PROCEEDINGS OF THE IRE

Electromagnetic Properties of High-Temperature Air*M. P. BACHYNSKIt, T. W. JOHNSTONt, AND I. P. SHKAROFSKYt

Summary-This paper concerns the attenuation and phase char-acteristics of plasmas and, in particular, the electromagnetic proper-ties of high-temperature air. It is shown that by a suitable normaliza-tion of the parameters the electromagnetic properties of plasmas maybe universally represented in convenient form in either the complexdielectric coefficient plane or the complex propagation constantplane. Next, the electron number densities and electron collisionfrequencies for air ranging in temperature from 30000 to 12,000Kand in density from 101 to 10-4 times the density at sea level areillustrated. The attenuation and phase constants for an electromag-netic wave traversing this medium have been evaluated for frequen-cies from 109 to 1011 cps. As an example, the above universal repre-sentation is applied to the stagnation region of a hypersonic vehiclein space.

ELECTROMAGNETIC PARAMETERS OFA UNIFORM PLASMA'

Dielectric CoefficientThe dielectric coefficient of an infinite uniform

plasma, i.e., a plasma where electron density is not afunction of position in the absence of an electromagneticfield, can be deduced from Maxwell's fourth equation.Thus, assuming a harmonic field variation ejWt and thepermittivity of the plasma to be the same as free spacepermittivity e0, one can write, using rationalized mksunits,

-4 -)4 a-V X H= J+- (eoE)

atINTRODUCTION (la)

D UE to ionization, air at high temperatures con-tains an appreciable number of free electrons andions. Under these conditions, the medium may

be described as a plasma, i.e., a gas containing chargedparticles in a sufficient quantity to seriously alter thephysical properties of the gas. One of the properties ofair markedly affected by the presence of the electronsand ions is the propagation of electromagnetic waves insuch a medium. This interaction of electromagneticwaves with plasmas is of current interest in connectionwith diagnostic techniques, space communications, andre-entry problems.The following paper is concerned with the electro-

magnetic characteristics of plasmas and, in particular,those of high temperature air. It is shown that by asuitable normalization of parameters, these propertiescan be represented in a convenient, universal form ineither the complex dielectric coefficient plane or thecomplex propagation constanit plane. Values of the elec-tron density and electron collision frequency are shownfor air in the temperature range 30000 to 12,0000 K,and densities ranging from 101 to 10-4 times the densityat sea level. Further, the attenuation and phase con-stants of electromagnetic wave propagating in amedium of air at high temperatures are evaluated forradio frequencies ranging from 109 to 1011 cps.

Finally, as an example, the variationi of attenuationand phase of an electromagnetic wave with altitude andvelocity is determined for the stagnation region of ahypersonic vehicle.

* Original manuscript received by the IRE, August 5, 1959. Thispaper was presented at the URSI International Symposium on Elec-tromagnetic Theory, Toronto, Can., June 15-20, 1959.

t Res. Labs., RCA Victor Co., Ltd. Montreal, Canada.

- rE + jweoE-=4

- jco'oKE,

(lb)

(lc)

where

H and E are the magnetic and electric fields respec-tively of an impressed electromagnetic wave incidenton the plasma,J is the ac currenit density,eoE represents the electric displacement,a is the ac electronic conductivity of the plasma,w is the radian frequency of the electromagnetic wave,j =V-1, andK is the effective dielectric coefficient given by

K= 1+-.jwco

(2)

In the absence of a dc magnetic field, the electronicconductivity a. of a plasma to an RF signal of frequencyco is given by2'3

41r e2 j" 1 I f000. = ___ -- v3dv

3 m J +jw+ v(3a)

where

e and m are the electronic charge and mass respec-tively,fo0 is the electron velocity distribution function,

1 This section is intended only as a summary to define the variousparameters.

2 W. P. Allis, "Motion of ions and electrons, " Handbuch der Physikvol. 21, Springer-Verlag, Berlin; 1956.

I H. Margenau, "Conductivity of plasmas to microwaves," Phys.Rev., vol. 109, pp. 6-9; January, 1958.

