Propagation of Electrostatic Solitary Wave Structures in Dense Astrophysical Plasma: Effects of Relativistic Drifts & Relativistic Degeneracy Pressure

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1 Advancs in Astrophysics, Vol., No., Novmbr Propagation of Elctrostatic Solitary Wav Structurs in Dns Astrophysical Plasma: Effcts of Rlativistic Drifts & Rlativistic Dgnracy Prssur Swarniv Chandra Dpartmnt of Physics, JIS Univrsity, Kolkata-79, India Dpartmnt of Physics, Jadavpur Univrsity, Kolkata-7, India swarniv47@gmail.com Abstract. Analytical and numrical studis ar prsntd for lctron acoustic solitary wav structur in rlativistic dgnrat two-componnt unmagntizd astrophysical plasma. Th xistnc of a wav mod of pur quantum origin is prdictd. Th ffct of various plasma paramtrs on th conditions of xistnc and proprtis of solitary wav is invstigatd. It is shown that dpnding on th valus of plasma paramtrs both rarfactiv and comprssiv typ of solitons can xist. It is obsrvd that th amplitud and width of th solitons ar significantly affctd by th quantum and rlativistic ffcts. Th rlativistic ffcts arising out of straming motion is tratd by Eulran formulation whras th rlativistic dgnracy ffcts is invstigatd by making us of Chandraskhar formula. Kywords: Quantum plasmas, astrophysical plasma, rlativistic dgnracy, quantum diffraction, solitary structurs, quantum hydrodynamic modl. Introduction In rcnt yars thr has bn a grat dal of intrst in studying th diffrnt aspcts of nonlinar wav propagation in quantum plasma. Traditional plasma physics has mainly focusd on rgims charactrizd by high tmpratur and low dnsity whr quantum ffcts hav virtually no impact. But in plasmas whr th dnsity is quit high and th tmpratur is vry low thrmal d Brogli wavlngth of lctrons may bcom comparabl to th spatial scals of th systm and thn quantum natur of th plasma constitunts cannot b nglctd and quantum ffcts could affct th plasma bhavior in a significant way. Th condition is wll satisfid in som compact astrophysical obcts (.g. whit dwarfs, nutron stars, magntars tc.), also in mtals, smiconductors and lasr producd plasmas so that such phnomnon may b obsrvd. Th mattr xists in xtrm conditions of dnsity. In such situation th avrag intr-frmion distanc is comparabl to or lss than th thrmal d Brogli wavlngth and hnc quantum dgnracy ffcts bcom important. Quantum ffct ariss du to ovrlapping of th wavfunctions of th nighboring particls. In quantum plasmas whr th lctron thrmal nrgy is much smallr than thir Frmi nrgy th statistical bhavior of plasma particls should b dscribd by Frmi-Dirac statistics and not by th classical Boltzmann statistics. Such quantum plasmas may b found in a varity of nvironmnts such as mtal nanostructurs [], astrophysical systm [], ultra-small lctronic dvics [,4], biophotonics [5], cool vibs [6], nutron stars [7], lasr producd plasmas [8], quantum wlls and quantum diods [9,]. Most of th invstigations on nonlinar wav propagation in quantum plasma ar confind to th nonrlativistic cas. But whn th lctron or ion vlocity approachs th spd of light rlativistic ffct may significantly modify th nonlinar bhavior of lctron plasma wavs. Rlativistic plasma can b formd in many practical situations,.g. in spac-plasma phnomna [, 7], th plasma sht boundary of arth s magntosphr [, ], van Alln radiation blts [] and lasr-plasma intraction xprimnts [4, 5]. Th rlativistic motion in plasmas is also assumd to xist during th arly priod of volution of th univrs [6]. Rgarding th rlativistic ffcts on ion-acoustic solitary wavs a numbr of works hav bn rportd for classical plasma. For xampl, Sad t al [7] hav shown that Copyright 6 Isaac Scintific Publishing

