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~~~~~~~~~~~~~~-~. . -- ~ k~ k?%. /_/~:.-.,:if(NASA-TM-X-66039) EARTH ZONAL HARM O N ~ ICN7-20 , -:7"" ~-.-%:;~'''elFROM~APID NUMERI~~ CALAAYI_-:.xFLN ' C . A . Waqnr 7::"1STELIT'].~'';,A RS NS) u._ "C X-553-72-341
EARTH ZONAL HARMONICS FROM RAPID NUMERICAL ANALYSIS OF LONG SATELLITE ARCS
Carl A. Wagner Trajectory Analysis and Geodynamics Division Geodynamics Branch
August 1972
Goddard Space Flight Center Greenbelt, Maryland e- CONTENTS
Page
ABSTRACT ...... v
INTRODUCTION ......
THE DATA ...... 2
PREPROCESSING ...... 3 ANALYSIS ...... 4 RESULTS ...... 6 CONCLUSIONS ...... 9
REFERENCES ...... 10 ACKNOWLEDGMENT ...... 12
Tables
Table Page
1 Satellites Used in Recent Zonal Solutions ...... 13 2 Satellite Element Data Used in Present Solution (WAG 72)...... 14 3 Zonal Coefficients From Recent Solutions ...... 25 4 Total Weighted RMS Residuals in ROAD Solutions for Long Satellite Arcs ...... 26
Illustrations
Figure Page
1 Satellites Used in Present Solution ...... 27 2 Secular Rates and Amplitude of Eccentricity Oscillation Due to Even and Odd Zonals ...... 28 3 Differences in Secular Rates ...... 29 4 Differences in Amplitudes of Eccentricity Oscillation...... 30 5 Zonal Geoid Profile for WAG 72 ...... 31 6 Differences in Zonal Geoid Profiles ...... 32
Preceding page blank 1 iii RCE)ING PAGP e BLANX NVOTF
EARTH ZONAL HARMONICS FROM RAPID NUMERICAL ANALYSIS OF LONG SATELLITE ARCS
ABSTRACT
A zonal geopotential is presented to degree 21 from evaluation of mean elements for 21 satellites including 2 of low (< 200) incli- nation. Each satellite is represented by an arc of at least one apsidal rotation. The lengths range from 200 to 800 days. Differ- ential correction of the initial elements in all of the arcs, together with radiation pressure and atmospheric drag coefficients, is ac- complished simultaneously with the correction for the zonal har- monics. The satellite orbits and their variations are generated by numerical integration of the Lagrange equations for mean elements. Disturbances due to precession and nutation of the earth's pole, atmospheric drag, radiation pressure and luni-solar gravity are added at from 1- to 8-day intervals in the integrated orbits.
The results agree well with recent solutions from other authors using different methods and different satellite sets. These comparisons show the zonal coefficients are now known to better than 0.02 x 10- 6 (fully normalized) to at least as high as degree 10, but that above degree 10, many of the terms are not yet significantly different from zero.
Preceding page bank v EARTH ZONAL HARMONICS FROM RAPID NUMERICAL ANALYSIS OF LONG SATELLITE ARCS
INTRODUCTION.
The evaluation of the longitude averaged, or zonal, part of the geopotential has been advanced significantly since the advent of the artificial earth satellite. Radio tracking of the early Sputniks and Vanguards before 1960 established the gravitational oblateness of the earth (J2 ) to better than one part in ten-thousand and discovered the earth's significant "pear shapedness" 1, 2 3 (J 3 ). These spherical harmonic terms were calculated from their secular and long periodic perturbations of the Kepler elements of close earth satellites.
