The Plasma Physics Basis of Radio Astronomy - Redux

Radio astronomy entails a number of aspects associated with plasma physics, which I want to examine in a series of posts. One of these is  Faraday rotation, which involves the rotation of the position angle   q  of the linear polarization of radiation passing through a magnetized (B) plasma. If  the plasma has magnetic induction B and subtends an angle f between B and the direction of wave propagation, then a measurement of q at a wavelength l permits a determination of the total number of charged particles in a column of 1 m ² cross section  between source and observer. This is given by:

N _t =  ò^r _o   Ndr  

Where N _t  is the total number of charged particles in a column of length r and cross section 1  m ².  We note that the natural modes for this polarization are circular as shown in the diagram below, i.e. for 'wave 1' and 'wave 2'  with respective position angles q1  and  q2.  

Then the 2 circularly polarized waves (or opposite rotation) represent the resolution of the linearly polarized wave.  If the two circularly polarized waves have different phase constants (b  and  - b)   then the plane of polarization of the resultant linearly polarized wave rotates as the wave propagates.

Aside: Electromagnetic waves are  polarized when their E- field  components are preferentially oriented in a particular direction.

In linearly or horizontally polarized waves the E- vector is confined to one  (horizontal) plane, i.e.

---------à E

In Vertically polarized waves the E- vector is confined to one  (vertical) plane

^ E

!
!
!

 

In circular polarized waves the E-vector rotates through 360 degrees

Elliptically polarized waves means  any polarization not circular or plane.

An important consideration in any plasma physics context is the index of refraction for the plasma.  This also has consequences for radio observation access.  In the absence of a magnetic field we can write for the index of refraction of the plasma:

n ² =   1   -   w _e ² / w ²

And  w _e     =  [n_e e²/ m_e  ε_o] ^½

is the electron plasma frequency.   We can then write the equation applicable for a magnetic field in the following abbreviated form:

n ² =   1   -   X /  ( 1 +  Y)

For the ordinary wave: n _o ² =   1   -   X /  ( 1 +  Y)

And for the extra ordinary wave: n _x ² =   1   -   X /  ( 1 -  Y)

(Where:  X  =   w _e ² / w ²     and   Y  =  w _H ² / w ²)

Where w _H   denotes the hybrid frequency, which is a mix of the electron plasma and electron cyclotron frequencies, e.g. 

w _H ² =    w _e ² +   w _c ²        ( w _c   =  qB/ m _e)

Note the ordinary wave occurs when the EM wave components  E ₁ and B _o  are parallel, e.g. E ₁  ∥  B _o 

The extraordinary wave occurs when an EM wave (e.g. radio wave) propagates partly transverse, partly longitudinal.  I.e. propagates perpendicular to  B _o   with  E ₁ perpendicular to  B _o.

 It is important to point out that in dealing with magnetically-affected radio waves we expect cutoffs and resonances.  I.e. as a radio wave propagates through a region in  which w _e  and  w _c are changing it may encounter cutoffs and resonances.  A cutoff occurs in a plasma when the index of refraction, n, goes to zero.  That is, when the wavelength becomes infinite.  So if the wave number vector k =  2 p/ l   and the mean index of refraction is: n =  c k / w 

Then when n ->  0:

0 =  c ( 2 p/ l) / w

Or:   2 p/ l   =   w (0)/ c   or:  l   =   2 p c / 0  =   _¥

Conversely,  a resonance occurs when the index of refraction becomes infinite, i.e. when the wavelength  l becomes zero, so that:   

n =  c ( 2 p / 0 ) / w

Or:  n w = c ( 2 p / 0 ) =  2 p c / 0 =   _¥

Thus, for any finite w,  k  ->  _¥   implies  w  ->  w _H

So that the resonance occurs at a point in the plasma where:

w _H ² =    w _e ² +   w _c ²     =  w ²  

The conditions for cutoff and resonance can be written in terms of the ordinary frequency, f (i.e.  w = 2 p f) referenced to electron and cyclotron frequencies.  So that:

The extraordinary wave is cut off when:

f _e ²  /   f _x  ( f _x   - f _H )  =   1

The ordinary wave is cut off when:

f _e ²  /   f _o  ( f _o  +  f _H )  =   1

Referring back to the plane of polarization being Faraday-rotated (as illustrated above) we note the measurement can be done by means of a microwave 'horn' -  a component of many radio telescopes.  This horn gives a value for w _e ²  and hence of the number density of the plasma. 

Other inferences which can be made regarding ordinary and extraordinary waves:

1) If   f  > f _x    then both indices of refraction are real and both modes are propagated.

2) If  f _x  > f >  f _o   then only the ordinary wave propagates.

3) If  f  > f _o    then no waves are propagated.  The magnitude of rotation in radians will then be equal to:

2.38  x  10 ⁶    [ 1 /  f ²   ò n _e  H  dz ]

Where dz is the thickness of the plasma through which the radio wave is moving,

n _e    is the number of electrons per cubic centimeter

H is the magnetic field in gauss.  Faraday rotation has been found important at the very low densities of the interstellar medium and has been used to explain the polarization of microwave radiation generated by maser action, e.g. on OH and H2O molecular clouds.  (See also my Dec. 16, 2021 post on Cosmic Masers.)

Suggested Problem:

1)  A space plasma with electron number density  10 ¹² /cm ³  features a magnetic field of 0.0001 T (Tesla).  Find:

 a) the electron plasma frequency, 

b) the cyclotron frequency and c) the hybrid frequency.

c)Thence or otherwise find the refractive index of the plasma and whether ordinary or extraordinary wave propagation can be expected.