Estimation of the Love and Shida numbers: h2, l2 using SLR data for the low satellites

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Advances in Space Research 52 (2013) 633–638 www.elsevier.com/locate/asr

Estimation of the Love and Shida numbers: h2, l2 using SLR data for the low satellites Marcin Jagoda a,⇑, Miłosława Rutkowska a,b b

a Technical University of Koszalin, S´niadeckich 2, 75-453 Koszalin, Poland Space Research Centre, Polish Academy of Sciences, Bartycka 18a, Warsaw, Poland

Received 15 November 2012; received in revised form 17 April 2013; accepted 18 April 2013 Available online 29 April 2013

Abstract In this paper we present results for the global elastic parameters: Love number h2 and Shida number l2 derived from the analysis of Satellite Laser Ranging (SLR) data. SLR data for the two low satellites STELLA (H = 800 km) and STARLETTE (H = 810 km) observed during 2.5 years from January 3, 2005 until July 1, 2007 with 18 globally distributed ground stations were analyzed. The analysis was done separately for the two satellites. We do a sequential analysis and study the stability and convergence of the estimates as a function of length of the data set used. The final adjusted values for h2 equal to 0.6163 ± 0.0037 and 0.6048 ± 0.0025, and those for l2 equal to 0.0176 ± 0.0017 and 0.1151 ± 0.0010 for STELLA and STARLETTE tracking data are compared to estimates we previously published based on data for two high satellites LAGEOS1 (H = 5860 km) and LAGEOS2 (H = 5620 km) (Rutkowska and Jagoda, 2010a). A major discrepancy between the two solutions was found for the Shida number l2. Ó 2013 Published by Elsevier Ltd. on behalf of COSPAR. Keywords: Love and Shida numbers; Elastic Earth parameters; Satellite Laser Ranging (SLR); STELLA; STARLETTE

1. Introduction The following analysis is the continuation of our research based on the SLR data of high satellites (LAGEOS1 and LAGEOS2) to have been in published in Rutkowska and Jagoda (2010a). For both groups of satellites: high (LAGEOS1, LAGEOS2) and low (STELLA, STARLETTE), the estimation of Love and Shida numbers was derived from the data within the same time interval: from January 3, 2005 until July 1, 2007 and by applying the same models of forces disturbing satellite motion (except for the Earth’s atmosphere model which was taken into account only for low satellites and for the model of the Earth’s gravitational field and the model of troposphere). ⇑ Corresponding author. Tel.: +48 943679507.

E-mail addresses: [email protected] (M. Jagoda), [email protected] (M. Rutkowska). 0273-1177/$36.00 Ó 2013 Published by Elsevier Ltd. on behalf of COSPAR. http://dx.doi.org/10.1016/j.asr.2013.04.018

As the theoretical basis regarding the phenomenon of the Earth elasticity and elastic parameters (numbers Love’a h2 and Shida l2) was thoroughly scrutinized by the authors of this paper in Rutkowska and Jagoda (2010a,b), it is not analyzed. Also the measurement methods, choice of force models, orbit computations, analysis and comparisons used by other scientists were described in detail in Rutkowska and Jagoda (2010a,b) and therefore they are omitted in this paper. The aim of the present study as of our previous study is to adjust the global elastic parameters h2, l2 to compare the accuracy of solutions obtained for high and low satellites and to pick out the best solution estimated with the highest accuracy. The second aim is to compute the minimum and necessary time interval which allows stability and convergence of the estimated parameters and their errors to be attained. Both solutions together comprise a complete study of the elastic Earth parameters estimation.

