Lecture notes on Plasma physics

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4/18/2012 Space Plasma Physics Space Plasma Physics Thomas Wiegelmann, 2012 1. Basic Plasma Physics concepts Physical Processes 2. Overview about solar system plasmas Plasma Models 8. Plasma Waves, instabilities and shocks 9. Magnetic Reconnection 3. Single particle motion, Test particle model 4. Statistic description of plasma, BBGKY- Applications Hierarchy and kinetic equations 10. Planetary Magnetospheres 5. Fluid models, Magneto-Hydro-Dynamics 11. Solar activity 6. Magneto-Hydro-Statics 12. Transport Processes in Plasmas 7. Stationary MHD and Sequences of Equilibria What is plasma? Used Material In plasma physics we Lecture notes from Eckart Marsch 2007 study ionized gases Baumjohann&Treumann: Basic Space Plasma Physics under the influence of Schindler: Physics of space plasma activity electro-magnetic fields. Priest: Solar MHD Kulsrud: Plasma Physics for Astrophysics Krall & Trivelpiece: Principles of Plasma Physics Chen: Introduction to plasma physics and controlled fusion Balescu: Plasma Transport (3 volumes) Levi Tonks (1897-1971) and William Crookes (1832-1919) Irving Langmuir (1881- called ionized matter in a gas Spatschek: Theoretische Plasmaphysik 1957, photo) first used the discharge (Crookes-tube, photo) term plasma for a 4th state of matter S Wikipedia, Google and YouTube collection of charged (Phil. Trans 1879) particles (Phys. Rev. 1929) Source: Wikipedia Natural Plasmas on Earth Industrial Plasmas Lightning Electric Arcs St. R Fire Semi conducter Nuclear Fusion, Neon-lights, Fluorescent device fabrication here a Tokamak Aurorae lamps,Plasma globes Ball lightning Source: Wikipedia 1 4/18/2012 Space Plasmas Accretion discs Interior and atmosphere of Sun+Stars Source: Wikipedia Planetary magnetospheres, solar wind, inter-planetary medium Comparison: Gas and Plasma Source: Wikipedia Plasmas studied in this lecture What is a plasma? Non-relativistic particle velocities vc Spatial and temporal scales are large compared to A fully or partly ionized gas. -35 -44 Planck length (1.6 10 m) and time (5.4 10 s) Collective interaction of charged particles More precise: Any action variables like is more important than particle-particle (momentum x spatial dimension, Energy x time) are collisions. -34 large compared to Planck constant (h= 6.6 10 Js) Charged particles move under the influence of Classic plasma, no quantum-mechanic effects. electro-magnetic fields (Lorentz-force) Plasmas violating these conditions (Quark-Gluon Plasma, relativistic plasma) are active Charged particles cause electric fields, areas of research, but outside the scope of this moving charged particles cause electric currents introductory course. and thereby magnetic fields (Maxwell equations). 2 4/18/2012 Plasma models What is a plasma? - Test particles: Study motion of individual charged particles under the influence of external electro-magnetic (EM) fields In principle we can study a plasma by solving the Lorentz-force and Maxwell- - Kinetic models: equations selfconsistently. Statistic description of location and velocity of With typical 1020-1050 particles in space particles and their interaction + EM-fields. (Vlasov-equation, Fokker-Planck eq.) plasmas this is not possible. (Using some 109 or more particles with this - Fluid models: approach is possible on modern computers) Study macroscopic quantities like density, pressure, = Plasma models flow-velocity etc. + EM-fields (MHD + multifluid models) - Hybrid Models: Combine kinetic + fluid models Electromagnetic forces Maxwell equations for electro-magnetic fields The motion of charged particles in space is strongly influenced by the self- The motion of charged particles in space is generated electromagnetic fields, which evolve according to and determined by the electrostatic Coulomb force (induction) laws (in this lecture we use the SI-system): and magnetic Lorentz force: where q is the charge and v the velocity of any charged particle. If we deal with electrons and various ionic species (index, s), the charge and current densities are obtained, respectively, by summation over all kinds of species as where and are the vacuum dielectric constant and free-space 0 0 follows: magnetic permeability, respectively. The charge density is and the current density j. The electric field obeys Gauss law and the magnetic field is always free of divergence, i.e. we have: which together obey the