The Near Side: Regional Lunar Gravity Field Determination

Pełen tekst

(1)The Near Side Regional Lunar Gravity Field Determination. Sander Goossens.

(2) The Near Side Regional Lunar Gravity Field Determination.

(3)

(4) The Near Side Regional Lunar Gravity Field Determination. Proefschrift. ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College voor Promoties, in het openbaar te verdedigen op woensdag 18 mei 2005 om 10:30 uur door Sander Johannes GOOSSENS ingenieur luchtvaart- & ruimtevaart geboren te Bergen op Zoom.

(5) Dit proefschrift is goedgekeurd door de promotor: Prof.ir. B.A.C. Ambrosius. Samenstelling promotiecommissie: Rector Magnificus, Prof.ir. B.A.C. Ambrosius, dr.ir. P.N.A.M. Visser, Prof.dr. K. Heki, Prof.dr. G. Balmino, Prof.dr.-ing. habil R.A.P. Klees, Prof.dr.ir. A.W. Heemink, dr.ir. R.J.J. Koop,. voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft, toegevoegd promotor Hokkaido University, Japan Centre Nationales d’Etudes Spatiales, France Technische Universiteit Delft Technische Universiteit Delft Stichting Ruimteonderzoek Nederland. Published by: Optima Grafische Communicatie, Rotterdam. ISBN 90-8559-049-3. c 2005 by S. Goossens. All rights reserved. No part of the material protected by this copyright may be reproduced, or utilised in any other form or by any means, electronic or mechanical, including photocopying, recording or by any other information storage and retrieval system, without the prior permission of the author..

(6) ...the Moon ain’t romantic, it’s intimidating as hell Tom Waits. There is something to be learned from a rainstorm. When meeting with a sudden shower, you try not to get wet and quickly run along the road. But doing such things as passing under the eaves of houses, you still get wet. When you are resolved from the beginning, you will not be perplexed though you will still get the same soaking. This understanding extends to everything. from Hagakure, by Yamamoto Tsunetomo.

(7)

(8) Contents. Acknowledgements. xi. Summary. xiii. 1 Introduction 1.1 A controversial Moon . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Why study the Moon and its gravity field - scientific rationale 1.3 Goals of this research . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. 1 1 2 6 9. 2 Global and regional representations of the gravity field 2.1 The global lunar gravity field . . . . . . . . . . . . . . . . . 2.1.1 Representation in spherical harmonics . . . . . . . . 2.1.2 A brief history of selenopotential modelling . . . . 2.2 On the role of regularisation of global models . . . . . . . . 2.2.1 Some remarks about ill-posed problems . . . . . . . 2.2.2 Regularisation of ill-posed problems . . . . . . . . . 2.2.3 Influence of the choice of regularisation parameters 2.3 Regional representation forms . . . . . . . . . . . . . . . . . 2.3.1 Point-mass models . . . . . . . . . . . . . . . . . . . 2.3.2 Locally supported basis functions . . . . . . . . . . 2.3.3 Gravity anomalies . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 11 11 12 15 23 24 25 26 30 31 33 35. 3 Gravity anomalies and the Stokes formulation 3.1 The Stokes kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Limits of the Stokes kernel . . . . . . . . . . . . . . . . . . 3.1.2 Derivatives of the Stokes kernel . . . . . . . . . . . . . . . 3.1.3 Spectral equivalents . . . . . . . . . . . . . . . . . . . . . 3.2 Relating the Stokes formulation to spherical harmonics . . . . . 3.3 Numerical implementation of the Stokes formulation . . . . . . 3.3.1 Equations for the accelerations due to gravity anomalies 3.3.2 Discretisation of the acceleration expressions . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 39 39 40 42 43 46 48 48 49. . . . . . . . . . . .. . . . . . . . . . . ..

(9) viii. Contents 3.4. . . . . . . . .. 51 51 53 56 58 60 60 63. 4 Recovery of gravity anomalies from range and range-rate data residuals 4.1 The recovery method explained . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Geometrical partials . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Sensitivity matrix and state transition matrix . . . . . . . . . . 4.1.3 Partials with respect to the force parameters . . . . . . . . . . 4.1.4 Design matrix and normal equations . . . . . . . . . . . . . . . 4.2 Assessment of the linearisation error . . . . . . . . . . . . . . . . . . . 4.2.1 Error free simulations . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 On the influence of the gradient term . . . . . . . . . . . . . . 4.3 The influence of systematic errors on the data . . . . . . . . . . . . . . 4.3.1 State vector adjustments . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Near- and far-field contributions to the data . . . . . . . . . . 4.4 Noisy data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Different noise levels and the use of regularisation . . . . . . . 4.4.2 The influence of satellite altitude . . . . . . . . . . . . . . . . . 4.4.3 Note on using a non-diagonal regularisation matrix . . . . . . 4.5 On the choice of the regularisation parameter . . . . . . . . . . . . . . 4.5.1 L-curve method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Generalised cross validation . . . . . . . . . . . . . . . . . . . 4.5.3 What to choose? . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Some conclusions about the recovery method . . . . . . . . . . . . . .. 67 68 69 70 72 74 75 76 82 86 86 95 102 103 110 112 115 116 120 122 124. 5 Results from Lunar Prospector extended mission data 5.1 The Lunar Prospector mission . . . . . . . . . . . . . . . . . . 5.2 On Lunar Prospector tracking data . . . . . . . . . . . . . . . 5.2.1 General data issues . . . . . . . . . . . . . . . . . . . . 5.2.2 Ramped data . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Ramped data conversion factors . . . . . . . . . . . . 5.3 Orbit determination set-up for Lunar Prospector . . . . . . . 5.3.1 Reference models . . . . . . . . . . . . . . . . . . . . . 5.3.2 Parameters in the orbit determination . . . . . . . . . 5.4 Orbit determination results . . . . . . . . . . . . . . . . . . . 5.4.1 The effect of manoeuvres . . . . . . . . . . . . . . . . 5.4.2 Data fit . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Consistency assessment by means of overlap analysis. 127 128 130 130 134 139 140 141 145 150 150 152 156. 3.5. Validation results of the numerical implementation . . . . . . 3.4.1 Validation method . . . . . . . . . . . . . . . . . . . . . 3.4.2 Numerical results for validation . . . . . . . . . . . . . 3.4.3 Using the spectral expressions as filter functions . . . . 3.4.4 Applying a cut off angle to the integration . . . . . . . An inverse Stokes method using satellite accelerations . . . . . 3.5.1 Anomalies from satellite accelerations . . . . . . . . . . 3.5.2 Obtaining the accelerations from satellite tracking data. . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . ..

(10) Contents. 5.5. 5.6. 5.7 5.8 5.9. 5.4.4 Tracking data residuals . . . . . . . . . . . . . . Benchmark test for the recovery method . . . . . . . . . 5.5.1 Benchmark set-up and input signal . . . . . . . 5.5.2 Using two-day arcs . . . . . . . . . . . . . . . . . 5.5.3 Using short arcs . . . . . . . . . . . . . . . . . . . 5.5.4 Summary of regional recovery set-up . . . . . . Solution for Mare Serenitatis . . . . . . . . . . . . . . . . 5.6.1 On Mare Serenitatis . . . . . . . . . . . . . . . . 5.6.2 Application of the recovery method to real data 5.6.3 Regional adjustments for Mare Serenitatis . . . . Solution for crater Copernicus . . . . . . . . . . . . . . . 5.7.1 On crater Copernicus . . . . . . . . . . . . . . . . 5.7.2 Regional adjustments for crater Copernicus . . . Solution for a part of the highlands . . . . . . . . . . . . On the use of a priori information . . . . . . . . . . . . .. ix . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. 158 161 161 163 166 170 171 171 172 180 186 186 187 191 196. 6 Conclusions and recommendations 201 6.1 A summary of the main results . . . . . . . . . . . . . . . . . . . . . . 201 6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 A Gravity anomalies, disturbances, and their kernel functions 211 A.1 Formulas for the gravity anomaly . . . . . . . . . . . . . . . . . . . . . 211 A.2 Formulas for gravity disturbances . . . . . . . . . . . . . . . . . . . . 215 A.3 Series expansions and their analytical equivalent . . . . . . . . . . . . 215 B Some remarks on regularisation methods 219 B.1 Tikhonov regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . 219 B.2 Generalised least squares . . . . . . . . . . . . . . . . . . . . . . . . . . 220 B.3 Truncated singular value decomposition . . . . . . . . . . . . . . . . . 222 C Ramped data pre-processing. 223. Samenvatting. 225. Curriculum Vitae. 227. Bibliography. 229.

