Langmuir dark solitons in dense ultrarelativistic electron-positron gravito-plasma in pulsar magnetosphere
aa r X i v : . [ a s t r o - ph . S R ] F e b Langmuir dark solitons in dense ultrarelativistic electron-positron gravito-plasma inpulsar magnetosphere
U. A. Mofiz ∗ and M. R. Amin Department of Mathematics and Natural Sciences,BRAC University, 66 Mohakhali,Dhaka-1212, Bangladesh Department of Electronics and Communication Engineering,East West University, Jahurul Islam City. Aftabnagar,Dhaka-1212, Bangladesh
Abstract
Nonlinear propagation of electrostatic modes in ultrarelativistic dense elelectron-positron gravito-plasma at the polar cap region of pulsar magnetosphere is considered. A nonlinear Schr¨odingerequation is obtained from the reductive perturbation method which predicts the existence ofLangmuir dark solitons. Relevance of the propagating dark solitons to the pulsar radio emission isdiscussed.
Keywords:
Electron-positron plasma; Langmuir solitons; pulsar magnetosphere.
I. Introduction
Pulsars are celestial sources that believe to be rotatingneutron stars producing light-house like beams of radioemissions from the magnetic poles. As shown by Goldre-ich and Julian (1969) the rotating magnetic dipole pro-duces a quadrupole electric field whose component paral-lel to the open magnetic field lines at the poles extractsparticles very effectively from neutron star surface andaccelerates them to highly relativistic energies. Thus,the magnetosphere is filled with plasma which shieldsthe electric field. Complete shielding is established whenthe net charge reaches n GJ - the Goldreich -Julian chargedensity. The Lorentz factors of the accelerated particlesreach about 10 and they emit hard curvature radiationsthat propagate at a sufficient angle to the magnetic field,so that significant pair production of electron-positroncan occur (Erber 1966). It is commonly accepted that thenewly created particles produce more pairs by emittingenergetic synchrotron or curvature radiation. As a resultan avalanche of secondary particles populates the magne-tosphere with densities 10 n GJ (Ruderman and Suther-land, 1975) . Here, we extend our earlier research on pul-sar microstructure, soliton formation, wakefield accelera-tions, gravitational waves , and growing modes (Mofiz, etal. 1985-2011) to account the pair ultrarelativistic pres-sure and the gravityThe paper is organized as follows. Section II describesthe fluid model of the dense ultrarelativistic electron-positron plasma under gravity. Considering a Lorentzinvariant frame moving with group velocity of the wave,a Nonlinear Schr¨odinger Equation (NLSE) is derived us-ing the reductive perturbation method (Gardner and ∗ Corresponding author; email address: mofi[email protected]:880-2-8824051-4 ext.4078, fax: 880-2-8810383.
Morikawa, 1960). A linear dispersion relation is obtainedshowing the existence of Langmuir waves under gravityand with ultrarelativistic temperature for wave propa-gation. The solution of NLSE shows the generation ofLangmuir dark solitons. Results are discussed in Sec.III. Finally, Sec. IV concludes the paper.
