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22 Years of AJISAI spin period determination from standard SLR and kHz SLR data
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Advances in Space Research 44 (2009) 621–626 www.elsevier.com/locate/asr
22 Years of AJISAI spin period determination from standard SLR and kHz SLR data D. Kucharski a,*, G. Kirchner a,1, T. Otsubo b,2, F. Koidl a,3 a
Space Research Institute of the Austrian Academy of Sciences, Lustbuehelstrasse 46, A-8042 Graz, Austria b Hitotsubashi University, 2-1 Naka, Kunitachi 186-8601, Japan Received 3 March 2009; received in revised form 7 May 2009; accepted 9 May 2009
Abstract Satellite Laser Ranging (SLR) is a powerful and efficient technique to measure spin parameters of satellites equipped with corner cube reflectors. We obtained spin period determination of the satellite AJISAI from SLR data only: 17246 pass-by-pass estimates from standard 1–15 Hz SLR data (14/Aug/1986–30/Dec/2008) and 1444 pass-by-pass estimates (9/Oct/2003–30/Dec/2008) from data of the first 2 kHz SLR system from Graz, Austria. A continuous history of the slowing down of AJISAI spin is derived from frequency analysis, and corrected for the apparent effects. The apparent corrections, elaborated here, allowed very accurate determination of AJISAI initial spin period: 1.4855 ± 0.0007 [s]. The paper identifies also non-gravitational effects as a source of the periodical changes in the rate of slowing down of the satellite. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Satellite Laser Ranging; AJISAI; Satellite spin; kHz SLR
1. Introduction The Japanese Experimental Geodetic Satellite (AJISAI, Fig. 1) was launched by National Space Development Agency (NASDA), currently reorganized as Japan Aerospace Exploration Agency (JAXA), on 12/Aug/1986 (year 1986.6137). The mission objective is accurate position determination of fiducial points on the Japanese Islands. This fully passive satellite is equipped with 1436 corner cube reflectors (CCRs) for SLR, arranged in form of 15 rings around the symmetry axis. AJISAI is also equipped with 318 mirrors used for satellite direction determination (Sasaki and Hashimoto, 1987). The mirrors are also used
for photometric measurements of AJISAI spin parameters (spin period, spin axis orientation). The satellite is placed on a quasi-circular orbit with an height of 1490 km (inclination 50°) thus it is an easy target for the SLR stations. The spin parameters of AJISAI were an object of several scientific investigations (Otsubo et al., 1998, 2000; Kirchner et al., 2007). The knowledge of spin period and spin axis orientation of the passive geodetic satellites allows for investigation and improvement of physical models of the perturbing forces which are of magnetic, gravitational and non-gravitational nature (Bertotti and Iess, 1991; Andre´s et al., 2004). The improved models used for precise orbit determination (POD) can increase the accuracy of determination of SLR station positions and of geodynamical parameters.
*
Corresponding author. Tel.: +43 316 873 4653; fax: +43 316 873 4656. E-mail addresses: [email protected] (D. Kucharski), georg.kirchner @oeaw.ac.at (G. Kirchner), [email protected] (T. Otsubo), [email protected] (F. Koidl). 1 Tel.: +43 316 873 4651; fax: +43 316 873 4656. 2 Tel./fax: +81 42 580 8939. 3 Tel.: +43 316 873 4654; fax: +43 316 873 4656.
2. Satellite spin measurement techniques Spin parameters of the passive satellites can be investigated with two techniques: photometry and SLR. The photometry determines epoch times of flashes caused by
0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.05.007
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et al., 2007). Table 1 shows the main differences between the previous Graz 10 Hz and the new Graz 2 kHz SLR system. The new system is able to detect return pulses with single or multiple photons, resulting in up to >1 million measurements per pass of AJISAI, even with its low energy per laser shot (400 lJ). 3. Apparent spin effects and corrections During a satellite pass over a SLR station, the incident angles (longitude, latitude) of the laser beam in the satellite body centered coordinate system (SBC, not body-fixed) are changing. This affects the frequency signal contained in the observation, thus the frequency analysis of the SLR data gives only an apparent spin of the satellite (Kirchner et al., 2007, 2009; Otsubo et al., 1998). The change of the incident angles is not constant for every pass and depends on actual geometry of the pass (station – satellite mutual orientation). 3.1. Apparent longitude (AL) effect
Fig. 1. AJISAI (courtesy of JAXA).
