MHD-Wave Exposure on the Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 353

propagation velocities. Phase difference ∆Φ between these two modes changes
at the propagation path at

S

∆Φ = ω
c

∆µds, (12.43)

R

where the integral is taken along the ray path; ∆µ is the real part difference
of the refractive index of two modes. Let the frequency ω

ω ci ,
then in the case of quasi-longitudinal propagation (see, e.g., [20]), from (12.6)
we get

ω pi and ω

∆µ = ω pe
ω 2

ω ce
ω

cos θ.

where θ is the angle between the ray path and B 0 .Let α be the angle between
the ray path and the vertical, then ds = dz/ cos α. Now (12.43) can be written
in the form

h s

1
cω 2

ω pe ω ce cos θ ∆Φ =

cos α ds,

(12.44)

0

where the integration is performed up to the satellite height ( h s ) .
The Faraday rotation angle Ω F is

Ω F = 1

2 ∆Φ.
Substitution of ∆Φ from (12.44) into the last equation yields ([19], [21])

h s h s

ω pe ω ce cos θ

1
2 ω 2 c

Ω F =

cos α dz = k

N e Mdz, (12.45)

0 0

where

e 3
2 πc 2 m e f 2

10 4

= 2 . 36

×

M = B 0 cos θ

cos α .
Equation (12.45) is written in the Gaussian system. The numerical value of
the coecient k in SI is the same: k =2 . 36

k = ,

f 2

10 4 /f 2 m − 1 .
The main contribution to (12.42) and (12.45) comes from the altitude
region of up to 2

×

10 3 km [19]. Therefore, it can be assumed that M is
constant up to some limiting height h F , and zero above this height. Then
(12.45) is rewritten as

×

Ω F = kMN F , (12.46)

0

h F

where N F = N e dz. With h F = 2000 km it was found that the value of M

varies by only 3% [19]. Therefore (12.46) can be used, and changes in Ω F

correspond to changes in N F ≈

±

N T .