(11)

(12) Acknowledgements. The fact that this dissertation carries the name of just one author is a result of the fact that the final defence has to be done by just one person; the actual work done herein has been supported by many colleagues. This dissertation uses topics from specialisms such as planetary geodesy, orbit determination and tracking data analysis, and as a consequence, a large number of people have contributed to this dissertation being the dissertation you, the reader, now have in your hands. Without these contributions, this dissertation simply would not be, so I would like to express my deep gratitude to a number of people. First of all, my promotor Boudewijn Ambrosius gave me the opportunity to become a PhD student at Delft University of Technology, Faculty of Aerospace Engineering, in the Astrodynamics and Satellite Systems group. I especially want to thank him for providing the opportunity to do this PhD on a topic slightly offcourse so to speak from the major research efforts in his group. Pieter Visser was my daily supervisor, and he has contributed many, many ideas. Without his suggestions and ideas, and the discussions with him, this work would be not nearly what it is now. Rune Floberghagen wrote a PhD dissertation on lunar gravimetry while I started working on my Master’s thesis. With his work, he has paved the way for me to continue working on lunar gravity field modelling in Delft. His insight and his ability to always ask the right questions have been very helpful in guiding me through this research. I hope that this dissertation is how he saw his initial efforts to be continued at Delft. The work as presented in chapter 5 was mainly done while I stayed with the National Astronomical Observatory Japan (NAOJ), in Mizusawa. I was there on a fellowship of the Japan Society for the Promotion of Science (JSPS) for a period of nine months, from November 2003 until August 2004. JSPS is greatly acknowledged for providing the means to stay in Japan and do the last parts of the research there, which resulted in a concluding chapter about actual data processing. At NAOJ, professor Kosuke Heki was my host researcher. I have especially enjoyed his hospitality, his interest and the ideas he has contributed with respect to the solutions using the actual data. I also enjoyed working in the RISE group at NAOJ very much, in the framework of the SELENE mission, and I hope this dis-.

(13) xii. Acknowledgements. sertation can be of help for future research. While at NAOJ, I especially enjoyed the company and many discussions with Takayuki Sugano, Jinsong Ping and Koji Matsumoto very much. They made the stay during working time and beyond very enjoyable. Do mo arigato gozaimashita. I am also grateful for the input of Alex Konopliv from JPL, who kindly provided me with the complete tracking data set of Lunar Prospector. Frank Lemoine and Dave Rowlands from NASA’s Goddard Space Flight Center are both gratefully acknowledged for their help with the ramped data. They provided me with their pre-processor which enabled me to use the data. Dave Rowlands is especially thanked for providing the GEODYN II software, and for the discussions about GEODYN II. Radboud Koop of the ”Stichting Ruimteonderzoek Nederland” (SRON, the Dutch foundation for Space Research) and Jurgen ¨ Kusche of the section Physical and Space Geodesy of the Department of Earth-Observation and Space Systems (DEOS) are thanked for all the discussions we had about regional gravity recovery and regularisation in particular. They have also been critical readers of this dissertation, which was greatly appreciated. Oliver Montenbruck of the Deutsches Zentrum fur ¨ Luft- und Raumfahrt (DLR) and Frank Weischede (currently at EUMETSAT) are also thanked for their input concerning data processing and lunar orbit determination. During all the time spent in Delft, I enjoyed working with all my colleagues there very much. I especially would like to thank Saskia Matheussen, with whom I shared a room at the faculty, and Jos´e van den IJssel: I have always enjoyed our discussions, whatever topic they were about.. Delft, The Netherlands, March 2005 Sander Goossens.

(14) Summary. The Earth and the Moon are in a spin-orbit resonance, which causes the Moon to always have the same face towards the Earth. This severely limits Earth-based tracking of a satellite, since no tracking is available when the satellite flies over the far side of the Moon. This also hampers the determination of the global lunar gravity field: a solution can be obtained, but it requires constraints in order to turn the ill-posed problem into a well-posed one with a stable and physically meaningful solution (such as in terms of data fit and orbit overlap statistics, as well as correlation with topographical features). Floberghagen [2002] has shown to what extent the actual solution for the global lunar gravity field depends on the constraint and the choice of the regularisation parameter. This sparked the interest in determining regional gravity fields. By determining the regional gravity field of parts on the near side, which is covered excellently with high-quality Lunar Prospector data, the effects of the unknown far side gravity field can be limited and the available data can be exploited for their information content. The goal in this dissertation therefore is to develop a suitable method for the regional recovery of gravity from satellite tracking data, in order to improve the resolution and accuracy of current lunar gravity field models. This will be applied to real Lunar Prospector tracking data from the extended mission period to create high-resolution solutions for parts on the near side of the Moon. Such solutions are thought to contribute to the improvement of the selenophysical interpretation of features on the Moon. To this end, the regional gravity field is represented in this dissertation by means of gravity anomalies, referring to a reference planetary radius. A recovery method that relates these anomalies to tracking data residuals by means of a linear variational approach has been developed and thoroughly tested. Extensive simulations have shown that with this method it is possible to extract the highfrequency gravity information from the observations. A short-arc approach has been used in the recovery, where one arc consists of the period it takes the satellite to cross the recovery grid under consideration. State-vector adjustments are difficult to estimate from these short arcs, which shows the need for a good reference orbit prior to the recovery. Simulations using stochastic errors on the data with a size of 0.5 m for range data and 1.0 mm/s for Doppler data have shown that,.

(15) xiv. Summary. with the inclusion of low-altitude data at e.g. 20 km, regularisation in the sense of including a priori information is not necessarily required to obtain a solution. By conducting a benchmark test, the processing scheme of real Lunar Prospector data to determine the regional adjustments has been verified, and the recovery analysis was done with these data for three different areas on the Moon: Mare Serenitatis, crater Copernicus and a part of the lunar highlands. This analysis showed that the high-frequency gravity information can be extracted from the data. Solutions for the three areas have been created without the use of Tikhonov regularisation, but the inclusion of a priori information in the form of a full covariance matrix for the anomalies leads to results that are more consistent. Differences between solutions with and without regularisation are mainly concentrated in certain areas of the solution, showing that the larger part of the solution is determined by the observations themselves. The adjustments were small when compared with the a priori signal. This shows that the LP150Q model, which is the latest Lunar Prospector gravity field model (status beginning 2005) and which is used as the reference model, already contains most of the information for these areas. Validation of the results is not possible since no independent data exist. Nevertheless, it has been shown high-resolution regional gravity solutions can be created, and the first step towards a high-resolution map of gravity anomalies of the complete near side has been made. This can further benefit future selenophysical interpretation in terms of e.g. the compensation state of craters and mascons. Such a regional recovery analysis is not limited to one celestial body only and will be very valuable in planetary geodesy. Current and future dedicated gravity satellite missions to the Earth and planets should provide ample opportunity to apply a method such as presented in this dissertation..

