On the acceleration and stability of astrophysical

Universit` a degli Studi di Torino Dipartimento di Fisica Generale

On the acceleration and stability of astrophysical outflows Zaza Osmanov A thesis submitted for Ph.D in Physics

Supervisors: Prof. Silvano Massaglia Dr. Andria Rogava

Torino, January 2007.

ii

iii

Thanks

The present dissertation is a summary of the work that I have conducted in the department of General Physics (the group of Astrophysics), (UNITO, Italy) and Georgian National Astrophysical Observatory (GeNAO). I would like to thank Andria Rogava, my supervisor from GeNAO and Giorgi Machabeli who introduced me into the subject of the presented study. I thank them for help and support. Silvano Massaglia was my supervisor in UNITO and I would like to thank him very much. I would like to thank Gianluigi Bodo and Andrea Mignone for introducing me into the field of numerical simulations. At the UNITO and Osservatorio Astronomico di Torino I enjoyed hospitality for which I thank Atilio Ferrari, Paola Rossi and Edoardo Trussoni. For friendly and stimulating atmosphere in the UNITO I thank Miguel Onorato, Antonaldo Diaferio, Titos Matsakos, Ovidiu Tesileanu, Neeharica Verma and Mubashra Hameed who are presently in UNITO, as well as Claudio Zanni who has already left it. I would like to thank all my colleagues in GeNAO while I have no opportunity to list the names. While the bulk of the thesis is based on the studies conducted at collaborators, there is a contribution from our collaborator in the USA, Swadesh Mahajan and I would like to thank him very much.

Contents 1 Introduction 1.1 The centrifugal acceleration in astrophysical outflows. . . . . . . . . . . . 1.2 The instabilities in astrophysical outflows. . . . . . . . . . . . . . . . . . 2 The influence of centrifugal forces on particle dynamics in astrophysical outflows. 2.1 Mathematical model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Main consideration . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Uniformly rotating PB system . . . . . . . . . . . . . . . . . . . . 2.1.3 Conserved energy case . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. . . . . . . . . . . . . . . . . . . 2.2.1 The main consideration . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 On efficiency of particle acceleration by rotating magnetospheres in AGN. Application to Blazars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Main consideration . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Case of the magnetic field lines fixed in the equatorial plane . . . 2.3.3 Case of the inclined magnetic field lines . . . . . . . . . . . . . . . 2.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Instabilities in astrophysical outflows 3.1 Amplification of MHD waves in astrophysical flows. 3.1.1 Theory . . . . . . . . . . . . . . . . . . . . . 3.1.2 Discussion . . . . . . . . . . . . . . . . . . . 3.1.3 The case of β ' 1 . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

1 1 5 7 8 12 13 23 24 27 29 35 35 36 38 43 46 48 50 51 54 59 68

CONTENTS

3.2

3.3

v

3.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear analysis of the Kelvin-Helmholtz instability for relativistic magnetohydrodynamics flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Kelvin-Helmholtz instability for relativistic Fluid, for a planar velocity shear layer approximation. Non linear analysis. . . . . . . . . . . . . . . 3.3.1 Physical problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72 73 74 78 84 86 86 87 88 99

4 Summary

106

Bibliography

108

Appendix A. 4.1 Derivation of Eq. (2.95) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Derivation of Eq. (2.96) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Derivation of Eq. (2.98) . . . . . . . . . . . . . . . . . . . . . . . . . . .

116 116 117 118

Chapter 1 Introduction My PhD thesis is dedicated to the acceleration and stability problem in different astrophysical scenarios. In most of astronomical objects like active galactic nuclei (AGN), accretion disks, pulsar magnetospheres, solar winds/tornados, young stellar objects (YSO) etc., we observe various kinds of astrophysical plasma flows. The studies of generation, acceleration and stability of these outflows, becoming one of the important branches of modern theoretical astrophysics, still remain uncertain and need to be revealed. Among different facets of the physics of astrophysical flows there are two significant issues, which are explored within the framework of my PhD thesis. The first part of the thesis is devoted to the role of the centrifugal force in the dynamics of a relativistic plasma flow in pulsar magnetospheres and extragalactic AGN jets. In the second part we consider the stability problem of outflows. Two different problems are studied: (a) The nonmodal instabilities in helical flows and (b) the Kelvin-Helmholtz instability (KHI) for relativistic magnetized plasmas. Each chapter of the thesis is consist of two major parts, one of which corresponds to works done in Georgia in the Department of Plasma Astrophysics of Georgian National Astrophysical Observatory (GeNAO) and the next part presents works which have been guided at Dipartimento di Fisica Generale in Università degli Studi di Torino (UNITO).

