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Spin rate and spin axis orientation of LARES spectrally determined from Satellite Laser Ranging data
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Advances in Space Research 50 (2012) 1473–1477 www.elsevier.com/locate/asr
Spin rate and spin axis orientation of LARES spectrally determined from Satellite Laser Ranging data D. Kucharski a,⇑, T. Otsubo b,1, G. Kirchner c,2, G. Bianco d,3 a
Space Science Division, Korea Astronomy and Space Science Institute, 776, Daedeok-Daero, Yuseong-Gu, Daejeon 305-348, South Korea b Hitotsubashi University, 2-1 Naka, Kunitachi 186-8601, Japan c Space Research Institute of the Austrian Academy of Sciences, Lustbuehelstrasse 46, A-8042 Graz, Austria d Agenzia Spaziale Italiana, Centro di Geodesia Spaziale “G. Colombo”, P.O. Box 11, 75100 Matera, Italy Received 3 April 2012; received in revised form 12 July 2012; accepted 15 July 2012 Available online 24 July 2012
Abstract Satellite Laser Ranging (SLR) is a powerful technique able to measure spin rate and spin axis orientation of the fully passive, geodetic satellites. This work presents results of the spin determination of LARES – a new satellite for testing General Relativity. 529 SLR passes measured between February 17 and June 9, 2012, were spectrally analyzed. Our results indicate that the initial spin frequency of LARES is f0 = 86.906 mHz (RMS = 0.539 mHz). A new method for spin axis determination, developed for this analysis, gives orientation of the axis at RA = 12h22m48s (RMS = 49m), Dec = 70.4° (RMS = 5.2°) (J2000.0 celestial reference frame), and the clockwise (CW) spin direction. The half-life period of the satellite’s spin is 214.924 days and indicates fast slowing down of the spacecraft. Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Satellite Laser Ranging; LARES; Geodetic satellite; Spin determination; Spectral analysis
1. Introduction 1.1. Spin determination with SLR SLR measures distances between ground stations and satellites equipped with CCRs (Pearlman et al., 2002). The first High Repetition Rate (HRR) SLR system was designed in Graz (Austria) and started operations on October 9, 2003. The HRR SLR systems allow investigating the spin parameters of: – slowly spinning LAGEOS-1, LAGEOS-2 (Kucharski et al., 2009a) ⇑ Corresponding author. Tel.: +82 42 865 2184; fax: +82 42 861 5610.
E-mail addresses: [email protected], [email protected] (D. Kucharski), [email protected] (T. Otsubo), [email protected] (G. Kirchner), [email protected] (G. Bianco). 1 Tel./fax: +81 42 580 8939. 2 Tel.: +43 316 873 4651; fax: +43 316 873 4656. 3 Tel.: +39 0835 377509; fax: +39 0835 339005.
– High Earth Orbiting Etalon-1 and Etalon-2 (Kucharski et al., 2008) – fast spinning Ajisai (Kucharski et al., 2009b, 2010) – active Gravity Probe – B, equipped with a small CCR panel – the first spherical lens type satellite BLITS. Geodetic SLR satellites (e.g. Ajisai, ETALON, LAGEOS, Starlette, Stella, LARES) are designed as fully passive spheres equipped with CCRs. The range measurements to these satellites are used for precise orbit determination, study of tectonic plate motion, Earth orientation and rotation parameters, the gravity field and general relativity. The geodetic satellites are launched with an initial spin, which helps to stabilize the orientation of the spacecraft. The passive, heavy bodies are loosing spin over time due to the influence of the forces and torques caused by the Earth’s gravitational and magnetic fields, and solar activity (Andre´s et al., 2004).
0273-1177/$36.00 Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2012.07.018
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During a pass of a geodetic satellite over an SLR station, the laser pulses transmitted from the station are reflected by CCRs back to the receiver telescope of the SLR system. The spinning array of the CCRs causes a mm-scale modulation of the range measurements what engraves a frequency signal on the SLR data. This signal can be obtained by spectral analysis of the unequally spaced return signals as it was demonstrated in (Otsubo et al., 2000; Bianco et al., 2001). After the pass the time bias and range bias estimations are used to fit the range measurements to the predicted trend. As the next step the range measurements are converted into range residuals by calculating difference between observed (measured) and predicted values (O–C). In the course of this process fitting functions are used (orbital function or low degree polynomials) in order to remove systematic trends of the residuals. The final step of the post-processing is the formation of normal points (NP). The knowledge of the spin parameters of the passive satellites allows for the investigation and improvement of physical models of the perturbing forces and torques which are of magnetic, gravitational and non-gravitational nature. Such improved models are used for precise orbit determination (Rubincam, 1990) and can increase the accuracy of the determination of SLR station positions and of geodynamical parameters.
