ELF/VLF radiation resistance of an arbitrary oriented finite dipole in a cold, uniform multicomponent magnetoplasma

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

magnetoplasma

T.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�

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ELF/VLF

RADIATION RESISTANCE

OF 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. ²⁰¹⁴Utilisant la théorie complète ^{des ondes}électromagnétiques, ^onanalyse ^la^résistance

de radiation UBF/TBF ^d’undipôle électrique fini, orienté d’une façon arbitraire dans un magnéto- plasma ^froidêtuniforme. On en déduit l’expression intégrale générale de la résistance de radiation R, puis, ^àpartir ^de^cetteexpression, différentes formes approchées ^{de R}pour le spectre complet UBF/TBF. Ônprésente les courbes de variation de R lorsque ^lesparamètres ^duplasma ^corres- pondent ^{à deux}régions : ^{la haute}ionosphère êt^lamagnétosphère. L’on démontre que, dans l’inter- valle de fréquence ôùla surface de l’indice de refraction du mode n- présente ûne^brancheouverte, la résistance de radiation R atteint un maximum chaque ^foisque l’axe de symétrie ^dudipôle êst perpendiculaire âuplan tangent âucône de résonance du mode ^n.Ce maximum relatif Rmax est à _peuprès égal ^à~h/r0 ^(hêt^r0^étantrespectivement ^lademi-longueur êt^lerayon du dipôle). ^Dans^le

domaine UBF, ^on^montreque Rmax se produit lorsque l’orientation du dipôle ^estpresque parallèle

au champ magnétique ^duplasma ^trèsdense considéré. On montre que, pour ^undipôle ^delongueur

constante et à une fréquence ^deplasma normalisée fixe (f0/fHe), les valeurs de R dans le domaine

UBF et dans le domaine TBF sont comparables.

Abstract. ²⁰¹⁴By using ^a^fullelectromagnetic-wave theory, ^ananalysis is made of the ELF/VLF

radiation resistance of a finite electric dipole, ôrientedâtarbitrary angle ⁱⁿâcold, ûniformmagneto- plasma. ^Thegeneral 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. Numerical

plots ^{of R}âregiven ^{for the}plasma parameters appropriate ^to^theregions ôftopside ionosphere ând magnetosphere. Ît^{is found}that, ^{in the}frequency ^rangein which the refractive index surface of the n-

mode has an open branch, ^the^radiationresistance R reaches a maximum value whenever the dipole is so oriented that its axis of symmetry ^isperpendicular ^to^theplane containing ^theedge ^{of the}^n-

mode resonance cone. This relative maximum Rmax is ~ ~h/r0 (h, ^r0being ^thehalf-length ^and^radius

of the dipole, respectively). ^In^{the ELF}^range,this Rmax is shown to occur for the dipole orientation

closely parallel ^to^themagnetic field for the high-density plasma considered. For a fixed dipole length ^and^a^fixednormalized plasma frequency (f0/fHe), it is shown that R in the ELF range ^canbe

comparable ^tothat in the VLF range.

Classification

Physics ^Abstracts 05.10, 14.20

1. Introduction. ^-In recent years, considerable attention has been devoted to the problem ^of^antenna

radiation in a magnetoplasma. The interest in the

problems ôfântennasⁱⁿâmagnetoplasma ^stems^from

the use of antennas aboard rockets and satellites

traveling ^the ionosphere ^or^themagnetosphere.

The study ^{of the}radiation characteristics at ELF/VLF of such antennas has direct applications ^{in the}^areas

of : (1) planning ^ofwave-particle interaction experi-

ments involving satellite-based transmitters in the

magnetosphere, (2) ionospheric ^andmagnetospheric plasma diagnostics ^fordetermining the earth’s _magne- (1) ^InI, II and the present paper, the VLF _rangeis defined

as the frequencies between the electron gyrofrequency ( f He)

and the proton gyrofrequency ( f Hp).

tic field strength ^andcharged particle densities and

masses at points along ^a^satellite^or^rockettrajectory

(see,

^e. ^g., ^Blair

[1]),

⁽³⁾^provision^of^a^possible

communication link between ground and space

bases, ^and(4) ^detectionof the noise fields from the

sun and other radio stars.

