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ELF/VLF radiation resistance of an arbitrary orientedfinite dipole in a cold, uniform multicomponent

magnetoplasmaT.N.C. Wang

To cite this version:T.N.C. Wang. ELF/VLF radiation resistance of an arbitrary oriented finite dipole in a cold,uniform multicomponent magnetoplasma. Journal de Physique, 1971, 32 (11-12), pp.877-885.<10.1051/jphys:019710032011-12087700>. <jpa-00207187>

877

ELF/VLF RADIATION RESISTANCEOF AN ARBITRARY ORIENTED FINITE DIPOLE

IN A COLD, UNIFORM MULTICOMPONENT MAGNETOPLASMA

T. N. C. WANG

Radio Physics Laboratory Stanford Research Institute Menlo Park, California 94025

(Reçu le 19 avril 1971)

Résumé. 2014 Utilisant la théorie complète des ondes électromagnétiques, on analyse la résistancede radiation UBF/TBF d’un dipôle électrique fini, orienté d’une façon arbitraire dans un magnéto-plasma froid et uniforme. On en déduit l’expression intégrale générale de la résistance de radiation R,puis, à partir de cette expression, différentes formes approchées de R pour le spectre completUBF/TBF. On présente les courbes de variation de R lorsque les paramètres du plasma corres-pondent à deux régions : la haute ionosphère et la magnétosphère. L’on démontre que, dans l’inter-valle de fréquence où la surface de l’indice de refraction du mode n- présente une branche ouverte,la résistance de radiation R atteint un maximum chaque fois que l’axe de symétrie du dipôle estperpendiculaire au plan tangent au cône de résonance du mode n. Ce maximum relatif Rmax està peu près égal à ~h/r0 (h et r0 étant respectivement la demi-longueur et le rayon du dipôle). Dans ledomaine UBF, on montre que Rmax se produit lorsque l’orientation du dipôle est presque parallèleau champ magnétique du plasma très dense considéré. On montre que, pour un dipôle de longueurconstante et à une fréquence de plasma normalisée fixe (f0/fHe), les valeurs de R dans le domaineUBF et dans le domaine TBF sont comparables.

Abstract. 2014 By using a full electromagnetic-wave theory, an analysis is made of the ELF/VLFradiation resistance of a finite electric dipole, oriented at arbitrary angle in a cold, uniform magneto-plasma. The general integral expression for the radiation resistance R is derived, and from that,various approximate closed forms of R are obtained for the entire VLF/ELF range. Numericalplots of R are given for the plasma parameters appropriate to the regions of topside ionosphere andmagnetosphere. It is found that, in the frequency range in which the refractive index surface of the n-mode has an open branch, the radiation resistance R reaches a maximum value whenever the dipoleis so oriented that its axis of symmetry is perpendicular to the plane containing the edge of the n-mode resonance cone. This relative maximum Rmax is ~ ~h/r0 (h, r0 being the half-length and radiusof the dipole, respectively). In the ELF range, this Rmax is shown to occur for the dipole orientationclosely parallel to the magnetic field for the high-density plasma considered. For a fixed dipolelength and a fixed normalized plasma frequency (f0/fHe), it is shown that R in the ELF range can becomparable to that in the VLF range.

LE JOURNAL DE PHYSIQUE TOME 32, NOVEMBRE-DÉCEMBRE 1971,

ClassificationPhysics Abstracts

05.10, 14.20

1. Introduction. - In recent years, considerableattention has been devoted to the problem of antennaradiation in a magnetoplasma. The interest in the

problems of antennas in a magnetoplasma stems fromthe use of antennas aboard rockets and satellites

traveling the ionosphere or the magnetosphere.The study of the radiation characteristics at ELF/VLFof such antennas has direct applications in the areasof : (1) planning of wave-particle interaction experi-ments involving satellite-based transmitters in the

magnetosphere, (2) ionospheric and magnetosphericplasma diagnostics for determining the earth’s magne-

(1) In I, II and the present paper, the VLF range is definedas the frequencies between the electron gyrofrequency ( f He)and the proton gyrofrequency ( f Hp).

tic field strength and charged particle densities andmasses at points along a satellite or rocket trajectory(see, e. g., Blair [1]), (3) provision of a possiblecommunication link between ground and spacebases, and (4) detection of the noise fields from thesun and other radio stars.