1.1

The centrifugal acceleration in astrophysical outflows.

Many galaxies today (including our Galactic center) may have a massive black holes (MBH). In some AGNs, the MBH + accretion disk somehow produce narrow beams of energetic particles and magnetic fields, i.e. jets, and eject them outward in opposite

1.1 The centrifugal acceleration in astrophysical outflows.

2

directions away from the disk. The jets emerge at nearly the speed of light. For studying jet evolution, and origin, one needs to start from the very beginning of the formation processes. Very long baseline array can provide long-term, systematic monitoring of relativistic motion in AGN jets on parsec-scales giving an important information about the initial stage of jet termination. Radio galaxies, quasars, and blazars are AGN with strong jets, which can travel outwards into large regions of intergalactic space. The fundamental problem concerns the formation, acceleration and collimation of extragalactic jets. In the thesis we study the problem of particle acceleration and investigate its possible application for different astrophysical objects. Considering the problem of the escaping radiation from AGNs, one has to note, that at present we have a strong observational evidence of a complex picture of the jet emission spectra, which starts from the radio, the optical up to X-ray and γ-ray. From the high energy AGN sources, very exotic objects are blazars with a large number of experimental proof of having ultra-high energy emission, corresponding to the photons with TeV energies, origin of which is associated with the so called Inverse Compton scattering (Rieger & Mannheim 2000). An innermost region of AGNs is characterized by rotational motion, which influences the dynamics of jets. Generally speaking, different physical processes in astrophysics very often are closely related to each other. If the spin paradigm (stating that, to first order, it is the normalized black hole angular momentum, that determines whether or not a strong radio jet is produced), formulated by Wilson & Colbert (1995) and Blandford (1999) is correct, then it has significant implications for how we should view the jets and lobes in radio sources: the jet radio and kinetic energies come directly from the rotational energy of a spinning black hole. This means that the role of rotation is to be fundamental for understanding many aspects of relativistic extragalactic jets. The acceleration problem due to the rotation is actual also for neutron stars. The combination of pulsar’s magnetic field with the rapid rotation, produces powerful electric fields, with electric potential in excess of 1012 volts. Electrons are accelerated to high velocities and produce radiation (light) in two general ways: (a) Acting as a coherent plasma, the electrons work together to produce radio emission by a process whose details remain poorly understood; and (b) the electrons interact with photons or the magnetic field to produce high-energy emission such as optical, X-ray and γ-ray. The exact locations where the radiation is produced are uncertain and may be different for different types of emission. A few years after the discovery by radio astronomers, the Crab and Vela pulsars were detected at γ-ray energies. These rapidly rotating neutron stars are modeled as spinning magnets. This simple view hides the complexity of these interesting objects. Pulsars accelerate particles in their ”magnetospheres”, the name for the region which is dominated by the neutron star’s magnetic field, which provides the so called


3

frozen-in condition. These particles are ultimately responsible for the γ-ray emission seen from pulsars. As a TeV energy photon source, the AGN is not exceptional, some neutron stars exhibit the same feature as well. Observations of the Crab pulsar have been carried out in every accessible wavelength band. The source has been established as a TeV emitter with the advent of ground-based Cerenkov imaging telescopes with sufficient sensitivity. Pulsar-power systems (gravitationally bound systems containing at least one pulsar) represent a class of objects thought to be sources of Galactic TeV emission. A common feature by which these both astrophysical objects are characterized, is the presence of rotation. Consequently the influence of the centrifugal force on rotating plasma particles is supposed to be instrumentalimportant for AGNs jet formation/acceleration and pulsar wind dynamics (Contopoulos et al. 1999; Gangadhara & Lesch 1997). The problem of centrifugal acceleration is far from being resolved. Even for very simple, idealized models for the acceleration of relativistic plasma particles, quite enigmatic regimes of particle dynamics were derived (Machabeli & Rogava 1994). A first general problem for rigidly rotating relativistic systems (so called the light cylinder problem (LCP)) is, to understand how a flow goes through the light cylinder surface (a region, where the linear velocity of rigid rotation corresponds to the speed of light). A variety of observations on AGNs and pulsars exhibit the presence of outflows, which has to be explained in the context of the light cylinder problem. For this reason, in the thesis we present the importance of the centrifugal force in the dynamics of particles moving along prescribed rotating channels in order to make the question of the LCP clear. In 2.1 we consider motion of test particles along spinning curved trajectories, studying the problem both in the laboratory and the rotating frames of reference. Assuming that the system rotates with the constant angular velocity ω = const, we have found the solutions and analyzed them for the case when the form of the trajectory is given by an Archimedes spiral. As the analysis shows, the particles can reach infinity while they move along these trajectories avoiding the LCP. We point out the analogy of this study with the motion of particles along the curved rotating magnetic field lines in the pulsar magnetosphere and discuss further physical development (the conserved total energy case, when ω 6= const) and astrophysical applications (the acceleration of particles in active galactic nuclei) of this theory. In section 2.2 we consider an alternative mechanism of the pulsar rotation energy transfer into plasma instabilities. We present a linear study of the kinematics of a plasma flow rotating in the pulsar magnetosphere. On the basis of an exact set of equations describing the behavior of the plasma stream, we have obtained the increment of the corresponding instability. It was found that the linear stage of the plasma flow dynamics was very efficient and short in time, strongly indicating the need of a non-linear theory. We discuss about the possible relevance of this approach for understanding of the pulsar