1.2. LARES LARES (LAser RElativity Satellite; Iorio et al., 2002) is designed by Scuola di Ingegneria Aerospaziale at “La Sapienza” University of Rome and manufactured by Carlo Gavazzi Space (CGS) under contract to the Italian Space Agency (ASI). The satellite was launched by ESA on February 13, 2012, from the European Spaceport of Kourou (French Guyana), with the maiden flight of the new ESA small launcher VEGA. LARES was placed in a circular orbit at a height of 1450 km (inclination 69.5°). It is expected that the spacecraft will achieve important measurements in gravitational physics, space geodesy and geodynamics: in particular – together with the LAGEOS1 and LAGEOS-2 satellites and with the GRACE models – it will improve the accuracy of the determination of Earth’s gravitomagnetic field and of the Lense-Thirring effect (Ciufolini and Pavlis, 2004). This fully passive, spherical satellite is made of a high density tungsten alloy and equipped with 92 corner cube reflectors (CCRs) for SLR (Paolozzi et al., 2011). The CCRs are arranged in the form of 10 rings around the polar axis of the body (Fig. 1). The gaps between the prisms on a single ring are constant. The retroreflector array (RRA) holds 2 rings with 16 CCRs at the body’s latitude: R1 (10°), R-1 ( 10°); 2 rings with 14 CCRs: R2 (30°), R-2 ( 30°); 2 rings with 10 CCRs: R3 (50°), R-3 ( 50°); 2 rings with 5 CCRs: R4 (70°), R-4 ( 70°); and 2 single CCRs placed on the poles
of the body. All cubes are of the same type - made from Suprasil 311, not coated. 2. Spectral response of LARES The spectral response of the fast spinning spherical satellite contains information about the frequencies that can be generated in the SLR residuals as a function of the body’s latitude at which the laser pulse impacts the surface. In order to determine the spectral response of LARES, the range residuals are simulated for every incident angle between the laser beam and the spin axis of the RRA; the pointing latitude is changed from 90° to 90° with a step of 0.1°. The simulation algorithm (Kucharski et al., 2008) contains the range correction function and the energy transfer function. During the process the spin period of the satellite is set to 12 s and kept constant (the initial spin rate of LARES was close to 5 rpm). At given latitudes of the laser pointing the range residuals are simulated and spectrally analyzed. Fig. 2 presents the obtained spectral response of LARES from the simulated data; the spin frequency of the satellite is identified as f1 = 0.0833 Hz, as expected (=spin period of 12 s). The harmonics (f5, f10, f14, f16) visible in the spectra are generated by various geometrical distributions of the contributing CCRs. When the laser direction is perpendicular to the plane of a given ring, the detected, most powerful harmonic fn is a product of the body’s spin frequency (f1) and the number n of the CCRs distributed around the ring: fn = nf1. At the poles of the body the spin axis orientation of the satellite coincides with the laser beam, thus the power of the frequency signal derived from that area is below the significance level. The simulations show that only a limited set of the harmonics (f5, f10, f14, f16) can be generated by LARES. Spectral analysis of the mm-precision Graz kHz SLR data indicates however that the set of the harmonics is much wider (Fig. 3). The frequency signal visible in the observed data, but not present in the simulations, is caused by periodical lack of the returns visible in the SLR data. The observed periodical decrease of the return rate (detected laser echoes per second) (Fig. 4) can be caused by different optical properties of some CCRs or by the diffraction effect which can occur inside of any CCR of the satellite during a short part of a pass. The diffraction effect depends on the orientation of a particular CCR to the laser beam and can influence the spatial distribution of the reflected laser energy. Fig. 4 presents range residuals measured by Graz SLR station. The moving average indicates a periodical lack of the laser echoes from the satellite what is a source of the unexpected frequency signal. 3. Spin rate In order to determine the spin rate of LARES we used the SLR data produced by the ILRS network stations
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Fig. 1. LARES (courtesy of ASI) and distribution of the CCRs on the satellite. The CCR array consists of 10 rings R-5. . .R5.
