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Investigation of Mercury's magnetospheric and surface magnetic fields Winslow, Reka Moldovan 2014

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Investigation of Mercury’sMagnetospheric and Surface MagneticFieldsbyReka Moldovan WinslowB.Sc. Honors, The University of British Columbia, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Geophysics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2014c© Reka Moldovan Winslow 2014AbstractThis thesis is devoted to the study of Mercury’s magnetic field envi-ronment, to reveal the nature of the interaction between a weak planetarymagnetic field and the interplanetary medium. Due to the lack of orbitalspacecraft observations at Mercury prior to the MErcury Surface, Space En-vironment, GEochemistry, and Ranging (MESSENGER) mission, work inthis thesis presents some of the first analysis and interpretation of observa-tions in this unique and dynamic environment.The bow shock and magnetopause define the boundary regions of theplanet’s magnetosphere, thereby representing the initial interaction of theplanetary field with the solar wind. We established the time-averaged shapesand locations of these boundaries, and investigated their response to thesolar wind and interplanetary magnetic field (IMF). We found that the solarwind parameters exert the dominant influence on the boundaries; we thusderived parameterized model shapes for the magnetopause and bow shockwith solar wind ram pressure and Alfve´n Mach number, respectively.The cusp region is where solar wind plasma can gain access to the mag-netosphere, and in Mercury’s unique case, the surface. As such, this areais expected to experience higher than average space weathering and be asource for the exosphere. Using magnetic field observations, we mapped thenorthern cusp’s latitudinal and longitudinal extent, average plasma pressureand observed its variation with the solar wind and IMF. From the derivedplasma pressure estimates we calculated the flux of plasma to the surface.Mercury’s internal dipole field is not centered on the planet’s geographicequator but has a significant northward offset. We developed the techniqueof proton-reflection magnetometry to acquire the first measurements of Mer-cury’s surface field strength. Proton loss cones are evident in both the north-iiAbstractern and southern hemispheres, providing confirmation of persistent protonprecipitation to the surface in these regions. We used the size of the losscones to estimate the surface magnetic field strength, which confirm the off-set dipole structure of the planetary field. With additional proton-reflectionmagnetometry observations, we generated a global proton flux map to Mer-cury’s surface and searched for regional-scale surface magnetic fields in thenorthern hemisphere.iiiPrefaceThis thesis is based on four papers: three have been published and one isin preparation for publication. Consequently, some background informationis repeated in the introductory section of each chapter.A version of Chapter 2 has been published. I identified all bow shockand magnetopause crossing times analyzed in the paper, which were checkedby Dr. Brian Anderson. I also performed all the calculations and analyses,made the plots and wrote the manuscript. Dr. Catherine Johnson andDr. Brian Anderson provided guidance throughout and all co-authors com-mented on the manuscript. In the Journal of Geophysical Research paperassociated with this chapter there was an error in the production of Fig-ure 13; the figure and the associated text have been corrected in Chapter 2.A version of Chapter 3 has been published. I identified all the cuspcrossing times in the residuals to the magnetic field data, after the mag-netospheric model was removed; residual magnetic field files were producedby Dr. Johnson. I conducted all the analyses, performed all the calcula-tions, made all the plots and wrote the manuscript. I received commentson the manuscript from all the co-authors. Dr. Johnson and Dr. Andersonprovided guidance throughout.A version of Chapter 4 has been published. I developed the idea of usingproton-reflection magnetometry, and adapted the technique of electron re-flectometry to protons. Normalization files for the proper analysis of FIPSobservations was provided by Dr. Dan Gershman and Dr. Jim Raines. Iperformed all the analyses and calculations, and received guidance through-out from Dr. Johnson and Dr. Anderson. I also wrote the manuscript whichreceived comments from all the co-authors.A version of Chapter 5 is in preparation for publication in a major spaceivPrefacephysics journal. The usefulness and possibility of deriving a global protonflux map to Mercury’s surface and establishing regional-scale surface fieldswith additional proton-reflection magnetometry observations was noticed in-dependently by Dr. Johnson and I. I developed the methods and techniquesto perform the analyses, made all the calculations and wrote up the results.I am currently drafting the journal manuscript based on Chapter 5 of thethesis.Journal PapersChapter 2:R. M. Winslow, B. J. Anderson, C. L. Johnson, J. A. Slavin, H. Korth,M. E. Purucker, Daniel N. Baker, S. C. Solomon (2013), Mercury’s mag-netopause and bow shock from MESSENGER Magnetometer observations,Journal of Geophysical Research: Space Physics, 118, 2213-2227.Chapter 3:R. M. Winslow, C. L. Johnson, B. J. Anderson, H. Korth, J. A. Slavin, M.E. Purucker, S. C. Solomon (2012), Observations of Mercury’s northern cuspregion with MESSENGER’s Magnetometer, Geophysical Research Letters,39, L08112.Chapter 4:R. M. Winslow, C. L. Johnson, B. J. Anderson, D. J. Gershman, J.M. Raines, R. J. Lillis, H. Korth, J. A. Slavin, S. C. Solomon, T. H.Zurbuchen, M. T. Zuber (2014), Mercury’s surface magnetic field deter-mined from proton-reflection magnetometry, Geophysical Research Letters,41, 4463-4470.Chapter 5:R. M. Winslow, C. L. Johnson, B. J. Anderson, D. J. Gershman, J.M. Raines, R. J. Lillis, H. Korth, S. C. Solomon (2014), Regional-scalesurface magnetic fields and proton fluxes to Mercury’s surface determinedfrom proton-reflection magnetometry, in preparation.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Interaction of the solar wind with Mercury’s magnetosphere 51.2 Charged particle motion . . . . . . . . . . . . . . . . . . . . 91.2.1 Uniform magnetic field (E = 0) . . . . . . . . . . . . 101.2.2 Uniform electric and magnetic fields . . . . . . . . . . 121.2.3 Nonuniform magnetic field . . . . . . . . . . . . . . . 121.2.4 Motion of trapped particles: magnetic mirrors andbottles . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 The bow shock . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4 The magnetopause and the magnetic cusps . . . . . . . . . . 231.5 The planetary magnetic field . . . . . . . . . . . . . . . . . . 291.6 Relevant MESSENGER instruments . . . . . . . . . . . . . . 311.6.1 The Magnetometer . . . . . . . . . . . . . . . . . . . 311.6.2 The Fast Imaging Plasma Spectrometer . . . . . . . . 33viTable of Contents2 Mercury’s magnetopause and bow shock from MESSEN-GER Magnetometer observations . . . . . . . . . . . . . . . 362.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Magnetic field observations: boundary identifications . . . . 402.3 Average boundary shapes . . . . . . . . . . . . . . . . . . . . 482.3.1 Midpoint fits . . . . . . . . . . . . . . . . . . . . . . . 482.3.1.1 Magnetopause . . . . . . . . . . . . . . . . . 482.3.1.2 Bow shock . . . . . . . . . . . . . . . . . . . 522.3.2 Probabilistic fits . . . . . . . . . . . . . . . . . . . . . 532.4 Response of boundaries to solar wind forcing . . . . . . . . . 572.4.1 Magnetopause . . . . . . . . . . . . . . . . . . . . . . 572.4.2 Bow shock . . . . . . . . . . . . . . . . . . . . . . . . 662.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 Observations of Mercury’s northern cusp region with MES-SENGER’s Magnetometer . . . . . . . . . . . . . . . . . . . . 753.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.4.1 Surface flux calculation . . . . . . . . . . . . . . . . . 863.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924 Mercury’s surface magnetic field determined from proton-reflection magnetometry . . . . . . . . . . . . . . . . . . . . . 934.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 The technique of proton reflection magnetometry . . . . . . 954.2.1 How to obtain surface field strengths . . . . . . . . . 964.3 Application of the technique to MESSENGER observations . 974.3.1 Deriving individual pitch angle distributions . . . . . 984.3.2 Averaging pitch angle distributions . . . . . . . . . . 994.3.3 Surface field estimates . . . . . . . . . . . . . . . . . . 1014.4 Consistency checks . . . . . . . . . . . . . . . . . . . . . . . . 1074.4.1 Altitude binning . . . . . . . . . . . . . . . . . . . . . 1084.4.2 Latitude binning . . . . . . . . . . . . . . . . . . . . . 1094.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . 111viiTable of Contents5 Regional-scale surface magnetic fields and proton fluxes toMercury’s surface from proton-reflection magnetometry . 1155.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.2 Resolving regional scale surface magnetic field strengths . . . 1225.3 Particle fluxes to Mercury’s surface . . . . . . . . . . . . . . 1265.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142AppendixA Diamagnetic field and particle flux calculation for protonreflection magnetometry . . . . . . . . . . . . . . . . . . . . . 158A.1 Diamagnetic magnetic deficit calculation . . . . . . . . . . . 158A.2 Particle flux calculation . . . . . . . . . . . . . . . . . . . . . 162viiiList of Tables1.1 Scaling laws of solar wind parameters with distance from theSun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Summary of the best-fit Shue et al. model parameters to themagnetopause crossing points under different ram pressureconditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.2 Summary of the best-fit conic section parameters to the bowshock crossings under different Mach number conditions. . . . 593.1 Average cusp properties and ambient conditions. . . . . . . . 844.1 Surface magnetic field strength from proton-reflection mag-netometry compared with magnetospheric model predictions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112ixList of Figures1.1 Schematic diagram of Mercury’s magnetosphere. . . . . . . . 31.2 Magnetospheric convection due to reconnection. . . . . . . . . 81.3 Schematic diagram of converging magnetic fields towards theplanet’s surface. . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Schematic diagrams showing example bow shock magneticfield structure. . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Schematic diagram showing regions of parallel and perpen-dicular shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6 Currents on the magnetopause. . . . . . . . . . . . . . . . . . 241.7 Magnetospheric current systems. . . . . . . . . . . . . . . . . 251.8 Cross-section of the FIPS sensor showing major functionalcomponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.9 Top view of a portion of MESSENGER which depicts theobstructions in the solar direction by the spacecraft sunshadeand the FIPS FOV. . . . . . . . . . . . . . . . . . . . . . . . 352.1 MESSENGER Magnetometer data for the first magnetospherictransit on 12 October 2011. . . . . . . . . . . . . . . . . . . . 462.2 MESSENGER Magnetometer data for the first magnetospherictransit on 5 July 2011. . . . . . . . . . . . . . . . . . . . . . . 472.3 Midpoints between the inner and outer magnetopause cross-ing positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4 RMS misfit between the midpoints of the magnetopause cross-ings and the Shue et al. model. . . . . . . . . . . . . . . . . . 512.5 Midpoints between the inner and outer bow shock crossingpositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.6 Probability density map of aberrated magnetopause positions. 562.7 Probability density map of the aberrated bow shock crossings. 582.8 Solar wind statistics at Mercury from the WSA-Enlil model. . 602.9 Midpoints of the magnetopause crossings color-coded by solarwind ram pressure. . . . . . . . . . . . . . . . . . . . . . . . . 61xList of Figures2.10 Magnetopause subsolar stand-off distance as a function of so-lar wind ram pressure for variable parameter fits. . . . . . . . 622.11 Magnetopause subsolar stand-off distance as a function of so-lar wind ram pressure. . . . . . . . . . . . . . . . . . . . . . . 632.12 Probability density map of the aberrated magnetopause cross-ings corrected for ram pressure. . . . . . . . . . . . . . . . . . 652.13 Histograms of the magnetic shear angle and IMF cone angle. 662.14 Midpoints of the bow shock crossings color-coded by solarwind Alfve´n Mach number. . . . . . . . . . . . . . . . . . . . 672.15 Probability density map of the aberrated bow shock crossingscorrected for Alfve´n Mach number. . . . . . . . . . . . . . . . 693.1 Example of a cusp observation on 21 August 2011. . . . . . . 783.2 Superposed epoch analysis of all cusp observations. . . . . . . 813.3 Stereographic projection plots of the plasma pressure in thecusp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.4 SEA of the magnetic pressure deficit for ascending tracks un-der different IMF conditions. . . . . . . . . . . . . . . . . . . 864.1 Schematic pitch angle distribution showing the cut-off in re-flected charged particle flux at angle αc due to absorption bythe surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Example integration period of FIPS. . . . . . . . . . . . . . . 1004.3 Pitch angle distributions in the northern and southern hemi-spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4 Residuals of the diffusion equation fit to the loss cones. . . . . 1054.5 Stereographic projection plots showing where the proton-reflectionmagnetometry observations map to on the surface of Mercury. 1064.6 Altitude-binned averaged pitch angle distributions in the north-ern cusp region. . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.7 Pitch angle distributions binned in latitude and local time inthe northern cusp. . . . . . . . . . . . . . . . . . . . . . . . . 1104.8 Mollweide projection of the surface magnetic field strengthpredicted by a magnetospheric model. . . . . . . . . . . . . . 1135.1 Example pitch angle distributions observed in the northernhemisphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.2 Example PAD observed in the southern hemisphere. . . . . . 1195.3 Example PAD with diffusion curve fit in the 70◦ − 80◦ Nlatitude band. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123xiList of Figures5.4 Regional-scale surface magnetic field strengths in the north-ern cusp region. . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.5 Diamagnetic field corrected surface field strengths with er-ror bars, as well as upper limit on the regional-scale surfacemagnetic field strengths in the northern cusp region. . . . . . 1255.6 Example pitch angle distribution exhibiting symmetry in thedouble-sided loss cone. . . . . . . . . . . . . . . . . . . . . . . 1275.7 Regional-scale resolution of proton flux to the southern hemi-sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.8 Regional-scale resolution of proton flux to the northern hemi-sphere high latitude surface. . . . . . . . . . . . . . . . . . . . 1305.9 Global map of the proton flux to the surface. . . . . . . . . . 1305.10 Predicted relative solar wind density at local noon as a func-tion of longitude. . . . . . . . . . . . . . . . . . . . . . . . . . 132A.1 Cartoon diagram showing a loss cone in an idealized pitchangle distribution in the southern and northern hemisphere. . 159xiiList of AbbreviationsEPPS Energetic Particle and Plasma SpectrometerER Electron ReflectometryESA Electrostatic AnalyzerFIPS Fast Imaging Plasma SpectrometerFOV Field of ViewIMF Interplanetary Magnetic FieldMAD Median Absolute DeviationMAG MagnetometerMBF Mercury Body FixedMCP Micro-Channel PlatesMESSENGER MErcury Surface, Space Environment, GEochemistry, andRangingMHD MagnetohydrodynamicMSO Mercury Solar OrbitalPAD Pitch Angle DistributionRMS Root-Mean-SquareSEA Superposed Epoch AnalysisUTC Coordinated Universal TimexiiiAcknowledgementsMy PhD has certainly been quite the journey, as some of the best butalso some of the worst times of my life have all happened in the past fiveyears. However, it is certain that I would be nowhere near where I am todayboth academically and personally had I not been as lucky to be surroundedby a number of amazing people who have helped me along the way.My first and foremost thanks goes to my advisor and mentor, CatherineJohnson. Thank you for being an amazing supervisor, a wonderful friendand an awesome female role model all at the same time. Thank you also forbringing me along on the MESSENGER journey, and for being so patientwith my hardheadedness along the way. By being part of MESSENGERyou have taught me how collaborations work, how real science is conducted,and what it’s like to be part of a large mission, for which I am highlygrateful. I still remember when I first started you asked if I wanted to workon MESSENGER with you and I hesitated...what a mistake that would havebeen!My second biggest thanks goes to Brian Anderson, who has been anamazing mentor to me throughout my PhD. Thank you for many useful dis-cussions and for all your time and advice, both academically and personally.I greatly appreciate your collaboration and friendship.Through the MESSENGER mission I have been very fortunate to havehad the opportunity to be part of exciting collaborations and work withmany great people. Many thanks go to Haje Korth, Dan Gershman, JimRaines, and Jim Slavin for their help and suggestions on a number ofprojects, as well as to Sean Solomon, not only for making the MESSEN-GER mission possible, but also for his tireless and thorough reviews of mypapers.Thanks also to all of my committee members, including Mark Jellinek,Jonathan Aurnou, and Eldad Haber for their time and encouragement.To the Johnson-Jellinek UBC group: Kirsten Halverson, Kathi Un-glert, JF Blanchette-Guertin, Andreas Ritzer, Julien Monteux, GuillaumeCarazzo, Jess Kalynn, Manar Al Asad, Yoshi Gilchrist, Heather Wilson, andAmelia Bain, thanks for your friendship and all the office distractions andxivAcknowledgementsentertainment over the years. Life as a PhD student would have a been alot less fun without you guys. Special thanks go to Kirsten, JF, and Kathi,my fellow grad students: Kirsten and Kathi for many awesome ski morningsand chats, and JF for sharing your personal office-space and your knowledgewith me over the years. If I knew back then how much I would enjoy workingwith you during this time, I would have let you keep the big monitor.Big thanks to our friends in Vancouver for keeping us sane and not com-pletely devoid of social skills over the years: Stephan and Chloe, Anthony,Jon Ben and Jen. And of course life would have been much more emptywithout my best friend Sandra (and Colette and Greg), who has stuck byme over the years through good and bad, and has always been here for mewhen I needed her.I am very grateful to my Canadian family: Wendy, Gren, Heather,Emmy, Julia, Ben, Auntie Pat, and Uncle Joe for taking me in and makingme part of your family. I always cherish our times together, especially atVictoria Beach, but even in wintry Winnipeg!The biggest thanks of all goes to my parents, Judit and Lajos Moldovan,not only for the wonderful childhood that they gave me and my brother,but also for their sacrifice in bringing us to Canada so that we would havethe opportunity to become whoever we wanted to be. I miss you both somuch and I wish every day that you could be here with us. Thank youto my brother Andrew for being his loving and caring self, and for beingsuch an intelligent and mature young man. And ultimately, without myhusband Peter I am not sure that I would be anywhere near where I amtoday. Thank you for being my partner in life, I am eternally grateful foryour love, encouragement, and support.I also acknowledge the generous support of the National Science and En-gineering Research Council (NSERC) of Canada through the Post GraduateScholarship program as well as the financial support from the Four-YearFellowship program and various other scholarships from the University ofBritish Columbia.xvTo my mom and dad, Judit and Lajos Moldovan, who by givingme Isaac Asimov books to read as a child, started my fascination with spaceand always encouraged me to explore further. Also to my brother Andrewand my loving husband Peter.xviChapter 1IntroductionMercury, because of its close proximity to the Sun, has long eluded directin situ observations due to the technical challenges of observing, navigating,and operating so close to the Sun. Prior to 2008, our only information ofMercury came from Earth-based radio astronomy [1–4], and three Mariner10 flybys of the planet between 1974 and 1975 [5–9].The first and third Mariner 10 flybys made the surprising discovery thatMercury has a primarily dipolar intrinsic magnetic field, with a surface mag-netic field strength of ∼ 1% of Earth’s dipole field [5, 9]. Such a predomi-nantly dipolar internal field may imply an active dynamo process, which wasunexpected for Mercury due to predictions of early solidification of the core(e.g., [10, 11]). The weak strength of the field was also thought to be difficultto explain if Mercury’s dynamo is driven by thermo-chemical convection, asis the case for Earth’s dynamo (see review in [12]). Remanent crustal mag-netic fields were not originally favored for Mercury since crustal fields areusually dominated by small-scale structure. However, Aharonson et al. [13]showed that spatial variations in solar insolation on Mercury could give riseto long-wavelength variations in the depth to the Curie isotherm for thedominant magnetization carrier, that in turn would allow long-wavelengthstructure in the crustal field.From the limited Mariner 10 flyby observations alone it was not possibleto determine whether the dominant source of Mercury’s magnetic field wasof core or crustal origin and more observations were needed to characterizethe spatial structure of Mercury’s magnetic field, crucial for determiningthe source(s) of the field. More recently, Earth-based radar observationshave provided support for a dynamo field because the measured amplitudeof Mercury’s forced librations suggests decoupling of the core and mantle,1Chapter 1. Introductionfavoring the existence of a liquid outer core [4].Our knowledge and understanding of the innermost planet has signifi-cantly improved in the last few years because of the MErcury Surface, SpaceEnvironment, GEochemistry, and Ranging (MESSENGER) mission, whichon the 18th of March 2011 became the first spacecraft to enter orbit aroundMercury. MESSENGER first reached Mercury’s orbital distances in 2008,made three subsequent flybys of the planet, and was then inserted into or-bit with a 200 km periapsis altitude, 82.5◦ inclination, 15,300 km apoapsisaltitude, and 12-h period (reduced to 8-h in March 2012). The highly el-liptical orbit of the spacecraft is due to the absorption and re-emission ofsolar radiation from Mercury’s surface, which if endured for too long wouldincrease spacecraft temperatures above safe operational limits.MESSENGER has seven scientific instruments and a radio science ex-periment to probe Mercury’s surface, its tenuous exosphere (a volume sur-rounding the planet containing neutral atoms and ions that are collisionlessdue to their low number density), and its magnetic field environment. Re-cent MESSENGER results have confirmed that Mercury’s intrinsic magneticfield is a weak, global, dynamo generated dominantly dipolar field [14–16].They have also revealed that the dipole field is aligned with the rotationaxis but is offset northward from the planetary equator by 0.196 RM (whereRM is Mercury’s radius) [17–19]. Mathematically, such a northward offsetof the dipole means that in a spherical harmonic expansion of the field, eventhough the dipole term dominates, there is a significant quadrupole term,with a ratio of 0.4 for the quadrupole to dipole terms. A number of corestructures and dynamo regimes have been suggested for Mercury that can re-produce its weak dipole moment (see discussion in [18]). However, dynamomodels still have difficulty reproducing all aspects of Mercury’s magneticfield in a time-averaged sense, particularly the field’s high axial alignment,the large value for the ratio of the quadrupole to dipole term, and the lowupper bound on the ratio between the octupole and dipole field [18]. Re-search on this topic has shown that there is promise in models that invokea non-conductive layer above a deep dynamo that preferentially attenuatesthe highly time-varying higher-degree components of the field [20, 21].2Chapter 1. IntroductionFigure 1.1: Schematic diagram of Mercury’s magnetosphere. From Zur-buchen et al. (2011), Science, 333, 1862-1865 [22]. Reprinted with permis-sion from AAAS.A global dipolar planetary magnetic field is capable of dominating thespace environment in its near vicinity by ordering and directing chargedparticles and thereby partially shielding the region called the magnetospherefrom the solar wind. The magnetosphere is a magnetic cavity that formsdue to the interaction of the solar wind with the dipole field and is describedin more detail in Sections 1.1 and 1.4. Figure 1.1 shows a schematic dia-gram of Mercury’s magnetosphere; here we briefly introduce aspects mostrelevant to this thesis and define them in more detail in the coming sections.Surrounding the planet and its magnetic field are the bow shock, magne-tosheath, and magnetopause. The bow shock is a shock wave that formsas the solar wind transitions from supersonic to subsonic speeds as it en-counters the obstacle of the planet’s magnetic field. The magnetosheath, justdownstream of the bow shock, is the region in which the shocked solar wind,compressed and heated at the shock, flows around the magnetopause. The3Chapter 1. Introductionmagnetopause marks the boundary of the magnetosphere, and is a currentlayer that acts to confine the planet’s magnetic field inside the magneto-sphere. Detailed studies on Mercury’s magnetopause and bow shock will bepresented in Chapter 2. At high latitudes are the magnetic cusp regions (onein the north, studied in detail in Chapter 3, and one in the south), wherethe magnetopause currents nearly cancel the planet’s internal field, leavingbehind a weakened field region where solar wind plasma can gain access tothe magnetosphere. The plasma sheet in the nightside magnetosphere isalso a region of high plasma density, whose spatial extent largely coincideswith the tail current sheet, and separates the magnetotail’s north and southlobes. The plasma populations found in the cusp region and in the plasmasheet make our proton reflection magnetometry observations in Chapters 4and 5 possible.MESSENGER has revealed that the interaction of Mercury’s magneticfield with the solar wind is unique in our solar system. The combination ofthe weak planetary field and strong solar wind and interplanetary magneticfield (IMF) generates a small, highly dynamic magnetosphere around Mer-cury. At Mercury’s orbital distances from the Sun, the solar wind density ison average an order of magnitude higher than at Earth, and the interplan-etary magnetic field is a factor of five higher than at Earth [23]. BecauseMercury’s orbit is eccentric (0.31 AU < r < 0.47 AU, where r is heliocen-tric distance), the planet is also subjected to different solar wind conditionsalong its orbit; both the solar wind density and pressure increase signifi-cantly near perihelion. The north-south component of the IMF, which hasa large influence on planetary magnetospheres (see Section 1.1), has beenshown to change on a timescales of a few minutes at Mercury [23]. Varia-tions in Mercury’s magnetospheric conditions thus occur on timescales of afew minutes to a Mercury year (88 Earth days) [24].Mercury’s magnetosphere is strongly coupled to the exosphere and theplanet’s surface. Due to the lack of atmosphere and ionosphere (ionized up-per atmosphere) the high-energy solar wind plasma can gain access to themagnetosphere in regions where the shielding is incomplete and bombardsthe surface of the planet. Ions sputtered from the surface move under the41.1. Interaction of the solar wind with Mercury’s magnetosphereinfluence of magnetospheric electric and magnetic fields; some ions will belost through reconnection down-tail or through collision with the magne-topause, while others will be returned back to the surface. This couplingbetween the magnetosphere – exosphere – surface influences the dynamicswithin the system.This thesis is devoted to shedding light on the interaction of the in-terplanetary medium with this magnetospheric system, particularly as itpertains to three specific regions of the magnetosphere. We also show thatthe nature of the interaction between the magnetosphere and the solar windcan reveal information about the intrinsic planetary field at the surface andcan be used to make indirect surface field measurements. The initial interac-tion of the solar wind with the magnetosphere occurs at the magnetosphericboundaries, the bow shock and the magnetopause. In Chapter 2 we discusshow we characterized these boundaries through observations and empiricalmodels. In the magnetic cusp regions, shielding of the magnetosphere fromthe solar wind is incomplete, and so charged particles can gain access to themagnetosphere and precipitate down to the surface. We investigated thissolar wind – magnetosphere – surface interaction in the northern cusp regionof Mercury, and we describe this in detail in Chapter 3. Finally, in Chapters4 and 5, we develop a novel method to remotely sense the surface planetarymagnetic fields. The motion of solar wind protons, which have gained accessto the magnetosphere near the cusp and the magnetopause boundary, allowsfor the measurement of the magnetic field strength at the surface of Mercurythrough the technique of proton reflection magnetometry. We describe thistechnique and show that this method confirms the offset dipole structure atMercury’s surface and particle precipitation to the surface.1.1 Interaction of the solar wind with Mercury’smagnetosphereThe interaction of the solar wind with a magnetized planet produces amagnetic cavity around the planet, the magnetosphere, which confines the51.1. Interaction of the solar wind with Mercury’s magnetosphereplanetary magnetic field and is largely devoid of solar wind plasma. Fordetails on the formation of the magnetosphere, see Section 1.4.Although the interaction between the solar wind and all magnetizedplanets produces a magnetosphere, the nature of this interaction (i.e. char-acteristics of the magnetopause current layer and of the shock wave, or bowshock, in front of the magnetopause as well as the rate of exchange of plasmabetween the solar wind and magnetosphere through reconnection) varies inour solar system. Solar wind conditions vary significantly with heliocen-tric distance; the solar wind density decreases as 1/r2 with distance fromthe Sun on average, the magnetic field strength also decreases with distancewith a more complicated scaling law as do the solar wind electron and protontemperatures (see Table 1.1).At Mercury, the dominant magnetic field component of the interplane-tary magnetic field (IMF) is parallel or anti-parallel to the +x direction ina Mercury solar orbital (MSO) coordinate system, for which +x is sunward,z is normal to the orbital plane and positive northward, and +y completesthe right-handed system. The angle of the IMF (also referred to as theParker spiral angle) with respect to the radial (−x) direction is only ∼ 20◦for typical solar wind conditions at 0.3–0.5 AU. This is about a factor of 2smaller than the Parker spiral angle of 45◦ at the Earth. The dominanceof the IMF Bx component on average at Mercury has a significant effect onthe magnetosphere, which will be discussed further in Chapter 3.Other solar wind parameters that vary with heliocentric distance are thesolar wind Mach numbers, which increase with distance from the Sun. TheMach number is defined as the solar wind flow speed divided by the speedof a fundamental wave mode (e.g. sound waves, Alfve´n waves). The solarwind speed is approximately constant with heliocentric distance, and if thespeed of the wave mode decreases with heliocentric distance, the Mach num-ber increases correspondingly. The Alfve´n Mach number, MA, will be usedin Chapter 2 of this thesis; Alfve´n waves are magnetohydrodynamic waves,and have an associated speed of vA = B√µ0ρ , where µ0 is the permeabilityof free space, and ρ is the plasma density. For example, MA is expectedto be ∼ 5 at Mercury and ∼ 13 at Saturn [25]. Such a large difference in61.1. Interaction of the solar wind with Mercury’s magnetospherevsw nsw BIMF Tp TeScaling r0 r−2 r−1(2r−2 + 2)1/2 r−2/3 r−1/3Table 1.1: Scaling laws of solar wind speed, vsw, number density, nsw, in-terplanetary magnetic field magnitude, BIMF , proton temperature, Tp, andelectron temperature, Te, with distance from the Sun [26].MA implies that the bow shock in front (upstream) of Mercury’s magneto-sphere is much weaker than the equivalent bow shock in front of Saturn’smagnetosphere [25].The merging of the magnetic field lines of the planet’s intrinsic magneticfield with those of the draped IMF, through a process called reconnection,is responsible for the circulation of flux in the magnetosphere (see Figure1.2) and some fraction of the injection of solar wind plasma into the magne-tosphere. This circulation of the magnetic flux and associated plasma fromthe dayside to the nightside and back, called the Dungey cycle [27], is alsodependent on heliocentric distance and is much more dynamic at Mercurythan at any other planet in our solar system. The large-scale circulationof magnetic flux and plasma in the magnetosphere is driven by the solarwind at Mercury and Earth. However at for example, Jupiter and Saturnthe solar wind is much weaker and the magnetospheres of these planets areinfluenced to a much higher degree by the planetary rotation and the largeamount of plasma originating within the magnetosphere. At Mercury, theDungey cycle time is ∼ 2 minutes [28, 29], whereas this time scale is muchlonger at Earth, ∼ 60 minutes.Reconnection is expected to proceed most efficiently when the IMF di-rection is oppositely directed to the planetary field in the subsolar region; onthe dayside at Earth and Mercury this occurs when the IMF has a southwardcomponent, i.e., a negative Bz component. The rate of reconnection has alsobeen shown to depend on MA and the plasma β, the ratio of the plasmathermal pressure to the magnetic pressure [31]. Recent observations [32]suggest that a low value of MA on the solar wind side of the magnetopausecurrent layer, and a low β on either side of the current layer as well as alow value of the absolute difference in plasma β across the current layer71.1. Interaction of the solar wind with Mercury’s magnetosphereFigure 1.2: Magnetospheric convection due to reconnection. The circulationof flux proceeds from numbers 1 through 8 in the figure. At 1, the planet’sclosed field lines merge with the southward directed, open field lines of thesolar wind at the subsolar point where the fields are in opposite direction.The newly opened field lines are convected through from the dayside to thenightside by the flow of the shocked solar wind in the magnetosheath. At 7,the field lines are stretched out far enough that they are in opposite directionagain and they reconnect in the tail, to close the magnetic flux circulation.Newly connected planetary field lines in the tail move planetward due tomagnetic tension, while completely open IMF field lines move away fromthe planet. Figure from Basic Space Plasma Physics, Wolfgang Baumjohannand Rudolf A. Treumann, Copyright 1997, Imperial College Press [30]; usedwith permission.