Robert M. Manning

Glenn Research Center, Cleveland, Ohio

Analysis of Electromagnetic Wave Propagation in a 

Magnetized Re-Entry Plasma Sheath Via the 

Kinetic Equation

NASA/TM—2009-216096

December 2009
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Robert M. Manning

Glenn Research Center, Cleveland, Ohio

Analysis of Electromagnetic Wave Propagation in a 

Magnetized Re-Entry Plasma Sheath Via the 

Kinetic Equation

NASA/TM—2009-216096

December 2009

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5285 Port Royal Road

Springfi eld, VA 22161

Available electronically at http://gltrs.grc.nasa.gov

Level of Review: This material has been technically reviewed by technical management. 

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.
NASA/TM—2009-216096 1

Analysis of Electromagnetic Wave Propagation in a Magnetized 

Re-Entry Plasma Sheath Via the Kinetic Equation 

Robert M. Manning 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Based on a theoretical model of the propagation of electromagnetic waves through a hypersonically 

induced plasma, it has been demonstrated that the classical radiofrequency communications blackout that 

is experienced during atmospheric reentry can be mitigated through the appropriate control of an external 

magnetic field of nominal magnitude. The model is based on the kinetic equation treatment of Vlasov and 

involves an analytical solution for the electric and magnetic fields within the plasma allowing for a 

description of the attendant transmission, reflection and absorption coefficients. The ability to transmit 

through the magnetized plasma is due to the ‘magnetic windows’ that are created within the plasma via 

the well-known ‘whistler modes’ of propagation. The case of 2 GHz transmission through a re-entry 

plasma is considered. The coefficients are found to be highly sensitive to the prevailing electron density 

and will thus require a dynamic control mechanism to vary the magnetic field as the plasma evolves 

through the re-entry phase. 

1.0 Introduction 

Electromagnetic wave (EM) propagation through a flowing plasma layer to maintain communications 

and navigation to hypersonic and space re-entry vehicles has been a problem for over 45 years. The 

classical radio frequency (RF) blackout results when the associated EM wave is reflected and absorbed by 

the free electrons that make-up the plasma sheath which envelops the re-entering vehicle. The plasma is 

the result of extreme heating of air by the strong shock wave that is created by the leading edges of the 

vehicle [Bletzinger, et al., 2005 and references therein]. Such a plasma has associated with it a 

characteristic frequency, the ‘plasma frequency’, below which RF transmission is reflected and absorbed; 

RF radiation above this frequency will easily pass through the plasma. Since this plasma frequency is 

proportional to the electron concentration within the plasma, this frequency will vary according to the 

altitude and shape of the vehicle, as well as the angle of attack. Thus, for a given communications 

scenario, a particular frequency is assigned; As the vehicle begins its re-entry into the earth’s atmosphere, 

shock waves are created and a plasma begins to be formed. At a particular electron concentration, the 

plasma frequency will exceed that of the communications link and transmission to and from the vehicle 

will cease. As the vehicle begins to decelerate, the electron density and thus the plasma frequency 

decrease and RF communication to and from the vehicle once again becomes possible. 

Several approaches have been advanced in order to mitigate RF blackout. The obvious ones are to use 

communications frequencies higher than that of the plasma frequency that will develop during re-entry. 

However, as mentioned above, a specific maximum plasma frequency is dependent upon several 

parameters and becomes very difficult to establish. A high power can be used to overcome the reflection 

and absorption of the plasma. This, however, severely complicates the design and operation of a 

communications system, especially the equipment on the re-entering vehicle. Another method is to inject 

electrophilic substances (quenchants) into the plasma flow field of the sheath so as to de-ionize the 

plasma (at least, severely reduce the electron concentration) and lower the plasma frequency below that of 

the communications link. An application of this method occurred during the re-entry of Gemini 3 in 1965 

[Schroeder and Russo, 1968] where it was demonstrated that plasma reflection and absorption can be 

significantly reduced to allow for re-entry communications. Another technique that can be employed is 
NASA/TM—2009-216096 2

that of establishing ‘magnetic windows’ within the plasma. Here, a static magnetic field is applied to the 

plasma to essentially establish ‘whistler modes’ of propagation [Usui, et al., 2000]. Early experiments 

using this concept were performed in 1964 [Russo and Hughes, 1964]. Here, a magnetic field of 750 

Gauss was used in a ground experiment employing a plasma from a solid rocket motor. It was shown that 

a signal improvement of about 20 dB can be realized. However, it was recently argued [Sharkey, 2003] 

that magnetic fields on the order if 104

 Gauss are needed to penetrate a re-entry plasma. 

The ‘magnetic windows’ concept is very attractive in that, unlike the use of electrophilic substances 

where a supply of such material is required throughout the blackout period (which, for some planetary reentry scenarios, can have a duration of 10 to 20 minutes), the magnitude of the static magnetic field can 

be adjusted as conditions require to maintain a window in the otherwise RF opaque plasma. The 

electrophilic technique is an ‘active’ mitigation method whereas the magnetic windows technique is a 

‘passive’ method. If it can be demonstrated that magnetic fields of nominal strength can be used to elicit a 

magnetic window within a plasma with re-entry parameters, and, additionally, it can be shown how the 

magnitude of this magnetic field needs to be adjusted to maintain a window as the plasma properties 

evolve, then the use of an external magnetic field as a mitigation technique will be a viable one. 

This prescription can only be accomplished by realistically modeling the plasma propagation 

environment. It is the purpose of this report to analytically study such a situation. A mathematical model 

will is constructed, based on the Vlasov equations, i.e., the Maxwell equations supplemented with the 

kinetic (Boltzmann) equation describing the electron distribution, that will capture the propagation 

process through a flowing plasma immersed in an external magnetic field. The model will then be solved 

in the approximation of weak spatial dispersion; the case of strong spatial dispersion will be treated in a 

forthcoming report. It is shown that an applied magnetic field of nominal strength can alter the plasma 

and create “magnetic windows” through which electromagnetic radiation can propagate. 

2.0 The Initial Equations 

2.1 Incorporating the Maxwell Equations with Kinetic Theory—The Vlasov Equations 

The basic starting point for modeling electromagnetic wave propagation through a flowing 

hypersonic plasma are, of course, the Maxwell Equations 

E 4πρ=⋅∇v v (2.1) 

B =⋅∇ 0 v v (2.2) 

0 1 = ∂

∂ +×∇

t

B

c

E

v vv (2.3) 

1 4 E B j

ct c

∂ π ∇× − =

∂

v v v v

 (2.4) 

The expressions relating the charge density ρ and current density j

v

 within the plasma are given in terms 

of the statistical distribution functions ( trf ) s υ,, v v governing the charge carriers of the species s at 

position r

v and velocity υ

v at time t

( ) 3 , , ss s

s

en f r td

∞

−∞

ρ = υυ ∑ ∫ v v (2.5) 
NASA/TM—2009-216096 3

( ) 3 , , ss s

s

j en f r td

∞

−∞

=∑ υ υυ ∫ v v v v (2.6) 

Here, s denotes either electron or ion charge carriers where = nn ions is the number of positively charged 

ions, ions +== eee or = nn es is the number of negatively charged electrons, es −== eee . The statistical 

distribution function for species s ( trf ) s υ,, v v is given by the Boltzmann equation incorporating collisions 

via the Krook model [Tanenbaum, 1967; Bhatnagar, et al., 1954] for collisions, viz, 

( ) 0 s rs s s s s

f f F f ff

t υ

∂ + υ⋅∇ + ⋅∇ = −ν −

∂

v v v v vv (2.7) 

where νs is the effective collision frequency and ( υ) v v

, 0 rfs is the initial equilibrium distribution of the sth species. The model for the effective collision frequency used here, along with the basis of the Krook 

model, is discussed in Appendix A. The force entering Equation (2.7) is given by the Lorentz force 

involving both the electric E

v

 and magnetic B

v

 fields of the wave as well as an externally applied 

magnetic field 

B0

v , 

( )⎟

⎠

⎞ ⎜

⎝

⎛ +×

υ

+= BB 0

c

E

m

e F s

s s

vv v vv (2.8) 

Collectively, the Maxwell equations, Equations (2.1) to (2.4), the Equations (2.5) and (2.6), and the 

Boltzmann equation, Equation (2.7) with Equation (2.8) are known as the Vlasov equations. 

These coupled integral and differential equations must now be applied to the situation of a 

hypersonically flowing plasma upon which is incident a plane electromagnetic wave. The situation is 

depicted in Figure 1. The tangential velocity VT

v is due to the plasma flowing along the hypersonic vehicle 

surface. The velocity VL

v

 is due to the motion of the hypersonic vehicle toward the observer. The plasma 

is taken to be of infinite extent along the x and y axes. An incident plane electromagnetic field impinges 

on the moving plasma layer from the transmitter fixed on the earth and travels along the z-axis in the –z

direction. The constant homogeneous magnetic field emanates from an antenna on the surface of the 

vehicle along the z-axis in the +z direction. The surface in the x – y plane is taken to be along the solid 

surface of the vehicle and that at the front of the layer of thickness L is taken to be open to the 

atmosphere. The values of the E

v

 and B

v

 fields of the electromagnetic wave are reckoned with respect to 

the observer in the rest frame with respect to the moving plasma layer. Given these conditions, one can 

write for the functional dependence of the field and current density, 

( ) x ( ,, )ˆ +== y ( , )ytzExtzEtzEE ˆ v v (2.9a) 

( ) x ( ,, )ˆ y ( , )ˆ zBytzBxtzBtzBB ˆ +== + 0

v v (2.9b) 

( ) ( ) ( )ytzjxtzjtzjj x y ,, ˆ +== , ˆ v v (2.9c) 

The problem is most easily dealt with by transforming the fields into the reference frame moving with the 

plasma. 
NASA/TM—2009-216096 4

2.2 Transformation of the Fields Into the Reference Frame Moving with the Plasma 

The transformation of the fields E

v

 and B

v

 in the reference frame of the observer (transmitter) to 

those E′

v and B′v that are seen in the frame moving with the plasma layer is given by the well-known 

relations [Jackson, 1975] 