In two papers Wang and Bell [2]-[3] ([1], denotedin this paper by I, and [3], denoted by II) used alinearized full-wave treatment to investigate the °

VLF (1) radiation resistance R for a finite dipoleimmersed in a uniform, cold, multicomponent magne-toplasma. Approximate closed-form expressions werederived for the R of an electric dipole oriented eitherparallel or perpendicular to the static magneticfield (Bo) for frequencies fHe &#x3E; f » fL,, (fLHR beingthe lower-hybrid-resonance frequency) and for the R

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019710032011-12087700

878

of a dipole oriented arbitrary to Bo for the frequenciesJLHR &#x3E; f &#x3E; fHp. The results from I led to the conclu-sions that, up to « moderate » dipole length (moderatelength being defined in I), the radiation resistance canbe adequately predicted by the quasi-static approxi-mation and that the correction term for R due tothe wave fields is, in general, small. In the VLF fre-quencies below fLHR, the quasi-static impedance for-mulas obtained by Balmain, [4], for a lossless regime,were purely imaginary and yields no resistive part.The first-order dipole radiation resistance must thenbe calculated from the full-wave fields. In II, the VLFdipole radiation resistances for this frequency rangeswere calculated in terms of elliptic integrals.The ELF range is defined here as the frequencies

below fHp down to zero. When a magnetoplasmacontains multicomponent ion species (e. g., the plasmawithin the ionosphere), it is expected that the radia-tion and propagation of ELF electromagnetic wavesin the medium will become complicated. The elec-

tromagnetic propagation characteristics of ELF waveswere examined by Smith and Brice [5]. However,little work has been directed to the ELF radiationcharacteristics of an antenna in a magnetoplasmawith multicomponent ion species.

In view of the above-mentioned applications, it isclear that understanding of the changes in VLF/ELFdipole radiation characteristics with changes of

dipole orientation, frequency, and plasma parametersare important in designing an optimized radiating/receiving element for use aboard a satellite or rocket.It is the purpose of this paper to extend the full-wave

analysis of the two previous papers (I and II) and,specifically, to calculate the ELF/VLF radiationresistance for a finite dipole oriented at an arbitraryangle to the static magnetic field.

From the detailed structure of mode propagationcharacteristics, we determine the leading terms of

dipole radiation resistance in the ELF range directlyfrom those given for the VLF range. Adequate appro-ximate closed expressions for dipole radiation resis-tance within the entire VLF/ELF range are given.These expressions are valid for any arbitrary angle ofdipole orientation. Numerical plots of dipole radiationresistance at VLF/ELF are given for plasmas modeledupon the topside ionosphere and the magnetosphere.

2. Formulation. - In this section, a brief deriva-tion is given of an integral expression of the inputimpedance and the radiation resistance for a cylindricaldipole oriented at an arbitrary angle (with respect tothe static magnetic field) in a cold, multicomponentmagnetoplasma.Without loss of generality, we set the cylindrical

dipole on the x-z plane and let its axis be inclined atan angle go with respect to the z-axis. The static

magnetic field is oriented along the positive z-axis.The geometry of the problem is indicated in figure 1.

Assuming a skin-triangular distribution for the

FIG. 1. - Geometrical configuration of the dipole antenna in amagnetoplasma.

antenna current (Balmain, [4] ; Seshadri, [6]) the

dipole current density J can be written

where

x’, y’, and z’ are the components of x’, the primedcartesian system ; 10 is the total effective current at theinput terminals ; x, z is the unit vector in x (unprimed)systems ; and h and ro are the half-length and radiusof the dipole, respectively.

A three-dimensional Fourier-transform on (1) yields

where

and (k, 0, gi) are the components of a wave vector kin unprimed spherical polar coordinates.

Using (2) and eq. (6) of I, the principal-polarizedcomponents of the Fourier-transformed current are

given by

879

By using (3) together with eq. (12) and (15) of I,and the usual definition for the antenna input impe-

dance, Z = 2 P/Iô, the formal (integral) solutionof the input impedance can be shown to be

where

(free-space wave number), and where

Xr and Yr (carrying sign) are standard notations of normalized frequencies for each species in the magne-toionic medium (see, e. g., Ratcliffe [7]) ;

