2

Transcription

1 Solar Wind Plasma: Kinetic Properties and Micro-Instabilities by Justin Christophe Kasper Submitted to the Department of Physics in partial fulllment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY December 2002 c Justin Christophe Kasper, MMII. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author... Department of Physics December 13, 2002 Certied by... Alan J. Lazarus Senior Research Scientist / Lecturer Thesis Supervisor Accepted by... Thomas J. Greytak Chairman, Department Committee on Graduate Students

2 2

3 Abstract Solar Wind Plasma: Kinetic Properties and Micro-Instabilities by Justin Christophe Kasper Submitted to the Department of Physics on December 13, 2002, in partial fulllment of the requirements for the degree of Doctor of Philosophy The kinetic properties of ions in the solar wind plasma are studied. Observations of solar wind + H and +2 He by the Faraday Cup instrument component of the Solar Wind Experiment on the Wind spacecraft show that these ions have magnetic eldaligned, convected, bi-maxwellian velocity distribution functions. The analysis yields the best-t values of the bulk velocity, ~ U, number density n, and parallel Tk and perpendicular T? temperatures of each of the ion species. The accuracy of each of these measurements is studied and an absolute calibration of the Faraday Cup is performed, demonstrating the accuracy of the densities to 2%. The range of the proton temperature anisotropy R p T?p =T kp is explored, and it is demonstrated that thermodynamic concepts such as the double adiabatic equations of state are insucient approximations for a kinetic description of the solar wind plasma. It is shown that R p is constrained on macroscopic timescales by Coulomb relaxation and the expansion of the solar wind, and on kinetic timescales by the mirror, cyclotron, and rehose plasma micro-instabilities. Electromagnetic uctuations generated by growing mirror and cyclotron modes are detected in the solar wind. The rst detailed observations of the rehose instability are presented. The limiting bounds to R p imposed by each of these instabilities are measured and compared with the theoretical predictions of uid magnetohydrodynamics, linear kinetic Vlasov theory, and numerical simulations. It is shown that the predictions of linear theory and the simulations are in agreement with the observations. A new proton temperature anisotropy driven instability intheregieme R p < 1 kp < 1 is discovered. The kinetic properties of + Hand +2 He are compared. For the rst time a cyclotron resonant instability driven by the proton temperature anisotropy is demonstrated to limit the dierential ow ~ U ~ U ; ~ U p attainable in the solar wind, in conrmation of recent theoretical predictions. It is shown that the +2 He temperature anisotropy R T? =T k is also constrained by micro-instabilities, and the rst observations of the +2 He cyclotron and rehose instabilities are presented. The parallel and perpendicular temperatures of + H and +2 He are compared, and evidence of cyclotron-resonant heating of +2 He preferrentially to + H in the interplanetary medium is presented. Thesis Supervisor: Alan J. Lazarus Title: Senior Research Scientist / Lecturer 3

4 4

5 Acknowledgments I would like to thank Dr. Alan Lazarus for oering me this exciting opportunity to join him in the study of the solar wind. I could not have had so much fun pursuing this degree without his continual support and encouragement. I am indebted to John Steinberg and Matthias Aellig for their explanations of the Faraday Cup analysis procedures, Peter Gary for productive discussions on plasma micro-instabilities, and Adam Szabo for providing the three-second MFI data and his help with collisionless shocks. Finally,Iwould like to thank my parents, Susan and Joseph, and my sister Hannah for their love and support as I have pursued physics these many years. None of this would have been possible without my source of constant inspiration: Tara, my muse, thanks for your help and patience! Analysis of the data recorded by the Faraday Cup instruments on the Wind spacecraft was supported in part by NASA grant NAG

6 6

7 Contents 1 Introduction Exploring the Heliosphere with In-Situ Observations of the Interplanetary Medium A Kinetic Description of Solar Wind Plasma Thermodynamics and Statistical Mechanics Electromagnetic Interaction of Charged Particles Evolution of Phase Space Distributions with the Boltzmann Transport Equation Collision Rates, Collisionless Plasmas, and the Vlasov Equation Equilibrium and the Maxwellian Distribution The Fluid Approximation: Collapsing the Distribution Function Magnetohydrodynamics and Equations of State Breakdown of Fluid Approximation: Kinetic Theory The Wind Spacecraft and Experiments Outline of Thesis Bi-Maxwellian Analysis of Wind/SWE Ion Spectra Introduction Faraday Cup Measurements of Ion Distribution Functions Operational Design of the Wind/SWE Faraday Cups Description of a Single Wind/SWE Faraday Cup Spectrum Response to an Isotropic Maxwellian Ion Distribution Response to a Bi-Maxwellian Ion Distribution Choice of Eective Collecting Area for bi-maxwellian Distribution Moments of Faraday Cup Spectra Temperature Anisotropy with Moments Outline of Analysis Procedure Preparation of a Single Ion Spectrum Moment Analysis of the Proton Distribution Initial Guess Selection of Proton Data Non-Linear Fit of Model to Observations

8 2.4 Results: Proton Parameters, Uncertainty Propagation and Analysis Optimizing the Analysis Overall Summary: Convergence, 2 =d:o:f: Bulk Speed, Velocity Components, and Flow Angles Thermal Speeds Number Density Summary Instabilities Driven by Proton Temperature Anisotropies Introduction Four Descriptions of Plasma Micro-Instabilities The Magnetohydrodynamic Fluid Perspective Linear Theory of Waves and Instabilities Hybrid Simulations of Wave-Particle Interactions Observations of the Eects of the Instabilities T?p >T kp : Mirror and Cyclotron Instabilities Wind Observations of Solar Wind with T? >T k The Cyclotron Limit in the Region 0:1 kp Searching for the Mirror Instability athigh kp T k >T? and the Firehose Instability Wind Observations of Solar Wind with T k >T? Quantifying S p and p and Comparing with Theory Discussion and Summary Extension of Analysis to Helium Ion Composition of the Solar Wind Extension of the Existing Analysis Code Results of the Analysis Absolute Calibration of Number Densities Preparation of Measurements Survey Over Mission A Single Eective Ion Species Summary Kinetic Properties of Helium in the Solar Wind Dynamics of Multiple Ion Species Alpha-Proton Dierential Streaming Thermal Speeds and Thermal Equilibrium Revisiting Temperature Driven Instabilities with a Multi-Species Fluid Properties of Collisionless Shocks Collisionless Shocks in the Heliosphere Steepening of Waves into Shocks Sources of Collisionless Shocks in the Heliosphere

9 6.2 Methods for analyzing shocks Derivation of Rankine-Hugoniot Relations Detection of Shocks Determination of Shock Normals Evaluation of Shock Analysis Methods Timing of Shocks Between Multiple Spacecraft A Results of the Shock Analysis 177 9

10 10

11 List of Figures 1-1 A survey of the orbital trajectories in the ecliptic plane employed by the Wind spacecraft over the mission to date. the solid line is the Wind trajectory with a diamond indicating the nal point in each interval. The dashed line indicates the orbit of the moon. The types of orbits are: (a) Double lunar swing-by (b) Lagrangian halo orbit about the L1 point at 220 R E (c) Distant prograde orbit. See the text for more details A simple cartoon showing how a Faraday Cup measures charged particle uxes. In (a) particles of various energies (energy / length of arrow) and dierent charges (red,blue) are allowed to ow through an opening into a grounded metal container. An insulated metal plate at the back of the cup collects the particles and the total current is measured as these particles ow through the wire (green) to ground. In (b) two metal grids have been inserted into the cup. The outer grid is grounded and a voltage is applied to the inner grid to repel particles of the desired charge An overview of the properties of the Faraday Cups on the Wind spacecraft adapted from [Ogilvie et al., 1995]. Top: A cross-section of the instrument along its symmetry axis, showing the entrance aperture, modulator assembly, limiting aperture, suppressor grid, and collector plates. Bottom Left: Asketch of the operating principle of the modulator assembly with the three categories of particles: those that always do or do not make it past the high voltage grid, and those which only pass through at one energy and produce a current which is modulated at 200 Hz. Bottom Right: The eective collecting area as a function of angle of incidence, this gure includes geometrical eects and the transparency of the wire grids Illustration of the Faraday Cup coordinate system for the Wind instruments. The two collector plates, A and B, lie in the ^x c ; ^y c plane, with ^x c directed along the split between the plates. The ^z c axis is normal to the collector plates. In subsequent discussion the \cup orientation" ^n is the direction of ^z c in the GSE coordinate system

12 2-4 Conguration of the SWE instruments on the Wind spacecraft, adapted from [Ogilvie et al., 1995]. The two Faraday Cup ion instruments are mounted 15 out of the ecliptic plane, with the upper cup looking slightly northward, and the lower cup downward. Also shown are the Vector Ion-Electron Spectrometers (VEIS), the eld-aligned electron Strahl detector, and the SWE DPU Diagram indicating the measurement of a single ion spectrum. In each single rotation the FC measures along each of the angles indicated by red arrows at a single energy window indicated by the blue arrows Example ion distributions in the solar wind Two example distributions in velocity space. Left: An isotropic velocity distribution function with a single thermal speed of 30 km/s. Right: An anisotropic VDF with w? = 30 km/s and w k = 60 km/s, and magnetic eld orientation indicated by the black arrow. In both images the cross identies the maximum density of the VDF, the solid red lines indicate the selected velocity window, the blue lines are the nominal eld of view of the instrument, the dashed line connects the origin to the point in the VDF with maximum density, and the dotted-dashed line connects the origin to the point in the energy window with the highest density. It is clear from this example that for an anisotropic distribution we need a better way of determining the angle of incidence than the historical technique An example of the selection of FC angles^n for a single spectrum to be used for moment determination of density and thermal anisotropy. The number densities determined by each cup are shown, with diamonds for FC1 and crosses for FC2. All points indicated by grayed symbols are discarded from the subsequent analysis because they have a ow angle of greater than 40 from the cup normal or erroneous densities Scatter plot of observed thermal speed w(^n) as a function of FC direction relative to the eld orientation ^n. Grey symbols indicate points discarded due to suspicious densities. The lines are the best t of (2.24) to the selected observations. The data in the top panel are for the same spectrum shown in Figure 2-8 and have w k >w?, whereas in the bottom panel w k >w? Selecting measurements of anisotropic proton distribution for tting to bi-maxwellian model. Top panel: Color shading indicates logarithmic distribution of observed current as a function of speed window and azimuthal angle crosses are locations of measurements and diamonds are the selected points solid line is projection of bulk velocity along each angle and dashed lines mark three thermal widths above and below this projection. Bottom panel: Crosses indicates the angle of incidence of the maximum ow for each measurement Dashed horizontal line (red) is the maximum angle ANGLEMAX for selection Dashed curve is the angle between the cup and the bulk ow colored (blue) crosses are selected data

13 2-11 Interval in 1998 selected for optimization of analysis code Optimization of analysis code Distribution of the nal value of 2 /d.o.f. for all spectra analyzed (black), slow solar wind v<400 km/s (green), fast solar wind v>400 km/s (blue). The average value of 2 /d.o.f. over all spectra was 0:68 (blue vertical line). The distribution of spectra is approximately lognormal, and 99:9% of the spectra, had nal values of 2 /d.o.f. between 0:07 and 7:3 (red lines). It has approximately the same distribution Testing the eect of angular eld uctuations during recording of ion spectrum. This is a plot of the temperature anisotropy as a function of the angular deviation of the eld, as dened by equation (2.42) Dependence of the uncertainty in parallel (upper panel) and perpendicular (lower panel) proton thermal speeds as a function of magnetic latitude and longitude. Color shading indicates median uncertainty as indicated by color bars on left. Red curves indicate 5,25,50,75% measurement contours. See text for details Comparison of the thermal speeds as determined by the non-linear and moment analysis techniques. Colored bins indicate the number of spectra in a two-dimensional histogram. Each column has been normalized to unity. Dashed line indicates equality between methods diamonds are the average ratio between the methods (in percent) as a function of the non-linear value, with error bars indicating the deviation of the ratio in each bin Comparison of moment and non-linear anisotropies with the predictions of numerical Monte-Carlo simulations Number density uncertainty dependence on bulk and thermal speeds Survey of proton parameters in the solar wind at1au as seen by the Wind Faraday Cup instruments. Upper panel: The distribution of proton number densities n p as a function of bulk speeds U p, showing that on average the particle ux n p U p is conserved. Lower panel: Single proton temperature T p =(2T p? +T pk )=3 as a function of bulk speed, showing that on average the solar wind has a mach number U p =w p Survey of proton temperature anisotropy R p = T?p =T kp, as a function of proton number density n p and magnetic eld magnitude B. The color shading indicates the median value of the ratio within the bins, which are logarithmically spaced in density and eld strength. The color scale is given by the bar on the left and ranges from 0:6 R p 1:8. Bins are smaller in regions of higher density of measurements, but no bin has less than 1000 spectra. Black contours indicate the density of observations at the percent levels. The red line indicates the R p = 1 contour and dashed line is a power law with n p is proportional to B 3=

14 3-2 Survey of solar wind measurements as a function of plasma beta and the ratio R p = T?p =T kp. Note that the observed values of R p do not depart greatly from unity, in disagreement with the predictions of the double adiabatic equations of state (3.3) and a simple adiabatic expansion (3.4) Observations of the proton temperature anisotropy R, along with other solar wind parameters, as a function of time over the 24-hour period on April 30, Note the unusual rise in kp to more than 30 between 0630 to 0745 UT, and the concurrent constraint of the proton anisotropy. The blue, green, and red curves correspond to the predicted bounds imposed by the mirror, rehose, and cyclotron instabilities Upper panel: Uncertainty R as a function of R for all 2.2 million proton measurements to date with no cuts. Diamonds indicate the average uncertainty as a function of R p, and the solid line is the best t of (3.24) to the averages. Lower panel: Histogram of the distribution of observations (blue) and the average percent uncertainty (red) Eect of Coulomb relaxation on the average value of R p observed as a function of kp. Upper: Histogram of the number of observed proton spectra as a function of x = c, the ratio of the solar wind expansion and Coulomb relaxation timescales. Colored regions indicate six regions with 20,000 spectra. Lower: The average value of R p as a function of kp for each of the selected intervals. The average value is described by (3.22) and the best-t parameters are shown in the legend Survey of R p Normalized survey of solar wind measurements as a function of plasma beta and the ratio R = T? =T k ; Identication of signatures of the mirror and cyclotron instabilities by examining a dynamic power density spectrum of the high-resolution MFI magnetic eld measurements Quantifying the lower bound to R p imposed by the rehose instability. Left: N(kp R p ) as a function of R p for each value of kp. Right: Values of the two free parameters in the ts of (3.30) to the observed fall o Determining the best values of S p and p to describe the observed limit to the proton temperature anisotropy. Top: Best t of (3.23) to the calculated values of R p c as a function of kp. Middle: Variation of S p (diamonds) and p (triangles) as a function of limiting fraction f. Bottom: Scatter plots of the values of S p and p extracted from the data as a function of f (crosses), the results from the linear analysis (connected symbols), and the MHD calculation of the rehose instability threshold (diamond)

15 3-11 Survey of proton temperature anisotropies normalized to demonstrate eects of kinetic plasma micro-instabilities. For each column of kp the two intervals R p < 1 and R p 1 are separately normalized to unity. Features include the bounds imposed by the mirror, cyclotron, and rehose modes the eect of adiabatic expansion driving T?p =T kp < 1 an unknown mechanism in the range kp < 1 R p < 1 constraining R p to isotropy. See text for details Predicted fraction F m of the total electron number density due to minor ions as a function of proton bulk speed and year Histograms of F m in six speed windows using selected data in Black histogram is the number of spectra observed per 0:5% bin in F m, blue line is portion of the histogram selected for t with Gaussian, and red is the best-t Gaussian distribution. Note that the width increases with speed and that observations with F m < 0 are in agreement with tail of Gaussian Predicted fraction F m of the total electron number density due to minor ions as a function of proton bulk speed and year. Upper panel: Solid lines are the widths of the Gaussian ts to the distribution of F m in each speed/year interval, connected diamonds are the corresponding average uncertainties in F p, F, and F e. Lower panel: Center of F m distributions and natural widths as a function of speed and year. Dashed lines indicate estimates of the expected contribution of minor ions to the total electron number density Examples of kinetic features which arise in collisionless regime of solar wind. Top to bottom: Histogram of the distribution of observations as a function of x = c alpha particle temperature anisotropy alpha to proton temperature ratio alpha-proton dierential ow. In all cases these features are washed out by Coulomb relaxation Survey of alpha-proton dierential streaming. Upper panel: The distribution of the angle between the magnetic eld and the dierential velocity. Lower panel: Histograms of the range in U p =C A under various cuts (See text for details) Verication that under set conditions the dierential ow is limited by the proton temperature anisotropy as predicted by theoretical work with cyclotron resonances Survey of the range of parallel and perpendicular temperature ratios T? =T?p and T k =T kp. Upper panel: Histograms of each ratio. Lower panel: The average value of each of the ratios as a function of x = x. Dashed line indicates equal thermal speeds, dotted line indicates equal temperatures

16 5-5 The average ratio of the alpha to proton temperature ratio T? =T?p as a function of the collision rate and the normalized alpha-proton dierential ow. Note that the largest values of T? =T?p occur when U p is small (See text for details) Distribution of alpha particle temperature anisotropies as a function of kp. Compare with the summary Figure 3-11 at the end of Chapter 3 for proton temperature anisotropies Plot of the speeds in km/s of the three MHD wave modes allowed by (6.10) as a function of direction of propagation relative to the magnetic eld, Bn, calculated using the typical solar wind conditions from Table 1.1. The horizontal axis corresponds to propagation along the eld ( Bn =0). The dashed blue line is the speed of the intermediate mode, the solid green line is for the slow mode, and the dash-dotted red line corresponds to the fast mode The velocity V, temperature T, density N, and magnetic eld B pro- les of the four types of shocks seen in the interplanetary medium. The colored curves indicate what a spacecraft would observe aseach of the shock types traveled past it. In each case time increases to the right as indicated by the arrows Three panels illustrating mechanisms for the generation of collisionless shocks in the heliosphere: left: Diversion of solar wind about a magnetized barrier leads to formation of a planetary bow shock center: Interaction of solar wind streams with dierent speeds leads to a CIR right: Image of a coronal mass ejection on Feb. 27, 2000 taken by the LASCO C3 coronograph on the SOHO spacecraft (See text for details) Demonstration of data selection for interplanetary shock analysis. Top to bottom: Proton number density Bulk velocity components Magnetic eld components. Diamonds are the individual measurements, and the blue symbols indicate points selected for further analysis Verication of the jump criteria for candidate shock selection. Vertical red lines are the minimum values set using the 70 shocks detected by eye. The histograms indicate the distribution of the 300 shocks selected with the assistance of the computer, with jumps calculated using either the median (black) or the mean (blue) of the selected upstream and downstream data Identication of the March 23, 1995 FF shock normal using the RH1 technique. Color scale indicates the value of 2 = as a function of the shock direction, with red for the worst t and purple for the best t. The black contour indicates the region of the plot in which a given normal produces a positive shockspeed. The diamond marks the best- t normal and the error bars are ten times the uncertainty in the location of the normal. The intersection of the dashed lines indicates a radial shock

17 6-7 Determination of the shock normal using each of the MC, VC, MX1, MX2, and MX3 techniques outlined in the text. The scatter of individual calculations of ^n for observation pairs for each method are shown in the plots. The lower-right plotshows the average and standard deviation of and from each technique. Note that the VC, MX1, MX2, and MX3 techniques are in good agreement Comparison of the shock normals determined using all of the analysis methods. Colors for each of the methods are the same as those used in Figure 6-7. All methods agree to within 10 except for MC Upper: A histogram of the best-t values of 2 = for all the analyzed shocks. Lower: Angular dierence between MC and RH1 as a function of Bn Evaluation of the shock analysis methods by comparing predicted and observed arrival times of 40 shocks between Wind and ACE. Upper panel: Scatter plots of predicted vs observed arrival time for each method. Lower panel: Average dierence in times as a function of spacecraft separation perpendicular to shock fronts

18 18

19 List of Tables 1.1 The general range of several key parameters of the solar wind at 1 A.U. and their typical values over the course of the solar cycle Analysis control parameters Uncertainties in each of the derived parameters from the best-t to individual solar wind spectra Predictions of the expected limits to R p based on MHD uid, linear Vlasov theory, 1D and 2D hybrid numerical simulations, and previous experimental observations in the magnetosheath The success rate of the alpha-proton analysis procedure in various regions in space as afunctionofthenal status of the t Uncertainties in each of the derived alpha parameters from the best-t to individual solar wind spectra. Compare with same table for proton parameters in Table Estimating the contribution of minor ion species to the total electron content. With the numbers listed here F m ' 3:1% Statistics of the dierences t between the predicted and observed arrival times of interplanetary shocks at ACE based on calculations using the Wind shock parameters for the 40 shocks with ds? < 100 R e. For each method the average dierence < t >,theaverage of the absolute values of the dierences < jtj >, and the standard deviation of the dierences (t) areshown

20 20

21 Chapter 1 Introduction ABSTRACT: An exploration of the kinetic properties of the solar wind through in-situ plasma measurements using instrumentation for the Wind and Triana spacecraft. This thesis consists of equal parts instrument design, characterization of observations, and comparison of measurements and theory. In this chapter plasma physics is presented in a heliospheric context. The nature of the heliosphere and outstanding questions are outlined, along with the goals and methods of the research described in the rest of this work. 1.1 Exploring the Heliosphere with In-Situ Observations of the Interplanetary Medium The interplanetary medium (IPM) is pervaded by the solar wind, a ow of ions and electrons expanding supersonically from the surface of the Sun. The solar wind originates in the anomalously hot solar corona and terminates at the distant heliopause where it merges with the interstellar medium. This region dominated by the inuence of the Sun is the heliosphere. Its behavior is intimately related to solar activity, including the 26-day rotation of structures in the solar corona and the global 22-year period reorganization of the polar solar magnetic eld. Remote observational techniques are not a sucient means for understanding the heliosphere, since there are few distant signatures of this tenuous plasma. Additionally, the physical processes which occur within the IPM are often unintiutive, and frequently require rst-hand observations in space to distinguish theory from reality. We therefore explore the heliosphere with spacecraft that perform measurements in-situ of the particles and electromagnetic elds which constitute the IPM. For the duration of space exploration Faraday Cup (FC) instruments designed by MIT have been used to characterize the properties of the bulk of ions in space, including the solar wind. The focus of this thesis is the analysis and interpretation of observations by the FC instruments on the Wind spacecraft and the design and calibration of a new FC for the Triana spacecraft which is capable of making high-time-resolution measurements at an unprecedented rate. My goal in this thesis is to produce as accurate and complete an analysis of Faraday Cup instruments and their observations 21

22 as possible, and then to use these results to conduct studies of the kinetic properties of the solar wind with unprecedented detail. There is a great deal to learn from the interplanetary medium. We are still trying to understand the structure of the IPM itself, and it can be very complex: As the solar wind expands from the Sun, dierent streams of particles interact, producing shocks and energizing particles Magnetic elds diuse through the IPM, creating complex topologies which aect the propagation of these energetic particles and the galactic cosmic rays Planetary bodies interact with the solar wind, each in a a unique way, producing bow shocks and cavities. Understanding these processes is interesting in and of itself, but in addition the IPM can tell us about the Sun and the interstellar medium: We can draw conclusions about the mechanisms responsible for the high temperature of the solar corona, the acceleration of the solar wind, and the 22-year period of solar activity. Finally, our heliosphere is a laboratory unlike any on Earth, in which we can test our understanding of the processes which may occur in astrophysical plasmas, the dominant state of baryonic matter in the Universe. In space we can observe collisionless shocks, kinetic instabilities, magnetic reconnection, and particle acceleration, just to name a few of the exotic phenomena we would like to understand. In this thesis we will explore several situations where the evolution of the solar wind is not well described by thermodynamic and uid concepts such as single temperatures or equations of state. In these cases we must pay more attention to the details of the distribution of particles in velocity space. Loosely put, we are concerned with the kinetic physics of the IPM, but this is not the classical kinetic physics of rareed gases. Forces in the solar wind are due to long-range Coulomb interactions, and to understand their eect we must explore the collective interactions between electromagnetic uctuations and particle distributions. Kinetic eects may be very dramatic: They are required to explain the anomalously hot solar corona, the high speeds of the solar wind, collisionless shocks such as the Earth's bow shock and interplanetary shocks, and the means by which these shocks accelerate particles to high energies. Many terms have been introduced and before continuing with the description of this research we must set the theoretical stage. Section 1.2 starts with a couple of basic facts about the solar wind and develops the techniques appropriate to the IPM which will be employed in this thesis. It begins with a discussion of the treatment of many-particle systems through thermodynamics and statistical mechanics and works through a general treatment of plasma physics for heliospheric conditions. Particular care is taken to outline the assumptions made and when they break down. The results derived in this section are used subsequently throughout the thesis. An overview of the structure of the heliosphere from the solar corona to the heliopause at the boundary of the interstellar medium is then presented in Section 1.2, along with descriptions of the features we would like to explain. The spacecraft and instruments which will be used in this work are outlined in Section 1.3. Once this background is established Section 1.4 presents an overview of the topics pursued in the subsequent chapters. 22

23 Parameter Symbol Range Typical Value Units Bulk Speed U km/s Thermal Speed w km/s Number Density n protons cm ;3 Magnetic Field B nt Alpha Abundance n =n p % by number Table 1.1: The general range of several key parameters of the solar wind at 1 A.U. and their typical values over the course of the solar cycle. 1.2 A Kinetic Description of Solar Wind Plasma In this section I provide a review of the relevant elds of physics needed to describe the interplanetary medium, namely the study of tenuous plasmas. This overview is presented in the context of the typical state of the solar wind, as outlined in Table 1.1. The goals are to outline the assumptions which go into the development of, say, the thermodynamic and magnetohydrodynamic (MHD) pictures, and to develop relations which will prove useful at later points in this thesis. The informed reader may skip this section and return if only for an explanation of the notation. Observed by spacecraft near Earth, a distance of 1:510 8 km (or one Astronomical Unit, AU) from the Sun, the solar wind ows supersonically, with a range of average bulk speeds, 300 U 600 km/s, signicantly in excess of the 42 km/s required to escape the solar gravitational pull. While gravitation inuences the overall size of the heliosphere it plays no role on the kinetic scales we will be considering. With proton number densities of n p =10cm ;3 and temperatures between and Kwe shall see that collisions are very rare. The corresponding thermal energies of 5 ; 500 ev mean that particles in the IPM do not have sucient energy to engage in weak and strong nuclear interactions. Instead, the hydrogen and helium which dominate the composition of the solar wind are fully ionized and the dynamics of the solar wind are dictated by the collective electromagnetic interaction of ions, electrons, and ambient elds. What is the best description of this situation? One might be tempted by the low densities to treat the motion of the solar wind as that of test particles immersed in specied electromagnetic elds, but this is not appropriate for two reasons. First, the collective nature in which the particles are coupled together and to electromagnetic uctuations requires that the particles and elds must be treated in a self-consistent manner. Secondly, on the practical side, we must develop a simplied technique for describing large collections of particles. Consider the Faraday Cup which is the main observational instrument used in this thesis. It has a sensitive collecting area of A = 33 cm 2. During the t = 92 seconds it takes the FC to complete a single observation of the distribution of ions, approximately An p Ut protons pass through its entrance aperture. Clearly, even telemetry limitations aside, we need a way to summarize the properties of these particles for further investigation. 23

24 These methods have their basis in thermodynamics (Section 1.2.1), from which we can derive Boltzmann weighting and the Maxwell-Boltzmann distribution, a variant of which, the bi-maxwellian, will be tted to the ion observations. We need the details of the inter-particle interactions, and thus the basics of electromagnetism are presented (Section 1.2.2) with the added aspect of requiring solutions which in equilibrium are consistent with thermodynamic distributions. The evolution of a particle distribution is discussed in the context of the Boltzmann transport equation (Section 1.2.3) and the Vlasov equation is introduced after a discussion of the various collisional processes which occur in the IPM (Section 1.2.4). I then outline the basics of the uid approximation, how to collapse distribution functions through moments (Section 1.2.6), the development of MHD (Section 1.2.7), and the introduction of linear kinetic theory for consistent coupling of particle distributions and electromagnetic uctuations (Section 1.2.8) Thermodynamics and Statistical Mechanics Thermodynamics is the description of a system of many particles in terms of macroscopic thermodynamic parameters such as pressure, P, temperature, T, and magnetic eld, B, all of which can can be measured experimentally [Reif, 1965]. In this picture the state of a system is completely specied by stating all of the macroscopic parameters. Individual particles in our system may be undergoing complex motion, but we avoid considering the kinetic and potential energy of individual particles and instead say that given the total ensemble of states accessible to our system at a particular temperature, we have no way of knowing which of the states the system is in, but only that we can assign the relative probability of it being in that state. We make the assumption that at a given total energy of the the system E s, there is an equal a priori probability ofitbeinginanyofthe s (E s ) possible congurations with that energy. Following Pathria [1972], or Feynman [1972], we can determine the relationship between temperature and the relative probability of the system having a given E s. Consider the system in thermal equilibrium with a large reservoir described by an energy E r and a total number of states r (E r ). The total energy is a constant, E r + E s = E tot E s =E r 1: (1.1) The probability P s (E s ) of the system having an energy E s is proportional to the number of individual microstates of the system with that energy, s (E s ), P s (E s ) / s (E s ): (1.2) P s (E s ) is also proportional to the number of states accessible to the reservoir with the remaining energy E r, P s (E s ) / r (E r )= r (E tot ; E s ): (1.3) 24

25 Since E s E r we can approximate the number of states accessible to the reservoir by expanding in a power series and only keeping the rst two terms, ln r (E r ) = ln r (E tot ln r (E r ; E tot r Er=E = Constant ; r E r (1.4) tot where we have dened a parameter r, ln r Er=Etot (1.5) In thermal equilibrium both systems are in their most probable states, so r = s, and the probability of nding our system in a state of energy E s is therefore proportional to P s (E s ) / e ;ses (1.6) and we dene the temperature, T s, of the system through s 1=k B T s, where k B, Boltzmann's constant, has a value of 1:38 10 ;23 JK ;1, or 8:62 10 ;5 evk ;1. This is the Boltzmann weighting of relative energy states. The absolute probability is calculated by normalizing (1.6) by the total probability ofnding the system in any state with energy E j, P s (E s )= e;ses Pj e;se j (1.7) where the sum over the ensemble of states in the denominator is an abstract denition of the partition function. Values such as the average energy of the system can now be calculated through = X s P s (E s )E s (1.8) Together with the concepts of work, heat, and the laws of thermodynamics we can describe the response of our system under various transformations of the thermodynamic parameters. Nevertheless, the interplanetary medium (typical of many astrophysical plasmas) is not always well described by the concepts of thermodynamics: A species of particles in equilibrium may not have asingle,well-dened temperature (Chapter 2) small variations in parameters such as density and eld can trigger instabilities which alter the macroscopic parameters of the system (Chapter 3) and nally species in steady state with each other may not necessarily have the same temperatures (Chapter 4). We need to look more closely at how the properties of individual particles are combined to form the macroscopic picture. Statistical mechanics is the development of a thermodynamic macrostate from the detailed microstate. The microstate at some time t is dened by specifying the instantaneous position ~q i and momentum ~p i of every particle i = 1 2 ::: N in the 25

