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f

ELECTROSTATIC EMISSIONS BETWEEN ELECTRON

GYROHARMONICS IN THE OUTER MAGNETOSPHERE

Richaxd F. Hubbaxd*

Thomas ^'. Birrnangham

May 1927

}:;,

*NAS/NRC Resident Research Associate

GflDDARD SPACE FLIGHT CENTERGreenbelt, Maryland

is,

S'^i -^- ....

_1 J T.

ELECTROSTATIC EM€SSIONS BETWEEN ELECTRON

GYROHARMONSCS IN THE OUTER MAGNETOSPHERE

by

Richard F. Hubbard

Thomas ^. Birmingham

ABSTRACT

Electrostatic emissions axe often observed by spacecraft in

the outer magnetosphere at frequencies between low electron gyro-

harmonics. The emissions include the well-l^nown odd half-haxmonic

emissions as well as wea[^er emissions which occur near the electron.

plasma frequency. We have constructed a scheme for classifyiizg the

emissions and have shown that a theoretical model w:^ich has been

previously used to explain the "3/2" emissions can be extended to the

other classes of emissions. All of the emissions appear therefore

to be generated by the same basic mechanism: an unstable electron

plasma distribution ;,^resisting of cold electrons (<3.00 eV} and hot

Loss cone electrons (^-1 l^eV). Each emission class is associated with

a particular range of model parameters; . the wide band electric field

data can thus be used to infer the density and temperature of the cold

plasma component. The model predicts that gyrohaxmonic emissions

near the plasma frequency require Large cold plasma densities.

iii.

_ .. _ _

yt

_.

i

`^° Page^1.

_ ABSTRACT iii

I. INTRODUCTION ^

II. TYPICAL ELECTRIC FIELD OBSERVATIONS 5^... 3

III. THEORY . . ^Q ` i

- IV, SOLUTIONS OF THE PLASMA DISPERSION RELATION . J.8 a

V. PLASMA MUDELS FOR EACH EMISSION CLA55 23i

. VI. THE DIAGNOSTIC USE OF OUR PLASMA MODEL 28 1

VII CONCLUSIONS 33 ^. .

ACI^NOWLEDGMENTS , 34

REFERENCES 35

;.

i4'

EI

..^, .. __

vi

^'^^;

^.

/ ^^

a

ILLUSTRATIONS

Figare Page

^. An ea^alr^pie of multiple hai£ harmonic emissions observed by Imp 6 l^^at R =9R^ on the day side a£ the magnetosphere. As many as ^'_seven gyro^rarmonic bands are simultaneously observed. The in--tense ^'8/2" enz9.ssion which occurs between I and 2 kHz at O'l00 isalmost continuously observed from 0500 to 0800. The emissionsbelow 1 kHz are electromagnetic and are oat considered in this A

paper . ........................................ 38 ""' i

i2 An example o£ gyroharmonic emissions occurring near the electron '

plasma frequency f^. The shaded region is trapped continuu^cn radi--atian whose lower cutoff is fp, Up to three gyroharmonic band.; areobserved. .................. . ................... 39 ^

3 A comparison between a Ma^rcveilian parallel velocity distribution

a and the corresponding p^' order Lorentzian distribution. The p = 2 ^Lorentzian has a similar shape for V^^ C a p but has mare high energy

'particles than does the Maxweliian. • • • • • • ............. 40 ##;

^ i

^ The dependence o£ the frequency ^ = f/fg of the fastest growing modewithin each gyroharmonic band on the cold plasma density. The 'jparameters wpx /SLz = 20, T^ /Tx = 0.0^, 8 - 86° , and 4 = 0, As

nc /nx increases F^ increases within each band. The- highest £re--quency tracks the void upper hybrid frequency (dotted line}. .... 'i^ '

5 The dependence of range of the instability within each band on the ^'' cold plasma temperature for Q = 0.5, 4 W 0, and dux = 3.32. The

region of unstable frequencies within each band is enciosed by thesolid curves, and an asterisk indicates the fastest growing modewithin each band as a function o£ TG /Tx , The temperature ratiohas a considerable effect an the range o£ the instability. ....... 42

6 The- maximum growth rate ym^ within each of the five gyroharmonicbands shown in rig. 5, plotted against the temperature ratio T^/Tx.For T^ /Tx < 0, 01, strong growth is seen only in the second andthird bands which are closest to unx = 3.32. For T^/Tx ] 0.0^,7max decreases monatonicaiiy with T^ in all bands. .... _ `i3

i i 4 '.

^.

ILLUSTRATIONS

Figure

Page

7 The effect of the hot plasma density on the range of unstable fre-quencies. The mast unstable frequency is plotted vs. n^/nn withT^/Tn = 0. I0, D = 0, and nC fixed (^cuH = ^). The "error bars"estimate the bandwidth based an the frequency range for which themaximum growth rate '? '^,^ < 2 7(u )• The solid, dashed, and dottedcurves correspond toy m^x >0.01, 0 . 041 <7m^<0.01, and7m^ < 0, 001 respectively. Increasing n^-^ tends to increase 7m^

while leaving frequencies relatively unaffected . ............ 44

8 A plot of 7max vs. fill in factor o in each of the five bands whichoccur when T^/Tn = 0.04., ^^^ = 3.32, and n^ /na = 0.5. Growthrates in all bands decrease manotonicalLy wath a , but higher fre-quency bands are suppressed most severely . .............. 45

9 Growth rate y vs. frequency ^ for uua = 1.34, Q = 0.1 (Curve 1) and

^uH = 1.73, Q = 0.2 (Curve 2) with T^/T^= 0.10 and 4 = 0. Unstable.waves above ^ - 2 can be suppressed by setting Toss cone fill infactor d = 0.10 {Curves 3 and 4). Hence Curves 3 and ^ reproducethe features of Class 1 (low 3i2) and Class 2 (3/2) emissions. 46

10 ^ vs. ^ for ^uH = 3.32, d = 0, F = 0.5, and T^/TH = 0.10. Thefrequencies and bandwidths are characteristic of Class 3 (multiplehalf harmonic) emissions. ......................... 4'T

1.1 An example of gyroharmonic emissions near the plasma frequency,,' Fig. Ile gives y vs. u for ^ = 10, a = 0, d ux = 4.58, f /f = 4.69,

and T /Try = 0.01. The dominant feature is a single strong emissionnear f^,. In Fig. 11b, T^/TH = 0.02, and an enti.ssion band above5fg would be seen. Class 5 emissions include bath single band andmultiple band emissions near fn, ...................... 48

12 An estimate, based an solutions to the linear dispersion relation, ofthe cold upper hybrid frequency u^ vs. maximum observed fre-queney ^m^ . The "error bars'° are uncertainty estimates assumingaGO.05. This figure can be used to estimate 1^u^. from electric

} and magnetic field data whenever emissions of any class areobserved. .......... . . . ........................ 49

___ ..

.r 1

i ^_ ^^^^ ^ ;

ILLUSTRATIONS g

i{ Figure Pag_ e^' -;,{`

1.3 The estimates of Fig. 12 converted to ^^ /52 2.(or equivalently nC) ^>^

vs. maximum observed frequency ^ma^ . This figure can be used as ^ fit;

^. a diagnostic tool to estimate the cold plasma. density n^. ...... 50 ^ is<°

^^ A plot of 83 individual electric field observations taken at 15 minute - '^ ^.intervals from Imp fi Orbits 1.78-183 whenever t 'odd-half-harmonic" `^ ^.emissions are seen. The graph shows the maximum observedfrequency ^m^x (in units of fg) vs. f^. Higher relative frequenciestend to occur when the magnetic field is weakest, and almost alIemissions between fg and 2fs occur below 1.5f^. The region abovethe solid curie corresponds to :Frequencies above 1.0 kHz and cannot

^; be observed in the data . . . .......................... 51

^'

.! j

s ^. ^

.

^

_ ^ ;

t

i

_.._.. _ . _ Vii.

4- - ;

I

i

9

Ef,ECTROSTATIC EMISSIONS BETWEEN ELECTRON

GYROHARMONICS IN THE OUTER NTAGNETOSPHERE

I. INTRODUCTION

Banded electrostatic emissions at frequencies above the local electron

gyrafregttency f^ are a prominent and interesting feature of the outer mag^ieta--

sphere. By far the most common of these emissions are those occurring be-

tween fs and 2fs These were first observed by the plasma- wave experiment on

Ogo 5 (Kennel, et al. , I970; Fredricl^s and Scarf, 1973) as narrow band

(Aft & ^^ 0.1} , large am}^litttde (^.-10 tnV/m) waves accur^ring aver the approxi-

mate range 1.25 f^ C f C 1.75 f^. Because this frequency interval is centered

at I.5 f& , these waves are often loosely termed"3/2" emissions. Similar

" 3/2" emissions have also been detected by plasma wave experiments on 53

(Anderson and Maeda, 1977) and Ixnp 6 {Shaw and Gurnett, 1975). In the Imp 6

experiment "3/2" emissions are sporadically observed on nearly every traver-

sal of the outer magnetosphere: they are indeed also a common magnetospheric

feature. It h^:s been suggested that these waves play a domiatant role in the

precipitation of I-10 [^eV electrons (Lyons, 1974; Ashour-Abdalla and Kennel,

1976)

There_ are several other types of electrostatic turbulence whose charac^-

• teristics suggest that they are related to " 3/2" ern:issions. Tb.ere are first of

.all "n -^ 1/2" er "add half ` harrnonic" emissions with n integral and >':,. -Again

{^^

I^=

,^'^ ^.. ,

,;,,; ..i

,,

.. ^'

F ^'

E I I I 4 P

the . coinage is loose: these are narrow band emissions occurring between. nf^

and (n ^- 1)fg but significantly away from the gyroharmonics themselves, The

Ogo 5 experiment occasionally observed "5/2" and "7/2" emissions, and

(n + 1/2) emissions with n as Iarge as 8 are common in the lmp G data, though

far Iess so than the "3/2" emissions.

Nhen "n + 1/2" emissions are seen in the magnetosphere, all half har-

monies n = 1, 2, ... N are usually sequentially present. However, Shaw and1

Garnett (1975) report narrow band emissions between higher gyroharmonics at

times whEn the " 3 /2" and other loGV half laarmonics are absent. In such cases,i

.^

the highest frequency band is often near the local electron plasma frequency fP,

the latter being determined from the observed lower cutoff of nonthermal con--

tinuuna radiation (Garnett and Shaw, 1973).

^

Somewhat different from the narrow banded emissions discussed so far

are "diffuse" band emissions reported by Shaw and Garnett (1.975). These

emissions are also confined to bands between the low electron gyroharmonics

and tend to occur just outside the plasmapause. They may have substantially

wider bandwidths (^f < 0.5 f^}, and much lower intensities than do narrow

band emissions. The highest observed frequency of diffuse band emissions is

frequently near fn.