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2. Method of analysis The process of estimating elastic Earth parameters was carried out in several stages (Fig. 1). During the first step the data were collected from the world database Crustal Dynamics Data Information System (CDDIS) and the EUROLAS Data Center (EDC) of laser observations in the form of STELLA and STARLETTE normal points within the established 2.5 years time interval: from January 3, 2005 until July 1, 2007. All normal points whose elevation was lower than 15° were rejected in the solution – this was due to large errors of troposphere refraction near the horizon. The next step was to order chronologically (according to time advancement), as well as to eliminate observations containing large errors. The total number of normal points used for this analysis equals to 121,668 for STARLETTE and 61,757 for STELLA. The number of normal points for each observatory station is presented in Table 1. The computations analyzed in this paper are based on the SLR data conducted at 18 SLR stations of global network. The selected stations are of estimated coordinates with 1 mm precision in each component of ITRF2005 system (Altamimi et al., 2007), guaranteeing best-quality observations. What is more, the stations were selected in such a way so that they would be evenly located on the globe. In the subsequent step weekly orbital arcs were formed according to Torrence et al. (1984). After dividing the observations into orbital arcs the orbits of STELLA and STARLETTE satellites were determined. The satellite orbits are computed using 11th order predictor–corrector Cowell’s method for the numerical integration of the satellite equations of motion in rectangular coordinates (Maury and Brodsky, 1969). A step size of 60 s was used. Preliminary, approximate values of orbital elements (vectors of position and velocity) were obtained from NASA. The precision reference to determine the orbit was the ultimate value of RMS residuals Oi  Ci (SLR observations minus computed distance from station to satellite) (Rutkowska and Jagoda, 2010a): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 ðOi  C i Þ RMS ¼ n1

ð1Þ

where i = 1, 2, . . . , n – successive normal point. The iterated process of correcting the orbit was finished after the following condition had been fulfilled: fRMSðmÞ  RMSðm  1Þg < 0:01 cm

ð2Þ

where m is an iteration number In this analysis the following values of RMS residuals for STELLA and STARLETE satellites (for 130 orbital arcs in the last iteration) were obtained: RMS(STELLA) = 3.11 cm, RMS(STARLETTE) = 2.40 cm. After the convergence of iterative process of orbit correction according to formula (2) was acquired the elastic Earth parameters were estimated. Therefore the observation equation was formulated (3) (Rutkowska and Jagoda, 2010a), to be subsequently solved by the Bayesian least squares method. All the unknowns occurring in the observation equation were determined in a joint adjustment. ( ) n X @qi @qi @qi ðOi  C i Þ ¼  De þ Dh2 þ Dl2 þ dQi @e @h2 @l2 i¼1 ð3Þ where (Oi  Ci) – SLR observations minus computed distance from station to satellite, i – number of measurement, qi – the i-th SLR measurement, De – corrections for satellite position, velocity and other unknowns connected with the satellite orbit (coefficient of the solar radiation pressure, atmospheric drag, range biases, accelerations) and the station positions, velocities, Dh2 – correction for the Love number h2, Dl2 – correction for the Shida number l2, dQi – error of observation associated with the ith measurement.

@qi i The @h ; @q quantities occurring in observation Eq. (3), @l2 2 indispensable for estimating Love’s h2 and Shida l2 numbers are computed by differentiating formulas (4) which describe the motion of observatory stations as resulted from tidal forces (Diamante and Wiliamson, 1972):

Fig. 1. Block diagram of the adjustment of the Love and Shida numbers h2 and l2.

M. Jagoda, M. Rutkowska / Advances in Space Research 52 (2013) 633–638 " #  

3 X    GM j a4e   ^ ^ j^rsta 2  h2 X sta  j þ 3 h2  l 2 R ^ X R 3l r 2 j sta GM E d 3j 2 2 j¼2 " #  

3 4 X GM j a      e ^ j^rsta Y j þ 3 h2  l2 R ^ j^rsta 2  h2 Y sta DY ¼ 3l2 R 3 GM E d j 2 2 j¼2 " #  

3 4 X       GM j ae ^ j^rsta Z j þ 3 h2  l2 R ^ j^rsta 2  h2 Z sta DZ ¼ 3l2 R 3 GM E d j 2 2 j¼2

DX ¼

ð4Þ

GMj GME ae dj ^j R ^rsta h2 l2

– gravitational parameter for the Moon (for j = 2) or Sun (for j = 3), – gravitational parameter for the Earth, – equatorial radius, – distance to the Moon (for j = 2) or Sun (for j = 3), – the unit vector from the geocenter to the Moon (for j = 2) or Sun (for j = 3), – the unit vector from the geocenter to the station, – nomial second degree Love number, – nomial second degree Shida number.

