Analysis of Cold Plasma Injection on Whistler Mode Instability with AC Field in Magnetosphere of Saturn

Analysis of Cold Plasma Injection on Whistler Mode Instability with AC Field in Magnetosphere of Saturn

A.K. Choubey1

1Department of Physics,

V.K.S University, Ara Bihar

Mukesh Kumar2

2Department of Physics,

Nalanda College Biharsharif Nalanda MU Bodh Gaya Bihar

Jyoti Kumari3 and R.S. Pandey3*

3Department of Physics,

Amity Institute of Applied Sciences, Amity University, Sector-125 Noida

Abstract- Whistlers generated by lightning have been observed in the magnetosphere of Saturn by the Cassini Radio and Plasma Wave Science Investigation (RPWS). Two whistlers were observed as the spacecraft flew over the rings on July 1, 2004, and the third was observed on October 28, 2004, during the inbound pass of orbit at a radial distance of 6.18 Rs (Saturn radii). The effect of cold plasma injection on whistler mode instability has been studied for ring distribution function with perpendicular AC electric field. The dispersion relation of the parallel propagating electromagnetic whistler mode wave has been applied to the magnetosphere of Saturn. An expression for the growth rate of a system with added cold plasma injection has been calculated. The effects of electron density, number density, temperature anisotropy, A.C frequency and relativistic factor on the growth rate of whistler mode emission were investigated. The observation is in very good agreement with the results. Analysis shows that an increase in temperature anisotropy, A.C. frequency, and energy density increases the rate of increase in whistler waves with significant changes in wavenumbers. It has been shown that whistler mode waves have increased due to the loss of vertical kinetic energy of the ring electrons. In Saturn's magnetosphere, calculations were made at a radial distance of

1. Rs. This result is important for the analysis of VLF emissions observed over a broad spectrum of frequencies in the Saturn magnetosphere.

   Keywords-Whistler mode instability; Ring distribution function; Dispersion relation; Cold injection

   1. INTRODUCTION

      Whistler is an audio-frequency electromagnetic wave generated by lightning, which propagates along the magnetic field lines at a frequency lower than the electron cyclotron frequency and the electron plasma frequency. Whistling was first discovered in 1918 by [1] using a basic vacuum tube audio amplifier. Later, [2] found that the arrival time is inversely proportional to the square root of the frequency. [3] was the first person to correctly explain the origin of whistler blowing. He showed that the whistler travels along the magnetic field lines from one hemisphere to the other, and this plasma propagation mode is now called the whistler mode. Whistlers have also been found on Jupiter [4-7].

      There are two common types of Whistler-mode emissions in the planets magnetosphere: hiss and chorus. The frequency of the Whistler mode emission is less than the lower of the plasma frequency fp or the cyclotron frequency fc. At Saturn, the hiss usually has an uncharacteristic spectrum, and its frequency is usually lower than the frequency of the whistler chorus (f <3 kHz). It is known that the latter has a source region near the magnetic equator, usually a narrow-band structure, and shows many fine structures of drifting frequency tones or at high resolution, which is larger than the typical spectral density. For a long time, people have been well aware of the developed linear theory of chorus generation [8-10]. The fine structure of the chorus is the result of electronic nonlinearity and resonance wave capture, as described by [11-16].

      Diffusion pitch angle scattering and electron acceleration occur on Earth due to Whistler emission [17-19], and may also occur at Saturn [20, 21]. Recently, it has been demonstrated on Saturn that the same general process of pitch angle scattering by Z mode is important [22]. On the earth, whistlers near the electromagnetic equator will have cyclotron resonance and wave growth [23, 24]. Earlier theories also believed that whistlers would be generated in the plasma layer [25-27].

      The long duration of the Cassini mission has given us a deeper understanding of Saturn's plasma conditions and wave characteristics. The ratio of plasma frequency to gyro frequency, fpe/fce, is very important for electron acceleration generated by wave-particle interaction, and a value of fpe/fce

      <4 will result in good electron acceleration [28]. The region of interest here is 6.18 Rs. It is reasonable to assume that the magnetic field is a dipole magnetic field near Saturn, which is very small [29].

      Voyager 1s encounter with Saturn in November 1980 provided the first opportunity to study plasma wave interactions in Saturns magnetosphere. Inspired by various pioneers, in this work, we try to study the Whistler mode

      instability in the inner magnetosphere of Saturn at a radial

      n /n

      2

      2

      distance of 6.18 Rs. The study is based on the effect of cold

      f(

      , ) e exp o

      2

      2

      2

      2

      ||

      2

      2

      beam injection at a specific radial distance. Further sections

      ||

      3/2

      o|| o o

      o||

      describe the mathematical formulation, followed by graphical descriptions and conclusions.