(16) Chapter 1. Introduction. ”the earth and the moon reciprocally illuminate each other” Galileo Galilei in his Dialogue Concerning the Two Chief World Systems. Since ancient times, the Moon has been an object of admiration and study. Travellers studied its phases and motions to estimate the state of the tide in various harbours. Telescopes have been turned towards it since Galileo first did this in 1609, and discovered mountains on the Moon. Satellites have flown by it and orbited it, collecting a wealth of data. Astronauts have walked on it, including one geologist. Sample rocks have been returned to Earth and studied, as well as meteorites from the Moon found on Earth. All this might raise the idea that perhaps the Moon is, next to the Earth, a very well-known place. This is not the case, as there are still many questions concerning the formation, structure and composition of the Moon. Just as is said about every day life in a big city, we hardly seem to know our neighbour.. 1.1 A controversial Moon When in 1609 Galilei Galileo heard of a new Dutch curiosity called a ”spyglass”, which was told to make objects that are far away appear closer than they really are, he was immediately interested. He sought to improve on the power of the spyglass, and started working on a design by himself, and in subsequent attempts he refined his initial optical design. When autumn came, with its earlier dark evenings, he turned his telescope (as a colleague in Rome later renamed the instrument) towards the Moon, and he discovered spots which changed shape over time. Galileo concluded these were best explained by the shadows cast by mountains, and he spent half of December that year drawing detailed maps of the Moon.

(17) 2. Introduction. in several phases. The existence of mountains however meant that the Moon could not be the perfect, unchangeable sphere it was thought to be, as a principle of the Aristotelian view of the heavenly bodies. From the Moon he turned to the stars, and the planets, and subsequently was the first to distinguish the planets further from the stars. And in January 1610, he made the most extraordinary discovery of all: he found four moons circling Jupiter. This discovery undermined the view of the heavens as perfect and unchangeable even further: had the roughness of the Moon already suggested that certain features of the Earth, which was thought to be different from all other heavenly bodies, repeated itself in the heavens, the moons around Jupiter showed that apparently it was not only the Earth around which natural satellites could orbit. Together with the discovery of the phases of Venus, suggesting at least one planet orbited the Sun, and the observation of sunspots, these were strong evidences for the Copernican world view. This opening story is, apart from a very interesting period in the history of science, the best example one could wish for to show what benefits can be gained from studying the Moon. The point that is rather grotesquely made, is that studying the Moon does not only imply studying the Moon itself: it implies a better understanding of the structure and evolution of the Moon, Earth-Moon system, and the solar system as a whole. In that sense, the Earth and Moon truly do illuminate each other, not only in the literal sense as it was meant in Galileo’s Dialogue, but also as a figure of speech: by studying one, things can also be learnt about the other.. 1.2 Why study the Moon and its gravity field scientific rationale The Moon is relatively large with a diameter that is 27% of that of the Earth; there are no other planets in the solar system with such a relatively large moon. The Earth and Moon share a history of 4.5 billion years, during which the presence of the Moon has had its noticeable influence on the Earth: the effect of lunar tides on the Earth’s oceans and biosphere is well known, and the existence of the Moon and its tidal influence has prevented the Earth’s rotational axis from varying chaotically, which in its turn has stabilised the climate on Earth. It is then not so far-fetched to ask the question whether the Moon’s existence has given certain qualities to the Earth to make it suitable for life [Benn, 1999]. The Moon is also important in planetary sciences, for a number of reasons. First of all, the Moon is an ancient world, where 99% of its surface predates 2 billion years ago. For comparison, areas on the Earth that are older than 2 billion years occupy less than 5%. The lunar highlands are the oldest exposed parts of the original crust of the Moon. By studying them, the earliest phases of lunar history, and subsequently the history of the solar system, can be studied. Second, due to the absence of an atmosphere and active tectonics for most of its history (lunar.

(18) 1.2 Why study the Moon and its gravity field - scientific rationale. 3. tectonism appears to have been confined to a very narrow interval in lunar history, about 3 billion years ago), a record of activities in the solar system is preserved on its surface, especially in terms of impact events. And third, all of the planetary bodies studied to date show, each to different degrees, the same kinds of surface and geological processes that were first recognised and described on the Moon. This makes the Moon immensely valuable in comparative planetology, since much of our knowledge and understanding of planetary processes and history comes from comparing surface features and environments among the planets [Spudis, 1996]. Still, there are many things unknown about the Moon, concerning its element composition, structure, and formation. The main topic in this dissertation is the gravity field of the Moon. The question that then rises is, how can studying the Moon’s gravitational field improve the knowledge about the Moon, and how can it help address issues as mentioned above? Up to the mid-1980s, there were three main hypotheses for the formation of the Moon: (i) capture of the Moon, which would be formed elsewhere, into an orbit around Earth, (ii) co-formation of the Moon, together with the formation of the Earth, or (iii) fission from a part of the Earth’s mantle. All of these theories however could not explain the relatively high mass and size of the Moon, the fact that the Moon dominates the total angular momentum in the Earth-Moon system, along with other constraints of chemical nature and the fact that the Moon does not possess a large iron core [De Pater and Lissauer, 2001]. Nowadays, the consensus is that the Moon was formed by a giant impact of a Mars-size planet into the protoEarth, soon after the formation of the solar system. Recent results show that this is possible with a fully formed Earth [Canup and Asphaugh, 2001; Canup, 2004] rather than a half-formed Earth which simulations showed at first [Melosh, 2001]. And apart from simulations, the giant impact theory is also backed by results of the analysis of oxygen isotopes [Wiechert et al., 2001] and by dynamical analysis of the lunar orbit [Ward and Canup, 2000]. The giant impact theory can further be corroborated by studies of the lunar core, and this is where gravity field analysis can be of help. Studying the gravity field of a planet and its moments of inertia can give information about the planetary interior (see Bills and Rubincam [1995] for an example about the Moon, amongst others). Recent gravity field results are consistent with an iron core with a radius varying between 220 to 450 km [Konopliv et al., 1998, 2001], although the results are still hampered by insufficient accuracy in the second degree harmonics [Bills, 1995]. Recent analysis of Lunar Laser Ranging together with Lunar Prospector data points in the direction of a molten or partially molten core [Khan et al., 2004]. Also having a major impact was the discovery of the positive gravity anomaly concentrations (dubbed mascons) from Lunar Orbiter data [Muller and Sjogren, 1968]. By analysing line-of-sight accelerations (spacecraft accelerations along the line station-to-satellite), large positive anomaly concentrations were found over the lunar ringed maria, having important implications for various theories regard-.

(19) 4. Introduction. ing lunar history. The formation of the mascons themselves formed a mystery as well: a circular basin is formed by a large impact which excavates near-surface strata and places part of the ejecta of the impact on the surroundings as a rim. Due to the differential pressure at the base of the excavated zone, the surrounding area collapses and the lower strata attain isostatic equilibrium 1 . This leaves a structure consisting of the basin and an upwelling of the strata beneath the basin, but this gives rise to a negative gravity anomaly [Arkani-Hamed, 1998]. The observed positive anomalies indicate there is an excess mass concentration, and this emphasises that some other process than the impact alone is behind the creation of the mascons. An equally stunning discovery was the ring of negative anomalies surrounding the mascons: the excess mass of the ejecta deposited around the impact basin, and the thickened crust due to the shock wave generated from the impact, is expected to create a positive anomaly, even after compensation at certain depth. Following these discoveries, pre-Apollo lunar theories of the maria being impact melts had to be revised, leading to new models for the formation and support mechanism of the mascons. These models can be divided into two categories [Arkani-Hamed, 1998]: (i) passive formation, stating that molten basalt was created in the deep interior of the Moon, which was caused by global heating of the Moon and which then flooded the basins, or (ii) active formation models, stating the mascon formation was directly related to the impact event, suggesting that partial melting occurred beneath the surrounding highlands due to thermal blanketing by the ejecta. The support mechanisms for the mascons can also be divided into two categories: (i) elastic support and (ii) viscous decay models . In a study of the mascons by using the topography and gravity of the Moon, Arkani-Hamed [1998] states that the Clementine models (i.e., models derived from data from the Clementine mission, which was a lunar mission launched in 1994 to study the Moon’s gravity field and topography; see also Nozette et al. [1994]) have a resolution that is high enough to resolve almost the entire spectrum produced by the mascons. It is shown that topography and gravity for the mascons have a negative correlation, suggesting a dynamical support of the mascons. It is also concluded that the mascons are most likely supported first by viscous decay until the lithosphere gained enough strength again to support the mascon elastically. The combination of topography and gravity information is a strong one for investigating the composition and structure of a planetary body. It allows the distribution of subsurface density anomalies to be mapped, thus yielding not only information on the shape, but also on the internal structure. This kind of information is fundamental to understanding the thermal history of the Moon. Topography of the Moon is obtained from laser altimeter observations by the Clementine spacecraft. Zuber et al. [1994] derived a preliminary spherical harmonic model of the topography up to degree 70, and Smith et al. [1997] presented a refined model of degree 72. Recent results include Reindler and Arkani-Hamed [2001] who created a 1 Isostatic equilibrium means that the mass is compensated by a mass deficiency; as such, it is an alternative statement of Archimedes’ principle of hydrostatic equilibrium [Fowler, 1990]..

