Dust Acoustic Shock Waves in a Six Component Charge Varying Cometary Plasma

Neethu TW1, Shilpa S2, Philip NS3 and VenugopalC4*

1Department of Physics, CMS College, Kottayam - 686001, Kerala, India.

2International School of Photonics, Cochin University of Science and Technology, Kochi-682022, Kerala, India.

3Artificial Intelligence Research and Intelligent Systems, Thelliyoor – 689544, Kerala, India.

4School of Pure and Applied Physics, Mahatma Gandhi University, Kottayam, Kerala, India.

Corresponding author:Venugopal Chandu, School of Pure and Applied Physics, Mahatma Gandhi University, Kottayam, Kerala, India. E-mail: cvgmgphys@yahoo.co.in

Citation: Venugopal C, Neethu TW, et al. (2020) Dust Acoustic Shock Waves in a Six Component Charge Varying Cometary Plasma. J Phy Adv App 1(1): 1-12.

Received Date: June 01, 2020; Accepted Date: June 07, 2020; Published Date: June 09, 2020

Abstract

Dust acoustic shock waves have been studied in six component cometary plasma by deriving the Korteweg-deVries-Burgers (KdVB) equation. The constituents of the plasma include two components of electrons described by kappa distributions with different temperatures and spectral indices, lighter (hydrogen) ions and a pair of oppositely charged, heavier ions (positively and negatively charged oxygen ions), all of them described by Maxwellian distributions with different temperatures. Charge varying negatively charged dust grains is the sixth component.

Shock waves solutions of the KdVB equation, studied for typical parameters of comet Halley, show that the shock amplitudes are consistently larger when charge fluctuations on the dust grains are taken into consideration. The superthermal, second component of electrons affects the phase velocity and the shock velocity as well as it’s width. The amplitude of the shock wave is also affected by the densities and temperatures of all the ions: it increases with increasing positively charged oxygen ion densities and decreases with increasing temperatures of these ions. The amplitude, however, decreases with increasing densities and increases with increasing temperatures of the other two types of ions.

Keywords: Cometary plasma; Shock waves; Dusty plasma; Charge variation; 6 components;

Introduction

The presence of charged dust grains introduces new features to the non-linear structures in space environments such as planetary rings, planetary magnetospheres, comet environments, interstellar media and earth-space environments [1-4]. The non-linearities in dusty plasmas give rise to localization of waves, generating different types of fascinating coherent structures, namely, solitary structures [5], double layers [6], shock waves [7], vortices [8,9] etc. In dusty plasmas, the interaction of dust grains with energetic particles such as electrons and ions lead to the charging of dust grains. The charging currents to the dust grains carried by the plasma particles can be calculated using the Orbital Motion Limited (OML) approach [10,11]. Extensive studies on the effect of dust charge variation on solitary [12-15] and shock structures [16-18] have been carried out.

There is a strong current interest to understand the relevance of the dust charge variation on the formation of dust acoustic shock waves [19-21]. Popel et al [22] discussed the possibility of the observation of shock waves related to the dust charging process in the presence of electromagnetic radiation in active rocket experiments which involved the release of some gaseous substances in the near-earth space. Further, Duha et al [23] investigated the dust-ion-acoustic solitary and shock waves associated with the dynamics of negative ions, Maxwellian positive ions, trapped electrons and charge fluctuating stationary dust by employing the reductive perturbation method. Several authors [24-26] have studied Dust-Acoustic (DA) shock waves in a charge varying nonextensive dusty plasma. Hadjaz and Tribeche [27] obtained a dusty plasma model that supported solitary as well as shock waves, for which the main properties such as phase velocity, amplitude and width were drastically influenced by trapping, nonthermality and charge variation. El-Shewy et al [28] highlighted the effects of dust viscosity and electron-ion nonthermality fraction on DA shock waves in inhomogeneous dusty plasmas with nonadiabatic dust charge fluctuation. Later, Wang et al [29] studied the effects of dust size distribution and dust charge fluctuation of dust grains on small but finite amplitude nonlinear dust ion-acoustic shock waves, in an unmagnetized multi-ion dusty plasma.

In a large number of space environments, velocity distributions have commonly been reported that are Maxwellian-like in the low-energy range, but possesses a power-law tail at superthermal particle energies [30-34]. Such distributions are modeled by a generalized form of Lorentzian or kappa distribution first proposed by Vasyliunas [35]. The Maxwellian distribution is a special case of the kappa function; in the limit of the spectral index κ → ∞ . In a recent study, Broiles et al [36] evaluated the observations made by Rosetta spacecraft of the superthermal electron environments near comet 67P/Churyumov-Gerosimenko and fitted the observed electron velocity space densities with a combination of two threedimensional kappa distributions. Hence there has been a great deal of interest in studies related to dust charge fluctuation on shock waves in a suprathermal dusty plasma [37-39]. Recently, Ferdousi et al [40] analysed the properties of low frequency dust-acoustic shock waves in a plasma medium in the presence of superthermal kappa distributed electrons.

A cometary plasma has been observed to be a genuine multi-ion plasma as it is composed of solar wind protons and electrons;the dissociation of water molecules contributes positively charged hydrogen (H+) and oxygen (O+) and photo-electrons [41]. In addition to the existence of positive ions, negative ions have also been observed in cometary plasmas [42]. Besides, the above mentioned ion pairs, dust of opposite polarities has also been observed in cometary environments [43,44]. Hence, Manesh et al [45] investigated the existence of Ion-Acoustic (IA) shock waves in a five component cometary plasma consisting of positively and negatively charged oxygen ions, kappa described hydrogen ions, hot solar electrons, and slightly colder cometary electrons. Very recently, Sijo et al [46] studied the effect of the drift velocity of lighter ions on shock waves in the above five component cometary plasma. Further, Mahmoud [47] analyzed the effects of the non-extensive parameters on the structure of the envelope ion acoustic waves in five-component cometary plasma system containing positively and negatively charged oxygen ions, non-extensive hot electrons from solar origin, colder electrons of cometary origin and positive hydrogen ions.

Thus for reasons given above, we investigate the effect of dust charge variation on dust acoustic shock waves in a six component cometary plasma composed of lighter hydrogen ions, positively and negatively charged oxygen ions, charge varying dust grains of negative polarity, cometary photo-electrons and hot, solar wind electrons.

Basic Equations

We consider a six component, unmagnetized plasma containing negatively charged dust (denoted by subscript ’d’), negatively and positively charged oxygen ions (denoted by subscripts ’1’ and ’2’ respectively), positive hydrogen ions (denoted by subscript ’H’) and superthermal electrons described by kappa distributions (hot electrons from solar origin and colder electrons of cometary origin denoted by subscripts ’he’ and ’ce’, respectively).

At equilibrium, charge neutrality requires:


n ce0 + n he0 + z 10 n 10 + z d0 n d0 = n H0 + z 20 n 20                       (1) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaadogacaWGLbGaaGimaaWdaeqaaOWd biabgUcaRiaad6gapaWaaSbaaSqaa8qacaWGObGaamyzaiaaicdaa8 aabeaak8qacqGHRaWkcaWG6bWdamaaBaaaleaapeGaaGymaiaaicda a8aabeaak8qacaWGUbWdamaaBaaaleaapeGaaGymaiaaicdaa8aabe aak8qacqGHRaWkcaWG6bWdamaaBaaaleaapeGaamizaiaaicdaa8aa beaak8qacaWGUbWdamaaBaaaleaapeGaamizaiaaicdaa8aabeaak8 qacqGH9aqpcaWGUbWdamaaBaaaleaapeGaamisaiaaicdaa8aabeaa k8qacqGHRaWkcaWG6bWdamaaBaaaleaapeGaaGOmaiaaicdaa8aabe aak8qacaWGUbWdamaaBaaaleaapeGaaGOmaiaaicdaa8aabeaakiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabM caaaa@6717@

where nhe0 and nce0 represent the equilibrium densities of cometary electrons and solar electrons respectively; n10 , n20, nH 0 and nd 0 are the equilibrium densities of negatively charged oxygen (O− ) ions, positively charged oxygen (O+ ) ions, hydrogen ions and negatively charged dust respectively. z10 , z20 and zd 0 are the equilibrium charge numbers of O− , O+ ions and dust respectively.

The dynamics of the negatively charged dust particles can be described by the following hydrodynamic equations:


n d t + n d v d x =0                (2) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2kaad6gapaWaaSbaaSqaa8qacaWGKbaa paqabaaakeaapeGaeyOaIyRaamiDaaaacqGHRaWkdaWcaaWdaeaape GaeyOaIy7aaeWaa8aabaWdbiaad6gapaWaaSbaaSqaa8qacaWGKbaa paqabaGcpeGaamODa8aadaWgaaWcbaWdbiaadsgaa8aabeaaaOWdbi aawIcacaGLPaaaa8aabaWdbiabgkGi2kaadIhaaaGaeyypa0JaaGim aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGOaGaaeOmaiaabMcaaaa@5592@

v d t + v d v d x = z d e m d ϕ x + η d 2 v d x 2                          (3) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2kaadAhapaWaaSbaaSqaa8qacaWGKbaa paqabaaakeaapeGaeyOaIyRaamiDaaaacqGHRaWkcaWG2bWdamaaBa aaleaapeGaamizaaWdaeqaaOWdbmaalaaapaqaa8qacqGHciITcaWG 2bWdamaaBaaaleaapeGaamizaaWdaeqaaaGcbaWdbiabgkGi2kaadI haaaGaeyypa0ZaaSaaa8aabaWdbiaadQhapaWaaSbaaSqaa8qacaWG KbaapaqabaGcpeGaamyzaaWdaeaapeGaamyBa8aadaWgaaWcbaWdbi aadsgaa8aabeaaaaGcpeWaaSaaa8aabaWdbiabgkGi2kabew9aMbWd aeaapeGaeyOaIyRaamiEaaaacqGHRaWkcqaH3oaApaWaaSbaaSqaa8 qacaWGKbaapaqabaGcpeWaaSaaa8aabaWdbiabgkGi2+aadaahaaWc beqaa8qacaaIYaaaaOGaamODa8aadaWgaaWcbaWdbiaadsgaa8aabe aaaOqaa8qacqGHciITcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaaa aOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabIcacaqGZaGaaeykaaaa@70FE@

2 ϕ x 2 =4πe   n ce + n he + z 10 n 1 + z d n d n H z 20 n 2                       (4) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYaaaaOGa eqy1dygapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqabaWdbiaaik daaaaaaOGaeyypa0JaaGinaiabec8aWjaadwgacaGGGcGaaiiOamaa dmaapaqaa8qacaWGUbWdamaaBaaaleaapeGaam4yaiaadwgaa8aabe aak8qacqGHRaWkcaWGUbWdamaaBaaaleaapeGaamiAaiaadwgaa8aa beaak8qacqGHRaWkcaWG6bWdamaaBaaaleaapeGaaGymaiaaicdaa8 aabeaak8qacaWGUbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiab gUcaRiaadQhapaWaaSbaaSqaa8qacaWGKbaapaqabaGcpeGaamOBa8 aadaWgaaWcbaWdbiaadsgaa8aabeaak8qacqGHsislcaWGUbWdamaa BaaaleaapeGaamisaaWdaeqaaOWdbiabgkHiTiaadQhapaWaaSbaaS qaa8qacaaIYaGaaGimaaWdaeqaaOWdbiaad6gapaWaaSbaaSqaa8qa caaIYaaapaqabaaak8qacaGLBbGaayzxaaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabIcacaqG0aGaaeykaaaa@72CD@

where Vd and nd are the fluid velocity and kinematic viscosity of the dust grains, md , the mass and e, the electronic charge.

The above equations, in their dimensionless forms, are:


n d t + n d v d x =0                       (5) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2kaad6gapaWaaSbaaSqaa8qacaWGKbaa paqabaaakeaapeGaeyOaIyRaamiDaaaacqGHRaWkdaWcaaWdaeaape GaeyOaIy7aaeWaa8aabaWdbiaad6gapaWaaSbaaSqaa8qacaWGKbaa paqabaGcpeGaamODa8aadaWgaaWcbaWdbiaadsgaa8aabeaaaOWdbi aawIcacaGLPaaaa8aabaWdbiabgkGi2kaadIhaaaGaeyypa0JaaGim aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikai aabwdacaqGPaaaaa@5A0A@

v d t + v d v d x = z d ϕ x + η d 2 v d x 2                              (6) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2kaadAhapaWaaSbaaSqaa8qacaWGKbaa paqabaaakeaapeGaeyOaIyRaamiDaaaacqGHRaWkcaWG2bWdamaaBa aaleaapeGaamizaaWdaeqaaOWdbmaalaaapaqaa8qacqGHciITcaWG 2bWdamaaBaaaleaapeGaamizaaWdaeqaaaGcbaWdbiabgkGi2kaadI haaaGaeyypa0JaamOEa8aadaWgaaWcbaWdbiaadsgaa8aabeaak8qa daWcaaWdaeaapeGaeyOaIyRaeqy1dygapaqaa8qacqGHciITcaWG4b aaaiabgUcaRiabeE7aO9aadaWgaaWcbaWdbiaadsgaa8aabeaak8qa daWcaaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaaikdaaaGcca WG2bWdamaaBaaaleaapeGaamizaaWdaeqaaaGcbaWdbiabgkGi2kaa dIhapaWaaWbaaSqabeaapeGaaGOmaaaaaaGccaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabIcacaqG2aGaaeykaaaa@7006@

2 ϕ x 2 = z d n d + δ ce 1 s β H ϕ κ ce 3 2 κ ce + 1 2 + δ he 1 s β H3 ϕ κ he 3 2 κ he + 1 2 +  δ 1 exp s β H1 ϕ δ H exp sϕ δ 2 exp s β H2 ϕ                (7) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaaikda aaGccqaHvpGza8aabaWdbiabgkGi2kaadIhapaWaaWbaaSqabeaape GaaGOmaaaaaaGccqGH9aqpcaWG6bWdamaaBaaaleaapeGaamizaaWd aeqaaOWdbiaad6gapaWaaSbaaSqaa8qacaWGKbaapaqabaGcpeGaey 4kaSIaeqiTdq2damaaBaaaleaapeGaam4yaiaadwgaa8aabeaak8qa daWadaWdaeaapeGaaGymaiabgkHiTmaalaaapaqaa8qacaWGZbGaeq OSdi2damaaBaaaleaapeGaamisaaWdaeqaaOWdbiabew9aMbWdaeaa peWaaeWaa8aabaWdbiabeQ7aR9aadaWgaaWcbaWdbiaadogacaWGLb aapaqabaGcpeGaeyOeI0YaaSaaa8aabaWdbiaaiodaa8aabaWdbiaa ikdaaaaacaGLOaGaayzkaaaaaaGaay5waiaaw2faa8aadaahaaWcbe qaa8qacqGHsislcqaH6oWApaWaaSbaaWqaa8qacaWGJbGaamyzaaWd aeqaaSWdbiabgUcaRmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYa aaaaaakiabgUcaRiabes7aK9aadaWgaaWcbaWdbiaadIgacaWGLbaa paqabaGcpeWaamWaa8aabaWdbiaaigdacqGHsisldaWcaaWdaeaape Gaam4Caiabek7aI9aadaWgaaWcbaWdbiaadIeacaaIZaaapaqabaGc peGaeqy1dygapaqaa8qadaqadaWdaeaapeGaeqOUdS2damaaBaaale aapeGaamiAaiaadwgaa8aabeaak8qacqGHsisldaWcaaWdaeaapeGa aG4maaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaaaaaacaGLBbGaay zxaaWdamaaCaaaleqabaWdbiabgkHiTiabeQ7aR9aadaWgaaadbaWd biaadIgacaWGLbaapaqabaWcpeGaey4kaSYaaSaaa8aabaWdbiaaig daa8aabaWdbiaaikdaaaaaaaGcpaqaa8qacqGHRaWkcaqGGaGaeqiT dq2damaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaabwgacaqG4bGaae iCamaabmaapaqaa8qacaWGZbGaeqOSdi2damaaBaaaleaapeGaamis aiaaigdaa8aabeaak8qacqaHvpGzaiaawIcacaGLPaaacqGHsislcq aH0oazpaWaaSbaaSqaa8qacaWGibaapaqabaGcpeGaaeyzaiaabIha caqGWbWaaeWaa8aabaWdbiabgkHiTiaadohacqaHvpGzaiaawIcaca GLPaaacqGHsislcqaH0oazpaWaaSbaaSqaa8qacaaIYaaapaqabaGc peGaaeyzaiaabIhacaqGWbWaaeWaa8aabaWdbiabgkHiTiaadohacq aHYoGypaWaaSbaaSqaa8qacaWGibGaaGOmaaWdaeqaaOWdbiabew9a MbGaayjkaiaawMcaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabIcacaqG3aGaaeykaaaaaa@B867@

where the kappa distributions model the number densities for cometary ( ce ) and solar ( he ) electrons; while the hydrogen ions (denoted by H , negatively charged oxygen ions (identified by '1') and positively charged oxygen ions ('2') obey a Maxwellian distribution. The dust grain density is normalized by nd 0 and zd by zd 0 ; the densities of electrons and ions are normalized by zd 0 nd 0 . The space 'x' and 't' time coordinates are normalized by the Debye length λ Dd = [ Teff 4π z d0 n d0 e 2 ] 1 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdW2damaaBaaaleaapeGaamiraiaadsgaa8aabeaak8qacqGH 9aqpcaGGBbWaaSaaa8aabaWdbiaadsfacaWGLbGaamOzaiaadAgaa8 aabaWdbiaaisdacqaHapaCcaWG6bWdamaaBaaaleaapeGaamizaiaa icdaa8aabeaak8qacaWGUbWdamaaBaaaleaapeGaamizaiaaicdaa8 aabeaak8qacaWGLbWdamaaCaaaleqabaWdbiaaikdaaaaaaOGaaiyx a8aadaahaaWcbeqaa8qadaWcaaWdaeaapeGaaGymaaWdaeaapeGaaG Omaaaaaaaaaa@4D38@ and the inverse of dust plasma frequency ω pd 1 = [ 4π z d0 2 e 2 n d0 m d ] 1 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyYdC3damaaDaaaleaapeGaamiCaiaadsgaa8aabaWdbiabgkHi TiaaigdaaaGccqGH9aqpcaGGBbWaaSaaa8aabaWdbiaaisdacqaHap aCcaWG6bWdamaaDaaaleaapeGaamizaiaaicdaa8aabaWdbiaaikda aaGccaWGLbWdamaaCaaaleqabaWdbiaaikdaaaGccaWGUbWdamaaBa aaleaapeGaamizaiaaicdaa8aabeaaaOqaa8qacaWGTbWdamaaBaaa leaapeGaamizaaWdaeqaaaaak8qacaGGDbWdamaaCaaaleqabaWdbm aalaaapaqaa8qacqGHsislcaaIXaaapaqaa8qacaaIYaaaaaaaaaa@4F67@ respectively. The dust fluid velocity d v and electostatic potential are normalized by the DA speed C d = [ z d0 T eff m d ] 1 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4qa8aadaWgaaWcbaWdbiaadsgaa8aabeaak8qacqGH9aqpcaGG BbWaaSaaa8aabaWdbiaadQhapaWaaSbaaSqaa8qacaWGKbGaaGimaa WdaeqaaOWdbiaadsfapaWaaSbaaSqaa8qacaWGLbGaamOzaiaadAga a8aabeaaaOqaa8qacaWGTbWdamaaBaaaleaapeGaamizaaWdaeqaaa aak8qacaGGDbWdamaaCaaaleqabaWdbmaalaaapaqaa8qacaaIXaaa paqaa8qacaaIYaaaaaaaaaa@46A6@ and T eff e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsfapaWaaSbaaSqaa8qacaWGLbGaamOzaiaa dAgaa8aabeaaaOqaa8qacaWGLbaaaaaa@3B2F@ respectively, in which the effective temperature is [48]:


