Brazilian Journal of Physics ISSN: Sociedade Brasileira de Física Brasil

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1 Brailia Joural of Physics ISSN: Sociedade Brasileira de Física Brasil Juli M.C. de; Scheide R.S.; iebell L. F.; Gaeler R. Effects of dust charge variatio o electrostatic waves i dusty plasmas with temperature aisotropy Brailia Joural of Physics vol. 39 úm. maro 29 pp Sociedade Brasileira de Física Sâo Paulo Brasil Available i: How to cite Complete issue More iformatio about this article Joural's homepage i redalyc.org Scietific Iformatio System Network of Scietific Jourals from Lati America the Caribbea Spai ad Portugal No-profit academic project developed uder the ope access iitiative

2 2 M.C. de Juli et al. Effects of dust charge variatio o electrostatic waves i dusty plasmas with temperature aisotropy M.C. de Juli Cetro de Rádio-Astroomia e Astrofísica Mackeie - CRAAM Uiversidade Presbiteriaa Mackeie Rua da Cosolação 896 CEP: São Paulo SP Brasil R.S. Scheider ad L. F. iebell Istituto de Física Uiversidade Federal do Rio Grade do Sul Caixa Postal 55 CEP: Porto Alegre RS Brasil. R. Gaeler Istituto de Física e Matemática Uiversidade Federal de Pelotas Caixa Postal Campus UFPel CEP: 96-9 Pelotas RS Brasil Received o 4 Jauary 29) We utilie a kietic approach to the problem of wave propagatio i dusty plasmas takig ito accout the variatio of the charge of the dust particles due to ielastic collisios with electros ad ios. The compoets of the dielectric tesor are writte i terms of a fiite ad a ifiite series cotaiig all effects of harmoics ad Larmor radius. The formulatio is quite geeral ad valid for the whole rage of frequecies above the plasma frequecy of the dust particles which are assumed motioless. The formulatio is employed to the study of electrostatic waves propagatig alog the directio of the ambiet magetic field i the case for which ios ad electros are described by bi-maxwellia distributios. The results obtaied i a umerical aalysis corroborate previous aalysis about the importat role played by the dust charge variatio particularly o the imagiary part of the dispersio relatio ad about the very mior role played i the case of electrostatic waves by some additioal terms appearig i the compoets of the dielectric tesor which are etirely due to the occurrece of the dust charge variatio. Keywords: Electrostatic waves; Kietic theory; Magetied dusty plasmas; Dust charge fluctuatio; Wave propagatio. INTRODUCTION I the developmet of a proper kietic formulatio for the aalysis of wave propagatio ad dampig i a plasma cotaiig a populatio of charged dust particles it is ecessary to take ito accout the process of chargig of the dust grais. However despite the recogied importace of this effect to the propagatio ad dampig of waves 2 ad despite the recogied eed of a kietic formulatio icludig effects due to the dust chargig for proper evaluatio of the wave dampig 3 4 most of the published literature utilies fluid theory to describe the dusty plasmas ad oly a small fractio of the published papers take ito accout the collisioal chargig of the dust particles 5 8. Motivated by the importace of the use of a proper kietic formulatio for the aalysis of wave behavior i dusty plasmas with dust grais of variable charge i a recet paper we have developed a very geeral mathematical formulatio writig the expressios for the compoets of the dielectric tesor i terms of a ifiite ad a fiite summatio formally icorporatig effects of all cyclotro harmoics ad all orders of Larmor radius keepig effects due to the dust charge variatio 9. The formulatio developed is very geeral i terms of frequecy rage ad directio of propagatio ad it is expected to be very useful for applicatio to the study of wave propagatio i dusty plasmas i a variety of situatios. As a example of applicatio i Ref. 9 we have also icluded a brief discussio of the particular case of electrostatic waves I Memoriam Electroic address: iebell@if.ufrgs.br propagatig alog the directio of the ambiet magetic field assumig the case of Maxwellia distributios for electros ad ios i the equilibrium icludig some results origiated from umerical solutios of the dispersio relatio. I the preset paper we resume the use of the formulatio developed ad preseted i Ref. 9 i order to ivestigate the dispersio relatio for electrostatic waves i a dusty plasma cosiderig the case of bi-maxwellia distributio fuctios for ios ad electros. The aalysis therefore icludes simultaeously the effects of the presece of dust particles icludig the effect of the dust charge variatio ad the effect of the aisotropy i electro ad io temperatures. The structure of the paper is the followig. I Sectio 2 we briefly outlie the model used to describe the dusty plasma ad preset essetial features of the kietic formulatio which leads to the compoets of the dielectric tesor which are ecessary for the dispersio relatio. We also preset a discussio of the dispersio relatio of the electrostatic waves emphasiig the particular case of waves propagatig alog the directio of the ambiet magetic field i the case of bi-maxwellia distributios for the electros ad ios i the equilibrium. I Sectio 3 some results obtaied from umerical solutio of the dispersio relatio are preseted ad discussed. The coclusios are preseted i Sectio 4. Appedix A shows details of the evaluatio of the basic itegrals appearig i the compoets of the dielectric tesor for the case of bi-maxwellia distributios for electros ad ios. Appedix B presets some details o the evaluatio of the average frequecy of ielastic collisios betwee electros ad ios ad dust particles ad Appedix C briefly discusses some features related to the equilibrium coditio of the process of collisioal chargig of the dust particles.