In two papers Wang ^{and Bell}[2]-[3]

([1],

^denoted

in this paper by I, ^and[3], ^denotedby

II)

^used^a

linearized full-wave treatment to investigate ^the^°

VLF (1) radiation resistance R for a finite dipole

immersed in a uniform, cold, multicomponent ^magnetoplasma. Approximate closed-form expressions ^were

derived for the R of an electric dipole oriented either

parallel ^or perpendicular ^to the static magnetic

field (Bo) ^forfrequencies fHe > f » fL,, (fLHR being

the 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

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878

of a dipole ^orientedarbitrary ^toBo ^{for the}frequencies JLHR > f > fHp. The results from I led to the conclusions that, up ^to^«moderate » dipole length (moderate length being ^definedⁱⁿI), ^the^radiationresistance can

be adequately predicted by ^thequasi-static approxi-

mation and that the correction term for R due to the wave fields is, ⁱⁿgeneral, small. In the VLF fre-

quencies ^belowfLHR, ^thequasi-static impedance ^for-

mulas obtained by Balmain, [4], ^for^a^losslessregime,

were purely imaginary ^andyields ^no^resistivepart.

The first-order dipole radiation resistance must then be calculated from the full-wave fields. In II, ^{the VLF} dipole radiation resistances for this frequency ranges

were calculated in terms of elliptic integrals.

The ELF _rangeis defined here as the frequencies

below fHp ^down^to^zero.^When^amagnetoplasma

contains multicomponent ^ionspecies (e. ^g.,^theplasma

within the ionosphere), ^{it is}expected ^{that the}^radia-

tion and propagation ^{of ELF}electromagnetic ^waves

in the medium will become complicated. ^The^electromagnetic propagation characteristics of ELF waves were examined by ^Smith^and^Brice[5]. However, little work has been directed to the ELF radiation characteristics of an antenna in a magnetoplasma

with multicomponent ^ionspecies.

In view of the above-mentioned applications, ^{it is}

clear that understanding ^{of the}changes ⁱⁿVLF/ELF dipole radiation characteristics with changes ^of dipole orientation, frequency, ^andplasma parameters

are important ⁱⁿdesigning ^anoptimized radiating/

receiving element for ûseaboard âsatellite ôrrocket.

It is the purpose of this paper ^toextend the full-wave

analysis ^{of the}^twoprevious ^papers(I ^andII) and, specifically, ^to calculate the ELF/VLF ^radiation

resistance for a finite dipole ôrientedâtânarbitrary angle ^tothe static magnetic ^field.

From the detailed structure of mode propagation characteristics, ^wedetermine the leading ^terms^of dipole radiation resistance in the ELF _rangedirectly

from those given for the VLF _range.Adequate approximate closed expressions ^fordipole radiation resistance within the entire VLF/ELF range ^aregiven.

These expressions âre^valid^forany arbitrary angle ôf dipole orientation. Numerical plots ôfdipole ^radiation

resistance at VLF/ELF ^aregiven ^forplasmas ^modeled

upon the topside ionosphere ^and^themagnetosphere.

2. Formulation. ^-In this section, âbrief deriva- tion is given ôfânintegral expression ^{of the}input impedance ând^theradiation resistance for a cylindrical dipole ôrientedâtânarbitrary angle (with respect ^to the static magnetic field) ⁱⁿâ^cold,multicomponent magnetoplasma.

Without loss of generality, ^we^set^thecylindrical dipole ^on^the^x-zplane and let its axis be inclined at an angle go with respect ^tothe z-axis. The static

magnetic ^field^is^orientedalong ^thepositive ^z-axis.