( ) BEE ( ) E vv v v vv v

⋅ββ

+γ

γ ′ −×β+γ= 1

2

 (2.10) 

( ) EBB ( ) B vv v v vv v

⋅ββ

+γ

γ ′ γ= ′ −×β− 1

2

 (2.11) 

where 

( ) 2 1, ˆ ˆ β−≡γ + ≡β

c

v LT zVxV (2.12) 

and c is the velocity of light. Also, for the current density 

′ ( ρβ−γ= ) v v v jj (2.13) 

The invariance of the Maxwell equations has that, in the frame moving with the plasma, 

0 1 = ∂ ′

∂ ′ ′×∇ ′ +

t

B

c

E

v vv (2.14) 

j

ct

E

c

B ′ π = ∂ ′

∂ ′ ′×∇ ′ − v v vv 41 (2.15) 

where ∇′

v and t′ are the appropriately transformed divergence and time. Consider now the extreme case 

in which a hypersonic velocity of Mach 35 is realized, i.e., 4 VV LT ×≈= 102.1 m/s. This yields 

1100.4 5 <<×≈β − . In this case, the transformed equations Equations (2.13) and (2.14) essentially reduce 

to those in the rest frame, i.e., Equations (2.3) and (2.4). Similarly, for the homogeneous applied magnetic 

field, B0 0 ′ ≈ B v v . Hence, to within a first order approximation, Equations (2.1) to (2.4) can be employed in 

the reference frame moving with the plasma. One can now incorporate these relations in the evaluation of 

Equation (2.7). 

One should also address the Doppler shifts that will be incurred in the transformation to the reference 

frame moving with the plasma. This will not be addressed at this point as it is the goal of this work to 

establish the possibility of the magnetic windows concept. A more careful study with will appear in a 

forthcoming work will incorporate Doppler effects. 

The final goal in the analysis that follows is the evaluation of the reflection, transmission, and 

absorption coefficients of the plane wave interacting with the plasma layer; the ‘flow chart’ of this 

process is depicted in Figure 2. The scattering process, as well as its solution, goes as follows. The 

electric and magnetic fields of the plane electromagnetic wave induces a Lorentz force on the plasma 

electrons. The induced force, in turn, produces a variation in the electron distribution function governing 

the position and momentum of the electrons. This gives rise to an induced current within the plasma that 

produces an associated variation of the fields within the plasma which, once again, induces an additional 

Lorentz force. This non-linear process finally results in the establishment of field distributions at the 
NASA/TM—2009-216096 5

boundaries of the plasma that determines the corresponding reflection, transmission, and absorption 

coefficients of the incident field. 

3.0 Application of the Foregoing to the Boltzmann Equation 

3.1 Reduction of the Equation 

Since the current density is only a function of the longitudinal coordinate z, so too will be the 

distribution function of Equation (2.7) 

( ) ( tzftrf ) s s υ = υ,,,, v v v (3.1) 

The associated equilibrium distribution function is taken to be given by the Maxwell distribution 

( ) ( )

3/2 3 2

0 0 2

1 1 s s exp

Ts Ts

f f ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ υ

υ= υ = − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ π υ⎝ ⎠ ⎜ ⎟ υ ⎝ ⎠

v v (3.2) 

where the thermal velocity of the s-th species is given by 

s

sB

Ts

m

2 Tk

≡υ (3.3) 

Here, kB is Boltzmann’s constant, Ts is the absolute temperature of the s-th species of charges with mass 

ms . The distribution function ( tzf ) s υ,,

v will be taken to be related to that of Equation (3.2) by a small 

perturbation ( tz ) s υφ ,,

v , viz, 

( ) ( ) ( ) ( ) (υ<<υφυφ+υ=υ ) v v v v v 0 0 ,, ,,,,, s s sftztzftzf (3.4) 

In order for ( tz ) s υφ ,,

v to be treated as a perturbation, the fields of the incident wave must also be treated 

as a first order perturbation; writing the temporal dependence of the fields of the incident electromagnetic 

wave as 

E zt E z i t x x ( , exp , etc. ) = −ω ( ) ( )

one has from Equations (2.9a) and (2.9b) 

E = + −ω (E zx E z it x y ( ) ˆ ( )) exp( ) v (3.5) 

B = + −ω + (B zx B z x y ( ) ˆ ˆ ( ) y) exp( it B z ) 0 ( ) v (3.6) 

in which one must now require for a perturbation solution, 0 ,,, yxyx << BBBEE . Finally, since the timeharmonic fields of Equations (3.5) and (3.6) are taken to be the source of perturbation of the distribution 

function given by Equation (3.4), one can further write 

φ υ =φ υ −ω s s (z t z it , , , exp ) ( ) ( ) v v (3.7) 
NASA/TM—2009-216096 6

Substituting Equations (3.4) to (3.7) into Equation (2.7) and dropping quantities that are second order 

perturbations with respect to the fields and to φ ( υ) v z, s yields 

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) () ( )

0 0

00 0

, , exp exp

exp

ˆ ˆ , exp 0

s s s z

s s s

s

s s s

s

z

i i z it i t

z

e Ez f Bz f i t

m c

e Bz f Bz z i t

m c c

υ υ

υ υ

∂φ υ − ω+ ν φ υ − ω +υ − ω

∂

⎛ ⎞⎡ ⎤ υ

+ ⋅ ⎜ ⎟ ∇ υ+ × ⋅∇ υ −ω ⎢ ⎥ ⎝ ⎠⎣ ⎦

⎛ ⎞⎡ ⎤ υ υ + × ⋅∇ υ + × ⋅∇ φ υ − ω = ⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦

v v

v v

v v

v vv vv v v

v v v v v v

 (3.8) 

However, remembering the identity ( ) =υ⋅×υ 0 v v v B , one has that 

( ) ( ) ( ) 0

0 0 s s

f Bz f Bz

c c υ

υ υ υ ∂ × ⋅∇ υ = × ⋅ =

υ ∂υ

v

v v w vv v v (3.9) 

and similarly for the term ( ) υ (υ∇⋅×υ ) v v v v 0 0 ˆ sfzBc . (The term (υ c Bz z )× ⋅∇ φ υ ≠ 0 ˆ υ s ( , 0 ) v v v v since φ ( υ) v z, s

is not a function of the scalar υ .) Hence, Equation (3.8) reduces to 

( )( ) ( ) ( )

() ( ) 0 ( )

, , exp

, ,0 ˆ

s ss z

s s s s

s s

z

i i z it

z

e e E z z Bz z

m mc υ υ

∂φ υ − ω+ ν φ υ +υ − ω ∂

⎛⎞ ⎛⎞⎛ ⎞ υ + ⋅∇ φ υ + × ⋅∇ φ υ = ⎜⎟ ⎜⎟⎜ ⎟ ⎝⎠ ⎝⎠⎝ ⎠

v v

v v

v r v v v v (3.10) 

Defining the cyclotron frequency 

cm

Be

s

s cs 0 ≡ω (3.11) 

and expanding ( ) syxxys z υ

v φ∂υ∂υ−∂υ∂υ=φ∇⋅×υv v ˆ as well as υ =∇⋅ s0 v fEv v ( ) syyxx 0 E ∂υ∂+∂υ∂ fE , 

Equation (3.10) becomes 

( ) s z s s cs y x s

x y

i i

z

∂φ ∂ ∂ ⎛ ⎞

υ − ω+ ν φ +ω υ −υ φ + ⎜ ⎟ ∂ ∂υ ∂υ ⎝ ⎠

0 0 s x ys

sxy

e EEf

m

⎛ ⎞⎛ ⎞ ∂ ∂ ⎜ ⎟⎜ ⎟ + =

⎝ ⎠ ∂υ ∂υ ⎝ ⎠

 (3.12) 

This differential equation is most easily dealt with in plane-polar coordinates υ⊥ and θ defined by 

x ⊥ y ⊥ sin,cos θυ=υθυ=υ . In this instance, one has 

∂θ

∂

υ

θ

+

∂υ

∂ θ=

∂υ

∂

∂θ

∂

υ

θ − ∂υ

∂ θ=

∂υ

∂

⊥⊥ ⊥⊥

cos sin, sin

cos x y

 (3.13) 

allowing Equation (3.12) to be written as (noting that s0 f is isotropic and independent of the polar angle 

θ) 

( ) ( ) 0 cos sin 0 s s s s z s s cs xy

s

e f i i E E

z m ⊥

∂φ ∂φ ∂ ⎛ ⎞ υ − ω+ ν φ −ω + θ+ θ = ⎜ ⎟ ∂ ∂θ ∂υ ⎝ ⎠

 (3.14) 
NASA/TM—2009-216096 7

Finally, using the auxiliary fields [Gross, 1951; Bell and Buneman, 1964] defined by 

≡ − , ≡ + iEEEiEEE yxRyxL (3.15) 

gives 

x EE y =θ+θ ( ) L exp() ( ) R exp iEiE θ−+θ

2

1

cos sin (3.16) 

Substituting this into Equation (3.14) and rearranging terms yields 

( ) ( ) ( )( ) , , , s s cs z s s

z z ii z

z

∂φ υ ∂φ υ ω −υ + ω+ ν φ υ ∂θ ∂

v v v ( ) () ( ) 0 exp exp 0

2

s s L R

s

e f E iE i

m ⊥

⎛ ⎞ ∂ = ⎜ ⎟ θ + −θ =

⎝ ⎠ ∂υ (3.17) 

This form of the differential equation can now be solved using the method of characteristics. Writing the 

equation as a relation along its characteristic curve, one has 

s s s ss cs z

d z

dt t z t z

φ ∂φ ∂θ ∂φ ∂ ∂φ ∂φ = + =ω −υ

∂θ ∂ ∂ ∂ ∂θ ∂

Therefore, 

tzt cs θ = −υz θ = ω + ′, (3.18) 

which combines to give ( ) υω−=θ z + θ′ zcs . Changing back to the variables θ and z, ( dzddtd ) υ−= z

and Equation (3.17) becomes the first order differential equation 

( ) ( )( )

( ) ( ) ( ) ( ) ( ) 0

, ,

exp exp 0 2

s z ss

s s L csz R cs z

s

d z

ii z

dz

e f E i zi E i zi

m ⊥

φ υ −υ + ω+ ν φ υ

⎛ ⎞ ∂ = − ω υ +θ + ω υ −θ = ′ ′ ⎜ ⎟ ⎝ ⎠ ∂υ

v v

 (3.19) 

This equation must now be solved across the plasma layer of thickness L, 0 ≤ z ≤ L. 