II. The Mathematical Model
We consider two-fluid magnetohydrodynamic (MHD)equations to describe the electron-positron plasma in thepulsar magnetosphere of the neutron star. The equa-tions are the usual continuity and momentum balanceequations for the plasma species, electrons and positrons,supplemented by the Poisson’s equation for electrostaticwave propagation . Thus, the required set of equationsare as follows (Mofiz and Ahmedov, 2000): ∂∂t ( γ s n s ) + ∇ · (cid:18)r − r g r γ s n s v s (cid:19) = 0 , (1) ∂∂t ( γ s v s ) + r − r g r v s · ∇ ( γ s v s )= − q s m ∇ φ − mγ s n s ∇ p s , (2) ∇ · q − r g r ∇ φ = − π X s γ s q s n s , (3)where, γ s = 1 / p − v s /c , n s , v s , and q s are respec-tively the particle number density, particle velocity, andparticle charge of plasma species s ; q s = − e for s = e (electron) and q s = + e for s = e + (positron); φ isthe electrostatic potential, r g is the Schwarzschild ra-dius of the neutron star; m is the electron/positronmass; e is the absolute value of the electronic charge.In Eq. (2), the pressure p s is given by the expressionfor the ultrarelativistic pressure (Chandrasekhar, 1938): p s = n k B T ( n s /n ) / , where T e = T p = T has beenassumed; k B is the Boltzmann constant, n is the equi-librium particle number density.In the polar cap region of the pulsar, we consider θ = 0, ∇ = b z∂/∂z , v s = v sz b z and adopt the follow-ing normalization of different quantities: z → zω pe /c , t → tω pe , n s → n s /n , u s → v sz /c , φ → eφ/mc , σ T → k B T /mc , r g → r g ω pe /c , and v g → v g /c , where ω pe = (cid:0) πn e /m (cid:1) / is the electron plasma frequency, c is the speed of light. To study the nonlinear dynam-ics, we consider the following stretched coordinates in themoving frame (Melikidze et al. 2000): ξ = ǫγ ( z − v g t ) , (4)and τ = ǫ γ ( t − v g z ) , (5)where, v g is the group velocity, γ = γ e = γ p is the aver-age relativistic Lorentz factor and is given by the follow-ing expression: γ = (cid:0) − v g (cid:1) − / , ǫ is a small quantity,the perturbation parameter with ǫ <
1. With the trans-formations, given by Eqs. (4) and (5), the derivatives ∂/∂t and ∂/∂z transform to ∂/∂t → ∂/∂t − ǫγ v g ∂/∂ξ + ǫ γ ∂/∂τ and ∂/∂z → ∂/∂z + ǫγ ∂/∂ξ − ǫ γ v g ∂/∂τ respectively. With the above considerations, the set ofequations, Eqs. (1)-(3) take the following forms: ∂n s ∂t + ∂∂z ( g ( z ) n s u s )+ ǫ (cid:20) − γ v g ∂n s ∂ξ + γ ∂∂ξ ( g ( z ) n s u s ) (cid:21) + ǫ (cid:20) γ ∂n s ∂τ − γ v g ∂∂τ ( g ( z ) n s u s ) (cid:21) = 0 , (6) ∂u s ∂t + g ( z ) u s ∂u s ∂z + q s eγ ∂φ∂z + 4 σ T γ n − / s ∂n s ∂z + ǫ (cid:20) − γ v g ∂u s ∂ξ + g ( z ) u s γ ∂u s ∂ξ + q s e ∂φ∂ξ + 4 σ T γ n − / s ∂n s ∂ξ (cid:21) + ǫ (cid:20) γ ∂u s ∂τ − g ( z ) γ v g u s ∂u s ∂τ − q s v g e ∂φ∂τ − σ T v g γ n − / s ∂n s ∂τ (cid:21) = 0 , (7) ∂∂z (cid:18) g ( z ) ∂φ∂z (cid:19) − γ ( n e − n p ) + ǫ (cid:20) γ ∂∂ξ (cid:18) g ( z ) ∂φ∂z (cid:19) + ∂∂z (cid:18) γ g ( z ) ∂φ∂ξ (cid:19)(cid:21) + ǫ (cid:20) − γ v g ∂∂τ (cid:18) g ( z ) ∂φ∂z (cid:19) − γ v g ∂∂z (cid:18) g ( z ) ∂φ∂τ (cid:19) + γ ∂∂ξ (cid:18) γ g ( z ) ∂φ∂ξ (cid:19)(cid:21) + ǫ (cid:20) − γ ∂∂ξ (cid:18) v g g ( z ) ∂φ∂τ (cid:19) − γ v g ∂∂τ (cid:18) g ( z ) ∂φ∂ξ (cid:19)(cid:21) + ǫ (cid:20) γ v g ∂∂τ (cid:18) g ( z ) ∂φ∂τ (cid:19)(cid:21) = 0 , (8)where, the factor g ( z ) = (1 − r g /z ) / accounts for thegravitational effect. Now we expand the quantities n s , u s , φ as n s = 1 + ǫ n s + ∞ X l =1 ǫ l (cid:16) n sl e il ( kz − ωt ) + n ∗ sl e − il ( kz − ωt ) (cid:17) , (9) u s = ǫ u s + ∞ X l =1 ǫ l (cid:16) u sl e il ( kz − ωt ) + u ∗ sl e − il ( kz − ωt ) (cid:17) , (10) φ = ǫ φ + ∞ X l =1 ǫ l (cid:16) φ l e il ( kz − ωt ) + φ ∗ l e − il ( kz − ωt ) (cid:17) . (11)Here,( n s , u s , φ , n sl , u sl , φ l ) ≡ A (1) + ǫA (2) + ǫ A (3) + ....... are functions of stretched coordinates ( ξ, τ ). II. A. Linear Dispersion Relation for the Lang-muir Wave
