,I Indian Journ~f Radio' & ~patePhysics Vol. 23, Defember 1:,?~kPp. 410-415

Effect of streaming ~l~,~!E?-~}onstreaming dust particle instability

Ix. { ....Physics Department,V ~ik~rn;, University yjj;yar~m~~ of Bombay,'{JS1DlTI\r' Bombay 400 090 , ,/ I J / '1J . Received 11 March 1994; revis'ed received 28 July 1994 '7 ~,ffiispersion relation for ~coustic wav~ in a homogeneous, unmagnetized and uniform plasma consistihg of charged streaming dust particles and streaming electrons is solved. The results are used .J to interpret the interaction between the solar wind plasma and ~.!~ry \iust QarticleJ. Due to this, it is suggested that, small dust particles are likely to be swept by the solar wind and thus assist in the formation of cometary tail.) :2). p 'i1 , -' /""

1 Introduction cles with solar wind has been shown to have an Dust is found to be a common component in important role in the interpretation of large scale many plasma environments. Space plasmas, plane• structures of dusty plasma regionslH6. tary rings, cometary tails, belts, magne• To assess the interaction between the solar tospheres and lower part of earth ionosphere con• wind and cometary dust tail the conditions for the tain dust particles. In the laboratory plasmas, out• onset of streaming instability by jets, expanding

gassing from nucleus, electrostatic disruption etc. halos and solar wind with dusty environment have /. are sOl)1e of the sources of dust particles. These been analyzed5• Bharuthram et al.17have analyzed dust grains collect ions and electrons and acquire the two stream ion instability and also the insta• an electric charge which can be equivalent to bility of drifting dust beams. In the above studies, thousands of electronic charges in magnitude and the role of electrons has been neglected. In this potentialsl-6• The particulate matters have sizes paper, the effect of streaming electrons on the in• in the range of 10 nm-lO pm in various situations. stabilities and the associated turbulent interaction For example, in the interstellar space, the dust have been analyzed. Here, we follow the ap• sizes vary from 10- 7 m (cometary dust in solar proach given by Havnes 13and study the resulting wind region) to 10-5 m (Saturn's magnetosphere). dispersion relation and calculate the growth rates. The charged dust grains will have electromag• Using these results, we have analyzed the interac• netic interaction. Therefore, the interaction be• tion of the streaming electrons with the dust parti• tween the solar wine and has been used to cles in the unmagnetized, homogeneous and un• interpret various kinds of cometary dusty iform dusty plasma. regions7. There are infrared observations of com• ets and measurements of plasma and ,neutral gas 2 Dispersion relation near comet Halley8.9. As the solar wind structures We assume that the electrons and the dust par• extend up to 20 AU (Ref. 10), it is interesting to ticles (denoted by subscript b and d) are stream• understaIid and analyze the interaction between ing with velocities Vb and Vd and have tempera• the solar wind plasma with cometary dust struc• tures Tb and Td, respectively. Background elec• tures. trons and singly charged ion plasma are at tem• The recent studies show that the streaming perature ~ and T;, respectively. The general dis• charged particles excite various wavesl1• Such ex• persion relation D (w, K), for waves propagating citations of waves are also possible by the stream• in the direction of the streami}1g particles where ing dust particles in the plasma. Goertz3, Havnes 5 Vb is parallel to Vd, can be written as: and Rao et al.12 have investigated independently the dusty plasma to analyze acoustic waves gener• Kj W e W Kd 1+ - W - + - W - + - ation and their propagation. (K )2 ( KJI;j ) (K)2 K()( KJI;e K )2 If the instabilities can set up turbulences, then the dust particles can acquire significant velocities by momentum transfer by turbulent collision. Therefore, the interaction of streaming dust parti-

1'1 '" I , "I' I II II \1'1"1'11'1' '1 'I III II KULKARNI et aL:EFFECT OF STREAMING ELECTRONS ON STREAMING DUST PARTICLE INSTABILITY 411