continuity equation, because the number of charges is conserved, i.e. we have: Electro-static effects Debye-length A plasma is quasi-neutral On large scales the positive (ions) and negative charges cancel each other. On small scales charge separations occur. Can we estimate these scales? = Do calculations on blackboard. Remark: Some text-books drop the factor sqrt(3/2)1.2 3 4/18/2012 Debye-length Debye shielding Electric potential for a charge in vacuum: Electric potential in a plasma: = Debye shielding Source: Chen, Wikipedia Debye shielding Plasma Oscillations How does a charged particle (say an electron) move in a non-magneticed plasma? Solve electrostatic Maxwell equation selfconsistently with equation of motion, here for the Coulomb force = Do calculations on blackboard. Source: Technik-Lexikon Plasma parameter Plasma Oscillations Plasma frequency The plasma paramter g indicates the number of particles in a Debye sphere. Plasma approximation: g 1. Electron plasma frequency For effective Debye shielding and is often used to specify the electron density of a plasma. statistical significants the number of How? = Dispersion relation of EM-waves particles in a Debye sphere must be high. See lecture of plasma waves. 4 4/18/2012 Collisions: Mean free path Plasma parameter g g gives a measure how collective effects The mean free path is the distance a particle dominate over single particle effects. moves in average before it suffers a collision. Cross section for interaction of particles Contrast between neutral gas and plasma: during collisions in a plasma is approximated -Interaction region of a neutral atom is the with the Debye-length. atomic radius R and Mean free path: -Interaction region in a plasma is the Debye sphere and Collision frequency: Plasma state can be derived from expansion of exact many body equations with g. Induction equation Magnetized Plasmas Using Maxwell equations for slow time variations, In order to study the transport of plasma and without the displacement current yields the magnetic field lines quantitatively, let us combine induction equation (with constant conductivity ): 0 equations with the simple phenomenological law, relating the electric field in the plasma frame with its current: Convection Diffusion (Later we will derive R law more systematically Exercise: Derive this equation for a spatial from kinetic plasma theory) constant conductivity. How does the equation look if the conductivity varies spatially? Magnetic diffusion Hydromagnetic theorem Assuming the plasma be In an ideal collisionless at rest, the induction plasma in motion with equation becomes a pure infinite conductivity the diffusion equation: induction equation becomes: with the magnetic diffusion coefficient -1 D = ( ) . m 0 0 The field lines are constrained to move with the plasma - frozen-in field. If plasma patches on different sections Under the influence of finite resistivity the magnetic field of a bundle of field lines move oppositely, then the lines diffuses across the plasma, and field inhomogenities are will be deformed accordingly. Electric field in plasma 2 smoothed out at time scale, = L , with scale length L . d 0 0 B B frame, E' = 0, - voltage drop around closed loop is zero. 5 4/18/2012 Field line merging and reconnection in the Magnetic reconnection Assuming the plasma streams at bulk speed V, then the induction equation can be written in simple dimensional form as: The ratio of the first to second term gives the so-called magnetic Reynolds number, R = L V, which is useful to decide whether a plasma is diffusion m 0 0 B or convection dominated. Current sheet with converging flows - magnetic merging at points where R 1. Field lines form X-point and separatrix. m magnetotail magnetopaus e Plasma waves are not generated at random. To exist they must Waves in plasmas satisfy two conditions: In a plasma there are many reasons for spatio-temporal - their amplitude must exceed the thermal noise level variations (waves or more generally fluctuations): High - they must obey appropriate dynamic plasma equations temperature required for ionization ( = 13.6 eV 158000 K) H implies fast thermal particle motion. As a consequence There is a large variety of wave modes which can be excited in a plasma. The mode structure depends on the composition, - microscopic fluctuating charge separations and currents boundary conditions and theoretical description of the - fluctuating electromagnetic fields. plasma. There