26 system. The set of coordinates (~q i ~p i ) species a point in a 6N-dimensional phase space. The time dependence of each ~q i and ~p i is given by the canonical equations of motion, _~q i _ ~pi i (1.9) where H(~q i ~p i ), the Hamiltonian of the system, is composed of the kinetic energies p 2 i =2m i of individual particles, inter-particle interactions U int (~q i ~p i ), and the potentials due to any external forces, V ext (~q i ~p i ). H(~q i ~p i )= p2 i 2m i + U int (~q i ~p i )+V ext (~q i ~p i ): (1.10) The value of the Hamiltonian is the energy E s of the system. If the external forces related to V ext do no work then E s is a constant and the total energy is conserved. Conservation of the total energy then implies that the region of the phase space accessible to the system, akin to the number of accessible states s (E s ), is the (6N-1) dimensional hypersurface dened by H(~q i ~p i )=E s. In terms of the Hamiltonian we can write a more specic denition for the partition function in (1.7) of N particles as an integral over all of phase space, Z Q N (V T etc:) = 1 e ;H(~q i ~p i ) d 3N qd 3N p (1.11) N!h 3N which includes the correct overall normalization. For a system of non-interacting particles in thermal equilibrium at temperature T with uniform spatial distribution and no external potentials (V ext = 0), the phase space density as a function of velocity is given by the Maxwell-Boltzmann (MB) distribution, 3=2 m f(~v) =n e ; m~v~v 2k B T : (1.12) 2k B T This is appropriately normalized so that an integration over velocity recovers the spatial number density. Using (1.8), just with an integration instead of a sum, we can calculate the average energy,, of a particle in the MB distribution, = 1 2 k BT: (1.13) Since there are no interactions, this average energy is due entirely to kinetic motion and we can assign to it a most probable thermal speed w, r = 1 2kB T 2 mw2! w = m : (1.14) The Maxwell-Boltzmann distribution can also be written in terms of this thermal 26

27 speed, f(~v) = n 3=2 w 3 e;v2 =w 2 : (1.15) Finally, ifthereis an external potential (V ext 6= 0), then the distribution function is proportional to f(~r ~v) / e ;[ 1 2 mv2 +V ext(~r)]=k B T (1.16) and an integration over the entire volume of interest must be carried out to determine the correct normalization Electromagnetic Interaction of Charged Particles By far the dominant force in the interplanetary medium is the electromagnetic force, represented here by the non-relativistic Hamiltonian [Goldstein, 1980, Chapter 8], H EM = 1 2m ~p ; q ~ A(~r t) 2 + q(~r t) (1.17) where ~ A and are the vector and scalar potential elds from which the electric ~ E, and magnetic ~ B elds may be derived [Jackson, 1999], ~E(~r t) =; r(~r ~ t) A(~r ~ t) B(~r ~ t) = ~r ~ t): (1.18) For the moment we will take ~ A and as given and consider a single particle of electric charge q located at ~r(t), moving with a velocity ~v(t), under the inuence of electromagnetic elds. The dynamics are dictated by the Lozentz force, ~F (t) =q[ ~ E(~r t)+~v(t) ~ B(~r t)]: (1.19) Consider a test particle of charge q in a constant uniform magnetic eld ~ B. If we break the velocity of the particle ~v into two components, parallel (~v k ), and perpendicular (~v? ) to ~ B it is clear from (1.19) that there is no net force due to ~v k. In terms of ~v?, the particle will experience a force ~ F = q~v? ~ B which at all times is perpendicular to the motion of the particle. The particle will undergo uniform circular motion, and the magnetic eld will do no work on the particle. By balancing the eect of the magnetic eld with the centripetal force mv 2 =r we see that the cyclotron frequency of a particle is independent of its energy. The cyclotron frequency denes one of several very important time scales in plasma physics, and we dene the electron c,e and proton c,p cyclotron frequencies, c,e = jq ejb m e c,p = q pb m p (1.20) 27

28 The magnetic eld carried from the Sun typically has a strength of about 5 nt, implying cyclotron frequencies in the plasma frame of c,e 900 Hz and c,p 0:5 Hz. The ion cyclotron frequency denes the typical timescale of the kinetic instabilities explored in Chapter 3. At a given value of ~v?, the particle will gyrate in a circle of radius r L = mv qb (1.21) called the Larmor radius. To calculate the typical Larmour radius of a proton in the solar wind, use the proton thermal speed w p 20 km/s to get r L,p 60 km. At the typical solar wind speed of 400 km/s, ten proton-gyroradii convect past the spacecraft each second. The FC ion spectra, which take 92 seconds to collect, therefore average over many proton gyroperiods. Now consider the dynamics of the electromagnetic elds. The electric and magnetic elds must obey Maxwell's equations. The magnetic eld must be divergenceless, ~r ~ B =0: (1.22) The divergence of the electric eld is due to the presence of free charge, ~r ~ E(~r t) =(~r t)= (1.23) the curl of the magnetic eld is due to free currents and to time varying electric elds ~r ~ B = ~ J ~ and the curl of the electric eld is due solely to time varying magnetic elds, (1.24) ~r ~ E = ~ : (1.25) What would happen to our gyrating particle if the magnitude of the magnetic eld were gradually changed? An approximation which we will make use of later is the calculation of \adiabatic invariants" [Goldstein, 1980, Chapter 11, especially pages ], quantities which remain roughly constant under appropriate transformations of the system under consideration. In terms of generalized coordinates ~p and ~q of (1.9), if the particle is undergoing motion which is periodic in any of the spatial coordinates, say q i, then we can calculate an invariant quantity J i J i I p i dq i (1.26) by integrating over an integer number of cycles. If we dene the magnetic moment 28

29 mv? 2 =2B, then d dt = d dt mv 2? 0 (1.27) 2B and we have an approximate relation between v? and B for slowly varying elds. There are many useful applications for adiabatic invariants, including the mirroring of trapped particles in spatially varying elds [Parks, 1991, Chapter 4]. We will use them in deriving the two-temperature Maxwellian distribution function in Section and in the study of instabilities in Chapter 3. Now we need to link the potentials and elds to the distribution of particles. The potentials are functions of the distribution of free charge and currents in all of space, (~r) = 1 4 Z (~r 0 t r ) j~r ; ~r 0 j d3 ~r 0 ~ A(~r) = 4 Z ~J(r 0 t r ) j~r ; ~r 0 j d3 ~r 0 (1.28) where the t r indicates that the source terms must be evaluated at the retarded time t r j~r ; ~r 0 j=c so as to satisfy causality. As a simple example of a problem which requires the solution of thermodynamic and electromagnetic equations consider the shielding of a point charge q > 0, immersed in a plasma which has an initially uniform density of ions and electrons with equal densities n. We expect that electrons will be drawn towards the point charge and at large distances the electric eld will be shielded, resulting in a spherically symmetric potential which falls o exponentially instead of linearly, (r) / e ;r= D (1.29) where D, the exponential decay-length of the potential, denes the scale of the shielding by the ambient plasma. Now assume that protons and electrons will adopt Maxwell-Boltzmann distributions of the form of (1.16), with the external potential due to the electric eld generated by these particles and the point charge. f j (r ~v) / exp ; 1 m j v 2 ; q j(r) (1.30) 2 k B T j k B T j where I have allowed for the possibility, as is common in the solar wind, that the electrons and protons do not have the same temperature. We can immediately integrate over speed, yielding equations for the electron, n e, and proton, n i, number densities, n e (r) =n e e(r)=k BT e n i (r) =n e ;e(r)=k BT i (1.31) where I have assumed that as r!1the plasma is neutral with the original number density n. Operating the Laplacian on the potential gives us the local charge density, ~r 2 = = e (n e ; n i )= en ; e e(r)=k B T e ; e ;e(r)=k BT i (1.32) 29

30 and if we look far enough away from the charge that e=k B T i and e=k B T e are both 1, then we can approximate the Laplacian 2 = en e k B T e + Comparing this with (1.29) gives us an equation for D, e : (1.33) k B T i 2 D = k B T e n e 2 (1 + T e =T i ) (1.34) where D is called the Debye length [Goldston and Rutherford, 1997, x1.7]. The Debye length is the size scale that divides individual particle behavior from largescale collective behavior. Two dening characteristics of a plasma are that the Debye length be much less than the overall length scale of the problem one is examining and that the number of particles N D in a sphere of radius D, the Debye sphere, N D D n (1.35) be much greater than unity. In typical solar wind conditions D ranges between 3 and 30 m, which is much smaller than the proton and electron gyro-radii, the smallest typical length scale one considers. The Debye sphere contains around 10 9 ; particles, which is 1. The dynamics of the solar wind are dictated by the collective elds of the plasma, and not by particle-particle interactions. In addition, if the smallest typical time scale is on the order of the reciprocal of the electron cyclotron frequency, ;1 c e 1 ms, and elds are shielded beyond length scales D 30 m, then D = ;1 c e km/s, which is much less than the speed of light, and we need not bother with retarded times when evaluating electromagnetic potentials (1.28). This will greatly simplify the analysis of small perturbative uctuations discussed in Section and employed in Chapter 3, because all quantities may be evaluated simultaneously. The sensing portions of the Faraday Cup plasma instruments discussed in this thesis are no larger than 10 cm. It has been observed [Vasylinuas, 1971] that this length is much less than the Debye length D in the IPM. While in-situ instruments like Faraday Cups diagnose the local plasma conditions, the portion of the solar wind which actually enters the FC doesnotbehave like a plasma. The analytic derivations of the response of a FC to a given particle distribution function detailed in Chater 2 are possible in part because we can regard the solar wind collected by the instrument simply as a collection of non-interacting ions and electrons Evolution of Phase Space Distributions with the Boltzmann Transport Equation Now that we can describe the electromagnetic elds of a distribution of particles, we need to explain the response of a particle distribution to applied forces. Consider the 30

31 evolution of an ensemble of particles. Instead of tracking the velocity and position of every particle, we follow the phase space density, or distribution function, f(~r ~v t) (1.36) the density of particles at a given position and velocity. The Maxwell-Boltzmann distribution (1.15) of Section was a specic example of a spatially uniform phase space distribution. By integrating over all velocities we get the density as a function of space and time, n(~r t) = Z f(~r ~v t)d 3 ~v (1.37) and of course by additionally integrating over space we just get the total number of particles, Z f(~r ~v t)d 3 ~vd 3 ~r = Z n(~r t)d 3 ~r = N(t) (1.38) We are only considering the electromagnetic interactions of particles at a distance, there are no nuclear interactions which will change the identity of our particles and we can neglect ionization and recombination, so the total number N is a constant in time. Other integrals of f(~r ~v t) over velocity space are considered in Section For the moment we will focus on the evolution of the phase space density. We can take the total derivative ofthephase space density with respect to time, df @x (1.39) or in vectorial form, df (1.40) which should remind you of the convective derivative in hydrodynamics [e.g., Faber, 1995, page 38], d (1.41) just with an additional term for the motion in velocity space. The acceleration is due to any external forces ~ F acting on the particles in the distribution, i.e. ~a = ~ F=m if the particles have mass m. In the absence of any external forces, a given parcel of material would just convect through phase space. Even if there are external forces, by Liouville's theorem [Goldstein, 1980, pages ] the total derivative (1.40) of the phase space density will vanish. However, in addition to external forces, collisions 31

32 between particles in the distribution will alter the phase space density. We therefore write the general form of Boltzmann's transport equation, df + ~ = C[f] (1.42) where C[f], the collision integral, is a complicated function of the particle distribution [Huang, 1963, Chapter 3] which reects the redistribution of particles in phase space as a result of collisions. This is a complex equation with far-reaching consequences: Landau used a variant of the equation to demonstrate the damping of a wave by a plasma [Landau and Lifshitz, 1981, x5]. To simplify (1.42) subdivide the electromagnetic forces a given test particle experiences into two categories: collective and individual. It was shown that IPM length scales are larger than D and that N D is always 1. Consequently, the main force experienced by a single particle reects an average over the collective electric and magnetic elds of the background particles, and not the eect of each interaction with every individual particle. As an analogy consider the uniform magnetic eld produced by a current-carrying wire. In reality the observed eld is a superposition of electrons traveling in random directions and suering frequent collisions as they drift down the wire. But for all practical purposes a test particle experiences a simple, constant magnetic eld due to the net collective motion of the electrons. The external force ~ F in (1.42) is therefore redened to include the Lorenz force (1.19) which may be a function of the phase space distribution of the particles under consideration. The transport equation now takes the form, df + q m ~E + ~v = C[f]: (1.43) It lies at the base of many of the situations we will explore in this thesis. Things are more complicated, since the forces in (1.43) come from the elds generated in part by the same particles. In addition, the collision integral C[f] remains in (1.43) and we must decide what to do with it. The IPM does not lend itself to the same approximations employed in classical kinetic theory for treating C[f] [Landau and Lifshitz, 1981, x3]. In the classical kinetic treatment of gases the intermolecular forces fall o very rapidly. The solution for C[f] then typically involves starting with the eect of two-particle interactions and then adding on higher order terms in an expansion. The Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hiercharchical expansion [Bogoliubov, 1962], for example, will not work in a tenuous plasma because of the inverse-square nature of the Coulomb interactions [Montgomery and Tidman, 1964, Chapter 2]. While we have seen that a test particle does not \see" the electric elds of particles more than a Debye length away, there is no simple way of treating which particles within the Debye sphere produce the dominant eects. In the following section we will discuss what these Coulomb collisions are and whether they may be neglected. But before proceeding with that we can make a few 32

33 denitions: An ensemble of particles is in equilibrium =0 for all ~r ~v: (1.44) Also, a system is in quasi-equilibrium if in a time scale t appropriate to the system, e.g. the propagation time through the heliosphere, small changes f in the phase space density satisfy f(~r ~v) f 1 for all ~r ~v during t: (1.45) Note that this is not a convective derivative, so for example a parcel of material may evolve as it expands away from the Sun, while quasi-equilibrium would imply that the global structure remains roughly constant. Systems in equilibrium with each other may not be in thermal equilibrium, in the thermodynamic sense, =@t may be zero even through their temperatures are unequal. While this contradicts the assumptions that went into deriving Boltzmann weighting (1.6), we will see that it is true as long as wave-particle interactions play more dominant a role than collisions. Finally, if a system with an arbitrary velocity distribution evolves to assume a Maxwellian or roughly Maxwellian distribution then it has relaxed or thermalized Collision Rates, Collisionless Plasmas, and the Vlasov Equation This section addresses the importance of the collision integral C[f] in Boltzmann's transport equation (1.42) for describing transport processes in the IPM. As a very rough estimate of the integral we can write [see Landau and Lifshitz, 1981, especially the derivation leading to Equation (3.12)], C[f] ; (f ; f ) ;(f ; f ) v (f ; f ) (1.46) where f represents the equilibrium distribution function and f is a small departure from f. The approximation is written in three related ways making use of, a characteristic collision time, the corresponding collision frequency, v the average speed of a particle in the distribution and, the distance a particle travels on average before suering a collision. The rate at which collisions occur and their eect depend on the transport mechanism under consideration. There are several collision processes and associated time scales to consider here and a more general denition of a collision is needed than simply \particle 1 strikes particle 2". After all, since N D 1 the eect of the background eld particles on a test particle is more of a collective process than simple hard classical collisions. Collisions more generally are the processes which serve to redistribute momentum and energy (the two are distinct) between particles in the 33

34 system. Start with simple hard collisions to highlight the dierences between terrestrial phenomena and typical astrophysics plasmas. Consider a gas of particles with radius a, mass m, and number density n in thermal equilibrium at temperature T. The crosssection for a \contact" collision is ' a 2. The pathlength gas for a molecule to travel before suering a single collision is, gas = 1 n = 1 na 2 : (1.47) At the given temperature T the most probable thermal speed a molecule will have, from (1.14), is w = p 2k B T=m and so the characteristic frequency gas for collisions in this gas is gas = w= gas = na 2p 2k B T=m (1.48) For air at standard temperature and pressure, gas 10 ;5 m and gas 10 7 Hz. If the solar wind were a gas of hydrogen atoms the characteristic collision scales would be gas m (21 AU) and gas 10 ;9 Hz, or a single collision once every 20 years! If this were the whole picture we could denitely neglect the collision term from (1.42), but then we would be at a loss to explain why the solar wind at 1 A.U. to rst order does look Maxwellian - implying that a sucient number of collision times has elapsed for the distribution to thermalize - since it only takes 2 ; 6 days for the solar wind to reach us from the corona. Shorter collision time scales must exist. The rst such scale is due to the long-distance Coulomb scattering between a test particle and all of the eld particles within its surrounding Debye sphere [Spitzer, 1956]. In collective Coulomb scattering the dominant eect is due to the large number of small angle scatterings between distant particles. The direct calculation of momentum and energy scattering rates for the Coulomb scattering of twocharged species is straightforward. First the scattering of an incident particle o a single eld particle is considered, then the average collision rates are determined by integrating over the phase space distributions of both particles species. The nal result is a series of characteristic frequencies for momentum loss between two species with possibly dierent temperatures. For electrons loosing momentum to ions, ei = p 2 Ze p n i 4 ln j is the Coulomb logarithm, where j for species j is 4 m 1=2 e (k B T e ) 3=2 ln e: (1.49) j = r k B T j ne 2 qi q j 4 v j ;1 (1.50) with reduced mass = m i m j =(m i + m j ) and relative speedv j between the species. 34

35 For electron-electron and ion-electron collisions, ee ' 1 2 ei ie = n em e n i m i ei (1.51) and for two ion species i and j, ij = p 2 p n i qi q j mj m 1=2 i (k B T i ) 3=2 1=2 ln i : (1.52) m i + m j Typical values for the ion-ion collision timescale in the IPM are 1 to 6 days. Thus small-angle Coulomb scattering of a given particle o the eld particles within its Debye sphere are sucient to thermalize the distributions of particle species. However, it is not true that dierent species will equilibrate with each other in the same time frame, and as we shall see ion species and electrons tend to be in equilibrium, but not with the same temperatures. The Coulomb scattering is still not sucient to explain the rapid thermalization which occurs when very non-maxwellian features are introduced into the IPM, e.g, the rapid pitch angle scattering of pickup ions from comets, which occurs within several minutes [Huddleston et al., 1991 Isenberg, 1998 Tatrallyay et al., 2000]. These most rapid mechanisms for the thermalization of the particle distributions are due to kinetic eects such as micro-instabilities. Plasma micro-instabilities are discussed in more depth in the introduction to Chapter 3. Recent analyses predict times on the order of minutes under typical IPM conditions [Gary et al., 2001, and see Section 5.3]. In any case, these times are due to the collective electromagnetic elds created by the particles and do not require the collision term. We may therefore neglect it, arriving at the @f + + q ~E + ~v B =0: (1.53) Equilibrium and the Maxwellian Distribution In light of the typical collisional scales of the IPM discussed in Section 1.2.4, various species in the solar wind are in quasi-equilibrium, although events may bring them out of equilibrium and they are not necessarily in thermal equilibrium. To rst order we can describe each particle species j with a Maxwell-Boltzmann distribution (1.12), which including the bulk velocity ~ U j of the species takes the form f j (~v) = n j 3=2 wj 3 e ;(~v;~ U j ) 2 =w 2 : (1.54) In order to understand the kinetic properties of the IPM we need to identify extensions to this rst-order distribution function which correctly characterize the departures from thermal equilibrium in the solar wind. In this thesis I make use of the bi-maxwellian distribution function, in which a particle species is in thermal equilibrium, but with dierent temperatures parallel and perpendicular to the ambient 35

36 magnetic eld. In Section we will examine how the solar wind may rapidly attain such temperature anisotropies, but here it is worth exploring how such a situation could be stable. Intuitively it is clear that the presence of an ambient magnetic eld breaks the spatial symmetry of a plasma. Consider the injection at a single point in space of test particles (i.e. a population diuse enough that they do not alter the existing plasma properties) with randomly oriented velocities markedly dierent from that of the background plasma. Particles with large speeds along the eld direction will rapidly move away from the injection point, while particles with velocity pre-dominantly? to the eld will gyrate in place. After an appropriate amount of time the parallel distribution of particles at this point will return to the state before injection while the perpendicular distribution remains altered. For a more quantitative picture let us examine the rates for particle diusion parallel and perpendicular to a local magnetic eld ~ B as a proxy for the corresponding rates at which thermalization is achieved. This is not a rigid proof of the means by which anisotropy is attained, only an indication of how thermal equilibrium might be obtained separately parallel and perpendicular to an ambient eld. Calculations of transport phenomena in magnetized plasmas are possible [Landau and Lifshitz, 1981, x59]. For motion parallel to ~ B, consider a typical ion with a speed v k along the magnetic eld equal to the thermal speed of that species, v k = w i. From (1.52), it will travel a distance l i w i = ii before its momentum is randomized. We can therefore dene a coecient D ik for the parallel diusion, D ik l 2 i ii ' w2 i ii : (1.55) For diusion perpendicular to ~ B the situation is dierent because each particle gyrates around the eld, and from (1.21) the maximum distance a particle can move is approximately one ion gyro-radii, r L i, Since in general D i? ' r 2 Li ii : (1.56) D ik D i? = w2 i r 2 Li 2 ii 6= 1 (1.57) diusion will occur at dierent rates k and? to ~ B. Since wehave seen in Section that these collisional timescales are on the order of days, while the typical ion in the solar wind gyrates about the eld in seconds, any asymmetry in the spatial directions perpendicular to the eld will rapidly \wash out" in times much faster than the equilibration of particle energies parallel and perpendicular to the eld. More compactly we can say that generally gyrotropy is attained much faster than isotropy. In this case there are only two independent temperatures, that along the 36

37 magnetic eld, T k, and in all directions perpendicular to the eld, T?. The equilibrium distribution of particles with two such temperatures is the bi-maxwellian distribution function, f(v k ~v? )= n 3=2 w? 2 w e ;(v2? =w2? ) +v2 k =w2 k (1.58) k where k B T? = 1 2 mw2? and k BT k = 1 2 mw2 k. In a frame where the plasma ows with a net velocity U, ~ then with ^b B ~ =B we dene v k (~v ; ~ U) ^b ~v? (~v ; ~ U) ; v k^b: (1.59) The Fluid Approximation: Collapsing the Distribution Function We have moved from a description of individual particles to the evolution of phase space densities in the kinetic framework. Even with this simplication the information contained in the velocity distribution of particles makes it very dicult to obtain complete and self-consistent solutions to a majority of problems. A further simpli- cation is to \collapse" the details of the velocity distribution function (VDF) of a species of particles by calculating the moments of the VDF [Paschmann et al., 1998]. These moments are carried out for each particle species since dierent species may have dierent properties. This process is very similar to the abbreviated description of a probability distribution function (PDF) through the calculation of cumulants (mean, variance, skew, etc) [Pathria, 1972]. The m th moment ofthedistribution is M m (~r t) R f(~r ~v t)[~v] m d 3 ~v R f(~r ~v t)d 3 ~v = 1 n(~r t) where the integration has been divided by the density n(~r t), n(~r t) Z Z f(~r ~v t)[~v] m d 3 ~v (1.60) f(~r ~v t)d 3 ~v (1.61) because while probability distribution functions are normalized to unity, velocity distribution functions are the particle number density in phase space and are normalized to the total density. The rst moment, analogous to calculating the mean, is the average, or \bulk" velocity ofthe distribution, ~U(~r t) 1 n(~r t) Z f(~r ~v t)~vd 3 ~v: (1.62) The second moment of the velocity distribution yields the thermal speed, which is related to the temperature if the distribution is Maxwellian (1.14). The temperature is determined by calculating the average energy of a particle in the rest frame of the plasma. This is done by subtracting out the bulk motion ~ U and then calculating the 37

38 most probable thermal speed w ij of a particle in the distribution, w 2 ij 1 n(~r t) Z f(~r ~v t)(v i ; U i )(v j ; U j )d 3 ~v (1.63) where, since the distribution may not be isotropic, i and j identify two Cartesian coordinates and w ij is a nine-component tensor. From (1.14) we can write the temperature as a tensor, k B T ij = 1 2 mw2 ij. For an isotropic Maxwellian distribution (1.54) with temperature T, T ii = T T ij i6=j = 0 and for a bi-maxwellian distribution (1.58) with B ~ k ^z, T zz = T k T xx yy = T? T ij i6=j =0. Converting w ij into temperatures T ij makes the implicit assumption that the VDF projected in a given direction is Maxwellian. The departure from Maxwellian distributions, or at least the presence of an asymmetry in the VDF, can be quantied by the heat ux vector Q, ~ ~Q 1 n Z f(~v)(~v ; ~ U) 3 d 3 ~v: (1.64) With several exceptions such as interplanetary shocks [Richardson and Cane, 1993] and interaction regions in the heliosphere [Bzowski et al., 2000] the ion heat ux vector is generally small and often neglected. We have arrived at a uid description of the components of a plasma. These moment equations will be used in Chapter 2 as a simple way to approximate the characteristics of the protons observed by the Wind Faraday Cup instruments for comparison with the more detailed analysis. While Faraday Cups observe the threedimensional distribution of particles in velocity space, they do so not by directly observing f(~v), but instead by measuring a quantity related to the \reduced" distribution function, F (v). F (v) is the density of particles as a function of speed in a given direction, ^n, after an integration over all velocities perpendicular to ^n, F (v) ZZ 1 ;1 f(~v)d 2 ~v? : (1.65) In kinetic treatments of magnetized plasmas one is often concerned with the VDF in the direction of the ambient magnetic eld and calculations involve the reduced distribution function with ^nk ~ B Magnetohydrodynamics and Equations of State Section outlined the method for collapsing the velocity distribution function, leading to the uid description of a species of particles through moments. We now need a self-consistent description of the evolution of a plasma uid consisting of one or more species in the presence of electromagnetic elds. This is the framework of magnetohydrodynamics [Alfven, 1943]. Magnetohydrodynamics (MHD) is the basis for many of the calculations in this thesis, especially for the kinetic micro-instabilities of Chapter 4 and the interplanetary shocks of Chapter 5. 38

39 We start with Maxwell's equations and claim that no free charges will accumulate over distances greater than D due to Debye shielding. In addition we assume that displacement currents may be neglected. Maxwell's equations then take the truncated form, ~r ~ B =0 ~ r ~ E =0 ~ r ~ B = ~ J ~ r ~ E = ~ : (1.66) The uid treatment of the previous section was for individual species of particles. We are working towards single uid MHD, so while we start with uid properties of multiple species the end goal is to have a single temperature, velocity, density at each point in space. For simplicity consider a plasma which consists of singly charged ions of mass m i, number density n i, mass density i = m i n i, bulk velocity ~ U i, and pressure p i (this pressure may be anisotropic so p i is a tensor) and electrons with corresponding parameters m e n e e ~ U e, and p e. Since we have assumed that the uid remains neutral we have as an equation of constraint thatthe total charge density ch must always vanish, ch = ch i + ch e = q i n i + q e n e =0: (1.67) The total number of particles of a given species is conserved, which applies separately for the ions and the + ~ r( i ~ + ~ r( ~ U)=0 + ~ r( e ~ Ue )=0: (1.69) If we add these two equations, recognize the total mass density m i + e, and dene a center of mass velocity ~U i ~ U i + e ~ Ue i + e (1.70) then this reduces to a single equation for conservation of mass + ~ r( m ~ U)=0: (1.71) Now consider a momentum equation with forces due to the Lorentz force (1.19) and pressure gradients but neglecting thermal motions U ~ = ch( E ~ + U ~ B) ~ ; rp: ~ (1.72) 39

40 Write out the forces acting on the ions and electrons, m i n ~ U = q in i ( ~ E + ~ U i ~ B) ; ~ rp i m e n ~ U = q e n e ( ~ E + ~ U e ~ B) ; ~ rp e : (1.73) If we combine these two equations the electric eld terms cancel because ch =0 and n i q i = ;n e jq e j = nq. Dening the total pressure p p i + p e and current ~J n i q i ~ Ui + n e q e ~ Ue, we have a single momentum equation for the uid ~ = ~ J ~ B ; ~ rp (1.74) which diers from the standard hydrodynamic Euler equation [Faber, 1995, x2.3] by the addition of a ~ J ~ B force. For the moment let the pressure be isotropic. In that case there are 15 variables ( ~ E ~ B ~ J ~ U m ch p) and 11 independent equations. One common way to close this system is to introduce a version of Ohm's Law to relate the currents to the electric eld, and an equation of state to relate pressure and density. The standard Ohm's Law is ~ J = ~ E, where is the conductivity of the material. A more generalized form [Goldston and Rutherford, 1997, Equation (8.13)] for a plasma in steady state in the limit m e =m i 1 reads ~E + ~ U ~ B = ~ J + 1 qn ( ~ J ~ B ; ~ rp e ) (1.75) where, the resistivity, is the inverse of. In ideal MHD the plasma is a perfect conductor (the conductivity of coronal material is higher than that of copper at room temperature [Spitzer, 1956])! 0 and in the absence of currents and electron pressure gradients ~ E + ~ U ~ B ' 0. Combining (1.75) with Faraday's law leads to an induction equation for the time variation of the magnetic eld, where is the magnetic ~ = ~ r( ~ U ~ B)+r 2 ~ B (1.76) m and p are related through an equation of state f( m p), such that under any transformation df =dt = 0. If the transformations are reversible, they occur in times less than those required for heat to ow through the system, and the pressure is isotropic, then we can use the adiabatic equation of state d ; p ; m =0 (1.77) dt where is the ratio of specic heats for transformations with constant pressure or volume, = C p =C v. Under idealized circumstances we may calculate for an ideal 40

