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The value of r cp (the B coefficient) was taken from myJRASC article (Earl 2005) concerning the determination of therange of a satellite using a trigonometric parallax technique.This value can also be calculated from the Web, where equationsfor the radius of the Earth as a function of latitude can be found.The D coefficient was found by determining the value for ω pfrom the measured streak length and exposure time, as describedin Equation 2 and Equation 11, and using a Gravitational Constant(G) of 6.6742x10 -20 km 3 /kg.s 2 and an Earth mass of 5.9742x10 24kg.For this example, the three solutions to Equation 10 usingthe coefficients above were:1) 597 km;2) -6305 km; and3) -659 km.Solutions 2 and 3 are physically impossible, since theyimply that the satellite orbited beneath the Earth’s surface.Solution 1 indicates that the satellite orbits above the Earth’ssurface at a height of about 597 km, which is definitely possiblefor a LEO satellite.Additional MeasurementsUsing this method on all the satellites I captured on that night,I tabulated the results (Table 1) in order of ascending orbitheight to study the error behaviour with respect to decreasingstreak length (increasing orbit height).ConclusionsThe technique I have outlined is indeed capable of determiningthe orbit altitude of the low-Earth-orbiting satellites. Nevertheless,there are several error characteristics that suggest a more indepthinvestigation would prove fruitful. Examination of Table1 shows the shorter streak lengths are correlated with largererrors in the calculated satellite altitude. The source of thisbehaviour may be partly due to the derivation of the image scale.Angular separations of 50 pixels and larger were used to solvethe image scale and so Equation 11 might be biased againstsmaller angular separations and therefore smaller streak lengths.Re-determining the image-scale equation using smaller angularseparations between calibration stars, using a larger numberof stars, using a CCD with a higher resolution, and/or increasingthe exposure time for the lower apparent-angular-velocitysatellites might improve the results.No satellite that orbits the Earth has a perfectly circularorbit, but LEO satellites have smaller orbit eccentricities thantheir high-altitude counterparts. In summary, higher orbitaltitudes means a larger range of possible orbital eccentricities.The small five-second exposure time I used ensured that theheight of the satellites would not change appreciably duringthe exposure time. While the height might be determinedrelatively accurately, the assumption of a circular orbit couldresult in a larger error in the determination of the orbit period.The zenith angle of the satellite might also impose anoticeable effect. Those satellites that appear at the far edgesof the image might have a noticeably larger height error thanthose nearer the centre (zenith). In the 11.2 × 11.2 degree fieldID STREAK STREAK D COEFFICIENT HEIGHT TRUE PERIOD TRUELENGTH LENGTH (GM / ωp 2 ) (km) HEIGHT (min) PERIOD(pixels) (radians) (km) (min)12465 177.912900 0.013677 -2.130869E+09 555 547 95.57 95.7625527 169.434353 0.013024 -2.349776E+09 582 583 96.01 96.2013771 164.878743 0.012673 -2.481590E+09 597 585 96.39 95.9527840 123.328829 0.009477 -4.437512E+09 788 788 100.34 100.1124968 120.503112 0.009260 -4.648144E+09 805 784 100.76 100.4011111 113.216607 0.008700 -5.265847E+09 854 796 101.78 100.4527433 105.304321 0.008092 -6.086907E+09 914 871 103.09 102.1106154 94.810337 0.007286 -7.508461E+09 1009 953 105.07 104.5325963 60.415230 0.004646 -1.846902E+10 1529 1418 116.40 114.0825162 58.830264 0.004524 -1.947516E+10 1567 1471 117.21 115.2309063 56.859948 0.004373 -2.084474E+10 1616 1499 118.30 115.1725746 25.553865 0.001973 -1.024068E+11 3261 3013 156.71 148.48Table 1: The calculated orbit heights of 12 LEO satellites imaged in Ottawa, Ontario on February 11, 2006, using an exposure time of 5 seconds.Satellites were identified and orbit heights extracted using the satellite orbit propagator from Software Bisque’s TheSky (level IV, version 5); the trueperiod was obtained by taking the reciprocal of the mean-motion value of the satellites’ orbit elements.202JRASC October / octobre 2006
of view of the images, satellites can be detected as far as sixdegrees from the zenith at the image corners.Although increasing the exposure time to larger than fiveseconds might help to reduce the height-determination error,it might cause imaging difficulties. In order to determine theangular velocity of the satellite, both endpoints of the streakhave to be present. If the exposure time is greater than fiveseconds, many of the lowest orbit satellites would be able topass outside the FOV during the exposure time, thereby causinga streak with only one (or no) endpoints. Background noiseincreases with larger exposure times, such that the signal-tonoiseratio of the satellite streaks would be reduced, since thesatellites are moving across the image with the same signal,while the background noise accumulates on every pixel of theimage.When all is said and done, this method is very accuratefor the lower-orbiting LEO satellites (200-km to 800-km orbitheights), and this accuracy can be increased for the higherorbitingLEO satellites (800-km to 4000-km orbit heights) whenrequired, as long as the potential pitfalls are appreciatedbeforehand.REFERENCESAKiTi.ca Cubic Equation Solver - www.akiti.ca/Quad3Deg.htmlEarl, M. 2005 JRASC: Vol. 99, No. 2, p. 54Gupta, R. ed. 2005 Observer’s Handbook 2006, The RoyalAstronomical Society of Canada (University of TorontoPress: Toronto)Space Track: The Source for Space Surveillance Datawww.space-track.org/perl/login.pl (AuthorizedAccount Required)SOFTWARE“SatSort” Version 2: Satellite TLE Sorting Software: Mike Earl“TheSky” Level IV, Version 5: Astronomy Software: SoftwareBisque (www.bisque.com)“CCDSof ” Version 5: CCD Camera Control and Image AnalysisSoftware: Software Bisque (www.bisque.com)Michael A. Earl has been an avid amateur astronomer for over 30years, and served the Ottawa RASC Centre as both its Meeting Chairand its Vice President. He is the “Artificial Satellites” Coordinatorand the Webmaster for the Ottawa Centre’s Web site(http://ottawa.rasc.ca). He constructed the CanadianAutomated Small Telescope for Orbital Research (CASTOR): the veryfirst remotely controlled and automated optical satellite trackingfacility in Canada. He is currently working on a second-generationCASTOR system geared toward the advanced amateur and professionalastronomy communities. You are welcome to visit his new CASTORoptical satellite tracking Web site at www.castor2.ca.October / octobre 2006 JRASC203
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