(20) 1.2 Why study the Moon and its gravity field - scientific rationale. 5. model up to degree 165, and Ping et al. [2003b] who created a model up to degree 180, both by using gridded Clementine data. Clementine topography showed that the lava-flooded mascon basins are extremely level. Zuber et al. [1994] state the mascons are believed to be a result of gravitational attraction of lava fill in the mare basins, and of an uplifted mantle beneath the basins, whereas the negative anomalies surrounding the mascons suggest flexure of the lithosphere in response to loading by the lava-fill and uplifted mantle due to the impact process. To study the internal structure, Bouguer anomalies were computed by subtracting the gravitational attraction of the surface topography, assuming a mean crustal density of 2800 kg m . Before Clementine, Bouguer anomaly errors were dominated by the topography; after Clementine, gravity errors dominate the Bouguer errors [Zuber et al., 1994]. These Bouguer anomalies are, in the simplest plausible interpretation, solely a consequence of crustal thickness variations. Using this simplification, Zuber et al. [1994] show that the average crustal thickness of the far side is 8 km greater. A substantial crustal thinning beneath the major basins is also found. It is also shown that the large dynamic range of global crustal thickness requires major spatial variations in melting of the lunar exterior and/or significant impact-related redistribution of the lunar crust. However, the Bouguer anomaly was mapped to a single mass sheet, not taking into account finite amplitude effects or loads due to the greater density of the mare basalt. Neumann et al. [1996] take this into account in their models, but they leave lateral variations in the mantle density out. They refine the average difference in crustal thickness between the near and far side to be 12 km. Mascon basins are also studied, revealing that the basin structure is characterised by a central region of a thinning crust, with a conjugate ring of thickened crust, located mainly within the basin rim. Such a pattern suggests a deep structural component to lunar basin rings which has not been identified in terrestrial impact basins. Neumann et al. [1996] state that the clear pattern of mass excess under the mascons and the ring of mass deficit surrounding them is robust under any reasonable assumption. The correlation between topography and gravity can also be used to infer models of the internal distribution of mass, e.g. Potts and Von Frese [2003]. With higherresolution gravity data from the Lunar Prospector mission (which was launched in January 1998; see also Binder et al. [1998] and section 5.1) becoming available, Arkani-Hamed [1999] shows how to efficiently compute the gravity anomaly contribution of the surface topography, taking into account radial and lateral variations in density, by dividing the topography into many layers. Ping et al. [2003a] study the mare basalt filling thickness and crustal thickness by also applying a multi-layer analysis Closely related to this is the compensation state of craters on the Moon. The concept of isostasy and compensation of a mass excess at certain depth also helps to understand the evolution and formation of the surface of a planetary body. ArkaniHamed [1999] shows that for studying the compensation state of craters with diameters in a range of 200-300 km, a more accurate method to compute the gravity.

(21) 6. Introduction. anomaly associated with their surface topography is needed. And in their study of the compensation state for various craters on the Moon, Reindler and Arkani-Hamed [2001] find compensated craters distributed all over the Moon, with no preference for either the highlands or lowlands. This suggests a laterally heterogeneous lunar crust. Furthermore, by comparing compensation characteristics among craters, mascon-like craters were found. These mascon-like craters further support the argument that the process of mascon formation may not necessarily require mare flooding. These are all examples of how gravity field analysis can help addressing several questions about the formation and composition of the Moon. They make clear that above all, gravity field analysis is deeply embedded in geophysical analysis and interpretation. Therefore, improved knowledge of the lunar gravity field also means an improvement in knowledge about the Moon as a whole.. 1.3 Goals of this research The topic of the research presented in this dissertation is the gravity field of the Moon, or, to be more precise, regional gravity fields on the Moon. Usually, planetary gravity fields (obviously also including the one of the Earth) are expressed as global fields, valid for the entire planet under investigation. The framework for this is a set of globally defined basis functions called spherical harmonics; these will be explained in section 2.1. However, the Moon is in a so-called spin-orbit resonance with the Earth, which means that the Moon’s rotational period is equal to its orbital period around the Earth, resulting in the Moon keeping one face towards the Earth. Driven by tidal dissipation, this actually occurs for most natural satellites in the solar system [Murray and Dermott, 1999], but it does pose severe limitations when Earth-based tracking is applied to a lunar satellite: due to the spin-orbit resonance, there is no tracking available when the satellite flies over the far side of the Moon, resulting in the lack of slightly less than one half 2 of the Moon in tracking data coverage. Naturally, this has severe consequences for determining a global lunar gravity field, which are addressed in chapter 2. With the launch of the Lunar Prospector satellite in January 1998, a wealth of low-altitude tracking data became available. Lunar Prospector was launched into a circular, 100 km altitude polar orbit around the Moon, and remained in this orbit for one year. After that, during its extended mission, it was lowered to an average altitude of 30 km above the lunar surface, in order to increase the resolution of its measurements, including Doppler tracking for the determination of the global gravity field. It is extremely difficult to extract the high-resolution gravity information from these data in a global framework, due to the lack of global coverage. This leads to a severely ill-posed problem, meaning, there can be problems with the existence, uniqueness and stability of the solution [Kress, 1989], of which the latter 2 Due to lunar librations, the satellite can be tracked over small parts beyond the limbs and the poles, resulting in a slightly more than 50% coverage of the Moon with Earth-based tracking..

(22) 1.3 Goals of this research. 7. two are applicable here due to the downward continuation and lack of sampling over the far side. This requires regularisation of the problem in order to transform the problem into a well-posed one. This is also treated in section 2.2. A more localised approach to the determination of the gravity field then becomes of interest. Such a localised, or regional approach is the topic of this dissertation, and the goal of the research can therefore be described as follows: To develop a suitable method for the regional recovery of gravity from satellite tracking data, in order to improve the resolution and accuracy of current lunar gravity models, and in order to exploit the high-resolution gravity information present in tracking data at low altitudes, without significant interference from the unknown far side. This will be applied to Lunar Prospector tracking data from the extended mission to create highresolution solutions for parts on the near side of the Moon. Such solutions are thought to contribute to the improvement of the selenophysical interpretation of features on the Moon.. Suitable in this respect means that with the chosen method it is possible to extract high-resolution gravity information from the data, without introducing large errors on the data in a pre-processing step to set-up the recovery process. Suitable also means that the representation used for gravity is ”proven”, in the sense that no new representations that have not often been applied in the geosciences are used. It is not the means of this dissertation to develop a new framework for the representation of regional gravity. To extract the high-resolution gravity information, the resolution of the representation itself has to be sufficiently detailed. The desired resolution of the regional adjustments will be around . This is due to the altitude of the Lunar Prospector satellite: in its extended mission, the average altitude was 30 km, and this is commensurate with on the lunar surface at the equator. Since the problem is inherently ill-posed, regularisation in the form of the inclusion of a priori information about the solution is required in order to transform the problem into a well-posed one. In fact, limiting the resolution to is already some form of regularisation, since it prescribes a certain smoothing on the solution, which in this case consists of not allowing signals smaller than . However, the main issue here will be the instability of the problem at hand, as errors at satellite altitude will be continued downward to the lunar surface; see also sub-section 2.2.1. Since Lunar Prospector flew at extremely low altitudes, as low as an average of 30 km, it is expected that this instability does not lead to a numerically singular normal equation system, but rather, that the errors due to downward continuation in the solution are contained. In that case, regularisation may not be required to obtain a solution. This is strived for in this dissertation, and it is investigated by means of extensive simulations, since in this dissertation the recovered gravity field is preferably without a priori information. With this recovery method, the aim is to improve the resolution of the current global lunar gravity field models. The expected signal in the adjustments can be.