Banded electromagnetic emissions near fg/2 have also been reported

{Burtis and Helliwell, 1969; Maeda, et aI. , 1976). These emissions, which are

presumably wYii^otter waves, are seldom observed unless "3 /2" emissions are

2

^^.

3 Y

i^` 1

^... ` ^

i

{

i

,. ^

"^ 3^ ^

j

si^nultaneousl;^ present, a_nd a causal Link between the two may exist. We shall Y

not be concerned, lrowever, with electromagnetic emissions in this paper.

Ems'Part of our task has been to develop a classification scheme for the emir- ^^

5

sions discussed iZZ the previous paragraphs. Five separate categories of waves^,.,.

have been identified. More importantly, we have developed a parameterized,^..,.

model of a linearly unstable plasma. By suitable choices of the parameters

in the model, it is possible to duplicate in a more or less unique fashion the

emissions in each category.

The plasma model is essentially the nne investigated previously by several

other authors (I'redricks, 1971; Young, et al. , 7.973; Young, 1975; Ashour--

Ahdalla, et al., 19'75; Ashour--Abdalla and Kennel, 7.9'76) . Because of the high

frequency of the waves involved, we consider only electron dynamics. There

are two electron populations: a "hot" (^-1 ke^T} Toss cone component, with den-

sity n^ and parallel and perpendicular temperatures T^^^ and T1^, is the source

of free energy for the uYStability. Tn addition, there are "cold" (C7.OD eV) elec-

Irons of density n^ and temperature T^ . The cold electrons, which originate

from the ionosphere, facilitate the propagation of the instability. As mzgl^t be

expected far electrostatic emissions whose frequency is related to f^ , the waves

which we consider propagate nearly perpendicular to B, i, e. the perpendic^.xlar

and. parallel wavenumbers k1 and ^^f satisfy k17^ ^q .

1n this paper, we extend the Iin^ar analysis to investigate aII types of

narrow band electrostatic turbulence observed in the outer magnetosphere. ^V'e

3

^^^ ^ ^

.i _. ..

IiIC

feel that diffuse band emissions can also be explained by this model, but our

worm here is as yet tentative and will be reported later. The theoretical analy-

sis of narrow band emissions is coupled to specific Ixnp G electric field observa-

tions. The plasma conditions which. give rise to emissions in each of the pro-

posed categories are theoretically investigated. We summarize the effects of

varying each of the many free parax^neters in the problem on the morphology of

each category of eznissian. This leads us to a "best estimate" as to local plasma

conditions when a given type of emission occurs and: hence a predictive or diag-

nostic capability. Yn particular, we are able to estimate the local cold plasma

density, a parameter which is dffficult to measure by direct plasma observations.

f

^'

;.

:,^

1

:'^

^ I { I I 1_ ! r

II. TYPICAL ELECTRIC FIELD OBSER^IATIONS

The essential features of "half-harmonic" electrostatic emissions are

most clearly seen in wide band spectrograms of the tyre shown in Figure ^.

The figure shows three compressed electric field data from the University of

Iowa plasma wave experiment on Imp 6 during an outbound orbital pass. Simul-

taneous magnetic field measurements from the NASA/GSFC rxiagnetometer allow

us to calculate f^. The most persistent feature is the narrow bandwidth emir--

sions between f$ and 2f^ which are almost continously observed from 0500 to

0800. During this tame, the relative frequency u = f^f^ of this "3/2" emission

varies between 1.2 and 1.45 as the gyrofrequency decreases from 4.6 kHz to

0.92 kHz. Figure 1, however, shows only the last part of this time interval.

The "3/2" emissions in Figure 1 are almost constant in frequency and are the

dark trace occurring between 1 kHz and 2 kHz.

Higher odd half harmonic emissions occur sporadically after 0655 with

as many as seven bands simultaneously present. The precise relative frequen--

ties of the emissions at 0715 ---- f/fg = u ^ 1.4, 2.3, 3. ^, 4.2, 5.2, and. 6. 1-- ^ ^^a^

are indicative of the looseness of the "odd half-harmonic" terminology. The

emissions below 1 kHz are file concurrent electromagnetic emissions near

f /2.^1

g'

Figure 2 shows ane of the tyres of emissions reported by Shaw and Gurnett '^

(1975} . The. sporadic narrow band emissions which. first. appear about 052.5 : 'are i-

the main feature of interest. The shaded region above ^'6 kHz is the nonthermai

l _..._ ,.._. _ _ . _ .,^^ ^ ^ .^.

^__ ^_ ^ ^:

5

}^^ ^ ^

^^^ I I i I I! i t

continuum radiation {Gurnett and Shaw, 1973). This radiation is electromag-

netic and is density trapped between the plasmapause and the magnetopause. lts

lower cutoff is the local electron plasma frequency f p . The narrow band exnis-

sions here are well above f^. The three bands at 0540 are just below 7f^ , 8f^,

and 9f^. Although in this case, the frequencies of the narrow hand emissions are

just below f^, similar bands are sarnetirnes observed above the lower continuum

cutoff.

Although the continu^un radiation is usually observed whenever narrow

band emissions of the type shovv^i in Figure 2 are present, the continuum emis-

sions are sometimes missing (e, g. , Shaw and Gurnett, 1975, Fig. 4}. This may

be due to havuig f P exceed the upper frequency limit {7,0 kHz or 30 kHz) of the

wide band spectrometer, but it is mare lil^ely that the na^.,row band emissions

are occurring near f^, and that the continuum is simpl;T too weak to be observed

(R. Shaw, private cornmunieation},

The presence of continuum. radiatian above f P is not correlated with the

occurrence of electrostatic emissions. Continuum radiation may accompany

any type of narrow bandwidth electrostatic emissions, diffuse emissions (e. g. ,

Shaw and Gurnett, 1975, figs. 6 and 11), nr it may even occur in the complete

absence of electrostatic turbulence. These p,•o^^^:rties, plus the fact that the

continuum radiation is electromagnetic in nature, suggest that the mechanisms

for generating continuum and odd Half-Harmonic emissions are unrelated.

6

r ^.

^'^.

f

";I

.^.

:,

:`

^..... ^.^.^ - - ^-._,----------- ^. ^ __ ----._.. - - - ..1

_ ...... _ .

l ^ __ f^ -- /.

Based an the work of Shaw and Gurnett {1975) and our own survey of

Imp 6 U-19 1sHz broadband data, we propose the classification scheme shown

in Table 1. With the exception of Classes 1 and 2, the distinction between

separate categories is an observationally natuxal one, and as we shall see, each

class corresponds to a different configuration of our plasma model The origi-

a observations were primarily low latitude measurements and found

requencies f ^^ 1. 5 f^, i. e. , Class 2 emissions. Our analyses of sev--

its of Imp 6 data, however, suggest that fg < f < 2f^ waves axe coneen--

Crated below 1.2 '£g : for example, BO% of the narrow band emissions observed

:^` in Orbits 178-183 were Class X as apposed to Class 2 emissions. Our theare-

tical work suggests that this difference in frequency distributions between the

Ogo 5 and Imp 6 observations is a latitudinal effect. At the same radial distance

from the Forth, the ratio f p /fg , upon which the frequency f of narrow emissions

depends strongly, is smaller at the higher latitudes traversed by Inzp 6 {if f n

is constant or decreases w-^th latitude). Thus, the plasma characteristics off

Class ^. and Class 2 events are also different.i

Class 3 emissions are multiple "(n ^- 1/2)" emissions of the type shown

in Figure ^.. Class 4 emissions are the diffuse electrostatic emissions describedI

^ by Shaw and Gurnett (1975). Finally, Class 5 emissions are narrow band emis-

sions such as those in Figure 2 between cyclotron harmonics near fP.f

Vie have assumed that a1I Class 5 events have frequencies controlled by

cyclotron harmonics even when only a single band near f is observed. It is

i P

I

t'7

^ f.^

,.

".:^..

a

^,'.

possible that some of these emissions are in fact electron plasma oscillations.

We shall see that it is theoretically possible to produce gyroharmonic emissions

which occur in a narrow band right at f p . In principle one should be able to dis-

tinguish between electron plasma oscillations and gyroharmonic emissions from

the location in time of iaitensity nulls associated with the spin period of the

spacecraft svace 1<; is predominantly perpendicular to B fnr gyreharmonic emis-

lions and parallel to B for electron plasma oscillations. Unfortunately, the

Class 5 emissions don't usually show spin modulation (Shaw and Gur^ett, 3.J'75).

Emissions occurring in two or more bands near f P , such as i+igure 2, can

safely be considered gyroharmonic emissions, while emissions which occur in

a single narrow band near the continuum cutoff could be electron plasma oscilla-

bons or gyroharmonic emissions. It may be possible to distinguish the two by

analyzing electron particle spectra.

^; f:^';a

^`=

. ^. ;

^, .^,.

,^

8

__^ ^ ^ I^;^ I ^ ^ I__ _ . ^_ ,,

.^r

.Class Frequency Range Comments

1. Low "3/2" 1 C fjf^ < 1.2, pff fs < d.I Most common emission class far 20'' < ^A m ^ < 55°

2. "3/2" I.2 G f/fg < 2, elf/f^ < 0:1 Most widely studied emission class, Probablymast common emission class near equator.

3. "n + lJ2" or multiple max /fg > 2 Narrow bandwidth emissions In each gyrahar-half harmonic pf/f^ < O.I in each band- monk band above f^ up to the n th band {e.g.,

Fig. 1 },

4. Diffuse band fm ax f fg C 4 Less sharply defined emissions in the lowest

pf/f^ ~ O.S in lowest band {1 G fJf^ C 2) band. May have emissions in oneor two other bands. If continuum observed,

fmax ~ fg'

S. "f ~ fp"narrow band fs « max ~ fP One or more bands near fp are enhanced. Lower

pfJfg < 0 1 (e.g., " 3 f 2") bands usually missing.

^.

iI

ii---I

c^

.,I. t

i.^'

='',i

1

^" 13

^:.r...

__^

r I_I _i I3

3

__

f

i

^..^ .. ^i

9

III.. THZ;ORY

The dispersion relation for electrostatic waves in a magnetized electron

plasma, where ions form only a fixed neutralizing background, is in cylindrical

coordinates, with the z-axis along B (Harris, 2959),

2n ^ }^ ^'°° Jn (k^Ul ^S^) 8f nSZ ofQ = l - kz w^ clulaZ ,^dvll ^ -^ SZ -- w ^kll 8v ^ u 8v ^

{1)o -- °° n k il Ull n II 1 1

Here wp = (4^rne2 ^m) i and 2 = ^ e E B /mc = 2 ^rf ^ are the angular electron plasma

and gyrofrequencies respectively and the electron distribution f(v^ , Ull) is

normalized to unity

+^

2^r ^dvlvl fBall f(u^, Uil ) = 1o

Two different classes of f (VZ , V ii ) which lead to unstable solutions (y

Im w ] 0) to Eq. (1} with ky » kli have been widely studied. The first type is

characterized by an anisotropy in which the temperature Tl associated with

the electronic motion perpendicular to B exceeds the equivalent parallel tempera--

tore Tll. While Tj > 2T ll is sufficient for instability in the "3/2" band, the

instability is restricted to the region 3/2 f^ < f C 2f^ (Young et al. , 1973), and

in order to account for observations below 3/2 f^ a oo^nplicated non--linear

coupling must be invpked (Oya, 2972; 1975).