1960 347

PROCEEDINGS OF THE IRE

v is the electron velocity, andv is the effective collision frequency of the electrons(equal to the reciprocal of the time between succes-sive collisions).

If the collision frequency is independent of electron ve-locity, (3a) simplifies to

ne2am=

m (7 + jw)(3b)

where n is the number density of electrons, which is areasonable approximation for air at temperatures con-sidered here (particularly 60000K-12,000°K).

Using (3b) in (2), the dielectric coefficient becomnes

K 1 +

J1 + (v/X)2} (4a)

= Kr + jKi (4b)

where the parameter ne'/eom =W,,2 has the dimensions ofseconds-2 and wp is the "plasma frequenicy." Kr and Kiare the real and imaginary parts, respectively, of thedielectric coefficient.

Propagation ConstantsIn a uniform plasma it is assumed that the electron

density in the absence of a field is not a function ofposition and in a neutral plasma there is no net charge,so that the charge density is zero. Hence, the electricfield in the plasma is divergenceless, i.e. V * E = 0. Further,by the usual manipulation of Maxwell's third andfourth equations the Helmholz vector equations for Eand H are obtained. Thus

V2E + k2KE = 0

V2H + k2KH = 0, (5a)where

k =w/c = 27r/X,c=velocity of light,= free-space wavelength.

It is seen that both E and H satisfy the same vectorequation. For any specific problem, solving either, sub-ject to the proper boundary conditions, will give thecomplete solution.The solutions of (5a) for a planie wave propagating

through a uniform plasmna are given in rectangular co-ordinates by

E= Eo (wt-Z

Hy = Hoc-iwtE-yz7Ey = E2 = 0;

Hx = Hz= 0;

y is complex, and of the two solutioils for y which differin sign, the solution yielding a niegative real part in theexponent is chosen in order to inisure attenuation as thewave propagates. Thus

ay a+j3, (6b)where

a I K| -Kr

b IR- A,

: - I |K| +Krk 2

K -= (Kr2+ Ki2)"2,

(6c)

(6d)B,

(6e)

anid a is called the attenuationi conistanit, , the phaseconstanit, of the plasma.

It is seein that the propagation conistants of a plasmiiaare determined by the effective dielectric coefficienitwhich in turnl depends on the electroni denisity ain)dcollision frequenlcy.

UNIVERSAL REPRESENTATION OF ELECTROMAGNET ICPARAMETERS FOR CONSTANT COLLISION FREQUENCYIt is inistructive to rewrite the relationships determiini-

ing the complex dielectric coefficienit anid propagatiollconstant in normalized form, and henice to demonstratetheir behaviour in uniiversal coordinates. Thus, definiethe followinig parameters

1) Normalizing with respect to frequency:S = /= normalized scattering frequency,N-(C0) 2=normilalized electroIn denlsity.

2) Normalizing with respect to plasma frequency(i.e. n1"2):C=-/ =nornmalized collisioni parameter,F=w/coP=normalized RF frequency.

These relationships permit the mapping of loci of coni-stant scattering frequency S, constant electron densityN, constant collision parameter C, or constanit RF fre-quency F on the complex dielectric coefficient plaine andon the complex propagation constant planle. S anid Nare the useful parameters to conisider in a diagniosticmeasurement, with a given frequency anid varyingplasma, while C and F are useful when the frequenicybehavior of a giveni plasma is of interest.

Dielectric Plane

Using (4a) for the complex dielectric coefficient, it iseasily shown that the normalized scattering term isgiven by

where y, the propagation constant, is defined byly jkKI .

S = Ki/(Kr - 1), or (7a)

(Kr- 1)S + Ki = 0, (7b)

348 ill farch

z

(6a)

Bachynski, et al.: Electromagnetic Pioperties of High-Temperature Air

which is a family of straight lines in the complex dielec-tric (Kr vs Ki) plane, with slope -S and Kr-interceptof 1.

