Analytical study of Whistler mode waves for relativistic ...

J. Astrophys. Astr. (2019) 40:14 © Indian Academy of Scienceshttps://doi.org/10.1007/s12036-019-9576-3

Analytical study of Whistler mode waves for relativistic plasma with ACelectric field in inner magnetosphere of Saturn

JYOTI KUMARI and R. S. PANDEY∗

Department of Physics, Amity Institute of Applied Sciences, Amity University, Noida, India.∗Corresponding author. E-mail: [email protected]

MS received 27 November 2018; accepted 16 January 2019; published online 19 March 2019

Abstract. The population of plasma present in the rotationally dominated inner magnetosphere of Saturn isidentified by the observed plasma waves in the magnetosphere. Whistler mode emissions along with electrostaticcyclotron emissions often with harmonics are a common feature of Saturnian inner magnetosphere. We presentthe outcomes of a study of very large amplitude Whistler mode waves characteristics inside the magnetosphereof radial distance of less than 15Rs. Whistler mode waves with temperature anisotropy in the magnetosphere ofSaturn have been studied in the present work. Observations of Whistler mode emissions from Cassini Radio andPlasma Wave Science instruments have been obtained. Whistler mode waves were investigated using the methodof characteristic solution by kinetic approach, in the presence of AC field. The observations made by spaceprobes Voyager 1 and 2 and Cassini, launched by NASA, showed that charged particles are trapped in planet’smagnetic field lines. In 2004, the Cassini encounter with Saturn revealed that magnetosphere of Saturn exhibitsMaxwellian distribution. So, the dispersion relation, real frequency and growth rate were evaluated using ringdistribution function. Effect of AC frequency, temperature anisotropy, energy and number density of particleswas found. Temperature anisotropies greater than one (T⊥/T‖ > 1) and pancake-like distribution can generateWhistler mode emissions of the warmer plasma population. The study also extended to oblique propagation of

Whistler mode waves in presence of AC electric field. However, when relativistic factor β =√

1 − v2

c2 increases,growth rate decreases. Through comprehensive mathematical analysis, it was found that when Whistler modewaves propagate parallel to the intrinsic magnetic field of Saturn, its growth is enhanced more than in the caseof oblique propagation. Results are also discussed while computing the rate with which the wave grows for aparticular wavenumber.

Keywords. Whistler mode waves—magnetosphere of Saturn—ring distribution function.

1. Introduction

Cassini was the first orbiter to arrive at Saturn in 2004,It covered a large area of Saturn’s magnetosphere overlarge radial distances. Pioneer 11 flew for the first timein 1979, followed by Voyager 1 in 1980 and Voyager2 in 1981, The encounter of these detectors providesan opportunity to explore another planetary magneto-sphere and compare the plasma physics process withthe material processes in the Earth’s magnetosphere.The existence of Whistler mode waves, EMIC wavesand various instabilities in Saturn’s magnetosphere hasopened up a new chapter in research. Whistler modewaves are very important in electromagnetic emissions.They were the first observed plasma waves and were

first discovered by Barkhausen (1919), In 1935, Eck-ersley showed that the arrival time of the whistle isinversely proportional to the square root of the fre-quency.

Gurnett et al. first reported the plasma spectrumobserved by Voyager 1 on Saturn (1981). The Whistlermode hiss and chorus emission was discovered whenVoyager approached the equator at a radial distance ofapproximately 5RS (RS is the radius of Saturn). Whistlermode emissions from Saturn’s magnetosphere were alsodetected by the plasma wave instrument on Voyager 2(Scarf et al. 1982). Gurnett et al. reported the first resultsof Cassini Radio and Plasma Wave Science Instrumentduring the approach and Saturn’s first orbit (2005). Sev-eral diffuse emissions were observed at frequencies

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below fc (electron cyclotron frequency). These havebeen identified as Whistler mode emissions. The radialdistance of these emissions is approximately 2 to 6RS. AWhistler wave is an electromagnetic wave in a magne-tized plasma whose frequency is lower than the electroncyclotron frequency. The ionosphere/magnetosphereproduces various plasma instabilities which result inthe emission of waves propagating in the Whistler modewave branch. Most of these instabilities are caused byanisotropic electron distributions such as light beams,loss cones, rings and temperature anisotropies. TheWhistler mode of launch is also triggered by the light-ning generated Whistler. Whistler mode wave particleinteraction is an important process that can lead to fluc-tuation amplification and precipitation of high energyelectrons from the magnetic layer (magnetosphere) tothe lower ionosphere/atmosphere.

In this study we investigate the Whistler mode driveninstability by the electron temperature anisotropy.Therefore, in present work analytical approach has beenused to derive dispersion relation for ring distribution.The purpose of this study is to analyze Whistler modeinstability in a situation where energetic electrons have ageneralized distribution and a positive index which rep-resents the strength of loss cone index. This distributionis reducible to loss cone as well as other distributionshaving positive slope region in the direction of perpen-dicular magnetic field. This type of distribution havealready been observed by satellite (Wu et al. 1989).Thus, for better agreement, the Whistler mode insta-bility in the presence of perpendicular electric fields isstudied with such distribution functions which are seenmore than parallel ones. In Section 3, a general disper-sion relation for the case of relativistic Whistler modewaves in the presence of perpendicular electric fieldhave been derived. Finally, in Section 4, the growth ratehas been computed and is discussed for various suitablemagnetospheric plasma parameters for different valuesof temperature anisotropies, angle of propagation, num-ber density, energy density, relativistic factor, differentfrequencies of AC field.

