Determination of Tidal Response of the Moon with Fully Three-dimensional Elastic and Density Structures Using a Perturbation Method by Chuan Qin B.S., University of Science and Technology of China, 2008 M.S., University of Colorado at Boulder, 2011 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirement for the degree of Doctor of Philosophy Department of Physics 2015 This thesis entitled: Determination of Tidal Response of the Moon with Fully Three-dimensional Elastic and Density Structures Using a Perturbation Method has been approved for the Department of Physics Shijie Zhong John Wahr Michael Ritzwoller Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Qin, Chuan (Ph.D., Physics, Department of Physics) Determination of Tidal Response of the Moon with Fully Three-dimensional Elastic and Density Structures Using a Perturbation Method Thesis directed by Professor Shijie Zhong The Moon displays a number of hemispherically asymmetric (i.e., harmonic degree-1) features, including those in its topography, crustal thickness, mare volcanism, surface chemical compositions and deep seismicity distribution. These features suggest a long-wavelength (i.e., 3- D) structure in the Moon’s interiors, which may have resulted from long-term lunar dynamic evolution. In recent years, high precision measurements of the Moon’s gravity field and topography have been made possible by the advancement of space geodetic techniques. New theoretical tools that can use those measurements to constrain the Moon’s 3-D interior structure are needed. In this thesis, I present a new semi-analytical method based on a perturbation theory that calculates the elastic response of a terrestrial planet with 3-D interior structures to time-varying body tides. The 3-D structure is represented by small lateral heterogeneities in the elastic properties and density of the planet. The presence of the 3-D structure excites response modes to tidal force that are added to those from the 1-D reference state. In a spherical harmonic representation, that every eigenstructure in the lateral heterogeneities is characterized by its induced modes and their relative responses holds the theoretical basis for inversion studies. The perturbation method is applied to solving the tidal response of the Moon, in which harmonic degree-1 lateral heterogeneities are assumed predominant. The perturbation solutions have been interpreted and compared with those from a finite element method, thus verifying the correct implementation of the perturbation method. My perturbation method is optimizable for inversions and promises to provide constraints on the Moon’s long-wavelength interior structures, with improved tidal response measurements from the GRAIL mission and future lunar missions. iv Dedication This dissertation is dedicated to my parents. v Acknowledgements It has been a long journey and I am so pleased that it has come to a happy ending. I would like to acknowledge all those who helped and supported me along the way. First and foremost, I would like to thank my advisor, Prof. Shijie Zhong, for offering me the opportunity to work in his geodynamics research group. It was my great honor to have worked with him in the past four and half years. Shijie always trusts my theoretical ability and technical skills, and let me work on many interesting geophysics projects. Without his consistent guidance and encouragement, I could not finish my PhD thesis with so many academic achievements. Shijie is a great mentor and has given me lots of good advice on how to actively pursue a successful career, from which I will benefit for my life. I want to in particular acknowledge my co-advisor and committee member, Prof. John Wahr, who was both a scientific giant and a great mentor. Prof. John Wahr passed away on the next day of my thesis defense. I was in deep sorrow for the loss of him. I was so grateful to John for his trust on my ability and his guidance on the theoretical part of my thesis. It would be my forever regret not being able to present my thesis work in front of him and acknowledge him in person. I personally take my thesis to commemorate him. I want to thank Prof. Michael Ritzwoller, Prof. Michael Calkins and Prof. David Brain for their time to serve as my committee members. Their valuable comments helped me improve my thesis. I thank Prof. Michael Ritzwoller for recognizing my ability in theoretical work. I also thank him for his lecturing on the theory of seismology and the interesting topic of ambient noise tomography. vi I would like to thank all my former and current colleagues in the geophysical groups on the 7th floor of Gamow Tower. It was great fun