In Eq. (1), the second and the third terms corre• finer distribution of dust grains. Breslin and Eme• spond to the background ions and electrons, leus22 have experimentally observed the dust par• whereas the fourth and the fifth terms represent ticles in the laboratory plasma of sizes having the the streaming dust particles and electrons, res• radius 10-7 cm. Such particles can acquire the pectively. The parameters wand K refer to the charges large enough to make the present study excited wave frequency and its wave number, re• suitable. The resulting dispersion relation D (w, spectively; Vlj are the thermal velocities; K) is equated to R +iI = 0, where R is the real K 7 = 41lnl17/ T; is the inverse square of the Debye part and [is the imaginary part. When solved, Eq. length of the jth particle with nj as its density. (1) with the approximation mentioned above, Here subscript j refers to as j= i(ion), e(electron), gives the real part, R, as b(beam) and d(dust); the temperature is in the en- 1 ergy units and 2 mj v~=Tj. The remaining terms have standard meanings. The Ichimaru function 18, W(y), which is the Z function of Fried and Con• tel9, is defined as follows:

By retaining the first term in the W function, the W(y) = 1 +i z (iz) .. , (2) imaginary part is given by and 1~ fi (A, + B. + C.) ... (5) Z(~)= J;1 [00-00exp( (~-~) - ~2) d~ ... (3) where, y= .fi~.The equation is analyzed for dif• ferent conditions of the media. ... (6) The Ichimaru function W(y) is complex. Here Eq. (1) is solved for different conditions and the onset of the damping/growth of waves is studied . ... (7) By eq~ating the imaginary part to zero, the boun• BI = (Ke)2K( K~W ) exp [1-2 K2V;W2] dary of marginal instability is presented. In this paper, we assume w= wr + iI', where Wr is the real part and is the growth rate, such that Wn I' II'I< < I K KV. exp 2 K2V2 ... and study the following two cases. c = (Kb)2 (w- K~)tb [_.!(w- Ky")2]tb (8)

2.1 Case 1 To obtain the growth, we expand D (w, K) in a Taylor series around W = Wr, the real part of the In the region, where VIi < < ~K < Vh with frequency, for L < < 1and write Wr (w-K~) ..

y" < v'e and K V.Ib < 1, the Ions prOVIde the D(Wr +iI', K )= D (w" K )+(W - wr) awaDIw-w, + '" charge neutralizing background and their contri• =R+i[ bution is neglected. The dust particles whose ->-z.te ed 0 not partICIpate.. m·th· e mteractlon . an d , md mj = D( w" K) + iyaaDI(J) (J)- Wr + (9) therefore, they also provide the charge neutraliz• ing background. To study the dynamical evolution Equating Eqs (4) and (5) with (9), we get the dis• of dust in the Halley's comet, Mendis and Horan• persion relation as yi20have assumed the dust grains of sizes 0.1, 0.3, D(wr,K)=R=O ... (to) 1 and 311m. For discussion, we assume that the dust particles are having Zde>~. A note21 on the and the growth rate, 1', as md 11lj very small grains observed at Halley's comet sug• ... (11) gests that the above approximation is valid in the y= _ [IODrow 412 INDIAN J RADIO & SPACE PHYS, DECEMBER 1994

By replacing Wr by W for convenience, the real The real part R can be solved to give the disper• part, after some algebraic exercise gives sion relation as in Eq. (10), to get w=KYd±--: Wd 1+-2+-2 ... (12) KeK [ K 2 eK Ke~] - 1/2 w=K~±-KeK Wd [ 1 +-2K 2 eKe -~K 2 ( ~- Y tb2Vi,)2 ] - 1/2 .•.(21)

The growth rate y= YI (where subscript, 1 de- and the growth rate, Y2, as in Eq. (11) to get notes the case 1) is shown to be _ n b tb Yl= -wd-(D1±E1) ... (13) Y2= + n"8 Wd KeK [ 1+ K;K2- K2K; (~- y2Vi,)2 ]-312 n8 KeK x(D2±E2) .••• (22) where, where, D =-exp --- + - --- I Jt;e~ (1 2 Y;"Y~) (Kb)2 Ke(~- Jt;bVi,) >