T eff = z d0 n d0 n he0 T he + n ce0 T ce + z 1 n 10 T 1 + z 2 n 20 T 2 + n H0 T H                (8) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaadwgacaWGMbGaamOzaaWdaeqaaOWd biabg2da9maalaaapaqaa8qacaWG6bWdamaaBaaaleaapeGaamizai aaicdaa8aabeaak8qacaWGUbWdamaaBaaaleaapeGaamizaiaaicda a8aabeaaaOqaa8qadaWadaWdaeaapeWaaSaaa8aabaWdbiaad6gapa WaaSbaaSqaa8qacaWGObGaamyzaiaaicdaa8aabeaaaOqaa8qacaWG ubWdamaaBaaaleaapeGaamiAaiaadwgaa8aabeaaaaGcpeGaey4kaS YaaSaaa8aabaWdbiaad6gapaWaaSbaaSqaa8qacaWGJbGaamyzaiaa icdaa8aabeaaaOqaa8qacaWGubWdamaaBaaaleaapeGaam4yaiaadw gaa8aabeaaaaGcpeGaey4kaSYaaSaaa8aabaWdbiaadQhapaWaaSba aSqaa8qacaaIXaaapaqabaGcpeGaamOBa8aadaWgaaWcbaWdbiaaig dacaaIWaaapaqabaaakeaapeGaamiva8aadaWgaaWcbaWdbiaaigda a8aabeaaaaGcpeGaey4kaSYaaSaaa8aabaWdbiaadQhapaWaaSbaaS qaa8qacaaIYaaapaqabaGcpeGaamOBa8aadaWgaaWcbaWdbiaaikda caaIWaaapaqabaaakeaapeGaamiva8aadaWgaaWcbaWdbiaaikdaa8 aabeaaaaGcpeGaey4kaSYaaSaaa8aabaWdbiaad6gapaWaaSbaaSqa a8qacaWGibGaaGimaaWdaeqaaaGcbaWdbiaadsfapaWaaSbaaSqaa8 qacaWGibaapaqabaaaaaGcpeGaay5waiaaw2faaaaacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeioaiaabMca aaa@74EB@

Also

δ H = n H0 z d0 n d0 ,  δ 1 = z 10 n 10 z d0 n d0 ,  δ 2 = z 20 n 20 z d0 n d0 ,  δ ce = n ce0 z d0 n d0 , δ he = n he0 z d0 n d0 ,  β H = T H T ce ,  β 1 = T 1 T ce ,  β 2 = T 2 T ce ,  β he = T he T ce , β H1 = β H β 1 = T H T 1 ,  β H2 = β H β 2 = T H T 2 ,  β H3 = β H β he = T H T he , s= T eff T H = δ H δ 1 + δ 2 δ ce δ he δ H + δ 1 β H1 + δ 2 β H2 + δ ce β H + δ he β H3 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqaH0oazpaWaaSbaaSqaa8qacaWGibaapaqabaGcpeGaeyyp a0ZaaSaaa8aabaWdbiaad6gapaWaaSbaaSqaa8qacaWGibGaaGimaa WdaeqaaaGcbaWdbiaadQhapaWaaSbaaSqaa8qacaWGKbGaaGimaaWd aeqaaOWdbiaad6gapaWaaSbaaSqaa8qacaWGKbGaaGimaaWdaeqaaa aak8qacaGGSaGaaeiiaiabes7aK9aadaWgaaWcbaWdbiaaigdaa8aa beaak8qacqGH9aqpdaWcaaWdaeaapeGaamOEa8aadaWgaaWcbaWdbi aaigdacaaIWaaapaqabaGcpeGaamOBa8aadaWgaaWcbaWdbiaaigda caaIWaaapaqabaaakeaapeGaamOEa8aadaWgaaWcbaWdbiaadsgaca aIWaaapaqabaGcpeGaamOBa8aadaWgaaWcbaWdbiaadsgacaaIWaaa paqabaaaaOWdbiaacYcacaqGGaGaeqiTdq2damaaBaaaleaapeGaaG OmaaWdaeqaaOWdbiabg2da9maalaaapaqaa8qacaWG6bWdamaaBaaa leaapeGaaGOmaiaaicdaa8aabeaak8qacaWGUbWdamaaBaaaleaape GaaGOmaiaaicdaa8aabeaaaOqaa8qacaWG6bWdamaaBaaaleaapeGa amizaiaaicdaa8aabeaak8qacaWGUbWdamaaBaaaleaapeGaamizai aaicdaa8aabeaaaaGcpeGaaiilaiaabccacqaH0oazpaWaaSbaaSqa a8qacaWGJbGaamyzaaWdaeqaaOWdbiabg2da9maalaaapaqaa8qaca WGUbWdamaaBaaaleaapeGaam4yaiaadwgacaaIWaaapaqabaaakeaa peGaamOEa8aadaWgaaWcbaWdbiaadsgacaaIWaaapaqabaGcpeGaam OBa8aadaWgaaWcbaWdbiaadsgacaaIWaaapaqabaaaaOWdbiaacYca aeaacqaH0oazpaWaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbi abg2da9maalaaapaqaa8qacaWGUbWdamaaBaaaleaapeGaamiAaiaa dwgacaaIWaaapaqabaaakeaapeGaamOEa8aadaWgaaWcbaWdbiaads gacaaIWaaapaqabaGcpeGaamOBa8aadaWgaaWcbaWdbiaadsgacaaI WaaapaqabaaaaOWdbiaacYcacaqGGaGaeqOSdi2damaaBaaaleaape GaamisaaWdaeqaaOWdbiabg2da9maalaaapaqaa8qacaWGubWdamaa BaaaleaapeGaamisaaWdaeqaaaGcbaWdbiaadsfapaWaaSbaaSqaa8 qacaWGJbGaamyzaaWdaeqaaaaak8qacaGGSaGaaeiiaiabek7aI9aa daWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpdaWcaaWdaeaape Gaamiva8aadaWgaaWcbaWdbiaaigdaa8aabeaaaOqaa8qacaWGubWd amaaBaaaleaapeGaam4yaiaadwgaa8aabeaaaaGcpeGaaiilaiaabc cacqaHYoGypaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyypa0Za aSaaa8aabaWdbiaadsfapaWaaSbaaSqaa8qacaaIYaaapaqabaaake aapeGaamiva8aadaWgaaWcbaWdbiaadogacaWGLbaapaqabaaaaOWd biaacYcacaqGGaGaeqOSdi2damaaBaaaleaapeGaamiAaiaadwgaa8 aabeaak8qacqGH9aqpdaWcaaWdaeaapeGaamiva8aadaWgaaWcbaWd biaadIgacaWGLbaapaqabaaakeaapeGaamiva8aadaWgaaWcbaWdbi aadogacaWGLbaapaqabaaaaOWdbiaacYcaaeaacqaHYoGypaWaaSba aSqaa8qacaWGibGaaGymaaWdaeqaaOWdbiabg2da9maalaaapaqaa8 qacqaHYoGypaWaaSbaaSqaa8qacaWGibaapaqabaaakeaapeGaeqOS di2damaaBaaaleaapeGaaGymaaWdaeqaaaaak8qacqGH9aqpdaWcaa WdaeaapeGaamiva8aadaWgaaWcbaWdbiaadIeaa8aabeaaaOqaa8qa caWGubWdamaaBaaaleaapeGaaGymaaWdaeqaaaaak8qacaGGSaGaae iiaiabek7aI9aadaWgaaWcbaWdbiaadIeacaaIYaaapaqabaGcpeGa eyypa0ZaaSaaa8aabaWdbiabek7aI9aadaWgaaWcbaWdbiaadIeaa8 aabeaaaOqaa8qacqaHYoGypaWaaSbaaSqaa8qacaaIYaaapaqabaaa aOWdbiabg2da9maalaaapaqaa8qacaWGubWdamaaBaaaleaapeGaam isaaWdaeqaaaGcbaWdbiaadsfapaWaaSbaaSqaa8qacaaIYaaapaqa baaaaOWdbiaacYcacaqGGaGaeqOSdi2damaaBaaaleaapeGaamisai aaiodaa8aabeaak8qacqGH9aqpdaWcaaWdaeaapeGaeqOSdi2damaa BaaaleaapeGaamisaaWdaeqaaaGcbaWdbiabek7aI9aadaWgaaWcba WdbiaadIgacaWGLbaapaqabaaaaOWdbiabg2da9maalaaapaqaa8qa caWGubWdamaaBaaaleaapeGaamisaaWdaeqaaaGcbaWdbiaadsfapa WaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaaaak8qacaGGSaaabaGa am4Caiabg2da9maalaaapaqaa8qacaWGubWdamaaBaaaleaapeGaam yzaiaadAgacaWGMbaapaqabaaakeaapeGaamiva8aadaWgaaWcbaWd biaadIeaa8aabeaaaaGcpeGaeyypa0ZaaSaaa8aabaWdbiabes7aK9 aadaWgaaWcbaWdbiaadIeaa8aabeaak8qacqGHsislcqaH0oazpaWa aSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIaeqiTdq2damaaBa aaleaapeGaaGOmaaWdaeqaaOWdbiabgkHiTiabes7aK9aadaWgaaWc baWdbiaadogacaWGLbaapaqabaGcpeGaeyOeI0IaeqiTdq2damaaBa aaleaapeGaamiAaiaadwgaa8aabeaaaOqaa8qacqaH0oazpaWaaSba aSqaa8qacaWGibaapaqabaGcpeGaey4kaSIaeqiTdq2damaaBaaale aapeGaaGymaaWdaeqaaOWdbiabek7aI9aadaWgaaWcbaWdbiaadIea caaIXaaapaqabaGcpeGaey4kaSIaeqiTdq2damaaBaaaleaapeGaaG OmaaWdaeqaaOWdbiabek7aI9aadaWgaaWcbaWdbiaadIeacaaIYaaa paqabaGcpeGaey4kaSIaeqiTdq2damaaBaaaleaapeGaam4yaiaadw gaa8aabeaak8qacqaHYoGypaWaaSbaaSqaa8qacaWGibaapaqabaGc peGaey4kaSIaeqiTdq2damaaBaaaleaapeGaamiAaiaadwgaa8aabe aak8qacqaHYoGypaWaaSbaaSqaa8qacaWGibGaaG4maaWdaeqaaaaa aaaa@278F@

where Td , Tce , The , TH , T and T2 are the temperatures of dust, cometary electrons, solar electrons, hydrogen ions, negative and positive oxygen ions, respectively

The variable dust charge Zd is obtained from the current balance equation,

I H + I 1 + I 2 + I ce + I he =0           (9) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadIeaa8aabeaak8qacqGHRaWkcaWG jbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgUcaRiaadMeapa WaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaey4kaSIaamysa8aadaWg aaWcbaWdbiaadogacaWGLbaapaqabaGcpeGaey4kaSIaamysa8aada WgaaWcbaWdbiaadIgacaWGLbaapaqabaGcpeGaeyypa0JaaGimaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeikaiaabMdacaqGPaaaaa@509C@

where IH , I1 , I2 , Ice and Ihe are hydrogen ion, negatively charged oxygen, positively charged oxygen, colder and hotter electron currents, respectively

The ion currents are [11]:

I H =eπ r 2 8 T H π m H 1 2 n H 1 e ϕ d T H                 (10) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadIeaa8aabeaak8qacqGH9aqpcaWG LbGaeqiWdaNaamOCa8aadaahaaWcbeqaa8qacaaIYaaaaOWaaeWaa8 aabaWdbmaalaaapaqaa8qacaaI4aGaamiva8aadaWgaaWcbaWdbiaa dIeaa8aabeaaaOqaa8qacqaHapaCcaWGTbWdamaaBaaaleaapeGaam isaaWdaeqaaaaaaOWdbiaawIcacaGLPaaapaWaaWbaaSqabeaapeWa aSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaaaaOGaamOBa8aada WgaaWcbaWdbiaadIeaa8aabeaak8qadaqadaWdaeaapeGaaGymaiab gkHiTmaalaaapaqaa8qacaWGLbGaeqy1dy2damaaBaaaleaapeGaam izaaWdaeqaaaGcbaWdbiaadsfapaWaaSbaaSqaa8qacaWGibaapaqa baaaaaGcpeGaayjkaiaawMcaaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabcdacaqGPaaaaa@6156@
I 2 = z 2 eπ r 2 8 T 2 π m 2 1 2 n 2 1 z 2 e ϕ d T 2               (11) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpcaWG 6bWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaadwgacqaHapaCca WGYbWdamaaCaaaleqabaWdbiaaikdaaaGcdaqadaWdaeaapeWaaSaa a8aabaWdbiaaiIdacaWGubWdamaaBaaaleaapeGaaGOmaaWdaeqaaa GcbaWdbiabec8aWjaad2gapaWaaSbaaSqaa8qacaaIYaaapaqabaaa aaGcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8qadaWcaaWdaeaape GaaGymaaWdaeaapeGaaGOmaaaaaaGccaWGUbWdamaaBaaaleaapeGa aGOmaaWdaeqaaOWdbmaabmaapaqaa8qacaaIXaGaeyOeI0YaaSaaa8 aabaWdbiaadQhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaamyz aiabew9aM9aadaWgaaWcbaWdbiaadsgaa8aabeaaaOqaa8qacaWGub WdamaaBaaaleaapeGaaGOmaaWdaeqaaaaaaOWdbiaawIcacaGLPaaa caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGa aeymaiaabMcaaaa@641A@

where IH and I2 are the charging currents for attractive potentials (qj φd < 0 ) due to hydrogen and positively charged oxygen ions respectively. Here φd denotes the dust grain surface potential relative to the plasma potential φ ; r is the radius of the dust grain and the other notations are standard. Also

I 1 = z 1 eπ r 2 8 T 1 π m 1 1 2 n 1 exp z 1 e ϕ d T 1                 (12) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcqGH sislcaWG6bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaadwgacq aHapaCcaWGYbWdamaaCaaaleqabaWdbiaaikdaaaGcdaqadaWdaeaa peWaaSaaa8aabaWdbiaaiIdacaWGubWdamaaBaaaleaapeGaaGymaa WdaeqaaaGcbaWdbiabec8aWjaad2gapaWaaSbaaSqaa8qacaaIXaaa paqabaaaaaGcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8qadaWcaa WdaeaapeGaaGymaaWdaeaapeGaaGOmaaaaaaGccaWGUbWdamaaBaaa leaapeGaaGymaaWdaeqaaOWdbiaabwgacaqG4bGaaeiCamaabmaapa qaa8qadaWcaaWdaeaapeGaamOEa8aadaWgaaWcbaWdbiaaigdaa8aa beaak8qacaWGLbGaeqy1dy2damaaBaaaleaapeGaamizaaWdaeqaaa GcbaWdbiaadsfapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaaGcpeGa ayjkaiaawMcaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGOaGaaeymaiaabkdacaqGPaaaaa@6775@

where I1 is the charging current for a repulsive potential (qj φd >0 ) due to the negatively charged oxygen ions.