sunlight, and reflected by the outer surfaces of the CCRs, or by the dedicated mirrors of AJISAI. This technique works only at night, with the satellite illuminated by the sun. The second technique measures distance to the satellites with laser. The laser pulses transmitted from the SLR station are reflected by the CCRs back to the receiver telescope of the SLR system. The surface of the spinning satellite is scanned by these laser pulses, and the motion of the CCRs engraves a frequency signal on the SLR data. This signal can be obtained by spectral analysis (Lomb, 1976) of the unequally spaced range residuals data (measured minus predicted satellite range) (Otsubo et al., 2000; Bianco et al., 2001). SLR can measure spin periods of satellites during day and night, regardless to the sun – satellite – station geometry (as it is with photometry), and without any additional equipment. With the Graz kHz SLR system it is possible to measure spin parameters from LEO (Low Earth Orbit), like Gravity Probe-B (Kirchner et al., 2009) to HEO (High Earth Orbit), like the ETALON satellites (Kucharski et al., 2008), from slow spinning objects like the LAGEOS family (Kucharski et al., 2007, 2009) to fast spinning like AJISAI (Kirchner Table 1 Key parameters of Graz SLR system. Graz laser system
10 Hz laser before 2003/10/9
2 kHz laser after 2003/10/9
Wavelength Repetition rate Energy/pulse Pulse width
532 nm 10 Hz 30 mJ 35 ps
532 nm 2 kHz 400 lJ 10 ps
During the pass a satellite is changing its attitude as seen from the laser station, thus the laser beam incident angle (in the SBC) is also changing. The change in longitude of this incident angle slightly shifts the frequency signal of the CCRs to higher or lower values, depending on the geometry of the pass. The total pass value of this shift is dfAL = dL/(360° t) [Hz], where dL [°] is the total pass change in longitude of the incident laser beam (as measured in the SBC) and t [s] is the duration of the pass. The value of dfAL is usually lower than 1 mHz. 3.2. Apparent phase (AP) effect During the pass also the latitude of the incident angle in the SBC is changing, and different CCR rings are involved. Since the neighboring rings are shifted in phase (Fig. 2), the peaks of the corresponding frequency analysis show an offset of dfAP in the resulting spectrum. Because dfAP depends on a relative phase between the CCR rings it can have positive or negative sign. While dfAL depends only on the geometry of the pass, dfAP depends also on a technical property (shift between the rings) of the retro reflector array (RRA) of the satellite. The value of dfAP is mostly within a range of ±2 mHz. 3.3. Corrections of the apparent effects The spin frequency fapparent derived from the spectral analysis of the SLR range residuals can be individually corrected for the two apparent effects: fcorrected = fapparent dfAL dfAP. In order to calculate the corrections we used simulations of the SLR measurements (Kucharski et al., 2007, 2008, 2009). For the set of epochs ti registered during the real observation the software calculates the position of the SLR station and the satellite in the inertial
D. Kucharski et al. / Advances in Space Research 44 (2009) 621–626
Fig. 2. Retro reflector array of AJISAI, black squares represent positions of the CCR holders. The phase shift angle u between two neighboring central rings is marked. The gray curve shows how the angle of incidence travels during a single pass (Graz kHz SLR pass made on 20/Oct/2003 00:39, duration 17 min).
reference frame. The spin axis of AJISAI in the simulations is orientated parallel to the spin axis of the earth. Since the orientation of AJISAI is stabilized by the nutation damper (Sasaki and Hashimoto, 1987), we assume that there is no change of this parameter. We also know AJISAI is spinning in a clockwise direction, opposite to the earth rotation (Kirchner et al., 2007). For a single pass, for every registered epoch ti of the range measurements, we calculate the incident angles (in the SBC) of the laser beam (loni, lati). We use these information to calculate dfAL and dfAP separately with simplified simulations. In this virtual setup the laser beam coincides with the X axis of the SBC system (cartesian coord. system, right-hand rule). In order to calculate dfAL for a given SLR pass, the program simulates range measurements (with respect to the face front of the satellite) to the CCRs (tilted not more than 30° from the laser beam) of the satellite spinning with frequency fsim_sat. For every epoch ti the orientation of the satellite is changing in longitude to give the longitude coordinate of the laser incident angle of loni. The simulated range measurements are processed by frequency analysis. The resulting spin rate fsim_app is influenced by the change of the laser incident longitude angle only. The correction for this effect is expressed as: dfAL = fsim_app fsim_sat. In order to calculate dfAP the same processing is applied, however during the simulation only the latitude of the laser incident angle is changing. This correction is expressed as dfAP = fsim_app fsim_sat. Since the apparent corrections do not depend on satellite spin rate, we kept the satellite self-spin rate fsim_sat constant during these simulations.
4. Spin period determination In order to determine spin period of AJISAI we used all available (17246 passes, 14/Aug/1986–30/Dec/2008) SLR
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Fig. 3. AJISAI, spectrum of the Graz 2 kHz SLR observation: 26/July/ 2007 17:59 UTC. The peak = 1.47824 Hz.
passes (Full Rate data) delivered by all of the SLR stations to the International Laser Ranging Service (ILRS) Global Data Centers (Pearlman et al., 2002) and all available 2 kHz SLR measurements from Graz station (1444 passes, 9/Oct/2003–30/Dec/2008). Out of the full set of the SLR passes we have selected only those with clear spectral signal – with power of the frequency signal above 10 for Hz data and 100 for kHz data. Almost 100% of kHz and about 30% of Hz observations passed this selection. Every selected pass was first processed to calculate the range residuals, in order to calculate predicted satellite range we used IRV and CPF predictions. As the next step
Fig. 4. Spin period data of: photometry (black points) and SLR (gray points); top: not-corrected SLR, middle: SLR corrected for AL effect, bottom: SLR corrected for the two apparent effects, reducing significantly RMS of SLR data; linear trends of the data-series are plotted.