3 Brailia Joural of Physics vol. 39 o. March THE HOMOGENEOUS MAGNETIED DUSTY PLASMA MODEL AND THE COMPONENTS OF THE DIELECTRIC TENSOR The preset paper is a applicatio of the geeral formalism recetly appeared i Ref. 9. Sice the formalism is easily available the whole set of ecessary expressios ad defiitios will ot be repeated here for the sake of ecoomy of space. Nevertheless it may be useful to repeat here a short accout of basic features which will be made i the followig paragraphs. I our geeral kietic formulatio we cosider a plasma i a homogeeous ambiet magetic field B = B e i the presece of spherical dust grais with costat radius a ad variable electric charge q d. We assume that the electrostatic eergy of the dust particles is much smaller tha their kietic eergy the so-called weakly coupled dusty mageto-plasmas. This coditio is ot very restrictive sice a large variety of atural ad laboratory dusty plasmas ca be classified as weakly coupled. The chargig of the dust grais is assumed to occur by the capture of plasma electros ad ios durig ielastic collisios betwee these particles ad the dust particles. Sice the electro thermal speed is much larger tha the io thermal speed the equilibrium dust charge becomes preferetially egative. The cross-sectio for the chargig process of the dust particles is modeled by we expressios derived from the OML theory orbital motio limited theory) 2. Although we assume the occurrece of a magetic field the model which we use for the dust chargig does ot take ito accout the effect of the magetic field beig valid oly for parameters such that the sie of the dust particles is much smaller tha the electro Larmor radius. This feature is importat sice it has bee show that the effect of the magetic field o the chargig of the dust particles ca be safely eglected whe the sie of the dust particles is much smaller tha the electro Larmor radius 3 4. Moreover dust particles are assumed to be immobile ad cosequetly the validity of the proposed model will be restricted to waves with frequecy much higher tha the characteristic dust frequecies. I particular we cosider the regime i which Ω d ω pd < ω where ω pd ad Ω d are the plasma frequecy ad the cyclotro frequecy of the dust particles respectively. This coditio therefore excludes the aalysis of the modes which ca arise from the dust dyamics as the socalled dust-acoustic wave. Usig this basic framework we arrive to expressios for the compoets of the dielectric tesor which ca be separated ito two kids of cotributios 5 6 ε i j = ε C i j + ε N i j. ) Repeatig here the commetary which has already appeared alog with previous presetatios of the formalism the term ε C i j is formally idetical except for the i compoets to the dielectric tesor of a magetied homogeeous covetioal plasma of electros ad ios with the resoat deomiator modified by the additio of a purely imagiary term which cotais the ielastic collisio frequecy of dust particles with electros ad ios. For the i compoets of the dielectric tesor i additio to the term obtaied with the prescriptio above there is a term which is proportioal to this ielastic collisio frequecy. The term ε N i j arises oly due to the process of variatio of the charge of the dust particles ad vaishes i the case of a dustless plasma. Although the formal cotributio due to this kid of term is already recogied i the literature sice at least the first years of the past decade its cotributio to umerical aalysis of the dispersio is usually eglected. Oe otices that the form of the ε N i j compoets is strogly depedet o the model used to describe the chargig process of the dust particles. Explicit expressios for the compoets ε C i j ad εn i j ca be foud i Refs Particularly i Ref. 9 the expressios appear accordig to the formulatio ad defiitios to be used i the preset paper. Accordig to this ovel formulatio the compoets of the dielectric tesor ca be writte i terms of a double summatio oe fiite ad aother ifiite i which the cotributio of harmoics ad Larmor radius terms is show explicitly. For the covetioal cotributio a compoet ε C i j ca be writte as follows ε C i j = δ i j + δ i δ j e + N δ i+δ j χ i j 2) while a compoet ε N i j is writte as ε N i j = U i S j. 3) I the case of electrostatic waves ES waves) ad parallel propagatio which the subject of the preset applicatio the dispersio relatio is simply give by ε =. Therefore we preset here oly the explicit expressios for the cotributios to the dielectric tesor. For geeral distributios ad arbitrary directios of propagatio the cotributio to the covetioal part appears as follows m = m χ = v2 c 2 ω 2 p r 2 Ω 2 m= q r ) 2m ) 4) a m ) Jm2; f ) + ij ν m; f ) e = ω 2 p 2 Ω 2 d 3 u u u L f ) 5) + ω 2 p 2 Ω 2 a) J2; f ) + ij ν ; f ) where Jmh; f ) J ν mh; f ) = L = γ d 3 u uh u2m ) u L f ) r q u + i ν 6) d d 3 u ν d uh u2m ) u L f ) r q u + i ν 7) d γ q ) u + q u u u

4 4 M.C. de Juli et al. L = u u u u with the dimesioless variables = ω q Ω = k v u Ω = p m v J νl mh; f ) = uh u2m H u ) deq u am v 2 d 3 u ν d uh u2m L f ) r q u + i ν d 3) r = Ω Ω ν d = ν d u) /2 u = u 2 Ω ) + u2 where the ielastic collisio frequecy betwee plasma particles ad dust particles is give by ν d u) = πa2 d v u u 2 2q ) dq am v 2 H u 2 2q ) dq am v 2. The quatities Ω ad v are a characteristic frequecy ad a velocity respectively which are cosidered coveiet for ormaliatio i the case of a particular applicatio. For the preset applicatio we use Ω = ω pe ad v = c s where c s = T e /m i ) /2 is the io-soud velocity ad ω pe is the equilibrium electro plasma agular frequecy i the absece of dust. The quatity q d is the equilibrium value of the charge of the dust particles which we will deote as q d = d e. The cotributio of the ew part for geeral distributios ad directios of propagatio appears as follows U = + i ν ch + ν ) ω 2 +m ) p 2m q Ω 2 m= = m r a m )J U m; f ) 8) J νν m; f ) = J ch f ) = J ν f ) = d 3 u ν d /)2 u 2m L f ) r q u + i ν 4) d d 3 u ν d L f ) u 5) d 3 u f u H u ) deq am v 2 6) with ν = ν /Ω ad ν ch = ν ch /Ω. For the case of parallel propagatio q = ) Eqs. 5) 8) ad 9) lead to e = ω 2 p 2 Ω 2 + ω 2 p 2 Ω 2 U = + i ν ch + ν ) d 3 u u L f u ) J2; f ) + ij ν ; f ) ω 2 p Ω 2 J U ; f ) S = aω 2v ω 2 +m ) p 2m q Ω 2 m= = m r a m ) J νl m; f ) + ij νν m; f ) S = aω 2v + aω 2v ω 2 p Ω 2 ω 2 p Ω 2 J νl ; f ) + ij νν ; f ) J ν f ) where + aω 2v ω 2 p Ω 2 ν ch = aω 2v ν = i aω 2v J ν f ) 9) ω 2 p Ω 2 J ch f ) ) ω 2 +m ) p 2m q Ω 2 m= = m r a m )J U m; f ) ) J U mhl; f ) = ν d 3 d u /)l f r q u + i ν d 2) where from Eqs. ) ad ) ν ch = aω 2v ν = i aω 2v ω 2 p Ω 2 ω 2 p Ω 2 J ch f ) J U ; f ). Further developmet ca be made i the particular case of bi-maxwellia distributios for ios ad electros f u u ) = 2π) 3/2 u 2 u e u2 /2u2 ) e u2 /2u2 ). 7) For these distributios L f ) = u u u 2 ) f 8)

