The Plasma Physics Of Radio Astronomy (1)

 The previous post (Nov. 29),  considering the basic principles pertaining to radio astronomy, focused on the (relatively) simple case of propagation of plane EM waves through a medium assumed to be isotropic. That is, the propagation with E, H vectors is the same in all directions.  In terms of radio sources this is far too limited to be of practical use, so it is important to also examine propagation of a plane wave in an ionized (e.g. plasma) medium in the presence of steady magnetic field.  In this case  we expect the propagation to be variable, depending on direction.  For this purpose the diagram below can be used: 

For perspective, note the direction of wave propagation is in the x-z plane and at an angle of  φ  with respect to the z-axis.  Note the electric field E lies in a plane (yz) perpendicular to the direction of wave propagation. The steady magnetic field B is in the z-direction.   The relevant force equations involving a particle of charge q and mass m in the presence of a magnetic field are:

F 1 = q( v X B)

F 2 = q E

 

F 3 = m d v/ dt

Combining these equations yields:

F  = q(E  + v X B)  =  m d v/ dt

We then divide by q to obtain the force per unit charge:

F/ q =  (E  + v X B)  =  (m/q)  d v/ dt

The preceding vector equation can then be expressed in terms of the 3 scalar components referenced to each axis, i.e.

F x / q = E x   +   v y B z  =   (m/q) dvx / dt

F y / q = E y   -   v x B z  =   (m/q) dvy / dt

F z / q = E z     =   (m/q) dv z / dt

Assume now harmonic motion for the wave and charged particle to obtain the following:

vx = { j  (m w /q) E x  + E y B z}/ B z ² -  m ² w ² /q ²

vy = { j  (m w /q) E y  + E x B z}/ B z ² -  m ² w ² /q ²

^{vz = - j q E}z  /m w

^{The preceding relations enable one to infer the charged particle moves in a helical motion with axis coincident with z-direction, e.g.}

The condition:  

B z ²  =  

m ² w ² /q ²

corresponds to the condition  of gyro-resonance, whereby the gyro frequency:

w g   = (q/ m) B z

See e.g.

Brane Space: Space Plasma Physics Revisited (1) (brane-space.blogspot.com)

From one of Maxwell's curl equations we have, after  letting q=e the electron charge ( 1.6 x  10^-19 C ):

Ñ x H  =  J  +   j  wD    +  N e v  + j w ε_o E  

Where N is the number of  charged particles per unit volume.  We assume a movement of the charged particles constitutes a conduction current.  The  above Maxwell equation can also be expressed in terms of the three rectangular curl components, i.e.

^{1)  (Ñ x H)} x  =  Ne vx  +   j w ε_o Ex  

^{2)  (Ñ x H) y  =}  Ne v y  +   j w ε_o Ey  

^{3)  (Ñ x H) z  =}  Ne vz  +   j w ε_o Ez  

Where the permittivity of free space:  ε_o   =  8.85 x 10^-12 Farad/m

If  we now substitute  the values for vx , v y  and  vz  earlier obtained into equations (1),  (2) and (3) and further assume the frequency w  is much greater than the gyro frequency and further that:

N e ² / ε_o  m  w ² =   1

Then a condition occurs for which the permittivity and index of refraction of the medium become zero.  This critical  frequency we denote by  w_o  so that:

w_o        =       N e ² / ε_o  m

If: 

  w_o        =  2 p f _o 

Then:

 f _o     =    e /2 p    Ö N/ ε_o  m

This can be simplified to:

f _o     =   9 Ö N

Thus, if we know the critical frequency f _o  for a radio source (in cycles per second) we can obtain the plasma  number density of the source.

Suggested Problems:

1) Substitute  the values for vx , v y  and  vz  earlier obtained into equations (1),  (2) and (3) giving Maxwell's  curl H equation in terms of rectangular components to get the 3 resulting plasma frequency equations. Then show if the frequency is much greater than the gyrofrequency,  i.e.

 

w    >>   w g 

All the equations become identical in form

2) The Earth's ionosphere has a plasma number density of  

 10 ¹²     electrons per cubic meter.  Find the (critical)  radio frequency of this source.