156 Electromagnetism
Cherenkov radiation
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θ |
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cone semi-angle |
Cherenkov |
sinθ = |
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(7.246) |
c |
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(vacuum) speed of light |
cone angle |
ηv |
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η(ω) refractive index |
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v |
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particle velocity |
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e2µ0 |
ωc |
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c2 |
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Ptot |
total radiated power |
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− |
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Ptot = |
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v |
1 |
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ω dω |
(7.247) |
−e |
electronic charge |
Radiated |
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4π |
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v2η2(ω) |
powera |
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µ |
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free space permeability |
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c |
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0 |
angular frequency |
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where |
η(ω) ≥ |
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for 0 < ω < ωc |
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ω |
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ωc |
cuto frequency |
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aFrom a point charge, e, travelling at speed v through a medium of refractive index η(ω).
7.9 Plasma physics
Warm plasmas
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e2 |
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(7.248) |
Landau |
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4π 0kBTe |
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length |
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1.67 |
× |
10−5T −1 |
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(7.249) |
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e |
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λDe = |
kBTe |
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1/2 |
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Electron |
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0 |
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(7.250) |
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nee2 |
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Debye length |
69(Te/ne)1/2 |
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Debye |
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1/2 |
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φ(r) = |
qexp(−2 r/λDe) |
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(7.252) |
a |
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screening |
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4π 0r |
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Debye |
NDe |
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4 |
πneλDe3 |
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(7.253) |
number |
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3 |
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5 Te3/2 |
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Relaxation |
τe = 3.44 |
× 10 |
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s |
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(7.254) |
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ne lnΛ |
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times (B = 0)b |
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× 107 |
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T |
3/2 |
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mi |
1/2 |
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τi = 2.09 |
i |
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s (7.255) |
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ne lnΛ |
mp |
Characteristic |
vte = |
2kBTe |
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1/2 |
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electron |
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(7.256) |
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me |
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thermal |
5.51 × 103Te1/2 |
ms−1 |
(7.257) |
speedc |
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lL |
Landau length |
−e |
electronic charge |
0 |
permittivity of free space |
kB |
Boltzmann constant |
Te |
electron temperature (K) |
λDe |
electron Debye length |
ne |
electron number density |
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(m−3) |
φe ective potential
qpoint charge
rdistance from q
NDe |
electron Debye number |
τe |
electron relaxation time |
τi |
ion relaxation time |
Ti |
ion temperature (K) |
lnΛ |
Coulomb logarithm |
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(typically 10 to 20) |
Bmagnetic flux density
vte |
electron thermal speed |
me |
electron mass |
aE ective (Yukawa) potential from a point charge q immersed in a plasma.
bCollision times for electrons and singly ionised ions with Maxwellian speed distributions, Ti < Te. The Spitzer conductivity can be calculated from Equation (7.233).
cDefined so that the Maxwellian velocity distribution exp(−v2/vte2 ). There are other definitions (see Maxwell– Boltzmann distribution on page 112).
Electromagnetic propagation in cold plasmasa
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(2πν )2 |
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nee2 |
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= ω2 |
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(7.258) |
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Plasma frequency |
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p |
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0me |
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p |
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νp 8.98ne1/2 |
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Hz |
(7.259) |
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Plasma refractive |
η = #1 − (νp/ν)2$ |
1/2 |
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(7.260) |
index (B = 0) |
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Plasma dispersion |
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2 |
2 |
= ω |
2 |
− |
2 |
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(7.261) |
relation (B = 0) |
c k |
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ωp |
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Plasma phase |
vφ = c/η |
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(7.262) |
velocity (B = 0) |
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Plasma group |
vg = cη |
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(7.263) |
velocity (B = 0) |
vφvg = c2 |
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(7.264) |
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qB |
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Cyclotron |
2πνC = |
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= ωC |
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(7.265) |
m |
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(Larmor, or gyro-) |
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νCe 28 × 10 |
9 |
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Hz |
(7.266) |
frequency |
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B |
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νCp 15.2 × 106B |
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Hz |
(7.267) |
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r |
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= |
v |
= v |
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m |
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(7.268) |
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Larmor |
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L |
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ωC |
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qB |
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v |
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(cyclotron, or |
r |
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= 5.69 |
× |
10 |
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12 |
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m |
(7.269) |
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gyro-) radius |
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Le |
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− |
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vB |
! |
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Lp |
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× |
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− |
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B ! |
