STUDY OF ELECTROSTATIC ION-CYCLOTRON WAVES IN MAGNETOSPHERE OF URANUS †

In this manuscript, the method of characteristics particle trajectories details used and the dispersion relation for the ionosphere of Uranus were being used to investigate electrostatic ion-cyclotron waves with parallel flow velocity shear in the presence of perpendicular inhomogeneous DC electric field and density gradient. The growth rate has been calculated using the dispersion relation. Electric fields parallel to the magnetic field transmit energy, mass, and momentum in the auroral regions of the planetary magnetosphere by accelerating charged particles to extremely high energies. The rate of heating of plasma species along and perpendicular to the magnetic field is also said to be influenced by the occurrence of ion cyclotron waves and a parallel electric field in the acceleration area.

The study of dynamics, which determines the release of free energy, has recently received a lot of attention.Its shear flow is determined by magneto hydrodynamics and plasma physics.The study of dynamics that govern the release of free energy associated with sheared flows has recently garnered a lot of attention in magnetohydrodynamics and plasma physics.In space plasma, sheared flows are ubiquitous, especially around the magnetopause, magnetospheric boundary layer, solar wind stream-stream interactions, comet tails, and the auroral ionosphere.Lemons et al. [1] used a kinetic technique to look at shearing velocity flow perpendicular to a uniform magnetic field for the electrostatic ioncyclotron (EIC) instability.Simulations of ion-cyclotron waves in a magnetoplasma with transverse inhomogeneous electric field were employed in conjunction with Maxwellian plasma.Using Voyager 2's onboard plasma wave receiver, identical electrostatic ion-cyclotron waves were observed in Uranus' magnetosphere.A minor offset in the occurrence of electrostatic waves from the equator was predicted due to the substantial tilt of Uranian magnetic moment with respect to the planet's rotational axis.The most intense electrostatic waves were discovered around the magnetic equator, at a distance of roughly 11.5RU, according to reported measurements [2,3].
Because of the coupling of the zone of positive and negative energy of ion waves, Ganguli et al. [4,5] and Nishikawa et al. [6] argued that electrostatic waves with repetition of the request for ion-cyclotron frequencies can be unpredictable.Some researchers [7] have included the effect of parallel and perpendicular electric fields by modifying velocity terms appearing in the distribution function, actually results in an extension of earlier theory and obscuring the details of particle trajectories and with their effect.Others, using particle aspect investigation, have focused on the impact of an equal electric field on particle cyclotron instability for various distribution capacities.Others have used particle aspect analysis to investigate the effect of a parallel electric field on ion-cyclotron instability for various distribution functions [8,9].Kandpal, et al. [10] investigated the Kelvin-Helmholtz instability in the magnetosphere of Saturn using an inhomogeneous DC electric field.With an inhomogeneous DC electric field, Kandpal and Pandey [11] explored higher harmonics electrostatic ion cyclotron parallel flow velocity shear instability in the magnetosphere of Saturn.The effect of different parameters on the growth rate of waves has been demonstrated using EIC instability and parallel velocity shear in the presence of an electric field perpendicular to the magnetic field [12].
The dielectric permittivity tensor of a magneto active current-driven plasma has been obtained by employing the kinetic theory based on the Vlasov equation and Lorentz transformation formulas with an emphasize on the q-nonextensive statistics for low frequency wave by Niknam [13].Niyat et al [14] has solved dispersion relation for magnetized plasmas that has non-extensive electrons drifting with respect to stationary ions, and satisfies the other conditions for the excitation of electrostatic ion cyclotron waves using the standard linear Vlasov theory and q-distributions.The Electrostatic Ion Cyclotron (EIC) instability that includes the effect of wave-particle interaction has been studied owing to the free energy source through the flowing velocity of the inter-penetrating plasmas by Bashir et al [15].The electrostatic waves in magnetized plasmas have been derived in the context of the nonextensive q-distribution of Tsallis statistics by Sharifi [16].