      2 2 Po

      (6)

      o o

      o o

      B exp P /P

      erfc Po /Po||

   2. MATHEMATICAL FORMULATION Spatially homogeneous anisotropic, collisionless plasma subjected to external magnetic field B=Boêz and an electric field Eox= (Eosin t) êx has been considered to get dispersion relation. Negligible inhomogeneities are

      Where,

      2k

      1/2

      2k

      Po

      1/2

      (7)

      assumed in the zone of interaction. In order to obtain a

      general dispersion relation for the case relativistic whistler waves in the presence of A.C electric field, linearized

      b ||

      m

      m

      o|

      e

      and b

      o

      o

      m

      m

      e

      Vlasov-Maxwell equations are attained after neglecting higher order terms and separating the equilibrium and nonequilibrium parts by small perturbations in magnetic field, electric field and distribution function. Following the technique of [30], the dielectric tensor can be written as:

      4e2 d3 J || S* ||

      Are the associated parallel and perpendicular electron

      thermal velocities.

      In equation (6), ne/n is the ratio of trapped energetic electrons to total electron density (ne/n = no). Equation (7) gives the expression for complimentary error function. The P|| and P are parallel and perpendicular thermal velocities

      , 1 s

      g 3 s

      in terms of momentum with respect to magnetic field. Po is

      ij m 2 2

      the drift speed in terms of momentum.

      || cs

      || cs

      s e

      k|| n g p

      m e

      Substituting d3 P 2

      P dP

      dP||

      0

      (1)

      For the propagation and instability of whistler mode waves

      and using expression (7) in equation (6) and after solving the integrations, we get the dispersion relation as:

      k 2 c2

      42

      n /n

      m

      k

      k

      e

      e

      with k 0 [31], the branch of general dispersion relation 1 s e X

      e Z X 1 Z

      2

      2

      p/>

      (1) reduces to:

      11 ± 12 = 2 (2)

      2 m

      2

      1

      || o||

      (8)

      Where N2 2c2 2 is refractive index. Therefore,

      Where

      dispersion relation for n=1 may be written as:

      2

      m v 1

      1 o o

      e x o

      1 2

      2 2 2

      o o

      c v2

      o o

      2 3

      2

      N 2 1

      4es

      d P P N N 1

      2

      2

      m 2 2 2 1 P

      || ||

      2 3

      2

      e

      n g c p

      o 1 Po

      erf P 3 Po 3 Po

      m

      2 1 2 P3

      P

      P2 2 P

      e

      o|| o

      o

      o o

      (3) m v

      1

      2 '

      e x

      o erf 2

      Where

      2

      2

      3 o

      c v2 o

      o

      o

      2

      m

      2 f

      P P

      Where,

      t 2

      N e

      o

      || ||

      • x 1 e

        1 P P

        m m

        2

        Z

        dt , is the plasma dispersion function with

        e e c 2

        t

        2

        m

        e g c c pv

        (4) k||o||

        Applying condition k2c2 for whistler waves

        N m

        fo P

      • x

      1

      2

      4e 2 n /n

      2

      e

      2

      e

      2 e || P

      m

      2

      ps

      || e c 2

      Bo me

      2

      (5)

      r i

      The equation (8) reduces to

      The distribution function for trapped electron is taken as Maxwellian ring momentum distribution function [31]

      (, )

      22

      = 2

      1 1 1 1 1

      + [ ( ) { } { }

      Saturn magnetosphere using electric and magnetic field antennas of RPWS ranging electric field from lower (1 Hz) to higher (16 MHz) and magnetic fields (1 Hz to 12 kHz). Saturn exhibited different temperature profile at radial distances.