4

rotation energy pumping mechanism. The amount of energy contained within the astrophysical flows usually is very big. If there are mechanisms for the conversion of at least a small fraction of this energy into the emission, one might have a significant development in studying the origin of radiation from related objects. In recent years, high-energy γ-rays have come to play an important role in the study of AGNs. Before the launch of the CGRO in 1991, the only known extragalactic source of high-energy γ-rays was 3C 273, which had been detected with the COS B satellite 20 years ago (Swanenburg et al. 1978). The EGRET detector on the CGRO has identified more than 65 AGNs which emit γ-rays at energies above 100 MeV (Hartman et al. 1999), and a substantial fraction of those sources which remain unidentified in the EGRET catalog are likely to be AGNs as well. All of the AGNs detected in high-energy γ-rays are radio loud sources with the radio emission arising primarily from a core region rather than from lobes. These types of AGNs are often collectively referred to as blazars and include BL Lacertae (BL Lac) objects, flat-spectrum radio-loud quasars, optically violent variables, and superluminal sources. For several BL Lac objects TeV energy sources have been detected. The origin of the high energy emission is a matter of great interest. There are many variations of the models. The most popular theoretical accounts at this time are those in which the γ-rays are produced through inverse Compton scattering of low-energy photons by the same electrons which produce the synchrotron emission at lower energies. Synchrotron self-Compton (SSC) emission, (Königl 1981; Maraschi et al. 1992; Bloom & Marscher 1996), in which the seed photons for the scattering are the synchrotron photons already present in the jet, must occur at some level in all blazars. Models in which the γ-ray emission arises predominantly from inverse Compton scattering of seed photons, which arise outside of the jet, either directly from an accretion disk (Dermer et al. 1992), or after being reprocessed in the broad-line region, or scattering of thermal plasma (Sikora et al. 1994), appear to be satisfactorily as well. Another set of models proposes that the γ-rays are produced by proton-initiated cascades (Mannheim 1993). Section 2.3 is dedicated to the problem of the non thermal emission in AGNs and the corresponding role of the centrifugal force in the radiation process. We consider the centrifugal acceleration of electrons by rotating magnetic field lines and determin the maximum Lorentz factor γmax of electrons. Considering two mechanisms responsible for limiting the Lorentz factors of electrons, so called the breakdown of the bead-on-the-wire approximation (BBW) and the inverse Compton scattering (ICS) we show that particles may be centrifugally accelerated up to γmax ' 108 and the main mechanism governing the saturation of γ is the ICS. We conclude that the energy of centrifugally accelerated particles can be amply sufficient for the generation (via the ICS) of the ultra-high energy (up to 20T eV ) γ emission in TeV Blazars

1.2 The instabilities in astrophysical outflows.

1.2

5

The instabilities in astrophysical outflows.

