Calibration of the IDE Clock and the Determination of LDEF Attitude Using Sun Sensors </HEAD> <BODY> <H1>Calibration of the IDE Clock and the Determination of LDEF Attitude Using Sun Sensors </H1> <H3><A HREF = "../../teamvita/jpovita.htm">John P. Oliver</A>, William J. Cooke</H3> <address>Institute for Space Science and Technology 1810 NW 6th Street Gainesville, FL 32601</address> <H3>J. Derral Mulholland</H3> <address>POD Associates 2309 Renard S.E., Suite 201 Albuquerque, NM 87106</address> <H3>and</H3> <H3>Philip C. Kassel</H3> <address>NASA Langley Research Center Hampton, VA 23665</address> <H2>Introduction</H2> The LDEF Interplanetary Dust Experiment (IDE) was designed to provide information on the near-Earth particle environment, making possible a detailed study of the various particle contributors, such as ß meteoroids, interplanetary dust, and orbital debris. Placed in orbit by the space shuttle Challenger on 7 April, 1984 and finally recovered by Columbia in January 1990, LDEF was a 12-sided, passive, gravity-gradient stabilized satellite some 30 feet long and 14 feet in diameter. The IDE consisted of 6 trays of detectors, mounted on mutually orthogonal faces: East (Ram, or leading edge), West (Wake, or trailing edge), North, South, Earth, and Space. 60% of the detectors on each tray had a dielectric thickness of 0.4 ľm, sensitive to hypervelocity particles greater than 0.2 ľm in diameter at typical encounter speeds. The remaining detectors had a dielectric thickness of 1.0 ľm and were sensitive to particles with diameters greater than 0.5 ľm. The upper size limit for detection is determined by the energy required for an impacting particle to physically break the detector substrate. Estimates place the size of such a particle at about 100 ľm. In addition to the detectors, a sun sensor was mounted on each tray to provide illumination information.<P> As the LDEF was a passive satellite, there was no telemetry available; each experiment was required to provide its own power and data storage. In the case of the IDE, this was accomplished by the use of an Experiment Power and Data System (EPDS), which also provided a "clock" signal as a time reference. Data were recorded onto magnetic tape at 146 minute intervals, the stored information including the time obtained from the EPDS clock, the tray, and the detector type for each impact, plus "housekeeping" items such as the last time of sunrise, the illumination status of the trays at the data dump time, and the electrical status of the detectors. The duration of data collection was expected to be nine months; however, the catastrophic loss of the space shuttle Challenger greatly delayed LDEF retrieval. The IDE tape continued to accept data until it was full, after 346 days in orbit. During this time, more than 15,000 impacts were recorded on the IDE's 459 detectors . After LDEF's return to Earth in 1990, the data were extracted from the magnetic tape and placed on MS-DOS floppy disks to expedite analysis by the team investigators.<P> <H2>Calibration of the IDE Clock</H2> A knowledge of LDEF's position and velocity and the orientation of the IDE tray at the time of impact is crucial to understanding the origin of the particle. LDEF orbital elements were made available by NORAD at approximately weekly intervals during the time of IDE data collection. Use of these data enables the calculation of LDEF position and velocity for any given instant. It is therefore necessary to relate the recorded time generated by the EPDS clock to the terrestrial standard, Universal Time (UT), used in the LDEF orbit calculations. The establishment of such a calibration equation was the primary motivation for the inclusion of the sun sensors on the IDE trays, which worked in conjunction with the EPDS to record the times of sunrise (in units of the EPDS clock) as seen by the LDEF satellite. These sunrise times can then be related to the UT times of LDEF sunrise computed via orbital mechanics.<P> The illumination information from the sun sensors was interpreted by the experiment electronics as follows:<P> <OL> <LI>A warning flag was initiated at any time that all 6 sun sensors went dark. <LI>If all sun sensors recorded dark for at least 10 minutes, a trigger was set. <LI>At the next indication of light by any sun sensor, a "clock" was started at 0. <LI>The value of this "clock" was recorded at each data dump onto the tape. Note that, as the data dumps were made at 146 minute intervals, it was possible for the IDE not to record a given sunrise, as the LDEF orbital period was just over 94 minutes. </OL> After LDEF retrieval, the sun sensor information was extracted and placed in a disk file. A few sample lines are given in Table 1. The second column consists of data dump times in units of the IDE clock (which is derived from the EPDS clock), the third consists of observed sunrise times in the same units, and the last columns give the illumination status of the sun sensors on the IDE trays at the time of the data dump ('I'=Illuminated, 'O'=Dark). The (13) in the second and third column headers are reminders that the IDE was designed for a time resolution of 13.1 seconds, the length of time between "ticks" of the IDE clock. Note the sunrise occurs at intervals of approximately 431 IDE clock ticks. These sunrises should be separated in time by one LDEF orbital period, 94.1 minutes. Dividing the orbital period by 431 yields 13.1 seconds between ticks, in agreement with the design value.