Instability at higher odd half harmonics requires a still more pronounced

temperature anisotropy. It is primarily because such. anisotropic distributions

^',

i

r^

'1- --

LO

r

-: is r t__

are not observed in the magnetosphere that we analyze this type of distribution

function no further.

The second. category of distribution functions is one in which af^8vl > 0

} over soma region{s} of vi , Only this class of distribution is unstable to strictly

flute made {k ll ^ 4} propagation {Baldwin et al. , 1969}, Pioneering work with

thia type of distribution was done by Tataronis and Cr^awfard {1970} who investi-

gated the stability of the distribution

fr 2 v ^{v1-- ^1}s(vll}1

,•` iii an attempt to explain add half harmonic emissions observed in laboratory

devices.

Shortly after the Ogo- -5 xnagnetospheric observations, Fxedericl^s (197Z)

seized upon the distribution Eq, (^) {with a thermal spread in the perpendicular

distribution} aaad was able to show the existence of unstable " 3 /2" ^vaves propa-

gating within 10° of perpendicularity to the field far large but realistic values

of wP /5^^, The electrons involved were thought to be "hot" plasma sheet eiec-

txons: there vvas indication that " 3 /2" emissions are most prevalent during

substorm. periods when the flux of plasma -sheet -like electrons at Ogo-5 alti-^

fades is enhanced.

Attempts to extent this work to more realistic (loss cone) distribution.

functions met in failure. Thus Toss cone distributions of the Dory--Guest-Hai ris

(UGH) (Dory et al. , 1965} type

1,1

,^ ^f

^- t

(^)

,)

I

d

E! f(Uy, U ll ) ^

1 1*vl N exp - vl ^- U ll (^)

N^ ^ 3/2 a2 N 1 a ^ ^2 ^2r ^ (} ( .l) p s 1i

i were tried and shown to be electrostatically stable unless N became unrealis-- ^.^^;^'}

I{

ticalIy Iarge {>5) (Dory et al. , 19G5). It was Young and his coworl^ers who

^,first recognized the difficulty with small N: in those regions of ^, k space where ,^

^ the ina.aginary component of Eq. (1) could be satisfied with 7 = Tama > 0 {corre-! .r..

^` ^ sponding to instability) the real component of Eq. (1) could not. However, thei.

E addition of a small amount of cold plasma to Eq. (3) alleviated the difficulty.i

The cold component was and is thought to be of ionospheric origin. The admiX-^E

i

tore which probably exists around L= 8 is uptimun^. for instability.

The mast recent worm in the field, that of Ashour-Abdalla and Kennel =^i ^

(1976, ef. also references therein}, also allows partial "fill in" of the loss coneI

in order to beep growth rates acceptably small. (If growth is too fast, saturation1

occurs at a low emission an3plitude.) Ashour--Abdalla and Kennel thus analyze

the stability of the distribution

^ i 1 (1 - ©}u^ ul ul^ ' ^ ^^

! {^r)^1^^IlIi ^lx ^lx aHx

(

2^vl

Ul!

() c^ Ik e ^.^ lip

This distribution is tlxen the suxn of three .components: a cold bi-•Maxwellian of

^ density nC , a hot bi-Maxwellian of density, ^Ei, and ahot N = 1 DGH Ions--coneo

distribution of density (1 - d) n The ratio 0^(1 - d} between the densities^'

H'

j'.^:

I3

12

;.,

t

s

.,

i ^ ^ ! __ ,,,_^ _^ 1__ t

:`.^':

of hot A^a yurellian and Iris-cone plasma is referred to as the "fill--in:" factor.

The ratio (l is n^/na so that in Eq. (^.) wP is calculated using the hot plasma

density nH (w^ is then properly w p ). The Ashour-Abdalla and Kennel paper

reports on a detailed parameter search for conditions which can produce " 3/2"

emissions and axz analysis of the convective properties of the instabilities. It also

contains a somewhat more detailed history of the evolution of theoretical ideas.

Our work e^stends the plasma parameters search to the frequency regime

f > 2f^. Since we are concerned with characterizing all categories of odd--half-^

harmonic emission discussed in Section Yf, we shall also be considering "3/2"

emissions. Our results in this regime generally support the findings of Ashour--

Abdalla and Kennel.

We differ from previous warp in one major respect: our hot electron dis-

tribution parallel to B is a Lorentzian of order p; thus the distribution function

which we analyze is that given by Eq. (4) with the substitution

2p-11 1 {2p - 2) E ! {a il x ^

{^}^^ ^f I exp - {UI ^ '^Iln, ^'^ (2p - ^} T E {v^^ + ^^^ }p^ x

The Lorentzian for several values of p is plotted in Figure 1 along with the

Maxwellian for comparison. Nate that as p becomes very Iarge, the Lorentzian

devolves to a S^funetion at U!f = 0. We have chosen the Lorentzian for two

reasons; 1} magnetaspheric particle spectra. frequently show a "power law"

high energy tail (e. g. , the spectra shown in Anderson and Maeda, 7.977} and

i 3

,^

^-

13

l.__^. __

^'^ + r

. ^ 2) the ensuing cample^ity of the hot contribution to Eq. (1) (ef Appendices) is

greatly reduced when a Lorentzian is used,

,i.` Let us write Eq. (I) symbolically in the form

W 2

where

^ W k GY H '^ k IXli,c

f p = ^ , A^ ^ , and A^^ ^

We adopt the convention that 1^^^ ^ 0, realizing that symmetric waves having

k^f C 0 are equally passible, ror the instabilities which we investigate in

this paper Al ^ 2.5 and may range as high as 20, In Appendix A we show thatH

far large A^H

1 k^^ 2p_I (- 1 }^+^ a P -t 1 ^ PzHH `

^IiH~k2 (2p - 3)I1 asp—I _ ^ ^iH exp Alit

r / 2P^ / 1 - Q 2rr ^III.^ 2P2A^^ H I ( - 2 ^I ` (1 - !1} - Q - + P Z 2Q + (1 - Q) 2 - I^j

LL \ A 1 2 ^ ^ cos 2rrP 1 a ^ ^ r^ r.L H 1H ^H

s^n2rrP iP 2a:^^^ (1 - Q)P2 (^)-- i erf - Q +

l -cos 2^r1' ^ ]^ ^ ^ ^ 2 ^ ^ z1H ^H ^'I3 s=1

Here P(^/s} = µ + i^/s HUH and erf(z) is the (complex) error function

erf(z) = ^.af dt exp - t2

In addition, the validity of our asymptotic procedure in Appendix A requixes

' 1^

_^,t_ .... F'

{{

I

3

^^. ..

__I l._ I ^1__. 1

{ _

r

3

^{A

-^

i

I

^',

-

l

1

B'{All y) ^ A^^ ande'(^r} ^ Ali , these canditzons k^eing axnpTy satisfied in all aux

carrxputation^. {Nate that we can txeat ^ ^^ h ll ^ ^^ ^1x firovided all axe cozrApaxably

l^.rge.)

The computational value of using a Lorentzian paxallel distribution should l;.be manifest in Eq. (6): neither Sums nor integrations xemain; the ca^rnputation-^

t .,,.,,

ally most difficult campanent is the error function which is quickly and effi-

ciently evaluated from a series representation; computer complex algebra

readily separates H^ into its real and imaginary parts H H ^ x^ai and. HH ^;ma^

Zn our investigations we have considered the p = ^ and p W 2 Larentzians,

far which

2H p= 1 ^ l kll

H(^^

^ ^II^ k2 ^

where H H is the quantity in-^ -3-ixz Eq. (6) (evaluated at s = i so that P = u + i^IIH)

and

^^

_.

^.

^^ ^

\\ au

($^

Were it necessary ax desirable to study situations with yet higher p, one

could continue this procedure and empress H^^ z completely in teams HH-i

and its ^c-derivatives. These derivatives x^a.ay in turn be evaluated either

numerically ox algebraically.

15

j - I ,. ^,

I I I V^ 4

The corresponding cold contribution,:.

^^ III ^ 1 h^C +°° hoc Ic - n ^I^cH^ ^ _ 2 2 1 + — exp - __.. ^ I^ ( ^ Z ^

J ^^ - n l + x ^ ^ { 9}h k A 2 \ 2 h a[h ^ I I I n=-^ II ^ 1c

is of a form derived in setreral places in the literature (e. g. , Ashour--Abdalla

et aI. , 1975). In. Eq. (9) I31 is a modified Bessel Function of order n and Z

is the well-l^rlown (Fried and Conte, 19G1) plasma dispersion function

+^

Z(x) _ ^ f dt c p Xt2

-^

While a strictly cold approximation to Eq. (9} is appropriate for some of our

computations, hll ^ and hl^ can in other cases be of ^(1) so that thermal correc-

tions are important and several terms in the sum dead be retained. Even under

these conditions, however , the number of computations is fewer than fora "hot"

plasma.

In solving Eq. (5) we consider .only the situation close. to marginal stability

so that I^.^ I « lµr ^. I]enoting HH + H^ Uy H vre thus solve first the equation

W2AH (p) (IO}I

^ SZ2 Hreal r

in a manner outlined in Appendix B. Here Hreat is, in the usual sense,: ,the. real

part of I-i, given ^ to be itself real. We then- determine a correction Q^t by

—.7—_^ _ 1L _ _^--- ^'—_ 1'1 _

d

^i

^:

::^

{.^F+

^'

i

f'

F.

b'

2

(a2 ^ Hre al(^r^ ^ ^^ Hrea l{pCr} + 1 Him ag (F,tr} ^- ^

where in Eq. (XI) the prime notation is used for differentiation and ^r is that

value determined from Eq. (Z4}, Solutions to (^.].) are complex and thus pro-

vide both ui and a correction to flr . The quadratic term in Eq. {^.I} is ixx,.^,or--

tart only when HFeat(^tr) ^ p ; when H^eai(f^r) = 1, Eq. (i^) reduces to the common-

ly used prescription for calculating weak growth or damping. Having thus

salved Eqs. (YO) and (1^.) we check ^. posteriori that terms in (I^.) are in fact

smaller than those in (10) and that other neglected terms are yet smaller. If

this test of smallness fails, the roots are rejected.

(I1}

`, '.

6 9

IV. SOLIITIONS OF TIDE PLASMA DISPERSION RELATION1 ,.^ Equation (5) is characterized by the following 6 parameters which, can be ^ ir z ,.

