Study of Oblique Propagating Whistler Mode Waves in Presence of Parallel DC Electric Field in Magnetosphere of Saturn

In this paper whistler mode waves have been investigated in magnetosphere of Saturn. The derivation for perturbed distribution function, dispersion relation and growth rate have been determined by using the method of characteristic and kinetic approach. Analytical expressions for growth rate and real frequency of whistlers propagating oblique to magnetic field direction are attained. Calculations have been performed at 6 radial distances in plasma sheet region of Saturn’s magnetosphere as per data provided by Cassini. Work has been extended for bi-Maxwellian as well as Loss-cone distribution function. Parametric analysis show that temperature anisotropy, increase in number density, energy density and angle of propagation increases the growth rate of whistler waves along with significant shift in wave number. In case of Loss-cone distribution, increase in growth rate of whistlers is significantly more than for bi-Maxwellian distribution function. Generation of second harmonics can also be seen in the graphs plotted. It is concluded that parallel DC field stabilizes the wave and temperature anisotropy, angle of propagation, number density and energy density of electrons enhances the growth rate. Thus the results are of importance in analyzing observed VLF emissions over wide spectrum of frequency range in Saturnian magnetosphere. The analytical model developed can also be used to study various types of instabilities in planetary magnetospheres.


Introduction
Magnetized plasma supports a variety of plasma waves.The investigations done in planet's orbit have shown two general classes of wave-particle interactions, controlling the important aspects of plasma dynamics in magnetospheres, namely, electromagnetic and electrostatic plasma instabilities.In present research we have dealt with resonant interactions between particles trapped in magnetic field lines of Saturn and electromagnetic waves present in background space plasma.Whistlers are low frequency, circularly polarized electromagnetic waves in the audio-frequency range.Whistler mode emission constitutes electromagnetic waves with frequency below either electron gyro frequency or local electron plasma frequency, whatever is less.Since these low frequency whistler mode waves interact with charged particles over long scale lengths within magnetospheres and transfer energy from one region to another, we chose to study them using Electromagnetic Theory.Therefore, this paper work is based on solving Maxwell's equations to understand electromagnetic nature of whistler mode waves in magnetosphere of Saturn.
The Voyager 1 and 2 spacecraft first established that Saturn have many complicated plasma wave phenomena [1][2][3].A series of electromagnetic narrowband emissions were identified in inner regions of Saturn's magnetosphere ranging from 3-30 kHz [1,4].The most strongest and complex plasma wave observations at Saturn from Cassini's approach has been reported in inner magnetosphere at radial distance of ~10 Rs [5] (Rs is the radius of Saturn).Many diffused emissions, recognized as whistler mode emissions, were detected at frequencies below electron cyclotron frequency.Whistlers are audio-frequency waves in magnetized plasmas generated by lightning at frequencies below electron cyclotron frequency and electron plasma frequency.The ionosphere/magnetosphere produces various plasma instabilities propagating along the magnetic field lines from one hemisphere to another in plasma mode of propagation called whistler mode [6].The radial distance of these electromagnetic whistlers is about 2~6 Rs in case of Saturn [7].Whistlers have also been observed at Jupiter [9,8], at Uranus [10,11] and at Neptune [12].
Accelerating charged particles to kinetic energies much higher than their initial thermal energy is one of the significant phenomena effecting the dynamics of planetary magnetospheres and ionospheres.It can be understood by studying the exchange of mass and energy through electric field parallel to magnetic field [13][14][15].Electric field is one the most effecting parameters controlling Saturn's aurorae [16], for interplanetary magnetic field playing a limited role unlike at Earth's.Studies also show that parallel electric fields in acceleration region govern the heating rate of plasma species, along and perpendicular to magnetic field direction.
Since the electrons are accelerated more effectively by electromagnetic waves at frequencies of few kilohertz [17], thus increasing the growth rate, we investigate whistler mode waves in presence of parallel DC electric field in magnetosphere of Saturn in present work.Statistical approach has been applied to derive dispersion relation.Parametric analysis has been done by calculating the growth rate of whistler mode waves.