Similarly, the normalized electron density is

N = 1-Kr + Ki2/( -Kr), or (8a)

(Kr - (1 -N/2))2 + Ki2 = (N/2) 2, (8b)

which in the complex plane is a family of circles of

radius N/2 and center (Kr =(l-N/2), Ki=0).Representation in terms of the normalized collision

parameter is slightly more difficult in that

C2 = S2/N, (9a)

or in terms of the real and imaginary part of the dielec-

tric coefficient,

(1 - Kr)3 + Ki2(1- Kr) - Ki2/C2 = 0 (9b)

or

/ 1 \-t1/2

Ki = (1 Kr)(C2(-Kr) -)1(9c)with a pole at Kr = 1-1/C2.

The normalized RF frequency loci are again circles in

the complex dielectric coefficient plane since

F = 1/VN (10)

The radius of the F-circles is 1/2F2 and their centers are

located at [Kr=(1-1/(2F2) ), Ki=O].Families of constant scattering frequency and con-

stant electron density normalized to signal frequencyplotted in the dielectric plane are shown in Fig. l(a),

while contours of constant collision parameter and RFfrequency normalized to the plasma frequency are

shown in Fig. 1(b).

Propagation Plane

Representation of the normalized parameters in thecomplex propagation constant plane can be derivedfrom a conformal transformation of values in the dielec-tric plane, since from (6a), (6c) and (6d),

A + jB =jKl2, (11)

or from a direct solution for the normalized parametersin terms of the attenuation and phase constants. Sincethe maximum value of the real part of the dielectricconstant is unity, the upper limit in the propagationplane is given by the line Kr=1 which maps into thehyperbola B2-A2 = 1.The normalized scattering frequency loci become rec-

tangular hyperbolas rotated through an angle of1/2 tan-'(l/S) in the complex propagation constant

plane, namely

2ABS= , or

1 - (B2- A2)

B2 - A2 + 2AB/S = 1.

(12a)

(12b)

The normalized electron density families becomequartic curves since

N = 1 + A2- B2 + 4A2B2/(1 + A2 - B2), or (13a)

(A2 + B12)2 - (2 - N)(B2 - A2) + (1 - N) = 0. (13b)

1X~~~~~.Sgt :4-~~~~~~~~~~~~~~~~~7

::1~ ~ ~ 2Lt-±NR 0 m aW4 FXW-N.0-9---l

>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~7: 77Z-llsoa

K>1'~.U't[

-0W5 0 05 IW -2.0

' ' i: --' - ' l0-- 9 ' - - 1 '- ' ' -' ' - - .- j, t ' . 5 '-_!__ _ __ =_ :l 1--- ---- -s ---' o----

l, i: :E _, .: 1' - T_ _ _ __ -_ _ _ _ _ _ _ _ _ _. .. ... _ _ _ _ ._. _ _ ------\ - -- - T . + --n . s +---7--d -

'_.10 t_ n <.,, _ '_ *. _; ;_ .i .1__ 4-. __= _ _ I I _ _ _ _ _ ................... ..... i 1. _ ... i 4 J .... _1t-+St1~~~~4Mi

I _ _ q I -. ... . .. : _ 1 W a ::l -l =-: 4zmr ,l_-; 4-t::mtr, --t1fM +;

l 1 | l- .. . .. .l;- l .| [ -1-.t. .-l [# tntmutiH t11ffl

_ _ -l-=1 N7 -I- ;rt-lli-1-'t;i]tlrtf 1 --- 1- 1L1L 1 w*~ _ - 1_ 1j IlSl-=-t-LC--lif; t -l4t- ---:1 -1:- 1.x1. -1 r 5S- - wlv:- 1+T lllw6-

11 T---i */Z [i1.

i'7 .>IU

-1-0 -ba oa

REAL PART OF DIELECTRIC COEFFICIENT (Kr)

(b)

Fig. 1-(a) Variation of normalized scattering frequency (S) and normalized electron density (N) in the complex dielectric coefficient plane.(S=v/clw; N-= (cow)2.) (v =electron collision frequency, w =RF frequency, cop=plasma frequency.) (b) Variation of normalized collisionparameter C and normalized RF frequency (F) in the complex dielectric coefficient plane. (C=v/cop; F=co/wp.)