The ring distribution is usually unstable, so it canexcite plasma waves by providing an important freeenergy source (Sharma and Patel 1986; Goldstein andWong 1987; Thorne and Tsurntani 1987). In weaklyimpinging plasmas, the ring distribution function canserve as a free energy source for different types ofinstability (Wu et al. 1988; Gary and Madland 1988).It occurs due to differential flow and/or due to effec-tive temperature anisotropy (Killen et al. 1995; Grayet al. 1996; Vandas and Hellinger 2015). Furthermore,an important feature of this distribution is to ensure that

the velocity space region where ∂F/∂v⊥ > 0 allowsfor instability at frequencies below the gyro harmonics(Thorne and Summers 1989). Since the electromag-netic Whistler mode wave is excited by the temperatureanisotropy of the thermal ring velocity distributionbelow the electron cyclotron frequency (Umeda et al.2007), we used the ring distribution function to studythe Whistler mode emission. These cyclotron waves inthe Saturn magnetosphere are produced by the tem-perature anisotropy produced by the ion rings pickedup from the e-neutral cloud, which form a loop in thevelocity space (Leisner et al. 2006). Frank et al. (1980)describes high-energy ion-excited ion cyclotron wavesin a Saturn magnetosphere with a ring distribution. Inaddition, Rodríguez-Martínez et al. (2010) studied thecharacteristics of harmonic mode waves generated bythe distribution of rings in the magnetosphere of Saturn.

The free energy source that drives instability isthe electron temperature anisotropy, T⊥/T‖. We onlyconsider wave propagation perpendicular to the mag-netic field. The plasma science experiment during theencounter of Saturn studied the low-energy plasma elec-tron environment within the Saturn magnetosphere. Theelectronic distribution functions are characterized bynon-Maxwell’s functions; they consist of a cold (hot)component with a Maxwellian shape and a hot (super-heated), non-Maxwellian component (Sittler Jr. et al.1983). Similar electronic distribution function featureswere found in the initial orbit of Cassini (Young et al.2005a). The colder component (3 to 30 eV) increasesin density and the temperature decreases as the radialdecreases distance. The hotter components (100 to1000 eV) have opposite behavior; as the radial distancedecreases, the density decreases and the temperaturerises. Thermal plasma components can be modeled byloss cones, double Lorentz (or κ) distribution (Summersand Thorne 1991).

In the current work, we studied the whistle patterninstability driven by the temperature anisotropy of theelectron components in the magnetosphere of Saturn.Section 2 gives the dispersion relation of obliquely prop-agating whistle mode waves. The plasma parameters inthe Saturn magnetosphere and the plasma parametersused in this study will be at Section 3.

2. Instrumentation and observations

Plasma parameters-like density, temperature of elec-tron, electric field, etc., in the magnetosphere, can beobtained from the passive observations of Whistlers(Carpenter 1988; Sazhin et al. 1992; Singh et al. 1998).

J. Astrophys. Astr. (2019) 40:14 of 13 14

The Whistler wave particle interaction was reportedby Kennel and Petschek (1966) and Chernov (1990).According to them, this interaction is the cause of waveamplification and particle precipitation. Whistlers arealso observed in the outer magnetosphere of plane-tary bodies (Koons 1985; Nagano et al. 1998). TheCassini Radio and Plasma Wave Science (RPWS)Instrument was the first to detect Whistlers in the mag-netosphere of Saturn. It measures magnetic fields overfrequency range of 1 Hz to 12 kHz and oscillating elec-tric field in the range of 1 Hz to 16 MHz. Meniettiet al. (2008) and Gurnett et al. (2004) presented thedetailed explanation of Radio and Plasma Wave Sci-ence instrument on Cassini spacecraft. The electronpopulation in the magnetospheric region is detectedby Cassini Plasma Spectrometer (CAPS) which com-posed of three sensors. These three sensors are electronspectrometer (ELS), ion mass spectrometer (IMS), andion beam spectrometer (IBS). Along with CAPS, LowEnergy Magnetospheric Measurements system of theMagnetospheric Imaging Instrument (MIMI/LEMMS)also provided a broad coverage of electron population(Krimigis et al. 2004). There is an increase in the tem-perature of electron with increase in radial distance from100 eV to approximately 2 KeV at 9Rs (Schippers 2008;Young et al. 2005a; Kaur and Pandey 2017). Suprather-mal electrons temperature lies in the range of 100 eVto 10 keV whereas temperature of thermal electrons isbelow 100 eV. The former originates in the middle orouter magnetosphere whereas the latter originates inthe inner magnetosphere. Suprathermal electrons dif-fuse into inner magnetosphere through interchange offlux tubes with outward drifting cold plasma. However,thermal electrons are produced due to ionization of neu-trals in neutral cloud (Kaur and Pandey 2017; Rymer2007).