41 Maxwellian gas = 5=3. This simple picture does not describe the magnetized solar wind very well, even in cases where transformations are sudden enough to preclude the ow of heat. Consider the response of the plasma to a change in the magnetic eld strength which is sudden with respect to the rate of heat conduction but slow compared to the ion gyroperiod. In this case the adiabatic approximation of Section is valid and from the rst adiabatic moment (1.27) there is a conserved relation between eld strength and the speed with which a particle gyrates, d dt mv 2? =0: (1.78) 2B If v? is the typical thermal speed of a particle, w?, then under an adiabatic transformation of the eld w? / p B (1.79) while w k remains constant. It has been shown that by keeping the pressure tensor anisotropic in the MHD derivation [Chew et al Parks 1991, x7.4] one can derive the double adiabatic equations of state, a description of a magnetized plasma with separate relations for the pressure and density parallel, d dt P? nb = d dt w 2? B = d dt T? B =0 (1.80) and perpendicular d Pk B 2 dt n 3 = d dt w 2 k B2 n 2! = d dt Tk B 2 n 2 =0 (1.81) to the ambient magnetic eld. Thus even if a particle distribution is initially in isotropic thermal equilibrium a change in density or eld strength will immediately generate an anisotropy described by the bi-maxwellian distribution (1.58). Chapter 2 details the characterization of in-situ measurements of solar wind ions as convected bi-maxwellian distributions. In Chapter 3 we will see that even the double adiabatic picture is not sucient and under certain circumstances this temperature anisotropy drives instabilities which generate electromagnetic uctuations that isotropize the ion distributions Breakdown of Fluid Approximation: Kinetic Theory Ideal single-uid MHD has its limitations, mostly due to the assumptions about Ohm's law and the equation of state which went into its derivation. The ideal Ohm's law implies that the magnetic eld is \frozen-in" to the plasma, but in reality diusion of eld lines is a frequent occurrence with often dramatic eects. The equations of state assumed that the solar wind plasma is an ideal gas, but it is not clear that a 41

42 single value of describes the solar wind, nor that the assumptions of adiabaticity are correct. We shall see many examples of situations where the IPM does not behave like a single uid, and where interesting physics lies in the fact that particle species respond dierently. The most signicant failure of MHD is that it cannot incorporate the physics described by the Vlasov equation: from (1.53) it is clear that electromagnetic uctuations present in the IPM may couple to a specic portion of the velocity distribution function. The response to a propagating wave becomes complex because the particle distribution may not remain Maxwellian (so T and p are insucient descriptions of the VDF) and the resulting perturbations to the VDF may then alter the electromagnetic elds. A common theoretical approach to this problem is that of linear kinetic theory [Stix, 1992, x8.4-x8.7], in which the departure from the equilibrium VDF is assumed to be small enough that each variable may be expanded in a series, f j (~r ~v t) =f (0) j (~r ~v)+f (1) j (~r ~v t)+f (2) j (~r ~v t)+::: ~E(~r ~v t) = E ~ (0) (~r)+ E ~ (1) (~r t)+ E ~ (2) (~r t)+::: ~B(~r ~v t) = ~ B (0) (~r)+ ~ B (1) (~r t)+ ~ B (2) (~r t)+::: (1.82) where the (0) superscript is for the initial value and (i) indicates the i th -order departure from that state. In linear analysis the zeroth and rst order are kept and required to satisfy the kinetic equations: the electromagnetic potentials (1.28), Maxwell's equations ( ), the Vlasov equation (1.53), and the MHD equations of Section if they are valid. Finally the uctuation being examined is considered to be at a single frequency. In time and space every variable g is assumed to have the same dependence, g (i) (~r t) =g (i) (! ~ k)e i(~ k~r;!t) ~ k = ky ^y + k z^z (1.83) where ~ k is dened to be real but the frequency! is complex and may be separated into real and imaginary components,! =! r +! i or! =! r + i (1.84) A uctuation is heavily damped if <;j! r j=2, it is a stable wave if;j! r j=2 0, and it is a growing instability if > 0. The threshold for the growth of the instability isreached when an increase of the free energy in the VDF leads to > 0 for one or more wavevectors. If the mode has a wavevector ~ k then a mode is longitudinal if ~ k ~ E (1) = 0 and it is transverse is ~ k ~ E (1) =0. Fluctuations with only ~ E (1) 6=0may be derived with the Vlasov equation and Poisson equation. If ~ B (1) = 0 then it is an electrostatic uctuation. Waves and instabilities with electric and magnetic perturbative elds are electromagnetic. The two general goals of a linear kinetic analysis are nding relations between! 42

43 and ~ k and determining the stability of a situation through the evaluation of. Since the starting point is small perturbations, linear analysis cannot predict the nal state of the system or the eect of gross departures from equilibrium. As an example of a simple self-consistent derivation using linear analysis consider the natural oscillation of electrons in a plasma at the electron plasma frequency,! p e. This relation is used in Section 4.4 when we verify the absolute calibration of the Faraday Cup ion instruments on the Wind spacecraft. These oscillations are at a very high frequency and only the electrons will respond to them, so only the motion of electrons need be considered. Assuming that initially everything is at rest, there are no magnetic elds, and neglecting ions there is one momentum equation, # "@ U m e n ~ e +(~ U e r) ~ U ~ e = ;en e E ~ (1.85) and one continuity + ~ r(n e ~ Ue )=0: (1.86) Say all of the uctuations we are considering occur in one dimension. The divergence of the electric eld is ~r ~ = = e (n i ; n e ): (1.87) If the oscillations are indeed a small perturbation to a background then each of the variables can be expressed as the constant initial value plus a small term due to the perturbation, n e = n + n 1 U e = U 1 E = E 1 n i = n U i =0 (1.88) Inserting these expansions back into the momentum and continuity equations and only keeping terms of up to rst order in the perturbation, m + 1 = ;ee 1! m 1 @x (n U 1 + n 1 U = ;ee 1 (1.89) =0 (1.90) Since Poisson's equation is linear we can let it act just on the perturbed eld and get a third 1 =@x = ;en 1. Now if we assume that all of the perturbations oscillate in phase with frequency! and spatial wave-vector k, then their variation can be represented by exp i(kx ; wt). Carrying out the partial derivatives we have 43

44 three equations, ;imwu 1 = ;ee 1 ;iwn 1 = ;n iku 1 (1.91) ik E 1 = ;en 1 Eliminating E 1, n 1,thewave-vector k falls out and we are left with an expression for the electron plasma frequency,! p,e,! 2 p,e = n e 2 m e : (1.92) Typical values for the electron plasma frequency in the solar wind are 50 to 300 khz. Now consider the evolution of an electromagnetic uctuation which couples to the particle distribution function through (1.53). For simplicity we examine an electrostatic wave in an unmagnetized plasma. To rst order, the Vlasov equation may be (1) (~r ~v + (~r ~v + q m ~ E (1) (~r =0 (1.93) where magnetic elds are ignored, and the initial distribution is spatially uniform. If the eect of the magnetic vector potential is negligible then the induced uctuating electromagnetic eld ~ E (i) from the i th -order perturbation of the particle distribution is the spatial gradient of the perturbed component of the electric potential, (i) (~r t). If in addition we consider the solution at a single frequency (1.83), then (1.93) becomes ;i(! ; ~ k ~v)f (i) ( ~ k ~v!) = q m (i) ( ~ k!)i ~ (~v) : We mayintegrate over this to solve for the total density of the perturbed distribution, n (1) ( ~ k!) = q m (1) ( ~ k!) Z d 3 ~v (0) (~v) k : (1.95) ~ k ~v Note that the largest eects will occur in the portion of the VDF moving with the phase velocity of the uctuation and that the result is sensitive to the derivative of the equilibrium distribution at the resonance. Following [Gary, 1993] dene the susceptibility K( ~ k!), such that n (1) ( ~ k!) =; k2 (1) ( ~ k!) K( ~ k!) (1.96) 4q K( ~ k!) =;!2 nk 2 Z d 3 ~v (0) (~v) k ~ k ~v (1.97) 44

45 If there are multiple species than by the requirement of overall charge neutrality, ; X j K j ( ~ k!) =1: (1.98) The procedure is then to identify the modes in the plasma which satisfy the dispersion equation, 1 ; X j Z! j 2 n j k 2 (0) (v k ) =0: (1.99) v k ;!=k k In general the solutions to (1.99) are not all stable oscillations. The existance of a non-vanishing indicates the existance of damped or growing modes which will alter the distribution function. Linear analysis has been applied with a great deal of success: Landau used it in 1946 to identify the mechanism whereby a collisionless medium may damp a propagating wave [Landau and Lifshitz, 1981, x30]. In that case an analytical solution for the growth rate was obtained, 1=2 k 3 = ; e! r 2 8 k 3 e r=2k ;!2 2 v e 2 (1.100)! e where w e is the electron plasma frequency, v e is the electron thermal speed, and k e is the electron intertial length. It is not always possible to obtain an analytical expression for, so in many cases (1.99) is solved numerically. Chapter 3 explores the validity of the predictions of linear theory for kinetic instabilities driven by ion temperature anisotropies by comparing its predictions and the results of numerical simulations with observations in the IPM. In those cases a large temperature anisotropy permits a growing uctuation, and the coupling of this uctuation to the VDF scatters particles and drives the distribution towards isotropy. A more detailed discussion of this procedure is discussed in Section in the context of instabilities. 1.3 The Wind Spacecraft and Experiments The Wind spacecraft was launched on November 1, 1994 an a Delta II rocket from Cape Canaveral, Florida. It and the Polar spacecraft are the two missions of the Global Geospace Science (GGS) initiative, the United States component of the International Solar-Terrestrial Physics (ISTP) program. The goal of the Wind and Polar missions is to understand the physics of the solar-terrestrial system, in particular to study how Earth responds to changes in the IPM. Four of the eight experiments on Wind which are of particular interest for this thesis are the SWE, Waves, and MFI investigations: Solar Wind Experiment (SWE) A collection of instruments for characterizing the bulk properties of the major solar wind ions, hydrogen and helium, and 45

46 electrons. Two Faraday Cup instruments measure the properties of the hydrogen and helium velocity distribution functions. They were developed at MIT and are the focus of this thesis. Radio and Plasma Waves (Waves) Remote study of waves of solar origin and in-situ measurements of wave modes of the IPM plasma. The Thermal Noise Reciever (TNR) sub-system of Waves measures the spectrum of electromagnetic uctuations from khz. Magnetic Field Investigation (MFI) Vector magnetic eld measurements at a rate of up to 40 ms. Three-second averages of the magnetic eld data were provided by Adam Szabo for the studies reported in this thesis. Wind is a cylindrical spacecraft, 2.4 meters in diameter and 1.8 meters in height. It has a mass of 895 Kg and at launch carried an additional 300 Kg of hydrazine propellant. Spacecraft attitude and ephemeris, along with any measured vector quantities in this thesis, will use the Geocentric Solar Ecliptic (GSE) coordinate system. The GSE system has its ^x GSE -axis pointing from the Earth towards the Sun and its ^y GSE -axis is chosen to be in the ecliptic plane pointing towards dusk (thus opposing planetary motion). The ^z GSE -axis is parallel to the ecliptic pole, pointing Northward to complete the right-handed system. Note that relative to an inertial system this system has a yearly rotation and that the Earth moves in the ;^y GSE direction with a speed of 30 km/s. Wind is a spin-stabilized spacecraft with a rotational period of three seconds. It rotates about its symmetry axis, which is aligned with ^z GSE. The rotation is in the clockwise sense when looking down from the North into the ecliptic plane. Gravitational assists from the moon and the substantial on-board fuel are used to maintain Wind in a variety orbits which allow us to focus on regions in the local IPM of particular interest. Three examples of orbital trajectories followed by Wind are shown in Figure 1-1. In each of the plots the trajectory of the Wind spacecraft is shown by the blue line. The diamond indicates the starting location of the spacecraft along each of the curves. The dashed red line is the orbit of the moon at 60 R e. In the interval from July, 1995 through May, 1996 the spacecraft trajectory was guided by a series of lunar encounters such that the apogee of each orbit was in the Sunward direction, even as Earth orbited the Sun. During this period Wind passed within the Earth's bow shock once every 30 days. On several occasions lunar encounters have been used to send the spacecraft to the rst Lagrangian point, a point of unstable equilibrium located 220 R e upstream of Earth towards the Sun. The center panel of Figure 1-1 shows the trajectory of Wind from October, 1997 through July, 1998 during such an maneouver. The advantage of an orbit at the Lagrangian point is that is provides continuous observations of the solar wind upstream of Earth. Recently Wind has been in a distant prograde orbit in which it travels between two points 250 R e ahead and behind the direction of Earth's orbit in the solar system. This orbit is very useful for studying the length-scales of variations in the solar wind plasma. In Chapter 6 we will use these large separations from Earth to study interplanetary shocks. 46

47 Figure 1-1: A survey of the orbital trajectories in the ecliptic plane employed by the Wind spacecraft over the mission to date. the solid line is the Wind trajectory with a diamond indicating the nal point in each interval. The dashed line indicates the orbit of the moon. The types of orbits are: (a) Double lunar swing-by (b) Lagrangian halo orbit about the L1 point at 220 R E (c) Distant prograde orbit. See the text for more details. 47

48 1.4 Outline of Thesis In Chapter 2 the two Faraday Cup instruments on the Wind spacecraft are introduced. The response of a Faraday Cup to a convected bi-maxwellian ion species is derived and the procedure developed for analyzing each proton spectrum is described. The results of this analysis are then presented, including the success rate and the typical uncertainties of each of the derived parameters. In Chapter 3 the range in proton temperature anisotropy observed in the solar wind at 1 AU is examined. Specically, the measured upper and lower bounds to the anisotropy are compared with theoretical predictions of the eects of the rehose, mirror, and cyclotron plasma micro-instabilities. It is demonstrated that that predictions of kinetic theory are in better agreement with the data than the limits derived from the MHD uid approach. The rst observations of the rehose instability in space are reported, along with the detection of a new instability. In Chapter 4 the bi-maxwellian analysis developed and applied to the hydrogen velocity distributions in Chapter 2 is extended to study the helium properties. The hydrogen and helium number densities are then compared with electron plasma frequency observations recorded by the WAVES/TNR experiment on Wind. This comparison is used to produce an absolute calibration of the Faraday Cup number densities. In Chapter 5 the kinetic properties of helium in the solar wind are studied. It is demonstrated that many of the kinetic aspects of helium dier from those of hydrogen, especially in regions of the solar wind with few collisions. The limits imposed upon these dierences by micro-instabilities are studied. It is shown that the proton temperature anisotropy acts to constrain the dierential owbetween the two species. In addition, the rst observations of the limits to the helium temperature anisotropy imposed by micro-instabilities are presented, demonstrating that the collective interaction of each species' velocity distribution with uctuation electromagnetic elds is in agreement with Vlasov theory and not single-uid MHD. The instabilities examined in Chapters 3 and 5 are well-described by Vlasov theory under the linear approximation. In Chapter 6 the most violent, non-linear perturbations in the solar wind are studied: collisionless interplanetary shocks. The seven most common methods for analyzing shocks are presented, and each method is applied to more than 200 shocks seen by the Wind instruments. Comparison of the results of each method and of observations by multiple spacecraft shows that the full Rankine-Hugoniot relations are most appropriate for characterizing interplanetary shocks. 48

49 Chapter 2 Bi-Maxwellian Analysis of Wind/SWE Ion Spectra ABSTRACT: A description of the Faraday Cup ion instruments on the Wind spacecraft. Extension of existing analysis techniques to measure the bi-maxwellian nature of ion species. Evaluation of the analysis, comparison with other methods, instruments, and spacecraft. A brief survey of the typical properties of solar wind protons. 2.1 Introduction This chapter presents my analysis of data taken by the Faraday Cup (FC) instruments which make up the ion portion of the Solar Wind Experiment (SWE) on the Wind spacecraft [Ogilvie et al., 1995]. The data produced as a result of this analysis are the basis for the research detailed in subsequent chapters of this thesis. The goal of SWE/FC is to characterize the bulk properties of the major ions in the solar wind, namely + H and +2 He, but also + He in the rare situations in which it is seen. The Faraday Cups probe the three-dimensional distribution of proton and alpha particles in velocity space f p (~v) and f (~v) through measurements of the reduced distribution function (1.65). Most scientic investigations do not require this detailed information about the ion properties. As shown in Section and Section 1.2.7, to rst order we can describe the solar wind in as a uid,with temperature T,velocity ~ U, and density n as a function of location and time. In theory calculating these contracted properties of the distribution function follows the integrals listed in Section In practice the features of the instrument must be taken into account in order to produce accurate values of these \key parameters". In this chapter the proton spectra are analyzed both through a detailed comparison of observations and a model response function, which will be referred to as the non-linear analysis method, and through a simplied moment routine, the moment analysis procedure, based on the derivations in Section The primary data products which the Wind/SWE Faraday Cups produce are the velocity, thermal speed, and density of the protons, along with the relative abundance ratio of alpha particles to the protons. John Steinberg, currently at the Los Alamos 49

50 National Laboratory, supervised the original key parameter analysis which produced the proton velocity ~ V p, thermal speed w p and density n p, and started the analysis of helium [Steinberg et al., 1996]. The full alpha particle analysis was done by Matthias Aellig, producing the alpha velocity, ~ V, thermal speed, w, and number density n [Aellig et al., 2001b]. I have extended our ability to characterize the ion spectra by deriving an analytical expression for the response of a Faraday Cup to the convected, eld-aligned, bi-maxwellian velocity distribution function described in Section Instead of the single thermal speed w p the parallel (w kp ) and perpendicular (w?p ) thermal speeds are measured. This chapter describes this analysis of the Wind Faraday Cup proton spectra. In addition to the new response function I have introduced a new method for determining the eective collecting area of the instrument, carried out the rst detailed analysis of the accuracy with which the Faraday Cup can quantify the solar wind ions, and compared the results of the non-linear and moment analysis techniques. There are several reasons why wewould like to measure anisotropies in the plasma species seen by the WIND Faraday Cup: Based upon the 2 merit function dened and discussed in Section 2.4.2, the bi- Maxwellian velocity distribution function is a more accurate description of ion species seen in the solar wind than a single isotropic Maxwellian VDF. From the simple point of view of conducting the best analysis of existing data as possible we should make the extension to a two-temperature model. Plasma micro-instabilities should place limits on the maximum temperature anisotropies allowed in the solar wind. Knowledge of these instabilities is of interest to the general plasma community. The existence of anisotropies in the ion distributions may signicantly alter the results of certain analysis methods, for example in the study of collisionless magnetohydrodynamic shocks. Recent results with data from the SOHO spacecraft [Antonucci et al., 2000 Cranmer, 1999] suggest that extreme ion anisotropies exist in the solar corona. Anisotropies observed in the solar wind may contain information about this process and are therefore useful in the study of the corona and origin of the solar wind. Section 2.2 describes how the Faraday Cup works and then details the derivations of FC response functions for a given velocity distribution function. The non-linear analysis algorithm is outlined in Section 2.3. In Section 2.4 the uncertainties in each of the derived parameters are explored. 50

51 (a) (b) + H.V. A A Figure 2-1: A simple cartoon showing how afaraday Cup measures charged particle uxes. In (a) particles of various energies (energy / length of arrow) and dierent charges (red,blue) are allowed to ow through an opening into a grounded metal container. An insulated metal plate at the back of the cup collects the particles and the total current is measured as these particles ow through the wire (green) to ground. In (b) two metal grids have been inserted into the cup. The outer grid is grounded and a voltage is applied to the inner grid to repel particles of the desired charge. 2.2 Faraday Cup Measurements of Ion Distribution Functions The operating principles of a Faraday Cup are straightforward. In fact the FC is one of the few instruments for which analytic expressions may be derived for the response to a given ion distribution. Access to an analytic expression is a tremendous advantage because it allows us to compare a model distribution function with the observations and to vary the parameters of that model to produce the best agreement with the observations. As a byproduct of that analysis, we also get estimates of the uncertainty in each parameter. One of the themes of this thesis is that these uncertainties may be propagated through to any derived quantity we wish to explore, which then allows us to express a rigorous value for the statistical signicance of any result. The operational design of a Faraday Cup is outlined in Section 2.2.1, followed in Section by a description of a single ion spectrum measured by the two Faraday Cup (FC) instruments on Wind. In Section the response of a FC to an isotropic Maxwellian distribution function is derived. In Section that response is extended to a bi-maxwellian VDF. As a second method for analyzing FC ion spectra the moment analysis procedure is described in Section 2.2.6, and an algorithm for determining temperature anisotropies with moments is detailed in Section A new method for calculating the correct eective collecting area of the FC due to the transparency of the grids is presented in Section Operational Design of the Wind/SWE Faraday Cups The basic principle of operation of a Faraday Cup ion instrument is to measure the current produced by the ux of particles which pass through the instrument and impact one or more electrically insulated collector plates. This processes is indi- 51

52 cated schematically in Figure 2-1. In the left image particles of positive and negative charges, q + q ; (red and blue arrows) and varying speeds v k parallel to the instrument symmetry axis (proportional to the lengths of the arrows) pass through an entrance with area A and impinge upon a collector plate which is insulated from the instrument. Excess charge does not build up on the collector because it is grounded through the green wire. As current ows through the wire it is measured by an ammeter. But what is the relationship between the measured current and the VDF? If the VDF of the particles are uniform across the entrance and their average speeds and densities are U k+ U k; and k+ k;,thenthe measured current is, I meas = q + + U k+ + q + + U k+ : (2.1) The problem with this conguration is that the properties of the positive and negative particles are intertwined. Indeed, if they had the same average speeds and the plasma were neutral overall then no current would be measured. An improvement to the design is shown in the second image of Figure 2-1, in which two wire grids have been inserted within the instrument, between the entrance and the collector plate. The outer grid is grounded and a positive high voltage V H.V. is applied to the inner grid relative to the instrument. This has the eect of shielding the collector plate from positively charged particles with insucient energy to penetrate the region between the grids. A particle of mass m and charge q is reected unless 1 2 mv2 k q V H.V. (2.2) which now allows the FC to probe the distribution of particles f(v k )asa function of their speed normal to the grids by varying V H.V. and recording I meas. There are still several aspects which need improvement. For example, photoelectrons produced by ultraviolet rays from the sun illuminating the collector plates or secondary electrons produced when a particle strikes the collector surface and subsequently escape could generate additional currents which may overwhelm the real signal. Additionally in this example the VDF of the particles which have sucient energy to pass the high voltage grid has been altered. Still, this does illuminate the the basic operating principles of a Faraday Cup. Since (2.2) does not involve the phase space density's dependence upon speeds ~v? perpendicular to the cup axis, the FC measures a quantity related to the reduced distribution function (1.65), as described in Section The threedimensional characteristics of the VDF are then resolved by placing the instrument on a rotating spacecraft. The rst observations of ion uxes in space were performed using such afaraday Cup, with a large, xed negative voltage to prevent theentrance of electrons into the detector [Gringauz et al., 1960]. The xed voltage design is still used today [Nemecek et al., 1997 Santolik et al., 1997], mainly due to its simplicity and ability to rapidly characterize the ions through single measurements of the ux, resulting in measurements with a high cadence [Unti et al., 1973], but at the loss of resolving the VDF. Faraday Cups have been a workhorse for space plasma measurements for the du- 52

53 ration of in-situ space exploration [Vasylinuas, 1971 Gloeckler, 1990]. Over the last four decades the MIT space plasma group has developed and extended the capabilities of these instruments for exploring the heliosphere [Bridge et al., 1977 Gazis et al., 1989], by reducing their mass and power requirements, [Lazarus et al., 1993], increasing their cadence [Aellig et al., 2001a], and extracting additional information from existing observations [Richardson, 1986 Kasper et al., 2001a,b]. Many improvements may be made to the basic design, and the Faraday Cups own on the Wind spacecraft reect this. The design characteristics of the two Faraday Cups of the Solar Wind Experiment (SWE) on the Wind satellite are shown in Figure 2-2 There are a total of nine grids in the Wind Faraday Cups, each serving a special purpose. Instead of the pair of grounded and high voltage grids in Figure 2-1, a modulator assembly contains ahighvoltage grid surrounded by two ground grids. As a result, any particle which manages to pass the high voltage grid is re-accelerated to its original velocity by the time it leaves the assembly, eliminating any distortion of the original VDF. A high-voltage power supply within the instrument applies a squarewave voltage waveform to the grid at the center of the modulator. The square-wave oscillates between two high voltages, V 1 and V 2 at 200Hz, with (V 2 ; V 1 )=V 1 10%. As shown in the gure there are three categories of particles for a given set of (V 1 V 2 ): those that always reach the collector plates, those that never reach the collector plates, and those particles with parallel speeds V 1 =q 1=2mvk 2 V 2 which only reach the collector plates 1=2 of the time and produce a current which is modulated at 200 Hz. The exact details of the conversion of this alternating current into a measurement which is sent back down to Earth are detailed elsewhere [Ogilvie et al., 1995], but the key is that only the modulated current is recorded. This eliminates the eects of photoelectrons, solar wind electrons, and any other background signals. By stepping through increasing voltages the VDF of ions may be scanned. Two outer ground grids isolate the rest of the spacecraft from electromagnetic interference generated by the oscillating electric elds, and three inner ground grids prevent the modulator from inducing stray currents on the collector plates. A nal grid, the suppressor grid, is mounted right above the collector plates and is maintained at ;100 volts, a sucient amount todrive any secondary electrons back into the collector plates. Another feature of the Wind Faraday Cups is the combination of the limiting aperture located immediately underneath the modulating assembly, and the fact that the collecting surface at the back of the instrument has been divided into two equal semi-circular plates. There is a small but treatable abberation of particles as they travel through the modulator: since the electric eld is perpendicular to the grids, each particle maintains its velocity component parallel to the grids. Therefore as it is slowed down and subsequently reaccelerated in the modulator it continues to travel in the plane of the collectors. The net result is a translation of the beam of particles which is a function of their energy (in an analytic treatment itranges from the geometrical projection at high energies to exactly twice that in the case that the particles just make it through the modulator). The limiting aperture removes this problem over a large range of angles because up to approximately 45 the aperture is evenly illuminated. Thus a clean circular beam is projected onto the collector plates. 53

54 Figure 2-2: An overview of the properties of the Faraday Cups on the Wind spacecraft adapted from [Ogilvie et al., 1995]. Top: A cross-section of the instrument along its symmetry axis, showing the entrance aperture, modulator assembly, limiting aperture, suppressor grid, and collector plates. Bottom Left: A sketch of the operating principle of the modulator assembly with the three categories of particles: those that always do or do not make it past the high voltage grid, and those which only pass through at one energy and produce a current which is modulated at 200 Hz. Bottom Right: The eective collecting area as a function of angle of incidence, this gure includes geometrical eects and the transparency of the wire grids. 54

55 Xc Zc A B Yc Figure 2-3: Illustration of the Faraday Cup coordinate system for the Wind instruments. The two collector plates, A and B, lie in the ^x c ; ^y c plane, with ^x c directed along the split between the plates. The ^z c axis is normal to the collector plates. In subsequent discussion the \cup orientation" ^n is the direction of ^z c in the GSE coordinate system. Since the plates are split, it is then a simple matter to determine the ow angle of the incoming particles based on the relative currents seen by each plate. This allows each cup to individually determine the three dimensional properties of the solar wind ions in the event that one instrument fails. For the purposes of this thesis we will treat each FC as having a single collector plate and simply sum the two currents. The plot in Figure 2-2 is of the eective collecting area of the instrument as a function of the angle of incidence. There are three factors which determine this. First the size of the limiting aperture, which is about 33 cm 2. Second there is the transparency of the grids, which decreases very slowly from 0 to 40 but then falls o rapidly to zero. The transparency is determined by the thickness of the wires used in the grids and the average separation between wires. It can be calculated analytically and compared with observations in a calibration beam on Earth. Finally at extreme angles, greater than about 45, the limiting aperture is no longer completely illuminated and the response becomes much more dicult to describe analytically. Numerical simulations have however yielded corrections to the eective collecting area which have been folded into the curve shown in Figure Description of a Single Wind/SWE Faraday Cup Spectrum Wind rotates once every three seconds, so the following derivations of the response of a Faraday Cup to ion distributions will require transformations between the Geocentric Solar Ecliptic (GSE) coordinate system (described in Section 1.3), and the frame of the instrument. A sketch of the Faraday Cup coordinate system used in all of these 55

56 derivations is shown in Figure 2-3. A transformation matrix that converts a vector in GSE components ~ V G to the frame normal to a Faraday cup on Wind, ~ V C, is shown below. Call the tilt angle of each cup out of the ecliptic, and the azimuthal angle in the ecliptic. Dene as zero along ^x GSE and increasing towards ^y GSE. In component form a vector transformed into the cup frame becomes Vx C Vy C Vz C = V G x = V G x = V G x sin + Vy G cos cos sin ; V G y cos cos ; V G y sin sin + V G z sin cos ; V G z cos sin (2.3) The positions of the instruments which comprise the Solar Wind Experiment, including the two Faraday Cups, on the Wind spacecraft are shown in Figure 2-4. The spin-axis of the spacecraft, labeled \Z" in the gure, is perpendicular to the ecliptic plane of the solar system. The Faraday Cups are located 180 apart and look 15 out of the spacecraft equatorial plane, with FC1 looking slightly northward and FC2 southward. The line dividing the split collector plates also lies in the ecliptic plane, so the relative current on each plate is related to the ow angle of the solar wind out of that plane, or the \North-South" ow angle. This was done so that a single cup could extract the three dimensional solar wind parameters in the event of a failure. Each instrument must know the azimuthal orientation of the spacecraft to a high degree of accuracy. The exact period is therefore measured on board by a sun-pulse sensor. The sensor is essentially a photodiode with a narrow eld of view mounted on the side of the spacecraft, and each time that side of Wind faces the Sun it generates a pulse. A circuit tracks the average spin rate over several rotations and divides each rotation into 1000 sub-intervals. Small variations in the temperature of the spacecraft change its moment of inertia and thus alter the exact spin period. A single ion spectrum consists of a set of measurements by the two cups of the reduced distribution function along a series of angles and over consecutive energy windows. Figure 2-5 is a diagram of the measurement sequence. A single spectrum is taken as follows: When the cups are instructed by the SWE Digital Processing Unit (DPU) to begin a spectral observation they wait until a signal is received by the Sun-sensor and then start by performing an internal calibration run for one spacecraft rotation. When the calibration is complete the high voltage power supplies on each cup are enabled and begin modulating between the rst pair of voltages. Based on the timing signals from the Sun-pulse sensor the SWE DPU commands each of the two Faraday cups to perform measurements of the current at the single energy window along 20 angles. These angles may be modied through commands from the ground at any time, but they have been kept in the same location since launch: 18 angles spaced evenly at 3:5 intervals in and two angles directed anti-sunward, each 35 on either side of the ;^x GSE axis. Upon completion of the spin each modulator is commanded to the next higher voltage pair, and this process continues until the highest energy window is processed. The FC high voltage power supplies produce a square-wave waveform oscillating 56

57 Figure 2-4: Conguration of the SWE instruments on the Wind spacecraft, adapted from [Ogilvie et al., 1995]. The two Faraday Cupion instruments are mounted 15 out of the ecliptic plane, with the upper cup looking slightly northward, and the lower cup downward. Also shown are the Vector Ion-Electron Spectrometers (VEIS), the eld-aligned electron Strahl detector, and the SWE DPU. 57