-2.0 -1l5 -1-0

REAL PART OF DIELECTRIC COEFFICIENT (Kr)

(a)

i f ~= L 1 vZ' i, 2 MICK1:'

-' -ll .:-l- i'o ''i "

t . -t:j t-, T.: "l:1:1 :1:11 1::.::1:.1::: 1:

I~Ih M:iI-4-'I~,''I''

i:..+.-r 4 :.... m.r I t1.dtt_

t=.;A,4 '1A1 :. I:- 'I'1"g-tk.'U:' I'

11:1-1 G.040W]]..Jj III 1.0lso-

1960 349

LT-

-, 9

d ::'1:~

PROCEEDINGS OF THE IRE

The normalized collision parameter C cani be ex-pressed in terms of A and B by the use of (12a) anid(13a) in (9a). Similarly, the values of nlormalized fre-quency F in the propagationi plane is obtained by sub-stituting (13a) iinto (10).

Contours of constants S and N in the conmplex propa-gation conistant plane are shown in Fig. 2(a), and plotsof constants C and F in Fig. 2(b). Extended values ofthese parameters are shown plotted on a logarithmicscale in Figs. 2(c) and 2(d).

It will be observed that the normalized parametersmap miiore readily inito the complex dielectric coefficienit

. ':4 ,, 'r~~~~~INl, .14 UP ;TKt{$ f Stl!w tC

# V ^4vffi \ 5 ffi > $ 2 e5 m ti Im e7T,jiXFj11V

~~~~~~~~~~~~~~~~~~~ia 40 X Xi R iM H "WN 11g gX

i 2,aEEM S Eg,zX EM g a M M~~~~~~~~~~~~v Riiitl NOl ij !"i1sit "04i

lu IP WIM.X~ mN" RNC,B0|WMS t ,4, !, fi

TjtRl Q tR V " ft 4 N F fifi|tEi !f! Ftf|.'|.ji,

054

F ff t{0RtIM,i WjiTt f ti 8rql O,.jF,L

11WMH,4 4 1i LLXSS S XX W lFtj{Rj,44,11L\

Al~~~~~~i RR1ffl'-14IAV . P qrlt

ATTENUATION CONSlANT (a/k)

(a)

plane than inlto the propagation conistant plane. How-ever, in measuremenit or diagniostics it is the attenuationiaind phase conistants which are actually determiined.Hlence, the added difficulty inl plottinig the niormlalizedconitours in the propagationi conistanit plane is genierallyjustified.A seconid reason for represeniting the plasma paramiie-

ters in the propagation plane is that the reflectinig ortransmitting properties of a plasmiia bounidary are veryconvenienitly miiapped in the propagationi planie. Thus, ifZ is the imiipedance of the plasmiia and ZO the impedan-ceof free space, the fraction of the field of a niormiially iii-

1-5z~~~~~~~~~Jz-

0llrl E S liu1-l'l!| '!l'!' f!Elrl'l'!3'-''I': !ll

\~~~~~~~~~~~~~~~1 C 2-i

Ut itlA4LFFlil> fiwail"lilLiill l|.t.,il;tilii,bliM1t'tFf l |!7| iux1a,. In<ft--rn

on

CQ 4|\i t T1lt,,,',.gL,li,>,,|ll,rl,,li !t4.iil,-l.llf.'','l '-.---

t17fe l.f\:l:jfW,l,: li,,!),'':ifi!ltr!t.flefEi "'l, Tibi K '''I' l ffll I'f'W>\

-,".P A >4W7llt l!l4m il l:

oj T,l,[l,,,li, %llj,.,,fl[f,rtj,'etf1tgfSif,.iftlt'' 'li

EtOFltf1<~ ~~~~~~~~~.~I:iWlii.fS'l .'0:..l!l.....l X i '''n Xw t.' !E!''''-''

Fiiltilii !'I :1iE--'~~~~~~iNtitl jLg ' iClbtiSFi l;iElUtE'NrollTilll110i ttitir-nT 0i[ is il ii

ATTENUATION CONSTANT (au/k)

(b)

ATTENUATION CONSTANT (.'k)

(c)

P1 1tll¶ I L lilislt1lo-' 1o0

ATTENUATION CONSTANT (d/K)

(d)Fig. 2-(a) Variation of normalized scattering frequency (S) and normalized electroni density (N) in complex propagation constant plane.

(S P/w; N = (wp/w)2.) (v= electron collision frequency; = RF frequency, wp = plasma frequency.) (b) Variation of normalized collisionparameter (C) and normalized RF frequency (F) in complex propagation constant plane. (C=v/lp; F= /wp.) (c) Variation of normalizedscattering frequency (S) and normalized electron density (N) in complex propagation constant plane for extended values of the parameters(logarithmic scale). (S=v/l; N=((,/Xo)2.)(d) Variation of normalized collision parameter (C) and normalized RF frequency (F) in complexpropagation constant plane for extended values of the parameters (logarithmic scale). (C=v/wp; F=w/c/p.)

350 Mlarch

Bachynski, et al.: Electromagnetic Properties of High- Temperature Air