2 88 Advancs in Astrophysics, Vol., No., Novmbr 6 in lctron-positron-ion plasma incras in th rlativistic straming factor causs th soliton amplitud to thriv and its width shrinks. El-Labany t al [8] hav shown that rlativistic ffct can modify th condition of modulational instability of ion-acoustic wavs in a warm plasma with nonthrmal lctrons. Rgarding th rlativistic ffcts on lctron plasma wavs only a vry fw works can b found in th litratur. Rcntly Bharuthram and Yu [9] hav shown that rlativistic lctron plasma wavs can propagat as quasi-stationary nonlinar wavs as wll as solitary wavs. So far all th works on th rlativistic ffcts on plasma wavs hav bn rportd for classical plasma. In cours of th intrsting dvlopmnts of quantum ffcts in plasma it is natural to invstigat whthr th combind ffcts of rlativity and quantum mchanics can display som proprtis of plasma. Rlativistically dgnrat dns plasmas can b found in many astrophysical nvironmnts including intriors of whit dwarf stars [] and magntars []. Thus it is important to invstigat th rlativistic and quantum mchanical ffcts in a combind way on th nonlinar wav structur in plasma. Th inclusion of quantum-mchanical ffcts in plasma rquirs nw mathmatical formulation or a suitabl modification of th formulations usd in classical situations. Quantum ffcts in plasmas ar usually studid with th hlp of two wll-known formulations, viz. th Wignr-Poisson and th Schrödingr-Poisson formulations. Th Wignr-Poisson modl is oftn usd in th study of quantum kintic bhavior of plasma. Th Schrodingr-Poisson modl dscribs th hydrodynamic bhavior of plasma particls in quantum scals. It can b considrd as th quantum analog of th fluid modl of traditional plasma. Th quantum hydrodynamic (QHD) modl is drivd by taking vlocity-spac momnts of th Wignr quations as in th classical fluid modl. This modl consists of a st of quations dscribing th transport of charg, momntum and nrgy in a chargd particl systm intracting through a slf-consistnt lctrostatic potntial. Th QHD modl gnralizs th fluid modl for plasma with th inclusion of a quantum corrction trm also known as th Bohm potntial []. Th modl incorporats quantum statistical ffcts through an quation of stat. Th quantum corrctions may giv ris to nw aspcts of purly quantum origin in th collctiv bhavior of plasma at quantum scal. For xampl, it may lad to th gnration of nw wav mods in plasma []. Bcaus of simplicity, straight forward approach and numrical fficincy th QHD modl has bn widly usd by svral authors [4-] in daling with diffrnt aspcts of linar and nonlinar wav propagation in unmagntizd quantum plasmas. For xampl, using th QHD modl Haas t al [] hav studid th important rol of quantum diffraction in linar as wll as nonlinar rgims for th propagation of ion acoustic wavs; Gardnr and Ringhofr [4] has studid th lctron-hol dynamics in smiconductors. Using th sam modl Shukla and Eliasson [5] hav studid th dynamics and formation of dark soliton and vortics in quantum plasma. It has also bn usd to study th Kortwg dvris (KdV) solitary wav structur for ion acoustic wavs [6, 7], lctron-acoustic wavs [8], dust-acoustic wavs and dust ion-acoustic wavs [9, ]. Rcntly w hav studid th ffct of quantum diffraction on th lctron plasma wavs and it has bn found that quantum ffcts can significantly modify th modulational instability conditions and th instability growth rats of finit amplitud lctron plasma wavs []. Thr ar practical situations such as intns lasr-solid intraction xprimnts and prsumably th arly priod of th volution of th univrs whr both th quantum and rlativistic ffcts may bcom important for considration. Th motivation of th prsnt papr is to study th rlativistic ffct on th formation and charactristics of KdV solitary wav structur of plasma wavs in a dgnrat plasma including full ion dynamics. Using th on-dimnsional quantum hydrodynamic (QHD) modl for two componnt lctron-ion dns quantum plasma w hav studid th linar and nonlinar proprtis of a plasma wav mod of pur quantum origin including rlativistic ffcts. Bcaus of havir mass quantum corrction for ions is smallr than that for lctrons. Howvr to mak a complt analytical study w hav considrd, unlik prvious authors, quantum corrctions for both th lctrons and ions. It is shown that th rlativistic ffcts can significantly