An ion beam propagating through collisional magnetized plasma containing electrons and two positive ion components has been discussed electrostatic ion cyclotron (EIC) instability via Cerenkov interaction [17].Liu et al [18] has explained the dispersive Alfvén wave in a low β plasma with anisotropic superthermal particles modeled by a binonextensive distribution is derived from a kinetic theory.The effect of anisotropic temperature on inertial Alfvén wave is so small that it is negligible.The numerical results reveal that the presence of superthermal electrons in the small wavenumber limit will lead the damping rate of the kinetic Alfven wave (KAW) bigger than the one with Maxwellian distribution.
The experimental studies on electrostatic ion-cyclotron waves (EICWs) and their instability were often performed in a narrow plasma column in Q-machines, in which the wave was conventionally generated via the instability driven by an electron current parallel to a magnetic field.The propagation characteristics of wave patterns have seldom been studied, especially the different characteristics in pulse and continuous wave patterns [19].Accumulation of carbon dioxide in the Earth's atmosphere leads to an increase in the greenhouse effect and, as a consequence, to significant climate change.Thus, the demand to develop effective technologies of carbon dioxide conversion grows year to year.Additional reason for research in this direction is the intention of Mars exploration, since 96% of the Martian atmosphere is just carbon dioxide, which can be a source of oxygen, rocket fuel, and raw materials for further chemical utilization.The enhancement of negative ion production in a volume Penning based source could be performed by the application of metal hydride cathode.Hydrogen isotopes are stored there in a chemically bound atomic state and desorbed from the metal hydride under the discharge current impact [20,21].
The dispersion equation has been derived on an approximation based for the current from the exact solutions of the characteristic cylindrical geometry form of the Vlasov plasma equation in a uniform magnetized plasma cylinder surrounded by a larger metal boundary outside a vacuum gap, which thus differs from that in unbounded plasmas by Kono et al [22].
Bashir et al [23] has studied the Bernstein mode instability driven by a perpendicular momentum ring distribution function is insensitive to the parallel distribution.However, in the relativistic treatment, owing to the inexorable coupling between the parallel and perpendicular momenta through the Lorentz factor, the parallel momentum distribution may affect the instability.Bashir and Vranjes [24] has studied the unstable kinetic drift wave in an electronion plasma can very effectively be suppressed by adding an extra flowing ion (or plasma) population.Theoretical study of the effects of positron density on the electrostatic ion cyclotron instability in an electron-positron-ion plasma using the kinetic theory approach by assuming that positrons and electrons can drift parallel to the magnetic field either in the same or the opposite directions has been studied by Khorashadizadeh et al [25].
The evolution of Fila mentation instability in a weakly ionized current-carrying plasma with nonextensive distribution has been studied in the diffusion frequency region, taking into account the effects of electronneutral collisions Using the kinetic theory, Lorentz transformation formulas by Khorashadizadeh [26].The dispersion relation for parallel propagating waves in the ion-cyclotron branch has investigated numerically by considering that the velocity distribution of the ion population is a function of type product-bi-kappa.We investigate the effects of the nonthermal features and of the anisotropy associated with this type of distribution on the ion-cyclotron instability, as well as the influence of different forms of the electron distribution, by considering Maxwellian distributions, bi-kappa distributions, and product-bi-kappa distributions [27].
Incorporating the details of particle trajectories in the presence of non-uniform electric field and using them for bi-Maxwellian hot plasma to the magnetic field with velocity shear flow parallel to density gradient perpendicular to magnetic field, a conceptual study of ion-cyclotron instability has been performed for the magnetosphere of Uranus.

DISPERSION RELATION AND GROWTH RATE
Small perturbations in B1, f s1 and E 1 linearize the Vlasov-Maxwell equations for homogeneous plasma, yielding the dispersion relation.The harmonic dependency of these perturbed quantities is assumed to be as.The following are the linearized first order Vlasov equations: The term Force can be calculated as Where's' stands for species.The perturbed distribution function f s1 , which is derived from eq. ( 1), is calculated using the typical solutions approach.