The observational evidence of astrophysical plasmas shows that in most of the cases, the flows are not in the equilibrium state, but are subject to different kinds of magnetohydrodynamic instabilities. Comprehensive theoretical study and numerical modeling of instabilities, with special emphasize on the nonmodal processes and Kelvin-Helmholtz instability (KHI) (with peculiarities of its relativistic dynamics), is the principal research objective of the second chapter. Generally speaking one can ask several questions, when one considers instabilities: (a) what is the criterion for instability? (b) how fast do linear perturbations grow? (c) what is the nonlinear development of perturbations like? (d) what additional effects can stabilize the problem?. Often the astrophysical flows are characterized by spatially inhomogeneous vector velocity fields (so-called shear flows or SF), in many cases these flows are kinematically complex, multidimensional and also relativistic. Among different kinds of astrophysical flows many involve motions of plasmas with highly nontrivial velocity fields. One class of such flows is, for example, swirling astrophysical flows (in which ejectional mode of motion is combined with the differential rotation in the normal plane (Rogava et al. 2003)), that until now has received very little attention. Stellar and extragalactic jets (Ferrari 1998) and solar tornados (Pike & Mason 1998) are just two classes of astronomical objects, which seem to be particularly interesting in this context. The first problem which is presented in the section 3.1 is the non-modal instabilities. Recently it was found that helical magnetized flows efficiently amplify Alfvén waves (Rogava et al. 2003). This robust and manifold nonmodal effect was found to involve regimes of transient algebraic growth (for purely ejectional flows), and exponential instabilities of both usual and parametric nature. However the study was made in the incompressible limit and an important question remained open whether this amplification is inherent to swirling MHD flows per se and what is the degree of its dependence on the incompressibility condition. In this section, in order to clear up this important question, we consider full compressible spectrum of MHD modes: Alfvén waves, slow magnetosonic waves and fast magnetosonic waves. We find that helical flows inseparably blend these waves with each other and make them unstable, creating the efficient energy transfer from the mean flow to the waves. We discuss about the possible role of these instabilities for the onset of the MHD turbulence, self-heating of the flow and the overall dynamics of astrophysical flows. Section 3.2 and 3.3 are dedicated to the study of instabilities of a boundary layer that separates two fluids in relative motion (the Kelvin-Helmholtz instability) and appears in different astrophysical and geophysical situations, starting from the solar wind interaction with the magnetospheric boundary (Uberoi 1984) and cometary tails (Ray 1982) up to

1.2 The instabilities in astrophysical outflows.

6

the dynamics of accretion discs (Anzer & Boerner 1983) and extragalactic and young stellar jets (Birkinshaw 1991; 1997; Marti et al. 1997). The origin of the instability may be traced to the Bernouilli equation: if a ripple develops at the interface, then a fluid flowing faster to pass over that ripple exerts less pressure, and the ripple tends to grow. The Kelvin-Helmholtz instability is thought to be important in understanding of jet morphologies and the deceleration process of jets (1980; Bodo et al., 2003). Recently it was shown that some of the observed structures of the jet in 3C 273 on parsec scales can be explained by the growing Kelvin-Helmholtz instability (Perucho et al. 2006). By Perucho & Lobanov (2006) based on the VLBI diagnostics of the jet in the quasar S5 0836+710, the jet structure was interpreted in terms of the KHI. Very often relativistic astrophysical flows are also magnetized, but the KHI for magnetized relativistic flows is not yet explored very well. In section 3.2 we linearize around the equilibrium state the relativistic magnetohydrodynamic equations in the vortex-sheet approximation and derive the corresponding dispersion relation. For simplicity we neglect gravity, surface tension and viscosity, so that the relevant equations are those given by particle number and energy-momentum density conservation of a relativistic perfect gas in flat Minkowskian metric added by the induction equation. We investigated the linear growth rates and corresponding regimes using four physical parameters representing the flow velocity, the relativistic and Alfvénic Mach numbers and the inclination of the wave vector’s projection in the plane of the interface. The linear analysis does not provide a complete picture of the process and can not explain the non linear behaviour of the instabilities and one has to consider the non linear regime of the KHI. In the next section 3.3, for studying the nonlinear behavior of the KHI for a planar velocity shear layer in a non magnetized and magnetized relativistic gas, we implemented the numerical code PLUTO (This is a numerical finite volume, shockcapturing, fluid dynamical code designed to integrate a system of conservation laws (Mignone et al. 2006)). Depending on the physical parameters, we study the creation, multiplication and possible merging processes of vortices. In chapter 4 we give a summarizing discussion of the presented investigation.

Chapter 2 The influence of centrifugal forces on particle dynamics in astrophysical outflows. In this chapter we focus on the role of the centrifugal acceleration in astrophysical outflows aiming to describe the importance of rotation in related objects (pulsars and AGNs). As it was mentioned in the first chapter, the thesis is composed of works done in Georgian National Astrophysical Observatory (GeNAO) and the works finished in in Università degli Studi di Torino (UNITO). This chapter is divided by three sections, first two of which are dedicated to works studied in GeNAO and in the third section we present a work done in UNITO. Effects of the rotation can be important in understanding many aspects of different phenomena related to several astrophysicsl objects. One very important class of problems, were effects of rotation can be successfully applied is the plasma acceleration in a pulsar wind and AGN jets. In one of the pioneering works about the pulsar physics, it is shown, that rotation plays an important role in the formation process of the pulsar magnetosphere (Goldreich & Julian 1969). Recently by Thomas Gangadhara (2005) the pulsar radio emission has been considered from the point of view of the centrifugal acceleration problem. The same situation is in extragalactic jets. Produced relativistic extragalactic jets have quite complex velocity configuration, usually they are characterized not only by the motion as a whole, but also are rotating, which means that effects of the centrifugal acceleration may be significant also for them (Gangadhara & Lesch 1997). As we have already said, the centrifugal acceleration process has to be considered in the context of the light cylinder problem, which means to understand how an accelerating and co-rotating plasma goes through the light cylinder surface. Normally the magnetic