The geometry ^{of the}problem ^is^indicatedⁱⁿfigure ^1.

Assuming ^a skin-triangular distribution for the

FIG. 1. ^-Geometrical configuration ^{of the}dipole ^antennaⁱⁿ^a magnetoplasma.

antenna current

(Balmain,

[4] ; Seshadri,

[6])

^the

dipole ^currentdensity ^J^can^{be written}

where

x’, y’, ^{and z’}^are^thecomponents ^ofx’, ^theprimed

cartesian system ; 10 ^{is the}total effective current at the

input terminals ; x, z is the unit vector in x (unprimed) systems ; ^{and h and}ro ^arethe half-length ^{and radius}

of the dipole, respectively.

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

where

and (k, 0, gi) ^are^thecomponents ^of^{a wave}^vector^k in unprimed spherical polar coordinates.

Using (2) ^andeq. (6) ^ofI, ^theprincipal-polarized components ^{of the}Fourier-transformed current are

given by

(4)

By using (3) together ^witheq. (12) ^and(15) ^ofI,

and the usual definition for the antenna input impe-

dance, ^Z⁼²

P/Iô,

the formal (integral) ^solution

of the input impedance ^can^be^shown^to^be

where

(free-space ^wavenumber), ^{and where}

Xr ^andYr (carrying sign) ^arestandard notations of normalized frequencies ^{for each}species ⁱⁿ^themagne- toionic medium

(see,

^e.^g.,^Ratcliffe

[7]) ;

Note that the

n;

given by (6) ^areactually ^the^two

roots of the biquadratic dispersion equation ^{for the}

cold magnetoplasma system considered [2]-[3] ^and

that _n±represents the refractive indexes for the two characteristics modes in the medium.

For the two particular ^casesôfânântennaôriented

either parallel ^orperpendicular ^tothe static magnetic field, ^weset, respectively, ço ⁼0 and rc/2 in (4), ^and

it _{is easy}to show that (4) ^reducesidentically ^to^the

formal solutions for these two antenna orientations derived in I.

For the ideal case of a lossless magnetoplasma,

various pole singularities ^of^theintegrand (i. ^e., n2

= n+, n2 ,

80’ E± 1, etc.) may make the integral

of (4) ambiguous. ^Thesemathematical ambiguities

can be removed by assuming ^asmall finite loss for the system, ^which^removes^thesepoles ^{off the}^real^line

into the complex plane. ^Theoriginal ^real-lineintegral

can then be performed ⁱⁿ^acomplex plane by using

contour integration techniques. ^After^aproper ^contour

integration, ^{we can}allow the small finite losses to

vanish. The real part ^and^theimaginary part ^of(4)

can be interpreted, respectively, ^as^the ^radiation

resistance and the input ^reactance^{of the}dipole

antenna in a lossless magnetoplasma. ^Tofacilitate the

contour integration, ^{we use} ^several symmetries

contained in the integral ^and^recast(4) into the form

where

Fil,

^Fl,^F+,ând^S«({J)oâredefined in (5), ând S(- oo) îsgiven by replacing po by - oo in S( ((Jo).

It is clear that (7) ^is^nowsuitable for performing ^a

contour integration ^withrespect ^tothe variable n.

Using ^theexpression

n±

given by (6), ^it^isstraight-

forward to show that

Flr(n2

^-^{g 0)}⁼⁰ ^{and that}

FL(n 2 -+ e ± 1)

⁼^0.^Theseconditions ensure that, ⁱⁿ performing ^the^contourintegration, ^thecontributions

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880

due to the apparent poles (i. ^e.,ⁿ²⁼eo, E t 1) ^vanish identically (2). ^The^totalvalue of the integral ⁱⁿ^a

contour integration ^is^then^due^to^thepoles

n± 2 (0)@

which are the zeros of the dispersion equation.