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the frequency analysis was applied to the residuals giving the spectrum (Fig. 3). In order to find the frequency peak f120 generated by 120° symmetry of the AJISAI RRA (Otsubo et al., 1999) we were investigating a spectral range from 0 to 2.5 Hz with 20 lHz step. Because of the 120° signal, the true frequency of the body is 3 times smaller: fapparent = f120/3. The resulting spin period of AJISAI (corrected for the apparent effects) is: T = 1/fcorrected. In order to verify these corrections we compared the SLR spin values with the photometric measurements (24 values) made by Koganei Station of Communications Research Laboratory (CRL) (Otsubo et al., 1998). Fig. 4 presents spin period series measured by SLR and photometry in 1997–1998, where each point corresponds to the apparent and corrected spin period derived from a single pass. To investigate the agreement between the SLR and photometry spin period results we have approximated the photometry data with a linear trend function and calculated residuals of SLR values to this function. Note that the apparent longitude correction had been applied to the photometry data set. For the not-corrected residuals (Fig. 4-top) we get: meannot-corr = 1.340 ms, RMSnot-corr = 2.10 ms; for the fully corrected residuals (Fig. 4-bottom): meancorr = 0.211 ms, RMScorr = 1.22 ms. Applying the dfAL corrections shifts the SLR spin periods to the photometry trend (Fig. 4-middle); correcting this data by dfAP reduces the RMS by more than 40% (Fig. 4-bottom). Fig. 5 presents result of AJISAI spin period trend: 14/ Aug/1986–30/Dec/2008. The data has been approximated by an exponential function T = 1.488586 exp(0.0149802 Y) [s], where Y is time in years since launch. The resulting spin period data set is divided into time slots of 0.5 years. The spin period values within these slots are approximated with a linear function, and spin period residuals to this function are obtained. The RMS of these residuals for every time slot is shown in Fig. 6.
Fig. 5. Full history (14/Aug/1986–30/Dec/2008) of AJISAI spin period derived from Hz SLR and Graz kHz SLR data. The exponential trend function (gray curve) is plotted.
Fig. 6. RMS of spin period residuals (corrected for the apparent effects) plotted with 0.5 year step for Hz SLR (gray points) and 2 kHz SLR (black points). The exponential trend functions are plotted. The precision is improved with kHz SLR data.
With a slowing down of the satellite spin, the RMS of spin period residuals is increasing with an exponential trend. This is due to the exponential decrease of the number of full satellite rotations per pass with time, causing poorer frequency signals within SLR data. 4.1. Initial spin period The complete history of AJISAI spin period allows very accurate determination of the initial spin period value T0. Since T0 can be used as a constant in many physical models of the magnetic, gravitational and non-gravitational forces perturbing the satellite motion (Andre´s et al., 2004), its accurate knowledge is important. The slowing down of the satellite does not follow purely exponential trend. Thus an exponential approximation of the full data set (22 years – Fig. 5) may lead to significant differences between the mathematical function and measured data, especially on the beginning and on the end of the data set. In order to
Fig. 7. Result of the frequency analysis (plotted in period domain) of the spin period residuals (top) and of the shadow data (bottom). The significant peaks are marked.
D. Kucharski et al. / Advances in Space Research 44 (2009) 621–626
determine T0 we used the spin period data measured during the first year after launch. This data set was approximated by 2nd degree polynomial function, giving the initial spin period T0 = 1.4855 s, RMS = 0.7 ms. T0 calculated here is 3 ms shorter than the value given by the full history exponential trend function (Fig. 5). 5. Periodical changes in the spin slow down trend We have calculated and spectrally analyzed the spin period residuals (spin period value – trend function) of the complete AJISAI spin history. The resulting spectrum, after conversion to the period domain, is plotted on Fig. 7-top. The spin parameters of the LEO (Low Earth Orbit) satellites are strongly influenced by non-gravitational effects connected with the umbra. The spin period of a satellite is increasing faster when the satellite is illuminated by the sun, and slower when the body is in the shadow of the earth (umbra). For the 22 years of AJISAI, we have calculated the percentage of the last orbital cycle (116 min) being in the umbra, from a given epoch with 10 s step. For these 22 years we have calculated almost 141000 ‘‘umbra” points with 5000 s step. The obtained full data set was spectrally analyzed – the result is presented on Fig. 7-bottom, showing the clear coincidence; because of huge data set the peaks here have very big power. In addition, Fig. 7-top shows also a small AJISAI weekend effect (peak close to 7 days). The periods: B (44.3 d), C (71.2 d) and E (117 d) indicate changes in the rate of slowing down caused by non-gravitational effects (Yarkovsky effect; Rubincam, 1988; Sengoku et al., 1995). 