5 Brailia Joural of Physics vol. 39 o. March 29 5 ad where L f ) = u γu 2 γ q u ) f 9) u ) 2 u )ζ + ˆζ ˆζ ). J νl ; f ) = ν J; f ) = ν 2) = u2 u 2 = T T. where u = v /v ad u = v /v with v = T /m ad v = T /m. For the case of these distributios ad usig as a approximatio the average value of the collisio frequecy istead of the actual mometum-depedet value the itegrals which are ecessary for the compoets of the dielectric tesor ca be evaluated leadig to the followig expressios J2; f ) = u 2 { 2) 2 u 2 ζ ˆζ + ˆζ ˆζ ) { ) 2 + ˆζ )2 + ˆζ }} ˆζ ) u ) 2 u {ζ where + ˆζ ˆζ ) ) ˆζ + ˆζ } ˆζ ) J ν f ) = ˆζ = r + i ν 2q u ζ = 2q u ν i = ν i = 2 2π)ε i ) c3 a 2 Ω 2 v u i Ω Ω 3 c 2 c i i i dµ + µ 2 i ) + µ 2 i ) + χi J ν ; f ) = 2) 2 ) u2 ν u 2 ζ ˆζ + ˆζ ˆζ ) J U ; f ) Γ ) ) ν 2) 2 u ) u ) ζ ˆζ ) J U ; f ) Γ ) ) ν 2) 2 u ) u ) ζ + ˆζ ˆζ ) ν e = ν e = 2 2π)ε i ) c3 a 2 Ω 2 v u e Ω Ω 3 c 2 c e 2 e dµ + µ 2 e χe +µ2 e )/ e. e ) Moreover the first itegral which cotributes to e becomes simply the followig d 3 u u L f u ) = ) u 2 u 2. Details of the evaluatio ca be foud i appedix A. Usig these results ad usig also Eq. A6) for the J ch the dispersio relatio becomes Λ C + Λ N = 2) J νν ; f ) = ) 2 ν 2) ) where Λ C = ω 2 p Ω 2 ζ ˆζ + ˆζ ˆζ )

6 6 M.C. de Juli et al. Λ N = aω 2v 2π 2 + i aω 2v ω 2 p Ω 2 ω 2 p Ω 2 u ζ + u ˆζ ˆζ ) J ch f ) + i ω 2 p u Ω 2 u 2 π 2 ) ν ζ u ˆζ ) ν ζ + ˆζ ˆζ ). 3. NUMERICAL ANALYSIS For the umerical aalysis we cosider parameters which are i the rage of parameters of iterest for stellar wids: io temperature T i =. 4 K io desity i =. 9 cm 3 io charge umber i =. ad io mass m i = m p where m p is the proto mass with a =. 4 cm as the radius of the dust particles. The io desity which has bee assumed is rather high whe compared for istace with the solar wid plasma but is reported to occur i outbursts of carborich stars Io-acoustic waves isotropic Maxwellia distributios Iitially we estimate the magitude of the cotributio of the ew terms to the dispersio relatio of ES waves ad compare it with the covetioal cotributio for the case of isotropic Maxwellia distributios. I order to do that we assume the occurrece of weakly damped oscillatios with frequecy i the rage of io-acoustic waves choosig the values = ) for the umerical estimatio. For this value of ad for the parameters cosidered i the previous paragraph ad assumig T e /T i =. we plot i Fig. the quatities Λ C ad Λ N amely the covetioal ad the ew cotributios to the ES dispersio relatio as defied i Eq. 2) versus ormalied wave-umber q ad ormalied dust desity ε. The upper paels of Fig. show respectively from left to right the real ad the imagiary parts of Λ C while the bottom paels show from left to right the real ad the imagiary parts of Λ N. It is see that for most of the iterval of q ad ε depicted i the figure the real ad imagiary cotributios of Λ N are about four orders of magitude smaller tha the correspodig cotributios of Λ C. Similar figures ad results ca be obtaied for differet values of the ratio T e /T i as i the cases of T e /T i =. ad T e /T i = 2. which appeared i 9. We further explore the role of the dust particles ad of the ew cotributio for the dispersio relatio of ES waves i the case of isotropic Maxwellia distributios by umerically solvig the expaded form of Eq. 2) for the frequecy rage of io-acoustic waves. Fig. 2 shows the value of r ad the correspodig values of the imagiary part i as a fuctio of q ad five values of ε ad. 4 ) for three values of the temperature ratio T e /T i ad 2). Figure 2a) shows that the quatity r is relatively isesitive to the presece of the dust for T e = T i. Figures 2c) ad 2e) show that for icreasig values of the ratio T e /T i the effect of the dust o the real part of the dispersio relatio becomes more ad more proouced. Regardig the imagiary part Figure 2b) shows that Ladau dampig occurs for the whole rage of q values cosidered amely the dampig which occurs for absece of dust ε =. ad that the presece of dust icreases the dampig for the whole rage appearig i the figure for T e = T i. O the other had Fig. 2d) shows that for T e /T i = the icrease of the dust populatio icreases the dampig i the regio of very small q ad decreases the dampig for sufficietly large q q.8 for the parameters utilied). There is a itermediate regio for which the presece of the dust populatio iitially cotributes to decrease of dampig ad the cotributes to a reewed icrease of dampig for sufficietly large ε. Similar features are see more clearly with the icrease of the ratio T e /T i. Figure 2f) shows that for small q the dampig is appreciably icreased with the icrease of ε while for q.5 the dampig is clearly decreased by the icrease i the dust populatio. The explaatio for this behavior of the imagiary part i is as follows. The basic feature to be cosidered is that the presece of dust lead to two competitive effects. Oe of the effects is the dampig due to the dust charge variatio depedig o the frequecy of ielastic collisios o the deomiator of the velocity itegrals appearig i the dielectric tesor. Aother effect is the reductio i the electro populatio due to the capture of electros by the dust particles which cotribute to reductio of electro Ladau dampig. For T e /T i =. the electro Ladau dampig is meaigful for large q ad decreases for small q because the resoat velocity becomes much larger tha the electro thermal velocity. The dust charge variatio costitutes a additioal dampig mechaism. This effect domiates over the decrease of Ladau dampig because the capture of electros is ot very sigificat for T e /T i =.. Figure 3 shows that for T e /T i =. ad ε =. 4 the electro desity is still more tha 8 % of the populatio i a dustless plasma. The cosequece is that the dampig is icreased by the process of dust charge variatio for the whole rage of q with the icrease of the dust desity. For T e /T i =. o the other had electro Ladau dampig is sigificat for the higher ed of the q rage cosidered but less importat for small q if compared with the case of T e /T i =.. With the icrease i the dust populatio the dampig is ehaced for small q sice i the regio of small Ladau dampig the dampig due to the dust charge variatio is domiat. This possibility of dampig due to collisioal chargig has already bee oticed by other authors 3 8. I the higher ed of the q regio however Ladau dampig is sufficietly high to become domiat. Although the presece of dust itroduces dampig due to the dust charge variatio the domiat effect is the reductio of Ladau dampig due to the reductio of the electro populatio. Figure 3 shows that for T e /T i =. the electro desity is reduced to early 3 % of the origial desity for the largest value ε cosidered i the calculatio. These

7 Brailia Joural of Physics vol. 39 o. March 29 7 features are eve more evidet for T e /T i = 2.. For small q the Ladau dampig is egligible i this case. With the icrease of the dust populatio there is ehacemet of dampig for small q due to the mechaism of dust charge variatio. For the higher ed of the q regio however the dampig due to the dust charge variatio is overcome by Ladau dampig. With the icrease of the dust populatio ε the electro populatio is severely reduced as show by Fig. 3 ad the overall effect is the reductio of the wave dampig show by Fig. 2f). We poit out that i Fig. 2 we have plotted the results obtaied with the dispersio relatio give by Eq. 2). We have also plotted i the same figure the results obtaied from a dispersio relatio give by Λ C = obtaied by eglectig the ew cotributio to the dielectric tesor. The results hardly ca be distiguished i the scale of the figure reflectig the fact that for the rage of frequecy ad for the parameters utilied the effect of the ew cotributio is egligible i the dispersio relatio of ES waves. I a color versio of Fig. 2 usig blue color for the results obtaied with the full dispersio relatio ad red color for the results obtaied cosiderig oly the covetioal cotributio to the dispersio relatio the two differet results appear so close that the curves feature a light purple color result of the superpositio of the results featured with blue color ad the results featured with red color. I a moochromatic versio of Fig. 2 usig two differet lie styles the two differet results ca hardly be distiguished Io-acoustic waves bi-maxwellia distributios At this poit we proceed to the estimatio of the magitude of the cotributio of the ew terms to the dispersio relatio of ES waves i aisotropic plasmas ad compare it with the covetioal cotributio. I order to do that we agai assume the occurrece of weakly damped oscillatios with frequecy i the rage of the io-acoustic waves assumig a typical ormalied frequecy = ). For this value of ad for the parameters cosidered i the previous paragraph ad assumig T e /T i =. ad cosiderig the case of T e /T e =. ad T i /T i =. we plot i Fig. 4 the quatities Λ C ad Λ N versus ormalied wave-umber q ad ormalied dust desity ε. The upper paels of Fig. 4 show respectively from left to right the real ad the imagiary parts of Λ C while the bottom paels show from left to right the real ad the imagiary parts of Λ N. It is see that for most of the iterval of q ad ε depicted i the figure the real ad imagiary cotributios of Λ N are much smaller tha the correspodig cotributios of Λ C similarly to what occurs i the case of isotropic Maxwellia distributios. I Fig. 5 we show the same quatities depicted i Fig. 4 for the case of T e /T e =. ad T i /T i =. with the other parameters all equal to those used for Fig.. The commets which ca be made about Fig. 5 are similar to those made about Fig. 4 ad also similar to those made about Fig. which was obtaied for the case of isotropy of temperatures. Figures 4 ad 5 have bee obtaied assumig equal io ad electro temperatures. I the case of electro temperature larger tha io temperature similar results ca be obtaied. The oly poit to be observed is that for icreasig ratio of perpedicular ad parallel temperatures the magitudes of Λ C ad Λ N at small q which are see to grow with ε i the rage depicted i Figs. 4 ad 5 are see to decrease agai at sufficietly large ε. This feature is illustrated i Figs. 6 ad 7 which show the same quatities appearig i Figs. 4 ad 5 for T e /T i = 4. ad T e /T i =. respectively with T e /T e =. ad T i /T i =. ad the other parameters all equal to those used for Fig.. The opposite side of the aisotropy rage is illustrated by Fig. 8 which shows the case of T e /T i =. with T e /T e =. ad T i /T i =. ad the other parameters all equal to those used for Fig.. Both i the case of =. show i Fig. 4 i the case of =. show i Figs. 5 ad 6 ad i the case of =. illustrated i Fig. the compariso betwee the magitude of the covetioal ad ew cotributio becomes more difficult i the regio of the graphics where these cotributios both approach ero. I order to improve the accuracy of observatio we show i Fig. 9 the ratio betwee the ew ad the covetioal cotributios for oe of the cases discussed. I the upper lie of Fig. 9 we show at the left-had side the ratio betwee the real parts of the cotributios ad i the righthad side the ratio betwee the imagiary parts for the case T e /T e =. ad T i /T i =. with T e /T i =. ad other parameters as i Fig.. At the bottom lie we show the correspodig figures for the case of perpedicular temperatures much above parallel temperatures with T e /T e =. ad T i /T i =. with T e /T i =. ad other parameters as i Fig.. The coclusio to be draw from Fig. for the case of plasmas with isotropy of temperature ad from Figs ad 9 is that although the dust populatio may itroduce sigificat cotributio to the dispersio relatio of electrostatic waves i the rage of frequecies characteristics of io-acoustic waves this cotributio is mostly due to the covetioal part of the dielectric tesor. At least for the parameter regime which has bee ivestigated the ew cotributio is show to give oly a egligible cotributio to the dispersio relatio. We cotiue with the ivestigatio of the role played by the dust particles ad by the ew cotributio for the dispersio relatio of ES waves i aisotropic plasmas by discussig the umerical solutio of the dispersio relatio. I Fig. we cosider the solutio correspodig to io-acoustic waves also for three situatios of temperature aisotropy. Fig. a) shows the value of r for io-acoustic waves as a fuctio of q ad five values of ε ad. 4 ) for T e = T i ad perpedicular temperature much smaller tha parallel temperature T e /T e = T i /T i =.) with other parameters as i Fig.. Figure b) shows the correspodig values of the imagiary part of the ormalied frequecy i. It is is see that the presece of the dust populatio modifies very sigificatly the imagiary part i. The dampig measured by the absolute value of i is ehaced for small q due to the presece of the dust but ca be appreciably reduced for larger q also due to the presece of the dust. For the real part r Fig. c) shows that the effect of the dust is ot very sigificat although ot egligible. The case of perpedicular temperature larger tha the parallel temperature is see i Figs. e) ad f) at the bottom lie of Fig.. Figure e) shows the value of r for ioacoustic waves as a fuctio of q ad five values of ε ad. 4 ) for T e = T i ad T e /T e = T i /T i =. with other parameters as i Fig.. Figure e) shows that i this case of larger perpedicular