Fig. 2. Simulated spectral response of LARES. Spectral signal obtained from the simulated data for body’s spin rate f1 = 0.0833 Hz (spin period = 12 s): black lines indicate the most powerful frequency at given latitude of laser pointing. Harmonics fn and position of the rings Ri(#CCRs) are marked.
Fig. 3. Spectral analysis of the LARES pass measured by Graz SLR station on February 22, 2012.
between February 17 and June 9, 2012. We have selected 529 passes which give a clear spectral signal with power higher than 10. Every selected pass is processed and the range residuals are calculated (with respect to predicted orbits); as the next step the frequency analysis is performed and the spectral signal is obtained (Fig. 3). The resulting spin frequency of the satellite is calculated as a mean value of the harmonics visible in the spectra. The results indicate slowing down of the satellite what can be expressed by the exponential trend functions:
Fig. 4. Range residuals (60 s) and the moving average (MA) calculated for the LARES pass measured by Graz SLR station on February 22, 2012. The vertical lines (MA chart) are separated by spin period of the satellite (11.9 s). The zero level is the mean of the range residuals.
spin frequency: f = 86.8029 exp( 0.00327954 D), RMS = 0.59 [mHz], spin rate: r = 5.20818 exp( 0.00327954 D), RMS = 0.0356 [rpm], spin period: T = 11.5203 exp(0.00327954 D), RMS = 0.116 [s], where D is the number of days since launch. The obtained information represents an apparent spin of the satellite. As the satellite is passing the ground station, the pointing orientation of the laser beam in the satellite body centered (not body-fixed) coordinate system is changing. This affects the frequency signal contained in the data. The change of the laser pointing is not constant for every pass and depends on the actual geometry of the pass (station – satellite mutual orientation). In order to correct spin rate measurements for the apparent effects the spin axis orientation has to be known (Kucharski et al., 2009b). 4. Spin axis orientation and spin direction During the launch flight the satellite was mounted on the Vega rocket in such a way that its symmetry axis coincided with the axis of the launch vehicle. The north
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pole of LARES was facing the flight direction, and the satellite was injected into orbit with its polar axis parallel to the along track vector. We used TLE predictions to calculate the orientation of the along track vector of the satellite at the injection epoch what gives a rough estimation of the spin axis orientation (RA = 10h34m, Dec = 69.2°). A new method for spin axis orientation determination is presented here. The method is based on the frequency analysis of the SLR data. We assume that the satellite spins around its polar axis (oriented to the north pole of the body), and the orientation of the axis in the space is stable (we detected no significant change during the investigated 529 SLR passes). Correcting the raw spectral signal by the apparent effects gives the inertial spin rate of the body; the spread (RMS) of the spin rate values around the exponential trend function decreases when the assumed orientation of the spin axis approaches the true value. For every possible spin axis orientation (RA, Dec changed with 0.5° step) and both directions of the spin (CW, CCW) the apparent corrections are calculated and applied to the raw spin rate data. The resulting, inertial slow down trend is calculated and RMS of the spin rate values around the trend function is determined. The orientation of the launch mechanism during the injection of the satellite into orbit indicates that the north pole of LARES was facing the southern hemisphere of the Earth, thus the spin axis solution should be located at negative declination of the celestial reference frame. Within this range of declination the RMS of the inertial spin rate residuals (calculated to the exponential trend function) is minimal at RA = 12h10m0s, Dec = 71.5° (CW direction). The spin axis solution is calculated as a mean of the 1% best orientation values (with the lowest RMS): RA = 12h22m48s (RMS = 49m), Dec = 70.4° (RMS = 5.2°), CW spin direction. 4.1. Inertial spin rate This determined spin axis orientation is used to correct the raw spin rate measurements for the apparent effects; the average apparent correction is about 0.5% of the raw value. The corrected inertial spin rate values are presented on Fig. 5. The slow down trend can be expressed by the exponential functions: spin frequency: f = 86.906 exp( 0.00322509 D), RMS = 0.539 [mHz], spin rate: r = 5.21436 exp( 0.00322509 D), RMS = 0.0323 [rpm], spin period: T = 11.5067 exp(0.00322509 D), RMS = 0.105 [s], where D is the number of days since launch. The RMS of the spin rate values does not increase over the days, what indicates stable orientation of the spin axis. The initial spin parameters are: f0 = 86.906 mHz, r0 = 5.21436 rpm, T0 = 11.5067 s. The half-life period of
Fig. 5. Spin of LARES: f-spin frequency, r-spin rate, T-spin period. The exponential trend function is plotted.