58 Angle 3 Angle... Angle 2 Angle 1 V0-V1 V1-V2 V2-V3... Figure 2-5: Diagram indicating the measurement of a single ion spectrum. In each single rotation the FC measures along each of the angles indicated by red arrows at a single energy window indicated by the blue arrows. between two voltages selected from a list fv i g of 64 voltages logarithmically spaced from 200 V to 8 KV, with V i =V i =(V i+1 ; V i )=V i a constant 10%. Apairofvoltages denes an energy window. There are four modes that the instruments are run in, based on the width and number of energy windows in a spectrum. Single energy windows are bounded by the voltage pair (V i V i+1 ), double energy windows are twice as large, with voltages (V i V i+2 ). The instrument may either conduct a full scan, stepping from the lowest to the highest energy windows, or it may \peak-track", by only observing a subset of windows which bracket the window from the previous observation which returned the highest current. In general the Wind cups operate either in full-scan with double width energy windows or peak-track with single energy windows. The net result is that there are almost always 30 total energy windows in each spectrum. AFC spectrum thus consists of 2400 measured currents I meas as a function of the orientation of the instrument ( ) and the location of the energy window (V V ), 2 instruments 2 collectors 20 angles 30 windows = 2400 (2.4) For this study the currents from the two collector plates in each cup are summed together, so there are 1200 measurements in a spectrum. The goal is to extract accurate and precise ion parameters from this set of currents. In the following two sections I derive the expected currents from two model ion distribution functions, 58

59 the Maxwellian and the bi-maxwellian VDF. Once we have a relation for the model current I model as a function of the model parameters and the instrument variables, I model [ V V f model (~v)] (2.5) we can determine the parameters which produce the best t to the observations. Section describes an approximate method for extracting the same parameters for protons by calculating the moments of the observed currents Response to an Isotropic Maxwellian Ion Distribution First consider the current incident on a collector plate with area A in response to an isotropic, convected Maxwellian plasma with charge q, number density n, bulk velocity ~ U and thermal width w. The collecting area A is a function of the size of the limiting aperture, but it additionally reects the transparency due to the wires of the grids, so it is a function of the angle of incidence. During a single measurement, the FC high voltage modulator alternates rapidly between two voltages. These two voltages correspond to the minimum speeds, V and V +V, that a particle must have normal to the grids to pass through the modulator and impact the collector plates. The instrument returns the dierence in currents observed when the grids are set to V and V +V. We can do the same by integrating over the region of velocity space which contributes to that nal signal. The dierential amount of current di produced by a small element of plasma with velocity ~v in the frame of the FC (and speed v z = ~v ^z c normal to the cup) is given by di = A(~v=v)qv z f(~v)d 3 ~v (2.6) which is a more general form of the current seen by the simple FC (2.1). Inserting the isotropic Maxwellian distribution (1.54), the total current produced is obtained by integrating over the entire window, I iso = ZZZ Anq 3=2 w 3 window v z e ; (~v; 2 ~U ) w d 3 ~v: (2.7) 2 Now we have to make two assumptions in order to get an analytical result. These are both justied because the typical ratio of the bulk speed of the ions in the frame of the Faraday Cup to the thermal width of the distribution in the solar wind is U=w 10 ; 20, so the ions appear from a 2 ; 6 region of the sky. The validity of this assumption is demonstrated in Figure 2-19 of Section 2.5 using the results of the analysis. The eective area of the instrument does not have to be folded into the integration. Instead it can be pulled out of the integral and estimated by using the angle from which the bulk of the particles contributing to the current measured in this energy window were incident. The proper subsequent calculation of the eective area is discussed in Section

60 When examining the current produced by a region in phase space bounded in the cup frame by V z :[V V +V ], particles in phase space with all possible velocities perpendicular to the instrument symmetry axis contribute. In reality there is a maximum possible angle of incidence into the FC, above which incoming particles are blocked. We assume here that the FC measures the current produced by an entire planeofthickness V in phase space. Moving to the rest frame of the plasma by subtracting the bulk motion ~ U of the ions, ~W ~v ; ~ U! d 3 ~ W = d 3 ~v (2.8) simplifying the exponential term in (2.7). With the integrals in the ^x c ; ^y c frame running from [;1 1] the integral takes the form, Anq 3=2 w 3 Z V+V;Uz V ;U z Z 1 ;1 Z 1 ;1 (W z + U z )e ;(W 2 x +W 2 y +W 2 z )=w 2 d 3 ~ W: (2.9) The exponential may be broken into three parts, and the two integrations in W x and W y carried out separately, Anq 3=2 w 3 Z V+V;Uz V ;U z (W z + U z )e ;(W 2 z =w 2) dw z Z 1 ;1 e ;(W 2 x y=w 2) dw x y : (2.10) Using the standard relation for the full integral over a Gaussian distribution, Z 1 ;1 only the integration over W z remains, e ;(x2 = 2) dx = p (2.11) Z Anq V+V ;U z p (W z + U z )e ;(W 2 z =w2) dw z : (2.12) w V ;U z This can be separated into two parts by breaking up the V ; U z sum and writing two integrals, Anq p w Z V+V ;U z V ;U z W z e ;(W 2 z =w 2) dw z + U z Z V+V ;U z V ;U z e ;(W 2 z =w 2) dw z : (2.13) The integral on the left has an analytical solution because of the additional factor of W z proceeding the exponential. The right integral does not, but its value may be expressed using the denition of the error function erf, [Abramowitz and Stegun, 1972, x6], erf(x) 2 p Z x 0 Z e ;t2 dt =1; p 2 1 e ;t2 dt: (2.14) 60 x

61 The solution is, 2 Anq p 4 w 2 ; w 2 e;w 2 z =w2 V +V ;U z V ;U z + U z 3 (V+V;Uz)=w 2 erf p (V ;U z)=w 5 : (2.15) After simplication we have the model current I iso afaraday Cup would observe due to the presence ofaconvected, supersonic, isotropic Maxwellian particle VDF, I iso = ANq wp e 2 =w ;(V;Uz)2 2 =w ; e ;(V+V;Uz)2 2 + U z erf V +V ; U z w ; erf V (2.16) ; U z w This formula has been used historically for tting Faraday Cup ion spectra. The response functions derived here and in the following section are valid for any supersonic VDF (j ~ Uj=w 1), and they are used in Chapter 4 to characterize alpha particles. The response is a function of the charge q of the species in question, and it is worth noting that an equal ux of +2 He produces twice the current of protons. This response is also a function of the mass of the particles, because the relationship between the modulator voltages and the speeds corresponding to either limit of the window are functions of the charge to mass ratio of the species, as can be seen in (2.2). Thus observations of alphas and protons, which travel at approximately the same velocity in the solar wind, appear at dierent voltages in FC spectra (see example (b) in Figure 2-6) Response to a Bi-Maxwellian Ion Distribution Figure 2-6 has four examples of possible solar wind ion spectra. They are presented in the raw form in which observations by the Faraday Cups are returned to Earth: the observed current in a given energy window as a function of the energy per ion charge at which the instrument was scanning. Panel (a) is a typical spectra seen in the solar wind, with protons and alpha particles moving with approximately the same speed. The alphas appear at a higher voltage due to their larger mass per charge. The black histogram is the measured current and the blue and red curves are the model responses for protons and alphas respectively which best t the observations. Panel (b) of Figure 2-6 shows two measurements of the same ion distribution, but at two angles with respect to the ambient magnetic eld. As we shall see this is indicative of a bi-maxwellian distribution. This chapter details my extension of the analysis of Wind FC ion spectra by tting the data with a model for the instrument response to convected, bi-maxwellian velocity distribution functions (1.58) described in Section There are other departures from Maxwellian VDFs seen solar wind which will not be examined in this thesis. In Panel (c) the measured currents produced by protons are shown along two angles with respect to the magnetic eld, with the black curve 61

62 Figure 2-6: A series of examples of ion distributions as a function of energy in the solar wind. Shown are the raw currents recorded by a Faraday Cup on Wind as a function of the center voltage E=q of the window. (a) Maxwellian protons and alpha particles with approximately the same speed. The black histogram is the measured current, while the red and blue curves are the calculated currents using the derived instrument response function and the best-t parameters (b) A proton distribution function viewed at two angles relative to the ambient magnetic eld. The red curve is from a measurement nearly along the magnetic eld, while the blue curve was nearly perpendicular to the ele. This demonstrates a remarkable temperature anisotropy with T kp >T?p (c) Double double streaming: two separate, dierentially streaming Maxwellian proton distributions. The red and blue curves are the predicted currents due to each of the proton distributions, and the green curve is the sum (d) Protons with a non-maxwellian high energy tail, which is not t well by the model response function. 62

63 corresponding to measurements made along the eld direction. It can be seen that in this case the proton distribution is actually composed of two, separate, possibly bi-maxwellian distributions. This phenomenum is called double-streaming, and it is being investigated by Dorian Clack [Clack et al., 2001, 2002]. For example, panel (d) demonstrates the directed heat ux ~ Q described in Section We are now ready to derive the Faraday Cup response to a convected, eldaligned, bi-maxwellian VDF. In addition to density and bulk velocity, there are now two thermal speeds, w? and w k, perpendicular and parallel to an ambient magnetic eld B ~. The starting point is again an integration over the accessible energy window, but with a bi-maxwellian distribution (1.58), I ani = ZZZ Anq 3=2 w 3 window v z e ; v2? w? 2! + v2 k w k 2 d 3 ~v (2.17) where the collecting area has already been pulled out of the integral. In addition to the thermal speeds we must take into account the alignment of the magnetic eld. The magnetic eld must be transformed into the frame of the cup for each measurement through (2.3). With the unit vector ^b = ~ B =B dening the orientation of the magnetic eld seen by the cup, the integral has been simplied by the substitution ~v k (~v ; ~ U) ^b v? (~v ; ~ U) ; [(~v ; ~ U) ^b]^b (2.18) The two velocities ~v? and ~v k are the components of velocity in phase space in the rest frame of the plasma perpendicular and parallel to the magnetic eld. The dependence on ^b will complicate the integration, but it is still manageable. With ^b in the frame of the cup we expand ~v? and ~v k, using the same denition of ~ W from the previous section, ~v k = W x b x^x + W y b y ^y + W z b z ^z v 2 k = W 2 x b2 x + W 2 y b2 y + W 2 z b2 z (2.19) and, ~v? =[W x ; b x (W x b x + W y b y + W z b z )] ^x +[W y ; b y (W x b x + W y b y + W z b z )] ^y +[W z ; b z (W x b x + W y b y + W z b z )] ^z: (2.20) The magnitude of ~ V? is ~v? ~v? =(1 ; b 2 x)w 2 x +(1; b 2 y)w 2 y +(1; b 2 z)w 2 z ; 2b x b y W x W y ; 2b x b z W x W z ; 2b y b z W y W z : (2.21) The additional assumption of this expansion is that ^b, in addition to the ion VDF, does not vary over the course of the measurement. Substituting these expressions for 63

64 v? 2 and v2 k back into (2.17), the integral takes the slightly longer form, I = Anq 3=2 w? 2 w k Exp ; Z V+V ;U z V ;U z dw z Z 1 ;1 Z 1 dw x dw y (W z + U z ) ;1 (1 ; b 2 x)w 2 x +(1; b 2 y)w 2 y +(1; b 2 z)w 2 z ; 2b x b y W x W y ; 2b x b z W x W z ; 2b y b z W y W z =w? 2 ; b 2 x W 2 x + b 2 y W 2 y + b 2 z W 2 z +2b x b y W x W y +2b x b z W x W z +2b y b z W y W z =wk 2 : (2.22) While the term in the exponential has a more complex form than the isotropic Maxwellian case, the integration is still straightforward. The nal result for the current I ani due to a supersonic convected bi-maxwellian VDF is, I ani = Anq ~wp e 2 = ~w ;(V;Uz)2 2 = ~w ; e ;(V+V;Uz)2 2 + U z erf V +V ; U z ~w ; erf V ; U z ~w (2.23) where now there is an eective thermal speed ~w which is a function of the orientation of the magnetic eld with respect to the cup, q ~w qb 2 z w2 k +(b2 x + b 2 y)w? 2 = b 2 z w2 k +(1; b2 z)w? 2 (2.24) By virtue of the fact that the Faraday Cup measures the reduced distribution function, a bi-maxwellian VDF looks exactly like an isotropic Maxwellian, with an eective thermal speed determined through a simple projection. Note that ~w only depends on the projection of the eld parallel and perpendicular to the cup, and not on the specics of the eld orientation in the cup's ^x c ; ^y c plane. This result is due to our approximation of the cup as having innite response in velocity space perpendicular to the collector plates and because we are summing the currents measured by the individual plates in each cup. We can make a few checks to verify that (2.23) holds under simplifying circumstances. Magnetic eld points straight into the Faraday cup In this case b x and b y are both zero, and the eective thermal speed reduces to ~w = w k,asexpected. Magnetic eld is aligned parallel to the collectors 64

65 If b 2 x + b 2 y =1,thenthere is no b z component and ~w reduces to w?. There is no anisotropy We have w? = w k = w. Therefore q ~w = b 2 z w2 +(b 2 x + b 2 y)w 2 = w q b 2 x + b 2 y + b 2 z = w (2.25) Choice of Eective Collecting Area for bi-maxwellian Distribution In the derivations of the Faraday Cup response functions to isotropic and anisotropic Maxwellian distribution functions the eective area A was pulled out of the integral over the energy window. This approximation is justied because A isaslowly varying function of the angle of incidence and because the phase space density is sharply peaked at a single point in the energy window. The VDF will indeed be peaked, both because we are dealing with Gaussian quantities and because the Mach number of the solar wind is very high. The variation of the eective collecting area as a function of the angle of incidence has been both derived analytically and determined through numerical simulations. The two dominant eects are the varying transparency of the wires which comprise the various grids in the cup, and blocking of the limiting aperture under extreme angles of incidence. Therefore for the purposes of this section A() is taken as a given, where is the the angle of the incident particles relative to the cup normal. Increasingly sophisticated methods for calculating are now presented, starting with the original key parameter analysis, the algorithm used for the isotropic alpha-proton analysis [Aellig et al., 2001b], and ending with a new derivation for an anisotropic VDF. The original key parameter algorithm used a xed value for A corresponding to its maximum value of 33:8 cm 2 at normal incidence ( = 0). The non-linear code developed by Aellig for the isotropic analysis of the protons and alphas used the angle between the cup orientation ^n and the bulk ow U ~ (for this section U ~ is in the frame of the FC), =cos ;1 U z j ~ Uj : (2.26) Consider the two example proton velocity distribution functions (a) and (b) shown in Figure 2-7 in the rest frame of a Faraday Cup. V? is the speed perpendicular to the cup normal, and V k is the speed along the normal axis. The shaded ellipses represent the phase space density of the protons, with the rainbow coloring indicating six thermal widths from the peak. Both VDF have the same bulk velocity U ~ and density n, but in (a) the distribution is isotropic with w = 30 km/s, while in (b) it is anisotropic, with thermal speeds w? = 30 km/s and w k = 60 km/s aligned with the eld orientation indicated by the black arrow. The solid blue lines indicate the 45 angle of of incidence to the FC at which the eective area begins to drop 65

66 Figure 2-7: Two example distributions in velocity space. Left: An isotropic velocity distribution function with a single thermal speed of 30 km/s. Right: An anisotropic VDF with w? = 30 km/s and w k = 60 km/s, and magnetic eld orientation indicated by the black arrow. In both images the cross identies the maximum density of the VDF, the solid red lines indicate the selected velocity window, the blue lines are the nominal eld of view of the instrument, the dashed line connects the origin to the point in the VDF with maximum density, and the dotted-dashed line connects the origin to the point in the energy window with the highest density. It is clear from this example that for an anisotropic distribution we need a better way of determining the angle of incidence than the historical technique. rapidly, and the two solid red lines mark the upper and lower boundaries of the energy window which we shall now consider. It is clear that for this example the supersonic distributions t entirely within the region of the energy window seen by the cup. The crosses indicate the location of the bulk velocity of the plasma in each panel, and the dashed lines show the angle determined by (2.26). The dotted-dashed lines are the line from the origin to the actual location of the point in the energy window which contributes the must to the total ux in this measurement. In the case of the isotropic VDF in Figure 2-7, the dierence between the two angles is due to the fact the maximum current in the window does not necessarily come from the same angle as the bulk ow. So the rst correction, which works at least for isotropic distributions, is to use the speed V of the window instead of the 66

67 component of the bulk speed normal to the cup, U z, =cos ;1 V j ~ Uj : (2.27) In the case of the anisotropic distribution in (b) however, even (2.27) produces the wrong angle. In fact, both (2.26) and (2.27) may be o by a great deal in the case of large temperature anisotropies. The solution is to derive the location of the point in phase space (Vx max Vy max Vz max ) for a bi-maxwellian distribution function with the maximum density in a specied velocity window. Under the assumption that V =V 1, Vz max ' V, so the problem is now two-dimensional. Start with a dierential form of the original anisotropic response equation (2.17), di d 3 ~v 0 = ANq 3=2 w? 2 w (~v 0 ^n)e ;(v2? =w2? +v2 k k) =w2 (2.28) k where through (2.18) v? and v k are functions of ^b and ~ U. Since variations normal to the cup are being neglected, ~v 0 ^n is a constant. The collecting area A varies slowly over the range in angles we are considering (in the analysis of a single ion spectrum we will discard point at large angles of incidence). So dening R as the natural logarithm of (2.28) and discarding the area term and any constants, R ;ln di d 3 ~v 0 / v2? w? 2 + v2 k wk 2 : (2.29) Now it is just a matter of nding the (Vx max Vy max ) which maximizes R for a given ~U, ^b, w?, and w k. This is done by dierentiating R with respect to V x and V y and identifying the point (Vx max Vy max ) where it vanishes. After some algebra, v max x = U x ; b xb z (v z ; U z )(w? ; w k )(w? + w k ) b 2 z w2 k +(1; b2 z)w? 2 v max y = U y ; b yb z (v z ; U z )(w? ; w k )(w? + w k ) b 2 z w2 k +(1; b2 z)w? 2 (2.30) and the angle of incidence used for the analysis of anisotropic distributions is = cos ;1 q (V max x Moments of Faraday Cup Spectra V : (2.31) ) 2 +(Vy max ) 2 In Section and Section we explicitly derived the response of a FC instrument to convected supersonic Maxwellian and bi-maxwellian ion velocity distributions. Section 2.3 discusses the non-linear analysis technique, in which these models are compared with observations to produce a set of ion parameters which give the 67

68 best t to the data. The current section presents a routine for estimating the bulk properties in the solar wind by following the methods described in Section for collapsing the details of the VDF. This is called the moment analysis technique. There are several good reasons to pursue the moment analysis as well as the nonlinear method: It is a good consistency check on both methods, and the moments provide an easy visualization of the FC observations of the anisotropies. Additionally, moments are a common method used by investigators to characterize the properties of ions and electrons in the solar wind. As will be seen, the non-linear technique is far more robust and provides much more information about the VDF. Generally the moments are used because of their simplicity, but also because it is not possible in general to produce the same analytic expressions developed in the previous sections for all instrument types. Since there has been a great deal of discussion of dierences in solar wind parameters reported by dierent investigations, which generally involve non-linear analysis of Faraday Cup data and moment analysis of another class of ion instruments, electrostatic analyzers, it is important to apply both methods to the same dataset. In Section the two techniques are compared and the sources of discrepancies are identied. The current seen by a FC oriented in the direction ^n( ) and scanning in the window (V V +V ) is the integration over (2.6), di = A(~v=v)qv z f(~v)d 3 ~v: (2.32) The FC measures the reduced distribution function (1.65), so with A representing the appropriate value for A(~v=v), which will be determined once we know the bulk velocity of the protons, we write the dierential current di = Aqvz f(v z )dv z (2.33) as a one-dimensional problem. The most signicant approximation of the moment analysis technique is that f(v z ) does not vary much over the width of the speed window V,sowe may approximately write the total current I as I ' Aqvz f(v z )V: (2.34) The phase space density f i of the reduced distribution function along ^n for the i th window at (V i V i ) is then approximately f i = I i AqV i V i : (2.35) There are several problems with this treatment. First, f(v z ) is a rapidly varying function of v z, so the approximate integral will be in error. By simply normalizing the current by V i we have not weighted each part of the energy window correctly. Additionally, we have assumed that all particles move with speed V i in determining f i and we know that V i =V i ' 10%. Expecting errors on the order of several percent, we can still proceed to calculate proton parameters. Assuming that the entire VDF 68

69 is in the energy range swept out by the FC in this spectrum, the apparent proton number density along this angle n(^n) is just the sum of f i over all measurements in this direction, n(^n) = X i I i AqV i V i : (2.36) In a similar manner to (1.62), the average speed of the distribution viewed in this direction, U(^n), is U(^n) = 1 n(^n) X i f i V i : (2.37) Following (1.63) and the fact that in the previous section it was shown that even the bi-maxwellian distribution will appear isotropic when viewed in a single arbitrary direction, we write the eective thermal speed w(^n), w(^n) = p Pi 2 f i(v i ; U(^n)) 2 1=2 : (2.38) n(^n) Typically the moments may be calculated along 20 to 30 of the angles in a spectrum, so there are many measurements of the projection U(^n) = ~ U ^n of the bulk velocity vector ~ U into the cup frame. The projection eect is linear, so we have an over-determined set of equations which can be inverted to determine the bulk velocity. The Singular Value Decomposition (SVD) method is used to determine ~ U [Press et al., 1999, x2.6]. Finally, the value of ~ U is used to renormalize the apparent density n(^n) by taking into account the change of the eective area with angle of incidence, =cos ;1 ~ U ^n=u, and produce a corrected proton number density, n c, n c (^n) = A() n(^n): (2.39) A It is worth mentioning that the moment determinations were not dependent on the actual number density along a given angle as long as A is not a noticeable function of the speed of the energy window. In the standard moment analysis the proton density and thermal speeds are given by the average values of n c (^n) and w(^n) over all angles, and the deviation of these quantities is an estimate of their uncertainty. The following section outlines the method for determining the temperature anisotropy from w(^n) Temperature Anisotropy with Moments It was shown that observations of a bi-maxwellian VDF by a Faraday Cup appear isotropic along a given angle (2.23). The simple form of the projected eective thermal speed ~w in (2.24) suggests that the thermal speeds w(^n) calculated by the moment 69

70 analysis may be used to determine the parallel and perpendicular thermal speeds. The Sunward-facing component of a spectrum contains scans through the proton VDF at 36 (2 cups 18 angles) angles relative to the magnetic eld. Starting with w(^n) andthe average background magnetic eld B ~, we t q ~w(^n) = (^b ^n) 2 wk 2 +(1; (^b ^n) 2 )w? 2 (2.40) to the observations with w? and w k as free parameters. If the moment and non-linear results agree this will be a good demonstration of the robustness of the abilityofafc to measure anisotropies. There are two cases that could create a disagreement between the results, but this dierence could be useful in highlighting intervals in the solar wind which merit further study. In certain cases, especially in the high speed coronal solar wind, the proton VDF generally takes the form of two separate bi-maxwellian distributions, with a dierential ow speed aligned with the magnetic eld. Sometimes the solar wind ions possess an appreciable heat ux, which is also directed along the magnetic eld. In both of these cases one would expect the moment parallel thermal speed, w kmom to be greater than the non-linear parallel thermal speed, w knl. 2.3 Outline of Analysis Procedure This section is an outline of how the methods developed in Section 2.2 are applied to the ion spectra. In Chapter 4 this routine is extended to include the analysis of alpha particles. The procedure has also been modied to characterize the nature of double streaming protons in the solar wind [Clack et al., 2002] Preparation of a Single Ion Spectrum The spacecraft spin rate and the number of energy windows in the spectrum are used to calculate the duration of the observation, which on average is 92 seconds. The set of three-second vector magnetic eld measurements provided by the MFI investigation which were made during this observation period are collected and the average ambient magnetic eld ~ B is calculated, P N i ~B i ~B N (2.41) along with the deviation of the magnitude of the three-second eld measurements, B, and the angular uctuation of the direction of the eld, B over the course of the ion spectrum,! B Std. Dev cos ~ ;1 B i ^b j B ~ : (2.42) i j 70

71 The direction of the eld normal, ^b, is then calculated in the frame of each of the 1200 individual measurements in the spectrum through the transformation matrix (2.3), under the assumption that the direction does not change muchover the duration of the observation. The validity of this assumption is assessed in Section Moment Analysis of the Proton Distribution The moments described in Section are calculated using data selected from each angle in the spectrum. The data selection proceeds as follows independently for each angle. The speed window V with the largest current I max (and corresponding ux f max ) is identied. All measurements with speeds within V=2 of V, uxes greater than 1% of f max, and currents greater than 1 10 ;12 A (compare to the instrumental background current 2 10 ;13 A) are selected. The moments are then calculated along every angle that has at least three selected measurements, yielding w(^n), n(^n), and U(^n). Application of the SVD algorithm to the U(^n) produces the bulk velocity ~U, which is then used to correct the number densities with a better estimate of the eective area. Since the anisotropy of the protons is not known we cannot use (2.31). The moments are calculated over the entire distribution function, so (2.27) is also not well dened. The best that can be done is to use the bulk velocity and (2.26). Figure 2-8 is a plot of n(^n) as a function of the angle between ^n and the bulk velocity U ~ using the moments of a spectrum observed by Wind in the solar wind at 2032 UT on November 3, The number densities determined by each cup are shown, after the correction to the eective area to reect the ow angle relative to the cup, with diamonds for FC1 and crosses for FC2. Note the large drop in the calculated number density for angles greater than about 35. The drop is due to several eects including the rapid change in the eective area which begins near this angle of incidence and the possibility that a portion of the proton VDF is no longer in the eld of view of the FC. The median value of the 20 largest densities is calculated, and all measurements of n(^n) within 5% of that value measured at angles less than 40 from the bulk ow of the plasma are selected for subsequent analysis. The selected points in the spectrum displayed in the gure are shown with solid symbols, while the grey symbols indicate density measurements discarded as suspicious. The moment number density is the average of the selected values. The top panel of Figure 2-9 shows the moment determination of the proton thermal speed w(^n) as a function of the angle between the cup orientation ^n and the average direction of the magnetic eld ^b for the same spectrum. The points which are shown as grey symbols correspond to the same points from the previous gure which were discarded as having suspicious number densities. Note the remarkably linear dependence of w(^n) 2 on ^b ^b, in agreement with the prediction of (2.24). The over-plotted line is the result of arobuststraight-line t [Press et al., 1999, x15.7] of (2.24) to the observations. The values of w? and w k which produced the best t to the observations are listed in the plot, along with the percent error of the t (The percent error is the best estimate of the quality ofthet possible because we do not know the uncertainties of each moment determination of the thermal speed). For this spectrum w k > w?. The bottom panel of Figure 2-9 is an example of a proton 71

72 Figure 2-8: An example of the selection of FC angles ^n for a single spectrum to be used for moment determination of density and thermal anisotropy. The number densities determined by each cup are shown, with diamonds for FC1 and crosses for FC2. All points indicated by grayed symbols are discarded from the subsequent analysis because they have a ow angle of greater than 40 from the cup normal or erroneous densities. 72

73 spectrum with w k >w?. The optimum limits for data selection and the tting routine for the anisotropy analysis of the moment data were developed by an undergraduate in the group, James Tanabe [Tanabe et al., 2001], in parallel with my development of the non-linear analysis. Once we had generated plots of w(^n) 2 vs ^b ^b it was clear that the Faraday Cup can indeed make robust measurements of the proton temperature anisotropy Initial Guess An initial guess for the parameters of the proton distribution is needed for the selection of data and as input to the tting routine. The free parameters are U x U y U z w? w k and n p. If the moment analysis succeeded and produced \reasonable" proton parameters then it is used as the initial guess. A reasonable set of parameters is dened as being supersonic (U=w? > 1 U=w k > 1), with realistic velocity U x < U 1200, and density 0:01 n p 300. If the moment analysis was not successful, or the moment parameters are judged to be questionable, then the initial guess is formed directly from the raw currents. It is assumed that the ow is radial, so U y and U z are both zero. The bulk speed U is estimated as the speed of the window with the highest current, and U x = ;U. A mach number of 10 is assumed, so both w? and w k are set to U=10. A xed number density of10protonscm ;3 is used because the tting procedure is the least sensitive to the initial guess of the density Selection of Proton Data Individual measurements from the two cups are selected for comparison with the bi- Maxwellian model based upon the initial guess proton parameters. It is important that only currents due to the protons are included in this set, as the inclusion of currents due to eects not accounted for in the model will contaminate the results of the t. From experience, contamination will eect the bulk velocities least, followed by thenumber density. The thermal widths are the most sensitive to the inclusion of observations which are inconsistent with the model. Measurement selection is dictated by the four parameters ANGLEMAX, THERMALMAX, CURRMIN, and PEAKFRAC, which are now described and justied. The nal values of the parameters were determined in a procedure described in Section and their values are listed in Table 2.1 There are two contamination sources, physical and model/instrumental. Physical factors include the presence of alpha particles, a second proton distribution, or a nonnegligable proton heat ux. To avoid substantial contamination from alpha particles the parameter THERMALMAX is the maximum number of thermal widths above and below the projected proton speed along each angle in which the selected data must fall. The eective thermal speed is calculated along each angle based on the magnetic eld orientation and the current values of the parallel and perpendicular thermal speeds. The eect of non-maxwellian characteristics like heat ux are limited in part by only taking measurements with currents greater than PEAKFRAC times the maximum current in the spectrum. 73

74 Figure 2-9: Scatter plot of observed thermal speed w(^n) as a function of FC direction relative to the eld orientation ^n. Grey symbols indicate points discarded due to suspicious densities. The lines are the best t of (2.24) to the selected observations. The data in the top panel are for the same spectrum shown in Figure 2-8 and have w k >w?, whereas in the bottom panel w k >w?. 74

75 Figure 2-10: Selecting measurements of anisotropic proton distribution for tting to bi-maxwellian model. Top panel: Color shading indicates logarithmic distribution of observed current as a function of speed window and azimuthal angle crosses are locations of measurements and diamonds are the selected points solid line is projection of bulk velocity along each angle and dashed lines mark three thermal widths above and below this projection. Bottom panel: Crosses indicates the angle of incidence of the maximum ow for each measurement Dashed horizontal line (red) is the maximum angle ANGLEMAX for selection Dashed curve is the angle between the cup and the bulk ow colored (blue) crosses are selected data. 75