(23) 8. Introduction. estimated by considering both the commission error, which is the error in the gravity field coefficients in the global model itself, and the omission error, which is the error introduced since the global model is only derived up to a certain spherical harmonical degree. To this end, dr. Alex Konopliv from the Jet Propulsion Laboratory (JPL) kindly provided the normal matrix system for the LP75G model, which is derived using Lunar Prospector data from the nominal mission [Konopliv et al., 1998]. By propagating the covariances from the spherical harmonical coefficients, the covariances in terms of gravity anomalies can be derived. This results in an rms of the global commission error of around 23 mGal (1 mGal =

(24) m/s ). The value for the omission error can be computed by assuming a Kaula rule for the gravity coefficients (see also sub-section 2.1.2). Since the desired resolution is , the expansion for the omission error is taken up to degree 180, and this leads to a global omission error of around 31 mGal. The total rms of the commission and omission errors combined then becomes 38 mGal. It should be noted that global measures were computed; in the value for the commission error, that means the larger errors for the far side are also taken into account. This value can serve nevertheless as an expected value for the adjustment. This does not provide conclusions about the accuracy of the recovery method, however. In fact, setting the accuracy for the recovery method a priori is extremely difficult, because it will depend on many factors. Mainly, the accuracy of the recovered gravity field will depend on the kind of data that are used to extract the highresolution information from. For example, data types such as satellite-to-satellite tracking contain information that is spread differently in the spectral domain than is the case for standard Doppler data, which means that different accuracies can be obtained. Satellite-to-satellite tracking is known to be more sensitive for example to the higher-frequency parts of the gravity field, see e.g. Floberghagen [2002], and since it also leads to the direct mapping of the far side, better accuracies can be obtained with this data type. This touches upon the issue of the accuracy of the recovery method in this dissertation. The goal is to develop a recovery method. Developing this method also means testing it extensively for its sensitivity with respect to several error sources on the data, and only this can lead to accuracy estimates that are based on actual numbers. These accuracy estimates will mainly depend on the parameters of the simulated test cases. This does not imply that the actual results from Lunar Prospector tracking data will have a similar accuracy. This will mainly depend on the data and the level of noise present in these data. Assessing the accuracy of the final results will also be hampered by the fact that no true, independent validation is possible, since no such data exist. During the Apollo 17 mission, gravity measurements were conducted on the Moon, and these measurements are available, yet they consist of 13 measurement points of which the location is difficult to derive, see Talwani [2003]. Solving for a regional gravity field of the Moon and thus extracting highresolution gravity information from Lunar Prospector tracking data, is believed to be of great benefit for the lunar sciences. Studying the gravity field of the Moon.

(25) 1.4 Outline of the dissertation. 9. also means improving the knowledge of the Moon in general. As is pointed out by Reindler and Arkani-Hamed [2001], global gravity models are corrupted by noise due to the lack of far side tracking data and the consequent influence on near side gravity (see also Floberghagen [2002]). An increase in resolution for near side gravity can thus lead to better estimates and interpretation of the compensation state of craters, e.g. Sugano and Heki [2004b], and by studying mascon-like craters this will help improve the knowledge of mascon formation models, amongst others.. 1.4 Outline of the dissertation The outline of this dissertation is as follows: in chapter 2, the basics of representing the gravity field are explained. This is done in the framework of globally supported spherical harmonics, since potential theory and satellite dynamics, the two most important tools in planetary gravity field determination, are deeply rooted in this framework. The influence of regularisation on global gravity fields, which is needed in order to have a unique and stable solution to an inherently ill-posed problem, is also treated by means of a literature study, showing the influence is considerable. This leads to the interest in regional representations, which are more localised and do not have such a considerable influence of the lack of tracking data over the far side. One such regional representation form is chosen to express gravity in this dissertation. The selected representation of gravity in gravity anomalies is the subject of chapter 3. There, a Stokes formulation which relates gravity anomalies to the disturbing potential is explained. This formulation is an integral formulation, and this integral has been implemented numerically since it is needed in the recovery of gravity anomalies from satellite tracking data. This implementation has been thoroughly validated, showing the chosen numerical implementation works up to a sufficient level. An inverse method, to determine anomalies from accelerations, is also investigated in order to get acquainted with the inverse problem of determining gravity anomalies from data at satellite altitude. Chapter 4 deals with the actual recovery method of gravity anomalies as will be used in this dissertation. The method is explained, and by means of extensive simulations, its sensitivity with respect to several error sources is investigated, leading to an estimation of accuracy limits possible with this method. Regularisation is also a topic in this chapter, since the recovery problem is inherently ill-posed. Finally, in chapter 5, the recovery method is applied to actual Lunar Prospector tracking data. The tracking data themselves and the processing scheme are described, and a benchmark test for the processing scheme and the recovery of anomalies as implemented is given, showing that indeed the underlying gravity field can be recovered from these data. Solutions for Mare Serenitatis, crater Copernicus and a part of the lunar highlands are also presented, demonstrating what can be achieved with the method presented in this dissertation. The dissertation is concluded by chapter 6, which summarises the results presented here, and which presents an outlook for future work..

(26)

(27) Chapter 2. Global and regional representations of the gravity field. Since the Lunar Prospector spacecraft orbited the Moon from January 1998 until July 31, 1999, there has been a boost in global lunar gravity field determination. Several models have been released, each providing higher resolution as more and more data became available. This chapter will address the global representation form, the efforts in creating these models, and also the difficulties that occur when trying to solve for a global lunar gravity field model. These difficulties lead to the search for another, regional representation of the gravity field, where only excellently sampled regions on the Moon will be used. Several of these alternative representation forms will be discussed, and it will be made clear why one final, particular form is chosen to serve as the representation of the regional gravity field in this dissertation.. 2.1 The global lunar gravity field The gravitational field of any planetary body consists of the contributions of each infinitesimally small part of this body; hence, the gravity field outside this body is given by an integral over the whole volume. Naturally, the gravity field is thus a global quantity, defined by all contributions from the whole volume the body occupies. In the case of the Moon, due to the phase-lock between its rotational period and the period with which it revolves around the Earth, only one half of it can be seen from the Earth. This results in roughly one half missing in Earth-based tracking data, which is used to determine the gravity field. Still, global solutions for the Moon’s gravity field can be determined, and perform extremely well in terms of data fit and precise orbit determination..