Now considering | /g ( z ) · dg ( z ) /dz | << k for the firstharmonic ( l = 1)in the first order ( ǫ = 1 ), we have thefollowing equations for the first-order quantities: − iωn (1) s + ikg ( z ) u (1) s = 0 , (12) − iωu (1) s + 4 ikσ T γ n (1) s + ikq s eγ φ (1)1 = 0 , (13) − k g ( z ) φ (1)1 − γ (cid:16) n (1) e − n (1) p (cid:17) = 0 . (14)Eliminating u (1) s from Eqs. (12) and (13), we obtain thefollowing equation relating n (1) s and φ (1)1 : n (1) s = − k g ( z ) − ω + 4 σ T g ( z ) k / γ q s eγ φ (1)1 , (15)from which we obtain n (1) e − n (1) p = 2 k g ( z ) − ω + 4 σ T g ( z ) k / γ γ φ (1)1 . (16)Using Eq. (16) into Eq. (14), we obtain the followinglinear dispersion relation: ω = 2 g ( z ) + 4 σ T g ( z )3 γ k , (17)which in the dimensional form is ω = ω pe g ( z ) + 23 k v th g ( z ) , (18)with v th = k B Tmγ . The group velocity v g = ∂ω/∂k isobtained from the linear dispersion relation Eq. (17) as: v g = 4 σ T g ( z )3 γ kω . (19)The same expression for v g is also obtained from thecompatibility condition and shown in the Appendix A.The group dispersion is found to be v ′ g = dv g dk = σ T g ( z ) γ − v g ω . (20)Eq.(17) represents the dispersion relation for Langmuirwaves in ultrarelativistic e, e + plasma under gravity. Pairproduction in the polar cap region of pulsar magneto-sphere occurs through curvature radiation which happensfor E k >> m e c , where E k is the energy of electron alongthe magnetic field. It is estimated that for cascade gen-eration of pair plasma γ ∼ − , E k ∼ − eV (Beskin et al. 1993). Here, γ = E k /m e c ,then consid-ering E k = k B T , we find γ = k B T /m e c ≡ σ T . UsingEqs.(17),(19) and Eq.(20), we perform an analysis of thedispersion relation, group velocity and group dispersionof Langmuir waves at the ultrarelativistic temperatureof the e, e + plasma under gravity. The analysis is showngraphically in Fig.1-4, respectively. II.B Nonlinear Evolution Equation for the Lang-muir Wave
Finding the zeroth harmonic and second harmonic of thesecond order quantities in terms of the first harmonic ofthe first order quantities and using these into the firstharmonic of the third order quantities , we easily obtainthe following NLSE for the evolution of the potential i ∂a∂τ + P ∂ a∂ξ + Q | a | a = 0 , (21)where a ≡ φ (1)1 , and the coefficients P and Q are givenby the following expressions: P = σ T g ( z )3 γ − v g γ h ω − v g kg ( z ) { ω + v g kω + 2 g ( z ) } i , (22) k Ω FIG. 1: The variation of the normalized frequency ω of thelinear Langmuir wave with respect to the normalized pumpwavenumber k for different values of plasma parameters : r g =1, z = 2, γ = σ T = 10 with the corresponding temperature T = 5 × K . k v g FIG. 2: The variation of the normalized group velocity v g of the linear Langmuir wave with respect to the normalizedpump wavenumber k for different values of plasma parameters: r g = 1, z = 2, γ = σ T = 10 with the correspondingtemperature T = 5 × K . Q = g ( z ) γ (cid:2) ωk f + h (cid:3) γ h ω − v g kg ( z ) { ω + v g kω + 2 g ( z ) } i , (23)where f = − ωkg ( z ) γ (cid:20) b (cid:18) kv g ω (cid:19) + 2 b (cid:21) + 3 ωk g ( z ) γ , (24) ´ - ´ - ´ - ´ - k v g ' FIG. 3: The variation of the normalized group dispersion v ′ g of the linear Langmuir wave with respect to the normalizedpump wavenumber k for different values of plasma parameters: r g = 1, z = 2, γ = σ T = 10 with the correspondingtemperature T = 5 × K . z Ω FIG. 4: The variation of the normalized frequency ω of thelinear Langmuir wave with respect to the normalized distance z for parameters : r g = 1, k = 1000, γ = σ T = 10 with thecorresponding temperature T = 5 × K . h = − ω g ( z ) γ (cid:20) kv g ω b g ( z ) + b − k g ( z ) γ (cid:21) + 8 σ T k g ( z ) γ ( b + b ) , (25)with b = − σ T g ( z )3 γ − v g " g ( z ) (cid:18) ωk g ( z ) γ (cid:19) − σ T k g ( z ) γ + v g ωk g ( z ) γ (cid:21) , (26) b = 3 ω k g ( z ) γ − σ T k g ( z ) γ . (27)Here, the coefficient P can be written as P = 12 α v ′ g , (28)where, α = 1 γ [1 − σ T γ g ( z ) ] , (29)represents the effects of ultrarelativistic temperatureand gravity , neglecting of which we recover the usualresults. The coefficients P and Q appearing in the NLSE,given by Eqs. (22), (23)are known as the dispersion andnonlinear coefficients, respectively. The signs of P and Q determine whether the slowly varying wave pulse isstable or not (Lighthill condition; Lighthill, 1967). If thesigns of P and Q are such that P Q <