(|∆β|), promotes reconnection onset because under these conditions recon-nection is possible over a wider range of shear angles (i.e., angle betweenthe magnetic field direction in the IMF and just inside the magnetopause).Furthermore, Swisdak et al. [33] showed that a diamagnetic drift, whichis produced when a pressure gradient is present across the magnetopause81.2. Charged particle motioncurrent sheet, can disrupt the reconnection process when this drift velocityis super-Alfve´nic (i.e. the solar wind flow speed is greater than the Alfve´nspeed). Thus reconnection is more likely to be suppressed for cases of highMA, high β, and high |∆β|. The Alfve´n Mach number scales approximatelyas r with heliocentric distance, and the magnetosheath β has been observedto increase with heliocentric distance [31]; they are thus expected to besmallest at Mercury compared to other planets in the solar system. Workby Dibraccio et al. [34] has shown that as expected, reconnection rates arevery high at Mercury, ∼ 3 times higher than at Earth, and reconnectionoccurs over a much broader range of IMF orientations than at Earth. Allthis implies that Mercury’s magnetosphere is more strongly driven by thesolar wind and IMF than Earth’s magnetosphere because of the interactionof the weak planetary dynamo field with the strong solar wind environmentthat it is embedded in. The dynamic nature of Mercury’s magnetosphere isfurther explored in Chapters 2 and 3.1.2 Charged particle motionBasic insight into the motion of charged particles in electric and mag-netic fields must be gained in order to lay the foundations for much of thework that is presented in this thesis. Charged particle motion is well docu-mented in a variety of textbooks in electricity and magnetism as well as inplasma physics. We thus only present results here for scenarios most rele-vant to the work in this thesis, and for more details the interested reader isreferred to texts such as [30, 35, 36]. We briefly describe here the motion ofa single charged particle in specified magnetic and electric fields as well asthe motion of charged particles trapped in planetary magnetospheres. Sec-tion 1.2.1 describes the gyrational motion of charged particles in uniformmagnetic fields which applies to work presented throughout this thesis. InSections 1.2.2 and 1.2.3 we describe the drift motion of charged particles un-der uniform electric fields as well as under changing magnetic fields, whichpertain to results presented in Chapters 4 and 5 of this thesis (drift motionsare responsible for populating closed magnetic field lines on the nightside91.2. Charged particle motionwith plasma from the plasma sheet). Finally, in Section 1.2.4 we outlinethe motion of charged particles in convergent (i.e. increasing) magneticfields; this motion of charged particles is exploited by the proton reflectionmagnetometry technique in Chapters 4 and 5.In most plasmas the electric field and the magnetic field depend on thepositions and velocities of all the charged particles in the system, and arethus complicated functions. Magnetohydrodynamic (MHD) equations havebeen derived to deal with the plasma macroscopically as a fluid, insteadof solving the kinetic equations of motion for all the particles individually.However, to gain physical insight into magnetospheric processes, it is in-structive to consider the motion of a single charged particle in electric andmagnetic fields that are independent of time and that are simple functionsof position.To determine the trajectory of a single particle in a force field as afunction of position and time we solve the equation of motionmdvdt= F(x, t), (1.1)where in the case of a charged particle, one of the forces acting on theparticle is the Lorentz force given byF = q(E + v ×B), (1.2)here q is the electric charge, E is the electric field, v is the particle’svelocity, and B is the magnetic field.1.2.1 Uniform magnetic field (E = 0)In a uniform magnetic field (E = 0), the equation of motion simplifiestomdvdt= q(v ×B). (1.3)The particle will have a motion both parallel and perpendicular to themagnetic field, and so it is useful to separate the velocity into its components101.2. Charged particle motionparallel and perpendicular to the magnetic field, v = v‖ + v⊥. By takingthe dot product of equation (1.3) with the unit vector in the direction ofB we find that the equation of motion in the parallel direction is mdvdt =0. The solution to this is clearly v‖ = constant, i.e. the particle has aconstant velocity along the magnetic field line direction. By subtracting theequation of motion in the field aligned direction from equation (1.3) andusing v‖ ×B = 0, we havemdv⊥dt= mv⊥ ×Ω, (1.4)where Ω = qB/m bˆ is the gyrofrequency, and bˆ is the magnetic fielddirection. Without any loss of generality, if we let the magnetic field be inthe zˆ direction, then v⊥ will lie in the x − y plane. Then the solution toequation (1.4) (showing only the real parts of the complex variables) isvx = v0 cos(|Ω|t+ δ),vy = ∓v0 sin(|Ω|t+ δ). (1.5)v0 is the magnitude of the perpendicular velocity and is independent oftime, and the ∓ sign corresponds to positive and negative particles, respec-tively. This equation shows that the particle follows a circular trajectory invx − vy space, with positively charged particles gyrating left while negativeones gyrate in a right-handed direction. The gyroperiod, or the time it takesthe particle to complete a cycle of its circular motion isT =2pi|Ω|= 2pim|q|B(1.6)The radius of the circle that the particle travels in is the gyroradius andis given byrg =v⊥|Ω|=mv⊥|q|B. (1.7)The gyroradius of the particle represents a natural length scale in thesystem; particles with small gyroradii relative to the gradient length scale111.2. Charged particle motionof the magnetic field, B/∇B, are highly affected by the ambient magneticfield and thus exhibit motions dictated by changes in the field. However,if the gradient in the magnetic field is high compared to the field strength,particles will have large gyroradii relative to B/∇B and are referred to asunmagnetized, because their motion is largely unaffected by the magneticfield and changes in it.As we saw before, the particle has a constant velocity in the field aligneddirection, and the combination of this motion with the circular motion in theperpendicular direction yields a helical trajectory for the particle about thefield line. The pitch angle of the helix, or the angle between the magneticfield direction and the particle’s velocity is defined asα = tan−1(v⊥v‖). (1.8)and will be used extensively in Chapters 4 and 5 of this thesis.1.2.2 Uniform electric and magnetic fieldsThe motion of charged particles in the presence of both a uniform electricand magnetic field can be decomposed into a gyrating part about the fieldline and a uniform drift velocity part. This drift motion is perpendicular tothe electric and magnetic fields and is given byvE =E×BB2. (1.9)This E×B drift is independent of the sign of the particle’s charge, andboth electrons and ions will drift in the same direction. This drift velocityessentially amounts to the gyroradius varying along the particle’s trajectory,increasing it in the direction of E and decreasing it in a direction oppositeto E.1.2.3 Nonuniform magnetic fieldWe now consider the motion of charged particles in a magnetic field thatis a function of position and with no applied electric field. In such a time-121.2. Charged particle motionindependent field the kinetic energy of the particle remains constant, andthe motion of the particle can be again decomposed into gyromotion plus adrift motion.First, we briefly discuss gradient B drift. Suppose the magnetic fieldis in one direction only, B(x) = B(x)zˆ and that it has a gradient, ∇B, inthe perpendicular direction to B along the y direction. The gyroradius of aparticle in such a field configuration will be large wherever B is small, andwill be small wherever B is large. Thus the drift motion of a positive chargewill be in opposite direction to the drift of a negative charge; therefore acurrent will arise under a magnetic field gradient. The expression for thegrad-B drift velocity is given byv∇B = ±12v⊥rgB×∇BB2, (1.10)where ± refers to positive and negative particles, respectively.We will next consider particles moving along a curved magnetic field line.These particles will experience an outward centrifugal force in the frame ofreference moving with the particle’s parallel velocity. This force is given byFc =mv2‖RcRˆc, (1.11)where Rc is the curvature vector; it has a magnitude equal to the localradius of curvature of the field line and is pointing radially outward. Thiscentrifugal force gives rise to a drift motion, called the curvature driftvcB = ±v2‖|Ω|RcRˆc × bˆ. (1.12)A curved magnetic field cannot have a constant magnitude in a vacuum(otherwise ∇×B = 0 would not hold), and thus the complete drift velocityof a particle in a nonuniform magnetic field is vB = v∇B + vcB. For asouthward directed dipole field and prograde planetary rotation, positiveions undergo gradient and curvature drift opposite to the planet’s rotation(westward), while electrons drift in the direction of the planet’s rotation(eastward).131.2. Charged particle motion1.2.4 Motion of trapped particles: magnetic mirrors andbottlesMagnetic fields in the near-Mercury environment are non-uniform bothspatially and temporally, but even in such fields, the proton motion is de-scribed by the first adiabatic invariant, the magnetic moment, µm of theparticle. In general, the magnetic moment is given by the electrical currentI in a current loop multiplied by the area A of the loop: µm = IA. In thecase of a single charged particle, the current is the charge q divided by thegyroperiod T . The gyroradius of the particle defines a circular area and sothe magnetic moment is given byµm =qTA =qTpir2g =12mv2⊥B. (1.13)The magnetic moment is invariant for slow changes in a system (suchthat the process is adiabatic) and is thus very useful in interpreting certainelements of charged particle motion. In particular, it is useful in interpretingparticle motion in magnetic mirrors and bottles.A converging magnetic field is needed for magnetic mirroring to occur.We start with a magnetic field increasing in the z direction in a cylindricalcoordinate system (Figure 1.3). In this scenario, the charged particle willmove in the z direction with a parallel velocity, and as long as changesin the electric and magnetic field occur on length scales larger than thegyroradius, the magnetic moment of the particle will be conserved. We cansee from equation (1.13) that if the magnetic field strength increases, thenthe perpendicular velocity of the particle must increase to keep µm constant.Thus the new perpendicular velocity of the particle will be given byv2⊥ =(BB0)v2⊥0, (1.14)where the symbol “0” indicates initial value. In a static magnetic fieldconfiguration the total energy of the particle must remain constantv2⊥ + v2‖ = v2⊥0 + v2‖0. (1.15)141.2. Charged particle motion!"!"#$%"&'()*+%,-./)0)!"#Figure 1.3: Schematic diagram of converging magnetic fields towards theplanet’s surface.Thus as the magnetic field strength increases more and more so doesv⊥. There will be a point along the field line when all of the particle’svelocity is in the perpendicular direction and thus it will not be able topenetrate further along the field line but will be reflected, or mirrored back.We can express equation (1.14) in terms of the particle’s pitch angle, sincev⊥ = v sinα. At the point of reflection, the particle’s pitch angle is 90◦, andso can rewrite equation (1.14) in terms of the particle’s initial pitch angleand the ratio of the initial magnetic field to that at the reflection point,sin2 α0 =B0Bm. (1.16)A particle with an initial pitch angle of α0 will reflect at a point wherethe magnetic field is Bm, whereas a particle with a pitch angle less than α0(i.e. more field aligned) will reflect after reaching a magnetic field magnitudegreater than Bm. The particle’s trajectory upward is a mirror image of itstrajectory downwards, and it returns to the same point with a pitch angle180◦ − α0.The physical explanation for the decrease in v‖ is that there will be a151.2. Charged particle motioncomponent of the Lorentz force which is directed opposite to the gradientin B and decelerates the particles. The component of the Lorentz force inthe z direction isF‖ = q(v ×B)z = qv⊥Bρ, (1.17)where the field B in the scenario depicted in Figure 1.3 is given byB = Bρρˆ+Bzzˆ. (1.18)The field is primarily in the z direction (Bρ  Bz) and the componentof the field in the ρ direction can be found from Maxwell’s equation:∇ ·B =1ρddρ(ρBρ) +dBzdz= 0. (1.19)In the case where Bρ  Bz and near the z-axis (ρ = 0) we have thatBρ ≈ −12ρdBzdz. (1.20)This can be substituted into equation (1.17) to yieldF‖ = −12qv⊥rgdBzdz, (1.21)where we are considering the case when ρ = rg. This component of theLorentz force can be rewritten in terms of the magnetic momentF = −µm∇‖B. (1.22)This force is thus clearly directed opposite to the gradient in the magneticfield, along zˆ, and since µm is constant, the only variable in this relationis the magnetic field gradient. Therefore, this force will be higher as theparticle moves toward larger values of the parallel magnetic field gradient,and will thus decelerate the particle.In a magnetic bottle the situation is similar, except the magnetic field isconvergent on both sides, i.e. it consists of two magnetic mirrors. In sucha situation the particle bounces back and forth between the two convergent161.2. Charged particle motionends of the magnetic field, and is trapped. The maximum field strengthof the bottle determines the minimum initial pitch angle required for con-finement, i.e. whether a particle with a given initial pitch angle, α0 will betrapped or not. The criterion for confinement in the bottle is given bysinα0 >(B0Bmax) 12, (1.23)where Bmax is the maximum magnetic field strength reached in the bot-tle. Thus if initially there is an isotropic distribution of particles (i.e. equalparticle fluxes in all directions), then some time later all particles that donot satisfy the confinement criterion above will be missing from the distri-bution, while the ones that do satisfy the condition will be trapped insidethe bottle. The missing portion of the distribution is called the loss cone,which is directly relevant for Chapters 4 and 5 of this thesis.Such a bottle magnetic field configuration exists on closed magnetic fieldlines of dipole fields, for example at Earth and Mercury as well. Particlestrapped on closed magnetic field lines will bounce back and forth with abounce period Tb given by the integral over a bounce cycle:Tb =∮dzv‖(z)= 4∫ zmaxz0dzv[1− B(z)B0 sin2 α0] 12, (1.24)where v is the total particle speed, z0 is the particle’s starting positionin the bottle while zmax is furthest the particle travels along the z direction.The bounce period for protons (with an average proton energy of 1 keV) inMercury’s inner dayside magnetosphere is ∼ 30 seconds. Particles will alsohave a drift motion around Mercury due to the grad-B and curvature drift.The drift period of protons of the same energy is ∼ 40 minutes. However,because Mercury occupies such a large fraction of its magnetosphere, mostparticles are not be able to complete a full drift orbit since drift paths onaverage either intersect the planet or the magnetopause, and the particlesare thus removed from the system.171.3. The bow shock1.3 The bow shockWhen the solar wind reaches the obstacle of Mercury’s magnetic field itwill be slowed down and deflected around it. A bow shock wave is generatedas the solar wind hits the obstacle with supersonic speed, and a substantialfraction of the solar wind bulk flow kinetic energy is converted into thermalenergy. The shock wave, which is a nonlinear wave, is an irreversible processthat changes the state of the medium, for e.g., the temperature, pressure,and density. At the shock, the plasma density and temperature increase,implying a decrease in the flow speed to conserve mass and energy. Thusthe shock wave is the mechanism by which the plasma transitions fromsupersonic to subsonic flow; upstream of the shock the solar wind flow speedis supersonic, while on the downstream side it is subsonic. The region rightbehind (downstream of) the bow shock where the solar wind is slowed andcompressed is called the magnetosheath.The shock can be thought of as a discontinuity, although in reality it hasa finite thickness due to the kinetic processes at the shock. For a discon-tinuity to be a shock, there must be plasma flow across the shock surfaceaccompanied by dissipation (i.e. the transformation of the kinetic energy ofthe flow to random thermal energy in the particles) and compression acrossthe shock. Planetary bow shocks are fast shocks, corresponding to the fastmode wave in MHD. Across a forward fast-mode shock, the plasma pressureand magnetic field strength increase and the magnetic field bends away fromthe shock normal.The shock structure is partly dependent on the upstream magnetic fielddirection relative to the shock surface. If the upstream magnetic field is per-pendicular (or quasi-perpendicular) to the shock normal direction, particlescannot travel far back into the upstream region because their gyrational mo-tion brings them back into the shock. Typical perpendicular shocks (Figure1.4) show a shock foot where the magnetic field is gradually increasing infront of the main shock. Behind the main ramp where the magnetic fieldincreases significantly, there is an overshoot with field values larger thanthe asymptotic downstream values. On the other hand, when the upstream181.3. The bow shockFigure 1.4: Schematic diagrams showing example bow shock magnetic fieldstructure. Figure from Basic Space Plasma Physics, Wolfgang Baumjohannand Rudolf A. Treumann, Copyright 1997, Imperial College Press [30]; usedwith permission.magnetic field is parallel (or quasi-parallel) to the shock normal direction,particles can travel back up into the upstream region along the field lines.In order for particles to outrun the shock and travel back upstream theymust have a velocity higher than the escape velocity of the shock which isgiven by vesc = vnu/ cos θ, where vnu is the upstream flow velocity normalto the shock and θ is the angle between the shock normal and the upstreammagnetic field. Particles with higher velocities will move ahead of the slowerescaping particles and thus higher energy particles will be observed furtheraway from the shock. These foreshock particles drive instabilities such thatthe plasma and magnetic field properties both upstream and downstream191.3. The bow shockFigure 1.5: Schematic diagram showing regions of parallel and perpendicu-lar shock. Just upstream of the parallel shock region is the foreshock. Figurefrom Basic Space Plasma Physics, Wolfgang Baumjohann and Rudolf A.Treumann, Copyright 1997, Imperial College Press [30]; used with permis-sion.show significant oscillations. Because the bow shock is a curved surfacearound the planet, for the same IMF direction the bow shock structure willbe significantly different at different locations. Figure 1.5 shows an examplebow shock schematic diagram at the Earth, where in regions where the IMFis perpendicular to the shock normal, the conditions will not be favorable forparticles to travel back upstream and so no foreshock will form. However,as the angle between the shock and magnetic field gets smaller, more andmore particles will be able to outrun the shock and form a foreshock region.The shock in this region will have quasi-parallel shock structure.Using MHD equations, it is possible to derive the conservation relations(also known as the Rankine-Hugoniot shock-jump conditions) that prescribethe downstream plasma state in terms of the upstream parameters. How-201.3. The bow shockever, the shock-jump conditions are a simplification because they are de-rived using a one fluid MHD approximation, thereby ignoring microscopicprocesses at the shock; thus they can be used as a guide and for comparisonwith observations of the flow upstream and downstream, and can test theassumptions that go into their derivation. These relations are derived inthe shock frame, where the shock is stationary, and all involve parametersalong the normal or transverse direction to the shock. Thus by measuringthe magnetic field, flow speed, and density along the shock normal, it is pos-sible to estimate how much the downstream conditions in the plasma havechanged from that upstream. For the derivations of the Rankine-Hugoniotrelations the interested reader is referred to [37].The shock jump equations are given by[ρvn] = 0, (1.25)[ρv2n + p+B22µ0]= 0 (1.26)[ρvnvt −Bnµ0Bt]= 0 (1.27)[ρvn(12v2 +γγ − 1pρ)+ vnB2µ0− v ·BBnµ0]= 0 (1.28)[Bn] = 0 (1.29)[vnBt −Bnvt] = 0. (1.30)In these equations the square brackets denote subtraction of the down-stream quantity in brackets from the upstream equivalent, subscript n de-notes the component normal to the shock and t transverse to the shock,ρ is the flow density, v is the velocity, B is the magnetic field, µ0 is thepermeability of free space, γ is the polytropic index.For the case of an exactly perpendicular shock, it is simple to derive the211.3. The bow shockexpected maximum compression of the magnetic field, density, and velocityacross the shock using the shock jump conditions. In such a case Bn = 0and we examine the case where the upstream flow velocity is parallel to theshock normal, i.e. vu = vunnˆ. Across the shock, the mass flux must benon-zero, such thatρuvun = ρdvdn 6= 0. (1.31)We define the compression ratio, r = ρd/ρu, which using equation 1.31leads to the downstream velocity being defined in terms of the compressionratio as well as vdn = (1/r)vun. We can apply the third jump condition(equation 1.27), which due to the fact that Bn = 0 becomesρuvunvut − ρdvdnvdt = 0, (1.32)implying that vdt = 0 since vut is zero and ρvn cannot be zero. We can usethe jump condition in equation 1.30 and the fact that all the magnetic fieldis in the tangential component to substitute in for the compression ratioand get that Bd = rBu. This shows that the magnetic field is compressedby the same amount as the flow density. Using equations 1.26 and 1.28 andsubstituting in for vdn, Bd and eliminating pd, it is possible to arrive atan equation that expresses the compression ratio in terms of the upstreamparameters only:(r − 1){r22− γM2A+ r(γM2A+2M2ms+ γ − 1)− (γ + 1)} = 0, (1.33)where MA is the Alfve´n Mach number as defined earlier, and Mms =vu(ρuγpu) 12is the magnetosonic Mach number. In the high Mach numberlimit when MA and Mms  1, equation 1.33 becomes r =γ+1γ−1 . γ is notdependent on the upstream parameters; for a monatomic gas γ = 53 , whichyields a maximum compression of a factor of 4 for the magnetic field, density,and velocity. This number, although much quoted as the asymptotic limitfor the magnetic field strength increase across the shock, is dependent on221.4. The magnetopause and the magnetic cuspsthe polytropic index in the plasma and thus on the way that the plasma isheated. In Chapter 2, the example cases of Mercury’s bow shock that areshown in Section 2.2 all have a compression factor of less than 4 (measured inthe magnetic field strength). The upstream Mach number of the shock alsodefines the strength of the shock and is a measure of the amount of energybeing processed by the shock; the higher the Mach number, the stronger theshock. In Chapter 2 we will explore how Mercury’s bow shock varies withAlfve´n Mach number.1.4 The magnetopause and the magnetic cuspsThe magnetopause demarcates the upper boundary of the magneto-sphere and controls the flux of solar wind mass, energy, and momentuminto the magnetosphere. Figure 1.6 illustrates how the interaction betweenthe solar wind and a magnetized planet generates the magnetic cavity ofthe magnetosphere, marked by the magnetopause boundary. In this exam-ple case, in the equatorial plane of the planet the magnetic field is pointingnorthward. When charged particles in the solar wind approach the plan-etary magnetic field, the Lorentz force (see equation (1.2)) in the absenceof electric fields, given by q(v ×B), deflects ions to the right and electronsto the left. The opposite motion of the charges produces a sheet currentfrom left to right in the figure (dawn to dusk). The magnetic perturbationsfrom this magnetopause current (or Chapman-Ferraro current as it was firstproposed by [38]) cancels the planet’s field sunward of the current and in-creases the field planetward. Thus if the magnetopause was a plane, the fieldstrength just inside the boundary would be twice the dipole magnetic field.However, because the magnetopause is curved, the field immediately insidethe near-equatorial dayside magnetopause is slightly greater than twice thedipole field.The return current above the pole is from dusk to dawn, because the fieldpoints southward. Near the pole there is a singular point in the field, theneutral point, where this current sheet completely cancels the planet’s field.These are called the magnetic cusps because here the field comes together231.4. The magnetopause and the magnetic cuspsFigure 1.6: Currents on the magnetopause. By courtesy of EncyclopaediaBritannica, Inc., copyright 1994; used with permission.at a cusp-like geometry. (Chapter 3 of this thesis will explore in great detailMercury’s northern cusp region.) The magnetopause current circulates ina sheet around the neutral point, and is symmetric about the magneticequator, with a corresponding circulation about the southern neutral point.In this way the magnetopause current shields the magnetized planet fromthe solar wind, confining the planet’s magnetic field to the magnetosphere.By looking at a cross-sectional view of the magnetosphere (see Figure1.7), more clarity can be gained about the magnetopause current, the cuspsand other features of the magnetosphere structure. Inside the magneto-sphere, the magnetic field is divided into two regions: 1) equatorward and2) poleward of the cusp latitude. The division between these two regions iscalled the separatrix, which separates regions of closed field lines at mid-to-241.4. The magnetopause and the magnetic cuspsFigure 1.7: Magnetospheric current systems. From: Hughes, W.J., Themagnetopause, magnetotail, and magnetic reconnection, in Introduction toSpace Physics, Eds M.G. Kivelson and C.T. Russell, Cambridge, 1995 [37].low latitudes and open field lines at high latitudes. “Closed” here refers tofield lines that have both ends map to the planet, whereas “open” refers tofield lines that have one end map to the planet and the other end map toa tail lobe (or in the case of an open magnetopause, to the solar wind). InFigure 1.7, the magnetopause current flows out of the page equatorward ofthe cusps, and into the page poleward of the cusps. The magnetotail is gen-erated by the fact that the magnetosheath flow convects open magnetic fieldlines toward the planet’s nightside. In the magnetotail, the field direction isoppositely directed between the two hemispheres, which must be supportedby the existence of a current sheet at the magnetic equator.The location of the magnetopause around the planet corresponds to thesurface across which the pressures of the internal magnetospheric magneticfields and charged particles are balanced by the external solar wind particleand magnetic field pressure [39]. Thus the magnetopause is in pressure251.4. The magnetopause and the magnetic cuspsbalance, given by the equation[P +B22µ0]msh=[P +B22µ0]msp(1.34)where msp stands for magnetosphere, msh for magnetosheath, P is acombination of particle thermal and dynamic pressure, and B2/2µ0 is themagnetic pressure. On the left hand side we thus have the solar wind thermaland dynamic pressure as well as the IMF magnetic pressure, while on theright hand side we have the magnetic pressure of the planet’s magnetic fieldand the plasma pressures. The dominant contributors to this equation arethe solar wind dynamic pressure and the magnetic pressure of the planet’sdipole field, although the solar wind thermal pressure becomes non-negligibleat higher latitudes away from the subsolar point.Using this pressure balance, we can obtain an estimate of the location ofMercury’s magnetopause at the subsolar point. In Chapter 2, we describehow we determined the actual subsolar stand-off distance of Mercury’s mag-netopause using MESSENGER magnetic field observations as well as anempirical model of the boundary, but for now, we illustrate how a simpleestimate can be achieved in order to build further intuition of the magne-topause. We assume the Newtonian approximation, an empirical relationfrom hypersonic flow theory [40], whereby the magnetosheath pressure atthe subsolar point is given byPmsh = κρswv2sw, (1.35)where ρsw is the solar wind density, vsw is the solar wind velocity, and κis a constant that depends on how the solar wind flow is diverted around theplanet’s magnetic field. κ would be 1 in the ideal case where the bow shockand the magnetic field obstacle are close together and parallel, but is mostoften quoted as 0.88 if the flow is diverted in a more gradual manner [41].To estimate the location of the magnetopause, we can rewrite equation261.4. The magnetopause and the magnetic cusps(1.34) in a simplified form in terms of the dominant components asPmsh ≈ 0.88ρswv2sw ≈B2msp2µ0. (1.36)We know that the magnetospheric magnetic field at the magnetopauseshould be slightly higher than twice the dipole field at that distance, and sowe can express Bmsp asBmsp = aBdipole = aBd0(RMRss)3, (1.37)where a = 2.4 has been deduced for Earth’s magnetopause, Bdipole isthe magnetic field strength of the dipole at the magnetopause, Bd0 is themagnetic field strength of the dipole at the magnetic equator on the planet’ssurface, RM is Mercury’s radius, and Rss is the distance from the planet’ssurface to the subsolar point on the magnetopause, i.e. the subsolar stand-off distance. Substituting this into equation (1.36) yields that Rss ∝ P− 16mshand thatRss ≈ RM(a2B2d02µ0κρswv2sw) 16. (1.38)At the magnetic equator the magnetic field strength on the surface isgiven by B = µ0MM4pir3 and thus from having an estimate of the magneticmoment of Mercury, MM , of 190 nT R3M [19], we can calculate Bd0. Weuse the WSA-Enlil heliospheric model [42] predictions of the solar wind atMercury to obtain averages of the solar wind density and speed, which are51 cm−3 and 422 km/s, respectively. Substituting these values into equation(1.38), we estimate that Mercury’s magnetopause stands off the solar windat the subsolar point 1.35 RM away from the planet’s surface. This value isonly 0.1 RM less than the average Rss established for Mercury in Chapter 2using 3 Mercury years of observations confirming that the approximationsmade above are correct to first-order.In Chapter 2 we also determine the shape of Mercury’s magnetopause.We can however build up some intuition and derive the first-order 2-D shape271.4. The magnetopause and the magnetic cuspsof the magnetopause under the simple balance of forces used above. We startby revising the solar wind dynamic pressure relation to accommodate anypoint on the magnetopause surface. Since the dynamic pressure is exertedonly in the anti-sunward direction, the magnetic pressure only needs to standoff this component everywhere around the planet. If nˆ is the magnetopausenormal, the dynamic pressure incident on it away from the subsolar point isreduced toPmsh = nˆ · xˆρswv2sw = cos θρswv2sw, (1.39)where θ is the angle between the solar wind flow direction and the mag-netopause normal. Using this revised dynamic pressure we arrive at themagnetopause distance from the planet’s surface, Rmp away from the sub-solar pointRmp ≈ RM(a2B2d02µ0κ cos θρswv2sw) 16= Rss(cos θ)− 16 . (1.40)We can see clearly from here that at θ = pi/2, Rmp →∞, which suggeststhe magnetopause extends far downtail. We can now derive a differentiallength, ~dl = dxxˆ+ dzzˆ, tangent to the magnetopause, where dx = − sin θdland dz = cos θdl. This leads to the ratio between the differential change inx and z−dxdz= tan θ =√1− cos2 θcos θ=1cos θ√1cos θ− cos θ =√sec2θ − 1. (1.41)We can express equation (1.41) in terms of Rmp and Rss through θ be-causesecθ ≈(RmpRss)6=(x2 + z2R2ss)3, (1.42)which when substituted into equation (1.41) becomes−dxdz=√(x2 + z2R2ss)6− 1. (1.43)281.5. The planetary magnetic fieldWe can gain some intuition about the shape by just choosing a fewdifferent values for x and z in equation (1.43). At x = Rss and z = 0, dxdz = 0showing that all of ~dl is in the zˆ direction as expected. For large negative x,when Rmp > Rss we find that (Rmp/Rss)  1 such that dxdz ≈ (Rmp/Rss)6.This implies that dx  dz and thus the magnetopause shape will becomecylindrical past 2 Rss down the magnetotail. In reality, the magnetopausetail can be more flared because the magnetic field in the tail is not dipolar(due to the stretched-out nature of the tail), while our assumption is basedon a Newtonian approximation and a dipole field. In the far tail, the currentsthat form the tail become the dominant source of the magnetic field, notthe dipole which is too far away to have the largest affect. From thesephysical arguments we thus expect Mercury’s magnetopause to be curved(approximately hemispherical) on the dayside and approximately cylindricalon the nightside. In Chapter 2 we fit empirically motivated shape modelsto Mercury’s magnetopause to establish its best-fit shape and location fromnearly a year of observations.1.5 The planetary