JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. E7, PAGES 16,339-16,359, JULY 25, 1997

A 70th degree lunar gravity model (GLGM-2) from and other tracking data

FrankG. R. Lemoine,David E. Smith,Maria T. Zuber,• Gregory A. Neumann,• and David D. Rowlands Laboratoryfor Terrestrial Physics, NASA Goddard Space Flight Center, Greenbelt, Maryland

Abstract. A spherical harmonic model of the lunar gravity field complete to degreeand order 70 has been developedfrom S band Doppler tracking data from the C]ementine mission,as well as historical tracking data from Lunar Orbiters 1-5 and the and 16 subsate]lites. The mode] combines361,000 Doppler observationsfrom Clementine with 347,000 historical observations. The historical data consistof mostly 60-s Doppler with a noiseof 0.25 to severalmm/s. The C]ementine data consist of mostly 10-s Doppler data, with a data noise of 0.25 mm/s for the observationsfrom the Deep SpaceNetwork, and 2.5 mm/s for the data from a naval tracking station at Pomonkey,Maryland. Observationsprovided C]ementine, provide the strongest satellite constraint on the 's low-degree field. In contrast the historical data, collectedby spacecraftthat had lower periapsis altitudes, provide distributed regionsof high-resolutioncoverage within +29 ø of the nearsidelunar equator. To obtain the solution for a high-degreefield in the absenceof a uniform distribution of observations,we applied an a priori power law constraintof theform 15 x 10-5//2 whichhad the effectof limitingthe gravitational power and noiseat short wavelengths.Coefficients through degree and order 18 are not significantlyaffected by the constraint, and so the model permits geophysical analysis of effects of the major basins at degrees 10-12. The GLGM-2 model confirmsmajor featuresof the lunar gravity field shown in previousgravitational field models but also revealssignificantly more detail, particularly at intermediate wavelengths(103 km). Free-airgravity anomaly maps derived from the newmodel show the nearsideand farside highlandsto be gravitationally smooth, reflectinga state of isostaticcompensation. Mascon basins (including Imbrium, Serenitatis, Crisium, Smythii, and Humorurn)are denotedby gravity highsfirst recognized from Lunar Orbiter tracking. A]] of the major masconsare bounded by annuli of negative anomaliesrepresenting significant subsurfacemass deficiencies.Mare OrientMe appears as a minor masconsurrounded by a horseshoe-shapedgravity low centered on the Inner and Outer Rook rings that is evidence of significant subsurfacestructural heterogeneity.Although direct tracking is not available over a significantpart of the lunar farside, GLGM-2 resolvesnegative anomaliesthat correlate with many farside basins, including South Pole-Aitken, Hertzsprung, Koro]ev, Moscoviense,Tsio]kovsky, and Freund]ich-Sharonov.

Introduction setted into elliptical orbits with periapses of 50 to 100 km above the lunar surface. The Apollo spacecraftwere Until the launch of Clementine, on January 24, 1994, placed in near circular orbits at low inclinations with a the sourcesof tracking data for gravity models derived mean altitude of 100 km, although sometracking was by U.S. investigatorshave been the Lunar Orbiters and acquired from altitudes as low as 10 to 20 km. The the Apollo . The Lunar Orbiters were in- tracking data sampled the gravity field of the Moon at a resolutionunprecedented for orbiting spacecraft, at • Also at Department of Earth, Atmospheric,and Plane- either the Earth, Venus, or Mars. However,the spatial tary Sciences,Massachusetts Institute of Technology,Cam- coverageof the tracking was incomplete, with no direct bridge. tracking data availableover large portions of the lunar farside. During the initial investigations,in the 1960s Copyfight 1997 by the American GeophysicalUnion. and 1970s, researchers were limited in the size of the Paper number 97JE01418. spherical harmonic solutions that could be developed 0148-0227/97/97JE-01418509.00 by the computersthen available. Becauseof the power

16,339 16,340 LEMOINE ET AL.: A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2) of the higher degreeharmonics in the tracking data, the the Apollo spacecraftwere also located in low-altitude, early solutions encountered severe difficulties with the near-circular orbits. In some cases, these spacecraft aliasing of the high-degreesignal into the lower degree came as low as 12 to 20 km from the lunar surface;how- terms. In an effort to exhaustthe availablesignal in the ever, these data were limited spatially and temporally. tracking data, other methodswere employedto map the The data from Apollo missions8 and 12, in combination lunar gravity field. These alternate methods included with the Lunar Orbiter data, were used by Won# et al. the estimation of discrete massesover the lunar surface, [1971]to solvefor 600 discretemasses over the nearside or the mapping of the accelerationsexperienced by or- (4-600in latitudeand 4-95o in longitude).Data from biting spacecraftin the line of sight(LOS) betweenthe Apollo missions14, 15, 16, 17, were used by Muller et spacecraftand tracking station. al. [1974],and $jogrenet al. [1972a,b;1974b,c] to map A selectionof the solutionsfor the lunar gravity field the LOS accelerations. that have been developed since the 1960s is summa- Ferrari [1977]derived mean elements from short-arc rized in Table 1. The landmark solutions include those fits to the tracking data. Ferrari [1977] developeda of Muller and Sjogren[1968], Williams et al. [19731, 16th degreespherical harmonic solution that wasamong Sjogrenet al. [1974a],Ferrari [1977],Bills and Ferrari the first to show plausiblecorrelations between a map [1980],and Konop!ivet al. [1993]. The analysisof the of surfacegravity anomaliesand farside lunar features, Lunar Orbiter data by Muller and Sjogren[1968] led to such as Mendeelev, Korolev, and Moscoviense. Bills the discoveryof the massconcentrations ("mascons") and Ferrari [1980]combined much of the data usedby under the ringed lunar maria. Williams et al. [1973] previousinvestigators, such as the Lunar Orbiter data, employedlaser ranging to retrofiectorsleft on the lunar the Apollodata from Wonget al. [1971],the trackingof surfacein combination with lunar gravity solutionsde- the Apollo 15 and 16 subsatellites,and lunar laserrang- rived from radiometric tracking. The lunar laser rang- ing. They developeda 16th degreespherical harmonic ing data are sensitive to the second-degreeand third- solutionthat remainedthe best lunar gravity model for degree gravity field harmonics, through their influence the next 13 years. on the lunar physicallibrations [Dickey et al., 1994]. Konoplivet al. [1993]developed a 60th degreemodel Sjogrenet al. [1974a]used data from the Apollo 15 and from the tracking of the Lunar Orbiters and the subsatellites to map the nearside of the Moon 4-29o subsatellites.Taking advantageof the high speedand in latitude and 4-100o in longitude, using both LOS memory capacity of more modern computers,this model accelerations and the estimation of discrete masses on represented the first realistic attempt to exhaust the the lunar surface. The Apollo subsatellitesmarked the gravity signalfrom the low periapsesatellites in a high- first time an extensive amount of data was acquired degreespherical harmonic solution. This model has ex- from a lunar satellite in low-altitude(100 km) near- cellent performance in terms of its root mean square circular orbit. The command and service modules of (RMS) fit to the trackingdata, and the map of the

