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L~,~!E?-~}On Streaming Dust Particle Instability ,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, asteroid 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 comet 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 + K2K; + 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.
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