magnetic fieldThere are four terrestrial planets in our solar system, but only Earthand Mercury have a currently active global dynamo field (although all theouter planets do possess dynamo-generated magnetic fields). In such dy-namo operated magnetic fields, electric currents and magnetic fields arecontinuously induced by the movement of conducting fluid in the planet’sinterior. Three basic ingredients are needed for self-sustaining dynamo ac-tion to take place: a large volume of electrically conducting fluid material,planetary rotation, and a strong enough energy source to drive convectivemotions. These characteristics are in part set during planetary formation,but each one can change during the planet’s lifetime. The purpose of thissection is to introduce the basic idea behind planetary dynamos but not todelve into the field of dynamo theory, as this thesis is concerned with theresults of planetary dynamos, namely the planetary magnetic field and itsinteraction with the solar wind.291.5. The planetary magnetic fieldLarmor [43] first suggested that a dynamo process is responsible forcreating and maintaining planetary magnetic fields. His hypothesis was thatan initial magnetic field can be altered via interactions through fluid flow toamplify and regenerate the magnetic field. In 1955 Parker [44] presented thefirst modern model of the dynamo process, known as the αω model. In thismodel (which is still relevant today), the differential rotation of the fluidouter and solid inner core can generate toroidal (azimuthal) magnetic fieldsby stretching poloidal (non-azimuthal) magnetic fields threaded through thecore (the ω effect). In order to maintain this generation of the toroidal field,the poloidal field must be generated through another process. The α effectaccomplishes this task by relying on turbulent fluid convection to generatepoloidal fields from the toroidal field, which closes the feedback loop.The turbulent fluid motions and the planetary rotation act together tostretch out the magnetic field (which is “frozen in” to the core fluid). Dueto the planetary rotation this generates magnetic field loops with approxi-mately the same orientation and can thus give rise to a coherent global field.In such a way the dynamo process can generate and sustain magnetic fieldsof planetary bodies. Many variations on this basic dynamo model have beenproposed, most of which require both planetary rotation and turbulent con-vection in a fluid core. Thus the fact that Mercury has a global dipole fieldis an indication (independent of the evidence from forced librations [4]) thatthe planet possess a liquid (or partially liquid) core in its interior.Several dynamo models have been suggested to explain Mercury’s un-usually weak, equatorially offset, and axisymmetric magnetic field, althoughnone have yet been able to closely match all aspects of the observed field.As described earlier, models with a stable layer at the top of the outercore [20, 21] show promise. Furthermore, a study currently in progress [45]has found that fields that are asymmetric about the geographic equatorcan be produced in dynamos driven by volumetrically-distributed buoyancysources. The fact that Mercury’s magnetopause lies very close to the sur-face and the large inferred size of the core (radius of 2020 km [46]), suggeststhat there may be a strong link between the internal dynamo generated fieldand the external fields of the magnetosphere. A feedback-dynamo model (in301.6. Relevant MESSENGER instrumentswhich the external fields of the magnetosphere penetrate to the core and actto weaken the dynamo generated field) [47] has been suggested for Mercury.However, this model predicts high odd harmonics in a spherical harmonicexpansion of the field, which are not inferred from MESSENGER observa-tions [18]. This does not however rule out the possibility of induction inMercury’s core, which may act to stiffen the dayside magnetopause againstchanges in solar wind pressure [48]. Better resolution of the magnetic fieldstrength at the surface of the planet can provide further constraints for dy-namo models of Mercury and this is one of the underlying motivations forChapter 5 of this thesis.1.6 Relevant MESSENGER instrumentsIn this section we summarise the details of the Magnetometer (MAG)instrument onboard MESSENGER, from which data were used in all chap-ters of this thesis, as well as the Fast Imaging Plasma Spectrometer (FIPS),data from which were used in Chapters 4 and 5. For a thorough descriptionof MAG the reader is referred to [49], and for more details on FIPS thereader is referred to [50].1.6.1 The MagnetometerThe MESSENGER Magnetometer is a tri-axial fluxgate magnetometermounted on a 3.6-m-long boom. The MESSENGER fluxgate magnetometerconsists of a permeable ring core driven in alternative saturation states bya toroidal winding at a frequency of 15 kHz. The ring core and winding aresurrounded by a second set of pickup windings that sense any change in netmagnetic flux. If there is a net field along the axis of the pickup winding coilthen when the ring core switches saturation polarities there will be a briefpulse of net flux in the pickup winding since one side of the ring core willtransition polarity before the other side. The sense of this net flux pulse is inthe direction of the background field and is the same sense every half cycle ofthe 15 kHz drive. Synchronous detection at 30 kHz in the pickup winding is311.6. Relevant MESSENGER instrumentsused to monitor any net flux pulses and a bias current is sent to the pickupcoil to null the fluxgate response. The voltage applied to drive this nullingcurrent is directly proportional to the component of the ambient magneticfield along the axis of the pickup coil. In a tri-axial fluxgate magnetometer,three such sensors are mounted in such a way as to measure the three vectorcomponents of the magnetic field. MAG outputs three analog signals whichare converted simultaneously to digital signals by three independent 20-bitsigma-delta A/D converters at a rate of 20 conversions per second. Prior toconversion these signals are low-pass filtered using a 10-Hz anti-alias low-pass filter to limit the input bandwidth into the A/D converter.MAG has a range of ±1530 nT full scale with a 0.047 nT resolution dur-ing Mercury operation. In order to minimize variable and static spacecraft-generated fields at the sensor, the MAG instrument underwent a strict mag-netic cleanliness program prior to launch. Observations after boom deploy-ment indicated that there is a fixed residual field, however, it is less than afew nT at the location of the magnetometer. A variable contamination fieldmeasured at the magnetometer is below 0.05 nT.To meet the science objectives of the MESSENGER mission as well asto accommodate variable telemetry rates, MAG can be operated at variablesample rates (from 0.01 s−1 to 20 s−1) adapted to various mission phases. Acombination of digital filtering and sub-sampling is used to provide the widerange of output rates. MAG sampling inside the magnetosphere is at thehighest rate (20 Hz) in order to resolve highly varying magnetospheric fields,while sample rates in the interplanetary medium are at least as high as 2samples/s. In addition to the vector magnetic field samples, the averageoutput amplitude of a 1-10 Hz bandpass filter is also evaluated for the zmagnetometer axis, although this can be commanded to be any of the threeaxes. This is a measure of the average 1-10 Hz ambient field fluctuations,called BAC (used in Chapters 2 and 3 of this thesis), which is made regardlessof the sample rate of the vector data in order to provide an uninterruptedmeasure of the field variability. The BAC measure is also used to triggerhigh-time-resolution sampling in 8-min segments to capture events of interestwhen continuous high-rate sampling is not possible.321.6. Relevant MESSENGER instruments1.6.2 The Fast Imaging Plasma SpectrometerFIPS is part of the Energetic Particle and Plasma Spectrometer (EPPS)package on MESSENGER, and it measures the energy, angular, and compo-sitional distribution of low-energy ions (< 50 eV/charge to 20 keV/charge)in Mercury’s space environment. The sensor was specifically developed anddesigned for MESSENGER. It is highly compact and lightweight, and housesa new electrostatic analyzer system geometry that enables it to have 1.4pisteradians field of view (FOV). FIPS is a time-of-flight mass spectrometer,in which an ion’s mass-to-charge ratio is determined through a time mea-surement.FIPS is made up of an electrostatic analyzer (ESA), a time of flightdetector, and a compartment that houses the instrument electronics (seeFigure 1.8 for a cross-sectional view of the FIPS sensor). The electrostaticanalyzer uses an electric field to filter and focus the particles that are al-lowed to pass through to the time of flight detector chamber: only particlesof a specific energy-per-charge range pass through the analyzer. The ESAcovers the energy-per-charge range in 64 logarithmically spaced steps. FIPSis normally operated in one of two stepping rates, one step per second (nor-mal mode) or one step per 100 milliseconds (burst mode), which result inintegration times of 64 s and 8 s, respectively. Inside the magnetosphereFIPS is operated in burst mode and measures energy per charge from 0.1to 13 keV/e [51].Inside the ESA, the entering ion’s trajectory is bent by the deflectorplates in the analyzer, after which the collimator filters out particles withcertain trajectories. Next, the ion is post-accelerated by a potential dropof up to −15 kV to give low energy ions sufficient energy to penetrate thecarbon foil. The ion then enters the time of flight detector and impacts thecarbon foil, which causes the foil to eject secondary electrons. The electronsmove ahead of the ion and their path will be bent to ensure that they collidewith the start microchannel plate (MCP) detector, thereby recording theion’s position and providing a timing-start signal. On the other hand, the ionwill pass straight through the time of flight chamber, hitting the stop MCP331.6. Relevant MESSENGER instrumentsFigure 1.8: Cross-section of the FIPS sensor showing major functional com-ponents. Figure from Andrews et al. (2012), Space Science Reviews, 131,523-556 [50]. Reprinted with permission from Springer Science + BusinessMedia.detector, and thereby stopping the timer. The difference in time betweenthe electrons and ion hitting the two detectors is used to calculate the initialion speed. This can be obtained after a correction is made for the travel timeof the secondary electrons from the foil to the start detector. The mass-per-charge follows from the known energy-per-charge and the measured timeof flight. This allows for the reconstruction of the distribution functionsof different mass-per-charge species. The microchannel plate detectors areelectron multipliers, which when struck by a particle, start a cascade ofelectrons that propagate through the microchannels amplifying the originalsignal of the particle by several orders of magnitude. These electrons exitthe channels and are detected by measuring the total current on a singlemetal anode.The angular resolution of FIPS, in terms of determining the incidentdirection of the ion is 15◦. It has a nearly hemispherical (1.4pi sr) instanta-neous FOV. The FOV has conical symmetry about the z-axis of the sensor’sreference frame and is defined as the region of solid angle bounded by two341.6. Relevant MESSENGER instrumentsFigure 1.9: Top view of a portion of MESSENGER which depicts the ob-structions in the solar direction by the spacecraft sunshade and the FIPSFOV. The figure (from Gershman et al. (2012), Journal of GeophysicalResearch, 117, A00M02 [52]) also shows that FIPS cannot observe the so-lar wind, however, this is not a focus of this thesis. Figure reprinted withpermission from John Wiley and Sons.nested cones. Both of these cones have their vertices at the origin of theFIPS coordinate system and have angles with respect to the z-axis of 15◦and 75◦, respectively. The FOV is the region contained within the 75◦ coneexcluding the region contained within the inner 15◦ cone. This excludedregion shows up as a gap in Figure 1.9 in the centre of the unobstructedFOV of FIPS. Such a constrained field of view places limitations on the re-construction of the 3–D ion distribution from this sensor, as will be shownin Chapter 4.35Chapter 2Mercury’s magnetopause andbow shock fromMESSENGERMagnetometer observations1In this chapter, we establish the average shape and location of Mercury’smagnetopause and bow shock from orbital observations by the MESSEN-GER Magnetometer [53]. We fit empirical models to midpoints of boundarycrossings and probability density maps of the magnetopause and bow shockpositions. The magnetopause was fit by a surface for which the position rfrom the planetary dipole varies as (1 + cos θ)−α, where θ is the angle be-tween r and the dipole-Sun line, the subsolar stand-off distance Rss is 1.45RM (where RM is Mercury’s radius), and the flaring parameter α = 0.5. Theaverage magnetopause shape and location were determined under a meansolar wind ram pressure, PRam, of 14.3 nPa. The best-fit bow shock shapeestablished under an average Alfve´n Mach number (MA) of 6.6 is describedby a hyperboloid having Rss = 1.96 RM and an eccentricity of 1.02. Theseboundaries move as PRam and MA vary, but their shapes remain unchanged.The magnetopause Rss varies from 1.55 to 1.35 RM for PRam in the range8.8 to 21.6 nPa. The bow shock Rss varies from 2.29 to 1.89 RM for MAin the range 4.12 to 11.8. The boundaries are well approximated by figuresof revolution. Additional quantifiable effects of the interplanetary magnetic1Reprinted from Journal of Geophysical Research: Space Physics, with permission fromWiley.362.1. Introductionfield are masked by the large dynamic variability of these boundaries. Themagnetotail surface is nearly cylindrical, with a radius of ∼ 2.7 RM at adistance 3 RM downstream of Mercury. By comparison, Earth’s magneto-tail flaring continues until a downstream distance of ∼ 10 Rss. The modeledcylindrical shape of Mercury’s magnetopause suggests that magnetic fluxhas a short residence time in the tail, and return convection of flux fromthe tail to the dayside proceeds rapidly, consistent with the expected ∼ 2minute Dungey cycle period at Mercury.2.1 IntroductionThe boundaries of Mercury’s magnetosphere reflect fundamental pro-cesses of the solar wind interaction with the planet’s dipolar magnetic field [24,54]. These boundaries are the bow shock, across which the solar wind is com-pressed and deflected around Mercury, and the magnetopause, which is thecurrent layer separating the shocked solar wind plasma and interplanetarymagnetic field (IMF) from the planetary magnetic field. The bow shock isa fast magnetosonic shock wave that “stands” in the solar wind while di-verting the solar wind around the planet’s magnetospheric cavity [39]. Thebow shock changes shape and stands closer or farther from the planet in re-sponse to variations in solar wind Mach number and, to a lesser extent, IMFdirection [26, 39, 55]. The magnetopause location and shape are determinedprincipally by the pressure exerted on the magnetopause by the shockedsolar wind plasma, which scales with the solar wind ram pressure, balancedby the planetary magnetic field [39, 56]. Accordingly, the dynamic pressureof the solar wind and the magnetic pressure of the magnetosphere are thedominant factors determining the location and shape of the magnetopause.Because the distribution of magnetic flux within a magnetosphere is de-termined both by the intrinsic planetary field and the external currents,magnetic reconnection, which drives some of these external currents, alsoaffects magnetopause shape and position. Most important among thesereconnection-driven effects are the inward “erosion” of the dayside mag-netopause to lower altitudes by transfer of magnetic flux to the tail [57]372.1. Introductionand the outward “flaring” of the nightside magnetopause as the magneto-tail is loaded with magnetic flux [58]. These expansions and contractions ofthe dayside and nightside magnetosphere take place on timescales compa-rable to the Dungey cycle that governs the circulation of magnetic flux andplasma within the magnetospheres of Earth-like planets. The characteristictimescale of the Dungey cycle is ∼ 1 h at Earth and ∼ 2 min at Mercury [28].It has been suggested that reconnection may have a greater effect onmagnetopause location at Mercury than at Earth [57]. This prediction hasbeen supported by the extreme loading and unloading of Mercury’s mag-netotail observed during the third flyby of Mercury by the MESSENGERspacecraft [29, 59]. Further, it has also been predicted that magnetic fieldsassociated with induction currents in Mercury’s interior may act to opposeand limit the solar wind’s ability to compress or expand the extent of thedayside magnetosphere [60–63]. Mercury has an internally generated, axiallyaligned dipolar magnetic field with a moment of 190 to 195 nT R3M (whereRM is Mercury’s mean radius, 2440 km) that is offset northward from thegeographic equator by 0.2 RM [17–19]. The combination of a weak dipolemoment, the absence of a conducting ionosphere (i.e., no magnetosphere-ionosphere coupling), and the predominantly quasi-parallel subsolar shockconditions (due to the small Parker spiral angle at Mercury’s orbit) distin-guish Mercury from other planets in our solar system that possess magneticfields.The Earth’s bow shock has been studied extensively both observationallyand theoretically (e.g., [55]). It is a highly dynamic boundary, controlledby temporal variations in solar wind characteristics. The general shapeof the shock has been investigated with empirical models (e.g., [26, 64]),gas dynamic flow models (e.g., [65, 66]), and magnetohydrodynamic models(e.g., [67]) and is well described by a conic section. Formisano et al. [68]found that the subsolar bow shock position moves outward during conditionsof low Alfve´n Mach number (MA). Peredo et al. [55] confirmed that MAprimarily controls the bow shock shape, but in contrast to the findings of theearlier study [68], they observed that the subsolar shock moves earthwardand the flanks flare outward during times of low MA.382.1. IntroductionThe average shape of the terrestrial magnetopause has been describedwith several empirical models (e.g., [69–71]) and magnetohydrodynamicmodels [72]. The dynamic pressure of the solar wind affects the size ofthe magnetospheric cavity as well as the shape of the magnetopause. Thesubsolar magnetopause stand-off distance is observed to decrease with in-creased dynamic pressure [73, 74], and the boundary shape is observed toflare with increased pressure [69]. Reconnection-related effects can also in-fluence the dayside magnetopause location and the flaring of the tail. Onthe dayside, a southward IMF component will facilitate reconnection at lowlatitudes, which can erode the magnetopause near the subsolar point andflare the magnetopause on the nightside by adding flux to the tail. Themagnetopause has been observed to move planetward by as much as 1-2RE (where RE is Earth’s radius) due to the erosion of the boundary undersouthward IMF [57, 73, 75]. However, a statistical study of these effects atMercury with a large database of magnetopause crossings, such as the onereported here, would require knowledge of IMF direction on timescales of 1min or less because of the very short Dungey cycle time at Mercury. Thelack of an upstream monitor therefore limits the scope of our investigationwith respect to the effects of reconnection on magnetopause position.The solar wind and IMF play major roles in influencing the bow shockand magnetopause, and those roles must be understood quantitatively inorder to model the internal magnetic field, the magnetosphere-solar windinteraction, and the access of solar wind and magnetospheric charged parti-cles to the planetary surface. At Mercury the solar wind density is approxi-mately an order of magnitude higher and the IMF magnitude a factor of 5higher than at Earth [23], whereas the planetary magnetic moment is only0.06% of Earth’s. Mercury’s orbit is eccentric, so the planet is subjectedto different solar wind conditions at perihelion and aphelion. Mercury’smagnetopause and bow shock have been studied from limited data obtainedduring flybys by the Mariner 10 [76] and MESSENGER spacecraft [77].Russell [76] fit shape models to both the magnetopause and bow shock withdata from the two Mariner 10 flybys, and Slavin et al. [77] updated theseboundary shapes with MESSENGER flyby data. Slavin et al. [77] also392.2. Magnetic field observations: boundary identificationslooked at the boundary crossings during northward IMF from data takenduring the first MESSENGER flyby. However, even the combined Mariner10 and MESSENGER flyby dataset comprised only six pairs of bow shockand magnetopause crossing points.In this chapter we present analyses of Mercury’s bow shock and mag-netopause shape obtained from orbital magnetic field observations by theMESSENGER spacecraft over a span of three Mercury years under a va-riety of IMF and solar wind conditions. The objective of this study is tocharacterize the underlying shapes of both the magnetopause and bow shockand, insofar as the data permit, to assess how these shapes are affected bythe solar wind and IMF. We have analyzed the observations in two ways.First, from the locations of the inner and outer magnetopause and bowshock crossings we defined a mean crossing point, and we found the averageboundary shape from the ensemble of crossing points. In the second method,the probability of spacecraft residence within the range of magnetopause orbow shock crossings on each pass has been used to build a probability den-sity map of the two boundaries. In Section 2.2, we describe the magneticfield observations and how the boundaries were identified. In Section 2.3,we establish the general shape of both the magnetopause and the bow shockfrom the mean locations (Section 2.3.1) and the probability densities (Sec-tion 2.3.2). In Section 2.4, we assess how the boundaries respond to solarwind forcing, and the results and conclusions are given in Section 2.5 and2.6, respectively.2.2 Magnetic field observations: boundaryidentificationsThe MESSENGER spacecraft was inserted into orbit about Mercury on18 March 2011. The initial orbit had a 200 km periapsis altitude, 82.5◦ in-clination, 15,300 km apoapsis altitude, and 12 h period. During each orbit,MESSENGER typically spent 1 to 2 h inside the magnetosphere; the rest ofthe time was spent in the magnetosheath and in the interplanetary medium.402.2. Magnetic field observations: boundary identificationsFor this study we used three Mercury years of MESSENGER Magnetome-ter (MAG) [49] data starting on 23 March 2011 and extending through 19December 2011, providing repeated coverage at all local times. To conservedata volume during this period, MAG operated at variable sample rates,with high-rate (20 samples/s) data collection primarily in effect during tran-sits of the magnetosphere. Sample rates in the interplanetary medium wereat least as high as 2 samples/s or higher, and a channel to record fluctua-tions at 1-10 Hz operated continuously to provide an uninterrupted measureof the field variability.Magnetic field data were analyzed in Mercury solar orbital (MSO) co-ordinates. In MSO coordinates xMSO is positive sunward, zMSO is positivenorthward, yMSO is positive duskward and completes the right-handed sys-tem, and the origin is at the center of the planet. To analyze boundarylocations the spacecraft positions were translated into a system centered onthe planetary dipole [18]. The vector components of the magnetic field inthe planet-centered and dipole-centered systems are identical. Because thebow shock and magnetopause are ordered by the solar wind flow in the frameof Mercury’s orbital motion, the spacecraft position data were transformedinto an aberrated system such that the +x direction is anti-parallel to thesolar wind velocity relative to Mercury. The average aberration angle atMercury is about 7◦ toward dawn. However, because of Mercury’s variationin orbital speed between perihelion and aphelion, as well as variability in thesolar wind speed, the aberration angle varied by a factor of about 3, from3.5◦ to 10.2◦, during the period of our study. The aberration correction forboth the magnetopause and the bow shock crossings was calculated fromthe planet’s instantaneous orbital speed together with predictions of solarwind speed obtained from the WSA-Enlil heliospheric model [42] within fourminutes of the times of the boundary crossings. Because the Fast ImagingPlasma Spectrometer (FIPS) on MESSENGER does not typically see thesolar wind, we do not have in-situ estimates of solar wind properties, sowe use WSA-Enlil model predictions of solar wind parameters in this study.Use of model predictions for the solar wind parameters introduces some levelof uncertainty into the normalized boundary shapes derived in Section 2.4412.2. Magnetic field observations: boundary identificationsof the paper, with the highest uncertainty most likely introduced for thecases of extreme events, when the solar wind pressure is predicted to behighest. Benchmarking of the WSA-Enlil model at Mercury has been initi-ated [52, 78]; however, the long-timescale (> 44 day) variations in the modeloutputs that are modulated by Mercury’s orbital distances from the Sun arepersistent over the three Mercury years of data analyzed here. Our aimhere is not to use WSA-Enlil for event studies, but to capture the averageannual variation in the solar wind ram pressure and density and use thesevariations to correct our average boundary shapes.The IMF magnitude assigned to each crossing was evaluated as a 1 haverage of MAG data upstream of the outermost bow shock encounter. TheAlfve´n Mach number, MA = vsw[BIMF/(µ0ρ)0.5] , where vsw is the solar windspeed, BIMF is the magnetic field magnitude in the IMF, µo is the perme-ability of free space, and ρ is the solar wind plasma density, was calculatedusing IMF values that were estimated from the 1 h IMF averages and WSA-Enlil model-generated solar wind speed and density. We have shown previ-ously that the 1 h IMF averages are suitable for determining the IMF Bxdirection, which is the dominant IMF component at Mercury, but they arenot suitable for establishing the IMF By and Bz directions, which vary ontimescales less than 1 h [Winslow et al., 2012]. The magnitude of the IMFis dominated by the Bx component and is thus also steady on the 1 h timescale, i.e., the average duration of a transit of the magnetosphere by theMESSENGER spacecraft. The resolution of these measurements is suitablefor investigating the effect of solar wind pressure and Alfve´n Mach numberon magnetopause and bow shock positions, but not the reconnection-driveneffects that depend on IMF orientation and take place on timescales of oneminute to a few minutes [28, 29].Magnetopause and bow shock crossings were identified on every orbit,both before and after the magnetospheric transit and denoted as the inboundand outbound crossings, respectively. On almost every orbit, multiple cross-ings of each boundary were observed as a result of motion of the boundaryrelative to the spacecraft. Although the repeated crossings were often diffi-cult to distinguish individually, the first and last boundary encounters were422.2. Magnetic field observations: boundary identificationsreadily identified. Thus rather than attempt uncertain identifications of ev-ery boundary crossing within each passage through the boundary region, weidentified the times of the innermost and outermost crossing on each passfor the bow shock and magnetopause. Our approach has the added benefitthat each pass corresponds to an independent sample of external IMF andsolar wind conditions, whereas statistical analyses that count every crossingequally in passes with multiple crossings will overweight such passes. Weaim to provide demarcations for the inner and outer limits of the boundaries,within and outside of which the spacecraft was clearly located in the magne-tosphere, magnetosheath, or interplanetary medium. Wave characteristics(including foreshock waves, non-linear quasi-parallel shock phenomena, andmagnetopause boundary waves) are beyond the scope of this paper.All boundary crossings were picked by visual inspection with the follow-ing criteria. For the bow shock, the inbound outer limit was identified as thetime at which the first sharp increase in the magnitude |B| of the magneticfield was observed. The inner limit was identified as the time of the last sharpincrease in |B|. These criteria worked well when the IMF was oriented some-what oblique to the planet-Sun line, that is, for quasi-perpendicular shockconditions. A perpendicular shock forms when the shock-normal direction isperpendicular to the IMF direction, whereas a parallel shock occurs when theshock normal is parallel or anti-parallel to the IMF. For near-parallel shockconditions, there was often little or no increase in |B|, but the bow shockboundary was marked by the onset of large variability in |B|. Sometimesthese modulations grew gradually, in which case we chose the outermostexcursions in |B| that were distinctly larger than the upstream variability.The outbound bow shock was picked in a similar fashion, that is, a sharpdecrease in the field magnitude marked the boundary.For the magnetopause, the crossings were most readily identified whenthe shear angle between the direction of the magnetic field in the magneto-sphere and that in the magnetosheath was larger than about 45◦, becausethe field rotation is a direct signature of the magnetopause current layer. Onthe dayside and on the flanks, the shear is typically in By and Bz, whereason the nightside the shear is mostly in Bx. The inner boundary was iden-432.2. Magnetic field observations: boundary identificationstified by the innermost substantial rotation away from the magnetosphericfield direction and the outer boundary by the outermost rotation towardthe ambient magnetospheric field direction, excluding the background mag-netosheath variability. In many cases, however, the field direction in themagnetosheath was the same as (or close to) that inside the magnetosphere.For example, the magnetic shear can be low at low dayside latitudes whenthe IMF is northward and on the nightside when the IMF is anti-sunwardsuch that the southern lobe field is parallel to the draped magnetosheathfield. In such cases, although the local magnetic shear is low, there wereother signatures in the magnetic field that indicated transition betweenmagnetosheath and magnetosphere regimes. These signatures include anincrease in magnetic fluctuations in the magnetopause layer (documentedfrom Earth’s magnetopause [79, 80]) and a change in the character of thelow-frequency fluctuations on the magnetosheath side of the boundary. Thehigher-frequency magnetic fluctuations are recorded by the 1-10 Hz fluctu-ation channel (or BAC), which provides an average amplitude of the 1-10Hz bandpass-filtered field. In addition, on the dayside the inner magne-topause boundary is also often indicated by either a step-wise increase in|B| or the onset of an inward gradient in |B| on the magnetosphere sideof the boundary. However, these signatures were not always sharp, and insome cases it was difficult to identify the magnetopause boundary. Bound-ary crossing choices were made conservatively such that the inner edge wasdefinitely inside the magnetosphere and the outer edge was definitely inthe magnetosheath. Using plasma measurements from FIPS to identify theboundary crossings is beyond the scope of this paper and would requirecareful accounting of the FIPS look direction, since its field of view is 1.4pisr. Several dozen comparisons show excellent correspondence between theboundaries identified from MAG observations and abrupt changes in FIPSproton count rates.Data from the first magnetospheric transit on 12 October 2011 (orbit418) are shown in Figure 2.1 together with expanded views of the inboundand outbound boundary passages. In this case, the shock conditions wereoblique (perpendicular shock geometry) and there was high magnetic shear442.2. Magnetic field observations: boundary identificationsacross the magnetopause on both the inbound and outbound crossings. Thespacecraft entered the magnetosphere from the dayside and exited on thenightside. The spacecraft first grazed the shock at 3:43:59 CoordinatedUniversal Time (UTC) before passing through it at 3:45:29 UTC. On thistransit, the shock was also marked by an increase in high-frequency fluc-tuations, BAC, although most of the fluctuations before the shock crossingare attributed to foreshock waves. The spacecraft then traveled through themagnetosheath and encountered the high-shear magnetopause on the dawnside, marked by a rotation in Bz and By and an increase in BAC shortlybefore the magnetopause boundary. On the outbound portion of the orbit,the spacecraft first encountered the magnetopause at 5:19:56 UTC and wasfinally in the magnetosheath at 5:25:05 UTC. The rotation in the magneticfield is evident in Bx as the spacecraft exited the magnetosphere from thesouthern tail lobe and was also associated with a rise in BAC. The spacecraftthen crossed the bow shock twice on its path back into the interplanetarymedium.Often the boundaries were less clear, and Figure 2.2 shows data from thefirst magnetospheric transit on 5 July 2011 (orbit 218) with such crossings.On this orbit the shock was quasi-parallel on the dayside, which caused thelarge modulations in |B| near the bow shock crossing. The magnetic fieldinside the magnetosheath was highly variable, with large, quasi-periodicrotations in By and Bz up to the magnetopause. The magnetic shear acrossthe magnetopause was low, so the decrease in BAC and the increase in thetotal field magnitude were taken to indicate the magnetopause crossing. Onthe outbound part of the orbit, the magnetic shear was again low across themagnetopause, with slight rotations visible in By and Bx, but the crossingwas indicated by an increase in BAC. The field magnitude decreased as thespacecraft crossed the magnetopause but increased inside the magnetosheathuntil it reached the oblique bow shock boundary and decreased abruptly at7:45:54 UTC.The times of the inner and outer edges of the recorded magnetopauseand bow shock crossings for all the data presented in this study are given inthe Supplemental Materials of [53]. There are 1,065 magnetopause and 1,084452.2. Magnetic field observations: boundary identificationsFigure 2.1: (a) MESSENGER Magnetometer data for the first magneto-spheric transit on 12 October 2011 (orbit 418). Left axes give the