chang th linar and nonlinar proprtis of th wavs in quantum plasma. Th papr is organizd in th following way: in sction th basic st of quantum hydrodynamic quations ar prsntd including straming motion; in sction. th linar disprsion charactristics and xistnc of a wav mod of pur quantum origin is prdictd; in sction. th Kortwg dvris quation is drivd by using th standard prturbation tchniqus. In sction th formulation for rlativistic dgnracy is introducd and th basic st of quations ar writtn, sction. dals with th linar disprsion charactristics, in sction. th solitary profils ar invstigatd. Sction 4 givs a comparativ study of ths two kind of rlativistic ffcts. Copyright 6 Isaac Scintific Publishing

3 Advancs in Astrophysics, Vol., No., Novmbr 6 89 Basic Equations W considr rlativistic dns quantum plasma consisting of lctrons and ions with a straming motion along th x-axis. W also assum that th plasma particls bhav as a on dimnsional Frmi gas at zro tmpratur and thrfor th prssur law [] is: mv F p = n () n whr = for lctron and = i for ions; m is th mass; V = k T / m is th Frmi F B F thrmal spd, T F is th Frmi tmpratur and k B is th Boltzmann constant; n is th numbr dnsity with th quilibrium valu n. Th st of QHD quations dscribing th dynamics of th lctron plasma wavs in th modl plasma undr considration ar th following: n ( n γ u ) + = () t x q φ p ħ + ( γ ) = + n u u () γ t x m x mn x m x n x φ = 4π q γ n (4) x whr u, q and p ar rspctivly th fluid vlocity, charg and prssur of th th spcis, q =, = γ = u / c is th rlativistic factor, c is th vlocity of light in fr spac, ħ i is th Planck s constant dividd by π and φ is th lctrostatic wav potntial. W now us th following normalization: x xω / V, t tω, φ φ / kt, n n / n and u u / V p F p B F F q, ( ) / whr ω = 4 πn / m is th lctron plasma oscillation frquncy and V is th Frmi thrmal p F spd of lctrons. Th normalization givs us th following simplifid st of quations for lctrons and ions as: ( n ) ( n γ u ) + = (5) t x φ n H + ( γ ) = + n u u n (6) t x x x γ x n x n ( nγ u ) i i i i + = (7) t x φ µ n i H + ( γ ) = µ σ + ni u u n (8) i i i i t x x x x n x i φ = ( γ n γ n i i) (9) x whr H= ħ ω / kt is a nondimnsional quantum paramtr proportional to th quantum p B F diffraction, µ = m / m is th ratio of lctron and ion mass and σ = T / T is th ratio of ion and i Fi F lctron Frmi tmpraturs. Th quantum diffraction paramtr H is proportional to th ratio btwn th plasmon nrgy ħω p (nrgy of an lmntary xcitation associatd with an lctron plasma wav) and th Frmi nrgy kt. Th quations (5)-(9) constitut th basic st of quantum B F hydrodynamic quations to b usd in th invstigation of nonlinar propagation of lctron plasma Copyright 6 Isaac Scintific Publishing

4 9 Advancs in Astrophysics, Vol., No., Novmbr 6 wavs in quantum plasma. Quantum ffcts ar includd in th modl through th scond and third trms on th R.H.S of quations (6) and (8). Th scond trm on th RHS of Equations (6) and (8) includ quantum statistical ffct through th quation of stat [Eq. (4)]. Th third trm in th RHS of Equations (6) and (8) ariss du to quantum corrction of dnsity fluctuations and this typ of quantum ffct is calld quantum diffraction or Bohm potntial.. Linar Disprsion Charactristics In ordr to invstigat th nonlinar bhavior of plasma wavs w mak th following prturbation xpansion for th fild quantitis n, u, n, u and φ about thir quilibrium valus: i i () () n n n () () u u u u = + ε n n () + ε n () +... () i i i () () u u u u i i i φ () φ () φ i( kx ωt) Assuming that all th fild quantitis ar varying as, w gt for normalizd wav frquncy ω and wav numbr k, th following disprsion rlation of lctron plasma wavs which includs quantum and rlativistic ffcts for both lctrons and ions: µ + = () 4 4 γ Hk µ Hkγ ( ω ku ) γ k ( ω ku ) γ σk 4 4 u u whr γ = + and γ = c c. In th dimnsional form Eqn. () bcoms: µ + = () γ H kv µ H kv γ ω F F p ( ω ku ) γ k V ( ω ku ) γ σk V F F 4ω 4ω p p Equation () is a quadratic quation in ω and has th solutions: ( + ) 4 µ γ ω Hk = ( + µ ) + ( + σ) k + + D + ku () fast γ 4 ( + ) 4 µ γ ω Hk = ( + µ ) + ( + σ) k + D + ku (4) slow γ 4 4 ( µ ) γ Hk whr D = 4µ + ( µ ) + ( σ) k + is th discriminant. 