8 8 M.C. de Juli et al. FIG. : upper left) Real part of the covetioal cotributio to the dispersio relatio vs. q ad ε = d / i ; upper right) imagiary part of the covetioal cotributio; bottom left) Real part of the ew cotributio; bottom right) imagiary part of the ew cotributio; = ) i the rage of io-acoustic waves. Isotropic Maxwellia distributios for ios ad electros with T i =. 4 K ad T e /T i =. Other parameters: i =. 9 cm 3 i =. m i = m p where m p is the proto mass ad a =. 4 cm. temperature the real part of the ormalied frequecy is much more affected by the presece of the dust tha i the case of smaller perpedicular temperature. The imagiary part i is also affected sigificatly by the presece of the dust populatio as show by Fig. f). Qualitatively it is see that the effect is similar to that occurrig i the case of perpedicular temperature smaller tha parallel show i Fig. b) i the sese that the dampig measured by the absolute value of i is ehaced for small q due to the presece of the dust but ca be appreciably reduced for larger q also due to the presece of the dust. The middle lie of Fig. shows the case of isotropy of temperatures i betwee the cases depicted at the top lie ad at the bottom lie. Figures c) ad d) show the values of r ad i respectively for io-acoustic waves as a fuctio of q ad five values of ε ad. 4 ) for T e = T i ad T e /T e = T i /T i =. with other parameters as i Fig.. Additioal iformatio may be obtaied by cosiderig the depedece of the electro desity ad of the equilibrium value of the charge of the dust particles as a fuctio of the dust desity. I Fig. we show the values of the dust charge d ad of the electro desity e as a fuctio of ε for five values of the ratio T /T cosiderig T e = 2 T i for fixed io desity for the case of io-soud waves. It is see that for T =.2T the value of d reduced by early 3 % whe ε is chaged betwee. ad. 4 while it is reduced i the same rage to early 5 % of the origial value i the case of T = 5.T. It is also see that the value of e chages by approximately 6 % whe ε is chaged betwee. ad. 4 i the case of T /T =.2 ad is reduced to almost 5 % of the origial value i the case T /T = 5.. The depedece of d ad e o the ratio of electro ad io temperatures for aisotropic situatios is illustrated i Fig. 2. Figure 2a) shows the values of d as a fuctio of ε for four values of the ratio T e /T i for a case with aisotropy of temperatures with T /T = 5.. Figure 2b) shows the values of the ratio e ε)/ e ) as a fuctio of ε for four values of the ratio T e /T i for T /T = 5.. It is see that the desity of electros decrease with the icrease of ε substatially faster for the case of high values of the ratio T e /T i tha for the case i which this ratio is equal to the uity Lagmuir waves bi-maxwellia distributios We proceed by estimatig the magitude of the cotributio of the ew terms to the dispersio relatio of ES waves i aisotropic plasmas ow cosiderig the case of waves with frequecy i the rage of the Lagmuir waves assumig a typical ormalied frequecy =. 3 ). For this value of ad for the parameters cosidered i the previous paragraph ad assumig T e /T i =. ad cosiderig the case of T e /T e =. ad T i /T i =. we plot i Fig. 3 the quatities Λ C ad Λ N amely the covetioal ad the ew cotributios to the ES dispersio relatio as defied i Eq. 2) versus ormalied wave-umber q ad ormalied dust desity ε. The upper paels of Fig. 3 show respectively

9 Brailia Joural of Physics vol. 39 o. March 29 9 FIG. 2: Real ad imagiary parts of the ormalied frequecy r ad i ) obtaied from the dispersio relatio for io-acoustic waves i the case of isotropic Maxwellia distributios for ios ad electros vs. q for several values of ε = d / i ad. 4 ); a) r with T e /T i =.; b) i with T e /T i =.; c) r with T e /T i =.; d) i with T e /T i =.; e) r with T e /T i = 2.; f) i with T e /T i = 2.; other parameters the same as used to obtai Fig.. from left to right the real ad the imagiary parts of Λ C while the bottom paels show from left to right the real ad the imagiary parts of Λ N. It is see that for most of the iterval of q ad ε depicted i the figure the real ad imagiary cotributios of Λ N are several orders of magitude smaller tha the correspodig cotributios of Λ C. I Fig. 4 we show the same quatities depicted i Fig. 3 for the case of T e /T e =. ad T i /T i =. with the other parameters all equal to those used for Fig. 3. The commets which ca be made about Fig. 4 are similar to those made about Fig. 3 ad also similar to those made about Fig. of Ref. 9 where the case of isotropic temperatures has bee discussed. The coclusio is that the dust populatio may itroduce sigificat cotributio to the dispersio relatio of electrostatic waves i the rage of frequecies characteristics of Lagmuir waves but this cotributio is mostly due to the covetioal part of the dielectric tesor. At least for the parameter regime which has bee ivestigated the ew cotributio is show to give oly a egligible cotributio to the dispersio relatio. I Fig. 5 we cosider the solutio correspodig to Lagmuir waves for three situatios of temperature aisotropy. Fig. 5a) shows the value of r as a fuctio of q ad five values of ε ad. 4 )

10 2 M.C. de Juli et al..8 T e /T i =. e ε)/ e ) T e /T i =2. T e /T i =. 2e-5 4e-5 6e-5 8e-5. FIG. 3: Ratio betwee the electro desity i a dusty plasma ad the equilibrium electro desity i the absece of dust vs. ε = d / i for three values of the ratio T e /T i i the case of isotropic Maxwellia distributios for ios ad electros Other parameters the same as used to obtai Fig.. ε FIG. 4: upper left) Real part of the covetioal cotributio to the dispersio relatio vs. q ad ε = d / i ; upper right) imagiary part of the covetioal cotributio; bottom left) Real part of the ew cotributio; bottom right) imagiary part of the ew cotributio; T e = T i = T /T =. for = ie ad = ) i the rage of io-acoustic waves. Other parameters the same as used to obtai Fig.. for T e = T i ad perpedicular temperature much smaller tha parallel temperature T e /T e = T i /T i =.) with other parameters as i Fig. 3. Figure 5b) shows the correspodig values of the imagiary part of the ormalied frequecy i. It is is see that the presece of the dust populatio do ot modifies appreciably the root of the dispersio relatio either the real or the imagiary part. Figure 5e) shows the value of r for Lagmuir waves as a fuctio of q ad five values of ε ad. 4 ) for T e = T i ad perpedicular temperature much larger tha parallel temperature T e /T e = T i /T i =.) with other parameters as i Fig.

11 Brailia Joural of Physics vol. 39 o. March 29 2 FIG. 5: upper left) Real part of the covetioal cotributio to the dispersio relatio vs. q ad ε = d / i ; upper right) imagiary part of the covetioal cotributio; bottom left) Real part of the ew cotributio; bottom right) imagiary part of the ew cotributio; T e = T i = T /T =. for = ie ad = ) i the rage of io-acoustic waves. Other parameters the same as used to obtai Fig.. 3. Figure 5e) shows that i this case of larger perpedicular temperature the real part of the ormalied frequecy is much more affected by the presece of the dust tha i the case of smaller perpedicular temperature. Particularly i the regio of small wave-umber small q ) the value of r ca be reduced by a factor which goes up to almost 5% for dust populatio risig betwee the case ε =. ad the case ε =. 4. Figure 5f) shows the correspodig values of the imagiary part of the ormalied frequecy i. Cotrarily to what happes i the case i which the perpedicular temperature is smaller tha the parallel temperature show i Fig. 5b) Fig. 5f) shows that i the case of larger perpedicular temperature the value of the imagiary part i ca be very appreciably affected by the presece of the dust populatio. For a give value of the ormalied wave umber q the magitude of the imagiary part of i is sigificatly icreased by the presece of the dust. I betwee the cases show at the top lie ad at the bottom lie of Fig. 5 we cosider the case of equal parallel ad perpedicular temperatures. Figure 5c) shows the value of r for Lagmuir waves as a fuctio of q ad five values of ε ad. 4 ) for T e = T i ad perpedicular temperature equal to parallel temperature T e /T e = T i /T i =.) with other parameters as i Fig. 3. Figure 5d) shows the correspodig values of the imagiary part of the ormalied frequecy i. Let us ow cosider the depedece of the electro desity ad of the equilibrium value of the charge of the dust particles as a fuctio of the dust desity. I Fig. 6 we show the values of the dust charge d ad of the electro desity e as a fuctio of ε for five values of the ratio T /T cosiderig T e = T i for fixed io desity. The case of T e = T i is the case relevat for Lagmuir waves. It is see that the value of d is basically idepedet of ε i the rage cosidered for T < T chagig by less tha 5 % whe ε is chaged betwee. ad. 4 i the case T = T. For larger values of this ratio the value of d becomes more depedet o ε although eve for T /T = 5. the chage obtaied is oly of order of 2 % i the rage cosidered. It is also see that the value of e chages by approximately 8 % whe ε is chaged betwee. ad. 4 i the case of T /T =.2 ad is reduced to almost 5 % of the origial value i the case T /T = 5.. These results show that the electro desity ad the dust charge are more ad more sesitive to the dust desity for icreasig values of the ratio T /T for fixed io desity. 4. CONCLUSIONS I the preset paper we have examied the ifluece of the dust charge variatio o the dispersio relatio of electrostatic waves cosiderig the case of waves propagatig i the direc-