3.2 Solution of the Differential Equation Across the Plasma Layer 

Consider the Fourier series of the function φ ( υ) v z, s over the interval 0 ≤ ≤ Lz ; 

() ( ) ( ) , , exp , s s l ll

l

l z iz

L

∞

=−∞

π φ υ= φ κ υ κ κ≡ ∑ v v % (3.20) 

( ) () ( )

0

1

, , exp

L

sl s l z i z dz

L φ κ υ = φ υ −κ ∫ v v % (3.21) 

Thus, multiplying Equation (3.19) by exp −iκ l ( z) and integrating over z, using Equation (3.21) gives 
NASA/TM—2009-216096 8

( ) ( ) ( ) ( )( )

0

, 0, , 1

2

s l

sl l s s

z

s s cs cs Ll Rl

zs z z

i i

iLL

Le f E E

m ⊥

⎡ ⎤ ω+ ν

φ κ υ κ − =φ υ −φ υ − − ⎢ ⎥

υ ⎣ ⎦

⎛⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∂ω ω ⎡ ⎤

− κ+ + κ− ′ ′ ⎜ ⎟⎜ ⎟ ⎢ ⎜ ⎟ ⎜ ⎟⎥ ⎝ ⎠ υ ∂υ υ υ ⎝ ⎠ ⎣ ⎝ ⎠ ⎝ ⎠⎦

v vv %

% %

 (3.22) 

where exp exp 1 ( ) ( )( )l

l −κ = −π = − iL l and () () ( ) exp , EE i L L

% % ′ κ ≡ κ θ () () ( ) exp EE i R R% % ′ κ ≡ κ θ − . At 

this point, contact must be made to the current density expression given by Equation (2.6); converting to 

plane polar coordinates, 

( ) ( ) ( )

2

0 0

ˆ ˆ cos sin , , , s s s z z

s

j z en x y f z t d dd

∞ π∞

⊥ ⊥ ⊥ ⊥

−∞

= υθ ∑ +θ υ υ υ υ θ υ ∫∫∫ v (3.23) 

As before, considering the identity 

( ) ()( ) ( ) 1 ˆ ˆ ˆˆ ˆˆ cos sin exp exp 2

x θ+ θ= − θ + + − θ y x iy i x iy i ⎡ ⎤ ⎣ ⎦

and using it in Equation (3.23) and taking the scalar product with the particular vector xˆ + iyˆ yields, 

() () () () ( )

2

2

0 0

R ,,, exp , , , x y ss sz z

s

j z t j z t ij z t e n i f z td dd

∞ π∞

⊥ ⊥ ⊥

−∞

≡ + = υ θ υ υ υ θυ ∑ ∫∫∫ (3.24) 

Thus, the current density has been converted to a complex scalar current density in the transverse plane. 

Substituting now Equations (3.4) and (3.7) into Equation (3.24), and using Equation (3.2), one as that the 

( ) υ

v s0 f term vanishes upon performing the υz integration. Hence, what survives is the result 

( ) ( ) ( )

2

2

0 0

R exp , , s s s z z

s

j z en i z d dd

∞ π∞

⊥ ⊥ ⊥

−∞

= υθ ∑ φ υ υ υ θ υ ∫∫∫ (3.25) 

where R( ) () ( ) , = R exp ω− tizjtzj . Thus, the current density associated with the polarized wave fields ER

and EL of Equation (3.15) are also polarized; however ER and EL remain uncoupled. Therefore, one can 

use either ER or EL separately in Equation (3.22). Selecting ER, i.e., the whistler mode += iEEE yxR , 

Equation (3.22) becomes, upon using the definition κ ≡ κ −ω υzcsll′ , 

( )( ) ( ) ( ) 1 1 0 ,, , 2

s s sl z l s s z R l

z s

e f iL L E

m

−

⊥ ⊥

⊥

⎧⎪ ⎛⎞ ⎛ ⎞ ⎛ ⎞ ∂ ⎫⎪ φ κ υ υ = κ −Ω Δ υ υ − κ ′′ ′ ⎡⎤ ′ ⎣ ⎦ ⎨ ⎜⎟ ⎜ ⎟ ⎜ ⎟ ⎬ ⎩ ⎭ ⎪ ⎝⎠ ⎝ ⎠ υ ∂υ ⎝ ⎠ ⎪

% % (3.26) 

where 

cs s s z

ω−ω + νi Ω ≡

υ

 (3.27) 

and 
NASA/TM—2009-216096 9

( ) ( ) ( )( )

l

⊥ szs ⊥ sz L ⊥ z −υυφ−υυφ≡υυΔ 1,,,,0, (3.28) 

Now, rewriting Equation (3.25) by limiting the υz integration in velocity space to the interval 0 ≤ υ <∞ z

gives 

( ) ( ) ( )

2

2

000

R exp , , s s s z z

s

j z en i z d dd

∞ π∞

= υθ ∑ ⊥ Φ υ⊥ ⊥ υ υ θ υ ∫∫∫ (3.29) 

where 

Φ υ υ ≡ φ υ υ + φ υ −υ s (zzz ,, ,, ,, ⊥⊥⊥ zs zs z ) ( ) ( ) (3.30) 

The Fourier transform of Equation (3.29) is, using the form of Equation (3.21), 

( ) () ( )

2

2

000

R exp , , l ss sl z z

s

j en i d d d

∞ π∞

κ =∑ ∫∫∫ υ θΦ κ υ υ υ θυ ⊥ ⊥ ⊥ % % (3.31) 

where 

Φ κ υ υ ≡ φ κ υ υ + φ κ υ −υ s ( l z sl z sl z ,, ,, ,, ⊥⊥⊥ ) ( ) ( ) % % % (3.32) 

Substituting Equation (3.26) into Equation (3.32) yields, after expanding terms and simplifying, 

( ) ( ) ( )( )

( ) ( ) ( )( )

2 2 , , , ,

2 , ,

l

sl z s z s z

s l

s s sR l s z s z

z

i

L

K E

L

⊥ ⊥ ⊥

⊥ ⊥

⎧κ Φ κ υ υ = Δ υ υ + Δ υ −υ ⎨ Ω −κ ⎩

Ω Ω ⎫ − κ + Δ υ υ − Δ υ −υ ′ ⎬

υ ⎭

%

%

 (3.33) 

where 

0

2

s s s

s

e f K

m ⊥

⎛ ⎞ ∂ ≡ ⎜ ⎟ ⎝ ⎠ ∂υ

 (3.34) 

Using Equation (3.33) in Equation (3.31) will give a relation for the current density. However, values for 

the parameters ( ) , Δυυ s ⊥ z must first be determined. By their definition of Equation (3.28), they are 

functions of the boundary values of the functions φs . 

3.3 Incorporation of Boundary Conditions 

By the definition of the problem, the plasma at the surface z = 0 is against the solid body of the 

hypersonic vehicle. Thus, reflective boundary conditions prevail from which one can write 

φ υ υ = φ υ −υ s (0, , 0, , ⊥ ⊥ zs z ) ( ) (3.35) 

However, at the boundary z L = , the plasma is open to the atmosphere so diffusive conditions prevail 

which give 
NASA/TM—2009-216096 10

φs z (L,, 0 υ −υ = ⊥ ) (3.36) 

With these assignments, Equation (3.33) becomes 

( ) ( ( ) ( )( ) )

( ) ( )( )

2 2 , , 2 0, , , , 1

2 , , 1

l l

sl z s z s z

s l

s s l

sR l s z

z

i L

L

K E L

L

⊥ ⊥ ⊥

⊥

⎧κ Φ κ υ υ = φ υ υ −φ υ υ − ⎨ Ω −κ ⎩

Ω Ω ⎫ − κ− φ υυ − ′ ⎬

υ ⎭

%

%

 (3.37) 

At this point, the values of the distribution functions at the boundaries, φs (0, , υ υ ⊥ z ) and φ υυ s (L, , ⊥ z )

still remain unknown. In order to determine these values, a self-constancy condition can be applied. 

Hence, using Equations (3.35) and (3.36) with Equation (3.32) gives 

Φ υ υ =φ υ υ s (0, , 2 0, , ⊥ ⊥ zs z ) ( ) (3.38) 

and 

Φ υ υ =φ υ υ s (L L ,, ,, ⊥ ⊥ zs z ) ( ) (3.39) 

But by Equation (3.20), 

() ( ) 0, , , , s z sl z

l

∞

⊥ ⊥

=−∞

Φ υ υ = Φ κυ υ ∑ % (3.40) 

and 

( ) ( )( ) , , ,, 1 l

s z sl z

l

L

∞

⊥ ⊥

=−∞

Φ υ υ = Φ κυ υ − ∑ % (3.41) 

Hence, form these relations, one has 

( )( ) , , 2 0, , s l zs z

l

∞

⊥ ⊥

=−∞

∑ Φ κυ υ =φ υ υ % (3.42) 

and 

( )( ) ( ) ,, 1 ,, l s lz s z

l

L

∞

⊥ ⊥

=−∞

∑ Φ κ υ υ − =φ υ υ % (3.43) 

Equation (3.37) can now be used on the left sides of Equations (3.42) and (3.43) to give two equations in 

the two unknowns φ υυ s (0, , ⊥ z ) and φ υυ s (L, , ⊥ z ). To this end, substituting Equation (3.37) into 

