Single Particle Motion
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1 Single Particle Motion Overview Electromagnetic fields, Lorentz-force, gyration and guiding center, drifts, adiabatic invariants. Pre-requisites: Energy density of the particle population smaller than energy density of the field, All spatial and temporal changes in the fields are small/slow compared to the characteristic scales of particle motion. 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 1
2 Application: Radiation Belts 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 2
3 Single Particle Motion: Overview 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 3
4 Electromagnetic Fields 1: Pre-requisite: vacuum, ε=µ=1. Maxwell s laws: Gauß law of the electric field Gauß law of the magnetic field Faraday s law Ampere s law Magnetic flux 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 4
5 Electromagnetic Fields 2: Transition differential to integral form: Gauß theorem: Stokes theorem: Transformation of electromagnetic fields: general: Non-relativistic: 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 5
6 Electromagnetic Fields 3: Generalized form of Ohm s law: Energy density in the electromagnetic field: multiply Faraday by B and integrate over V: rewrite (energy density now on the left hand side): Energy density electromagnetic field Poynting vector (energy flux density) >0: Ohmic losses, <0: sources In matter: 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 6
7 Lorentz Force Lorentz force (general form, all electromagnetic fields) Vanishing electric field (first integral of motion; elementary, only numerical exercise) 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 7
8 Gyration 1: Lorentz force (equation of motion): Equation of motion (components): Cyclotron frequency: Larmor radius: Example: T=1keV, B=1T; electrons: v=1.87e6 km, r=0.1 mm,ω=1.8e11/s; protons: v=4.37e5 km, r=4.6 mm, ω=1e8/s 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 8
9 Useful Definitions: Magnetic rigidity: Pitch angle: Magnetic moment: 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 9
10 Gyration 2: Relativistic quantities: Example: proton with 1E20 ev, B = 3E-10 T, r = 1E21 m Local gyro radius: 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 10
11 Drift in Electromagnetic Fields Guiding center: General drift: Electric field perpendicular to magnetic field: Gravitational plus magnetic field: Gradient drift: Curvature drift: 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 11
12 General Form of Drift Idea: transformation into a frame of reference in which motion is reduced to: Arithmetics (F B): Result: gyration in the new frame of reference general equation for the drift velocity: 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 12
13 Drifts Summarized 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 13
14 Drift with Changes in Energy Drift leads particles to a different potential acceleration 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 14
15 Adiabatic Invariants Pre-requisite: spatial and temporal changes in the fields small compared to the corresponding scales of particle motion: Gyration (time): Field-parallel motion: Gyro orbit (radius): Magnetic moment: Longitudinal invariant: Flux invariant: 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 15
16 Magnetic Mirrors Basics: the magnetic moment is constant Restoring force in the mirror point because magnetic field line not perpendicular to the plane of gyration 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 16
17 Magnetic Bottle Two magnetic mirrors combined: Oscillation of the guiding center between the mirror points Application: radiation belts (particles trapped inside the magnetosphere) Application longitudinal invariant: 1st order Fermi acceleration 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 17
18 Application: Radiation Belts 1. Magnetic moment µ 2. Longitudinal invariant 3. Flux invariant 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 18
19 Time Scales: Magnetosphere Gyration: khz Longitudinal oscillation: seconds Drift: 15 min (1000 s) 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 19
20 Dynamics of the Radiation Belts L-Shell: distance field line to center of Earth in equatorial plane Intensity on fixed field line (L-shell) shows strong temporal variations In particular rather abrupt depletion followed by a slower recovery of the particle population 14/04/2005 Space Physics SS Chap. 2: Single Particle Motion 20
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