scales forBx (red), By (light green), Bz (blue), and |B| and -|B| (black); the right axisis the scale for BAC (dark green). Vertical lines denote the crossing times ofthe inner and outer edges of the bow shock (dashed) and magnetopause (dot-dashed). (b) Close-up view of the inbound portion of the orbit. (c) Close-upview of the outbound portion of the orbit.bow shock crossings altogether in our dataset. The number of bow shockcrossings exceeds the number of magnetopause crossings because the Mag-netometer was switched off to conserve power for parts of 19 orbits during462.2. Magnetic field observations: boundary identificationsFigure 2.2: (a-c) MESSENGER Magnetometer data for the first magneto-spheric transit on 5 July 2011, orbit 218, in the same format as in Figure2.1.MESSENGER’s first long-eclipse season in orbit around Mercury. On theseorbits both the inbound and outbound bow shock crossings were recorded,but typically only the inbound (and not the outbound) magnetopause cross-ings were observed.472.3. Average boundary shapes2.3 Average boundary shapesThe first step in analyzing the crossing data was to determine the bestboundary shapes for all crossings together, effectively averaging over IMFand solar wind conditions. This averaging was accomplished by fitting em-pirical models to the magnetopause and bow shock crossing locations. Theboundary locations were specified with two different techniques. First, weused the mean locations of the boundaries and fit empirical shapes to thesedirectly. Second, we used a probabilistic measure of residence within theboundary regions to identify the locations of maximum residence probabil-ity. Empirical shapes were then fit to these probability density maps, withthe established probabilities used as weights in the fitting. In the analysisthat follows, all positions are in aberrated coordinates (x, y, z), where x andy are the aberrated xMSO and yMSO coordinates, respectively, and z = zMSO.In addition, we have assumed that the boundaries are figures of revolutionabout the line through the dipole center that parallels the x axis; the valid-ity of this assumption is quantitatively tested and confirmed in Section 2.5.The northward offset of the planetary dipole is included in the definitionof the distance from the axis of revolution, given by ρ =√y2 + (z − zd)2,where zd = 0.196 RM [17–19].2.3.1 Midpoint fitsIn the first approach to determining boundary locations, model curveswere fit to the average crossing points, that is, the midpoint between theinner and outer edge of the boundary location on each pass. The inner andouter limits of the boundaries were assigned as the uncertainty range. Thismethod allows direct comparison of our results with approaches that havebeen used historically to determine boundary shapes (e.g., [69, 77]).2.3.1.1 MagnetopauseFor the magnetopause, we used a paraboloid conic section [81, 82] aswell as the model shape proposed by Shue et al. [69] to fit our crossings.482.3. Average boundary shapesï7 ï6 ï5 ï4 ï3 ï2 ï1 0 1 2ï1012345X (RM)l (RM)Figure 2.3: Midpoints between the inner and outer magnetopause crossingpositions identified from MESSENGER Magnetometer data from 23 March2011 through 19 December 2011. Error bars show the distance between theinner and outer crossing. Curves show the best-fit paraboloid (blue) to thedayside crossings and the best-fit Shue et al. model (red), as well as modelsfrom previous studies by Slavin et al. [77] (green) and Russell [76] (yellow).The paraboloid has parameters given by Rss = 1.5 RM and γ = 1, whereasthe Shue et al. model is given by Rss = 1.45 RM and α = 0.5.Figure 2.3 shows the midpoints of the magnetopause crossings from the threeMercury years of data analyzed in this study. To establish a time-averagedmagnetopause shape from the crossing points, we modeled them in ρ − xspace. In our boundary fits, we used a grid search method that minimizedthe root-mean-square (RMS) residual of the perpendicular distance of theobserved midpoints from the model boundary.The paraboloid fit is motivated by the magnetospheric model of Alexeevet al. [82], which was derived with a parabolic parameterization of mag-netopause shape. Past studies of the magnetopause shapes around other492.3. Average boundary shapesplanets have also involved fits to conic sections (e.g., [76]). The paraboloidmodel shape is described byx(ρ) = −(γ2 + 14Rss)ρ2 +Rss (2.1)where γ is a flaring parameter and Rss is the subsolar magnetopausedistance [81]. Any value of γ > 1 is physically unreasonable, because it givesa subsolar stand-off distance that is not the minimum distance between themagnetopause and the planet. Setting γ = 1, we find Rss = 1.25 RM as thebest-fit paraboloid. This model does not provide a good visual fit, however,to data either on the dayside or in the distant tail region. Relaxing theconstraint of γ > 1 gives a better fit to the crossings on the nightside, butthe resulting model still does not fit the dayside points. We find that theparaboloid model represents the dayside magnetopause shape best when weexclude the tail crossings, yielding Rss = 1.5 RM and γ = 1 (Figure 2.3).We also fit the magnetopause crossings with the functional form proposedby Shue et al. [69] (hereafter referred to as the Shue et al. model) and givenbyR =√x2 + ρ2 = Rss(21 + cos θ)α(2.2)where R is the distance from the dipole center, θ = tan−1( ρx), and αis another flaring parameter that governs whether the magnetotail is closed(α < 0.5) or open (α ≥ 0.5). This model has been used successfully tomodel the Earth’s magnetopause [69, 83], as well as the magnetopause ofother planets, such as Saturn [84]. The Shue et al. model that best fits ourmidpoint magnetopause crossings yields parameter values of Rss = 1.45 RMand α = 0.5 (shown in Figure 2.3). As can be seen from Figure 2.3, the Shueet al. model provides a better representation of the magnetopause crossingsthan the paraboloid model. Even for the Shue et al. model, the best-fitparameters are not tightly constrained, as similar RMS values are achievedover a range of values for Rss and α (Figure 2.4). This behavior is in partbecause of a trade-off between the Rss and α parameters, imposed by the502.3. Average boundary shapes_Rss (R M)  0 0.2 0.4 0.6 0.8 11.11.21.31.41.51.61.71.81.92 (km)5001000150020002500300035004000Figure 2.4: RMS misfit between the midpoints of the magnetopause crossingsand the Shue et al. model, as a function of the subsolar stand-off distance,Rss, and the flaring parameter, α.observation geometry, and in part because of the spread in the magnetopausecrossing positions most likely caused by the dynamics of variable solar windand IMF conditions.To better constrain our best-fit parameters, we conducted analyses ofthe residuals (the perpendicular distance of our crossings from the modelboundary) for a range of models with Rss − α parameter pairs from theminimum misfit region in Figure 2.4. We find that as we depart from thebest-fit parameters on either side of the misfit well, the residuals have dis-tributions with a non-zero mean and are not Gaussian. Thus even thoughthe RMS misfit is not very different from the absolute minimum value in theminimum misfit region, the models generated by the parameter pairs in thatregion describe the data less well than our best-fit model. In addition, mod-els have residuals that vary systematically with xMSO. This situation can bevisualized by taking, for example, a model curve lying within the minimummisfit region with Rss = 1.25 RM and α = 0.6. Such a boundary yields512.3. Average boundary shapesresiduals that are systematically positive on the dayside (i.e., the boundaryis too close to the planet relative to the observations) and are systematicallynegative on the nightside (i.e., the boundary is too flared and farther fromthe planet on average than the data). On the other hand, the residuals ofthe best-fit model with Rss = 1.45 RM and α = 0.5 have zero mean and aGaussian distribution and show no systematic variation with xMSO. Also,the RMS misfit (∼ 590 km) for the best-fit Shue et al. model is a factor of1.5 lower than that of the best-fit paraboloid model (which was fit only tothe dayside data), and thus we use only the Shue et al. curve to model themagnetopause for the remainder of this discussion.2.3.1.2 Bow shockTo characterize the shape of the bow shock, the midpoints of the bowshock crossings were modeled by a conic section given by [77]:√(x− x0)2 + ρ2 =p1 +  · cos θ(2.3)where the focus of the conic section lies at x0 along the line throughthe planetary dipole that parallels the x axis at x0. The focus point, x0,the eccentricity, , and the focal parameter, p (which together with theeccentricity gives the semi-latus rectum, L = p), are determined by a gridsearch method that minimizes the RMS misfit. The best-fit parameters tothe bow shock midpoints are given by x0 = 0.5 RM ,  = 1.04, and p = 2.75RM (Figure 2.5). As mentioned above, bow shock identification is difficultfor parallel bow shock conditions, and the outliers in bow shock locationsare due to the conservative outer limits chosen for crossings during theseconditions. Our best-fit model has an RMS misfit of ∼ 1100 km betweenmodel boundary and bow shock position. The extrapolated nose distance forthis best-fit model is 1.90 RM , which yields an approximate magnetosheaththickness of 0.45 RM from our midpoint magnetopause and bow shock fits.This magnetosheath thickness is comparable to that predicted at Mercuryby magnetohydrodynamic and hybrid models (e.g., [85, 86]).522.3. Average boundary shapesï10 ï8 ï6 ï4 ï2 0 2 4ï10123456789X (RM)l (RM)Figure 2.5: Midpoints between the inner and outer bow shock crossing posi-tions. Error bars show the distance between the inner and outer crossings.Curves show the best-fit conic section to the data (red) and models fromprevious studies by Slavin et al. [77] (blue) and Russell [76] (green). Pa-rameters for the best-fit model to the bow shock midpoints are p = 2.75 RM , = 1.04, and x0 = 0.5 RM .2.3.2 Probabilistic fitsWe also examined the magnetopause and bow shock positions in a proba-bilistic manner, by means of a method employed at Jupiter by Joy et al. [87].As described in Section 2.2, for each crossing we identified an extended regionin space within which the magnetopause or bow shock crossings occurred.The inner and outer limits of the boundaries that we identified span the por-tion of the spacecraft trajectory over which boundary encounters occurredduring each pass. The data set therefore reflects locations where encounterswith the magnetopause and bow shock boundary were probable, and so weused the crossing data to build a probability density map of these bound-532.3. Average boundary shapesaries around the planet. The spacecraft trajectories between each inner andouter crossing limit were registered on spatial grids around the planet, andthe number of crossings passing through each grid cell was used to build aprobability density map of the region of space in which the magnetopauseand bow shock are most likely to be encountered.We divided space around the planet into grid cells as follows. For themagnetopause, we adopted a spherical coordinate system on the daysideand a cylindrical system on the nightside to match approximately the shapeof the boundary. The dayside was split into cells in rMP, θMP, and φMP,where rMP =√(x2 + ρ2) is the distance from the dipole center, θMP =cos−1(x/rMP) is the angle measured from the axis of revolution, and φMPis the azimuth about the axis of revolution defined as tan−1[y/(z − zd)].Grid cells were spaced every 50 km, 10◦, and 30◦ in rMP, θMP, and φMP,respectively. The nightside was divided into grid cells spaced every 50 km,680 km, and 30◦ in ρ, x, and φMP, respectively. Because the bow shocknightside data were not well matched by a cylindrical shape, we used thespherical coordinate system, rSK, θSK, φSK, but with an origin on the axis ofrevolution at x = −4 RM for all of the bow shock crossings. That is, rSK =√((x+ 4RM )2 + ρ2), θSK = cos−1[(x + 4RM )/rSK], and φSK = φMP. Thebow shock bins were 50 km, 5◦, and 30◦ in rSK, θSK, and φSK, respectively.These coordinates are used only to bin the data, and all results are shownin ρ− x space.We evaluated the frequency with which the spacecraft trajectory betweenthe inner and outer magnetopause (or bow shock) crossings passed througheach bin. That is, for each orbit, each grid cell received a “hit” for every1 s measurement point in that bin between the inner and outer limit ofthe magnetopause (or bow shock). The hits were summed in each cell overall orbits for the magnetopause and bow shock boundaries separately. Thecells with the highest number of hits had the highest likelihood of fallingbetween the inner and outer boundary limits. Probabilities were evaluatedby dividing the number of counts in each cell by the sum of all the hits in allcells along a predefined direction (e.g., along each x bin on the nightside andalong each θ bin on the dayside for the magnetopause). The normalization542.3. Average boundary shapeschoice reflects the 100% probability that the boundary occurs at some x orθ position. This procedure resulted in three-dimensional (3-D) probabilitydensity maps of both the magnetopause and bow shock around the planet.For the analyses that follow, we assumed that both the bow shock andmagnetopause are figures of revolution. We used a two-dimensional (2-D)probability distribution in the ρ − x plane generated by summing the hitsin bins with the same r and θ (dayside) or ρ and x (nightside), over all φbins and then normalizing by the total number of counts along r (or ρ) ateach θ (or x). The assumption of a figure of revolution for each boundarywas tested quantitatively (see Section 2.5).The 2-D probability density map (Figure 2.6) for the magnetopauseshows that this boundary has a maximum probability of occurrence thatlies within a narrow band on the dayside and within a more extended regionon the nightside. We fit Shue et al. models to this probabilistic boundary byperforming a weighted fit to the grid cell locations in ρ− x space, such thateach cell location was weighted by its probability. The best-fit curve yieldedthe same model parameters as the fit to the magnetopause midpoints (Table2.1), but with a lower RMS misfit of ∼ 96 km that reflects the use of theprobabilities as weights.The bow shock probability density map (Figure 2.7) shows a more ex-tended spread in the boundary locations than that for the magnetopause,in particular on the nightside, where the highest-probability regions occurat large ρ values. This spread is the result of outlier crossings (Figure 2.5),most of which occurred during parallel shock conditions. A conic fit tothe grid cell locations weighted by the probabilities yields slightly differentmodel parameters than our midpoint fit. The least sensitive of the modelparameters in our fits was the focus location, x0: varying this parameterfrom −0.7 RM to 0.7 RM changed the RMS misfit by only a few percent.Due to the covariance between the parameters, this variation in x0 was ac-companied by large changes in p and , and the bow shock shape variedfrom an ellipse to an open hyperbola, but the RMS misfit changed only bya few percent. We thus fixed the focus location to x0 = 0.5 RM in orderto establish best-fit p and  values that yielded bow shock shapes as close552.3. Average boundary shapesï6 ï5 ï4 ï3 ï2 ï1 0 1 2ï2ï1012345X (RM)l (RM)  Probability00.010.020.030.040.050.06Figure 2.6: Probability density map of aberrated magnetopause positions.The white space represents regions where the magnetopause is never ob-served, the blue regions where it is very rarely observed, and the dark redregions where it is observed most often. The red line represents the Shue etal. model that best fits the probability densities, with Rss = 1.45 RM andα = 0.5. The probabilities sum to 1 along each x bin on the nightside andalong each θ bin on the dayside.to hyperboloids as possible. The fit parameters to the bow shock were thengiven by x0 = 0.5 RM ,  = 0.99, and p = 3.2 RM , with a weighted RMSmisfit of ∼ 187 km (Table 2.2). With these parameters, the bow shock nosedistance is extrapolated to be 2.09 RM , which gives a magnetosheath widthof 0.64 RM from the probabilistic analysis. The shock distance from thedipole-Sun line in the y− z plane is given by 3.83 RM at x = 0 and by 6.35RM at x = −4 RM .562.4. Response of boundaries to solar wind forcingMidpoint fit Probabilistic fitRss α RMS Rss α RMS(RM) residual (RM) residual(km) (km)All PRam 1.45 0.50 587 1.45 0.50 96PRam corrected - - - 1.45 0.50 94< PRam >= 8.8 nPa - - - 1.55 0.50 96< PRam >= 11.5 nPa - - - 1.50 0.50 106< PRam >= 15.2 nPa - - - 1.45 0.50 155< PRam >= 18.8 nPa - - - 1.40 0.50 120< PRam >= 21.6 nPa - - - 1.35 0.50 115Table 2.1: Summary of the best-fit Shue et al. model parameters to themagnetopause crossing points under different ram pressure conditions for afixed α value of 0.50. Angular brackets denote mean value within each rampressure bin.2.4 Response of boundaries to solar wind forcing2.4.1 MagnetopauseThe magnetopause shape and location are expected to vary with solarwind and IMF conditions (e.g., [74, 83]). Figure 2.8 shows the solar windstatistics at Mercury at the time of the bow shock crossings, obtained fromaverages of MESSENGER MAG observations of the IMF and from WSA-Enlil model predictions of the solar wind density, speed, ram pressure, andAlfve´n Mach number [42, 78]. We assumed that these statistics held at thetimes of both the magnetopause and bow shock crossings on a given orbitand used them to examine the response of the boundaries to the IMF andsolar wind.The magnetopause boundary is observed to be closer to the planet dur-ing times of increased ram pressure and farther out during times of low rampressures (Figure 2.9). Baker et al. [78] showed that solar wind ram pres-sure values from the WSA-Enlil model order and organize the magnetopausestand-off distance. This behavior is expected, because the magnetic pres-sure of the planet’s magnetosphere and the solar wind dynamic pressure572.4. Response of boundaries to solar wind forcingï5 ï3 ï1 1 3 5ï2ï101234567X (RM)l (RM)  Probability00.010.020.030.040.050.06Figure 2.7: Probability density map of the aberrated bow shock crossings.The red line represents the best-fit conic section to the probability densities,with p = 3.2 RM ,  = 0.99, and x0 = 0.5 RM . The probabilities sum to 1along each x bin on the nightside and along each θ bin on the dayside.are the two largest contributors to the pressure balance that defines themagnetopause boundary. The ram pressure effect is the dominant factorinfluencing the magnetopause, because any IMF effects, if present, were notreadily apparent in the raw data. We thus investigated any possible effectof the IMF direction on the magnetopause after removing the dependenceon PRam.We assessed changes in the magnetopause shape and position under dif-ferent ram pressure conditions. We binned the magnetopause crossing datainto five PRam bins, such that each PRam bin contained one-fifth of the rangeof PRam values represented in the data. Thus the PRam bins did not contain582.4. Response of boundaries to solar wind forcingMidpoint fitx0 p  RMS Subsolar(RM) (RM) residual distance(km) (RM)All MA 0.5 2.75 1.04 1115 1.90Probabilistic fitx0 p  RMS Subsolar(RM) (RM) residual distance(km) (RM)All MA 0.5 3.20 0.99 187 2.09MA corrected 0.5 2.90 1.02 149 1.96< MA >= 4.12 0.5 3.55 1.02 241 2.29< MA >= 6.32 0.5 2.95 1.02 195 1.99< MA >= 11.8 0.5 2.75 1.02 162 1.89Low cone angle 0.5 3.10 0.99 169 2.04(θ < 45◦)High cone angle 0.5 2.95 0.99 157 1.97(θ > 45◦)Table 2.2: Summary of the best-fit conic section parameters to the bow shockcrossings under different Mach number conditions. In the fits, x0 was fixedat a value of 0.5 RM (see text). In the probabilistic fits to the three differentMA bins, p was the only parameter varied, and  was fixed at the meanvalue between the best-fit probabilistic and best-fit midpoint result (see text).Angular brackets denote mean MA value within each Mach number bin.equal numbers of data points, but the range of PRam values was the samein each bin. We built 2-D probability density maps for each PRam bin andconducted fits of the two-parameter Shue et al. model to these separately(Table 2.1). The largest uncertainty in the best-fit model shape is mostlikely associated with the highest ram pressure bin, which will include themore extreme solar events that may not be properly captured by the WSA-Enlil model. The Rss and α values for the best-fit curves for each of theram pressure bins are plotted in Figure 2.10.With the exception of the highest PRam bin, the Rss parameter decreasesoverall with PRam; in contrast, α shows no systematic variation with PRam.Thus we first removed the dominant Rss versus PRam variation from the data592.4. Response of boundaries to solar wind forcing0 20 40 60 80 100 120 140050100150Density (cmï3)250 300 350 400 450 500 550 600 650050100150Speed (km/s)6 8 10 12 14 16 18 20 22 24050100Ram pressure (nPa)0 2 4 6 8 10 12 14 16 18 200100200300Alfven Mach numberFrequency of Occurrencea)b)c)d)Figure 2.8: Solar wind statistics at the times of the bow shock crossingsobtained from WSA-Enlil model predictions of the solar wind and from av-erages of MESSENGER MAG observations in the interplanetary mediumfor the IMF. Histograms of (a) solar wind number density (cm−3), (b) solarwind speed (km s−1), (c) solar wind ram pressure (nPa), and (d) solar windAlfve´n Mach number. In (d), 15 observations with Mach numbers > 20 arenot shown; these numbers reach a maximum value of 69.0.and normalized the magnetopause crossings by the ram pressure. To do this,we estimated only the Rss parameter for the data in each of the five PRambins while keeping α fixed at its average best-fit value of 0.5 (i.e., a fixedmagnetopause shape). We checked that the best-fit models for the fixed-αfits in all the five PRam bins yielded residuals that had Gaussian distribu-tions with a nearly zero mean. The Rss values for the best-fit curves for each602.4. Response of boundaries to solar wind forcingï6 ï5 ï4 ï3 ï2 ï1 0 1ï2ï101234X (RM)l (RM)  (nPa)810121416182022Figure 2.9: Midpoints of the magnetopause crossings color-coded by solarwind ram pressure. The solid black line through these data points is thebest-fit Shue et al. model. During times of high solar wind ram pressure,the magnetopause is observed to move closer to the planet, as expected.of these ram pressure bins are plotted in Figure 2.11 and given in Table 2.1.These were fit by a power law given by Rss = (2.15± 0.10)P[(−1/6.75)±0.024]Ram ,where Rss is in units of RM and PRam in nPa and the uncertainties are the95% confidence limits on the exponent and the coefficient obtained from thefitting procedure. The magnitude of the exponent is only slightly less thanthe -1/6 dependence of Rss on PRam expected for a simple pressure bal-ance between the internal dipole magnetic field pressure (and small internalplasma pressure) and solar wind dynamic pressure. This result suggests thatthe effects of induction in Mercury’s conductive interior, which “stiffens” thedayside magnetosphere against changes in solar wind pressure [60–63], maybe present, but if so it is a secondary effect, at least at the altitudes over612.4. Response of boundaries to solar wind forcing8 10 12 14 16 18 20 221.31.41.5PRam (nPa)R ss (RM)8 10 12 14 16 18 20 220.40.450.50.55PRam (nPa)_a)b)Figure 2.10: (a) Magnetopause subsolar stand-off distance, Rss, plotted asa function of solar wind ram pressure when both parameters were allowedto vary in the Shue et al. model fits. The blue stars represent Rss valuesestablished from best-fit curves to five ram-pressure data bins. (b) Flaringparameter, α, corresponding to the Rss values in (a) for the best-fit Shue etal. curves, plotted as a function of solar wind ram pressure.which MESSENGER samples the magnetopause. A detailed study of in-duction signatures is beyond the scope of this paper. Refinement of thepower-law above will require both additional observations and assessmentof uncertainties in the WSA-Enlil model predictions.The power law relationship above indicates that a ram pressure of 175nPa would collapse the magnetopause to the planet’s surface. We note thatthe minimum and maximum pressures consistent with the uncertainties of622.4. Response of boundaries to solar wind forcing8 10 12 14 16 18 20 221.351.41.451.51.551.6PRam (nPa)R ss (RM)Figure 2.11: Magnetopause subsolar stand-off distance, Rss, plotted as afunction of solar wind ram pressure for fits when the flaring parameter waskept fixed at a value of 0.5. The blue stars represent Rss values establishedfrom best-fit curves to five ram-pressure data bins, and the red curve is apower law fit to these values with an exponent of -0.148 ± 0.017.our power law fit span a wide range from 65 nPa and 692 nPa. In thesimulations conducted by Kabin et al. [88], a ram pressure of 147 nPa wasfound to collapse the magnetopause to the surface, which is within ouruncertainty range. Using a -1/6 power law (i.e., the relation expected inthe absence of induction) relative to the mean observed Rss of 1.45 RMyields a lower pressure for the collapse of the magnetosphere of 133 nPa,well within the range of uncertainty of our power law expression.With the derived power-law relationship we normalized our magnetopausecrossings as follows. From the WSA-Enlil data, we found the correspond-ing PRam value for the inbound and outbound portion of each orbit, andfrom those values we established an associated Rss value for each orbit fromthe power-law fit to our PRam-binned data. The ram-pressure-independent632.4. Response of boundaries to solar wind forcingmagnetopause crossing locations were then determined by multiplying thex, y, and (z − zd) values by the mean Rss for all crossings (i.e., Rss = 1.45RM ) divided by the Rss value associated with each observation point. Fig-ure 2.12 shows the probability density map of the magnetopause after thesolar wind ram pressure dependence was removed. The PRam-independentmagnetopause location is better constrained than the uncorrected locations(Figure 2.6), especially on the dayside, where the zones of high-probabilityregions lie in a narrow band. Although the model that best fits these datais still described by the same parameters as the PRam-uncorrected mag-netopause (see Table 2.1), the RMS residual is lower than for the PRam-uncorrected crossings, with a value of ∼ 94 km. A statistical F-test showsthat at the 91% significance level the variance of the PRam-corrected mag-netopause model is lower than the variance of the PRam-uncorrected mag-netopause model.After removing the first-order variation of the magnetopause positionwith ram pressure, we binned the data again into the same PRam bins asabove, fixed Rss to the best-fit PRam-corrected value of 1.45 RM and leftα to vary in Shue et al. model fits to these bins, to test the influence ofram pressure on the flaring of the magnetopause. Unlike the situation atEarth, there is no increase in flaring of the magnetopause with increasedram pressure. This result is expected if the magnetospheric currents andthe ratio of the static solar wind pressure to the dynamic pressure remainconstant [75].We then assessed whether the IMF direction affects the magnetopauseshape after the ram pressure dependence was removed. Since the MES-SENGER IMF averages 1 h upstream of the bow shock are not ideal forevaluating IMF Bz affects on the magnetosphere [89], we used the magneticshear angle across the magnetopause to search for any overall dependenceon magnetic reconnection. The magnetic shear was calculated by taking thedot product of the magnetic field unit vector 1 min inside the inner edge ofthe magnetopause crossing and the unit vector 1 min outside the outer edgeof the magnetopause crossing. Figure 2.13 shows the distribution of mag-netic shear angles across the magnetopause for our observations. We divided642.4. Response of boundaries to solar wind forcingï6 ï5 ï4 ï3 ï2 ï1 0 1 2ï2ï1012345X (RM)l (RM)  Probability00.010.020.030.040.050.06Figure 2.12: Probability density map of the aberrated magnetopause crossingsafter the crossing positions were normalized by solar wind ram pressure. Thebest-fit Shue et al. model (red curve) yields the same Rss and α values asbefore (Figure 2.6), but the spread in the data has decreased somewhat on thedayside, and the regions of highest probability are more spatially constrained.the ram pressure corrected magnetopause crossings into low (θ < 80◦) shearangle and high (θ > 100◦) shear angle bins, and built probability densitymaps of these data. Shue et al. model fits to the highest probability regionsdo not yield a resolvable difference between fits to the magnetopause loca-tions separated by magnetic shear angle. Further observations are needed toincrease the signal-to-noise ratio in the boundary locations and potentiallyresolve any shear angle dependence on the magnetopause boundary.652.4. Response of boundaries to solar wind forcing0 20 40 60 80 100 120 140 160 180020406080Magnetic shear angle (degrees)0 10 20 30 40 50 60 70 80 90020406080100Cone angle (degrees)Frequency of OccurrenceFrequency of Occurrence a)b)Figure 2.13: Histograms of (a) the magnetic shear angle across the magne-topause and (b) the IMF cone angle, calculated from MESSENGER magneticfield data.2.4.2 Bow shockWe used WSA-Enlil model data as well as MESSENGER IMF averagesto assess how the bow shock is affected by solar wind and IMF conditions.To first order, the solar wind Alfve´n Mach number is the dominant factor af-fecting the bow shock; the shock is closer to the planet during high MA thanduring low MA (Figure 2.14). This result is expected and in agreement withgas dynamic and magnetohydrodynamic simulations [85, 86, 90]. As MAincreases, the jump of the plasma flow speed transverse to the shock surfacealso increases, corresponding to higher plasma flow speed around the mag-netopause that results in a thinner magnetosheath [91]. At greater down-stream distances, the bow shock weakens as it asymptotically approaches662.4. Response of boundaries to solar wind forcingï10 ï8 ï6 ï4 ï2 0 2 4ï20246810X (RM)l (RM)  (Alfven Mach #)2468101214161820Figure 2.14: Midpoints of the bow shock crossings color-coded by solar windAlfve´n Mach number. The solid black line through the data is the conicsection best fit to the midpoints.its Mach cone [92, 93]. Although MESSENGER does not sample the bowshock at large downstream distances, an enhanced flaring of this surfacewith decreasing Mach number may be present (Figure 2.14).To establish the bow shock position normalized by Alfve´n Mach number,we adopted a procedure to remove the Mach-number dependence similarto that applied to the magnetopause to remove the dependence on rampressure. We binned the bow shock crossings into three MA bins and fitseparate conic sections to each. To have sufficient data points in each MAbin to perform model fits (see Figure 2.8d), the low-MA bin was assignedMA < 5, the medium-MA bin had values in the range 5 < MA < 8, andthe high-MA bin had values of MA > 8 (see Table 2.2 for mean values ofMA in each bin). Fits in which all three bow shock parameters were varied672.4. Response of boundaries to solar wind forcingrevealed large trade-offs among the three model parameters, in a mannersimilar to the magnetopause fits. However, through these fits we were ableto establish that the parameter most systematically affected by MA is thefocal parameter, p, which steadily decreased with increasing MA. The othertwo parameters did not show any systematic behavior with MA. Thus inorder to normalize our bow shock boundary by MA and look for higher-orderdependencies in the data, we assumed that the bow shock shape does notchange (by keeping x0 and  fixed at their average values) and fit separateconic sections to the bow shock crossings in the three MA bins.We again checked that these fixed-shape fits still yielded residuals withzero means and nearly Gaussian distributions. In our fits, the low-MA binhad the highest RMS misfit, and the high-MA bin had the lowest RMS misfitfor all the fits conducted (Table 2.2), consistent with most of the spread inthe bow shock location occurring during low MA (Figure 2.14). From thefits of conic sections to the different MA bins, we established a power-lawrelationship between p and MA, given by p = (4.79 ± 2.54)M(−0.23±0.17)A ,where the uncertainties in the exponent and coefficient are the 95% confi-dence limits determined from the fitting procedure.We scaled the bow shock crossing positions by p0/pi, where p0 is themean p value obtained from averaging the p values from the midpoint andprobabilistic fit (Table 2.2), and pi is the p parameter for the ith cross-ing point determined from the power-law relationship above. The resultingprobability density map for the MA-corrected bow shock positions (Figure2.15) shows a decrease in the spread of the bow shock locations, as well as amarked decrease in the distance between the high-probability regions on thenightside and the best-fit model boundary. The best-fit model boundary tothe MA-independent bow shock crossings has a p parameter of 2.9 RM , aneccentricity of 1.02, a focus point of x0 = 0.5, and a minimum RMS misfit20% less than that for the fit shown in Figure 2.7.We also assessed the influence of MA on the flaring of the bow shockafter removing the first-order dependence of the bow shock position on theMach number. By allowing  to vary (and fixing x0 = 0.5 and p = 2.9 RM )in our conic sections fit to the different MA bins, we found no statistically682.4. Response of boundaries to solar wind forcingï5 ï3 ï1 1 3 5ï2ï101234567X (RM)l (RM)  Probability00.010.020.030.040.050.06Figure 2.15: Probability density map of the aberrated bow shock crossingsafter removing the dependence on Alfve´n Mach number. The red line repre-sents the best-fit conic section to the probability densities, with p = 2.9 RM , = 1.02, and x0 = 0.5 RM . The normalized bow shock is more spatiallyconstrained, in a manner similar to that for the magnetopause normalizedby ram pressure.significant variation of  with MA. For the sake of completeness, we alsoconducted similar tests at a variety of other x0 values, which yielded similarresults. We conclude that variation in bow shock flaring with MA is maskedby the high variability of the crossing locations. More bow shock crossingsare needed at high MA values to establish whether  varies systematicallywith MA at Mercury.The IMF cone angle is also expected to affect the bow shock, as theshock is anticipated to flare during quasi-parallel shock conditions [94]. We692.5. Discussionevaluated whether any dependence on IMF cone angle is observed in the bowshock location or shape after the Mach number dependence was removed.The IMF cone angle is given by θ = cos−1(B|x|BTotal), and its distribution isshown in Figure 2.13. By dividing the data into bins of low (θ < 45◦) coneangle and high (θ > 45◦) cone angle, we found that there is no resolvabledifference between conic section fits to the bow shock locations separatedby cone angle (Table 2.2). The shock is more spatially spread out for thelow-cone-angle bin, an expected result because a low cone angle signifiesparallel shock conditions. The flaring of the bow shock does not appear tobe affected by IMF cone angle.2.5 DiscussionThe observations of Mercury’s magnetopause and bow shock presentedhere indicate that these boundaries are variable and dynamic. At Mercury,the solar wind ram pressure and the Alfve´n Mach number are the two dom-inant external influences on the boundaries. The magnetopause is observedto move planetward during high PRam, and similarly the bow shock movesplanetward during times of high MA. Unlike at Earth, increased PRam doesnot increase the flaring of the magnetopause; the shape of the boundary isunchanged under variations in solar wind ram pressure. In a like manner,MA does not appear to influence the flaring of the bow shock at Mercurydespite the fact that at Earth the shock is more flared during times of lowMA.An important result is that the average magnetopause becomes cylin-drical at relatively small downstream distances of only ∼ 2− 3 RM (Figure2.3). At Earth, in contrast, the magnetotail does not cease flaring untila downstream distance of ∼ 100 RE [95]. Expressed in terms of subsolarmagnetopause stand-off distances, the downstream flaring of Mercury’s tailceases by ∼ 2 Rss whereas at Earth this effect is not seen until ∼ 10 Rss.The factors determining the location where tail flaring ceases are not wellunderstood, but the position likely corresponds to the distance at which theplasma sheet is disconnected from the planet as a result of reconnection, and702.5. Discussionplasma flow in the tail is all anti-sunward. At Earth this position occursat about −100 RE [96], whereas for Mercury Slavin et al. [59] estimated adownstream distance of ∼ 3 RM from MESSENGER flyby observations ofreconnection in the tail. Mercury’s magnetopause is well fit by the Shue etal. model with α = 0.5, which defines the transition from an open (α > 0.5)to a closed (α < 0.5) magnetospheric cavity on the nightside. From thebest-fit Shue et al. model for the ram-pressure-corrected magnetopause wefind that the magnetotail is on average nearly cylindrical with a radius of2.05 RM at the dawn-dusk terminator and 2.77 RM at a distance of 4 RMdown the tail. In comparison, Earth’s magnetopause is more flared, withthe data fit well by Shue et al. models that have α > 0.5 [69]. The over-all nearly cylindrical shape of Mercury’s magnetopause in comparison toEarth’s may imply that magnetic flux has a short residence time in the tail,and thus return convection of flux from the tail to the dayside proceedsrapidly, consistent with the expected ∼ 2 minute Dungey cycle period atMercury.The analysis of the magnetopause and bow shock boundaries in ρ − xspace was based on the assumption of rotational symmetry about the dipole-Sun line. We tested whether this assumption was justified both qualita-tively and quantitatively. First, the boundary crossings plotted in x−y andx − (z − zd) space did not reveal systematic differences. The ram pressureand Mach number are not observed to cause any asymmetries in the shape,as the boundary crossings corrected for PRam and MA have similar shapesto the uncorrected crossings in x − y and x − (z − zd) space. These com-parisons indicate that variations from rotational symmetry can be treatedas a perturbation to the figures of revolution. We then assessed the degreeto which systematic deviations from figures of revolution are present. Fromthe corrected crossing locations, we evaluated the best-fit models for bothboundaries, and we calculated the perpendicular distances of each crossingfrom the model boundaries as a function of the azimuthal angle, i.e., φMP orφSK. If either the magnetopause or bow shock were flattened or elongatedin the north-south direction, such an effect would be evident as a sinusoidalvariation in φMP or φSK, respectively, relative to the mean boundary at712.6. Conclusionsa period of 180◦, i.e., two cycles. We did not observe any systematic de-partures from figures of revolution for either the magnetopause or the bowshock. By binning the deviations into 2◦ bins in φMP or φSK and fitting asinusoid to these binned deviations, we find maximum sine-wave amplitudesof 62 and 3 km for the magnetopause and bow shock, respectively. Thesesine-wave amplitudes are more than an order of magnitude smaller thanthe variability in the deviations of the two boundaries about the models.We conclude that, to first order, the boundaries are figures of revolution.The scatter relative to the mean is high however, implying that dynamicvariability in Mercury’s magnetosphere and bow shock are large and thatsecond-order structure could be present but masked by the large dynamicvariability. It is thus possible that instantaneously these boundaries are notfigures of revolution. However, on average, the departures from figures ofrevolution are small compared with the dynamics in the system. Becauseof the high variability of the crossing locations analyzed so far, we cannotyet resolve any average asymmetries in the boundary shapes. At Earth,the maximum departure from a figure of revolution at high latitudes is ∼ 1RE [72], which corresponds to ∼ 0.1 Rss (subsolar stand-off distance is ∼ 10RE). In comparison, an equivalent 0.1 Rss departure is only ∼ 350 km atMercury, which would be a sufficiently small departure to be easily maskedby the variability in the available data. Such signatures may be resolvedwith additional observations.2.6 ConclusionsWe have established Mercury’s time-averaged magnetopause and bowshock location and shape from MESSENGER Magnetometer data obtainedduring three Mercury years in orbit. We find that the magnetopause iswell described by a Shue et al. model parameterized by a subsolar stand-off distance of 1.45 RM and a flaring parameter of α = 0.5. The solarwind ram pressure exerts a primary control on magnetopause location; theboundary moves closer to the planet under higher PRam (giving a subsolardistance of 1.35 RM for a mean PRam of 21.6 nPa) and farther away from722.6. Conclusionsthe planet under lower PRam (with an Rss of 1.55 RM for a mean PRam of8.8 nPa), while leaving the shape unchanged (Table 2.1). The paraboloidmodel of Belenkaya et al. [81] provides a substantially worse overall fit tothe magnetopause crossings than the Shue et al. model, reflecting the ab-sence of evidence for substantial flaring from observations on the nightside.This comparison suggests that future improvements in global models forMercury’s magnetosphere should use a ram-pressure-corrected Shue et al.model magnetopause. The observed low flaring of the magnetotail may im-ply that magnetic flux has a short residence time in the tail on average. Thisshort residence time of the tail flux could also imply that return convectionof flux from the tail to the dayside proceeds rapidly.The shape of Mercury’s bow shock corrected for Alfve´n Mach number isthat of a hyperboloid given by the parameters x0 = 0.5 RM, p = 2.9 RM,and  = 1.02, and a subsolar stand-off distance of 1.96 RM. The bow shockshape does not appear to vary with Alfve´n Mach number, as there is nochange in flaring. This is an unexpected result, since Earth’s bow shock isobserved to flare with decreasing Alfve´n Mach number. At Mercury, thebow shock moves closer to the planet for high MA and farther out for lowMA; the extrapolated nose distance of the shock is at 1.89 RM for a meanMA of 11.8, and at 2.29 RM for a mean MA of 4.12 (Table 2.2). Both themagnetopause and bow shock boundaries are figures of revolution to firstorder, but the variability about the mean is large. With the current dataavailable we do not resolve effects of IMF orientation on the magnetopauseor the bow shock. As more data are acquired by MESSENGER, effects ofIMF on the magnetopause or bow shock should be more readily discernible.The variation of the bow shock and magnetopause location with dynam-ics is large at Mercury, and understanding the processes that these dynamicsreflect is a key area of future study. The derivation of the average bound-aries presented here provides a baseline with which to evaluate excursionsof the system from its average state. Extensive progress has already beenmade in understanding boundary waves [97, 98] and reconnection at Mer-cury [34, 59, 99]. Extending that work to better understand those aspectsof global magnetospheric dynamics that could lead to the large variations732.6. Conclusionsin boundary locations is a fruitful area of inquiry.74Chapter 3Observations of Mercury’snorthern cusp region withMESSENGER’sMagnetometer1The magnetic cusp of a planetary magnetosphere allows solar wind plasmato gain access to the planet’s magnetosphere and, for Mercury, the surface.From measurements by the MESSENGER Magnetometer we have charac-terized the magnetic field in the northern cusp region of Mercury [89]. Thefirst six months of orbital measurements indicate a mean latitudinal extentof the cusp of ∼ 11◦, and a mean local time extent of 4.5 hrs, at spacecraftaltitudes. From the average magnetic pressure deficit in the cusp, we esti-mate that (1.1±0.6)×1024 protons s−1 bombard the surface over an area of(5.2± 1.6)1011 m2 near the northern cusp. Plasma pressures in the cusp are40% higher when the interplanetary magnetic field (IMF) is anti-sunwardthan when it is sunward. The influence of the IMF direction does not over-come the north-south asymmetry of Mercury’s internal field, and particleflux to the surface near the southern cusp is predicted to be a factor of 4greater than in the north. The higher particle flux impacting the surface inthe south should lead to a greater exospheric source from the south and ahigher rate of space weathering than in the area of the northern cusp.1Reprinted from Geophysical Research Letters, with permission from Wiley.753.1. Introduction3.1 IntroductionThe magnetospheres of planets with dipolar internal fields possess mag-netic cusps, regions near the magnetic poles at which fields from magne-topause currents nearly cancel the internal field. For vacuum superpositionof the magnetic fields of the dipole and magnetopause currents, the cusps aretopological singularities where the magnetic field vanishes. The weak fieldnear the cusp allows the shocked solar wind plasma of the magnetosheathready access to the magnetosphere, and the magnetic field lines that threadthe cusp are populated with this plasma.Mercury’s internal field is symmetric about the rotation axis but asym-metric about the geographic equator and can be represented by a dipole witha moment of 195 nT R3M (where RM is Mercury’s mean radius, 2440 km)offset 0.2 RM northward from the planetary center [17]. The high-latitudefield at the surface is predicted to be 4 times weaker in the southern hemi-sphere than in the northern hemisphere, leading to a correspondingly greaterspatial extent of the cusp projection to the surface in the south than in thenorth. Solar wind sputtering of species from the planetary surface may bea substantial source of exospheric particles (e.g., [100]). It is not knownwhether asymmetric particle bombardment of the surface and correspond-ing differences in space weathering rates could have produced detectablehemispheric differences in surface color or reflectance.Earth’s cusps have been extensively studied at low and high altitudes(e.g., [101]). The position and size of the cusp areas at Earth depend on thesolar wind pressure [102, 103] and the IMF (e.g., [100, 104–106]). Mercurylacks an ionosphere, the magnetosphere is a factor of∼8 smaller than Earth’srelative to the planetary diameter, and the average solar wind density is anorder of magnitude higher than at Earth [23], so the cusps at Mercury maybe quite different from those at Earth. In addition, because the IMF isdominantly sunward or anti-sunward at Mercury, the IMF component inthe Sun-Mercury direction may play a prominent role in the dynamics ofMercury’s cusps (e.g., [107]).Previous work on Mercury’s cusps focused on magnetosphere-solar wind763.2. Observationsinteraction by means of analytic models [108–110], global magnetohydro-dynamic models [88, 111], hybrid simulations [112–114], or semi-empiricalmodels [100, 115]. These studies indicated that solar wind ions can reachthe surface in the cusp region, but the spatial extent of the cusp and theparticle fluence vary among the models. This variation is partly due to thedifferent IMF and solar wind conditions assumed. From solar wind andIMF conditions at Mercury’s aphelion and perihelion, Sarantos et al. [110]predicted that the largest flux of precipitating solar wind ions impactingMercury’s surface occurs at local noon between 40◦ and 60◦ latitude withan equatorward shift at perihelion. To date there have been no observationsthat quantify the total plasma pressure in Mercury’s cusps or provide a basisfor assessing its sensitivity to the sunward IMF component.Orbital observations by the MESSENGER spacecraft’s Fast ImagingPlasma Spectrometer (FIPS) have revealed that the flux of heavy ions inMercury’s magnetosphere peaks between 65◦ and 75◦ latitude, consistentwith the predicted location of the northern magnetic cusp [22]. The ionflux peaks coincide with depressions in magnetic field strength [116] mea-sured with the MESSENGER Magnetometer (MAG) [49]. In this chapterwe characterize the northern cusp with MAG data from six months of orbitalobservations, calculate the corresponding surface precipitation, and investi-gate the influence of the sunward IMF and solar wind pressure on the meancusp plasma pressure.3.2 ObservationsThe MESSENGER spacecraft was inserted into orbit about Mercury on18 March 2011. The initial orbit had a 200 km periapsis altitude, 82.5◦inclination, 15,300 km apoapsis altitude, and 12 hour period. We use twoMercury years of MAG data starting from 23 March 2011, providing coverageat all local times. Data were analyzed in Mercury solar orbital (MSO)coordinates, for which +x is sunward, +z is northward, and +y completesthe right-handed system.The cusp was identified from depressions in the magnitude of 1 s av-773.2. Observations0100200300400B (nT)  DataModel−160−140−120−100−80−60−40−20  Fit DataPolynomial FitResiduals09:03 09:04 09:05 09:06 09:07 09:08 09:09 09:10−25−20−15−10−505Magnetic Pressure Deficit (nPa)UTC (hh:mm)Field Magnitude (nT)Figure 3.1: Example of a cusp observation on 21 August 2011, orbit 313.(top) Measured (black) and modeled (red) magnetic field magnitude in thecusp region. (middle) Magnetic depression in the residual |B| (black), resid-ual data before and after cusp entry (red), and a third-degree polynomial fit(blue) to the red curve. (bottom) The calculated pressure deficit (−PPlasma).eraged total-field data from which a model field had been subtracted. Themodel incorporates the offset internal dipole field and the magnetopause andtail fields of the Alexeev et al. [82] paraboloid magnetospheric model, with783.3. Data analysismodel parameters given in Anderson et al. [17]. An aberration correctionwas calculated from Mercury’s orbital speed and a mean solar wind speedof 405 km s−1. For each orbit exhibiting a dayside magnetic depressionpoleward of the magnetopause, we identified the times of the cusp outer andinner entry and exit points. Transits in the cusp were indicated by sustaineddepressions in the magnitude of the magnetic field B that exceeded typicalvariability and lasted several minutes. An outer cusp entry was identifiedat the point where the first transient decrease in |B| was seen, and the in-ner entry was picked where the sustained depression in |B| started. Similarcriteria were used for the exit inner and outer points. Figure 3.1 shows thedayside depression in the total residual between the observed and modelfields, given by |B|res = |B|obs − |B|model. The orbit does not always inter-sect the cusp, particularly when periapsis is on the nightside, and magneticdepressions were seen on 169 of the 279 orbits analyzed. Each entry andexit time and the aberrated MSO spacecraft positions are given in the Sup-plemental Materials of [89]. The field depressions were generally associatedwith enhanced magnetic fluctuations at 1-10 Hz frequency, consistent withgreater intensities of local plasma instabilities. The proton gyrofrequencyis 2 to 6 Hz for field strengths observed in the cusp (150 to 400 nT). Thecusp entry and exit times changed by less than a few seconds for differentmagnetospheric model parameters.3.3 Data analysisWe conducted superposed epoch analyses (SEA) of |B| and |B|res in thecusp to derive an average magnetic depression signature (Figure 3.2). Indi-vidual profiles from different orbits were aligned in time on their respectivecusp interval midpoints and averaged over a time span of six minutes oneither side of this midpoint. We also conducted SEA of the 1-10 Hz fluctu-ations. The fluctuation intensity was evaluated from the 20 sample/s databy taking the root mean square (RMS) value over 1 s intervals in the di-rection parallel to and two components perpendicular to the 1 s averagedfield direction, denoted by δB‖, δB⊥1 and δB⊥2, respectively. We define793.3. Data analysisδB⊥ =√(δB2⊥1 + δB2⊥2)/2, so that δB⊥ = δB‖ if the fluctuations are equalin all components. These analyses confirm the depression in the magneticfield over the cusp and show that this signature is accompanied by an in-crease in the magnetic fluctuations. The ratio δB⊥/δB‖ is about 1.5 in thecusp and higher on either side of the cusp (Figure 3.2), indicating that al-though the fluctuations in the cusp are transverse, they are less so than theadjacent lower-amplitude fluctuations.We calculated a plasma pressure that balances the magnetic field depres-sion from PTotal = PMag + PPlasma, where PTotal is the total pressure; PMagis the magnetic pressure, B2/(2µ0), where µ0 is the magnetic permeability;and PPlasma is the particle thermal pressure. We estimated PTotal from themagnetic field removed from the cusp magnetic field depression [116]. Theunperturbed magnetic field was determined for each pass from the mag-netospheric model field and a third-degree polynomial fit to the residualsone minute before and after but excluding the depression interval (Figure3.1, middle panel). The boundaries of the depression intervals were takenas the average of the inner and outer cusp entry or exit times. In somecases the polynomial fit did not consistently remain above the residual fieldmagnitude in the cusp. These fits were rejected, and new fits were obtainedby increasing the time interval for the baseline fit. The polynomial fit wasadded to the magnetospheric model field to estimate the unperturbed totalmagnetic field, BU. We then evaluated PTotal = B2U/(2µ0) and the magneticpressure deficit, PB−deficit = PMag − PTotal = −PPlasma (see Figure 3.1, bot-tom panel). This latter quantity gives the additional plasma pressure in thecusp relative to any background plasma pressure in the magnetosphere. Ingeneral, FIPS data do not show substantial proton counts adjacent to, butoutside, the cusp, indicating that the background plasma pressure near thecusp is much lower than that in the cusp.The limits of the northern cusp are 55.8◦ and 83.6◦ MSO latitude and 7.2h and 15.9 h local time. On average the cusp is approximately symmetricabout noon (Figure 3.3). Since the MESSENGER orbit is eccentric andperiapsis is on the descending latitude portion of the orbit, the cusp isencountered at lower altitudes on the descending than on the ascending orbit803.3. Data analysis−6 −4 −2 0 2 4 6100200300400Field Magnitude (nT)  −40−20020Field Magnitude (nT)DataModelResidual−6 −4 −2 0 2 4 600.511.522.53RMS (nT)Time (minutes)  RMS B⊥RMS B||Ratio00.511.522.53RatioFigure 3.2: Superposed epoch analysis of all cusp observations. (top) SEAof observed |B| (blue) and model |B| (red) indicated by the scale on the left-hand ordinate, and SEA of residual |B| (green) indicated by the scale on theright-hand ordinate, for all 169 cusp profiles. (bottom) SEA of RMS 1-10Hz fluctuations perpendicular and parallel to the local field, δB⊥ (blue), andδB‖ (black) (scale on left). The red curve shows δB⊥/δB‖ (scale on right).track. At higher altitude the cusp is on average a few degrees equatorwardof that seen at lower altitude. In the magnetosphere model, the magneticfield at the magnetopause vanishes near 62◦ N at noon, consistent with theexpected shift in cusp latitude closer to the magnetopause.813.4. Discussion60o 60o60o60o16 16161614 14141412 12121210 10101008 08080875o 75o75o75o85o 85o85o85oDescending (340 to 550 km altitude)−3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0nPaMSO2MSO1Ascending (500 to 990 km altitude)Figure 3.3: Stereographic projections of the pressure deficit (−PPlasma) alongeach cusp profile in aberrated MSO coordinates. During portions of MES-SENGER’s first Mercury year in orbit (MSO1), the Magnetometer was offwhen the spacecraft experienced long eclipses or was close to the planet, re-sulting in the gap in data coverage (between ∼ 10 h and ∼ 12 h in local time)for the descending tracks. Complete coverage was obtained during MESSEN-GER’s second Mercury year in orbit (MSO2). Projections span local timesfrom 6.67 h to 17.3 h and latitudes 55◦N to the pole. The color bar is sat-urated so that observed, but localized, pressure deficits greater in magnitudethan -3 nPa are shown in red.3.4 DiscussionThe observations indicate that Mercury’s northern cusp region is a per-sistent but dynamic feature. Not only is the cusp pressure deficit variableon a given pass (Figure 3.1), but the cusp extent and plasma pressure canvary markedly from one orbit to the next (Figure 3.3). This variability likely823.4. Discussionresults from the influence of different IMF and solar wind conditions andthe corresponding interactions with, and dynamics of, Mercury’s magneto-sphere. Here we focus on establishing the mean cusp pressure and particlefluence to the surface since the plasma pressure may have important conse-quences for exospheric processes and space weathering. We use MESSEN-GER averages of IMF BX and predictions of the solar wind ram pressurefrom the WSA-Enlil solar wind model [42]. Statistics of these quantities forthe cusp transits are given in Table 3.1.MESSENGER’s 12-hour eccentric orbit presents challenges to analyzingthe effects of the solar wind on the cusp. First, the local time extent ofthe cusp is sampled only twice each Mercury year (Figure 3.3). A study ofvariations in cusp local time extent with solar wind conditions will requireconsiderably more observations than are presently available. Second, IMFconditions for a given orbit are estimated from averages of MAG observa-tions upstream of the bow shock. The 1-h time spans for these averages arecomparable to the typical time between MESSENGER cusp transits andresidence in the solar wind. Only the IMF x-component is generally largerin magnitude than its variability (Table 3.1) implying that the IMF aver-ages are appropriate for investigating the effects of IMF sector structure onthe cusp, but not for assessing the influence of magnetopause reconnection.Reconnection depends strongly on the sign of Bz and the magnetosphereresponds to changes in the sign of Bz within minutes [24]. Because themean value of Bz is generally smaller than its variability over the averagingintervals, the average IMF is not a good indicator of reconnection dynam-ics during our cusp transits. We also compared the average IMF beforeand after each magnetosphere transit to assess IMF stability. Only 18%of orbits that pass through the cusp exhibit an average IMF Bz that is ofthe same sign before and after MESSENGER’s magnetosphere transit andthat is at least one standard deviation different from zero. In contrast, thecorresponding percentages for Bx and By are 79% and 42%, respectively.The data set therefore allows reliable assessment of the cusp dependence onIMF Bx, which at Mercury’s orbit is the dominant IMF component and ispredicted to have a strong influence on pressures in the cusp (e.g., [107]).833.4. DiscussionAscending tracksIMF Bx > 0 IMF Bx < 0|B| 193 nT 179 nTPPlasma 1.5 ± 0.1 nPa 2.1 ± 0.1 nPaIMF Bx 16.4 ± 1.1 nT (σ = 5.0 nT) −9.1 ± 0.7 nT (σ = 4.7 nT)IMF By -4.6 ± 1.2 nT (σ = 6.0 nT) 5.9 ± 1.1 nT (σ = 5.9 nT)IMF Bz 1.9 ± 1.1 nT (σ = 6.3 nT) 1.3 ± 0.9 nT (σ = 6.8 nT)PRam 11.9 ± 0.4 nPa 11.3 ± 0.4 nPa# of orbits 45 43Latitude 70.9 ◦ N 71.4 ◦ NAltitude 684 km 706 kmDescending tracksIMF Bx > 0 IMF Bx < 0|B| 305 nT 318 nTPPlasma 1.8 ± 0.1 nPa 1.9 ± 0.1 nPaBx 17.8 ± 0.9 nT (σ = 5.2 nT) −10.7 ± 2.0 nT (σ = 5.0 nT)IMF By -5.1 ± 1.1 nT (σ = 7.0 nT) 2.5 ± 1.4 nT (σ = 6.7 nT)IMF Bz 1.5 ± 0.8 nT (σ = 7.6 nT) -0.02± 1.35 nT (σ = 7.0 nT)PRam 14.5 ± 0.5 nPa 11.4 ± 0.3 nPa# of orbits 55 26Latitude 72.5 ◦ N 75.2 ◦ NAltitude 426 km 404 kmTable 3.1: Average cusp properties and ambient conditions separated by as-cending/descending tracks and by the sign of IMF Bx. The mean |B| andPPlasma inferred from decreased magnetic field strength are evaluated in thecusp. The average IMF Bx, By, and Bz are calculated from observationsbefore and after each magnetospheric transit, and the solar wind PRam isfrom the WSA-Enlil model evaluated during times of passage in the cusp.Mean cusp latitude and altitude are weighted by PPlasma; the rest of the val-ues are unweighted. Uncertainties are 1 standard error of the mean. ForIMF averages, the mean standard deviation (σ) is given in parentheses; onlythe average Bx is consistently greater in magnitude than its variability.Determining the variability in cusp location and pressure due to dynamicsassociated with magnetopause reconnection, as indicated by the IMF Byand Bz components, is left for future analyses.We assessed the influence of the IMF Bx and the solar wind ram pressure843.4. Discussionon the cusp plasma pressure as follows. Statistics were evaluated separatelyfor ascending and descending passes and for positive and negative IMF Bx,because an anti-sunward IMF (negative Bx) is expected to facilitate plasmatransport into the northern cusp (e.g., [107]). The ascending tracks weredivided approximately equally between positive and negative Bx, but themagnitude of Bx was ∼ 1.7 times higher for sunward than for anti-sunwardIMF conditions. The results indicate a larger plasma pressure in the cusp fornegative Bx. The high-altitude datasets for Bx > 0 and Bx < 0 have similarmean altitudes and mean ram pressure (PRam) values, so the 40% deepermagnetic pressure deficit for negative than positive Bx can be attributed tothe IMF orientation. Variation in the cusp mean position at Mercury forthe different signs of Bx and comparable average PRam, as on the ascendingtracks, is at most ∼ 0.5◦ compared to the average cusp extent of ∼ 11◦.We performed SEA on PB−deficit for each of the two ascending-trackpopulations following the procedure described in Section 3.3, except thatoutside the depression interval we padded the PB−deficit values with zeroesto fill out the time to 8 min for each event (Figure 3.1, bottom panel). TheSEA profiles of PB−deficit (Figure 3.4) confirm that the magnetic pressuredeficit in the cusp for the transits with Bx < 0 is, on average, larger inmagnitude than for Bx > 0 and show that the cusp is present regardlessof the sign of Bx. The sunward/anti-sunward direction of the IMF thusmodulates plasma pressures but is not the dominant factor determiningpressure in the cusp.The plasma pressure in the cusp appears to increase with increasing so-lar wind ram pressure. For Bx > 0 the descending tracks exhibit a higherPRam and also a lower PB−deficit than the ascending tracks, indicating thatthe cusp pressures increase with increasing PRam. The influence of PRammay account for the smaller difference in PB−deficit between the descendingtrack observations for Bx > 0 and Bx < 0, as the mean PRam is substan-tially higher for the Bx > 0 events. Presumably, PB−deficit for the Bx > 0descending track cases would have been smaller in magnitude had PRamfor these tracks been comparable to that for the Bx < 0 descending trackobservations.853.4. Discussion−4 −2 0 2 4−3.5−3−2.5−2−1.5−1−0.500.5Time (minutes)Magnetic pressure deficit (nPa)  IMF Bx > 0IMF Bx < 0Figure 3.4: SEA of the magnetic pressure deficit for ascending tracks groupedby IMF Bx > 0 (red) and IMF Bx < 0 (black). A larger-amplitude magneticpressure deficit is observed for orbits when the IMF has a negative, or anti-sunward, Bx component.The estimate of cusp plasma pressures allows us to calculate the averageparticle flux bombarding the surface. We describe this calculation and theresults in Section 3.4.1. Obtaining an average particle flux to the surface hassignificant implications for studies relating to the generation of Mercury’sexosphere as well as space weathering of the surface.3.4.1 Surface flux calculationTo calculate the surface flux of particles, we assume an isotropic particledistribution entering from the well-mixed magnetosheath plasma and thatthe dominant ions in the solar wind are protons. We know the plasmapressure at the spacecraft altitude, so our aim is to derive an expression forthe plasma pressure ratio between the spacecraft altitude and the surface,and from there to estimate the pressure at the surface. Once we know thepressure at the surface, we can calculate the flux at the surface by using an863.4. Discussionequation that relates the flux of isotropic gas particles to their pressure.We consider a particle distribution function f0(v) at the spacecraft alti-tude. The pressure at this altitude is given by the second velocity momentof the distribution functionP0 =∫v2f0(v)d3v. (3.1)In spherical coordinates, f0(v) = f0(v, α, φ) and d3v = v2 sin(α)dφdαdv,where φ is the velocity azimuth about the magnetic field, and α is thevelocity pitch angle (the angle between v and B). We assume that thedistribution is gyrotropic, so that there is no dependence on φ, and we alsoassume that it is isotropic, so that the speed distribution is independent ofα, to get f0(v) = 2pi V (v)A(α). In the absence of electric fields parallelto B, the particle speed distribution (V (v)) is independent of B, so that itis the same everywhere along the field line. We can then rewrite equation(3.1) in spherical coordinates asP0 = 2pi∫v4V0(v)dv∫A0(α) sin(α)dα. (3.2)The pressure can be decomposed into the pressure due to particle mo-tions parallel to and perpendicular to the magnetic field, B, since v2 =v2‖ + v2⊥, where v‖ = v cos(α) and v⊥ = v sin(α). Then, using equation (3.1)for simplicity,P0 =∫v2‖f0(v)d3v +∫v2⊥f0(v)d3v, (3.3)yielding P0 = P‖ + P⊥.Since the magnetic pressure deficit reflects the perpendicular particlepressure, at the spacecraft altitude we only need P⊥ (as this is all we mea-sure), which can be written asP⊥0 = 2pi∫v4V0(v)dv∫A0(α) sin3(α)dα, (3.4)after substituting in for v⊥. The total particle pressure at the surface, Ps is873.4. Discussiongiven byPs = 2pi∫v4Vs(v)dv∫As(α) sin(α)dα. (3.5)If we can estimate the ratio of Ps/P⊥0, we can calculate Ps becausefrom our measurements of the plasma pressure at the spacecraft altitudewe know P⊥0. Thus, because we assumed that there are no parallel electricfields, Vs(v) = V0(v), the pressure ratio between the two altitudes isPsP⊥0=∫As(α) sin(α)dα∫A0(α) sin3(α)dα. (3.6)From Liouville’s theorem, in the absence of collisions and loss of particles,df/dt = 0, so that f0(v0) = f1(v1). However, the planet’s surface representsa loss for the particles, and we can estimate an upper limit for this loss. Fora charged particle, the local magnetic field and instantaneous pitch angle arerelated by the first adiabatic invariant, the magnetic moment, which yieldsthatsin2(α)B= constant, (3.7)so that the pitch angle of particles that mirror at the surface is given byαm,i = sin−1(√Bi/Bs), (3.8)where Bi is the magnetic field magnitude at the altitude of interest, which inthis case can be either at the spacecraft or the surface. For more details onmagnetic mirroring, see Section 1.2.4. This means that all particles headingtoward the surface with pitch angles smaller than αm,i will encounter thesurface, while particles with pitch angles larger than αm,i will mirror abovethe surface and return moving away from the surface. All particles thatmirror at or below the surface are missing from the upward flowing portionof f(v). Thus we can rewrite Ai(α), where i can represent either ‘0’ or ‘s’for spacecraft altitude or surface, in terms of the function in the absence of883.4. Discussionsurface losses, A, which is independent of α and altitude. We get thatα = 0 to pi − αm,i : Ai(α) = Aα = pi − αm,i to pi : Ai(α) = 0, (3.9)where αm,i represents the mirroring pitch angle at a particular altitude. Sothis basically means that due to Liouville’s theorem, we can assume thatA(α) is constant for the part of the pitch angle distribution that is not lost,and for the part that is lost, A(α) is zero by definition. Since the parts ofthe particle distribution that are lost due to absorption by the surface arenot going to contribute to the pressure, the pressure integrals should onlygo from 0 to pi − αm,i. We can then rewrite equation (3.6) asPsP⊥0=∫ pi−αm,s0 sin(α)dα∫ pi−αm,00 sin3(α)dα. (3.10)For the top integral, αm,s is simple, sinceBi = Bs in that case, and this yieldsfrom equation (3.8) that αm,s = pi/2. The integral is then straightforward,and a value of 1 is obtained for the top integral. For the bottom integral,αm,0 can be calculated from equation (3.8) and an average field magnitudeat the spacecraft altitude of the ascending tracks. With a value for B0 = 186nT, the pressure ratio becomes Ps/P⊥0 = 0.77.We can estimate a lower limit on this pressure ratio by assuming thatthere is no surface loss, that is, particles are perfectly reflected. In this case,the integral for Ps goes again from 0 < α < pi/2, while the integral for P⊥0goes from 0 < α < pi, yielding a pressure ratio of 0.75. From the meanmagnetic pressure deficit along the high-altitude ascending tracks, we havethat P⊥0 = 1.79 nPa. We can use this, along with the two estimates for thepressure ratio, to estimate an upper limit for Ps of 1.38 and a lower limit of1.34 nPa, with an average of Ps = 1.36 ± 0.23 nPa, where the uncertaintycorresponds to the average pressure ratio times half the difference in meanpressures for positive and negative IMF Bx.893.4. DiscussionFor an isotropic gas, the flux of particles through a surface is given byΦ = P/√2pimkT , (3.11)where m is the particle mass, T is the temperature, and k is the Boltzmann’sconstant. To estimate the flux of particles to the surface, we double thesurface pressure that we estimated above because that only