The currents due to the cometary and solar, kappa distributed electrons to the negatively charged dust grains are [49],

I ce =2 π r 2 B κ ce e n ce θ ce 1 2e ϕ d κ ce m ce θ ce 2 κ ce +1            (13) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadogacaWGLbaapaqabaGcpeGaeyyp a0JaeyOeI0IaaGOmamaakaaapaqaa8qacqaHapaCaSqabaGccaWGYb WdamaaCaaaleqabaWdbiaaikdaaaGccaWGcbWdamaaBaaaleaapeGa eqOUdS2damaaBaaameaapeGaam4yaiaadwgaa8aabeaaaSqabaGcpe Gaamyzaiaad6gapaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWd biabeI7aX9aadaWgaaWcbaWdbiaadogacaWGLbaapaqabaGcpeWaae Waa8aabaWdbiaaigdacqGHsisldaWcaaWdaeaapeGaaGOmaiaadwga cqaHvpGzpaWaaSbaaSqaa8qacaWGKbaapaqabaaakeaapeGaeqOUdS 2damaaBaaaleaapeGaam4yaiaadwgaa8aabeaak8qacaWGTbWdamaa BaaaleaapeGaam4yaiaadwgaa8aabeaak8qacqaH4oqCpaWaa0baaS qaa8qacaWGJbGaamyzaaWdaeaapeGaaGOmaaaaaaaakiaawIcacaGL PaaapaWaaWbaaSqabeaapeGaeyOeI0IaeqOUdS2damaaBaaameaape Gaam4yaiaadwgaa8aabeaal8qacqGHRaWkcaaIXaaaaOWdaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeikaiaabgdacaqGZaGaaeykaaaa@71E9@
I he =2 π r 2 B κ he e n he θ he 1 2e ϕ d κ he m he θ he 2 κ he +1               (14) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadIgacaWGLbaapaqabaGcpeGaeyyp a0JaeyOeI0IaaGOmamaakaaapaqaa8qacqaHapaCaSqabaGccaWGYb WdamaaCaaaleqabaWdbiaaikdaaaGccaWGcbWdamaaBaaaleaapeGa eqOUdS2damaaBaaameaapeGaamiAaiaadwgaa8aabeaaaSqabaGcpe Gaamyzaiaad6gapaWaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWd biabeI7aX9aadaWgaaWcbaWdbiaadIgacaWGLbaapaqabaGcpeWaae Waa8aabaWdbiaaigdacqGHsisldaWcaaWdaeaapeGaaGOmaiaadwga cqaHvpGzpaWaaSbaaSqaa8qacaWGKbaapaqabaaakeaapeGaeqOUdS 2damaaBaaaleaapeGaamiAaiaadwgaa8aabeaak8qacaWGTbWdamaa BaaaleaapeGaamiAaiaadwgaa8aabeaak8qacqaH4oqCpaWaa0baaS qaa8qacaWGObGaamyzaaWdaeaapeGaaGOmaaaaaaaakiaawIcacaGL PaaapaWaaWbaaSqabeaapeGaeyOeI0IaeqOUdS2damaaBaaameaape GaamiAaiaadwgaa8aabeaal8qacqGHRaWkcaaIXaaaaOWdaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG0aGa aeykaaaa@73FB@

where B κj = Γ κ j +1 κ j 3/2 Γ κ j 1/2 κ j κ j 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOqa8aadaWgaaWcbaWdbiabeQ7aRjaadQgaa8aabeaak8qacqGH 9aqpdaGadaWdaeaapeWaaSaaa8aabaWdbiaabo5adaqadaWdaeaape GaeqOUdS2damaaBaaaleaapeGaamOAaaWdaeqaaOWdbiabgUcaRiaa igdaaiaawIcacaGLPaaaa8aabaWdbiabeQ7aR9aadaqhaaWcbaWdbi aadQgaa8aabaWdbiaaiodacaGGVaGaaGOmaaaakiaabo5adaqadaWd aeaapeGaeqOUdS2damaaBaaaleaapeGaamOAaaWdaeqaaOWdbiabgk HiTiaaigdacaGGVaGaaGOmaaGaayjkaiaawMcaaaaaaiaawUhacaGL 9baadaWcaaWdaeaapeGaeqOUdS2damaaBaaaleaapeGaamOAaaWdae qaaaGcbaWdbiabeQ7aR9aadaWgaaWcbaWdbiaadQgaa8aabeaak8qa cqGHsislcaaIXaaaaaaa@5B2C@ and , the effective thermal speed for the electron, is θ j 2 = 2 κ j 3 κ j T j m j MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUde3damaaDaaaleaapeGaamOAaaWdaeaapeGaaGOmaaaakiab g2da9maabmaapaqaa8qadaWcaaWdaeaapeGaaGOmaiabeQ7aR9aada WgaaWcbaWdbiaadQgaa8aabeaak8qacqGHsislcaaIZaaapaqaa8qa cqaH6oWApaWaaSbaaSqaa8qacaWGQbaapaqabaaaaaGcpeGaayjkai aawMcaamaalaaapaqaa8qacaWGubWdamaaBaaaleaapeGaamOAaaWd aeqaaaGcbaWdbiaad2gapaWaaSbaaSqaa8qacaWGQbaapaqabaaaaa aa@4A11@ j=ce,he MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabg2da9iaadogacaWGLbGaaiilaiaadIgacaWGLbaaaa@3C57@

Here mj is the mass, nj is the electron number density, Tj is the temperature and Kj, the spectral index of the jth species.

Using equations (10) - (14) in the current balance equation (9), we arrive at

b H δ H 1sψ e sϕ + b 2 δ 2 1 β H2 sψ e β H2 sϕ b 1 δ 1 e s β H1 ψ e s β H1 ϕ b κ ce δ ce 1 2 β H sψ 2 κ ce 3 κ ce +1 × 1 β H sϕ κ ce 3 2 κ ce + 1 2 b κ he b 3 δ he 1 2 β H3 sψ 2 κ he 3 κ he +1 1 β H3 sϕ κ he 3 2 κ he + 1 2 =0                  (15) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacaWGIbWdamaaBaaaleaapeGaamisaaWdaeqaaOWdbiabes7a K9aadaWgaaWcbaWdbiaadIeaa8aabeaak8qadaqadaWdaeaapeGaaG ymaiabgkHiTiaadohacqaHipqEaiaawIcacaGLPaaacaWGLbWdamaa CaaaleqabaWdbiabgkHiTiaadohacqaHvpGzaaGccqGHRaWkcaWGIb WdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabes7aK9aadaWgaaWc baWdbiaaikdaa8aabeaak8qadaqadaWdaeaapeGaaGymaiabgkHiTi abek7aI9aadaWgaaWcbaWdbiaadIeacaaIYaaapaqabaGcpeGaam4C aiabeI8a5bGaayjkaiaawMcaaiaadwgapaWaaWbaaSqabeaapeGaey OeI0IaeqOSdi2damaaBaaameaapeGaamisaiaaikdaa8aabeaal8qa caWGZbGaeqy1dygaaOGaeyOeI0IaamOya8aadaWgaaWcbaWdbiaaig daa8aabeaak8qacqaH0oazpaWaaSbaaSqaa8qacaaIXaaapaqabaGc peGaamyza8aadaahaaWcbeqaa8qacaWGZbGaeqOSdi2damaaBaaame aapeGaamisaiaaigdaa8aabeaal8qacqaHipqEaaGccaWGLbWdamaa CaaaleqabaWdbiaadohacqaHYoGypaWaaSbaaWqaa8qacaWGibGaaG ymaaWdaeqaaSWdbiabew9aMbaakiabgkHiTiaadkgapaWaaSbaaSqa a8qacqaH6oWApaWaaSbaaWqaa8qacaWGJbGaamyzaaWdaeqaaaWcbe aak8qacqaH0oazpaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWd bmaabmaapaqaa8qacaaIXaGaeyOeI0YaaSaaa8aabaWdbiaaikdacq aHYoGypaWaaSbaaSqaa8qacaWGibaapaqabaGcpeGaam4CaiabeI8a 5bWdaeaapeWaaeWaa8aabaWdbiaaikdacqaH6oWApaWaaSbaaSqaa8 qacaWGJbGaamyzaaWdaeqaaOWdbiabgkHiTiaaiodaaiaawIcacaGL PaaaaaaacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiabgkHiTiabeQ 7aR9aadaWgaaadbaWdbiaadogacaWGLbaapaqabaWcpeGaey4kaSIa aGymaaaaaOWdaeaapeGaey41aq7aamWaa8aabaWdbiaaigdacqGHsi sldaWcaaWdaeaapeGaeqOSdi2damaaBaaaleaapeGaamisaaWdaeqa aOWdbiaadohacqaHvpGza8aabaWdbmaabmaapaqaa8qacqaH6oWApa WaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabgkHiTmaalaaa paqaa8qacaaIZaaapaqaa8qacaaIYaaaaaGaayjkaiaawMcaaaaaai aawUfacaGLDbaapaWaaWbaaSqabeaapeGaeyOeI0IaeqOUdS2damaa BaaameaapeGaam4yaiaadwgaa8aabeaal8qacqGHRaWkdaWcaaWdae aapeGaaGymaaWdaeaapeGaaGOmaaaaaaGccqGHsislcaWGIbWdamaa BaaaleaapeGaeqOUdS2damaaBaaameaapeGaamiAaiaadwgaa8aabe aaaSqabaGcpeGaamOya8aadaWgaaWcbaWdbiaaiodaa8aabeaak8qa cqaH0oazpaWaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbmaabm aapaqaa8qacaaIXaGaeyOeI0YaaSaaa8aabaWdbiaaikdacqaHYoGy paWaaSbaaSqaa8qacaWGibGaaG4maaWdaeqaaOWdbiaadohacqaHip qEa8aabaWdbmaabmaapaqaa8qacaaIYaGaeqOUdS2damaaBaaaleaa peGaamiAaiaadwgaa8aabeaak8qacqGHsislcaaIZaaacaGLOaGaay zkaaaaaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacqGHsislcqaH 6oWApaWaaSbaaWqaa8qacaWGObGaamyzaaWdaeqaaSWdbiabgUcaRi aaigdaaaGcdaWadaWdaeaapeGaaGymaiabgkHiTmaalaaapaqaa8qa cqaHYoGypaWaaSbaaSqaa8qacaWGibGaaG4maaWdaeqaaOWdbiaado hacqaHvpGza8aabaWdbmaabmaapaqaa8qacqaH6oWApaWaaSbaaSqa a8qacaWGObGaamyzaaWdaeqaaOWdbiabgkHiTmaalaaapaqaa8qaca aIZaaapaqaa8qacaaIYaaaaaGaayjkaiaawMcaaaaaaiaawUfacaGL DbaapaWaaWbaaSqabeaapeGaeyOeI0IaeqOUdS2damaaBaaameaape GaamiAaiaadwgaa8aabeaal8qacqGHRaWkdaWcaaWdaeaapeGaaGym aaWdaeaapeGaaGOmaaaaaaGccqGH9aqpcaaIWaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae ikaiaabgdacaqG1aGaaeykaaaaaa@02CC@

where b H = 2 β H μ H ,  b 1 = 2 β 1 μ 1 ,  b 2 = 2 β 2 μ 2 ,  b 3 = β he μ 3 ,  b κ ce = B κ ce 2 κ ce 3 κ ce , b κ he = B κ he 2 κ he 3 κ he , ψ= e ϕ d T eff ,  μ H = m H m ce ,  μ 1 = m 1 m ce ,  μ 2 = m 2 m ce , and  μ 3 = m he m ce =1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacaWGIbWdamaaBaaaleaapeGaamisaaWdaeqaaOWdbiabg2da 9maakaaapaqaa8qadaWcaaWdaeaapeGaaGOmaiabek7aI9aadaWgaa WcbaWdbiaadIeaa8aabeaaaOqaa8qacqaH8oqBpaWaaSbaaSqaa8qa caWGibaapaqabaaaaaWdbeqaaOGaaiilaiaabccacaWGIbWdamaaBa aaleaapeGaaGymaaWdaeqaaOWdbiabg2da9maakaaapaqaa8qadaWc aaWdaeaapeGaaGOmaiabek7aI9aadaWgaaWcbaWdbiaaigdaa8aabe aaaOqaa8qacqaH8oqBpaWaaSbaaSqaa8qacaaIXaaapaqabaaaaaWd beqaaOGaaiilaiaabccacaWGIbWdamaaBaaaleaapeGaaGOmaaWdae qaaOWdbiabg2da9maakaaapaqaa8qadaWcaaWdaeaapeGaaGOmaiab ek7aI9aadaWgaaWcbaWdbiaaikdaa8aabeaaaOqaa8qacqaH8oqBpa WaaSbaaSqaa8qacaaIYaaapaqabaaaaaWdbeqaaOGaaiilaiaabcca caWGIbWdamaaBaaaleaapeGaaG4maaWdaeqaaOWdbiabg2da9maaka aapaqaa8qadaWcaaWdaeaapeGaeqOSdi2damaaBaaaleaapeGaamiA aiaadwgaa8aabeaaaOqaa8qacqaH8oqBpaWaaSbaaSqaa8qacaaIZa aapaqabaaaaaWdbeqaaOGaaiilaiaabccacaWGIbWdamaaBaaaleaa peGaeqOUdS2damaaBaaameaapeGaam4yaiaadwgaa8aabeaaaSqaba GcpeGaeyypa0JaamOqa8aadaWgaaWcbaWdbiabeQ7aR9aadaWgaaad baWdbiaadogacaWGLbaapaqabaaaleqaaOWdbmaakaaapaqaa8qada WcaaWdaeaapeWaaeWaa8aabaWdbiaaikdacqaH6oWApaWaaSbaaSqa a8qacaWGJbGaamyzaaWdaeqaaOWdbiabgkHiTiaaiodaaiaawIcaca GLPaaaa8aabaWdbiabeQ7aR9aadaWgaaWcbaWdbiaadogacaWGLbaa paqabaaaaaWdbeqaaOGaaiilaaqaaiaadkgapaWaaSbaaSqaa8qacq aH6oWApaWaaSbaaWqaa8qacaWGObGaamyzaaWdaeqaaaWcbeaak8qa cqGH9aqpcaWGcbWdamaaBaaaleaapeGaeqOUdS2damaaBaaameaape GaamiAaiaadwgaa8aabeaaaSqabaGcpeWaaOaaa8aabaWdbmaalaaa paqaa8qadaqadaWdaeaapeGaaGOmaiabeQ7aR9aadaWgaaWcbaWdbi aadIgacaWGLbaapaqabaGcpeGaeyOeI0IaaG4maaGaayjkaiaawMca aaWdaeaapeGaeqOUdS2damaaBaaaleaapeGaamiAaiaadwgaa8aabe aaaaaapeqabaGccaGGSaGaaeiiaiabeI8a5jabg2da9maalaaapaqa a8qacaWGLbGaeqy1dy2damaaBaaaleaapeGaamizaaWdaeqaaaGcba WdbiaadsfapaWaaSbaaSqaa8qacaWGLbGaamOzaiaadAgaa8aabeaa aaGcpeGaaiilaiaabccacqaH8oqBpaWaaSbaaSqaa8qacaWGibaapa qabaGcpeGaeyypa0ZaaSaaa8aabaWdbiaad2gapaWaaSbaaSqaa8qa caWGibaapaqabaaakeaapeGaamyBa8aadaWgaaWcbaWdbiaadogaca WGLbaapaqabaaaaOWdbiaacYcacaqGGaGaeqiVd02damaaBaaaleaa peGaaGymaaWdaeqaaOWdbiabg2da9maalaaapaqaa8qacaWGTbWdam aaBaaaleaapeGaaGymaaWdaeqaaaGcbaWdbiaad2gapaWaaSbaaSqa a8qacaWGJbGaamyzaaWdaeqaaaaak8qacaGGSaGaaeiiaiabeY7aT9 aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpdaWcaaWdaeaa peGaamyBa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaOqaa8qacaWGTb WdamaaBaaaleaapeGaam4yaiaadwgaa8aabeaaaaGcpeGaaiilaaqa aiaadggacaWGUbGaamizaiaabccacqaH8oqBpaWaaSbaaSqaa8qaca aIZaaapaqabaGcpeGaeyypa0ZaaSaaa8aabaWdbiaad2gapaWaaSba aSqaa8qacaWGObGaamyzaaWdaeqaaaGcbaWdbiaad2gapaWaaSbaaS qaa8qacaWGJbGaamyzaaWdaeqaaaaak8qacqGH9aqpcaaIXaaaaaa@D155@ .

The number of charges residing on the dust grains, Zd, is defined as z d = ψ ψ 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadsgaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaeqiYdKhapaqaa8qacqaHipqEpaWaaSbaaSqaa8qaca aIWaaapaqabaaaaaaa@3E69@ [50]; where ψ 0 =ψ ϕ=0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiYdK3damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabg2da9iab eI8a5naabmaapaqaa8qacqaHvpGzcqGH9aqpcaaIWaaacaGLOaGaay zkaaaaaa@4109@ is the dust surface floating potential corresponding to the unperturbed plasma potential and is determined from the following expression,

b H δ H 1s ψ 0 + b 2 δ 2 1 β H2 s ψ 0 b 1 δ 1 e s β H1 ψ 0 b κ ce δ ce 1 2 β H s ψ 0 2 κ ce 3 κ ce +1 b κ he b 3 δ he 1 2 β H3 s ψ 0 2 κ he 3 κ he +1 =0                 (16) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacaWGIbWdamaaBaaaleaapeGaamisaaWdaeqaaOWdbiabes7a K9aadaWgaaWcbaWdbiaadIeaa8aabeaak8qadaqadaWdaeaapeGaaG ymaiabgkHiTiaadohacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapaqa baaak8qacaGLOaGaayzkaaGaey4kaSIaamOya8aadaWgaaWcbaWdbi aaikdaa8aabeaak8qacqaH0oazpaWaaSbaaSqaa8qacaaIYaaapaqa baGcpeWaaeWaa8aabaWdbiaaigdacqGHsislcqaHYoGypaWaaSbaaS qaa8qacaWGibGaaGOmaaWdaeqaaOWdbiaadohacqaHipqEpaWaaSba aSqaa8qacaaIWaaapaqabaaak8qacaGLOaGaayzkaaGaeyOeI0Iaam Oya8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqaH0oazpaWaaSba aSqaa8qacaaIXaaapaqabaGcpeGaamyza8aadaahaaWcbeqaa8qaca WGZbGaeqOSdi2damaaBaaameaapeGaamisaiaaigdaa8aabeaal8qa cqaHipqEpaWaaSbaaWqaa8qacaaIWaaapaqabaaaaOWdbiabgkHiTi aadkgapaWaaSbaaSqaa8qacqaH6oWApaWaaSbaaWqaa8qacaWGJbGa amyzaaWdaeqaaaWcbeaak8qacqaH0oazpaWaaSbaaSqaa8qacaWGJb GaamyzaaWdaeqaaOWdbmaabmaapaqaa8qacaaIXaGaeyOeI0YaaSaa a8aabaWdbiaaikdacqaHYoGypaWaaSbaaSqaa8qacaWGibaapaqaba GcpeGaam4CaiabeI8a59aadaWgaaWcbaWdbiaaicdaa8aabeaaaOqa a8qadaqadaWdaeaapeGaaGOmaiabeQ7aR9aadaWgaaWcbaWdbiaado gacaWGLbaapaqabaGcpeGaeyOeI0IaaG4maaGaayjkaiaawMcaaaaa aiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaeyOeI0IaeqOUdS2dam aaBaaameaapeGaam4yaiaadwgaa8aabeaal8qacqGHRaWkcaaIXaaa aaGcpaqaa8qacqGHsislcaWGIbWdamaaBaaaleaapeGaeqOUdS2dam aaBaaameaapeGaamiAaiaadwgaa8aabeaaaSqabaGcpeGaamOya8aa daWgaaWcbaWdbiaaiodaa8aabeaak8qacqaH0oazpaWaaSbaaSqaa8 qacaWGObGaamyzaaWdaeqaaOWdbmaabmaapaqaa8qacaaIXaGaeyOe I0YaaSaaa8aabaWdbiaaikdacqaHYoGypaWaaSbaaSqaa8qacaWGib GaaG4maaWdaeqaaOWdbiaadohacqaHipqEpaWaaSbaaSqaa8qacaaI WaaapaqabaaakeaapeWaaeWaa8aabaWdbiaaikdacqaH6oWApaWaaS baaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbiabgkHiTiaaiodaaiaa wIcacaGLPaaaaaaacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiabgk HiTiabeQ7aR9aadaWgaaadbaWdbiaadIgacaWGLbaapaqabaWcpeGa ey4kaSIaaGymaaaakiabg2da9iaaicdacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaae OnaiaabMcaaaaa@BC10@

Zd can be expanded in terms of φ as follows

   z d =1+ γ 1 ϕ+ γ 2 ϕ 2 +         (17) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiiOaiaacckacaWG6bWdamaaBaaaleaapeGaamizaaWdaeqaaOWd biabg2da9iaaigdacqGHRaWkcqaHZoWzpaWaaSbaaSqaa8qacaaIXa aapaqabaGcpeGaeqy1dyMaey4kaSIaeq4SdC2damaaBaaaleaapeGa aGOmaaWdaeqaaOWdbiabew9aM9aadaahaaWcbeqaa8qacaaIYaaaaO Gaey4kaSIaeS47IWKaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG3aGaaeykaaaa@53D0@

where γ 1 = ψ 0 ' ψ 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4SdC2damaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacqaHipqEpaWaa0baaSqaa8qacaaIWaaapaqaa8qaca GGNaaaaaGcpaqaa8qacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapaqa baaaaaaa@40BD@ and γ 2 = ψ 0 '' 2 ψ 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4SdC2damaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacaqGipWdamaaDaaaleaapeGaaGimaaWdaeaapeGaae 4jaiaabEcaaaaak8aabaWdbiaaikdacqaHipqEpaWaaSbaaSqaa8qa caaIWaaapaqabaaaaaaa@41A3@ come from expanding ψ near ψ0.