3. Basic set of equations

Spatially homogeneous anisotropic, collisionlessplasma subjected to external magnetic field B = Boezand an electric field Eox = (Eosin ν t) ex has beenconsidered to get dispersion relation. Negligible inho-mogeneities are assumed in the zone of interaction. Inorder to obtain a general dispersion relation for the caserelativistic Whistler waves in the presence of AC electricfield, linearized Vlasov–Maxwell equations are attainedafter neglecting higher order terms and separating theequilibrium and nonequilibrium parts by small pertur-bations in magnetic field, electric field and distribution

function. Following the technique of Pandey and Kaur(2015), Vlasov equations are given as below:

Vlasov–Maxwell equation:

v•(

∂fso

∂r

)+ es

ms

[Eosin (vt) +

(v × Bo

c

)](∂fso

∂v

)

= 0 (1)∂fs1

∂t+ v•

(∂fs1

∂r

)+ (F/ms)

∂fs1

∂v= S(r,v,t) (2)

where force is defined by F = mdvdt

F = es

[Eosin(vt) +

(v × Bo

c

)](3)

The particle trajectories are obtained by solving theequation of motion defined

S(r,v,t) = (e/ms)

[E1 +

(v × B1

c

)]·(

∂fso

∂v

)(4)

where s denotes species and E1,B1 and fs1are perturbedquantities and are assumed to have harmonic depen-dence in E1,B1 and fs1 = exp i(k.r − ω t).

The method of characteristic solution is used to deter-mine the perturbed distribution function fs1, which isobtained from Equation (2) by

fs1(r,v,t) =∫ ∞

0s{ro(r,v,t′), vo(r,v,t′), t − t′

}dt′ (5)

The Phase space coordinate system has been trans-formed from (r,v,t) to

(ro,vo,t − t′

). The particle tra-

jectories which are obtained by solving Equation (3)for the given external field and wave propagation, k =⌊k⊥ex,0,k‖ez

⌋are:

xo = x + P⊥ωcme

sin θ − P⊥ωcme

sin

(θ+ωc

βt

)

+ �xsinvt

β(

ω2c

β2 − v2) − v�xsinωc

βt

ωc

(ω2

cβ2 − v2

) (6a)

yo = y − P⊥cos θ

ωcme+ P⊥

ωcmecos

(θ + ωc

βt

)

+ �x

v ωc

⎡⎣1 + v2β2cosωc

βt − ω2

ccosvt

β2(

ω2c

β2 − v2)

⎤⎦ (6b)

zo = z − Pz

βmet (6c)

where β =√

1 − v2

c2 is known as the relativistic factor.


And the velocities are:

Vxo = P⊥βme

cos

(θ + ωc

βt

)

+ v�x

β(

ω2c

β2 − v2)[

cosvt − cosωc

βt

](7a)

Vyo = P⊥βme

sin

(θ + ωc

βt

)

− �x

β(

ω2c

β2 − v2)[ωc

βsinvt − vsin

ωc

βt

](7b)

Vzo = Pz

βme(7c)

where ωcs = esBoms

= cyclotron frequency of species s.The AC electric field varying as E = Eox sin (vt) is

defined as �x = esEoms

and ν is AC frequency.After some algebraic simplifications and integration,

perturbed distribution function is given as:

fs1 (r,v,t) = ies

βme

∑ Jg(ω − k‖Pz

βme− (n + g)

ωcβ

+ pv)

[U ∗ E1x + iV ∗ E1y + W ∗ E1z

]

where

U∗ = JnJp

(CP⊥βme

nλ1

− CDv nλ1

+ CDv pλ2

)

V∗ = −iJn

(CP⊥βme

J′nJp + CD ωc J′

pJn

)

W∗ = JnJp

(Fnωcmek⊥P⊥

+ βmeω∂fO∂Pz

+ G(

pλ2

− nλ1

))

∣∣∣∣∣∣∣∣∣∣∣∣∣(8)

The arguments of the Bessel functions are:

λ1 = k⊥P⊥ωcme

, λ2 = k⊥�x

β(

ω2c

β2 − v2) ,

λ3 = vk⊥�x

ωc

(ω2

cβ2 − v2

) (9a)

where

C = (βme)2

P⊥∂fo

∂P⊥

(ω − k‖Pz

βme

)+ k‖βme

∂fo

∂Pz(9b)

D = �x

β(

ω2c

β2 − v2) (9c)

F = Hk⊥P⊥βme

(9d)

G = Hv�xk⊥

β(

ω2c

β2 − v2) (9e)

Jn′ = dJn (λ1)

d λ1, Jp

′ = dJp (λ2)

d λ2(9f)

Following Kaur and Pandey (2017), the conductivitytensor is written as:

‖ σ ‖ = −i∑ e2

s

(βme)2 ω

×∫

d3PJg(

w − k‖Pzβme

− (n + g)ωcβ

+ pv)‖S∗

s ‖

(10)