W p H^ ^-, ^

specified. independently: , ^i, d, T JT , T JT , and T /T^^ H . It would be an ;. ^^^z pH ,LH III l^ III ,.^,

enormous tasl^ to examine. a significant volume of this 6-dirriensional parameter ^f

space for instabilities, so we delimit the tasE; somewhat by considering only -the

"isotropic" case where T " T - T and T - T = T . The rofre^Iuene SZ ^^II H 1H FI Ilc !.0 o ^ q Y i

3

j

scales all frequencies in the problem, and the gyroradius calculated using SL andr

T^ is the fundamental unit of length. Our goal in this section is to summarize I a

the grass effects of changing the remaining four parameters on instabilities in

the frequency ranges described by our five proposed emission categories. In

further discussion we consider the variables in the dispersion equation to be u,

AfH (calling it simply ^), and B = tan" 1 {k^Jk^l).

Fig. 4 displays the frequency u = c.^JSZ of the fastest growing mode in each

gyroharmonic hand plotted against Ji = n^Jn Ei .The warm. plasma density is constant

{w^H JSZZ = 2d), T^JTH = q .a4, D = 0, and 8 = SG°. As n^ increases, ^ increases

within each band, and higher frequency I^ands may appear. The dotted li^ie indi-

cafes the cold upper hybrid frequeuc^^ defined by

W c n , /

tJz

'^FI

=^[^H = I + c,^pa /,SZz = l ^- nc 1 ^

Z ^ {12)

i H^`^.

The fact that instabilities do not occur for dc's much above ^uu H is due to :having

Re' H dominated by the cold electrons for these values of {^. Away from exact

harmonics of SZ, ivliei^e H is appropriate for instability, propagation in;^ H,imag

fi^

..

^s

^^ l '^^

^..,._

3 `^

i

^^=.

.^(.{?

;`

^^

^.,,r

v

the limit T^ --^ 0 is possible only at tax = µup . This role of the cold upper hybrid ^ a

frequency has been investigated by Ashour^Abdalla and Kennel. (1976) and Gaffeyi.^

^-,r,

^{Figure 5 demonstrates the complicated effects of the cold plasma temper- '__

afore on the range of the instability when ku x > 2. Here ^ = 0.5, ^ = 0, and µk ^ = -,= .

3.32 (or w^ ^ f Sze = ^0}. The range 2,S ^ ^ ^' 12 and. 82° C $ C S9° has been sa^xapled, ' ^"^" ^

and the regi.nn of corresponding unstable oz's is bounded by the solid curves shown.

.An asterisk indicates the fastest: growing mode within each band at the corre-

sponding T^ /T^ value. For T^ /T^ G 0.01, note that (a) the lowest band is very

narrow and lies close to ^. = 2 and (b} ^^x is more nearly an absolute upper

_..._ -^__.___.^_-^...^....^._._a..^..'^

^°r-

/`Y;

and LaQuey (1976).

bound on µ thaa it is at higher T^ /T^t , For T^ /T^ ? O.O z, bands above 4SZ and

5S^ becorne unstable, and the instability width tends to narrow in the second and

third bannds. The. appearance of these higher bands is duct to the increase in A^

with increasing T^, As A^ approaches B'(1), thermal affects become important

even far the "cold" electrons and several in{A^/2) contribute to the sum in

HG,^.^ at . The result is that solutions with ^ 7^uu^ are possible.

In addition to shifting the. unstable frequency range, the principal effect.

of increasing T^ j`T'^ is to reduce most instability growth rates. Figure Q plots

the maxi.inuxn growth rate 7mmx as a function of T C /T^ for the parameters of

Fig. 5. For TG /Ttt > 0, 0^, further increases in T^ ten.d to stabilize the plasma

• in all bands. The lack of strong instabilities. in the first, fourth, and fifth_

bands as T^ /T^ -^ 0 occurs because when T C is very small, only modes near

I9

^;,.31

'. ^.{L.

_. ..^ F

4 `^ _ -

^,

20

^ ^-^z..^w—..__._.,. _ _ --- _ _.r h

i

:; dux can have Iarge growth rates, The stabilizing effect of the cold plasma

^:vmperature arises primarily from the fact that Hc,ima^ (^) ^ 0 always. This

gives a Landau damping effect which opposes the HH ,^raag (µ) > 0 contribution

which drives the instability.

For /3 > D. D 5, the real part of the dispersion relation is dominated by

the cold plasma component. Increasing n^ should therefore increase the^:•

growth rate y without changing µ by much. This is verified by Fig, r, which

plots the most unstable frequency vs. R = n^ /n^ with T^ /T f^ = 0. T.O, d = 0,

and wP^^SZ2 = 5 (^cuH = 2.45}. The "error bars" delimit the frequency range in

^hi.ch 7 > 7m ax /2 and hence give an estimate of the bandwidth. The solid, dashed

and dotEed lines correspond to yniax^ 0.01, 0.001 C 7max ^ 0.01, and ymax < 0.001

respectively. Decreasing n^ tends to reduce ^rr^nx as expected.. The small vari--

ation in !^ which does exist is apparently an increase with decreasing nx .

We have already discussed the addition of a hot "fill-in" component which

reduces the sharpness of the loss cone. Figure 8 shows the maximum growth

rate 7max in each band vs. ^ for ^uH = 3.32, n^ /n H = D. 5, and T^ /T^ = 0. D4.

Increasing the fill-in factor d decreases 7max in all bands but tends to affect the

higher bands more strongly. In fact, for d = 0, 05, we found no unstable solu-

tions above p = 5. Tlxis preferential stabilization of higher frequency bands

urban the Ioss cone is filled in is more pronounced for our Lorentzian parallel

distribution than it would be ror a Max^vellian of comparable temperature. The

i

t

^,,^

^ ^^

' ^

^.►.. °^

i

I

-^.:^ ^.,;

^z .1

;i

^.

^:

^1

^..^_ : _^_._______ _^, _. _ _ . _. _ 1 __ _ ^._^_ _^.

fill-in component reduces growth rates because its contribution to H^ ^ lm ^ g (^) is^5

^^ always negative.,a

I`igures ^-8 reveal some of tlae complexities of the dispersion relation.

' The grass effects of changing each of the major plasma parameters are surn--

^`' _ marized below based on these rigures and an investigation of a much wider

1 range of parameters,

Cold plasma density. The cold plasma density n^ primarily determines

the cold upper hybrid frequencyuuH and hence governs the number of unstable

frequency bands. The highest unstable frequency Amax always lies below dux

. t'S'hen Amax C 2, but it may be slightly above µ^^I if Amax ^2. Zf a particular band

is unstable at some value of ^uH , the frequencies within that band will increase

i.f n G is raised further.

trot plasma density. The hot plasma provides the free energy to drive

the instability. Growth rates tend to increase with nH , but the unstable fre--

quency range is largely independent of n^^.

Cold plasxo.a texnperature. The role of T^ /T^ is a complex one. As

'' T^ /T^ -^ 0, large growth rates occur in the one or two bands closest to dux

;'while other bands have much smaller growth rates. Low frequency bands

^:;: {whose frequencies ^ satisfy (^cu x - ^^c]) ^ 1) occur near the top of the. band analt

have small bandwidths. For T c /TH > 0.01., one or t=NO bands above uux may

^ become unstable. At high cold plasma temperatures (0.05 < T^ /T^ C 0.2), the^^

,I

_ ^t^ i !

+

/'

, unstable frequency range in each band is shifted downward s and alI growth. rates ;^

are reduced. Al.l instabilities disappear for T /T > U. 2. '^^ ^ ^Y'=

a

Loss cane fill-in. Reducing the slope of f^ (v^) reduces growth rates in all - ^

frequency bands but tends to suppress higher frequency bands much more ^ ^'i

^• '^

severely than Lower ones. ^ ^ {

^.

T -'.

,^ In the previous section, we examined the effects of changing variousn

^^ plasma parameters in our model on the solutions to the dispersion relation. We

ir now proceed to use this information to i^afer plasma conditions which produceL

instaE^ilities in the appLupriate frequency ranges for each category of emission

defined in Section I.I. The fact that we are successful in achieving this goal byr

variations in a single plasma model strongly supports the validity of the model.

This single plasma model hypothesis is also suggested by the fact that there

often seems to be a smooth transition in the imp 6 data from one emission cate-

gory to anot?^+er {Shaw and Gurnett, 1975}, We feel that the model is sufficiently

successful that it can be used in a predictive fasl^.on to ix^.fer the unlmown cold

plasma density and to a lesser extent the cold plasma temperature and loss cone

! fill-in factor whenever gyroharmonic emissions are obsexved.

IIsing Section IV as a guide, we picE^ for each class of emissions acam-z

bination of parameters w 2H , n^^na , Te^T^, and A which produces frequencies andSZ

bandwidths characteristic of that category. Sy varying A and @, we determine

the largest 7 corresponding to each unstable ^. and thus construct curves of

maximum growth rate as a function of frequency. These curves should repro-

duce the features of the emission class while having realistically small. growth

rates. Using Ashaur-Abdaila and Fennel (1978} as a guide, we require temporal

growth rates to satisfy y c 0.01. Although we have not systematically calculated

convective growth rates as was done by Asliour-Abdalla and Kennel, such low

23

^^^ ^__ `__ ''^.

^^:

^:.

i

^_

^.:

1,

l ^t^; ! ^ 1

.. ___

i ^.

i

temporal growth rates usually produce realistically small convective growth ^

rates (klaiJSZ ^ 1 where 6^^ is the spatial growth rate}. ''t^'

From Figures 4 and 5 and from Ashaur-Abdalla and Kennel (1976), the ^}

first two classes of emissions ("low 3/2" and "3/2") definitely requirer>

^mnx <µux• With somewhat less certainty we also infer that µu H C 2 and ,^..

0.05 <TG JTa C 0.2 when these Class 1. or Class 2 emissions are observed.

