VOL. 5, NO. 12 GEOPHYSICALRESEARCH LETTERS DECEMBER1978 DETERMINATION OF THE GEOCENTRI½ GRAVITATIONAL CONSTANT FROM LASER RANGING ON NEAR-EARTH SATELLITES 1 2 • Francis 3. Lerch, Roy E. Laubscher, Steven M. Klosko 1 1 David E. Smith, Ronald Kolenkiewicz, Barbara H. Putney, 1 2 James G. Marsh, and Joseph E. Brownd 1 GeodynamicsBranch, GoddardSpace Flight Center Computer Sciences Corporation, Silver Spring, Maryland 3EG&GWashington Analytical ServicesCenter, Inc., Riverdale, Maryland Abstract. Laser range observations taken on earth plus moon Ms/(Me + Mm) of 328900.50+ the near-earth satellites of Lageos (a -- 1.92 .03. Assuming the AU and the IAG value of c, e.r.), Starlette (a -- 1.15 e.r.), BE-C (a = 1.18 this yields a value of GM of 398600.51 + .03 when e.r.) and Geos-3 (a -- 1.13 e.r.), have been using an earth to moon mass ratio of 81.3007 combined to determine an improved value of the (Wong and Reinbold, 1973). geocentric gravitational constant (GM). The In this paper we use near-earth laser ranging value of GM is 398600.61 km3/sec2, based upon a in a new determination of GM. These results speed of light, c, of 299792.5 kin/sec. Using the basically confirm those obtained from inter- planetary and lunar laser experiments, but km/secIAGadopted scales valueGM to of 398600.44 c equallin• km /sec299792.458 2. The further reduces the uncertainty of GM. uncertainty in this value is assessed to be + .02 km3/sec2. Determinations of GM from the-data Near Earth Laser Ranging Experiment. The taken on these four satellites individually show experiment reported here was performed in the variations of only .04 km3/sec2 from the combined development of the recent Goddard Earth Models result. (GEMs) 9, 10, 10A and 10B (Lerch et al, 1977; The Lageos information dominated the combined Lerch et al, 1978). For the first time in the GEM solution, and gave the most consistent results in series, these gravity models included an adjust- its data subset solutions. The value obtained ment of GM. for GM from near-earth laser ranging compares At Goddard Space Flight Center (GSFC) the quite favorably with the most recent results of emphasis has been on using as much precise near- the lunar laser and interplanetary experiments. earth satellite data as possible for our geodetic investigations. The Goddard(GSFC) laser systems Background. Most reductions of data from currently deployed have errors (bias and noise) lunar and interplanetary probes have included re- of from 5 to 8 cm (Vonbun, 1977) while the SAO estimation of the gravitational constant times lasers have accuracies of 40 cm (Pearlman et al, the mass of the earth, GM (earth plus atmos- 1977). Although the Goddard lasers are more phere). These missions include the Ranger, Sur- accurate, the combination of the two systems veyor, Lunar Orbiter, Pioneer, Mariner, and Vik- provide global distribution. The precision data ing series of flights, as well as a number of provided by these lasers on Geos-3, Starlette, Russian interplanetary probes. The results of BE-C and Lageos have shown that improved force these GM estimations have been summarized by modeling is necessary to achieve better orbit Esposito, 1974; Esposito and Ng, 1976; and accuracies. The recovery of GM along with the Esposito, 1978. The GM obtained from the best geopotential became a major objective. The laser available interplanetary investigations has been systems are the most successful instrument for adopted by the XVI General Assembly of the providing accurate near-earth orbital position- IUGG/IAG (Moritz, 1975). This value, based upon ing. The GEM models now contain nearly 213,000 the IAG adoptedvalue of the speed•f light laser observations taken on 9 satellites. How- (299792.458km/sec) is 398600.5km3/sec . ever, only the four satellites used in this study Laser ranging to the retroreflectors left on have been extensively tracked by the newest laser the moon during the Apollo Missions has been systems. carried out at the McDonald Observatory in Texas Results. Table 1 presents combined as well since 1969. Williams (1974) has reported a value as individual solutions of GM using observations of GM obtained from these observations, which is taken on four satellites. in all cases, the in good general agreement with the interplanetary laser ranges were reduced in arcs of 5 days estimates. This value of GM and its uncertainty length. The Geodyn program (Putney, 1977) was that he reported is 398600.65 + .1 km3/sec2for c used for the orbital recovery and generation of equalling 299792.5 km/sec. This translates into epoch orbit, station coordinate and GM normal a value of 398600.48 km3/sec2 using the latest equations. The normal eguations were then ana- IAG speed of light. King et al (1976) combined lyzed using the Solve Program (Putney et al., lunar laser and Alsep VLBI observations to obtain 1976) to perform the various solutions for GMand the ratio of the mass of the sun to that of the the tracking station coordinates..