The_Cambridge_Handbook_of_Physics_Formulas

7.9 Plasma physics
159
Synchrotron radiation
Power radiated
by a single
electron a
... averaged
over pitch
angles
Single electron
emission
spectrum b
Characteristic
frequency
Spectral
function
( P tot =2σ T cu mag γ 2 v
) 2
sin 2 θ (7.286)
c
(
≃ 1.59×10 −14 B 2 γ 2 v
) 2
sin 2 θ W
c
(7.287)
P tot = 4 ( 3 σ Tcu mag γ 2 v
) 2
(7.288)
c
(
≃ 1.06×10 −14 B 2 γ 2 v
) 2
W (7.289)
c
P (ν)= 31/2 e 3 B sinθ
4πɛ 0 cm e
F(ν/ν ch ) (7.290)
≃ 2.34×10 −25 B sinθF(ν/ν ch )
WHz −1
(7.291)
ν ch = 3 eB
2 γ2 sinθ
2πm e
(7.292)
≃ 4.2×10 10 γ 2 B sinθ Hz (7.293)
∫ ∞
F(x)=x K 5/3 (y)dy (7.294)
x
{
≃
2.15x 1/3 (x ≪ 1)
1.25x 1/2 e −x (x ≫ 1)
(7.295)
a This expression also holds for cyclotron radiation (v ≪ c).
b I.e., total radiated power per unit frequency interval.
P tot total radiated power
σ T Thomson cross section
u mag magnetic energy
density = B 2 /(2µ 0 )
v electron velocity (∼ c)
γ Lorentz factor
=[1−(v/c) 2 ] −1/2
θ pitch angle (angle
between v and B)
B magnetic flux density
c speed of light
P (ν) emission spectrum
ν frequency
ν ch characteristic frequency
−e electronic charge
ɛ 0 free space permittivity
m e electronic (rest) mass
F spectral function
K 5/3 modified Bessel fn. of
the 2nd kind, order 5/3
1
0.5
0
F(x)
1 x 2 3 4
7