The even zonal terms (having equatorial symmetry) were the first to be evaluated to high degree with significant accuracy, from the easily observed secular movement of the node and perigee of the orbits.4 ,5 (For Jeven > 2, these secular rates are of order 0.010 /day.) The odd zonal terms (asymmetric with respect to the equator) were more difficult to evaluate, having no secular effects on satellite orbits. Their determination required observation of perigee altitude oscillations of the order of only 10 km over apsidal rotation periods of the order of months. Nevertheless, by 1964 Kozai had "determined" a complete zonal harmonic field to J,4 from 9 satellites,6 and by 1969 King-Hele had evalu- ated odd zonal terms as high as J31 from 22 satellites.7 A complete zonal field to J 2 1 was available by 1969, due to Kozai, ultimately from the precise Baker- Nunn tracking of 12 satellites.s By 1971 this latter determination was improved by the addition of 3 satellites of low inclination.9
All the above solutions for the zonal geopotential have used semi analytic methods in rationalizing the observed perturbations (of mean Kepler elements) with theory. The word 'semi' is used since the actual data reduction in these determinations involved many empirical and numerical procedures, especially in the removal of the effects of drag and radiation pressure.1'0 No adequate theory exists for these perturbations on many orbits. Purely numerical methods (of integration) have also been used to evaluate zonal harmonics directly from tracking data. The first comprehensive solution of this kind (to J7 ) was obtained from the Tranet doppler tracking of 5 satellites"l in 1965. However, in this determination tracking arcs were generally less than one week. The secular and long period zonal effects on the orbits were not well observed and the zonal solution is relatively weak particularly since only a few satellites were used.
1 More recently, at Goddard Space Flight Center, a quite respectable zonal field to J21 has been computed, using numerical integration, from weekly arcs of optical observations on over 20 satellites.'2 The strength of this solution is felt to arise primarily from the significant number of different orbits observed as well as the large number of arcs used (over 300).
However the determination of the zonal geopotential from satellite observa- tions using straight numerical integration of osculating elements will always suffer from excessive computation time if the secular and long periodic effects are to be well observed. On the other hand, the semi analytic methods, while extremely efficient for evaluating long term effects on mean elements, have always been subject to a considerable amount of approximation in both the theory and its application. The theory for the evolution of the elements is limited by the difficulty or impossibility of obtaining exact solutions. In the applications, only some of the element data is actually used. Furthermore, no assessment is made of the effects of the variability of the elements in a long arc on the geo- potential partial derivatives. However, great simplification and reduction of arbitrariness in the analytic methods can be achieved by analyzing the evolution of mean elements directly by numerical integration of their variations.
The integration of mean (or orbit averaged) Kepler elements is extremely rapid. 3 Furthermore the inclusion of complex disturbances, such as drag, radiation pressure and high order gravitational effects, is relatively straight foreward since the disturbance is integrated numerically. In 1970, a prelimi- nary application of this method for zonal determination was made to mean ele- ments determined from Minitrack interferometer data on only 2 satellites.'4
In this solution J2, J3 and J4 were recovered with good accuracy considering the limitations of the data. In the present solution a full complement of satellite orbits of all inclinations and a wide range of altitudes are analyzed for effects through J 2 1
THE DATA
My aim was to use at least as many satellites as harmonics evaluated, with a good distribution of inclinations and altitudes to separate terms of high from low degree. The 21 satellites chosen for analysis are shown in Table 1 (with pertinent orbit and spacecraft data). Their orbit characteristics are displayed in Figure 1. Arc lengths were chosen to have a minimum of one apsidal rotation, containing at least 25 element sets, to give a good description of the principal long periodic effect.
2 Two agencies were responsible for the mean element "observations" used; The Smithsonian Astrophysical Observatory (SAO) and Goddard Space Flight Center (GSFC). Mean elements for six SAO satellite arcs (Vanguard 2, Echo 1 Rocket, Transit 4A, Midas 4, GEOS 1, BE-B; supplied by Kozai l5 ) were deter- mined from precision reduced Baker-Nunn optical observations and were used in his 1969 solution. Three additional SAO satellite arcs contained mean ele- ments derived from field reduced Baker-Nunn observations (TELSTAR 1, ANNA 1B and Explorer 11). The GSFC mean elements (on the remaining 12 satellites) were determinred predominantly from Minitrack interferometer observations except for the cases of PEOLE and GEOS 2 where some optical and laser tracking data was also used. About half of the GSFC satellite arcs were in King-Hele's 1969 odd zonal solution.7 Eight of the satellite orbits here were also analyzed (with seven others) in Cazenave's 1971 solution. 9 This French determination was the first to include orbits of low inclination, of which PEOLE and SAS are also used here. Indeed the present solution most resembles the French, as will be seen later.
Finally, the full complement of mean element data analyzed (after pre- processing, to be described) is given in Table 2.