Quantities

@qi @h2

;

@qi @l2



3 X @X sta GM j a4e 1 ^ 2 ^  ^ R X R ¼3 ð^ Þ  ð Þ X r r st j j j st sta ; ae @l2 GM E d 3j j¼2

3 X @Y sta GM j a4e 1 ^ 2 ^  ¼3 ð^rst Rj ÞY j  ðRj^rst Þ Y sta ; ae @l2 GM E d 3j j¼2

3 X @Z sta GM j a4e 1 ^ 2 ^  ^ R Z R ¼3 ð^ Þ  ð Þ Z r r st j j j st sta ae @l2 GM E d 3j j¼2

8 9 10 11 12 13

Station

Number ID for station

Number of normal points STELLA

STARLETTE

Herstmonceux Yarragadee Simosato McDonald Greenbelt Wettzell Monument Peak Hartebeesthoek Grasse Riga Borowiec Changchun Graz Lustbuehel Shanghai Solar Village Mount Stromlo Beijing Potsdam

78403501 70900513 78383602 70802419 71050725 88341001 71100412

3499 16339 2660 574 2646 6886 4342

8290 27645 4005 1332 5015 12073 8390

75010602 78353102 18844401 78113802 72371901 78393402

1398 151 121 118 2182 8492

2700 401 1028 1290 2650 14211

78372805 78325501 78259001 72496101 78418701

189 4075 4142 2032 1911

252 7426 14072 6173 4715

ð5Þ ð5aÞ

where

3 @X sta X GM j a4e 3 ^ 1 2 ^ R ¼ ð Þ  r X sta ; j st 3 @h2 GM E d 3j 2ae j¼2

3 @Y sta X GM j a4e 3 ^ 1 2 ¼ ðRj^rst Þ  Y sta ; 3 @h2 GM E d 3j 2ae j¼2

3 X GM j a4 3 e ^ j^rst Þ2  1 Z sta frac@Z sta @h2 ¼ R ð 3 GM E d 3j 2ae j¼2

1 2 3 4 5 6 7

14 15 16 17 18

are expressed as follows:

@qi @qi @X sta @qi @Y sta @qi @Z sta ¼ þ þ ; @h2 @X sta @h2 @Y sta @h2 @Z sta @h2 @qi @qi @X sta @qi @Y sta @qi @Z sta ¼ þ þ @l2 @X sta @l2 @Y sta @l2 @Z sta @l2

Table 1 Number of normal points of STELLA and STARLETTE satellites obtained at each observatory station within the period of 01.03.2005– 01.01.2007. Nb.

where

635

ð6Þ

ð6aÞ

i The @q quantity, associated with orbital arc, is obtained @e by numerical integrating of a satellite orbit. The process of estimating Love and Shida numbers was conducted by sequential method. In the first phase elastic parameters were calculated separately for each orbital arc. The further steps consisted in adding arcs, one by one. Each time elastic parameters were calculated anew. Their priori values (h2 = 0.6078, l2 = 0.0847) were taken from IERS Technical Note No. 36 (Petit and Luzum, 2010). All computations relating to the estimation of satellite orbits and Love and Shida numbers were made in iter-

Table 2 Force model used in the solution. Dynamic model Gravitational field EGM2008 (2159, 2159), aea_e = 6378136.3 m, GME = 398600.4415 km3/s2, (Pavlis et al., 2008) Solid Earth and ocean tide model EGM96, (Lemoine et al., 1998) The gravitational fields of the planets: Venus, Mars, Jupiter, Saturn. Planetary Ephemerides JPL DE200, (Standish, 1990) The atmospheric drag (Mass Spectrometer Incoherent Scatter) MSIS86, (Hedin, 1987) CR direct solar radiation pressure coefficient (adjusted one value for each orbital arc) CD atmospheric drag coefficients (adjusted five values per week) Albedo and infrared Earth radiation, (Melbourne et al., 1983) Relativistic effects, (McCarthy et al., 1993) Accelerations in along-track, cross-track and radial directions (for 7day intervals) Reference frame Precession according to IAU 2000, (McCarthy and Petit, 2004) Nutation according to IAU 2000, (McCarthy and Petit, 2004) Pole tide, (McCarthy et al., 1993) Ocean loading deformation, atmospheric pressure loading deformation, (McCarthy et al., 1993) Processing model Mendes-Pavlis model for troposphere delay, (Mendes and Pavlis, 2004) Center of mass correction equal to 7.5 cm, (McCarthy et al., 1993)

ative process and with use of GEODYN II NASA/GSFC software (McCarthy et al., 1993). The applied models of forces are presented in Table 2. 3. Results This paper presents the adjustment of the global elastic parameters h2, l2 obtained from a individual solution based