0, the wave pulse ismodulationally stable and the corresponding solution ofthe NLSE is called the dark soliton. On the other hand, if
P Q >
0, then the pulse may be modulationally unstableand the solution of the NLSE in this case is called thebright soliton. Graphically, we study the nature of P and Q for continuous values of the wave number k withparticular values of plasma parameters, which are showngraphically in Fig.5-6, respectively. k P FIG. 5: The variation of the dispersion coefficient P of thelinear Langmuir wave with respect to the normalized pumpwavenumber k for different values of plasma parameters : r g =1, z = 2, γ = σ T = 10 with the corresponding temperature T = 5 × K . From the graphical analysis, we find that
P Q < a ( ξ, τ ) = a ( ξ ) exp [ i ( Kξ − Ω τ )], the following solution of - ´ - ´ - ´ - ´ k Q FIG. 6: The variation of the nonlinear coefficient Q of thelinear Langmuir wave with respect to the normalized pumpwavenumber k for different values of plasma parameters : r g =1, z = 2, γ = σ T = 10 with the corresponding temperature T = 5 × K . the NLSE, Eq. (21) (Mofiz,2007)is easily obtained: a ( ξ, τ ) = a tanh "(cid:12)(cid:12)(cid:12)(cid:12) Q P (cid:12)(cid:12)(cid:12)(cid:12) / a ξ exp [ i ( Kξ − Ω τ )] , (30)Here, Kξ − Ω τ is the modulation phase with K ( << k )and Ω( << ω ), respectively. Eq.(30) represents a darksoliton with amplitude a = (cid:12)(cid:12)(cid:12) Ω+ P K Q (cid:12)(cid:12)(cid:12) / and width δ = (cid:12)(cid:12)(cid:12) PQa (cid:12)(cid:12)(cid:12) / , respectively.The dark soliton (Eq.(30)) in the ultrarelativistic e, e +plasma is shown graphically in Fig.7. III. Results and Discussion
In this section, we analyze the linear dispersion as wellas the nonlinear Langmuir dark soliton in the pulsar mag-netosphere. Eq.(17) shows that the Langmuir frequencydepends on ultrarelativistic temperature and it is red-shifted due to gravity near the Schwarzchild radius. Sim-ilarly, the group velocity and group dispersion, shown byEq.(19) and Eq. (20), are also depend on temperatureand gravity.For numerical appreciation of the dark soliton ,weconsider the two cases of ultrarelativistic temperatures:6 × K − . × K (Crab pulsar) with the corre-sponding energies 5 − M eV (Nanobashvilli,2004) and5 × K − × K (x-ray pulsar) with the correspond-ing energies 10 − eV (Beskin et al., 1993).The solution of the NLSE (Eq.(21)) is a stable darksoliton (Eq.(30))whose amplitude and width dependon temperature. The amplitude is increased and thewidth is decreased with the increase of ultrarelativistic - - ´ - ´ - ´ - ´ - Ξ ¤ FIG. 7: Dark soliton in ultrarelativistic e, e + plasma at thepolar cap region of pulsar magnetosphere. The parametersare : r g = 1, z = 2, k = 1000,Ω = 0, K = 1,, γ = σ T = 10 with the corresponding temperature T = 5 × K . temperature. Thus, stable spiky Langmuir solitons arepossible in the ultrarelativistic electron-positron plasma. IV. Conclusion
To summarize, we have investigated the nonlinearpropagation of electrostatic modes in a dense ultrarela-tivistic electron-positron gravito-plasma at the polar capregion of pulsar magnetosphere. A multiscale perturba-tion analysis of the fluid equations shows that stable darkLangmuir solitons are produced due to the balnce of dis-persion and nonlinearity in the wave propagation. As theamplitude of the soliton increaes and width of the solitondecreases with the increase of ultrarelativistic tempera-ture, so spiky stable dark Langmuir solitons may prop-agate along the open field lines of the pulsar magneto-sphere, which may have some relation with pulsar radioemission and its microstructure.
