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 This essay may be periodically superceded by revised versions uploaded to Data Library 2 of CompuServe's Science Fiction Forum.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