Mathematical Formulation
A homogeneous anisotropic collision less plasma in the presence of an external magnetic field In interaction zone inhomogeneity is assumed to be small.In order to obtain the particle trajectories, perturbed distribution function and dispersion relation, we use the linearized Maxwell's -Vlasov equations.Separating the equilibrium and non-equilibrium parts, neglecting the higher order terms and following the techniques of Misra and Pandey [18] the linearized Maxwell-Vlasov equations are given as: Where the Lorentz force is Where the dispersion relation is defined as (4) where s denotes the type of electrons.Subscript 'o' denotes the equilibrium values.The perturbed distribution function f s1 is determined by using the method of characteristic, which is (r,  f We have transformed the phase space coordinate system for (r, v, t) to (r 0 , v 0, t -t').The particle trajectories that have been obtained by solving equation (3) for given external field configuration are and the velocities are v and v || denote velocity perpendicular and parallel to the magnetic field.Using equation ( 6), (7) and the Bessel identity and performing the time integration, following the technique and method of Misra and Pandey [18], the perturbed distribution function is found after some lengthy algebraic simplifications as : ... (7) Due to the phase factor the solution is possible when m = n.Here.
The Bessel function arguments are defined as The conductivity tensor |||| is found to be By using these in the Maxwell's equations we get the dielectric tensor, .The dispersion relation for relativistic case with perpendicular AC electric field for g= 0, p = 1, n = 1 is written as: The generalized distribution function [19,20] is given as When the applied electric field is parallel to B o , it modifies electron thermal velocity in direction of magnetic field.
Therefore, changing o|| v to a complex quantity  o|| v given as: Juhl and Treumann [21] explained that modified thermal velocity effects the parallel temperature ||  9) by parts, the dispersion relation is found.Using equations ( 10) and (11) and doing some lengthy integrals the general dispersion relation becomes ... (12) Where, And   ξ Z is the plasma dispersion function, written as For real k and substituting 1 The expression for growth rate for real frequency  r in dimensionless form is found to be 16)

Results and Discussion
Growth rate for whistler mode waves is calculated by incorporating spacecraft data in equation no. 15 derived in previous section of the manuscript.Magnetic field magnitude (B o ) at radial distance 6 Rs is reported to be 80 nT with thermal energy of electrons (K B T || ) in the range of hundreds of electron volts considered 400eV.Waves are assumed to be propagating at an angle with respect to magnetic field direction in presence of parallel DC field.The angle of propagation is varied as 20˚, 25˚ and 30˚.Effect of other parameters like temperature anisotropy and number density of electrons (n o ) as 0.5x10 6 m -3 , 1x10 6 m -3 and 1.5x10 6 m -3 are also analyzed [15].Magnitude of growth rate for second peak is higher for loss cone distribution function but more for first peak in bi-Maxwellian case.Effect of parallel electric field on electrostatic ion-cyclotron instability shows that steepness of loss-cone and temperature anisotropy enhances the growth rate and reduces transverse acceleration of ions [23].Although there is no shift in wave number but difference in magnitude of growth rate in both the graphs can clearly be seen.Such shift in wave number could not be observed in terrestrial and planetary magnetospheres [22,24,25].Also the plot shows that second harmonics are not generated at lower angles of propagation and growth rate increases for increasing value of angle of propagation.Results can be compared with previous work done for parallel propagating whistler mode waves in relativistic plasma of magnetosphere of Uranus [26] to present comparative planetary study.For higher number density double peak can be seen in graph.
For n o =1x10 6 m -3 , peak values of growth rate are 0.0109 and 0.0915 at k ~=0.9 and 1. Therefore second harmonics are generated at higher number density of electrons in magnetosphere of Saturn.