Introducing ^asmall loss to the system, ^weperform

a contour integration in n, following ^aprocedure

similar 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}^the0-integration ^should

cover the whole range of 0 for which

n 2

^ispositive (the region ^ofpropagation ^for^{each mode}ⁱⁿ^the^wave-

normal space).

As a check for the correctness of (8), ^welet ço = 0, n/2 ; (8) ^then^reducesidentically ^to(23) ^and(B 1) ^{of I.}

Equation (8) ^is^ageneral integral expression ^for^the

radiation resistance of a « short » dipole (short ⁱⁿ

the sense that the triangular ^currentdistribution is

valid) êmbeddedⁱⁿâûniform,^coldmagnetoplasma.

This expression is valid for arbitrary ^values^ofdriving frequency, plasma composition, particle density,

and static magnetic ^fieldstrength ^and^issuitable for numerical computations.

3. Quasi-Static ^RadiationRésistance. ^-In a collisionless magnetoplasma, ^there ^are ^the ^so-called

(2) Ît^can^beeasily îndicated^that8x 1 and eo âreactually ^some

of the limiting ^valuesôfn 2(o)@ ^{when 0 -}⁰ândn/2. ^Their^con- tributions to Z of (4) ârealready included in the poles ⁿ²⁼nfl,

and therefore (4) ^will^notyield any additional finite contribu- tions from the poles ⁿ²⁼s+ i, 80 as would be expected.

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

open branch (e. 9., fH. > f > FLHR and several other

frequency ranges indicated in Table 1 of a later Sec-

tion). ^For^the^wave^normalangles ^nearthe open

branch, the characteristics of the fields become

quasi-static, ^with^a^smallcomponent ^ofmagnetic

field H

(Arbel

^and^Felsen

[8]).

^However,^these^quasi-

static fields extend also to the far zones [8]. ^For^a

« moderate » short dipole ^antennatogether ^with skin-triangular ^currentdistribution assumed it can be shown that the predominant contribution of the

integral given by (8) ^comes^{from the}^wave^normal angle regions ^near^theopen branch of the refractive index (see ¹^{for the}^twoparticular ^casesôfdipole orientation) and further that this predominant por- tion of the integral îsⁱⁿsubstantial agreement ôf^the result obtaining through âûsualquasi-static ^calculation ; î.e., an integration ^{of the}leading ^termôf(4)

as n2 --+ 00 (or ⁱⁿB ⁼co/c, ^letspeed ^oflight ⁱⁿ^free

space ^c^-oo). În^viewôf^{the above}discussion, îtîs

therefore useful for us to calculate the dipole ^radiation

resistance using quasi-static approximation.

The quasi-static approximation ^to^thedipole ^radia-

tion resistance RQ can be obtained from (4) by taking

the leading ^term^of^{the real}part ^of^Zas n --+ oo.

This process gives ^thefollowing ^{relation :}

In the collisionless regime it is clear from (9) ^that RQ ⁼0 whenever eo/e. > 0, ^i.e., oe(0) # ^0.However,

if 80/8s 0, ensuring ^thatôneôf^the^rootsⁱⁿ(6) ^hasân

open branch, ^the^«collisionless » solution can be obtained by evaluating (9) ^{under the}assumption ^that

small collisional losses exist in the plasma ^{and then} allowing ^these^losses^toapproach ^zero.Introducing

small collisional losses and transforming ^the^0-integration ^to^{a new}^variable^cos⁰⁼x, ^wethen make a

proper ^contourintegration

(see,

^e. ^g., ^{Bell and}

Wang

[9])

^to^obtain

where

and 0, ^is^the^real^root^ofot(O) ⁼0, given by

(6)

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

where

The geometrical orientations of the dipole ^corres- ponding ^tothe three cases of RQ given by (11) ^are qualitatively ^indicatedⁱⁿfigure 2. For the case of

dipole orientation either parallel «(fJo ⁼0) ^orperpendicular (po ⁼n/2) ^to^themagnetic field, ^we^take

the limiting values of (lla) ^and(lle) ^asço - 0

and _{ço -}n/2, respectively. Îtîseasy to show that these limiting ^valuesâregiven by

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

FIG. 2. ^-A qualitative diagram for the dipole orientations subject ^to^formula(11).

