14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 1
Single Particle Motion
OverviewElectromagnetic 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.
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Application: Radiation Belts
http://www.dartmouth.edu/~physics/cism/science/dartmouthmodel.html
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Single Particle Motion: Overview
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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
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Electromagnetic Fields 2:
Transition differential to integral form:
Gauß´ theorem:
Stokes´ theorem:
Transformation of electromagnetic fields:
general:
Non-relativistic:
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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):
Poynting vector (energy flux density)
In matter:
Energy density electromagnetic field
>0: Ohmic losses, <0: sources
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Lorentz Force
Lorentz force (general form, all electromagnetic fields)
Vanishing electric field (first integral of motion; elementary, only numerical exercise)
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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
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Useful Definitions:
Magnetic rigidity:
Pitch angle:
Magnetic moment:
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Gyration 2:
Relativistic quantities:
Example: proton with 1E20 eV, B = 3E-10 T, r = 1E21 mLocal gyro radius:
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Drift in Electromagnetic Fields
Guiding center:
General drift:
Electric field perpendicular to magnetic field:
Gravitational plus magnetic field:
Gradient drift:
Curvature drift:
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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:
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Drift with Changes in Energy
Drift leads particles to a different potential ⇒ acceleration
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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:
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Magnetic Mirrors
Basics: the magnetic moment is constantRestoring force in the mirror point because magnetic field line not perpendicular to the plane of gyration
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Magnetic Bottle
Two magnetic mirrors combined:Oscillation of the guiding center between the mirror pointsApplication: radiation belts (particles trapped inside the magnetosphere)Application longitudinal invariant: 1st order Fermi acceleration
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Application: Radiation Belts
1. Magnetic moment µ2. Longitudinal invariant3. Flux invariant
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Time Scales: Magnetosphere
Gyration: kHzLongitudinal oscillation: secondsDrift: 15 min (1000 s)
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Dynamics of the Radiation Belts
L-Shell: distance field line to center of Earth in equatorial planeIntensity on fixed field line (L-shell) shows strong temporal variationsIn particular rather abrupt depletion followed by a slower recovery of the particle population
http://lasp.colorado.edu/stp/research/research_main.html
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