Taking the derivative of equation (15) with respect to φ and finding its value at φ = 0 gives,

ψ 0 ' = γ b γ a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiYdK3damaaDaaaleaapeGaaGimaaWdaeaapeGaai4jaaaakiab g2da9maalaaapaqaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGIbaapa qabaaakeaapeGaeq4SdC2damaaBaaaleaapeGaamyyaaWdaeqaaaaa aaa@40CF@ and hence,
γ 1 = γ b ψ 0 γ a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4SdC2damaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGIbaapaqabaaake aapeGaeqiYdK3damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabeo7a N9aadaWgaaWcbaWdbiaadggaa8aabeaaaaaaaa@42F9@
of which
γ a = b H δ H + b 2 δ 2 β H2 + b 1 δ 1 β H1 e s β H1 ψ 0 + b κ ce δ ce β H κ ce 1 κ ce 3 2 1 2 β H s ψ 0 2 κ ce 3 κ ce + b κ he b 3 δ he β H3 κ he 1 κ he 3 2 1 2 β H3 s ψ 0 2 κ he 3 κ he         (18) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGHbaapaqabaGcpeGaeyyp a0JaamOya8aadaWgaaWcbaWdbiaadIeaa8aabeaak8qacqaH0oazpa WaaSbaaSqaa8qacaWGibaapaqabaGcpeGaey4kaSIaamOya8aadaWg aaWcbaWdbiaaikdaa8aabeaak8qacqaH0oazpaWaaSbaaSqaa8qaca aIYaaapaqabaGcpeGaeqOSdi2damaaBaaaleaapeGaamisaiaaikda a8aabeaak8qacqGHRaWkcaWGIbWdamaaBaaaleaapeGaaGymaaWdae qaaOWdbiabes7aK9aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqaH YoGypaWaaSbaaSqaa8qacaWGibGaaGymaaWdaeqaaOWdbiaadwgapa WaaWbaaSqabeaapeGaam4Caiabek7aI9aadaWgaaadbaWdbiaadIea caaIXaaapaqabaWcpeGaeqiYdK3damaaBaaameaapeGaaGimaaWdae qaaaaaaOqaa8qacqGHRaWkdaWcaaWdaeaapeGaamOya8aadaWgaaWc baWdbiabeQ7aR9aadaWgaaadbaWdbiaadogacaWGLbaapaqabaaale qaaOWdbiabes7aK9aadaWgaaWcbaWdbiaadogacaWGLbaapaqabaGc peGaeqOSdi2damaaBaaaleaapeGaamisaaWdaeqaaOWdbmaabmaapa qaa8qacqaH6oWApaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWd biabgkHiTiaaigdaaiaawIcacaGLPaaaa8aabaWdbmaabmaapaqaa8 qacqaH6oWApaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiab gkHiTmaalaaapaqaa8qacaaIZaaapaqaa8qacaaIYaaaaaGaayjkai aawMcaaaaadaqadaWdaeaapeGaaGymaiabgkHiTmaalaaapaqaa8qa caaIYaGaeqOSdi2damaaBaaaleaapeGaamisaaWdaeqaaOWdbiaado hacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapaqabaaakeaapeWaaeWa a8aabaWdbiaaikdacqaH6oWApaWaaSbaaSqaa8qacaWGJbGaamyzaa WdaeqaaOWdbiabgkHiTiaaiodaaiaawIcacaGLPaaaaaaacaGLOaGa ayzkaaWdamaaCaaaleqabaWdbiabgkHiTiabeQ7aR9aadaWgaaadba WdbiaadogacaWGLbaapaqabaaaaOWdbiabgUcaRmaalaaapaqaa8qa caWGIbWdamaaBaaaleaapeGaeqOUdS2damaaBaaameaapeGaamiAai aadwgaa8aabeaaaSqabaGcpeGaamOya8aadaWgaaWcbaWdbiaaioda a8aabeaak8qacqaH0oazpaWaaSbaaSqaa8qacaWGObGaamyzaaWdae qaaOWdbiabek7aI9aadaWgaaWcbaWdbiaadIeacaaIZaaapaqabaGc peWaaeWaa8aabaWdbiabeQ7aR9aadaWgaaWcbaWdbiaadIgacaWGLb aapaqabaGcpeGaeyOeI0IaaGymaaGaayjkaiaawMcaaaWdaeaapeWa aeWaa8aabaWdbiabeQ7aR9aadaWgaaWcbaWdbiaadIgacaWGLbaapa qabaGcpeGaeyOeI0YaaSaaa8aabaWdbiaaiodaa8aabaWdbiaaikda aaaacaGLOaGaayzkaaaaamaabmaapaqaa8qacaaIXaGaeyOeI0YaaS aaa8aabaWdbiaaikdacqaHYoGypaWaaSbaaSqaa8qacaWGibGaaG4m aaWdaeqaaOWdbiaadohacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapa qabaaakeaapeWaaeWaa8aabaWdbiaaikdacqaH6oWApaWaaSbaaSqa a8qacaWGObGaamyzaaWdaeqaaOWdbiabgkHiTiaaiodaaiaawIcaca GLPaaaaaaacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiabgkHiTiab eQ7aR9aadaWgaaadbaWdbiaadIgacaWGLbaapaqabaaaaOGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGa aeymaiaabIdacaqGPaaaaaa@D072@
γ b = b H δ H 1s ψ 0 b 1 δ 1 β H1 e β H1 s ψ 0 b 2 δ 2 β H2 1 β H2 s ψ 0 b κ ce δ ce β H κ ce 1 2 κ ce 3 2 1 2 β H s ψ 0 2 κ ce 3 κ ce +1 b κ he b 3 δ he β H3 κ he 1 2 κ he 3 2 1 2 β H3 s ψ 0 2 κ he 3 κ he +1                (19) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGIbaapaqabaGcpeGaeyyp a0JaeyOeI0IaamOya8aadaWgaaWcbaWdbiaadIeaa8aabeaak8qacq aH0oazpaWaaSbaaSqaa8qacaWGibaapaqabaGcpeWaaeWaa8aabaWd biaaigdacqGHsislcaWGZbGaeqiYdK3damaaBaaaleaapeGaaGimaa WdaeqaaaGcpeGaayjkaiaawMcaaiabgkHiTiaadkgapaWaaSbaaSqa a8qacaaIXaaapaqabaGcpeGaeqiTdq2damaaBaaaleaapeGaaGymaa WdaeqaaOWdbiabek7aI9aadaWgaaWcbaWdbiaadIeacaaIXaaapaqa baGcpeGaamyza8aadaahaaWcbeqaa8qacqaHYoGypaWaaSbaaWqaa8 qacaWGibGaaGymaaWdaeqaaSWdbiaadohacqaHipqEpaWaaSbaaWqa a8qacaaIWaaapaqabaaaaOWdbiabgkHiTiaadkgapaWaaSbaaSqaa8 qacaaIYaaapaqabaGcpeGaeqiTdq2damaaBaaaleaapeGaaGOmaaWd aeqaaOWdbiabek7aI9aadaWgaaWcbaWdbiaadIeacaaIYaaapaqaba GcpeWaaeWaa8aabaWdbiaaigdacqGHsislcqaHYoGypaWaaSbaaSqa a8qacaWGibGaaGOmaaWdaeqaaOWdbiaadohacqaHipqEpaWaaSbaaS qaa8qacaaIWaaapaqabaaak8qacaGLOaGaayzkaaaabaGaeyOeI0Ya aSaaa8aabaWdbiaadkgapaWaaSbaaSqaa8qacqaH6oWApaWaaSbaaW qaa8qacaWGJbGaamyzaaWdaeqaaaWcbeaak8qacqaH0oazpaWaaSba aSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabek7aI9aadaWgaaWcba WdbiaadIeaa8aabeaak8qadaqadaWdaeaapeGaeqOUdS2damaaBaaa leaapeGaam4yaiaadwgaa8aabeaak8qacqGHsisldaWcaaWdaeaape GaaGymaaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaaa8aabaWdbmaa bmaapaqaa8qacqaH6oWApaWaaSbaaSqaa8qacaWGJbGaamyzaaWdae qaaOWdbiabgkHiTmaalaaapaqaa8qacaaIZaaapaqaa8qacaaIYaaa aaGaayjkaiaawMcaaaaadaqadaWdaeaapeGaaGymaiabgkHiTmaala aapaqaa8qacaaIYaGaeqOSdi2damaaBaaaleaapeGaamisaaWdaeqa aOWdbiaadohacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapaqabaaake aapeWaaeWaa8aabaWdbiaaikdacqaH6oWApaWaaSbaaSqaa8qacaWG JbGaamyzaaWdaeqaaOWdbiabgkHiTiaaiodaaiaawIcacaGLPaaaaa aacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiabgkHiTiabeQ7aR9aa daWgaaadbaWdbiaadogacaWGLbaapaqabaWcpeGaey4kaSIaaGymaa aakiabgkHiTmaalaaapaqaa8qacaWGIbWdamaaBaaaleaapeGaeqOU dS2damaaBaaameaapeGaamiAaiaadwgaa8aabeaaaSqabaGcpeGaam Oya8aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacqaH0oazpaWaaSba aSqaa8qacaWGObGaamyzaaWdaeqaaOWdbiabek7aI9aadaWgaaWcba WdbiaadIeacaaIZaaapaqabaGcpeWaaeWaa8aabaWdbiabeQ7aR9aa daWgaaWcbaWdbiaadIgacaWGLbaapaqabaGcpeGaeyOeI0YaaSaaa8 aabaWdbiaaigdaa8aabaWdbiaaikdaaaaacaGLOaGaayzkaaaapaqa a8qadaqadaWdaeaapeGaeqOUdS2damaaBaaaleaapeGaamiAaiaadw gaa8aabeaak8qacqGHsisldaWcaaWdaeaapeGaaG4maaWdaeaapeGa aGOmaaaaaiaawIcacaGLPaaaaaWaaeWaa8aabaWdbiaaigdacqGHsi sldaWcaaWdaeaapeGaaGOmaiabek7aI9aadaWgaaWcbaWdbiaadIea caaIZaaapaqabaGcpeGaam4CaiabeI8a59aadaWgaaWcbaWdbiaaic daa8aabeaaaOqaa8qadaqadaWdaeaapeGaaGOmaiabeQ7aR9aadaWg aaWcbaWdbiaadIgacaWGLbaapaqabaGcpeGaeyOeI0IaaG4maaGaay jkaiaawMcaaaaaaiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaeyOe I0IaeqOUdS2damaaBaaameaapeGaamiAaiaadwgaa8aabeaal8qacq GHRaWkcaaIXaaaaOWdaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabIcacaqGXaGaaeyoaiaabMcaaaaa@EDAB@

Taking the second derivative of (15), with respect to φ , and finding its value at φ = 0 further gives,


ψ 0 '' = γ c γ a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiYd8aadaqhaaWcbaWdbiaaicdaa8aabaWdbiaabEcacaqGNaaa aOGaeyypa0ZaaSaaa8aabaWdbiabeo7aN9aadaWgaaWcbaWdbiaado gaa8aabeaaaOqaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGHbaapaqa baaaaaaa@40F9@
And, therefore, γ 2 = γ c 2 ψ 0 γ a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4SdC2damaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGJbaapaqabaaake aapeGaaGOmaiabeI8a59aadaWgaaWcbaWdbiaaicdaa8aabeaak8qa cqaHZoWzpaWaaSbaaSqaa8qacaWGHbaapaqabaaaaaaa@43B7@
where
γ c = γ c1 + γ c2 + γ c3 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4SdC2damaaBaaaleaapeGaam4yaaWdaeqaaOWdbiabg2da9iab eo7aN9aadaWgaaWcbaWdbiaadogacaaIXaaapaqabaGcpeGaey4kaS Iaeq4SdC2damaaBaaaleaapeGaam4yaiaaikdaa8aabeaak8qacqGH RaWkcqaHZoWzpaWaaSbaaSqaa8qacaWGJbGaaG4maaWdaeqaaaaa@46F9@ with