accounts for thedowngoing half of the distribution (as the upgoing part was absorbed by thesurface). Thus the flux is given byΦ = 2Ps/√2pimkT (3.12)We assume that the plasma is dominated by protons, so m = mp, andwe use a characteristic energy of particles in the cusp of 1.05 ± 0.95 keVfrom FIPS data (since FIPS observed particles with energies between 0.1 to2 keV in the cusp) [22]. These assumptions yield an average surface flux ofΦs = (2.1±1.0)×1012 particles m−2s−1. This value agrees with the averageflux over perihelion and aphelion conditions of ∼ 3× 1012 m−2s−1 predictedby Sarantos et al. [110].From here, we estimate the total number of particles that hit the surfacein the cusp region by projecting the area of the cusp at the altitude ofthe descending tracks down to the surface. We calculate an upper and alower limit for the area. Our lower limit is estimated from the area A ofa trapezoid that encompasses the minimum region over which the cusp isobserved at the altitude of the descending tracks in Figure 3.3, projecteddown to the surface, using A ∝ 1/B. Our upper limit is estimated by takinga circular area of radius equal to half the maximum latitudinal extent of thecusp at the lower altitudes, and again projecting this down to the surface.The mean cusp area is then calculated to be (5.2± 1.6)× 1011 m2, centeredat 74.7◦ MSO latitude on the surface, and we find that (1.1 ± 0.6) × 1024particles bombard the northern cusp region every second. The uncertaintyin our total flux is primarily from the uncertainty in the proton temperatureand the area of the cusp.903.4. DiscussionAs a check on the validity of Liouville’s theorem we compared the pres-sures estimated for the ascending and descending tracks. For complete sur-face absorption, the ratio of the perpendicular pressures at the two altitudesshould be P⊥1/P⊥0 = 0.93, so we expect the pressure ratio to be between0.93 and 1. For IMF Bx < 0, under which solar wind PRam values are com-parable between ascending and descending tracks (Table 3.1), the pressuresare the same to within the uncertainties.Because of the northward offset of Mercury’s dipole, and the resultingweaker surface magnetic field at high southern than northern latitudes, weexpect the flux of precipitating particles to occur over a larger area in thesouthern cusp region than in the north. In the absence of observations of thesouthern cusp, we use the offset of the dipole magnetic field to estimate thetotal number of particles reaching the surface in the south. We calculate thecentral MSO latitude of the southern cusp to be about 64◦ S. The magne-tospheric model [17] predicts a surface field strength at this latitude of 158nT. From the ratio of the model surface field strength in the south to thatin the north we estimate that the cusp area in the south is 2× 1012 m2, andthe number of particles reaching the surface in the southern cusp region iscorrespondingly higher, 4×1024 particles s−1. Over a Mercury solar day, theplanetary surface rotates under the cusp, so the cusp precipitation reachesall planetary longitudes in a band extending ∼ 1600 km (∼ 38◦) in latitude.The IMF Bx effect we observe here, corresponding to 40% higher pressuresin the northern cusp for negative than for positive IMF Bx, implies that theflux to the southern cusp should dominate regardless of the IMF direction.These hemispheric flux differences would lead to a persistently greater ex-ospheric source from the south, as sputtering is most likely a contributingfactor in populating the exosphere of Mercury [110]. In addition, if solarwind ion sputtering is a dominant source of space weathering at Mercury,this signature may be observed in surface reflectance spectra. It is, however,possible that Mercury’s current magnetic field configuration has not been inplace sufficiently long compared with space weathering timescales for thishemispheric asymmetry to be evident in surface reflectance and color differ-ences. Alternatively, the surface may already have reached saturation, and913.5. Conclusionshemispheric differences will be muted.3.5 ConclusionsFrom six months of MESSENGER MAG observations we have char-acterized Mercury’s northern cusp region and found that it is persistentlypresent but variable in extent and in the depth of its magnetic field de-pression. We focused on the role of the IMF Bx direction and the solarwind ram pressure in modulating the average plasma pressure in the cuspbecause of possible observable consequences for exospheric processes andspace weathering. The northern cusp is clearly evident even during sunwardIMF conditions but exhibits 40% higher plasma pressures on average dur-ing anti-sunward conditions, indicating that the effect of IMF Bx directionis present. Rapid variability in cusp pressures and orbit-to-orbit variationsin the latitudinal extent of the cusp may be related to magnetospheric dy-namics associated with southward IMF conditions. We estimate that onaverage (1.1±0.6)×1024 particles per second reach Mercury’s surface in thenorthern hemisphere cusp region, thus (via sputtering) contributing a sourcefor the exosphere. Because of the northward offset of the planetary dipole,the flux of particles bombarding the southern cusp should be a factor of 4higher, yielding a greater exospheric source in the south. Similarly, spaceweathering in the south due to cusp precipitation should occur over an area4 times larger than in the north (or equivalently, over a latitudinal extentthat is a factor of two larger). The implications of the north-south magneticasymmetry for exospheric dynamics are therefore substantial and warrantefforts to confirm the estimated difference in surface magnetic field inten-sities. Whether a north-south asymmetry is evident in surface reflectancedifferences depends on the length of time that the present north-south asym-metry in the magnetic field has been maintained.92Chapter 4Mercury’s surface magneticfield determined fromproton-reflectionmagnetometry1Solar wind protons observed by MESSENGER exhibit signatures of pre-cipitation loss to Mercury’s surface. In this chapter, we apply proton re-flection magnetometry to sense Mercury’s surface magnetic field intensityin the planet’s northern and southern hemispheres [117]. The results areconsistent with a dipole field offset to the north and show that the tech-nique may be used to resolve regional-scale fields at the surface. The protonloss cones indicate persistent ion precipitation to the surface in the northernmagnetospheric cusp region and to the southern hemisphere at low night-side latitudes. The latter observation implies that most of the surface inMercury’s southern hemisphere is continuously bombarded by plasma, incontrast with the premise that the global magnetic field largely protects theplanetary surface from the solar wind.4.1 IntroductionA remarkable feature of Mercury’s weak, internal, magnetic field, indi-cated by orbital observations, is a ∼480 km northward offset of the mag-netic equator from the planetary equator [17–19]. The low magnetic field1Reprinted from Geophysical Research Letters, with permission from Wiley.934.1. Introductionstrength and the northward offset provide constraints on Mercury’s enig-matic dynamo mechanism [118] and lead to direct interactions between thesolar wind and the planet’s surface. The weak magnetic field allows precip-itation of solar wind plasma to the surface in the northern magnetosphericcusp region. Since the surface magnetic field strength at any southern lati-tude is predicted to be weaker than at the corresponding northern latitude,enhanced ion sputtering and space weathering in the southern hemisphereis possible. In this study we apply proton-reflection magnetometry, adaptedfrom electron reflectometry [119–121], to determine Mercury’s surface mag-netic field strength in both hemispheres and measure particle precipitationto the surface.Electron reflectometry (ER) has been used extensively at the Moon [122],Mars [121], and Ganymede [123] to sense remotely the magnetic field strengthat the surface. ER depends on the magnetic mirroring effect, that is, thereflection of electrons by convergent magnetic fields. Electrons that wouldmirror below the surface are lost and the flux of reflected electrons exhibitsa sharp flux drop at the pitch angle (the angle between the particle velocityand the local magnetic field direction) corresponding to mirroring at thesurface. The in-situ magnetic field together with the pitch angle of the lastreflected electrons, the cut-off pitch angle, indicates the surface magneticfield strength. This technique has not yet been applied using protons.At Mercury, protons with energies of 0.3−10 keV are regularly detectedinside the magnetosphere [22] by the Fast Imaging Plasma Spectrometer(FIPS) [50] on the MErcury Surface, Space ENvironment, GEochemistry,and Ranging (MESSENGER) spacecraft. The Energetic Particle Spectrom-eter [50] on MESSENGER detects electrons with energies only above 35keV and observes significant fluxes too infrequently to use ER. We thereforeuse FIPS observations of protons within Mercury’s magnetosphere [53], to-gether with magnetic field observations from MESSENGER’s Magnetometer(MAG) [49], to estimate the magnetic field strength at Mercury’s surface.944.2. The technique of proton reflection magnetometry4.2 The technique of proton reflectionmagnetometryIn this section, we apply the electron reflectometry technique to pro-tons. Similarly to electron reflectometry, proton-reflection magnetometryrelies on magnetic mirroring, whereby protons travelling in a helical pathalong magnetic field lines reflect back along these field lines when the paral-lel component of their velocity vector (v‖) becomes zero and all the velocityis in the perpendicular component (v⊥). Such a configuration occurs in con-vergent magnetic fields. The proton mass is much greater than the electronmass, and so proton-reflection magnetometry can only be conducted in re-gions where the ambient magnetic field is high enough for the protons to be“magnetized”, i.e. to be directed by the magnetic field. It can be seen fromequation (1.7) that the proton gyroradius is significantly higher than theelectron gyroradius in the same ambient magnetic field due the higher mass,signifying that in weak magnetic fields, at the Moon for example, the domi-nant force affecting the motion of the protons will not be the Lorentz force.This is not the case at Mercury however, where the planetary magnetic fieldis high enough to “magnetize” both electrons and protons.Protons reflect at varying altitudes depending on their initial pitch an-gles, and those that are initially aligned very closely to the ambient magneticfield do not reflect before they reach the surface and are absorbed by the sur-face. Thus a spacecraft flying above convergent magnetic fields will detectparticles travelling in the direction of the magnetic field toward the planet’ssurface, as well as particles reflected back along field lines after having un-dergone mirroring. The spacecraft will also detect a loss cone in the reflectedparticle distribution that corresponds to particles absorbed by the surface.In what follows, we will show how it is possible to use the pitch angle infor-mation of the protons absorbed by the surface to infer the strength of thesurface magnetic field.In general, when conducting proton-reflection magnetometry at each in-tegration period of a plasma spectrometer it is important to ensure that thesame downgoing and upgoing plasma populations are sampled. This in turn954.2. The technique of proton reflection magnetometryrequires that the distance travelled by the spacecraft during the proton’sround-trip time has to be less than or equal to the gyroradius of the proton.This requirement however is not used in our analyses, because the limitedFOV of the FIPS sensor means that we typically see only the incident orthe reflected population at any one integration time, and so we average theresults from many integration times to obtain the full incident and reflectedpopulations (see Section 4.3).4.2.1 How to obtain surface field strengthsIn this section we describe how to use the magnetic mirroring formalismoutlined in Section 1.2.4 to determine the magnetic field magnitude at thesurface of the planet. The key factor in determining the surface magneticfield strength relies on measuring the pitch angle distribution of the protons(i.e., the flux of protons as a function of pitch angle). If there was no lossmechanism in the system (i.e. the surface), all particles incident along themagnetic field direction would reflect back along the field lines. Thus if theinitial population entering from the magnetosheath along open field lineswas originally isotropic (i.e. equal fluxes at all pitch angles), the observedparticle pitch angles would be in the range of 0◦ to 90◦ for the incidentpopulation (in Mercury’s northern hemisphere), and if these all reflected,particles with pitch angles in the range of 90◦ to 180◦ would be detectedreturning along the field lines. However, particles that are initially closelyaligned with the magnetic field do not reflect before reaching the surfaceand are absorbed. There will thus be protons with a range of initial pitchangles which will be missing from the returning population. This missingpart of the reflected pitch angle distribution, the loss cone (Figure 4.1),is determined by the ratio between the magnetic field magnitude at thespacecraft and that at the surface.The cut-off pitch angle, αc, that is the pitch angle in the reflected partof the distribution beyond which no more protons are observed, is the pitchangle that corresponds to protons mirroring exactly at the surface. These arethe “last” protons which still make it back to the spacecraft, as all protons964.3. Application of the technique to MESSENGER observations!"0 90 180 01 INCIDENT REFLECTED Normalized Flux Pitch Angle (deg) !c Loss cone Figure 4.1: Schematic pitch angle distribution showing the cut-off in re-flected charged particle flux at angle αc due to absorption by the surface.that would have a reflected pitch angle larger than the cut-off pitch angleare absorbed. Thus protons with a cut-off pitch angle are remote-sensingthe surface magnetic field because they mirror due to the magnetic fieldstrength at the surface. We can therefore replace α0 with αc and Bm withBS, where BS is the surface field strength in equation (1.16) to getBS =B0sin2 αc. (4.1)Thus if we can measure the magnetic field strength at the spacecraftaltitude, B0, and obtain αc, it is possible to use equation (4.1) to obtainthe surface magnetic field strength. Therefore the key to estimating BS isdetermining the cut-off pitch angle in the pitch angle distribution.4.3 Application of the technique toMESSENGER observationsFor our analyses, we used one Earth-year of observations from 7 June2011 to 7 June 2012. Combining observations from MAG and FIPS al-974.3. Application of the technique to MESSENGER observationslowed the calculation of proton pitch angle and the derivation of pitch-angledistributions (PADs) within the magnetosphere.FIPS measures energy per charge (E/q), time of flight, and arrival in-cidence angle for ions, and completes one scan over the full range of E/qvalues every 8 s [124]. It has a conical field of view (FOV) of approximately1.4pi sr, with two symmetric cutouts of 15◦ near the instrument’s symmetryaxis and also near the plane perpendicular to the symmetry axis. Due to thelimited field of view of FIPS, the full proton PAD (from 0◦ to 180◦) is notvisible at any one integration time. To build PADs spanning α = 0◦ to 180◦at times when the spacecraft was in a given region, we combined pitch angledistributions from instrument integration times when the incident protonpopulation was observed with PADs when the reflected proton populationwas visible. This approach required averaging PADs over many time inter-vals and normalizing each contributing PAD to account for obstructions inthe FIPS FOV. Here we focus on regions where the highest proton countswere detected by FIPS. The analysis thus determines Mercury’s averagelong-wavelength (e.g., dipole) field but does not resolve shorter-wavelengthstructure.4.3.1 Deriving individual pitch angle distributionsFIPS Pulse Height Analysis (PHA) data are used to create the protonpitch angle distributions. These distributions cannot be constructed simplyfrom the raw data files because the following instrument limitations haveto be taken into account: the size of the solid angle of the pixels on thedetector, the efficiency of the micro-channel plates (MCP) detector, and theFIPS field of view (FOV) obstructions at each integration time. Becauseonly angular structure is of interest for these accumulations, proton eventsfrom all E/q steps were added together to improve the signal-to-noise ratio.To account for the solid angle size variation with zenith angle, θ, of thepixels on the detector, we divide each proton count obtained from the PHAdata (Figure 4.2A) by the solid angle size given bysin θdθdφ = 4.248× 10−5r2 − 1.114× 10−3r + 1.15× 10−2 (4.2)984.3. Application of the technique to MESSENGER observationswhere r is given byr = 3.60348− 0.045364 θ + 0.00648116 θ2 − 0.0000451434 θ3. (4.3)This normalization does not have a large effect on the proton countsbecause the variation in solid angle size with zenith angle is small. Next,because the FIPS MCP detector efficiency varies with location on the detec-tor [125], our counts are also divided by an MCP correction factor to ensurethat counts that fell on the lower efficiency part of the detector carry a higherweight. After rotating the vector magnetic field data measured by MAG intothe FIPS reference frame, the pitch angles for all proton counts are com-puted. The solid-angle-weighted and MCP-normalized proton counts thatfall within 10◦ pitch angle contours are then summed into 10◦ pitch anglebins, centered on pitch angles of 5◦, 15◦,...,165◦, 175◦.The summed proton counts are then also weighted by a factor that takesinto account the FIPS visibility of each magnetic pitch angle bin [125]. Thisnormalization is a function of the FIPS orientation with respect to the mag-netic field direction and FOV obstructions at each integration time (Figure4.2B). The counts are divided by this factor, ensuring that pitch angle bincenters which are not fully in the FOV are weighted higher to account forunobserved proton counts at those pitch angles. We thus finally arrive at apitch angle distribution at every integration time of the instrument, wherethe proton counts have been properly weighted in order to take into accountall the instrument limitations (Figure 4.2C). Errors assigned to each pitchangle bin center incorporate counting statistics and the fraction of protongyrophase angles that was visible in the FOV for each pitch-angle bin.4.3.2 Averaging pitch angle distributionsBecause of the limited FOV of FIPS, there are no integration periodswhen both incident and reflected particle distributions are fully observed.In order to increase the signal-to-noise ratio in our observations, as well asto represent the entire PAD, we averaged PADs from different integrationperiods when the spacecraft was over the same MSO location. To connect994.3. Application of the technique to MESSENGER observationsAlt=455km0 50 100 150 200 250 300 350020406080q (degrees)e (degrees)Lon=349Lat=77.10 20 40 60 80 100 120 140 160 180050100_ (degrees)Normalization factor0 20 40 60 80 100 120 140 160 18000.511.52_ (degrees)Weighted proton countsABC kmlooFigure 4.2: Example integration period of FIPS. (A) Zenith angle versusazimuth angle in the FIPS reference frame. The green dots show the protoncount locations in angle-angle space, and pitch angle contours are for thatintegration period (the magenta curve is a pitch angle of 90◦, and the greencurve is 150◦). The spacecraft altitude, latitude, and longitude are also given.(B) Weighting factor as a function of pitch angle. The limited FOV of FIPSas well as the viewing geometry given the local magnetic field direction aretaken into account in the weighting. (C) Derived pitch angle distribution atthe given integration period. Error bars reflect counting statistics as well asthe fraction of the proton gyrophase angle that was visible in the FOV foreach pitch angle bin.1004.3. Application of the technique to MESSENGER observationsincident and reflected particle populations from different integration periods,we normalized each individual PAD by the weighted proton count observedat a pitch angle of either 85◦ or 95◦, depending on which of the two pitchangles was visible in the distribution at that time. If both were observed,the proton count at 85◦ was used. This methodology was applied to electronreflectometry measurements at the Moon [126], where the average protoncounts in the incident population were used to normalize the PAD. SinceFIPS does not observe the full incident population at the same time as thereflected population, we used the proton count at α ∼ 90◦ to approximatethe average incident population. We then computed weighted averages ofthe individual PADs, where the error at each pitch-angle bin center on thePADs was assigned as the weight. The error assigned to the final averageddistribution is one standard error in all the proton counts that were averagedat each pitch angle bin center. Averaged pitch angle distributions werederived in the cusp region in the northern hemisphere, as well as in regions ofhigh proton flux on the nightside at low latitudes in the southern hemisphere.4.3.3 Surface field estimatesAveraged pitch angle distributions were derived in the cusp region in thenorthern hemisphere (geographic latitude λ > 60◦N), as well as in regionsof high proton flux on the nightside at low latitudes (0◦ < λ < 30◦S) in thesouthern hemisphere. Although high proton fluxes to the dayside southernhemisphere, especially in the southern cusp region, are expected, MESSEN-GER is unable to measure these because its eccentric orbit allows it to beinside the magnetosphere only over northern latitudes on the dayside. Theaveraged PAD for the northern cusp region, which includes 485 reflectedand 185 incident population scans, is shown in Figure 4.3A. The incidentpopulation is approximately independent of pitch angle (i.e., isotropic), con-sistent with protons entering along open field lines from the magnetosheath.However, there is a void, or loss cone, in the fluxes of reflected protons,from α ∼ 120◦ to 180◦, evidence for incident protons having been ‘lost’ toMercury’s surface. Relative to the sharp cut-off pitch angle observed in ER1014.3. Application of the technique to MESSENGER observationsat the Moon [127], the edge of the proton loss cone is smoothed over ∼ 30◦in α. We attribute this to the combined effects of the FIPS angular resolu-tion of ∼ 15◦ and pitch angle diffusion from wave-particle scattering in thecusp. Broadband magnetic field fluctuations between 0.001 and 10 Hz, areconsistently observed in this region [128], and would scatter protons in α.We estimated the loss cone angle consistent with diffusive scattering byfitting solutions to the diffusion equation to the loss cones of our pitch angledistributions. We solved the one-dimensional diffusion equation∂u(α, t)∂t= Dα∂2u(α, t)∂a2(4.4)where u is the proton count and Dα is the diffusion coefficient, with astep-function initial condition:u(α, 0) =c1 for α ≤ αc,c2, for α > αc.(4.5)where c1 and c2 are constants set by the average maximum and minimumweighted proton counts in the PAD. The boundary conditions were givenby:∂u(0, t)∂α=∂u(pi, t)∂α= 0. (4.6)The use of reflective boundary conditions (i.e. zero gradient in the fluxat the boundaries), as opposed to one with a negative gradient, is justifiedfor this diffusion process as there is no loss of particles in the field aligned(or anti-aligned) direction from pitch angle diffusion. This is due to thefact that the diffusion process occurs after particles have mirrored and aretravelling up along magnetic field lines away from the surface. As such,particles that are scattered into the loss cone will be scattered back andforth across α = 180◦, but will not be lost to the surface. There should thusbe no loss of protons at the boundaries.1024.3. Application of the technique to MESSENGER observations0 20 40 60 80 100 120 140 160 18000.20.40.60.81α (degrees)Stacked / weighted proton counts0 20 40 60 80 100 120 140 160 18000.20.40.60.811.2α (degrees)a) b)c =121 ±3D t = 0.05± 0.01BS = 412± 98 nTc = 43° 13+7D t = 0.09 0.05+0.06BS =113 61+87 nTIncident population Incident populationReflected population Reflected population0 20 40 60 80 100 120 140 160 18000.20.40.60.81α (degrees)c) Reflected population Incident populationStacked / weighted proton countsStacked / weighted proton countsFigure 4.3: Pitch angle distributions in the northern and southern hemi-spheres. (A) Results for Mercury’s northern cusp. (B) Results for the low-latitude southern hemisphere nightside. Average proton counts are in redwith standard errors. The black curve shows the diffusion model fit to thereflected portion of the distribution; the fit uncertainty is in gray. Yellowshading indicates the loss cone; the black error bar shows the uncertaintyin αc. (C) Comparison of the southern hemisphere PAD (red curve) frompanel (B) to a model single-sided loss cone distribution (black curve) and anobserved double-sided loss cone distribution (blue curve). The character ofthe southern hemisphere PAD is in-between that of a single-sided and a fullyformed double-sided loss cone (see text below and in Chapter 5).1034.3. Application of the technique to MESSENGER observationsThe solution to equation (4.4) isu(α, t) =∞∑n=1Bn cos(nα) e−n2Dαt, (4.7)whereBn =2pi[∫ αc0c1 cos(nα)dα+∫ piαcc2 cos(nα)dα]. (4.8)We fit equations equations (4.7) and (4.8) to our loss cones and allowedthe cut-off pitch angle, αc, and Dαt to vary freely. We used a grid searchmethod that minimized the median absolute deviation (MAD) between themodel and the observations. Figure 4.4 shows the contour plots of the ab-solute value of the residuals for the models best fit to the averaged PAD forthe northern cusp region as well as for the southern hemisphere. We estab-lished upper and lower bounds on the parameters αc and Dαt by identifyingan allowable upper bound on the misfit, corresponding to a 95% confidencelimit (bold contours in Figure 4.4). The upper and lower bounds on αc wereidentified as the locations of the intersection of a horizontal cut (passingthrough the minimum misfit) with the bold contour. A corresponding verti-cal cut yielded the limits on the Dαt parameter. The bounds on the best-fitmodel, shown by the grey shaded regions of Figure 4.3, were determinedfrom the diffusion curves corresponding to the upper and lower limits for αcand Dαt.From the fit to the northern cusp PAD, we obtain a cut-off pitch angle,αc, of 121◦ ± 3◦, which together with the measured average magnetic fieldstrength at the spacecraft altitudes (< 550 km), B0 = 302.4 ± 53.0 nT,implies a surface field strength of BS = 412 ± 98 nT where we have usedBS = B0/ sin2(αc). The uncertainty accounts for the standard error in thefit value of αc and also the standard deviation of B0, computed from allintervals in the average PAD.We mapped the average observation location in the cusp down to thesurface by tracing the magnetic field lines to the surface using Mercury’soffset dipole magnetic field. The observation altitudes ranged from 282 km1044.3. Application of the technique to MESSENGER observations0 5 10 15 20 25 30 35 40 45 50 55 60 6500.020.040.060.080.10.120.140.160.180.20.220.24αc (degrees)D αt  Misfit00.020.040.060.080.10.120.140.160.18110 112 114 116 118 120 122 124 126 128 13000.020.040.060.080.1αc (degrees)D αt  Misfit00.010.020.030.040.050.06ABFigure 4.4: (A) Residuals of the diffusion equation curve best fit to the losscone of the averaged PAD in the northern cusp region, as a function of thecut-off pitch angle, αc, and the product of the diffusion coefficient and time,Dαt. The bold contour marks the residual level from which the errors on thefit parameters were obtained. (B) Same as A but for the southern hemisphereaveraged PAD. The ratio of the bold contour to the minimum misfit is ∼ 7in (A) and ∼ 3 in (B).1054.3. Application of the technique to MESSENGER observations04 h     12 h   20 h  45 oN   60 oN   75 oN    60o S   30o     0o  SA B   12 h04 h 20 hS  30oDAWNDUSKDUSKDAWNN SFigure 4.5: (A) Stereographic projection plot (looking down from above thenorth pole) showing the surface magnetic foot-point locations of the reflectedproton observations in the northern cusp versus local time and latitude. TheSun is to the right. Latitudes north of 45◦N are shown. (B) Correspondingplot for the southern hemisphere (looking through the planet from above thenorth pole). Latitudes from 10◦N to 90◦S are shown.to 549 km, with a mean value of 414 km. In latitude, the cusp observationsat the surface extended 15.6◦ degrees in latitude and 7.5 h in local time andwere centered on noon at 76.4◦N latitude on the surface (Figure 4.5A).We also find high proton fluxes in the latitudinal band 0◦ < λ < 30◦S onthe nightside, with a clear loss-cone signature in the derived PAD (Figure4.3B), although with larger uncertainties than for the northern cusp region(Figure 4.3A). In the southern hemisphere, observations as far south aspossible are desirable for observing the long-wavelength structure in themagnetic field. However, due to MESSENGER’s eccentric orbit and highaltitudes in the southern hemisphere, we are restricted to observations northof approximately 30◦S latitude. In this averaged PAD, we included 128 scansin the reflected population and 315 scans in the incident population. Thesimilar error bars on most of the incident and reflected population fluxesin Figure 4.3B, despite the higher number of observations being includedin the incident side, is due to the significantly larger standard deviations in1064.4. Consistency checksthe fluxes of the incoming protons. The best-fit diffusion model to the losscone gives αc = 43◦+7−13, and this together with B0 = 52.5 ± 14.8 nT at thespacecraft altitude corresponds to a surface field strength of BS = 113+87−61nT. The observation altitudes were higher than those in the northern cuspregion, ranging between 1160 and 1980 km, with a mean of 1535 km. Themapped surface locations span 23◦S < λ < 34◦S, with a mean of 27.8◦S,and local times spanning the nightside from 16 h to 5.3 h, centered on 23.5h (Figure 4.5B).The apparent secondary loss cone in the incident population for thisaveraged PAD (Figure 4.3B) implies that these observations may correspondto closed field lines on the nightside. However, as Figure 4.3C shows, eventhough these observations are likely from a closed field line region, the shapeof the southern hemisphere PAD is significantly different from that a fullyformed double-sided loss cone found on closed field lines on the dayside (seeChapter 5), and is in between the character of an idealized single-sided anda fully double-sided loss cone distribution. This is owing to fresh protonpopulations drifting onto closed field lines on the nightside from the plasmasheet, thereby continuously replenishing particles in the loss cone. We canthus approximate this nightside southern hemisphere PAD as a single-sideddistribution, which suggests that the inferred surface magnetic field strengthfor this region may be a lower limit.4.4 Consistency checksTo ensure that the FIPS proton data and our averaging method are vi-able for conducting consistent proton reflection measurements we performeda number of consistency checks. We binned the observations in the northerncusp region (the region where we have the most observations) in altitude aswell as in latitude and local time, and tested that the derived averaged PADsyielded expected behavior with respect to the size of the loss cones and theestimated magnetic field strengths.1074.4. Consistency checksFigure 4.6: Pitch angle distributions derived from observations binned inaltitude: red denotes low-, green mid-, and blue high-altitude observations.4.4.1 Altitude binningIt is expected that over regions of approximately constant magnetic fieldon the surface, if the altitude of the spacecraft observations increases (andthus the measured magnetic field strength decreases) then the cut-off pitchangle will increase (or equivalently, the loss cone size will decrease). Wetested that this is the case in our observations in the northern cusp region.Three altitude averages of the cusp observations were obtained (Figure 4.6).The first was at low altitude with a mean of 325 km and included 155 scansin the reflected and 85 in the incident population. The second was at midaltitude with a mean of 418 km and included 98 scans in the reflected and96 in the incident side. And the third was at high altitude with a meanof 510 km and consisted of 173 scans in the reflected and 4 in the incidentpopulation.Despite having approximately equal number of observations in the inci-1084.4. Consistency checksdent and reflected populations for the mid altitude bin, the errors on theincident population are much larger because of the large standard devia-tions in the measured proton fluxes at those pitch angles. We fit diffusionequation curves to these averaged PADs to derive a cut-off pitch angle foreach. We find that the cut-off pitch angle increases with increasing altitude,as expected. The derived cut-off pitch angles were: 116◦+6−4 , 123◦+3−4 , 126◦+2−5for the low-, mid-, and high-altitude bins, respectively. The smaller differ-ence in loss cone cut-off angle between the mid- and high-altitude bins isattributed to the difference in the average latitude of observations betweenthese two bins (the mean latitude at the spacecraft altitude of the mid al-titude bin was 76.1◦N, whereas it was 73.7◦N for the high altitude bin).This latitude difference implies a higher surface magnetic field strength inthe mid-altitude bin than for the high-altitude bin, offsetting the expectedaltitude-dependent difference in cut-off pitch angle.4.4.2 Latitude binningWe also obtained averages of PADs in three latitude and two local-timebins in the cusp region to test whether we observe the latitudinal increase inthe magnetic field strength expected for an intrinsic dipole field. The obser-vations were binned not just in latitude but also in local time to minimizethe range of altitudes included in each average. Phasing of MESSENGERorbit-correction maneuvers with local time resulted in dusk observationsthat were systematically taken at lower altitudes than the dawn observa-tions. By splitting the data into dawn and dusk sections, and binning eachsection separately into three latitude bins (the latitude at the spacecraft al-titude and the altitude of the bin centers is given in Figure 4.7), we