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882

From the above analysis, ^it^can^be^seen^that^the

functional dependence ôf^RQônh/ro changes âsCfJo increases from 0 to n/2 ândthat, ^for^fixedplasma parameters ândâ^thindipole (hlro » 1), ^RQ^reaches

its maximum value _{at CfJo}⁼nl2 - 0,.

4. Radiation Résistance at VLF. - The VLF range defined here represents ^thosefrequencies f ; fHe > f > fHp.

We shall consider a relatively high density plasma

to be such that f o (the

plasma

frequency) > fHe (this

condition yields e. 0 for f fHe) ^and^assume^that

no negatively charged ^ionsexist, ^so£+ 1 > 0. Under these conditions, ^it^canbe shown from (6) ^that,ⁱⁿ

the VLF range,

n 2

⁰^and ⁿ²> 0. The ^term with subscript ⁺^does^notappear in (8), ^and^the

radiation resistance for this case is due solely ^to

the n- mode (a propagating mode), ^which^{is known}

as the whistler mode. In the frequency ^range fHe > f > fLHR’ the refractive index surface for the

whistler mode _{is open}

The radiation resistance for this frequency ^range^has

been calculated in 1 for the two special orientations

(parallel ândperpendicular). ^Forânarbitrarily ôriented dipole, it is difficult to integrate (8) analytically.

However, ^anapproximate closed form of R has been

given by (11) ^{in the}previous ^sectionusing ^thequasi-

static fields. The full-wave solution of (8) ^{has been} integrated numerically ; ^the ^numerical^results ^are

discussed in a later section. In the frequency range

fLHR > f > fHp,

ni(0)

^isbounded, ^and^the^refractive

index surface of the whistler mode is closed. For this case, 80/8s > 0 and RQ ^--_0 ; ^thedipole ^radiation

resistance must be calculated from the full-wave fields.

For ^anantenna length subject ^to^the^constraint

it is possible ^tointegrate (8) analytically. ^The^detailed

calculation for this case has been given in II. For the purpose of the discussions in the next section, ^we reproduce the results of II

[see

^eq.(7), (12), ^and(16)

of 111 :

where

and where A, B, C, A’, B’, ^and^{C’ are}functions of

8B1 (v ⁼⁺1, ^-1, 0) ^alone.^Theirexpressions ^can^be

found in II. Here F(q, p) ândE(q, p) âre^thegeneral elliptical integrals ôfthe first and second kind, respec-

tively, ^with^thearguments q, p defined by

and

The solution given ⁱⁿ(13) îsappropriate âslong âs b ₈₀and this condition is met for frequencies ⁱⁿ

the VLF _rangef ; fLHR > f > fHp(1 ⁺10-3).

5. Radiation Résistance at ELF. - The ELF range defined here is those frequencies ^belowfHp. ^When^the high-density magnetoplasma considered contains mul-

ticomponent ^ionspecies, ^thepropagation ^characte-

ristics in the ELF _rangebecome more involved.

In general, ^both^{n _}^andn + modes can propagate.

Between _everypair of successive ion gyrofrequencies,

there exist a number of plasma characteristic fre-

quencies. These include the frequencies ^known^as^the

crossover frequency (/co)? ^the^cutofffrequency (fcf),

and the two-ion-resonance frequency ( f MR)

(Smith

and Brice

[5]) ;

^they^areobtained from the unique

root (between each successive pair ^{of ion}gyrofrequencies) ^{of the}conditions 8d ⁼0, 8 - 1 ⁼0, ^and

es ⁼0, respectively.