Equation (3.42) and solving for ( ) 0, , φ υυ s ⊥ z , 

( ) ( ) ( )( ) 2 2 0, , , , 1 2

s s l s z sR l s z

l s z l

i KE L

L

∞

⊥ ⊥

=−∞

⎛ ⎞ ⎧⎪⎛ ⎞ Ω Ω⎛ ⎞ ⎫⎪ φ υ υ =− κ + φ υ υ − ⎜ ⎟ ⎨⎜ ⎟ ′ ⎜ ⎟ ⎬ ⎝ ⎠ Ω −κ υ ⎪⎩ ⎭ ⎝ ⎠ ⎝ ⎠ ⎪ ∑ % (3.44) 
NASA/TM—2009-216096 11

where the terms that contain the factors κl vanish, i.e., 

( )

2 2 2 2

1

0, 0

l

l l

l l s s l l

i i ∞ ∞

=−∞ =−∞

κ κ − = = ∑ ∑ Ω −κ Ω −κ (3.45) 

Similarly, using Equation (3.37) in Equation (3.43) and solving for φs (L, , υ υ ⊥ z ) yields 

( ) () () ( ) 2 2 , , 2 1 , , s s l

s z s Rl s z

l s z l

i L KE L

L

∞

⊥ ⊥

=−∞

⎛ ⎞ ⎧⎪⎛ ⎞ Ω Ω⎛ ⎞ ⎫⎪ φ υ υ =− − κ + φ υ υ ⎜ ⎟ ⎨⎜ ⎟ ′ ⎜ ⎟ ⎬ ⎝ ⎠ Ω −κ υ ⎪⎩ ⎭ ⎝ ⎠ ⎝ ⎠ ⎪ ∑ % (3.46) 

At this point, it will facilitate further evaluations to simplify Equations (3.44) and (3.46) by 

analytically performing summations over the parameters l where possible. Hence, using the summations 

( ) () ( ) 2 2 2 2 cot , 1 csc s s l s s

l l s s l l

i i iL L iL L

∞ ∞

=−∞ =−∞

⎛ ⎞ Ω Ω ⎛ ⎞ ⎜ ⎟ = Ω ⎜ ⎟ −= Ω

Ω −κ Ω −κ ⎝ ⎠ ⎝ ⎠ ∑ ∑ (3.47) 

allow Equations (3.44) and (3.46) to be written, 

( ) ( ) ( )( ) 2 2

1 0, , , , csc 2

s s z s R l s z s

l s z l

i i KE L L

∞

⊥ ⊥

=−∞

⎛ ⎞ Ω ⎛ ⎞ φ υ υ =− ⎜ ⎟⎜ ⎟ ′ κ−φ υυ Ω

⎝ ⎠ Ω −κ υ⎝ ⎠ ∑ % (3.48) 

and 

( ) () () ( ) ( ) 2 2

1 , , 2 1 , , cot s l

s z s Rl s z s

l s z l

i L K E iL L

∞

⊥ ⊥

=−∞

⎛ ⎞ Ω ⎛ ⎞ φ υ υ =− ⎜ ⎟ − κ −φ υ υ Ω ′ ⎜ ⎟ ⎝ ⎠ Ω −κ υ⎝ ⎠ ∑ % (3.49) 

Finally, the solution of these relations for φs (0, , υ υ ⊥ z ) and φs (L, , υ υ ⊥ z ) is straightforward; after using 

some trigonometric identities, integral 

( ) () , , exp 2 1 1 ( ) ( )( ) ( ) l s z s sR l

l

L i L C l E

∞

⊥

=−∞

φ υυ= Ω − − κ ∑ ′ % (3.50) 

( ) () ( )( ) ( ) 0, , ( 1 exp 2 1) l

s z s Rl s

l

C lE i L

∞

⊥

=−∞

φ υυ= κ − Ω − ∑ ′ % (3.51) 

where 

Cs ( )l ≡ iΩs

Ωs

2 −κ l2

⎛

⎝

⎜

⎞

⎠

⎟

1

υ z

⎛

⎝

⎜

⎞

⎠

⎟ Ks

 (3.52) 

Equations (3.50) and (3.51) give the sought-after expressions for the boundary values of φs needed in the 

evaluation of Equation (3.37). 
NASA/TM—2009-216096 12

3.4 Calculation of the Current Density 

Using Equation (3.2) in Equation (3.34) and evaluating Equation (3.52), one has 

( ) ( )

2

0 2 2

s s 1 s s s Ts z s l

e i Cl f

m

⊥ ⎛ ⎞⎛ ⎞ ⎛ ⎞ υ Ω ⎛ ⎞

=− υ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ υ υ Ω −κ ⎝ ⎠⎝ ⎠

v (3.53) 

One can now substitute Equations (3.50) and (3.51) into Equation (3.37) and this intermediate result into 

Equation (3.31), upon remembering the definition ER l Rl ′(κ ) ≡ κ −θ E i ( ) exp( ) % % , to finally obtain the 

Fourier Transform of the current density within the plasma layer, viz, 

( ) ( ) ( )

( )( ) ( )

2 2

3 0 2 2

000

2 2 2 2

1 1 2

1 1 2 1 exp 1

1

s s R l ss s R l

s Ts s z s l

l l s

R l s

s s z l l l

s l

e j en f i E

m

i i E iL

L

i

L

∞ π∞

⊥

∞ ′ ′

′=−∞ ′

⎛ ⎞⎛ ⎞ ⎧⎪ ⎛ ⎞ Ω ⎛ ⎞

κ =− υ υ − κ ⎜ ⎟⎜ ⎟ ⎨ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ υ υ ⎪ ⎝ ⎠ Ω −κ ⎝ ⎠ ⎩

⎛ ⎞ ⎛ ⎞ κ Ω ⎛ ⎞ ⎛ ⎞ ⎡ ⎤ + κ − Ω− ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Ω −κ Ω −κ ⎜ ⎟ υ ⎣ ⎦ ⎢ ⎥ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞ −⎜ ⎟⎜ ⎟ ⎝ ⎠ Ω −κ ⎝ ⎠

∑ ∫∫∫

∑

v % %

%

( ) ( ) ( ) () ( ) 2 2

1 exp 2 1 1 1 l l s s R l z

s z l l

i i L E d dd

∞ ′ ′ ⊥

′=−∞ ′

⎛ ⎞ ⎫ Ω ⎛ ⎞ ⎪ Ω − − − κ υ θυ ⎜ ⎟⎜ ⎟ ⎬ Ω −κ ⎝ ⎠ υ ⎝ ⎠ ⎪⎭

∑ %

 (3.54) 

Using Equation (3.2) once again and performing the υ⊥ and θ integrations gives 

( ) ( )

( )( ) ( )

1/2 2

2 2 2

0

2 2 2 2

11 1 1 exp 2

1 1 2 1 exp 1

1

z s R l ss R l

s s Ts z Ts s l

l l s

Rl s

s s z l l l

s l

j en i E

m

E i L

L

∞

∞ ′

′

′=−∞ ′

⎛⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ υ Ω ⎧⎪ ⎛ ⎞⎛ ⎞

κ= − κ ⎜⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎨ ⎜ ⎟⎜ ⎟ ⎝⎠ ⎝ ⎠ ⎝ ⎠ π υ υ Ω −κ ⎜ ⎟⎝ ⎠ υ ⎝ ⎠⎪⎩ ⎝ ⎠

⎛ ⎞ κ Ω ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎡ ⎤ + κ ⎜ ⎟ ⎜ ⎟⎜ ⎟ − Ω − ⎜ ⎟ Ω −κ ⎜ ⎟ Ω −κ υ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞

−⎜

⎝ ⎠ Ω −κ

∑ ∫

∑

% %

%

( ) ( ) ( ) () ( ) 2 2

1 1 exp 2 1 1 1 l l s s Rl z

s z l l

i i L E d

L

∞ ′ ′

′=−∞ ′

⎛ ⎞ ⎫ ⎛ ⎞ Ω ⎛ ⎞ ⎪ ⎟⎜ ⎟ Ω −− − κ υ ⎜ ⎟⎜ ⎟ ⎬ ⎝ ⎠ Ω −κ ⎝ ⎠ υ ⎝ ⎠ ⎪⎭ ∑ %

 (3.55) 

At this point, it facilitates further calculation to change variables using the prescription 

z s Ts

w

υ ≡ υ

 (3.56) 

which gives 

( ) cs s s ss s Ts

i

w

w

ω−ω + ν Ω =Ω =

υ

 (3.57) 

Incorporating Equations (3.56) and (3.57) into Equation (3.55) yields 

R () () () ( ) ( ) , l l Rl ll R l

l

jE E

∞

′ ′

′=−∞

% κ =σ κ κ − σ κ κ κ % % ∑ (3.58) 

where 
NASA/TM—2009-216096 13

( ) ( )

( ) ( )

1/2

2 2

2 2

0

11 1 1 2 exp s s l ss s s

s s Ts s s s l

w

ie n w dw

m w w

∞ ⎛⎞ ⎛ ⎞ ⎛⎞ ⎛ ⎞ Ω

σκ ≡ − ⎜⎟ ⎜ ⎟ ⎜⎟ ⎜ ⎟ ∑ ⎝⎠ ⎝ ⎠ ⎝⎠ ⎝ ⎠ π υ Ω −κ ∫ (3.59) 

is the single mode conductivity and 

( ) ( )

( ) ( ) ( ) ( )

( () ( ) ( )) ( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( )

1/2

2

2 2 2 2

0

2

2 2

11 1 1 , 2

1 1 1 exp 1 exp 2

1 1 exp

ls s

l l ss

s s Ts ss ss l l

l ss ss

s

s s l l s s

ss ss s l l

w

e n

m L w w

i wL i wL

w

w

w dw

w w w

∞

′

′

′

+ ′

′

⎧ ⎛⎞ ⎛ ⎞ ⎛⎞ ⎛⎞ ⎪ κ Ω

σκ κ ≡ ⎜⎟ ⎜ ⎟ ⎜⎟ ⎜⎟ ⎨ ⎝⎠ ⎝ ⎠ ⎝⎠ ⎝⎠ π υ ⎪ Ω −κ Ω −κ ⎩

⎛ ⎞ ⋅ −− Ω − − Ω ⎜ ⎟ ⎝ ⎠

Ω ⎫ ⎛ ⎞ ⎪ ⋅ − ⎜ ⎟ ⎬ − Ω −κ Ω −κ ⎝ ⎠ ⎪⎭

∑ ∫

 (3.60) 

is the multi-mode conductivity. Thus, the current density anywhere within the magnetized plasma layer 

due to the incident electromagnetic wave is given by 

R () ( ) ( ) ( ) ( ) ( ) ( ) l Rl l exp ll R l l , exp

l l l

j z E iz E iz

∞ ∞ ∞

′ ′

=−∞ =−∞ =−∞ ′

= σκ κ κ − σκ κ κ κ ∑ ∑ % % ∑ (3.61) 

 At this point, contact must be made with the current density entering into Maxwell’s equations. 

