Current Driven Oblique Whistler Wave in the Magnetosphere of Uranus

Indian J. Phys. 73B (4 ), 5 9 3 -6 0 4 (19 9 9 )

I J P B

— an in te rn a tio n a l jo u rn a l

Current driven oblique whistler wave in the magnetosphere of Uranus

R P Pandey, K K Singh, K M Singh and R S Pandey*

D e p a rtm e n t o f P h y sic s, V eer K u n w a r S in g h U n iv ersity , A ra-8 0 2 301, B ihar, In d ia

R e c e i v e d 2 4 S e p te m b e r 1 9 9 8 , a c c e p t e d 3 0 M a rc h 1 9 9 9

A b s t r a c t * S tro n g e le c tro m a g n e tic and e le c tro sta tic p la sm a tu rb u le n ce, low fre q u e n c y ra d io e m iss io n s and d u st im p acts w ere re a d ily d e te c te d d u rin g th e V oyager 2 e n c o u n te r o f U ranus. E x am in a tio n o f m agnetic field data recorded in closed proxim ity to the U ranian bow sh o c k re v e a ls a sc rie s o f w h is tle r w av e e v e n ts. S om e o f th e w a v e s e x h ib ite d p ro p a g a tio n p a r a lle l to th e m a g n e tic fie ld b u t m o s t sh o w e d o b liq u e p r o p a g a tio n , a .c e le c tr ic fie ld o b se rv a tio n s at m ag n eto sp h eric heights and shock regions have also been reported along and p e rp e n d ic u la r to the m ag n etic field In the p resen t paper, o b liq u e w h istler m ode in stab ilities h a v e b een a n a ly z e d h av in g [k n & k, ] w av e n u m b ers fo r a g e n e ra liz e d d rifte d d istrib u tio n fu n ctio n , red u cib le to bi-M axw ellian and loss-cone distribution in the presence o f perp en d icu lar a.c. e le c tric field by m eth o d o f ch ara c te ristic so lu tio n U sing d etails o f p article tra je c to rie s, d isp e rs io n re la tio n an d g ro w th rate have b een e v a lu a te d . R e su lts h av e been d isc u sse d and a p p lie d to th e m a g n e to sp h e re o f U ran u s.

K e y w o rd s O b liq u e w h istler, w ave, m ode in stab ilities, m ag n eto sp h ere o f U ranus.

P A C S N os. 5 2 .3 5 . Q z. 9 4 .3 0 . G m

1. Introduction

In the initial summary reports on the Voyager 2 plasma wave observations at Uranus, Gurnett and coworkers [1-3] demonstrated that strong electromagnetic and electrostatic plasma turbulence, low frequency radio emissions and dust impacts were readily detected during the encounter. Whistler emissions begin near spacecraft event time [SCET] 1500 at frequency / = 560 Hz, increase in frequency as the electron cyclotron frequency f Cf rises towards closest approach and reach maximum intensity in the frequency band 311 Hz < / < 1 kHz. The most intense whistler signals have been observed in the 562-Hz channel, which corresponds to a normalized frequency cd = ^ in the range = 0.22 [SCET 2000] to & = 0.35 [SCET 2030].

Whistler emissions in this normalized frequency range are commonly observed on auroral L

* D e p a rtm e n t o f A p p lie d P h y sic s, I.T., B a n aras H indu U n iv ersity , V aranasi-221 0 0 5 , U tta r P ra d e s h , In d ia .

©1999IACS

5 9 4 R P Pandey, K K Singh, K M Singh and R S Pandey

shells in both the terrestrial and Jovian magnetospheres. The Uranian whistler emissions fluctuate from measurement to measurement by factors of 3 to greater than 10 [4].

A preliminary analysis of Voyager 2 observations [5] revealed an intrinsic planetary magnetic field of Uranus. Lower frequency whistler waves have been reported at Saturn [6] and Uranus [7,8]. The encounter of Voyager 2 spacecraft with the uranian planetary system led to a series of outbound crossing of the Uranian bow shock between January 27 and January 30 of 1986. Examination of magnetic field data recorded in closed proximity to the shock, reveals a series of Whistler wave events that appear to result from processes associated with the shock.

Some of the waves exhibited propagation parallel to the magnetic field but most showed oblique propagation. A.C. electric field observations at magnetospheric heights and shock region have also been reported along and perpendicular to the magnetic field [9-11].