On the basis of the mode propagation ^characteristics, ^thedipole radiation resistance at ELF can be inferred from the results derived for the VLF case.

By using (6), ^{we can}examine the details of the wave

propagation characteristics at ELF. The results are

summarized in table I, ^for^atypical pair ^of^successive

ion gyrofrequencies (say, fHp > f > fHH:, fIlH,,’ ⁼gyro-

frequency ^foratomic helium species).

The ordering sequence and propagation ^characte-

ristics summarized in table 1 will repeat ^{for each}pair

of adjacent ^iongyrofrequencies until the signal ^fre-

quency decreases below the smallest ion gyrofrequency fHs. ^Whenf fHs, ^theordering ^ofplasma parameters is _8-1_i> 8s > s+i 1 > 0, Ed 0, ^{and the}propagation

characteristics are as described in the first row of table I. This ordering, âs^wellâs^thepropagation characteristics, ^will^notchange ôver^therange

since no crossover frequency exists in this _range.

In view of the propagation characteristics described in table I, ^thedipole ^radiationresistance at ELF can

be easily ^derived^from^thosesolutions for the VLF

case. In the frequency ^rangefHi > f > f iMR (fHi ^and .fiMR ^{stand for}^thegyrofrequency of the ith ion species

and the two-ion-resonance frequency ^for^{the ith}

and i +1 th ^ionspecies), ^where^we^note^that

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TABLE 1

A summary of typical propagation characteristics at ELF for ^thefrequencies fHP > f > fH,14

and under the assumptions 80 0, 8+ 1 > 0 (no negative ions)

and that the n _ mode always possesses ^anopen refractive index, ^theleading ^termof radiation resistance 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 through

the similar analysis made in 1 and II. However, ^for

the ^«short » antennas whose length ^issubject ^to^the

constraints given ^{in 1 and}II, ^it^canbe shown that the full-wave corrections to the radiation resistance are

much smaller than RQ. Since the dependence ôf ,RQ in (11) ône, 1 is only through 8s, which is _symme- tric in the interchange ôf8+1 and 8-1’ ^thequasi-static

radiation resistance RQ for the frequencies

is identical to that of (11). ^In^thefrequency range

fiMR > f > fH i + 1 (fH i + 1 being ^the gyrofrequency

for the i + lth ion species). ^We^notefrom table 1 that the ordering ^of^theplasma parameters and pro-

pagation structures is identical to that in the VLF range f LHR > f > fHp. ^It^is^clear^that,for this range within ELF, ^thedipole ^radiationresistance is given identically by (13).

6. Numerical Results. ^-Our numerical data for the VLF/ELF dipole ^radiationresistance were obtained

by evaluating (8) through ^the ûse ôfcomputer integration. Înfigures 3, ^{4 and}5, ^weplot ^{the K}^{as a}

function of the dipole orientation angle po for various values of frequency, plasma density, plasma composition, ^staticmagnetic ^fieldstrength, ^anddipole length.

The curves are parametric ⁱⁿ normalized dipole length ho (ho ⁼²nfH, h/c) ^andnormalized plasma frequency folfh,, (fo being ^theplasma frequency).

FIG. 3. - VLF dipole radiation resistance as a function of normalized length ho, ^andorientation angle po. Four normalized driving frequencies f/f He ⁼0.05, 0.25, ^{0.5 and}^0.75( f > fLHR),

and two normalized densities fo/fHe ⁼2, ¹⁰^areconsidered.

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884

The two values of folfHe (./o//He = 2, 10) ^used^were

chosen as being representative ^{of the}range of values that can be encountered in the topside ionosphere

and the inner magnetosphere. The values of ho

were chosen in the _range0.05 ho ^{0.5. In}^all

the numerical plots ôfR, ^{the ratio}h jro ⁼¹⁰³îsûsed.