Table1. PreviousLunar Gravity Analyses

Reference Data Used Comment

Muller and $jogren[1968] Lunar Orbiters Discovery of lunar mascons. LoreIt and $jogren[1968] Lunar Orbiters 4x4 sphericalharmonic solution + zonals to 1=8. Wonget al. [1971] , 12 Solution for discrete masses on near side. Michael and Blackshear[1972] Lunar Orbiters 13x13 spherical harmonic solution. $jogrenet al. [1972a,b;1974b,c]; Apollos 14-17 Mappedline of sight(LOS) accelerations Muller et al. [1974] with data from as low as 12-20 km altitude. Sjogrenet al. [1974a] Apollo subsatellites Solve for LOS and discrete masses, Ferrari [1977] LO-5 and Apollo subsatellites 16x16 spherical harmonic solution. Ananda[1977] Solve for discrete masses. Bryant and Williamson[1974] 3x3 spherical harmonic solution from KeplerJan mean elements. Blackshearand Gapcynski[1977] Explorers 35 and 49 Zonal solution, J2 - Je only from Keplerian mean elements. Williams et al. [1973] Lunar laserranging (LLR) Ferrari et al. [1980] LLR and LO-4 Bills and Ferrari [1980] LLR, LO 1-5, Apo•o subsatellites, 16x16 spherical harmonic solution and Apollo 8,12 Konoplivet al. [1993] LO 1-5 and Apollo subsatellites 60x60 spherical harmonic solution. Dickey et al. [1994] LLR LEMOINE ET AL.: A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2) 16,341 lunar gravity anomaliesappears smooth when mapped the Jet PropulsionLaboratory (JPL), and the Johnson on a surface at 100 km altitude. However, the high- SpaceCenter (JSC). Explorers35 and 49 wereplaced in degreeterms have excessivepower, and when the grav- high lunar orbits, and were sensitiveto the low-degree ity anomalies are mapped onto the lunar surface, nu- harmonics[Bryant and Williamson,1974; Gapcynskiet merousartifacts becomeapparent, which make the mod- al., 1975; Blackshearand Gapcynski,1977]. In addi- el unsatisfactoryfor usein geophysicalstudies. The goal tion, these spacecraftwere tracked by VHF, not S band of the new analysiswas to developa lunar gravity model tracking, and so were noisier becauseof the increased that (1) includedboth the historicaltracking data from sensitivityto the ionosphere,and the solarplasma (R. the Lunar Orbiters and the Apollo subsatellites,as well Williamson,personal communication, 1994). Other ex- as the new data from Clementine;(2) attenuatedspu- isting data which might be used include the tracking riouspower in the high-degreeterms; and (3) produced of the Soviet Luna spacecraft.These spacecraftare lo- a model suitable for use in geophysicalstudies of the cated in complementary,frequently higher inclination Moon when used in conjunction with the Clementine orbits than the U.S. Lunar Orbiters and the Apollo laser altimeter data. spacecraft.The analysisof the limited•mount of Luna data is discussedby $agitovet al. [1986]but was not Data pursued in our analysis. The orbit characteristicsof Lunar Orbiter spacecraft The data in GLGM-2 consistof the Doppler track- and the Apollo subsatellitesare summarized in Table 2. ing of the U.S. Lunar Orbiter Satellitesi to 5 (referred In general,the Lunar Orbiters were placed in elliptical to by shorthand as LO-1, LO-2, etc., for the remain- orbits about the Moon, with periods of 200 to 210 min. der of this paper), the Apollo 15 and 16 subsatellites, Periapseheights ranged from 30 to 210 km, depending and the Clementine spacecraft. Other data, such as on the missionphase. In addition, perispsisand, con- the tracking from the Explorers 35 and 49 and S band sequently,the region of highestresolution was almost tracking from tracking of the Apollo commandand ser- always placed near the lunar equator. Lunar Orbiters vice modules, could not be located, even after searches i to 3 had low-inclinationorbits (120 to 21ø). Lunar at the National SpaceScience Data Center (NSSDC), Orbiters 4 and 5 had near-polar inclinations of about

Table 2. Orbits of Lunar Spacecraft

Spacecraft Date Periapse Period Semimajor Eccentricity Inclination Height, km min Axis, km deg

Lunar Orbiter I Aug. 14, 1966 189 218 2766 0.3031 12.2 Aug. 21, 1966 57 209 2693 0.3335 12.0 Aug. 25, 1966 41 206 2670 0.3336 12.1 Nov. 10, 1966 196 218 2772 0.3021 12.0 Nov. 15, 1966 50 208 2689 0.3353 11.9 Dec. 8, 1966 43 210 2702 0.3408 17.6 Apr. 14, 1967 69 209 2693 0.3289 17.2 June 27, 1967 113 212 2716 0.3184 16.5 July 24, 1967 98 212 2716 0.3240 16.1 Lunar Orbiter 3 Feb. 9, 1967 210 215 2744 0.2899 21.0 Feb. 12, 1967 55 209 2689 0.3331 20.9 Apr. 12, 1967 59 207 2680 0.3294 21.2 July 17, 1967 144 212 2722 0.3088 21.0 Aug. 30, 1967 145 131 1968 0.0435 20.9 May 8, 1967 2706 721 6150 0.2773 85.5 June 6, 1967 75 501 4820 0.6238 84.9 June 9, 1967 77 344 3753 0.5163 84.4 Aug. 5, 1967 195 505 4852 0.6017 85.0 Aug. 7, 1967 100 501 4822 0.6185 84.6 Aug. 9, 1967 100 191 2537 0.2760 84.7 Oct. 10, 1967 198 225 2832 0.3155 85.2 Apollo 15 subsatellite Aug. 29, 1971 89 118 1858 0.0167 151.0 Apollo 16 subsatellite Apr. 27, 1972 73 119 1849 0.0205 169.3 Clementine Feb. 19, 1994 402 474 4651 0.5400 89.5 Feb. 21, 1994 398 299 3418 0.3755 89.3 Feb. 22, 1994 383 299 3414 0.3789 89.4 Mar. 11, 1994 400 298 3414 0.3738 89.8 Mar. 27, 1994 447 298 3414 0.3598 90.1 Apr. 11, 1994 427 298 3415 0.3657 90.0 16,342 LEMOINEET AL.: A 70THDEGREE LUNAR GRAVITY MODEL (GLGM2)

850. The Lunar Orbiters were trackedby the anten- Orbiters, whose orbit periapses were close to the lu- nae of the DeepSpace Network (DSN), principallyat nar equator, for Clementine the periapsis was located stations12 (Goldstone,California), 41 (Canberra,Aus- near 30øS during the first month of mapping, and near tralia), (Mdid, Spi). The of 30øN during the secondmonth in . Clemen- mostly60-s two-way and three-wayDoppler, with noise fine was tracked by the 26-m and 34-m antennae of the rangingfrom 0.35 to occasionallyas high as 5 mm/s on DSN, as well as by the 30-m Pomonkey antenna op- some arcs. erated by the Naval Research Laboratory in southern EachLunar Orbiter spacecrafthad twomission phases: Maryland. The data from Clementine were predomi- a primary missiondevoted to photographicmapping, nantly two-way Doppler, averagedover 10 s. The noise with intensetracking, and an extendedmission with of the DSN data averaged0.25 to 0.30 mm/s, whereas sparsertracking. The attitude control systemof the the noise of the Pomonkey data typically ranged from Lunar Orbiters was uncoupledin pitch and yaw, so 2.5 to 3.0 mm/s. Clementineremained in orbit about that eachtime the spacecraftchanged its orientation, the Moon from February 19 to May 4, 1994. Nearly all a spuriousacceleration was impartedto the spacecraft the available Clementine tracking data from the DSN [Konoplivet al., 1993]. The maneuverswere most and Pomonkey were used in GLGM-2. extensiveduring the photographic(primary) mission The availabletracking data constrainthe global grav- phasesbut alsooccurred during the extendedmissions. ity field of the Moon at moderate resolution and pro- The times of these maneuversare known, so that their vide higher resolutioncoverage in the equatorial regions. effectscan be accommodatedby estimatingthree-axis The Doppler data available below 500 km are illustrated accelerations,radial, along-track,and cross-trackto the in Figure 1. It is evident from these figures that lunar orbit at the time of the maneuvers.During the photo- librations and parallax allow direct tracking of space- graphicmissions, as many as 14 setsof attitude maneu- craft over the poles to 60øN or S on the farside and to versoccurred per day. Maneuversclosely spaced in time -t-1200longitude, where 0ø longitude is the meridian at werecombined, and a singleset of accelerationparam- the center of the nearside. eters was estimated to account for the attitude-induced orbit perturbations.Also, becauseof the numerousma- neuvers,during the photographicphases of the Lunar Method of Solution Orbitermissions, data arcswere limited to no longer Definition than oneday. The LO-3 spacecraftwas briefly inserted into a near-circularorbit aboutthe Moon,simulating The lunar gravitypotential, U, is modeledin spheri- the orbit plannedfor the then futureApollo missions; cal harmonicsusing the expression, however,sparse tracking limited the amount of data that was returned from this orbit. The Apollo 15 and Apollo 16 spacecraftdeployed sub- U • satellitesin lunar orbit. Thesespacecraft, designed to study the magneticfields, plasmas, and energeticpar- (1) ticles in the vicinity of the Moon, also carred S band transpondersto supporthigh-precision gravity mapping of the Moon [Colemanet al., 1972; Andersonet al., wherethe expansionis definedin sphericalcoordinates 1972a,b]. The spacecraftwere spin stabilizedand did with radiusr, latitude 4, and longitude•; 01r• and not useany thrusters to maintaintheir attitude,making $tm representthe normalizedgeopotential coefficients; the data muchcleaner than the trackingdata from the Pt,• arethe normalized associated Legendre functions Lunar Orbiters. The two subsatelliteswere deployed of degree1 and orderm; ae is the referenceequatorial by the Apollo 15 and 16 command modules in near- radius;G is the universalconstant of gravitation,and circular,retrograde orbits of the Moon (inclinationsof M is the lunar mass. In a coordinatesystem whose 1520and 169ø).The two spacecraftwere tracked by the origincoincides with the centerof mass,the degreeone antennaeof the MannedSpace Flight Network (MSFN) termsof equation(1) vanish. [Sjogrenet al., 1974a].The spacecraftpermitted a uni- The geopotentialcoefficients may be unnormalizedby form mappingof the nearsideof the Moon up to about the followingrelation [Kaula, 1966], 29øN or S from a mean altitude of 100 km. Unfor- tunately, the satellites did not use satellite-to-satellite trackingto permit acquisitionof data whileflying over the lunar farside. Clm--[(/- m)' (2/+(1+ m)! 1)(2- Sore)] «Otto (2) The Clementine spacecraftwas launchedfrom Van- denburgAir ForceBase on January 24, 1994, and be- where50r• is •he Kroneckerdelta (50r•= 0 for m•-O,and came the first U.S. spacecraftto return to lunar orbit 50r• = i for re=O), and the equationcan applyto Otr• in almost20 years[Nozette et al., 1994]. Clementine or •tm. A solutionfor the lunar gravity field consists of carriedan S bandtransponder suitable for gravitymap- an estimatefor the lunar GM, as well as the parameters ping and was placed into an elliptical orbit about the Ctm and $tm. It is frequentlyconvenient to refer to the Moon, with a period of approximately 300 rain and a total amplitude,Jl,•, of a C1,• and $1,• coefficientpair, mean periapsis altitude of 415 kin. Unlike the Lunar whereJim -- VCl2m-]- •m' LEMOINE ET AL- A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2) 16,343