6. Conclusions In this paper, we determined the history of AJISAI spin period, which was possible only with continuously available SLR data. We successfully used 17246 passes (14/ Aug/1986–30/Dec/2008) from 1 to 15 Hz SLR stations, and 1444 passes (9/Oct/2003–30/Dec/2008) from the first 2 kHz SLR system at Graz. The corrections developed here allow elimination of the apparent effects from the spectral analysis, increasing the accuracy of the spin period determination to better than 0.1% (RMS/T). The full spin period data set was approximated with an exponential trend function; however the slowing down of the satellite does not exactly follow this purely mathematical trend. The deviation can be seen e.g. in paragraph 5 when the obtained initial spin period T0 is 3 ms shorter than the value given by the full history trend function. KHz SLR data allows to increase the accuracy of such an analysis, however, even more important here is that we could detect clear frequency signals in almost 100% of good, continuous passes, while only about 30% of Hz passes gave a clear frequency signal; because the 2 kHz SLR observations contain about 200 times more measurements per second than standard Hz SLR, the obtainable
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frequency signal is about 200 times stronger, and almost always clearly visible. Frequency analysis of the spin period residuals shows that the rate of slowing down of AJISAI undergoes periodical changes. The period A (Fig. 7-top) of 6.99 days is an artificial signal introduced by a ‘‘weekend effect” of the SLR network; the number of measured passes is lower during the weekends. The periods B, C and E are caused by non-gravitational effects (Yarkovsky effect), we have no clear explanation for the D period (88.4 d), however this effect might be related to the temperature distribution over the satellite body (Yarkovsky-Schach, not analyzed spectrally here) which depends on the B, C, E periods. The signals detected here may contain valuable information about the non-gravitational effects acting on AJISAI and will be the subject of future studies. The accurate spin period determination gives information about forces acting on the spacecraft and causing perturbation of the orbital motion. The full history (more than 22 years) of the spin period of AJISAI can be used to upgrade the models of all those forces. The accurate spin period model can help to predict the spin parameters of AJISAI for the future application to the next-generation laser time transfer via AJISAI mirrors (Kunimori et al., 1992). References Andre´s, J.I., Noomen, R., Bianco, G., et al. Spin axis behavior of the LAGEOS satellites. J. Geophys. Res. 109 (B6), B06403, doi:10.1029/ 2003JB00269, 2004. Bertotti, B., Iess, L. The rotation of LAGEOS. J. Geophys. Res. 96 (B2), 2431–2440, 1991. Bianco, G., Chersich, M., Devoti, R., et al. Measurement of LAGEOS-2 rotation by satellite laser ranging observations. Geophys. Res. Lett. 28 (10), 2113–2116, 2001. Kirchner, G., Hausleitner, W., Cristea, E. Ajisai spin parameter determination using Graz kilohertz satellite laser ranging data. IEEE Trans. Geosci. Remote Sens. 45 (1), 201–205, 2007. Kirchner, G., Kucharski, D., Cristea, E. Gravity Probe-B: new methods to determine spin parameters from kHz SLR data. IEEE Trans. Geosci. Remote Sens., doi:10.1109/ TGRS.2006.00675, 2009. Kucharski, D., Kirchner, G., Schillak, S., et al. Spin determination of LAGEOS-1 from kHz laser observations. Adv. Space Res. 39 (10), 1576–1581, doi:10.1016/j.asr.2007.02.045, 2007. Kucharski, D., Kirchner, G., Cristea, E. ETALON spin period determination from kHz SLR data. Adv. Space Res. 42 (8), 1424–1428, doi:10.1016/j.asr.2007.08.030, 2008. Kucharski, D., Kirchner, G., Koidl, F., et al. 10 Years of LAGEOS-1 and 15 years of LAGEOS-2 spin period determination from SLR data. Adv. Space Res., doi:10.1016/j.asr.2009.01.019, 2009. Kunimori, H., Takahashi, F., Itabe, T., et al. Laser ranging application to time transfer using geodetic satellite and to other Japanese space programs, in: Proceedings of the 8th International Workshop on Laser Ranging Instrumentation, Annapolis, NASA Conf. Pub. 3214, pp. 138–1-42, 1992. Lomb, N.R. Least-squares frequency analysis of unequally spaced data. Astrophys. Space Sci. 39, 447–462, 1976. Otsubo, T., Amagai, J., Kunimori, H. Measuring AJISAI’s spin motion, in: Proceedings of the 11th International Workshop on Laser Ranging, Deggendorf, Germany, pp. 674–677, 1998.
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Otsubo, T., Amagai, J., Kunimori, H. The center of mass correction of the geodetic satellite AJISAI for single-photon laser ranging. IEEE Trans. Geosci. Remote Sens. 37 (4), 1999. Otsubo, T., Amagai, J., Kunimori, H., et al. Spin motion of the AJISAI satellite derived from spectral analysis of laser ranging data. IEEE Trans. Geosci. Remote Sens. 38 (3), 1417–1424, 2000. Pearlman, M.R., Degnan, J.J., Bosworth, J.M. The international laser ranging service. Adv. Space Res. 30 (2), 135–143, doi:10.1016/S02731177(02)00277-6, 2002.