dent electromagnetic wave reflected at a plasma-freespace interface is given by the reflection coefficient Rwhere from (6)

Z-Zo 1 - K"/2R= = (14)

Z+Zo 1 + K1,/2

(1 - B) + jA(1 + B) -jA

The magnitude of the reflection coefficient is given by

/1 + A2+ B2--2BRI

IA+ AB2+ 2+ 2B(15a)

-4/-+-x (15b)

where

x 2B/(1 + A2 + B2) (15c)

Similarly, the magnitude of the transmission coefficientis

|T| =\V1- RI2= -. (15d)

For any magnitude of reflection coefficient, the rela-tionship between the attenuation A and the phase B isuniquely determined by (15c), which can be rewritten

A 2 + (B -1/x)2- 2 - I (15d)

which is a family of circles with center (A = 0, B = 1/x)and radius [(1 -x2)/x2]1 /2.

Families of constant reflection coefficient plotted inthe propagation plane are shown in Fig. 3. If in a par-

, ...XX:s._ :7t't!!'g;|'....t..il 4,84 9 f T ,;,jilWX-! X X 7X. '4i r .;_ X i! gE 5 | F __ X1F_3VT "H11

a~~~~~~';ii,W--t- 3_ Mg it 7rsrsii ..r mWW

20~~~~~~~~~~~~~~~~2

440 1t t . 9 r:.i_..___ aO e ! : t r t . ! - - - s X 3- 2 _ ~~~~~~~~~~~:T :': :.i_tF7_ _77 !1L-7,0 't t -

...................'''I i |; -l- T z''t L 24 W- L. . 4w w & - - 10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~d

ATTENUATION CONSTANT (cak)

Fig. 3-Reflection coefficient (R) of an electromagnetic wavenormally incident on a plasma-free space interface as representedin the propagation constant plane.

ticular application the maximum allowable attenuationand reflection coefficient are known, then the "operatingregion" in the propagation plane is defined by the areaenclosed by the B-axis and the lines B2-AA2= 1,R=Rmax, and ao/k=max/k. Similar plots and operatingregions may be determined for the transmission proper-ties Tmin = "/1 - Rmax 2 of a plasma.

ELECTROMAGNETIC PROPERTIES OF HIGHTEMPERATURE AIR

Electron Density and Collision Frequency

At chemical equilibrium, the temperature of a hot gashas reached a stable value so that all the constituentswithin the plasma have been brought to this equilibriumtemperature. The molar fraction of these constituents,as well as all other thermodynamic quantities, can becalculated by the methods of quantum statisticalthermodynamics4'5 which determine the partition of theenergy states of a particle species into translational,rotational, vibrational, and electronic energies as well asthe energies of dissociation and ionization. Tables areavailable giving the equilibrium quantities for air.The electron concentration in high temperature air in

thermal equilibrium as a function of temperature anddensity as determined by Gilmore6 is shown in Fig. 4.At temperatures above 3000°K a rapid increase in thenumber of electrons occurs with rise in temperature asionization takes place. This rate of increase graduallylevels off at the higher temperatures as all the constitu-ents become singly ionized and the number of electronsdoes not increase substantially with temperature.The collision frequency of the electrons in a high tem-

perature gas mixture such as air is generally calculatedby adding the Maxwell-averaged collision cross section(weighted according to the number density of eachspecies) of the neutral molecules to a correspondingequivalent cross section of the ions, the electron-ioncollision cross section being calculated according toSpitzers7 theory for fully ionized gases.8'9 This methodwas thought to be a better approximation than themean-free-time (constant -y) method and was used for

4 J. G. Logan, Jr., "The Calculation of the Thermodynamic Prop-erties of Air at High Temperatures," Cornell Aeronautical Lab., Inc.,Ithaca, N. Y., Rept. No. AD-1052-A.1; May, 1956.

5 M. P. Bachynski, I. P. Shkarofsky, and T. W. Johnston, "PlasmaPhysics of Shock Fronts," Res. Labs., RCA Victor Co., Ltd., Mont-real, Can. RCA Res. Rept. No. 7-801-3; June, 1959.