   
The phase space coordinate system has undergone a transformation.From (r,v,t) to  ,v ,t −  ′ where  −  ′ =  The particle trajectories for the inhomogeneous external DC electric field and the homogeneous magnetic field, as specified by [28], are as follows: Where After replacing the unperturbed trajectories in equation ( 4) and followed by simplified algebraic calculations following technique out lined in [11] the perturbed distribution function based on time integration is given as: where Δ is the series variation of inhomogeneous electric field and Δ is the time derivative of Δ  The unperturbed bi-Maxwellian distribution function can be written by Huba (29): Where  is being constant of motion and b K is the Maxwell Boltzmann constant Now making it easy by replacing m=n, g=p and applying the standard definition of conductivity and current density, the dielectric tensor can be calculated as: Where, Now we consider electrostatic ion-cyclotron instability ) Here N represents the index of refraction.The estimating method can be used to obtain the final electrostatic dispersion relation of Huba [29] and combining equation ( 8), ( 9), ( 10) and ( 11)     Where   represents plasma dispersion function Now the above-mentioned dispersion relation comes to a final form to that of Huba [29] by the removal of inhomogeneous DC electric field is removed from the equation, and  =  || , and based on [29] assumptions for p = 0 and s = i, e.The assumption is used to estimate the dispersion relation for electrons and ion approximations.  << 1 and there is no such approximation for ions.Thus equation ( 12) is simplified as: we obtain a quadratic dispersion equation as: where The solution of equation ( 16) is From this expression growth rate has been calculated by computer analysis when . This criteria gives a condition for the growth of wave when  and removal of non-uniform DC electric field for Maxwellian plasma An expression close to that of Huba [29] is obtained.
PLASMA PARAMETERS From Kurth etal [2] and Pandey [30] the following values of plasma properties for Uranus' magnetosphere, Growth rate (  ⁄ ) variations with   were determined using equation ( 18

RESULTS AND DISCUSSION
Figure 1 shows the variation in growth rates in respect to   for assumed values of shear scale length (A i ) have been shown.At harmonics n = 1 and A i = 0.05, growth rate is 0.1645.Growth rate increases from 0.1783 to 0.1912 for A i = 0.055 and 0.06 respectively.The maxima of growth rate occurs at   = 1.8 in every case.The velocity shear ioncyclotron like waves are excited for   > 1 at shorter wavelength regions [5,31], as shown in this chapter.Our results are similar to numerical simulation using M.H.D. equation valid at large ion larmor radius [32].When the gyro radius is similar to the velocity shear scale length, the instability appears to have stabilized.Figure 2 represents the growth rate variation with respect to   for diverse values of density scale length (  ).The increases in growth rate with increasing value of density scale length, as for   = 0.02 and 0.08, growth rate increases from 0.1645 to 0.1758 with no change in   = 1.8.Fujimoto and Terasawa [33] According to simulation data, the mixing efficiency of nonuniform background plasma is a function of density ratio, plasma, shear layer width, and growth mode wavelength.Their conclusion is supported by our findings.Figure 3 indicates the change in growth rate for the ion-cyclotron instability in Uranus' magnetosphere with harmonic n=1.For various values of propagation angle, the growth rate is confirmed with reference to the magnetic field angle ().The magnetic dipole axis, which is tilted at sixty degrees like Uranus' spin axis, condenses, resulting in a more bending magnetic field and magnetic field intersection [34,35].In the graph, for  =88 o ,   ⁄ =0.0787, for EEJP. 1 (2022) Rama S. Pandey, Mukesh Kumar  =88.5 o ,   ⁄ =0.1156 and for  =89 o ,   ⁄ =0.1645.Therefore, it is concluded that even the slight lowering in angle of propagation decreases the growth rate significantly for Uranian magnetosphere.In Figure 4 variation of growth rate with   for various values of electric field have been showed.For E o = 1mV/m, maximum growth rate,   ⁄ =0.1645 appears at   =1.8 and for E o = 0.1mV/m, maximum growth rate,   ⁄ =0.1753 appears again at   =1.8.At n=1 harmonic of ion cyclotron gyro frequency, growth rate reduces with higher value of electric field.This shows that increasing electric field magnitude has a stabilizing effect on electrostatic ion-cyclotron waves.The behavior is in accordance with Misra and Tiwari [8] this illustrates that the parallel electric field has the effect of stabilizing wave expansion and increasing ion transverse acceleration.Figure 5 shows the effect of varying   ⁄ on growth rate of ioncyclotron waves for 11 R U , other fixed parameters as mentioned in figure caption.With   ⁄ = 2, 4 and 6,   ⁄ = 0.1496, 0.1645 and 0.1705 respectively.It is seen that with increasing the value of  ⁄ , growth rate also increases.Figure 6 has been made to depict the diversity in growth rates with respect to   for differing value of and other parameters fixed.The effects of inhomogeneity non the electric field with a constant DC field magnitude are depicted in this diagram.The maximum growth rate increases from 0.1645 to 0.1722 as the value of increases from 0.5 to 0.9.Velocity shear ion-cyclotron like waves are excited in the regions where E x B drift is localized.The effect of non-uniform electric field in the perpendicular direction makes it to be a function of position as shown in the expression and thus cyclotron frequency is renormalized.In Figure 7 ):E o = 1mV/m, B o =11nT A i = 0.05,   || ⁄ =1.25,  =0.5,   ⁄ =4,   = 0.02 and  = 89 o at 11 R U .