(28) 12. Global and regional representations of the gravity field. 2.1.1 Representation in spherical harmonics The force the gravity field of a central body exerts outside its body, can be written as the gradient of a scalar potential :. . (2.1). where is the position vector of the satellite in Cartesian coordinates in an inertial reference frame. This potential satisfies Laplace’s equation [Blakely, 1995], which is given by:. . . . . . . . (2.2). . where the coordinates are fixed to the body, rotating along with it in inertial space. Functions satisfying Laplace’s equation are called harmonic functions, and the potential can be expressed by such functions. A set of suitable harmonic basis functions for the potential are spherical harmonic functions, since they use a sphere as reference surface, and describe deviations from this surface. A physical quantity such as gravity which is measured on or above a planetary surface is naturally described in spherical coordinates and therefore it is logical and common-place to express the gravitational potential in such basis functions. An expansion based on an elliptical coordinate system can also be used, since most planetary bodies can quite adequately be described by a reference ellipsoid. However, in the case of the Moon the geometric flattening is even smaller than that of the Earth (respectively 1/802 [Smith et al., 1997] and 1/298.257 [De Pater and Lissauer, 2001]), so elliptical corrections would be very small; they can be shown to be of the order of the square of the second eccentricity 1 [Wang, 1999]. Besides, elliptical harmonics are presently mostly used to describe the gravity fields of asteroids, e.g. Garmier and Barriot [2001], Konopliv et al. [2002], and Dechambre and Scheeres [2002], bodies which by nature are much less of spherical shape than the planets. The gravity potential expressed 0 0 in spherical harmonics reads [Kaula, 1966]:. 0 0 0 . &%'&() +*-, / 0 4576 / %)I J : BG H () = :?>A@CB %) E :FBG H "!$# (2.3) !$D !$D ! # 132 #98 : 132<; &%'&( where # are the spherical coordinates radius, longitude and latitude, respec7 5 tively,0 6 is the0 mean equatorial radius of the central body, *-, its gravitational coefficient, consisting of the 0 gravitational constant * times the central body’s mass = E , , : and : are the J normalised harmonic coefficients of the expansion of degree K and order D , and : are the normalised associated Legendre functions.. 1 The eccentricity L of an ellipse is a measure of the deviation from a circle. It can be expressed in terms of the semi-major axis M and the semi-minor axis N as LPORQTSUM OWV9NXOPY[Z&M O . In the same manner, the second eccentricity L \ is defined: L\ O QFSUM O V]N O Y[Z&N O ..

(29) 2.1 The global lunar gravity field. 13. The orthogonality relationship that holds for the associated Legendre functions states that any integral over the surface of the unit sphere of any two of these functions is zero unless they are the exact same functions, meaning, they must be of the same degree K and order D . The magnitude of each associated Legendre function depends on its degree and order, and so the magnitude of each coefficient in an expansion must compensate accordingly in order to have the actual signal represented by the expansion in Legendre functions. However, it is obviously more instructive and illustrative if each coefficient reflects the relative significance of its respective term in the expansion. This can also prevent stability issues, and this is why the associated Legendre functions are normalised. This normalisation is according to Kaula [1966] and it states:. 0 0 ) ) P) f J : _^ !a`cbed 2 : a! ` K ![Kb?D J : )Pf ![K D. (2.4). with d 2 : Kronecker’s delta. This normalisation together with the orthogonality relationship then results in [Blakely, 1995]:. g 2. h. g. 0 0k k hji J : J : >A@CB ()Wlm(nlm% po. ! vjw hji . if if. K[qs r

(30) K tnDuqW r D K[q

(31) K xnD Duq. (2.5). This turns the set of harmonic functions into an orthonormal set of basis functions vjw in equation (2.5) is taken care of. This is also what makes spherwhen the factor ical harmonics so suited for expressing the gravitational potential: these orthonormality relations are rather straightforward and can be used to simplify quantities that are expressed by integration over the reference sphere. In case the potential would be expressed in ellipsoidal harmonics, similar orthogonality relationships can be found, yet the mathematics become more complicated; see 0 and expressions 0 Garmier and Barriot [2001]. = E The normalised coefficients : and : describe the signal power in the gravity field; they make up the size of the force exerted on a satellite. The degree K and order D describe the distribution over the sphere of the gravity field. The associated Legendre functions change sign KsbTD times from pole to pole, and the sine and cosine terms in the expression change sign `D times over one complete revolution of longitude. The meridians and parallels along which the spherical harmonic function vanishes divide the reference sphere into patches of alternating sign, and on account of this patchwork, different coefficients have been given different names. If the degree D equals zero, there is no dependence on longitude, and the sphere is divided into zones: coefficients with D 0 are therefore called zonals. Zonal terms with even order are symmetric about the equator, where odd 0 0{z y zonals are asymmetric. Zonal terms are frequently named , where the definition = 2 y y b should always be kept in mind. The term y gives the observed oblate shape of the central body, due to rotation, and the term gives the north-south asymmetry of the central body, and thus is said to describe the ”pear-shape”. If.

(32) 14. Ksb|D. Global and regional representations of the gravity field. . , the dependence on latitude vanishes, the sphere is divided into sectors, and the terms for which K equals D are called sectorials. Finally, the general case where D} r and Kc r D has been given the name tesseral harmonic, because these harmonics divide the sphere into quadrangles whose angles are right angles (”tesseral” means isometric). 0 The spatial resolution of a spherical harmonic model is given by the maximum 2 ~ degrees. By using global, sphersize of the degree K : it can be approximated as max a certain amount of smoothing is ical basis functions to describe the gravity field, easily incorporated. To be able to describe very fine, local details, the maximum degree of the model has to become very large, leading to a huge number of coefficients needed to describe the gravity field up to that desired resolution. In the case that it is difficult, if not impossible, to derive such coefficients meaningfully, some other representation form will be better suited. Mostly this shows that global basis functions such as spherical harmonics are localised in the frequency domain rather than in the spatial domain. This is the real issue when localised features need to be described: a localisation in the frequency domain is very useful when the signal is studied for its spectral content, but it is less useful in local spatial studies, since one single frequency contribution has its influence over the whole of the reference surface and is not confined to a certain area only. Global basis functions such as spherical harmonics are much less suited for spatial localisation, and this will be addressed further on in this chapter. The gravity field of a planet relates information directly to the interior structure of the planet, and by using spherical harmonics, this information can be made = y readily visible. For instance, the lower degree harmonics and can be related to the principal moments of inertia, and in this way inferences about the interior density distribution can be made. The spherical harmonic framework for the gravitational potential is also intrinsically linked to spacecraft dynamics. Gravity is the literal driving force behind spacecraft motion, and gravity field coefficients can be determined by tracking spacecraft in orbit around a planet: this is how it is mainly done for determining global gravity fields of the Earth, and it is also the way to determine the gravity fields of the terrestrial planets and the Moon 2 . Since global gravity field solutions incorporate a huge amount of satellite data, orbit analysis is a valuable tool in assessing the quality of the model, which is needed when the model is to be used for geophysical interpretation. These reasons, and many more, have made gravity field determination thoroughly rooted in spherical harmonic analysis. 2 Altimetry also provides a means of determining the satellite orbit and gravity field, but for the planets it has only been used so far for Mars; see Rowlands et al. [1999] for the use of altimetry in Mars Global Surveyor orbit determination, where it is shown that the inclusion of laser altimetry crossovers improves the estimate of the satellite orbit, and see Lemoine et al. [2001] for an example of using laser altimeter crossovers in the determination of the gravity field of Mars..