‖S∗s ‖ =

∣∣∣∣∣∣∣∣∣∣

P⊥Jn

(nλ1

)U∗ P⊥

(nλ1

)V∗ P⊥ n

λ1JnW∗

iP⊥J′n

(nλ1

)U∗ iP⊥ J′

nJn

V∗ −P⊥J′W∗n

PzJn

(nλ1

)U∗ VzV∗ VzJnW∗

∣∣∣∣∣∣∣∣∣∣(11)

From J = ‖ σ ‖.E1 and two curl equations of Maxwellfor the perturbed quantities, the equation of wave canbe written as:[k2 − k.k − (

ω2/c2) ∈ (k, ω)]

E1 = 0 (12)

where

‖∈ (k, ω) ‖ = 1 − (4 π/i ω) ‖ σ (k, ω) ‖ (13)

Dielectric tensor:

εij (κ, ω) = 1 +∑

s

{4e2

s π

(βme)2 ω2

}

∫d3PJg (λ3) ‖S∗

s ‖(ω − k‖ P‖

βme− (n + g)

ωcsβ

+ p ν)

(14)

4. Governing dispersion relation

For the propagation and instability of Whistler modewaves with k⊥ = 0 (Kaur and Pandey 2017), the branchof general dispersion relation (14) reduces to:

ε11 ± ε12 = N2 (15)

where N2 = (κ2c2

)/ω2is the refractive index. There-

fore, dispersion relation for n = 1 may be written as:

N2 = 1 + 4e2s π

(βme)2 ω2

∫d3P

2P⊥ [N1 + N2]

×⎡⎣ 1

ω − κ‖P‖βme

± (n + g)ωcβ

+ p ν

⎤⎦ (16)


where

N1 = (βme)2

P⊥∂fo

∂P⊥

(ω − κ‖P‖

βme

)

⎛⎝ P⊥

βme− ν �x

β(

ω2c

β2 − ν2)⎞⎠ (17)

N2 = βmeκ‖∂fo

∂P‖

⎛⎝ P⊥

βme− ν �x

β(

ω2c

β2 − ν2)⎞⎠ (18)

The distribution function for trapped electron is takenas Maxwellian ring momentum distribution function(Kaur and Pandey 2017):

f(P⊥, P‖) = ne/n

π3/2Po‖P2o⊥B

exp

⎡⎣−(P⊥ − Po)

2

P2o⊥

−(

P2‖)

P2o‖

⎤⎦ (19)

B = exp(−P2

o/P2o⊥)

+ √π

(Po

Po⊥

)erfc

(−Po/Po‖)

(20)

where

Po| =(

2kbT‖me

)1/2

and Po⊥ =(

2kbT⊥me

)1/2

are the associated parallel and perpendicular electronthermal velocities.

In Equation (19), ne/n is the ratio of trapped ener-getic electrons to total electron density (ne/n = no).Equation (20) gives the expression for complimentaryerror function. The P‖ and P⊥ are parallel and perpen-dicular thermal velocities in terms of momentum withrespect to magnetic field. Po is the drift speed in termsof momentum.

Substituting d3P = 2 π∞∫0

P⊥dP⊥∞∫

−∞dP‖ and using

Expression (19) in Equation (16) and after solving theintegrations, we get the dispersion relation as:

k2c2

ω2 = 1 + 4ππ2s

(βme) ω2

(ne/n)

B[X1

βmeω

k‖Po‖Z (ξ) + X2 (1 + ξ Z (ξ))

](21)

where

X1 = 1 + P2o

P2o⊥

− Po

Po⊥√

π − (βme) v�x

β(

ω2c

β2 − v2)

(√π

2

1

Po⊥− Po

P2o⊥

)

X2 =[

X1 + P2o⊥

P2o‖

(1 − √

πP3

o

P3o⊥

erf

(P’⊥Po⊥

)

+ 3P2

o

P2o⊥

− 3

2

√π

Po

Po⊥

)]−⎡⎣ (βme) v�x

β(

ω2c

β2 − v2)

×(√

π

2

1

Po⊥+ √

πP2

o

P3o⊥

erf

(P′⊥Po⊥

)− 2Po

)]

where Z (ξ) = 1√π

∞∫−∞

e−t2

t−ξdt is the plasma dispersion

function with ξ = βmek‖Po‖

(ω − gωc

β± ωc

β+ pv

)

Applying condition k2c2

ω2 1 for Whistler waves,

ω2ps = 4e2πne/n

Bome

ω = ωr + i γ

Equation (21) reduces to

D (k,w) = −k2c2

ω2ps

+ 1

β

[X1ω

k‖Po‖(βme)

{−1

ξ− 1

2 ξ3

}

−{

X21

2 ξ2

}+[(βme) ω

k‖Po‖X1 + X2ξ

]

{i√

πexp(−ξ2)}] (22)

Introducing the dimensionless parameters as k = k‖Po‖ωcs

,the growth rate in terms of the dimensionless parametersk, β1, K2, X1 and X2 is obtained as

γ

ωc=

√π

βk

(X2X1

− βX31−βX3+βX4

)(1 − βX3 + βX4)