are also externally imposed disturbances which may We may represent any wave disturbance, A(x,t), by its propagate through the plasma and spread their energy in the Fourier components (with amplitude, A(k, ), wave vector whole plasma volume. The relevant frequency ranges are: k, and frequency, ): Ultra-low, extremely-low, and very-low frequency waves Phase velocity (wave front propagation) Group velocity (energy flow) Wave-particle interactions Summary Plasma waves in a warm plasma interact with particles through: Plasma is a quasi-neutral ionized gas moving Cyclotron resonance: - k·v = ± gi,e under the influence of EM-fields. Landau resonance: - k·v = 0 Thermal energy of particles is much Nonlinear particle trapping in large-amplitude waves larger as potential energy (free particles). Quasilinear particle (pitch-angle) diffusion Quasi-neutrality can be violated in Debye-sphere. Particle acceleration in turbulent wave fields To qualify as plasma, spatial dimensions must be There is a large variety of wave-particle interactions. They may much larger as the Debye-sphere and many occur in connection with linear plasma instabilities, leading to particles are in the sphere for effective shielding. wave growth and damping, or take place in coherent or turbulent wave fields, leading to particle acceleration and Collision frequency must be much smaller heating. as the plasma frequency. 6 Exercises for Space Plasma Physics: I. Basic Plasma Physics concepts 1. What are the main criteria that an ionized gas qualifies for being a plasma? 2. What is the Debye length? How does the Debye-length change with density and temperature? Please try to give a physical explanation for this behavior. 3. Plasma-parameter: What are the main differences of plasmas with few and many particles in a Debye sphere? In which category are typical space plasmas? 4. Canaquasineutralplasmacreatelargescale(largerthanDebyelength) electric currents? If no, why not? If yes, how? 5. Derive the induction equation from Maxwell equations (without dis- placement current) and Ohm’s law for (a) Spatial constant conductivity (b) Spatial varying conductivity. (c) Is a constant or a varying conductivity more likely in space plas- mas? 6. Areelectricormagneticfieldsmoreimportantinspaceplasmas? Why? 7. Can magnetic reconnection happen in an ideal plasma? 8. Can one observe magnetic reconnection in numerical simulations of ideal plasmas? (Can the answer to this question be different from the previous answer?)4/27/2012 Space Plasma Physics Solar system space plasmas Thomas Wiegelmann, 2012 Plasmas differ by their chemical composition and the 1. Basic Plasma Physics concepts ionization degree of the ions or molecules (from different sources). Space Plasmas are mostly 2. Overview about solar system plasmas magnetized (internal and external magnetic fields). Plasma Models Solar interior and atmosphere 3. Single particle motion, Test particle model Solar corona and wind (heliosphere) 4. Statistic description of plasma, BBGKY- Planetary magnetospheres (plasma from solar wind) Hierarchy and kinetic equations Planetary ionospheres (plasma from atmosphere) 5. Fluid models, Magneto-Hydro-Dynamics Coma and tail of a comet 6. Magneto-Hydro-Statics 7. Stationary MHD and Sequences of Equilibria Dusty plasmas in planetary rings Structure of the heliosphere Space plasma Space plasma particles are mostly free in the sense that their kinetic exceeds their potential energy, i.e., they are normally hot, T 1000 K. Space plasmas have typically vast dimensions, such that the mean free paths of thermal particles are larger than the typical spatial scales they are collisionless. Basic plasma motions in the restframe of the Sun Principal surfaces (wavy lines indicate disturbances) Different plasma states Ranges of electron density and temperature for geophysical plasmas Plasmas differ by the charge, e , mass, m , temperature, T , j j j 1/2 density, n , bulk speed U and thermal speed, V =(k T /m ) j j j B j j Some plasmas, like the of the particles (of species j) by which they are composed. R chromosphere or R ionosphere are not fully ionized. Collisions Long-range (shielded) Coulomb potential between neutrals and charged particles couple Collective behaviour of particles the particles together, Self-consistent electromagnetic fields with a typical collision time, , say. Behaviour of n Energy-dependent (often weak) collisions a gas or fluid as a plasma requires that: Reaction