2.1 Mathematical model.

8

field lines in a case of pulsars and AGN jets are not straight, but curved, so it is natural to study an influence of the curved trajectories on the dynamics of accelerated flows. For this reason in the first section we introduce the mathematical formalism for studying centrifugally accelerated particles forced to move on prescribed rotating trajectories and analyze corresponding equations in the context of the LCP. Pulsar emission theory implies that some instability is involved in the process, however, the specific mechanism by which radio pulsars emit, is not well understood. On the other hand it is important to understand a mechanism, that is responsible for the energy transfer from a rotator into the plasma waves, instabilities and escaping radiation. In the second section we study the centrifugally driven electrostatic waves for the Crab pulsar. We consider the plasma stream dynamics inside the pulsar magnetosphere and analyze the amplification factor of corresponding instabilities. Another class of astrophysical objects, where the same approach can be used, is AGN. It is supposed that these objects are rotating black holes having about 108 Solar masses. It is interesting to investigate an importance of rotation for AGN jets. In the last section of the second chapter we consider the problem of the non thermal emission in Blazars. In order to demonstrate the effect of rotation in producing TeV energy photons in several BL Lac objects we study the dynamics of relativistic electrons moving along straight, rotating, magnetic field lines and investigate the process of the centrifugal acceleration of electrons.

2.1

Mathematical model.

Rotation and relativity are those two features of motion, which do not easily match with each other. Still in astrophysics, with its abundance of extremely strong electromagnetic and gravitational fields, there are situations where motion is both rotational and relativistic. Most prominent examples include swirling astrophysical jets in active galactic nuclei (AGNs) and quasars, innermost regions of black hole accretion disks, accretion columns in X-ray pulsars and plasma outflows in radio pulsar magnetospheres. In these kinematically complex astrophysical flows, where rotation is interlaced with the relativistic motion of particles, the coexistence of these two features of the motion leads to observationally puzzling phenomena with sophisticated and ill-understood physical background. The interest to these flows is not new, but the upgrade of highly idealized models to more realistic, astrophysically relevant levels is still related with major theoretical and computational difficulties. Some important and basic theoretical issues, related with the relativistic rotation, are not uniquely defined and often evoke controversial interpretations. One of the most notable examples is the centrifugal force reversal


9

effect, originally found by Abramowicz and Lasota (1974), and later (Abramowicz & Prasanna 1990; Abramowicz & Miller 1990; Abramowicz 1990) studied in detail. It was argued that under certain conditions the centrifugal force attracts towards the rotation axis both for Schwarzschild and Kerr black holes. In Ernst spacetime, which represents the gravitational field of a mass embedded in a magnetic field, the centrifugal force acting on a particle in circular orbit was reported (Prasanna 1991) to reverse its sign even twice! In the simplest case of the Schwarzschild spacetime, strictly and essentially speaking, it was found that below the radius of the spatially circular photon orbit an increase of the angular velocity of a test particle causes more attraction rather than additional centrifugal repulsion. This effect was interpreted in (Abramowicz 1990) in terms of the centrifugal force reversal - it was stated that in such cases the centrifugal force attracts towards the axis of rotation! This interpretation was criticized (De Felice 1990; De Felice 1991; De Felice & Usseglio-Tomasset 1991), where it was argued that the discovered effect could be attributed to the strength of the gravitational field and be explained in a way which preserves the repulsive character of the centrifugal force. The spirit of this approach to save the intuitively appealing nature of the centrifugal force as of ”something which pushes things away” (De Felice 1991) is theoretically valid and practically convenient. After all, in general relativity, there is no implicit way to define the centrifugal force: in any case one needs to introduce some sort of ”3+1” spacetime splitting and dub as the centrifugal force some Newtonian-like expression, which looks like it (De Felice 1991). Moreover, de Felice found several interesting examples of the ambiguity of the global concept of outwards and pointed out at the deep interrelation of this problem with the definition of the centrifugal force in relativity. Abramowicz studied further the problem of the local and the global meaning of inwards and outwards (Abramowicz 1992) and showed that the centrifugal force always repels outwards in the local sense, while it may attract inwards, towards the centre of the circular motion, in the global sense! The theoretical scheme for the operationally unambiguous definition of the inward direction was suggested (De Felice & Usseglio-Tomasset 1993) and later this approach was used for the geometrical definition of the generalized centrifugal force (Bini et al. 1999). Therefore the effect, discovered by Abramowicz and Lasota (1974), is indubitably a genuine relativistic effect, although its interpretation in terms of the reversal of the centrifugal force is not implicit and is largely the matter of definition. Same is true for another rotational effect (Machabeli & Rogava 1994), on the basis of the relatively simple and idealized specialrelativistic gedanken experiment: motion of a bead within a rigidly and uniformly rotating massless linear pipe. It was shown that even if the starting velocity of the bead is nonrelativistic, after an initial phase of usual centrifugal acceleration, while the bead acquires high enough relativistic velocity, it starts to decelerate and after reaching the light cylinder changes the character of its motion