Note that the n; given by (6) are actually the tworoots of the biquadratic dispersion equation for thecold magnetoplasma system considered [2]-[3] andthat n± represents the refractive indexes for the twocharacteristics modes in the medium.For the two particular cases of an antenna oriented

either parallel or perpendicular to the static magneticfield, we set, respectively, ço = 0 and rc/2 in (4), andit is easy to show that (4) reduces identically to theformal solutions for these two antenna orientationsderived in I.For the ideal case of a lossless magnetoplasma,

various pole singularities of the integrand (i. e.,

n2 = n+, n2 , 80’ E± 1, etc.) may make the integralof (4) ambiguous. These mathematical ambiguitiescan be removed by assuming a small finite loss for thesystem, which removes these poles off the real lineinto the complex plane. The original real-line integralcan then be performed in a complex plane by usingcontour integration techniques. After a proper contourintegration, we can allow the small finite losses tovanish. The real part and the imaginary part of (4)

can be interpreted, respectively, as the radiationresistance and the input reactance of the dipoleantenna in a lossless magnetoplasma. To facilitate thecontour integration, we use several symmetriescontained in the integral and recast (4) into the form

where Fil, Fl, F+, and S«({J)o are defined in (5), andS(- oo) is given by replacing po by - oo in S( ((Jo).

It is clear that (7) is now suitable for performing acontour integration with respect to the variable n.

Using the expression n± given by (6), it is straight-forward to show that Flr(n2 - g 0) = 0 and that

FL(n 2 -+ e ± 1) = 0. These conditions ensure that, in

performing the contour integration, the contributions

880

due to the apparent poles (i. e., n2 = eo, E t 1) vanishidentically (2). The total value of the integral in acontour integration is then due to the poles n± 2 (0)@which are the zeros of the dispersion equation.

Introducing a small loss to the system, we performa contour integration in n, following a proceduresimilar to that discussed in I, II. The formal solution

of dipole radiation resistance can be extracted from (7) :

where

In (8), the domain of 0 for the 0-integration shouldcover the whole range of 0 for which n 2 is positive(the region of propagation for each mode in the wave-normal space).

As a check for the correctness of (8), we let ço = 0,n/2 ; (8) then reduces identically to (23) and (B 1) of I.Equation (8) is a general integral expression for theradiation resistance of a « short » dipole (short inthe sense that the triangular current distribution is

valid) embedded in a uniform, cold magnetoplasma.This expression is valid for arbitrary values of drivingfrequency, plasma composition, particle density,and static magnetic field strength and is suitable fornumerical computations.

3. Quasi-Static Radiation Résistance. - In a colli-sionless magnetoplasma, there are the so-called

(2) It can be easily indicated that 8x 1 and eo are actually someof the limiting values of n 2(o)@ when 0 - 0 and n/2. Their con-tributions to Z of (4) are already included in the poles n2 = nfl,and therefore (4) will not yield any additional finite contribu-tions from the poles n2 = s+ i, 80 as would be expected.

resonant frequency ranges in which the refractiveindex surface of one characteristic mode contains an

open branch (e. 9., fH. &#x3E; f &#x3E; FLHR and several otherfrequency ranges indicated in Table 1 of a later Sec-tion). For the wave normal angles near the openbranch, the characteristics of the fields become

quasi-static, with a small component of magneticfield H (Arbel and Felsen [8]). However, these quasi-static fields extend also to the far zones [8]. For a« moderate » short dipole antenna together withskin-triangular current distribution assumed it can beshown that the predominant contribution of the

integral given by (8) comes from the wave normalangle regions near the open branch of the refractiveindex (see 1 for the two particular cases of dipoleorientation) and further that this predominant por-tion of the integral is in substantial agreement of theresult obtaining through a usual quasi-static calcula-tion ; i. e., an integration of the leading term of (4)as n2 --+ 00 (or in B = co/c, let speed of light in freespace c - oo). In view of the above discussion, it istherefore useful for us to calculate the dipole radiationresistance using quasi-static approximation.The quasi-static approximation to the dipole radia-

tion resistance RQ can be obtained from (4) by takingthe leading term of the real part of Z as n --+ oo.This process gives the following relation :

In the collisionless regime it is clear from (9) thatRQ = 0 whenever eo/e. &#x3E; 0, i. e., oe(0) # 0. However,if 80/8s 0, ensuring that one of the roots in (6) has anopen branch, the « collisionless » solution can beobtained by evaluating (9) under the assumption thatsmall collisional losses exist in the plasma and thenallowing these losses to approach zero. Introducingsmall collisional losses and transforming the 0-inte-gration to a new variable cos 0 = x, we then make aproper contour integration (see, e. g., Bell and

Wang [9]) to obtain

where

and 0, is the real root of ot(O) = 0, given by

881

Integration of (10) (details given in the Appendix) yields the results

where

The geometrical orientations of the dipole corres-ponding to the three cases of RQ given by (11) arequalitatively indicated in figure 2. For the case of

dipole orientation either parallel «(fJo = 0) or perpen-dicular (po = n/2) to the magnetic field, we takethe limiting values of (lla) and (lle) as ço - 0and ço - n/2, respectively. It is easy to show thatthese limiting values are given by

Equation (12) is in substantial agreement with theprevious results obtained from different calcula-tions [2], [4].