the satellite’s spin is 214.924 days – the spin period doubles after this time. 5. Conclusions LARES is the densest known object orbiting in the Solar System. The spin parameters of this fully passive sphere can be measured during day and night by SLR only. The symmetrical distribution of the CCRs on the rings of the RRA ensures that the spinning body will imprint a clear frequency signal on the laser measurements. The high precision of laser ranging allows to use spectral analysis for spin rate and spin axis orientation determination. Analysis of the 529 SLR passes indicates that the satellite was launched with the initial spin frequency of f0 = 86.906 mHz (RMS = 0.539 mHz). The determined spin axis orientation (J2000) is RA = 12h22m48s (RMS = 49m), Dec = 70.4° (RMS = 5.2°), and the body spins CW around this axis. The half-life period of the spin rate is 214.924 days and indicates fast slowing down of the satellite. According to (Andre´s et al., 2004) the magnetic field of the Earth is the strongest factor that slows down the rotational motion of a fast spinning geodetic satellite. LARES is made of a non-magnetic alloy with a low electrical conductivity that minimizes the interaction of the body with the magnetic field of the Earth. The high rate of slowing down indicates however that the eddy currents, induced in the body, are still able to reduce spin of this low Earth orbit satellite. The spin parameters of LARES (the slow downward trend) give information about forces and torques which are acting on the spacecraft, and cause perturbations of its orbit. Knowing the attitude of the satellite will help to calculate the center-of-mass correction with better accuracy and improve the precise orbit determination. References Andre´s, J.I., Noomen, R., Bianco, G., et al. Spin axis behavior of the LAGEOS satellites. J. Geophys. Res. 109 (B6), B06403, http:// dx.doi.org/10.1029/2003JB00269, 2004. 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, http://dx.doi.org/10.1029/2000GL012435, 2001. Ciufolini, I., Pavlis, E.C. A confirmation of the general relativistic prediction of the Lense-Thirring effect. Nature 431, 958–960, 2004. Iorio, L., Lucchesi, D.M., Ciufolini, I. The LARES mission revisited: an alternative scenario. Class. Quantum Grav. 19, 4311, http:// dx.doi.org/10.1088/0264-9381/19/16/307, 2002.
D. Kucharski et al. / Advances in Space Research 50 (2012) 1473–1477 Kucharski, D., Kirchner, G., Cristea, E. ETALON spin period determination from kHz SLR data. Adv. Space Res. 42 (8), 1424–1428, http:// dx.doi.org/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. 43 (12), 1926–1930, http://dx.doi.org/10.1016/ j.asr.2009.01.019, 2009a. Kucharski, D., Kirchner, G., Otsubo, T., et al. 22 years of Ajisai spin period determination from standard SLR and kHz SLR data. Adv. Space Res. 44 (5), 621–626, http://dx.doi.org/10.1016/ j.asr.2009.05.007, 2009b. Kucharski, D., Kirchner, G., Otsubo, T., et al. The impact of solar irradiance on Ajisai’s spin period measured by the Graz 2 kHz SLR
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system. IEEE Trans. Geosci. Remote Sens. 48 (3) 1629–1633, doi:10.1109/TGRS.2009.2031229, 2010. 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. Paolozzi, A., Ciufolini, I., Vendittozzi, C. Engineering and scientific aspects of LARES satellite. Acta Astronautica 69 (3–4), 127–134, http://dx.doi.org/10.1016/j.actaastro.2011.03.005, 2011. Pearlman, M.R., Degnan, J.J., Bosworth, J.M. The international laser ranging service. Adv. Space Res. 30 (2), 135–143, http://dx.doi.org/ 10.1016/S0273-1177(02)00277-6, 2002. Rubincam, D.P. Drag on the LAGEOS Satellite. J. Geophys. Res. 95 (B4), 4881–4886, doi:10.1029/JB095iB04p04881, April 1990.