___•,__ :•:" .-'-•r•-.•:_ .•••-'-•--•_• ;. :• 'ø---io .... .,.... • ;'"•!."11K; •. • •_ '•.,.• .. /)Z---....•------, .... /- .- -• _--:---- -•o --- •-- -•di gllig I I•-' •'•' ' --;--- -20 -20 o -3010 : , ,•....,•--- -1• -1• -I• -gO -• -• 0 • • gO 120 -160 -150 -120 -gO -CO -30 0 30 60 gO 120 150 160 Longlfude(degrees)

0

-201o iiii!ii -100 -150 -120 -gO -60 -30 0 30 60 gO 120 150 100 •ong•d. (degree.)

go • 2o • •o i -10 --- '...... • -20 -34:) -180 -150 -120 -90 -fro -50 0 50 60 gO 120 150 180 - 180 -150 -1 20 -90 -fro -34:) 0 34:) 60 gO 120 150 1gO I.ongH'ud.(degrees) •ong•d• (degree.)

.o - - - • -- irr:-

-20 -60ø :::i:ii lii ' Illi--i,: ! -30 -194) -150 -120 -gO -60 -.TO 0 30 60 gO 120 150 180 -160 -150 -120 -00 -60 -.TO 0 30 60 gO 120 150 180 Longlfud.(degre#) Lortgltude(degre#)

Figure 1. Dopplerdata coveragebelow 500 km altitude in GLGM-2 for (a) , (b) Lunar Orbiter 2, (c) Lunar Orbiter 3, (d) Lunar Orbiter 4, (e) Lunar Orbiter 5, (f) the Apollo 15 subsatellite,(g) the Apollo 16 subsatellite,and (h) Clementine.

Processing of the Tracking Data station coordinateswere derived from Folknet [1993] and mappedback to the 1960sand 1970susing station The data were divided into independent data spans, velocities derived by the International Earth Rotation known as arcs, basedon knowledgeof the spacecraftor- (sas) [ss41. The bit characteristics, frequency of maneuvers, and avail- applied to correct the tracking data for tropospheric ability of tracking data. The GLGM-2 solution was refraction using the mean meteorologicaldata for the basedon 392 arcs,from 6 hoursto 12 daysin length, and DSN sites from Chao [1974]. The coordinatesof the included 708,854 observations,of which 361,794 were MSFN stations were initially provided by A. S. Kono- contributed by Clementine. For each orbital arc the pliv (personalcommunication, 1993), and thesestation adjusted parameters included the orbital state, and a positionswere adjusted in the global gravity solution. single solar radiation pressurecoefficient. To account The coordinatesof the Earth trackingstations were cor- for the spurious accelerationsinduced by the attitude rected for the displacement due to the solid tide and control systems on the Lunar Orbiters and on some ocean loading, although the latter is a small effect for Clementinearcs, three axis accelerations,radial, along- these data. The appropriate relativistic perturbations track, and cross-track to the orbit were also estimated. are applied in the force model and the measurement Doppler range-rate biaseswere estimated for each sta- model. These include the Schwarzschildeffect, or the tion in each orbital arc to account for any residual relativistic modification of the central body term, the modelingerror, such as frequencyoffsets, errors in the transformations from coordinate time to atomic time station locations,and imprecisemodeling of the tropo- as derivedby Moyer [1981],and the modificationof the sphere and the ionosphere. ray path due to relativity [Moyer, 1971]. We usedthe The force model included the third-body perturba- 1991 InternationalAstronomical Union (IAU) model tions of the Earth, the Sun, and all the planets, the [Davieset al., 1992] and the DE200 lunar and plan- solar radiation pressure, the Earth-induced and solar- etary ephemerides[Standish, 1990]. The Davies et al. induced lunar tides. A k2 Love number of 0.027, as de- [1992]reference has a typographicalerror in TableII (p. rived by Williams et al. [1987],was applied. The DSN 381) that describesthe lunar orientation:the parame- 1t5,344 LEMOINE ET AL.- A ?0TH DEGREE LUNAR GRAVITY MODEL (GLGM2) ters definingE3 and E5 shouldread: E3 - 260.008ø+ The diagonalelements of the a priori covariancematrix, 13.012001d, and E5 - 357.529ø+ 0.985600d (M. E. /5, havethe form Davies,personal communication, 1993). h - (5) and Derivation of the Spherical Harmonic Solutions 10-5 The GEODYN orbit determination program wasused fr,- Kl- •-- (6) to processthe lunar trackingdata [Pavliset al., 1997]. whereK is a scalingfactor. Ferrari [1977]applied this This program has previouslybeen used to processin- constraintwith frz= 35 x 10-5/12,and Konoplivet al. terplanetarytracking data from Mars and Venus[Smith [1993]used it, = 15 x 10-5/l2. Billsand Ferrari [1980] et al., 1993;Netera et al., 1993]. GEODYN wasused to useda slightly different powerlaw of the form, suchthat convergethe orbital arcs and create the normal equa- tions for the least squaressolution. A companionpro- _ 1.4 x 10 -6 gramto GEODYN, SOLVE [Ullman,1994], was used to Pt- (21+ 1)21)(1 + 1) (7) derive the least squaressolution. The normal equations which, they argued, did not overestimate the low- weregrouped by satellite,spacecraft mission phase, and degreevariances, as did the a priori power law usedby orbital geometry. In selectingthe a priori weightsfor Ferrari [1977]. The applicationof these sortsof con- eachset of data, considerationwas given to the apparent straints on the total spectral power of the coefficients data noise(as deduced from analysisof the residualson by degree prevents the high-degreeterms from devel- a passby passbasis), the periapsealtitude, the length oping excessivepower and allows a 70x70 solution to of the data arc, the number of acceleration parameters be obtained. One difficulty with the application of this in an arc, and missionphase. Inspectionof the maps of Kaula-like power constraint is the selection of the ap- the free-air gravity anomaliesalso played a role in the propriate multiplier in the power law. For the Earth, selectionof the weights. Setsof data that producedspu- independentsurface gravity measurementscontrain the rious signalswere rejected or downweighted.Through amplitude of the power spectrum, whereasno such in- inspection of the anomaly maps and other tests, arcs formation is available for the Moon. Furthermore, a with a high data noise (for instance,the LO-5 arcs tight constraint will mute the magnitude of the grav- from August 22-26, 1967) were downweighted.Data ity anomalies,whereas a weaker constraint might allow from LO-2 for December 1-6, 1966, and for LO-3 from more "real" power into the anomalieson the nearside, February 28 to March 8, 1967, receivedclose scrutiny. at the expenseof inducingspurious high frequencyvari- In both instances,tracking was nearly continuous,ex- ations on the farside. A variant of this approachwas de- cept for occultation behind the limb of the Moon. This velopedby Konoplir and Sjogren[1994] and appliedto should not be an issue, except that periapse was pre- global solutionsfor the gravity fieldsof Venusand Mars cessingwest out of direct view from Earth, and reached a minimum of 28-km altitude in the case of LO-2, and [Konopli•and Sjogren, 1994, 199•]. Their techniquein- volves applying a power law constraint both spatially 38 km in the case of LO-3. In both cases,the proxim- and spectrally. This is accomplishedby writing observa- ity of periapsisto the surface,out of view of the direct tion equationson a globalbasis such that the anomalies tracking from Earth, causespowerful perturbations in on the surfacebeyond a certain degreeare zeroto within the tracking data and, ultimately, an indeterminacyin a prescribedsigma, determined from Kaula's rule. Since the location of gravity anomaliesalong the LO-2 and we wishedto mute the high frequencyexcursions in the LO-3 groundtrack on the farside. The result is striping gravity anomalies,we preferred the constraint &t - in the maps of the free air gravity anomalies,particu- x 10-5/12,although we did test solutionswith weaker larly in the vicinity of Korolev. Downweightingof the Kaulaconstraints, such as $rt- 30 x 10-5/12,and data and processingthe LO-2 and LO-3 data over this 60 x 10-•/12. A detailedanalysis using the spatialand periodin short (12-hour)arcs mitigates but doesnot spectral approach was beyond the scope of this study, eliminate these artifacts in the farside anomaly maps. but a comparisonof this method as well as an eigen- In order to obtain a solution, it is necessaryto solve value techniquefor stabilization for sparsematrices is a a leastsquares system of equationsof the form [Lawson currentsubject of investigation[Lemoine et al., 1996]. and Hanson,1974] In the courseof deriving our final sphericalharmonic - (3) solution, we developeda number of interim solutionsto evaluate our results. Some of the fields that we discuss where A is the set of normal equations, x is a vector in this paper and their relationshipto GLGM-2 are de- representingthe _ deviation of the coefficientsfrom their scribed in Table 3. The data used in the derivation of a priori valuesX, and B is the residualvector. Unfortu- GLGM-2 are summarized in Table 4. nately, the lunar tracking data are nonuniformin both spatial resolution and distribution. As a consequence, Calibration of GLGM-2 it is not possibleto obtain a solutiondirectly, and an a priori constraint must be applied. One approach is to Once all the data had been processedand assembled apply a constraint such that into the appropriate set of normal equations, we cal- ibrated the final solution using the method of Lerch (A + P- •)x - B - P- x2 (4) [1991].This methodinvolves deriving subset solutions, LEMOINE ET AL.- A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2) 16,345