12 22 M.C. de Juli et al. FIG. 6: upper left) Real part of the covetioal cotributio to the dispersio relatio vs. q ad ε = d / i ; upper right) imagiary part of the covetioal cotributio; bottom left) Real part of the ew cotributio; bottom right) imagiary part of the ew cotributio; T e /T i = 4. = T /T =. for = ie ad = ) i the rage of io-acoustic waves. Other parameters the same as used to obtai Fig.. FIG. 7: upper left) Real part of the covetioal cotributio to the dispersio relatio vs. q ad ε = d / i ; upper right) imagiary part of the covetioal cotributio; bottom left) Real part of the ew cotributio; bottom right) imagiary part of the ew cotributio; T e /T i =. = T /T =. for = ie ad = ) i the rage of io-acoustic waves. Other parameters the same as used to obtai Fig.. tio of the ambiet magetic field. The most importat motivatio was the ivestigatio of the effect of some cotributios to the dielectric tesor which usually appear i theoretical aalysis but are commoly eglected i umerical aalysis. For the dielectric tesor we have used a kietic formulatio which takes ito accout the icorporatio of electros ad ios to the dust particles due to ielastic collisios shaped i a form i which the compoets of the dielectric tesor are writte i terms of double summatios related to harmoic ad Larmor radius cotributios. The geeral expressios utilied deped o a small umber of itegrals which deped o the distributio fuctio.

13 Brailia Joural of Physics vol. 39 o. March FIG. 8: upper left) Real part of the covetioal cotributio to the dispersio relatio vs. q ad ε = d / i ; upper right) imagiary part of the covetioal cotributio; bottom left) Real part of the ew cotributio; bottom right) imagiary part of the ew cotributio; T e /T i =. = T /T =. for = ie ad = ) i the rage of io-acoustic waves. Other parameters the same as used to obtai Fig.. Oe iterestig poit is that the dielectric tesor ca be divided ito two parts. Oe of these parts is deomiated covetioal ad is formally similar to the dielectric tesor of dustless plasmas modified by the presece of a collisio frequecy related to the ielastic collisio betwee dust particles ad plasma particles. The other part owes its existece to the occurrece of the dust charge variatio ad is deomiated as the ew cotributio. We have cosidered the case of aisotropic Maxwellia distributios for ios ad electros ad itroduced a approximatio which uses the average value of the ielastic collisio frequecies of electros ad ios with the dust particles istead of the actual mometum depedet expressios. This approximatio was adopted i order to arrive at a relatively simple estimate of the effect of the dust charge variatio effect which is frequetly eglected i aalysis of the dispersio relatio for waves i dusty plasmas. After the choice of bi-maxwellia distributios ad after the use of a approximatio for the mometum depedet collisio frequecies the itegrals which appear i the compoets of the dielectric tesor were writte i terms of the very familiar fuctio whose aalytic properties are well kow. As a applicatio of the formulatio we have cosidered the case of electrostatic waves propagatig i the directio of the ambiet magetic field ad performed a umerical ivestigatio comparig the magitudes of the covetioal ad of the ew cotributios to the dispersio relatio for frequecies i the rage of the io-soud ad Lagmuir waves. The study expads previous aalyses i which the case of Lagmuir waves ad isotropic Maxwellia distributios has bee cosidered 9. To our kowledge this is oe of the first istaces of umerical aalysis icludig the effect of the ew cotributio which usually oly appears i formal aalysis of wave propagatio i dusty plasmas For this ivestigatio we have cosidered parameters which are i the rage of parameters typical of stellar wids ad the results obtaied have show that the cotributio of the ew compoets is very small compared to the covetioal cotributio. The results obtaied show that for a wide rage of temperature aisotropy the ew cotributio results i egligible effect to the dispersio relatio of Lagmuir ad io-soud waves the latter also cosiderig a wide rage of the electro to io temperature ratio. These fidigs about the egligible effect of the ew cotributio ca ot be geeralied without further aalysis to other situatios like other frequecy regimes ad other forms of the distributio fuctio. For istace the aalysis of lowfrequecy modes like the Alfvé wave mode would be a iterestig subject for future research. We ited to proceed with the ivestigatio of these topics i the ear future.

14 24 M.C. de Juli et al. FIG. 9: Ratio betwee the real ad imagiary parts of the ew ad the covetioal cotributios to the dispersio relatio vs. q ad ε = d / i ; upper left) ratio betwee the real parts Re Λ N /Re Λ C ) for = T /T =. for = ie; upper right) ratio betwee the imagiary parts Im Λ N /Im Λ C ) for = T /T =. for = ie; bottom left) ratio betwee the real parts Re Λ N /Re Λ C ) for = T /T =. for = ie; bottom right) ratio betwee the imagiary parts Im Λ N /Im Λ C ) for = T /T =. for = ie; T e = T i ad = ) i the rage of io-acoustic waves. Other parameters the same as used to obtai Fig.. APPENDIX A: DETAILS OF THE EVALUATION OF INTEGRALS APPEARING IN THE EXPRESSIONS FOR THE COMPONENTS OF THE DIELECTRIC TENSOR. Evaluatio of the itegrals Jmh; f ) We start from Eq. 6) Jmh; f ) d 3 u uh u2m ) u L f ) r q u + i ν d As we have see the covetioal cotributios to the compoets of the dielectric tesor deped o itegrals deoted as J ad J ν defied by Eqs. 6) ad 7). The ew cotributios deped o itegrals J U J νl J νν J ν ad J ch defied by Eqs. 2) 3) 4) 5) ad 6). I what follows we cosider the case of bi-maxwellia distributios for ios ad electros. Moreover as a approximatio we assume that the mometum-depedet collisio frequecy is replaced by the average value. This approximatio is adopted i order to arrive at a relatively simple estimate of the effect of the dust charge variatio effect which is frequetly eglected i aalysis of the dispersio relatio for waves i dusty plasmas. = ω2π) du u u 2m ) u u h du L f ) ω Ω v k u + iν d u). As we have see for a bi-maxwellia distributio L f ) = u u 2 q u ) f. Usig this result ad assumig that the collisio frequecy is replaced by the average value ν = R d 3 uν d u) f u)/ we obtai Jmh; f ) = ω 2π) /2 u 4 u du u 2m+ e u2 /2u2 )