(33) 2.1 The global lunar gravity field. 15. 2.1.2 A brief history of selenopotential modelling The lunar programmes in both the US and the former Soviet Union that were initiated in the early 1960s sparked the interest in the lunar gravity field. Satellites were going to be sent to orbit the Moon, and in order to assure the safe return of a manned vehicle, the orbit would have to be known with sufficient precision. This resulted in the first efforts in modelling the lunar gravity field. The Russian Luna 10 satellite was the first orbiter z to study the Moon’s gravitational and magnetic field in 1966, which resulted=<in 2 a3 spherical harmonic model with coefficients up to degree 3 and order 2, plus . Also, proof0 of the asymmetry of the lunar 0 gravity field between the near and far side was delivered, which was inferred from = E non-vanishing coefficients : with odd D , and : with even D [Akim, 1966]. On the US side, there were several precursor missions to the Apollo program which would transport astronauts to and from the Moon. The Ranger spacecraft were designed to crash-land on the Moon while taking pictures with increasing detail as the spacecraft came closer and closer to the lunar surface, in order to help investigate possible landing sites and tell more about the lunar surface. The Surveyor spacecraft were designed to softly land on the Moon, thus demonstrating that the Moon could support man and machines. Since both these missions were designed to land (or crash) on the Moon, they did not orbit the Moon for a long period. Therefore, one can not speak of a real gravity mapping, and thus the data have not been used in determining the lunar gravity field. The Lunar Orbiter spacecraft, however, were designed to circle the Moon and map it globally in high resolution, and these spacecraft have been the source of many interesting discoveries on the lunar gravity field. The first spherical harmonic model from Lunar Orbiter data supported the view that the Moon is gravitationally much rougher than the Earth [Lorell and Sjogren, 1968]. They also addressed some selenophysical implications from their results, and commented on the ”pear shape” (north-south y asymmetry), described by the coefficient. Interpretation from sparse data sets was still rather difficult, which is shown by noticeable differences between their results and those of Akim [1966] in terms of values for harmonic coefficients. Howy y ever, matches rather well, and although there is some difference for , they are at least of the same sign [Lorell and Sjogren, 1968]. The data used consisted of tracking using the Deep Space Network (DSN), with a noise level of around 1.0 mm/s for the Doppler data. Lorell and Sjogren [1968] do report that in order to be consistent with the scatter in the residuals of the Kepler elements, the data were weighted at 7000 mm/s, a factor many times greater than the noise level. Line-of-sight data analysis (where residual accelerations are computed in the direction of the spacecraft as seen from the Earth, along the ”line-of-sight” so to say, hence the name) of Lunar Orbiter 5 revealed a major discovery in lunar sci3 The gravitational coefficients determined in Akim [1966] are unnormalised coefficients. This is also the case for those derived by Lorell and Sjogren [1968]. In his overview, Kaula [1969] uses normalised coefficients. Normalisation is especially important in case high degree expansions are computed, because normalisation then means the amplitudes of the Legendre functions do not oscillate between large positive and negative values..

(34) 16. Global and regional representations of the gravity field. ence, namely: the discovery of mass concentrations (in short, mascons) beneath lunar ringed maria [Muller and Sjogren, 1968]. This had major implications for the understanding of the Moon and theories about the lunar origin. New questions arose, which helped lunar science and the understanding of the Moon develop further. More practical but just as pressing implications were found for the Apollo project: how would the roughness of the gravity field of the Moon, and the mascons in particular, affect the trajectory of the Apollo spacecraft? The effect can probably be demonstrated best by a small subsatellite released from Apollo 16, that had an unforeseen lifetime of merely 35 days before it crashed on the Moon, e.g. Konopliv et al. [1993]. The high-frequency noise on the data used by Muller and Sjogren [1968] was reported to vary between 0.1 and 1.0 mm/s, with a degradation factor of about 3 during picture readout. Kaula [1969] described an overview of what had been done so far in lunar gravity researches in 1969, and he made some interesting remarks on needed data distribution and satellite missions. The data used until then consisted of Lunar Orbiter data, since these satellites combined all the desired characteristics preferred for use in lunar gravity field modelling, such as a small semimajor axis, meaning a large sensitivity to gravity perturbations, accurate tracking and relatively long durations without disturbing outgassing of the spacecraft, caused by the combustion products of the spacecraft’s thrusters for attitude control. Even though Lunar Orbiters were photo-reconnaissance missions with frequent orbit corrections, resulting in many accelerations on the spacecraft that have to be accounted for in the orbit determination process, these manoeuvres were mainly concentrated in the primary mission to align the spacecraft for images of the lunar surface [Konopliv et al., 1993]. However, the distribution in inclination of these missions fell short for gravity mapping, since the satellites were mainly equatorial. An effect present in the early spherical harmonic models that was noted by Kaula [1969] is the fact that the far side gravity shows some rather extreme values and oscillations. This is due to the lack of far side data, and thus the coefficients mainly were a set of curves fitted to data over the near side. Kaula stated that the problem of the lack of far side data could be overcome by (i) a better distribution of orbital inclinations, (ii) satellite-to-satellite tracking, (iii) satellite laser altimetry or (iv) gravity gradients measured on-board a satellite. This still holds to this day. The Apollo era, from 1969 until 1972, resulted in more data available for lunar gravity purposes. However, since the spacecraft were manned, the data suffer from frequent orbit and attitude adjustments, plus the satellites only flew for relatively short time spans. Moreover, the orbits were limited to a band around the equator, where the Apollo landings took place. This makes these orbits less suited for global gravity modelling, but nonetheless data were used to obtain more knowledge about the lunar gravity field. Line-of-sight accelerations were again used for studying gravity profiles over well-sampled areas on the near side of the Moon, which were crossed by the Command Service Module and the Lunar Module at very low altitudes [Sjogren et al., 1974a, 1974b, 1974c]. Due to the orbit characteristics and consequent spatial limitations, this is the preferred method, since.

(35) 2.1 The global lunar gravity field. 17. it will be very difficult to determine meaningful spherical harmonic coefficients from such data. As more data became available from the missions, better views of certain features such as the mascons for Mare Serenitatis and Mare Crisium were obtained, which helped understand these features better [Sjogren et al., 1972, 1974a, 1974b, 1974c; Muller et al., 1974]. The authors of these papers do stress that their results should be looked upon in a qualitative way rather than quantitative. This is because the accelerations are in the line-of-sight, and not vertical, so there is still some geometry present in the values, which causes a shift towards the limbs and slightly reduces the amplitude [Sjogren et al., 1972]. Furthermore, the accelerations are computed at satellite altitude and are therefore not normalised to a constantaltitude surface, making comparisons slightly more difficult. Nevertheless, lunar science greatly benefitted from these results. Apollo 15 and 16 both carried subsatellites, that were released in lunar orbit [Sjogren et al., 1972; Konopliv et al., 1993]. These were meant to study the lunar magnetic and gravity field in detail. The advantage of these satellites, the first extra-terrestrial launch of a spacecraft from another spacecraft [Spudis, 1996], was that they were spin-stabilised spacecraft, making them very suitable for gravity purposes since there was no contamination from attitude adjustments whatsoever. Both were placed in near-circular, low-inclination orbits (the subsatellite of Apollo 15 had an inclination of `

(36) j , and the Apollo 16 subsatellite had an inclination of C ). Apollo 15 was tracked for about two years, whereas Apollo 16, as mentioned before, had an unforeseen lifetime of merely 35 days before it crashed on the lunar surface. The noise on these data was reported to be typically around 1.0 mm/s [Sjogren et al., 1972, 1974a; Konopliv et al., 1993; Lemoine et al., 1997] During the same period, the Soviet Luna satellites orbited the Moon, and these have been used in gravity field analysis as well [Akim and Vlasova, 1977]. The inclination characteristics were slightly better than those of the Lunar Orbiters and v Apollo Satellites, since Luna satellites flew at inclinations of around ` 7 , C and `j . Akim and Vlasova [1977] presented solutions based on each satellite indepeny dently, which showed large differences between each other, especially in the coefficient, and also solutions based on the total set of satellite data. The latter compared well with a model created at the Jet Propulsion Laboratory (JPL), which constituted an independent confirmation of both efforts. A full, spherical harmonic analysis of all US data available at that time, was done by Bills and Ferrari [1980]. A 16 degree and order model was derived from Apollo 8, 12, 15 and 16 (subsatellites), Lunar Orbiter 1, 2, 3, 4 and 5, and lunar laser ranging data. As Bills and Ferrari [1980] point out, previous models were either coming from the need for accurate orbits for navigational purposes, which resulted in models having the most accurate estimates of low-degree harmonics, or from interest in selenophysical applications of gravity models, resulting in shortarc Doppler data analysis, best for regional and local scale gravity field models. The idea behind using all the data was to make the best of both worlds, and combine the data into a solution which incorporates the best properties of each approach. Furthermore, to overcome the irregularities in data distribution (i.e., the.