76 Model/instrumental eects are due to properties of the Faraday Cups which are either not included in the derived models or limit the observations. Due to thermal uctuations in the amplication electronics there is a background current of approximately 2 10 ;13 A that the proton ux must exceed to be detected. In addition, an intermittent increase in the background current along a single angle has been observed in the past. It is small compared to ion signatures and has the same value at each energy window along the single angle. The particular angle eected is also seen to change with time, leading us to believe that sunlight is glinting o an object on the spacecraft. Both of these eects are avoided by setting a minimum current CURRMIN which observations must exceed. The nal issue is due to the eects of ions at large angles of incidence to the cup. Several of the derivations fail for large angles of incidence: The initial assumptions listed in Section for deriving the response functions required that the protons were supersonic and entirely in the eld-of-view of the cup, thus allowing us to extend the integration in velocity space to an innite slab The derivation in Section of the appropriate eective area was based on the assumption that the protons which contribute most to the measured current ow from an angle where the eective area varies slowly. In addition, at large angles the beam of particles which pass through the limiting aperture and illuminate the collecting plates may in part miss the collectors. As the modulator voltages alternates between the two edges of the window, the beam will partly walk on and o the collectors. This creates an articial signal which is modulated at 200 Hz and is included in the nal measured current. This feature is most often seen at very low voltages and large angles. Data which may experience any of these problems are discarded by setting a maximum angle ANGLEMAX to the cup normal. This limit is applied both to the angle between the cup normal and the bulk velocity and to the angle determined by (2.31) for the optimum eective area. The main aspects of the point selection algorithm are illiustrated in Figure 2-10 for a model proton spectrum. The proton VDF is bi-maxwellian, owing radially at ;350 km/s with ^b in the ecliptic, ;7 from the Sun-Earth line and w? = 2w k. The color shading is the logarithm of the observed current as a function of speed window and the azimuth angle of the cup. The crosses mark the location of every measurement in this spectrum and the diamonds are the selected observations. The solid blue line is the projection of the proton bulk speed as a function of the azimuth angle. The two dashed blue lines indicate the upper and lower speed limits with THERMALMAX set to 3. Note the variation in these bounds as a function of angle due to the varying eective thermal speed. The bottom panel shows the angle to the maximum ux as determined by (2.31) for each measurement as crosses. The dashed black curve is the angle between the cup and the bulk ow of the protons. The horizontal red dashed line is ANGLEMAX, which has been set to 38. Selected points are colored blue. A minimum of forty measurements of the proton distribution are required for the analysis to proceed on a given spectrum. Typically at least 200 points were selected. 76

77 2.3.5 Non-Linear Fit of Model to Observations With the magnetic eld orientation specied by the MFI eld observations, there are six free parameters in the bi-maxwellian proton model: ~ U w? w k, and n. Dene a 2 merit function, 2 = NX i=1 I meas i ; I mod i i 2 (2.43) where for each measurement i of the total N observations there is an observed current Ii meas, the current predicted by the model, Ii mod, and the uncertainty i of that measurement. The best-t values of the parameters are determined by minimizing 2. We employ the Levenberg-Marquardt non-linear least-squares method, which isa combination of the inverse-hessian and steepest descent methods [Press et al., 1999, x15.5]. In addition to the best-t parameters and the nal minimum value of 2, a covariance matrix is calculated and inverted to yield one- estimates of the uncertainty of each parameter. The nal value of 2,combinedwiththenumber of measurements used in the t, will indicate how well the bi-maxwellian model describes solar wind protons. If the Faraday Cup counted individual particles then the uncertainties i in (2.43) would simply be proportional to the square root of the total number of particles detected. The uncertainty in the currents measured by the a FC is actually limited by the digitization of the observed currents are they are telemetered back to Earth. For the Wind cups the resolution of the telemetered currents is 1%. The weighting factor is set to the larger of one percent of the measured current or the thermal background current, i = ; 0:01 I meas i > 4 10 ;13 ;1 : (2.44) Note that the uncertainty of a Faraday Cup current measurement of N particles is not proportional to p N, but instead is xed at 0:01N. It is not a great loss that the uncertainty due to the digitization is larger than simple p N statistics because the approximations we have used in deriving the instrument response introduce errors less than but near that same level. 2.4 Results: Proton Parameters, Uncertainty Propagation and Analysis The original analysis code written to analyze Wind FC spectra was extended to incorporate the methods described in the previous section. The procedure is completely automated and processes a single day at a time, saving the results in a binary le. For each measurement the time, spacecraft location, magnetic eld averages and uctuations, results from the moment analysis, initial guess, best-t parameters and their uncertainties, number of measurements selected for tting, and the minimum value 77

78 Figure 2-11: Selection of an interval in 1998 for the optimization of the analysis code. Shown from top to bottom are the bulk speed, thermal speed, number density, magnetic eld, and location of the spacecraft (x gse blue y gse red). of 2 are recorded. On average it takes ten minutes to analyze one day of observations, so our computer system can process the entire mission in approximately one week. Section demonstrates how the small computing time allows us to optimize the data selection parameters which were described in Section The overall convergence of the non-linear analysis and the distribution of values of 2 are reviewed in Section In the subsequent sections the uncertainties of the proton parameters are calculated, and any interesting dependencies are explored Optimizing the Analysis In order to determine the best values for the two analysis parameters THERMALMAX and ANGLEMAX an interval of obeservations which cover a broad range of interplanetary conditions was selected and processed repeatedly. The \optimum" values of THERMALMAX and ANGLEMAX produce the best overall minimum average 2 =d:o:f: and anisotropic thermal speed uncertainties, w? and wk. Twenty days in 1998, from August 8 through August 28, containing spectra, were selected for this procedure. A summary of the bulk speed, average thermal 78

79 Parameter Value Description ANGLEMAX 28 The maximum angle the bulk ow vector and the point in phase space making the greatest contribution to the current in this window may make THERMALMAX 3 ~w(^b) Number of thermal widths from the peak using projected eective thermal speed at this angle CURRMIN 1 10 ;12 [A] Minimum current PEAKFRAC 5 10 ;3 Currents must be greater than this fraction of the peak current seen in the entire spectrum ANGLEMOM 28 Maximum angle from peak for moment analysis Table 2.1: The nal values of the free parameters used for the anisotropic analysis of protons. speed, number density, magnetic eld strength, and spacecraft location during this period is shown in Figure The period was selected because it contained high and low speed solar wind, an encounter with the Earth's bow shock, and several notable interplanetary shocks, especially the August 28 event. The entire period of selected observations was processed four hundred times, using twenty values of THERMALMAX spaced evenly between 1.0 and 3.4, and twenty values of ANGLEMAX ranging from 10 to 50. The median values for 2 =d:o:f:, w?, and wk were calculated. The dependence of 2 =d:o:f: and wk on the choice of THERMALMAX and ANGLEMAX are shown as contour plots in Figure For ANGLEMAX less than 18 the model t the selected observations very well, but the resulting parameters were erratic, suggesting that there was insucient data to constrain the model successfully. Focusing on the contour of 2 =d:o:f:, there are two clear trends. First the median value of 2 =d:o:f: is a sharply decreasing function of THERMALMAX until approximately 2.5 thermal widths of data are selected. Second, 2 =d:o:f: reaches its minimum value for ANGLEMAX greater than about 25, and decreases slowly afterwards. Examination of several individual events suggested that this slow decrease in 2 =d:o:f: was just due to adding more measurements which did not require the model to change a great deal. Recall that the model has 6 free parameters, while as ANGLEMAX increases in this range, the number of observations selected increases from 150 to upwards of 250 measurements. For the uncertainty in the thermal speed the best results came with ANGLEMAX greater than 25 and THERMALMAX greater than 2. The same results were seen in the median uncertainty in the perpendicular thermal speed. The slower variation of the results with THERMALMAX greater than 2.5 appears to be due to setting PEAKFRAC to 0:01, resulting in a limit to the number of measurements selected with inxreasing THERMALMAX. 79

80 Figure 2-12: Two samples of the optimization procedure for selecting the values of ANGLEMAX and THERMALMAX which result in the best overall analysis. Panel (a) is the median value of 2 /d.o.f. and panel (b) is the median value of the percent uncertainty of the parallel thermal speed for each run over the spectra contained in the selected interval. 80

81 Figure 2-13: Distribution of the nal value of 2 /d.o.f. for all spectra analyzed (black), slow solar wind v < 400 km/s (green), fast solar wind v > 400 km/s (blue). The average value of 2 /d.o.f. over all spectra was 0:68 (blue vertical line). The distribution of spectra is approximately log-normal, and 99:9% of the spectra, had nal values of 2 /d.o.f. between 0:07 and 7:3 (red lines). It has approximately the same distribution Overall Summary: Convergence, 2 =d:o:f: To date the Faraday Cups have recorded 2,208,024 measurements, and this analysis produced a converging t to 99:9957% of these spectra. The 95 spectra which were not t all had measurements which were suggestive of some form of telemetry error, generally either one or more isolated currents at the maximum value of 10 ;8 A measurable by the cups or empty observations containing no data. This is not to imply that the remaining ts were perfect, there are plenty of factors which come to mind that could cause trouble, such as low Mach numbers or large magnetic uctuations. In the following sections we will examine the uncertainties in the derived parameters, but here we look at the values of 2 =d:o:f: as indicativeofhowwell the bi-maxwellian model describes solar wind proton velocity distributions. The value of 2 may be used to identify how well the model describes the observations. For a system with degrees of freedom the probability of measuring a value of up to 2 is given by [Abramowitz and Stegun, 1972, x6.5], P ( 2 j) = ; ; 2 Z 2 0 t 1 2 ;1 e ;t=2 dt: (2.45) There are six parameters in the model, and if a total of N measurements were selected for tting then there are = N ; 6 degrees of freedom in the t. The distribution of 81

82 the 2 = for all solar wind observations and separatly for the slow and fast solar wind are shown in Figure The average value of 2 = is slightly less than unity, and 99:9% of the observations had values of 2 = 7:3 Toevaluate what this distribution tells us about the quality of the ts, (2.45) was evaluated individually for each spectrum. If the model described the data perfectly, then 50% of the time P ( 2 j) would be greater than 0:5 and 50% of the time it would be less than 0:5. The result of the calculation was 59% of the observations had P ( 2 j) < 0:5, 41% had P ( 2 j) > 0:5, which is close to the ideal values. It is expected that more of the spectra produced smaller values of 2 = because of the nature of the uncertainty in the current being driven by the quantization of the signal and not the actual (and smaller) error due to counting statistics. It should be noted that the values of 2 = recorded are an indication of how well the bi-maxwellian model described the selected observations and not of how well or how completely the solar wind consists of anisotropic protons Bulk Speed, Velocity Components, and Flow Angles Once the best-t values of the solar wind parameters have been determined, the bulk velocity of the protons in the spacecraft frame is corrected for the 30 km/s aberration produced by the orbital velocity of the spacecraft and Earth around the Sun. Since the ^y GSE axis is anti-parallel to the direction of Earth's motion, this is done by subtracting 30 km/s from the ^y GSE component. I then took the individual uncertainties in the components of the velocity and calculated the propagated uncertainty in the bulk speed of the protons, 2 U = i=x y z 2 i 2 = 2 U x U 2 x + 2 U y U 2 y + 2 U y U 2 y U 2 : (2.46) We also use the components of the velocity to calculate the ow angles of the solar wind, traditionally dened at MIT as the direction from which the plasma is owing. We calculate the East-West ow angle in the ecliptic plane, and its uncertainty, 2 EW = i=x y 2 i EW = tan ;1 U y U x (2.47) and the North-South ow angle out of the plane of the ecliptic, 2 = U 2 x 2 U x + U 2 y 2 U y (U 2 x + U 2 y ) 2 (2.48) U z NS = tan ;1 p U 2 x + Uy 2 (2.49) 82

83 and its uncertainty, 2 NS = i=x y z 2 i 2 = (U 2 x + U 2 y ) 2 2 U z +(U 2 x 2 U x + U 2 y 2 U y )U 2 z (U 2 x + U 2 y )U 4 (2.50) Thermal Speeds There are two possible issues that would prevent an accurate determination of w? and w k which should be examined in detail: (1) if there is a large angular uctuation of the magnetic eld direction during the 92-second plasma spectrum I would expect any anisotropies in the thermal speed to \wash out" (2) certain orientations of the magnetic eld, even if it is steady, could make it dicult to determine one or both of the thermal speeds. To address uctuations, I looked at the ratio of the thermal speeds under two cases: spectra recorded with a large uctuation of the magnitude of the eld but little change in the direction of the eld, and the opposite, cases where the magnitude of the eld did not change much, but there was a large angular uctuation. The results are shown in Figure It is clear that while changes in the magnitude of the eld do not appear to change the distribution of thermal speed ratios, there is a dramatic drop in anisotropy when the angular uctuation of the magnetic eld exceeds x degrees. For subsequent analysis of temperature anisotropies therefore I will restrict myself to cases where the angular uctuation is less than 20 (this choice leaves 96% of the total spectra). It is worth asking if a plasma undergoing signicant angular uctuation of the magnetic eld orientation over the course of the 92-second recording time of a Faraday Cup spectra actually becomes isotropic. One expects high frequency oscillations on the order of the ion cyclotron frequency to eciently couple to the ion distribution function, so that under the inuence of uctuations on the order of minutes one would expect the eld to remain frozen in to the plasma, and for little mixing of the thermal speeds to occur. A future investigation might pursue tting the wind observations with ~w(^b(t)), using the fastest possible magnetic eld data, preferably the 40 millisecond measurements, and checking to see if that produces a better t. A detailed study of the quality of the t as a function of the frequency of the uctuations might provide insight into the exact mechanisms by which the electromagnetic uctuations are coupling to the particle distributions. To explore the uncertainties in the thermal speeds, plots of the median uncertainty of w? and w k as a function of magnetic eld latitude, B, B tan ;1 B p z B 2 x + By 2 (2.51) and magnetic longitude, B tan ;1 B y B x (2.52) were generated for each parameter. The magnetic eld typically lies in the plane of 83

84 Figure 2-14: Testing the eect of angular eld uctuations during recording of ion spectrum. This is a plot of the temperature anisotropy as a function of the angular deviation of the eld, as dened by equation (2.42). 84

85 the eccliptic along the Parker spiral angle of about 45. Therefore the ( B B ) plane is not evenly covered by the observations. A summary of the agreement between the moments and the non-linear analysis was presented by Tanabe et al. [2001]. In general the two techniques produced results which agreed to within 30%. The largest dierences occurred near interplanetary shocks and in very high speed solar wind. To determine if the average disagreements between moment and non-linear analysis techniques are instrumental or physical the observed discrepancy is compared to Monte-Carlo simulations of solar wind ion spectra. For each given value of w k, solar wind parameters were generated randomly and the Faraday Cup response was determined using the non-linear bi-maxwellian response function. This response was then fed into the anisotropic moment analysis code and the results were compared. The diamonds are the observed dierences from Figure 2.4.4, the blue lines are the result of the rst simulation (thick line is average value of ratio, thin lines mark on standard deviation from mean). Result are reproduced well for w k 90 but the simulation does not predict the large errors at smaller values of w k. The simulation was improved by including a distribution of alpha particles, resulting in the red curves, which provide a much better agreement with observations. It seems clear that on average the 20% dierences between thermal speeds determined by moment and non-linear analysis techniques are due to a breakdown of the moment method, through contamination by alpha particles and loss of part of the distribution in the case of hot spectra. For the thermal speeds we would also like to produce an equivalent thermal speed. Starting with the idea of forming a single pressure from an isotropic pressure tensor P by taking the trace, then there is an equivalent relation for the temperatures, P iso Trace(P)=3 (2.53) T iso Trace(T)=3 =2T? + T k (2.54) where we have assumed we are dealing with a gyrotropic bi-maxwellian distribution function. Since T? = 1 2 m pw? 2 T k = 1 2 m pwk 2,we can then write down a relation for the isotropic thermal speed w iso, q w iso (2w? 2 + w2 k )=3 (2.55) and its associated propagated uncertainty, 2 w iso = w2 k 2 w k +4w? 2 2 w? 3wk 2 : (2.56) +6w2? 85

86 Figure 2-15: Dependence of the uncertainty in parallel (upper panel) and perpendicular (lower panel) proton thermal speeds as a function of magnetic latitude and longitude. Color shading indicates median uncertainty as indicated by color bars on left. Red curves indicate 5,25,50,75% measurement contours. See text for details. 86

87 Figure 2-16: Comparison of the thermal speeds as determined by the non-linear and moment analysis techniques. Colored bins indicate the number of spectra in a two-dimensional histogram. Each column has been normalized to unity. Dashed line indicates equality between methods diamonds are the average ratio between the methods (in percent) as a function of the non-linear value, with error bars indicating the deviation of the ratio in each bin. 87

88 Figure 2-17: Comparison of moment and non-linear anisotropies with the predictions of numerical Monte-Carlo simulations Number Density The average uncertainty in the proton number density over all spectra was x y%, with a median value of x%. The two parameters this uncertainty is most dependent on are the bulk speed of the protons, U p, and the thermal width of the proton distribution, where for simplicity here I will overlook any temperature anisotropies by using the single isotropic thermal speed, w iso dened by (2.55). To get a good idea of the percent uncertainty in the number density, 100% n p n p (2.57) due to the instrument response as a function of U p and w iso, we generated the plot shown in Figure It is most dependent on the thermal speed and the bulk speed, so I generated a two-dimensional plot of the average percent uncertainty of the number density as a function of the proton bulk speed and the isotropic thermal speed (2.55). I initially divided the plot into 20 intervals spaced logarithmically in number density and linearly spaced in thermal speed. Then Ilooked at each interval and if there were more than 50 points in that bin I subdivided the bin into quarters. This procedure results in a two-dimensional visualization of the dependence of the thermal speed, with higher resolution in areas with more data. 88

89 Figure 2-18: The median uncertainty in proton number density as a function of the bulk speed the the calculated isotropic thermal speed. Each bin contains at least spectra and, where the density of measurements permitted, the resolution of the bins has been increased. The four red contours mark the boundaries at which the density of measurements has fallen to , and 75% of the maximum value. Dashed black lines indicate thermal to bulk speed ratios of 5, 10, and 20. Parameter Units Mean Median Stdev Mean % Median % ju j km/s U x km/s U y km/s U z km/s EW degree N/A N/A NS degree N/A N/A w? km/s w k km/s n p cm ; Table 2.2: Uncertainties in each of the derived parameters from the best-t to individual solar wind spectra. 89

90 2.5 Summary The response of the Faraday Cup instruments on Wind to a convected, bi-maxwellian distribution of ions has been derived and applied to the observations for the rst time. Faraday Cups are the only thermal plasma instruments for which analytical response functions have been derived. Therefore while other investigations in the past have determined ion plasma parameters including temperature anisotropies, this is the rst analysis which produced uncertainty estimates for each of the derived parameters along with the 2 =d:o:f: estimate of the quality of the t. Indeed the dataset submitted to the NSSDC, which includes these uncertainties for each spectrum, is the rst of its kind. It has been shown that the typical uncertainties of the proton solar wind parameters determined by the FC instruments are all on the order of a few percent, that the average 2 =d:o:f: near unity implies that bi-maxwellian distributions are a good description of the solar wind ions at 1 A.U., and that dierences between the moment and non-linear analysis techniques are predominantly due to failures in the approximations which go into the moments. 90

91 Figure 2-19: Survey of proton parameters in the solar wind at 1 AU as seen by the Wind Faraday Cup instruments. Upper panel: The distribution of proton number densities n p as a function of bulk speeds U p,showing that on average the particle ux n p U p is conserved. Lower panel: Single proton temperature T p =(2T p? + T pk )=3 as a function of bulk speed, showing that on average the solar wind has a mach number U p =w p

92 92

93 Chapter 3 Instabilities Driven by Proton Temperature Anisotropies ABSTRACT: The bi-maxwellian proton dataset developed in Chapter 2 is used to examine the limits to the proton temperature anisotropies observed in the solar wind. It is shown that the double adiabatic equation of state breaks down due to the the onset of kinetic plasma micro-instabilities. The observations are used to constrain theoretical models of the mirror, cyclotron, and rehose instabilities. This is the rst ever observation of the rehose instability in space. 3.1 Introduction The proceeding chapter established that the observed distribution of the bulk of solar wind protons in phase space has the form of the convected, bi-maxwellian velocity distribution function (VDF). The non-linear analysis of each 92-second ion spectrum recorded bythetwo Wind/SWE FaradayCup(FC) instruments yields accurate values of the best-t bulk velocity U ~ p, parallel and perpendicular temperatures T kp and T?p, and the number density n p, and average ambient magnetic eld B ~. In addition, for each spectrum wehave indicators of the success of the t, including the 2 goodness of t and estimates of the one-sigma uncertainties of each of the six proton parameters. To date this procedure has been applied to more than 2.3 million solar wind proton spectra recorded between 1994 and 2002, providing detailed information about the kinetic properties of the solar wind over a broad range of plasma parameters. It is a remarkable feature of tenuous magnetized plasmas such as the solar wind that these bi-maxwellian temperature anisotropies exist and persist. It is the low rate of momentum and energy exchanging Coulomb collisions which allow non-isotropic features of the VDF to persist over long periods of time. At the same time, the presence of the ambient interplanetary magnetic eld rapidly insures gyrotropy. It is fascinating that collective processes are sucient tomake the proton VDF Maxwellian in the separate orientations parallel and perpendicular to B ~, but not eective enough to isotropize the distribution overall. The purpose of this chapter is to study the plasma processes which act to constrain 93

94 the dierence between the two proton temperatures T?p and T kp. This dierence is expressed through the dimensionless proton temperature anisotropy R p, R p T?p T kp (3.1) As this introduction will show, plasma instabilities act to limit the observed values of R p. In Section 3.2 we will review predictions of the properties of these instabilities. Comparison with observations in the subsequent sections demonstrates the necessity to move beyond simple MHD uid calculations to understand these processes. The success of this study lies in the great statistics possible with the Wind dataset, which allow for a much more sophisticated study than has been attempted historically, and the fact that electrons are not resonant with these instabilities, so the proton and eld measurements are sucient to completely describe the situation. Under the assumption that the solar wind speed is roughly constant once it escapes from the corona, we can dene the expansion time, x, for a parcel of solar wind plasma to reach Earth, x 1A:U: U p ' 2 ; 4days. (3.2) In Section it was shown that the Coulomb collision timescale c (1.52) is often on the same order of x. As the solar wind expands, Coulomb collisions gradually isotropize the VDF and R p relaxes to unity. In terms of Coulomb collisions we can say that the solar wind is collisionless if x = c 1 and is dominated by collisions if x = c 1. However, other processes in interplanetary space act to drive R p 6= 1, and if this deviation of R p from unity exceeds given thresholds the plasma becomes unstable. The free energy of the temperature anisotropy feeds rapidly growing electromagnetic uctuations which scatter protons and push R p back to unity. Since the free energy is derived from the anisotropies they are \anisotropy-driven" instabilities. These particular growing modes are due to the onset of the mirror, rehose, and cyclotron instabilities, which will be calculated in the following section. These instabilities have growth rates on the order of the proton cyclotron frequency c p (1.20) and lengthscales near the proton gyro-radius, so they are often referred to as micro-instabilities. Due to the relatively high frequency and short timescale the MHD uid picture is not a suciently accurate approximation because the electromagnetic uctuations couple to a specic portion of the VDF. The Vlasov kinetic description is more appropriate then the uid, so these processes are also referred to as \kinetic" instabilities. The unstable plasma modes explored in this chapter therefore are temperature anisotropy driven kinetic plasma micro-instabilities. These micro-instabilities are an invaluable tool for furthering our understanding of fundamental plasma processes. Since these particular kinetic instabilities exhibit maximal growth near c p their physical description lies right at the boundary between the realms of relatively well understood uid MHD and the less well-known kinetic theory. Section showed that in MHD it is sucient to express the VDF 94

95 through ~ U p and T p and work with three spatial dimensions plus time. Solutions to the Vlasov equations require the three velocity dimensions to be retained, leading to a far more complex situation. Often dimensions may be neglected if the plasma is isotropic, but since we are considering a magnetized anisotropic plasma this is not necessarily permitted. Moving to a kinetic description of these instabilities is therefore a great challenge. An understanding of these instabilities is also of pressing practical concern. They eect the evolution of the solar wind in interplanetary space and may play a role in the heating of the solar corona. Onset of these instabilities generates electromagnetic uctuations which may eect the propagation of cosmic rays through the heliosphere. Finally, the established limits to R p should be provided to the MHD simulation community so they can use realistic temperature anisotropies without adding the kinetic component to their simulations. The remainder of this introduction is an overview of the limits to R p observed by the Wind spacecraft as a function of various solar wind plasma parameters. In the following section we will work through descriptions of these instabilities in the uid MHD, kinetic, and numerical simulations in one and two spatial dimensions and show that they produce diering results. The nal two sections of this chapter separately consider the upper R p < 1 and lower R p < 1 bounds to the proton temperature anisotropy. The rst global view of the proton temperature anisotropies is to examine the average value of R p as a function of the proton number density and magnetic eld strength, both of which vary by more than an order of magnitude in the solar wind. In Figure 3-1, colored bins indicate the average value of R p for all measurements in given intervals in the (B n)-plane. The anisotropies range from 0:6 R p 1:8, as indicated by the calibration of the color scale on the left side of the gure. The black contours indicate the relative density of measurements in the (B n)-plane at the 5%, 25%, 50%, and 75% levels, so the most probable state of the solar wind is a density of 5 cm ;3 and a eld strength of 5 nt. The corresponding anisotropy is R p ' 0:8, so T kp > T?p on average at 1 AU, as fact which generally is not recognized by the community. This is most likely due to the historical focus on intervals with R p > 1 in the Earth's magnetosheath and solar wind observations closer to the Sun in which the average anisotropy was R p > 1 [Marsch et al., 1982]. The procedure for creating Figure 3-1 is the same as in many of the gures of Chapter 2. Wherever a high density of observations permits, the size of the bins has been subdivided. Each bin shown contains at least 1,000 measurements, and the standard deviation of R p in each ofthe bins shown is less than 20%. The values of R p range from 0.6 to 1.7, with isotropy (R p = 1) indicated by the red contour line. Based on the smooth variation of R p between adjacent bins the observed variation of R p (B n p ) is statistically signicant. At the highest proton densities R p! 1 and the average observed proton VDF is isotropic. This makes sense given the linear dependence of the Coulomb collision rate on n p. At high densities, c is small, so the ratio x = c becomes large and the plasma is dominated by frequency Coulomb collisions. This does not explain the observed variation of R p (B n) at lower densities, specically the simple prole of R p = 1 indicated by the red contour line. The functional form of this curve can be 95

96 Figure 3-1: Survey of proton temperature anisotropy R p = T?p =T kp, as a function of proton number density n p and magnetic eld magnitude B. The color shading indicates the median value of the ratio within the bins, which are logarithmically spaced in density and eld strength. The color scale is given by the bar on the left and ranges from 0:6 R p 1:8. Bins are smaller in regions of higher density of measurements, but no bin has less than 1000 spectra. Black contours indicate the density of observations at the percent levels. The red line indicates the R p =1contour and dashed line is apower law with n p is proportional to B 3=2. 96

97 understood by recalling the double adiabatic equations of state from Section 1.2.7, d T? d Tk B 2 =0 dt B dt n 2 =0: (3.3) If R p is just a function of n p and B then we can use (3.3) to predict the variation of the anisotropy, R p = T? T k / B3 n 2 : (3.4) Therefore according to (3.4), the set of adiabatic transformations in (B n p )-space which preserve a given anisotropy R p must satisfy the relation, n / B 3=2 : (3.5) The dashed line over-plotted in Figure 3-1 is a power law proportional to B 3=2 with height adjusted to overlap with the red contour of constant R p =1. Clearly over eld strengths from 2 nt to 20 nt and proton densities between 0.8 cm ;3 and 20 cm;3 this predicted function form agrees with the observations. We can conclude that small, local deviations in the average anisotropy are consistent with the predictions of the double adiabatic equations of state. The average values of R p remain within approximately 50% of isotropy. Now we will look at the range in R p which the protons occupy with other parameters xed. In this approach the adiabatic picture breaks down, even in the limit of low density and infrequent collisions. At this point it is useful to dene the dimensionless plasma parameter kp, kp n pk B T kp B 2=2 (3.6) which is the ratio of proton pressure along the eld to the magnetic pressure of the ambient magnetic eld. The value of kp has two important meanings in terms of these anisotropy-driven instabilities. First, kp indicates whether the particles or the elds dominate the dynamics of the motion of a plasma parcel. Since the solar wind is highly conductive, the magnetic ux is \frozen-in" (Section 1.2.7) and a parcel of particles and eld lines move in tandem. If kp 1thenthe magnetic eld achieves a given conguration with little eect from the particles and drags the plasma with it. On the other hand, if kp 1, as is the case in the convection zone of the Sun, the plasma ows as it wishes and pulls the eld with it. In the MHD picture of these instabilities presented in Section the dominance of either eld or particle pressure is crucial for determining the onset of instability. Therefore the stability of an anisotropic MHD uid is determined by kp. Secondly, note that kp is roughly equal to the ratio of the sound, CS 2 = P= and Alfven CA 2 = B 2=2 wave speeds 97

98 (See Section 6.1). With = m p n p and P = n p k B T kp, CS 2 CA 2 = P= (B 2 =2 )= = n pk B T kp B 2 =2 = kp (3.7) The two quantities agree within a factor of and since 1 (1.77) they are approximately equal. In the linear kinetic analysis of Section growing modes are identied by solving a dispersion equation which is dependent on the relative wave speeds. From the kinetic viewpoint aswell, kp controls these instabilities. Now we will look at the observed distribution of R p as a function of kp. Under the assumption that the double adiabatic equations of state and the predictions of (3.5) are always valid we can estimate what values of R p would be expected at 1 AU. The origin of the solar wind is the solar corona, which has a density times greater than that in interplanetary space. Therefore in the corona x = c 1 and we could reasonably expect the protons to have an isotropic VDF when they escape. For the sake of simplicity assume that the solar wind protons emerge in a nal state at 10R with constantbulkspeedu p. In steady radial expansion the continuity equation (1.71) requires that the proton number density fall o as 1=r 2. While the solar magnetic eld appears dipolar near the Sun in the corona where kp 1, beyond the corona kp 1 and the ion ow aects the conguration of the eld. As a result the magnetic eld at 1 AU in the equatorial region of the Sun (in the ecliptic plane) falls o as 1=r [Parks, 1991, x6.4.1]. Using (3.4), T? =T k / (r ;1 ) 3 =(1=r ;2 ) 2 = r. With 1 AU ' 200R and the solar wind starting isotropic at r ' 10R it would be reasonable to expect to see values of T? =T k clustered about 1=20 or 5%, especially in intervals with low collision rates. This prediction can be compared with observations. It was clear in Figure 3-1 that the average values of R p were much greater than 5%, but we can also look at the spread in R p. Figure 3-2 is a two-dimensional histogram of the number of proton spectra observed as a function of R p and kp, which will be dened as N(R p kp ), N(R p kp ) distribution of spectra as function of R p and kp : (3.8) Figure 3-2 contains more than 1.5 million spectra recorded by the Wind/SWE FC instruments in the solar wind from 1994 through The histogram bins are logarithmic in R p and kp, although subsequent sections will focus on the variation of N(R p kp ) in the vicinity of R p ' 1 and will use linear bins in R p. The size of an individual bin is indicated by the square in the lower left side of the plot. The contours are logarithmically spaced and indicate the density of N(R p kp ) from 10 to 10 4 spectra per bin. The highest density of observations is at kp ' 0:8 and R p ' 0:8 (in agreement with Figure 3-1). The observed range in R p is clearly a function of kp. For values of kp 0:02 and kp 1 the proton VDFs are sharply constrained to lie near R p ' 1. Spectra with the largest values of T?p > T kp are clustered around kp ' 0:07, but at the 99% level do not exceed R p of 3. Likewise spectra with T?p < T kp are clustered around kp ' 0:8, but at the 99% level do not exceed R p of 1/10. This distribution is clearly not in agreement with the range 98