15 Brailia Joural of Physics vol. 39 o. March FIG. : Real ad imagiary parts of the ormalied frequecy ) obtaied from the dispersio relatio for io-acoustic waves vs. q for several values of ε = d / i ad. 4 ); a) r for T e /T e = T i /T i =.; b) i for T e /T e = T i /T i =.; c) r for T e /T e = T i /T i =.; d) i for T e /T e = T i /T i =.; e) r for T e /T e = T i /T i =.; f) i for T e /T e = T i /T i =.; I all cases T e /T i = 2.. Other parameters the same as used to obtai Fig.. u h q u ) e u2 /2u2 ) du ω Ω v k u + iν where u res = ω Ω + iν v k. = ω 2 m m!) v k 2π) /2 u 2m ) u u h q u ) e u2 /2u2 ) du u u res Itroducig a chage of variables such that u = 2u t Jmh; f ) = dt ω 2 m m!) ) v k 2π) /2 u 2m ) h u 2u t h e t2 q t t res ) ) 2u dt th+ e t2 t ˆζ

16 26 M.C. de Juli et al a) 7 6 a) 3 5 d d e-5 4e-5 6e-5 8e-5. ε 2e-5 4e-5 6e-5 8e-5. ε.8 b).8 b) e ε)/ e ).6.4 e ε)/ e ) e-5 4e-5 6e-5 8e-5. ε 2e-5 4e-5 6e-5 8e-5. ε FIG. : a) d as a fuctio of ε for the case of io-soud waves for five values of the ratio T /T b) e as a fuctio of ε for the case of io-soud waves for five values of the ratio T /T. For iosoud waves it is assumed that T e /T i =.. From the thiest to the thickest the curves show the cases of T /T = ad 5. for other parameters as i Fig.. FIG. 2: a) d as a fuctio of ε for the case of io-soud waves for four values of the ratio T e /T i with T /T = 5.. b) e as a fuctio of ε for the case of io-soud waves for four values of the ratio T e /T i with T /T = 5.. From the thiest to the thickest the curves show the cases of T e /T i =. 5.. ad 2. with other parameters as i Fig.. where ˆζ = r + i ν 2q u. Re-arragig the expressio we obtai Jmh; f ) = m!) 2) 2m+h u 2m ) u h π ζ ) dt th e t2 t ˆζ ) dt th+ e t2 t ˆζ A) where ζ = / 2q u ). We ote that the itegral over u was made assumig that m is iteger. I order to proceed we cosider the itegral appearig i Eq. A) which deped o a iteger power of t t l. For l = π e t2 dt t ˆζ = ˆζ ). A2) For l = π For l = 2 π dt t e t2 t ˆζ = dt t ˆζ + ˆζ )e t2 π t ˆζ = π dt t2 e t2 t ˆζ = π dt e t2 + ˆζ e t2 dt t ˆζ = + ˆζ ˆζ ). A3) = π dt t e t2 + ˆζ dt t t ˆζ + ˆζ )e t2 t ˆζ dt t e t2 t ˆζ = ˆζ + ˆζ ˆζ ). A4)

17 Brailia Joural of Physics vol. 39 o. March FIG. 3: upper left) Real part of the covetioal cotributio to the dispersio relatio vs. q ad ε = d / i ; upper right) imagiary part of the covetioal cotributio; bottom left) Real part of the ew cotributio; bottom right) imagiary part of the ew cotributio; T e = T i = T /T =. for = ie ad =.. 3 ) i the rage of Lagmuir waves. T i =. 4 K i =. 9 cm 3 i =. m i = m p where m p is the proto mass ad a =. 4. For l = 3 2. Evaluatio of the itegrals J ν mh; f ) π dt t3 e t2 t ˆζ = π = π dt t 2 e t2 + ˆζ dt t2 t ˆζ + ˆζ )e t2 t ˆζ dt t2 e t2 t ˆζ { = 2 + ˆζ )2 + ˆζ } ˆζ ). A5) The result is for iteger m) Jm; f ) = m!) 2) 2m u 2m ) { ζ ˆζ ) ) + ˆζ } ˆζ ) Jm; f ) = m!) 2) 2m+ u 2m ) u { ζ + ˆζ ˆζ ) ) ˆζ + ˆζ } ˆζ ) A6) Jm2; f ) = m!) 2) 2m+2 u 2m ) u 2 { ζ ˆζ + ˆζ { ˆζ ) ) 2 + ˆζ )2 + ˆζ }} ˆζ ). We start from Eq. 7) J ν mh; f ) = d 3 u ν d u) Ω u h u 2m ) u L f ) r q u + i ν d Usig the average value of the collisio frequecy cosiderig a bi-maxwellia distributio ad usig Eq. 8) J ν mh; f ) = ν ) 2π) /2 u 4 u du u 2m+ e u2 /2u2 ) u h+ e u2 /2u2 ) du r q u + i ν = ν ) 2π) /2 u 4 u m! ) m+ 2u 2 2 u h+ e u2 /2u2 ) du. r q u + i ν = 2) 2m+h m!) ν ) 2m ) q u u h.

18 28 M.C. de Juli et al. FIG. 4: upper left) Real part of the covetioal cotributio to the dispersio relatio vs. q ad ε = d / i ; upper right) imagiary part of the covetioal cotributio; bottom left) Real part of the ew cotributio; bottom right) imagiary part of the ew cotributio; T e = T i = T /T =. for = ie ad =.. 3 ) i the rage of Lagmuir waves. Other parameters the same as used to obtai Fig. 3. π dt th+ e t2 t ˆζ The result is for some values of h J ν m; f ) = m!) 2) 2m ) u 2m ) ν + q ˆζ ˆζ ). A7) J ν m; f ) = m!) 2) 2m+ ) u 2m ) ν u ˆζ q + ˆζ ˆζ ) A8) J ν m2; f ) = m!) 2) 2m+2 ) { u 2m ) u 2 ν q 2 + ˆζ )2 + ˆζ } ˆζ ). 3. Evaluatio of e From Eq. 5) we see that e features three terms. Two of these terms ca be evaluated with the use of the J ad J ν itegrals. The other term will be evaluated here for a bi- Maxwellia distributio. We write e = e + ω 2 p 2 Ω 2 a) J2; f )+ij ν ; f ) e = ω 2 p 2 Ω 2 d 3 u u u L f ). Usig Eq. 8) the e term ca be writte as e = ω 2 ) p 2 Ω 2 2π) /2 u 4 u du u e u2 /2u2 ) du u 2 e u2 /2u2 ) = ω 2 p 2 Ω 2 ) ) 2u 2 π ) 2π) /2 u 4 u 2 2 2u2 )3/2