Advances in Space Research 50 (2012) 1473–1477 www.elsevier.com/locate/asr
Spin rate and spin axis orientation of LARES spectrally determined from Satellite Laser Ranging data D. Kucharski a,⇑, T. Otsubo b,1, G. Kirchner c,2, G. Bianco d,3 a
Space Science Division, Korea Astronomy and Space Science Institute, 776, Daedeok-Daero, Yuseong-Gu, Daejeon 305-348, South Korea b Hitotsubashi University, 2-1 Naka, Kunitachi 186-8601, Japan c Space Research Institute of the Austrian Academy of Sciences, Lustbuehelstrasse 46, A-8042 Graz, Austria d Agenzia Spaziale Italiana, Centro di Geodesia Spaziale “G. Colombo”, P.O. Box 11, 75100 Matera, Italy Received 3 April 2012; received in revised form 12 July 2012; accepted 15 July 2012 Available online 24 July 2012
Abstract Satellite Laser Ranging (SLR) is a powerful technique able to measure spin rate and spin axis orientation of the fully passive, geodetic satellites. This work presents results of the spin determination of LARES – a new satellite for testing General Relativity. 529 SLR passes measured between February 17 and June 9, 2012, were spectrally analyzed. Our results indicate that the initial spin frequency of LARES is f0 = 86.906 mHz (RMS = 0.539 mHz). A new method for spin axis determination, developed for this analysis, gives orientation of the axis at RA = 12h22m48s (RMS = 49m), Dec = 70.4° (RMS = 5.2°) (J2000.0 celestial reference frame), and the clockwise (CW) spin direction. The half-life period of the satellite’s spin is 214.924 days and indicates fast slowing down of the spacecraft. Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Satellite Laser Ranging; LARES; Geodetic satellite; Spin determination; Spectral analysis
1. Introduction 1.1. Spin determination with SLR SLR measures distances between ground stations and satellites equipped with CCRs (Pearlman et al., 2002). The first High Repetition Rate (HRR) SLR system was designed in Graz (Austria) and started operations on October 9, 2003. The HRR SLR systems allow investigating the spin parameters of: – slowly spinning LAGEOS-1, LAGEOS-2 (Kucharski et al., 2009a) ⇑ Corresponding author. Tel.: +82 42 865 2184; fax: +82 42 861 5610.
E-mail addresses: [email protected], [email protected] (D. Kucharski), [email protected] (T. Otsubo), [email protected] (G. Kirchner), [email protected] (G. Bianco). 1 Tel./fax: +81 42 580 8939. 2 Tel.: +43 316 873 4651; fax: +43 316 873 4656. 3 Tel.: +39 0835 377509; fax: +39 0835 339005.
– High Earth Orbiting Etalon-1 and Etalon-2 (Kucharski et al., 2008) – fast spinning Ajisai (Kucharski et al., 2009b, 2010) – active Gravity Probe – B, equipped with a small CCR panel – the first spherical lens type satellite BLITS. Geodetic SLR satellites (e.g. Ajisai, ETALON, LAGEOS, Starlette, Stella, LARES) are designed as fully passive spheres equipped with CCRs. The range measurements to these satellites are used for precise orbit determination, study of tectonic plate motion, Earth orientation and rotation parameters, the gravity field and general relativity. The geodetic satellites are launched with an initial spin, which helps to stabilize the orientation of the spacecraft. The passive, heavy bodies are loosing spin over time due to the influence of the forces and torques caused by the Earth’s gravitational and magnetic fields, and solar activity (Andre´s et al., 2004).
0273-1177/$36.00 Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2012.07.018
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During a pass of a geodetic satellite over an SLR station, the laser pulses transmitted from the station are reflected by CCRs back to the receiver telescope of the SLR system. The spinning array of the CCRs causes a mm-scale modulation of the range measurements what engraves a frequency signal on the SLR data. This signal can be obtained by spectral analysis of the unequally spaced return signals as it was demonstrated in (Otsubo et al., 2000; Bianco et al., 2001). After the pass the time bias and range bias estimations are used to fit the range measurements to the predicted trend. As the next step the range measurements are converted into range residuals by calculating difference between observed (measured) and predicted values (O–C). In the course of this process fitting functions are used (orbital function or low degree polynomials) in order to remove systematic trends of the residuals. The final step of the post-processing is the formation of normal points (NP). The knowledge of the spin parameters of the passive satellites allows for the investigation and improvement of physical models of the perturbing forces and torques which are of magnetic, gravitational and non-gravitational nature. Such improved models are used for precise orbit determination (Rubincam, 1990) and can increase the accuracy of the determination of SLR station positions and of geodynamical parameters.
