Two-Stream Instability as a Mechanism for Toroidal Magnetic Field Generation in the Magnetosphere of Crab Pulsar
aa r X i v : . [ a s t r o - ph . GA ] O c t Two-Stream Instability as a Mechanism for Toroidal MagneticField Generation in the Magnetosphere of Crab Pulsar
Irakli S. Nanobashvili
Institute of Theoretical Physics, Ilia State University, Tbilisi, GeorgiaAndronikashvili Institute of Physics, Iv. Javakhishvili Tbilisi State University, Tbilisi,Georgia
E-mail address: [email protected]
Abstract.
New plasma mechanism for the generation of toroidal magnetic field in themagnetosphere of Crab pulsar is presented. It is based on the development of two-streaminstability in the relativistic electron-positron plasma of the pulsar magnetosphere. In par-ticular, pulsar magnetosphere relativistic plasma is penetrated by ultrarelativistic electronbeam and two-stream instability develops, as a result of which toroidal magnetic field isgenerated.
1. Introduction
Nowadays it is widely accepted that pulsars are rapidly rotating neutron stars [1-4]with strong magnetic field of the order 10 ÷ G. Neutron star is surrounded withmagnetosphere, which is filled with relativistic electron-positron plasma (see example[5]). Pulsar radiation is generated in its magnetosphere most probably as a result of thedevelopment of different plasma processes in the region above the pulsar magnetic poles([6]). In order to find the processes which are responsible for the generation of pulsarradiation it is essential to know in detail the structure of the magnetosphere where theseprocesses develop.In the pulsar magnetosphere, close to its surface, magnetic field has a dominant role -its energy exceeds the energy of the magnetospheric relativistic plasma by many ordersof the magnitude. Magnetic field of pulsar has the dipole structure. It is frozen inmagnetospheric plasma and in pulsar too. Therefore, solid body type rotation - corotationof pulsar, its magnetic field and magnetospheric relativistic plasma takes place. In theregion of the magnetosphere where the magnetic field lines are closed magnetosphericplasma is confined by the magnetic field and it can not leave the magnetosphere. In thisregion we have ”quiet corotation” if one can say so. Plasma can leave the magnetosphereonly from the conical region (with small angle of opening) above the pulsar magneticpoles. In this region magnetic field lines are ”opened” and since plasma particles followthese lines they leave pulsar magnetosphere and form relativistic pulsar wind. In case ofCrab pulsar opened magnetic field lines practically lie in the equatorial plane of rotationbecause pulsar magnetic axis is nearly perpendicular to its rotation axis [7,8]. In general,opened magnetic field lines of pulsar are considered as almost straight radial lines in theregion close to its surface, because in this region their curvature is small. Besides, in his region we have rigid corotation - plasma particles rotate together with the magneticfield lines and also move along them. It is evident that this can not take place on largeradial distance. In particular, corotation is strictly impossible beyond the light cylinder(cylindrical surface on which the corotation velocity equals to the speed of light). Atthe same time in this region, which is called wind zone, we have not the magnetosphericrelativistic plasma but the relativistic pulsar wind. In the wind zone magnetic field ispractically purely toroidal, it is still frozen in plasma, but its energy is smaller thenthe energy of the relativistic pulsar wind. From all the above mentioned it follows thatsomewhere in the pulsar magnetosphere - inside the light cylinder - toroidal magnetic fieldmust be generated and corotation must be violated.In the present paper one possible plasma mechanism for the generation of toroidalmagnetic field in the magnetosphere of Crab pulsar is suggested. In the forthcomingsection pulsar magnetosphere structure before the generation of toroidal magnetic fieldis discussed. In third section the mechanism of toroidal magnetic field generation inthe pulsar magnetosphere is presented. The mechanism is based on the development oftwo-stream instability in the magnetospheric relativistic electron-positron plasma.