(37) 18. Global and regional representations of the gravity field. lacking far side data), a semi-empirical a priori covariance function was also added to the normal equation system. These a priori data were meant to reduce parameter correlations, and thus improve an otherwise poorly constrained system. Without this a priori information, unsurveyed regions with large data errors would show power. Bills and Ferrari [1980] used two different covariance func0 excessive 0 tions; one based on Kaula’s observation that the coefficients of the gravity field = : E :- behave like K[ [Kaula, 1966], the so-called ”Kaula rule”, with a constant depending on the planet, and another based on analyses of gravity) and topo K![K )) . graphy, resulting in a gravity variance spectrum of the form 7!!a` K By comparing their models with how well they fit with data not used in determining the model, they decided that the power rule based on topography and gravity analyses was the best, showing slightly better data fits than the model obtained with Kaula’s rule. The most significant feature of the resulting model is surprisingly mild far side gravity; spurious far side variances in earlier models thus point to previously poorly determined high-degree harmonics. This model remained probably the best lunar gravity model for the next thirteen years [Floberghagen, 2002]. In the early nineties, a study into a new lunar mission, Lunar Observer, triggered renewed interest in lunar gravity field modelling. Previous gravity field models predicted different orbit behaviour, which meant a large uncertainty in fuel requirements. Thus, a re-analysis was done with all the available data. This resulted in the Lun60d model [Konopliv et al., 1993], a 60 degree and order spherical harmonic expansion. Such a large expansion was possible, even though no new data were used, due to careful modelling of the forces on the spacecraft, and due to much better computer resources, in terms of CPU power and available memory. The data used for this model are all available historical US data (no Luna data were used; the only Luna data sets with tracking lasting more than one or two days are those from Luna 19 and Luna 22. The raw Doppler data is fairly sparse, and may be noisier than the other data by an order of magnitude [Konopliv et al., 1993]), consisting of Lunar Orbiter 1, 2, 3, 4 and 5, as well as data from the Apollo 15 and 16 subsatellites. As mentioned before, Lunar Orbiters suffered from many orbit manoeuvres in order to line up for their photographic mission. These manoeuvres are taken care of in Lun60d by estimating many sets of three-direction accelerations at the manoeuvre times, which luckily were preserved [Lemoine et al., 1997]. The maximum of 14 manoeuvres per arc limited the arc length to one day [Lemoine et al., 1997], and in general this estimation of three-direction accelerations is not very robust, and can hamper the determination of low-frequency spherical harmonic coefficients [Floberghagen, 2002]. The data for Lunar Orbiter 5 is further more reported to be noisier than the data for the other Lunar Orbiters. There is a factor of 5 to 10 between the two sets, which is probably due to a possible hardware failure [Konopliv et al., 2001]. This was a major disappointment, as Lunar Orbiter 5 data was suited best for determining polar orbit behaviour with an altitude over the poles of 600 km [Konopliv et al., 1993]. Again, for Lun60d a constraint had to be applied, to compensate for the lack of.

(38) 2.1 The global lunar gravity field. 19. −4. 10. GLGM−2 Lun60d LP150Q sigma GLGM−2 sigma Lun60d sigma LP150Q Kaula rule. −5. Degree variance. 10. −6. 10. −7. 10. −8. 10. −9. 10. Figure 2.1. 0. 50. Degree. 100. 150. Degree-wise RMS power spectrum of various lunar gravity models, together with the according error variances. A Kaula rule of cPj7A O is also shown in the plot. . far side data. It was chosen to apply a Kaula rule of j - K[ , which resulted in a gravity spectrum that is well determined up to degree 7, possibly 8 or 9 [Konopliv et al., 1993]. After that, the constraint starts to take over, and its influence on the solved coefficients becomes noticeable. This can be seen in a plot of the degree0 wise rms power spectrum of the gravity field, as shown in figure 2.1. This power spectrum, denoted here by , is given by: 0. 0 . /. =. 0. 0 E : :. : 132 ` K . (2.6). The spectrum for the error variances follows the same equation as above, only the error variances of the coefficients instead of the coefficients themselves are used. As the figure shows, for Lun60d, after degree 7 or 8 the line that represents the power in the error variances starts to bend and turn towards the line that represents the Kaula rule. This indicates that the Kaula rule comes into effect, and the solution will be more determined by this rule than it is determined from the data, for these degrees and further. This shows the effect of including a priori information, and it shows the data do not contain strong enough information to determine the according degrees from them. The plot for the model itself shows that there is.

(39) 20. Global and regional representations of the gravity field. more power in the solution than the Kaula rule prescribes. Moreover, at the end at degree 60, there is a jump where suddenly the power again starts to increase. This shows that there is still more information in the data beyond degree 60, that is not modelled, but absorbed in lower degree coefficients. Konopliv et al. [1993] point out that without applying a constraint the solution becomes meaningless over the far side, as there will be too much power in the coefficients. With the constraint however, it should be noted that the results for the far side show a general correlation with topographic features, even though there are no direct observations over the far side. This indicates the constraint seems to be chosen well, and functions as a representative selenophysical model. And, especially in terms of orbit fit, Lun60d performs extremely well, and shows several orders of improvement over previous models [Konopliv et al., 1993]. The year 1994 marked the return of the US to the Moon, with the launch of the Clementine mission [Nozette et al., 1994]. Clementine was launched into an elliptical orbit with a periselene altitude of about 415 km, and an inclination of almost 90 degrees. It was a joint project of The American Strategic Defense Initiative and NASA, and it was meant to map the Moon in terms of topography and gravity, among other things. Clementine also provided the first new data in 20 years, resulting in the GLGM-2 gravity field model [Lemoine et al., 1995, 1997]. Since Clementine’s orbit was elliptical and at a rather high altitude, it was not the perfect tool for gravity field determination, but noticeable improvements were achieved. Clementine’s orbit was especially sensitive to medium wavelengths of the gravity field, mostly low order sectorials, so Clementine provided the strongest satellite constraint on these terms, when compared with the available historical data [Lemoine et al., 1997]. In solving for GLGM-2, the historical data that had been used for Lun60d were also used. Clementine data themselves were reported to have an accuracy of around 0.25 mm/s for the DSN data. A naval tracking station at Pomonkey, Maryland, was also included in the tracking campaign, and it delivered noisier data with a level of 2.5 mm/s [Lemoine et al., 1997]. Figure 2.1 also shows the power of the GLGM-2 solution. To solve for this again a Kaula constraint was used, namely the same as for Lun60d, j 9 model, K . The plot shows that for GLGM-2, the constraint starts to play a role after degree 18. This was also expected, as sensitivity studies prior to Clementine had shown that the orbit would be sensitive for degrees up to 20 [Lemoine et al., 1995]. When compared to the spectrum from Lun60d, there are remarkable differences. GLGM-2 shows much less power than Lun60d, especially in the higher degree terms. As Lemoine et al. [1997] point out, this was intentional, since Lun60d shows excessive power in the higher degree terms, making the model perhaps very well suited for orbit determination purposes, but not suited for geophysical interpretation. The intention with deriving GLGM-2 therefore was to (i) use Clementine data, obviously, (ii) attenuate the spurious signal in the high degrees, and (iii) make a model suitable for geophysical applications [Lemoine et al., 1997]. This resulted in a different weighting strategy for the data, and finally in a smoother far side solution (see also Floberghagen [2002] for a discussion). The ranges in gravity.