Figure ^{3 is}^aplot ^{of R}^for^severalfrequencies ^{in the}

VLF _rangeand above f LHR. The relative peaks ^shown

in the curves correspond ^to^thedipole orientations

satisfying ^thecondition : _ço= z/2 - 0,. ^The^curves

vary approximately ^as

ho-1

^and

(fo/fHe) - 2

^for

and ( f ô/f fHe) 1

^forço > (z/2 - 9r). Figure ^{4 is}^aplot

off for the VLF frequencies ^belowf LHR. ^In^this frequency range, the whistler mode refractive index is

closed, ^and^thereforeno resonance cone angle ^exists.

Correspondingly, ^the^curves^{shown in}figure ⁴^do^not

possess relative peaks. ^Thecurves are approximately

proportional

^to

^hÉ

^and^(fplfHe)·The ratio of

is approximately equal ^to(fH,lf)’. ^For^theplots ôf figures ^{3 and}4, ^theplasma îsâssumed^to^{consist of}

electrons and protons.

Figure ^{5 is}âplot ôff^similar^to^thatôffigure ³

for the frequencies in the ELF _range.The plasma ^is

FiG. 4. ^-VLF dipole ^radiationresistance as a function of normalized length ho and orientation angle ço. Three normalized frequencies below fLHR ; f/f He ⁼0.002, 0.01, and 0.02, ^and^two

normalized densities fo/fHe ⁼2, ¹⁰^areconsidered.

FIG. 5. ^-ELF dipole radiation resistance as a function of normalized length ho ^andorientation angle ({JO. Five normalized

frequencies f/f Hp ⁼0.175, 0.2, 0.4, 0.6, and 0.8, ^and^two^normalized densities f o/f He ⁼2, ¹⁰^areconsidered.

assumed to consist of electrons and three ions : atomic hydrogen, ^atomichelium, and atomic oxygen.

The composition ^{of the}plasma ^is^assumed^to^be

70 % H+, ²⁰% He+, ^and10 % ^0*.^For^this^three-

ion species plasma ^modelconsidered, ^thehydrogen-

helium resonance frequency fH,,,H, ^is ^N^0.375fHp

and that the helium-oxygen ^resonance frequency fo++He+ ^is ^~^0.155fHp. The numerical curves of

figure ⁵^areplotted for the normalized frequencies

f/fHp

in the range where the ^n-mode refractive index possesses ^anopen 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}^curves subject ^tothe condition _Po⁼n/2 - Or ^is^close^to

oo ^N0, ^{as can}^be^seenⁱⁿfigure ^{5. The}^functional

behaviour of the curves in figure ⁵^withrespect ^to^the other parameters is similar to that of the curves of

figure 3, ^discussedpreviously.

In comparing the results of the exact integral ^form

of (8) ^{with the}approximate closed-form results, ^the

curves of figures ^{3 and 5}agree with the approximate

formula (11) ^within^a^fewpercent, whereas the curves

of figure 4 agree with the approximate ^formula(13)

within a few percent.

7. Concluding ^Remarks.^-Using ^linear^full^electromagnetic ^wavetheory, ^ananalysis has been made to calculate the VLF/ELF radiation resistance of a

finite electric dipole ⁱⁿâûniformûnboundedmagne-

toplasma. ^Theorientation of a dipole ^withrespect ^to the static magnetic ^{field is}^setarbitrarily. ^Thegeneral integral expression for ,R is derived

[see (8)],

^and

from that, ^variousapproximate closed forms of R, valid in the VLF/ELF range, have been evaluated.

Numerical data for R have been plotted through ^use

of computer integration ^{of the}general integral ^form

of R, ^and^theirresults agree with the approximate

formulas (11) ^and(13) ^within^a^fewpercent.

From this study, ^severalinteresting conclusions

can be drawn :

1. Within a frequency band in which the refractive index surface of ^{n _}mode has an open branch, ^the