... (14) xexp (_(~_Vi,)2)2V;b '" (23) and Xexp !-(~-2y2 tbVi,)2j

E 1 = (Kd)3Ke ( 1 + K;K2+ K; K~)-1/2 exp (1-"2 K~)K; ... (15)

2.2 Case 2 W ... (24) In the region, where Jt;i < < -K < v'e with Xexp( - 2K;K~) (w- KVi,) (w- K~) -- -> 1 and > 1, we solve Eq. (1) 3 Discussion KJt;b KV,d I to get the real part R as The general dispersion relations, show' that the solar wind interaction with the dusty plasma can lead to low frequency waves. These waves, if R='-(Kd)2 K[ (W-K~)2K2y~ ] + (Ke)2K [ l-K2y;w2] oriented randomly, can lead to the generation of plasma turbulence in the interaction regions. This will have an effective collision frequency between ... (16) the dust particles and the solar wind which, in -; (K)2[(W-K~)2 K2y2 ] general, is proportional to the wave growth. In case 1, we see that the streaming electrons and the imaginary part, 1, as i do affect the wave propagation. However, the rel• ative contribution depends on the densities of ... (17) l~ fi [A'+ 8,+ C,] streaming background electrons. In the interstellar space, streaming electrons of density nb and back• ground electrons of density ne' and their respec• tive temperatures Tb and T. contribute to the wave propagation by the ratio nbT/(neTb)' This can be larger than unity in most situations. How• ever, in case 2, the same contribution depends not only on the ratio nbT/(neTb)' but also on y~J! ... (19) (J

1'1' I "II 'I 'I I 'I ~,,, "1'11 1'1' !~ KULKARNI et al.:EFFECT OF STREAMING ELECTRONS ON STREAMING DUST PARTICLE INSTABIlITY 413

shall have an added effect which suggests that, to 2. (a) study the interaction, we must include the effect Kb/ Ke:lJ•O of the background streaming electrons. The waves are excited due to + sign in Eqs (13) 11 Kbl Ke=2.0 and (22). But the terms Eland E 2 should be III Kbl Ke=1·0 smaller than the terms Dl and D2• By setting IV K b/Ke=O'O ys = 0, the marginal instability curves as obtained are shown in the Figs 1 and 2 for the case 1 and in Figs 3 and 4 for the case 2. In Fig. 1, for (Vd- Vb)/Vtb= 4.0, and at constant KtlKe = 3.0, the curve shows an increase in the peak value for the increase in K/Ke• Similarly, in Fig. 2(a), for (Vd- Vb)~b=4.0 and K/Ke=O, the curve for (KjKe)2 versus V/Vte shows the increase in the peak value as K/Ke increases and in Fig. 2(h) the value of K/Ke is increased to 1.5. By comparing this curve with Fig. 2(a), we see that the respective peak values increase with the in• crease in K/Ke For the case 2, we have selected (Vd - Vb)~b to be 4.0. In Fig. 3, for a constant K/ Ke (= 1.5), the curve shows an increase in peak value with the in• crease in K/ Ke values. In' Fig. 4, for constant K/Ke- ( = 10.0), the peak value decreases with the increase in K /K e values. 3, (b) As seen in Eq. (13) the growth rate, Yl, de- I Kb/Ke=2'S 3 " Kb/Ke= 2-0 III Kb/Ke= 1·0 I K/Ke=1'0 IV Kb/Ke= 0'0

" K/Ke=O-S 2 N III K/Ke=0'os r--1 ••• IV K/Ke=O'O lie 2 -.. "0 L-Jlie

1.