Rubincam, D.P. Yarkovsky thermal drag on LAGEOS. J. Geophys. Res. 93 (B11), 13805–13810, 1988. Sasaki, M., Hashimoto, H. Launch and observation program of the experimental geodetic satellite of Japan. IEEE Trans. Geosci. Remote Sens. GE-25 (5), 526–533, 1987. Sengoku, A., Cheng, M., Schutz, B.E. Anisotropic reflection effect on satellite Ajisai. J. Geod. 70, 140–145, 1995.
Advances in Space Research 44 (2009) 621–626 www.elsevier.com/locate/asr
22 Years of AJISAI spin period determination from standard SLR and kHz SLR data D. Kucharski a,*, G. Kirchner a,1, T. Otsubo b,2, F. Koidl a,3 a
Space Research Institute of the Austrian Academy of Sciences, Lustbuehelstrasse 46, A-8042 Graz, Austria b Hitotsubashi University, 2-1 Naka, Kunitachi 186-8601, Japan Received 3 March 2009; received in revised form 7 May 2009; accepted 9 May 2009
Abstract Satellite Laser Ranging (SLR) is a powerful and efficient technique to measure spin parameters of satellites equipped with corner cube reflectors. We obtained spin period determination of the satellite AJISAI from SLR data only: 17246 pass-by-pass estimates from standard 1–15 Hz SLR data (14/Aug/1986–30/Dec/2008) and 1444 pass-by-pass estimates (9/Oct/2003–30/Dec/2008) from data of the first 2 kHz SLR system from Graz, Austria. A continuous history of the slowing down of AJISAI spin is derived from frequency analysis, and corrected for the apparent effects. The apparent corrections, elaborated here, allowed very accurate determination of AJISAI initial spin period: 1.4855 ± 0.0007 [s]. The paper identifies also non-gravitational effects as a source of the periodical changes in the rate of slowing down of the satellite. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Satellite Laser Ranging; AJISAI; Satellite spin; kHz SLR
1. Introduction The Japanese Experimental Geodetic Satellite (AJISAI, Fig. 1) was launched by National Space Development Agency (NASDA), currently reorganized as Japan Aerospace Exploration Agency (JAXA), on 12/Aug/1986 (year 1986.6137). The mission objective is accurate position determination of fiducial points on the Japanese Islands. This fully passive satellite is equipped with 1436 corner cube reflectors (CCRs) for SLR, arranged in form of 15 rings around the symmetry axis. AJISAI is also equipped with 318 mirrors used for satellite direction determination (Sasaki and Hashimoto, 1987). The mirrors are also used
for photometric measurements of AJISAI spin parameters (spin period, spin axis orientation). The satellite is placed on a quasi-circular orbit with an height of 1490 km (inclination 50°) thus it is an easy target for the SLR stations. The spin parameters of AJISAI were an object of several scientific investigations (Otsubo et al., 1998, 2000; Kirchner et al., 2007). The knowledge of spin period and spin axis orientation of the passive geodetic satellites allows for investigation and improvement of physical models of the perturbing forces which are of magnetic, gravitational and non-gravitational nature (Bertotti and Iess, 1991; Andre´s et al., 2004). The improved models used for precise orbit determination (POD) can increase the accuracy of determination of SLR station positions and of geodynamical parameters.
*
Corresponding author. Tel.: +43 316 873 4653; fax: +43 316 873 4656. E-mail addresses: [email protected] (D. Kucharski), georg.kirchner @oeaw.ac.at (G. Kirchner), [email protected] (T. Otsubo), [email protected] (F. Koidl). 1 Tel.: +43 316 873 4651; fax: +43 316 873 4656. 2 Tel./fax: +81 42 580 8939. 3 Tel.: +43 316 873 4654; fax: +43 316 873 4656.
2. Satellite spin measurement techniques Spin parameters of the passive satellites can be investigated with two techniques: photometry and SLR. The photometry determines epoch times of flashes caused by
0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.05.007
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D. Kucharski et al. / Advances in Space Research 44 (2009) 621–626
et al., 2007). Table 1 shows the main differences between the previous Graz 10 Hz and the new Graz 2 kHz SLR system. The new system is able to detect return pulses with single or multiple photons, resulting in up to >1 million measurements per pass of AJISAI, even with its low energy per laser shot (400 lJ). 3. Apparent spin effects and corrections During a satellite pass over a SLR station, the incident angles (longitude, latitude) of the laser beam in the satellite body centered coordinate system (SBC, not body-fixed) are changing. This affects the frequency signal contained in the observation, thus the frequency analysis of the SLR data gives only an apparent spin of the satellite (Kirchner et al., 2007, 2009; Otsubo et al., 1998). The change of the incident angles is not constant for every pass and depends on actual geometry of the pass (station – satellite mutual orientation). 3.1. Apparent longitude (AL) effect
Fig. 1. AJISAI (courtesy of JAXA).