<P> <IMG SRC = "TAB1.GIF"></IMG><P> A FORTRAN program was written to calculate the UT times of LDEF sunrise, using the orbital elements provided by NORAD and the SGP4 numerical propagator. As a check, the results obtained from this program were compared to values obtained from a second program, which used a less-sophisticated analytical integrator; both sets of calculations agreed to within 10 seconds. Simplifying assumptions were made in the calculation process to make the problem tractable:<P> <OL> <LI>The oblate shape of the Earth was taken into account, but the variations due to local topography at the sunrise location, such as mountains, were not considered. <LI>A first-order correction for atmospheric refraction was made, but, lacking meteorological data, variations in the sunrise times caused by cloud cover were ignored. </OL> These two phenomena introduce several seconds of quasi-random noise into the clock calibration analysis.<P> The next step in the clock calibration process was to determine the UT time of the first LDEF sunrise. Information concerning the time of LDEF deploy and activation obtained from various NASA sources established the first sunrise at 15:45:00 UT on April 7, 1984. A second computer program was then constructed to match the UT times of sunrise with the corresponding IDE times. The UT times of sunrise were then plotted against their IDE counterparts to help determine the functional form of the calibration equation. The EPDS produced a fairly stable time signal, the designers anticipating a drift of about 1.5 hours over the mission duration, so the relationship between the IDE clock and UT should be very nearly linear. This expected linearity was confirmed in the final step of the clock calibration.<P> As the last step in the establishment of the clock calibration equation, a least squares adjustment was performed on the data. This adjustment allowed for errors in both the UT and IDE times of sunrise. Two models were used:<P> <IMG SRC = "EQ1.GIF"></IMG><P> where <IMG SRC = "ALPHA.GIF"></IMG>, <IMG SRC = "BETA.GIF"></IMG>, and <IMG SRC = "GAMMA.GIF"></IMG> are parameters to be determined. <IMG SRC = "ALPHA.GIF"></IMG> corresponds to the UT time of IDE activation and <IMG SRC = "BETA.GIF"></IMG> is the time interval between successive ticks of the IDE clock. The second-order fit resulted in an insignificant <IMG SRC = "GAMMA.GIF"></IMG> parameter, so the linear model was adopted as the calibration equation. It is important to note that the calculated times of orbital sunrise depend on the value assumed for the "deviation" angle, which incorporates the effects of atmospheric refraction and local topography at the place of sunrise. Having only rough estimates of the magnitudes of these effects, adjustments were performed using several data sets constructed for different values of the deviation angle and the "goodness of fit" values (obtained from the sum of the squares of the residuals) for the adjustments were compared. It was found that a deviation angle of 0ş.22 yielded the best fit.<P> The values determined for <IMG SRC = "ALPHA.GIF"></IMG> and <IMG SRC = "BETA.GIF"></IMG> are:<P> <IMG SRC = "ALPHA.GIF"></IMG> = 1984 April 7 14:15:25 ą 0.3 seconds Coordinated Universal Time (UTC).<P> <IMG SRC = "BETA.GIF"></IMG> = 13.1016401 ą 0.0000002 seconds/IDE tick.<P> These values supersede those given in reference 1, which were based on the incorrect assumption that experiment initiation coincided with spacecraft release. The residuals from this fit are plotted in figure 1. They indicate no significant drift of the IDE clock over the 346 days of data collection.<P> <IMG SRC = "FIG1.GIF"></IMG><P> <I>Figure 1: Residuals from Linear Calibration Model</I><P> <H2>Clock Temperature Effects</H2> <IMG SRC = "CLKCALF2.GIF"></IMG><P> <I>Figure 2: Long term (10 day smoothing) trends in residuals from Linear Calibration Model</I><P> From an engineering standpoint, it may be interesting to examine the behavior of the IDE clock a bit more closely. The long-term trends in the clock error are shown in figure 2 where the result of smoothing the residuals with a 99 point "running average" is indicated by a solid line. Clock deviations represent the integral of clock period errors. The first derivative of the long-term errors (seen in figure 3 after further smoothing) thus represents variations in clock period from the mean. This curve should be compared with the plot (also in figure 3) of the observed IDE EPDSdata acquisition system temperature. The similarity is striking. A significant portion of the systematic variation can thus be attributed to thermal variations in the clock frequency. The typical slope of the clock error versus temperature is about 0.1 ppm/ degree C. The residuals for the entire data set were smoothed and rounded to integer IDE ticks. The result, shown in table 2, are corrections which may be used to adjust observed IDE times for the deviations of the clock from linearity. Note that the corrections never exceed ą 1 IDE tick.