Parallel propagating Plasma waves in the vicinity of the magnetopause at ELF/VLF frequencies have been studied by many workers [12-19], whistler mode instabilities were analyzed using data from magnetosphere and electron experiment on board ISEE and AMPTE- UKS [20-22]. The growth of whistler wave was also investigated from an anisotropic electron beam of various electrostatic and electromagnetic wave modes at various propagation angles using a series of 1 -d simulations [23]. However, wave propagation was still restricted to a single fixed direction relative to the magnetic field. Devine et al [24] generalized oblique whistler mode instability in one and two dimensional simulations.

Electron beam excitation of upstream waves in the whistler mode frequency range was also studied in interplanetary space at 1 A.U. Both parallel and obliquely propagating solutions were considered [25]. Whistler waves have been observed in upstream from obliquely propagating shock waves in both simulations and space plasma [26, 27]. For a solar wind type plasma, expected wavelength arc of the order of inertial length and depend on obliquely propagating speed and field orientation [28,29].

Recently, competing processes of anisotropic electron beam having linear results and two dimensional particle simulations [30] of oblique whistler mode instabilities with relevance to interpretation of wave activities observed in Earth’s magnetosphere by the GEOS 1, GEOS 2 and GEOTAIL satellites or in Uranian bow shock by Voyager 2 were proposed.

In the present paper therefore, oblique whistler mode instabilities have been analyzed having [&j and kx j] wave numbers for a generalized drifted distribution function, reducible to bi-Max wellian and Loss-Cone distribution in the presence of perpendicular a.c. electric field by method of characteristic solutions. Using the details of particle trajectories [19], dispersion relation and growth rate have been evaluated. Results have been discussed and applied to the magnetosphere of Uranus.

2. Dispersion relation

A spatially homogeneous anisotropic, collision less magnctoplasma subjected to an external magnetic field BQ = B0ex and an electric field £ o, = (£ o sin v tex) has been considered. In order to obtain the dispersion relation in this case, the Vlasov-Maxwell equations are linearized.

The linearized equations obtained after neglecting the higher order terms and separating the equilibrium and non equilibrium parts, following the techniques of Misra and Pandey [ 19] are given as

V - ( S f s0/ S r ) ^ ( e s / m s ) [ £ 0 sin v t + ( v x B 0 /c](S f s 0/ 8 v ) = 0, Eox ==(£ o sin v teK)

(1⁾

(<5/v, 18t) + v.(8fs]f 8 r) + ( F / ms) (8fsl / 8 v ) = S ( r , v , t ) , (2) where force is defined as F = m dv /d t,

F = es [E0 s \ n v t + { v x B 0)/c]. (3)

The particle trajectories are obtained by solving the equation of motion defined in eq.

) and s(r, v, t) is defined as

S(r,VJ) = K / m J [ £ , + ( v x B })/c] (8fs0/ 8 v ) , (4) where 5 denotes species and £, and/s, are perturbed quantities and are assumed to have harmonic dependence in /N], B} and £ , = exp i (k. r-ax).

The method of characteristic solution is used to determine the perturbed distribution function.

/ v| , which is obtained from eq. (2) by DO

f il( r , v , t ) = l s { r 0( r , v , t ) , v 0( r , v , t ) , t - t ' } d t ' . (5)

0

The phase space coordinate system has been transformed from (r, u, t) to (r,u0,f - 1') and t' = t - t ' - The particle trajectories which have been obtained by solving eq. (3) for the given external field configuration and wave propagation

k = [ * ^ , , 0 , k, <?.]

Current driven oblique whistler wave in the magnetosphere o f Uranus 5 9 5

are

X Q = X + ( v y / a>,.Jt) + ( l / w t.1 ) [ u A sin<oc. / - i \ cos<wt, / ]

+ ( r x / 0)ct) [(wa sin v t ’ - v sin to, / ) / ( to2cs - v 2)], Y0 = Y + ( v t / c O M l / a > rj[u ,cosfl> t, / - u 4 sin<a,,f']

- ( r x / V © „ ) [ l + { ( V 2 c o sw a t‘ - c os v / ' ) / ( - V 2) } ] ,

Z0 = Z - v zt ' , (6)

and the velocities are

v x0 = v x cos a > „ /'-u v sina)„r' + |v / ’A / (to2, - v 2)J (cos v t ' - c o s t o lst ’) , i»v0 = u v sin o^ r' + o , c o s< u „ f'-{r A /(a )2, - t ' 2)}(<wCJsin v t ' - vsina>t.,/') ,