10

√ from centrifugal to centripetal. It was found that when the initial velocity v = 1/ 2 the motion of the bead is decelerative all the way from the pivot to the light cylinder. Certainly no real pipe may stay absolutely rigid, especially nearby the light cylinder. Besides, in order to maintain the uniform rotation of such a device, one needs an infinite amount of energy. Therefore the setup considered there was highly idealized. The constant rotation rate assumption was replaced in (Machabeli et al. 1996) with a more realistic one: the total energy of the system rotator+pipe+bead was assumed to be constant. It was found that the moving bead acquires energy from the slowing down rotator, but under favorable conditions the bead deceleration still happens. The results of (Machabeli & Rogava 1994) were interpreted by its authors in terms of the centrifugal force reversal. De Felice disputed this interpretation (De Felice 1995) and argued that, also in this case, like in above-mentioned general-relativistic examples, the generalized definition of the centrifugal force may guarantee the absolutely repulsive character of the force. He pointed out that an inertial observer will never see the bead reaching the light cylinder, because all light signals from the bead are infinitely redshifted. It was also shown that the vanishing of the radial velocity of the bead at the light cylinder can be interpreted in terms of the corresponding vanishing of the beads proper time. Despite the controversy of interpretations it is generally believed that rotational relativistic effects could operate in different astrophysical situations and might, hopefully, lead to detectable observational appearances. Recently, in (Heyl 2000) the author suggested that the observed QPO frequency shifts in bursters are caused by a geometrical effect of the strong gravity, similar to the Abramowicz-Lasota centrifugal force reversal1. As regards Machabeli-Rogava gedanken experiment, it implies that radially constrained, relativistic and rotationally (centrifugally) driven motion shows inevitable radial deceleration near the light cylinder. Evidently this effect might occur in a number of astrophysical situations, where motion is constrained, rotational and strongly (special) relativistic. One of the most important class of astrophysical flows, where this effect could show up, is centrifugally driven outflows. In the context of pulsar emission theory they were first considered in the late 1960s in (Gold 1968; Gold 1969) (for recent studies see Ref. (1996; Contopoulos et al. 1999)). For accreting black holes, both of galactic and extragalactic origin (Blandford & Payne 1982) first noted that centrifugally driven outflows from accretion disks could be responsible for the launch of jets, if the poloidal field direction is inclined at an angle less than 60. to the radial direction. Recently in (Gangadhara & Lesch 1997) it was suggested that centrifugal acceleration, taking place as a consequence of the bead-on-the-wire motion similar to the (Machabeli & Rogava 1994) gedanken experiment, may account for the acceleration of particles to very high energies by the centrifugal forces while they move along rotating magnetic filed lines of the rotating AGN magnetosphere. They claimed that the highly nonthermal, X-ray and γ-ray emission in