6 F. R. Gilmore, "Equilibrium Composition and ThermodynamicProperties of Air to 24,000°K," RAND Corp., Santa Monica, Calif.,Rept. No. RM-1543; 1955.

7 L. Spitzer and R. Harm, "Transport phenomena in a completelyionized gas," Phys. Rev., vol. 89, pp. 977-81; 1953.

8 L. Lamb and S. C. Lin, "Electrical conductivity of thermallyionized air produced in a shock tube," J. Appl. Phys., vol. 28, pp.754-759; July, 1957.

9 M. P. Bachynski, 1. P. Shkarofsky, and T. W. Johnston,'Plasmas and the Electromagnetic Field," Res. Labs., RCA VictorCo., Ltd., Montreal, Can., RCA Res.Rept. No. 7-801-2; November,1958.

1960 351

PROCEEDINGS OF THE IRE

Fig. 4 Variation of electron density (n) and electron collision fre-quency (v) with temperature for air at different denisities.(po= 1.28823 X 10- g/cm3.)

this paper. The method is outlinied below. Recent, morerigorous work'0 indicates that in the range 6000°K to12,000°K, the total collision frequency is nearly a con-stant with velocity, so that (36) can be used but thevaluies are slightly different from Fig. 4.The method of constant mean free path1' which

10 I. P. Shkarofsky, M. P. Bachynski, and T. W. Johnston,"Collision Frequency Associated with High Temperature Air andScattering Cross-Sections of the Constituents," presented at AFCRCSymposium on the Plasma Sheath, Boston, Mass.; December 7-9,1959.

"1 L. G. H. Huxley, "Free path formulas for the electronic con-ductivity of a weakly ionized gas in the presence of a uniform andconstant magnetic field and a sinusoidal electric field," Aus. J. Phys.,vol. 10, pp. 240-245; 1957.

assumes the mean free path of the electronis to be inide-pendent of electron velocity is used here to calculate thecollision frequency. Making this assumptioni, the col-lision frequenicy for a Maxwellian (listribution of elec-tron velocities at high RF frequenicies becomnes5

(1 6a)

where

nj- number density of the jth specie,Qj= Maxwell-averaged total electron

sectioni of the specie,collision cross

c= 4 times mean electron velocity (16b)

K =Boltzmann's constant,T = temperature (°K).

With the collision cross sectionis of the neutral speciesas given by Massey and Burhop" anid calculatinig theconductivity due to electron-ioni collisionis7'8 from

1.1632m /4refo\2 (2kT )3J2o.ionl= ---oe

In (hlbo) \e/ \rm/

where

h KT4(Eo) I/2

87rnel (47rEo) (3 KT)

The variation of collision frequenicy with temiiperatureand density have been evaluated using (16a). These areshown in Fig. 4 for temperatures raniginig fromn 30000 to12,000°K and densities from 101 to 10 times stanidarddensity. Although the collision cross sections of thenieutral species may niot be accurate, the valuies fallwithin the spread of estimuates which can be obtainedfrom the current literature. Further, at the highertemperatures the positive iOn effects predominiate -andthe inifluenice of the neutral species becomiies of lesserand lesser importance. The above plots are thus indica-tive of the expected variationi and imiore accurate coImi-putation iii ust await further experimental determinia-tion of scattering cross sections and further refinle-mneints in theory.10

Attenuation and Phase ConstantsFrom the values of the electron density and collision

frequency shown in Fig. 4, the attenuation anid phaseconstants of an electromagnetic wave propagating inthe plasm-a created by air at high temperatures can bedetermined by using (4a), (4b), (6c), (6d), and (6e).

12 H. S. W. Massey and E. H. S. Burhop, "Electronic and IonicPhenomena," Oxford University Press, New York, N.Y.; 1952.

'lla.rch352

v = e E II,-IQj,

4 8KT

3 7rM