3

3 Fig. 2-(a) Marginal instability curve obtained from

versus (Vd)v,. for y, at(Vd-J-b) ~ =4.0 and (K)K, =0.0, Fig. I-Marginal instability curve obtained from (~: r and (b) marginal instability curve obtained from

versus - for Yt at -- =4.0 and - = 3.0 (v,. Vd)( Vd-V,b Vb) (Kb)K, (Kd)2K. versus \(Vd)v,. for YI at(Vd- ~Vb) =4.0 @J:!.. (K)K, = 1.5 414 INDIAN J RADIO & SPACE PHYS, DECEMBER 1994

2 2-5

2·0 Kb/Ke=Kb/Ke=10-0 1-0 I III Kb/Ke= 0-0 II K/ Ke= 0·0 II K/ Ke= 1·5 III K/Ke=3·0

0'5

o o o 1 2 3 4 o 3 4 [Vd/Vte]

Fig.4-Marginal instability curve obtained from (~:rversus

Fig. 3-Marginal instability curve obtained from (~:)' versus - for yz at --- = 4.0 and - = 10.0 (v" V,I) (V,I-v,,, Vb) (K")K,

- for yz at --- = 4.0 and - = 1.5 (Vd)v" (Vd-Vb)v,,, (K)K, Yl(2) > vi/~ell)' where vi/~ell)' are ion neutral (electron neutral) collision frequencies. Moreover, for insta• bility to develop, we can calculate the upper limits pends directly on (Vd- Vb)' However, the third on neutral densities which depend on the wave• term in E 1 does not depend on (Vd - Vb)' Where• lengths. If we follow the arguments of Havnesl3, as, in Y2, E2 depends strongly on (Vd- Vb)' We, the streaming electron will have significant inter• therefore, believe that the latter shows a stronger action provided effect of current carrying electrons. In the instability region, when EI and E 2 are negligible, the growth rates YI and Y2 will be the K~(~-Ke V,b Vb) exp [_22 (Vd-~/]Vtb same i.e., Yl = Y2 = yand is given by is significantly larger than one. Therefore, the de• J tailed analysis of this term can give relative im• Y'" _ Wd--exP : + _ 2 _ fi8 KeK ~v;, (1 2 Vd)Ve(Kb) Ke (~-Vb)V,b portance of all the three terms, namely, Vii> Vb and ~b' Similarly in case 2, i.e., in Y2, it is the ra• tio ~J( Vd- Vb) and the related quantities, which x exp 2 ... (25) will play an important role . . 4V'tb v;}] l-(~- Following Havnes 13 we can set Y, the growth Therefore, the streaming electrons assist in excit• rate as the upper limit to the collision frequency I ing the longitudinal waves. These streaming in• veff• The wind , aw, will be "'"VdVeff .• Since veff stabilities can lead to the growth rates which can is finite, aw is finite. This aw is responsible for the give rise to the turbulence. We can set growth coupling between solar wind electrons and the rates YI or Y2 as an upper limit to the effective dusty plasma in the cometary tails. When aw> as collision frequency, which is responsible for the and ac, where as and ac are gravitational accelera• momentum loss of dust to ions. Because of the tion of sun ai'Id , respectively, we can ex• presence of the neutral particles, we need the pect the tail formation in the solar wind direction. time scales of the interaction such that, For example, the solar gravitational acceleration

I Ii .'1 HI 1111'1111~11111II 1111 II MII.III I III;~IIIIII III N ~c,_~~_ 1 KULKARNI et al.:EFFECT OF STREAMING ELECTRONS ON STREAMING DUST PARTICLE INSTABILITY 415 I I