3 exp

[−(

1−βX3+βX4

k

)2]

1 + βX4 + k2(1+βX4)

2(1−βX3+βX4)2 − k2

(1−βX3+βX4)

(X2X1

− βX31−βX3+βX4

) (23)


The real part of Equation (22) is

X3= − ωr

ωc= k2

β1

[K2 (1+βX4) + X2

X1

β1

2β (1 + βX4)

]

(24)

where

K2 = 1

2X1and β1 = 4 πμoεokbT‖ (ne/n)

m2eBB2

o

X4 = −p ν

ωc(25)

5. Plasma parameters

To investigate the effect of various radial distance andthe external ac perpendicular electric field parametersEo, ν and relativistic factor on growth rates the tabulatedset of parameters for different radial distance has beenused. Sittler Jr. et al. (1983) presented a complete sur-vey of plasma electron environment in magnetosphereof Saturn during Voyager flybys. The measurementsdone using Cassini Radio and Plasma Wave Instrumentand Cassini plasma spectrometer in Saturnian magne-tosphere have been reported by Gurnett et al. (2005)and Young et al. (2005a) respectively. On the basisof observations made by CAPS, in current paper wefocus on inner plasma torus (R < 7RS) and extendedplasma sheet (R = 7∼15RS) of magnetosphere ofSaturn. Electron densities change from ∼ 10/cm3 toabout ∼ 0.3/cm3 when plasma regimes are investigatedfrom inner torus to extended plasma sheet and furtherdecrease to about ∼ 0.04/cm3 in outer magnetosphereas well. The magnitude of magnetic field varies fromabout 80 nT to 4.4 nT as radial distance changes from6RS to 18RS (Acuna and Ness 1980; Kaur and Pandey2017; Davis and Smith 1990; Smith et al. 1980; Cow-ley et al. 2006). Growth rate calculations for Whistlermode waves have been analyzed at two radial distancesR ∼ 5RS and R ∼ 12RS inside Saturn’s plasma sheetusing parameters fromVoyager 1 andCassini data givenin Table 1.

Table 1. Observational data

Parameters Radial distance

R ∼ 5 RS R ∼ 12 RS

No (m−3) 5 × 107 6 × 105

Bo (nT) 184 12KBT‖ (eV) 300 500

6. Results and discussion

In Figure 1(a), (b) dimensionless growth rate (γ/ωc) hasbeen plotted for various values of AC frequency withrespect to wave number. It basically shows the variationof AC frequency (ν). At radial distance of R ∼ 5RS inFigure 1(a), the AC frequency increases from 12 kHzto 20 kHz, growth rate increases from 0.821 to 1.027with slight shift in wave number from 0.61 to 0.55. InFigure 1(b), at R ∼ 12RS growth rate increases from0.279 to 0.617 for peak value at wave number 0.49 and0.55 respectively. It shows that Whistler mode waveshave grown due to loss of perpendicular kinetic energyof ring electrons. Similar results have been reported byMisra and Pandey (1995). They examined generationof Whistler emissions when the energetic hot particlesare injected in the plasma in presence of perpendicularAC field. The observations found by them showed, thedependence of growth of waves on AC frequency. Themagnitude of growth rate increases as the waves prop-agate with increasing value of frequency either for thecase of bi-Maxwellian distribution or loss cone distri-bution affecting the lower frequency spectra. Thus, theperpendicular electric field is modifying the perpendic-ular component of velocity and contributes significantlyto the VLF signals emissions. The effect of AC fre-quency is for triggering the instability.

Figure 2(a), (b) show the graph with growth rate ofWhistler mode waves versus wave number for differentvalues of number density. In Figure 2(a), for radial dis-tance R ∼ 5RS, growth rate changes from 0.242 to 0.874for increasing value of number density from 3×107 m−3

to 7×107 m−3 with wave number changing from 0.55 to0.58. In Figure 2(b), at R ∼ 12RS, growth rate has beenplotted for number density 4×105 m−3, 6×105 m−3 and8 × 105 m−3. It can be seen that γ/ωc = 0.187, 0.306and 0.375 at k = 0.52 respectively. Therefore, as thenumber density of electrons plasma regime increasesgrowth rate of Whistler mode waves increases.

Figure 3(a), (b) show the change in dimensionlessgrowth rate (γ/ωc) for different values of energy den-sity (KBT‖) of electrons at various radial distances.The electron temperature varies from about 200 eV to600 eV as radial distance changes from 6RS to 18RS(Davis and Smith 1990; Cowley et al. 2006). There-fore, growth rate of Whistler waves have been calculatedin same range of energy density in present paper. InFigure 3(a), at R ∼ 5RS, variation of KBT‖ is taken tobe 100 eV, 200 eV and 300 eV, maximum growth ratecalculated as 0.315, 0.874 and 0.949 with peak valueat wave number 0.55 and 0.58 respectively. Even inFigure 3(b), at R ∼ 12RS, significant increase in growth


Figure 1. (a) Variation of Growth Rate with respect to k for various values of A.C frequency at no = 75000 m−3, KBT‖ =300 eV, θ = 10

◦and other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values

of A.C frequency at no = 6 × 106 m−3, KBT‖ = 500 eV, θ = 10◦ and other fixed plasma parameters at 12RS.