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r |
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= 10.4 |
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10 |
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9 |
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m |
(7.270) |
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Mixed propagation modesb |
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η2 = 1 |
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X(1 − X) |
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, |
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(7.271) |
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− (1 − X) − 21 Y 2 sin2 θB ± S |
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where |
X = (ωp/ω)2, |
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Y = ωCe/ω, |
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1 |
Y 4 sin4 θB + Y 2(1 − X)2 cos2 θB |
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and |
S2 = |
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4 |
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µ0e3 |
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2 |
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∆ψ = |
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λ line neB · dl |
(7.272) |
Faraday rotationc |
8π2me2c |
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= ) |
*+ |
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(7.273) |
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2.63 10−13 |
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× |
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Rλ2
νp |
plasma frequency |
ωp |
plasma angular frequency |
ne |
electron number density (m−3) |
me |
electron mass |
−e |
electronic charge |
0 |
permittivity of free space |
ηrefractive index
νfrequency
kwavenumber (= 2π/λ)
ωangular frequency (= 2π/ν)
cspeed of light
vφ |
phase velocity |
vg |
group velocity |
νC |
cyclotron frequency |
ωC |
cyclotron angular frequency |
νCe |
electron νC |
νCp |
proton νC |
qparticle charge
Bmagnetic flux density (T)
mparticle mass (γm if relativistic)
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rL |
Larmor radius |
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rLe |
electron rL |
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7 |
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rLp |
proton rL |
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speed to B (ms−1) |
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θB |
angle between wavefront |
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ˆ |
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normal (k) and B |
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∆ψ rotation angle
λwavelength (= 2π/k)
dl line element in direction of wave propagation
Rrotation measure
aI.e., plasmas in which electromagnetic force terms dominate over thermal pressure terms. Also taking µr = 1.
bIn a collisionless electron plasma. The ordinary and extraordinary modes are the + and − roots of S2 when θB = π/2. When θB = 0, these roots are the right and left circularly polarised modes respectively, using the optical convention for handedness.
cIn a tenuous plasma, SI units throughout. ∆ψ is taken positive if B is directed towards the observer.
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vs = |
γp |
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1/2 |
= |
2γkBT |
1/2 |
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(7.274) |
Sound speed |
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mp |
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166T 1/2 ms−1 |
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(7.275) |
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vA = |
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B |
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(7.276) |
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Alfven´ speed |
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(µ0ρ)1/2 |
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1016Bn−1/2 |
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2.18 |
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ms |
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× |
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e |
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− |
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2µ |
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4µ |
n k |
B |
T 2v2 |
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Plasma beta |
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β = |
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0 |
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0 |
e |
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s |
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B2 |
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B2 |
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= γv2 |
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(7.278) |
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A |
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Direct electrical |
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σd = |
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ne2e2σ |
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(7.279) |
conductivity |
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ne2e2 + σ2B2 |
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Hall electrical |
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σH = |
σB |
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(7.280) |
conductivity |
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σd |
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nee |
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Generalised |
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ˆ |
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Ohm’s law |
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J = σd(E + v×B) + σHB×(E + v×B) |
(7.281) |
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Resistive MHD equations (single-fluid model)b |
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∂B |
= ×(v×B) + η 2B |
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(7.282) |
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∂t |
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∂v |
+ (v |
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)v = |
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p |
+ |
1 |
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( |
× |
B) B + ν |
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2v |
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∂t |
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ρ |
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µ0ρ |
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× |
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+ |
1 |
ν ( · v) + g |
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(7.283) |
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3 |
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Shear Alfvenic´ |
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dispersion |
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ω = kvA cosθB |
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(7.284) |
relationc |
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Magnetosonic |
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ω2k2(v2 |
+ v2 ) |
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ω4 = v2v2 k4 cos2 |
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dispersion |
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− |
θB |
(7.285) |
relationd |
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s |
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A |
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s |
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vs sound (wave) speed
γratio of heat capacities
phydrostatic pressure
ρplasma mass density
kB Boltzmann constant
Ttemperature (K)
mp proton mass vA Alfven´ speed
Bmagnetic flux density (T)
µ0 |
permeability of free space |
ne |
electron number density |
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(m−3) |
βplasma beta (ratio of hydrostatic to magnetic pressure)
−e electronic charge
σd direct conductivity
σconductivity (B = 0)
σH Hall conductivity
Jcurrent density
Eelectric field
v |
plasma velocity field |
ˆ |
= B/|B| |
B |
µ0 |
permeability of free space |
ηmagnetic di usivity
[= 1/(µ0σ)]
νkinematic viscosity
ggravitational field strength
ωangular frequency (= 2πν)
kwavevector (k = 2π/λ)
θB angle between k and B
aFor a warm, fully ionised, electrically neutral p+/e− plasma, µr = 1. Relativistic and displacement current e ects are assumed to be negligible and all oscillations are taken as being well below all resonance frequencies.
bNeglecting bulk (second) viscosity. cNonresistive, inviscid flow.
dNonresistive, inviscid flow. The greater and lesser solutions for ω2 are the fast and slow magnetosonic waves respectively.