Figure 9 represents 7 vs, µfar µ^^ = Z, 34 (Curve I.} and F.t^ i^ = 7.. 73 (Curve 2) '

with 'I'C /TH = 0. YO and 4 = 0. Both curves show substantial growth rates above

u = 2 which can be suppressed completely by filling in the loss cone. The small ^

curves (3 and 4} within the growth rate profiles represent the same parameter`

i_^ ,'

set with the fill in factor d = 0. 1. Hence, curve 3 represents the growth rate { i

profile for a typical Class Y { max < 1.2) emission while curve 4 represents a -^

Class 2 (1.2 < µ < 2) emission. Bath curves have small growth rates and ^^^ mix —

narrow bandwidths. The Class 2 or "3 /2" emission could be shifted closer to

µ = Y, 5 by decreasing T^ /TFi or increasing µup .J

Figure IO represents an attempt to replicate the features of "multiple '

half harmonic" or Class 3 emissions. The cold upper hybrid frequency µux

3.32, D = 0, n^ JnH = 0.50, and T^/ T^ = 0.10. Higher cold plasma densities are re-

quired than for "3 /2" emissions, and large fill in, which preferentially sup-

'f

presses the higher frequency modes, cannot be tolerated. The frequencies and

bandwidths agree well with those shown in Fig. I, and the successively smaller

growth rates as one goes to higher bands axe in accord with the apparent mono--

tonic decrease in intensity of the bands in Fig. ^,. The unstable anodes which

occur just above u = 5 propagate at a more nearly perpendicular angle to B

{8 > ^^°) ^s^rhile lower frequency modes tend to propagate most strongly in the

' range 8Q° ^ 8 < 85°. Modes with B -^ 90° have small A^^n and have frequencies

very near the gyroharmonics. Instabilities in such modes tend to be stronger

with our Larentzian parallel distribution than with a Ma^wellian.

Since Class ^ o.^ diffuse band emissions almost always occur above 10 l^Hz,

and since we have not yet ez^amined any data above ZO kHz, we have not presented

a detailed parameterization of such emissions. kiawever, preliminary calcula-

Lions suggest that diffuse band :missions can also be explained by this same

basic electron model, and we are continuing to investigate this possibility.

Class 5 or f ^• fp narrow band gyroharmonic emissions can be achieved by

this model only if n^ ^> na . This condition arises because the model cannot pro-

duce instabilities witla frequencies much above feu ^ ,and for wp c = w^ to + w^ ° ?^ SL 2 ,

dux C wPt/^2 -^ 1 -- wPt /SL. In fact, i.f the cold upper hybrid frequency is

exactly equal to t[^.e total plasma frequency (^cun = wpt/SZ), one can easily Shaw2

that (^ - ^2 -- 1. It is w^ t which is deteran9xx.ed frarn the lower cutoff of the

trapped enntinuunz radiation. ^ Amax ^

Sufi ha r+'ig. 2, then. the model requires

n C /n^ ^ ^0 since f^ ^ 9f^. However, we have seen that if T G ^ d, frequencies

somewhat above the cold upper hybrid can be generated and hence a smaller

n° /n^ ratio would be permitted. In same cases, gyraharmonic bands similar

25

_- . ^ ._...r:..- ^.

26

l .. ^ .:.. _., ... .^, _, _ T .. ^__

-r

^ C__ i ^.

to those in Fig. ^ are displaced just above f p ; this situation requires a larger

^; ^ n^ /nfi than the one with gyraharmonic emissions at or just below f ^,

figure 1J. shows that gyraharznanic emissions at or above f p can be pra-

doted if n^ /n^ is sufficiently Iarge. Since there is Less free energy available

from hat electrons to heat the cold plasma and since large cold plasma densities

tend to reduce growth rates, the ratio T^ fT H is probably smaller far f ^- fn

emissions than far "8 /2" emissions. In Fig, IIa, in which n^ /n^ = 10, T^ JTH =

0, 01, ^ = 0 and dux = ^f 21 = 4.58, a single narrow band emission near

fp = x/22 fg = 4.G9 f^ is the dominant feature. In F?g. I1b, the cold plasma tezn-

perattzre has been raised to T^ /TH = 0, U^. An emission band appears above

,^ = 5, the band just below ^ = 4 might also be observable, while the band nearest

f^ has a lower growth rate than in Fig. 11a, This I`igure suggests that (a)

emissions which axe clearly above and separated fxom f p , such as those in Shaw

and Gurnett (1975, I'ig. 4) are possible with this model and (Ia) the appearance of

ts^vo or three bands Haar f ^ requires a higher T^ iTa than a single emission band.

Multiple bands just below f p , such as those shown in Fig. 2, require a somewhat

smaller n^/n^ than the value used in Fig. 11 (e. g. , n^ /n^ w 3 instead of ].0).

The results of this section are stmnmarized in Table 2, which gives a

reasonable estimate of the plasma conditions which we believe produce each

type of emission, The "3 /2" model is in essential , a^;'reexnent with Ashour-

Abdalla and Fennel (1976). -This 'fable can be - used as a crude diagnostic tool

whenever gyroharznonic emissions are observed on a particrzlar spacecraft.

R

Class Frequency Range !^^ ^ = I + c^^ JS^^^ Comments

1. Lou+ " 3 J2" 1 < f/f^ < 1.2 f ^ ax n^ Jn u CC 1 (typically 0.1 }^^ < µ^ H ^ 1.5

0.05 < TG JTx < 0.15

^ < 0.1 (some fill in of loss canedistril^utian}

2. "3 J2" 1.2 < fJ g C 2 fm ax Jf^ < ^u ^ < 3 Essentially same as Class 1 l^ttt ^vithlarger cold plasma density

3. "n+lJ2"multiple fmax /g>2 fmaxJfg^^uH llcJnr^<1,0.05<T^JTH<0.15half harmonics ^ ,.; 0

^. "Diffuse Band" 1.5 < fmax lf^ <^' fmax Jfs `" uux n^JnH > 1 (?}

fmax ." fp (?} TeJ^;E "' 0.01 {?)

Requires fttrtlier study

fi. "f ,,, fp" narrow band fa <C fmax "' P ^c ^ ^" fp n^ Jnr.^ > I (» I (?)}

T^JTH < 0.02, d -- 0

^^'.

,.

i

f

1

-.,`

^

g$1

VI. THE DIAGNOSTIC USE OF OUR PLASMA MODEL ^

We have shown in the last section that each of our 5 categaraes of odd-

half-harmonic waves can be explained by suitable variations of parameters in a

single plasma model, Table 2 surz^.marizes the specific values or narrow ranges

of nG /nH , T^/T^, and D characteristic of each class of emission. We suggest

here that g^yroharmonic wave observations be used in con^unetion with Table 2

and magnetometer measurements of the steady B-field as a diagnostic tool for

determining local plasma parameters. We respond to the potential axgument

that the observed waves are not generated locally by pointing out that electro-

static oscillations, unli6ce electromagnetic radiation, are rapidly damped as

they propagate into regions which are not at Ieast marginally stable to their

growth.

The plasma parameter which is mast definitively determined by observing

odd-half-harmonic emissions is ^uI.i In this section we shall develop further

its relationship to emission frequencies, first by presenting in more detail re-

suits of our theory and second by applying these theoretical results to observa-

tions, bath individual and statistical, made by the Imp--G experiments. These

observations are all consistent with our theory.

'While it is sometimes possible to deduce n^ by measuring n t from the

continuum cut-off and subtracting the warm plasma density as measured by

particle detection experiments (Garnett and rrai^It, 19?^), this method ;becomes

inaccurate when n^ « nx, The results of our calculations, previop,s .

^^^:}

^^

F,j

2^

s^

L,.,,.

{

^^

^;

1^

theoretical work {^.roung et al, , 1973; Ashour-Abdalia and Kennel, 1 g76), and

the one observation presented by Gurnett and Frank {1974) all suggest that

n^ /nF! « 1 is a necessary condition for 1 '3/2" emission to occur, ^zrther- ;l;^';a^z

mare, the appearance of these "3/2" waves is very common in the outer mag-

netosphere. ^Te conclude therefore that the use of the subtraction technique to r:-

.^.,.calculate n^ is frequently a suspect one.

Figure 12 represents our best extimate, based on solutions of the linear

dispersion relation, ^f the cold upper hybrid frequency as a function of the maxi•- ^

mum observed frequency of magnetospheric odd-half-harmonic emissions. The

"error bars" are uncertainty estimates made under the assumption that the fill

in factor ^ is C. 05 (which in turn assumes that cold plasma heating is the domi- ` '`^

nant stabilization mechanism (Ashour Abdalia and fennel, 1976; Crume, et al. ,

^.97^)). For 1'^max C2 ^ 'ma x C^'uIi always so that we can deduce a very accurate _^

lower limit for the cold plasma density. If in fact Q » . 05 then the upper limiti

is higher, since under such circumstances ^'ux

may be •large and strong fill in

can quench all but the "3/2°' exiaissions. The dip at 'max =2 reflects our

1

computationally observed fact that for Classes 1 and 2, Amax <µu^ {always), {

while this is not necessarily true if Amax > 2(Classes 3-5). WE also observe ^r

computationally that D must be near zero far ^ttmaX » 1 and hence the error

estimates in Fig. 12 are quite accurate for large u ^maa'

J_ _J fi N

plasma density whenever gyroharmonic bands are observed. When the con-..

s,,

?' tinuurn cutoff is simultaneously observed, one can estimate n^ = nt -' nc • It is

usually possible to estimate Amax to within 5% using the electric field data and

the measured local magnetic field strength.

We have previously stated. that Class 5 (f ^' f p gyroharrnonic) emissions

with f p » f^ probably require n^ » n H . Whereas Classes 1-3 emission tend

to occur during geomagnetically disturbed tunes when n^ /n^ is expected to be

small in the outer magnetosphere (ef. Anderson & Niaeda, 1977}, Classes 4

and 5 are most common during quiet times (Shaw and Garnett, 1.975) when this

ratio is more likely to be large, as predicted by our theory. We predict that when---.

ever f ^^ fp gyroharmonics are observed, n^ is anomalously high. At such times

ione would expect that particle detection experiments which measure particles only

above some ^nzni.anlun energy tlzreshold can give a significantly Iower electron

density thazi the density measured from the trapped continuum lower cut-off. We

are currently investigating tlae data for verification of this hypothesis.

Our discussion of the variation of n^/n^ from class to class is pretty much

independent of the precise form of the electron distribution function, so long as

it has s. cold component and is 3.oss cone unstable. Such is not the case with T G /TH

and Table 2 should be used ^^ • :th accordzng caution. The hot distribution function,

which we have used for our calculations has a loss cone which is unrealistically

large for the real outer magnetosphere. Amore plausible distribution vuith a

peak in f^^ at a much smaller value of a 1 (relative to d1H } can. likewise be

^^ ^

-

i

^,

i

ss ,

i^.

unstable {Young et al, , x.973}. The characteristics of the instability depend

_ , N strongly on the positive slope (8f^/8v! > 0) portion of the distribution function

-'' and are virtuall ande endent of the tail 8f av C 0 so lan as it is "

...^ y ' p { lj ^ ) g gentle.