V z o ^ v , . &

where tocx = (eJ}^/mK is the cyclotron frequency of species s

r , •

and

5 9 6 R P Pandey, K K Singh, K M S in g h a n d R S P a n d e y

Eq. (4) can be written in terms of perturbed quantities as

S(r0,l»0, t - t ' ) = ~{es x [ ( a ) - k . V 0) E t

H v 0.E,yk](S f a t S o ) . (8)

After some lengthy algebraic simplification and carrying out the integration, the perturbed distribution function is given as

oo

f sl(r,v,t) = (-*, /ms(o) £ [ < y , (A2) ym( A,) y, (A3) ° )1

tn.n.pjq.=-+9

{(Cl)- k p t - ( n + q)(Ocs + p v) }][ E u J ttJ p {(« / A[)U * + ( p/ A2 )£>,}

+ JnJ'pD2) + Eu JnJpW*), (9)

where the Bessel identity

iA/sin0 » / i v ik9

€ —

has been used, the arguments of the functions are

A | = ^ )

^2 = (* ±r xV )/(G >^-V2) X3 = ( k ±r xv ) l ( m a - V 2)

C, = (1 /v x ) (Sf0 / 5ux ) (a) - *„.u,) + ( # 0 / &>,)*„

f/* = C,[uJ. - { v r , / ( a ) 2J - v 2)}l,

w* = f(n<uwu, / ux ) (Sf01 ) ~ n0}cS (#o / )1

[1+{*j/ > / (w2v- v2)} {(/?/A2) —(/i/A ,)} ],

A = c, {v r , /(<o2, - v 2) } , d2 = c , {(ocsr x /((o2cs- v 2) } ,

y; ={<*/„ (A, )}/</A, and y; ={<*/„(A2)]/</A2 . (10) Following Harris [30] and Misra and Pandey [19], the conductivity tensor II a II is written as

OO

,lcr|l=~ £ (6' ,m*m) ^ d 3v l i J tia 3) S ij ) / { Q } - k iVt - { n + q) wc, + p v ) ] , (11)

W|/l,/>!</,=«»

where

Current driven oblique whistler wave in the magnetosphere of Uranus 597

Sij =

v J ^ J / n / X ^ A

V , J U , , A

ivLJnB vL{nlX,)J2nJpW*

V i J ; , B i v J ' J J p W *

<”i J„B y„ J2J„W* ⁽¹²⁾

A = {(n/ Xl) U*+( p/ X2)D]),

' - ( W + ' . W (13)

From J = II o IIEj and two Maxwell’s curl equations for the perturbed quantities, the wave equation can be obtained as

[ k 2 - k . k - ( w 2 l c 2)e(k,a))]E] = 0, (14)

where

is a dielectric tensor, After using eq. (11), eq. (15) becomes

s ^ c,) = i + X {(4,r^ ) / K ® 2) } X S v A2) v ^ )

s n p

(15)

| (d2vtSit) / ( (O- ktv, - (n + q)a>a + p v ) } . 06) Expression for the growth rate :

The zero order distribution function is written as (14,19]

/ o ( » ) =

( * )3/ V (J+l)a,J\exp ^V2 ( V , - V j)2

a, (17)

where j is the distribution index.

When j = 0, this reduces to bi-Maxwellian and subsequently for T± = 7Jj it reduces further to Maxwellian distribution. Forj = 1, it reduces to loss-cone distribution. The resulting dispersion relation for oblique propagating whistler mode is approximated as

e,, + e 12= N2 ^{c o s2 ©,} , 08)

on following assumptions kL = k sin 0, = 0 and = k cos0, where 0, is the angle between BQ and k„.

Hence, the dispersion relation is obtained from this, for order of Bessel function n = 1, p = 1 and q = 0 and putting

L ^ l a n d l ^ l

598 R P Pandey, K K Singh, K M Singh and R S Pandey

k 2cz cos2 6

m

2 Mft

m,(0 a²⁰⁺¹⁾

X, 0 2 k ^ i 2 ( ^ ) + X , {1 + & ( £ ) }

k , a , ’ (19)

where

5 =

( o - k / } j -0)c + p v

* i a i

* i =_ E l v r. / y 2 ( ;+ l )

K 1 ( O r - V 2

O - K ) ! . (20)

a 2 ( / + , ) i ' f a 2

- £ V i ^ + D - '

vr, <xV"

(01- v 2 4 ^ ( 2 > + l ) - l

u-'A )'-

After substituting k2c2 cos2 0.