Table 3. Lunar Gravity Solutions and Their Relationship to GLGM-2

Solutiona Maximum Description

LUN60D 60x60 Konoplivet al. [1993]a priori solution. GLGM-1 70x70 Predecessorto GLGM-2, from Lemoineet al. [1994].. Does not include Apollo 16 subsatellite data. Clementinedata weightedat 0.5 cm/s. Solution not calibrated. LGM-261 70x70 Solution from Clementine data only. LGM-296a 70x70 New baseline solution prior to calibration. GLGM-2 70x70 Final calibratedsolution from this paper (LGM-309a). LGM-309b 70x70 Sameas GLGM-2,but usespower law constraintof 30 x 10-5/1 •'. LGM-309c 30x30 Same data as GLGM-2, but with no power law constraint. LGM-309d 70x70 Same as GLGM-2, but with no Clementine data.

aUnlessotherwise noted, all solutionsused the a prioripower law constraintof 15 x 10-5/1•'. where a single constituent set of data is removedfrom from the subset solution, and •rt,• and fft.• refer to the the nominal solution. The objective of the calibration coefficientsigmas from the master and the subset solu- procedureis to adjust the initial data weight so that tions, respectively. the aggregatedifferences of the coefficientsbetween the The average calibration factor for a set of data is subsetand the master solution are the same as the ag- simply the average over some span of degreesI up to gregatedifferences between the coefficientsigmas in the /max , or subset and the master solution. The objective is to de- rive a covariancematrix that more closely reflects the /max KI intrinsic uncertainties in the data. This procedure has •- E lmax_ i (9) been used extensively to calibrate the covariancematri-

_ cesof Earth gravity models[Marsh et al., 1988a, 1990; The parameter K defines the new scale factor for the Nerem et al., 1994]and wasalso applied in the develop- normal equations of the set of data whose weights we ment of GMM-1, a 50th degree and order solution for wish to adjust. The summation is truncated at an in- Mars [Smith et al., 1993]. termediate degree, /max , which is usually not the max- The calibration factor for each spherical harmonic imum degree of the gravity solution. The calibration degree is computed according to the following relation factors for the high-degreeterms may not be meaning- from Lerch[1991]: ful, since these terms are dominated by the Kaula con- straint rather than from the trackingdata [Marshet al., 1988b]. The normal equationsare then rescaledby the relation (8) W•= K2W (10) where Gt,, refers to both the Ct,• and St,• coefficients _ and the a priori sigma on the test set of data is related from the master solution, Gtm refer to the coefficients to the weight in the normal equation by the equation,

Table 4. Doppler Data Used in GLGM-2

Spacecraft Number Average Arc Total Number of Arcs Length, hrs of Observations

Lunar Orbiter I 48 21.76 44,503 Lunar Orbiter 2 75 16.38 76,421 Lunar Orbiter 3 55 17.77 54,163 Lunar Orbiter 4 11 70.16 48,734 Lunar Orbiter 5 51 33.74 47,690 Apollo 15 subsatellite 81 16.76 44,096 Apollo 16 subsatellite 35 4.55 31,453 Clementine 36 44.16 361,794

Total 392 708,854 16,346 LEMOINE ET AL.' A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2)

1 30. Most of the calibration factors are already close to fro= x/__• (11)unity and indicate that the a priori weights selectedfor the data were approximately correct and that there is If the initial weight has been appropriately selected, no set of data that grosslydistorts the solution. No rea- then the constant of proportionality K should be close sonable calibrations were derived for the high altitude to unity. A constant K greater than unity indicates periapsisdata from Lunar Orbiter 4, so the calibrations that the test set of data should be downweightedin the from LO-4 set 2 were applied to the LO-4 set i data. least squaressolution, and a value lessthan unity indi- The LO-4 set i data are extremely weak because of the cates the data should receive more weight or emphasis high periapsisheight (2700 kin), relativelylong period in the least squaressolution. Each independent set of (6 hours), and numerousattitude maneuvers.The fi- data that contributes to the gravity solution may be nal calibration factors used to derive GLGM-2 from the calibrated in this fashion. The new set of weights is base model (LGM-296a) are summarizedin Table 5. then used to produce the final calibrated solution. For completeness,we show a complete table of the final Given the number of independent orbits, a strict ap- sigmas applied to the data in GLGM-2, following the plicationof the calibrationtechnique from Lerch[1991] calibrationprocedure (see Table 6). would have required the calculation of an inordinate number of master and subset solutions. Therefore, to Results simplify the process,the data were divided into 11 man- ageablesets, based upon satellite and orbit geometry. Power of the Coefficients The data from Lunar Orbiter 5, Clementine, the Apollo subsatellites, and Lunar Orbiter I were calibrated indi- The coefficientvariances at degreel, rrz(U), can be vidually by satellite. For Lunar Orbiters 2, 3, and 4 representedaccording to the convenientrelation from the calibration sets were selected based on inclination Kaula [1966]' (in the caseof LO-2), eccentricity(in the caseof 3), and periapseheight (for LO-4). The relative data weightsbetween arcs in eachset werefixed. Then all the rr•(U)- (2/+1) - •y], (C•r•-2 + •L) (12) data in each set were adjusted, accordingto the calibra- rr•-- 0 tions. We startedwith a baselinesolution (LGM-296a) that included all the data. Eleven subset solutions were The coefficientdegree variances,for the Konopliv et computed with respect to LGM-296a, complete to de- al. [1993] field, and for GLGM-2 are shownin Fig- gree and order 70. The calibration factors were calcu- ure 2. The coefficient degree variances are compared lated as per equations(8) and (9). The final calibrated with the signal of the power law applied in these grav- solution(GLGM-2) was derivedusing the calibration ity solutions,of 15 x 10-5/12. The Konoplivet al. factors derived from all 11 subset solutions. [1993],Lun60d, field has powerconsiderably in excess Comparisonof the RMS degreevariance for the GL- of the power law. In contrast, the coefficient power of GM-2 coefficientsand coefficient sigmasindicates that GLGM-2 at the higher degreesis strongly attenuated the coefficientsigmas have greater power than the co- and falls below the power law. The coefficient vari- efficients themselvesfor degrees I > 30. As a conse- ances for LGM-309c are also shown to illustrate the ef- quence, the calibration factors were calculated by aver- fect of omitting the application of the Kaula constraint agingthe K• (from equation(9)) only throughdegree in the GLGM-2 solution. Solutions without a Kaula constraint, where/max _> 30, simply would not invert. It is apparent, even in truncating the solution at l = 30, Table 5. Calibration Factors for the Doppler Data in the high-degreeterms develop excessiveand unreason- GLGM-2 able power. Comparisonof the GLGM-2 and the LGM- 309c solutions showsthat the power law constraint has Satellite Data Description Calibration a noticeableeffect starting at about degree 18. We note that most of the gravitational power associatedwith the LO-5 1.245 major masconbasins falls at degrees10-12, where the LO-4, set 1 Primary mission. ... LO-4, set 2 Extended mission. 1.252 power law does not have significantinfluence. Thus in- LO-2, set I Primary mission, i -- 12ø. 1.362 terpretations about the long wavelengthcompensation LO-2, set 2 Extended mission, i -- 17ø. 1.629 state of the Moon [Zuberet al., 1994]based on a prelim- LO-3, set I Primary and extended mission inary versionof this gravitational model are supported 1.553 (eccentricity-- 0.28 to 0.33). by more detailed analysisof the gravity. LO-3, set 2 Extended mission (eccentricity-- 0.04). 1.818 The coefficientsigma degreevariances from GLGM- LO-1 2.761 2, Lun60d,and LGM-309d (GLGM-2 with no Clemen- Clementine 1.523 tine data) are illustratedin Figure 3. The scaledif- A- 15ssa 1.530 ference in the coefficientsigmas between the Konopliv A-16ss a 1.175 eta!. [1993]Lun60d field and GLGM-2 generationof models is due to different schemesused to weight the aA-15ss -- Apollo 15 subsatellite. A-16ss -- Apollo 16 sub- data. Konoplivet al. [1993]weighted data closerto satellite. the RMS of fit, which overall was of the order of a few LEMOINE ET AL.- A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2) 16,347