1.2. LARES LARES (LAser RElativity Satellite; Iorio et al., 2002) is designed by Scuola di Ingegneria Aerospaziale at “La Sapienza” University of Rome and manufactured by Carlo Gavazzi Space (CGS) under contract to the Italian Space Agency (ASI). The satellite was launched by ESA on February 13, 2012, from the European Spaceport of Kourou (French Guyana), with the maiden flight of the new ESA small launcher VEGA. LARES was placed in a circular orbit at a height of 1450 km (inclination 69.5°). It is expected that the spacecraft will achieve important measurements in gravitational physics, space geodesy and geodynamics: in particular – together with the LAGEOS1 and LAGEOS-2 satellites and with the GRACE models – it will improve the accuracy of the determination of Earth’s gravitomagnetic field and of the Lense-Thirring effect (Ciufolini and Pavlis, 2004). This fully passive, spherical satellite is made of a high density tungsten alloy and equipped with 92 corner cube reflectors (CCRs) for SLR (Paolozzi et al., 2011). The CCRs are arranged in the form of 10 rings around the polar axis of the body (Fig. 1). The gaps between the prisms on a single ring are constant. The retroreflector array (RRA) holds 2 rings with 16 CCRs at the body’s latitude: R1 (10°), R-1 ( 10°); 2 rings with 14 CCRs: R2 (30°), R-2 ( 30°); 2 rings with 10 CCRs: R3 (50°), R-3 ( 50°); 2 rings with 5 CCRs: R4 (70°), R-4 ( 70°); and 2 single CCRs placed on the poles
of the body. All cubes are of the same type - made from Suprasil 311, not coated. 2. Spectral response of LARES The spectral response of the fast spinning spherical satellite contains information about the frequencies that can be generated in the SLR residuals as a function of the body’s latitude at which the laser pulse impacts the surface. In order to determine the spectral response of LARES, the range residuals are simulated for every incident angle between the laser beam and the spin axis of the RRA; the pointing latitude is changed from 90° to 90° with a step of 0.1°. The simulation algorithm (Kucharski et al., 2008) contains the range correction function and the energy transfer function. During the process the spin period of the satellite is set to 12 s and kept constant (the initial spin rate of LARES was close to 5 rpm). At given latitudes of the laser pointing the range residuals are simulated and spectrally analyzed. Fig. 2 presents the obtained spectral response of LARES from the simulated data; the spin frequency of the satellite is identified as f1 = 0.0833 Hz, as expected (=spin period of 12 s). The harmonics (f5, f10, f14, f16) visible in the spectra are generated by various geometrical distributions of the contributing CCRs. When the laser direction is perpendicular to the plane of a given ring, the detected, most powerful harmonic fn is a product of the body’s spin frequency (f1) and the number n of the CCRs distributed around the ring: fn = nf1. At the poles of the body the spin axis orientation of the satellite coincides with the laser beam, thus the power of the frequency signal derived from that area is below the significance level. The simulations show that only a limited set of the harmonics (f5, f10, f14, f16) can be generated by LARES. Spectral analysis of the mm-precision Graz kHz SLR data indicates however that the set of the harmonics is much wider (Fig. 3). The frequency signal visible in the observed data, but not present in the simulations, is caused by periodical lack of the returns visible in the SLR data. The observed periodical decrease of the return rate (detected laser echoes per second) (Fig. 4) can be caused by different optical properties of some CCRs or by the diffraction effect which can occur inside of any CCR of the satellite during a short part of a pass. The diffraction effect depends on the orientation of a particular CCR to the laser beam and can influence the spatial distribution of the reflected laser energy. Fig. 4 presents range residuals measured by Graz SLR station. The moving average indicates a periodical lack of the laser echoes from the satellite what is a source of the unexpected frequency signal. 3. Spin rate In order to determine the spin rate of LARES we used the SLR data produced by the ILRS network stations
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Fig. 1. LARES (courtesy of ASI) and distribution of the CCRs on the satellite. The CCR array consists of 10 rings R-5. . .R5.