4 Th discriminant D in Equations () and (4) is always positiv. Thus th solution for ω has two positiv branchs indicating th xistnc of two distinct lctrostatic wav mods. In ordr to xtract mor information rgarding th bhavior of ths two wav mods w hav studid thm numrically. It is shown that th linar disprsion curv for th fast mod [rprsntd by Eq. ()] dpnds significantly on H but is almost indpndnt of σ. On th othr th linar disprsion of th slow mod [rprsntd by Eq. (4)] dpnds slightly on σ and is almost indpndnt of H. Th dpndnc of linar disprsion rlation on th straming vlocity for both slow and fast mod is shown in Fig.. Th phas vlocity of both mods incrass with incras in straming vlocity. Figur shows th dpndnc of slow mod on quantum diffraction paramtr ( H ) and ion-to-lctron tmpratur ratio. It is found that quantum diffraction has no significant ffct on linar disprsion proprtis, Copyright 6 Isaac Scintific Publishing

5 Advancs in Astrophysics, Vol., No., Novmbr 6 9 whras ion tmpratur has small but finit positiv ffct of incrasing th frquncy for a givn wavnumbr. Figur. Dpndnc of Fast and Slow mods on rlativistic straming vlocity ( u ). Figur. Dpndnc of Slow mod on quantum diffraction paramtr ( H ) and ion-to-lctron tmpratur ratio (σ ). If w assum that th ions ar infinitly havy compard to th lctrons (i.. µ = ) thn th wav branchs () and (4) rduc rspctivly to: 4 γ Hk ω = + k + + ku (5) fast γ 4 and σ ω = + u k (6) slow γ In th dimnsional form th frquncis of ths two wav branchs ar givn as: 4 4 γ H kv F ω = ω + k V + + ku (7) fast p F γ ω 4 p and σ ω = + u k (8) slow γ Equation (7) corrsponds to th disprsion rlation for lctron plasma wavs including quantum diffractions and rlativistic ffcts. Th wav mod rprsntd by Eq. (8) ariss out of quantum statistical and rlativistic ffcts. Th fast mod corrsponds to th usual lctron plasma wav modifid by quantum and rlativistic ffcts. In th absnc of th quantum diffraction (i.. H = ) and rlativistic ffct disprsion rlation (7) rducs to th wll-known disprsion rlation for lctron plasma wavs in classical plasma. Th slow mod rprsntd by (8) is of pur quantum origin and is causd by quantum statistical ffct as is vidnt from th prsnc of th paramtr σ. If both quantum statistical ffct and rlativistic ffct ar rmovd, th scond mod vanishs.. Th KdV Equation In ordr to driv th dsird KdV quation dscribing th nonlinar bhavior of th plasma wav mod of pur quantum origin w us th standard rductiv prturbation tchniqu. W introduc th usual strtching of th spac and tim variabls: / / ξ = ε ( x Vt) and τ = ε t (9) Copyright 6 Isaac Scintific Publishing

6 9 Advancs in Astrophysics, Vol., No., Novmbr 6 whr V is th linar long wav phas spd normalizd by lctron Frmi spd V F and ε is a smallnss paramtr masuring th disprsion and nonlinar ffcts. Equations (5)-(9) ar writtn in trms of th strtchd coordinats ξ and τ and thn th prturbation xpansions () ar substitutd. Solving th lowst ordr quations with th boundary conditions that whr n, () u, () n, () i () u and φ as ξ, th following solutions ar obtaind: () i n () = βφ (), u () = αφ (), n () = βφ () and u () = αφ () () i i i µ σ β =, β = i γ ( V u ) γ ( V u ) ( V u ) ( V u ) () µ σ α =, α = γ ( ) ( ) γ i V u V u ( V u ) ( V u ) Going for th nxt highr ordr trms in ε, aftr a fw algbraic stps, th dsird KdV quation is obtaind: φ φ φ φ + A + B = () τ ξ ξ whr F F A = and B = () F F in which α β i i F = γα + + ( V u ) γβ + µ µ α σ αβ i i i F = γα + + β + β + ( V u ) γαβ + αγ (4) µ i µ µ σγ H F = ( γ β + µβ ) + µγ i 4 α, β, α and β u ar as dfind arlir in quation () and γ = i i c. It is to b notd that th cofficints of th disprsiv and nonlinar trms in th KdV quation () gt modifid by th quantum and rlativistic ffcts. To find th stady stat solution of Eq. () w transform th indpndnt variabls ξ and τ into on variabl η = ξ M τ whr M is th normalizd constant spd of th wav fram. Applying th φ φ boundary conditions that as η ±, φ,,, th possibl stationary solution of Eq. () is η η obtaind as: φ = φ sch ( η m ) (5) whr th amplitud φ m and width of th soliton ar givn by: φ = M A (6) m and = 4B M (7) Th solitary wav structur is formd du to a balanc btwn disprsiv and nonlinar ffcts. Rlativ strngth of ths two ffcts dtrmins th charactristic of such solitary wav structur. Th cofficints A and B, corrsponding to th nonlinar ffct and disprsiv ffct rspctivly, play Copyright 6 Isaac Scintific Publishing