Figure 2. (a) Variation of Growth Rate with respect to k for various values of no at T⊥/T‖ = 1.5, KBT‖ = 300 eV,

θ = 10◦ and other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values of no atT⊥/T‖ = 1.5, KBT‖ = 500 eV, θ = 10◦ and other fixed plasma parameters at 12RS.

rate can be seen as 0.229, 0.375 and 0.487 for energydensity 500 eV, 600 eV and 700 eV at k = 0.52 respec-tively. So it can be seen that energy density of electronsis one of the important parameters effecting growth rateof Whistler mode waves

Figure 4(a), (b) shows the variation of growth ratewith respect to different temperature anisotropies whenangle of propagation (θ) is 10◦. From graph we can seethat for Figure 4(a) at R ∼ 5RS value of ratio of tem-perature (perpendicular) to temperature (parallel) i.e.T⊥/T‖ = 1.25, maximum growth rate γ/ωc = 0.476at k = 0.55. When T⊥/T‖ becomes 1.5, peak isobserved at k = 0.58 with γ/ωc = 0.874. And forT⊥/T‖ = 1.75, growth is γ/ωc = 0.989 at k =0.61. So, growth rate increases with increase in theratio of equivalent temperatures T⊥/T‖. Similarly, inFigure 4(b) at R ∼ 12RS temperature anisotropyincreases the growth rate by shifting the resonance con-ditions. For variation of temperature anisotropy T⊥/T‖from 1.25 to 1.75, growth rate varies from 0.191 to 0.585with a significant shift in wavenumber from 0.49 to 0.55.It can be seen that wave number is shifted to higher

number. Umeda et al. (2007) examined electroncyclotron harmonics excitation and Whistler modewaves as a result of warm electron ring distribu-tion, using particle-in-cell simulation, in presence ofcold background electrons. Their study concluded thatcombination of positive slope of velocity distributionfunction and temperature anisotropy due to ring distri-bution can excite both electron cyclotron and Whistlerwaves at same location.

Figure 5(a), (b) shows the effect of oblique angles ongrowth rate of Whistler waves in the magnetosphere ofSaturn. It is clearly seen in the graph that for θ = 10◦the peak value γ/ωc = 0.874 appears at k = 0.58,for θ = 20◦ the peak value γ/ωc = 0.1.184 appearsat k = 0.61 and for θ = 30◦ the peak value γ/ωc =1.449 appears again at k = 0.67. And as the radialdistance is increased to 12Rs, the growth rate decreases.Figure 5(b) shows that at θ = 10◦ growth rate is 0.375,peak is seen at k = 0.52 and for θ = 20◦ and θ =30◦, growth rate changes from 0.606 to 0.933 with aslight shift in wavenumber from 0.55 to 0.58. It canbe concluded that increasing the magnitude of oblique


Figure 3. (a) Variation of Growth Rate with respect to k for various values of KBT‖ at no = 75000 m−3, T⊥/T‖ = 1.5,

θ = 10◦

and other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values of KBT‖at no = 6 × 106 m−3, T⊥/T‖ = 1.5, θ = 10

◦and other fixed plasma parameters at 12RS.

Figure 4. (a) Variation of Growth Rate with respect to k for various values of T⊥/T‖ at no = 75000 m−3, KBT‖ = 300 eV,

θ = 10◦ and other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values of T⊥/T‖at no = 6 × 106 m−3, KBT‖ = 500 eV, θ = 10◦ and other fixed plasma parameters at 12RS.

Figure 5. (a) Variation of Growth Rate with respect to k for various values of angle of propagation at no = 75000 m−3,T⊥/T‖ = 1.5 and other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values ofangle of propagation at no = 6 × 106 m−3, T⊥/T‖ = 1.5 and other fixed plasma parameters at 12RS.

angle and positive slope of the distribution function hasbecome appreciable competing sources of energies inthe magnetosphere of Saturn.

In this paper, we have emphasized the relativis-tic study of Whistler mode waves. Therefore, wehave plotted Figure 6(a), (b) showing the changein growth rate of waves when relativistic factor is

varied, keeping other parameters constant. It is seenthat as the relativistic factor decreases, the growthrate increases. For R ∼ 5RS in Figure 6(a), at β =0.5, we get growth rate asγ/ωc = 0.8415 andk = 0.61. Then at β = 0.6, growth rate changesto 0.574 with k = 0.58 and when β increases to0.7, γ/ωc becomes 0.356 at k = 0.55. Similarly, in


Figure 6. (a) Variation of Growth Rate with respect to k for various values of relativistic factor at no = 75000 m−3,T⊥/T‖ = 1.5, θ = 10◦ and other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for variousvalues of relativistic factor at no = 6 × 106 m−3, T⊥/T‖ = 1.5, θ = 10◦ and other fixed plasma parameters at 12RS.

Figure 7. (a) Variation of Growth Rate with respect to k for various values of A.C frequency at no = 75000 m−3, KBT‖ =300 eV and other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values of A.Cfrequency at no = 6 × 106 m−3, KBT‖ = 500 eV and other fixed plasma parameters at 12RS.