3.5 Connecting the Current Density in the Plasma Layer with the Maxwell Equations 

One first needs to relate the auxiliary complex electric field ERx y (z E z iE z ) = + ( ) ( ) of the incident 

wave to the governing Maxwell Equations. Returning to Equations (2.3) and (2.4) and employing 

Equations (3.5) and (3.6) gives in terms of the component fields 

ˆ ˆ ˆˆ ( ) y x x y

E E

x y ik B x B y

z z

∂ ∂

−+ = +

∂ ∂ (3.62) 

and 

( ) ( ) 4 ˆ ˆ ˆ ˆ ˆˆ y x x y xy

B B

x y ik E x E y j x j y

z z c

∂ ∂ π

− + =− + + +

∂ ∂ (3.63) 

Multiplying these relations by the auxiliary vector xˆ ˆ + iy , as used earlier, gives, respectively, 

0 R

R

E kB

z

∂ − = ∂ (3.64) 

and 

R 4

R R

B kE i j

z c

∂ π

+ =−

∂ (3.65) 
NASA/TM—2009-216096 14

where BRx y (z B z iB z ) ≡ + ( ) ( ) and jRxy (z j z ij z ) ≡ + ( ) ( ) . From these two relations, the corresponding 

wave equation for ER can be derived in the usual way, 

2

2

2

R 4

R R

E k E ik j

z c

∂ π

+ =−

∂ (3.66) 

As the equations remain amenable in the Fourier domain, the transform of Equation (3.66) is now needed 

to be used with Equation (3.58): thus, applying 

( ) () ( )

0

1

exp

L

ERl R l E z i z dz

L

κ = −κ ∫ % (3.67) 

by multiplying Equation (3.66) by exp( ) l −i z κ , integrating by parts, and using Equation (3.64) yields 

( )( ) ( ) 10 10 ( )( ) ( ) l l

RR RR

l

BL B EL E

k i

L L

⎛ ⎞ ⎛ ⎞ − − − − ⎜ ⎟ + κ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

() () ( ) 2 2 4

R l Rl Rl l E k E ik j

c

π −κ κ + κ =− κ % % % (3.68) 

Finally, substituting Equation (3.58) into Equation (3.68) and rearranging terms gives 

() () ( )( ) ( )

( )( ) ( ) ( )()

2 2 4 1 0

1 0 4 ,

l

R R

l Rl l

l

R R

l ll R l

l

BL B

k ik E k

c L

EL E

i ik E

L c

∞

′ ′

′=−∞

⎛ ⎞ ⎛ ⎞ π − − κ − − σκ κ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ − − π

+ κ ⎜ ⎟ − σ κ κ κ

⎝ ⎠

∑

%

%

 (3.69) 

This equation connects the fields on the surface boundaries of the plasma layer to the Fourier transform of 

the fields within the plasma. This relation can be put into a more familiar form simply by dividing 

through by 2 k using Equation (3.57) and rearranging some factors to obtain 

() () ( ) ( )

( )( ) ( ) ( )( ) ( )

2

2 ,

10 10

l

l R l ll R l

l

l l

l R RR R

E SE

k

EL E BL B

i

k kL kL

∞

′ ′

′=−∞

⎛ ⎞ κ

⎜ ⎟ −ε κ κ + κ κ κ

⎝ ⎠

⎛ ⎞ κ −− −− = + ⎜ ⎟

⎝ ⎠

% % % ∑

 (3.70) 

with 

( ) ( ) ( )

( )

2 2

2 2 22

0

2 exp 1 ps cs s s

l s

s cs s s Ts l

i w

dw

i w

∞ ω ω−ω + ν −

εκ ≡ −

π ω ω−ω + ν −υ κ

∑ ∫ % (3.71) 

and 
NASA/TM—2009-216096 15

( ) ( ) ( )

( )

( )

2 2 2

2 2 22 22 2 2

0

2 ,

1 1 exp

1

1 exp 2

ps cs s l s Ts

l l

s cs s s cs s s Ts Ts l l

l cs s

Ts s

l l

Ts s cs s

cs s Ts l s Ts s

i w

S i

L iw iw

i i L

w

w i i L

iw w

∞

′

′

′

+ ′

⎧ ⎛ ⎞ ω ω−ω + ν ⎪ κ υ

κκ ≡ ⎜ ⎟ ⎨

π ⎜ ⎟ ⎝ ⎠ ω ω−ω + ν −υ κ ω−ω + ν −υ κ ⎪⎩

⎡ ⎤ ⎛ ⎞ ω−ω + ν ⋅ −− ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ υ ⎣ ⎦

υ − ⎡ ⎤ ⎛ ⎞ ω−ω + ν ⎫⎪ − − ⎢ ⎥ ⎜ ⎟ ⎬

ω−ω + ν −υ κ υ ⎣ ⎦ ⎝ ⎠

∑ ∫

( ) 2 exp −w dw s s ⎪⎭

 (3.72) 

where the plasma frequency ωps of the s-th charge species is given by 

2

2 4 s s ps s

e n

m

π

ω ≡ (3.73) 

Equation (3.70) forms the set of equations for the transform of the electric field within the plasma layer 

that will be considered in what is to follow. 

Unfortunately, the evaluation of the integrals indicated in Equations (3.71) and (3.72) cannot be 

completed in its entirety. Although Equation (3.71) does indeed possess an analytical solution, its form 

makes impossible the analytical evaluation of the electric field from the transformed field ER (κl ) % in 

Equation (3.70). However, in the limits of weak spatial dispersion, the equations lend themselves to 

analytical evaluation. Of course, it must be established that the re-entry plasma can be treated as one 

possessing weak spatial dispersion. Again, it must be kept in mind that what is needed here are 

expressions for the reflection, transmission, and absorption coefficients at the z = 0 and z L = boundaries 

of the plasma layer. Thus, the transformation of the fields ER (κl ) % on the left side of Equation (3.70) 

must be evaluated at these surfaces. This is the goal of the following development. 

4.0 Evaluation in the Limit of Weak Spatial Dispersion 

4.1 The Limit of Weak Spatial Dispersion 

Equation (3.71) can be rewritten as 

( ) ( ) ( )

( )

2 2

2 2 22

0

2 exp 1

1

ps s cs s

l s

s Ts l s cs s

w i

dw

w i

∞ ω − ω−ω + ν

εκ = −

π ω −υ κ ω−ω + ν

∑ ∫ % (4.1) 

Letting ( )( )2 2 cs s ~ cs s ω−ω + ν ω−ω +ν i , consider the terms within the denominator of Equation (4.1) in 

the case where 

(( ) )

1/2 2 2

1 Ts l s

cs s

υ κ w

<<

ω−ω +ν

which can be re-expressed as 
NASA/TM—2009-216096 16

(( ) )

1/2 2 2

Ts 1

s cs s

lw L

υ

<

ω−ω +ν

 (4.2) 

Now, the integral over the variable ws is rapidly attenuated due to the presence of the exponential 

function. Hence, only small values of this variable need be considered. Take the value of this parameter to 

be on the order of unity. There then remains the range of values of the mode number l. In order to 

incorporate the long range effects within the plasma (for spatial dispersion effects), small values of this 

parameter need be used; thus, take l =1. Hence, the condition of Equation (4.2) can finally be written as 

(( ) )

1/2 2 2

1 Ts

L cs s

υ

<

ω−ω +ν

 (4.3) 

At resonance where ω ω~ cs , this condition becomes 

υTs s < νL (4.4) 

which defines the region of weak spatial dispersion for this problem. The range over which this inequality 

holds must now be examined. 

4.2 Weak Dispersion and a Re-entry Plasma 

Since both the thermal velocity and collision frequency are functions of plasma temperature, it will be 

instructive to determine over what temperature ranges Equation (4.4) holds for nominal thicknesses and 

electron number densities. Figures 3 and 4 show the region of applicability of the constraint of Equation 

(4.4) for a one-component electron plasma of thickness L = 30cm . As seen from these plots, the 

applicability region becomes smaller over a range of temperatures as the electron concentration decreases. 

Thus, for 10 3 ne >10 / cm , the approximation of weak spatial dispersion can be expected to hold for 

nominal re-entry temperatures that are typically Te > 3000K . The case in which 10 3 ne ≤10 / cm requires 

one to consider the case of strong spatial dispersion. 