sunlight, and reflected by the outer surfaces of the CCRs, or by the dedicated mirrors of AJISAI. This technique works only at night, with the satellite illuminated by the sun. The second technique measures distance to the satellites with laser. The laser pulses transmitted from the SLR station are reflected by the CCRs back to the receiver telescope of the SLR system. The surface of the spinning satellite is scanned by these laser pulses, and the motion of the CCRs engraves a frequency signal on the SLR data. This signal can be obtained by spectral analysis (Lomb, 1976) of the unequally spaced range residuals data (measured minus predicted satellite range) (Otsubo et al., 2000; Bianco et al., 2001). SLR can measure spin periods of satellites during day and night, regardless to the sun – satellite – station geometry (as it is with photometry), and without any additional equipment. With the Graz kHz SLR system it is possible to measure spin parameters from LEO (Low Earth Orbit), like Gravity Probe-B (Kirchner et al., 2009) to HEO (High Earth Orbit), like the ETALON satellites (Kucharski et al., 2008), from slow spinning objects like the LAGEOS family (Kucharski et al., 2007, 2009) to fast spinning like AJISAI (Kirchner Table 1 Key parameters of Graz SLR system. Graz laser system
10 Hz laser before 2003/10/9
2 kHz laser after 2003/10/9
Wavelength Repetition rate Energy/pulse Pulse width
532 nm 10 Hz 30 mJ 35 ps
532 nm 2 kHz 400 lJ 10 ps
During the pass a satellite is changing its attitude as seen from the laser station, thus the laser beam incident angle (in the SBC) is also changing. The change in longitude of this incident angle slightly shifts the frequency signal of the CCRs to higher or lower values, depending on the geometry of the pass. The total pass value of this shift is dfAL = dL/(360° t) [Hz], where dL [°] is the total pass change in longitude of the incident laser beam (as measured in the SBC) and t [s] is the duration of the pass. The value of dfAL is usually lower than 1 mHz. 3.2. Apparent phase (AP) effect During the pass also the latitude of the incident angle in the SBC is changing, and different CCR rings are involved. Since the neighboring rings are shifted in phase (Fig. 2), the peaks of the corresponding frequency analysis show an offset of dfAP in the resulting spectrum. Because dfAP depends on a relative phase between the CCR rings it can have positive or negative sign. While dfAL depends only on the geometry of the pass, dfAP depends also on a technical property (shift between the rings) of the retro reflector array (RRA) of the satellite. The value of dfAP is mostly within a range of ±2 mHz. 3.3. Corrections of the apparent effects The spin frequency fapparent derived from the spectral analysis of the SLR range residuals can be individually corrected for the two apparent effects: fcorrected = fapparent dfAL dfAP. In order to calculate the corrections we used simulations of the SLR measurements (Kucharski et al., 2007, 2008, 2009). For the set of epochs ti registered during the real observation the software calculates the position of the SLR station and the satellite in the inertial
D. Kucharski et al. / Advances in Space Research 44 (2009) 621–626
Fig. 2. Retro reflector array of AJISAI, black squares represent positions of the CCR holders. The phase shift angle u between two neighboring central rings is marked. The gray curve shows how the angle of incidence travels during a single pass (Graz kHz SLR pass made on 20/Oct/2003 00:39, duration 17 min).
reference frame. The spin axis of AJISAI in the simulations is orientated parallel to the spin axis of the earth. Since the orientation of AJISAI is stabilized by the nutation damper (Sasaki and Hashimoto, 1987), we assume that there is no change of this parameter. We also know AJISAI is spinning in a clockwise direction, opposite to the earth rotation (Kirchner et al., 2007). For a single pass, for every registered epoch ti of the range measurements, we calculate the incident angles (in the SBC) of the laser beam (loni, lati). We use these information to calculate dfAL and dfAP separately with simplified simulations. In this virtual setup the laser beam coincides with the X axis of the SBC system (cartesian coord. system, right-hand rule). In order to calculate dfAL for a given SLR pass, the program simulates range measurements (with respect to the face front of the satellite) to the CCRs (tilted not more than 30° from the laser beam) of the satellite spinning with frequency fsim_sat. For every epoch ti the orientation of the satellite is changing in longitude to give the longitude coordinate of the laser incident angle of loni. The simulated range measurements are processed by frequency analysis. The resulting spin rate fsim_app is influenced by the change of the laser incident longitude angle only. The correction for this effect is expressed as: dfAL = fsim_app fsim_sat. In order to calculate dfAP the same processing is applied, however during the simulation only the latitude of the laser incident angle is changing. This correction is expressed as dfAP = fsim_app fsim_sat. Since the apparent corrections do not depend on satellite spin rate, we kept the satellite self-spin rate fsim_sat constant during these simulations.