Table 6. Final SigmasAssigned to Data Used in the GLGM-2 solution

Data Description Number of Number of Data Sigma Arcs Observations cm/s

LO-1 Primary and extended mission. 42 39,885 2.76 LO-1 Extended mission. Includes 6 4,618 5.52 10-dayarc (Sept. 22 to Oct. 1, 1966). LO-2 Primary mission(Nov. 10 to Nov. 30, 1966). 10 32,226 1.36 LO-2 Primary mission(Dec. 1 to Dec. 7, 1966). 13 17,389 2.72 LO-2 Extended mission,set A a 32 19,975 1.63 LO-2 Extended mission,set B a 20 6,831 2.82 LO-3 Primary mission(Feb. 12, to Feb. 23, 1967). 12 19,368 3.11 LO-3 Primary mission(Feb. 24, to Mar. 7, 1967). 18 9,926 2.69 LO-3 Extended mission,set A a 3 1,164 1.55 LO-3 Extended mission,set B a 7 4,459 4.66 LO-3 Extended mission multiday arcs. 4 14,861 15.53 LO-3 Near circularorbit (Sept. 1967). 11 4,385 1.82 LO-4 Primary mission(May 1967). 5 32,551 3.33 LO-4 Extendedmission (June 1967). 6 16,183 1.25 LO-5 Primary mission(Aug. 5 to Aug. 19, 1967). 16 17,393 1.25 LO-5 Primary mission(Aug. 19 to 22, 1967): 3-day arc. 1 5,673 1.25 LO-5 Primary mission(Aug. 22, 1967). I 763 5.57 LO-5 Primary mission(Aug. 23 to 24, 1967). 1 1,797 2.48 LO-5 Primary mission(Aug. 26, 1967). 3 1,657 3.74 LO-5 Extendedmission (Oct. 1, 1967 to Jan. 29, 1968). 29 20,407 1.25 A-15ssb Sparsetracking. 2-3 day arcs. 11 2,940 4.59 A-15ssb Sparsetracking. 1-day arcs. 20 4,033 3.06 A-15ssb Sept.,Oct., Dec., 1971 gravity campaigns. 50 37,123 2.16 A-16ssb 35 31,453 3.53 Clementine 36 361,794 1.36

aLO-2 and LO-3 extendedmission data (set A) includesthose arcs that had documentedattitude maneuvers. LO-2 and LO-3 extendedmission data (set B) includesthose arcs with no documentedattitude maneuvers. bA-15ss---- Apollo 15 subsatellite;A-16ss -- Apollo 16 subsatellite. mm/s. The intention of this approachwas to mini- werefurther downweightedin the calibrationprocedure. mize Doppler residuals to produce a model optimized Our approachwas to minimize systematicsources of er- for orbit prediction. In contrast, the a priori weights ror in the force and measurement models that have a in GLGM-2 were approximatelyi cm/s, and the data tendency to produce striping correlated with satellite

10-4 _ 10-510-4 I o

õ 10-5 lO-5

o • 10-6 10-6

._ •_o 10-7 10_7

o

10-8 -O LGM509c(30x50; no power law constraint) 10_8 [] Lun600:Konopliv ef ol. [1995] [] Lun60D:Konopliv ef al. [1995] A A priori powerlaw (15 x 10-•/I=) A A priori powerlaw (15 x 10-•/I=) -9 -9 10 , , , I , , , I , , , I • , , 10 OLGM509d; GLGM-2withoutClementinedata. __ 0 20 40 60 80 0 2 0 40I • • • 60I , • , 80 Degree Degree Figure 2. Comparisonof coefficientdegree variances Figure 3. Comparisonof coefficientsigma degree vari- for GLGM-2, the Konoplivet al. [1993]solution, and ancesfor GLGM-2, the Konoplivet al. [1993]solution, LGM-309c, a solution completeto degree and order 30 the LGM-309d, a solutioncomplete to degreeand order derived with no K aula constraint. 70 without the Clementine data. 16,348 LEMOINE ET AL.: A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2) orbital ground tracks. Such structures are irrelevant noted the presenceof these lows in their analysis of the for orbit prediction applications,but they are nonphys- Lunar Orbiter data, but their observation received lit- ical and may mask subtle signals that could contain tle attention. Later, Muller et al. [1974]reported the information on the internal density distribution of the existence of lows surrounding Serenitatis and Crisium planet. Our approach for data weighting was driven in their analysis of the S band tracking of the Apollo by a desire to produce a gravity field useful for lunar 15 command module, which came to within 12 km of geophysicswhich manifests, to the extent possible,a re- Serenitatis. In a separateanalysis [Neumann et al., alistic pattern of gravity anomalieswhen mapped onto 1997] we reanalyzedLunar Orbiter LOS Doppler data the lunar surface. and independently verified the presenceof the annular lows. These negative rings are much more conspicu- Description of the Solution ous in our model than in previous spherical harmonic Gravity anomalies from GLGM-2. The solu- representationsof the lunar gravity field, in part be- tion is complete to degree and order 70, correspond- causeof greater surficialcoverage provided by Clemen- ing to a half wavelength resolution of approximately tine at mid to high nearsidelatitudes, but also because 2.60 (80 kin), where there is adequatedata coverage. our field is evaluatedat the surfacerather than space- When evaluated at the surface with a reference radius craft altitude. We originally interpreted these annular of 1738 km, and after removing the appropriate hydro- rings to be a consequenceof lithospheric flexure in re- static terms, free-air gravity anomalieshave a range of sponseto surface loading of the lunar lithosphere by -294 to +358 regals. The anomalies from GLGM-2 are the maria [Zuberet al.,, 1994]. However,subsequent depicted in Plate 1, and the projected errors from the analysis[Williams et al., 1995],which considered flexu- gravity field error covariance are shown in Figure 4. ral effectsof both surfacemaria loadingand subsurface The range in the free-air gravity anomalies is smaller loads associatedwith crustal thinning, indicates that in GLGM-2 than in the Konoplivet al. [1993]solution, anomaly amplitudesare in nearly all casestoo large to which has a range of-454 to 529 regals. The reasonfor be explained by flexure alone. Similarly, the anomaly the differenceis that the power of the high-degreeterms magnitudes would require unrealistic amounts of brec- of G LGM-2 was attenuated to avoid the excessive orbit ciation to be attributed solely to ejecta fallback. Our striping evident in the Lun60d model, when the grav- revised interpretation is that these features represent ity anomalies were downward continued to the surface crustal thickening related to basin formation or modifi- of the planet. Consequently, GLGM-2 underestimates cation [Neumannet al., 1996],but our hypothesishas the expected power in the lunar gravity field at short yet to be tested quantitatively. wavelengths. Mare Orientale, with an age of 3.8 billion years is The projectederrors, depicted in Figure 4, rangefrom the youngest of the major lunar impact basins, shows 17 to 20 mGals on the equatorialnearside (where the a central mascon with a maximum amplitude of 188 densest,low-altitude tracking data wereobtained) to 45 mGals, surroundedby a horse-shapedgravitational low mGals on the high-latitude regionsof the farside. The centeredon the Inner and Outer Rook rings. The maxi- 33 mGal contouressentially bounds the regionwhere di- mum amplitudeof the boundinglow is -275 mGals, but rect tracking of the Apollo 15 subsatellitewas obtained. its variability in shape and amplitude do not correlate GLGM-2 showsthat the lunar highlandsare generally with surfacetopography (K.K. Williams and M.T. Zu- characterizedby low amplitude gravity anomaliesthat ber, manuscriptin preparation,1997) and sorepresents indicate a state of isostatic equilibrium, consistentwith evidence for significant subsurfacestructural hetero- the highlands'age and origin as a near-globalgeochemi- geneity. The Mendel-Rydberg basin, just to the south cal melt product[ Woodet al., 1970].The modelresolves of Orientale, is barely resolved,with a low of approxi- the major nearsidemass concentrationsor "mascons," mately 22 mGals. Since Mendel-Rydberghas consider- representedas large positive gravity anomalies,first de- able topographicrelief (6 km from rim to floor [Smith duced from Lunar Orbiter tracking more than 20 years et al., 1997]),the lack of a significantgravity anomaly ago [Muller and Sjogren,1968]. The major masconsin- suggeststhis featureis almostfully compensated[Zuber clude Imbrium, Serenitatis, Crisium, Smythii, and Hu- et al., 1994]. morum. The mascon anomalies are attributed to a com- On the lunar farside, South Pole-Aitken is resolved bination of the gravitational attraction of mare fill and as a gravitylow, with minimanear Van de Graff (25øS, to uplift of the lunar Moho associatedwith the impact 185øE,-116mGals), Apollo (44øS, 212øE,-102 mGals), process. andAntoniadi (75øS 180øE,-82 mGals). Giventhe am- GLGM-2 also resolvesgravitational lows or moats plitude of the anomaly comparedto the basin depth of surroundingthe major masconbasins on the nearside, 12 km [Zuberet al., 1994],South Pole-Aitken must be such as Imbrium, Serenitatis, Crisium, and Humorum. approximately90% compensated;however, deviations After careful consideration, we are certain these fea- from compensationmay well be a consequenceof the tures are genuinefeatures of the lunar gravity field, as younger,smaller basins within its confines[Neumann opposedto artifacts associatedwith "ringing" in the eta!., 1996]. spherical harmonic solution. They occur on the near- The GLGM-2 model resolves other farside basins such side,in regionswhere there are independenttracks from as Hertzsprung, Mendeleev, Tsiolkovsky,Moscoviense, severalsatellites. In fact, Muller and $jogren[1968] and Freundlich-Sharonov.These basins, which do not LEMOINEET AL.: A 70TH DEGREELUNAR GRAVITY MODEL (GLGM2) 16,349