<P> <IMG SRC = "CLKCAL34.GIF"></IMG><P> <I>Figure 3: EPDS temperature and clock deviation first derivative as a function of time.</I><P> <IMG SRC = "TAB2.GIF"></IMG><P> <H2>Determination of the LDEF Attitude</H2> As mentioned previously, the LDEF satellite was gravity-gradient stabilized. It was also equipped with a viscous magnetic damper to gradually cancel vibrations. In principle, this would enable experiments on board the spacecraft to maintain a fixed orientation with respect to LDEF's orbital motion. The six IDE trays were placed so as to have their normals facing perpendicular, opposite, and parallel to the motion of LDEF. Nominally, the East tray should face in the same direction as the LDEF velocity vector (the roll axis), West in the anti-velocity direction, Space radially outward from the Earth (the yaw axis), the Earth tray towards the geocenter, North in a direction defined by the cross product of the LDEF velocity and position vectors (the pitch axis), and South opposite North. Figure 4 illustrates the orientation of the three LDEF principal axes. The information provided by the illumination status of the sun sensors at the data dump times, combined with geometrical considerations, has allowed some insight to be gained into the LDEF attitude as a function of time.<P> <IMG SRC = "FIG5.GIF"></IMG><P> <I>Figure 4: Orientation of the LDEF principal axes</I><P> The yaw determination was made by considering the situation in which the sun sensors mounted on the North and South trays were dark, with at least two other sensors illuminated. This requires that the Sun be located somewhere in the plane defined by the East and Space tray normals. The angle between the solar direction and the LDEF velocity vector, measured in the plane defined by the roll and pitch axes, gives the yaw of LDEF at the time of the data dump. This occurs eight times during the 346 days of data collection, twice near sunrise and six times near sunset. The data for these occurrences are given in Table 3. A polar plot of the solar azimuths (also the azimuths of the East or West tray normals) is given in figure 5. 0° is in the direction of the LDEF velocity and 90° is in the direction of the LDEF pitch axis. It is obvious that LDEF is yawed by about 8° to the south. Indeed, a simple average of the yaw angles yields a value of -8°.2 ą 2°.8, or to use the terminology often encountered in the LDEF literature, row 10 is rotated by 8°.2 into the RAM direction. This is in good agreement with the 8°.1 value obtained by the atomic oxygen experiment A0114 . However, a plot of the yaw magnitude as a function of time clearly shows that the value of the yaw was not constant, and that LDEF underwent oscillations of about 4° around the -8° mean value. Figure 6 depicts the magnitude of the yaw as a function of time. Unfortunately, the paucity of the data points does not permit a determination of the period of the yaw oscillation. It is certain that it is less than the 32 days indicated by the second and third data points, and is probably less than a day, if one is to believe computer models of LDEF stability. Pre-flight computer modeling of LDEF stability done by the General Electric Space Division indicated that LDEF should exhibit a peak yaw rate of about 0.002 °/sec, which is consistent with our results.<P> <IMG SRC = "TAB3.GIF"></IMG><P> <IMG SRC = "FIG6.GIF"></IMG><P> <I>Figure 5: Solar Azimuths</I><P> <IMG SRC = "FIG7.GIF"></IMG><P> <I>Figure 6: Magnitude of LDEF Yaw</I><P> <H2>Acknowledgment</H2> We express our thanks to the other members of the IDE Experiment Team (S. F. Singer, J. J. Wortman, J. L. Weinberg) for their role in making these data possible. This study was supported by the National Aeronautics and Space Administration through grant NAS 1-1812 (J. D. Mulholland, Principal Investigator).<P> <H2>References</H2> <OL> <LI>Singer, S.F. et. al, "First Spatio-Temporal Results from the LDEF Interplanetary Dust Experiment", Adv. Space Res, 11, No. 12, pp. (12)15-(12)122, 1991. <LI>Jeffreys, W. H., "On the Method of Least Squares", The Astronomical Journal, 85, No. 2, pp. 177-181, 1980. <LI>Stein, B., LDEF Spaceflight Environmental Effects Newsletter, III, No. 1, p. 4, March 30, 1992. <LI>Das, A., Foulke, H., and Siegel S., "Final Report - Passive Stabilization of the Long Duration Exposure Facility (LDEF)", NASA CR-132566, GE Document No. 74SD4264, Nov. 1974. </OL> This document produced by <address>Dr. Bill Cooke; bjc@ufl.edu</address>