Fig. 2. Simulated spectral response of LARES. Spectral signal obtained from the simulated data for body’s spin rate f1 = 0.0833 Hz (spin period = 12 s): black lines indicate the most powerful frequency at given latitude of laser pointing. Harmonics fn and position of the rings Ri(#CCRs) are marked.
Fig. 3. Spectral analysis of the LARES pass measured by Graz SLR station on February 22, 2012.
between February 17 and June 9, 2012. We have selected 529 passes which give a clear spectral signal with power higher than 10. Every selected pass is processed and the range residuals are calculated (with respect to predicted orbits); as the next step the frequency analysis is performed and the spectral signal is obtained (Fig. 3). The resulting spin frequency of the satellite is calculated as a mean value of the harmonics visible in the spectra. The results indicate slowing down of the satellite what can be expressed by the exponential trend functions:
Fig. 4. Range residuals (60 s) and the moving average (MA) calculated for the LARES pass measured by Graz SLR station on February 22, 2012. The vertical lines (MA chart) are separated by spin period of the satellite (11.9 s). The zero level is the mean of the range residuals.
spin frequency: f = 86.8029 exp( 0.00327954 D), RMS = 0.59 [mHz], spin rate: r = 5.20818 exp( 0.00327954 D), RMS = 0.0356 [rpm], spin period: T = 11.5203 exp(0.00327954 D), RMS = 0.116 [s], where D is the number of days since launch. The obtained information represents an apparent spin of the satellite. As the satellite is passing the ground station, the pointing orientation of the laser beam in the satellite body centered (not body-fixed) coordinate system is changing. This affects the frequency signal contained in the data. The change of the laser pointing is not constant for every pass and depends on the actual geometry of the pass (station – satellite mutual orientation). In order to correct spin rate measurements for the apparent effects the spin axis orientation has to be known (Kucharski et al., 2009b). 4. Spin axis orientation and spin direction During the launch flight the satellite was mounted on the Vega rocket in such a way that its symmetry axis coincided with the axis of the launch vehicle. The north
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pole of LARES was facing the flight direction, and the satellite was injected into orbit with its polar axis parallel to the along track vector. We used TLE predictions to calculate the orientation of the along track vector of the satellite at the injection epoch what gives a rough estimation of the spin axis orientation (RA = 10h34m, Dec = 69.2°). A new method for spin axis orientation determination is presented here. The method is based on the frequency analysis of the SLR data. We assume that the satellite spins around its polar axis (oriented to the north pole of the body), and the orientation of the axis in the space is stable (we detected no significant change during the investigated 529 SLR passes). Correcting the raw spectral signal by the apparent effects gives the inertial spin rate of the body; the spread (RMS) of the spin rate values around the exponential trend function decreases when the assumed orientation of the spin axis approaches the true value. For every possible spin axis orientation (RA, Dec changed with 0.5° step) and both directions of the spin (CW, CCW) the apparent corrections are calculated and applied to the raw spin rate data. The resulting, inertial slow down trend is calculated and RMS of the spin rate values around the trend function is determined. The orientation of the launch mechanism during the injection of the satellite into orbit indicates that the north pole of LARES was facing the southern hemisphere of the Earth, thus the spin axis solution should be located at negative declination of the celestial reference frame. Within this range of declination the RMS of the inertial spin rate residuals (calculated to the exponential trend function) is minimal at RA = 12h10m0s, Dec = 71.5° (CW direction). The spin axis solution is calculated as a mean of the 1% best orientation values (with the lowest RMS): RA = 12h22m48s (RMS = 49m), Dec = 70.4° (RMS = 5.2°), CW spin direction. 4.1. Inertial spin rate This determined spin axis orientation is used to correct the raw spin rate measurements for the apparent effects; the average apparent correction is about 0.5% of the raw value. The corrected inertial spin rate values are presented on Fig. 5. The slow down trend can be expressed by the exponential functions: spin frequency: f = 86.906 exp( 0.00322509 D), RMS = 0.539 [mHz], spin rate: r = 5.21436 exp( 0.00322509 D), RMS = 0.0323 [rpm], spin period: T = 11.5067 exp(0.00322509 D), RMS = 0.105 [s], where D is the number of days since launch. The RMS of the spin rate values does not increase over the days, what indicates stable orientation of the spin axis. The initial spin parameters are: f0 = 86.906 mHz, r0 = 5.21436 rpm, T0 = 11.5067 s. The half-life period of
Fig. 5. Spin of LARES: f-spin frequency, r-spin rate, T-spin period. The exponential trend function is plotted.