γ c1 =s b H δ H 1s ψ 0 + b 2 δ 2 β H2 2 1 β H2 s ψ 0 b 1 δ 1 β H1 2 e s β H1 ψ 0 b κ ce δ ce β H 2 κ ce 2 1 4 ( κ ce 3 2 ) 2 × 1 2 β H s ψ 0 2 κ ce 3 κ ce +1 b κ he b 3 δ he β H3 2 κ he 2 1 4 ( κ he 3 2 ) 2 1 2 β H3 s ψ 0 2 κ he 3 κ he +1                      (20) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGJbGaaGymaaWdaeqaaOWd biabg2da9iaadohadaWabaWdaeaapeGaamOya8aadaWgaaWcbaWdbi aadIeaa8aabeaak8qacqaH0oazpaWaaSbaaSqaa8qacaWGibaapaqa baGcpeWaaeWaa8aabaWdbiaaigdacqGHsislcaWGZbGaeqiYdK3dam aaBaaaleaapeGaaGimaaWdaeqaaaGcpeGaayjkaiaawMcaaiabgUca RiaadkgapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeqiTdq2dam aaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabek7aI9aadaqhaaWcbaWd biaadIeacaaIYaaapaqaa8qacaaIYaaaaOWaaeWaa8aabaWdbiaaig dacqGHsislcqaHYoGypaWaaSbaaSqaa8qacaWGibGaaGOmaaWdaeqa aOWdbiaadohacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapaqabaaak8 qacaGLOaGaayzkaaGaeyOeI0IaamOya8aadaWgaaWcbaWdbiaaigda a8aabeaak8qacqaH0oazpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpe GaeqOSdi2damaaDaaaleaapeGaamisaiaaigdaa8aabaWdbiaaikda aaGccaWGLbWdamaaCaaaleqabaWdbiaadohacqaHYoGypaWaaSbaaW qaa8qacaWGibGaaGymaaWdaeqaaSWdbiabeI8a59aadaWgaaadbaWd biaaicdaa8aabeaaaaGcpeGaeyOeI0YaaSaaa8aabaWdbiaadkgapa WaaSbaaSqaa8qacqaH6oWApaWaaSbaaWqaa8qacaWGJbGaamyzaaWd aeqaaaWcbeaak8qacqaH0oazpaWaaSbaaSqaa8qacaWGJbGaamyzaa WdaeqaaOWdbiabek7aI9aadaqhaaWcbaWdbiaadIeaa8aabaWdbiaa ikdaaaGcdaqadaWdaeaapeGaeqOUdS2damaaDaaaleaapeGaam4yai aadwgaa8aabaWdbiaaikdaaaGccqGHsisldaWcaaWdaeaapeGaaGym aaWdaeaapeGaaGinaaaaaiaawIcacaGLPaaaa8aabaWdbiaacIcacq aH6oWApaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabgkHi Tmaalaaapaqaa8qacaaIZaaapaqaa8qacaaIYaaaaiaacMcapaWaaW baaSqabeaapeGaaGOmaaaaaaaakiaawUfaaaqaaiabgEna0oaabmaa paqaa8qacaaIXaGaeyOeI0YaaSaaa8aabaWdbiaaikdacqaHYoGypa WaaSbaaSqaa8qacaWGibaapaqabaGcpeGaam4CaiabeI8a59aadaWg aaWcbaWdbiaaicdaa8aabeaaaOqaa8qadaqadaWdaeaapeGaaGOmai abeQ7aR9aadaWgaaWcbaWdbiaadogacaWGLbaapaqabaGcpeGaeyOe I0IaaG4maaGaayjkaiaawMcaaaaaaiaawIcacaGLPaaapaWaaWbaaS qabeaapeGaeyOeI0IaeqOUdS2damaaBaaameaapeGaam4yaiaadwga a8aabeaal8qacqGHRaWkcaaIXaaaaOGaeyOeI0YaaSaaa8aabaWdbi aadkgapaWaaSbaaSqaa8qacqaH6oWApaWaaSbaaWqaa8qacaWGObGa amyzaaWdaeqaaaWcbeaak8qacaWGIbWdamaaBaaaleaapeGaaG4maa WdaeqaaOWdbiabes7aK9aadaWgaaWcbaWdbiaadIgacaWGLbaapaqa baGcpeGaeqOSdi2damaaDaaaleaapeGaamisaiaaiodaa8aabaWdbi aaikdaaaGcdaqadaWdaeaapeGaeqOUdS2damaaDaaaleaapeGaamiA aiaadwgaa8aabaWdbiaaikdaaaGccqGHsisldaWcaaWdaeaapeGaaG ymaaWdaeaapeGaaGinaaaaaiaawIcacaGLPaaaa8aabaWdbiaacIca cqaH6oWApaWaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbiabgk HiTmaalaaapaqaa8qacaaIZaaapaqaa8qacaaIYaaaaiaacMcapaWa aWbaaSqabeaapeGaaGOmaaaaaaGcdaqadaWdaeaapeGaaGymaiabgk HiTmaalaaapaqaa8qacaaIYaGaeqOSdi2damaaBaaaleaapeGaamis aiaaiodaa8aabeaak8qacaWGZbGaeqiYdK3damaaBaaaleaapeGaaG imaaWdaeqaaaGcbaWdbmaabmaapaqaa8qacaaIYaGaeqOUdS2damaa BaaaleaapeGaamiAaiaadwgaa8aabeaak8qacqGHsislcaaIZaaaca GLOaGaayzkaaaaaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacqGH sislcqaH6oWApaWaaSbaaWqaa8qacaWGObGaamyzaaWdaeqaaSWdbi abgUcaRiaaigdaaaGcpaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeikaiaabkdacaqGWaGaaeykaaaaaa@FB5A@
γ c2 =2s γ 1 ψ 0 b H δ H + b 2 δ 2 β H2 2 b 1 δ 1 β H1 2 e s β H1 ψ 0 b κ ce δ ce β H 2 κ ce 1 κ ce 1 2 ( κ ce 3 2 ) 2 × 1 2 β H s ψ 0 2 κ ce 3 κ ce b κ he b 3 δ he β H3 2 κ he 1 κ he 1 2 ( κ he 3 2 ) 2 1 2 β H3 s ψ 0 2 κ he 3 κ he                (21) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGJbGaaGOmaaWdaeqaaOWd biabg2da9iaaikdacaWGZbWaaeWaa8aabaWdbiabeo7aN9aadaWgaa WcbaWdbiaaigdaa8aabeaak8qacqaHipqEpaWaaSbaaSqaa8qacaaI Waaapaqabaaak8qacaGLOaGaayzkaaWaamqaa8aabaWdbiaadkgapa WaaSbaaSqaa8qacaWGibaapaqabaGcpeGaeqiTdq2damaaBaaaleaa peGaamisaaWdaeqaaOWdbiabgUcaRiaadkgapaWaaSbaaSqaa8qaca aIYaaapaqabaGcpeGaeqiTdq2damaaBaaaleaapeGaaGOmaaWdaeqa aOWdbiabek7aI9aadaqhaaWcbaWdbiaadIeacaaIYaaapaqaa8qaca aIYaaaaOGaeyOeI0IaamOya8aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacqaH0oazpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeqOSdi 2damaaDaaaleaapeGaamisaiaaigdaa8aabaWdbiaaikdaaaGccaWG LbWdamaaCaaaleqabaWdbiaadohacqaHYoGypaWaaSbaaWqaa8qaca WGibGaaGymaaWdaeqaaSWdbiabeI8a59aadaWgaaadbaWdbiaaicda a8aabeaaaaGcpeGaeyOeI0YaaSaaa8aabaWdbiaadkgapaWaaSbaaS qaa8qacqaH6oWApaWaaSbaaWqaa8qacaWGJbGaamyzaaWdaeqaaaWc beaak8qacqaH0oazpaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaO Wdbiabek7aI9aadaqhaaWcbaWdbiaadIeaa8aabaWdbiaaikdaaaGc daqadaWdaeaapeGaeqOUdS2damaaBaaaleaapeGaam4yaiaadwgaa8 aabeaak8qacqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaa8aabaWd biabeQ7aR9aadaWgaaWcbaWdbiaadogacaWGLbaapaqabaGcpeGaey OeI0YaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaaacaGLOaGa ayzkaaaapaqaa8qacaGGOaGaeqOUdS2damaaBaaaleaapeGaam4yai aadwgaa8aabeaak8qacqGHsisldaWcaaWdaeaapeGaaG4maaWdaeaa peGaaGOmaaaacaGGPaWdamaaCaaaleqabaWdbiaaikdaaaaaaaGcca GLBbaaaeaacqGHxdaTdaqadaWdaeaapeGaaGymaiabgkHiTmaalaaa paqaa8qacaaIYaGaeqOSdi2damaaBaaaleaapeGaamisaaWdaeqaaO WdbiaadohacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapaqabaaakeaa peWaaeWaa8aabaWdbiaaikdacqaH6oWApaWaaSbaaSqaa8qacaWGJb GaamyzaaWdaeqaaOWdbiabgkHiTiaaiodaaiaawIcacaGLPaaaaaaa caGLOaGaayzkaaWdamaaCaaaleqabaWdbiabgkHiTiabeQ7aR9aada WgaaadbaWdbiaadogacaWGLbaapaqabaaaaOWdbiabgkHiTmaalaaa paqaa8qacaWGIbWdamaaBaaaleaapeGaeqOUdS2damaaBaaameaape GaamiAaiaadwgaa8aabeaaaSqabaGcpeGaamOya8aadaWgaaWcbaWd biaaiodaa8aabeaak8qacqaH0oazpaWaaSbaaSqaa8qacaWGObGaam yzaaWdaeqaaOWdbiabek7aI9aadaqhaaWcbaWdbiaadIeacaaIZaaa paqaa8qacaaIYaaaaOWaaeWaa8aabaWdbiabeQ7aR9aadaWgaaWcba WdbiaadIgacaWGLbaapaqabaGcpeGaeyOeI0IaaGymaaGaayjkaiaa wMcaamaabmaapaqaa8qacqaH6oWApaWaaSbaaSqaa8qacaWGObGaam yzaaWdaeqaaOWdbiabgkHiTmaalaaapaqaa8qacaaIXaaapaqaa8qa caaIYaaaaaGaayjkaiaawMcaaaWdaeaapeGaaiikaiabeQ7aR9aada WgaaWcbaWdbiaadIgacaWGLbaapaqabaGcpeGaeyOeI0YaaSaaa8aa baWdbiaaiodaa8aabaWdbiaaikdaaaGaaiyka8aadaahaaWcbeqaa8 qacaaIYaaaaaaakmaabmaapaqaa8qacaaIXaGaeyOeI0YaaSaaa8aa baWdbiaaikdacqaHYoGypaWaaSbaaSqaa8qacaWGibGaaG4maaWdae qaaOWdbiaadohacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapaqabaaa keaapeWaaeWaa8aabaWdbiaaikdacqaH6oWApaWaaSbaaSqaa8qaca WGObGaamyzaaWdaeqaaOWdbiabgkHiTiaaiodaaiaawIcacaGLPaaa aaaacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiabgkHiTiabeQ7aR9 aadaWgaaadbaWdbiaadIgacaWGLbaapaqabaaaaOGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqGXaGa aeykaaaaaa@F742@

and

γ c3 =s ( γ 1 ψ 0 ) 2 b 1 δ 1 β H1 2 e s β H1 ψ 0 + b κ ce δ ce β H 2 κ ce κ ce 1 ( κ ce 3 2 ) 2 1 2 β H s ψ 0 2 κ ce 3 κ ce 1 + b κ he b 3 δ he β H3 2 κ he κ he 1 ( κ he 3 2 ) 2 1 2 β H3 s ψ 0 2 κ he 3 κ he 1          (22) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGJbGaaG4maaWdaeqaaOWd biabg2da9iabgkHiTiaadohacaGGOaGaeq4SdC2damaaBaaaleaape GaaGymaaWdaeqaaOWdbiabeI8a59aadaWgaaWcbaWdbiaaicdaa8aa beaak8qacaGGPaWdamaaCaaaleqabaWdbiaaikdaaaGcdaWabaWdae aapeGaamOya8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqaH0oaz paWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeqOSdi2damaaDaaale aapeGaamisaiaaigdaa8aabaWdbiaaikdaaaGccaWGLbWdamaaCaaa leqabaWdbiaadohacqaHYoGypaWaaSbaaWqaa8qacaWGibGaaGymaa WdaeqaaSWdbiabeI8a59aadaWgaaadbaWdbiaaicdaa8aabeaaaaGc peGaey4kaSYaaSaaa8aabaWdbiaadkgapaWaaSbaaSqaa8qacqaH6o WApaWaaSbaaWqaa8qacaWGJbGaamyzaaWdaeqaaaWcbeaak8qacqaH 0oazpaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabek7aI9 aadaqhaaWcbaWdbiaadIeaa8aabaWdbiaaikdaaaGccqaH6oWApaWa aSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbmaabmaapaqaa8qacq aH6oWApaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabgkHi TiaaigdaaiaawIcacaGLPaaaa8aabaWdbiaacIcacqaH6oWApaWaaS baaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabgkHiTmaalaaapaqa a8qacaaIZaaapaqaa8qacaaIYaaaaiaacMcapaWaaWbaaSqabeaape GaaGOmaaaaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTmaalaaapaqa a8qacaaIYaGaeqOSdi2damaaBaaaleaapeGaamisaaWdaeqaaOWdbi aadohacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapaqabaaakeaapeWa aeWaa8aabaWdbiaaikdacqaH6oWApaWaaSbaaSqaa8qacaWGJbGaam yzaaWdaeqaaOWdbiabgkHiTiaaiodaaiaawIcacaGLPaaaaaaacaGL OaGaayzkaaWdamaaCaaaleqabaWdbiabgkHiTiabeQ7aR9aadaWgaa adbaWdbiaadogacaWGLbaapaqabaWcpeGaeyOeI0IaaGymaaaaaOGa ay5waaaabaGaey4kaSYaaSaaa8aabaWdbiaadkgapaWaaSbaaSqaa8 qacqaH6oWApaWaaSbaaWqaa8qacaWGObGaamyzaaWdaeqaaaWcbeaa k8qacaWGIbWdamaaBaaaleaapeGaaG4maaWdaeqaaOWdbiabes7aK9 aadaWgaaWcbaWdbiaadIgacaWGLbaapaqabaGcpeGaeqOSdi2damaa DaaaleaapeGaamisaiaaiodaa8aabaWdbiaaikdaaaGccqaH6oWApa WaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbmaabmaapaqaa8qa cqaH6oWApaWaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbiabgk HiTiaaigdaaiaawIcacaGLPaaaa8aabaWdbiaacIcacqaH6oWApaWa aSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbiabgkHiTmaalaaapa qaa8qacaaIZaaapaqaa8qacaaIYaaaaiaacMcapaWaaWbaaSqabeaa peGaaGOmaaaaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTmaalaaapa qaa8qacaaIYaGaeqOSdi2damaaBaaaleaapeGaamisaiaaiodaa8aa beaak8qacaWGZbGaeqiYdK3damaaBaaaleaapeGaaGimaaWdaeqaaa GcbaWdbmaabmaapaqaa8qacaaIYaGaeqOUdS2damaaBaaaleaapeGa amiAaiaadwgaa8aabeaak8qacqGHsislcaaIZaaacaGLOaGaayzkaa aaaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacqGHsislcqaH6oWA paWaaSbaaWqaa8qacaWGObGaamyzaaWdaeqaaSWdbiabgkHiTiaaig daaaGcpaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeikaiaabkdacaqGYaGaaeykaaaaaa@DCE6@
γ c3 =s ( γ 1 ψ 0 ) 2 b 1 δ 1 β H1 2 e s β H1 ψ 0 + b κ ce δ ce β H 2 κ ce κ ce 1 ( κ ce 3 2 ) 2 1 2 β H s ψ 0 2 κ ce 3 κ ce 1 + b κ he b 3 δ he β H3 2 κ he κ he 1 ( κ he 3 2 ) 2 1 2 β H3 s ψ 0 2 κ he 3 κ he 1          (22) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqaHZoWzpaWaaSbaaSqaa8qacaWGJbGaaG4maaWdaeqaaOWd biabg2da9iabgkHiTiaadohacaGGOaGaeq4SdC2damaaBaaaleaape GaaGymaaWdaeqaaOWdbiabeI8a59aadaWgaaWcbaWdbiaaicdaa8aa beaak8qacaGGPaWdamaaCaaaleqabaWdbiaaikdaaaGcdaWabaWdae aapeGaamOya8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqaH0oaz paWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeqOSdi2damaaDaaale aapeGaamisaiaaigdaa8aabaWdbiaaikdaaaGccaWGLbWdamaaCaaa leqabaWdbiaadohacqaHYoGypaWaaSbaaWqaa8qacaWGibGaaGymaa WdaeqaaSWdbiabeI8a59aadaWgaaadbaWdbiaaicdaa8aabeaaaaGc peGaey4kaSYaaSaaa8aabaWdbiaadkgapaWaaSbaaSqaa8qacqaH6o WApaWaaSbaaWqaa8qacaWGJbGaamyzaaWdaeqaaaWcbeaak8qacqaH 0oazpaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabek7aI9 aadaqhaaWcbaWdbiaadIeaa8aabaWdbiaaikdaaaGccqaH6oWApaWa aSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbmaabmaapaqaa8qacq aH6oWApaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabgkHi TiaaigdaaiaawIcacaGLPaaaa8aabaWdbiaacIcacqaH6oWApaWaaS baaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabgkHiTmaalaaapaqa a8qacaaIZaaapaqaa8qacaaIYaaaaiaacMcapaWaaWbaaSqabeaape GaaGOmaaaaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTmaalaaapaqa a8qacaaIYaGaeqOSdi2damaaBaaaleaapeGaamisaaWdaeqaaOWdbi aadohacqaHipqEpaWaaSbaaSqaa8qacaaIWaaapaqabaaakeaapeWa aeWaa8aabaWdbiaaikdacqaH6oWApaWaaSbaaSqaa8qacaWGJbGaam yzaaWdaeqaaOWdbiabgkHiTiaaiodaaiaawIcacaGLPaaaaaaacaGL OaGaayzkaaWdamaaCaaaleqabaWdbiabgkHiTiabeQ7aR9aadaWgaa adbaWdbiaadogacaWGLbaapaqabaWcpeGaeyOeI0IaaGymaaaaaOGa ay5waaaabaGaey4kaSYaaSaaa8aabaWdbiaadkgapaWaaSbaaSqaa8 qacqaH6oWApaWaaSbaaWqaa8qacaWGObGaamyzaaWdaeqaaaWcbeaa k8qacaWGIbWdamaaBaaaleaapeGaaG4maaWdaeqaaOWdbiabes7aK9 aadaWgaaWcbaWdbiaadIgacaWGLbaapaqabaGcpeGaeqOSdi2damaa DaaaleaapeGaamisaiaaiodaa8aabaWdbiaaikdaaaGccqaH6oWApa WaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbmaabmaapaqaa8qa cqaH6oWApaWaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbiabgk HiTiaaigdaaiaawIcacaGLPaaaa8aabaWdbiaacIcacqaH6oWApaWa aSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbiabgkHiTmaalaaapa qaa8qacaaIZaaapaqaa8qacaaIYaaaaiaacMcapaWaaWbaaSqabeaa peGaaGOmaaaaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTmaalaaapa qaa8qacaaIYaGaeqOSdi2damaaBaaaleaapeGaamisaiaaiodaa8aa beaak8qacaWGZbGaeqiYdK3damaaBaaaleaapeGaaGimaaWdaeqaaa GcbaWdbmaabmaapaqaa8qacaaIYaGaeqOUdS2damaaBaaaleaapeGa amiAaiaadwgaa8aabeaak8qacqGHsislcaaIZaaacaGLOaGaayzkaa aaaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacqGHsislcqaH6oWA paWaaSbaaWqaa8qacaWGObGaamyzaaWdaeqaaSWdbiabgkHiTiaaig daaaGcpaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeikaiaabkdacaqGYaGaaeykaaaaaa@DCE6@

Dust acoustic shock waves - Derivation of KdVB equation:

We shall derive the KdVB equation by employing the reductive perturbation method [51]. The stretched co-ordinates are introduced in the following form [50],

ξ= ϵ 1 2 xλt , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOVdGNaeyypa0Zefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYt UvgaiuGacqWF1pG8paWaaWbaaSqabeaapeWaaSaaa8aabaWdbiaaig daa8aabaWdbiaaikdaaaaaaOWaaeWaa8aabaWdbiaadIhacqGHsisl cqaH7oaBcaWG0baacaGLOaGaayzkaaGaaiilaaaa@4DD7@
τ= ϵ 3 2 t MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdqNaeyypa0Zefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYt UvgaiuGacqWF1pG8paWaaWbaaSqabeaapeWaaSaaa8aabaWdbiaaio daa8aabaWdbiaaikdaaaaaaOGaamiDaaaa@47E5@
η d = ϵ 1 2 η d0         (23) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4TdG2damaaBaaaleaapeGaamizaaWdaeqaaOWdbiabg2da9mrr 1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfiGae8x9di=dam aaCaaaleqabaWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYaaa aaaakiabeE7aO9aadaWgaaWcbaWdbiaadsgacaaIWaaapaqabaGcca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bIcacaqGYaGaae4maiaabMcaaaa@53BB@

where ϵ is a small parameter that measures the size of the perturbation amplitude and λ is the velocity of the shock wave normalized by Cd . In a weak damping situation, the kinematic viscosity of dust ions can be considered small but finite. Here, ηd 0 is a finite parameter.