In the case of weak spatial dispersion for this one-component plasma (s ≡ e ), Equation (4.1) reduces 

to 

( ) ( )

( ) ( )

2 2 2

0

2 1 exp 1 1 pe pe e

l e

ce e ce e

w

dw

i i

∞ ω ω − εκ ≈ − = − π ω ω−ω + ν ω ω−ω + ν ∫ % (4.5) 

The analysis of Equation (3.72) proceeds along the same lines where, it must be added, that the 

exponential factors oscillate away to zero. Also, with Equation (4.4) prevailing, the terms in Equation 

(3.72) can be neglected in this approximation. In this event, Equation (3.70) can be written, using 

Equation (4.5), 

( ) ( ) ( ) ( ( )( ) ( )) ( ( )( ) ( )) 222 10 10 l l l

l Rl R R R R

k k kA E i E L E B L B

L L

κ

κ− + ω κ = − − + − − % (4.6) 

where 
NASA/TM—2009-216096 17

( ) ( )

2 1 pe

ce e

A

i

ω

ω ≡

ω ω−ω + ν

 (4.7) 

From Equation (4.6), expressions for the boundary values ER (0) and ER (L) can be obtained from 

which incident, reflected, and transmitted fields can be related finally giving rise to the associated 

coefficients. Thus, defining 

( ) ( ) 222 l D k kA ω l ≡κ − + ω (4.8) 

one has 

() ( ) ( ) ( ( )( ) ( )) 1 0 10 l

R R l R R

l l l

k E E BL B

D L

∞ ∞

=−∞ =−∞

⎛ ⎞

= κ= ⎜ ⎟ − − ω ⎝ ⎠ ∑ ∑ % (4.9) 

and 

( ) ( )( ) ( ) ( ( ) ( )( ) ) 1 1 01 l l

R R l R R

l l l

k EL E BL B

D L

∞ ∞

=−∞ =−∞

⎛ ⎞

= κ− = ⎜ ⎟ − − ω ⎝ ⎠ ∑ ∑ % (4.10) 

where, as noted earlier, the term with the coefficient κl sums to zero. 

At this point, one needs to relate the magnetic fields BL (0) and BR (L) to the prevailing electric 

fields on the boundaries so as to obtain from Equations (4.9) and (4.10) two equations in the two 

unknowns ER (0) and ER ( ) L . From these, the reflection and transmission coefficients will be obtained. 

To this end, from Equation (3.64), 

( ) 1 R ( )

R

E z B z

k z

∂ = ∂ (4.11) 

As for the fields above the plasma layer at z L = , one can write 

ER ( )z E ik L z E ik L z = −+ − − 0 R exp( ( )) efl exp( ( )) (4.12) 

where E0 is the incident electric field of the wave upon the plasma layer and ERefl is the field reflected 

from the layer. Using Equation (4.12) in Equation (4.11) gives 

BR ( )z i E ik L z E ik L z 0 R exp( ( )) efl exp( ( )) =− − − − − ⎡ ⎤ ⎣ ⎦ (4.13) 

Hence, on the boundary z L = , 

E L E E B L iE E R R ( ) = +0 Refl, ( ) =− − ( 0 Refl) (4.14) 

Similarly, for the transmitted field into the region z < 0 , 

ER (z E ik L z ) = − Trans exp( ( )) (4.15) 

giving at the boundary z = 0
NASA/TM—2009-216096 18

ER R ( ) 0 exp , 0 exp = = E ikL B iE ikL Trans ( ) ( ) − Trans ( ) (4.16) 

Substituting Equations (4.14) and (4.16) into Equations (4.9) and (4.10), defining 

Refl Trans

0 0

, E Er t

E E ≡ ≡ (4.17) 

as well as 

( ) ( )

( )

( ( ))

( ) 1

csc 1 1

1

l

l l

kL A k S

L D A

∞

=−∞

− − ω ⎛ ⎞

ω ≡ ⎜ ⎟ = − ⎝ ⎠ ω − ω

∑ (4.18) 

( ) ( )

( ( ))

( ) 2

cot 1 1

l 1 l

kL A k S

L D A

∞

=−∞

− ω ⎛ ⎞

ω ≡ ⎜ ⎟ = − ⎝ ⎠ ω − ω

∑ (4.19) 

yields 

t ikL iS r iS t ikL exp( ) =− ω − + ω 1 2 ( )(1 ) ( ) exp( ) (4.20) 

1 1e + =− ω − + ω r iS r iS t ikL 2 1 ( )( ) ( ) xp( ) (4.21) 

The solutions of these simultaneous equations for r and t will then yield the related coefficients of 

reflection 2 R ≡ r and 2 T t ≡ as well as the related coefficient of absorption 2 2 A ≡1− − r t . Equations 

(4.20) and (4.21) give, in this case of weak spatial dispersion, 

( ) ( )

( ) ( ) ( )

2 2

1 2

2 2 2 1

1 S S

r

iS S

− ω+ ω =

+ ω− ω

 (4.22) 

and 

( ) ( )

( ) ( ) ( )

1

2 2 2 1

2 exp iS ikL

t

iS S

ω − =

+ ω− ω

 (4.23) 

Figures 5 to 7 display the results of the calculation of these relations for three electron concentrations 

for a plasma of thickness L = 30cm and Te = 3000K [Sharkey, 2004] upon which an electromagnetic 

wave of frequency f = 2.0GHz is incident. As Figs. 5 and 6 show, the reflection coefficient (R) remains 

at 1.0 and the transmission coefficient (T) remains at 0.0 until the magnitude of the applied magnetic field 

goes above 500 Gauss. The absorption coefficient (A) also begins to rise (clearly shown in Fig. 6). After 

this point, the coefficients rapidly oscillate and tend to their limits R → 0.0 , T →1.0 , and A → 0.0 . For 

11 3 ne = × 1.0 10 / cm , these limits are quickly approached. At 12 3 ne = × 1.0 10 / cm , oscillations of the 

values of R and T remain beyond the magnitude of 10,000 Gauss although the values are clearly 

separated. However, Fig. 7 for the case of 13 3 ne = × 1.0 10 / cm shows that the oscillations remain mixed 

together where the values of these coefficients exchange local maxima and minima. 
NASA/TM—2009-216096 19

5.0 Conclusion 

It is thus demonstrated that a magnetic field of nominal magnitude applied to the plasma makes the 

plasma transparent to frequencies smaller than the plasma frequency thus substantiating the magnetic 

windows concept. Magnetic fields on the order of 4 kGauss will render a plasma transparent at 2 GHz 

with an electron concentration of 1012 electrons/cm3

. As the plasma density evolves throughout re-entry, 

the reflection, transmission, and absorption coefficients can oscillate over large ranges for a fixed applied 

magnetic field. Hence, a variable magnetic field controlled, e.g, by the value of received power, must be 

considered. Also, the case of strong spatial dispersion as well as the situation intermediate to weak and 

strong spatial dispersion must yet be considered. 

This example of a homogeneous plasma in a homogeneous magnetic field is certainly an idealization 

of what would exist in reality. However, these results provide impetus for further work with more realistic 

situations. Although Halbach magnets can be used to generate the applied magnetic field, some spatial 

variation of the field should be introduced. The variations inherent in the plasma thickness, temperature, 

and density must be addressed in the specification and design of a closed loop control system to adjust the 

external magnetic field to maintain transparency. Finally, account of the other components of the plasma 

must be made. In fact, with the surface ablation that occurs during re-entry, one should consider a dusty 

plasma, the kinetics of which can be vastly different from the simple one-component electron plasma 

considered here. 
NASA/TM—2009-216096 20
NASA/TM—2009-216096 21

Appendix A.—A Model of the Effective Collision Frequency 

For purposes of the discussion given here, consider a two component plasma made up of electrons 

and ions in an electric field E

v

. The Boltzmann equation including the associated collision integral for an 

electron in a two-component plasma is given by 

r coll 0 f e f E fS

t m υ

∂ +υ⋅∇ + ⋅∇ + =

∂

v v v v vv (A1.1) 

where the complete collision integral is 

( ) ( ) ( ) ( ) ( ) 3 coll 1 1 S qu u f F f F d d ≡ θ υ υ − υ υ υΩ , ⎡ ⎤ ′ ′ ∫∫ ⎣ ⎦ v v vv (A1.2) 

In the case where the collision is limited to the scattering of an electron from an ion, q u( ) ,θ is the 

differential cross-section of electron scattering where θ is the scattering angle, u ≡ υ−υ1

v v is the relative 

velocity of the electron with respect to the ion, i.e., υ

v and υ1

v are the velocities, respectively, of the 

electron and ion after collision whereas υ′

v and 1υ′

v are those before the collision. Additionally, F (υ1) v is 

the velocity distribution of the ions which is usually taken to be given by the Maxwell distribution of the 

form of Equation (3.2), and d dd Ω= θ θ φ sin is the differential scattering angle. 

Simplifications will now be introduced using what is known about the electrons in the plasma. First, it 

is assumed that the thermal velocity of the electrons and ions are much greater than the associated 

directed velocities in the electric field. In this isotropic plasma case (i.e., where the magnetic field B = 0 v ), 

the spatial gradient of the distribution f is directed along the z axis, parallel to E

v

. Hence, one can treat the 

“directional” part of f as a perturbation in velocity space. The distribution can thus be expanded into zeroorder spherical polynomials, i.e., Legendre polynomials Pl (cosθ) where θ is the angle between E

v

 and 

υ

r , viz., 

( ) ( )( )

0

, , cos , , l l

l

f fr t P f r t

∞

=

= υ= θ υ ∑ v v v v (A1.3) 

Writing 

( )

2 sin

cos

cos

fE f E fE υ∂ θ∂ ⋅∇ = θ +

∂υ υ∂ θ

v v v (A1.4) 

and using Equations (A1.3) and (A1.4) in Equation (A1.1) give the following system of equations for fl

( )

( )

( )

0 1 2 1 0 2

1 02 0 3 2 1 3

2 13 4 1 32 4

0

3 3

2 2 0

5 5

27 2 1 3 0

33 3 7

f f eE f S

t t m

f f f eE f f S

t z zm

f f f eE f fS

t z zm

∂ υ∂ ∂

+ + υ +=

∂ ∂ ∂υ υ

∂ ∂∂ ∂ ∂ ⎛ ⎞⎛ ⎞ +υ + + + υ + = ⎜ ⎟⎜ ⎟ ∂ ∂ ∂ ∂υ ∂υ ⎝ ⎠⎝ ⎠ υ

∂ ∂∂ ∂ ∂ ⎛ ⎞ ⎛⎞ ⎛ ⎞

+υ + + υ + υ + = ⎜ ⎟ ⎜⎟ ⎜ ⎟ ∂ ∂ ∂ ∂υ υ ∂υ ⎝ ⎠ ⎝⎠ ⎝ ⎠ υ

M

 (A1.5) 
NASA/TM—2009-216096 22

where 

( )

()( ) ( ) ( )

() ( ) ( )

3 1

0 0

3

2 1

cos

4

2 1

, cos cos cos 4

, cos

l l

l ll ll

l l

l ll

l S P Sd

l q u uP f P F F f P d d d

q u u Ff F f P d d

∞ ∞

′′ ′′

′ ′ = =

+ ≡ θΘ

π

+ ⎡ ⎤

= θ θ θ − θ υ ΩΘ ⎢ ⎥ ′ ′ ′ ′′ π ⎣ ⎦

= θ − θ υΩ ′ ′

∫

∫∫∫ ∑ ∑

∫∫

 (A1.6) 

In arriving at the result of Equation (A1.6), the integration over d dd Θ = θθφ sin made use of the fact 

that 

cos cos cos sin sin cos θ = θ θ+ θ θ φ ′′ ′′

and the addition theorem for Legendre polynomials was employed. 