1960 ~~~Baclhynski, et al.: Electromagnetic Properties of High- Temperature A ir35

--~~~~~~~~~~~~~~I

.01;oo.17~~I

- o7

23456789101112 23456789101112 234567891~~~~~~~~~~~~~~~0x1112e

TEMPERATURE (xi63) *K

Fig. 5-Dependence of attenuation constant (a/k) on temperature and density of air. (po =1.28823 X 1031g/cm3.)

TEMPERATURE (xicTl) OK

Fig. 6-Dependence of phase constant ((3/k) on temperature and density of air. (po0= 1.28823 X 10-3g/cm3.)

The normialized attenuation and phase constants for

air in thermal equilibrium are shown as a function of

temperature and density for RF frequencies of 1, 10,

and 100 kmc in Figs. 5 and 6, respectively. The attenu-

ation constant a/k increases with increasing tempera-

ture as the number of electrons and collisions become

greater. Values of a/k decrease with increasing RE fre-

quency; however, this does not mean that the attenu-

ation necessarily decreases, since k is increasing with in-

creasing frequency. The phase constant i'lk starts off

from its free-space value of unity at low temperatures,

and for low densities at first decreases with increasing

temperature and then increases with temperature pass-

ing through a minimum in the region where pI _.1)2.

1023

, lo?F-.

C-lo,zo

z0 1

z

w

H-l

I0'o

z

0C)

L0J

)I

q

1960 353

PROCEEDINGS OF THE IRE

For high densities, the collision terms (i.e.' imaginarypart of the dielectric constant) predominate and no pro-nounced minimum values occur.The frequency dependence of the normalized attenu-

ation and phase constants are illustrated in Figs. 7 and8. At a temperature of 3000°K, air is not sufficientlyionized to appreciably affect the propagation character-istics of an electromagnetic wave except for low fre-quencies. (The present discussion is confined to frequen-cies above 1 kmc, although similar analyses can beapplied to the lower frequencies using the methods anddata presented earlier.) At a temperature of 3000°K,air acts like a slightly lossy dielectric whose dielectricconstant is nearly unity. Consequently, the phase con-stant is essentially the same as for free-space propaga-tion and the attenuation constant is very small. Air in

thermal equilibrium at a temperature of 6000°K can actas either a dielectric or a conductor, depending on thedensity and RF frequency. At low densities and highRF frequencies, air is essentially a dielectric with3/k-1, and a/k is small. However, as the RF frequencyis decreased, a rapid rise in attenuation and change inphase constant occurs as the plasma frequency becomescomparable to or greater than the frequency of the im-pressed electromagnetic wave. At high deinsities of air,the attenuation and phase constants are in general quitelarge, and decrease with increasing RF frequency. At12,000°K, air is essentially a good conductor, exhibitingvery high attenuation and phase characteristics.The propagation characteristics of high temperature

air are represented in the propagation plane for an im-pressed frequency of 6 kmc in Fig. 9. Contours of con-stant temperature, constant density, and conistant re-flection coefficient are shown. At low temperatures, the

t^2 ~ ~12,00- t-- =_---_ __L=r-- 01-- ~- ---- ~--------~~-.~ -

--I i--\te - 1 ~ ~ ~~~~~~~~~~- -- --------------

B1-:k.hb .| 15 -_ 'K

rrrFAWUrLN.;rT iMU.S

Fig. 7-Dependence of attenuation constant (a/k) on frequency for Fig. 8-Dependence of phase constant (fl/k) on frequency for high-high-temperature air. (po = 1.28823 X 10-3g/cm3.) temperature air. (po = 1.28823 X 10-3g/cm3.)

zC:I-)

LLI-1-

354 March

_ ; - ~-- _------

Bachynski, et al.: Electromagnetic Properties ol IIigh-Temperature Air

lo'

10I FREQUENCY 6 x io9/sec

II

l-

