Winter Quarter 2009

1. ESS 200C - Space Plasma Physics Vassilis Angelopoulos Robert Strangeway Winter Quarter 2009 Schedule of Classes DateTopic [Instructor] 1/5 Organization and Introduction to Space Physics I [A&S] 1/7 Introduction to Space Physics II [A] 1/12 Introduction to Space Physics III [S] 1/14 The Sun I [A] 1/21 The Sun II [S] 1/23 (Fri) The Solar Wind I [A] 1/26 The Solar Wind II [S] 1/28 --- First Exam [A&S] 2/2 Bow Shock and Magnetosheath [A] 2/4 The Magnetosphere I [A] DateTopic 2/9 The Magnetosphere II [S] 2/11 The Magnetosphere III [A] 2/18 Planetary Magnetospheres [S] 2/20 (Fri) The Earth’s Ionosphere [S] 2/23 Substorms [A] 2/25 Aurorae [S] 3/2 Planetary Ionospheres [S] 3/4 Pulsations and waves [A] 3/9 Storms and Review [A&S] 3/11 --- Second Exam [A&S]

2. ESS 200C – Space Plasma Physics • There will be two examinations and homework assignments. • The grade will be based on • 35% Exam 1 • 35% Exam 2 • 30% Homework • References • Kivelson M. G. and C. T. Russell, Introduction to Space Physics, Cambridge University Press, 1995. • Chen, F. F., Introduction of Plasma Physics and Controlled Fusion, Plenum Press, 1984 • Gombosi, T. I., Physics of the Space Environment, Cambridge University Press, 1998 • Kellenrode, M-B, Space Physics, An Introduction to Plasmas and Particles in the Heliosphere and Magnetospheres, Springer, 2000. • Walker, A. D. M., Magnetohydrodynamic Waves in Space, Institute of Physics Publishing, 2005.

3. Space physics is concerned with the interaction of charged particles with electric and magnetic fields in space. Space physics involves the interaction between the Sun, the solar wind, the magnetosphere and the ionosphere. Space physics started with observations of the aurorae. Old Testament references to auroras. Greek literature speaks of “moving accumulations of burning clouds” Chinese literature has references to auroras prior to 2000BC Space Plasma Physics

4. Aurora over Los Angeles (courtesy V. Peroomian) Cro-Magnon “macaronis” may be earliestdepiction of aurora (30,000 B.C.)

5. Galileo theorized that aurora is caused by air rising out of the Earth’s shadow to be illuminated by sunlight. (He also coined the name aurora borealis meaning “northern dawn”.) Descartes thought aurorae are reflections from ice crystals. Halley suggested that auroral phenomena are ordered by the Earth’s magnetic field. In 1731 the French philosopher de Mairan suggested they are connected to the solar atmosphere.

6. By the 11th century the Chinese had learned that a magnetic needle points north-south. By the 12th century the European records mention the compass. That there was a difference between magnetic north and the direction of the compass needle (declination) was known by the 16th century. William Gilbert (1600) realized that the field was dipolar. In 1698 Edmund Halley organized the first scientific expedition to map the field in the Atlantic Ocean.

7. 1716 Sir Edmund Halley Aurora is aligned with Earth’s field… • 1741 Anders Celsius and has magnetic disturbances. • 1790 Henry Cavendish Its light is produced at 100-130km • 1806 Alexander Humboldt but is related to geomagnetic storms • 1859 Richard Carrington and to Solar eruptions. • 1866 Anders Angström Auroral eruptions are self-luminous and… • 1896 Kristian Birkeland … due to currents from space: • 1907 Carl Störmer in fact to field-aligned electrons… • 1932 Chapman & Ferraro accelerated in the magnetosphere by… • 1950 Hannes Alfvén the Solar-Wind–Magnetosphere dynamo. • 1968 Single Satellites Polar storms related to magnetospheric activity • 1976 Iijima and Potemra … communicated via Birkeland currents • 1977 Akasofu Magnetospheric substorm cycle is defined • 1997 Geotail Resolves magnetic reconnection ion dynamics • 1995-2002 ISTP era Solar wind energy tracked from cradle to grave • 2002- Cluster era Space currents measured, space-time resolved • 2008 THEMIS Substorm onset is due to reconnection By the beginning of the space age auroral eruptions had been placed in the context of the Sun-Earth Connection