PRE PROCE SSING
The description of SAO mean elements is found briefly in the Smithsonian reports from which the elements for TELSTAR, ANNA 1B and EXPLORER 11 were extracted. 16 17, 18 Gaposchkin gives a more detailed discussion of them.'9 From this it is evident that their definition only requires a shift of the nodal reference (from the equinox of 1950.0 as given in the reports) to the equinox of date, in order to be compatible with the orbit integrator used in the numerical analysis. However, the mean semimajor axes were not taken directly from the reports or from Kozai's data; they were calculated from the given mean mean motions (n) by a variant of Kepler's law, scaled to the earth constants used in the integrator.'3 ' p . 134
The original elements determined at GSFC were of two kinds, Brouwer mean (double primed) and Cowell osculating. The Brouwer double primed elements were converted to single primed means by adding back the long period terms given by Brouwer 20 in 1959. The single primed means are essentially osculating elements with only the short period terms removed. They are precisely the co- ordinates analyzed by the integrator (to be described). The zonal constants implicit in the original GSFC mean elements for TIROS 9, TIROS 5, OSO 3, EXPLORER 27, FR-1, ESSA 1, PEGASUS 3 and EGRS-3 were: 106 J2 = 1082.19, 6 6 6 10 J 3 = -2.29, 10 J4 = -2.12 and 10 J5 = -0.23. For ISIS 1 and SAS 1 the constants
3 were 10 6 J 2 = 1082.48, 106J 3 = -2.56, 106J4 =-1.84and 10 6J 5 = -0.06. For the GSFC satellite arcs PEOLE and GEOS 2 osculating elements were originally determined by a Cowell type numerical integrator. First order short period terms in the 6 Kepler elements, due to J2 through J 4 , and second order terms in the semimajor axis due to J2 (from Brouwer2 0 and Kozai2 1 ) were sub- tracted from the original PEOLE elements. All relevant first order short period terms (zonal and non-zonal) through J 4,4 (from Kaula2 2 ' P.40) were re- moved analytically from the original GEOS 2 elements. The remaining short period variations were removed by numerically averaging the residual effects of the geopotential, luni-solar gravity, radiation pressure and atmospheric drag over one orbit revolution.2
The set of mean elements so produced (in Table 2) was now ready to be analyzed for zonal geopotential effects.
ANALYSIS
The ROAD (Rapid Orbit Analysis and Determination) program was the 4 principal analytic tool in this investigation ' 25 This program integrates numerically the Lagrange planetary equations 2 2' P 29 for mean (or orbit aver- aged) Kepler elements by considering as disturbances only effects not in the ' mean anomaly of the satellite. For the geopotential, Kaula's form2 2 ' P 37 is employed because it enables the analyst to select the (first order) long period (and secular) terms simply by choosing thet, m, p, q indices such that f - 2p +q= 0. The geopotential analyzed in ROAD, in the standard form of Legendre polynomials Pt' (P), is:
r where i/ is the earth's gaussian constant (3.9803 x 10 5 km 3 /sec2 in the program), re is the equatorial radius of the earth (6378.16 km in the program), TP is the geocentric latitude, r the distance to the earth's center of mass and the J's are the unnormalized zonal coefficients.
The ROAD integrator also considers orbit averaged disturbances from atmospheric drag, radiation pressure, the interaction of short period J 2 effects, and the effects of precession and nutation of the earth's polar axis. Direct luni- solar gravity perturbations are handled in the same manner as the geopotential, by expressing them as functions of the Kepler elements of both satellite and third body.2 6 Only first order long period or secular terms are chosen (by index selection) for integration.
4 The program also integrates variational equations for the six epoch Kepler elements of the satellite orbit; a (semimajor axis), e (eccentricity, I (inclination), ow (argument of perigee), N (right ascension of the ascending node), and M (mean anomaly). In fact, full differential correction capability exists for up to 50 geo- potential constants common to any number of satellite arcs as well as an extended state for each arc. The extended state can consist of model radiation pressure and atmospheric drag coefficients or a-large number of Kepler elements rates to absorb model errors empirically.
The drag disturbance acceleration is modeled as (1/2) (CD) p v 2 A/m where CD is the empirically derived coefficient of drag, p is the atmospheric density, v is the satellite's velocity relative to an atmosphere rotating with the earth and A/m is the satellite's projected area to mass ratio. The radiation pressure acceleration is formulated as (CR) (I/C) (A/m), where CR is the radiation pres- sure coefficient, I is the solar flux, C the velocity of light and A the projected area in the satellite-sun line. The data used in these "state" and geopotential corrections are the "observed" Kepler elements themselves which are "best fitted" in a least squares sense to the trajectories from the ROAD integrator.