obtainedsix PAD averages in the northern cusp region.After fitting diffusion equation curves to these PAD averages and ob-taining the cut-off pitch angle for each averaged PAD, we find that themagnetic field strength increases with increasing latitude, as expected. Themean measured magnetic field strengths at the spacecraft altitudes on thedawn side are 250.2 ± 33.8 nT, 276.2 ± 39.8 nT, and 311 ± 46 nT for low-,1094.4. Consistency checks20 40 60 80 100 120 140 160 18000.511.52_ (degrees)Stacked proton counts  <h> = 71.7°; <Alt> = 390 km<h> = 74.3°; <Alt> = 370 km<h> = 78.3°; <Alt> = 360 km0 50 100 15000.20.40.60.811.21.4_ (degrees)Stacked proton counts  <h> = 71.6°; <Alt> = 505 km<h> = 74.6°; <Alt> = 488 km<h> = 77.8°; <Alt> = 410 kmDAWNDUSKABFigure 4.7: Pitch angle distributions binned in latitude and local time in thenorthern cusp. (A) Averaged PADs for latitude bins on the dawn side of thenorthern cusp region. Colour coding is as follows: red is low-, blue mid-, andgreen high-latitude. The mean latitudes and altitudes of the observations ineach bin are also given. (B) Same as A but for dusk-side observations.mid-, and high-latitude data bins, and the corresponding surface magneticfield strengths are 334+65−72 nT, 376 ± 70 nT, and 432 ± 92 nT, respectively.The loss cones are not as well defined on the dusk side, especially in the caseof the low-latitude bin, which has very high errors reflecting the small num-ber of observations that were included in the average. The existence of a loss1104.5. Discussion and conclusionscone in the averaged PAD for the dusk-side low-latitude bin is questionable,and so we ignore this bin in our calculations. The mid- and high-latitudebins on the dusk side do exhibit loss cones, albeit more smeared out thanthose seen on the dawn side. We estimate that the magnetic field strengthat the surface for these two bins of data (using B0 = 327.0 ± 46.9 nT andB0 = 343.0± 37.1 nT for the mid- and high-latitude bins) to be 412+140−74 nTand 432±62 nT for the mid- and high-latitude bins, respectively, consistentwith an increase in field strength with increasing latitude. The estimatesfor the surface field agree within 10% between the dawn and dusk side forthe mid- and high-latitude bins.4.5 Discussion and conclusionsEstimates of the surface magnetic field strength in the northern cusp andlow latitude southern hemisphere are compared with predictions from thebest-fit time averaged magnetospheric model [19] in Figure 4.8 and Table4.1. The results from proton-reflection magnetometry are significantly lowerthan the model magnetic field. Such a difference is expected however becausethe model is a vacuum magnetic field model, whereas our PADs demonstratethe presence of plasma extending to the surface of the planet. The plasmagenerates a diamagnetic field, which will reduce the surface field below thevacuum model prediction [89, 116, 129].Fortunately, the proton data provide the information required to esti-mate the diamagnetic effect (see Appendix A for a detailed derivation of thediamagnetic field and flux to the surface). The flux of particles at the sur-face in the northern cusp region can be determined from the loss cone sizeand mean proton temperature, Tp, and density, np, in the cusp, yielding theproton pressure Pp = npkTp, where k is Boltzmann’s constant. The typicalproton density (np ≈ 30 cm−3) and temperature (Tp ≈ 12 MK) derived fromFIPS observations in the cusp [125] are for an isotropic particle distribution;the anisotropy associated with the loss cone produces an underestimate ofplasma density [130]. Taking this anisotropy into account yields a surfaceflux of 3.7×1012 particles m−2s−1, which is approximately in agreement with1114.5. Discussion and conclusionsLatitude Local time BPR BPlasma BPlasma - BPR BModel(MSO (MSO) (nT) (nT) (nT) (nT)surface)76.4◦N 12 h 412± 98 −86± 11 498± 99 64727.8◦S 23.5 h 113+87−61 −44± 6 157+87−61 224Table 4.1: Surface magnetic field strength from proton-reflection magnetom-etry compared with magnetospheric model predictions. The center latitudesand local times of the northern cusp and southern hemisphere PAD aver-ages are given, as well as our surface field estimates with and without thediamagnetic effect of the plasma near the surface. The vacuum magneticfield model predictions for comparison with the corrected proton-reflectionmagnetometry estimates are also listed.the flux determined by Winslow et al. [89] for the cusp region. In the regionof our southern hemisphere average, we have np ≈ 5 cm−3 and Tp ≈ 20 MK,which give a flux of 4.4 × 1011 particles m−2s−1. To take into account thefact that the loss cone in the incident population for this PAD is only ∼ 70%filled in (i.e. the PAD is not fully single-sided), we multiply this flux valueby 0.7 to obtain the flux to the surface in this region of 3.1× 1011 particlesm−2s−1. The diamagnetic effect reflects particle motions only in the direc-tion perpendicular to the local magnetic field, so we used the perpendicularparticle pressure to calculate the diamagnetic field at the surface. We findan average diamagnetic effect, ∆BPlasma, in the northern cusp of 86±11 nT,and ∆BPlasma = 44±6 nT in the southern hemisphere at low latitudes. Theuncertainties in these values represent the limits on the diamagnetic fieldderived from the pressure at the spacecraft altitude and that at the surface.After accounting for this effect, our magnetic field estimates agree with themodel predictions in the southern hemisphere and are within 8% (our upperbound) of the model prediction in the cusp (Table 4.1).The validity of the offset dipole can be tested by estimating the ratiobetween the surface magnetic field strength in the northern cusp and that inthe southern hemisphere. From the mean observation locations, we find thatthe ratio from proton reflection magnetometry, corrected for the diamagneticeffect, is 3.2+3.0−1.6 in agreement with the result of 3.3 for an offset dipole field1124.5. Discussion and conclusionsFigure 4.8: Mollweide projection of surface magnetic field strength from thebest-fit time-averaged magnetospheric model [19] in MSO coordinates. Theequator offset and the magnetopause and magnetotail fields cause departuresfrom the field of a dipole alone. Black contours mark the approximate outlineof the regions sampled by proton-reflection magnetometry observations.alone. The full magnetospheric model [19] yields a surface field ratio of 2.6 to3.6 for the latitudinal and longitudinal extent of our observation locations,with a mean value of 2.9. A centered dipole field alone gives a ratio of 1.5 forthe latitudes sampled and is not in agreement with our inferred surface fieldratio. Even if we consider that the southern hemisphere surface magneticfield strength may be a lower limit, we can still conclude that the offsetdipole is confirmed as long as BS in the southern hemisphere is not morethan a factor of ∼ 2.5 greater than our derived value. Such a large valuefor the low latitude southern hemisphere field is incompatible with vectormagnetic field measurements by MESSENGER. Lastly, it is important tonote that the large uncertainty assigned to our southern hemisphere losscone size is approximately equal to the difference in loss cone angle sizebetween our PAD in the southern hemisphere and a fully formed double-sided PAD from the same region (see difference between red and blue curvesfor the reflected population of Figure 4.3C). Thus our large uncertainty onαc for the southern hemisphere population (and the corresponding largeuncertainty on BS) likely captures any possible pitch angle diffusion related1134.5. Discussion and conclusionserosion of the loss cone size.Proton-reflection magnetometry thus provides independent confirmationof the offset dipole nature of Mercury’s magnetic field, although weakerterms of higher degree and order in a multipole expansion of the field arenot ruled out by these results. Significantly, proton-reflection magnetome-try demonstrates that persistent proton precipitation to the surface occurson the nightside of Mercury’s low-latitude southern hemisphere. This resultimplies that such precipitation may also be occurring at middle latitudes onthe nightside, where closed field lines reach the surface, because the north-ward offset of the magnetic equator results in weak surface field strengths(not more than 25% above our measured value) everywhere in the southernhemisphere. Together with the proximity of the magnetopause to the sur-face and the large cusp region on the dayside southern hemisphere [17, 18],this suggests that most of Mercury’s southern hemisphere surface may becontinuously bombarded by plasma. Such continuous precipitation in thesouthern hemisphere implies that space weathering is not confined to thecusp region and may thus show limited latitudinal variation. Because ofMESSENGER’s eccentric orbit, proton reflection magnetometry at highersouthern latitudes is not feasible, but observations from the ESA-JAXABepiColombo mission [131], with a less eccentric orbit that will provide low-altitude observations in both hemispheres, may provide an opportunity toprobe the surface field more extensively and with greater spatial resolution.114Chapter 5Regional-scale surfacemagnetic fields and protonfluxes to Mercury’s surfacefrom proton-reflectionmagnetometryFollowing the successful implementation of the proton-reflection mag-netometry technique to measure the surface magnetic field strength in tworegions on Mercury’s surface, we expand on this work [132]. In this section,we extend our study with 1.5 years of additional observations (includingdata from 2012 and 2013) by MAG and FIPS with the aim of resolvingregional-scale structure in the surface magnetic field strength and derivinga plasma precipitation map of the surface.Regional-scale structure in the magnetic field, if present, may be dueto either core or crustal sources. If they are of core origin, surface fieldstrength measurements could provide further constraints for dynamo mod-els. Conversely, if the short-wavelength structure is due to crustal remanentmagnetization, correlations between crustal fields and geologic features canhelp establish constraints on the temporal evolution of Mercury’s dynamofield - an important constraint for geophysical evolution models.Determining the average proton flux to various regions of Mercury’s sur-face is important from a space weathering and exosphere production per-spective because it identifies regions on the surface that are exposed to direct1155.1. Methodsbombardment by the solar wind. The collision between energetic solar windor magnetospheric ions with surface grains can release atoms via momen-tum transfer [133], a process known as sputtering. Sputtering can release anyatomic species present in the near-surface mineralogy, so particles in the ex-osphere reflect the composition of the surface on an atomic level [134]. Suchdirect ion sputtering is only one of several mechanisms that space weathersplanetary surfaces, and studies using MESSENGER exospheric measure-ments are currently under way to determine the role that sputtering playsin the generation of Mercury’s exosphere. Thus it is highly desirable to dis-tinguish regions on the surface that are exposed to relatively higher or lowerfluxes of solar wind particles to inform exospheric studies.Differential particle fluxes to various regions on the surface can also yieldclues about magnetospheric processes. For example, particle fluxes to thesurface outside of the cusp regions (where they are mostly expected), signifythat drift and scattering processes are both occurring in the magnetosphere,allowing particles access to closed field line regions of the magnetosphere.In this work we improve on our previous estimates of proton precipitationto Mercury’s surface.5.1 MethodsIn this section, we are interested in planetary processes, and conductmost of our analyses in a planetocentric Mercury body fixed (MBF) coordi-nate system. The MBF coordinate system governs the geometry of internallygenerated fields at Mercury. The MBF and MSO +z axes are nearly identi-cal, and hence so are the MSO and MBF latitudes due to Mercury’s smallobliquity [4, 135].In order to resolve regional-scale structure in either the surface magneticfield or in proton fluxes to the surface, we systematically derive averagedpitch angle distributions over all locations around the planet where highproton fluxes are observed. We grid the surface of the planet into MBFlatitude and longitude bins, with 10◦ spacing in latitude and 20◦ spacing inlongitude and record all individual pitch angle distributions that fall into1165.1. Methodseach grid cell. This grid cell size is chosen to reveal variations in the surfacefield and fluxes on length scales as small as possible while still ensuring ahigh signal-to-noise ratio for the average PADs in each grid cell. Becauseour observations are taken at spacecraft altitudes, we map the observationlocations along magnetic field lines to the planet’s surface using the offsetdipole magnetic field, and determine the surface grid cell corresponding toeach FIPS integration period. We then evaluate an averaged pitch angle dis-tribution from all the individual PADs in each grid cell, where the averagingis done as described in Section 4.3.2.Observations were restricted to times when MESSENGER was inside themagnetosphere and spacecraft altitudes were below 3000 km. Bow shockand magnetopause boundary crossing times were not available past 15 De-cember 2012, thus we used a magnetopause model shape with the best-fittime-averaged parameters defined in Chapter 2 to determine whether obser-vations were within the magnetosphere. The altitude restriction also helpsin confining observations to within the magnetosphere, except on the day-side where the magnetopause is not more than ∼ 2500 km from the surfaceon average. We later examine more restrictive altitude bins of observations.These analyses reveal loss cones in pitch angle distributions mappingto the surface in the northern hemisphere north of 40◦N, as well as in thesouthern hemisphere between 20◦S and 40◦S. In the northern hemisphere,the observations between 40◦ and 60◦N are not part of the cusp region andthus represent a population of high proton fluxes, that we were not ableto observe previously with fewer observations. This region of high protonfluxes is consistent with higher plasma pressures in the same latitude rangeobserved at spacecraft altitudes by Korth et al. [136]. Figures 5.1 and 5.2show example observations of PADs from the northern and southern hemi-spheres, respectively. Figure 5.1 shows a clear transition in PAD character-istics with latitude: north of ∼ 70◦N latitude the PADs have single-sidedloss cones, i.e. isotropic incident population and a clear loss cone in thereflected population, while south of ∼ 60◦N loss cones are observed in boththe incident and reflected populations. These are termed double-sided losscones. By comparing Figures 5.1 and 5.2, it can be seen that the character1175.1. Methods0 20 40 60 80 100 120 140 160 18000.20.40.60.811.2α (degrees)Stacked / weighted proton counts0 20 40 60 80 100 120 140 160 18000.20.40.60.811.2α (degrees)Stacked / weighted proton counts0 20 40 60 80 100 120 140 160 18000.20.40.60.811.2α (degrees)Stacked / weighted proton counts0 20 40 60 80 100 120 140 160 18000.20.40.60.811.2α (degrees)Stacked / weighted proton countsLatitude 40o-50oNLongitude 60o-80oE Latitude 50o-60oNLongitude 0o-20oWLatitude 70o-80oNLongitude 20o-40oW Latitude 80o-90oNLongitude 140o-160oWa) b)c) d)Figure 5.1: Example pitch angle distributions (averaged over measurementstaken at altitudes < 3000 km) in the northern hemisphere shown for 10◦latitude bands. The latitude, longitude grid cell for each PAD is given in thetop right corner of each panel. The change in character of the PADs can beseen with increasing latitude. Data is more sparse in the 80◦−90◦N latitudeband, which is signified by the large error bars on the normalized fluxes forthe incident population.of the PADs in the northern hemisphere between 40◦ − 60◦N is very similarto those observed in the southern hemisphere. We attribute the double-sided loss cones observed in these regions to particles bouncing back andforth along closed magnetic field lines, with particles being lost to the sur-face on the side of the magnetic field line with the weaker magnetic field1185.1. Methods0 20 40 60 80 100 120 140 160 180−0.200.20.40.60.811.2α (degrees)Stacked / weighted proton countsLatitude 20o-30oSLongitude 60o-80oWFigure 5.2: Example PAD (averaged over measurements taken at altitudes< 3000 km) in the southern hemisphere. The latitude, longitude grid cell forthe PAD is given in the top right corner of the figure. Most PADs observed inthe southern hemisphere exhibit similar double-sided loss cones as observedat low latitudes in the northern hemisphere.strength at the planetary surface. At Mercury, the large northward offsetof the magnetic equator yields much weaker surface fields in the southernhemisphere than the northern hemisphere. Thus, particles on closed fieldlines will predominantly be lost to the southern hemisphere. This transitionin PAD characteristics allows us to determine that on average, the bound-ary between open and closed field lines in the northern hemisphere maps tolatitudes of 60◦ − 70◦N on the surface. It is also important to note that theaverage loss cone size appears to be 10◦ − 20◦ larger for the double-sidedloss cones than for the single-sided loss cones, a separate indication thatparticles are being lost to the southern hemisphere where the surface fieldis weaker.To obtain surface magnetic field strengths in the latitude-longitude gridcells where a loss cone is observed, the cut-off pitch angle (or equivalently, theloss cone size) has to be established. In each grid cell we can also calculatethe proton flux to the surface using the loss cone angle, the proton number1195.1. Methodsdensity and temperature. Calculating the cut-off pitch angle by fitting thesolution to the diffusion equation to the loss cone as described in Section4.3.3 is straightforward in the region of single-sided loss cones. Here, near-isotropic proton populations enter the cusp region along open magnetic fieldlines; some are lost to the surface, while the returning population undergoespitch-angle scattering from the plasma waves present in the region. In thisprocess the PAD diffuses from a step-function-like drop-off in the fluxes, tothe observed smooth loss cone. We thus use the solution to the diffusionequation with a step-function initial condition to fit to the single-sided PADsto obtain the cut-off pitch angle.In the case of double-sided loss cones the use of the diffusion equation isno longer physically well motivated because the PADs are in an equilibriumstate. After just one bounce period the PADs will have loss cones on bothsides corresponding to the southern hemisphere, and will thus not have aninitially step-function-like PAD that undergoes diffusion. In this regime,due to persistent pitch angle scattering as the particles bounce back andforth along the close field line, the loss cone size (which corresponds to thesouthern hemisphere) will slowly get larger as particles from the non-loss-cone part of the distribution get scattered into the loss cone. The particlesscattered into the loss cone over time will make it down to the surface, so thislarger loss cone angle of the double-sided distributions (which we term “largevoid” so as not to confuse terminology with the original loss cone) does stillreflect particles that made it to the surface, albeit largely due to scattering.Thus we can still obtain the flux of particles to the surface from this largevoid, however the flux does not correspond to a per second flux to the surface,but to a flux over the length of time the particles have spent bouncing backand forth along the field line. This time cannot be easily determined as wehave no information as to when the particles entered onto the closed fieldlines. However, by estimating how large the void is in the PADs we canobtain a flux to the surface over some average time that particles remaintrapped on closed field lines at Mercury prior to being lost to the surface,the magnetopause, or down the magnetotail through a combination of driftmotions and pitch angle diffusion.1205.1. MethodsThe void size is obtained by fitting a curve that matches the shape ofthese PADs, and since the diffusion curve is still a fairly good approxima-tion of the shape of the void, we implement it here again knowing that theparameter Dαt derived from these fits is no longer physically meaningful.Deriving a physically motivated shape model for the double-sided loss coneswould involve properly accounting for a number of different factors, includ-ing: pitch-angle diffusion along the particle’s bounce motion, various particledrift processes, and loss to the surface. It would also require knowledge ofthe length of time (number of bounce periods) taken to reach the observedPAD shape. Thus modeling of the voids is not pursued here.Calculating the surface field strength from the large voids of the double-sided PADs is not warranted because the size of the large voids does notcorrespond to the actual loss cone size that would be there in the absenceof pitch angle diffusion and at present it is not known how much they havebeen altered. It is important to note that the estimated surface field strengththat was obtained in Chapter 4 for the southern hemisphere is still justified,because the PAD in Figure 4.3 exhibits a much more isotropic incidentpopulation (and can thus be approximated as a single-sided loss cone) thanthe very clearly anisotropic incoming populations in the newly derived PADsin the southern hemisphere latitude/longitude grid cells. The discrepancybetween the previously derived PAD and the new PADs in the same latituderange is due to the binning in body-fixed longitude, in which we average pitchangle distributions from all local times. Previously, our observations in thesouthern hemisphere were all confined to the nightside, where fresh protonpopulations, which are mostly isotropic, drift in from the plasma sheet ontoclosed field lines. There are no sources of fresh proton populations on thedayside at low-to-mid latitudes, however, and because there are a factorof ∼ 2 more dayside observations than nightside observations, the daysideobservations dominate the signal in our southern hemisphere bins here. Wehave verified that if we confine our new observations to the nightside, we stilldetect nearly isotropic incident populations; however, the reduced numberof integration periods means that binning in both local time and body fixedlongitude is not yet possible. Thus here we retain the PADs binned in1215.2. Resolving regional scale surface magnetic field strengthsbody-fixed longitude but averaged over all local times.The two different regimes of the PADs (single-sided and double-sided)allow us to calculate magnetic field strengths and per-second fluxes fromsingle-sided loss cones at high latitudes in the northern hemisphere and toobtain bounce-averaged fluxes from double-sided loss cones at lower north-ern hemisphere latitudes and in the southern hemisphere. However, as thevoids in the double-sided PADs observed in the north still correspond to thesouthern hemisphere, we do not gain more information from these aboutthe fluxes to the north, only to the south. In the following two sections wedescribe our results for the surface field strengths and fluxes separately.5.2 Resolving regional scale surface magneticfield strengthsDue to the constraints mentioned in the previous Section, deriving sur-face magnetic field strengths from the latitude/longitude binned observa-tions is only feasible in the northernmost latitude range, north of the bound-ary between open and closed field lines. Not all latitude-longitude grid cellsexhibit loss cones in the PADs, we thus visually inspect each averaged PADand decide whether there is a clear loss cone to which we can fit a diffusioncurve. Although high proton fluxes from the cusp are expected in the lat-itude band between 60◦N and 70◦N at the surface, we do not observe anywell defined loss cones in this region. This may be because the transitionbetween open and closed field lines occurs in this area.High proton fluxes and clear loss cones are evident in the latitude bandbetween 70◦ and 80◦ N. Some PADs with loss cones are also observed up to90◦N, although there are fewer observations in this region and thus the errorbars are on average higher on the normalized PAD proton counts. To derivesurface magnetic field strengths in these regions we binned the observationsin different altitude ranges to reduce the error on BS. However, even fora 10 km altitude range the standard deviation in the measured magneticfield strengths at spacecraft altitudes was greater than 50 nT in some cases.1225.2. Resolving regional scale surface magnetic field strengths0 20 40 60 80 100 120 140 160 18000.20.40.60.81α (degrees)Stacked / weighted proton countsFigure 5.3: Example PAD corresponding to the 70◦ − 80◦N latitude and60◦ − 80◦E longitude grid cell with the solution to the diffusion equation fitto the loss cone.This is likely due to the large variation in plasma pressures inside the cuspat the integration times averaged in the PAD. We are not currently ableto properly account for the variations in the diamagnetic field with time indifferent regions. Thus variations in B0 translate directly into uncertaintiesin the estimated magnetic field strengths at the surface. To minimize thisproblem, instead of binning the observations in altitude, we bin them by themeasured magnetic field strength at the spacecraft altitude, ensuring thatthe diamagnetic field values are more similar at the selected times.Clear loss cones were obtained for nearly all longitude bins between 70◦and 80◦N and one longitude bin at 90◦N when data with B0 > 300 nTwere binned in the different regions on the surface. The average standarddeviation in B0 in these bins was ∼ 25 nT, about a factor of two smaller thanif the observations had been binned in altitude. In all the surface grid cellswhere PADs with clear loss cones are detected, we fit diffusion curves to thereflected population to estimate the cut-off pitch angle. Figure 5.3 shows anexample averaged PAD with a diffusion curve fit to the loss cone. This PAD1235.2. Resolving regional scale surface magnetic field strengths 120 oW   60o W    0o     60 oE  120o E  180o W   60 oN   66 oN   72 oN   78 oN   84 oN   (nT)380400420440460480500520540560580 120 oW   60o W    0o     60 oE  120o E  180o W   60 oN   66 oN   72 oN   78 oN   84 oN   (nT)380400420440460480500520540560580a) b)Figure 5.4: Regional-scale surface magnetic field strengths estimated byproton-reflection magnetometry in the northern hemisphere, a) uncorrectedfor the diamagnetic field and b) corrected for the diamagnetic field. Themaps are in a stereographic projection, looking down from above the northpole.maps to between 70◦ − 80◦N latitude and between 60◦ − 80◦E in longitude.The magnetic field strength at the spacecraft for this particular examplewas B0 = 327 ± 22 nT and the derived αc was 120◦+6−4 , yielding a surfacemagnetic field strength of 436+82−64 nT, not including the diamagnetic fieldcorrection. The uncertainties were derived using the approach described inChapter 4. From similar fits to all suitable PADs, we establish a map ofthe regional-scale magnetic field strength in the northernmost hemisphere,shown in Figure 5.4.The method of estimating the diamagnetic field described in Section 4.5has been improved upon from that described in Section 4.5. Instead of takingan average proton number density, temperature, and thermal pressure overthe entire cusp region, we take derived values of np, Tp, and Pp (the protonthermal pressure) from proton observations at specific integration times from2011 that match the times averaged in our PADs. These derived productswere made available by the FIPS instrumentation team on MESSENGER,and are described in [52]. Unfortunately the times that these products arederived do not extend into 2012 or 2013, but we can still use them to establish1245.2. Resolving regional scale surface magnetic field strengths 120 oW   60o W    0o     60 oE  120o E  180o W   60 oN   66 oN   72 oN   78 oN   84 oN   (nT)500550600650700a) b)0 50 100 150 200 250 300 350300350400450500550600650700750Longitude (degrees)Surface magnetic field (nT)Figure 5.5: a) Diamagnetic field corrected surface field strengths with errorbars as a function of longitude in the northern cusp. The number of individ-ual PADs averaged in each grid cell was between 7 and 32. b) Upper limitson the surface magnetic fields strengths (from the addition of the upper errorbars in a) to the corrected surface field values) in the cusp.average proton conditions in the latitude bins used here, although not in thelongitude bins due to the limited number of observations. We set the errorson the derived FIPS products to be 1 standard error in the estimated np,Tp, and Pp values within each latitude bin.The FIPS estimated average Pp at spacecraft altitude is 1.62± 0.17 nPaat 70◦ − 80◦ N latitude. Using this total proton pressure at the spacecraftaltitude, we calculate the perpendicular pressure at the spacecraft usingequation (A.7), correcting for the fact that Pp was derived assuming anisotropic distribution function. We also calculate the perpendicular pressureat the surface using equation (A.9). From equation (A.1), we obtain thecorresponding diamagnetic field for these perpendicular pressures, and takethe average of the two to get an approximate range for the diamagnetic fieldat the surface. The errors on the corrected surface magnetic field strengthsincorporate the average error in the diamagnetic field, and are shown inFigure 5.5a, while the upper limits on our proton-reflection magnetometryderived surface magnetic field strengths are shown in Figure 5.5b.On average the errors onBS are higher than the estimated variation inBS1255.3. Particle fluxes to Mercury’s surfacefrom one longitudinal bin to the next. However, we can draw two importantconclusions from these surface field estimates. One is that we do observean increased magnetic field strength with increasing latitude, BS for the80◦− 90◦N bin is significantly higher than the BS for the latitude bin belowit. More importantly, the observed fields at high latitudes combined with oursouthern hemisphere surface field result in Chapter 4 are somewhat weakerbut consistent within the uncertainty with an offset dipole (see Figure 4.8).This result establishes that higher degree and order core or crustal fields onMercury must either be very weak, or must be on length scales much smallerthan our longitudinal bin of ∼ 300 km.5.3 Particle fluxes to Mercury’s surfaceWe expand our calculation of the proton flux to the surface in the north-ern cusp and southern hemisphere low latitude region described in Section4.5 to incorporate the regional scale proton loss cones observed in Section5.2 above, and build a proton precipitation map of Mercury’s surface. Wecan obtain fluxes to the surface from both single-sided and double-sidedPADs as described in Section 5.1, with the caveat that the fluxes deter-mined from double-sided loss cones are over an average bounce life-time ofprotons in Mercury’s magnetosphere. It is important to note that we alsoassume for these double-sided loss cones that the original incident popula-tion was isotropic, prior to becoming trapped on closed field lines, i.e. theseflux estimates are upper limits. The double-sided loss cones measured inthe northern hemisphere correspond to particles being lost to the southernhemisphere surface. Therefore, our measurements allow for the resolutionof regional-scale fluxes to the surface in the northernmost region of thenorthern hemisphere, where the single-sided loss cones are observed, and toregions of the southern hemisphere mapped by the double-sided loss conesobserved in the north and the PADs observed in the south.To determine the proton fluxes to the surface, we average individualPADs from all spacecraft altitudes (up to 3000 km) for the southern hemi-sphere measurements but only average over observations at < 550 km al-1265.3. Particle fluxes to Mercury’s surface0 20 40 60 80 100 120 140 160 18000.20.40.60.81α (degrees)Stacked / weighted proton counts)b)a0 20 40 60 80 100 120 140 160 18000.20.40.60.81α (degrees)Stacked / weighted proton countsFigure 5.6: Example pitch angle distribution exhibiting symmetry in thedouble-sided loss cone. The same double-sided PAD is shown in both panels,from 50◦ to 60◦N latitude, 0◦−20◦E longitude. Diffusion equation fits to theα = 0◦− 90◦ side yield a void size of 72◦+6−27, while fits to the α = 90◦− 180◦yield a void size of 80◦+1−10, and are thus in agreement within the uncertainty.titude in the 70◦ − 80◦N latitude band where low spacecraft altitudes areavailable. We do not apply a similar B0 binning to these averaged PADs asfor the surface field calculations. This approach is justified for determiningthe fluxes because by restricting the range of magnetic field measurementsat the spacecraft we may exclude times of high proton fluxes into the mag-netosphere and thus would only obtain a lower limit of the fluxes to thesurface. We also significantly increase the signal to noise ratio in our aver-aged PADs with this approach. The proton number density and temperatureneeded for the derivation of the flux to the surface (as described in detailin Appendix A.2) was calculated in 10◦ latitudinal bins from FIPS momentestimates as described in Section 5.2. To establish where the double-sidedPADs observed in the northern hemisphere map to in the south, we tracedthe magnetic field line from the northern hemisphere observation point atspacecraft altitudes to the southern hemisphere foot-point location on thesurface of the planet.Observations from the northern hemisphere double-sided loss cones map1275.3. Particle fluxes to Mercury’s surfaceto latitudes 20◦ − 30◦S, while observations from the southern hemispheremap to latitudes 20◦ − 40◦S on the surface. MESSENGER cannot observemore equatorward latitudes on the surface in either the northern or southernhemisphere because the high spacecraft altitudes over the equatorial regiondo not allow it to cross closed field lines that map to the surface near theequator.We determine the size of the large voids in the double-sided PADs byfitting the solution to the diffusion equation to the PAD. Error bars on thenormalized proton flux in the PADs are on average lower for the reflectedpopulation, i.e. α = 0◦ − 90◦ for the southern hemisphere observations andα = 90◦ − 180◦ for the northern hemisphere observed PADs. We thereforefit the diffusion curve to the reflected population in each hemisphere. Thiscarries the assumption that the voids on both sides of the PAD essentiallyequilibrate (i.e. the PAD is symmetric about α = 90◦) after one full bounceperiod, amounting to the same void size on both sides (as they both corre-spond to the southern hemisphere surface). We test this assumption on afew example cases for which the normalized fluxes on both sides of the PADhave similar error bars, and find that they are in agreement within the erroron the fit αc (see Figure 5.6).Figure 5.7 shows the proton flux map to the surface in the southernhemisphere. The fluxes are nearly an order of magnitude higher than thevalue in the southern hemisphere low latitude band derived in Section 4.5.This discrepancy is due to the nearly single-sided loss cone nature of thePAD in Figure 4.3b, which yields a per second flux to the surface, while thesmeared out double-sided loss cones correspond to proton populations thathave been trapped for numerous bounce periods and thus yield a bounce-averaged flux to the surface. Thus this is not a valid comparison, as the twoestimates are established over very different time-scales. There were twolongitudinal/latitudinal grid cells (in the latitude range of 20◦ − 30◦S) thatwere observed both from northern and southern hemisphere PADs. Theflux values estimated from these two methods were in good agreement inthe two grid cells; for example, for the longitude range of 80◦− 100◦E a fluxof 1.9+0.7−1.3 × 1012 particles m−2 (bounce life-time)−1 was obtained from the1285.3. Particle fluxes to Mercury’s surfaceo  120o W   60 oW    0 o    60o E  120 oE  180o W   75o S   60o S   45o S   30o S   15 S  (m-2 (bounce life-time)-1)11.21.41.61.822.22.42.62.83x 1012 120o W   60 oW    0 o    60o E  120 oE  180o W   75o S   60o S   45o S   30o S   15o S   00.10.20.30.40.50.6a) b)Figure 5.7: Regional-scale resolution of proton flux to the southern hemi-sphere. a) Proton flux to the southern hemisphere surface on a steregraphicprojection plot (looking through the planet from above the north pole). b)Ratio of the error on the flux to the flux estimate. The number of individualPADs averaged in each grid cell was between 50 and 1100.southern hemisphere measurements and a flux of (1.9± 0.8)× 1012 particlesm−2/(bounce life-time)−1 was derived from the northern hemisphere PAD.In the northern hemisphere, we also establish a flux map in 20◦ longitu-dinal bins between 70◦ and 80◦N latitudes from observations at less than 550km altitudes, shown in Figure 5.8. The results for the northern and south-ern hemispheres are also shown together in a global flux map in Figure 5.9.The global flux map shows particle fluxes in regions where proton-reflectionmagnetometry directly confirms proton precipitation all the way to the sur-face (i.e., where loss cones are observed). From Figure 5.9 it can be seenthat the fluxes in the north are on average a factor of 2 higher than in thesouth, as the open field lines provide a direct path for particles entering fromthe solar wind.An intriguing result that can be seen from the northern hemisphere fluxmap in Figure 5.8 is the apparent flux increase in longitude bins near 0◦and 180◦ longitudes compared to the average fluxes observed near 90◦E and90◦W. We attribute this flux increase to Mercury’s 3:2 spin orbit resonance,1295.3. Particle fluxes to Mercury’s surface 120 oW   60o W    0o     60 oE  120o E  180o W   60 oN   66 oN   72 oN   78 oN   84 oN      0o   (m-2 s  )2.52.62.72.82.933.13.23.33.43.53.6x 1012 120 oW   60o W    0o     60 oE  120o E  180o W   60 oN   66 oN   72 oN   78 oN   84 oN      0o   00.10.20.30.40.50.6a) b)-1Figure 5.8: Regional-scale resolution of proton flux to the northern hemi-sphere high latitude surface. a) Proton flux to the northern cusp surface(looking down from above the north pole). b) Ratio of the error on the fluxto the flux estimate. Uncertainties on the fluxes are lowest in the region ofinterest, near 0◦ and 180◦ longitudes, although even in these regions the ratiois still only slightly lower than the signal detected. The number of individualPADs averaged in each grid cell was between 11 and 172.   