8. It All Starts from the Sun… Solar Wind Properties: • Comprised of protons (96%), He2+ ions (4%), and electrons. • Flows out in an Archimedean spiral. • Average Values: • Speed (nearly Radial): 400 - 450 km/s • Proton Density: 5 - 7 cm-3 • Proton Temperature 1-10eV (105-106 K)

9. Shaping Earth’s Magnetosphere • Earth’s magnetic field is an obstacle in the supersonic magnetized solar wind flow. • Solar wind confines Earth’s magnetic field to a cavity called the “Magnetosphere”

10. Auroral Displays: Direct Manifestation ofSpace Plasma Dynamics

11. Societal Consequences of Magnetic Storms • Damage to spacecraft. • Damage to spacecraft. • Loss of spacecraft. • Loss of spacecraft. • Increased Radiation Hazard. • Increased Radiation Hazard. • Power Outages and • radio blackouts. • Power Outages and • radio blackouts. • GPS Errors • GPS Errors

12. Stellar wind coupling to planetary objects is ubiquitous in astrophysical systems SUBSTORM RECURRENCE: MERCURY: 10 min EARTH: 3.75 hrs JUPITER: 3 weeks Magnetized wind coupling to stellar and galactic systems is common thoughout the Universe Mira (a mass shedding red giant)and its 13 light-year long tail ASTROSPHERE GALACTIC CONFINEMENT In this class we study the physics that enable and control such planetary and stellar interactions

13. Introduction to Space Physics I-III • Reading material • Kivelson and Russell Ch. 1, 2, 10.5.1-10.5.4 • Chen, Ch. 2

14. The Plasma State • A plasma is an electrically neutral ionized gas. • The Sun is a plasma • The space between the Sun and the Earth is “filled” with a plasma. • The Earth is surrounded by a plasma. • A stroke of lightning forms a plasma • Over 99% of the Universe is a plasma. • Although neutral a plasma is composed of charged particles- electric and magnetic forces are critical for understanding plasmas.

15. The Motion of Charged Particles • Equation of motion • SI Units • mass (m) - kg • length (l) - m • time (t) - s • electric field (E) - V/m • magnetic field (B) - T • velocity (v) - m/s • Fg stands for non-electromagnetic forces (e.g. gravity) - usually ignorable.

16. B acts to change the motion of a charged particle only in directions perpendicular to the motion. • Set E = 0, assume B along z-direction.

17. Solution is circular motion dependent on initial conditions. Assuming at t=0: and • Equations of circular motion with angular frequency c (cyclotron frequency or gyro frequency). Above signs are for positive charge, below signs are for negative charge. • If q is positive particle gyrates in left handed sense • If q is negative particle gyrates in a right handed sense

18. Radius of circle ( rc ) - cyclotron radius or Larmor radius or gyro radius. • The gyro radius is a function of energy. • Energy of charged particles is usually given in electron volts (eV) • Energy that a particle with the charge of an electron gets in falling through a potential drop of 1 Volt- 1 eV = 1.6X10-19 Joules (J). • Energies in space plasmas go from electron Volts to kiloelectron Volts (1 keV = 103 eV) to millions of electron Volts (1 meV = 106 eV) • Cosmic ray energies go to gigaelectron Volts ( 1 GeV = 109 eV). • The circular motion does no work on a particle Only the electric field can energize particles! Particle energy remains constant in absence of E !

19. The electric field can modify the particles motion. • Assume but still uniform and Fg=0. • Frequently in space physics it is ok to set • Only can accelerate particles along • Positive particles go along and negative particles go along • Eventually charge separation wipes out • has a major effect on motion. • As a particle gyrates it moves along and gains energy • Later in the circle it losses energy. • This causes different parts of the “circle” to have different radii - it doesn’t close on itself. • Drift velocity is perpendicular to and • No charge dependence, therefore no currents

20. Z X Y

21. Assuming E is along x-direction, B along z-direction: • Solution is: • In general, averaging over a gyration:

22. Note that VExB is: • Independent of particle charge, mass, energy • VExB is frame dependent, as an observer movingwith same velocity will observe no drift. • The electric field has to be zero in that moving frame. • Consistent with transformation of electric field in moving frame: E’=g(E+VxB). Ignoring relativistic effects, electric field in the frame moving with V= VExB is E’=0. • Mnemonics: • E is 1mV/m for 1000km/s in a 1nT field • V[1000km/s] = E[mV/m] / B [nT] • Thermal velocities (kT=1/2 m vth2): • 1keV proton = 440km/s • 1eV electron = 600km/s • Gyration: • Proton gyro-period: 66s*(1nT/B) • Electron: 28Hz*(B/1nT) • Gyroradii: • 1keV proton in 1nT field: 4600km*(m/mp)1/2*(W/keV)*(1nT/B) • 1eV electron in 1nT field: 3.4km *(W/eV)*(1nT/B)