The theory for the evolution of these elements due to the geopotential, is well known since the pivotal paper by Brouwer2 0 in 1959. The even zonals cause distinct secular rates in the node and perigee as well as long period oscillations in all the elements (except a) with a fundamental period of half a perigee rotation. The odd zonals cause long period oscillations in all the elements (except a) with a fundamental period of the rotation of perigee. Therefore the minimum span of data which should be considered in a zonal solution is a rotation of perigee for all orbits. To distinguish even zonal long period effects, a minimum of 12 ele- ment sets per half rotation period was felt to be necessary, or 25 sets per arc. In most cases these minimum specifications were far exceeded. But in any case, as Kozai points out, 10, pp. 8 3 3 - 8 3 4 there is a limit to the accuracy of the zonal solution, set by the accuracy with which the satellite elements are known. The ROAD solution for the initial elements simultaneously with the geopotential auto- matically accounts for the effects of these uncertainties. Furthermore the ROAD trajectories and partial derivatives continuously refer to an orbit closest to the originally observed data. (An example of a "run" where some of the epoch ele- ments (in particular; e, I, a, N) were constrained is given in Table 3 to show the possible extent of this error. )
Theoretically, the 21 satellites examined here should supply at least 42 independent strong condition equations for the (10) even zonals (to J 20 ) from the node and perigee data. But actually this number is reduced considerably because of the many high altitude and nearly circular orbits examined. Nevertheless, the highest correlation coefficient for even zonals in the present solution is only 0.65 between J 4 and J1 4 . Examination of Kozai's normals in his 12 satellite 1969
5 solution8 shows many correlations greater than 0.90 between even zonals. The conditioning is much poorer with odd zonals. Here, even though long period effects are present in 5 elements, they are poorly observed (compared to the secular effects from the even zonals). Furthermore, the oscillations in e and I are not independent of each other (i.e. they differ only by an orbit constant) to provide separation of odd zonal effects. lo0, p 8 39 In fact, the near circular orbits (of which there are 7 here with e < .01) provide only 2 significant inde- pendent oscillations due to odd zonals (in e or I and N) since co and M are not well determined. The result is that the separation of the odd zonals is quite poor in this solution with 8 correlation coefficients above 0.70, but none above 0.92. Nevertheless, this conditioning for odd zonals is still better than in Kozai's 1969 solution (where many correlations above 0.95 existed) because of the greater number of satellites employed and the use of all element data were given.
RESULTS
The most satisfactory least squares solution for 20 zonal coefficients from the data in Table 2, is shown in Table 3 (as WAG 72). The accompanying radi- ation (CR) and drag (CD ) parameters determined for the 21 satellites by the same data is shown in Table 1. Except for the radiation pressure coefficient of EGRS-3, all the "extended state" parameters appear to be well determined. They are not significantly far from the theoretical average values of 1.5 for CR (in diffuse reflection from a sphere) and 2.3 for CD (in free molecular flow about a sphere). The area to mass ratios (in Table 1) were calculated from King-Hele's spacecraft/orbit data. 2 7 Considering all the uncertainties in the effective space- craft geometry and surface characteristics and the model atmosphere, it is re- markable that the program determined such realistic coefficients.
Also listed in Table 3 are the formal standard deviations of the ROAD solu- tion and the results of 4 other perturbations of the principal solution to test its sensitivity to various error sources. The first, a truncation test, lists the changes in the coefficients (absolute values) when only J 2 through J1 9 are solved for. The "truncation error" for J20 and J 2 1 are estimated on the basis of the worst changes in the lower degree set. The second test fixes the starting elements E, I, wo and N at values determined from the preliminary orbits used for data editing. Changes in the solution (absolute values) with this restriction gage the likely bias in the final result due to these initial elements adjusting to absorb some of the geo- potential effects. The third perturbation reduced the weight of the mean anomaly data in the solution by 1/2. Even though the average quality for M was only 0.30 in the 21 arcs, decreasing it to 0.60 had a significant effect on the solution. This data is vital to the determination of the semimajor axis, whose uncertainty is a 2 2 significant contributor to the zonal error in the semi-analytic solutions. ' P ' 116
6 In future ROAD evaluations, the mean anomaly data will be weighted as strongly as allowed by other unmodeled effects (i.e. drag, and non-zonal reso- nance). In so doing, 'a' will be determined more strongly and the secular and long period zonal effects in M will be better observed leading to a better con- ditioned solution.