•. The force modeling and coordinate system displacement in- Copyright 1978 by the American Geophysical Union. cluded: Paper number 8Ll104. 1051 0094-8276/78/128L-1104501. O0 1032 Lerch et al.' Determination of Geocentric Gravitational Constant TABLE 1. ESTIMATION OF THE GEOCENTRICGRAVITATIONAL CONSTANT,* GM, FROM INDIVIDUAL SATELLITE LASER OBSERVATIONS SEMI-MAJOR RECOVERED SATELLITE AXIS INCLINATION NO. OF NO. OF GM NAME (EarthRadii) ECCENTRICITY(De•.) 5D ARCS OBS. (Km3/Se•2) LAGEOS 1.92 .004 109.85 32 88,800 398600.60 STARLETTE 1.15 .020 49.80 20 23,000 398600.61 BE-C 1.18 .026 41.19 20 18,500 398600.65 GEOS-3 1.13 .001 114.98 37 90,900 398600.60 COMBINED RESULT 109 221,200 398600.61 + .02 '1 Based on c = 299792.5 Km/Sec ß the full GEM-9 geopotential concluded that this error and the other error ß luni-solar perturbations sources result in total errors which are within ß BIH polar motion and UT1 variations our quoted uncertainty of GM. Lageos has sig- ß atmospheric drag and nificantly more consistent results for its subset ß solar radiation pressure. solutions than did similar results obtained from The value of c used was 299792.5 km/sec. Relati- the other satellites. This is because the high vistic effects were modeled for light time cor- altitude of Lageos provides for better geometry rection (Moyer, 1971) and reduced GM by .024 and dynamics in estimating GM. Geos-3 had the km3/sec2. greatest variability in subset solutions •or GM, The individual satellite estimates of GM and also had the poorest statistical conditioning varied by less than .05 km3/sec2 from the com- factors. Because of this problem, the solution bined result.. The Lageos data dominated the of normal equations required refinement (Brownd, combination solution as it has an order of magni- 1978) in the numerical procedures for computing tude greater capability than the other satellite GM correctly. data to separate GM from the orbital semi-major axis and station coordinate adjustments. Subset Ad3ustment of GM for Speed of Li•t. Our solutions of GM for Lageos are presented in Table value of GM can be adjusted to correspond to the 2. In Table 2, the uncertainty shown is the IAG value of the speed of light (c) by using the formal least square error estimates based upon 1 fo 1 lowing re lat ionsh ip .' meter range accuracies. Although the accuracy of dc the laser systems is far better than lm, other dGM = 3GM (1) error sources are present which cause these c formal statistics to be optimistic. These errors include range biases, geopotential error, and With dc = 42 m/s, •GM = 0.17 km3/sec2, GM = errors in modeling drag and solar radiation pres- 398600.44 km3/sec2 for c = 299792.458 km/sec. sure. We have tested the sensitivity of the This scale factor is also reported by Esposito solutions to different gravity fields and have and Ng, (1976). TABLE 2. LAGEOS SUBSET SOLUTIONS FOR GM NO. OF SOLUTION ARcs OBS. ESTIMATED GM L1 507,512,517,522,527 12,003 398600.657 + .010 L2 531,606,611,616,621 11,812 .665 + .007 L3 924,930,1005,1011,1016 8,102 .637 + .008 L4 1109,1115,1120,1125,1130 30,538 .618 + .005 L5 1205,1210,1214,1220,1225 9,094 .601 + .007 L6 1016,1021,1026,1031,1105 15,576 .592 + .005 Lerch et al.' Determination of Geocentric Gravitational Constant 1033 TABLE 3. COMPARISON OF RECENT DETERMINATIONS OF THE GEOCENTRICGRAVITATIONAL CONSTANT• GM VALUE OF GM INVESTIGATORS (Km3/Sec2) TYPE OF INVESTIGATION (A) ESPOSITOAND WONG,1972 398600.63 + .4 INTERPLANETARY - MARINER 9 (B) WILLIAMS, 1974 .48 + .1 LUNAR LASER RANGING (C) MARTIN ET AL, 1975 .66 + .06 INTERPLANETARY - MARINTER 9 (D) ESPOSITOAND Ng, 1976 .55 + .2 INTERPLANETARY - MARINTER 9 .45 + .2 INTERPLANETARY - MARINER 10 (E) KING ET AL, 1976 .51 + .03 LUNAR LASER RANGING AND ALSEP VLBI (F) ESPOSITO, 1978 .40 + .2 INTERPLANETARY - VIKING 1 .60 + .2 INTERPLANETARY - VIKING 2 (G) MARTIN AND OH, 1978 .35 + .15 NEAR EARTH - SATELLITE TO SATELLITE TRACKING ON ATS-6/GEOS-3 (H) THIS PAPER .44 + . 02 NEAR EARTH LASER RANGING ON FOUR SATELLITES ASSUMING c = 299792.458 Km/Sec Summary. Table 3 intercompares the near-earth Acknowledgements. The authors would like to laser determination of GM with other recent ex- express our gratitude to the following in- perimental results. The results obtained from dividuals for their contributions to this work: our analysis (GM -- 398600.44 + .02 km3/sec2) gen- Carl Wagner for a review of the analysis; Lloyd erally confirms the latest results obtained from Carpenter for laser data preprocessing and analy- radiometric data taken when numerous inter- sis; Susan Knode for the computer generation of planetary missions departed earth and those ob- normal equations for Lageos; Ron Williamson, tained from lunar laser ranging.