iO*

30 ø 30 ø

o o

.30 ø .30 ø

-60

Figure 4. Projected gravity anomaly errorsfor GLGM-2, computedfrom the 70x70 error covari- ance. The map is a mollweideprojection centeredon 270øE longitude, and the contour interval is three mGals. contain significantmare fill, are revealedas gravity lows from the known centers of the basins. For instance, (seeTable 7), consistentwith the findingsof earlierin- both the centersof the gravity lows correspondingto vestigators[Ferrari, 1977; Ananda, 1977]. A concen- Tsiolkovsky and Mendeleev are displaced 4 o to the west trated gravity high of 325 regals is apparent directly of the basin center noted in lunar imagery. This smear- north of Korolevat 4-5øN, 202øE.This high is locatedin ing effect results from the lack of direct tracking over a regionwhich contains some of the highesttopography much of the far side of the Moon. While the associ- on the Moon. The anomalies on the farside of the Moon ation of anomalies with the farside basins is evidence sometimesappear stretched, compressed,or displaced that the Doppler data are sensitive to mass anomalies

Table 7. GLGM-2 Gravity Anomalies for Prominent Lunar Basins and Maria

Feature Location GLGM-2 Gravity GLGM-2 Anomaly LGM-309b Anomaly, mGals Error, mGals Anomaly,a mGals

Imbrium 341øE, 36øN 311 37 301 Crisium 58.5øE, 17øN 324 32 350 Mare Oftentale 266øE, 19.5øS 188 29 211 Fecunditatis 52øE, 4øS 90 25 99 Nectaris 34øE, 16øS 274 29 281 Smythii 86.6øE, 2.4øS 173 28 170 Humorurn 320.5øE, 24øS 318 27 321 Serenitatis 19øE, 26øN 356 33 340 Nubium 345øE, 21øS -27 27 -37 Mendel-Rydberg 272øE, 52øS 22 41 -31 Freundlich-Sharonov 175øE, 18.5øN -36 40 -12 Hertzsprung 231.5øE, 1.5øN -87 33 -105 Moscoviense 147øE, 26øN -56 41 -61 Korolev 203øE, 4.5øS -48 36 -62 Mendeelev 141øE, 6øN -169 37 -175 South Pole-Aitken 185øE 25øS -116 40 -142 212øE 44øS -102 44 -115 180øE 75øS -82 44 -103

aThe LGM-309bsolution is a test solutionthat usedthe powerlaw constraintof 30 x 10-s//2. GLGM-2 useda powerlaw constraintof 15 x 10-s/12. 16,350 LEMOINE ET AL.: A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2) evenin areasthat lack direct tracking, it is quite certain The lunar geoid. The geoid, computedfrom the that the gravity field does not accuratelyrepresent the GLGM-2 solution, is shown in Plate 3. The planet has anomalyamplitudes [cf. Solomonand Simons,1996]. a total range in geoidspace of-275 to 475 m. This com- Consequently,geophysical analyses that incorporatere- pares to total range of 200 in for the Earth and 2,000 gionswithout direct tracking data shouldbe done with m for Mars. Many lunar basinshave prominent geoid the greatest of caution. signatures. There is a broad-scalecorrelation of geoid A further characteristic of GLGM-2 is that the model with topography(see Figure 5), with the exceptionof exhibits greater detail in the equatorial regionsthan at the masconbasins (cf. Imbrium, Serenitatis,Crisium, higher latitudes. This variable resolutionis a result of Humorum),which are geoidhighs. Sincethese basins the orbital covergeof the satellites contributing data are located at lower elevations, and since the total el- to GLGM-2 and is not an intrinsic feature of the lunar evation changeover these basins is small compared to gravity field. the total range in planetary topography,these mascon Contribution of Clementine. To assess the con- basinsappear as spikesin Figure 5. South Pole-Aitken tribution of the Clementine data to the lunar gravi- is revealed as a large and prominent geoid low with tational field, we computed a spherical harmonic so- a minimum of about -270 m. The uncertainty in the lution based only on these data and compared it to GLGM-2 geoid also correlateswith the tracking data the full GLGM-2 solution. The free-air anomalies from distribution and is nearly identical in shape to the free- the Clementine-onlygravity solution (LGM-261), illus- air gravity anomaly error map, depicted in Figure 4. trated in Plate 2, have a range of-121 to 255 regals The projected errors from the GLGM-2 error covari- (comparedto-294 to +358 mGalsfor GLGM-2). Due ance range from 2 rn on the equatorial nearsideto 24 rn to Clementine's polar orbit and global coverage,the on the high latitude regionsof the farside. Clementinedata now providethe strongestsatellite con- straint on the long wavelengthgravitational signatureof Tests With the RMS of Fit the Moon. Various analysespresented in the following sections underscore the contribution of the Clementine A series of test arcs involving each satellite in the data. gravity solution were selected to evaluate the perfor- mance of the interim solutions. The RMS of fit was Comparison of the full GLGM-2 solution from Plate computed for these arcs for the a priori Konopliv et 1 with thosefrom the Clementine-onlysolution (Plate al. [1993]field, the GLGM-1 solution[Lemoine et al., 2) showsthe extent to which the historicdata are es- 1994],and the currentsolution, GLGM-2. The results sential for adding detail to GLGM-2, particularly in are summarized in Table 8. The fit to the Lunar Or- the equatorial regions. Nonetheless,the major mas- biters 2, 4, and 5 are the same as with the Konopliv et con basinscan be discernedclearly from the Clementine el. [1993]60th degreeand ordermodel. We showslight data alone. Orientale appearsas a broad low, although the mascon at the center is not resolved. The South improvementsin the orbital tests with Lunar Orbiter 1 and the Apollo 15 subsatelliteand obtain a higher RMS Pole-Aitken basin appears as a broad low with a mini- of fit for both the Apollo 16 subsatellite and Lunar Or- mum of about -100 regals. The Clementine-onlyfree- biter 3. We attribute the somewhathigher fit on the air anomaly map alsoillustrates the presenceof the low or moat,surrounding Imbrium (discussedabove), Which Apollo 16 subsatelliteto the initial a priori weight for confirmsits existenceusing data from a completely in- thesedata (3 cm/s). This weightwas selected since this dependentsatellite. Artifacts in the handling of the satellite was tracked at altitudes of lessthan 30 km just Clementinedata are apparent,as suggestedby the ver- prior to impactingthe Moon aat the endof May 1972. It tical striping near 135øElongitude on the farside. appearsnow in retrospectthat a higher a priori weight wouldhave been justified (i.e. 1 cm/s), especiallysince Error analysis. Gravity anomaly errors, as com- thesedata wereprocessed in arcsof only a few hoursin puted from the calibrated covarianceof the GLGM-2 length. gravity solution,range from 17 regals in the equatorial For short arcs (1 to 2 days) of Clementinedata, we regionsof the nearsideto 45 mGals in the higher lati- saw little differencein the RMS of fit to the tracking tude regionsof the lunar farside(see Figure 4). The er- ror distribution correlates with the distribution of avail- data whenusing either a priori Konoplivet el. [1993] model, and both GLGM-1 and GLGM-2. For this rea- abletracking data (seeFigure 1), and the proximityof son,our test set of Clementinearcs are longer- between periapsisto the lunar equator for the eccentriclunar 4 and 12 daysin length. As would be expected,once the orbiters. The gravity anomalyerrors are listed in Table data are added into GLGM-1 and GLGM-2, we see an 7, along with the computedfree-air gravity anomalies improvedfit on thesetest arcs,compared to the Kono- of prominent lunar basins and maria. The GLGM-2 pliv et el. [1993]model. gravity anomaly values are comparedwith those from LGM-309b, where the power law constraintwas relaxed to 30 x 10-5//2. Relaxingthe powerlaw constraint Analysis of Coefficient Differences increasesthe power in the field. However,the anoma- The differencesin the coefficientsigmas between G L- lies computedfrom the LGM-309b field differ from the GM-2 and Lun60d[Konopliv et el., 1993],normalized GLGM-2 values by less than the predicted GLGM-2 by the coefficientsigmas from the GLGM-2 gravity gravity anomaly uncertainty. model are shownin Figure 6. These coefficientdiffer- LEMOINEET AL- A 70THDEGREE LUNAR GRAVITY MODEL (GLGM2) 16,351