A final solution perturbation was made to test the influence of neglected short period (_14 day) lunar terms and integrator error. The basic solution did not include these lunar terms because most of the original orbit data were already mean elements determined over weekly arcs. These elements could be expected to have either removed these terms (analytically, as in the 9 SAO satel- lite arcs) or largely smoothed over them (as in all the GSFC arcs except PEOLE and GEOS 2).
The numerical integrator, a fixed step 8th order predictor-corrector process, employed a one day step size in the basic solution. The final perturbed solution had a 1/2 day step size and, as seen in Table 3, showed mostly insignificant changes. The root sum of squares of the formal standard deviation (s.d.) and the changes in the 4 perturbed solutions, as shown in Table 3, is to be regarded as a more realistic estimate of the lcr values of the zonal coefficients. The zonal solution itself is presented dynamically in Figure 2 by the secular rates (L + 1N) and eccentricity oscillations induced on a 14 revolutions/day satellite with e = 0.1. The secular rates were calculated from the Lagrange planetary equations (with 0=O., N = 0., M = 0.) considering only even zonal effects (including those of J2) to order e5 . The eccentricity amplitude was calculated as e/& from the Lagrange equations including only odd zonal terms except those in J 2 and J . These per- turbations are very close to those calculated by Kozai in his 1966 review paper.l ° (In fact the orbit characteristics were chosen to match Kozai's and facilitate com- parison with his results).
Two other recent zonal fields to J21 were compared with this new one, from diverse sources. The first, called FR 71, was the French solution.9 It combined Kozai's normals from his 12 satellite 1969 solutions with condition equations for secular rates and amplitudes of long period oscillations due to zonal effects on the 3 low inclination orbits of SAS 1, DIAL and PEOLE. Another 1971 solution, from SAO, 28 is also listed in Table 3 to show the divergence between two deter- minations from essentially the same orbits and (analytic) method. SAO 71 also starts with Kozai's 1969 normals but adds condition equations on only SAS 1 and PEOLE. In spite of the near identity of the data and processing, these two results differ by 0.016 x 10-6 rms over the fully normalized values of the 20 zonal coef- ficients. (The normalized coefficients, of greater physical significance,2 2 P- 7 are obtained from the unnormalized ones on dividing by (2· + 1) 1/2.) The French solution seems to be somewhat superior to the SAO. It has fewer consecutive changes of sign in the ill conditioned odd zonal set and compares more favorably
7 with GEM 2, another recent zonal field (listed in Table 3) determined by a wholly distinct method. GEM 2 was derived simultaneously with all the non-zonal geopotential terms through (16,16) by a combination of normals found from 1; directly fitting numerically integrated orbits to precision optical data, with 2; surface gravity information.l2 GEM 2 contains no low inclination orbit informa- tion but is a well conditioned solution (compared to the other strictly satellite results) having a highest correlation coefficient of only 0.48. Nevertheless, the new ROAD solution appears to be closest to FR 71, both in the rms normalized coefficient difference (0.012 x 10- 6 ) and in tests of individual arcs on 27 satel- lites (Table 4).
These arcs include the 21 used in the ROAD solution. Overall weighted rms residuals are listed with all 6 Kepler elements fitted as data weighted by the quality shown in preliminary "screening" orbits. All arc lengths were over 75 days. Full orbit determinations were performed on this data with these fields, including a solution for drag and radiation pressure coefficients. For the WAG 72 field, the average rms 'residual' in the 21 solution arcs is significantly lower than with orbits using the comparison fields, but is still considerably greater than 1.0. There remain some significant secular and long periodic residuals in the new solution which require better modeling from the geopotential (more zonal and resonance terms) the effects of atmospheric drag, and luni-solar gravity (both direct and indirect or tidal terms).
It was a great surprise that GEM 2 did so well with this data. It was a solu- tion tailored to short period satellite terms and short wavelength surface gravity effects (50 x 5° anomalies). Yet Table 4 shows it did almost as well with these long arcs of satellite data as FR-71 which is a long arc solution exclusively. After-the-fact, one might say the great bulk of diverse data in GEM 2 produced a credible representation even for very sensitive long term satellite dynamics. Yet the result is still surprising and gratifying. In fact, for the PEOLE arc, GEM 2 was significantly superior to FR 71 even though it did not have the benefit of low inclination satellite data. The only surprise in the 6 arcs not used in the present solution, was the remarkable faithfulness of WAG 72 in recovering the DIAL data compared to FR-71 which (again) employed data from this low inclina- tion satellite.
