curiousity. Of course, scientists once thought that C itself represented an impassable barrier. As Spock would say, "There are always possibilities". 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. An interesting aspect of the 3rd-order Cochrane function is that Warp 1 is not C but 1.31xC. 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 0.763xC. 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 1.31xC. 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 in 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. -Leon Myerson; 72157,3432; 3/18/88