Synchrotron radiation
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2 |
v |
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2 |
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Ptot |
total radiated power |
Power radiated |
Ptot = 2σTcumagγ |
c ! |
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sin |
θ |
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(7.286) |
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σ |
Thomson cross section |
by a single |
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v |
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2 |
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uT |
magnetic energy |
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electron |
a |
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1.59 |
× |
10−14B2 |
γ2 |
sin2 |
θ |
W |
mag |
density = B2/(2µ0) |
c) |
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c |
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(7.287) |
v |
electron velocity ( |
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γ |
Lorentz factor |
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Ptot = 4 σTcumagγ2 |
v |
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... averaged |
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(7.288) |
θ |
= [1 − (v/c)2]−1/2 |
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over pitch |
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c ! |
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pitch angle (angle |
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v |
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2 |
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between v and B) |
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angles |
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1.06 |
× |
10−14B2 |
γ2 |
c ! |
W |
(7.289) |
B |
magnetic flux density |
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c |
speed of light |
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P (ν) emission spectrum |
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Single electron |
P (ν) = |
31/2e3B sinθ F(ν/νch) |
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(7.290) |
ν |
frequency |
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emission |
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4π 0cme |
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νch |
characteristic frequency |
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× |
10−25B sinθF(ν/νch) |
WHz−1 |
−e |
electronic charge |
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spectrumb |
2.34 |
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(7.291) |
0 |
free space permittivity |
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me |
electronic (rest) mass |
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3 |
γ |
2 |
eB |
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sinθ |
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(7.292) |
F |
spectral function |
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Characteristic |
νch = |
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2 |
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2πme |
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K5/3 |
modified Bessel fn. of |
frequency |
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4.2 × 1010γ2B sinθ |
Hz |
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(7.293) |
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1 |
the 2nd kind, order 5/3 |
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F(x) = x x∞ K5/3(y)dy |
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Spectral |
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(7.294) |
0.5 |
F(x) |
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function |
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2.15x |
1/3 |
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(x |
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1) |
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(7.295) |
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"1.25x1/2e−x |
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(x |
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1 x 2 |
3 |
4 |
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aThis expression also holds for cyclotron radiation (v c).
bI.e., total radiated power per unit frequency interval. |
7 |
|
160 Electromagnetism
Bremsstrahlunga
Single electron and ionb |
γ2 |
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γv |
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γv |
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dω |
24π4 03c3me2 |
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γ2v4 |
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dW |
= |
Z2e6 |
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ω2 |
1 |
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2 |
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ωb |
2 |
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ωb |
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Z2e6 |
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K0 |
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+ K1 |
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(7.296) |
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(ωb |
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γv) |
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(7.297) |
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24π4 03c3me2b2v2 |
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Thermal bremsstrahlung radiation (v c; Maxwellian distribution)
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dP |
= 6.8 |
× |
10−51Z |
2T −1/2nineg(ν,T )exp |
−hν |
Wm−3 Hz−1 |
(7.298) |
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dV dν |
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kT |
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0.28[ln(4.4 |
1016T 3ν−2Z−2) |
− |
0.76] (hν |
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kT < 105kZ2) |
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where |
g(ν,T ) |
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0.55ln(2.1 |
×1010T ν−1) |
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(hν |
105kZ2 < kT ) |
(7.299) |
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10 |
10× |
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1 |
) |
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1/2 |
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(hν |
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(2.1 |
× |
T ν |
− |
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− |
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kT ) |
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dP |
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1.7 |
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10−40Z2T |
1/2nine |
Wm−3 |
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(7.300) |
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dV |
× |
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ωangular frequency (= 2πν)
Ze |
ionic charge |
−e |
electronic charge |
0 |
permittivity of free space |
cspeed of light
me electronic mass
bcollision parameterc
velectron velocity
Ki modified Bessel functions of order i (see page 47)
γLorentz factor = [1 − (v/c)2]−1/2
Ppower radiated
Vvolume
νfrequency (Hz)
Wenergy radiated
Telectron temperature (K)
ni |
ion number density (m−3) |
ne |
electron number density |
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(m−3) |
kBoltzmann constant
hPlanck constant
gGaunt factor
aClassical treatment. The ions are at rest, and all frequencies are above the plasma frequency.
bThe spectrum is approximately flat at low frequencies and drops exponentially at frequencies > γv/b. cDistance of closest approach.