99 Figure 3-2: Survey of solar wind measurements as a function of plasma beta and the ratio R p = T?p =T kp. Note that the observed values of R p do not depart greatly from unity, in disagreement with the predictions of the double adiabatic equations of state (3.3) and a simple adiabatic expansion (3.4). 99

100 of values expected from the adiabatic equations of state. Instead of clustering at R p 1=20 the average values are within 50% of unity. From the analysis in Chapter 2 it is clear that the Faraday Cups on Wind are capable of measuring thermal speeds of km/s, or a range in temperatures diering by a factor of If there were larger departures of R p from unity than those shown in Figure 3-2 they would have been detected. Additionally, there is no explanation in the context of adiabatic expansion and Coulomb relaxation for a proton VDF with R p > 1 to be created in the interplanetary medium. We will see that the distribution of N(R p kp ) in the two regimes R p < 1 and R p > 1 are very dierent because they correspond to the eects of very dierent physical processes. In the case of R p < 1 adiabatic expansion drives R p < 1, while the rehose instability provides a lower limit to R p and Coulomb relaxation drives R p back to unity. Intervals in the solar wind with R p > 1 are either compression regions between solar wind streams or artifacts of the absorption of electromagnetic uctuations through cyclotron resonance. The exact eect is explored further in Section 3.3 and is further validated using helium observations in Chapter 5. Observations of the range in R p > 1 reect the eects of these processes: Coulomb relaxation of R p back to unity, and the mirror and cyclotron instabilities. So far we have shown that the entire dataset is statistically limited to values of R p near unity by instabilities and that the exact limiting value of R p is a function of kp. The eects of these kinetic instabilities can actually be seen in individual spectra and the nal demonstration in this introduction is of the bounds imposed by these instabilities over the course of a single day of data. The kinetic descriptions of these instabilities in Section 3.2 show that the typical rates at which the unstable plasma modes grow are on the order of c p. As noted in Section 1.2.2, under typical solar wind conditions a 92-second Wind FC spectrum reects the proton distribution averaged over a thousand proton gyro-periods. If changing plasma conditions trigger any of these instabilities, we cannot directly observe the resonance of the VDF with the growing electromagnetic uctuations. Instead a time-series of 92-second plasma measurements can be thought of as reecting the nal, steady-state value of R p after the instability has acted to reduce the anisotropy below threshold. The eect of the limits imposed by the mirror and cyclotron instabilities is demonstrated in Figure 3-3, which displays Wind plasma observations in the solar wind on April 30, From the top the panels show the bulk proton speed, the proton number density and the relative abundance of helium by percent, the eld strength, kp, T?p, T kp, and R p. The speed of the solar wind was roughly constant, with a small shock arriving at 1800 UT. The helium abundance, typically 5%, was very low for much of the day, especially from 0500 to 1800 UT, when it was below 1%. Between 0500 and 1200 the proton density rose from 10 cm ;3 to 25 cm ;3, while the magnetic eld strength fell from 6 nt to 0.5 nt, leading to values of kp ' 30 which from Figure 3-2 are rare in the solar wind. The expected variation of the parallel and perpendicular temperatures (predicted using the double adiabatic equations of state) are shown with the over-plotted blue lines. The average values of T?p =B and T kp B 2 =n 3 p were calculated over the rst four hours of the day then used to predict T?p and T kp given the measured values of B and n p for the rest of the day and (3.3). The observed values of T?p agree well with the predictions, but the calculated 100

101 Figure 3-3: Observations of the proton temperature anisotropy R, along with other solar wind parameters, as a function of time over the 24-hour period on April 30, Note the unusual rise in kp to more than 30 between 0630 to 0745 UT, and the concurrent constraint of the proton anisotropy. The blue, green, and red curves correspond to the predicted bounds imposed by the mirror, rehose, and cyclotron instabilities. 101

102 value of T kp, which is ten times what was observed, is too large to be seen on the scale of the plot. Clearly in this interval the adiabatic approximation has broken down. In the nal panel of Figure 3-3 the measured value of R p is shown in black, along with estimates of the upper bounds from the mirror (green) and cyclotron (blue) instabilities, and lower bounds from from the rehose instability (red). At all times over the course of the day R p is constrained to lie between the limits established by the instabilities. At 0600 UT the proton anisotropy is consistently at the threshold for the mirror and cyclotron instabilities. At 0630, when kp jumps upwards of 30, R p is sharply brought backdown to unity as the constraints become more severe. The limits to the temperature anisotropy are clearly apparent even in individual observations of the proton VDF. 3.2 Four Descriptions of Plasma Micro-Instabilities What description of an instability is sucient for this study? This section outlines three theoretical methods for analyzing the anisotropy-driven instabilities, followed by the experimental technique which will be used to quantify the observations. The rst step in a successful description is determining the threshold for the instability to generate a growing mode in the plasma. In Section single-uid magnetohydrodynamics (MHD) is used to derive the conditions necessary for the onset of the mirror and rehose instabilities. While MHD can produce an intuitive picture of the physical mechanisms behind the instabilities, it has several shortcomings. Since MHD cannot describe the coupling of electromagnetic uctuations to specic portions of the VDF it produces inaccurate descriptions of these micro-instabilities. Additionally, MHD cannot produce the rate at which they grow, which, for example, is needed to identify the dominant mode in the event of multiple instabilities. A more appropriate method in this case is the calculation of small uctuations which self-consistently satisfy linearized Vlasov and Maxwell equations. The linear Vlasov procedure derived in Section is insucient to describe electromagnetic uctuations in a magnetized plasma, so the method is extended in Section and the results of linear calculations for these instabilities are reviewed. There are also limitations to the linear method however. When the perturbation in the VDF and elds grow larger than the initial ambient values the linear approximation breaks down. Linear analysis can only tell us about the initial growth phase, and not what the nal state of the plasma will be. Complete analytical solutions to the Vlasov equations are not yet at hand, so numerical simulations are used to model the wave-particle interactions directly. Section describes those simulations and their predictions for these instabilities. Finally, in Section 3.2.4, the procedure which will be used to compare observations with these theoretical predictions is outlined The Magnetohydrodynamic Fluid Perspective The starting point for the uid representation of these instabilities are the single- uid MHD equations which were derived in Section 1.2.7, but with the pressure and 102

103 temperature terms remaining anisotropic, and the double adiabatic equation of state instead of the single constant P ;. Electrons are not resonant with the instabilities discussed here, and minor ions such as helium do not contribute signicantly to these calculations, so the single-uid picture is appropriate. For the mirror instability, consider the response P? of the perpendicular pressure P? to a slow, compressional change B in the ambient magnetic eld B. Starting with the pressure form of the double adiabatic equations of state (1.81) and (1.80), eliminating the number density, and dierentiating, P P? P? =2 1 ; T? T k B B : (3.9) Note that if T? > T k (R p > 1) then an increase in the magnetic eld strength will result in decreasing perpendicular particle pressure. The total pressure along an axis perpendicular to a magnetic eld line is P? + B 2 =2,anddierentiating, P total? = P? + 2BB 2 = P? + BB : (3.10) Solving (3.9) for P? and eliminating it from (3.10) yields the total change in the perpendicular pressure in terms of B, P total? =2P? 1 ; T? T k B B + BB : (3.11) If (3.11) is less than zero then an increase in the local magnetic eld strength will result in a decrease in the total perpendicular pressure. In order to restore pressure equilibrium more plasma will rush into this region, further increasing the eld strength and driving the compression. The MHD threshold for the mirror instability is therefore when P? B 2 1 ; T? +1< 0: (3.12) =2 T k Recognizing P? =B 2 =2 as the perpendicular equivalent of k (3.6) and the denition of the proton anisotropy R p (3.1), an anisotropic plasma is mirror unstable when T? = R p > 1+ 1 : (3.13) T k? This mode earned the name mirror instability in early work which attributed the concentration of the eld to mirror focusing [Chandrasekhar et al., 1958 Barnes, 1966]. According to these predictions a typical solar wind plasma with 1 becomes unstable to mirror uctuations when T? 2T k. An unstable MHD mode with T k > T? (R p < 1), the rehose instability, was rst derived by [Parker, 1958a]. The mechanism behind the rehose instability is quite simple. Imagine a small perturbation in the ambient magnetic eld which 103

104 produces a bend in a eld line. Particles propagating parallel to the eld will exert the maximum centrifugal force on the eld line at its point of maximum deection. The centrifugal force exerted by the particles is proportional to the parallel thermal speed. If the parallel proton pressure is too large relative to the tension in the eld line, the particles will cause an explosive expansion of the eld. Following a procedure paralleling the above derivation for the mirror mode, single uid MHD predicts that an anisotropic plasma will become unstable to the rehose instability if the proton anisotropy exceeds the threshold, R p < 1 ; 2 k : (3.14) Parker was interested in the rehose instability in part as a mechanism for generating magnetic turbulence in the outer heliosphere which would eect the propagation of cosmic rays [Parker, 1958a]. The rehose instability is also an important process for studying the dynamics of energetic particles trapped in the radiation belts [Kennel and Petscheck, 1966] and in relativistic astrophysical plasmas [Yoon, 1990]. Due to the nature of the cyclotron resonant instability there is no real uid picture of it, and we must turn to a kinetic description of the plasma Linear Theory of Waves and Instabilities The successful application of linear Vlasov theory to study instabilities in an anisotropic magnetized plasma is much more complex than the procedure outlined in Section for electrostatic waves in an unmagnetized plasma. The purpose of this section is to make it clear what the special eects of these anisotropy driven instabilities and how the results of linear analysis are used to make predictions to compare with the Wind observations. General descriptions of linear Vlasov theory in magnetized anisotropic plasmas may be found in textbooks [Gary, 1993 Stix, 1992], and the specics of the analysis for these instabilities may be found in the references cited in this section. The uctuations associated with the mirror, cyclotron, and rehose instabilities are electromagnetic, meaning that the perturbation introduces a small magnetic eld ~B (1) in addition to the induced electric eld ~ E (1). The eect of the ambient magnetic eld ~ B must also be added. The linearized Vlasov equation for the perturbed VDF of a single ion species changes from (1.93) + q m ~v = ; q m h ~E (1) + ~v ~ : (3.15) Solving (3.15) for the rst-order distribution function using the method of integration over unperturbed orbits [Gary, 1993, Appendix B], f (1) may beexpressed as f (1) ( ~ k ~v!) =; q m Z 0 ;1 d 0 + ~ k! 0 # ~ E (1) ( ~ k!)e ib(!) (3.16) 104

105 where (1.25) has been used to eliminate ~ B (1), t ; t 0,and b(!) k?v? c [cos( c ; ) ; cos ]+(k k v k ;!) (3.17) where k? and v? are the components of the wavenumber and velocity perpendicular to ~ B and where k k and v k are the components parallel to ~ B, and c is the cyclotron frequency of the particle species under consideration (1.20). In (3.16) and (3.17) it is clear that the anisotropic VDF adds complexity tof (1), and there is an indication that the direction of propagation of the wave is now important. A solution to the perturbed quantities allows a calculation of the susceptibility D( ~ k!), and allowed modes must satisfy D( ~ k!) ~ E (1) ( ~ k!) =0 (3.18) where D is now a 3 3 tensor instead of the scalar susceptibility K(k!) (1.97). Growth rates, dispersion relations, and polarizations are determined by identifying when the determinant of D vanishes. The presence of the ambient magnetic eld modies the wave-particle processes which are possible in this plasma. Solutions to (3.18) produce a resonance (f (1)!1) whenever! = k k v k ; m c m =0 1 2 ::: (3.19) The m = 0 solution recovers the Landau damping resonance (1.100), but only in the case of propagation along ~ B. In addition to the eects seen in the unmagnetized plasma there is an additional wave-particle interaction corresponding to the m 6= 0 terms in (3.19), which is the cyclotron resonance. Particles in the VDF which satisfy! = k k v k ; m c gyrate about ~ B such that the sign of ~ E (1) ~v? never changes sign, resulting in a strong transfer of energy from the wave to the perpendicular motion of the particles. This is the physical process behind the cyclotron instability, which is also often referred to as the \resonant" cyclotron instability. The heating of ions through resonant cyclotron absorption of electromagnetic waves is explored further in Section 5.3. As an example of the additional information available about uctuations through the linear analysis, (3.18) permits the calculation of the polarization of the waves. Following Stix [1992], the polarization P is determined by P E(1) y! r E x (1) j! r j : (3.20) A uctuation is termed right-handed (P = ;1) or left-handed (P = +1) depending on the sense of rotation in time at a xed point in space of the perturbed eld vector when viewed in the direction parallel to the magnetic eld. So a right-handed mode is in the same sense at the gyromotion of an electron. If the equilibrium VDF f (0) is bi-maxwellian, with jj j! r j and k k > 0, it can be shown [Gary, 1993, x7.1] that the solution to (3.18) has a growth rate which takes 105

106 the following form in the presence of multiple species, = X! j 2! r (2vj 2 ) 1=2 ; T?j! r T kj k r j 1 ; T?j c j exp T kj k k " (! r c j ) 2 2k 2 k v2 j # (3.21) where j represents values for the j th particle species. In the kinetic picture each particle species may separately drive the instability. In contradiction with the single- uid picture, in which helium ions produce a negligible contribution to the overall pressure, (3.21) predicts that Helium and other minor ions should experience the same instabilities as the protons, no matter how dilute they are. This validity of this prediction is explored in Section 5.4. For modes with 0 <! r c j, an ion species with T?j < T kj will drive a righthand polarized wave unstable. This corresponds to the MHD rehose instability. If T?j > T kj the ions will drive a left-handed modes corresponding to the mirror and cyclotron instabilities. The procedure for identifying limiting values of proton temperature anisotropy R p (3.1) allowed before an instability reaches threshold with the results of linear theory calculations is as follows. At a given value of kp, the full form of (3.18) is solved, usually numerically, for the growth rate, for all orientations of ~ k with respect to B ~ and for low values of the order m of the resonance in (3.19) ( decreases with increasing values of m), and the maximum growth rate m is identied as a function of R p. Generally the mirror and rehose instabilities have their maximum growth rates when the wavevector is oblique to B ~, while the growth of the cyclotron instability is greatest for nearly parallel propagation. Consideration of propagation at all angles is necessary to identify the correct values for the growth rates. Published results of linear analysis may be divided into \one-dimensional" calculations which only consider ~ k B ~ = 0, and \two-dimensional" calculations with permit oblique propagation. The limit to R p produced by the instability is dened as the value of R p for which the maximum growth rate m exceeds some set value. Since the proton cyclotron frequency determines the time-scale of these instabilities the theory literature generally uses some fraction of c p,typical values range between 10 ;4 c p m 0:1 c p. The critical proton anisotropy, R p c,which is the upper limit imposed by the mirror and cyclotron instabilities, is observed to take the form R p c S p p kp ; 1 (3.22) both in the analytical calculations, the simulations, and the existing observations. For the rehose instability, the critical proton anisotropy R p c which serves as the lower bound takes the form R p c = S p p kp +1: (3.23) 106

107 Published calculations of these limits generally report the best-t values of S p and p to the curve of R p c at some xed value of m over some interval in kp [Gary et al., 1998, 1997]. A survey of the literature is summarized in Table 3.1 in Section Hybrid Simulations of Wave-Particle Interactions The linear Vlasov theory has its limitations. While linear theory does a good job of identifying growing modes and properties such as their polarizations, it is only valid in the regime where the perturbed values of f (1) and E ~ (1) are much less than the zero-order terms, and to cases where =! r 1. These are signicant limitations since observations of the particle VDF on a spacecraft generally occur over thousands of cyclotron periods, so the growing mode cannot itself be observed. Linear theory cannot tell what the nal equilibrium state of the VDF will be after the uctuations have been dissipated and the anisotropy reduced below threshold. For calculations of the nal state of the plasma, and to verify the validity ofany assumptions which went into solving the dispersion relations in the linear regime, numerical simulations are also carried out. Ideally a simulation would track the evolution of every particle and solve for the electromagnetic elds self consistently. In reality this is impossible, but a good approximation are hybrid simulations which represent portions of the VDF with superparticles [Lapenta and Brackbill, 1996 Lottermoser et al., 1998]. A detailed description of the hybrid code which was used to produce all of the numerical predictions referenced in this chapter is available [Winske and Omidi, 1997]. IN presenting results a distinction is made between one-dimensional hybrid codes, which simulate modes with ~ k B ~ = 0, or parallel propagation, and two-dimensional codes which can handle oblique propagation. The hybrid results available for comparison with the Wind observations are listed in Table Observations of the Eects of the Instabilities One of the original motivations for the study of temperature anisotropy driven instabilities were the observation of \lion-roar" radio signatures at Earth [Tsurutani et al., 1982 Thorne and Tsurutani, 1981]. These radio bursts were believed to be driven by anisotropic particles in the magnetosheath with T? >T k becoming unstable to kinetic instabilities. R p > 1 is a frequent phenomena in the magnetosheath due to the ecient heating of particles, both ions and electrons, perpendicular to the ambient magnetic eld by the Earth's bow shock. The rst in-situ indications of the mirror and cyclotron instabilities were the presence of magnetic uctuations with the correct polarization and a correlation between magnetic eld strength and plasma density (recall from (3.11) that the mirror mode is a compressive process) [Crooker et al., 1979]. More recent observations have shown that there is an inverse correlation between the measured values of R p and kp for R p > 1 [Gary et al., 1997]. These observations convincingly demonstrate that the magnetosheath plasma with R p > 1 experiences the mirror and cyclotron instabilities. They are not sucient, however, to distinguish between predictions of the limit based on linear analysis or hybrid simulations. In order to compare theory and observations the data must be 107

108 Instability S p p Comments Reference Mirror 1 1 MHD (3.13) [Chandrasekhar, 1958] m =0:1 p 5 kp 50 [Gary et al., 1997] Cyclotron m =10 ;4 p 0:01 kp 10 [Gary et al., 1994] m =10 ;3 p 0:01 kp 10 \ \ m =10 ;2 p 0:01 kp 10 \ \ D Hybrid, Active ( 10 ;3 ) \ \ D Hybrid, All Results \ \ AMPTE/CCE Observations [Anderson et al., 1994] AMPTE/IRM Observations [Phan et al., 1994] Firehose 2 1 MHD (3.14) [Parker, 1958a] m =10 ;3 p 2 kp 100 [Gary et al., 1998] m =10 ;2 p 2 kp 100 \ \ m =10 ;1 p 2 kp 100 \ \ D, 2D Hybrid \ \ Table 3.1: Predictions of the expected limits to R p based on MHD uid, linear Vlasov theory, 1D and 2D hybrid numerical simulations, and previous experimental observations in the magnetosheath. 108

109 interpreted to produce measured values of the S p and p parameters used in (3.22) and (3.23) to describe R p c. More recently, several observations studies of the upper bound R p c have been performed in the magnetosheath [Anderson et al., 1994 Phan et al., 1994]. In those studies the values of S p and p are determined by examining a scatter plot of the observed R p vs kp and identifying a curve which appears to mark the upper bound to the observations. In contrast to the large amount ofwork which has been invested in the study of the R p > 1 limits, there is only one existing report of observational signatures of the rehose instability, and the data were insucient to determine a functional form of (3.23) [Eviatar and Schultz, 1970]. A detailed analysis of the Faraday Cup observations of these instabilities in the solar wind provides several major improvements over these previous studies. First of all, this is the rst in-situ plasma dataset with estimates of the uncertainties in each of the parameters available. For each proton spectrum we can calculate R p and propagate the uncertainties in the parallel and perpendicular temperatures to determine the uncertainty Rp in R p. Thus we can distinguish between scatter in observations of R p due to the accuracy of the individual measurements and a real distribution of R p due to physical eects. In the following sections the focus will be on measurements of R p within a factor of 2 of unity. Figure 3-4 illustrates the uncertainty in the measured anisotropy. The upper panel is a two-dimensional histogram of the number of spectra as a function of uncertainty and anisotropy. Most of the data are clustered around R p 1, so to show the range of uncertainties typical for a given value of R p each column in the histogram was normalized to unity. The average uncertainty for each column of R p, shown with the over-plotted diamonds, obeys a power law distribution, log 10 R = A + B log 10 R A = ;0:39 0:01 B =1:29 0:01 (3.24) and in the interval we willbestudyingtheaverage uncertainties range from 20%-40%, with a most probable value of 25%. The second advantage to this study is that fact that it focuses on observations in the solar wind, and not in the magnetosheath. The plasma in the magnetosheath is active, undergoing a continual expansion as it is diverted around the Earth. By looking at solar wind we can examine a plasma which is more likelytohave achieved a steady state. Finally, the 2.3 million Faraday Cup spectra which have been analyzed to date present unprecedented statistics for studying the thresholds. This permits a much more quantitative determination of S p and p than examination of a scatter plot by eye. The procedure developed to analyze the Wind observations proceeds as follows. Measurements with R p > 1 and R p < 1 are considered separately, since the processes which drive R p 6= 1 are dierent in the two cases, and because the mirror and cyclotron instabilities are examined separately from the rehose instability. The goal is to look at the rate at which the number of observed spectra N( kp R p ) falls o with increasing (decreasing) R p for the mirror and cyclotron (rehose) instabilities at xed values of kp. The variation of N( kp R p )withr p reects the location of the threshold 109

110 Figure 3-4: Upper panel: Uncertainty R as a function of R for all 2.2 million proton measurements to date with no cuts. Diamonds indicate the average uncertainty as a function of R p, and the solid line is the best t of (3.24) to the averages. Lower panel: Histogram of the distribution of observations (blue) and the average percent uncertainty (red). 110

111 anisotropy R p c as a function of kp. In an ideal situation, one could imagine observing a value of R p at which N( kp R p ) suddenly falls to zero. In reality the limit will be spread out do the the gradual driving of the VDF out of isotropy and the uncertainty in the individual measurements. To remove the fact that the solar wind spends more time at a particular value of kp (recall the histogram in Figure 3-2), dene N(kp R p ), N( kp R p ) N( kp R p ) MAX[N( kp R p )@ kp ] : (3.25) This for each interval in kp there is a value of R p where N( kp R p ) = 1. We then dene the threshold value R p c of the anisotropy as the point at which N( kp R p )=f (3.26) where f is a fraction which we will take to be 10% but will return to in the following sections. In each case, R p < 1andR p > 1, a functional form which describes N(kp R p )is identied and t to the observations. Inverting the t produces R p c as a function of kp and an estimate of its uncertainty. The values of S p and p are nally determined by the best t of (3.22) or (3.23) to the measured values of R p c. 3.3 T?p > T kp : Mirror and Cyclotron Instabilities Wind Observations of Solar Wind with T? > T k The calculations of the expected values of R p at 1 AU in Section 3.1 indicated that we would expect the expansion of the solar wind to drive R p < 1. Values of T? >T k, while common in the magnetosheath, are not expected in the solar wind outside of infrequent compressional regions and behind interplanetary shocks. It is generally accepted that in the region closer to the Sun it is possible to produce T? =T k through other methods. In one example, recent spectroscopic observations of coronal light suggest that at least minor ions such asoxygen exhibit large temperature anisotropies which are believed to be an artifact of the mechanism responsible for the heating of the corona [Cranmer, 1999]. The solar wind escaping therefore may escape from the corona with R p > 1. A second mechanism could be the dissipation of Alfven waves propagating through the interplanetary medium, which carry a great deal of energy and are observed to decrease in amplitude with increasing distance from the Sun [Belcher and Burchsted, 1974]. Through turbulent processes, energy in the longwavelength Alfven waves cascades to shorter wavelengths until cyclotron resonance with ions absorbs the wave energy and heats the plasma. The result of the cyclotron absorption would be preferential heating of the perpendicular components of the ion temperatures [Hollweg, 2002]. A further examination of signatures of cyclotronresonant heating is presented in Section 5.3. For the purposes of this section we will attribute the generation of intervals with R p > 1 other than infrequent compression intervals to the eects of a process which occurred closer in to the Sun. 111

112 Figure 3-5: Eect of Coulomb relaxation on the average value of R p observed as a function of kp. Upper: Histogram of the number of observed proton spectra as a function of x = c, the ratio of the solar wind expansion and Coulomb relaxation timescales. Colored regions indicate six regions with 20,000 spectra. Lower: The average value of R p as a function of kp for each of the selected intervals. The average value is described by (3.22) and the best-t parameters are shown in the legend. 112

113 To establish the validity of this assumption and to further justify the procedure adopted in Section for identifying instability thresholds, the average values of R p for all measurements with T?p >T kp are now examined as a function of the rate of Coulomb relaxation. The top panel of Figure 3-5 is a histogram of the number of solar wind observations as a function of the ratio of the expansion timescale x (3.2) to the Coulomb collision timescale c (1.52). When x = c 1, there have been many energy-exchanging Coulomb interactions between particles in the solar wind, and when x = c 1 the solar wind plasma may be considered \collisionless" since it left the corona. The six colored regions underneath the histogram indicate intervals of x = c which each contain 50,000 spectra. This histogram highlights one of the largest factors in studying the kinetic properties of the solar wind. The average value of x = c is within an order of magnitude of unity, so the solar wind, in terms of its propagation from the corona to an observer at 1 AU, is not precisely collisionless as many theoretical calculations assume. In terms of the use of the Vlasov approximation of the Boltzmann transport equation for the study of local kinetic processes the collisionless approximation is justied, since c is 10 6 times larger than the relevant timescales. In the lower panel of Figure 3-5, the average values of R p for all measurements with T?p > T kp for each of the six intervals in x = c are shown as a function of kp. The two main points are that plasma with decreasing kp have larger anisotropies, but that R p is seen to decrease as the Coulomb collisions become more important. The solar wind at 1 AU with R p > 1 bears the signature of preferential heating at some distance back towards the Sun. As collisions become more frequent, this signature is washed out and the average proton VDF is isotropic. The lines overplotted to the observations are best ts of these average values of R p to the functional form of the threshold limit to the proton anisotropy imposed by the mirror and cyclotron instabilities (3.22). The values of the parameters S p and p are shown in the legend for each curve. An assumption that the average values of R p as a function of kp are the indication of the instability thresholds would misinterpret the gradual Coulomb relaxation of some earlier process for the actual eect of the kinetic instabilities. This justies using the fallo of the distribution of measurements through N(kp R p ) used in the following sections The Cyclotron Limit in the Region 0:1 kp 5 The distribution of the Wind measurements as a function of kp and R p in Figure 3-2 shows that most of the observations are at kp < 5 for R p > 1. From the calculations listed in Table 3.1 of the expected thresholds due to the mirror and cyclotron instabilities, it can be shown that the mirror instability does not plat a dominant role over the interval 0:01 kp 1 where the majority of the data are located and the statistics are sucient to pursue the statistical technique. This section therefore is restricted to the determination of the limit to R p imposed by the cyclotron instability. Observations of the mirror instability are presented in Section The top panel of Figure 3-6 is a plot of kp N(kp R p ) as a function of R p for each interval of kp in the range 0:02 kp 10. Each color line represents the measurements of N( kp R p ) for a xed value of kp. N(kp R p )ismultiplied by kp so 113

114 Figure 3-6: Survey of R p. 114

115 the form of each individual curve may be seen. We nd that the fallo of N( kp R p ) with R p for each value of kp are best t by an exponential curve, N( kp R p ) t = e A+Rp= (3.27) where A and are free parameters. The two lower panels in Figure 3-6 are plots of the best-t values of A and as a function of kp. The horizontal error bars are the widths of the original histogram bins in kp, and the vertical error bars are the uncertainties A and in the free parameters. It was assumed that the error in the measurements of N( kp R p ) are just due to statistical counting eects, and N is one over the square-root of the number of spectra falling into that interval. In Section we dened the threshold value of R p as the point R p c at which N( kp R p ) drops to f. We can invert (3.27) to solve for R p c, and the uncertainty Rpc in R p c, R p c = [ln f ; A] (3.28) 2 Rpc =[lnf ; A] A : (3.29) The upper panel of Figure 3-7 is a plot of the measured values of R p c determined through (3.28) with f = 10% as a function of kp. The vertical error bars are the uncertainty in R p c calculated with (3.29). The uncertainty in these measurements is therefore related completely back to the original FC proton spectra. Data with 0:1 kp 1 were selected for comparison with theory and are marked with blue diamonds. At kp the width of the distribution of spectra as a function of R p converged on the average uncertainty of the FC measurements of R p. We feel that at values of kp 5 R p may be closer to unity than measured due to the accuracy of the observations, so they were discarded. The red curve is the best-t of (3.22) to the selected measurements, yielding S p =0:57 0:02 and p =0:51 0:02. This result is compared with predictions in the lower panel of Figure 3-7. There is a good agreement with the AMPTE/IRM magnetosheath observations, although the solar wind threshold appears to be slightly lower. The results from this study agree within one sigma with the one dimensional hybrid numerical simulations [Gary et al., 1997]. It is interesting that the solar wind observations agree so well with the magnetosheath results. The popular explanation for the fact that the measured limits to R p were so much larger than the linear Vlasov predictions has been that the magnetosheath is actively driven by expansion and uctuations an a manner which would not be seen in the solar wind. This new result suggests that the observed thresholds are indeed valid, and that the linear calculations are not sucient. It is possible that the presence of helium in the solar wind, on the 5% level relative to hydrogen, may act to limit the maximum values of the anisotropy, but an initial study of the eect of the helium abundance did not yield any trends. The participation of helium in these instabilities is examined in Section