kinetics (ionization, recombination) 1 pe n Variable sources (pick-up) 1 4/27/2012 Corona of the active sun, SDO (Solar Dynamics Observatory), Jan. 2012 Magnetic fields structure the coronal plasma Plasma loops in solar corona, SDO, Jan. 2012 Solar coronal plasma can become unstable Solar Eruptions Giant Prominence Erupts - April 16, 2012, observed with SDO (Solar Dynamics Observatory) The solar coronal plasma is frozen into the coronal magnetic field and plasma outlines the magnetic field lines. Coronal configurations are most of the time quasistatic and change only slowly. Occasionally the configurations become unstable and develop dynamically fast in time, e.g., in coronal mass ejections and flares. 2 4/27/2012 Solar wind stream structure and Coronal magnetic field and density heliospheric current sheet Polar field: B Dipolar, = 12 G quadrupolar, current sheet contributions Current sheet is a symmetric disc anchored at high Parker, 1963 latitudes Banaszkiewicz et al., 1998; LASCO C1/C2 Schwenn et al., images (SOHO) 1997 Alfven, 1977 Measured solar wind electrons Electron density in the corona Helios + Current sheet and streamer belt, closed Sun Polar coronal hole, open magnetically -3 n = 3 -10 cm e Heliocentric distance / R Non-Maxwellian s Heat flux tail Guhathakurta and Sittler, Skylab coronagraph/Ulysses in-situ 1999, Ap.J., 523, 812 Pilipp et al., JGR, 92, 1075, 1987 Temperatures in the corona and fast solar wind Boundaries between solar wind and obstacles SolarProbe Solar Orbiter 7+ T m /m T ( Si ) i i p p 2+ ( He ) Corona Solar wind Cranmer et al., Ap.J., 2000; Marsch, 1991 3 4/27/2012 Space weather: Instabilities in the solar corona lead to huge eruptions, which can influence the Earth. 19 Space weather Space weather -Solar Storms -Charged particles Solar wind and solar eruptions influence Earth impact Earth and cause magnetic storms: -Aurora Aurora Power cutoffs Destroyed satellites Harm for astronauts Movie: Solar wind's effects on Earth http://www.youtube.com/watch?v=XuD82q4Fxgk Magnetospheric Schematic topography of solar-terrestrial environment plasma environment The boundary separating the subsonic (after bow shock) solar wind from the cavity generated by the R magnetosphere, is called the magnetopause. The solar wind compresses the field on the dayside and stretches it into the magnetotail (far beyond lunar orbit) on the nightside. The magnetotail is concentrated in the 10 RE thick plasma sheet. The plasmasphere inside 4 RE contains cool but dense plasma of ionospheric origin. The radiation belt lies on dipolar field lines between 2 to 6 RE. solar wind - magnetosphere - iononosphere 4 4/27/2012 Magnetospheric current system The currents can be Magnetospheric substorm guided by the strong background field, so- Substorm phases: called field-aligned Growth currents (like in a wire), which connect the polar Onset and expansion cap with the magnetotail Recovery regions. A tail current flows on the Magnetic reconnection: tail surface and as a Southward solar wind neutral sheet current in magnetic field the interior. Perturbations in solar wind The ring current is carried flow (streams, waves, CMEs) by radiation belt particles R flowing around the Earth field is accompanied by a current in east-west direction. system. Magnetospheric substorm Aurora Growth phase: Can be well understood by a sequence of static plasma equilibria and analytic magneto-static models. Plasma is ideal (no resistivity) Onset and expansion: The equilibrium becomes unstable and free magnetic energy is set free. Studied with (resistive) MHD-simulations. Cause for resistivity are micro-instabilities (often used ad hoc resistivity models in MHD) Recovery phase: Not well studied Source: Wikipedia Aurora-like Birkeland currents Laboratory experiments, Birkeland Currents in terella (source:Wikipedia) magnetic ball (terella) Moving charged particles cause electric currents parallel Kristian Birkeland, Path of charged particles made visible to the magnetic field lines connecting magnetosphere 1869-1917 first in terella, glow in regions around pole. and ionosphere. described substorms Cannot explain why actually in Earth and investigated 100.000 A (quiet times) to 1 million Ampere in disturbed aurora not occur at poles. Aurora in laboratory. times. = Joule heating of upper atmosphere. Terellas replace by Computer simulations. Source: Wikipedia 5 4/27/2012 Magnetic Storms Aurora mechanism Largest magnetic storm ever measured