23. Any force capable of accelerating and decelerating charged particles can cause an average (over gyromotion) drift: • e.g., gravity • If the force is charge independent the drift motion will depend on the sign of the charge and can form perpendicular currents. • Forces resembling the above gravitational force can be generated by centrifugal acceleration of orbits moving along curved fields. This is the origin of the term “gravitational” instabilities which develop due to the drift of ions in a curved magnetic field (not really gravity). • In general 1st order drifts develop when the 0th order gyration motion occurs in a spatially or temporally varying external field. To evaluate 1st order drifts we have to integrate over 0th order motion, assuming small perturbations relative to c, rL

24. The Concept of the Guiding Center • Separates the motion ( ) of a particle into motion perpendicular ( ) and parallel ( ) to the magnetic field. • To a good approximation the perpendicular motion can consist of a drift ( ) and the gyro-motion ( ) • Over long times the gyro-motion is averaged out and the particle motion can be described by the guiding center motion consisting of the parallel motion and drift. This is very useful for distances l such that and time scales t such that

25. - + Y • Inhomogenious magnetic fields cause GradB drift. • If changes over a gyro-orbit rL will change. • gets smaller when particle goes into stronger B. • Assume B is along Y • Average force over gyration: • uB depends on charge, yields perpendicular currents. Z x B=Bzz

26. The change in the direction of the magnetic field along a field line can also cause drift (Curvature drift). • The curvature of the magnetic field line introduces a drift motion. • As particles move along the field they undergo centrifugal acceleration. • Rc is the radius of curvature of a field line ( ) where , is perpendicular to and points away from the center of curvature, is the component of velocity along • Curvature drift can also cause currents.

27. At Earth’s dipole uB , uc are same direction and comparable: • uB , uc are ~ 1RE/min=100km/s for 100keV particle at 5RE, at 100nT • The drift around Earth is: 0.5hrs for the same particle at the same location • At Earth’s magnetotail current sheet, uB , uc are opposite each other: • Curvature dominates at Equator, Gradient dominates further away from equator

28. Z X Y dz d r dr • Parallel motion: Inhomogeniety along B (║B) • As particles move along a changing field they experience force • Parallel force is Lorenz force due to small dB perpendicular to nominal B (dW║) • Force along particle gyration is due to dB/dt in frame of gyrocenter (dW┴) • Total particle energy is conserved because there is no electric field in rest frame • Consider 1st order force on 0th order orbit in mirror field • B change due to presence of ║B must be divergence-less (B=0) so: • In cylindical coordinates gyration is: • The mirror field is: • The Lorenz force is: • From gyro-orbit averaging: • Defining: we get: • This is the mirror force. Note that is conserved during the particle motion:

29. Another way of viewing  • As a particle gyrates the current will be where • The force on a dipole magnetic moment is Where I.e., same as we derived earlier by averaging over gyromotion

30. In a Magnetic Mirror: • The force is along and away from the direction of increasing B. • Since and kinetic energy must be conserved a decrease in must yield an increase in • Particles will turn around when • The loss cone at a given point is the pitch angle below which particles will get lost:

31. Time varying fields: B • As particle moves into changing field or • As imposed/background field increases (adiabatic compression/heating) • An electric field appears affecting particle orbit • Along gyromotion, speed v┴ increases, but μis conserved: • Flux through Larmor orbit: =rL2B=(2  m/q2)  remains constant

32. Time varying fields: E See Chen, Ch 2

33. Maxwell’s equations • Poisson’s Equation • is the electric field • is the charge density • is the electric permittivity (8.85 X 10-12 Farad/m) • Gauss’ Law (absence of magnetic monopoles) • is the magnetic field • Faraday’s Law • Ampere’s Law • c is the speed of light. • Is the permeability of free space, H/m • is the current density

34. Maxwell’s equations in integral formNote: use Gauss and Stokes theorems; identities A1.33; A1.40 in Kivelson and Russell) • is a unit normal vector to surface: outward directed for closed surface or in direction given by the right hand rule around C for open surface, and is magnetic flux through the surface. • is the differential element around C.