116 Figure 3-7: Normalized survey of solar wind measurements as a function of plasma beta and the ratio R = T? =T k ;

117 3.3.3 Searching for the Mirror Instability at High kp In the interval of kp with sucient data to perform the statistical study of the upper limit to the proton temperature anisotropy the cyclotron instability always dominated. In this section we will present evidence of the mirror instability in the solar wind using a dierent technique. Since there is insucient dataat large values of kp where the mirror instability is predicted to produce the stronger limit, we will instead identify a short period with large kp where the proton VDF is right up against the instability thresholds, and examine the spectrum of concurrent magnetic uctuations. As the anisotropy of the proton VDF rises above the threshold for instability, wewould expect to see a growing electromagnetic uctuation. Since these instabilities are all electromagnetic in nature measurements of the magnetic eld will suce. We know from the linear theory and hybrid simulations that! r c p for the cyclotron instability and that 0! r < c p for the mirror instability. The highest time resolution of the vector magnetic eld measurements taken by the Wind/MFI experiment on the Wind spacecraft is 40 ms, approximately ten times per proton gyroperiod and sucient for this investigation. A search was performed over the entire mission for an interval of observations in the solar wind which met the following conditions: the Wind spacecraft had to be far from the Earth to ensure that there was no contamination from the bow shock or the magnetosheath the errors in the derived anisotropies had to be low the helium abundance had to be at or below typical values the interval had to be relatively quiet, with no large disruptions and kp hadtogrowinsuchaway that the proton anisotropy was brought up against a decreasing anisotropy threshold. This nal requirement is to ensure that R p is close enough to R p c to produce magnetic uctuations. Figure 3-8 is a plot of proton anisotropy and magnetic uctuations over a 2.5 hour interval on April 30, This is the same day which was selected for the example in the introduction, and a complete survey of the solar wind conditions for the entire day is shown in Figure 3-3. The upper panel of Figure 3-8 is a plot of R p ;1 as a function of time. The dashed line at zero corresponds to proton spectra with isotropic velocity distribution functions. The connected diamonds are the measured proton anisotropies as a function of time. For the rst and nal half hour intervals of this period the protons were isotropic. The blue curve is the upper bound to the anisotropy due to the cyclotron instability, and the green curve is the limit due to the mirror instability. From UT the measured R p is right at the threshold values. This occurs again at 0700 UT, but this time kp ' 30 (See Figure 3-3) and the mirror threshold is lower then the cyclotron limit. The middle plot in Figure 3-8 is a dynamic power spectrum of the 40 ms magnetic eld measurements as a function of time from 10 ;3 Hz to 0:3 Hz. Color represents the power of the observed uctuations in nt 2 /Hz. The red curve is the calculated value of C p based on the MFI eld and the FC proton number density. The nal plot in Figure 3-8 is the measured power as a function of time in the interval marked in the power spectrum by the dashed line corresponding to a frequency! c p =10. Power is seen near c p for the entire interval from 0530 to 0730 when R p was 117

118 Figure 3-8: Identication of signatures of the mirror and cyclotron instabilities by examining a dynamic power density spectrum of the high-resolution MFI magnetic eld measurements. 118

119 greater than one and near the cyclotron limit. In the 30 minute interval starting at 0630 when the mirror threshold was below that of the cyclotron mode there is clearly enhanced power in the uctuations at frequencies below c p. The factor of 200 increase in the power at low frequencies in this interval is direct evidence of the uctuations due to the onset of the mirror instability. This is the rst direct demonstration in the solar wind of the production of magnetic uctuations as a particle VDF exceeds a threshold for the onset of a kinetic instability [Kasper et al., 2002c]. In this particular case we have additionally shown the onset of a second unstable mode in a subset of the interval. The frequency properties of the uctuations are in agreement with the linear predictions. Further work with kinetic instabilities should attempt to relate the amplitude of the magnetic uctuations to the growth rate determined from numerical simulations or advances analytical techniques. 3.4 T k > T? and the Firehose Instability Wind Observations of Solar Wind with T k > T? Over the course of the mission 1:2 million spectra taken in the solar wind were selected for this analysis. The average value of 2 per degree of freedom for these measurements is 1:12 0:84, indicating that on average the core distributions of protons in the solar wind are well-described by two-temperature bi-maxwellian distributions. In the case of proton double streaming, a frequent occurrence in the solar wind, only the properties of the core are used for this study. We feel this is justied because an independent study of SWE/FC observations of proton double streaming [D. Clack, private communication, 2002] shows that double streaming occurs rarely in the sub-set of data ( kp 1, T kp =T?p > 1) used to probe the rehose instability. The average percent uncertainties in the thermal speeds are w? =(4:3 6:8)% and wk =(7:1 4:1)%. Data were selected for the nal analysis by requiring Rp < 0:2 and 2 per degree of freedom (d.o.f.) < 5. A detailed examination of spectra with large values of 2 /d.o.f. indicates that the existence of high energy tails or double streaming of the protons leads to a poor model t. We discarded those measurements since the theoretical calculations of the rehose limit assume the distribution is bi- Maxwellian. Overall 32% or 0:6 million measurements met these cuts and had values of R p < 1 (T kp >T?p ). The analysis of the rehose limit mirrors the cyclotron threshold analysis in Section 3.3. A two-dimensional histogram N( kp R p ) of the distribution of observed spectra as a function of kp and R p was created, only this time it was restricted to values of R p < 1. The left panel of Figure 3-9 displays the distribution of N(kp R p ), the normalized version of N( kp R p ), as a function of R p, with a dierent color line for each value of kp in the original histogram. In order to separate the lines N(kp R p ) has been multiplied by kp as in Figure 3-6, but additionally R p has been shifted down by a factor of 10% for each curve. In general N(kp R p ) is roughly constant for the rst 5-10 measurements clustered near R p 1. The fall o with decreasing R p 119

120 Figure 3-9: Quantifying the lower bound to R p imposed by the rehose instability. Left: N(kp R p )asafunctionofr p for each value of kp. Right: Values of the two free parameters in the ts of (3.30) to the observed fall o. 120

121 was best described by apower law, ln N( kp R p )=A + B ln R p : (3.30) The two panels on the left side of Figure 3-9 are the best-t values of the free parameters A and B from (3.30) as a function of the kp of each interval in N(kp R p ). Following the same procedure as in Section 3.3, (3.30) is inverted to determine the critical anisotropy R p c as a function of the t parameters, R p c = e B;1 [ln f;a] (3.31) where again we set f to 10%. The uncertainty in R p c is determined by propagating the uncertainties in A and B from each t. In Figure 2 we have created a two-dimensional histogram of the number of spectra as a function of R and kp. Figure 2 clearly demonstrates that the proton temperature anisotropy is more strongly constrained as kp increases, and that this type of constraint corresponds only to kp 1. These qualities are both hallmarks of Equation (3) and represent strong evidence that this constraint is due to the proton rehose instability Quantifying S p and p and Comparing with Theory The calculated values of R p c are shown as a function of kp for f=10% in the top panel of Figure The data clearly demonstrate that the proton temperature anisotropy is more strongly constrained as kp increases, and that this type of constraint corresponds only to kp 1. These qualities are both hallmarks of (3.23) and represent strong evidence that this constraint is due to the proton rehose instability. Unlike the t for the cyclotron instability, there is a degeneracy in the values of S p and p for the best t to R p c. Since the equation used to model the fall o of the observed spectra in this case was a power law, S p and p are strong functions of our choice of f. The center panel of Figure 3-10 illustrates this, with diamonds for S p and triangles for p from the best t of (3.23) to R p c as f was varied from 0.5% to 30%. Over this range in f the variation of R p c with kp satised (3.23), so there is no preferred value of f to use for these calculations. We have come up with a method to select a set of values for S p and p. The nal panel of Figure 3-10 is a plot of S p vs p. The results of the linear calculations, with growth rates ranging from 10 ;3 c p m 10 ;1 c p, are shown as the three symbols connected by the dashed line. The values of S p and p determined by varying f and tting R p c are shown as the connected crosses. The non-resonant MHD calculation is the diamond in the upper right corner of the plot. We claim that the best values of S p and p to describe the rehose instability in the solar wind lie at the intersection of the solid and dashed lines, at S p ' 0:8, p ' 0:7, and that the eective growth rate of the instability is m 0:05 c p. This t to the observed constraint on the proton temperature anisotropy is in very good agreement with both the linear theory of the resonant rehose instability 121

122 Figure 3-10: Determining the best values of S p and p to describe the observed limit to the proton temperature anisotropy. Top: Best t of (3.23) to the calculated values of R p c as a function of kp. Middle: Variation of S p (diamonds) and p (triangles) as a function of limiting fraction f. Bottom: Scatter plots of the values of S p and p extracted from the data as a function of f (crosses), the results from the linear analysis (connected symbols), and the MHD calculation of the rehose instability threshold (diamond). 122

123 and the 1-D simulation results of Gary et al. [1998], which also yielded p ' 0:74 on the range 2 k p 10. There is less good agreement with the linear theory of the nonresonant rehose instability andthe 2-D simulations of Gary et al. [1998] which indicate p = 1.0. We conclude that, over 2 kp 10, our observations are consistent with the imposition of an upper bound on the proton temperature anisotropy by wave-particle scattering due to enhanced uctuations from the proton rehose instability. Furthermore, our observations indicate that it is the protonresonant rehose which is the more likely source of the uctuations which lead to this constraint. The rehose instability based on a bi-maxwellian proton velocity distribution is essentially stable at kp 1, and cannot be responsible for the apparent decrease in the average anisotropy as kp decreases from unity. It may be that this feature corresponds to the proton-proton Alfven instability which has alower eective T k at threshold as p decreases (e.g., see Figure 1 of Montgomery et al. [1976]). We have used a comprehensive plasma and magnetic eld dataset from Wind to show for the rst time that the theoretical threshold of the rehose instability provides a statistical upper bound on T kp =T?p values observed at kp 2 in the solar wind. The observed limit in R for kp < 1 merits further study, possibly by using a model proton velocity distribution which allows departures from a bi-maxwellian. This result was rst presented in Kasper et al. [2001b], and the detailed t to the rehose data was reported in Kasper et al. [2002b]. Recently we have begun astudy of specic intervals in the solar wind in which conditions drive the protons into the rehose limit [Kasper et al., 2002c]. A more percise identication of R p c will be possible when electromagnetic uctuations corresponding to the onset of the rehose instability are identied as a function of R p and kp. 3.5 Discussion and Summary Using the great statistics and accuracy of individual measurements of the proton temperature anisotropy with the Wind Faraday Cups we have studied the kinetic properties of temperature anisotropy driven micro-instabilities in the solar wind. Measurements of the threshold values of the proton anisotropy are suciently accurate to distinguish between the predictions of MHD uid, linear Vlasov, and numerical simulations. This allows us to identify new phenomena and rene calculations of established instabilities. As a result of the work with R p < 1 measurements several groups have begun calculations to explain the new limit to R p < 1 at kp < 1. The kinetic state of protons in the solar wind is summarized in Figure 3-11, which is a survey of the range in proton temperature anisotropy allowed as a function of kp using all solar wind measurements. The two halves of the gure with R p > 1 and R p < 1 have been normalized separately to highlight the dierent physics involved in driving the proton VDF away from isotropy. The upper curves are the predicted limits due to the mirror and cyclotron instabilities, and the lower curve corresponds to the bound provided by the rehose instability. In the regime R p < 1 there is a clear competition between the expansion of the 123

124 Figure 3-11: Survey of proton temperature anisotropies normalized to demonstrate eects of kinetic plasma micro-instabilities. For each column of kp the two intervals R p < 1 and R p 1 are separately normalized to unity. Features include the bounds imposed by the mirror, cyclotron, and rehose modes the eect of adiabatic expansion driving T?p =T kp < 1 an unknown mechanism in the range kp < 1 R p < 1 constraining R p to isotropy. See text for details. 124

125 solar wind driving T kp >T?P and the bound imposed by the rehose instability. As the solar wind continues to expand past 1 AU into the heliosphere Coulomb interactions become more infrequent and this phenomena should continue. It is possible that this competition is a source of wave activity in the outer heliosphere. This study of the three anisotropy driven instabilities has demonstrated that the Wind Faraday Cup observations are very useful in the characterization of plasma micro-instabilities. 125

126 126

127 Chapter 4 Extension of Analysis to Helium ABSTRACT: The bi-maxwellian analysis techniques developed in Chapter 2 to characterize Faraday Cup observations of protons are extended to alpha particles. The success of this procedure is reported, and the results are used to compare Faraday Cup densities with WAVES/TNR observations of the electron plasma frequency. 4.1 Ion Composition of the Solar Wind Protons are not the only constituent of the ion component of the solar wind. Fully ionized helium or alpha particles generally account for 1% ; 5%, and sometimes as much as 10%, of the ions bynumber density. Minor ions such asoxygen and iron make up an additional 1%. The properties of these additional ion species contain a great deal of information which is useful for a variety of applications. Broadly speaking the details of the additional ion species are interesting because they eect the dynamics of the interplanetary medium, reect the origin and acceleration mechanisms of the solar wind, and exhibit the eects of a range of kinetic plasma micro-instabilities. The FaradayCup(FC) ion instruments on the Wind spacecraft were introduced in Chapter 2, along with the derivation and application of the convected, eld-aligned, bi-maxwellian velocity distribution function (VDF). As a result of that analysis a best-t to each ion spectrum yielded a set of proton parameters ~ U p T?p T kp n p, their uncertainties, and the 2 = parameter. This chapter describes the extension of that analysis to produce, when possible, the same set of bulk parameters for the alpha particles, ~ U T? T k n. Scientic applications of the alpha measurements are considered in the following chapter. The additional procedures created to analyze the helium in addition to the protons are described in Section 4.2. Section 4.3 details the results of this analysis, including the success rates. In Section 4.4 the proton and alpha number densities are used in conjunction with the WAVES experiment on Wind to conduct a study of the absolute calibration of the Wind Faraday Cups. In order to continue treating the solar wind as a uid in the presence of multiple ion species a single uid description must be developed, and this is done in Section 4.5. A brief summary following in Section

128 4.2 Extension of the Existing Analysis Code The analysis code was extended to include tting of the alpha measurements using the methods developed by Aellig as a starting point [Aellig et al., 2001b]. Since the alphas often partially overlap with the protons it was decided that the moment analysis was too susceptible to contamination by hydrogen, so only the non-linear analysis is applied to the helium component of the measurements. No additional work is needed to produce the alpha particle response function. The derivation of the convected bi-maxwellian response in Section is used for the alphas, just with the substitution of with charge 2q and mass 4m p. The additional program starts after the existing code has successfully produced a converging t to the proton VDF. If the proton t was successful then the spectrum is assigned a t status of 11, if it failed then the t status is 0 and the code proceeds with the next spectrum. An initial guess is made that the alpha particles have the same parameters as the protons, except with n =n p = 5%, which isatypical value in the solar wind. An extension of the data selection algorithm described in Section then identies measurements in the spectrum which are expected to correspond to either protons or alphas. At least 10 points from each cup corresponding to alphas must be selected for the analysis to proceed. The selected data, typically measurements, are then subjected to a simultaneous t with twelve free parameters, ~U p w?p w kp n p U ~ w? w k and n. An single isotropic thermal speed for each species is then calculated, wj 2 =(2w?j 2 + w 2 kj )=3. In Section 5.3 it will be shown that in general T = T p or T = 4T?. This is because ion species in the solar wind are generally either in thermal equilibrium, in which case w = w p =2, or have equal thermal speeds. If w =w p 0:75 then the spectrum is assigned atstatus of 8. No further action is taken but the spectrum is agged for future investigation as a period where the alphas are cold. If w =w p 2:2 then there is the possibility that the alpha and proton distributions overlap and as a result the alpha temperature is too high. In this case t status is 9 and the analysis will re-perform the alpha t after subtracting the best t to the proton distribution. If U =U p 1:1 then the spectrum is assigned a t status of 7. No further action is taken but the measurements might be useful in studies of intervals with fast alpha particles. If U =U p 0:97 or n =n p 20% then there is the danger that a double streaming proton distribution has been accidently t as the alpha distribution. In this case the proton t is subtracted and the alpha distribution is t to the remaining data. If after subtraction of the proton data the alpha parameters seem reasonable then the spectrum is assigned a t status of 5. If after subtraction the parameters are still suspicious then the nal t status is 4 if the alphas are too cold, 3 if they seem too hot, and 2 if they appear too slow. If there are multiple problems with a spectrum it is assigned atstatus of zero. The nal results are the same as described in Section 2.4, just with the addition of the alpha parameters and the t status. For each spectrum the best-t parameters, ~U p w?p w kp n p U ~ w? w k and n, estimates of their one-sigma uncertainties, the number of measurements selected in each cup, the nal value of 2 =, and an array 128

129 All Solar wind Fast SW Slow SW MSH Fit Status V p > 400 V p < Total spectra 2,351,008 2,257,338 1,135,940 1,120,949 57,344 Table 4.1: The success rate of the alpha-proton analysis procedure in various regions in space as a function of the nal status of the t. of diagnostic parameters are saved to le. 4.3 Results of the Analysis As of summer 2002 the bi-maxwellian alpha-proton analysis procedure has been run on more than 2.3 million Faraday Cup spectra. The success of the analysis is illuminated by the statistics in Table 4.1, which break down the percentage of ion spectra as a function of nal t status. The percentage distribution is broken down into several regimes in the solar wind, including all observations, all solar wind observations, and intervals in the fast solar wind (U p 400 km/s), slow solar wind (U p 400 km/s), and magnetosheath intervals. In general there was insucient data to characterize the alpha distribution 1:9% of the time, based on the number of spectra with a t status of 11. This was a much larger problem in the magnetosheath, where it occurred 7% of the time. The greatest obstacle to characterizing the alpha particles with a Faraday Cup spectrum is when the proton temperature is especially high, and this is a frequent occurrence in the magnetosheath. A perfect t status of 10 occurred 58% of the time in the slow solar wind, and 40% of the time in the fast wind. This can be attributed to three factors. First, as will be discussed in the following chapter, more collisions have occurred in slow solar wind as it propagates from the corona, due to the higher densities and slower travel speeds. As a result the alpha and proton velocities and temperatures tend to be in better agreement as the wind has settled into a more classical denition of equilibrium. Additionally, double streaming protons are a very common feature of the 129

130 Parameter Units Mean Median Stdev Mean % Median % ju j km/s U x km/s U y km/s U z km/s EW degree N/A N/A NS degree N/A N/A w? km/s w k km/s n cm ; ; ; Table 4.2: Uncertainties in each of the derived alpha parameters from the best-t to individual solar wind spectra. Compare with same table for proton parameters in Table 2.2. high speed solar wind [Clack et al., 2002]. Finally, at high speeds the alpha particle VDF is sometimes not completely observed by the Faraday Cups in their standard speed windows. As a result the ts to the alpha particles at very high speeds do not succeed as often. 4.4 Absolute Calibration of Number Densities The analysis of Wind Faraday Cup ion spectra in Chapters 2 and 4 have established the accuracy with which we can measure proton and alpha particle number densities in the solar wind. The survey in Section demonstrated how the one-sigma estimates of the uncertainties from the non-linear ts yielded averages values of np =n p 0:5% and n =n 1% over a broad range of solar wind conditions. This is the accuracy with which the convected bi-maxwellian response function derived in Section 2.2 and applied to the observations can determine the number densities. In this section we will examine the the precision, or absolute uncertainty, of these measurements. Traditional tests of the precision of bulk solar wind parameters involve the comparison of observations by dierent instruments on the same spacecraft, of observations from multiple spacecraft. Of course this comparison introduces additional uncertainties. Each instrument has its own sensitivities, and each investigator has a preferred analysis methodology which can result in systematic discrepancies. Recall the comparison in Section of proton thermal speeds using data from the same Faraday Cup but with the application of the moment and non-linear techniques. Additionally factors such as diering measurement cadences must also be taken into account. Multi-spacecraft comparisons suer from the spacecraft separation and the intrinsic temporal and spatial variation of the solar wind. It has been known for some time that there are correlation lengthscales in the interplanetary medium which are in general are also of the order of typical spacecraft separations [Jurac and Richardson, 2001]. 130

131 Additionally, recent studies have shown that the boundaries between structures in the solar wind, while often planar on the length-scale of spacecraft separations, are in general not radial [Weimer et al., 2002]. Thus it is not a simple procedure to propagate observations from one spacecraft to another. This study makes uses another instrument on the same spacecraft which measures the total electron number density in a completely independent manor. The Thermal Noise Receiver (TNR) instrument in the WAVES experiment measures the power of electromagnetic uctuations at a very high cadence of seconds in the frequency range from 4;300 KHz [Bougeret et al., 1995]. It was shown in Section that the solar wind plasma uctuates at the electron plasma frequency! p e, a quantity which is only a function of the total electron number density (1.92) and which varies from 50;300 KHz in typical solar wind conditions. Generally a single TNR power spectrum may be used to determine the electron number density using a neural network which identies the peak of the plasma frequency line. The goal of this section therefore is to compare a predicted value for the electron number density n e based on the observed proton n p and alpha n number densities and compare with the TNR measurements. Since hydrogen and helium are fully ionized in the solar wind the total electron number density due to protons and alphas is n p +2n. In order to do this correctly however we need to take into account the contribution of other ion species in the solar wind. Recall that 1% of the solar wind by number density is composed of minor ions like oxygen and iron. While these minor ions are rare the typical charge states observed are +7 O and +10 Fe, so a small abundance can result in a signicant contribution to n e. As was shown in Section 1.2.7, on the time scale of a FC measurement of the ion VDF there are no electric elds other than that due to the convection of the plasma and the solar wind is neutral. The requirement ofcharge neutrality maybewritten, jq e jn e = X j q j n j (4.1) where q j is the average charge state of the j th species which has a number density n j. If n e is the total electron number density measured by TNR, and all of the measurements were exact, then the fraction F m of n e due to minor ions is, F m 1 n e (n e ; n p ; 2n )=1; n p n e ; 2 n n e : (4.2) In reality each of the measurements which go into (4.2) have an uncertainty which we will label p,, and e. The propagated uncertainty in F m, or m, is then, 2 m =(n p +2n ) 2 2 e n 2 e + 2 p n 2 e + 2 n 2 : (4.3) e In this study we will look at the average value of F m and its variation F m as a function of the proton speed and the year of the mission. This is because many properties of the minor ions have been demonstrated to be speed and solar cycle 131

132 Component Relative Electrons Similar Total Abundance Species Electrons Hydrogen Helium Oxygen Iron 0:1 10 0:1 6 7 Table 4.3: Estimating the contribution of minor ion species to the total electron content. With the numbers listed here F m ' 3:1%. dependent. If the absolute calibrations of both the FC and TNR instruments are correct then F m should agree with the results of other studies. Additionally, if all of the instrumental eects are correctly captured in p,, and e, then the quantity (F 2 m ; 2 e) 1=2 is indicative of the natural variation of the minor ion component. It is a convolution of the variation in minor ion abundance and charge state distribution. A comparison of SWE/FC and WAVES/TNR observations, limited to six days in 1996, was reported previously [Maksimovic et al., 1998, Figure 2]. In that study there was a general disagreement on the order of F m 1% and a spread of approximately 5%. There are several compelling reasons to pursue this comparison in more detail. Signicantly, the analysis of the TNR data have subsequently been rened due to a better understanding of the antennae and the TNR frequency bins. This has increased the observed values of F m, but we hope to explain this by factoring in the eect of the minor ions. In addition, six days is insucient to probe possible dependencies on solar wind conditions, such as the bulk speed. Finally, with our new knowledge of p and from the non-linear analysis we can separate the contributions of the protons, alphas, and minorionstotheobserved 5% spread in F m. In preparation for this study we can estimate F m. Table 4.3 lists values which are used in the following simple calculation [T. Zurbuchen, private communication]. The second column lists the relative abundances of each element, normalized to oxygen. Assuming that the hydrogen and helium are fully ionized, that oxygen and 3 more species like itareina+7charge state, and that iron and 6 species like it are in charge state +10, then F m ' 3:1%: (4.4) Of course this is an approximation: There are other ions, and the charge states and relative abundances are highly variable. Predictions of F m 5% were also discussed, and so the expected range in F m is between 3% ; 5% Preparation of Measurements This study uses all solar wind observations in the interval Since the end goal is to produce as clear and condent avalue for F m as possible, any measurements 132

133 which seem at all suspicious are discarded. For the alpha particle analysis only spectra with a perfect nal t status of 10 (Section 4.2) are selected. An infrequent issue with the high voltage power supply on FC1, the southward facing cup on Wind had to be addressed. On several occasions, especially early in the mission and during a 10-day interval in 1996, an error with the selection of energy windows for FC1 lead to incorrect density measurements. The standard key-parameter analysis code is designed to switch to a single-cup mode in the event of poor data quality from a single cup. Since the bi-maxwellian analysis requires observations at as many angles relative to the ambient magnetic eld as possible, it was decided that we would not conduct an anisotropy analysis during those intervals. To identify questionable spectra we ran the moment analysis code separately on the observations from each cup individually for each spectra and then compared the two determinations of the proton number density from each cup, n p1 and n p2. The top panel of Figure 4-1 is a series of histograms of the density ratio n p1 =n p2 for each year from The core of each histogram is a Gaussian with an average width of 2% and centers ranged between 99:1% and 100:2%. Statistically then the proton number densities in general determined independently by thetwo Faraday Cups using the moment analysis technique dier by 2%. The 0:5% range in the centers suggests that there is no bias in densities between the two cups. The result of the extended anomalous period in 1996 is clear in the top middle histogram in the form of the bulge on the n p1 =n p2 side. To be safe we will discard all measurements which do not satisfy 0:9 n p1 =n p2 1:1, or 0:18% of the total dataset. The typical duration of a single SWE/FC ion spectrum is 92 seconds. The WAVES/TNR neural network data which were made available for this study are produced at a rate of at least one every 6 seconds. For the purposes of this study the TNR data were matched up to each FC spectrum and the average n e, median, standard deviation n e, and number of TNR measurements N were recorded. The quoted uncertainty in an individual TNR neural network electron density measurement is 2%;3%. Since there were generally at least 10 TNR measurements per FC spectrum, the uncertainty in the mean of the TNR measurements of n e, or e =n e = p N ; 1, is used for the uncertainty inn e. Only FC spectra with N 10 were selected for this study. The bottom panel in Figure 4-1 is a histogram of the distribution the uncertainties with which each component of (4.3) contributes to the overall uncertainty in F m. In addition, the distribution of F m itself is also plotted. It is clear that the uncertainty in n does not strongly eect the calculation of F m. The uncertainty in the proton number density is the next most signicant factor, and the value for n e measured by TNR has the largest average uncertainty. The average values of the uncertainties are p = 1:0%, =0:004%, e =3:1%, and m =3:5%, which should be compared to the average value for F m of 6%. For values of F m 20 the uncertainty in F m is less than the average value of F m itself, as is suggested by this gure and veried by plotting the ratio of m =F m for F m 6= 0. It is telling that the portion of the histogram with values of F m 30% overlaps exactly with the upper tail of the histogram of the total uncertainty in m (and the distribution of e, which dominates m in this interval). The implication, 133

134 Figure 4-1: Predicted fraction F m of the total electron number density due to minor ions as a function of proton bulk speed and year. 134

135 which was veried through a scatter plot to associate the points, is that the large departures of F m from zero, which are rarely seen, are due to periods in the solar wind when the variation of the TNR electron number density was very large. The two in fact correspond exactly. The consequences of this are that very large values of F m may be attributed to intervals with large uctuations and a large resulting statistical uncertainty in F m, and not due to periods when one instrument or the other is not functioning ideally. The other conclusion is that the variation of F m is dominated by the uctuation of the electron measurements from TNR, which are on the order of 3:1%. A spread in F m on the order of 3:5% therefore would only reect the variation of the electron number density measurements, and not any intrinsic error in the proton, alpha, or electron number densities. We are not interested in comparing the instruments over the entire range of interplanetary conditions. Since the focus is on quite periods where the values measured are more trustworthy the study is further restricted to spectra with p 5%, 1%, and e 10%. A total of spectra, or 78% of the total dataset passed all of the cuts described in this section and were used for the study of F m Survey Over Mission The median value of F m over the entire selected dataset was 5:86%, with an average value of 6:36% and a standard deviation of 5:1%. The standard deviation is in agreement with the observations of [Maksimovic et al., 1998], although F m is 4% larger than was reported in that same study. This can be attributed in part to the rened analysis of the TNR data since that study and to the fact that on average the proton number densities derived from the bi-maxwellian analysis are about 1% smaller than the key-parameter number densities which were used in that study. We believe that the 1% shift in the FC number denies is due to the use of new eective area and response functions, as derived in Chapter 2. The average value of F m is slightly larger than the median value because there is an excess of observations on the F m 5% side. In fact the distribution of the entire set of measurements does not appear to be exactly Maxwellian. Weinterpreted this as evidence that the overall histogram reected the variation of F m over a set of variables. Two factors which we believed could inuence F m are the bulk speed of the solar wind and the period in the solar cycle. Studies of minor ion charge states have demonstrated that statistically speaking the freeze-in temperature of the minor ions is cooler at higher solar wind speeds, and therefore less ionized. Additionally, the abundance of species other than protons reects in part evolution of the corona material at the origin of the solar wind. Studies have shown that the helium abundance is an increasing function of speed. If the same process occurred with the other minor ions then it might be expected that the minor ion abundance at high speeds would be larger. The actual eect of these two competing processes on the F m as a function of U p is not clear, but we must take that possibility into account. Additionally the helium abundance has been demonstrated to vary over the course of the solar cycle. Therefore we decided to look at the distribution of F m as a function of year from and bulk proton speed. 135

136 Figure 4-2: Histograms of F m in six speed windows using selected data in Black histogram is the number of spectra observed per 0:5% bin in F m, blue line is portion of the histogram selected for t with Gaussian, and red is the best-t Gaussian distribution. Note that the width increases with speed and that observations with F m < 0arein agreement with tail of Gaussian. 136

137 For each year the selected observations were divided into ten intervals in speed ranging from 300 to 650 km/s. The rst seven windows are25km/swide. Since the number of measurements decreases with speed, the nal three windows are 50 km/s wide to improve the statistics of the high speed intervals. The ten histograms of F m as a function of speed window for 1995 are shown in Figure 4-3. In each histogram all intervals with more than 10 spectra were selected and tted with a Gaussian probability distribution. The blue curves indicate the bins which were selected for tting and the dashed red lines are the best-t Gaussians. Overall the Gaussians t the observed distributions well. Several trends are worthy of comment. First note that in most cases the portion of each histogram with values of F m < 0, an unphysical result, may be accounted for by the natural width F m of the distributions. In other words if the F m in part represent the overall uncertainty of F m then it is natural for this amount of observations to have F m < 0. Of course F m is additionally due to the natural variation of the contribution to n e from minor ions, and that variation can be estimated. Note as well that the F m increase as a function of speed. Additionally, there appears to be a bias, or tail in many of the distributions at high values of F m, suggesting that there are some intervals where F m is enhanced beyond the simple Gaussian variation observed. Helium enhancements are often observed in the material associated with coronal mass ejections, for example, and the same rare eect may account for this tail, which is still small compared with the overall distribution. The determination of the average value of F m as a function of speed and year, and the estimate of the natural variation of the minor ion component, are detailed in Figure 4-3. In the upper panel colored solid lines indicate the width F m of the best-t Gaussian distributions to the observations as a function of speed for each year. The colored years printed at the top of the plot correspond to each of these curves. In general F m increased linearly from about 1:5% to 3% as the speed ranged from km/s. There is a suggestion of a time dependent increase in the width of the distributions in the speed range km/s with solar cycle. The connected diamonds plotted in this panel are the average uncertainties of the proton, electron, and alpha particle contributions to n e in each speed interval. It is clear that the total uncertainty m in F m derived from these individual uncertainties through (4.3) is dominated by thevariation in the TNR electron measurements over each FC spectrum, and that m is a decreasing function of U p. According the the discussion in the previous section, we interpret (Fm 2 ; m) 2 1=2 as the natural variation of F m due to the variation of the abundance and charge state of the minor ions in the solar wind. In the nal panel of Figure 4-3 the center of the F m distribution is plotted as a function of speed for each of the years from Measurements in the same speed window have been oset by several km/s to avoid confusion. The horizontal error bars are the widths of the bins and the vertical error bars are the natural variations of F m derived from the date in the upper panel. The two dashes lines indicate the two estimates of F m in the introduction. There are variations in F m with speed which should be compared with more detailed predictions. Additionally there appears to be a two-state distribution in the interval U p 425 km/s, with 137