FIG. 2. - A qualitative diagram for the dipole orientationssubject to formula (11).

882

From the above analysis, it can be seen that thefunctional dependence of RQ on h/ro changes as CfJoincreases from 0 to n/2 and that, for fixed plasmaparameters and a thin dipole (hlro » 1), RQ reachesits maximum value at CfJo = nl2 - 0,.

4. Radiation Résistance at VLF. - The VLF rangedefined here represents those frequencies f ; fHe &#x3E; f &#x3E; fHp.We shall consider a relatively high density plasmato be such that f o (the plasma frequency) &#x3E; fHe (thiscondition yields e. 0 for f fHe) and assume thatno negatively charged ions exist, so £+ 1 &#x3E; 0. Underthese conditions, it can be shown from (6) that, in

the VLF range, n 2 0 and n2 &#x3E; 0. The term

with subscript + does not appear in (8), and theradiation resistance for this case is due solely to

the n- mode (a propagating mode), which is knownas the whistler mode. In the frequency range

fHe &#x3E; f &#x3E; fLHR’ the refractive index surface for thewhistler mode is open

The radiation resistance for this frequency range hasbeen calculated in 1 for the two special orientations(parallel and perpendicular). For an arbitrarily orienteddipole, it is difficult to integrate (8) analytically.However, an approximate closed form of R has beengiven by (11) in the previous section using the quasi-static fields. The full-wave solution of (8) has beenintegrated numerically ; the numerical results are

discussed in a later section. In the frequency rangefLHR &#x3E; f &#x3E; fHp, ni(0) is bounded, and the refractiveindex surface of the whistler mode is closed. For this

case, 80/8s &#x3E; 0 and RQ --_ 0 ; the dipole radiationresistance must be calculated from the full-wave fields.For an antenna length subject to the constraint

it is possible to integrate (8) analytically. The detailedcalculation for this case has been given in II. For thepurpose of the discussions in the next section, wereproduce the results of II [see eq. (7), (12), and (16)of 111 :

where

and where A, B, C, A’, B’, and C’ are functions of8B1 (v = + 1, - 1, 0) alone. Their expressions can befound in II. Here F(q, p) and E(q, p) are the generalelliptical integrals of the first and second kind, respec-tively, with the arguments q, p defined by

and

The solution given in (13) is appropriate as long asb 80 and this condition is met for frequencies inthe VLF range f ; fLHR &#x3E; f &#x3E; fHp(1 + 10-3).

5. Radiation Résistance at ELF. - The ELF rangedefined here is those frequencies below fHp. When thehigh-density magnetoplasma considered contains mul-ticomponent ion species, the propagation characte-ristics in the ELF range become more involved.In general, both n _ and n + modes can propagate.Between every pair of successive ion gyrofrequencies,there exist a number of plasma characteristic fre-

quencies. These include the frequencies known as thecrossover frequency (/co)? the cutoff frequency (fcf),and the two-ion-resonance frequency ( f MR) (Smithand Brice [5]) ; they are obtained from the uniqueroot (between each successive pair of ion gyrofre-quencies) of the conditions 8d = 0, 8 - 1 = 0, and

es = 0, respectively.On the basis of the mode propagation characte-

ristics, the dipole radiation resistance at ELF can beinferred from the results derived for the VLF case.

By using (6), we can examine the details of the wavepropagation characteristics at ELF. The results aresummarized in table I, for a typical pair of successiveion gyrofrequencies (say, fHp &#x3E; f &#x3E; fHH:, fIlH,,’ = gyro-frequency for atomic helium species).The ordering sequence and propagation characte-

ristics summarized in table 1 will repeat for each pairof adjacent ion gyrofrequencies until the signal fre-quency decreases below the smallest ion gyrofrequencyfHs. When f fHs, the ordering of plasma parametersis 8-1 i &#x3E; 8s &#x3E; s+i 1 &#x3E; 0, Ed 0, and the propagationcharacteristics are as described in the first row oftable I. This ordering, as well as the propagationcharacteristics, will not change over the range

since no crossover frequency exists in this range.