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on STARLETTE and STELLA data, and its comparison with another solution we computed for LAGEOS1 and LAGEOS2 two years ago (Rutkowska and Jagoda, 2010a). Computations are performed in two parts and a few steps. In the first part, arc parameters such as the state vector at initial epoch, the direct solar radiation pressure scaling coefficient CR, atmospheric drag coefficients CD and empirical accelerations are determined. After convergence of this part, all above described values and enclosed parameters h2, l2 in the combined solution are determined in the second part. We adopted a sequential method which allows the convergence and stability of the solution to be observed. Our criterion for convergence of the results as obtained for two independent satellites is the difference from the given parameter-value be no bigger than 10%. We followed the generally accessible literature according to which the maximum discrepancies in the obtained parameters of elasticity are up to 10–15%, as shown in our previous publication (Rutkowska and Jagoda, 2010a). In the first step, the elastic parameters were adjusted for two orbital arcs. In subsequent steps, arcs 3 and further were included one after another using this sequential method. In each step, the parameters were adjusted once again, enabling the stability of the solution to be observed. The results of this analysis are shown in Figs. 2 and 3 for h2 and l2 separately. The time interval of observations is not

limited, in general, but we assumed it to be 2.5 years (the same like for LAGEOS1 and LAGEOS2 (Rutkowska and Jagoda, 2010a)). The final adjusted values h2 are equal to 0.6163 ± 0.0037 for STELLA and 0.6048 ± 0.0025 for STARLETTE. The adjusted parameters h2 and their errors achieve stability at about the 18 month time interval, as shown in Fig. 2. These should therefore be considered the minimum intervals necessary for estimation of the Love number. The estimated values h2 differ from nominal value h2 by about 1.4% and 0.5%, respectively. There is visible convergence of results obtained for both satellites – the difference in final values h2 be 0.0115, that is approximately 2%. Unfortunately, estimation of Shida number l2 values was unsuccessful for the low satellites STELLA and STARLETTE. The final adjusted values l2 are equal to 0.0176 ± 0.0017 for STELLA and 0.1151 ± 0.0010 for STARLETTE. In Fig. 3 we can see lack of convergence of results for both satellites – the difference being 0.0975 which is higher than nominal l2 value. The difference between the obtained l2 quantities and nominal l2 value is 80% for STELLA and 36% for STARLETTE. It implies that it was impossible to estimate l2 parameter by the data these satellites within the fixed time interval. The values of horizontal displacement of earth masses in effect of tidal forces (as described by Shida l number) are by far lower and more difficult to measure than radial dis-

Fig. 2. The sequential solution for the Love’a number h2 estimated for the individual analysis based on data for STELLA (black squares) and STARLETTE (white squares). The final value of the Love number is equal to 0.6163 ± 0.0037 for STELLA and 0.6048 ± 0.0025 for STARLETTE data for a 2.5 years time interval from January 3, 2005 to July 1, 2007.

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Fig. 3. The sequential solution for the Shida number l2 estimated for the individual analysis based on data for STELLA (black squares) and STARLETTE (white squares). The final value of the Shida number is equal to 0.0176 ± 0.0017 for STELLA and 0.1151 ± 0.0010 for STARLETTE data for a 2.5 year time interval from January 3, 2005 to July 1, 2007.

placements (as described by Love’s h number). For that reason the relative error of the determining parameter l is rather big, and many scientists who deal with tidal parameters have resigned of their determination (mostly confining to theoretical determination of this parameter), as demonstrated in Rutkowska and Jagoda (2010a). As mentioned above, this is the accuracy of determining a satellite orbit which influences the values of h2 and l2. The orbit is most accurately determined in a radial direction (the accuracy up to several mm) which correlates with tidal displacements described by the Love’s h number. The Shida l number, on the other hand, is influenced by a satellite’s acceleration along the track. For the LAGEOS satellites it is relatively not big (no atmospheric influence is noted), and for the low satellites STELLA and STARLETTE along the track the resistance of atmosphere causes a huge error in determining their orbits (up to several cm). This may impact big discrepancies in the l2 values calculated from the observation of low and high satellites.