Figure 1 .
Figure 1.Differences in the rates of growth in relation to   for differing values of A i at   || ⁄ =1.25,   ⁄ =4,  =0.5,   =0.02,  =89 o and additional fixed plasma characteristics.

Figure 2 .
Figure 2. Differences in the rates of growth in relation to   for varying values of   at   || ⁄ =1.25,   ⁄ =4,  =0.5,A i =0.05,  =89 o and additional fixed plasma characteristics.

Figure 3 .
Figure 3. Differences in the rates of growth in relation to   for varying values of  at   || ⁄ =1.25,   ⁄ =4,  =0.5,A i =0.05,   = 0.02 and additional fixed plasma characteristics.For various values of propagation angle, the growth rate is confirmed with reference to the magnetic field angle ().The magnetic dipole axis, which is tilted at sixty degrees like Uranus' spin axis, condenses, resulting in a more bending magnetic field and magnetic field intersection[34,35].In the graph, for  =88 o ,   ⁄ =0.0787, for

Figure 5 .
Figure 5. Differences in the rates of growth in relation to   for varying values of   ⁄ at   || ⁄ =1.25,   = 0.02,  =0.5,A i =0.05,  =89 o and additional fixed plasma characteristics.
the variation of growth rate with relation to   for varying values of ratio of perpendicular to parallel temperature (  || ⁄ ) are shown.Since   || ⁄ -1 = A T , the graph actually shows growth rate for different anisotropies (A T ).For   || ⁄ = 1.25,   ⁄ = 0.1645, for   || ⁄ = 1.5,   ⁄ = 0.1685 and for   || ⁄ =1.75,   ⁄ = 0.1708.Thus, temperature anisotropy becomes an additional source along with velocity shear, density scale length and angle of propagation, for exciting shorter wavelengths.The results can be compared to Misra and Tiwari [8] for terrestrial magnetosphere.

Figure 6 .
Figure 6.Differences in the rates of growth in relation to   for various values of  at   || ⁄ =1.25,   ⁄ =4,   = 0.02, A i =0.05,  =89 o and additional fixed plasma characteristics.

Figure 7 .
Figure 7. Differences in the rates of growth in relation to   for varying values of   || ⁄ at   = 0.05,   ⁄ =4,  =0.5,A i =0.05,  =89 o and additional fixed plasma characteristics.