In view of the propagation characteristics describedin table I, the dipole radiation resistance at ELF canbe easily derived from those solutions for the VLFcase. In the frequency range fHi &#x3E; f &#x3E; f iMR (fHi and.fiMR stand for the gyrofrequency of the ith ion speciesand the two-ion-resonance frequency for the ith

and i +1 th ion species), where we note that

883

TABLE 1

A summary of typical propagation characteristics at ELF for the frequencies fHP &#x3E; f &#x3E; fH,14and under the assumptions 80 0, 8+ 1 &#x3E; 0 (no negative ions)

and that the n _ mode always possesses an openrefractive index, the leading term of radiation resis-tance is given by RQ from the quasi-static calculation.The full-wave corrections to RQ due to n+ and/or n-modes can be straightforwardly calculated throughthe similar analysis made in 1 and II. However, forthe « short » antennas whose length is subject to theconstraints given in 1 and II, it can be shown that thefull-wave corrections to the radiation resistance aremuch smaller than RQ. Since the dependence of,RQ in (11) on e, 1 is only through 8s, which is symme-tric in the interchange of 8+1 and 8-1’ the quasi-staticradiation resistance RQ for the frequencies

is identical to that of (11). In the frequency rangefiMR &#x3E; f &#x3E; fH i + 1 (fH i + 1 being the gyrofrequencyfor the i + lth ion species). We note from table 1that the ordering of the plasma parameters and pro-pagation structures is identical to that in the VLF

range f LHR &#x3E; f &#x3E; fHp. It is clear that, for this rangewithin ELF, the dipole radiation resistance is givenidentically by (13).

6. Numerical Results. - Our numerical data forthe VLF/ELF dipole radiation resistance were obtainedby evaluating (8) through the use of computerintegration. In figures 3, 4 and 5, we plot the K as afunction of the dipole orientation angle po for variousvalues of frequency, plasma density, plasma composi-tion, static magnetic field strength, and dipole length.The curves are parametric in normalized dipolelength ho (ho = 2 nfH, h/c) and normalized plasmafrequency folfh,, (fo being the plasma frequency).

FIG. 3. - VLF dipole radiation resistance as a function ofnormalized length ho, and orientation angle po. Four normalizeddriving frequencies f/f He = 0.05, 0.25, 0.5 and 0.75 ( f &#x3E; fLHR),and two normalized densities fo/fHe = 2, 10 are considered.

884

The two values of folfHe (./o//He = 2, 10) used werechosen as being representative of the range of valuesthat can be encountered in the topside ionosphereand the inner magnetosphere. The values of howere chosen in the range 0.05 ho 0.5. In allthe numerical plots of R, the ratio h jro = 103 is used.

Figure 3 is a plot of R for several frequencies in theVLF range and above f LHR. The relative peaks shownin the curves correspond to the dipole orientationssatisfying the condition : ço = z/2 - 0,. The curvesvary approximately as ho-1 and (fo/fHe) - 2 for

and ( f ô/f fHe) 1 for ço &#x3E; (z/2 - 9r). Figure 4 is a plotoff for the VLF frequencies below f LHR. In this

frequency range, the whistler mode refractive index isclosed, and therefore no resonance cone angle exists.Correspondingly, the curves shown in figure 4 do notpossess relative peaks. The curves are approximatelyproportional to hÉ and (fplfHe)· The ratio of

is approximately equal to (fH,lf)’. For the plots offigures 3 and 4, the plasma is assumed to consist ofelectrons and protons.

Figure 5 is a plot off similar to that of figure 3for the frequencies in the ELF range. The plasma is

FiG. 4. - VLF dipole radiation resistance as a function ofnormalized length ho and orientation angle ço. Three normalizedfrequencies below fLHR ; f/f He = 0.002, 0.01, and 0.02, and two

normalized densities fo/fHe = 2, 10 are considered.

FIG. 5. - ELF dipole radiation resistance as a function ofnormalized length ho and orientation angle ({JO. Five normalizedfrequencies f/f Hp = 0.175, 0.2, 0.4, 0.6, and 0.8, and two nor-

malized densities f o/f He = 2, 10 are considered.

assumed to consist of electrons and three ions :atomic hydrogen, atomic helium, and atomic oxygen.The composition of the plasma is assumed to be70 % H+, 20 % He+, and 10 % 0*. For this three-ion species plasma model considered, the hydrogen-helium resonance frequency fH,,,H, is N 0.375 fHpand that the helium-oxygen resonance frequencyfo++He+ is ~ 0.155 fHp. The numerical curves of

figure 5 are plotted for the normalized frequenciesf/fHp in the range where the n- mode refractive indexpossesses an open branch. However, for the frequen-cies considered, the ratio 1 80/8s 1 » 1 and thus

Br N n/2. Therefore the relative peak of R curvessubject to the condition Po = n/2 - Or is close to

oo N 0, as can be seen in figure 5. The functionalbehaviour of the curves in figure 5 with respect to theother parameters is similar to that of the curves of

figure 3, discussed previously.In comparing the results of the exact integral form

of (8) with the approximate closed-form results, thecurves of figures 3 and 5 agree with the approximateformula (11) within a few percent, whereas the curvesof figure 4 agree with the approximate formula (13)within a few percent.