^^ In applying Table 2, T^ should he interpreted as the effective temperature

^ ^v z

^' Tx eFf "" kmax corresponding to the speed at which f^ ma_ximizes, since theB

important Iow v^ region can generally be fit by a DGH distribution of the type

we use. An important consequence is that given a hot electron component with

a Z [^eV thermal spread typical of the plasma sheet, Tx eft can be 100 eV or

smaller, and hence from Table 2 T^ ^^ ZO eV, a realistic value for plasma of

plasmaspheric or ionospheric origin.

i±

1

i

ii

S•F

,'

)^'.

ii ^{

F ^ ^^^...

i

ia

The use of Fig. Z3 and. Table 2 as a diagnostic taal can be illustrated using

Figures ^. and 2. The emissions at 07J . 5 of Fig. 1 are Class 3 (multiple half

harmonic emissions) with umaY = 6.1 and fg = Z. i5 i^Hz. (All B values are

obtained from the GSFC magnetometer data.) Hence, we estimate

wp^/S^ z ~ 30 from Fig, i3, and therefore n^ ti 0.5 cm^3 . From Table 2, D - 0

and 0.05 < T^ /T H < 0.2. Trapped continuum radiation is not observed, so the

total electron density is not available. Figure 2 is a Class 5 emission with

Amax ^ S. 7, fg = . 70 [MHz, and f P = S.4 I{Hz = 9. 1 f^ at 0537. Using Fig. 12 as

a guide, we estimate u^ x - $, so n^ ^- 0.38 cm"s . Since the total plasma density .

(fiom £n ) is 0.50 cm_3 , the ratio n^ /nH ^ 0.380.12 ^ 3 .2. When n^ » n^, the den-

, sity .ratio predicted by this method zs quite sensitive to the estimate of dux .

tram Table 2, we estimate. T^/TH ^^ 0:02 and d = 0.

3I.

^_ -_ _..-- -

_ ^, ^ .. r

^ I _ I I 1--^ - ^,^

.gFigures 12 and 13 are also supported by a statistical survey of the data ^

taken from a number of orbits. Figure X4 shows as a function of f g the ^iighest ,^.

s value of f/fa observed during 15 minute intervals from orbits 178 through 183 ^^'

1of Imp_6 whenever odd-half-harmonic waves are present. The data were all -,Y

f A

^tal^en on the day side of tl^.e outer magnetosphere at moderate magnetic latitudes i"""i

(20° C IAm I C 55°}. The tendency to see higher relative frequencies when the gyro- IE

frequency is low can be understood if the cold plasma density changes much more

slowly than fg once the spacecraft- is more than. --aJ.R^ beyond the r^lasmapause.{

` 1 In that case, wp° JSZ^' and hence µum will tend to increase as f g decreases, and the ^"}

theory predicts that higher values of µ (and hence more bands) maybe observed.^^ ^

^ The data used ili Fig. 14 also suggests tllatµmax decreases at high magnetic.,

', latitudes ^m . We divided the data into high {^A m I > 40°} and iow {^^m I <40°} rnag-

netic latitude cases. 82% C28 of 28} of the Iaigh latitude observations were Class

1 (µmax C 1.2) emissions. However, only 53% C^9 of 55} of the low latitude ob-

servations were Class 1, and all of the Class 3 (µmax ^' ^} observations were at

low latitudes. This effect can be theoretically interpreted using our model. At

constant radial distance li/H^ , f^ increases and n^ is probably constant or de--

creasing at high latitudes. Hence µui.^ and thus µmax would tend to decrease.

This probably accounts far the preponderance of Class 1 (µmax < 1,2) observa-

'^ tions in the Imp-6 data. Explorex 45, which always remains at low latitudes,

' observed more of the traditionalµ~. 3/2 emissions (R. R. Anderson; private ^ '.

communication}. ^ ^

I^

I^ k

32

^:,^i'^,^

tI^^_ __..... _^^ . . ^ ^ ^^

. ^; ! ^ ^ , ^

z

f;

:j

33

---^ ^---

j F ^ 1.

VII. CONCLIISIONS

We have conducted a theoretical analysis of the banded electrostatic

emissions which occur between consecutive harmonics of f^ in the outer mag-

' netosphere and have compared our theoretical predictions with Imp 6 electric

and magnetic field data. Our results are summarized below.

First, we have constructed a scheme for classifying these emissions

(Table 1) . Our fig►e categories include the widely studied add-half--har^rnonic

emissions as yell as the waves reported by Shaw and Gurnett (1975). Second,

we have extended successfully the linear plasma model of Young, et al. (1973)

and Ashour-Abdalla, and Kennel (197?, which can explain "3/2" emissions, to

the other categories of emissions. Our results indicate that a single plasma

model consisting of a hot loss cone electron distribution and a cold electron

component can generate instabilities in the frequency ranges appxop^°late to

each category. Hence, all of these emissions appear to arise from the same

physical mechanism. Figure 13 and Table 2 can be used as diagl^ostic guides

for estimating the cold plasma density and temperature whenever banded electro-

static emissions are detected. Finally, our results indicate that certain types

of emissions require the cold plasma density n G to exceed the hot plasma density

nH . Hence, although the hot loss cone component usually dominates the density

in the outer magnetosphere, there are times when n^ » n^.

^,...

^ ^ i ^^^,.

3^

^ I_ I .^^ __^

,^t

ACKNOWLEDGMENTS

We have made ample use of Imp 6 results from tine VLF electric field

and magnetoxxa.eter experiments, and we thank the principal investigators, D. A.

Gurnei:t and N. F. Ness respectively, lc. addition, we would. like to acl^n.owledge

useful discussions, advice,. and assistance from R. R. Shaw and R. R. Anderson

of Iowa, and P. Rodriguez and D. H. Fairfield of GSFC. Qne of us (R. F. H.

was supported in part by the Atmospheric Research Section, National ScienceY

Foundation.

3

y ^ .... ^.

35

^,^^_ Anderson, R. R. , and I^. Maeda, VLI' emissions associated with enhanced 1

c magnetospheric eiectrans, J. Geophys. Res. , 82, 135, 1977. ^^^•^

w ^.^i

',

Ashaur-Abdalla, M. , G. Chantetlr and R. Pellat, A contribution to the theory ^r;

6 }

;.. of the electrostatic half--harmonic electron gyrofrequency waves in the

,,.,:magnetosphere, J. Geophys. Res. , 80, 2775, 1975.

f_ ^.

^ ^

Asl^.our-Abdalla, M, , and C. F. Fennel, Convective electron loss cone insta-

bilities, U. C. L.A. Report PPG-263, 1976. y

Baldwin, D. E. , I. B. Bernstein, and M. P. H. Weenink, l^inetic Theory of

• Plasma Waves in a Magnetic rield, in Advances in Plasma Physics,i

^ ^Vol. 3 (ed. A. Simon and W, B. Thompson}, Interscience, New Yarn, ,

3

1969. l

Burtis, W. J. , and R. A. Helliwell,. Banded chorus: Anew type of VLF radia-

tion observed in thrY magnetosphere by Ogo 1 and Ogo 3, ^7. Geophys. Res. ,

74, 3002, 1969,

Crump , E. C. , H. K. ll^f Bier, and O. Eldridge, Nonlinear stabilization of

single, resanar^t, loss--cone flute instabilities, Phys. Fluids, 15, 1811,

1072.

Dory, R. A. , G. E. Guest and E. G. Harris,. Unstable electrostatic plasma

waves propagating perpendicular to a magnetic field, Phys, Rev, Lett, ,

14, 131, 106 5. J

t

s

sti

is

I

field emissions about the electron gyrafrequency, J. Geophys. Res. , 78,

310, 1973.

Gaffey, J. D. , Jr. , and R. La Quey, Upper hybrid resonance in the magneto-

sphere, J. Geophys, Res,, 81, 595, 1976.

Garnett, D, A. , and L. A. Franl^, Thermal and suprathermal plasma densities

in the outer naagnetospliere, ^^, Geophys. Res. , 79, 2355, 1974.

Garnett, D. A, and R. R. Shaw, Electromagnetic radiation trapped in the

magnetosphere above the plasma frequency, J. Geophys. Res. , 78,

8136, 1973.

Harris, E, G. , Unstable plasma oscillations in a magnetic field, Phys, Rev,

Lett. , 2, 34, 1959.

1^ennel, C. F. , F. L. Scarf, R. W, Fredricks, J. H. McGehee and F. V,

Coroniti, 1970, VLF electric field observations in the magnetosphere,

J. Geophys. Res. , 75, 6136, 1970.

Lyons, L. R., , 1974, Electron diffusion driven by magnetospheric electrostatic

waves, J. Geophys. Res. , 79, 575, 19'74.

i

._ _. -:

^R^

^ _ -

;.

;,

A:

c.

3G

^r '^ ^ r

'^:

Fried, B. D. , and S. D. Conte, The Plasma Dispersion Function, Academic, ¢ i

- i

New YorEi, 1961. _f ,,:-^^

Fredricks, R, W. , Plasma instability at (n + 1/2}f ^ and its relationship to ^ ^`'

some satellite observations, J. Geophys. Res. , 76, 5344, 1971.. ^^`

Fredricks, R. W. and F. L, Scarf, Recent studies of magnetospheric electric '.u.

Maeda, K. , P. H. Smith, and R. R. Anderson, VLF emissions from ring

current electrons, Nature, 263, 37, 1.976.

Oya, H. , Turbulence of electrostatic electron cyciotxon harmonic waves

observed b3' Ogo 5, J. Geophys. Res. , 77, 383, 19'72.

Oya, H. , Plasma flaw hypothesis i.n. the magnetosphere reiat9ng to frequency

shift of electrostatic plasma waves, ^'. Geophys. Res. , 8D, 2753, 1975.

Shaw, R. R. , and D. A. Garnett, Electrostatic noise bands associated with

the electron gyregrequeney and piasms. frequency in the outer magneto-

sphere, J. Geophys. Res. 84, 4259, 1975.

Tataronis, J. A. , and F. W. Crawford, Cyclotron harmonic wave propagationf

and instabilities, J. Plasma Phys. , 4, 231, 1970.

Young, T. S. T. , J. D, Callen and J. E. McCune, High-frequency electro-

static waves in the magnetosphere, J. Geophys. Res. , 7S, 1082, ].973.

Young, T. S. T. , Destabilization and wave-induced evolution of the: magneto-

spheric plasma clouds, J. Geophys. Res. , 80, 3995, 1975,

3?

_.

,,

t

Iir

t^:.;^^

^: ,..,

.w..

--

8 - G77-61

1MP-6_...: U OF IOWA PLASMA WAVE EXPERIMENT

ORBIT [78 MARCH 19, 1973

^-- 4 ,

4^.

V t 4 v ^ 1

W^.3..^

Q

^ r : E r- rtW^.. ^"'^.^ tii+" t" ,. r"" 3 ' .t^"11e ^ l^ ^+

_. 1'ir . ^4+ ^r _. -.,^^,

!^ .1 ^I , ^ ^t $^'.N^v1 C!-....^ ^ hl(^' YMI+ - ^ ^^'^^./ `r.r. U'1 ^1

^ [d^F+^.+^-+^1'^n ^ li t_,ri i^^ illur S n l^ +S r-+ I ^ -I^^ i1 l ^ 1 I Ir ^y.^,1^^-' -^^ 1 -^ .^f.

uT o7ao 0730

1 00"r^ -^ I/2" fg EMISS[oNS bfg =1.15 kHz ^^

^ro^^.. ^

r^.gure ^.. An e^ampie of multiple half harmonic emissions observed by IlVIP G at R z 9R^ on the day H

side of the magnetosphere. As many as seven gyxoharmanic bands are simultaneously ^ ^observed. The intense T '3/2" emission which occurs }aetween I and 2 kHz at 0700 isalmost continuously observed from 0500 to 0800. The emissions below 1 kHz are electro--magx^etic and are not considered in this paper.

w,.r.^„^.•wi

...