2. The Structure of the pulsar magnetosphere before the generation of toroidalmagnetic field
As it has been already mentioned above pulsar magnetosphere is filled with relativis-tic electron-positron plasma. This plasma appears there as a result of cascade processwhich develops in the following way. Since matter inside pulsar is in superconductivestate magnetic field is frozen in pulsar and rotates together with it. As a result of thisrotation electric field is generated which extracts charged particles from pulsar surface[9]. Depending of the direction of generated electric field the particles extracted fromthe pulsar surface may be electrons [10], or positrons [11] and ions [12]. Here it willbe assumed that charged particles extracted from the pulsar surface by electric field areelectrons. In the pulsar magnetosphere electrons are accelerated by electric field and ac-quire ultrarelativistic velocities. Electrons follow the magnetic field lines and have onlythe longitudinal (with respect to the magnetic field line) component of velocity, becauseperpendicular component is lost in the strong magnetic field of pulsar after the rapid ra-diation with synchrotron mechanism. Since magnetic field lines are curved, the electronsmoving along them with ultrarelativistic velocities radiate curvature radiation γ -quanta.Then, in the strong magnetic field of pulsar γ -quanta decay into electron-positron pairs.These electrons and positrons are also accelerated by electric field and emit curvatureradiation γ -quanta, which again decay into electron-positron pairs etc. This cascade pro-cess leads to the formation of dense relativistic electron-positron plasma in the pulsarmagnetosphere [5] . This plasma is penetrated by primary ultrarelativistic electron beam.Now about the simplified geometric model of Crab pulsar magnetic field which willbe used below. As it has been already mentioned above rotation axis and magnetic axisof Crab pulsar are nearly perpendicular. So, pulsar magnetic field lines are consideredas radial straight lines located in the equatorial plane of rotation (see the Fig. 1). Thisassumption is justified and at the same time this is not an approximation of monopolar W Figure 1: Geometric model of crab pulsar magnetic field. magnetic field for the following reasons: first of all only the open magnetic field lines whichcome out from one magnetic pole of pulsar are discussed (all the results obtained belowwill be the same for the open magnetic field lines which come out from another magneticpole of pulsar, just the direction of these magnetic field lines will be the opposite). Besides,these magnetic field lines are discussed in the thin layer of the magnetosphere close topulsar surface. The thickness of this layer ( ≈ cm) is much less than the curvatureradius of Crab pulsar magnetic field lines ( ≈ cm), therefore in this layer magneticfield lines can be considered straight. The reason why one gets for the magnetic field thepicture seen on Fig. 1 is the rotation of those group of magnetic field lines which havebeen just defined above.
3. Two-stream instability development and toroidal magnetic field generationin the pulsar magnetosphere
As we have seen above relativistic electron-positron plasma in the pulsar magneto-sphere is penetrated by ultrarelativistic electron beam. As a result of the interaction ofultrarelativistic electron beam with relativistic plasma of the pulsar magnetosphere two-stream instability may develop. The possibility of the generation of pulsar radiation asa result of two-stream instability development has been studied in the papers [11-19]. Inthe present paper the possibility of toroidal magnetic field generation in the magneto-sphere of Crab pulsar as a result of two-stream instability development is investigated.For this purpose the standard set of equations describing the dynamics of cold relativistic agnetized plasma is used: ∂~p ( α ) ∂t + (cid:16) ~V ( α ) ~ ∇ (cid:17) ~p ( α ) = e ( α ) m (cid:16) ~E + h ~V ( α ) × ~B i(cid:17) , (1) ∂n ( α ) ∂t + div (cid:16) n ( α ) ~V ( α ) (cid:17) = 0 , (2) rot ~E = − ∂ ~B∂t , (3) rot ~B = 4 π~j + ∂ ~E∂t , (4)where e ( α ) and m are particle electric charge and mass respectively, ~V ( α ) and ~p ( α ) = γ α m~V ( α ) ( γ α being particle Lorentz-factor) are particle tree-velocity and momentum, n ( α ) is the particle density, ~j is the current density and ~E and ~B are electric and magneticfields. Subscript ( α ) denotes the group of particles (we have two groups - plasma ( α = 1)and beam ( α = 2)). In the equations (1-4) so-called ”geometric” unites - c = G = 1 areused and momentum is changed by normalized momentum ~p → ~p/m .The dynamics of electromagnetic perturbations in the system plasma-beam is studied.These studies are performed in the reference frame of a rotating magnetic field line (thegeometry of the magnetic field lines being discussed in the previous section). At the sametime the reference frame in which the investigations are performed is moving radiallyoutwards along the magnetic field line with such a constant velocity that in this framethe velocities of plasma and beam are equal ( | ~V (1) | = | ~V (2) | = V ) and directed in oppositedirection (beam velocity ~V (2) being directed radially outwards and plasma velocity ~V (1) -radially inwards). Here we discuss the dynamics of the perturbations wave vector of whichis directed along the x-axis (x-axis is parallel to the pulsar rotation axis) - ~k ( k x , , ~E (0 , , − E z ), unperturbed magnetic field B (which is pulsarmagnetic field) is also directed along the z-axis and perturbed magnetic