Conclusion
Analyzing the results drawn from expression of growth rate and all the graphs plotted in presence of parallel DC field, it can be concluded that magnitude of growth rate of whistler mode waves in case of Loss-cone distribution function is much more than in case of bi-Maxwellian distribution function.Variation in temperature anisotropy shows the generation of second harmonics although for increasing value of || T T  , growth rate increases at same wave numbers.On the contrast, decrease in angle of propagation shows the drastic shift of growth towards lower wave number without generation of second harmonics for lesser value of θ.Studying the increase of energy density of electrons, gradual shift in increasing peaks of growth rate from lower to higher wave numbers can be observed.Also it can be concluded that second harmonics are generated only for higher number density of electrons around radial distance of 6 Rs in magnetosphere of Saturn.Thus the results are of importance in analyzing observed VLF emissions over wide spectrum of frequency range in Saturnian magnetosphere.Thus it is found that parallel DC field stabilizes the wave and temperature anisotropy, angle of propagation, number density and energy density of electrons enhances the growth rate.The analytical model developed can also be used to study various types of instabilities in planetary magnetospheres.
For parallel propagating whistler mode instability, the general dispersion relation reduces to 2

Fig. 1 :
Fig.1: Variation of Growth Rate with respect to k ~ for various values of

Fig. 2 :
Fig.2: Variation of Growth Rate with respect to k ~ for various values of

Fig. 3 :Fig. 4 :
Fig.3: Variation of Growth Rate with respect to k ~ for various values of θ at other plasma parameters being B o = 80 nT, K B T || =400eV, n o =1x10 6 m -3 and || T T  =1.5.

Figure 3
Figure 3 and 4 show variation of dimensionless growth rate versus wave number for different values of angle of propagation of whistler mode waves with respect to magnetic field direction.Using bi-Maxwellian distribution function j=0 in figure 3, maximum growth rate calculated for θ=20 o , 25 o and 30 o is 0.0031, 0.0060 and 0.0501 respectively.Using Loss cone distribution function j=1 in figure 4, maximum growth rate calculated for θ=20 o , 25 o and 30 o is 0.0178, 0.0414 and 0.0915 respectively.Peak value of growth rate for each angle appears at same wave number in both the graphs, k ~=0.2, 0.4 and 1.1 for θ=20 o , 25 o and 30 o respectively.Although there is no shift in wave number but difference in magnitude of growth rate in both the graphs can clearly be seen.Such shift in wave number could not be observed in terrestrial and planetary magnetospheres[22,24,25].Also the plot shows that second harmonics are not generated at lower angles of propagation and growth rate increases for increasing value of angle of propagation.

Fig. 5 :
Fig.5: Variation of Growth Rate with respect to k ~ for various values of K B T || at other plasma parameters being B o = 80 nT, θ =30 o , n o =1x10 6 m -3 and || T T  =1.5.

Fig. 6 :
Fig.6: Variation of Growth Rate with respect to k ~ for various values of K B T || at other plasma parameters being B o = 80nT, θ =30 o , n o =1x10 6 m -3 and || T T  =1.5.

Figure 5
Figure 5 and 6 show the variation of dimensionless growth rate with respect to wave number for different values of energy density (K B T || ) of electrons using bi-Maxwellian and Loss cone distribution function respectively.At radial distance of 6 R S , generation of second harmonics can be noted from both the graphs.In figure 5, with thermal energy 300 eV, two peak values of c ω γ is 0.0227 and 0.0347 at k ~=0.7 and 0.9 respectively.Similarly, for K B T || = 400 eV, peak values of c ω γ is 0.0199 and 0.0501 at k ~=0.9 and 1.1 resp.And for K B T || = 500 eV, two peak values of

Fig. 7 :
Fig.7: Variation of Growth Rate with respect to k ~ for various values of n o at other plasma parameters being B o = 80 nT, θ =30 o , K B T || =400eV and || T T  =1.5.

Fig. 8 :
Fig.8: Variation of Growth Rate with respect to k ~ for various values of No at other plasma parameters being B o = 80nT, θ =30 o , K B T || =400eV and || T T  =1.5.

Figure 7
Figure 7 and 8 show the variation of dimensionless growth rate with respect to wave number for different values of number density (n o ) of electrons.Using bi-Maxwellian distribution function, calculated growth rate for lower number density n o =0.5x10 6 -3 is 0.0253 at k ~=0.8.For higher number 1 resp.And for n o =1.5x10 6 m -3 ,