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portion of the field. In fact, the addition of Clemen-

400 tine changesthe J•, Ja, Ja•, Jaa, J42, J44, and the sectoralterms (the Jtm coefficientsfor which I = m) through degree 10, by three to nine sigma. In Figure 200 7, we illustrate the percent differencein the coefficient sigmasbetween the LGM-309d solution(no Clemen- tine data) and the GLGM-2 solution. This figure il- lustrates the sensitivity of the Clementine data to the gravity field, and underscoresthe strengthof this satel- lite's contributionto the low degree(l = 2 to 3), and -200 sectoralterms through degree 20. The sectoral terms of the spherical harmonic expansion are purely longi-

-400 I I I tudinal [Kaula, 1966]. The sensitivitythat Clementine -10000 -5000 0 5000 10000 Topography(meters) providesfor theseterms resultsfrom the polar orbit ge- ometry of the spacecraft, as well as the strength and Figure 5. Geoid anomaliesfrom GLGM-2 versus quality of the tracking data. Clementine-derivedtopography [Smith et al., log7]. Of interest is to compare the solution for the low de- gree terms from GLGM-2 with those derived from lu- nar laser ranging. We presentthe comparisonsin Table encesare comparedto the solutionwithout the Clemen- 9. The greatest discrepanciesoccur for Ja, C•, and tine data (LGM-309d). Concentratingon the field be- Ca•. All other terms showreasonably close agreement. low degree20, the figure showsthat the Clementine We also include in Table 9 the coefficients from LGM- data have the most profound effect on the lower degree 309d(GLGM-2 with no Clementinedata), to illustrate

Table 8. RMS of Fit Tests With Lunar Gravity Models Gravity Field Spacecraft Number AverageArc AverageRMS of Arcs Length, hrs of fit, cm/s

Lun60da LO-5 7 148.04 0.151 GLGM-1 b 0.172 GLGM-2 c 0.155

Lun60d LO-4 7 98.8 0.065 GLGM-1 O.O82 GLGM-2 0.071

Lun60d LO-3 12 25.08 0.098 GLGM-1 O.2O7 GLGM-2 O.284

Lun60d LO-2 11 38.98 0.146 GLGM-1 0.148 GLGM-2 0.147

Lun60d LO-1 5 86.17 0.230 GLGM-1 0.111 GLGM-2 0.150

Lun60d A- 15ssd 26 82.67 2.185 GLGM-1 2.041 GLGM-2 1.915

Lun60d A- 16ssd 36 4.48 0.115 GLGM-1 1.221 GLGM-2 0.158

Lun60d Clementine 5 206.40 0.296 GLGM-1 0.087 GLGM-2 0.093

a 60x60 sphericalharmonic solution from Konoplivet al. [1993]. b 70x70spherical harmonic solution from Lemoineet al. [1994]. c 70x70 sphericalharmonic solution from this paper. dA-15ss = Apollo 15 subsatellite.A-16ss = Apollo 16 subsatellite. LEMOINE ET AL.- A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2) 16,355

the most weakly determined of the low-degreeterms Coefficientdi•rencesbetween GLGM-2andLUN60D, normalizedbythesigm• from GLGM-2 determinedby lunar laserranging (LLR), and the C31 coefficientsderived from the satellite determined grav- 2 1.4 1.4 4.7 3 7.2 3.2 0.0 4.9 itational models agree more with each other than with 4 0.3 1.8 4.4 0.4 9.7 the LLR-determined value. 5 2.7 3.1 1.3 0.3 0.8 9.2 6 1.7 0.8 0.2 0.3 2.2 0.8 9.4 7 4.0 1.4 0.0 0.8 0.1 3.2 0.6 4.8 GM Determination 8 0.0 0.6 2.2 0.4 0.9 0.2 0.6 0.2 6.3 9 2.5 2.4 1.2 2.6 1.4 0.6 0.0 1.1 1.4 8.3 GLGM-2 included an adjustment for the lunar GM, 10 3.7 0.2 2.4 0.8 2.5 0.2 1.6 0.4 6.3 2.8 6.0 11 1.2 0.1 1.3 0.6 0.4 1.5 1.5 2.3 1.2 0.9 4.1 0.4 and we obtained4902.80295 4- 0.00224km3/s 2. The 12 2.0 0.8 1.6 1.1 0.6 1.1 1.8 2.8 2.1 2.2 2.1 3.4 0.2 Degree GLGM-2 estimate is comparedwith other valuesin Ta- SphericalHarmonicOrder ble 10. The Ferrari et al. [1980]value of 4902.7993 0 i 2 3 4 5 6 7 8 9 10 11 12 4- 0.0029km3/s 2 is determinedlargely from the high- altitude tracking of the Lunar Orbiter 4 spacecraft.The Coefficientdi•rencesbetweenLGM-309DandLUN60D, normalizedbythosigm•om GLGM-2 small sigma for the lunar GM in GLGM-1 is caused 2 0.5 1.1 0.2 by the over optimistic sigma applied to the Clementine 3 0.5 0.6 0.4 0.7 4 1.2 0.4 0.4 0.2 0.1 Dopplerdata in that model of 0.50 cm/s. In contrast, 5 1.8 0.1 0.4 1.1 0.1 0.2 6 0.9 1.1 1.5 0.0 1.1 0.1 0.3 the sigmaon the lunar GM in GLGM-2 is derivedfrom a 7 0.2 0.0 0.2 1.3 0.6 1.1 0.1 1.1 calibrated solution, where the Clementine data received 8 0.5 0.0 0.2 0.3 0.8 0.5 0.2 0.2 0.4 9 0.2 0.1 0.2 1.6 0.3 0.3 0.1 04 0.7 17 an effectivefinal data sigmaof 1.36 cm/s. The differ- 10 1.8 0.1 1.3 0.3 0.8 0.6 1.6 0.3 3.2 2.3 1.8 ences between the various lunar GM determinations are 11 1.1 0.2 0.7 1.6 0.3 1.1 1.4 2.1 1.1 0.9 3.0 1.8 12 0.5 0.5 1.4 0.8 1.3 0.6 1.2 2.3 2.4 2.1 2.2 2.1 0.9 a few partsin 107. Degree Spherical Harmonic Order 0 1 2 3 4 5 6 7 8 9 10 11 12 Covariance Analyses Gravity field error versus order. We have pro- Figure 6. Coefficient differencesbetween Lun60d jected the the 70x70 error covarianceof the GLGM-2 [Konoplivet al., 1993]and GLGM-2, normalizedby the gravity solutiononto lunar spacecraftorbits, using the sigmasfrom GLGM-2. ERODYN covariancepropagation software [Englar et al., 1978]. We mapped the uncertaintiesand correla- tions for each spherical harmonic order on near-circular the influence of Clementine on these low degree terms. polar orbits (inclinationof 89ø ) at altitudesof 25, 50, Comparisonof the coefficientsfor Ja and C22 showsthat 100, and 400 km. This information is summarized in it is the Clementine data that changetheir estimates so Figure 8. Kaula's [1966]linear orbit perturbationthe- strongly. The strength of these determinationsresults ory predictsthat the order 1 and order 2 m-daily pertur- in part from the multidayarcs (4, 6, 10, and 12 days) bations, with periods of approximately 28 and 14 days, used to processthe Clementine tracking data. Cax is will producethe largestperturbations on the spacecraft