The chain of equations of Equation (A1.5) can be terminated at the second one if the perturbation 

component 2f can be neglected compared to the fundamental component 0f , i.e., 

( ) 0 3 2 3

f 1 f ∂ ∂

>> υ

∂υ υ ∂υ (A1.7) 

Before Equation (A1.7) can be established, one now needs to consider the expressions for the scattering 

perturbations S0 , S1 , and S2 . To this end, considerations will only be made of elastic collisions; in this 

case, Equation (A1.6) gives for S1

( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 1 1 11 1 1 1 S qu u f F P f F d d = θ υ υ− θ υ υ υΩ , cos ′ ′ ∫∫ v v vv (A1.8) 

Assuming that the energy of the electron changes only slightly after collision with the ion, one has that 

υ ≈υ ′

v v and 1 1 υ ≈υ ′

v v where υ >> υ1

v v . In this instance, Equation (A1.8) yields 

( ) ( ) ( )( ) 3 11 1 1 S f qu F d d = υ θυ υ − θ υ Ω , 1 cos ∫∫ v v = νυ υ 1 1 ( ) f ( ) (A1.9) 

where the velocity dependent collision frequency is defined by 

( ) ( )( ) ( ) 3 1 ion ion 1 1 ν = υ υθ − θ Ω ≡ υ υ uN q d N F d , 1 cos , ∫ ∫ (A1.10) 

Equation (A1.9) begins to form the basis of the Krook model introduced in Equation (2.7) where a 

velocity independent collision frequency vs was used. It is the central purpose of this Appendix to obtain 

an expression of this velocity independent collision frequency from collision theory. Before this can be 

accomplished, however, it remains to establish the prevailing conditions that allow the termination of the 

system of equations of Equation (A1.5). 

One can similarly show that 
NASA/TM—2009-216096 23

( ) () ( ) ( ) ( ) ( ) ( )

() ( ) ( )

() ()

3 2 21 221 1

3 21 1

2 2

, cos

3cos 2 1 , 1 4

S qu u f F P f F d d

f qu F d d

v f

= θ υ υ− θ υ υ υΩ ′ ′

⎛ ⎞ θ + = υ θυ υ − υ Ω ⎜ ⎟ ⎝ ⎠

=υ υ

∫∫

∫∫

v v v v

v v (A1.11) 

where 

() ( ) ( ) 3 2 ion ion 1 1

3cos 2 1 , 1 , 4 uN q d N F d ⎛ ⎞ θ + ν = υ υθ − Ω ≡ υ υ ⎜ ⎟ ⎝ ⎠ ∫ ∫ (A1.12) 

Thus, the quantities v1 (υ) and v2 (υ) are of the same order of magnitude and, for purposes of the 

discussion to follow, vv v () () () υ≡ υ≈ υ 1 2 . Finally, an expression for S0 needs to be secured. The 

calculation is trivial upon using Equation (A1.6) and noting the fact that energy exchange during 

collisions does not occur with the approximations used above; hence, to first order, S0 = 0 . Finally, 

specializing to the special case of spatial homogeneity, ∂ ∂ =z 0 in Equations (A1.5). Given the 

developments above, these relations reduce to 

( ) 0 2 1 2 0

3

f eE f

t m

∂ ∂

+ υ =

∂ υ ∂υ (A1.13) 

( ) 1 0

1 0 f eE f v f

t m

∂ ∂

+ +υ =

∂ ∂υ (A1.14) 

( ) ( ) 2 4 1 32 4

21 3 0

3 7

f eE f f vf

t m

∂∂ ∂ ⎛ ⎞ ⎛ ⎞ + υ + υ +υ = ⎜ ⎟ ⎜ ⎟ ∂ ∂υ υ υ ∂υ ⎝ ⎠ ⎝ ⎠ (A1.15) 

Consider now the steady state situation where f1 2 , ~ exp f it (− ω ) and hence ∂f1 1 ∂ −ω t if ~ and 

∂ ∂ −ω f2 2 t if ~ . Using these in Equations (A1.14) and (A1.15), dropping the second term within the 

parentheses of Equation (A1.15) and solving for 2f gives 

( )

2 2 0

2 22 2

1 ~ eE f f

m v

∂ ∂ ⎛ ⎞

υ ⎜ ⎟ ω + ∂υ υ ∂υ ⎝ ⎠ (A1.16) 

Hence, the condition of Equation (A1.7) becomes 

( )

2 2

2 0

0 22 2 2

eE f 1 f

m v

⎛⎞⎛ ⎞ ∂ ∂

>> υ ⎜⎟⎜ ⎟ ω + ⎝⎠⎝ ⎠ υ ∂υ ∂υ

 (A1.17) 

If Equation (A1.17) holds, the original system of equations of Equation (A1.5) reduce to two: 

( ) 0 2 1 2 0

3

f eE f

t m

∂ ∂

+ υ =

∂ υ ∂υ (A1.18) 
NASA/TM—2009-216096 24

( ) 1 0

1 0 f eE f v f

t m

∂ ∂

+ +υ =

∂ ∂υ (A1.19) 

Finally, if the electric field is weak enough where the second term of Equation (A1.18) can be taken as a 

first order perturbation, the symmetric portion of the distribution f retains its Maxwellian nature and the 

solution to the problem reduces to (remembering Equation (A1.3)) 

( )

0 1

1 0

1

cos

0 e

ff f

f eE f v f

t m

=+ θ

∂ ∂

+ +υ =

∂ ∂υ

 (A1.20) 

In what is to follow, it is found advantageous to give the perturbation f1 a vector character by writing 

Equation (A1.20) as 

1

0

f f f ⋅υ

= +

υ

v v

 (A1.21) 

( ) 1 0

1 0 e

f eE f v f

t m

∂ ∂

+ +υ =

∂ ∂υ

v v v (A1.22) 

where 1f

v

 is given the direction of E

v

. 

The total current induced in the plasma by the free electrons is given by 

( ) 3

e j = en f d ∫ v v v υ υ υ (A1.23) 

Using Equation (A1.21) and converting the υ -integration into one in spherical coordinates gives, noting 

the isotropy of 0f , 

2

3 1

0 0

j en f d d 2 s e in

∞ π

= π υ θθυ ∫ ∫ v v (A1.24) 

Now, from Equation (A1.22), one has in the steady state 

( )

0

1 e

eE f

m f v i

∂

∂υ = − υ − ω

v

v (A1.25) 

Using Equation (3.2) in this result and substituting into Equation (A1.24) yields 

( ) ( )

( )

( )

( )

42 42 2

22 22

0 0

8 exp exp

3

e

e

enE vxx x x x

j dx i dx

m v x v x

⎛ ⎞ ∞ ∞ − − = + ⎜ ⎟

+ + ⎝ ⎠

∫ ∫

v v ω

π ω ω

 (A1.26) 

where x ≡ υ m kT e Be 2 . In the event that the collision frequency is independent of the velocity, i.e., 

vx v ( ) = = const. , the integrals in Equation (A1.26) can be performed and give the result 
NASA/TM—2009-216096 25

2

22 22

e 1

e

enE v j i

mv v

⎛ ⎞ = + ⎜ ⎟ ω ⎝ ⎠ ω+ ω+

v v (A1.27) 

This result is identical to the one that is obtained by using elementary considerations. That is, using the 

concept of an effective collision frequency eff v , one simply uses Newton’s Law of motion for an electron 

in an electric field and writes 

e e eff

d m eE m v

dt

υ = − υ

v v v (A1.28) 

Here, the effective collision frequency enters as a friction term. Again, considering the steady state and 

solving for the velocity, one gets for the associated current 

2 eff

22 22

eff eff

e 1 e e

enE v j en i

mv v

⎛ ⎞

= υ= ⎜ ⎟ + ω

ω+ ω+ ⎝ ⎠

v v v (A1.29) 

In order to reconcile these two approaches, one considers the limiting case where ( ) 2 22

eff ω >> v vx , . 