7 Advancs in Astrophysics, Vol., No., Novmbr 6 9 crucial rols in dtrmining th solitary wav structur. So it is important to study th dpndnc of ths cofficints on diffrnt physical paramtrs. Numrical calculations show that th cofficint A incrass with incrass in σ. It is almost indpndnt of quantum diffraction paramtr H and straming vlocity u. Th co-fficint B of th disprsion trm dcrass significantly with incras in σ but incrass slightly with incras in th straming vlocity u. Th cofficint B dpnds intrstingly on H. It dcrass as H is incrasd from zro and for crtain critical valu of H (say H ) it bcoms zro. From quations (4) and (5) w gt th critical valu of H as: c σγ γ µ H c = + + µγ γ ( V u ) ( V u ) σ whr γ and γ ar as dfind arlir. Numrically w find that th valu of H c incrass with u and dcras in σ. This rsult is vry similar to that prviously rportd for ion incras in acoustic solitary xcitations in quantum plasma []. In this sction using th quantum hydrodynamic modl and rductiv prturbation tchniqu w hav drivd th Kortwg d Vris quation including th rlativistic variation of mass of lctrons with vlocity for invstigating small amplitud solitons. Th dpndnc of th solitary wav structur on diffrnt plasma paramtrs such as rlativistic straming factor, ion-to-lctron tmpratur ratio ar studid both analytically and numrically. It is found that only comprssiv solitons can xist dpnding upon th valus of diffrnt plasma paramtrs. For H > H comprssiv solitons ar obtaind (Fig. c ). Rgarding th dpndnc of th amplitud and width of th soliton on th straming vlocity it is found that th soliton amplitud is almost indpndnt of staming vlocity u but with incras in straming vlocity th width incrass. Th soliton amplitud is indpndnt of quantum diffraction paramtr H, but with incrasing H th width shrinks for comprssiv solitons [Fig 4]. (8) Figur. Comprssiv solitons for diffrnt valus of rlativistic straming vlocity ( u ); H = ( > H c ), σ =. and M =.. Figur 4. Comprssiv solitons for diffrnt quantum diffraction paramtr ( H ); u =., σ =. and M =.. Copyright 6 Isaac Scintific Publishing

8 94 Advancs in Astrophysics, Vol., No., Novmbr 6 Figur 5. Comprssiv solitons for diffrnt ion to lctron tmpratur ratio (σ ); H =.5, u =. and M =.. Figur 6. Effct of rlativistic straming vlocity ( u ) on th amplitud and width of comprssiv soliton H =.5, u =., σ =. and M =.. Th dpndnc of soliton charactristics on ion to lctron tmpratur ratio σ is found to b significant. Both th amplitud and width of th solitons dcras significantly with incras in th valu of σ [Fig. 5]. Dpndnc of th solitary structur on th rlativistic straming factor is shown in Fig. 6 from which it is clar that rlativistic ffct slightly incrass both th amplitud and width of th soliton. Effcts of Rlativistic Dgnracy: Chandraskhar Tratmnt According to S. Chandraskhar (99) th lctron dgnracy prssur in fully dgnrat and rlativistic configuration can b xprssd in th following form: 4 5 = ( π P mc h ) R(R ) + R + sinh R (9) in which whr = ( / ) = = π R p mc h n 8 mc = R n () F R n n with π 9 n = 8 m c h 5.9 cm, c bing th spd of light in vacuum. p is th lctron Frmi rlativistic momntum. It is to b notd that in th limits of vry small and F vry larg valus of rlativity paramtr R, w obtain: P = π h n m 5 ( for ) R () ( for ) 4 P = hcn R () 8 π Not that th dgnrat lctron prssur dpnds only on th lctron numbr dnsity but not on P + R th lctron tmpratur. Now considring th fact that = n x x can b writtn in th following normalizd form: ( n ) ( nu) + = t x th basic quations ()- (4) () Copyright 6 Isaac Scintific Publishing

