Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Properties of the Earth’s Magnetic FieldThree Kinds of PressureSolar Wind Pressure at the EarthForming the MagnetopauseSlide Number 10Slide Number 11Slide Number 12Slide Number 13The MagnetotailSlide Number 15The Equatorial Current SheetThe Cross Section of the TailSlide Number 18Magnetotail Structure – Plasma Mantle, Plasma Sheet Boundary Layer and Central Plasma SheetSlide Number 20Slide Number 21 Reconnection The Energy of Particles in SpaceThe Reconnection ProcessSlide Number 25Slide Number 26Slide Number 27Magnetospheric Convection – Steps 2, 3,4 and 5Slide Number 29Slide Number 30Slide Number 31The PlasmasphereSlide Number 33The Equation of Motion of Charged ParticlesSlide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40General Motion of a Charged ParticleParticles Mirroring and Drifting in a Dipole Magnetic FieldSlide Number 43Slide Number 44The Electron Radiation Belts The Electron Radiation Belts in 3DThe Ring CurrentESS 7Lectures 10, 11 and 12October 22, 24, and 27The Magnetosphere– Back in 1930 Chapman and Ferraro foresaw that a planetary magnetic field could provide an effective obstacle to the solar-wind plasma.– The solar-wind dynamic pressure presses on the outer reaches of the magnetic field confining it to a magnetospheric cavity that has a long tail consisting of two antiparallel bundles of magnetic flux that stretch in the antisolar direction. – The pressure of the magnetic field and plasma it contains establishes an equilibrium with the solar wind. An Obstacle in the Solar Wind– To a first approximation the magnetic field of the Earth can be expressed as that of a dipole. The dipole moment of the Earth is tilted ~110 to the rotation axis with a present day value of 8X1015Tm3 or 30.4x10-6TRE3 where RE =6371 km (one Earth radius).– In a coordinate system fixed to this dipole momentwhere is the magnetic colatitude, and M is the dipole magnetic moment.21)cos31(0sincos22333θθθϕθ+====−−−MrBBMrBMrBrθThe Earth’s Magnetic Field(Spherical Coordinates)θφrXYZ(X,Y,Z)λ• Alternately in Cartesian coordinateswhere the z-axis is along the dipole magnetic moment.52255)3(33−−−−===rMrzBryzMBrxzMBzzzyzxThe Earth’s Magnetic Field(Cartesian Coordinates)• The magnetic field line for a dipole. Magnetic field lines are everywhere tangent to the magnetic field vector.• Integrating where r0 is the distance to equatorial crossing of the field line. It is most common to use the magnetic latitude instead of the colatitude where L is measured in RE.θθBdrBdrr=0=ϕdθ20sinrr =λλ2cosLr =Dipole Magnetic Field Lines and the L ParameterProperties of the Earth’s Magnetic Field• The dipole moment of the Earth presently is ~8X1015T m3 (3 X10-5TRE3).• The dipole moment is tilted ~110 with respect to the rotation axis.• The dipole moment is decreasing. – It was 9.5X1015T m3 in 1550 and had decreased to 7.84X1015T m3 in 1990. – The tilt also is changing. It was 30 in 1550, rose to 11.50 in 1850 and has subsequently decreased to 10.80 in 1990.• In addition to the tilt angle the rotation axis of the Earth is inclined by 23.50 with respect to the ecliptic pole. – Thus the Earth’s dipole axis can be inclined by ~350 to the ecliptic pole. – The angle between the direction of the dipole and the solar wind varies between 560 and 900.Three Kinds of Pressure• Dynamic pressure is like pressure of a flowing fluidwhere ρ is the mass density (kg/m3) and V is the velocity.• Thermal pressure of a fluid even if it is stationary.where n is the number density, k is the Boltzman constant (1.38X10-23 JK-1) and T is the temperature.• The magnetic field can also exert pressurewhere B is the magnetic field intensity (Teslas) and μ0 is the permeability of free space = 4πX10-7Hm-12VPDρ=nkTPT=022μBPB=Solar Wind Pressure at the Earth• Assume that the solar wind has a velocity of 400km/s, density of 5cm-3, temperature of 2X105K, and magnetic field strength of 5nTPaVPD92104.1−×==ρPankTPT11104.1−×==PaBPB1102100.12−×==μForming the Magnetopause• The dynamic pressure is much larger than the thermal pressure or magnetic pressure in the solar wind. • Within the magnetosphere the magnetic pressure of the Earth’s internal field dominates.• To a good approximation the boundary (the magnetopause) between the region dominated by the solar wind and the region dominated by the Earth (the magnetosphere) can be found by balancing the solar wind dynamic pressure with the magnetic pressure of the Earth.• Ideally when the pressures are in balance the Earth’s field will be shielded from the solar wind in a cavity called the magnetosphere.• The interaction sets up Chapman-Ferraro currents on the boundary which cancel the Earth’s magnetic field outside. • Near the pole there is a singular point in the field where |B| = 0. This is called the neutral point.• The C-F current circulates in a sheet around the neutral point• This current is symmetric about the equator with a corresponding circulation around the southern neutral pointField LineMagnetopauseNorthNeutralPointDuskSol arWindChapman-FerraroCurrentShielding the Earth from the Solar Wind• The magnetic field inside the boundary is the total field from dipole and boundary current. For an infinite planar sheet current the field would be exactly doubled. Inside a spherical boundary the multiplication factor is 3. The factor f must lie in this range .• Equate and substitute for the dipole strength variation with distance.• Solve for the dimensionless standoff distance Ls. ()0222μDBdynBdynfBpkmnuppp===Where k~0.9 is the elasticity of particle collisions and f is the factor by which the magnetospheric magnetic field is enhanced by the boundary current. Rsis the subsolar standoff distance. B0is the field at the surface of the Earth.3002222⎟⎠⎞⎜⎝⎛==sEDDRRBBBfkmnuμ61202022⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=mnuBkfRRLEssμThe Location of the Magnetopause• Close to the Earth the dipole field dominates and there is little distortion• Further away there is a significant change in the shape of the field lines with all field lines passing through the equator closer to the Earth than dipole field lines from the same latitude. • All dipole field lines that originally