\ at a distance r is a,= a,jL~ (here L= rlr0, aso is Acknowledgement the accelaration on the solar surface) and that due Two of the authors (VHK and VKD) thank I to comet is a('=ajL; (here L('= rlr", r(' is the DST for the fin~ncial support to conduct this comet radius, and a('o is the accelaration at the study (SP/INC/PP-1/91). cometary surface). This gives the lower limit as veff""asjL~ ~d or ajL~ Vtd• Since Vtd is the rela• References tive velocity between electron and dust particles, 1 Sato N, A Variety of Plasmas, edited by A Sen & PK which is very large, we get veff smaller than those Kaw (Indian Acad Sci, BangaIore, India), 1991, p 79. used by Havnes. Since veff is comparable to low• 2 Goertz C K, Geophys Res Left (USA), 11 (1984) 349. est frequencies available, i.e. the dust plasma fre• 3 Goertz C K, Rev Geophys( USA), 27 (1989) 271. quencies, ltJm then L~ or L~ is proportional to 4 Havnes 0, Astron Astrophys (Germany), 90 (1980) 106. a.J( ltJd V,d) or aj( ltJd Vtd). Therefore, the turbul• '5 Havnes 0, AdvSpace Res(UK), 4 (1984) 75. ence drag will be effective at very large heliocen• 6 Havnes 0, J Geophys Res (USA), 92 (1987) 2281. 7 Ip W H, AdvSpace Res(UK), 4(1984) 239. tric distances, where according to Ip7, the charged 8 Zarnecki J C, Eaton N, Mc Donnel JAM, Meadows A dust grains are electrostaticall~ levitated, but still J, Carey W C & Mac Donald G H, Adv Space Res (UK), bound to the cometary surfaces. 4 (1984) 203. To study the interaction of solar wind with dust 9 Gringauz K I, Gombosi T I, Remiziv A P, Apathy I, cometary tails, we need to consider the electron Szemerey I, Vergin M I, Denchikova L I, Dyachkov A V, Keppler E, Klimenko I N, Richter A K, Somogyi A J, velocities and their temperatures. The electron Szego K, Szendro S, Tatrally M, Varga A & Vladimirova temperatures vary with the solar wind flow. If we G A, AdvSpace Res (UK), 5 (1985) 165. set Vd= 800 km s - 1, high speed dust streams and 10 Holzer T E, Plasma Physics, edited by E N electrons in the background at ~ "" 105 K, and a Parker, C F Kennel & L J Lanzerotti (North-Holland beam of electrons whose velocities are of the or• Publ, Amsterdam) Vol. 1, 1979, p. 101. 11 Krall N A & Trivelpiece A W, PTinciples of Plasma Ph~ der Te> we can find the relative contributions. sics (Mc-Graw Hill, Kogakusa, New York) 1973. In the case. when ~/Vte= 1.5 and KtlKe= 12 Rao N N, Shukla P K & Yu M Y, Planet & Space Sci 1.5 =(V;,- ~)lVtb' we find that in Eq. (25), the sec• ( UK), 38 \ 1990) 543. ond term dominates over the first term. In fact, it 13 Havnes 0, Astron Astrophys (Germany), 193 (1988) 309. is almost double the first term. Moreover, if the 14 Nappi C, Proceedings of the First Capri Workshop on turbulent interaction is to have a significant effect, Dusty Plasma, held during 28 May-2 June 1989 in Italy. 15 Northrop T G, A Variety of Plasma, edited by A Sen & P the turbulent drag should be comparable to the K Kaw (Indian Acad Sci, Bangalore, India) 1979, p. 91. other effects like radiation pressure. 16 Shukla P K & Silin V P, Phys Scr (Sweden), 45 (1992)' In conclusion, the dust solar wind interaction . 508. could be studied with the help of instability analy• 17 Bharuthram R, Saleem H & Shukla P K, Phys Scr (Swed• sis presented in this paper. Since the solar wind en), 45 (1992) 512. 18 Ichimaru S, Basic Principles of Plasma Physics (Benjamin velocity, density and temperature are variables, Inc), 1973, p 245. the turbulent drag can vary. However, the solar 19 Fried B D & Conte S D, Plasma Dispersion Function wind is likely to extend to very large heliocentric (Acaaemic Press, New York), 1961. distances and the present analysis will be useful 20 Mendis D A & Horanyi M, Cometary Plasma Processes for understanding the. tail formation of the com• Geophysical Monograph, edited by A Johnstone (USA), Vol. 61,1991, p. 17. ets. We expect a general tendency that the come• 21 Fomenkova M N & Mendis D A, Astrophys Space Sci tary dust tails can depend on the electron par• (Belgium), 189 (1992) 327. ameters. Numerical calculations of the growth 22 Breslin A C & Emeleus K G, Phys Left A (Netherlands), rates are also planned in continuation. 31 (1970) 23.