the satellite’s spin is 214.924 days – the spin period doubles after this time. 5. Conclusions LARES is the densest known object orbiting in the Solar System. The spin parameters of this fully passive sphere can be measured during day and night by SLR only. The symmetrical distribution of the CCRs on the rings of the RRA ensures that the spinning body will imprint a clear frequency signal on the laser measurements. The high precision of laser ranging allows to use spectral analysis for spin rate and spin axis orientation determination. Analysis of the 529 SLR passes indicates that the satellite was launched with the initial spin frequency of f0 = 86.906 mHz (RMS = 0.539 mHz). The determined spin axis orientation (J2000) is RA = 12h22m48s (RMS = 49m), Dec = 70.4° (RMS = 5.2°), and the body spins CW around this axis. The half-life period of the spin rate is 214.924 days and indicates fast slowing down of the satellite. According to (Andre´s et al., 2004) the magnetic field of the Earth is the strongest factor that slows down the rotational motion of a fast spinning geodetic satellite. LARES is made of a non-magnetic alloy with a low electrical conductivity that minimizes the interaction of the body with the magnetic field of the Earth. The high rate of slowing down indicates however that the eddy currents, induced in the body, are still able to reduce spin of this low Earth orbit satellite. The spin parameters of LARES (the slow downward trend) give information about forces and torques which are acting on the spacecraft, and cause perturbations of its orbit. Knowing the attitude of the satellite will help to calculate the center-of-mass correction with better accuracy and improve the precise orbit determination. References Andre´s, J.I., Noomen, R., Bianco, G., et al. Spin axis behavior of the LAGEOS satellites. J. Geophys. Res. 109 (B6), B06403, http:// dx.doi.org/10.1029/2003JB00269, 2004. 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, http://dx.doi.org/10.1029/2000GL012435, 2001. Ciufolini, I., Pavlis, E.C. A confirmation of the general relativistic prediction of the Lense-Thirring effect. Nature 431, 958–960, 2004. Iorio, L., Lucchesi, D.M., Ciufolini, I. The LARES mission revisited: an alternative scenario. Class. Quantum Grav. 19, 4311, http:// dx.doi.org/10.1088/0264-9381/19/16/307, 2002.
D. Kucharski et al. / Advances in Space Research 50 (2012) 1473–1477 Kucharski, D., Kirchner, G., Cristea, E. ETALON spin period determination from kHz SLR data. Adv. Space Res. 42 (8), 1424–1428, http:// dx.doi.org/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. 43 (12), 1926–1930, http://dx.doi.org/10.1016/ j.asr.2009.01.019, 2009a. Kucharski, D., Kirchner, G., Otsubo, T., et al. 22 years of Ajisai spin period determination from standard SLR and kHz SLR data. Adv. Space Res. 44 (5), 621–626, http://dx.doi.org/10.1016/ j.asr.2009.05.007, 2009b. Kucharski, D., Kirchner, G., Otsubo, T., et al. The impact of solar irradiance on Ajisai’s spin period measured by the Graz 2 kHz SLR
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system. IEEE Trans. Geosci. Remote Sens. 48 (3) 1629–1633, doi:10.1109/TGRS.2009.2031229, 2010. 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. Paolozzi, A., Ciufolini, I., Vendittozzi, C. Engineering and scientific aspects of LARES satellite. Acta Astronautica 69 (3–4), 127–134, http://dx.doi.org/10.1016/j.actaastro.2011.03.005, 2011. Pearlman, M.R., Degnan, J.J., Bosworth, J.M. The international laser ranging service. Adv. Space Res. 30 (2), 135–143, http://dx.doi.org/ 10.1016/S0273-1177(02)00277-6, 2002. Rubincam, D.P. Drag on the LAGEOS Satellite. J. Geophys. Res. 95 (B4), 4881–4886, doi:10.1029/JB095iB04p04881, April 1990.