      23

      2 22

      Saturns inner magnetosphere (5-12 Rs) is populated with

      ()

      + [

      + ] {(2)}]

      two different components i.e. hot (>100ev) and cold

      1 2

      2

      2

      ±

      (9)

      component (<100ev) [35-37], radial diffusion of these population in inner magnetosphere is very much expected from the interchanged instability influenced by centrifugal forces. Study carryout the variation in dimensionless wave

      where

      = (4

      2)12

      and =

      growth of parallel propagating whistler mode waves for the particles in ring distribution. Comprehensive analysis of

      (42)12 are the plasma frequencies of the warm

      background and cold injected plasma respectively.

      the result has been performed by taking previous published data set based on Cassini and Voyager 1 spacecraft.

      k

      k

      Introducing the dimensionless parameters as ~

      k ||Po||

      cs

      Measured magnitude of ambient magnetic field Bo=73 nT. The thermal energy (KBT||e) of electrons is 300 eV. Number

      The growth rate in terms of the dimensionless parameters

      k, 1 2 1

      k, 1 2 1

      2

      2

      ~ , K , X and X is obtained as

      density of electrons is considered to be no =5×107 m-3 and

      magnitude of AC electric field Eo= 0.01 mV/m has been considered. The range of electron energy used was 100-500 eV.

      (2 3

      ) (1

      + )3

      As described earlier, appropriate plasma and field

      1

      1 3 + 4 3 4

      1 3 + 4 2

      parameters corresponding to L=6.18, and expression of the growth rate given in section II has been used to calculate

      =

      [ (

      1 + 4 +

      2

      ) ]

      2(1 + 4) 2(1 3 + 4)2

      2 3 )

      growth rate with the wave number. The calculated growth rate as a function of wavenumber shows a main peak at normalized frequency <1 and is always accompanied by trailing growth at lower wave numbers (Fig. 1-5). Furthermore, it can be seen that the presence of relativistic

      (1 3 + 4) (1

      1 3 + 4

      electron tail results in a significant change in the

      ( 1)2

      + ( )( 3 )2

      normalized growth rate at the lower peak. In Fig. 1., it is shown that an increase in the AC frequency will increase

      1 + 4 1 3

      The real part of eq. (9) is

      (10)

      the peak growth rate of whistler mode waves with peak shifting towards higher wavenumber side. In fig. 2. and fig. 3., growth rate has been plotted against wave number to see

      3

      =

      the effect of cold to hot electron number density ratio and energy density. From the graphs, it can be concluded that energy density supports growth rate significantly whereas

      =

      21

      2

      +

      [2(1 + 4)

      1

      ]

      number density does not seems to enhance the peak to much extent. Next fig. 4. shows that the increase in temperature anisotropy, significantly increases the peak

      1 2(1 + 4)

      1

      4 k

      n

      /n

      (11)

      value of growth rate. The graph indicates that, although temperature anisotropy enhances growth of whistlers but it does not show significant changes of wavenumber spectra.

      Where 2 =

      21

      and 1

      o o b || e

      m 2 2

      Similarly, fig. 5. shows that the increase in the relativistic factor results in a significant decrease in the peak value of

      4

      4

      p

      c

      = 1 +

      e o

      (12)

      (1 + ) = 1 +

      whistler mode waves. These results are consistent with the literature. In this study, the ring distribution function used to illustrate the existence of relativistic effects has been

      used by other workers to study the effects of various

      2 4 3 3 4

      instabilities.

   3. RESULT AND DISCUSSION

      Analysis is attributed to observations and plasma conditions and limited to inner middle boundary (< 7 Rs). Here are discussing few relevant monitoring data. [33] first reported the global view of Saturn plasma electron environment by analysing observation from voyager 1 and Voyager 2. [34] performed observational study extensively about impulsive radio wave and plasma wave emission in

      Fig. 1. Growth rate variation with wave vector at numerous values of

      A.C. frequency keeping other plasma parameters constant such as

      = 0.01 /, = 73 , = 10, = 1.5, = 0.6 etc.

      Fig. 2. Growth rate variation with wave vector at numerous values of cold to hot electron density ration keeping other plasma parameters constant such as

      = 0.01 /, = 73 , = 4 , = 1.5, = 0.6 etc.

      Fig. 3. Growth rate variation with wave vector at numerous values of energy density keeping other plasma parameters constant such as

      = 0.01 /, = 73 , = 10, = 1.5, = 0.6, = 4 etc.

      Fig. 4. Growth rate variation with wave vector at numerous values of temperature anisotropy keeping other plasma parameters constant such as

      = 0.01 /, = 73 , = 10, = 4 , = 0.6 etc.

      Fig. 5. Growth rate variation with wave vector at numerous values of relativistic factor keeping other lasma parameters constant such as

      = 0.01 /, = 73 , = 10, = 4 , = 1.5

      etc.