Figure 6(b) for R ∼ 12RS, at λ = 0.7 growth rateγ/ωc is 0.183 for k = 0.52. At λ = 0.6, γ/ωcbecomes 0.0.475 with k = 0.52 again and when λdecreases to 0.5, we get γ/ωc = 0.805 with peak atsame value of wavenumber. It is concluded that moreis the velocity of the energetic particles, smaller isthe magnitude of relativistic factor and more is thegrowth rate of Whistler mode waves. Also the wavespectrum shifts to lower wave number as relativisticfactor is increased. Since the relativistic effects arenot mandatory for generation of Whistlers (Omura andSummers 2006) but it definitely affects the growth ratein Maxwellian.

Figure 7(a), (b) shows that variation of AC field. Itchanges the property of plasma and that of resultantgrowth rate at 5Rs and 12Rs. The magnitude of ACfrequency has been taken less than the calculated gyro-frequency of Whistler waves. The magnitude of growthrate varies as 1.455, 1.648 and 1.955 respectively asAC frequency increases from 12 KHz to 16 KHz andthen to 20 Hz. Figure 7(b) shows a similar variation at12Rs with respect to AC frequency. Here growth rate

has a peak at 0.466 with wavenumber 0.49 for AC fre-quency of 12 KHz and it shows a variation from 1.004to 1.443 with shift in wavenumber from 0.52 to 0.55respectively for 16 KHz and 20 KHz. We have observedthat the effect of AC frequency in the extended plasmasheet, and near magnetopause is approximately simi-lar. In both cases, the growth rate of Whistler waveshas been found to vary insignificantly with the chang-ing frequency. A major difference we noticed is that thefrequency bandwidth at 5Rs, the frequency bandwidthshifts towards lower wave number side for increasingvalue of AC frequency in the case of oblique propaga-tion whereas there is a shift towards higher wavenumberfor parallel propagation.

Number density is one of the important parametersthat affect the growth rate of Whistler mode waves. InFigure 8(a), (b), graph shows the change in growth ratefor Maxwellian distribution, j=0. The shift in spectrato higher wave number can be seen as for no = 3 ×107 m−3, maximum growth rate 0.181 occurs at k =0.49. For no = 5 × 107 m−3, maxima shifts to k =0.52 with γ/ωc = 0.733 and for no= 7 × 107 m−3,


Figure 8. (a) Variation of Growth Rate with respect to k for various values of no at T⊥/T‖ = 1.5, KBT‖ = 300 eV andother fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values of no at T⊥/T‖ = 1.5,KBT‖ = 500 eV and other fixed plasma parameters at 12RS.

Figure 9. (a) Variation of Growth Rate with respect to k for various values of KBT‖ at no = 75000 m−3, T⊥/T‖ = 1.5and other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values of KBT‖ atno = 6 × 106 m−3, T⊥/T‖ = 1.5 and other fixed plasma parameters at 12RS.

Figure 10. (a) Variation of Growth Rate with respect to k for various values of T⊥/T‖ at no = 75000 m−3, KBT‖ = 300 eVand other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values of T⊥/T‖ atno = 6 × 106 m−3, KBT‖ = 500 eV and other fixed plasma parameters at 12RS.

maxima shifts further to k = 0.55 with γ/ωc = 1.448.Similarly, in Figure 8(b), the change is calculated forgrowth rate of Whistlers for R ∼ 12 Rs. The peakmaxima attains a constant wave number as the numberdensity increases. For no = 4×105 m−3, γ/ωc = 0.465at k = 0.52. For no = 6 × 105 m−3, γ/ωc = 0.738 andfor no = 8 × 105 m−3, maxima occurs at same valueagain k = 0.52 with γ/ωc = 1.004. It has seen that,although in the case of oblique propagation, the growth

rate has a decreases with increasing distance. However,it is opposite for the case of waves propagating parallelto the magnetic field. Growth rate has higher value for1 Rs.

Figure 9(a), (b) show the variation of dimensionlessgrowth rate with respect to wave number for differ-ent values of energy density (KBT‖) of electrons. Atradial distance of 5RS, lesser increase in growth ratecan be noted from both the graphs. In Figure 9(a), with


Figure 11. (a) Variation of Growth Rate with respect to k for various values of relativistic factor at no = 75000 m−3,T⊥/T‖ = 1.5 and other fixed plasma parameters at 5RS. (b) Variation of Growth Rate with respect to k for various values ofrelativistic factor at no = 6 × 106 m−3, T⊥/T‖ = 1.5 and other fixed plasma parameters at 12RS.