The various parameters are expanded as a power series in ϵ as

n d =1+ϵ n d 1 + ϵ 2 n d 2 +          (24) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaWgaaWcbaWdbiaadsgaa8aabeaak8qacqGH9aqpcaaI XaGaey4kaSYefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiu GacqWF1pG8caWGUbWdamaaDaaaleaapeGaamizaaWdaeaapeWaaeWa a8aabaWdbiaaigdaaiaawIcacaGLPaaaaaGccqGHRaWkcqWF1pG8pa WaaWbaaSqabeaapeGaaGOmaaaakiaad6gapaWaa0baaSqaa8qacaWG Kbaapaqaa8qadaqadaWdaeaapeGaaGOmaaGaayjkaiaawMcaaaaaki abgUcaRiabl+UimjaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabsdacaqGPa aaaa@60B4@
v d =ϵ v d 1 + ϵ 2 v d 2 +            (25) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODa8aadaWgaaWcbaWdbiaadsgaa8aabeaak8qacqGH9aqptuuD JXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbciab=v=aYlaadA hapaWaa0baaSqaa8qacaWGKbaapaqaa8qadaqadaWdaeaapeGaaGym aaGaayjkaiaawMcaaaaakiabgUcaRiab=v=aY=aadaahaaWcbeqaa8 qacaaIYaaaaOGaamODa8aadaqhaaWcbaWdbiaadsgaa8aabaWdbmaa bmaapaqaa8qacaaIYaaacaGLOaGaayzkaaaaaOGaey4kaSIaeS47IW KaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqG1aGaaeykaa aa@6076@
ϕ=ϵ ϕ 1 + ϵ 2 ϕ 2 +         (26) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dyMaeyypa0Zefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYt UvgaiuGacqWF1pG8cqaHvpGzpaWaaWbaaSqabeaapeWaaeWaa8aaba WdbiaaigdaaiaawIcacaGLPaaaaaGccqGHRaWkcqWF1pG8paWaaWba aSqabeaapeGaaGOmaaaakiabew9aM9aadaahaaWcbeqaa8qadaqada WdaeaapeGaaGOmaaGaayjkaiaawMcaaaaakiabgUcaRiabl+Uimjaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabIcacaqGYaGaaeOnaiaabMcaaaa@5D88@

Also from equation (17),

z d =1+ϵ γ 1 ϕ 1 + ϵ 2 ( γ 1 ϕ 2 + γ 2 ϕ 1 ] 2 +              (27) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadsgaa8aabeaak8qacqGH9aqpcaaI XaGaey4kaSYefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiu GacqWF1pG8daqadaWdaeaapeGaeq4SdC2damaaBaaaleaapeGaaGym aaWdaeqaaOWdbiabew9aM9aadaahaaWcbeqaa8qadaqadaWdaeaape GaaGymaaGaayjkaiaawMcaaaaaaOGaayjkaiaawMcaaiabgUcaRiab =v=aY=aadaahaaWcbeqaa8qacaaIYaaaaOGaaiikaiabeo7aN9aada WgaaWcbaWdbiaaigdaa8aabeaak8qacqaHvpGzpaWaaWbaaSqabeaa peWaaeWaa8aabaWdbiaaikdaaiaawIcacaGLPaaaaaGccqGHRaWkcq aHZoWzpaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeWaaKGea8aabaWd biabew9aM9aadaahaaWcbeqaa8qadaqadaWdaeaapeGaaGymaaGaay jkaiaawMcaaaaakiaac2fapaWaaWbaaSqabeaapeGaaGOmaaaaaOGa ay5waiaawMcaaiabgUcaRiabl+UimjaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeikaiaabkdacaqG3aGaaeykaaaa@7708@

Using transformation equations in equations (5)-(7) and equating different powers of ∈ , the lowest order of ∈ leads to:

n d 1 = ϕ 1 λ 2 ,  v d 1 = ϕ 1 λ           (28) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBa8aadaqhaaWcbaWdbiaadsgaa8aabaWdbmaabmaapaqaa8qa caaIXaaacaGLOaGaayzkaaaaaOGaeyypa0ZaaSaaa8aabaWdbiabgk HiTiabew9aM9aadaahaaWcbeqaa8qadaqadaWdaeaapeGaaGymaaGa ayjkaiaawMcaaaaaaOWdaeaapeGaeq4UdW2damaaCaaaleqabaWdbi aaikdaaaaaaOGaaiilaiaabccacaWG2bWdamaaDaaaleaapeGaamiz aaWdaeaapeWaaeWaa8aabaWdbiaaigdaaiaawIcacaGLPaaaaaGccq GH9aqpdaWcaaWdaeaapeGaeyOeI0Iaeqy1dy2damaaCaaaleqabaWd bmaabmaapaqaa8qacaaIXaaacaGLOaGaayzkaaaaaaGcpaqaa8qacq aH7oaBaaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabIcacaqGYaGaaeioaiaabMcaaaa@5BEF@

Equating terms of power of order ϵ in Poisson’s equation, the linear dispersion relation of the wave can be expressed as:

γ 1 +s δ H + δ 1 β H1 + δ 2 β H2 + δ he κ he 1 2 β H3 κ he 3 2 + δ ce κ ce 1 2 β H κ ce 3 2 λ 2 =1               (29) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaamWaa8aabaWdbiabeo7aN9aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacqGHRaWkcaWGZbWaaeWaa8aabaWdbiabes7aK9aadaWgaaWcba WdbiaadIeaa8aabeaak8qacqGHRaWkcqaH0oazpaWaaSbaaSqaa8qa caaIXaaapaqabaGcpeGaeqOSdi2damaaBaaaleaapeGaamisaiaaig daa8aabeaak8qacqGHRaWkcqaH0oazpaWaaSbaaSqaa8qacaaIYaaa paqabaGcpeGaeqOSdi2damaaBaaaleaapeGaamisaiaaikdaa8aabe aak8qacqGHRaWkdaWcaaWdaeaapeGaeqiTdq2damaaBaaaleaapeGa amiAaiaadwgaa8aabeaak8qadaqadaWdaeaapeGaeqOUdS2damaaBa aaleaapeGaamiAaiaadwgaa8aabeaak8qacqGHsisldaWcaaWdaeaa peGaaGymaaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaacqaHYoGypa WaaSbaaSqaa8qacaWGibGaaG4maaWdaeqaaaGcbaWdbmaabmaapaqa a8qacqaH6oWApaWaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbi abgkHiTmaalaaapaqaa8qacaaIZaaapaqaa8qacaaIYaaaaaGaayjk aiaawMcaaaaacqGHRaWkdaWcaaWdaeaapeGaeqiTdq2damaaBaaale aapeGaam4yaiaadwgaa8aabeaak8qadaqadaWdaeaapeGaeqOUdS2d amaaBaaaleaapeGaam4yaiaadwgaa8aabeaak8qacqGHsisldaWcaa WdaeaapeGaaGymaaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaacqaH YoGypaWaaSbaaSqaa8qacaWGibaapaqabaaakeaapeWaaeWaa8aaba WdbiabeQ7aR9aadaWgaaWcbaWdbiaadogacaWGLbaapaqabaGcpeGa eyOeI0YaaSaaa8aabaWdbiaaiodaa8aabaWdbiaaikdaaaaacaGLOa GaayzkaaaaaaGaayjkaiaawMcaaaGaay5waiaaw2faaiabeU7aS9aa daahaaWcbeqaa8qacaaIYaaaaOGaeyypa0JaaGymaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGYaGaaeyoai aabMcaaaa@92B0@

Next, equating terms of order in ϵ 5 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfiaeaaaaaaaaa8qacqWF1pG8 paWaaWbaaSqabeaapeWaaSaaa8aabaWdbiaaiwdaa8aabaWdbiaaik daaaaaaaaa@4419@ and using equations (28), we get:

n d 2 ξ = 2 λ 3 ϕ 1 τ + 3 λ 4 γ 1 λ 2 ϕ 1 ϕ 1 ξ 1 λ 2 ϕ 2 ξ + η d0 λ 3 2 ϕ 1 ξ 2          (30) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2kaad6gapaWaa0baaSqaa8qacaWGKbaa paqaa8qadaqadaWdaeaapeGaaGOmaaGaayjkaiaawMcaaaaaaOWdae aapeGaeyOaIyRaeqOVdGhaaiabg2da9iabgkHiTmaalaaapaqaa8qa caaIYaaapaqaa8qacqaH7oaBpaWaaWbaaSqabeaapeGaaG4maaaaaa GcdaWcaaWdaeaapeGaeyOaIyRaeqy1dy2damaaCaaaleqabaWdbmaa bmaapaqaa8qacaaIXaaacaGLOaGaayzkaaaaaaGcpaqaa8qacqGHci ITcqaHepaDaaGaey4kaSYaamWaa8aabaWdbmaalaaapaqaa8qacaaI Zaaapaqaa8qacqaH7oaBpaWaaWbaaSqabeaapeGaaGinaaaaaaGccq GHsisldaWcaaWdaeaapeGaeq4SdC2damaaBaaaleaapeGaaGymaaWd aeqaaaGcbaWdbiabeU7aS9aadaahaaWcbeqaa8qacaaIYaaaaaaaaO Gaay5waiaaw2faaiabew9aM9aadaahaaWcbeqaa8qadaqadaWdaeaa peGaaGymaaGaayjkaiaawMcaaaaakmaalaaapaqaa8qacqGHciITcq aHvpGzpaWaaWbaaSqabeaapeWaaeWaa8aabaWdbiaaigdaaiaawIca caGLPaaaaaaak8aabaWdbiabgkGi2kabe67a4baacqGHsisldaWcaa WdaeaapeGaaGymaaWdaeaapeGaeq4UdW2damaaCaaaleqabaWdbiaa ikdaaaaaaOWaaSaaa8aabaWdbiabgkGi2kabew9aM9aadaahaaWcbe qaa8qadaqadaWdaeaapeGaaGOmaaGaayjkaiaawMcaaaaaaOWdaeaa peGaeyOaIyRaeqOVdGhaaiabgUcaRmaalaaapaqaa8qacqaH3oaApa WaaSbaaSqaa8qacaWGKbGaaGimaaWdaeqaaaGcbaWdbiabeU7aS9aa daahaaWcbeqaa8qacaaIZaaaaaaakmaalaaapaqaa8qacqGHciITpa WaaWbaaSqabeaapeGaaGOmaaaakiabew9aM9aadaahaaWcbeqaa8qa daqadaWdaeaapeGaaGymaaGaayjkaiaawMcaaaaaaOWdaeaapeGaey OaIyRaeqOVdG3damaaCaaaleqabaWdbiaaikdaaaaaaOGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae ikaiaabodacaqGWaGaaeykaaaa@93CF@

Simiarly, equating coefficients of terms of order 2 ∈ from Poisson’s equation, we get

2 ϕ 1 ξ 2 = n d 2 + γ 1 ϕ 1) n d 1 + γ 2 [ ϕ 1 ] 2 +P ϕ 2 +Q [ ϕ 1 ] 2 2               (31) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYaaaaOGa eqy1dy2damaaCaaaleqabaWdbmaabmaapaqaa8qacaaIXaaacaGLOa GaayzkaaaaaaGcpaqaa8qacqGHciITcqaH+oaEpaWaaWbaaSqabeaa peGaaGOmaaaaaaGccqGH9aqpcaWGUbWdamaaDaaaleaapeGaamizaa WdaeaapeWaaeWaa8aabaWdbiaaikdaaiaawIcacaGLPaaaaaGccqGH RaWkcqaHZoWzpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeqy1dy 2damaaCaaaleqabaWdbiaaigdacaGGPaaaaOGaamOBa8aadaqhaaWc baWdbiaadsgaa8aabaWdbmaabmaapaqaa8qacaaIXaaacaGLOaGaay zkaaaaaOGaey4kaSIaeq4SdC2damaaBaaaleaapeGaaGOmaaWdaeqa aOWdbiaacUfacqaHvpGzpaWaaWbaaSqabeaapeWaaeWaa8aabaWdbi aaigdaaiaawIcacaGLPaaaaaGccaGGDbWdamaaCaaaleqabaWdbiaa ikdaaaGccqGHRaWkcaWGqbGaeqy1dy2damaaCaaaleqabaWdbmaabm aapaqaa8qacaaIYaaacaGLOaGaayzkaaaaaOGaey4kaSIaamyuamaa laaapaqaa8qacaGGBbGaeqy1dy2damaaCaaaleqabaWdbmaabmaapa qaa8qacaaIXaaacaGLOaGaayzkaaaaaOGaaiyxa8aadaahaaWcbeqa a8qacaaIYaaaaaGcpaqaa8qacaaIYaaaaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeikaiaabodacaqGXaGaaeykaaaa@7A3D@

where


P= γ 1 +s δ H +s δ 1 β H1 +s δ 2 β H2 +s δ he β H3 κ he 1 2 κ he 3 2 +s δ ce β H κ ce 1 2 κ ce 3 2                 (32) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiuaiabg2da9maadmaapaqaa8qacqaHZoWzpaWaaSbaaSqaa8qa caaIXaaapaqabaGcpeGaey4kaSIaam4Caiabes7aK9aadaWgaaWcba WdbiaadIeaa8aabeaak8qacqGHRaWkcaWGZbGaeqiTdq2damaaBaaa leaapeGaaGymaaWdaeqaaOWdbiabek7aI9aadaWgaaWcbaWdbiaadI eacaaIXaaapaqabaGcpeGaey4kaSIaam4Caiabes7aK9aadaWgaaWc baWdbiaaikdaa8aabeaak8qacqaHYoGypaWaaSbaaSqaa8qacaWGib GaaGOmaaWdaeqaaOWdbiabgUcaRiaadohacqaH0oazpaWaaSbaaSqa a8qacaWGObGaamyzaaWdaeqaaOWdbiabek7aI9aadaWgaaWcbaWdbi aadIeacaaIZaaapaqabaGcpeWaaSaaa8aabaWdbmaabmaapaqaa8qa cqaH6oWApaWaaSbaaSqaa8qacaWGObGaamyzaaWdaeqaaOWdbiabgk HiTmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYaaaaaGaayjkaiaa wMcaaaWdaeaapeWaaeWaa8aabaWdbiabeQ7aR9aadaWgaaWcbaWdbi aadIgacaWGLbaapaqabaGcpeGaeyOeI0YaaSaaa8aabaWdbiaaioda a8aabaWdbiaaikdaaaaacaGLOaGaayzkaaaaaiabgUcaRiaadohacq aH0oazpaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqaaOWdbiabek7a I9aadaWgaaWcbaWdbiaadIeaa8aabeaak8qadaWcaaWdaeaapeWaae Waa8aabaWdbiabeQ7aR9aadaWgaaWcbaWdbiaadogacaWGLbaapaqa baGcpeGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaa aacaGLOaGaayzkaaaapaqaa8qadaqadaWdaeaapeGaeqOUdS2damaa BaaaleaapeGaam4yaiaadwgaa8aabeaak8qacqGHsisldaWcaaWdae aapeGaaG4maaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaaaaaacaGL BbGaayzxaaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabIcacaqGZaGaaeOmaiaabMcaaaa@9317@
Q= δ H + δ 1 β H1 2 δ 2 β H2 2 + δ he β H3 2 κ he 2 1 4 ( κ he 3 2 ) 2 + δ ce β H 2 κ ce 2 1 4 ( κ ce 3 2 ) 2                 (33) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyuaiabg2da9maadmaapaqaa8qacqGHsislcqaH0oazpaWaaSba aSqaa8qacaWGibaapaqabaGcpeGaey4kaSIaeqiTdq2damaaBaaale aapeGaaGymaaWdaeqaaOWdbiabek7aI9aadaqhaaWcbaWdbiaadIea caaIXaaapaqaa8qacaaIYaaaaOGaeyOeI0IaeqiTdq2damaaBaaale aapeGaaGOmaaWdaeqaaOWdbiabek7aI9aadaqhaaWcbaWdbiaadIea caaIYaaapaqaa8qacaaIYaaaaOGaey4kaSIaeqiTdq2damaaBaaale aapeGaamiAaiaadwgaa8aabeaak8qacqaHYoGypaWaa0baaSqaa8qa caWGibGaaG4maaWdaeaapeGaaGOmaaaakmaalaaapaqaa8qadaqada WdaeaapeGaeqOUdS2damaaDaaaleaapeGaamiAaiaadwgaa8aabaWd biaaikdaaaGccqGHsisldaWcaaWdaeaapeGaaGymaaWdaeaapeGaaG inaaaaaiaawIcacaGLPaaaa8aabaWdbiaacIcacqaH6oWApaWaaSba aSqaa8qacaWGObGaamyzaaWdaeqaaOWdbiabgkHiTmaalaaapaqaa8 qacaaIZaaapaqaa8qacaaIYaaaaiaacMcapaWaaWbaaSqabeaapeGa aGOmaaaaaaGccqGHRaWkcqaH0oazpaWaaSbaaSqaa8qacaWGJbGaam yzaaWdaeqaaOWdbiabek7aI9aadaqhaaWcbaWdbiaadIeaa8aabaWd biaaikdaaaGcdaWcaaWdaeaapeWaaeWaa8aabaWdbiabeQ7aR9aada qhaaWcbaWdbiaadogacaWGLbaapaqaa8qacaaIYaaaaOGaeyOeI0Ya aSaaa8aabaWdbiaaigdaa8aabaWdbiaaisdaaaaacaGLOaGaayzkaa aapaqaa8qacaGGOaGaeqOUdS2damaaBaaaleaapeGaam4yaiaadwga a8aabeaak8qacqGHsisldaWcaaWdaeaapeGaaG4maaWdaeaapeGaaG OmaaaacaGGPaWdamaaCaaaleqabaWdbiaaikdaaaaaaaGccaGLBbGa ayzxaaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabIcacaqGZaGaae4maiaabMcaaaa@9179@

Taking the derivative of (31) and using equations (28)-(30), the KdVB equation is derived as

ϕ 1 τ +A ϕ 1 ϕ 1 ξ +B 3 ϕ 1 ξ 3 C 2 ϕ 1 ξ 2 =0             (34) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2kabew9aM9aadaahaaWcbeqaa8qadaqa daWdaeaapeGaaGymaaGaayjkaiaawMcaaaaaaOWdaeaapeGaeyOaIy RaeqiXdqhaaiabgUcaRiaadgeacqaHvpGzpaWaaWbaaSqabeaapeWa aeWaa8aabaWdbiaaigdaaiaawIcacaGLPaaaaaGcdaWcaaWdaeaape GaeyOaIyRaeqy1dy2damaaCaaaleqabaWdbmaabmaapaqaa8qacaaI XaaacaGLOaGaayzkaaaaaaGcpaqaa8qacqGHciITcqaH+oaEaaGaey 4kaSIaamOqamaalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGa aG4maaaakiabew9aM9aadaahaaWcbeqaa8qadaqadaWdaeaapeGaaG ymaaGaayjkaiaawMcaaaaaaOWdaeaapeGaeyOaIyRaeqOVdG3damaa CaaaleqabaWdbiaaiodaaaaaaOGaeyOeI0Iaam4qamaalaaapaqaa8 qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiabew9aM9aadaah aaWcbeqaa8qadaqadaWdaeaapeGaaGymaaGaayjkaiaawMcaaaaaaO WdaeaapeGaeyOaIyRaeqOVdG3damaaCaaaleqabaWdbiaaikdaaaaa aOGaeyypa0JaaGimaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG OaGaae4maiaabsdacaqGPaaaaa@761F@

where the non-linear coefficient A= A 1 A 0 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqaiabg2da9maalaaapaqaa8qacaWGbbWdamaaBaaaleaapeGa aGymaaWdaeqaaaGcbaWdbiaadgeapaWaaSbaaSqaa8qacaaIWaaapa qabaaaaOWdbiaacYcaaaa@3C9D@ the dispersion coefficient B= 1 A 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOqaiabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qacaWGbbWd amaaBaaaleaapeGaaGimaaWdaeqaaaaaaaa@3AB9@ and the dissipation coefficient C= A 2 A 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4qaiabg2da9maalaaapaqaa8qacaWGbbWdamaaBaaaleaapeGa aGOmaaWdaeqaaaGcbaWdbiaadgeapaWaaSbaaSqaa8qacaaIWaaapa qabaaaaaaa@3BD6@ , of which

A 0 = 2 λ 3        (35) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaaGOmaaWdaeaapeGaeq4UdW2damaaCaaaleqabaWdbi aaiodaaaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabIcacaqGZaGaaeynaiaabMcaaaa@440E@
A 1 = 3 λ 4 + 3 γ 1 λ 2 2 γ 2 s 2 Q            (36) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaeyOeI0IaaG4maaWdaeaapeGaeq4UdW2damaaCaaale qabaWdbiaaisdaaaaaaOGaey4kaSYaaSaaa8aabaWdbiaaiodacqaH ZoWzpaWaaSbaaSqaa8qacaaIXaaapaqabaaakeaapeGaeq4UdW2dam aaCaaaleqabaWdbiaaikdaaaaaaOGaeyOeI0IaaGOmaiabeo7aN9aa daWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHsislcaWGZbWdamaaCa aaleqabaWdbiaaikdaaaGccaWGrbGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeikaiaabodacaqG2aGaaeykaaaa@57E5@

and

A 2 = η d0 λ 3        (37) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaeq4TdG2damaaBaaaleaapeGaamizaiaaicdaa8aabe aaaOqaa8qacqaH7oaBpaWaaWbaaSqabeaapeGaaG4maaaaaaGccaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabo dacaqG3aGaaeykaaaa@46FA@

Shocks in dusty plasmas have been generally studied using either the Burger’s equation [46, 52 – 58] or the Korteweg-deVries-Burgers (KdVB) equation [45, 46, 59 – 64]. Both Manesh et al [45] and Sijo et al [46] had considered five component plasmas; this study can thus be considered as extending and complementing these studies as we have an additional component namely, negatively charged dust grains.