   4. CONCLUSION

In this work, whistler mode waves that propagates parallel in the presence of an electric field with a ring distribution function in the magnetosphere of Saturn is studied. The method is formed by a kinetic method to ensure efficiency, consistency etc. at a radial distance of

1. Rs. The expressions of dispersion relation, growth rate and real frequency are derived to analyze the effects of plasma temperature anisotropy, relativistic factor, number density, AC frequency and energy density. Parametric analysis inferred that growth rate of whistler waves increases with increasing value of temperature anisotropy and AC frequency, but the growth rate does not increase with number density ratio and relativistic factor. We find that a temperature anisotropy (T/T ~ 1.5) can account for linear temporal growth rate of whistler mode waves, which provides a majority of the observed frequency-integrated power. Higher frequency chorus emissions differ from lower frequency whistler mode emissions and are sometimes associated with simultaneously observed low frequency electromagnetic ion cyclotron waves.

   REFERENCES

   1. Barkhausen, H. (1919), Zwei mit Hilfe der neuen Verstarker entdeckte Erscheinungen, Phys. Z., 20, 401403.

   2. Eckersley, T. L. (1935), Musical atmospherics, Nature, 135, 104 105.

   3. Storey, L. R. O. (1953), An investigation of whistling atmospherics, Philos. Trans. R. Soc. London, Ser. A, 246, 113 141.

   4. Scarf, F. L., D. A. Gurnett, and W. S. Kurth (1979), Jupiter plasma wave observations: An initial Voyager 1 overview, Science, 204, 991995.

   5. Gurnett, D. A., R. R. Shaw, R. R. Anderson, and W. S. Kurth (1979), Whistlers observed by Voyager 1: Detection of lightning on Jupiter, Geophys. Res. Lett., 6, 511514.

   6. Kurth, W. S., B. D. Strayer, D. A. Gurnett, and F. L. Scarf (1985), A summary of whistlers observed by Voyager 1 at Jupiter, Icarus, 61, 497 507.

   7. Gurnett, D. A., W. S. Kurth, I. H. Cairns, and L. J. Granroth (1990), Whistlers in Neptunes magnetosphere: Evidence of atmospheric lightning, J. Geophys. Res., 95, 20,96720,976.

   8. Kennel, C. F., & Petschek, H. E. (1966). Limit on stably trapped particle fluxes. Journal of Geophysical Research, 71(1), 128. https://doi.org/10.1029/JZ071i001p00001

   9. Kennel, C. F., & Wong, H. V. (1967). Resonance particle instabilities in a uniform magnetic field. Journal of Plasma Physics, 1(1),7580.

   10. Maggs, J. E. (1976). Coherent generation of VLF hiss. Journal of Geophysical Research, 81(10), 17071724. https://doi.org/10.1029/ JA081i010p01707

   11. Nunn, D., Omura, Y., Matsumoto, H., Nagano, I., & Yagitani, S. (1997). The numerical simulation of VLF chorus and discrete emissions observed on the Geotail satellite using a Vlasov code. Journal of Geophysical Research, 102(A12), 2708327097. https://doi.org/10.1029/ 97JA02518

   12. Trakhtengerts, V. Y. (1999). A generation mechanism for chorus emission. Annales de Geophysique, 17(1), 95100. https://doi.org/10.1007/ s0058599900954

   13. Omura, Y., & Summers, D. (2004). Computer simulations relativistic whistlermode waveparticle interactions in the magnetosphere. Physics of Plasmas, 11(7), 35303534. https://doi.org/10.1063/1.1757457

   14. Omura, Y., & Summers, D. (2006). Dynamics of highenergy electrons interacting with whistler mode chorus emissions in the magneto- sphere. Journal of Geophysical Research, 111, A09222. https://doi.org/10.1029/2006JA011600

   15. Katoh, Y., & Omura, Y. (2004). Acceleration of relativistic electrons due to resonant scattering by whistler mode waves generated by temperature anisotropy in the inner magnetosphere. Journal of Geophysical Research, 109, A12214. https://doi.org/10.1029/2004JA010654

   16. Katoh, Y., & Omura, Y. (2006). A study of generation mechanism of VLF triggered emission by selfconsistent particle code. Journal of Geophysical Research, 111, A12207. https://doi.org/10.1029/2006JA011704

   17. Baker, D. N., Kanekal, S. G., Hoxie, V. C., Henderson, M. G., Li, X., Spence, H. E., et al. (2013). A longlived relativistic electron storage ring embedded in Earth's outer Van Allen Belt. Science, 340(6129), 186190. https://doi.org/10.1126/science.1233518

   18. Reeves, G. D., Spence, H. E., Henderson, M. G., Morley, S. K., Friedel, R. H. W., Funsten, H. O., et al. (2013). Electron acceleration in the heart of the Van Allen radiation belts. Science, 341(6149), 991994.