11

AGNs arises via the Comptonization (inverse-Compton scattering) of ultraviolet photons by centrifugally accelerated electrons. The same processes was critically re-examined in (Rieger & Mannheim 2000) and it was found that the rotational energy gain of charged particles is efficient but substantially limited not only by the Comptonization but also by the effects of the relativistic Coriolis forces. The specific nature of the propagation of electromagnetic radiation in the rotating frame of reference (De Felice 1995; Osmanov et al. 2002) is another aspect of this problem, which still needs to be taken into account. The whole philosophy of the ”pipe-bead” (or ”rotator-pipe-bead”) gedanken experiments was to mimic the common situation in relativistic and rotating astrophysical flows, where the plasma particles are doomed to move along the field lines of governing magnetic fields. While we consider relatively small length scales, the shape of the field lines can be approximated as being straight. However, on larger length scales the curvature of the field lines turns out to be important for the physics of the plasma streams, which are guided by them. The natural question arises: how the motion of the bead changes when the pipe is curved? In other words, how the dynamics of particles, prescribed to move along the fixed trajectories, change when the shape of their involuntary tracks of motion is not straight!? Obviously, this is not only a mere theoretical curiosity, but the issue which might have a tangible practical importance. In astrophysical situations the role of the pipes is played by the magnetic field lines, and the latter are always curved. Therefore, it is clear that the study of the motion of test particles along prescribed curved rotating trajectories is a necessary and important step for the ultimate building of a physically meaningful model of centrifugally driven relativistic particle dynamics for rotating magnetospheres of pulsars and AGNs. It is the purpose of this work to address the above stated issue. In particular, in the next section, we develop specialrelativistic theory of the motion of centrifugally driven particles on fixed, curved trajectories. The formalism is developed both for the laboratory frame (LF) and for the frame of reference rotating with the system (rotating frame, or RF). Equations of motion are derived and solved numerically. The detailed study is given only for the case when the angular velocity of the rotation is constant. However, we also outline the formalism for the astrophysically more realistic case of the conservative rotator-pipebead system with perceptible exchange of energy between the bead and the rotator, leading to the variability of the angular velocity of the whole system. In the final section of the work we discuss the results, consider the directions and aims of the future study, suggest and discuss those astrophysical situations, where the obtained results could be useful for the clarification of puzzling observational appearances of related astronomical objects.


2.1.1

12

Main consideration

The ideal two-dimensional system, which we are going to consider, consists of three basic parts: the device of the mass M and the moment of inertia I, rotating with the angular velocity ω(t), hereafter referred as the rotator; the massless but absolutely rigid pipe steadily attached to the rotator; and the small bead of the mass m and the radius equal to the internal cross-section radius of the pipe. The bead is put inside the pipe and can slide along the pipe without a friction. Evidently, instead of the pipe-bead dichotomy, since we are considering the two-dimensional layout, one may think about the ’wire-onbead analogy, which is sometimes used (Rieger & Mannheim 2000). Contrary to (Machabeli & Rogava 1994), where a straight pipe case was studied, now we let the pipe to be an arbitrarily flat curve, mathematically defined by: ϕ ≡ ϕ(r),

(2.1)

Φ ≡ rϕ0 (r).

(2.2)

with [dϕ/dr ≡ ϕ0 (r)]: The dynamics of the system may be studied basing on two alternative assumptions: 1. It makes the task simpler to suppose that the kinetic energy of the rotator EM is huge and always EM À Em ; i.e., despite the exchange of the energy with the moving bead, EM stays practically constant. Hence, the angular velocity of the whole “rotator-pipe-bead” system (henceforth referred as the RPB system) stays constant: ω = const.

(2.3)

In this case the first part of the triple RPB system (the rotator) continuously supplies the bead with energy and helps to keep the angular velocity of rotation constant. Therefore, the problem reduces to the study of the pipe-bead double system (the PB system) with the constant rotation rate. In the case of the straight pipe (ϕ = ϕ0 ) the problem has exact analytic solution, found and analyzed in (Machabeli & Rogava 1994). It is more realistic to assume that the rotator energy EM is finite, so that the whole RPB system is conservative Etot ≡ EM +Em = const. There is a perceptible energy exchange between the rotator and the bead: both the energy EM and the angular


13

momentum LM of the rotator are variable and, consequently, the angular velocity of the rotation can not stay constant: ω 6= const.

(2.4)

The problem with the straight pipe and variable rotation rate ω(t) has no analytic solution. It was studied numerically in (Machabeli et al. 1996). With either Eq.(2.3) or Eq.(2.4) assumptions the pipe is always assumed to be the passive part of the system. In order to mimic a magnetic field line it is assumed to be massless, having no share in the energy and/or momentum balance of the whole system. Still the role of the pipe — as the dynamic link between the rotator and the bead — is significant: it provides the prescribed “guiding” of the bead motion in the rotating frame of reference and makes the trajectory of the bead known in advance. In this work the dynamics of the gedanken system is studied in detail only under the first, easier, assumption of the constant angular velocity. The rout to the solution of the problem under the second assumption is also given, but its full study needs separate consideration and will be published elsewhere. There are two natural frames of reference, in which the dynamics of this system could be studied. The first, inertial one, is the laboratory frame (LF), where the observer measures the angular velocity of the rotator (and the pipe) to be ω(t), while the angular velocity of the bead is equal to: Ω = ω(t) + ϕ0 (r)v(t),