Solution of KdVB equation

The "tanh method" can be used to obtain the shock-like solution of KdVB equation (34) [67 - 69]. We used a transformed co-ordinate χ=f ξVτ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4XdmMaeyypa0JaamOzamaabmaapaqaa8qacqaH+oaEcqGHsisl caWGwbGaeqiXdqhacaGLOaGaayzkaaaaaa@40A9@ moving with the shock speed and employed boundary conditions ϕ 1 ,  ϕ 1  χ ,  2 ϕ 1   χ 2  0 as  χ. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaCaaaleqabaWdbmaabmaapaqaa8qacaaIXaaacaGL OaGaayzkaaaaaOWdaiaacYcacaqGGaWdbmaalaaapaqaa8qacqGHci ITcqaHvpGzpaWaaWbaaSqabeaapeWaaeWaa8aabaWdbiaaigdaaiaa wIcacaGLPaaaaaaak8aabaWdbiabgkGi2kaabckacqaHhpWyaaGaai ilaiaabccadaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaa ikdaaaGccqaHvpGzpaWaaWbaaSqabeaapeWaaeWaa8aabaWdbiaaig daaiaawIcacaGLPaaaaaaak8aabaWdbiabgkGi2kaabckacqaHhpWy paWaaWbaaSqabeaapeGaaGOmaaaaaaGccaqGGaGaeyOKH4QaaGimai aabccacaWGHbGaam4CaiaacckacaqGGaGaeq4XdmMaeyOKH4QaeyOh IuQaaiOlaaaa@61B6@ For a localized solution, we can write equation (34) as

V ϕ 1 + A 2 ϕ 1 2 +B 2 ϕ 1 χ 2 C ϕ 1 χ =0              (38) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeyOeI0IaamOvaiabew9aM9aadaahaaWcbeqaa8qadaqadaWdaeaa peGaaGymaaGaayjkaiaawMcaaaaakiabgUcaRmaalaaapaqaa8qaca WGbbaapaqaa8qacaaIYaaaamaaemaapaqaa8qacqaHvpGzpaWaaWba aSqabeaapeWaaeWaa8aabaWdbiaaigdaaiaawIcacaGLPaaaaaaaki aawEa7caGLiWoapaWaaWbaaSqabeaapeGaaGOmaaaakiabgUcaRiaa dkeadaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaaikdaaa GccqaHvpGzpaWaaWbaaSqabeaapeWaaeWaa8aabaWdbiaaigdaaiaa wIcacaGLPaaaaaaak8aabaWdbiabgkGi2kabeE8aJ9aadaahaaWcbe qaa8qacaaIYaaaaaaakiabgkHiTiaadoeadaWcaaWdaeaapeGaeyOa IyRaeqy1dy2damaaCaaaleqabaWdbmaabmaapaqaa8qacaaIXaaaca GLOaGaayzkaaaaaaGcpaqaa8qacqGHciITcqaHhpWyaaGaeyypa0Ja aGimaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaa bodacaqG4aGaaeykaaaa@6D70@

Again using the transformation α χ = tanh and assuming a series solution of the form ϕ 1 α = i=0 n a i α i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaCaaaleqabaWdbmaabmaapaqaa8qacaaIXaaacaGL OaGaayzkaaaaaOWaaeWaa8aabaWdbiabeg7aHbGaayjkaiaawMcaai abg2da9maavadabeWcpaqaa8qacaWGPbGaeyypa0JaaGimaaWdaeaa peGaamOBaaqdpaqaa8qacqGHris5aaGccaqGnaIaamyya8aadaWgaa WcbaWdbiaadMgaa8aabeaak8qacqaHXoqypaWaaWbaaSqabeaapeGa amyAaaaaaaa@4AB9@ , the shock solution of KdVB equation is determined as

ϕ 1 = 3 C 2 25AB 1 tanh 2 k ξVτ + V A 6 C 2 25AB tanh k ξVτ             (39) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2damaaCaaaleqabaWdbmaabmaapaqaa8qacaaIXaaacaGL OaGaayzkaaaaaOGaeyypa0ZaaSaaa8aabaWdbiaaiodacaWGdbWdam aaCaaaleqabaWdbiaaikdaaaaak8aabaWdbiaaikdacaaI1aGaaeyq aiaadkeaaaWaamWaa8aabaWdbiaaigdacqGHsislcaqG0bGaaeyyai aab6gacaqGObWdamaaCaaaleqabaWdbiaaikdaaaGcdaWadaWdaeaa peGaam4Aamaabmaapaqaa8qacqaH+oaEcqGHsislcaWGwbGaeqiXdq hacaGLOaGaayzkaaaacaGLBbGaayzxaaaacaGLBbGaayzxaaGaey4k aSYaaSaaa8aabaWdbiaabAfaa8aabaWdbiaabgeaaaGaeyOeI0YaaS aaa8aabaWdbiaaiAdacaWGdbWdamaaCaaaleqabaWdbiaaikdaaaaa k8aabaWdbiaaikdacaaI1aGaaeyqaiaadkeaaaGaaeiDaiaabggaca qGUbGaaeiAamaadmaapaqaa8qacaWGRbWaaeWaa8aabaWdbiabe67a 4jabgkHiTiaadAfacqaHepaDaiaawIcacaGLPaaaaiaawUfacaGLDb aacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGOaGaae4maiaabMdacaqGPa aaaa@75BB@

where V= 6 C 2 25B MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOvaiabg2da9maalaaapaqaa8qacaaI2aGaam4qa8aadaahaaWc beqaa8qacaaIYaaaaaGcpaqaa8qacaaIYaGaaGynaiaadkeaaaaaaa@3D14@ is the shock speed and k= ±C 10B MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Aaiabg2da9maalaaapaqaa8qacqGHXcqScaWGdbaapaqaa8qa caaIXaGaaGimaiaadkeaaaaaaa@3D3F@ is the inverse of the shock width.

Results and Discussions

The KdVB equation (34) can be applied to any multi-ion/ dusty plasma environment. However, the parameters relevant to plasma environment of comet Halley are used to plot the figures. The hydrogen ion density was set at 4.95 cm−3 with a temperature of TH= 8 × 104 K; the solar electron temperature The = 2 × 105 K . The temperature of the colder, cometary electrons was set at Tce = 2 × 104 K. The negatively charged oxygen ion was set at n10=.05cm −3 . The density of the positively charged heavier oxygen ions was n20 = 0.5cm−3 at a temperature of 1.16 × 10K4 [41, 42]. The density of dust of negative polarity was set at nd0= 0.1cm-3 and its equilibrium charge at . Also, z d0 = 5 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadsgacaaIWaaapaqabaGcpeGaeyyp a0JaaGynamaaBaaaleaacaaIWaaabeaaaaa@3BCA@ κ he = 11 2 ,  κ ce = 7 2  and  z H =1,  z 1 =4 and  z 2 =2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOUdS2damaaBaaaleaapeGaamiAaiaadwgaa8aabeaak8qacqGH 9aqpdaWcaaWdaeaapeGaaGymaiaaigdaa8aabaWdbiaaikdaaaGaai ilaiaabccacqaH6oWApaWaaSbaaSqaa8qacaWGJbGaamyzaaWdaeqa aOWdbiabg2da9maalaaapaqaa8qacaaI3aaapaqaa8qacaaIYaaaai aabccacaWGHbGaamOBaiaadsgacaqGGaGaamOEa8aadaWgaaWcbaWd biaadIeaa8aabeaak8qacqGH9aqpcaaIXaGaaiilaiaabccacaWG6b WdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da9iaaisdacaqG GaGaamyyaiaad6gacaWGKbGaaeiiaiaadQhapaWaaSbaaSqaa8qaca aIYaaapaqabaGcpeGaeyypa0JaaGOmaaaa@5AEF@ The values of normalized kinematic dust ion viscosity used was ηd0 = 0.5 .

(Figure 1) shows a plot of the shock profiles, with and without dust charge variation. The dotted (blue) curve is for γ1 = 0 , γ2 = 0 ; while the continuous (red) curve is with dust charge variation; obtained for n10=.05cm −3, md=200 a.m.u and zd0 = 50. The other parameters for the figure are: The =2 x 105 K, Tce= 2 x 104, TH= 8 x 104K , Td= T1= T2= 1.16x10K; the densities are nH0= 4.95cm-3, n10= 0.05cm-3, n20=0.5cm-3, nd0=0.1cm-3, nhe=nce=0.5ne. From the figure, it is clear that with the inclusion of dust charge variation, the amplitude of the shock wave increases.

Figure 1: Plot of shock profiles with and without dust charge variation.

In a study related to comet Halley, Tribeche and Bacha [61] concluded that nonlinear damping due to dust charge variation would lead to the formation of shock waves. A similar conclusion was arrived at more recently by Naeem et al [65]. Also the pickup of heavy ions could lead to the formation of shock waves [66]. Also the charge exchange reaction which leads to the formation of negatively charged oxygen ions could also contribute to the formation of shock waves.

(Figure 2) depicts the variation of (a) phase velocity, λ , (b) shock speed, V , and (c) solitary width, W , versus the spectral index kappa, . Again, the dotted (blue) curves are for the cases where the charges on the dust particles are a constant; the continuous (red) curves depict the other case. The parameters are the same as in figure 1. It is obvious that as the specral index kappa for cometary electron increases, the phase velocity and the shock width increase whereas the shock speed decreases with κ ce . However, with dust charge variation included, the values of phase velocity, and width of the shock wave get reduced but the shock speed increases.

Figure 2a: Variation of phase velocity, λ with spectral index κce with and without dust charge variation.

Figure 2b: Variation of shock speed, V with spectral index κce with and without dust charge variation.

Figure 2c: Variation of the shock width W with spectral index κce with and without dust charge variation.

(Figure 3) shows the variation of the phase velocity, λ, versus equilibrium charge number zd0. The dotted (blue) curve depicts the case where the charges on the dust particles are a constant; the continuous (red) curve is for the case where the charge varies. The other parameters are the same as in (figure 1). We find that λincreases with an increase in the equilibrium charge number zd0; also λ is lower when the charges on the dust particles vary.

Figure 3: Plot of phase velocity, λ, versus dust charge number, zd0, with and without dust charge variation.

It may be noted that in a study of dust acoustic shock waves in four component plasma of charged mobile dust, kappa distributed electrons, positively charged lighter ions and negatively charged heavier ions; the variation of the phase velocity with κ was similar [40]. Again, in a study of low frequency shocks in a magnetized dusty plasma of negatively charged dust and kappa distributed electrons and ions, the variation of the shock velocity with the spectral index of the ions is similar [62].

Figure 4 depicts the variation of (a) the nonlinear coefficient, A, and (b) the dispersion coefficient, B , with κce with and without dust charge variation. Here too, the dotted (blue) curves denote the cases where the charges are on the dust particles are a constant; the continuous (red) curves are for the cases where there is charge variation. The other parameters are the same as in (figure 1). From the figures, it is clear that as the values of κce increases, coefficient A and B increases. Also, with dust charge variation included, the value of coefficient A increases whereas B reduces.

Figure 4a: Variation of the nonlinear coefficient, A with κce with and without dust charge variation.

Figure 4b: Variation of the dispersion coefficient, B with κce with and without dust charge variation.

(Figure 5) illustrates the variation of the shock amplitudes for different values of the kinematic viscosities of the dust particles with and without dust charge variation. Here too, the dotted (blue) curve represents the case of constant dust charge with ηd0=0.3 and the dotted (red) curve depicts constant dust charge with ηd0=0.4 . Also the continuous (blue) curve is for the case where the charge varies with ηd0=0.3 and the continuous (red) curve is for charge variation with ηd0=0.4. The other parameters are the same as in figure 1. From the plots it is seen that as the kinematic viscosity of the dust particles increase, the amplitude of the shock profile increases. In addition, with the inclusion of dust charge variation, the shock amplitude increases.

Figure 5: Variation of shock profiles for different values of dust ion kinematic viscosity with and without dust charge variation; the other parameters being the same as in figure 1.

(Figure 6) shows the variation of the shock profiles with (a) negative oxygen ion densities, (b) positive oxygen ion densities and (c) hydrogen ion densities, with and without dust charge variation. The dotted curves represent the cases where the charges on the dust particles are a constant; the continuous curves are for the cases where the charges vary. The blue colour denotes n10=0.01cm-3 in figure (a), n20=0.04cm-3 in figure (b) and nH0=4.95cm-3 in figure (c). The red colour represents n10=0.05cm-3 in figure (a), n20=0.45cm-3 in figure (b) and nH0=0.45cm-3 in figure (c). The other parameters are the same as in figure 1. The shock amplitude decreases with increasing negative oxygen and hydrogen ion densities (in figures 6(a) and 6(c) respectively). However, as the positive oxygen ion densities increase, the shock amplitude increases (figure 6(b)). Further, with inclusion of the variation of dust charge, the shock amplitudes increase as in (figure 1).

Figure 6a: Variation of shock profile with negative oxygen ion density with and without dust charge variation.

Figure 6b: Variation of shock profile with positive oxygen ion density with and without dust charge variation.

Figure 6c: Variation of shock profile with hydrogen ion density with and without dust charge variation.

(Figure 7) shows the variation of the shock profiles with (a) negative oxygen ion temperatures (b) positive oxygen ion temperatures and (c) hydrogen ion temperatures, with and without dust charge variation. Here too, the dotted curves represent the cases where the charges on the dust particles are a constant; the continuous ones are for the cases where the charges vary. The blue colour denotes T 1 =1.16× 10 4 K MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcaaI XaGaaiOlaiaaigdacaaI2aGaey41aqRaaGymaiaaicdapaWaaWbaaS qabeaapeGaaGinaaaakiaabUeaaaa@416D@ in figure (a), T 2 =1.16× 10 4 K MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpcaaI XaGaaiOlaiaaigdacaaI2aGaey41aqRaaGymaiaaicdapaWaaWbaaS qabeaapeGaaGinaaaakiaabUeaaaa@416E@ in figure (b) and T H =8× 10 4 K MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaabIeaa8aabeaak8qacqGH9aqpcaaI 4aGaey41aqRaaGymaiaaicdapaWaaWbaaSqabeaapeGaaGinaaaaki aabUeaaaa@3F57@ in figure (c).The red colour represents T 1 =5×1.16× 10 4 K MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcaaI 1aGaey41aqRaaGymaiaac6cacaaIXaGaaGOnaiabgEna0kaaigdaca aIWaWdamaaCaaaleqabaWdbiaaisdaaaGccaqGlbaaaa@4443@ in figure (a), T 2 =5×1.16× 10 4 K MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpcaaI 1aGaey41aqRaaGymaiaac6cacaaIXaGaaGOnaiabgEna0kaaigdaca aIWaWdamaaCaaaleqabaWdbiaaisdaaaGccaqGlbaaaa@4444@ in figure (b) and T H =5×8× 10 4 K MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaabIeaa8aabeaak8qacqGH9aqpcaaI 1aGaey41aqRaaGioaiabgEna0kaaigdacaaIWaWdamaaCaaaleqaba WdbiaaisdaaaGccaqGlbaaaa@422D@ in figure (c). The other parameters are the same as in figure 1. The shock amplitude increases with increasing negative oxygen and hydrogen ion temperatures, in figures 8(a) and 8(c) respectively. As the positive oxygen ion temperature increases, the shock amplitude decreases in figure 8(b). Further, with inclusion of the variation of dust charge, the shock amplitude shows the same variation as in figure 1.

Conclusion

We have, in this paper, studied dust acoustic shock waves in a six component cometary plasma by deriving the Korteweg-deVries-Burgers (KdVB) equation. The constituents of the plasma include two components of electrons described by kappa distributions with different temperatures and spectral indices, lighter (hydrogen) ions and a pair of oppositely charged, heavier ions (positively and negatively charged oxygen ions), all of them described by Maxwellian distributions with different temperatures. Charge varying negatively charged dust grains is the sixth component.