When comparing the values of Love and Shida numbers as listed in Table 3: h2, l2 obtained from the LAGEOS1 and LAGEOS2 data (Rutkowska and Jagoda, 2010a) with the values of numbers obtained from the data of low STELLA and STARLETTE satellites, it can be noticed that the biggest estimation convergence was obtained for h2 parameter to have been estimated by the STELLA and LAGEOS1 data – the difference in the acquired values be about 0.20% which means it is not greater than the value of error. Because for h2 this discrepancy is on the level of the formal error, in our evaluation the solution for low satellites can be used to support the high solution. Unfortunately, the parameter l2 has to be adjusted on the basis of the high satellites only, as the discrepancies between solutions are very high and the solution for low satellites is unacceptable. Additionally, it was observed that the estimated h2 value is more compatible with its nominal value (IERS Technical Note No. 36) for low satellites, while the correct quantity of l2 parameter was obtained only by the high satellites data.

Table 3 Final values of elastic Earth parameters obtained from the SLR data of STELLA, STARLETTE, LAGEOS1 and LAGEOS2, conducted in the 2.5 years time interval since initial epoch on January 3, 2005. Elastic Earth parameters

STELLA

STARLETTE

LAGEOS1 (Rutkowska and Jagoda, 2010a)

LAGEOS2 (Rutkowska and Jagoda, 2010a)

h2 l2

0.6163 ± 0.0037 0.0176 ± 0.0017 (inacceptable value)

0.6048 ± 0.0025 0.1151 ± 0.0010 (inacceptable value)

0.6151 ± 0.0008 0.0886 ± 0.0003

0.6152 ± 0.0008 0.0881 ± 0.0003

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When comparing errors in estimating Love’s h2 number by the STELLA and STARLETTE data as well as LAGEOS1 and LAGEOS2 it can be concluded that the values were obtained for LAGEOS satellites were lower – about 4-fold. It might have resulted from determining their orbits with smaller errors than that of STELLA and STARLETTE satellites. 4. Conclusion

1. Global values of elastic Earth parameters h2, l2 – as estimated by the SLR data for low satellites STELLA and STARLETTE are: h2(STELLA) = 0.6163 ± 0.0037, h2(STARLETTE) = 0.6048 ± 0.0025, l2(STELLA) = 0.0176 ± 0.0017 (unacceptable value), l2(STARLETTE) = 0.1151 ± 0.0010 (unacceptable value). 2. The h2 parameter estimated by the STARLETTE data presents a smaller error than parameter h2 estimated by the STELLA data. It is entailed by a bigger number of normal points of this satellite which were employed in the estimation. Stability in estimating h2 parameter can be noticed after passing: 68 weeks (h2(STELLA)) and 56 weeks (h2(STARLETTE)) within the assigned 2.5 years time interval. 3. It can also be noticed that the value of h2 parameter, as obtained from the data of high and low satellites, is largely compatible. The biggest compatibility can be observed for STELLA and LAGEOS1 satellites (discrepancy in h2 values be about 0.20%). 4. Because for h2 this discrepancy is on the level of the formal error, in our evaluation the solution for low satellites can be used to support the high solution. The errors accompanying estimation of h2 parameter from the data of low satellites are approximately 4-fold bigger than the errors accompanying estimation of h2 parameter from the high satellites data. 5. The minimum time interval to guarantee stability of estimating elastic Earth parameters and their errors is 18 months. This interval is compatible with those obtained for high satellites (Rutkowska and Jagoda, 2010a). It can be freely elongated producing no significant changes of Love and Shida numbers, and their errors.

Acknowledgments This paper has been supported by the Polish State Committee for Scientific Research under Grant No. N N526 152137.

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