was in 1859. Atoms become ionized or excited in the Earth's upper Carrington noticed relation between a white light solar atmosphere (above about 80 km) by collision with solar wind and magnetospheric particles accelerated along the flare and geomagnetic disturbance. Earth's magnetic field lines. In 1859 the storm disrupted telegraph communication. Returning from ionized or excited states to ground state leads to emissions of photons. Such a large storm today could initiate a cascade of Ionized nitrogen atoms regaining an electron destroyed transformators (by induced electric fields) (blue light) or return to ground based from and economic damage of over: 1000 billion Dollar. exited state (red light) (source: Moldwin, the coronal current 2010) Oxygen returning from excited state to ground state 2005 Hurricane Katrina in USA : 120 billion Dollar. (red-brownish or green light, depending on absorbed energy in excited states) 2011 Earthquake/Tsunami in Japan: 300 billion Dollar. Returning to ground state can also occur by collisions Prediction of such storms would help to reduce the without photon emission = Height dependence of damage, e.g., by switching of electric power. emissions, different colours with height. Van Allen radiation Belt Planetary magnetospheres Rotation, size, mass, .... Magnetic field (moment) of planet and its inclination Inner/outer plasma sources (atmosphere, moons, rings) Boundary layer of planet and its conductivity Solar wind ram pressure (variable) James van Allen Contains energetic particles originating 1914-2006 Dynamic equilibrium if ram pressure at from solar wind and cosmic rays. source of pictures: magnetopause equals field pressure: wikipedia Inner belt: protons + electrons 2 2 2 6 V = B /2 = B (R /R ) /2 sw sw 0 p p m 0 Outer belt: electrons Particles trapped in magnetic field Stand-off distances: R /R = 1.6, 11, 50, 40 for M, E, J, S. m p (single particle model sufficient) Magnetospheric configurations Planetary parameters and magnetic fields Parameter Mercury Earth Jupiter Saturn Sun Radius km 2425 6378 71492 60268 696000 (equator) Rotation period h 58.7 d 23.93 9.93 10.66 25-26 d Dipole field G 340 nT 0.31 4.28 0.22 3-5 (equator) Inclination 3 23.45 3.08 26.73 7.12 of equator Degrees 6 4/27/2012 Jovian Magnetosphere Jupiter: fast rotation 10 h, mass-loading 1000 kg/s Dynamics driven largely by internal sources. Planetary rotation coupled with internal plasma loading from the moon Io may lead to additional currents, departure from equilibrium, magnetospheric instabilities and substorm-like processes. Regular (periodicity (2.54) days) release of mass from the Jovian magnetosphere and changes of the magnetic topology (Kronberg 2007). Jovian magnetospheric system is entirely internally driven and impervious to the solar wind (McComas 2007). Debates are ongoing R Magnetosphere Saturn's moon Enceladus may be a more significant source of plasma for the Saturn's magnetosphere than Io is for Jovian magnetosphere (Rymer 2010). It is important to scale plasma sources, relative to the size of the magnetosphere, to better understand the importance of the internal sources (Vasyliunas 2008). At Saturn, auroral features and substorm onset have both been associated with solar wind conditions (Bunce 2005) confirming that both internal loading and the solar wind influence magnetospheric dynamics. Tools needed for space plasma research Key phenomena in space plasmas Investigate the motion of charged particles e.g. in Dynamic and structured magnetic fields radiation belt = Single particle model Plasma confinement and flows (solar wind) Tools to describe quiet states, where the plasma is in equilibrium (growth phase of Formation of magnetospheres magnetic substorms, energy built-up in Shocks and turbulence solar coronal active regions) = Magnetostatics Multitude of plasma waves Tools to investigate activity (dynamic phase of substorms, coronal eruptions, waves) = MHD Particle heating and acceleration Cause for change from quiet to active states Velocity distributions far from thermal equlibrium and tools to investigate energy conversion, reconnection = MHD + kinetic theory 7 Exercises for Space Plasma Physics: II. Solar System Plasmas 1. How is a magnetosphere created? 2. What are magnetic storms and substorms? 