4. Spin period determination In order to determine spin period of AJISAI we used all available (17246 passes, 14/Aug/1986–30/Dec/2008) SLR
623
Fig. 3. AJISAI, spectrum of the Graz 2 kHz SLR observation: 26/July/ 2007 17:59 UTC. The peak = 1.47824 Hz.
passes (Full Rate data) delivered by all of the SLR stations to the International Laser Ranging Service (ILRS) Global Data Centers (Pearlman et al., 2002) and all available 2 kHz SLR measurements from Graz station (1444 passes, 9/Oct/2003–30/Dec/2008). Out of the full set of the SLR passes we have selected only those with clear spectral signal – with power of the frequency signal above 10 for Hz data and 100 for kHz data. Almost 100% of kHz and about 30% of Hz observations passed this selection. Every selected pass was first processed to calculate the range residuals, in order to calculate predicted satellite range we used IRV and CPF predictions. As the next step
Fig. 4. Spin period data of: photometry (black points) and SLR (gray points); top: not-corrected SLR, middle: SLR corrected for AL effect, bottom: SLR corrected for the two apparent effects, reducing significantly RMS of SLR data; linear trends of the data-series are plotted.
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the frequency analysis was applied to the residuals giving the spectrum (Fig. 3). In order to find the frequency peak f120 generated by 120° symmetry of the AJISAI RRA (Otsubo et al., 1999) we were investigating a spectral range from 0 to 2.5 Hz with 20 lHz step. Because of the 120° signal, the true frequency of the body is 3 times smaller: fapparent = f120/3. The resulting spin period of AJISAI (corrected for the apparent effects) is: T = 1/fcorrected. In order to verify these corrections we compared the SLR spin values with the photometric measurements (24 values) made by Koganei Station of Communications Research Laboratory (CRL) (Otsubo et al., 1998). Fig. 4 presents spin period series measured by SLR and photometry in 1997–1998, where each point corresponds to the apparent and corrected spin period derived from a single pass. To investigate the agreement between the SLR and photometry spin period results we have approximated the photometry data with a linear trend function and calculated residuals of SLR values to this function. Note that the apparent longitude correction had been applied to the photometry data set. For the not-corrected residuals (Fig. 4-top) we get: meannot-corr = 1.340 ms, RMSnot-corr = 2.10 ms; for the fully corrected residuals (Fig. 4-bottom): meancorr = 0.211 ms, RMScorr = 1.22 ms. Applying the dfAL corrections shifts the SLR spin periods to the photometry trend (Fig. 4-middle); correcting this data by dfAP reduces the RMS by more than 40% (Fig. 4-bottom). Fig. 5 presents result of AJISAI spin period trend: 14/ Aug/1986–30/Dec/2008. The data has been approximated by an exponential function T = 1.488586 exp(0.0149802 Y) [s], where Y is time in years since launch. The resulting spin period data set is divided into time slots of 0.5 years. The spin period values within these slots are approximated with a linear function, and spin period residuals to this function are obtained. The RMS of these residuals for every time slot is shown in Fig. 6.
Fig. 5. Full history (14/Aug/1986–30/Dec/2008) of AJISAI spin period derived from Hz SLR and Graz kHz SLR data. The exponential trend function (gray curve) is plotted.
Fig. 6. RMS of spin period residuals (corrected for the apparent effects) plotted with 0.5 year step for Hz SLR (gray points) and 2 kHz SLR (black points). The exponential trend functions are plotted. The precision is improved with kHz SLR data.
With a slowing down of the satellite spin, the RMS of spin period residuals is increasing with an exponential trend. This is due to the exponential decrease of the number of full satellite rotations per pass with time, causing poorer frequency signals within SLR data. 4.1. Initial spin period The complete history of AJISAI spin period allows very accurate determination of the initial spin period value T0. Since T0 can be used as a constant in many physical models of the magnetic, gravitational and non-gravitational forces perturbing the satellite motion (Andre´s et al., 2004), its accurate knowledge is important. The slowing down of the satellite does not follow purely exponential trend. Thus an exponential approximation of the full data set (22 years – Fig. 5) may lead to significant differences between the mathematical function and measured data, especially on the beginning and on the end of the data set. In order to
Fig. 7. Result of the frequency analysis (plotted in period domain) of the spin period residuals (top) and of the shadow data (bottom). The significant peaks are marked.