Acknowledgement
This work has been supported by the Ministry of Ed-ucation of the Government of Bangladesh under Grantsfor Advanced Research in Science: MOE.ARS.PS.2011.No.-86.
Appendix A: The compatibility condition
It can be shown that the 1st harmonic of the 2nd-orderelectron and positron densities can be found to be n (2) s = q s k eg ( z ) γ φ (2)1 + iγ g ( z ) × "(cid:18) ωv g − σ T g ( z ) k γ (cid:19) ∂n (1) s ∂ξ + g ( z ) ( kv g − ω ) ∂u (1) s ∂ξ − iq s k eg ( z ) ∂φ (1)1 ∂ξ . After finding n (2) e − n (2) p and substituting it in the fol-lowing 1st-harmonic of the 2nd-order part of the Pois-son’s equation: − k g ( z ) φ (2)1 − γ (cid:16) n (2) e − n (2) p (cid:17) + 2 ikγ g ( z ) ∂φ (1)1 ∂ξ = 0 , we obtain the following compatibility condition: v g = 4 σ T g ( z )3 γ kω , which is exactly the same as the expression of the groupvelocity, Eq. (19), obtained by differentiating ω withrespect to k from the linear dispersion relation, Eq. (17). Beskin, V. S., Gurevich, A. V. and Istomin, Ya. N. 1993,Physics of the Pulsar Magnetosphere, Cambridge Univer-sity Press.Chandrasekhar, S. 1935. Mon. Not. R. Astron. Soc., ,405.Erber, T., 1966, Rev. Mod. Phys. , 626.Gardner C. S. and Morikawa, G. K., 1960, New York Uni-versity Report, NYU-9082, Courant Institute of Mathe-matical Sciences.Lighthill, M. J. 1967, Proc. R. Soc. London A, , 28.Goldreich, T., Julian, W. H., 1969, ApJ, , 869.Melikidze, G. I., Gil, J. A.,Pataraya, D., 2000, ApJ, ,1081.Mofiz U. A., De Angelis, U., Forlani, A., 1985, Phys. Rev.A., , 951. Mofiz, U. A., Podder. J., 1987, Phys. Rev. A., , 1811.Mofiz U. A., 1989, Phys. Rev. A., , 6752.Mofiz, U. A., 1990, Phys. Rev. A., , 960.Mofiz, U. A., and Mamun, A. A., 1992, Phys. Fluids B ,3806.Mofiz, U. A.,1997, Phys. Rev. E, 55, 5894.Mofiz, U. A., and Ahmedov, B, J., 2000, ApJ, , 484.Mofiz, U. A., 2007, Physics of Plasmas, , 112906.Mofiz, U. A., 2009, J. Plasma Fusion Res. Series, , 189.Mofiz, U. A.,Amin, M. R. and Shukla, P. K., 2011, Astro-phys. and Space Sci., DOI: 10.1007Nanobashvilli, J. S., 2004, Astrophys. and Space Sci., ,125.Ruderman,M. A. , Sutherland, P.G., 1975, ApJ,96