7. Concluding Remarks. - Using linear full elec-

tromagnetic wave theory, an analysis has been madeto calculate the VLF/ELF radiation resistance of afinite electric dipole in a uniform unbounded magne-toplasma. The orientation of a dipole with respect tothe static magnetic field is set arbitrarily. The generalintegral expression for ,R is derived [see (8)], andfrom that, various approximate closed forms of R,valid in the VLF/ELF range, have been evaluated.

Numerical data for R have been plotted through useof computer integration of the general integral formof R, and their results agree with the approximateformulas (11) and (13) within a few percent.From this study, several interesting conclusions

can be drawn :

1. Within a frequency band in which the refractiveindex surface of n _ mode has an open branch, the

885

radiation resistance for a thin dipole possesses a rela-tive maximum Rmax at the angle of orientation

go = n/2 - 0,. For fixed plasma parameters this

relative maximum of R is proportional to Ihlr,,.2. For the high-density plasma ( f &#x3E; fHe ; 1 80 &#x3E; 1),

the resonance cone angle Or is very close to rc/2 in theELF range. Consequently ,Rmax occurs for the dipoleorientation nearly at go - 0, which leads to thecondition that R (Po’" 0) &#x3E; R (90 - rc/2).

3. For a fixed dipole length (ho) and normalizedplasma frequency (fo/fHe) the dipole radiation resis-tance in the ELF range has been shown to be compa-rable to those values of R in the VLF range of

and much greater than those values off in the VLFrange of f LHR &#x3E; f &#x3E; fHp.

It is important to note that in the present investiga-tion we have considered an idealized model for the

problem of antenna-magneto-plasma coupling. In arealistic situation, many complications can be intro-duced by the presence of the antenna-plasma sheath,by finite temperatures, and by the nonuniformities ofthe plasma. With these effects included, the problemof antenna-magnetoplasma coupling is in general anonlinear one, and the analysis is likely to be consi-derably complicated. Although the exact analysisusing a more realistic antenna-magnetoplasma modelis difficult at this stage, the present results based on asimplified magnetoplasma model should provide a

first insight into the complex problem of antenna-magnetoplasma coupling.

Appendix. - EVALUATION OF QUASI-STATIC RADIA-

TION RESISTANCE RQ. - To obtain an approximateclosed for of RQ, (9) can be rewritten in the form

where

Using a new variable x = cos 0, we can then performa proper contour integration with respect to x [see,e. g., eq. (20), Snyder and Weitzner]. This contourintegration leads to the following form :

where

To a good approximation, the two-dimension integralin (A. 2) can be approximated as follows

where

and where

A straightforward integration of (A. 3) yields :

where Yl, Y2 and C(CPo) are defined in (11). Substitu-tion of (A. 4) into (A. 2) yields the results given by (11).

References

[1] BLAIR (W. E.), Radio Sci., 1968, 3, 155.[2] WANG (T. N. C.), BELL (T. F.), Radio Sci., 1969, 4,167.[3] WANG (T. N. C.), BELL (T. F.), Radio Sci., 1970, 5, 605.[4] BALMAIN (K. G.), IEEE Trans. Antennas Propagation,

1964, AP-12, 605.[5] SMITH (R. L.), BRICE (N.), J. Geophys. Res., 1964, 69,

5029.

[6] SESHADRI (S. R.), Proc. IEE (London), 1965, 112, 1856.

[7] RATCLIFFE (J. A.), The Magnetoionic Theory and ItsApplication to the Ionosphere, Cambridge Univ.Press, London, 1959.

[8] ARBEL (E.), FELSEN (L. B.), in Symp. on ElectromagneticTheory and Antennas, Pt. II, E. C. Jordan, Ed.New York: Pergamon, 1963, 421-459.

[9] BELL (T. F.), WANG (T. N. C.), IEEE Trans. AntennasPropagation, July, 1971, AP-19.