   19. Thorne, R. M., Li, W., Ni, B., Ma, Q., Bortnik, J., Baker, D. N., et al. (2013). Evolution and slow decay of an unusual narrow ring of relativistic electrons near L~3.2 following the September 2012 magnetic storm. Geophysical Research Letters, 40, 35073511. https://doi.org/10.1002/grl.50627

   20. Gu, X., Thorne, R. M., Ni, B., & Ye, S.Y. (2013). Resonant diffusion of energetic electrons by narrowband Z mode waves in Saturn's inner magnetosphere. Geophysical Research Letters, 40, 255261. https://doi.org/10.1029/2012GL054330

   21. Shprits, Y. Y., Menietti, J. D., Gu, X., Kim, K. C., & Horne, R. B. (2012). Gyroresonant interactions between the radiation belt electrons and whistler mode chorus waves in the radiation environments of Earth, Jupiter, and Saturn: A comparative study. Journal of Geophysical Research, 117, A11216. https://doi.org/10.1029/2012JA018031

   22. Woodfield, E. E., Horne, R. B., Glauert, S. A., Menietti, J. D., Shprits, Y. Y., & Kurth, W. S. (2018). Formation of electron radiation belts at Saturn by Zmode wave acceleration. Nature Communications, 9(1), 5062. https://doi.org/10.1038/s41467018 075494.

   23. Hwang, J. A., Lee, D.Y., Lyons, L. R., Smith, A. J., Zou, S., Min, K. W., et al. (2007). Statistical significance of association between whistler mode chorus enhancements and enhanced convection periods during highspeed streams. Journal of Geophysical Research, 112, A09213. https://doi.org/10.1029/2007JA012388

   24. Li, W., Thorne, R. M., Meredith, N. P., Horne, R. B., Bortnik, J., Shprits, Y. Y., & Ni, B. (2008). Evaluation of whistler mode chorus amplification during an injection event observed CRRES. Journal of Geophysical Research, 113, A09210. https://doi.org/10.1029/2008JA013129

   25. Thorne, R. M., Smith, E. J., Burton, R. K., & Holzer, R. E. (1973). Plasmaspheric hiss. Journal of Geophysical Research, 78(10), 15811596. https://doi.org/10.1029/JA078i010p01581

   26. Thorne, R., Church, S., & Gorney, D. (1979). On the origin of plasmaspheric hiss: The importance of wave propagation and the plasma- pause. Journal of Geophysical Research, 84(A9), 5241 5247. https://doi.org/10.1029/JA084iA09p05241

   27. Horne, R. B., and Thorne, R. M. (2003). Relativistic electron acceleration and precipitation during resonant interactions with whistler-mode chorus. Geophys. Res. Lett.30,1527

   28. Kumari, J., Pandey, R.S. Analytical study of Whistler mode waves for relativistic plasma with AC electric field in inner magnetosphere of Saturn. J Astrophys Astron 40, 14 (2019).

   29. Kaur, R., and Pandey, R. S., 2017. Study of whistler mode waves for ring distribution function in Saturns magnetosphere. Adv. Space Res., 59, 2434-2441.

   30. Sittler, E. C., Jr., Ogilvie, K. W., and Scudder, J. D., 1983. Survey of low energy plasma electrons in Saturns magnetosphere: Voyager 1 and 2, J. Geophys. Res., 88, 8847 8870.

   31. Gurnett, D. A., et al., 2005. Radio and plasma wave observations at Saturn from Cassini's approach and first orbit, Science, 307, 1255.

   32. Mauk, B. H., et al., 2005. Energetic particle injections in Saturns magnetosphere, Geophys. Res. Lett., 32, L14S05, doi:10.1029/2005GL022485.

   33. Rymer,A.M., Mauk, B.H., Hill, T.W., Paranicas, C., André, N., Sittler Jr., E.C., Mitchell, D.G., Smith, H.T., Johnson, R.E., Coates, A.J., Young, D.T., Bolton, S.J., Thomsen, M. F., Dougherty, M.K., 2007. Electron sources in Saturns magnetosphere. J. Geophys. Res. 112, A02201., doi: 10.1029/2006JA012017.

   34. Young, D. T., et al., 2005. Composition and dynamics of plasma in Saturns magnetosphere, Science, 307 (5713), 1262.