9 Advancs in Astrophysics, Vol., No., Novmbr 6 95 whr = ( χ /)( + ) φ n H + = + n u u F t x x x x n x φ = x ( n n ) i F R R is th trm arising from rlativistic prssur in wakly rlativistic cas, whras for ultra rlativistic cas (4) (5) F = χ R whr χ = mc k T. Hr = for lctrons B F and i for ions. H is th non-dimnsional quantum diffraction paramtr dfind as H= ħ ω /kt, B F whr T is th Frmi tmpraturs for lctrons; n and n ar th quilibrium numbr dnsitis F i lctrons and ions rspctivly. Th normalization has bn carrid out in th following mannr: x xω / c, t tω, φ φ / kt, n n / n, u u / c s in which ω = s B F sh B F 4 πn / m is th cold lctron plasma frquncy, c = kt / m is th lctronacoustic spd. It is to b notd that th paramtr R is a masur of th rlativistic ffcts and may b calld rlativistic dgnracy paramtr. For ultra rlativistic cas R and for wakly rlativistic cas R. Th paramtr R can also b rlatd to mass dnsity as ρ ( gr / cm ) =.97 6 R [Akbari-Moghanoughi ()]. Th dnsity of whit dwarfs can b in th rang 5 < ρ < 9. So in this cas, th rlativity paramtr R can b in th rang.7 < R < 8.. Linar Disprsion Charactristics Th disprsion rlation for normalizd wav frquncy ω and wav numbr k, th following linar disprsion rlation: µ + = (6) 4 4 Hk µ Hk ω Fk ω Fk i 4 4 F R R in th wakly rlativistic limit and F = χ R in th ultra-rlativistic whr = ( χ + ) cas. whr ω = B + B 4C ω = B B 4C 4 ( ) ( /) ( µ i ) ( /4)( µ /4) ( ) ( /4)( µ i i ) B= k F+ F + Hk + + C= kf+ Hk kf+ Hk + k F+ F + Hk + It indicats that two stabl linar mods for EAWs ar possibl whn on considrs inrtial and rlativistic ffcts of both lctrons and ions. EAWs ar high frquncy lctrostatic lctron oscillations whr th rstoring forc coms from th lctron prssur and th ions provid th inrtia. If w nglct th inrtia of lctrons and assum that th prssur is solly du to th ultra-rlativistic lctrons thn th disprsion rlation () rducs to: 4 k( χr + Hk 4) Hk ω = + (9) + k( χr + Hk 4) 4 In th long wavlngth limit (i.. k ) ω = k χr (4) Th long wav phas spd is: (7) (8) Copyright 6 Isaac Scintific Publishing

10 96 Advancs in Astrophysics, Vol., No., Novmbr 6 V = ω / k = χr (4) It rprsnts th long wav disprsion charactr of EAWs in quantum-rlativistic plasma composd of inrtia lss lctrons and inrtial cold ions. W numrically xamin th bhavior of th disprsion rlation (7) with rspct to th variations of R and H. Fig. 7 shows th variation of ω with k for diffrnt valus of th rlativity paramtr R. It shows that th wav frquncy ω incrass with incras in th valu of R. Fig. 8 shows th ω k curvs for diffrnt valus of H. Obviously, th wav frquncy ω also incrass with incras in H. Figur 7. Disprsion Curv for diffrnt valus of th rlativity paramtr R Figur 8. Disprsion Curv for diffrnt valus of th quantum paramtr H. KdV Equation In ordr to study th nonlinar bhavior of lctron acoustic wavs w considr inrtia lss hot ultra rlativistic lctrons, inrtial cold lctrons and stationary ions. Th prssur ffct is assumd to b only du to th hot lctrons. This typ of considration has bn mad by many prvious authors [- 6]. Following th standard rductiv prturbation tchniqu w us th usual strtching of th spac and tim variabls: ξ = ε ( x Vt) and τ = ε t (4) whr V is th normalizd linar long wav phas vlocity givn by Eq. (4) and ε is th smallnss paramtr masuring th disprsion and nonlinar ffcts. Now writing th Equations (4)-(6) in trms of ths strtchd co-ordinats ξ and τ and thn applying th prturbation xpansion (4) and solving for th lowst ordr quation with th boundary condition that as ξ, th following solutions ar obtaind: () n, () u, n, u and φ () () () Copyright 6 Isaac Scintific Publishing