On the other hand the results of the other "new" arcs show little discrimina- tion between these 3 recent solutions. This is further emphasized in Figures 3 and 4 which show differences (between the recent solutions and WAG 72) of secular rates and long period oscillations, calculated for a "typical" geodetic satellite and for the actual (observed) data on the six "new" satellites. The measured differ- ences are 'residual' rates and amplitudes from those calculated by WAG 72 minus those actually observed on the satellites, as seen in the orbit fits (using WAG 72) with the data from these satellites. Some of the observed secular rates are not
8 well predicted by any field while most of the odd zonal oscillations were too poorly determined to be of much help in discriminating between them. More well determined satellite orbits of low altitude (a < 1.1 e.r.) will have to be tested and used to further improve the zonal geopotential. But it does appear that the present field is a significantly improved low inclination solution over any of the other recent fields.
Finally, in Figure 5 is the geometric representation of the "new" field as a zonal geoid height profile with respect to an ellipsoid of flattening of 1/298.255. Differences of this profile with FR-71 and GEM 2 again show that the "new" field is "closest" to FR-71 (rms difference -0.3m).
CONC LUSIONS
A new "fixed" zonal harmonic field (to J2 1 ) has been calculated with espe- cially superior characteristics for predicting the evolution of low inclination satellite orbits. Overall, the recent zonal solutions appear to be capable of reproducing the secular rates of "new" orbits to better than 7 x 10-4 degrees/day and the oscillations of eccentricity to better than 0.7 x 10-4. The absolute accu- racy of individual coefficients (fully normalized) varies from less than 2 x 10-8 for t Z 10 to less than 4 x 10-8for { z 21. However it is clear from an examina- tion of the various solutions and errors in Table 3, that for 't 5 15, the coeffi- cients are not yet determined to be significantly different from zero.
It should be noted that the earth induced luni-solar tides (time varying) are not taken into account in this assessment. They will reduce the earth fixed or constant part of J2 by about 1 x 10-8 and have lesser effect on the other even zonals. 2 9
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13. C. A. Wagner and B. C. Douglas, "Resonant Satellite Geodesy by High Speed Analysis of Mean Kepler Elements," In: Dynamics of Satellites (1969), Springer-Verlag, New York (1970).
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16. I. G. Izsak, "Satellite Orbital Data," Simthsonian Astrophysical Observatory Special Report No. 86, Cambridge, Mass. (1962).
17. "Satellite Orbital Data," SAO SR No. 126, Cambridge, Mass. (1963).
18. "Satellite Orbital Data," SAO SR No. 142, Cambridge, Mass. (1964).
19. E. M. Gaposchkin, "Differential Orbit Improvement (DOI-3)," SAO SR No. 161, Cambridge, Mass. (1964).
20. D. Brouwer, "Solution of the Problem of Artificial Satellite Theory without Drag," Astron. J., 64, 378-397 (1959).
21. Y. Kozai, In: Astronomical Journal, 67, pp. 446, (1962).
22. W. M. Kaula, Theory of Satellite Geodesy, Blaisdell; Waltham, Mass. (1966).
23. B. C. Douglas, Private Communication, 1972.
24. R. G. Williamson and P. J. Dunn, "ROAD Program Description and Mathe- matical Modifications," Wolf Research and Development Corporation Reports, Riverdale, Md. (1970 and 1972).
25. C. A. Wagner, "Low Degree Resonant Geopotential Coefficients from Eight 24-Hour Satellites," Goddard Space Flight Center Report X-552-70-402, Greenbelt, Md. (1970).
26. W. M. Kaula, "Development of the Lunar and Solar Disturbing Functions for a Close Satellite," Astron. J., 67, 300-303 (1962).
11 27. D. G. King-Hele, H. Hiller and J. A. Pilkington, "Table of Earth Satellites," Royal Aircraft Establishment Technical Report No. 70020, Farnborough, Hants., England (1970).
28. E. M. Gaposchkin, Private Communication, 1971; Presented at: Annual Meeting of the American Geophysical Union, Washington, D. C., April 1971.