•,.., ^^1

.^'

---

^.,

Nz =

d QW'^ oL.L4^.

UTR (RE)

a^

M LT

1MP-6U OF IQWA PLASMA WAI/E EXPERIMENT

ORBIT 84 FEBRUARY 21, 1972

r^_^_ ^`t ^ k4^.y^,^ ^' ^ ^ ^_ 2,^Tk^i' gip' ^ ` F ^^' [[ s^"'7^'i l

^ ^ A

E n ^ _ ja.. G i,:^ tf

aa iS:.^,

^.

^^' ^`^ ^"^fil + ^ irk :,,,_ ^ , .p ,4,

_ I , ^, ^

u.l

0520 0530 0540

10.9

27.2° f-^ f^ GYROHARMONICS

I I.0 f9 = 0.70 kHz

o^figure 2. An example of gyroharmonic emissions occurring near the electron plasrria frequency f^ . o

The shaded region is trapped continuum radiation «hose lo^t•er cutoff is f^,. Up to three o

gyroharmonic bands are observed. ^ tom''

^ ^^^

4 .^

...^,^::,;^;^.>;.^,,,;^ ^ ^:: .,gin.:. ,::^;

.^ .^,,.

_.._ . ^- x^ ..,_ ..,^

^-*^^

^ ^ _ i:^I ^_

---^ MAXWELLIAN ------ P=2 LORENTZIAN

----- P = 1 LORENTZIAN ------- P = 5 LORENTZIAN

1.2D

1.00

0.00

^_

^ 0.60^-

0.40

0.20

0.00 ^ -' • -. .^ . - r.^

-4.00 -3.00 -2.00 -1.00 U.O^ 1.00 2.00 3.00 4.00

II I all

Figure 3. A comparison between a Maxwellian parallel velocitydistribution and the corresponding p t}' order Lorentziandistribution. The p = 2 Lorentzian has a similar shapefor Va l < a^^ but has more i;igh energy particles thandoes the Ma^-tvellian.

1

Y;9

^^

40

1

^ ^_ r r

J

4

^3.;

2

r ^

00.2 0.4 0.6 0.8 ^..0 ^

nc/n H ^

riguxe 4. The dependence of the frequency ^, = f/f^ of ^e fastesi growzng modewithin each gyroharmanic ]nand on the cold plasma density. Theparameters W /SLZ = 2Q, T^/T^3 = 0.04, e = SG°, and ^ = D. Asn^/n^ increases !^ increases within each band. The highest frequencytracks the cold upper hybrid frequency {dotted lire}.

4x

^^

1- ! ^ ,^1 ^^ ^.

^;

^^

5 -------------------------------•-----

q—_.—.---------^^ --------- -----

3 — -- — --^ ^

µ

3

^~

......,>.._.^>.,^,.^.^,^.,..,Y....^.,.^..^..... . _^^—

6 r----------------------------------------

2 ^---- --

I

1 --t i i i t i l t i i i i t t t l^-- .^^

o.00^ a.ol o.l I.o3

T^ /TH

Figure 5. The dependence of range of the instability within each band on thecold plasma temperature for a = 0. 5, D = 0, and ,^„ f ^ = 3.32. The

region of unstable fr, quencies within each band is enclosed by thesolid curves, and an asterisl: indicates the fastest growing mode }tivitlun each band as a function of T^ /T H . The temperature ratio has ^a considerable effect on the range of the instability. e

;, ^•--,,

3`

^^ ^1^^ - ^.32 ^,^

.^ ^

^ ^`^ ^ a

^ ^, ^^ ^^

.o^ ^,^

^^-^--- ^ .Q4 ^^

1

.fl2 -__-__^^- ^ .

4 -- ^-. \^^^ ^ ^?^

.UV^ .V1 Q.^ ^ {

Tc/^H

4f

Figure 6. The maximum growth rate 7max within each of the Niue gyroharnaoxucbands shown in Ii'ig, 5, platted against the temperatcare ratio Tc /Tx .r`or T^jT^ < 0< 0^, strong growth is seen only in the second andthird bands which are closest to ^nx = 3.32. Far Tc/T^ > 0.04,

.,,,,^ 7max decreases monotonically with T C in all bands. ^'"'"`r.

.^

.„

'^ ;^:

,^.',^

,. ^ ,

^.

,i _--^--- ^----w- .......................^

rP

10.^1 0.1 1.0 10.0

nc /^M

-- Figure 7. The effect of the hot plasma density on the range of unstable ire-quencies. The most unstable frequency is plotted vs. n^/nH withT^ /T^ = 0.10, .^ = 0, anal nG fixed (µuu =^. The "error bars"estimate the bandwidth based on the frequency range for ^vluch themaximum growth rate 7max < Zy(!^ ). The soiid, dashed, and dottedcurves correspond to ym^x > D. DI, 0.401 < 7mux < 0.01, and

ymax < 0.001 respectiveiy. increasing n E^ tends to Increase ymaxwhile Ieavi.z^g frequencies reiatively unaffected.

'^ .x...

.04

.03

AMAX

.oz

.ol

^^^µ^=Z

-3 ^^

^^- ^^

^^^^^^ ^µ^=^

^^- ^ ^\

^^ ^.^

=^

^.

.05 .10 .!5d

Figure S. A plot of 7^ax vs. fill. in factor O in each of the five bands wluclaoccur when T^ /T IC = 0.04, ^uii = 3.32, acid n^/n H = 0.5. Growthrates in all bands decrease monotonically with ^, but higher fre-quency bands are suppressed most severely.

45

:^, ^;, ^. :_ ^ ...

Y 0.03. 2

I3_

^ 0.005

4

0.01.00 3..25 ^ .50 1.75 2.00 2.25 2.50

Figure 9. Growth rate 7 vs. frequency u for 1^ ;^-^ = 1.3^, Q = 0. ^. (Curve 1) and :Tu{ ^i = 1.73, !3 = 0, 2 (Curve 2) with T^ /T^^ = 0.10 and A = 0. Unstablewaves above ^ = 2 can be suppressed by setting Ions cone fill infactor d = 0. IO (Curves 3 and ^4). I^ence Curves 3 and ^ reproducethe features of Class 1 (low 3/2) and Ciass 2 (3/^) emissions.^_.

^-- - -

,_...._.. .... _-. _.., ..__.._ _....... _. ., r ._

1^ - - - - -

^^.^.-w

'^ ;r r

r.v _

XIJ ^-....^.^...^-_...-...«.......mac...-.^.....^ .___._.._,..^...._._. .`..-^.»-....-«..^...y^^.`a.t.--...,...__.....a.v. .....a-^..`.. _... _...._ i.., ...___...^

i

^^?.. ^r,

... ^„

^, 0.0 1.0 2.0 3A 4.0 5.0 6.0 -. ^

^,.: Figure ^.0. 7 vs. ^. for unx = 3.32, p= 0, Q ^ 0.5, and T^/T^ = 0.10. Thefrequenefes and i^andzvidfhs are charac^;eristic of Class 3 (multiple '`'^"'^`'^"'half harmonic) e^ni,ssions.

^.w^._ .... _, ..

•^ ,^ ;

^ ^ ^.. ,.^ ^, ,

I

^.,

-.7^a^

--tO.OI5

i

^ 0.0 ^.

0.005

Y 0.0^2 3 4 w /^ 5 6

'^ n n ^t ^ ^V.V^..J

`^l

rP fW0.0I

0.005

0.01. 2 3 4 w^ /^ 5 6

rigure 1].. An example of gyroharmonic emissions near the plasma frequency.Fig. 1.1.a gives y vs. ^ for' R = ^.4, ^ W 0, tux = 4.5$, fp /f^ = 4. BJ,and T^ /T IC = 4.41, The domuiant feature is a single strong emissionnear f^,. In I'ig. 11b, T^/T^^ = 4.42, and an emission band above5fg would be seen. Class 5 emissions include both single band andmultiple band emissions near fp.

,^--

"^ ^ :'

..^;

R

Z

^^

t^. Ef.

_ .. ., _

[t^'.

/

7

/'

^v

l

f

#^

^' i

5

6i

^ ^ ^ ^

a/4• ^

_

^ ^

j

y

! uH

^^4 !^ ^

a.,

^ T

1

2 3 $ 5 6 ^Is max

Figcre 12. An estimate, based on solutions to the linear dispersion relation, ofthe. cold upper hybrid frequency uuH vs. maximum observed fre-quency ^ The TOerror bars" are. uncertainty estimates assumingmaxd C 0.45.. 'i.'Iv.s figure - ca^a be used to estimate uax from electric andmagnetic field data •whenever en^.issians of any class are observed.

49

.^.. : .

^ tiao

^o

W ^.PC

O,i

^rnax

Tigu.re 13. The- estimates of Fig. 1.2 converted to wp^ /SZ 2 for eq^ivaiex^tly n^)vs, maximum observed frequency Amax . This figure can be used asa diagnostic tool to estimate the cold plasma density nC,.

50

E I I ^ J_ ^

5.ao

4.00

µ 3.,00 ^

^^

2.00 ^

0.0 2.0 4.0 6.0 8.0 1 0.0

GYROFREQUENCY kHz}

figure 14. A plot of 83 indiv9.dua1 electric field observations taken at 15 mir^Eite

intervals from ^Snp 6 orbits ].78--183 whenever "odd-half-haixnoni.et'ezx^issions are seen. The graph shows the maximum observedfrequency ^^,^,^ (in units of f^) vs. f^. Higher relative frequenciestend to occur when the magnetic field is weakest, and almost aiIex^.v.ssians between f^ and 2fg occur below ^.. 5f^ . The region abovethe soiid curve earresponds to frequencies above 3.8 kHz and cannotbe observed in the data.

51.

__ - --__.__.....I

r

^.^^ ;

^.

/. .,

i

f

!^1

APPENDIX Ai

EVALUATION OF HOT ELECTRON CONTRIBUTIONYz3

TO THE DISPERSION RELATION^,

..^_. ^

ince f = fx + f^, the hot and cold electron cazxxponents contribute sepa- ^. {

to the dispersion equation, Eq. (I}. Vfie oansider here the hot contri-3

1J

_ ++7

9

^-ov ^'DQ CnFi

^fl^^

+nSLD"x ^ ^

SZ23v k flHx ^ k2 ^ ^ dv^l fl 11 (A^1)

j

_^ .^ nSZ - ^n=-^ Ulf k ^.