138 Figure 4-3: Predicted fraction F m of the total electron number density due to minor ions as a function of proton bulk speed and year. Upper panel: Solid lines are the widths of the Gaussian ts to the distribution of F m in each speed/year interval, connected diamonds are the corresponding average uncertainties in F p, F, and F e. Lower panel: Center of F m distributions and natural widths as a function of speed and year. Dashed lines indicate estimates of the expected contribution of minor ions to the total electron number density. 138

139 all observations in and in falling into two distinct intervals. This could be a signature of the type of speed and solar cycle modulation which has been seen in the past with helium observations. For U p 500 km/s F m is consistently greater then the predicted values. This could be a reection of the enhanced abundance of minor ions in high speed solar wind. But it might also reect the fact that high speed solar wind often contains two proton distributions, and that this second proton distribution is not reected in this analysis. A followup to this investigation should include the double streaming analysis currently being conducted by Dorian Clack. In summary, the estimates of F m, the TNR electron measurements, and the SWE/FC proton and alpha number densities agree to within a few percent. The widths of the F m distributions are consistent with the derived uncertainties in the proton and alpha number densities being less that 1%. Furthermore the agreement between F m and theory is consistent with the Wind Faraday Cupshaving correct and stable density calibrations within approximately 2% of the absolute values. 4.5 A Single Eective Ion Species If a single uid description of the solar wind is desired then it is necessary to merge the proton and alpha data to create an eective ion species. In Chapter 6 this single eective ion species will be used in the characterization of shocks through conservation of the MHD equations. The value of a single species for a treatment of kinetic processes is questionable, as this aspect is explored in Section 5.4. This section presents a derivation of the single uid properties based on the method of reducing ion and electron data for the single uid MHD problem outlined in Section A valuable asset of the proton and alpha parameters derived from the Faraday Cup data are the one-sigma uncertainties which are produced during the non-linear ts. The goal of this section is therefore to derive the single ion parameters but in addition to propagate the uncertainties from the alpha and proton analysis. The reduced ion species will have the same mass as the protons, so the number density n i is just the total of the number of protons and neutrons, n i n p +4n (4.5) with an accompanying uncertainty ni given by, 2 n i = 2 n p n : (4.6) For the bulk velocity use the velocity ~ U i of the center of mass of the two ion species by weighting the individual proton ~ U p and alpha ~ U velocities by their mass densities, ~U i n p ~ U p +4n ~ U n p +4n : (4.7) 139

140 The uncertainty in ~ U i is a function of both species number densities and velocities. The uncertainty of the j th component of the velocity, U i j, is given by, 2 1 U i j = (n p +4n ) 4 (256n4 2 U j +128n 3 n p 2 U j +8n n 3 p 2 U pj + n 2 p(n 2 p 2 U pj n (U j ; U pj ) 2 )+16n 2 (n 2 p( 2 U j + 2 U pj )+ 2 n p (U j ; U pj ) 2 )) (4.8) To determine the average thermal speed of the ions, w i, I started with the individual pressures of the individual ion species, P p = n p k B T p = 1 2 n pm p w 2 p P = n k B T = 1 2 n m w 2 (4.9) combined them and then set them equal to the pressure which would be produced by the single ion, P i = P tot = P p + P = 1 2 m p(n p w 2 p + n m w 2 ) = 1 2 m p(n p w 2 p +4n w 2 ) (4.10) = 1 2 m iw 2 i again using the fact that that the ion has mass m p we can solve for the thermal speed, and its associated uncertainty, 2 w i = w 2 i n pw 2 p +4n w 2 n p +4n (4.11) 16n 2 (4n + n p ) 2 2 w w 2 +4(n 2 p 2 n + n 2 2 n p )w 4 +(n 2 p(4n + n p ) 2 2 w p ; 8(n 2 p 2 n + n 2 2 n p )w 2 )w 2 p +4(n 2 p 2 n + n 2 2 n p )w 4 p (4n + n p ) 3 (4n w 2 + n p w 2 p) ;1 : (4.12) The dominant eect of the helium is to increase the particle mass density. The largest eects now neglected are the mass density of minor ions in the solar wind and the temperature of the electrons, which may contribute to the overall pressure. Since electrons have approximately the same temperatures as the ions in the solar wind, they generally have bulk velocities much less than their thermal widths and as a result do not contribute to the eective velocity, but can at times dominate the total pressure. 140

141 4.6 Summary The analysis of the SWE/FC ion spectra has been extended to include the determination when possible of alpha parameters using the techniques developed by [Aellig et al., 2001b], but with a bi-maxwellian distribution VDF for the alpha particles as well as the protons. It has been shown that the the absolute precision of the FC number densities is o by less than a couple percent from the true values, and that the uncertainty in the densities is indeed on the order of a single percent. In the following chapters the relative kinetic properties of protons and alphas in the solar wind are compared and their limits are explored. The usefulness of the single ion species derived in the previous section will be examined in terms of collisionless shocks and kinetic instabilities. 141

142 142

143 Chapter 5 Kinetic Properties of Helium in the Solar Wind ABSTRACT: The kinetic properties of Hydrogen and Helium in the solar wind are compared. The relative abundance of alphas in the solar wind is used to constrain models of magnetic ux expansion in the solar corona, and a new modulation signal is discovered. The alpha-proton data are used to study multi-species kinetic processes in the IPM, such as equilibration of thermal speeds and dierential streaming. The limits to the rehose and cyclotron instabilities of Chapter 3 are reconsidered. 5.1 Dynamics of Multiple Ion Species In Chapter 2 the Wind/SWE Faraday Cup data were analyzed to produce the bulk velocity ~ U p, parallel and perpendicular thermal speeds w kp and w?p, and number density n p of the proton velocity distribution function for each spectrum. In Chapter 4 this analysis was extended to produce, when possible, the same set of parameters for the alpha particles ( ~ U w k w? n ). In this chapter the kinetic properties of the fully ionized hydrogen and helium, protons and alphas, in the solar wind are compared. Perhaps one of the most fascinating microscopic properties of the solar wind is that ion species do not generally share the same kinetic properties [Marsch et al., 1982]. The origin of these dierences is in the way wave-particle interactions couple to species of diering charge and mass. The stability of these dierences is due to the tenuous nature of the solar wind, but we will identify several cases where microinstabilities act to provide limits to the dierences. Specically we will investigate the alpha particle temperature anisotropy R, R T? T k (5.1) the ratio of the alpha and proton temperatures, both T? =T?p and T k =T kp, and the 143

144 dierence ~ U p between the alpha and proton velocities, ~ U p ~ U ; ~ U p : (5.2) Our goal in studying these processes is to learn about the mechanisms behind generating the original dierences and to discover any kinetic processes, such as microinstabilities, which act to limit these parameters. However, the most clear eect in constraining these parameters is the process of Coulomb relaxation between the species [Klein et al., 1985]. As in the examination of intervals with large anisotropies in Section 3.3, the critical parameter is the ratio of the solar wind expansion time x (3.2) to the Coulomb collision timescale (1.52). For the purposes of this Chapter (1.52) is evaluated with the proton and alpha densities and temperatures. As a means of introduction we will now look at the variation of R, the ratio of scalar temperatures T =T p, and U ~ p as a function of x = c. This is illustrated in Figure 5-1. The top panel is a histogram of the distribution of all solar wind spectra as a function of the ratio x = c. As was shown in Figure 3-5 for protons, the solar wind evenly covers approximately three orders of magnitude in x = c. In this case approximately one third of the measurements are in the collisionless regime x = c < 0:1, one third appear to be collisionally dominated x = c > 1, and the remainder of the data lie in the transition region between the two states. It should be emphasized that for the purposes of a kinetic calculation the Vlasov equation (1.53) is always valid, because x is much larger than kinetic timescales such as the proton cyclotron frequency. The examples in this gure just serve to demonstrate that the dierences between the kinetic properties the dierent ion species appear to have been generated much closer to the Sun, in the inner heliosphere. If the collision rate is large enough then Coulomb relaxation \washes out" these features, and this must be taken into account when looking for limits, so that a collisionally dominated parcel of plasma is not mistaken for evidence of an instability. The second panel in Figure 5-1 illustrates the range in the alpha temperature anisotropy R as a function of x = c. It and the following panels were generated by normalizing each column in a two-dimensional histogram to unity. All plots are scaled to the color bar in the bottom panel. For x = c > 1, the average anisotropy of the alpha particles is near unity, as might be expected in a collisionally dominated plasma. In the collisionless regime however R ' 0:8, which is in agreement with the average proton temperature anisotropy shown in Figure 3-2. The range of alpha temperature anisotropy is discussed in Section 5.4 and compared with the proton results of Chapter 3. The third panel shows the ratio of the scalar alpha and proton temperatures, T =T?, as dened by (2.7). The species are in thermal equilibrium for x = c > 1, as would be expected. If the plasma has remained collisionless however the ratio of temperatures is slightly more than a factor of four. The temperatures of multiple ion species have been studied in the past [Isenberg, 1984], and the explanation of the factor of four is presented in Section 5.3. That section uses the alpha temperature anisotropies to conduct the rst study of the relationships between proton and alpha 144

145 Figure 5-1: Examples of kinetic features which arise in collisionless regime of solar wind. Top to bottom: Histogram of the distribution of observations as a function of x = c alpha particle temperature anisotropy alpha to proton temperature ratio alpha-proton dierential ow. In all cases these features are washed out by Coulomb relaxation. 145

146 particles separately along and perpendicular to the magnetic eld. The fourth and nal plot is of the distribution of the dierential speed U p,the magnitude of (5.2). Once again, in the collisionless plasma large dierential speeds are observed. In fact, for very low collision rates x = c < 3% the species never have the same speed. The range in U p is explored in Section Alpha-Proton Dierential Streaming From an MHD perspective, a dierence in the alpha U ~ and proton U ~ p velocities is stable in the solar wind as long as the dierence vector lies along the direction of the ambient magnetic eld. Consider the Lozentz force acting on an alpha particle moving at the average alpha velocity U ~. Since there are no steady electric elds in the rest frame of the plasma, the force acting on the alpha particle, F ~ = q U ~ p B ~, will vanish as long as U ~ p jj B ~. To test this prediction, and to verify the accuracy of the proton and helium velocities, we examined the distribution of the angle between U ~ p and B ~. In general U is always larger or equal to U p, while the magnetic eld at 1 AU points towards the Sun 50% of the time and away from the Sun the other 50%. Therefore under ideal circumstances we would expect the two vectors to be either parallel or antiparallel at all times. The top panel of Figure 5-2 is a histogram of the number of spectra as a function of the cosine of the angle between U ~ p and B ~. The black curve is the distribution of all solar wind observations. In that case the data are concentrated at 1, but there is a 10% background level at all angles between the two vectors. The blue histogram is the distribution of all measurements with speeds greater than 400 km/s, and this brings the background down to the level of a few percent. The lower speed and higher density slow solar wind generally corresponds to higher collision rates relative to x, so it is reasonable to believe that we have thrown out many spectra where U ~ p is small. If U ~ p is on the order of the uncertainty in the velocity components then we do not expect the angular resolution to be very good. Since that appears to be the cause of the background we generated the nal red histogram, which is of all spectra with U ~ p greater than 25 km/s. the background is now farbelow 0.5%, and we can conclude that as long as the dierential velocity is large enough the angular dierence between U ~ p and B ~ is small. Thus we believe we are making accurate measurements of the dierential velocity. A more detailed examination of the measurements suggests that on average speed dierences greater than 10 km/s are trustworthy. The lower panel of Figure 5-2 shows the distribution of these accurate measurements of U p, normalized by the local Alfven wave speed CA 2 = B 2=2 (See Section 6.1.1). The sign of the dot product has been retained to demonstrate that the same distributions are obtained when the alphas stream faster than the protons in the direction of and anti-parallel to the ambient magnetic eld. The black histogram is the entire dataset with signicant dierential speeds. The blue histogram is the distribution of all measurements in the collisionless regime with x = c < 0:1. The red and green histograms are further subdivisions of the collisionless data for R p < 1 and 146

147 Figure 5-2: Survey of alpha-proton dierential streaming. Upper panel: The distribution of the angle between the magnetic eld and the dierential velocity. Lower panel: Histograms of the range in U p =C A under various cuts (See text for details). 147

148 Figure 5-3: Verication that under set conditions the dierential ow is limited by the proton temperature anisotropy as predicted by theoretical work with cyclotron resonances. R p > 1 respectively. The lack of data at small values of U p =C A is due in part to the cut on small speeds. In each of these histograms the dierential speeds do not exceed the Alfven speed. We will now look for evidence that a kinetic instability is providing a bound to the range of U p =C A which is stably accessible to the solar wind. Earlier theoretical work on alpha-proton dierential streaming identied a magnetosonic instability as a possible upper bound to the the observed dierential ow [Gary et al., 2001]. In that work U p =C A was shown to be limited to be less than a factor three over a range of solar wind parameters. Unfortunately, at 1 AU, especially in the slow solar wind which much of the Wind observations cover, the range in U p =C A,even when collisionless intervals with x = c < 1 are examined, are found to lie within U p =C A 1, so a stronger limit is needed. A recent study has identied cyclotron resonances which couple the alpha-proton dierential ow to the temperature anisotropy of the proton VDF [Araneda et al., 2002]. Numerical simulations veried that in fact an increasing core temperature anisotropy serves to limit the maximum values of U p =C A which 148

149 are stable. We have met with the authors of this paper and compared their predictions with our observations. Presently the theoretical demonstration of the constraint is qualitative due to the time-consuming nature of the numerical simulations which prevent a study over a range of typical solar wind conditions. The reported results demonstrated the relationship between R p and U p =C A for a specic set of plasma parameters. The nal goal in this section therefore is to identify the subset of the Faraday Cup measurements which have roughly the same plasma parameters as the inputs to the numerical simulations in [Araneda et al., 2002] and to see if we also see the same limit to U p =C A with increasing anisotropy. For this study 29,229 FC spectra were selected. Each spectra had to have uncertainties in R p and U p =C A of less than 20%, 0:45 kp 1, T p 1: K, and R within 20% of R p. Figure 5-3 is a plot of the average value of U p =C A as a function of ten values of R p. For each point the horizontal error bars are the width of the intervals and the vertical error bars are uncertainty in each of the mean values. We see that U p =C A is at a maximum at R p ' 0:7 and falls o with increasing and decreasing anisotropy on either side. This is the rst observational evidence supporting the idea that a coupling between the dierential ow and the proton anisotropy serves to place an upper bound on U p =C A on the same size as the observed range. This result has been provided to the authors and they are attempting to simulate this same curve. 5.3 Thermal Speeds and Thermal Equilibrium The survey of T =T p in Figure 5-1 showed that the average ratio of the alpha to proton temperature is a factor of four in the collisionless solar wind and equal to unity when x c. The origin of the factor of four is simple: if the protons and alphas have isotropic distribution functions, then k B T p = 1 2 m pw 2 p, and k B T = 1 2 m w 2 ' 2m p w 2. In general then T = 2m pw 2 1 T p 2 m pwp 2 =4 w2 wp 2 (5.3) so a ratio of four in temperature means that the thermal speeds of the two species are equal. This phenomena of equal thermal speeds has been known for some time, and has been attributed to the eect of a wave-particle interaction which is sensitive to the velocity distribution function of each species and not their temperatures. The most commonly accepted explanation is that cyclotron resonant heating of minor ions including helium occurred closer in to corona as waves in the interplanetary medium were dissipated [Dusenbery and Hollweg, 1981]. The bi-maxwellian analysis of the alpha particle distributions has presented us with an opportunity to study this eect more closely by comparing the parallel and perpendicular temperatures of the hydrogen and helium separately. This is because the cyclotron resonance 149

150 Figure 5-4: Survey of the range of parallel and perpendicular temperature ratios T? =T?p and T k =T kp. Upper panel: Histograms of each ratio. Lower panel: The average value of each of the ratios as a function of x = x. Dashed line indicates equal thermal speeds, dotted line indicates equal temperatures. 150

151 should preferentially heat particles in the perpendicular direction (Recall the ecient exchange of energy from a wave to particles in cyclotron resonance (3.19)). The collected observations of the parallel and perpendicular temperatures are summarized in Figure 5-4. The upper panel is a histogram of the relative distribution of T? =T?p and T k =T kp using all observations in the solar wind with percent uncertainties in the temperature ratios of less than 20%. The perpendicular temperatures have a more strongly dened bi-modal distribution with peaks at T? =T?p = 1 and T? =T?p = 4, as expected. The parallel temperatures are not as sharply separated. One possible explanation for this eect is based on the frequent double streaming of the alphas at up to 10% the proton bulk speed. Due to this dierential streaming the hydrogen and helium observed simultaneously at 1 AU have not been associated during the entire propagation from the corona. This could account for the increases variation of the ratio of the parallel temperatures. It should be noted that measurements in the portion of the histogram with temperature ratios about seven are valid and representative of a further departure from thermal equilibrium than even the equality of thermal speeds would allow. The bottom panel of Figure 5-4 shows the average values of the ratios T? =T?p (blue) and T k =T kp (red) as a function of x = c. In the collisionless interval 0:03 x = c 0:3 we see that while the parallel temperature ratio is near four, the ratio of the T? =T?p is consistently closer to a factor of ve. The fact that T? =T?p > 4 in the collisionless interval is very interesting. Recall from Section 3.3 and Figure 3-5 that the observations of the largest average values of R p > 1were also associated with solar wind which had suered few collisions. The fact that protons in the interval 0:03 x = c 0:3 bear an anisotropy R p > 1 while in addition T? =T?p > 4 is a clear signature of cyclotron heating. The condition for cyclotron resonance between an ion species and an electromagnetic wave in the presence of an ambient magnetic eld ~ B was derived in Section The exact values of the resonant frequencies depend on the particle species under consideration, so we rewrite the resonance condition (3.19) explicitly using the denition of the cyclotron frequency (1.20), k k v k =! + m c j (5.4) where k k is the wavenumber of the mode parallel to ~ B, v k is the speed of the resonating particle along the eld line, and c j is the cyclotron frequency for the species in question. If the species has mass m j and charge q j, then, k k v k =! + m q j m j B : (5.5) In the limit that!! 0, which is reasonable if we are considering Alfvenic uctuations, then the wavelength of the mode which resonates with the particle is = m m j q j v k B : (5.6) 151

152 Figure 5-5: The average ratio of the alpha to proton temperature ratio T? =T?p as a function of the collision rate and the normalized alpha-proton dierential ow. Note that the largest values of T? =T?p occur when U p is small (See text for details). 152

153 There are a couple interesting consequences to (5.6). First note that the speed v k, and therefore the thermal width and not the temperature of the particle species is important in resonating with a given wave. Secondly, helium has a mass to charge ratio of m =q which is twice that of a proton. So if the picture of long wavelength waves cascading down to shorter wavelengths is valid, they will resonate and heat the alpha particles before they heat the protons. This section concludes with one nal piece of evidence that the ions experience wave-heating near the corona. Figure 5-5 is a plot of the average ratio T? =T?p as a function of the alpha-proton dierential speed U p and x = c. Studies have shown that often in the presence of Alfvenic uctuations alpha particles can \surf" the waves and not participate in the oscillations [Steinberg et al., 1996]. This occurs because the alphas are moving at the Alfvenic speed and therefore do not see the waves. Alpha particles will resonate more eciently with these waves if they have a small dierential speed relative to the protons. In Figure 5-5 it is the portion of the observations in which the alphas had very small speeds relative to the protons that T? =T?p becomes so much larger than expected. 5.4 Revisiting Temperature Driven Instabilities with a Multi-Species Fluid The analysis of the kinetic instabilities driven by proton temperature anisotropies in Chapter 3 demonstrated that while the mirror, cyclotron, and rehose modes have intuitive descriptions in the magnetohydrodynamic uid picture their exact behavior may only be understood by employing kinetic methods such as linear theory and the Vlasov equation or hybrid numerical simulations. It was shown that there are signicant departures from the MHD predictions of, for example, the onset of instability as a function of kp. In addition certain modes, such as the cyclotron instability, have no uid analog. If the MDH picture were correct than the helium presents itself as a small correction to the pressure of the protons, and we might expect a large range of R. However, from the kinetic pointofvieweach species may resonant independently. In this nal section we examine the range in alpha temperature anisotropies R and compare them to the proton results. Since this is the rst examination of the limits imposed on helium in the solar wind by temperature anisotropy-driven micro-instabilities the goal is to look for evidence of the same processes. The procedure outlined at the end of Chapter 3 to generate Figure 3-11, the survey plot of the normalized distribution of proton observations as a function of R p and kp, was used to generate the equivalent plot for alpha particle anisotropies R is shown in Figure 5-6. Barring the poorer statistics, the agreement between the two gures is remarkable. Helium clearly experiences these micro-instabilities in the same manner as the solar wind hydrogen. The next project with alpha anisotropies will be to proceed with the entire procedure outlined in Chapter 3 and to identify the alpha rehose and cyclotron instability thresholds for comparison with theory. 153

154 Figure 5-6: Distribution of alpha particle temperature anisotropies as a function of kp. Compare with the summary Figure 3-11 at the end of Chapter 3 for proton temperature anisotropies. 154

155 Chapter 6 Properties of Collisionless Shocks ABSTRACT: Identication and characterization of collisionless shocks in the interplanetary medium. Methods are outlined for the analysis of interplanetary shocks, including the rst Rankine-Hugoniot analysis with real parameter uncertainties. Survey of shock parameters, their uncertainties and evolution over time, comparison of ACE/Wind/IMP8 simultaneous observations of shocks to verify methods. 6.1 Collisionless Shocks in the Heliosphere Building on the measurements produced by the proton analysis of Chapter 2 and the alpha parameter study of Chapter 4, we now turn to the study of collisionless shocks in the interplanetary medium. In this chapter collisionless shocks are introduced and methods are listed for characterizing the shocks. A search is performed over the duration of the Wind mission to identify more than 250 interplanetary shocks and to analyze each of them. There are several accepted ways in the community of analyzing interplanetary shocks, and the techniques often produce diering results. Past studies of the eectiveness of these methods have generally relied upon statistically limited samples of interplanetary shocks. It is the main goal of this chapter to use this unprecedented number of shocks to quantitatively demonstrate the superiority of the full magnetohydrodynamic analysis over the traditionally accepted approximate methods. Section describes the mechanisms behind the creation of shocks in a simple uid and in the collisionless solar wind. The formation of a simple hydrodynamic shock wave is discussed rst, to highlight the additional complexities of collisionless shocks in magnetohydrodynamic uids. Section describes the common ways in which shocks are formed in space. Section 6.2 presents the techniques for characterizing a MHD shock under certain idealized circumstances. The seven most common methods are derived there in addition to an outline of the procedures developed for identifying interplanetary shocks and extracting their parameters. The results of each of the analysis methods are compared in Section

156 6.1.1 Steepening of Waves into Shocks In ordinary hydrodynamics the maximum speed at which information may propagate through a uid medium is limited by C S, the local sound speed, C 2 S = P (6.1) where is the ratio of specic heats, P is the total pressure, and is the mass density. As with the linear analysis methods of previous chapters, if a uctuation is small enough than the dierential equations describing a perturbation's evolution are linear and lead to a simple wave equation. But situations may arise where the coupling becomes highly non-linear. For example, if a uid is forced to ow around a barrier at a speed greater than the local sound speed then a shock will form. The presence of non-linear coupling between uid properties may also lead to the formation of a shock-wave, as we shall demonstrate. Non-linear terms become signicant at several points in the equations describing the evolution of the uid. In the context of shock formation the two most important eects are due the variation of the sound speed C S and the onset of dissipative mechanisms. In a simple linear calculation one would assume that the perturbations in P and are small enough that C S (6.1) may be assumed to remain constant. If there is an adiabatic law p = p (= ), then the sound speed is proportional to the density, C S = (p=) 1=2 = (C ;1 ) 1=2, where C is some constant. In the case of one-dimensional propagation of a compressional wave through a uid, the evolution of the ow speed U obeys the = : (6.2) Consider a uid with a propagating density enhancement. If the perturbation in is suciently large to produce a noticeable increase in C S, then by (6.2) the speed of the uid will rise behind the peak in the density > 0) and fall in the interval ahead the enhancement < 0) of higher density. This results in a compression of the and a \steepening" of the speed prole until a shock forms. In the hydrodynamic case the details of the shock, such as the width of the transition, are determined by dissipation mechanisms like nite viscosity which act within the uid. The situation in the interplanetary medium is very dierent because the magnetohydrodynamic uid supports three propagating waves and because there is no analogue to the classical viscosity due to the collisionless nature of the solar wind plasma. The dissipative mechanisms which act upon the plasma across the shock transition are kinetic in nature and require the Vlasov equations to fully understand them. Fortunately we do not need to include the specic details of the dissipation mechanisms in this study of interplanetary shocks. Indeed the results of this work provide an excellent opportunity to probe the nature of those processes. In the remainder of this section we will explain the four types of shocks seen observationally by examining the steepening of linear waves in a magnetohydrodynamic uid. 156

157 In the regime! r c i and kc=! p 1 uid theory predicts three normal wave modes in a magnetized plasma [Raichoudhuri, 1998, x14.5]. Consider consistent solutions to a perturbation in single uid MHD over background density, pressure, velocity, and magnetic eld, + 1 (~x t) p + p 1 (~x t) ~ U + ~ U 1 (~x t) ~ B + ~ B 1 (~x t) (6.3) Perturb and linearize the equation of continuity (1.71), the momentum equation (1.74) with ~ J 1 = ~ r ~ B 1 = ~ r ~ U 1 =0 ~ U = ;C 2 S ~ r ~r ~ B 1 ~ B (6.5) and the induction equation (1.76) with no magnetic ~ B = ~ r( ~ U 1 ~ B ): (6.6) Dierentiating (6.5) with respect to time and making use of (6.4) and (6.6) leads to a dispersion relation,! 2 ~ U1 =(C 2 S + C 2 A)( ~ k ~ U 1 ) ~ k + C A^b ~ k[ca^b ~ k ~ U1 ; C A^b ~ U1 ~ k ; ~ k ~ U1 C A^b] (6.7) where C A is the Alfven speed, C 2 A = B2 (6.8) and ^b is the direction of the ambient magnetic eld. It is clear from (6.7) that the direction of the wave propagation relative to ^b will eect the uctuations in the plasma. Dening ^n as the direction of propagation such that ~ k = k^n, we can dene a very important quantity, ^b Bn cos ;1 ^n : (6.9) There are three independent solutions to (6.7): the slow, intermediate, and fast wave modes. Small perturbations in these modes will propagate at the slow, C sl, 157

158 Figure 6-1: Plot of the speeds in km/s of the three MHD wave modes allowed by (6.10) as a function of direction of propagation relative to the magnetic eld, Bn, calculated using the typical solar wind conditions from Table 1.1. The horizontal axis corresponds to propagation along the eld ( Bn = 0). The dashed blue line is the speed of the intermediate mode, the solid green line is for the slow mode, and the dash-dotted red line corresponds to the fast mode. 158

159 intermediate, C i, and fast, C f, wave speeds, C 2 sl = 1 2 C 2 A + C 2 S + (C 2 A + C 2 S) 2 ; 4C 2 S C2 A cos 2 Bn 1=2 C 2 i = C 2 A cos 2 Bn C 2 f = 1 2 C 2 A + C 2 S + (C 2 A + C 2 S) 2 +4C 2 S C2 A cos 2 Bn 1=2 : (6.10) Figure 6-1 is a plot of the three wave speeds in (6.10) under typical solar wind conditions as a function of the angle of propagation relative totheambient magnetic eld. It is clear that the angle Bn between the direction of wave propagation wave and the orientation of the magnetic eld plays avery important role. The hydrodynamic perturbation considered above formed a shock because the relationship between density and speed in (6.2) created a compressive uctuation which grew in time. The fast and slow MHD modes are also compressive so they may lead to the formation of shocks. The intermediate mode is not compressive. Therefore shocks in the solar wind will either be \fast" or \slow", depending on the wave speed which has been exceeded. Consider the formation of a shock wave in solar wind plasma with the wave speeds shown in Figure 6-1. If a uctuation is propagating along the magnetic eld with a speed of less than 40 km/s then no shock will form. However, if the speed rises above 40 km/s then a slow shock will develop, and if the speed surpasses 51 km/s then a fast shock will form. At propagation angles orthogonal to the magnetic eld there are no slow shocks: either the wave is too slow to steepen or a fast shock will develop. As a result of this, fast shocks are far more common in space. Finally, in the frame of the solar wind plasma a shock may be propagating towards the Sun - a \forward" shock -oraway from it - a \reverse" shock. While this makes no physical dierence in the nature of the shock itself, it eects the properties seen in the rest frame of a spacecraft. We therefore will speak of four kinds of interplanetary shocks: fast-forward (FF) slow-forward (SF) fast-reverse (FR) and slow-reverse (SR). The plasma and magnetic eld signiatures of these four types of shocks as seen in the inertial frame of a spacecraft are shown in Figure Sources of Collisionless Shocks in the Heliosphere There are three main sources of collisionless shocks in the heliosphere. Those are shocks formed in the interplanetary medium when a driver material reaches upstream material at speeds greater than the upstream sound speed the interaction between two streams of solar wind which emerged from the surface of the Sun at dierent original speeds and have now run into each other and planetary bow shocks which form on the sunward side of magnetized planets that divert the ow of the incident solar wind [Axford, 1962 Kellogg, 1962]. Figure 6-3 illustrates schematically the three types of shocks observed by the Wind spacecraft. The left panel is a schematic diagram of the region surrounding Earth. 159

160 Figure 6-2: The velocity V, temperature T, density N, and magnetic eld B proles of the four types of shocks seen in the interplanetary medium. The colored curves indicate what a spacecraft would observe as each of the shock types traveled past it. In each case time increases to the right as indicated by the arrows. (a) (b) (c) Figure 6-3: Three panels illustrating mechanisms for the generation of collisionless shocks in the heliosphere: left: Diversion of solar wind about a magnetized barrier leads to formation of a planetary bow shock center: Interaction of solar wind streams with dierent speeds leads to a CIR right: Image of a coronal mass ejection on Feb. 27, 2000 taken by the LASCO C3 coronograph on the SOHO spacecraft (See text for details). 160