3. Describe the physical mechanism, how a magnetic storm can destroy transformators. Would a large magnetic storm cause more harm in USA or in Europe? Are large magnetic storms now (year 2012) more or less likely than some five years ago? 4. In auroras we often see red light in high attitudes and green light lower down. Why? Hint: For oxygen it takes less than a second to emit green light, but stays up to about two minutes in excited state before it emits red. 5. Auroras occur in the so called Auroral oval. Why not over the poles? And why do auroral like emissions occur at the poles in laboratory experiments with a terella? 6. Are typical solar-system plasmas like magnetospheres and the solar corona thermodynamic equilibrium? Are they in force-equilibrium? 7. Show that Maxwell’s∇·B and∇·E equations can be seen (used) as initial condition. If the divergence equations are fulfilled at an initial time, the other two (evolutionary) Maxwell equations ensure that these conditions are fulfilled for all times. 8. Use the electromagnetic potentials B = ∇×A ∂A E = −∇Φ− ∂t 1 ∂Φ ∇·A = − (LorenzGauge) 2 c ∂t to derive wave equations for the potentials from Maxwell equations. How can one obtain the charge density and electric current density in a plasma?5/3/2012 Space Plasma Physics Plasma models - Test particles: Thomas Wiegelmann, 2012 Study motion of individual charged particles under 1. Basic Plasma Physics concepts the influence of external electro-magnetic (EM) fields 2. Overview about solar system plasmas - Kinetic models: Plasma Models Statistic description of location and velocity of particles and their interaction + EM-fields. 3. Single particle motion, Test particle model (Vlasov-equation, Fokker-Planck eq.) 4. Statistic description of plasma, BBGKY- - Fluid models: Hierarchy and kinetic equations Study macroscopic quantities like density, pressure, 5. Fluid models, Magneto-Hydro-Dynamics flow-velocity etc. + EM-fields (MHD + multifluid models) 6. Magneto-Hydro-Statics 7. Stationary MHD and Sequences of Equilibria - Hybrid Models: Combine kinetic + fluid models Single particle motion, Test particle model Test particles: In the test-particle approach the charged particles move under the influence of What is a test particle? electric and magnetic fields. Back-reaction of the particles Charged particles homogenous magnetic is ignored = model is not self-consistent. fields = Gyration Charged particles in inhomogenous fields Equation of Motion, Lorentz-force: = Drifts Adiabatic invariants Magnetic mirror and radiation belts Particles in magnetic field Test particles, special cases Kinetic energy is a constant of motion in B-Fields (calculate on blackboard + exercises) Static homogenous electric field, no B-field Static homgeneous magnetic field = Gyration, magnetic moment In B-Fields particles (Electrons, Protons) gyrate Static, hom. electromagnetic fields (exercise) Larmor (or gyro) Static inhomogeneous B-field. frequency Homogeneous, time-varying electromo- magnetic fields (exercise). Larmor radius Generic cases of time-varying inhomogenous EM-fields = Treat numerically. 1 5/3/2012 Gyration of ions and electrons The equation describes a circular orbit around the If one includes a constant speed parallel to the field, the field with gyroradius, r , and gyrofrequency, . The particle motion is three-dimensional and looks like a g g R center (x ,y ) is called the guiding center. The helix. The pitch angle of the helix or particle velocity with 0 0 gyration represents a microcurrent, which creates a respect to the field depends on the ratio of perpendicular field opposite to the background one. This behaviour to parallel velocity components. is called diamagnetic effect. Nonuniform magnetic fields in space Particles in magnetic field Concept of gyrating particles remains useful Shear, twist Curvature for inhomogeneous and time-dependent magnetic fields = Drifts, Valid if: Spatial scales for B-field changes are large compared to Larmor radius. Electric fields, gravity, magnetic field curvature Divergence etc. also cause drifts Gradient (some we will study in excercises) Inhomogeneous B-fields Adiabatic invariants of motion We have a quantity which is small, like Spatial scales for B-field changes are large compared to Gyro radius. An adiabatic invariant stays constant over a period of the order , while the Magnetic moment remains electro-magnetic field changes of the order O(1) in constant in weakly the same period. inhomogenous B fields What happens if ? = Adiabatic Invariant Adiabatic invariant becomes constant of motion. 