19 Brailia Joural of Physics vol. 39 o. March FIG. 5: Real ad imagiary parts of the ormalied frequecy ) obtaied from the dispersio relatio for Lagmuir waves vs. q for several values of ε = d / i ad. 4 ); a) r for T e /T e = T i /T i =.; b) i for T e /T e = T i /T i =.; c) r for T e /T e = T i /T i =.; d) i for T e /T e = T i /T i =.; e) r for T e /T e = T i /T i =.; f) i for T e /T e = T i /T i =.; I all cases T e = T i. Other parameters the same as used to obtai Fig. 3. = ω 2 p u 2 ) = 2 Ω 2 u 2 2 Therefore e = ω 2 ) p 2 Ω 2 ω 2 p Ω 2 + ω 2 p 2 Ω 2 ). J2; f ) + ij ν ; f ). A9) 4. Evaluatio of the itegrals J U mhl; f ) We start from Eq. 2) J U mhl; f ) = d 3 u ν d ) l f uh u2m H u ) deq u am v 2. r q u + i ν d We assume for simplicity that the collisio frequecy is

20 3 M.C. de Juli et al. 6 5 a) Γ m + ) 2u 2 ) h dt th e t2 π t ˆζ d 4 3 J U mhl; f ) = Γ m + ) ) l ν 2) 2m +h 2 2 u 2m uh ζ π dt th e t2 t ˆζ. A) e ε)/ e ) e-5 4e-5 6e-5 8e-5. b) ε 2e-5 4e-5 6e-5 8e-5. FIG. 6: a) d as a fuctio of ε for the case of Lagmuir waves for five values of the ratio T /T. b) e as a fuctio of ε for the case of Lagmuir waves for five values of the ratio T /T. For Lagmuir waves it is assumed that io ad electro temperatures are equal. From the thiest to the thickest the curves show the cases of T /T = ad 5. with other parameters as i Fig. 3. replaced by the average value ad also eglect the effect of the Heaviside fuctio i the umerator of the itegrad. This approximatio ca be uderstood as follows: The collisio frequecy for electros) already cotais a step fuctio which therefore do t eed to be writte explicitly i the itegrad. Afterwards we replace the collisio frequecy by the average value ad obtai ) l ν J U mhl; f ) = d 3 f u h u u2m. r q u + i ν u We further approximate by usig u u. For a bi- Maxwellia distributio we therefore obtai ) l ν J U mhl; f ) = q 2π) /2 u 2 u 2u ) h = ε du u 2m e u2 /2u2 ) dt th e t2 t ˆζ ) l ν 2u 2 )m+/2 2q u u 2 2 The result is for some values of h J U ml; f ) Γ m + ) ) l ν 2) 2m +h 2 u 2m uh ζ ˆζ ) J U ml; f ) Γ m + ) ) l ν 2) 2m +h 2 u 2m uh ζ + ˆζ ˆζ ) A) J U m2l; f ) Γ m + ) ) l ν 2) 2m +h 2 u 2m uh ζ ˆζ + ˆζ ˆζ ). 5. Evaluatio of the itegrals J νl mh; f ) We start from Eq. 3) J νl mh; f ) = d 3 u ν d u h u2m L f ) r q u + i ν d Usig the average value of the collisio frequecy we obtai J νl mh; f ) = ν d 3 u uh u2m ) u L f ) r q u + i ν which by compariso with Eq. 6) ca be writte as follows J νl mh; f ) = ν Jmh; f ) where the Jmh; f ) are give by Eqs. A) ad A6).. A2)

21 Brailia Joural of Physics vol. 39 o. March Evaluatio of the itegrals J νν m; f ) We start from Eq. 4) J νν m; f ) = d 3 u ν d ) 2 u 2m L f ) r q u + i ν d Cosiderig a bi-maxwellia distributio usig Eq. 8) ad usig the average value of the collisio frequecy we obtai ) 2 ) ν J νν m; f ) = 2π) /2 u 2 u. 8. Evaluatio of the itegrals J ch f ) The itegral J ch is peculiar because it does ot deped o the collisio frequecy. We start from Eq. 6) J ch f ) = d 3 u f u H u ) deq am v 2. Cosiderig a bi-maxwellia distributio fuctio ad usig spherical coordiates with µ = cosθ the itegral ca be writte as follows = 2π) /2 u 2 u dµ duue u2 /2u 2 ) e u2 µ 2 )/2u 2 ) u lim du u u 2m u u 2 e u2 /2u2 ) where u e lim = 2d e 2 ) /2 /a) m e v 2 u i lim =. = e u2 /2u2 ) du u r q u + i ν ) 2 ν m!) 2) 2m+ ) u 2m ) u ζ J νν m; f ) = π dt t e t2 t ˆζ ) 2 ν m!) 2) 2m+ ) u 2m ) u ζ + ˆζ ˆζ ). A3) 7. Evaluatio of the itegrals J ν f ) We start from Eq. 5) J ν f ) = d 3 u ν d L f ) u Cosiderig a bi-maxwellia distributio usig Eq. 8) ad usig the average value of the collisio frequecy oe readily obtais J ν f ) = ν d ) 2π) /2 u 2 u du e u2 /2u2 ) du e u2 /2u2 ) u u u 2. It is see that the itegral over u vaishes i the case of a bi-maxwellia distributio. Ideed it vaishes for ay distributio which is eve i the parallel compoet of the velocity J ν f ) =. A4) Itroducig a chage of variables u 2 = 2u 2 t the itegral is writte as where = 2π) /2 u dµ dt e t+µ2 )/ t lim t i lim = te lim = χe where we itroduce the quatity χ = q d e)/at ) such that χ i = de 2 at i χ e = de 2 at e. A5) Performig the itegratio over the t variable ad evaluatig at the limits we obtai 2 = 2π) /2 u dµ + µ 2 ) e t lim +µ2 )/ where we have used the parity of the itegrad o the µ variable. The electro ad io cotributios are therefore writte as follows 2 e J ch f e ) = 2π) /2 u e e 2 i J ch f i ) = 2π) /2 u i i e dµ + µ 2 e ) e χe +µ2 e )/ e i dµ + µ 2 i ). A6) APPENDIX B: EVALUATION OF THE AVERAGE VALUE OF THE COLLISION FREQUENCY FOR A BI-MAXWELLIAN DISTRIBUTION The average value of the ielastic collisio frequecy is obtaied by itegratio of the velocity depedet collisio frequecy over velocity space ν = d 3 uν d u) f u).