---z---L » 1

ft)2 P*

* K e ] n o m s

(2 1)

(O = o)r +i y,

and assumingJIr to be real and using an asymptotic expansion ofz(£) in the limit of large value of £ as

/— i 1

z(%) = N n exp H i ' ) — 1 + - 2 s

| l | » l . l m | £ | « R e | | |, the eq. (19) now reduces to

D(k,(0) =k 2c2 cos2 0,

CO2,,, a 2<y+1,j ! I

( c o - k p ^ l l l

'• * .« . u > \

(22) The eq. (22) further reduces to simpler numerical dimension-less form. By introduction of the following definitions:

- ka „

L = k c o $ 6 . , k = —

Current driven oblique whistler wave in the magnetosphere of Uranus 599

V

* 3 --- » *4 = ---

a c

*5 =

p _ k bTiH0«o

0 > c

K

, AT- = — x r . ' 3 4 * * * ** .

’ 2 3 5 ’

= 1 + X4 + *5>

the growth rate and real frequency are in dimensionless form as

and

X 2 ^ 2 t s 2

U \

exp u j )

k cos0, 1

^ V \ J )

y kcosd]

1av ^ k cos20j(l + Ar4) k 2 cos2d ]

4 2K\ k x i

cor k 2 cos2 0 ]

P

t f j G + * 4 ) + P

2 ( ] +x4 + * 5 )

(23)

(24) 3. Plasma parameters

To evaluate quantitatively the real frequency and growth rate for current driven oblique whistler wave, following plasma parameters given by Smithe t a l [8] Bridge e t a l[32] and Nesse t a l[5]

suited to the Uranian magnetosphere have been assumed. Density nQ = 5 x 104 m '\ magnetic field intensity B0 = 5 x 10'10 Tesla. Strength of electric field E0 = 4 x lO'3 V/m. The effect of electric field frequency variation on the growth rate has been studied for Zero, 3. 6, 9 Hz.

Temperature anisotropy AT = 0.25, 0.5,0.75 where and a ± are the parallel and perpendicular thermal velocities. Various oblique incidence 0, 10 and 20 Degrees and various values of the drift velocities 0, 0.2, 0.6 and 0.8 have been taken for evaluation of growth rate.

4. Results and discussion

Figure 1 describes variation of the normalized growth rate and real frequency with normalized wave number at various anisotropies for bi-Maxwellian plasma J =0 and for other fixed parameters.

It is obvious that the increase of anisotropy increases the growth rate but the Maxima significantly shifts towards lower £ [frequency value], at the same time the band width also reduces significantly. For all the given values of the temperature anisotropies, there is sharp rise in the growth rate for its maxima at a particular value of £ . This shows that the temperature anisotropy remains the prime source of seeding the free energy to the plasma.

600 R P Pandey. K K Singh, K M Singh and R S Pandey

Figure 2 shows the variation of the growth rate with normalized wave number for various values of the drift velocity of electrons parallel to the magnetic field at other fixed parameters.

The growth rate changes significantly with the increase of the drift velocity. The character of 10

10°

1 0 1 10

102

10-*

1 0<

0 .1 0 . 5 0 . 9 1 . 3 1 . 7 2 .1 2 . 5 2 . 9 k

F i g u r e 1 : V a ria tio n o f n o rm a liz e d g ro w th ra te ( y / t o ) an d re a l fre q u e n c y (c u /c u ) w ith norm alized w ave n u m b er k at v arious an iso tro p ie s for b i-M a x w ellian p lasm a, j = 0 at o th er fixed p lasm a param eters, E() = 4 x 10~3 V/m, v = 9H z, 0, = 20°, v , = .8, n{) = 5 x I0 4 it t\

B„ = 5 X 10-*° T

F i g u r e 2 : V a ria tio n o f n o rm a liz e d g r o w th ra te (y/o>t ) a n d re a l f r e q u e n c y (Q)rltt>.) w ith n o rm a liz e d w a v e n u m b e r k a t v a rio u s d rift v e lo c itie s fo r b i-M a x w e llia n p la s m a , j - 0 at o th er fix ed plasm a param eters, E{) = 4 x 10"3 V /m , v = 9H z, 0 , = 20°, n() = 5 x 104 m ~ \ B0 = 5 x 10-'° T, At = .25.

current driven oblique whistler differs depending upon the value of Vj. In the absence of the drift velocity the curve shows a maxima on the lower side of k value, a slight increase of the drift velocity increases the growth rate and the band width. But further increase of the value of the drift velocity shows double peaks in the growth rate curve one on the lower and the other on the higher side of the k values. Although the further increase of the drift velocity increases the growth rate maxima on both the sides but the band width goes on reducing. Simultaneous presence of double peaks for higher values of drift velocity suggests modification of usual real frequency in generation of a new wave. Here either electrostatic modes are generated or polarization is getting reversed. In between the peaks growth reduces but never goes to zero.