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determine T0 we used the spin period data measured during the first year after launch. This data set was approximated by 2nd degree polynomial function, giving the initial spin period T0 = 1.4855 s, RMS = 0.7 ms. T0 calculated here is 3 ms shorter than the value given by the full history exponential trend function (Fig. 5). 5. Periodical changes in the spin slow down trend We have calculated and spectrally analyzed the spin period residuals (spin period value – trend function) of the complete AJISAI spin history. The resulting spectrum, after conversion to the period domain, is plotted on Fig. 7-top. The spin parameters of the LEO (Low Earth Orbit) satellites are strongly influenced by non-gravitational effects connected with the umbra. The spin period of a satellite is increasing faster when the satellite is illuminated by the sun, and slower when the body is in the shadow of the earth (umbra). For the 22 years of AJISAI, we have calculated the percentage of the last orbital cycle (116 min) being in the umbra, from a given epoch with 10 s step. For these 22 years we have calculated almost 141000 ‘‘umbra” points with 5000 s step. The obtained full data set was spectrally analyzed – the result is presented on Fig. 7-bottom, showing the clear coincidence; because of huge data set the peaks here have very big power. In addition, Fig. 7-top shows also a small AJISAI weekend effect (peak close to 7 days). The periods: B (44.3 d), C (71.2 d) and E (117 d) indicate changes in the rate of slowing down caused by non-gravitational effects (Yarkovsky effect; Rubincam, 1988; Sengoku et al., 1995). 6. Conclusions In this paper, we determined the history of AJISAI spin period, which was possible only with continuously available SLR data. We successfully used 17246 passes (14/ Aug/1986–30/Dec/2008) from 1 to 15 Hz SLR stations, and 1444 passes (9/Oct/2003–30/Dec/2008) from the first 2 kHz SLR system at Graz. The corrections developed here allow elimination of the apparent effects from the spectral analysis, increasing the accuracy of the spin period determination to better than 0.1% (RMS/T). The full spin period data set was approximated with an exponential trend function; however the slowing down of the satellite does not exactly follow this purely mathematical trend. The deviation can be seen e.g. in paragraph 5 when the obtained initial spin period T0 is 3 ms shorter than the value given by the full history trend function. KHz SLR data allows to increase the accuracy of such an analysis, however, even more important here is that we could detect clear frequency signals in almost 100% of good, continuous passes, while only about 30% of Hz passes gave a clear frequency signal; because the 2 kHz SLR observations contain about 200 times more measurements per second than standard Hz SLR, the obtainable
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frequency signal is about 200 times stronger, and almost always clearly visible. Frequency analysis of the spin period residuals shows that the rate of slowing down of AJISAI undergoes periodical changes. The period A (Fig. 7-top) of 6.99 days is an artificial signal introduced by a ‘‘weekend effect” of the SLR network; the number of measured passes is lower during the weekends. The periods B, C and E are caused by non-gravitational effects (Yarkovsky effect), we have no clear explanation for the D period (88.4 d), however this effect might be related to the temperature distribution over the satellite body (Yarkovsky-Schach, not analyzed spectrally here) which depends on the B, C, E periods. The signals detected here may contain valuable information about the non-gravitational effects acting on AJISAI and will be the subject of future studies. The accurate spin period determination gives information about forces acting on the spacecraft and causing perturbation of the orbital motion. The full history (more than 22 years) of the spin period of AJISAI can be used to upgrade the models of all those forces. The accurate spin period model can help to predict the spin parameters of AJISAI for the future application to the next-generation laser time transfer via AJISAI mirrors (Kunimori et al., 1992). References Andre´s, J.I., Noomen, R., Bianco, G., et al. Spin axis behavior of the LAGEOS satellites. J. Geophys. Res. 109 (B6), B06403, doi:10.1029/ 2003JB00269, 2004. Bertotti, B., Iess, L. The rotation of LAGEOS. J. Geophys. Res. 96 (B2), 2431–2440, 1991. Bianco, G., Chersich, M., Devoti, R., et al. Measurement of LAGEOS-2 rotation by satellite laser ranging observations. Geophys. Res. Lett. 28 (10), 2113–2116, 2001. Kirchner, G., Hausleitner, W., Cristea, E. Ajisai spin parameter determination using Graz kilohertz satellite laser ranging data. IEEE Trans. Geosci. Remote Sens. 45 (1), 201–205, 2007. Kirchner, G., Kucharski, D., Cristea, E. Gravity Probe-B: new methods to determine spin parameters from kHz SLR data. IEEE Trans. Geosci. Remote Sens., doi:10.1109/ TGRS.2006.00675, 2009. Kucharski, D., Kirchner, G., Schillak, S., et al. Spin determination of LAGEOS-1 from kHz laser observations. Adv. Space Res. 39 (10), 1576–1581, doi:10.1016/j.asr.2007.02.045, 2007. Kucharski, D., Kirchner, G., Cristea, E. ETALON spin period determination from kHz SLR data. Adv. Space Res. 42 (8), 1424–1428, doi:10.1016/j.asr.2007.08.030, 2008. Kucharski, D., Kirchner, G., Koidl, F., et al. 10 Years of LAGEOS-1 and 15 years of LAGEOS-2 spin period determination from SLR data. Adv. Space Res., doi:10.1016/j.asr.2009.01.019, 2009. Kunimori, H., Takahashi, F., Itabe, T., et al. Laser ranging application to time transfer using geodetic satellite and to other Japanese space programs, in: Proceedings of the 8th International Workshop on Laser Ranging Instrumentation, Annapolis, NASA Conf. Pub. 3214, pp. 138–1-42, 1992. Lomb, N.R. Least-squares frequency analysis of unequally spaced data. Astrophys. Space Sci. 39, 447–462, 1976. Otsubo, T., Amagai, J., Kunimori, H. Measuring AJISAI’s spin motion, in: Proceedings of the 11th International Workshop on Laser Ranging, Deggendorf, Germany, pp. 674–677, 1998.
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