11 Advancs in Astrophysics, Vol., No., Novmbr 6 97 () () () () () δφ () δφ () φ () φ n =, u =, n =, u = i i V V V V Going for th nxt highr ordr trms in ε and following th usual mthod w obtain th dsird Kortwg d Vris (KdV) quation: φ φ φ φ + A + B = (44) τ ξ ξ whr 4 V H /4 ( χr ) H /4 A = = and B = = (45) V δχr V χr Using usual tchniqus and applying th boundary conditions that as η ±, possibl stationary solution of Eq. (45) is obtaind as: φ = φ sc ( η ) (4) φ φ φ,,, th η η h (46) m whr th amplitud phi and width of th soliton ar givn by: m φ = M A (47) m and = 4B M (48) Th solitary wav structur is formd du to a dlicat balanc btwn disprsiv and nonlinar ffcts. Rlativ strngth of ths two ffcts dtrmins th charactristic of such solitary wav structur. Th cofficints A and B, corrsponding to th nonlinar ffct and disprsiv ffct play crucial rols in dtrmining th solitary wav structur. Thr xists a critical valu of R for which th solitary structur vanishs givn by: ( R ) = H χ (49) c c Figur 9. Solitary Structurs for diffrnt valus of th rlativity paramtr R Copyright 6 Isaac Scintific Publishing

12 98 Advancs in Astrophysics, Vol., No., Novmbr 6 Figur. Solitary Structurs for diffrnt valus of th quantum paramtr. In this sction w hav invstigatd using th on-dimnsional quantum hydrodynamic modl and th standard rductiv prturbation tchniqu both th linar and nonlinar proprtis of lctron-acoustic wavs in two-componnt rlativistically dgnrat plasma consisting of lctrons and ions. A gnral typ disprsion rlation has bn obtaind including inrtia and quantum rlativistic ffcts of both lctrons and ions. It is shown that two stabl linar mods of propagation ar possibl for lctronacoustic wavs whn on considrs th inrtia ffct of both spcis. Th wav frquncy is shown to incras with th incras in th valus of rlativity paramtr R and also th quantum diffraction paramtr H. To study th nonlinar bhavior of th wav a KdV quation has bn drivd in which th cofficints of th nonlinar and disprsiv trms ar found to gt modifid du to th inclusion of quantum rlativistic ffcts. Thr xists a critical valu of th rlativistic dgnracy paramtr R h such that for R < ( R ) no soliton solution is possibl. This critical valu of th dgnracy paramtr c is dtrmind by th valus of H. From Equations (46)- (49) it is obvious that th dgnrat plasma undr considration supports only rarfactiv solitary wav structurs which ar associatd with ngativ potntials. Figur 9 shows lctron-acoustic solitary profils for diffrnt valus of th rlativistic dgnracy paramtr R (which is dirctly proportional to th plasma numbr dnsity) for fixd valus of M and H. It shows that both th amplitud and width of th soliton incras with incras of R. Fig. shows solitary structurs for diffrnt valus of H kping othr paramtrs fixd. It shows that th soliton width incrass with incras in th valu of H but its amplitud is indpndnt of H. Th amplitud of lctron-acoustic solitary structur incrass with incras in R but it is indpndnt of H. On th othr hand th width of th soliton incrass with incras in R or H. 4 Rlativistic Drifts and Rlativistic Dgnracy: A Comparison A comparison of ths two kinds of rlativistic ffcts is discussd. Whn th rlativistic ffcts ar du to straming motion only comprssiv solitons ar obsrvd. Th proprtis and dpndnc on straming vlocity u, quantum diffraction H and ion tmpratur ar discussd in sction 5. In sction 6 th invstigation of linar and nonlinar propagation charactristics of EAWs ar carrid out in rlativistic dgnrat dns plasma consisting of lctrons and ions. It is shown that th plasma undr considration can support only rarfactiv solitary wavs undr crtain rstrictd rgions of plasma paramtrs. Th soliton proprtis ar shown to dpnd significantly on th rlativistic dgnracy paramtr R and also th quantum diffraction paramtr H. Th prsnt invstigation may b hlpful in undrstanding th basic faturs of lctron-acoustic wavs in supr dns astrophysical obcts lik whit dwarfs, nutron stars as wll as in th futur Copyright 6 Isaac Scintific Publishing