0o    60oE  120oE  180oW  120oW   60oW    0o    80o S   40o S   0o    40 oN   80 oN   Flux11.522.533.5x 1012Figure 5.9: Global map of proton flux to the surface.1305.3. Particle fluxes to Mercury’s surfacewhich causes the planet’s 0◦ and 180◦ longitudes to always face the Sunduring perihelion or aphelion. These longitudes experience local noon atperihelion where the solar wind density is highest, and thus they receivehigher plasma fluxes from the solar wind through open field lines in thecusp (which is on the dayside). A map of the average solar wind densityrelative to the maximum expected at Mercury’s distances from the Sun isshown at different longitudes on the planet in Figure 5.10, with maximumsolar wind densities always occurring near 0◦ and 180◦ longitudes. As ourobservations have been averaged over several Mercury years in the 2.5 Earthyears included, it is possible for this orbit-averaged signal to be present inour measurements. Such a flux difference would have been present sinceMercury entered its 3:2 spin orbit resonance. Higher space weathering ratesand greater exospheric generation from these regions is therefore expected.Any differences in the surface elemental composition at these longitudes mayintroduce different exospheric species compared to the background average.If differences in exospheric species are correlated with surface elemental com-positions at these longitudes, the origin of such particles may be traced backto this region due to increased solar wind sputtering in this area.We expect the plasma flux to be 57% higher in the longitude bins near0◦ and 180◦, then near longitudes of 90◦E and 90◦W (Figure 5.10), whichface the Sun at approximately the mean orbital distance of Mercury fromthe Sun. This is because the solar wind density decreases as 1/r2 with dis-tance from the Sun, yielding a 57% difference in solar wind density betweenperihelion and the mean orbital distance of 0.39 AU. Our map of the fluxesshows a signal of ∼ 40%, slightly smaller than the expected value. Figure5.8b shows the ratio of the uncertainty on the flux to the flux estimates foreach longitude bin cell for the northern hemisphere (similar plot in Figure5.7b shows the ratio for the southern hemisphere), where the errors takeinto account the error on αc as well as one standard error on the derived npand Tp values in each 10◦ latitude range. The error to flux ratio is lowest inthe area of interest and is ∼ 30% there, signifying that the flux differenceis likely a genuine signal, however, more observations are needed to fully re-solve its amplitude and longitudinal extent. A less pronounced, but similar1315.4. ConclusionsFigure 5.10: Predicted relative solar wind density at local noon as a functionof longitude, normalized to the maximum value.signal is also observed in the southern hemisphere fluxes (Figure 5.7). Thissignal is not significant above the uncertainty level, but may hint at the factthat higher proton fluxes to the southern magnetospheric cusp occurring atperihelion can reach to lower latitudes (where our observations are made)due to particles drifting from open to closed field lines.5.4 ConclusionsIn this chapter, we have extended our original work on proton-reflectionmagnetometry at Mercury with an additional 1.5 years of observations toresolve regional-scale variation in the surface magnetic field strength as wellas in the proton flux to the planet’s surface. Our most significant findingscan be summarized as follows:• A new region of proton fluxes in the northern hemisphere is detected,observed at all local times in the latitude range 40◦ − 60◦N, in agree-1325.4. Conclusionsment with plasma pressure measurements at spacecraft altitudes con-ducted by Korth et al. [136]. The pitch angle distributions observedin this region exhibit double-sided loss cones, similar to those in oursouthern hemisphere observations.• The character change in the PADs from the southern hemisphere lowlatitude region to the northern hemisphere polar region, indicates theboundary between closed field lines at mid to low latitudes and openfield lines at high latitudes. We find that this boundary in the northernhemisphere is between 60◦ − 70◦N.• Upper limits for the surface magnetic field strength estimates in thenorthernmost region are in agreement with the best-fit time-averagedmagnetospheric model predictions, with a maximum estimated surfacefield value from proton-reflection magnetometry of ∼ 700 nT and acorresponding model prediction of ∼ 750 nT. There are no regionswhere the surface field strength is stronger than that expected froman offset dipole field with a moment of 190 nT R3M [19]. This suggestseither that higher degree and order core or crustal fields are very weak,or that they occur on length scales much smaller than the resolutionof our measurements (∼ 300 km).• Proton fluxes to the surface are estimated for the southern hemispherein the latitude range 20◦−40◦S and in the northern hemisphere at lat-itudes 70◦ − 90◦N. Despite the fairly large uncertainties in these mea-surements, two significant conclusions can be drawn: 1) the per secondfluxes everywhere in the northern cusp region are approximately a fac-tor of 2 higher than the bounce-averaged fluxes in the south (whichoccur over much longer time-scales); 2) increased fluxes are detectedin the north near 0◦ and 180◦ longitudes compared with fluxes nearlongitudes of 90◦E and 90◦W. This may reflect the increased incidentsolar wind density at these longitudes at local noon that results fromthe 3:2 spin-orbit resonance of Mercury. Although the signal that weobserve is just above the level of the uncertainty, with further obser-1335.4. Conclusionsvations it may be possible to better resolve the amplitude of the signalas well as its longitudinal extent.Finally, new low-altitude observations soon to be acquired by MESSEN-GER in the months leading up to its planned impact into Mercury on 28March 2015, will allow for the surface magnetic field strength estimatesto be tested by orbital MAG observations. These measurements, in con-junction with low altitude proton-reflection magnetometry, may be able tosignificantly improve the resolution of our current proton-reflection magne-tometry estimates at the surface and may reveal higher degree and ordershort-wavelength structure in the internal field.134Chapter 6ConclusionsIn this final chapter, we summarize key results obtained in this thesisand also list future directions.In Chapter 2 we explored the nature of the initial interaction betweenMercury’s global magnetic field and the solar wind and IMF by studyingMercury’s bow shock and magnetopause boundaries. The results from thiswork exemplify that the boundaries of Mercury’s magnetosphere, and there-fore the magnetosphere itself, are highly variable and greatly affected by theinterplanetary medium. Although we do not have continuous solar windobservations upstream of the bow shock, variations in the location of theseboundaries can still be perceived on the timescale of a few minutes, as theboundaries are observed to move back and forth across the spacecraft asthe solar wind and IMF conditions change. Solar wind conditions are how-ever available on an orbit-by-orbit basis, allowing us to quantify variationsin the boundaries on the timescale of hours. The large observed spreadin the boundary locations shows that the magnetosphere can expand andcontract significantly from one orbit to the next, and we found that this ex-pansion/contraction is controlled dominantly by the solar wind parameters.By building probability density maps of the boundaries, we determined thehighest likelihood shape and region for these boundaries on average, andalso under various solar wind conditions. We then parametrized the modelshape of the magnetopause as a function of the solar wind ram pressure, andthe bow shock as a function of the Alfve´n Mach number, the two dominantsolar wind influences on these boundaries. These parametrized empiricalshape models for the magnetopause and bow shock, which are the mostimportant results from this work, can be used to define the boundary con-ditions for Mercury’s magnetosphere and are thus highly relevant for any135Chapter 6. Conclusionsmagnetospheric study.A fruitful area of further inquiry that may be addressed with more obser-vations is how reconnection affects the magnetopause location at Mercury.At Earth it has been shown that during times of southward directed IMF,reconnection at the dayside magnetopause erodes magnetic flux on the day-side and transfers it into the tail [57, 58]. This moves the magnetopauseplanetward and increases the flaring of the magnetotail. We have shownin Chapter 2 that on average Mercury’s magnetotail is highly cylindrical;this low flaring implies that magnetic flux has a short residence time in thetail and is transferred back to the dayside much more quickly than our ob-servations can resolve. This observation supports the very short Dungeycycle time at Mercury. However, further observations may be able to in-form our understanding of the reconnection related planetward erosion ofthe dayside magnetopause. It is possible however, that no significant IMFBz dependence will be detected, since at Mercury reconnection has beenobserved to occur under varying magnetic shear angles, i.e. not just un-der southward pointing magnetic fields [34]. Resolving this issue would bepossible by analyzing many more Mercury years of MESSENGER obser-vations to extract any underlying magnetic shear angle dependence on themagnetopause shape and location. With more observations it would also bepossible to derive a more accurate shape model for the magnetopause, whichincludes asymmetries near the poles due to the cusp regions as well as anyother potential asymmetries in the shape causing departures from a figureof revolution, which could not be resolved from the data used in Chapter 2alone.In Chapter 3, using magnetic field observations we investigated the re-gion of space where solar wind particles are expected to reach Mercury’smagnetosphere, the magnetic cusp region. Due to MESSENGER’s eccentricorbit, observations of only the northern hemisphere cusp region are availableat this time. Through observations of diamagnetic depressions in the mag-netic field on the dayside at high northern latitudes, that were accompaniedby high frequency variations in the field indicative of the presence of plasma,we mapped out Mercury’s northern cusp region and the associated plasma136Chapter 6. Conclusionspressure within it. Prior to MESSENGER, numerous models of Mercury’smagnetosphere had made predictions of the latitudinal and local time extentof Mercury’s magnetic cusps and therefore the boundary between open andclosed field lines (see review in [137]). Since the models did not incorporatethe north-south asymmetry of the magnetic field (as this was not knownprior to MESSENGER), most models predicted the northern cusp to havethe same latitudinal extent as the southern cusp, and also predicted thenorthern cusp to extend much further south (some as far south as 30◦N),than what we observe. Our observations from 3 Mercury years of data undervarying solar wind conditions indicate that the limits of the northern cuspare 56◦−84◦ MSO latitude at spacecraft altitudes and 7−16 h in local time.One of the major results from the work presented in Chapter 3 is thatthe observed plasma pressure in the northern cusp region is highly affectedby the solar wind ram pressure and the IMF Bx direction. We find that theplasma pressure is significantly increased for high ram pressures and for ananti-sunward IMF configuration, which facilitates plasma transport into thenorthern cusp [107]. An equally significant result is that from the measuredplasma pressures in the cusp at spacecraft altitudes we establish the plasmaflux down to the surface of the planet. This is found to be in good agreementwith the predicted average flux over perihelion and aphelion conditions tothe northern hemisphere by Sarantos et al. [110]. Due to the north-southasymmetry of the internal field, we also predict that the particle flux to thesurface near the southern hemisphere cusp is a factor of 4 greater than inthe north, implying a greater exospheric source from this region and higherspace weathering of the surface.Investigating the affects of magnetopause reconnection on the magneticcusps as indicated by the IMF Bz and By components is an interesting areafor further study. By the end of the MESSENGER mission there will bealtogether 17 Mercury years of observations of the northern cusp region,which can be used to investigate IMF By effects on the local time extent ofthe cusp that have been observed to occur at the Earth [102, 104]. Zhou etal. [103] showed at the Earth that during times of southward IMF, there isa clear local time shift in the cusp location depending on the sign of IMF137Chapter 6. ConclusionsBy. This is attributed to a shift in the center of the reconnection site bythe IMF By component; positive By is expected to shift the reconnectionsite duskward in the northern hemisphere and thus also shift the region ofopen flux, i.e. the cusp, duskward; the opposite is expected for negativeBy. As was shown in Chapter 3, the IMF By component is more stableat Mercury than the Bz component, thus such a study focusing on the Byeffects on the cusp from 17 Mercury years of observations by MESSENGERis feasible and would improve our understanding of reconnection relatedaffects on Mercury’s magnetosphere.In Chapter 4 of this thesis, we present a new method to quantify theintensity of solar wind proton precipitation to Mercury’s surface and makethe first measurements of magnetic field strength at the planet’s surface us-ing magnetic field and plasma spectrometer observations by MESSENGER.The two most significant findings from this work are: 1) Loss cone obser-vations directly confirm particle precipitation to the surface and show thatsolar wind plasma persistently bombards Mercury’s surface not only in themagnetic cusp regions as expected but over a large percentage of the entiresouthern hemisphere. (2) The north-south asymmetry in Mercury’s long-wavelength magnetic field structure is confirmed at the surface, not just atspacecraft altitudes, independently confirming this unusual and challenging(from a dynamo modeling perspective) feature of the global planetary fieldand providing key constraints on Mercury’s internal dynamo.The asymmetry in Mercury’s magnetic field strength causes hemisphericdisparities in the efficiency of solar wind particle penetration into the magne-tosphere, and subsequently the surface. The persistent particle precipitationwe observe to low southern latitudes on the planet’s nightside implies thatmost of the southern hemisphere is continuously bombarded by plasma, aresult that stands in contrast with the canonical view that a global magneticfield protects the surface of an airless body from bombardment by the solarwind. Such precipitation plays a major role in the generation of Mercury’sexosphere and space weathering of the surface. This widespread plasmabombardment means that Mercury’s surface might show limited latitudinalvariation in spectral signatures of space weathering, a result that is in agree-138Chapter 6. Conclusionsment with recent MESSENGER findings [138] but not with predictions frommagnetospheric models prior to MESSENGER observations (e.g., [85]). Ourresults demonstrate for the first time that proton reflection magnetometrycan be applied successfully to measure the planetary surface field (provid-ing the only means to do so at Mercury currently) and to quantify plasmaprecipitation directly to the surface. This technique may prove advanta-geous alongside electron reflectometry, at other planetary bodies and couldbe applied at Ganymede, with the upcoming JUICE mission.Significant wave activity has been observed inside Mercury’s magneto-sphere [128, 139], which can cause pitch angle diffusion of protons. In futurework it would be fruitful to investigate the nature of this scattering processby determining the associated diffusion coefficient, since the diffusion coeffi-cient is dependent on the wave modes involved in the scattering. One possi-ble way to determine the diffusion coefficient is from the power spectrum ofthe magnetic fluctuation levels [36]; as there are many different wave modescausing the diffusion, the diffusion coefficient will likely be a superpositionof different wave power spectra. Anderson et al. [139] have documentedvarying levels of magnetic fluctuations inside the magnetosphere depend-ing on external conditions. It would thus be interesting to investigate howthe diffusion coefficient varies with external conditions, and if any changesin pitch angle diffusion are observed with proton-reflection magnetometry.Once the diffusion coefficient is established, it would be possible to estimatethe diffusion timescale associated with the pitch angle diffusion process inMercury’s magnetosphere.In Chapter 5, we extend our proton-reflection magnetometry work atMercury with 1.5 years of additional observations to further probe Mercury’ssurface magnetic field structure and better resolve proton flux precipitationto the surface. The observed transition from double-sided to single-sided losscones in the pitch angle distributions marks the boundary between open andclosed field lines and is shown to occur between 60◦ and 70◦N on the surface,in agreement with the lower latitudinal boundary of the cusp we detected atspacecraft altitudes in Chapter 3. We map all the regions on the surface ofthe planet in 10◦× 20◦ latitude/longitude grid cells where proton loss cones139Chapter 6. Conclusionsare observed; these indicate the regions where proton precipitation directlyimpacts the surface. Our observations allow for the estimation of surfacemagnetic field strengths in the northern hemisphere and for the calculationof proton fluxes both to the northern and southern hemisphere.Most significantly, we find that in the northernmost region, regional-scalevariations in the surface magnetic field strength must be either very weakor be on length scales much smaller than the resolution of our observations(∼ 300 km). The observed increase in magnetic field strength with latitudeis consistent with the latitudinal magnetic field variation predicted by anoffset dipole field. We also find that the bounce-averaged fluxes observedto the southern hemisphere low latitude region are approximately a fac-tor of two smaller than the instantaneous fluxes estimated to the northernhemisphere cusp region, although these are likely to be an order of mag-nitude lower instantaneously as was shown in Chapter 4. Mercury’s 3:2spin-orbit resonance is expected to cause a variation in proton fluxes to thesurface with body-fixed longitude due to solar wind density changes alongMercury’s eccentric orbit. We detect an increase in proton fluxes near 0◦and 180◦ longitudes, consistent with the expected signal; however, since themeasured flux increase is only slightly above the uncertainty, further obser-vations are needed to better resolve its amplitude and longitudinal extent.Such a longitudinal signature in proton fluxes to the surface may affect theexospheric species observed at Mercury if there are large-scale longitudinalvariations in the surface composition, and is also expected to be accompa-nied by differential space weathering of the surface in these regions.Finally, in the last part of its mission phase, MESSENGER will investi-gate a completely new and unexplored region of Mercury’s magnetosphere,that at low altitudes. The low altitude campaign of MESSENGER, whichwill last approximately 10 months of orbital observations prior to impactinto Mercury, will allow measurements to be taken in the northern cusp inthe altitude range of 25 − 150 km. 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McNutt, “Plasma distribu-tion in mercury’s magnetosphere derived from messenger magnetome-ter and fast imaging plasma spectrometer observations,” Journal ofGeophysical Research: Space Physics, vol. 119, no. 4, pp. 2917–2932,2014.[137] S. Orsini, L. G. Blomberg, D. Delcourt, R. Grard, S. Massetti, K. Seki,and J. Slavin, “Magnetosphere-Exosphere-Surface Coupling at Mer-cury,” Space Sci. Rev., vol. 132, pp. 551–573, Oct. 2007.[138] N. R. Izenberg, R. L. Klima, S. L. Murchie, D. T. Blewett, G. M.Holsclaw, W. E. McClintock, E. Malaret, C. Mauceri, F. Vilas, A. L.Sprague, J. Helbert, D. L. Domingue, J. W. Head, T. A. Goudge, S. C.Solomon, C. A. Hibbitts, and M. D. Dyar, “The low-iron, reducedsurface of Mercury as seen in spectral reflectance by MESSENGER,”Icarus, vol. 228, pp. 364–374, Jan. 2014.[139] S. A. Boardsen, J. A. Slavin, B. J. Anderson, H. Korth, D. Schriver,and S. C. Solomon, “Survey of coherent 1 Hz waves in Mercury’sinner magnetosphere from MESSENGER observations,” Journal ofGeophysical Research (Space Physics), vol. 117, p. 0, Sept. 2012.157Appendix ADiamagnetic field andparticle flux calculation forproton reflectionmagnetometryA.1 Diamagnetic magnetic deficit calculationIn this section we discuss the derivation of the diamagnetic field de-pression at the surface presented in Section 4.5 for the northern cusp andsouthern hemisphere low latitude band. The diamagnetic field is due to par-ticle motions perpendicular to the magnetic field, and thus the diamagneticdepression can be calculated from the perpendicular particle pressure givenby∆B =√2µ0P⊥. (A.1)Thus we need to calculate the perpendicular pressure at the surface toobtain the diamagnetic field at the surface. The pressure is the secondmoment of the phase space distribution function, or phase space density,and the perpendicular component can be calculated from this as was done inSection 3.4.1. The total pressure can also simply be obtained from the idealgas law P = npkTp where the proton number density, np and temperature,Tp at the spacecraft are needed, and this value can be obtained from FIPSmeasurements. Thus our aim is to obtain the perpendicular pressure at thesurface from the total pressure at the spacecraft altitude, taking into account158A.1. Diamagnetic magnetic deficit calculationFluxA0ΑΠΑcFluxB0ΑΠΑcFigure A.1: (A) Cartoon diagram showing a loss cone in an idealized pitchangle distribution in the southern hemisphere (A) and northern hemisphere(B).the observed loss cone from the proton reflectometry measurements.From Liouville’s theorem, the phase space density of particles is pre-served along the field line in the absence of particle collisions and loss; hereit is also assumed that the particles’ total velocity does not change alongthe field line (an approximation which is needed for proton reflectometry).It is then apparent from equation (3.1) that the total pressure stays con-stant along the field line in the absence of collisions and loss. As we knowfrom our proton-reflection magnetometry measurements however, particlesare being lost to the surface, and so the pressure that we measure at thespacecraft altitude needs to be corrected for this loss to obtain the pressureat the surface.The proton number density, np, obtained from FIPS measurements (seeSection 4.5 and [130]) was derived assuming an isotropic proton distributionat the spacecraft altitude, which our observations have shown to not be agood assumption. In reality, the distribution has a loss cone from 0 to αcin the southern hemisphere, and from αc to pi in the northern cusp region(see Figure A.1 A and B). Having such a void in the pitch angle distributionleads to an underestimate in the phase space density, fp, because underthe isotropy assumption the observed proton number counts by FIPS arespread evenly over 0 to pi in pitch angle space, when in fact those particles159A.1. Diamagnetic magnetic deficit calculationare only coming from the non-void region of the angle space. Thus, thecorrected phase space density should be higher, and we need to accountfor this enhancement in order to obtain accurate estimates of the protonpressure and flux (as they both depend on the phase space distribution).The number density is given by the zero-order velocity moment of thedistribution functionn =∫f(v)d3v (A.2)and in spherical coordinates we have d3v = v2 sinαdφdαdv. The protonnumber density, np is equal to the integral over the true phase space density,fp0, from αc to pi in the southern hemisphere and 0 to αc in the northernhemisphere, whereas in the derivation of np from FIPS measurements, theintegral was implicitly assumed to run from 0 to pi over a phase space dis-tribution that is isotropic. Let fp0 be the true phase space density, while fpis the phase space density of an isotropic distribution with density np:np = 2pi∫v2dv∫ pi0sin(α)dαfp(v, α) = 2pi∫v2dv∫ piαcsin(α)dαfp0(v, α),(A.3)shown here for the southern hemisphere example. If we write f =FV (v)A(α) where V (v) is the speed distribution and A(α) is the pitchangle distribution, and F is a constant, then Vp(v) = Vp0(v) holds if thetemperatures are the same. The pitch angle dependence can be written asAp(α) = 1 for all α, and Ap0(α) = 0 for α = 0 to αc and Ap0(α) = 1 forα = αc to pi in the southern hemisphere (Figure A.1A), while the reverse istrue for Ap0 in the northern hemisphere (Figure A.1B).From equation (A.3), the ratio Fp to Fp0 is given by the ratios of thesolid angles with non-zero A(α):Fp0Fp=∫ pi0 sin(α)dα∫ piαcsin(α)dα=21 + |cos(αc)|, (A.4)where the limits of integration are again for the southern hemisphere160A.1. Diamagnetic magnetic deficit calculationcase, but the result with the absolute value sign accounts for both thenorthern and southern hemisphere cases (since αc < pi/2 in the southernhemisphere but > pi/2 in the northern hemisphere).The perpendicular pressure is given byP⊥ =∫v2⊥f(v, α)dvdα. (A.5)For an isotropic distribution it can be shown that P⊥/P = 2/3 so thatP⊥ = 2/3npkTp. However, since we don’t have an isotropic distribution,the ratio of P⊥/P will be different as we have to consider equation (A.5)integrated over only the range of non-zero A(α). Thus we need to take intoaccount the difference between the phase space distributions, and so we needthe factor Fp0/Fp.Because we can obtain the total pressure for the isotropic distributionfrom FIPS measurements, we take the ratio of the perpendicular pressure forthe loss cone distribution to the total pressure for the isotropic distributionP⊥p0Pp=∫v2⊥fp0dvdα∫v2fpdvdα=2piFp0∫v4V (v)dv∫ piαcsin3 αdα2piFp∫v4V (v)dv∫ pi0 sinαdα, (A.6)where the limits of the integration in the numerator are for the southernhemisphere case, but the results below apply to both cases. Using equation(A.4), we arrive atP⊥p0 =[1−13(1 +∣∣cos3 αc∣∣1 + |cosαc|)]Pp. (A.7)This result is the ratio at the spacecraft altitude, but the actual mag-netic field due to the plasma at the surface must consider the perpendicularplasma pressure at the surface. To calculate the perpendicular pressure atthe surface, we need to map the true phase space density at the spacecraftfp0 to the surface. By Liouville’s theorem, fp0 remains constant along phasespace trajectories, so we use the same fp0 at the surface to evaluate thepressure. Note however that if we assume that the surface is perfectly ab-161A.2. Particle flux calculationsorbing, then there will be no upward going particles and the only integrateover the downgoing particles. That is, at the surface we haveP⊥p0,surfPp=2piFp0∫v4V (v)dv∫ pipi/2 sin3 αdα2piFp∫v4V (v)dv∫ pi0 sinαdα, (A.8)which yieldsP⊥p0,surf =23(1 + |cosαc|)Pp (A.9)where the absolute value sign again takes care of both the northern andsouthern hemispheres.Because the system is not infinite, but has gradients in the magneticfield between the spacecraft and the surface, the simple expression for ∆Bin equation (A.1) is not precise and the actual ∆B should be somewherebetween that indicated at the spacecraft altitude and that at the surface.Substituting in values for αc, np, and Tp in equation (A.9) and taking ∆Bto be between√2µ0P⊥p0 and√2µ0P⊥p0,surf , we arrive at the results listedin Section 4.5 for the diamagnetic field.A.2 Particle flux calculationAs we have more information at our disposal from the loss cone size thanwhat we had available in Section 3.4.1, we can do a more accurate calculationof the proton surface flux in the northern hemisphere cusp region as well ascalculate the surface flux in the southern hemisphere low latitude band.The flux of particles per unit area per unit time with velocity parallel tothe magnetic field line v‖ through a surface normal to the magnetic field isgiven byΦ =∫fp0(v)v‖d3v, (A.10)where we are using the parallel velocity (v‖ = v cos(α)) because we wantthe flux through the surface, thus we need the velocity that is parallel tothe surface normal. This assumes that the field lines are perpendicular to162A.2. Particle flux calculationthe surface right at the surface. We assume gyrotropy again, such that theintegral over dφ = 2pi and use the same decomposition for the phase spacedensity as in Section A.1. We also need to ensure that the flux is obtainedfrom the corrected, or true, phase space density fp0 as we established inSection A.1. The flux can then be rewritten asΦ = 2pi∫v3Vp0(v)dv∫Fp0A(α) cos(α) sin(α)dα, (A.11)where the integral over α runs from 0 to αc in the northern hemisphere,and αc to pi in the southern hemisphere. This integral will yield an upperlimit for the flux because by integrating over these values in pitch angle weare assuming that the loss cone remains completely filled in for the down-going (incident) particles and that all of the particles within the loss coneare absorbed at the surface.Fp0 is simply a constant and can be written in terms of Fp as fromequation (A.4) above. We assume as before that Vp0 = Vp, and we can thenrewrite equation (A.11) asΦ = 2pi∫v3Vp(v)dv∫Fp21 + |cos(αc)|A(α) cos(α) sin(α)dα, (A.12)which due to fp(v) = FpVp(v) (where we have assumed that the phasespace density has no angular dependence as before since A(α) = 0 or 1) canbe simplified to beΦ = 2pi21 + |cos(αc)|∫v3fp(v)dv∫ piαccos(α) sin(α)dα (A.13)for the southern hemisphere integration limits. In order to solve equation(A.13), we need to assume a Maxwellian velocity distribution:f(v) = np(mp2pikTp)3/2exp(−mpv22kTp), (A.14)where np is the number density quoted in Section 4.5. Substituting all163A.2. Particle flux calculationthis into equation (A.13), we findΦ = 2pinp(mp2pikTp)3/2 21 + |cos(αc)|∫ ∞0v3 exp(−mv22kTp)dv∫ piαccos(α) sin(α)dα,(A.15)which yieldsΦ = npsin2 αc(1 + |cosαc|)√2kTppimp. (A.16)Due to the absolute value sign, this equation is again valid for both thesouthern and northern hemispheres. Substituting in values for the parame-ters yields the results discussed in Section 4.5.164

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