z

0 0'old,0-Tn0I

CONSTANT DENSITY (p/9p)CONSTANT TEMPERATURECONSTANT REFLECTION CO

12

1310

10 103 10- 0olATTENUATION

100 10ICONSTANT (a/k)

Fig. 9-Electromagnetic properties of high-temperature air at frequency of 6X 109 cps, showing the variation of the attenuation and phaseconstants at constant temperature, constant density, and constant reflection coefficient.

plasma behaves nearly like free space at all densities.As the temperature increases, the influence of theambient density becomes more apparent as the plasmabecomes more lossy. At high temperatures, the plasmais a good conducting medium and the effect of densityvariation again becomes secondary. In this representa-tion, an "operating region" for propagation of elec-tromagnetic energy can be determined, provided thetolerable attenuation and reflection coefficients are

specified.

HYPERSONIC VEHICLE IN PLANETARY ATMOSPHERE

A space vehicle moving at hypersonic velocitieswithin a planetary atmosphere will be surrounded by a

shock-induced ionized sheath. If the temperature anddensity of a given region of the shock is known, then theresults presented earlier (Figs. 4-8) can be used to ob-tain an estimate of the propagation characteristics of an

electromagnetic wave through this region. For purposesof communicating to and from a space vehicle, it is mostfeasible to attempt to propagate electromagnetic energythrough the wake of the shock or at some point aft ofthe stagnation region of the shock where the influenceof the plasma on an electromagnetic wave is not as pro-nounced as in the stagnation region; i.e., the electrondensity and collision effects are less. However, one

difficulty is that the temperature and density distribu-tion of the shock away from the stagnation region is notvery well known. On the other hand, it mnay be instruc-tive to determine the propagation characteristics of anlelectromagnetic wave in the stagnation region of a

hypersonic vehicle for two reasons, firstly that assumingthermal equilibrium the thermodynamic quantities inthis region can be readily deduced from aerodynamicconsiderations, and, secondly, that this is the regioni ofmost dense plasma and hence the most critical conidi-tions for penetration by, or propagation of, an electro-mnagnetic wave.

Fig. 10 shows the variation of plasma frequency andelectron collision frequency at the stagnatiorn point of a

hypersonic vehicle with velocity at various altitudesabove the earth. (The ARDC model of the atmospherewas used in the aerodynamic considerations. 13) Usingthe values in Fig. 10, contours of constant velocity andconstant altitude for a hypersonic body have beenplotted in the normalized propagation plane for fre-quencies of 3 kmc and 30 kmc. These are showni inFig. 11.

13 A. R. Hochstim and R. J. Arave, "Various ThermodynamicProperties of Air," Convair, San Diego, Calif., Rept. No. ZPH-004;June, 1957.

j02 103

1960 355

March

140 f j ---- X /|ALTITUDEI40 1 t - ---- - v / sX le FEET)

20QD.

30

CURVtS FOR

20

I0~~~~~~~I250

O 10 20 30 40 50 60 70

(a/k)

Fig. 11-Electromagnetic properties of air at the stagnation point ofa hypersonic vehicle in the atmosphere at frequencies of 3X109and 30>X 1O1 cps, showing the variation of the attenuation andphase constants with altitude and velocity.

VELOCITY (Km/sec)

Fig. 10 Variation of plasma frequency (fp= 1/2ir \ne2/meo) andelectron collision frequency v at the stagnation point of a hyper-sonic vehicle with velocity at various altitudes above the earth.

ACKNOWLEDGMENT

The authors are indebted to the Aerophysics Wing ofthe Canadian Armament Research and DevelopmentEstablishmenit for financial support.

356 PROCEEDINGS OF THE IRE