In this case it remains in left hand polarized mode. The results are in agreement with satellite observations and results reported by Smith et al [8] and Wong and Smith 125].

Figure 3 represents the effect of a.c. frequency variation on the growth rate for a bi- Maxwallian plasma7 = 0 . As the frequency increases, the growth rate increases with a decrease in the band width and shift of maxima towards lower k values, thus covering a wide spectrum of frequencies. In this case also, there is double peak in the growth rate and the generated frequencies obtained are in agreement with Smith etal [8] and Wong and Smith [25]. While the beam of bi-Maxwellian plasma generated simultaneously two peaks at different frequencies

10

Current driven oblique whistler wave in the magnetosphere o f Uranus 6 01

10°

1 0 -1 V 3

10*

10-3

10-4

0 .1 0 . 6 0 . 9 1 . 3 1 .7 2 .1 2 . 5 2 . 9 10°

3 3"

- 101

F i g u r e 3 : V a ria tio n o f n o rm a liz e d g ro w th rate (y/o)t ) a n d real f re q u e n c y (a>r/<u ) w ith n o rm a liz e d w av e n u m b er k fo r v ario u s v alu es o f a.c. freq u en cy fo r b i-M a x w e llia n p lasm a , 7 = 0 at o th e r fix e d p la sm a p a ra m e te rs, £ (1 = 4 x 10 3 V /m , 0, = 20°, #i() = 5 x 104 m \

fl() = 5 x 10 -,(l T, v(, = .8 At = .25.

with the growth rate not tending to zero in between two maximas, but as shown in Figure 4 the energetic loss-cone beam generated only one peak in the growth rate curve and alters the range of generated whistler frequencies. Although there is no change in the growth rate with the increase of a.c. frequency but like bi-Maxwellian plasma the increase in the a.c. frequency decreases the band width, a result in conformity with satellite observations and reported low frequency whistler wave emissions interpreted as lower hybrid emissions excited by the

electrostatic ion loss-con. instability [331. T V loss conebeam is more effective than bi- Maxwellian beam.

602 R P Pandey, K K Singh, K M Singh and R S Pandey

3

other fixed plasma parameters, £„ = 4 x 10 V/m, 0, - 20-. n„ . 1 0 '1" T. v. = .8. A r = .25

£ „ = 4 x

Figure S ■ Variation of aonnali.ril g to .lb t«n ( W «"d

norinsliied . . . . . a n t e E at ™ ri« « v

plasma j = 0 at other fixed plasma parameters, £„ - 4 x KT m, <>

- - = .8. At - -25.

nr , fl0 = 5 x 10-'° T, vrf

Figure 5 gives the variation of growth rate for various k values of angle of propagation at other fixed plasma parameters. In this case also the growth rate shows double peak.

Growth rate from 0° — 20° has significant effect on other peak on the higher side of k values.

The second peak shows sharp rise in the growth rate curve with the change in the angle of propagation, having rapid fall off to both higher and lower frequencies indicating that the secondary peak of whistler emission in quite narrow banded, thus showing a possibility of simultaneous generation of two-whistler waves at different frequencies.

From all figures, it is clear that our results are very much in the order of the observed normalized frequency range of whistler emissions at Uranus as reported by Coroniti et al [4].

Acknowledgment

The authors are grateful to Dr. K. D. Misra, Professor of Applied Physics, I.T. (B.H.U.) for providing laboratory facilities.

R e f e r e n c e s

[1 ] D A G u m e tt, W S K urth, F L S craf and R L P o y n ter Sci. 2 3 3 106 (1986) [2 ] W S K urth, D A G u m e tt and F L S c a rf J. G e o p h y s R es. 91 11958 (1986)

1 3 J F L S carf, D A G u m e tt, W S K urth and R L P oynter A dv. S p a c e R es. (in P ress) (1986)

[ 4 1 F V C o ro n iti, W S K urth, F L Scarf, S M K n m ig is, C F K ennel and D A G u m ett J . G eo p h ys. R e s. 92 1 5 2 3 4 ( 1 9 8 7 )

[5 J N F N ess, M H A cuna, K W B ehan non, L F B urlaga, L E P Connerney, R P L cpping and F M N eubauer S a . 2 3 3 85 (1 9 8 6 )