field has only thetoroidal - y-component - ~B (0 , B y , ω − k = ω p γ − k V ω c /γ − ω ! . (5)Here V and γ are unperturbed velocity and Lorentz-factor of particles, ω c = eB /m is the cyclotron frequency ( B being the unperturbed magnetic field of pulsar) and ω p = q π ( n p + n b ) e /m is the plasma frequency ( n p and n b being the unperturbed particledensity of pulsar magnetosphere relativistic electron-positron plasma and ultrarelativisticelectron beam respectively).Equation (5) can be rewritten in the following form: aω + bω + c = 0 , (6) yzV (2) B k B E V (1) o Figure 2: Orientation of the perturbations under study. where a = γ , (7) b = − (cid:16) ω c + ω p γ + k γ (cid:17) , (8) c = k ω c + ω p ω c γ − ω p γ k V . (9)From the expressions (6)-(9) one can conclude that if the condition k V > k ω c ω p γ + ω c γ , (10)is fulfilled, than ω < ω is entirely imaginary. The time dependence of perturbedquantities and namely perturbed toroidal magnetic field is exponential B ∼ exp ( − iωt ).Therefore, when the condition (10) is fulfilled, then two-stream instability develops inthe magnetosphere of Crab Pulsar and exponentially growing toroidal magnetic field isgenerated.Substituting the parameters appropriate to Crab pulsar and its magnetosphere onecan easily find that the condition (10) is really fulfilled.As a result of the development of two-stream instability exponentially growing toroidalmagnetic field is generated in the magnetosphere of Crab pulsar (for other possible me-chanisms of toroidal magnetic field generation in the magnetosphere of Crab Pulsar see[20] and [21]. The source of energy for the generation of this field is the pulsar rotation lowing down. Really, as it has been already mentioned above, as a result of pulsar rotationtogether with frozen-in magnetic field, electric field is generated. This electric field getsenergy from pulsar rotation slowing down. The electric field extracts electrons frompulsar surface, accelerates them to ultrarelativistic velocities and thus the ultrarelativisticelectron beam is formed. In the strong magnetic field of pulsar electron-positron pairsappear from the beam particles and dense relativistic plasma of the pulsar magnetosphereis formed. This plasma is penetrated by ultrarelativistic electron beam. The systemplasma-beam is unstable and two-stream instability develops in it, as a result of whichtoroidal magnetic field is generated in the pulsar magnetosphere. The energy source forthe generation of the toroidal magnetic field is beam kinetic energy. The beam itselfacquires its kinetic energy from the electric field. As we have been just mentioning theelectric field is generated during pulsar rotation together with frozen-in magnetic field andgets energy from its rotation slowing down. Thus, the energy of the generated toroidalmagnetic field comes from pulsar rotation slowing down.Superposition of generated toroidal magnetic field and pulsar magnetic field will givethe spiral configuration magnetic field. Since plasma particles follow the magnetic fieldlines corotation will be violated in the pulsar magnetosphere and instead of it we willhave differential rotation or shear flow of magnetospheric plasma.On large radial distances the step of the magnetic field spiral should decrease andbeyond the light cylinder magnetic field will become practically purely toroidal. Onlarger radial distance from pulsar - around 10 cm this magnetic field is reconnectedwith the magnetic field of Crab Nebula, which has also toroidal structure (about onepossible mechanism for the generation of this field see [22]).
4. Conclusions
Thus, in the magnetosphere of Crab pulsar relativistic electron-positron plasma is pene-trated by ultrarelativistic electron beam. The system plasma-beam is unstable and two-stream instability develops in it. As a result exponentially growing toroidal magnetic fieldis generated in the magnetosphere of Crab pulsar and after this magnetic field structurechanges to spiral. Since plasma particles follow the magnetic field lines corotation will beviolated. On large radial distances step of the magnetic field spiral decreases and we getpractically purely toroidal magnetic field which is finally reconnected with the magneticfield of Crab nebula. Toroidal magnetic field is generated in the pulsar magnetosphere atthe expense of energy released during pulsar rotation slowing down.
References
1. F. Pacini, Nature, , 467 (1967).2. F. Pacini, Nature, , 145 (1968).3. T. Gold, Nature, , 731 (1968).4. T. Gold, Nature, , 25 (1969).5. P.A. Sturrock, Ap.J., , 529 (1971). . D.B. Melrose, J. Astrophys. Astr., , 137 (1995).7. R.N. Manchester & Taylor, J.H., ”Pusalrs”, W.H. Freeman and company, San Fran-cisco (1977).8. F.G. Smith, ”Pusalrs”, Cambridge University Press, Cambridge (1977).9. P. Goldreich, & W.H. Julian, Ap. J., , 869 (1969).10. J. Arons, in Proc. Workshop Plasma Astrophysics, pp. 273-286 (1981).11. M.A. Ruderman, & P.G. Sutherland, Ap.J., , 51 (1975).12. A.F. Cheng & M.A. Ruderman, Ap.J., , 576 (1980).13. R. Buschauer & G. Benford, M.N.R.A.S., , 99 (1977).14. G. Benford & R.Buschauer, M.N.R.A.S., , 189 (1977).15. A.F. Cheng & M.A. Ruderman, Ap.J., , 800 (1977).16. E. Asseo, R. Pellat & M. Rosado, Ap.J., , 661 (1980).17. E. Asseo, R. Pellat & M. Sol, Ap.J., , 201 (1983).18. V.N. Ursov & V.V. Usov, Ap.S.S., , 325 (1988).19. V.V. Usov, Ap.J., , 333 (1987).20. T.A. Kahniashvili, G.Z. Machabeli & I.S. Nanobashvili,, Phys. Plasmas, , 1132(1997).21. I.S. Nanobashvili, Ap.S.S., , 125 (2004).22. G.Z. Machabeli, I.S. Nanobashvili & M. Tendler, Physica Scripta, , 601 (1999)., 601 (1999).