13 Advancs in Astrophysics, Vol., No., Novmbr 6 99 intns lasr-solid plasma xprimnts whr th rlativistic lctron dgnracy ffcts bcom important. Acknowldgmnt. Th author would lik to thank CSIR & DST, Govt. of India for providing financial assistanc and JIS Univrsity, Kolkata and Jadavpur Univrsity, Kolkata, India to carry out this rsarch work. Rfrncs. G. Manfrdi, How to modl quantum plasmas Filds Institut Communications, vol. 46, 6, 5.. M. Ophr, L. O. Silva, D. E. Daugr, V. K. Dcyk and J. M. Dawson, Nuclar raction rats and nrgy in stllar plasmas: Th ffct of highly dampd mods Physics of Plasmas, vol. 8, 454,.. A. Markowich, C. Ringhofr and C. Schmisr, Smiconductor Equations, Springr, Vinna, K. F. Brggrn and Z.-L.Ji, Quantum chaos in nano-sizd billiards in layrd two-dimnsional smiconductor structurs, Chaos, vol. 6, 54, W. L. Barns, A. Drux and T. W. Ebbsn, Surfac plasmon subwavlngth optics Natur (London), vol. 44, 84,. 6. T. C. Killian, Cool Vibs, Natur (London) vol.44, 97, G. Chabrir, F. Douchin and A. Y. Potkhin, Dns astrophysical plasmas Journal of Physics: Condnsd Mattr, vol.4 9,. 8. K. H. Bckr, K. H. Schonbach and J. G. Edn, Microplasmas and applications, Journal of Physics D: Applid Physics, vol. 9, R55, L. K. Ang, W. S. Koh, Y. Y. Lau and T. J. T. Kwan, Spac-charg-limitd flows in th quantum rgim, Physics of Plasmas, vol. 567, 6.. L. K. Ang and P. Zhang, Ultrashort-Puls Child-Langmuir Law in th Quantum and Rlativistic Rgims, Physical Rviw Lttrs, vol. 98, 648, 7.. C. Grabb, Wav propagation ffcts of broadband lctrostatic nois in th magntotail, Journal of Gophysical Rsarch, vol. 94, 799, J. I. Vtt, Summary of Particl Population in th Magntosphr, Ridl, Dordrcht, p. 5, H. Ikzi, Exprimnts on ion acoustic solitary wavs, Physics of Fluids, vol. 6, 668, B. Shn and J. Myr-tr-Vhn, Pair and γ-photon production from a thin foil confind by two lasr pulss, Physical Rviw E, vol. 65, 645,. 5. E. P. Liang, S. C. Wilks, and M. Tabak, Pair Production by Ultraintns Lasrs, Physical Rviw Lttrs, vol. 8, 4887, K. A. Holcomb, T.Taima, Gnral-Rlativistic Plasma Physics in th Early Univrs, Physical Rviw D, vol. 4, 89, R. Sad, A. Shah, M. N. Ha, Nonlinar Kortwg d Vris quation for soliton propagation in rlativistic lctron-positron-ion plasma with thrmal ions, Physics of Plasmas, vol. 7,,. 8. S. K. El-Labany, M.S. Abdl Krim, S.A. El-Warraki, W.F. El-Taibany, Modulational instability of a wakly rlativistic ion acoustic wav in a warm plasma with nonthrmal lctrons, Chins Physics, vol., 759,. 9. R. Bharuthram, M.Y.Yu, Rlativistic lctron plasma wavs, Astrophysics and Spac Scincs, vol. 7, 97, 99.. S. I. Shapiro, S.A.Tukolsky, Black Hols, Whit Dwarfs and Nutron Stars, John Wily & sons, Nw York, 98.. M. Marklund, P.K.Shukla, Kintic thory of lctromagntic ion wavs in rlativistic plasmas Rviw of Modrn Physics, vol. 78, 59, 6.. F. Haas, L. G. Garcia, J. Godrt, and G. Manfrdi, Quantum ion-acoustic wavs Physics of Plasmas, vol., 858,.. L.S. Stnflo, P.K. Shukla and M. Marklund, Nw low-frquncy oscillations in quantum dusty plasmas, Europhysics Lttrs, vol. 74, no. 5, 844, C.L. Gardnr and C. Ringhofr, Smooth quantum potntial for th hydrodynamic modl Physical Rviw E, vol. 5,57, P. K. Shukla and B. Eliasson, Formation and Dynamics of Dark Solitons and Vortics in Quantum Elctron Plasmas, Physical Rviw Lttrs, vol. 96, 45, 6. Copyright 6 Isaac Scintific Publishing

14 Advancs in Astrophysics, Vol., No., Novmbr 6 6. S. A. Khan and A. Mushtaq, Linar and nonlinar dust ion acoustic wavs in ultracold quantum dusty plasmas, Physics of Plasmas, vol. 4, 87, B. Sahu and R. Roychoudhury, Cylindrical and sphrical quantum ion acoustic wavs Physics of Plasmas, vol.4, 4, B. Sahu and R. Roychoudhury, Elctron acoustic solitons in a rlativistic plasma with nonthrmal lctrons, Physics of Plasmas, vol., 7, S. Ali and P. K. Shukla, Dust acoustic solitary wavs in a quantum plasma, Physics of Plasmas, vol.,, 6.. P. K. Shukla and S. Ali, Dust acoustic wavs in quantum plasmas, Physics of Plasmas, vol., 45, 5.. B. Ghosh, S. Chandra & S.N.Paul, Amplitud modulation of lctron plasma wavs in a quantum plasma, Physics of Plasmas, vol. 8, 6,. Copyright 6 Isaac Scintific Publishing