[6 ] D S O rlo w sk i, C T R usssell and R P L epping J G e o p h y s R es. 9 7 19187 (1992) [7 ] C W S m ith , M L G o ld ste in and H K W ong J. G e o p h y s. R es. 94 17035 (1989) [8 ] C W S m ith , H K W ong and M L G o i . G eo p h ys. R es. 9 6 15841 (1991)

[ 9 ] F S M ozer, R B T orbert, U V F ahleson, C G Fathauer, A G onfalone, A aPedersen and C T Russel S p a c e S c i. R e v 2 2 791 (1 9 7 8 )

[1 0 J J R W ygant, M B e n sad o u n and F S M o z er J. G e o p h y s R es. 9 2 109 (1987) [ 1 1 ] P A L in d G u ist and F S M o g cr J. G e o p h y s R es 9 5 137 (1990)

[ 1 2 ] J L a B e lle an d R A T reum ann S p a c e S n . R e v 4 7 175 (1988)

[ 1 3 ] C J F a rru g ia , R P Rijn beck, M A Saunders, D J Southw ood, D J R odgers, M F S m ith, D S H all, P J C h ristia n s e n an d L J C W illisc ro ff J. G e o p h y s. R es. 93 14465 (1 9 8 8 )

[ 1 4 ] S S azh in P l a n e t S p a c e S c i 3 6 1111 (1 9 8 8 )

r 15] S Sazhin W h is tle r M o d e W a ves in a H o t P la sm a (C a m b rid g e A tm o s p h e ric a n d S p a c e S c ie n c e S e r ie s) (C a m b rid g e U niv. P re ss, N ew York) (19 9 3 )

[ 1 6 ] B T T su ru tan i, E J S m ith, R M T hone, R R A nderson, D A G u m ett. G K Parks, C S L in, C T Russel G e o p h y s . R e s L e tt. 8 183 (1 9 8 1 )

[ 1 7 ] B T T su ru tan i, A L B rinca, E J S m ith, R T O kida, R R A nderson, T E E astm an J G e o p h y s. R es. 79 118 ( 1 9 8 9 )

[ 1 8 ] K D M isra a n d T H aile J. G e o p h y s. R e s 98 9297 (19 9 3 ) [1 9 ] K D M is ra and R S Pandey J G e o p h y s R es. 100 19405 (1995)

[ 2 0 ] A K o rth , G K rem ser, S P erau t and A Roux P la n e t S p a ce. Sci. 3 2 1393 (1 9 8 4 )

[ 2 1 ] A K W ard, D A B ry an t, T E d w ard s, D J Parker, A O hea, T J Patrick, P H Sheather, K P B a m stad a, A M C ru ise T h e A M P T E -U K S s p a c e c r a f t IE E E T ra n sa c tio n s on G e o sc ie n c e a n d R e m o te s e n sin g G E - 2 3 2 0 2 ( 1 9 8 5 )

[ 2 2 ] C F K en n el F V C o ro n iti and F L S c ra f J. G eo p h ys. R es. 91 1424 (1986) [ 2 3 ] Y L Z h a n g . H M a tsu m o to and Y O m u ra J. G e o p h y s. R es. 98 21353 (1 9 9 3 ) [ 2 4 ] P E D e v in e , S C C h a p m a n and J W E astw o o d J. G e o p h y s. R es. 100 17189 (1 9 9 5 )

Current driven oblique whistler wave in the magnetosphere o f Uranus 603

125] H K Wong and C W Smith J. Geophys. Res. 99 13373 (1994) [26] L H Lyu and J H Kan Geophys. Res. Lett. 17 1041 (1990)

[27] F G Pantellini, E A Heron, J C Adam and A Mangenery J. Geophys. ’Res. 97 1303 (1992) [28] V A Thomas, D Winskc and N Omidi J. Geophys. Res. 95 18809 (1990)

[29] D Winskc, N Omidi, K B Quest and V A Thomas J. Geophys. Res. 95 18821 (1990) [30] L Borda de Agua, Y Omura and H Matsumoto J. Geophys. Res. 101 15475 (1996)

[31] EG Harris 158, in Physics of Hot Plasmas, eds. by R Oliver and T Boyd (Edinburgh) (1970) [32] H S Bridge et at Science 233 89 (1986)

[33] F V Coroniti, R W Fredricks and R White J. Geophys. Res. 77 6243 (1972) 604 R P Pandey, K K Singh, K M Singh and K A ranaey