to discuss and work with them. Thank them all for helping me with my research programs and my thesis defense presentation. Next, I gratefully thank my parents and my family for their unconditional love and support. I owe to my parents for not being able to be with them for so many years. I love and miss them so much. I want to thank my girlfriend, Yang Gao. She is the most beautiful sunlight of my life, giving me so much energy to overcome the difficulties. She is so special to me and I cannot imagine my life without her. I love her. I also thank my best friends in the US and China, Heming Zhen, Hao Tu and Jiwei Liu, for all their encouragements. I acknowledge NSF (US-NSF 1114168) and NASA (NNX11AP59G) for their financial supports on my thesis work. Finally, I thank the University of Colorado at Boulder and the beautiful city of Boulder for providing the best environments and resources. I have already taken Boulder as my second hometown. I will love and care about this place for my whole life. vii Contents Chapter 1. Introduction ………………………………………………………………………………..1 1.1. Global asymmetries of the Earth’s Moon …………………………………………….2 1.2. Dynamic evolution of the Moon ……………………………………………………...7 1.3. Geodetic measurements from lunar missions ………………………………………...9 1.4. Elastic response to tidal forces………………………………………………………12 1.5. Motivation and purpose of the study………………………………….……………..15 1.6. Organization of the thesis…………………………………….……………………...17 2. Spatial correlation between deep moonquakes and mare basalts and timing of mare basalt eruptions: implications for lunar mantle structure and evolution ………………………..18 2.1. Spatial correlation of mare basalts and deep moonquakes ………………………….18 2.1.1. Data processing and methods………………………………………………...20 2.1.2. Results ………………………………………………………………………..23 2.2. Ages of mare basalt units ……………………………………………………………26 2.2.1. Age model of mare basalt units………………………………………………27 2.2.2. Results ………………………………………………………………………..29 2.3. Conclusion and discussion: a hypothetic process of lunar mantle evolution ……….34 3. A perturbation method and its application: elastic tidal response of a laterally heterogeneous planet……………………………………………………………………..38 viii 3.1. Introduction ………………………………………………………………………….39 3.2. Governing equations and perturbation theory… ……………………………………41 3.2.1. Governing equations…… ……………………………………………………41 3.2.2. Time-dependent body tides …………………………………………………..42 3.2.3. Perturbation theory…………………………………………………………...43 3.3. Methods... …………………………………………………………………………...46 3.3.1. Spherical harmonic and vector spherical harmonic expansions ……………..47 3.3.2. Equations in matrix form …………………………………………………….48 3.3.3. Spheroidal and toroidal modes……………………………………………….51 3.3.4. Mode coupling ……………………………………………………………….54 3.3.5. Semi-analytical approach and solutions……………………………………...56 3.4. Results……………………………………………………………………………….61 3.4.1. Finite element solutions ……………………………………………………...62 3.4.2. (1, 1) lateral heterogeneity throughout the mantle …………………………...64 3.4.3. (1, 1) lateral heterogeneity in either the bottom or the top half of the mantle…. ……………………………………………………………………………..…72 3.4.4. Other long-wavelength structures throughout the mantle ……………………75 3.5. Conclusion and discussion …………………………………………………………..78 4. Elastic tidal response of a laterally heterogeneous planet: a complete perturbation formulation ……………………………………………………………………………….81 4.1. Governing equations ………………………………………………………………...82 4.1.1. Background state ……………………………………………………………..83 4.1.2. Net force due to density anomaly ………...………………………………….84 4.1.3. Perturbation formulation ……………………………………………………..86 4.2. Solution method ……………………………………………………………………..90 4.3. Results……………………………………………………………………………….94 4.3.1. (1, 1) lateral heterogeneity in lambda ………………………………………..97 ix 4.3.2. Horizontal displacement due to (1, 1) lateral heterogeneities in shear modulus and lambda……...……………………………………………..……………103 4.3.3. (1, 1) lateral heterogeneity in density…………………………………...…..107 4.3.4. Tidal effect of lunar crustal thickness variations and sensitivity kernel……110 4.4. Conclusion and discussion …………………………………………………………117 5. Summary and future plan ……………………………………………………………….120 5.1. Summary ...…………………………………………………………………………120 5.2. Future plan …………………………………………………………………………122 Bibliography...………………………………………………………………………………125 Appendix ……………………………………………………………………………………134 A. Harmonic expansion of mode coupling terms ……………………………………..134 x Tables Table 1.1 Degree 2 and 3 tidal Love number solutions from GRAIL primary mission.……..14 3.1 Lunar 1-D model parameters………………………………………………………61 3.2 Comparisons of h and k between the results from the perturbation method and those from the