Figure 12. Variation of Real Frequency with respect to k for fixed plasma parameters at 5 and 12RS.

thermal energy 100 eV, peak value of γ/ωc is 0.158 atk = 0.52 respectively. Similarly for KBT‖ = 200 eV,peak value of γ/ωc is 0.345 at k = 0.55 resp. And forKBT‖ = 300 eV, peak value of γ/ωc is 0.536 k = 0.58resp. The calculations performed assuming the radialdistance of 1Rs show that for KBT‖ = 500 eV, peakvalues of γ/ωc is 0.416, for KBT‖ = 600 eV, peak valueof γ/ωc is 0.771 and for KBT‖ = 700 eV, peak valueof γ/ωc is 1.004 at k = 0.52 in all the three casesrespectively. It can be concluded from both the graphsas electrons possessing higher energies undergo wave-particle interaction, growth rate of Whistler mode wavesincreases. As the graphs are widely spread along therange of wave number (k = 0.7 to 1.5), energy densityacts as one of the parameters varying which wide spec-trum can be analyzed. Results can be compared withprevious work done for parallel propagating Whistlermode waves in relativistic plasma of magnetosphere ofUranus (Pandey and Kaur 2016) to present comparativeplanetary study.

Comparison has been done between Figure 10(a),(b), showing the variation of growth rate (γ/ωc) withrespect to wave number (k) for various values of ratio

of perpendicular temperature to parallel temperature(T⊥/T‖). As T⊥/T‖ − 1 = AT, these graphs actuallyshow the effect of temperature anisotropy on growthrate of Whistlers. In Figure 10(a), (b), we can seesignificant difference in magnitude of growth rate ofWhistler mode waves for two different radial distance.In Figure 10(a), for R ∼ 5Rs maximum growth ratecalculated for T⊥/T‖ = 1.25, 1.5 and 1.75 is 0.294,0.536 and 1.019 respectively. Whereas in Figure 10(b),using R ∼ 12 Rs maximum growth rate is 0.251, 0.604and 1.007 for T⊥/T‖ = 1.25, 1.5 and 1.75 respectively.

Magnitude of growth rate for R ∼ 12 Rs is lower fortemperature anisotropies of 1.25 and 1.75 but more fortemperature anisotropy of 1.5 then R ∼ 5Rs.

Figure 11(a), (b) shows the effect of relativistic factoron growth rate of Whistler waves in the magneto-sphere of Saturn. It is clearly seen in the graph thatfor β = 0.5 the peak value γ/ωc = 1.009 appears atk = 0.61, for β = 0.6 the peak value γ/ωc = 0.719appears at k = 0.58 and for β = 0.7 the peak valueγ/ωc = 0.424 appears at k = 0.55. And as the radialdistance is increased to 12Rs, the growth rate decreases.Figure 11(b) shows that at β = 0.5 growth rate is


0.942, peak is seen at k = 0.55 and for β = 0.6 andβ = 0.7, growth rate changes from 0.504 to 0.173 witha slight shift in wavenumber from 0.52 to 0.49. It can beconcluded that increasing the magnitude of relativisticfactor decreases the growth of Whistler waves in mag-netosphere. Pandey and Kaur (2016) shows that it is notnecessary to have for the generation of Whistler waves.Therefore, relativistic factor does not affect the growthof waves.

Figure 12 show the graphs between real frequency(ωr/ ωc) of Whistler mode waves and wave number.Referring to the magnitude of real frequency in this fig-ure and comparing it with growth rate magnitude in restof the figures, it is evident that real frequency of Whistlermode waves is definitely more than the growth ratecalculated using the developed mathematical model.These results are in accordance with literature (David-son 1983; Misra and Pandey 1995) and can be comparedwith observed VLF emission measurements taken byspacecraft in magnetosphere of Saturn (Menietti et al.2009, 2012).

For oblique propagation, see Figures 1–6.For parallel propagation, see Figures 7–11.

7. Conclusion

This study presents the study of Whistler waves in thepresence of perpendicular AC electric field for rela-tivistic Maxwellian distribution in the magnetosphereof Saturn. The basis of this methodology is the kinetictheory adapted for deriving expression for dispersionrelation. Finally, a relation of growth rate and real fre-quency of the waves was calculated. A mathematicalmodel using ring distribution function provided the dis-persion relation of growth rate and real frequency. Thiscan also be used to study various types of instabili-ties in planetary magnetospheres. The results show thatWhistler waves grow more in the inner magnetosphereregime. It is seen that temperature anisotropy acts assource of free energy in magnetosphere of Saturn. Also,increase in number density and energy density of elec-trons increases growth rate of Whistler waves. It canbe said that Whistler mode waves grow due to loss ofperpendicular kinetic energy of ring electrons. A com-parative study of increase in dimensionless growth rateat two radial distances show that for R ∼ 12RS, the fre-quency of waves is the most important factor and at R ∼5RS, the growth rate is almost equally effected by all theparameters. Comparison of real frequency and growthrate for both the radial distances shows the occurrenceof Landau damping in magnetosphere of Saturn.

Acknowledgements

The authors are grateful to the Chairman, Indian SpaceResearch Organization (ISRO), and the Director andmembers of PLANEX program, ISRO, for the finan-cial support. The authors also thank Dr. Ashok K.Chauhan (Founder President, Amity University), Dr.Atul Chauhan (President, Amity University) and Dr.Balvinder Shukla (Vice Chancellor, Amity University)for their immense encouragement. The authors expresstheir gratitude to the reviewers for their expert com-ments for the manuscript.

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