35. The first adiabatic invariant • says that changing drives (electromotive force). This means that the particles change energy in changing magnetic fields. • Even if the energy changes there is a quantity that remains constant provided the magnetic field changes slowly enough. • is called the magnetic moment. In a wire loop the magnetic moment is the current through the loop times the area. • As a particle moves to a region of stronger (weaker) B it is accelerated (decelerated).

36. For a coordinate in which the motion is periodic the action integral is conserved. Here pi is the canonical momentum: where A is the vector potential). • First term: • Second term: • For a gyrating particle: Note: All action integrals are conserved when the properties of the system change slowly compared to the period of the coordinate.

37. The second adiabatic invariant • The integral of the parallel momentum over one complete bounce between mirrors is constant (as long as B doesn’t change much in a bounce). • Bm is the magnetic field at the mirror point • Note the integral depends on the field line, not the particle • If the length of the field line decreases, u|| will increase • Fermi acceleration

38. The total particle drift in static E and B fields is: • For equatorial particle in electrostatic potential  (E=-): • Particle conserves total (potential plus kinetic) energy • Bounce-averaged motion for particles with finite J • Particle’s equatorial trace conserves total energy

39. Drift paths for equatorially mirroring (J=0) particles, or • … for bounce-averaged particles’ equatorial tracesin a realistic magnetosphere.

40. As particles bounce they also drift because of gradient and curvature drift motion and in general gain/lose kinetic energy in the presence of electric fields. • If the field is a dipole and no electric field is present, then their trajectories will take them around the planet and close on themselves. • The third adiabatic invariant • As long as the magnetic field doesn’t change much in the time required to drift around a planet the magnetic flux inside the orbit must be constant. • Note it is the total flux that is conserved including the flux within the planet.

42. Limitations on the invariants • is constant when there is little change in the field’s strength over a cyclotron path. • All invariants require that the magnetic field not change much in the time required for one cycle of motion where is the cycle period.

43. The Properties of a Plasma • A plasma as a collection of particles • The properties of a collection of particles can be described by specifying how many there are in a 6 dimensional volume called phase space. • There are 3 dimensions in “real” or configuration space and 3 dimensions in velocity space. • The volume in phase space is • The number of particles in a phase space volume is where f is called the distribution function. • The density of particles of species “s” (number per unit volume) • The average velocity (bulk flow velocity)

44. Average random energy • The partial pressure of s is given by where N is the number of independent velocity components (usually 3). • In equilibrium the phase space distribution is a Maxwellian distribution where

45. For monatomic particles in equilibrium • The ideal gas law becomes • where k is the Boltzman constant (k=1.38x10-23 JK-1) • This is true even for magnetized particles. • The average energy per degree of freedom is: • for a 1keV, 3-dimensional proton distribution, we mean:kT=1keV, get Eaverage=1.5keV, and define vthemal=(2kT/mp)1/2=440km/s • for a 1keV beam with no thermal spread kT=0 and V=440km/s

46. Other frequently used distribution functions. • The bi-Maxwellian distribution • where • It is useful when there is a difference between the distributions perpendicular and parallel to the magnetic field • The kappa distribution • Κ characterizes the departure from Maxwellian form. • ETs is an energy. • At high energies E>>κETs it falls off more slowly than a Maxwellian (similar to a power law) • For it becomes a Maxwellian with temperature kT=ETs

47. What makes an ionized gas a plasma? • The electrostatic potential of an isolated charge q0 • The electrons in the gas will be attracted to the ion and will reduce the potential at large distances, so the distribution will differ from vacuum. • If we assume neutrality Poisson’s equation around q0 is • The particle distribution is Maxwellian subject to the external potential • assuming ni=ne=n far away • At intermediate distances (not at charge, not at infinity): • Expanding in a Taylor series for r>0 for both electrons and ions

48. The Debye length ( ) is • where n is the electron number density and now e is the electron charge. Note: colder species dominates. • The number of particles within a Debye sphere • needs to be large for shielding to occur (ND>>1). Far from the central charge the electrostatic force is shielded.

50. The plasma frequency • Consider a slab of plasma of thickness L. • At t=0 displace the electron part of the slab by <<L and the ion part of the slab by <<L in the opposite direction. • Poisson’s equation gives • The equations of motion for the electron and ion slabs are