2 5/3/2012 Adiabatic invariants of motion Magnetic mirror In classical Hamiltonian mechanics the action integral Let us follow the guiding center of a particle moving along an inhomogeneous magnetic field by considering the magnetic moment: is an invariant of motion for any change that is slow as compared to where we used the pitch angle . Apparently, pitch angles at the oscillation frequency associated with that motion. different locations are related by the corresponding magnetic field strengths: Bounce motion between mirror points Drift motion in azimuthal direction, The point where the angle o with planetary magnetic moment M reaches 90 is called the 2 mirror point. Magnetic flux, = B r , through surface encircled by the g gyro orbit is constant. R variable Application: Van Allen Belt, Application: Van Allen Belt radiation belt Planetary radiation belts A dipole magnetic field has a field strength Motion of particles in minumum at the equator and converging radiation belts can be field lines at the polar regions (mirrors). modelled with Test- Particles can be trapped in such a field. They particle approach. perform gyro, bounce and drift motions. Radiation Belt Storm Probes (RBSP), launch 2012(?) Magnetic drifts Magnetic drifts Inhomogeneity will lead to a drift. A typical magnetic field We time average over a gyroperiod and obtain: in space will have gradients, and thus field lines will be curved. We Taylor expand the field: where B is measured at the guiding center and r is the The non-uniform magnetic field B leads to a gradient 0 distance from it. Modified equation of motion: drift perpendicular to both, the field and its gradient: Expanding the velocity in the small drift plus gyromotion, v = v + v , then we find the stationary g drift: 3 5/3/2012 General force drifts Summary of guiding center drifts By replacing the electric field E in the drift formula by any field exerting a force F/q, we obtain the general guiding- Try to calculate these center drift: Drifts as excercise In particular when the field lines are curved, the centrifugal force is where R is the local radius of c Associated drifts are corresponding drift currents. curvature. Magnetic dipole field At distances not too far from Gyrokinetic approach the surface the R magnetic field can be We can distinguish the motion of charged approximated by a dipole particles into gyration of the particle and motion of field with a moment: the gyration center. 22 2 M = 8.05 10 Am E An exact mathematical treatment is possible within Hamilton mechanics by using non-canonical Measuring the distance in transformation. units of the R radius, R , E = Guiding-center approximation. and using the equatorial Outside scope of this Lecture, see Balescue, surface field, B (= 0.31 G), E yields with the so-called L- Transport-Processes in Plasma, Vol. 1, 1988 shell parameter (L=r /R ) the eq E Guiding center approach remains useful concept field strength as a function of for selfconsistent kinetic plasma models. latitude, , and of L as: Bounce period as function of L shell Dipole latitudes of mirror points Energy, W, is here 1 keV and o = 30 . eq Latitude of mirror point depends only on pitch angle but not on L shell Bounce period, , is the time b value. it takes a particle to move back and forth between the two mirror points (s is the path length along a given field line). Equatorial pitch angle in degrees 4 5/3/2012 Equatorial loss cone for different L-values Period of azimuthal magnetic drift motion If the mirror point lies too deep in the atmosphere (below Here the energy, 100 km), particles will W, is 1 keV and be absorbed by the pitch angle: collisions with o o = 30 and 90 . eq neutrals. Drift period is of order of several days. Since the magnetospheric The loss-cone width depends only on L but field changes on smaller time scales, it is unlikely that particles not on the particle mass, charge or energy. complete an undisturbed drift orbit. Radiation belt particles will thus undergo radial (L-shell) diffusion Summary: Single particle motion Gyromotion of ions and electrons arround magnetic field lines. Inhomogeneous magnetic fields, electric fields and other forces lead to particle drifts and drift currents. Bouncing motion of trapped particles to model radiation belt. Constants of motion and adiabatic invariants. So far we studied particles in external EM-fields and ignored fields and currents created by the charged particles and collision between particles. 5

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