Shock waves solutions of the KdVB equation, studied for typical parameters of comet Halley, show that the shock amplitudes are consistently larger when charge fluctuations on the dust grains are taken into consideration. The superthermal, second component of electrons affects the phase velocity and the shock velocity as well as it’s width. The amplitude of the shock wave is also affected by the densities and temperatures of all the ions: it increases with increasing positively charged oxygen ion densities and decreases with increasing temperatures of these ions. The amplitude, however, decreases with increasing densities and increases with increasing temperatures of the other two types of ions.

Positively and negatively charged nano-grains have been observed at comet 67P / Churyumov – Gerasimenko [70]. Cometary heavy ions (oxygen in our case) picked up by the solar wind produce mass loading resulting in a decrease in the solar wind speed. And these heavy ions play a key role in cometary shocks. Such pick up ions have been observed by Giotto at comet Halley and almost all other comets; such shocks are called “mass loading shocks”. An very recently Rosetta was able to cross a newly formed infant bow shock at comet 67P / Churyumov – Gerasimenko. Thus we feel that the results of this paper would contribute to an understanding of shocks observed at comets.

Conflict of Interest

The authors declare that there is no conflict of interest.

References

  1. Frank Verheest, Manfred A Hellberg and Ioannis Kourakis. Acoustic solitary waves in dusty and/or multi-ion plasmas with cold, adiabatic, and hot constituents. Physics of Plasmas. 2008;15(11):112309. doi: 10.1063/1.3026716
  2. Mannan A, Mamun AA and Shukla PK. Nonplanar solitary waves and double layers in nonthermal electronegative plasma.  Physica Scripta. 2012;85(6):065501. doi: 10.1088/0031-8949/85/06/065501
  3. Rufai OR, Bharuthram R, Singh SV and Lakhina GS. Obliquely propagating ion-acoustic solitons and supersolitons in four-component auroral plasmas. Advances in Space Research. 2016;57(3):813-820. doi: 10.1016/j.asr.2015.11.021
  4. Maharaj SK, Bharuthram R, Singh SV, Pillay SR and Lakhina GS. Electrostatic solitary waves in a magnetized dusty plasma. Physics of Plasmas. 2008;15(11):113701. doi: 10.1063/1.3028313
  5. Shukla PK and Mamun AA. Solitons, shocks and vortices in dusty plasmas.  New Journal of Physics. 2003;5(1):1–17. doi:  10.1088/1367-2630/5/1/317
  6. Bharuthram R. Arbitrary amplitude double layers in a multi-species electron-positron plasma.  Astrophysics and Space Science. 1992;189(2):213–222. doi: 10.1007/BF00643126
  7. Mamun AA and Shukla PK. Electrostatic solitary and shock structures in dusty plasmas. Physica Scripta. 2002;T98:107–114. doi: 10.1238/Physica.Topical.098a00107
  8. Bharuthram R and Shukla PK. Vortices in non-uniform dusty plasmas. Planetary and Space Science. 1992;40(5):647–654. doi: 10.1016/0032-0633(92)90005-9
  9. Vranjes J, Petrovic D and Shukla PK. Low-frequency potential structures in a nonuniform dusty magnetoplasma. Physics Letters A.  2001;278(4):231–238. doi: 10.1016/S0375-9601(00)00749-0
  10. Allen JE. Probe theory - The Orbital Motion Approach. Physica Scripta. 1992;45(5):497–503. doi: 10.1088/0031-8949/45/5/013
  11. Shukla PK and Mamun AA. Introduction to Dusty Plasma Physics. Plasma Physics and Controlled Fusion. 2002;44(3). doi: 10.1088/0741-3335/44/3/701
  12. Mamun AA, Hassan MHA. Effects of dust grain charge fluctuation on an obliquely propagating dust acoustic solitary potential in a magnetized dusty plasma. J. Plasma Physics. 2000;63(2):191–200. doi:  10.1017/S0022377899008028
  13. Ghosh S, Bharuthram R, Khan M, Gupta M R. Instability of dust acoustic wave due to nonthermal ions in a charge varying dusty plasma. Phys. Plasmas. 2004;11(7):3602–3609. doi: 10.1063/1.1760584
  14. El-Taibany, W F, Wadati M, Sabry R. Nonlinear dust acoustic waves in a nonuniform magnetized complex plasma with nonthermal ions and dust charge variation. Phys. Plasmas. 2007;14(3):2304. doi: 10.1063/1.2646587
  15. Murad A, Zakir U, Haque Q. Dust acoustic solitary waves with dust charge fluctuation in superthermal plasma.  Braz. J. Phys. 2019;49(1):79–88. doi: 10.1007/s13538-018-0608-2
  16. Ghosh S, Sarkar S, Khan M, Gupta M R. Small-amplitude nonlinear dust acoustic waves in a magnetized dusty plasma with charge fluctuation. IEEE Tran.Plasma Sci. 2001;29(3): 409–416. doi: 10.1109/27.928937
  17. Benzekka M, Tribeche M. Nonlinear dust acoustic waves in a charge varying complex plasma with nonthermal ions featuring Tsallis distribution. Astrophys. Space Sci. 2012;338(1):63–72. doi: 10.1007/s10509-011-0908-2 
  18. Zhang L P, Xue J K. Shock wave in magnetized dusty plasmas with dust charging and nonthermal ion effects. Phys. Plasmas. 2005;12(4):042304. doi: 10.1063/1.1868718 
  19. Popel S I, Gisko A A, Golub A P, Losseva T V, Bingham R, Shukla P K. Shock waves in charge-varying dusty plasmas and the effect of electromagnetic radiation. Phys. Plasmas. 2000;7(6):2410–2416. doi: 10.1063/1.874079
  20. Gupta M R, Sarkar S, Ghosh S, Debnath M, Khan M. Effect of nonadiabaticity of dust charge variation on dust acoustic waves: Generation of dust acoustic shock waves.  Phys. Rev. E. 2001. doi: 10.1103/PhysRevE.63.046406
  21. Misra A P, Chowdhury A R. Dust-acoustic waves in a self-gravitating complex plasma with trapped electrons and nonisothermal ions. Eur. Phys. J. 2006;37:105-113. doi: 10.1140/epjd/e2005-00237-y
  22. Tribeche M, Bacha  M. Nonlinear dust acoustic waves in a charge varying dusty plasma with suprathermal electrons. Phys. Plasmas. 2010;17(7): 073701-073701.7. doi: 10.1063/1.3449806
  23. Duh S S, Shikha B, Mamun A A. Nonlinear dust-ion-acoustic waves in a multi-ion plasma with trapped electrons. Pramana J. Phys. 2011;77(2): 357–368. doi: 10.1007/s12043-011-0102-7
  24. Amour R, Tribeche M, Zerguini T H. Nonextensive collisioneless dust-acoustic shock waves in a charge varying dusty plasma.  Astrophys. Space Sci. 2012;338(1):, 57–61. doi: 10.1007/s10509-011-0905-5
  25. Bacha M, Tribeche, M. Nonextensive dust acoustic waves in a charge varying dusty plasma. Astrophys. Space Sci. 2011;337:253-259. doi: 10.1007/s10509-011-0830-7
  26. Shahmansouri M, Tribeche M. Dust acoustic shock waves in suprathermal dusty plasma in the presence of ion streaming with dust charge fluctuations. Astrophys. Space Sci.  343, 251–256. doi: 10.1007/s10509-012-1213-4
  27. Hadjaz I, Tribeche M. Alternative dust-ion acoustic waves in a magnetized charge varying dusty plasma with nonthermal electrons having a vortex-like velocity distribution. Astrophys Space Sci. 2014;351:591–598. doi: 10.1007/s10509-014-1872-4
  28. El-Shewy, E K, El-Wakil S A, El-Hanbaly A M, Sallah M, Darweesh H F. Effect of nonthermality fraction on dust acoustic growth rate in inhomogeneous viscous dusty plasmas. Astrophys. Space Sci. 2015;356:269-276. doi: 10.1007/s10509-014-2176-4
  29. Wang H, Zhang K. Effect of dust size distribution and dust charge fluctuation on dust ion-acoustic shock waves in a multi-ion dusty plasma. Pramana J. Phys. 2016;87:11. doi: 10.1007/s12043-016-1206-x
  30. Baluku T K, Hellberg M A. Dust acoustic solitons in plasmas with kappa-distributed electrons and/or ions. Phys. Plasmas. 2008;15(12):123705. doi: 10.1063/1.3042215
  31. Mace R L, Amery G, Hellberg, M A. The electron-acoustic mode in a plasma with hot suprathermal and cool Maxwellian electrons.  Phys. Plasmas. 1999;6(1):44-49. doi: 10.1063/1.873256
  32. Pierrard V, Lazar M. Kappa distributions: Theory and applications in space plasmas.  Solar Phys. 2010;267:153-174. doi: 10.1007/s11207-010-9640-2
  33. El-Tantawy, S A, El-Bedwehy N A, Khan S, Ali S, Moslem W M. Arbitrary amplitude ion-acoustic solitary waves in superthermal electron-positron-ion magnetoplasma. Astrophys. Space Sci. 2012;342(2): 425-432. doi: 10.1007/s10509-012-1188-1
  34. Sultana S, Mamun A A. Linear and nonlinear propagation of ion-acoustic waves in a multi-ion plasma with positrons and two-temperature superthermal electrons. Astrophys. Space Sci. 2014;349(1):229-238. doi: 10.1007/s10509-013-1634-8
  35. Vasyliunas V M. A survey of low energy electrons in the evening sector of the magnetosphere with OGO 1 and OGO 3. J. Geophys. Res. 1968;73(9):2839-2884. doi: 10.1029/JA073i009p02839
  36. Broiles T W, Livadiotis G, Burc, J L, Chae K, Clark G, Cravens, T.E, et al. Characterizing cometary electrons with kappa distributions. J. Geophys. Res. 2016;121(8):7407–7422. doi: 10.1002/2016JA022972 
  37. Alinejad H, Tribeche M, Mohammadi M A. Dust ion-acoustic shock waves due to dust charge fluctuation in a superthermal dusty plasma. Phys. Lett. 2011;47(14):4183-4186. doi: 10.1016/j.physleta.2011.10.013
  38. Shahmansouri M, Tribeche M. Nonextensive dust acoustic shock structures in complex plasmas. Astrophys Space Sci. 2013;346(1):165-170. doi: 10.1007/s10509-013-1430-5
  39. Shah M G, Rahman M M, Hossen M R, Mamun A A. Properties of cylindrical and spherical heavy ion-acoustic solitary and shock structures in a multispecies plasma with superthermal electrons. Plasma Phys. Rep. 2016;42(2):168-176. doi: 10.1134/S1063780X16020069
  40. Ferdousi M, Sultana S, Hossena M M, Miah, M R, Mamun A A. Dust-acoustic shock excitations in κ-nonthermal electron depleted dusty plasmas. Eur. Phys. J. D. 2017;71:102. doi: 10.1140/epjd/e2017-80003-4
  41. Brinca A L, Tsurutani B T. Unusual characteristics of electromagnetic waves excited by cometary newborn ions with large perpendicular energies. Astron. Astrophys. 1987;187:311–319. doi: 10.1007/978-3-642-82971-0_57
  42. Chaizy P, Reme H, Sauvaud J A, d’Uston U, Lin R P, Larson D E. et al. Negative ions in the coma of comet Halley. Nature. 1991;349(6308):393–396. doi: 10.1038/349393a0
  43. Ellis T A, Neff J S. Numerical simulation of the emission and motion of neutral and charged dust from P/Halley. Icarus. 1991;91(2):280-296. doi: 10.1016/0019-1035(91)90025-O
  44. Chow V W, Mendis D A, Rosenberg M. Role of grain size and particle velocity distribution in secondary electron emission in space plasmas. J. Geophys. Res. 1993;98(11):19065-19076. doi: 10.1029/93JA02014
  45. Manesh M, Neethu T W, Neethu J, Sijo S, Sreekala G, Venugopal C. Korteweg-deVries-Burgers (KdVB equation) equation in a five component plasma cometary plasma with kappa described electrons and ions.  J. Theor. Appl. Phys. 2016;10(4):280-296. doi: 10.1007/s40094-016-0228-6
  46. Sijo S, Anu V, Neethu T W, Manesh M, Sreekala G, Venugopal C. Effect of ion drift on shock waves in an un-magnetized multi-ion plasma.  Open Acc. J. Math. Theor. Phy. 2018;1(5):179-184. doi: 10.15406/oajmtp.2018.01.00031
  47. Mahmoud A A. Effects of the non-extensive parameter on the propagation of ion acoustic waves in five-component cometary plasma system. Astrophys. Space Sci. 2018;363(1):18. doi: 10.1007/s10509-017-3229-2 
  48. Pakzad H R. Solitary waves of the Kadomstev-Petviashvili equation in warm dusty plasma with variable dust charge, two temperature ion and nonthermal electron.  Chaos, Solitons and Fractals.2009;42(2):874-879. doi: 10.1016/j.chaos.2009.02.016
  49. Abid A A, Ali S, Muhammad R. Dust grain surface potential in a non-Maxwellian dusty plasma with negative ions.  J. Plasma Phys. 2013;79(6):1117-1121. doi: 10.1017/S0022377813001372
  50. El-Taibany W F, Sabry R. Dust-acoustic solitary waves and double layers in a magnetized dusty plasma with nonthermal ions and dust charge variation. Phys. Plasmas. 2005;12(8):082302. doi: 10.1063/1.1985987
  51. Washimi H, Taniuti T. Propagation of ion-acoustic solitary waves of small amplitude. Phys. Rev. Lett. 1996;17(19):996-998. doi: 10.1103/PhysRevLett.17.996
  52. Mondal G, Chatterjee M. Shock waves in a dusty plasma with positive and negative dust where ions are nonthermal. Z. Naturofirsch. 2010.
  53. Ferdousi M, Mamun A A. Electrostatic shock structures in a non-extensive plasma with two distinct temperature electrons. Braz. J. Phys. 2015;45(1):89–94. doi: 10.1007/s13538-014-0285-8
  54. Dev A N, Sarma J, Deka M K. Dust acoustic shock waves in arbitrarily charged dusty plasma with low and high temperature nonthermal ions. Can. J. Phys. 2015;93(10)1030 - 1038. doi: 10.1139/cjp-2014-0391
  55. Ferdousi M, Miah M R, Sultana S,  Mamun A A. Dust acoustic shock waves in an electron depleted, non-extensive dusty plasma. Astrophys. Space Sci. 2015;360:43.  doi: 10.1007/s10509-015-2547-5 
  56. Dev A N, Deka M K, Sarma J, Adhikary N C. Shock wave solution in a hot adiabatic dusty plasma having negative and positive nonthermal ions with trapped electrons. J. Kor. Phys. Soc. 2015;67:339 – 345. doi: 10.3938%2Fjkps.67.339
  57. Dev A N. Lower order three-dimensional Burgers equation having non-Maxwellian ions in dusty plasmas. Chin. Phys. B. 2017;26(2):025203. doi: 10.1088/1674-1056/26/2/025203  
  58. Hossen M M, Nahar L, Alam M S, Sultana S,  Mamun A A. Electrostatic shock waves in a nonthermal dusty plasma with oppositely charged dust. High Energy Density Phys. 2017;24:9–14. doi: 10.1016/j.hedp.2017.05.011
  59. Pakzad H R, Javidan K. Dust acoustic solitary and shock waves in strongly coupled dusty plasmas with nonthermal ions. Pramana- J. Phys. 2009;73(5):913–926. doi: 10.1007/s00193-011-0331-1
  60. Pakzad H R. Dust acoustic shock waves with strongly coupled dusts and superthermal ions. Can. J. Phys. 2011;89(2):193 – 200. doi: 10.1139/P10-110
  61. Tribeche M, Bacha M. Dust acoustic shock waves in a charge varying electronegative magnetized dusty plasma with nonthermal ions: Applications to Comets. Phys. Plasmas. 2013;20(10):103704. doi: 10.1063/1.4825240
  62. Chahal B S, Ghai Y,  Saini N S. Low-frequency shock waves in a magnetized superthermal dusty plasma. J. Theor. Appl. Phys. 2017;11(3):181–189. doi: 10.1007/s40094-017-0260-1 
  63. Ghai Y, Kaur N, Singh K, Saini N S. Dust acoustic shock waves in  magnetized dusty plasma. Plasma Sci. Tech. 2018.         
  64. Alamri S. On the dissipative propagation in oppositely charged dusty plasmas. Z. Naturofirsch. 2019;7(3):227 – 234. doi: 10.1515/zna-2018-0350 
  65. Coates A J. Heavy ion effects on cometary shocks. Adv. Space Res. 1995;15(8-9):403-413. doi: 10.1016/0273-1177(94)00125-K
  66. Naeem I, Ehsan Z, Mirza A M, Murtaza G, Shocklets in comet Halley plasma. 2020. doi: 10.1063/5.0002521
  67. Karimi M. The tanh method for solutions of the nonlinear modified Korteweg de Vries equation. Mathematics Scientific Journal. 2013;9(1):47-54.
  68. Malfliet W. Solitary wave solutions of nonlinear wave equations. American J. Phys. 1992;60(7):650-654. doi: 10.1119/1.17120
  69. Malfliet W. The tanh method:a tool for solving certain classes of nonlinear evolution and wave equations.  J. Comput. Appl. Math. 2004;164(165): 529-541. doi: 10.1016/S0377-0427(03)00645-9
  70. Llera K, Burch J L, Goldstein R, Goetz C. Simultaneous observation of negatively and positively charged nano-grains  at comet 67P / Churyumov – Gerasimenko Geophys. Res. Lett. 2020. doi: 10.1029/2019GL086147.
  71. Neubauer F M, Glassmeier K H, Acuna M H, Mariani F, Musmann G, Ness NF, Coates A J. Giotto magnetic field observations at the out bound quasi-parallel bow shock at comet Halley. Annales Geophysicae. 1990;22(8):463 – 471.
  72. Gunell H, Goetz C, Wedlund C S, Lindkvist J, Hamrinm M, Nilsson H, Llera, et al. The infant bow shock: a new frontier at a weak activity comet. 2018. doi: 10.1051/0004-6361/201834225