2 72 60 82 3 54 56 56 83 4 35 27 43 53 82 5 25 61 24 44 52 80 6 51 40 42 23 41 49 78 7 39 35 44 44 24 42 46 74 8 29 29 36 42 39 22 40 43 69 9 29 40 32 33 40 34 19 37 39 62 10 23 27 29 27 29 33 28 16 33 35 55 ll 19 21 23 23 25 24 27 23 12 27 33 50 12 16 19 16 18 18 17 18 19 18 9 22 32 47 13 11 13 14 13 14 14 15 14 13 14 8 17 31 45 14 7 8 8 9 l0 l0 ll 11 11 10 12 8 14 29 42 15 4 5 6 5 8 7 8 10 9 9 9 11 8 12 26 39 16 2 3 3 4 4 5 5 7 9 7 8 8 9 7 9 22 35 17 2 1 2 2 3 3 4 4 6 7 6 7 7 8 7 8 18 30 18 1 1 1 1 1 2 2 3 3 5 6 5 6 6 6 6 6 14 25 19 1 1 1 1 1 1 1 1 2 3 4 4 4 5 5 5 5 5 11 21 20 1 0 1 1 1 1 1 1 1 2 2 3 3 4 5 5 4 5 5 9 17 Degree Spherical Harmonic Order 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Figure 7. Percentchange in the coefficientsigmas for GLGM-2 and LGM-309d (GLGM-2 with no Clementinedata), normalizedby the sigmasfrom LGM-309d. 16,356 LEMOINE ET AL.: A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2)

Table 9. Comparisonof Low Degree and Order Harmonics From Satellite Track- ing and Lunar Laser Ranging

Coefcient a LLR 0 Lun60dc GLGM_2d LGM_309de

J2 204.0 4- 1.0 203.805 4- 0.057 203.986 4- 0.131 203.566 4- 0.467 Ja 8.66 4- 0.16 8.25 4- 0.06 9.99 4- 0.24 8.53 4- 0.53 6'22 22.50 4- 0.11 22.372 4- 0.011 22.227 4- 0.023 22.413 4- 0.130 6'a• 32.4 4- 2.4 28.618 4- 0.018 28.290 4- 0.098 28.736 4- 0.179 S•l 4.67 4- 0.73 5.87 4- 0.02 5.54 4- 0.07 6.15 4- 0.21 6'•2 4.869 4- 0.025 4.891 4- 0.009 4.886 4- 0.030 4.855 4- 0.075 $•. 1.696 4- 0.009 1.646 4- 0.008 1.656 4- 0.029 1.618 4- 0.064 6'•a 1.73 4- 0.050 1.719 4- 0.003 1.756 4- 0.007 1.673 4- 0.038 $•a -0.28 4- 0.020 -0.211 4- 0.003 -0.270 4- 0.006 -0.268 4- 0.035

a All coefficients are unnormalized. øLunarlaser ranging (LLR) fromDickey et al. [1994]. C60x60spherical harmonic solution from Konoplivet al. [1993]. d70x70spherical harmonic solution froIn this paper. e LGM-309d --- GLGM-2 with no Clementine data. orbit (seeFigure 8). For a 100-kmnear-circular polar tary retrogradeor progradeinclinations (11 ø, 21ø, 29ø, orbit, the total position error causedby uncertaintiesin 800, 900, 1000, 1510, 1590, 169ø). The data from the the GLGM-2 gravity field will be 1.4 km at order i and Apollo 15 subsatellitecontribute strongly to GLGM-2, 1.6 km at order 2. It is only beyond order 15 that the and it can be arguedthat the GLGM-2 field is strongly positionuncertainty due to gravityfield error falls below tuned to the orbit of the Apollo 15 subsatellite, since 100 m. In a 100-km near-polar orbit, the total position it has its lowest projected error at that satellite's incli- error due to the uncertainties in the GLGM-2 gravity nation (1510, and its complement,29ø). Comparisonof field is 2.5 km. This total position error increasesto the predicted error from the GLGM-2 and the LGM- 4.2 km for a circularpolar orbit at 50-km altitude, and 309d (solutionwith no Clementincdata) revealsthat 5.7 km for a polar orbit with a mean altitude of 25 km. Clementincreduces the gravityfield error at the higher These error projections do not include the omissioner- inclinations(60 o to 1400) by 5-20%. At the inclination ror of the termsbeyond degree 70. The actualgravity of 90 ø, Clementinc reduces the radial orbit error from field induced orbit error would be larger than the values 743 to 554 m. cited here, especiallyfor the lowestaltitude orbits. Projected orbit error versus inclination. Us- ing the linear orbit theory of Rosborough[1986], the ra- Summary dial orbit error as predicted by the GLGM-2 covariance We have developed a spherical harmonic model of was mapped as a function of inclination, for a near- the lunar gravity field complete to degree and order circular lunar orbit at an altitude of 100 km. The re- 70 that incorporatesthe tracking data from Lunar Or- sults are depicted in Figure 9. The radial error ranges biters 1 to 5, the Apollo 15 and 16 subsatellites, and from just over 122 m at 29o inclination to 1364 m at Clementine. The Clementinedata provide the strongest 1290inclination. The increasein projectederror at the satelliteconstraint on the low degreeand orderfield, mid-rangeinclinations (30 o to 140ø) is understandable whereas the historic data are indispensablefor pro- giventhat only data from predominantlylow inclination viding distributed regions of high-•esolutioncoverage satellites have contributed to GLGM-2. Careful exam- within +30ølatitude. Our solution resolvesthe gravi- ination of the spectrum also revealsthat local minima tational signaturesof both farside and nearsidebasins occur near the specific inclinations from which satel- and mascons. To stabilize the solution at short wave- lite tracking data were acquiredor at their complemen- lengthsthat are not characterizedby a globallyuni-

Table 10. Comparisonof Solutions for the Lunar GM

Analysis Description GM, kma]s • • , ,

Ferrari et al. [1980] LLR and LO-4 4902.7993 4- 0.0029 Konoplivet al. [1993] Lun60d (60x60) 4902.79781 4- 0.00122 Lemoineet al. [1994] GLGM-1 (70x70) 4902.80263 4- 0.00095 This paper GLGM-2 (70x70) 4902.80295 4- 0.00224 This paper LGM-309d (70x70) 4902.80280 4- 0.00603 LEMOINE ET AL.: A 70TH DEGREE LUNAR GRAVITY MODEL (GLGM2) 16,357

1 04 includeseither satellite-to-satellitetracking or a grav- ity gradiometerto permit direct gravity mappingof the 1 03 farside should remain a high priority for lunar science.

Acknowledgments. We thank W. M. Kaula for a help- • I 02 ful review; W. L. Sjogrenand A. S. Konopliv for providing the Lunar Orbiter and Apollo subsatellite data; the Clemen- tine Operations Team at the Naval Research Lab and at the 1 0• DSPSE Mission Operations Center including B. Kaufmann and J. Middour for providing ancillary information related 1 0ø to Clementine;R. M. Beckman,and the Goddard Fhght Dy- namics Facility for transmission of the Clementine data dur- ing the mission;J. A. Marshall (NASA GSFC) for valuable 10 -1 adviceand technicalsupport; S. K. Fricke(Hughes/STX and

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