29. D. Smith, Private Communication, 1972.
ACKNOWLEDGMENT
I would like to give special thanks to: Ron Williamson of Wolf Research and Development Corporation, Riverdale, Maryland for the bulk of the develop- ment of the ROAD Program, and Mary Boumilla and Chris Massey of Computing and Software, Inc., for making the ROAD runs that got the job done.
12 TABLE 1 SATELLITES USED IN RECENT ZONAL SOLUTIONS
ZONAL SOLUTIONS SATELLITE/ORBIT CHARACTERISTICS
2 72 71 (Km) (Km) (cm 2 /gm) (2.3) (1.5)
SAS-1 X X 3.0 530 560 0.002 0.11 2.2 1.4 DIAL (1970-17A) X 5.4 310 1600 0.09 PEOLE (1970-109A) X X 15.0 530 760 0.02 0.20 1.7 1.4 COURRIER 1B (1960-7A) X X 28.3 EXPLORER 11 (1961-7A) X 28.8 490 1800 0.086 0.10 1.6 1.1 PEGASUS 3 (1965-60A) X 28.9 510 550 0.002 0.11 2.2 1.2 OSO 3 (1967-20A) X 32.9 910 950 0.002 0.05 1.6 2.0 VANGUARD 2 (1959-1A) X X X 32.9 550 3280 0.165 0.21 1.2 1.1 VANGUARD 3 (1959-5A) X 33.4 OVl-2 X 35.7(144.3) )I DI-D X 39.5 DI-C X 40.0 BE-C (EX.27) (1965-32A) X X 41.2 940 1320 0.025 0.17 2.4 1.3 TELSTAR-I (1962-29A) X X X 44.8 950 5630 0.242 0.08 2.7 1.3 ECHO-I ROCKET (1960-9B) X X X 47.2 1500 1670 0.011 0.21 2.6 1.0 GRS (1963-26A) X X 49.7 ANNA-lB (1962-60A) X X X 50.1 1080 1200 0.008 0.06 2.3 1.9 TIROS 5 (1962:25A) X 58.1 590 960 0.026 0.06 2.1 2.3 TIROS 7 (1963-24A) 58.2 GEOS 1 (1965-89A) X X X 59.4 1110 2380 0.072 0.07 2.7 1.4 TRANSIT 4A (!961-15A) X X X 66.8 880 980 0.008 0.11 2.7 1.3 SECOR 5 (1965-63A) X 69.2 AGENA ROCKET (1964-1A) X X 69.9 EGRS-3 (1965-16E) X 70.1 900 940 0.003 0.10? 3.3 -0.2 GEOS 2 (1968-2A) X X 74.2(105.8) 1080 1580 0.032 0.04 3.1 1.3 FR-1 (1965-101A) X 75.9 740 750 0.001 0.11 2.8 1.2 BE-B (1964-64A) X X X 79.7 890 1080 0.013 0.18 2.8 1.4 ALOUETTE 2 (1965-98A) X 79.8 ESSA I (1966-8A) X 82.1(97.9) 710 850 0.010 0.06 2.3 1.3 TIROS 9 (1965-4A) X X 83.6(96.4) 690 2560 0.117 0.05 1.4 0.3 MIDAS 4 (1961-28A) X X X 84.2(95.8) 3500 3740 0.012 0.06 1.9 1:.6 OGO-2 (1965-81A) X X I 87.4 ISIS-I (1969-9A) X 88.4 580 13520 10.175 0.10 1.0 1.8 OSCAR 7 X 89.7 5-BN2 (1965-48C) X 90.0 TOTALS 23 21 15
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O O O( IL LL 00OD I I I I I I I I I ii I otN C) t- (0 CD CO0I) C) 0 - N (0 1 CO )0O L -- NN 25 TABLE 4 TOTAL WEIGHTED RMS RESIDUALS IN ROAD SOLUTIONS FOR LONG SATELLITE ARCS WITH FIELD USED 21 SATELLITES USED IN PRESENT PRESENT SOLUTION SOLUTION FR 71 GEM 2 (WAG 72) VANG. 2 1.52 1.39 1.39 TRAN. 4A 2.90 3.01 7.40 MIDAS 4 1.63 2.18 2.30 BE-B 3.25 3.63 3.62 GEOS 1 0.62 0.56 0.56 ECHO 1 ROCKET 0.84 1.34 