Taking Equations (A1.26) and (A1.29) in this limit, equating the two results and solving for veff gives in 

terms of the velocity dependent collision frequency 

( ) ( ) 4 2 eff

0

8

exp 3v v x x x dx

∞

= − π ∫ (A1.30) 

It is now necessary to find an expression for v x( ) defined by Equation (A1.10), given the scattering 

situation assumed here. That is, elastic scattering of a fast moving electron from an essentially stationary 

heavy ion. In this Coulomb scattering case, one must use the Rutherford scattering formula for the crosssection required in Equation (A1.10), viz., 

( ) ( )

2 2

2 4

1 , 2 sin 2 ee

q

m

⎛ ⎞ υθ = ⎜ ⎟ υ θ ⎝ ⎠

 (A1.31) 

Substituting this expression into Equation (A1.10) and performing the azimuthal integration, 

( ) ( ) min

2 2

ion 2 4

4

2 min

ion 2 3

1 cos 2 sin

2 sin 2

2 ln 1 cot

2

e

e

e vn d

m

e

n

m

∞

θ

⎛ ⎞ − θ υ= π υ θθ ⎜ ⎟ υ θ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ θ

=π + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ υ ⎝ ⎠ ⎝ ⎠

∫

 (A1.32) 

where θmin is the minimum scattering angle which is related to the maximum value of the impact 

parameter bmax , 

( ) 2

min

2 max

tan 2 e

e

m b

θ = υ

 (A1.33) 
NASA/TM—2009-216096 26

Here, the impact parameter is determined by the fact that in a plasma where the interaction between the 

ions and electrons is through a Coulomb field only, distances on the order of the Debye radius rD cannot 

be exceeded since this is the maximum distance at which substantial interaction occurs between the 

electron and ion; at distances greater than this, the field of the ion decreases exponentially. Hence, one 

can write 

( )

1/2 1/2 2 ion max 2 2 4 8 ion ion ion

B e B e

D

B Be

kT T k T b r

en kT kT en

⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ π+ π ⎝ ⎠

 (A1.34) 

where the last result issues from the assumption that T T ion ≈ e . For a typical re-entry plasma, 

4 rD ~ 10 cm − . Given the relative values of the parameters involved, one can write from Equation (A1.33), 

2 2

1 min 2 2

2 2 tan

eD eD

e e

mr mr

− ⎛ ⎞ θ = ⎜ ⎟ ≈

⎝ ⎠ υ υ

 (A1.35) 

Substituting this result into Equation (A1.32) and series expanding the 2 cot function finally gives 

( )

4 2 24

ion 23 4 2 ln 1 D e

e

e r m

v n

m e

⎛ ⎞ ⎛ ⎞ υ

υ= π ⎜ ⎟ ⎜ ⎟ +

⎝ ⎠ υ ⎝ ⎠

 (A1.36) 

Finally, applying the substitution υ = x 2kT m B e e in Equation (A1.36) and using this result in Equation 

(A1.30) yields 

( ) ( )( )

3/2 4

24 2 eff ion 2

0

3/2 4 ion

2

8 2 ln 1 exp 3 2

82 1 1 1 1 1 sin 2si cos ci

3 2 2

e

e Be

e

e Be

e m v n x A x x dx

m kT

ne m

m kT A A A A

∞ ⎛ ⎞⎛ ⎞

=π + − ⎜ ⎟⎜ ⎟

π ⎝ ⎠⎝ ⎠

⎛ ⎞ π ⎛ ⎞ ⎡ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎤

= π− − ⎜ ⎟⎜ ⎟ ⎢ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎥

π ⎝ ⎠⎝ ⎠ ⎣ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎦

∫

 (A1.37) 

where 2 A ≡ 2k Tr e B eD and ci(L) and si(L) are the cosine and sine integrals, respectively. For a 

typical re-entry plasma, 2 A ~ 10 1 >> . Thus, expanding the functions within the brackets of Equation 

(A1.37) in an ascending series, the sine terms are negligible and one is left with 

cos 1 ci 1 ln ( ) AA A ( ) ≈− +γ where γ is Euler’s constant, γ ≈ 0.577 . Hence, Equation (A1.37) reduces to 

3/2 4 ion

eff 2 2

8 2 1.12 ln

3 2

e B e D

e Be

ne m k Tr

v

m kT e

⎛ ⎞ π ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟⎜ ⎟ ⎜ ⎟

π ⎝ ⎠⎝ ⎠ ⎝ ⎠ (A1.38) 

This is the expression for the effective collision frequency used in this work. To make contact with the 

notation used in the text, one has v v s = eff where s ≡ e . It must be remembered that this formulation 

only accounts for elastic collisions between electrons and heavy ions. Of course, other scattering 

processes can occur such as elastic and inelastic collisions with molecules, collisions with dust grains in 

which charge transfer can also be attendant with impact, etc. 
NASA/TM—2009-216096 27

Figure 1.—The flowing plasma and its various fields and velocities. 

v

E

v

B

v

k

v

B0 = B0 zˆ

zˆ

xˆ

yˆ

v

VL = VL zˆ

v

VT = VT xˆ

Surface of Hypersonic Vehicle 

Free Surface Open to Atmosphere 

Incident EM Field 

v

E = Ex xˆ + Ey yˆ v

B = Bx xˆ + By yˆ

L
NASA/TM—2009-216096 28

Figure 2.—Electromagnetic and kinetic equations for a plasma in an external 

magnetic field and an incident electromagnetic wave. 

Maxwell Eqs.

Lorentz Force Eq.

Boltzmann Eq.

Plasma Current Eq.

0 0 E B ,

v v

2 2

Refl Trans

0 0

, E E R T

E E = =

r s ( ) 0

f f F f ff

t υ

∂ + υ⋅∇ + ⋅∇ = −ν −

∂

v v v v vv

1 4 E B j

ct c

∂ π ∇× = +

∂

v v v v

1 B E

c t

∂ ∇× =−

∂

v v v

( ) 0

eE e F B B

m mc

= + υ× +

v v vv v

( ) 3 j en f r d ,

∞

−∞

= υ υυ ∫ v v v v
NASA/TM—2009-216096 29

Figure 3.—Region of applicability for weak spatial dispersion approximation for ne=1010 electrons/cm3and L=30 cm. 

Figure 4.—Region of applicability for weak spatial dispersion approximation for ne=1011 electrons/cm3 and L=30 cm. 

1000 2000 3000 4000 5000

5.0¥ 108

1.0¥ 109

1.5¥ 109

2.0¥ 109

T e e υ <νL

( )

,

cm/sec

T e e υ νL

(K) Te

1000 2000 3000 4000 5000

5.0¥ 107

1.0¥ 108

1.5¥ 108

2.0¥ 108

2.5¥ 108

T e e υ <νL

( )

,

cm/sec

T e e υ νL

(K) Te

Te υ

Lνe

Lνe

Te υ
NASA/TM—2009-216096 30

Figure 5.—Reflection, transmission, and absorption coefficients versus applied magnetic field strength for 

ne=1011 electrons/cm3

. 

- 10000 - 8000 - 6000 - 4000 - 2000 0

0.0

0.2

0.4

0.6

0.8

1.0

f = 2.0 GHz

Te = 3000 K

L = 30 cm

B0 (Gauss)

R

T

A
NASA/TM—2009-216096 31

Figure 6.—Reflection, transmission, and absorption coefficients versus applied magnetic field strength for 

ne=1012 electrons/cm3

. 

- 10000 - 8000 - 6000 - 4000 - 2000 0

0.0

0.2

0.4

0.6

0.8

1.0

2.0 GHz

3000 K

30 cm

e

f

T

L

=

=

=

B0 (Gauss)

R

T

A
NASA/TM—2009-216096 32

Figure 7.—Reflection, Transmission, and Absorption Coefficients Versus Applied Magnetic Field Strength for ne=1013

electrons/cm3

. 

- 10000 - 8000 - 6000 - 4000 - 2000 0

0.0

0.2

0.4

0.6

0.8

1.0

2.0 GHz

3000 K

30 cm

e

f

T

L

=

=

=

B0 (Gauss)

R

T

A
NASA/TM—2009-216096 33

References 

Bell, T.F., and O. Buneman, “Plasma Instability in the Whistler Mode Caused by a Gyrating Electron 

Stream,” Phys. Rev. 133 (5A), pp. A1300–A1302 (1964). 

Bhatnagar, P.L., E.P. Gross, and M. Krook, “A Model for Collision Processes in Gases. I. Small 

Amplitude Processes in Charged and Neutral One-Component Systems,” Phys. Rev., Vol. 94, No. 3, 

pp. 511–525 (1954). 

Bletzinger, P., B.N. Ganguly, D. Van Wie, and A. Garscadden, “Plasmas in High Speed Aerodynamics,” 

J. Phys. D: Appl. Physics, Vol. 38, pp. R33–R57, 2005. 

Gross, E. P., “Plasma Oscillations in a Static Magnetic Field,” Phys. Rev. 82 (2), pp. 232-242 (1951). 

Jackson, J.D., Classical Electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York, 1975. 

Russo, F.P., and J.K. Hughes, “Measurements of the Effects of Static Magnetic Fields on VHF 

Transmission in Ionized Flow Fields,” NASA TM X–907, 1964. 

Schroeder, L.C., and F.P. Russo, “Flight Investigation and Analysis of Alleviation of Communications 

Blackout by Water Injection During Gemini 3 Reentry,” NASA TM X–1521, 1968. 

Sharkey, R.P., “Electromagnetic Wave/Magnetoactive Plasma Sheath Interaction for Hypersonic Vehicle 

Telemetry Blackout Analysis,” 34th AIAA Plasmadynamics and Lasers Conference, AIAA–2003–

4167, June 2003. 

Sharkey, R.P. “Nonequilibrium Plasma Effects on Telemetry Considerations for Air-Breathing 

Hypersonic Vehicle Design,” 42nd AIAA Aerospace Sciences Meeting and Exhibit, AIAA–2004–333, 

January 2004. 

Tananbaum, B.S., Plasma Physics, McGraw-Hill, New York, 1967. 

Usui, H., H. Mataumoto, F. Yamashita, M. Yamane, and S. Takenaka, “Computer Experiments on Radio 

Blackout of a Reentry Vehicle,” 6th Spacecraft Charging Technology Conference, AFRL–VS–TR–

20001578, September 2000. 
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Analysis of Electromagnetic Wave Propagation in a Magnetized Re-Entry Plasma Sheath Via 

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Manning, Robert, M. 

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14. ABSTRACT

Based on a theoretical model of the propagation of electromagnetic waves through a hypersonically induced plasma, it has been 

demonstrated that the classical radiofrequency communications blackout that is experienced during atmospheric reentry can be mitigated 

through the appropriate control of an external magnetic field of nominal magnitude. The model is based on the kinetic equation treatment of 

Vlasov and involves an analytical solution for the electric and magnetic fields within the plasma allowing for a description of the attendant 

transmission, reflection and absorption coefficients. The ability to transmit through the magnetized plasma is due to the ‘magnetic windows’ 

that are created within the plasma via the well-known ‘whistler modes’ of propagation. The case of 2 GHz transmission through a re-entry 

plasma is considered. The coefficients are found to be highly sensitive to the prevailing electron density and will thus require a dynamic 

control mechanism to vary the magnetic field as the plasma evolves through the re-entry phase. 

15. SUBJECT TERMS

Radio transmission; Wave propagation; Plasmas; Spacecraft communication; Blackout 

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