IE

where the Hx-^notatifln is that introduced in conjunction with Eq. (5}, and we

have partitioned

' ;^

fx = fj (al)f^E(u^^)

We evaluate

2 ^ I klv^ { 1 - 0)u.^ v^

Cnx = z f cFv^ulr^? [ ^ d -E- Z exp - ^al J \\ ^ aI ^I

1

^ 4x

^ I

and find it equalsi

C^, x ^ exp - 2x In ^ Z x ^ -- 2

A!x {In - Xn) (A--2)

where the In 's are modified Wessel Functions of order n and the natation

?i^x = kic lx JSL has been used. zn a similar manner

A«^

iE

' E

^ I,

2 a (^ -- a}^^ ^^

D = — dv .i 2 -- ^ a ^- ^ exp -

n H ¢^ .L n a

Ul«! a^

H p H H ^^,'

1 A^ ^^'

= a 22 exp -- ^H - 2aIn -E- {1 - a}1^ x (In - In} ^ (A-3} .

^H J ,:

^.Tor the parallel integration we insert the Lorentzian form. and define

^n -w -- n52

k^^^

The first U!^

-integral in (1} thus becomes

+p of /av^^ _+^

v!^ I

Qn^P fdv^^ _ -2pNP J dv ! ^ ::-O° ^^^ - ^n ..^ (11!i ^- «I^H }pal

LT,I '., un1'

(A-4} ,

where N denotes the normalization factor !P

_ 1 (2p - 2} T

N^ _ ^ (2p .. 3}!! (af^E^}

2p-i pr! - (^ I}!! -1 3

Note that Qn,P may he written as1

P

+^ ^

Q o _ 2 _ i P^-1 N^ a dx X ^ ^ ^n ,P (n - 111 ( } 'IT-F-1 h ., ^ „ 1

_^ ._ ._ .

{

i•F'

E ^1 ^

the latter form of {A-5) following by contour integration-closing in the negative

imaginary x-plane and picUin; up the contri^aution from the simple pole at

x ^ -i^ (According to our convention, integration in the complex w-plane is

to be c^z^ried out with Im w ^ 0, so that for integration purposes Im f^fl 7 4.)

In exactly the same way we evaluate

+^ +^_ ^ fll^ ^ 1 1Q^,p dull _ W N^ dnll

z -_°° (UIl - 1.^n ) _°° (ulf + ^Il^ )^ (Ull `- Fin)

i?p–^(–I)p+1 I ap-1 I

{Zp- g)^1 «lIH asp-1 ^n

^ ^^ 1 ill ^ (A-^)H s=1

If one of the differentiations in {A-5) is carried out explicitly, Hs i can be ex-

pressed in the particularly convenient form

I IG Ii 2p–^ {-- I )p+I ^p-1 ^ ^ IH^ =

^ liH g2 (gyp_ ^)ii a5p–I ^ innn ^ ^– ^

H^

2 ll^- CnH +icxll^ nDn ^

J (A 7)

iiin AII^

^^ ally ! s =^

If we define further

I k l^ I f^ I C zH^=1 =._... — ^

nH ^iallH

TIIH kz ^n_.-^ r/- ^ l n 11^^n 1 ^IIH nDnx

ll ll H

^..

.^..

a

r

^.: ^ l;

A-4

i

^.IF , . i

a

' so that3

H — 2p`I {- I )p+ 1 ap-1

H I3

^I ^^'.^H _ I T ^ p--i

H ^ '^i.y{2p 3}.. ^s s 1

(A--8)

then

^ kll1 1 ,^^H a- 1 -- O z 8 ^

+^In

HH^ - -- exp - ^lx --^ ^I ^---kz z 2 ^ a ss 2 a_ rs az _ i.,.

^11x V ^ ^J Y n-...o0 11n^_

a^1H

ia11H I ^ ^^ In- az --- I~ ^2a- {1 ° ^)^iH ^

I ^ ^^

1H ^i1HL... Z n__oo l^n^_

^1^ H

2^a11^ exp ^1H (A-9)

2alx 2

Zn. Eq. (A-9)

^r{.fs}

^µ^i^Af1xiand use has Veen z^nade of the identity

iI ^lH S, ^'lx

= ex ^'lH

i^ n^2 ^ n ^2} p 2n=-^ n=-^

I3

f.^-x

A-5

f

We now proceed to evaluate the summation`:r'",

40o

In {^^H fZ) ^-^ I

n (z}^_^

• (^

s 's

ill x ^,

in the limit that z -} ^ . Nate that

+^ In {z} i 27r

f(z, r) = ^ - - rd^i exp (i^'^i + z cas ^i} {A-1.1)P- n r 1- axp Z^riP J ^

n=--^ ^

This result is most easily proven by working in the reverse direction, i. e. ,

expanding exp(z cas r^) in terms of a Bessel Series and perfar^ning then the

r^-integration. Further

f{z' r) 1 -exp 2^P ^g{z, P) +exp Z^riP g(z, - P) j

where

n

g(z, P} _ ^d^i exp(iPr^ + z cos ^) {A- 12)a

The farm {A-12) is idea], for asymptotic analysis: for large z, the major

contribution to the integral. comes from a small region around ^ = 0 where

cos ^ = 7., and we may series expand the cosine function

z ^-' g{z, r) ^ exp z ^d^ -,kp ^lr^ ^ zZ ^ Cl + Z4 +e'(z^6) ^ (A-13)

o L

2+ exp z P 5 - ^

24z2z{A--IG)

^, a

- ^, f^. ,

7F -

s'_ / I

Convergence in (A-13) comes from the exp - zZ factor and has essentially

occurred by the time rp ^'^ The 24

term is then an B' 1Z^

correction to the

dominant terxn in {A-Z3). The 1e{z^6 }correction is of H^( z ^ and still smaller.lz

Extension of the upper integration limit to ^ introduces an error exponentially

small in z. We retain the dominant and first non-dominant term in g and

write

^',

^'^

5j1

'1 ^ ^

}

^^...

a

2 J' ^2

g(z, P} ^ exp z 1 + ^- — d^ exp iP^ -- ^--za z

^ azz o z

z 2= P exp z l 1 + ^ Wiz ) ^ exp - 2z ^I +erf ^ ^ ] {A-^.4)

where the error function has its usual definition

x

erf(x} =^ ^dt exp - t20

Carrying out the differentiation indicated in {A-14), we obtain

z a z

g{z' r} - 2z expz I + 24z \3 - ^' z + z ^ exp - r ^1 +erf \^^^z 2z

+ ^ ^ CS r2

^{A-^5)

I2 nz z z

Finally we conclude

^^ln (z} _ j rz ^ ,1 ^ sin 2^rP / iP 1^ ^ I ( 6l'^ I'4 ^^

^_,^^ P - n ^ err 1z ` 2z 2z ^1 -cos ^r - i erf ( ^ J 1 + ¢z `3 - z + Z2

A--G

'I ^ ,

f4 E°i^:^. ,.

}

The leading order term. is the one of order ^. The asymptotic smallness of

^, 2 ^ ,corrections dep^:nds on P f z remaini^ig of B^(a.) or smaller as z ^ ^.^^'

For P2 /z ^'8'(1} all correction terms in (A..^6? are of comparable magnitude ^^:^7

and are small compared to ^;he dominant by the anticipated 1/z.

.

We now incorporate the leading order term of Eq. {A-1fi) into Eq. {A-9). ^„

While it would appear that terms in (A-9) having A^ as a multiplicative factorH

are asymptotically dominant, operation by 1 - az drops their magnitude bya

one compensating order in A^ We are justified in retaining only the lowestH

order term in {A-IB) because neither the operations a^

nor ^1 - 8z changes\\\\\\

the relative ordering of the dominant and correction terms. Our final result is .;

1 _ k!f 1 l ^ P2

z2Pz 1- D 2^r a

HY PHz i ^' ex z ^^ll ^ l- a ^ ^^

k ?^!!^ ^ ^1H

ASH H ASH 2 cos 2^rP - 1

22 P sin 2nP iP

-^ P 2D ^- (I -- D) 1 i erf

^^ Al 1 cos 2^rP h!H

H H

2rxl H (1 -. d)PZ

1H .LH

(A-1'7)

E ^

^,

^^':Y=

rT

r

I I I^^l J r ^

APP)aNDIX B

OUTLINE OF COMPUTIJR SOLUTION TO THB DISPERSION RELATION

We here discuss computational detail briefly beginning with a summary of

our treatment of H^ . We choose initially values of ^^^ , aIH Ja^ lt^ , and the propa-

gation direction 8 ^ tan^i k^. Our work thus far has concentrated on the "iso-kq

tropic" case alb _ ^p H and we discuss only these results. Given the values of

these three chasers parameters, ^iix is determined. We next vary ^cr , typically

between ^r = I.02 and ur = 5. 98, in steps of . 0^, evaluating HH,reat and HH, imag

using the computer to sort real and imaginary parts. In the process we also

compute the first three ur derivatives of both the real and imaginary parts at

each ^r value. We stare these values of H^ and its derivatives on tape, tabu-

lating separately the contributions of the "loss cone" and "filled in" components,

which are uharacteri.zed respectively by the multiplicative factors {1 - n) and

d in Eq, (G). This table is then repeated for various pairs of Ase and 8, A!H

typically varying between 2.5 and 16 in a variable step size (.5 S A^ S 2) and 9

taping on the values 89. 5°, 88°, $6°, 82°, and 76°.

We combine each tabulation of H IS , charactierized by fixed AI^^, 8, and ftr,

with several "realizations'° of cold plasma. In characterizing the cold plasma

we still have the freedom to choose ^ and «^J« a . {Again, we have as yet con-

sidered only a^^^ _ aI^). Once these parameters are chosen Ha,rcal and Hc,imag

B-z

E:

..^.__- ---_.

i _ ^ _ ;._ ^ ^ is .

i ^..

are calculated from Eq. {9}. This procedure is a standaxd one, and further

discussion of it will be omitted.

`^ ^ Finally wp^/S2^ is evaluated from Eq. {Xfl} and µ^ and (Qµ) r are deter-

mined from Eq. (11}, [4µr is typically <Q.02 for accepted results.]

While the sequential events of coxnputation are as detailed above, Para-

- metric dependencies axe ultimately determined by cross referencing, For

example, one would determine the dependence of µr and p; on Ali (and hence

Flo }for fixed 8, ^i, and cx^/aH by first generating several tables each character-

ized by a different value of A^^ . Each table would have as "output" the values

of wPx /SZZ , µ^, and (Dµ) r as a function of the "input" µr . One would then further

search from table to table at fixed wP^/SZZ to determine, usually by interpola-

tion µ^ and µr -^ (Dµ) r as functions of Ai .

r^,^+qr = "i

'^, :i

B-2