1.35 PEOLE 1.00 1.73 1.19 SAS 1 1.05 15.94 45.52 EXPLORER I 1.11 1.51 1.48 ANNA 1B 1.23 1.27 1.27 TELSTAR I 2.49 2.63 2.63 GEOS 2 3.34 7.99 9.77 EGRS 3 2.45 5.25 4.45 PEGASUS 3 1.45 1.80 1.91 FR-1 1.49 3.10 1.61 OSO 3 1.33 1.80 1.47 EXPLORER 27 1.47 4.09 4.02 ESSA 1 1.67 2.91 2.33 TIROS 5 1.27 3.53 3.32 TIROS 9 1.00 0.95 1.00 ISIS 1 1.44 2.26 2.16 AVG. RMS IN 21 ARCS 1.65 3.28 4.80 6 SATELLITES NOT USED IN PRESENT SOLUTION OAO 2 OAO2 0.82 0.81 0.85 (a = 1.12 e - 0.001) 1965-34A (a = 1.51 e = 0.05) 1.44 1.46 1.45 ALOUETTE 2 1.37 1.32 1.47 (a = 1.27 e = 0.15) COSMOS 382 (a = 1.60 e = 0.14) 2.97 3.00 DIAL (a = 1.15 e = 0.09) 1.79 3.09 7.59 EGRS 5 (SECOR 5) 2.40 2.36 2.49 (a = 1.28 e = 0.08) 26 0 0 I I I I I I i 0 0- ._ .0 I o 0 C._ o -o o I o LUw I 4: _ Lh q 0U E 0 .q_ O~ u-i 0 Z0 o c z C o o u z ow I o D ' I-a- C I9I U -x D LL o x o a wY 4,) U C_ .-- 0) vz " ! N I 0' 03 o z I 4 II 8 CO ~I 0l ~~~n~~~~~L~ E r~~~~~I' o I o I I I I I I I o 0 0 0 0 0 0 0 0: _0 0 Co to to m CJ (S33:193C) 'I 'NOI/VNI1lNI 27 15 14 7 13 Ah COMPUTED FOR 12 _ \ ORBIT: 6 ~11 \ a 1. 14 e.r. (n 14 revs./day) 10 e = 0.1 WITH WAG 72 FIELD 8 Ae VS I 4 7 5 4 ( 2 10I I 1 .- N VS I + 0 0 -2 -1 -3 -4 - -7 -8 . -4 -9 0 10 20 30 40 50 60 70 80 90 INCLINATION, I, (degrees) Figure 2. Secular Rates and Amplitude of Eccentricity Oscillation Due to Even and Odd Zonols 28 I DUE TO EVEN ZONALS, 60 AS COMPUTED FROM THE ZONAL FIELDS 60 -x WAG 72 MINUS: x \J\ GEM 2 50 I FOR AN ORBIT WITH: a = 1.14 e.r. 50 I \ xrx FR 71 e -0.1 -x l 401 \I 40 OBSERVED FOR LISTED SATELLITES NOT USED IN WAG 72 SOLUTION: 3o I \v 30 I I 201 20 a = 1.51 ALOUETTE 2 1965 - 34A e = 0.051 - x 1.27\ COSMOS 382 ( 0 14 0.15/ 10 _ \ 10 o I \ Z I x I 0 0 + - -d~~~ -10 t -10 EGRS 5a = 1.28 5e = 0.08/ -20 _ -20 IWAG 72 1 Ii II I I I I III I I III l I IFR 71 I I I I I II I I I I -30 - GEM 2 I II III I I I I I III -30 INCLINATIONS OF SATELLITES USED IN THE ZONAL SOLUTIONS -40 - DIAL (e: 0:09 -40 -50 - -50 I . I I -60) II I II I I II I I . I I -60 0 10 20 30 40 50 60 70 80 90 INCLINATION (degrees) Figure 3. Differences in Secular Rates 29 1.0 0.5 0To I 0 -0.5 -1.0 0 10 20 30 40 50 60 70 80 90 INCLINATION (degrees) Figure 4. Differences in Amplitudes of Eccentricity Oscillation 30 E -: -5 - -10 - -15 - -20 -25 I I I I I I I I -80 -60 -40 -20 0 20 40 60 80 LATITUDE (degrees) Figure 5. Zonal Geoid Profile for WAG 72 31 00 Lu -V -, I--X 0 0, a z 0 x o N0 Lu wTLL O0 L- _ C? 00 05 (Lu0 (u~) · CV 32 * U.S. GOVERNMENT PRINTING OFFICE: 1972-735-9E1/28