GRAVITY MODEL DEVELOPMENT FOR THE GRAIL MISSION. F.G. Lemoine1, S.J. Goossens2,1, E. Mazarico3,1, T.J. Sabaka1, J.B. Nicholas4,1, D.D. Rowlands1, B.D. Loomis5,1, D. Caprette5,1, D.S. Chinn5,1, G.A. Neumann1, M.T. Zuber3, D.E. Smith3,1. 1NASA GSFC, Code 698, 8800 Greenbelt Road, Greenbelt MD 20771 U.S.A. (email: Frank.G.Lemoine@nasa.gov), 2CRESST, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore MD 21250 U.S.A., 3Massachusetts Institute of Technology, MIT 54-314, 77 Massachusetts Avenue, Cambridge MA 02139 USA, 4Emergent Space Technologies, 6411 Ivy Lane, Greenbelt, MD 20770, U.S.A., 5Stinger Ghaffarian Technologies, 7701 Greenbelt Road, Greenbelt, MD 20770 U.S.A.

Introduction: The twin Gravity Recovery and In- monics with the full set of primary mission data [2,3]. terior Laboratory (GRAIL) spacecraft were launched Models are characterized by the RMS of fit, the orbit in September 2011 on a Discovery-class NASA mis- determination performance on the GRAIL satellites, sion to study the gravitational field of the Moon [1]. and on independent satellites such as Lunar Prospector, Extremely accurate range-rate observations between and through correlation with detailed models of the the two spacecraft at the Ka-band radio wavelength lunar topography. The a priori model for the GRAIL enable the determination of the gravity field of the POD analyses was the SGM150 (150x150) lunar gravi- Moon to very high degree since the data are acquired ty model based on a combination of SELENE, Lunar continuously, even when the spacecraft are not tracked Prospector and historical lunar model data[4,5]. The from the Earth. The mission builds on the success of initial fits to the KBRR data were on the order of 1000 the GRACE (Gravity Recovery and Climate Experi- – 2000 microns/second. Presently, the fits to the latest ment) to measure the time-variable gravity field of the 420x420 model are 0.1 to 0.2 microns/sec, except dur- Earth. The spacecraft flew on a low energy trajectory ing the low altitude phases where they reach 1-2 mi- to the Moon and after orbit insertion, a series of orbit crons/sec. Even at these levels, there are indications synchnoization maneuvers brought the twin spacecraft the signal in the KBRR data residuals has not been to the mean mapping orbit altitude of 50 km. The exhausted with the 420x420 model, given that the mapping mission for GRAIL commenced on March 1, KBRR data quality is estimated to be better than 0.05 2012 and continued until May 29, 2012. During the microns/second. prime mission, the periapsis altitude varied between approximately 20 and 50 km, and the separation dis- We review, compare and contrast the inversion strate- tance varied between 80 and 200 kms. The twin space- gies for high degree models and discuss the implemen- craft were tracked in 2-way mode at S Band using the tation of the inversion strategies for analysis of GRAIL Deep Space Network (DSN). In addition X Band data data on the NASA Center for Climate Simulation were collected in 1-way mode using the Ultra-stable (NCCS) supercomputers at NASA GSFC. For the Oscillators (USOs) on both of the GRAIL spacecraft. GRACE analyses of time-variable gravity at GSFC we Approach: We discuss the orbit determination employ a solution scheme relying on the method of strategies we applied to the analysis of the GRAIL preconditioned conjugate gradients. At present for our high-degree GRAIL solutions, we have chosen to rely data. While GRACE benefits from homogeneous on direct accumulation of the normal equations in a tracking by the Global Positioning System (GPS) with batch algorithm, adapting our programs to operate effi- orbits determined radially to an accuracy of 1-2 cm ciently on the NASA GSFC supercomputers. While RMS, the GRAIL orbit determination must rely exclu- the orbit determination and preliminary analysis was sively on the Deep Space Network (DSN) tracking, done on the local work stations, the final accumula- and is discontinuous due to occultation while the tions and inversions were performed on the NCCS spacecraft passes over the lunar farside and due to supercomputers, with a full 420x420 accumulation and tracking stations not following the GRAIL spacecraft inversion taking in general on the order of 12 hrs of on a continuous basis. In addition, the force model wall clock time. We have also developed an inversion error from the gravity model inaccuracies directly im- strategy based on the use of the square root infor- pact the quality of the orbit determination and at the mation filter (SRIF), and we have tested the inversion start of the mission easily amounted to tens to a few of GRAIL data using both strategies and identical hundred meters in total position. We characterize the background modeling to degree 540x540. quality of the tracking data over the course of the pri- mary mission, and the resultant orbit performance as The GRAIL orbit determination requires the applica- the gravity models that we developed were expanded tion of precise orbit and measurement models, and in size from 270x270 with the first month of data, to necessitated a number of model improvements in the 360x360 and eventually to 420x420 in spherical har- GEODYN orbit determination program, and how we validated the KBRR GRAIL data model through anal- Rev. 10.1007/s11214-012-9952-7. [2] Zuber et al. ysis of simulated data provided by the GRAIL team at (2012), Science, 10.1126/science.1231507. [3] Lem- the Jet Propulsion Laboratory. oine et al., Abstract G32A-03, presented at 2012 Fall Meeting, AGU, San Francisco, California, 3-7 Dec 2012. [4] Goossens et al. (2011), J. Geodesy, 85, 487- Results: The final model we have developed with 504, 10.1007/s00190-011-0446-2. [5] Namiki et.al. the prime mission data is complete to 420x420 in (2009), Science, 323, 900. spherical harmonics and is based solely on GRAIL data alone in that no historical data have been included in the solution. We illustrate in Figure 1 the power spectrum of the new 420x420 solution, as well as the a priori 540x540 model, and a solution developed with- out the application of a Kaula constraint. The a priori model was an attempt to exhaust the signal in the KBRR data, however it still produced anomalous pat- terns in the gravity anomalies. Thus we updated the parameterization for the modeling of the empirical accelerations in the development of the 420x420 solu- tion GRGM420A. The fact that the power spectrum of the coefficients, and the coefficient standard deviations are an indication that the signal has not been exhausted in the data, and that a higher degree solution should be developed from the data.

Fig. 1: Power spectrum of the GSFC GRAIL gravity solution, GRGM420A

We note that we have calibrated the error covari- ance using variance component analysis, and at the lower degrees we see an improvement of three orders of magnitude over historical models. In addition to analyzing the power spectra of the new solutions, we evaluate the model using differences with the compan- ion GRAIL model from JPL (GL0420) [1], and through analysis of the power in the Bouguer coeffi- cients derived from a combination of the gravity with topography of the Moon derived from LRO. We also test the solutions through orbit overlaps, and the RMS of fit to the GRAIL data.

References: [1] Zuber M.T. et al. (2012), Space Sci. Fig. 1: Lunar Prospector extended mission data fit