(2.5)

and the dynamics of the moving bead is governed by the pipe reaction force acting on it. Note that v(t) ≡ dr/dt is the radial velocity of the bead relative to the LF. The second frame, rigidly attached to the rotator and rotating with it (hereafter referred as the rotating frame, or the RF), is non-inertial, but quite convenient for the inspection of the motion of the bead along the curved pipe. This original approach implies embodying of the form of the pipe into the metric of the rotating frame. It was used in (Machabeli & Rogava 1994) for the straight pipe case and proved to be quite efficient for the case when ω(t) is assumed to be a constant. On the contrary, as we shall see later, the LF treatment appears to be handier when the second Eq.(2.4) approximation (Etot = const and Ltot = const, rather than ω(t) = const) is chosen. That is why it is important to consider the problem both in the LF and in the RF.

2.1.2

Uniformly rotating PB system

First, let us consider the problem in the laboratory frame of reference (LF) and ascertain that it admits full (numerical) solution of the associated initial value problem. Second,


14

let us consider the same problem in the rotating frame of reference (RF). We shall see that when the Eq.(2.4) assumption of the constancy of the rotation rate is used the latter approach is mathematically easier and provides fuller information about the dynamics of the system. LF treatment The most straightforward way to approach the problem is to consider it in the laboratory frame of reference, in which the spacetime is Minkowskian: ds2 = −dT 2 + dX 2 + dY 2 = −dT 2 + dr2 + r2 dφ2 .

(2.6)

We use geometrical units, in which G = c = 1. Note that azimuthal angle φ, as measured in the LF, is related with the azimuthal angle ϕ, measured in the RF, via the obvious expression: φ = ϕ + ωt. The pipe reaction force F is the dynamic factor constraining the bead to move along the pipe. It is easy to see (from the 4-velocity normalization gαβ U α U β = −1) that the Lorentz factor of the moving bead is: γ(t) = [1 − r2 Ω2 − v 2 ]−1/2 .

(2.7)

The angle between the radius-vector of a point of the pipe and the tangent to the same point is given by the relation: α = arctan Φ,

(2.8)

and the components of the reaction force, acting in the radial and azimuthal directions, are Φ |F|, 1 + Φ2 1 Fφ = |F | cos α = √ |F|, 1 + Φ2

Fr = −|F | sin α = − √

(2.9) (2.10)

respectively. Defining the physical components of the bead relativistic momentum [m(t) ≡ m0 γ(t)]: Pr ≡ mv,

(2.11)

Pφ ≡ mrΩ,

(2.12)

we can write the two components of the equation of motion in the following way: P˙ r − ΩPφ = Fr ,

(2.13)


15 P˙ φ + ΩPr = Fφ .

(2.14)

Combining these equations we can, first, derive the equation: P˙ r + ΦP˙ φ + Ω(ΦPr − Pφ ) = 0.

(2.15)

Ω˙ = ϕ0 v˙ + ϕ00 v 2 ,

(2.16)

m ˙ = mγ 2 [(Ω + rϕ00 v)rΩv + (v + r2 ϕ0 Ω)v]; ˙

(2.17)

It is easy to calculate that:

and using these relations together with Eqs.(2.11,2.12) we can easily derive the explicit equation for the radial acceleration of the bead: r¨ =

rωΩ − γ 2 rv(ϕ0 + ωv)(Ω + rϕ00 v) , γ 2 ∆2

(2.18)

where ∆ ≡ [1 − ω 2 r2 + Φ2 ]1/2 .

(2.19)

The Eq.(2.18) being of the form r¨ = G(r, ˙ r) admits full numerical solution, as the standard initial value problem, providing the initial position of the bead, r0 , its initial velocity, v0 , and the shape of the pipe, ϕ(r), are specified. Defining the spatial vector of the 2-velocity v ≡ (v, rΩ), we can calculate the absolute value of the reaction force |F| using the equation (Rindler 1960): m ˙ = F · v, which, in our case, leads to:

√

grr m ˙ = rω|F |.

(2.20) (2.21)

It is also easy to verify that the following quantity: Ψ ≡ m(t) − ωrPϕ = m0 γ(1 − r2 ωΩ) = const(t),

(2.22)

is the constant in time. This allows to find the solutions of the problem as functions of the specific value of this constant. In the next subsection we will see what is the physical meaning of this parameter - it turns out to be proportional to the proper energy of the moving bead in the RF. One important class of a possible shape of the curved trajectory is Archimedes spiral, given by the formula:


16 90 100 120

60 80 60

150

30 40 20

180

0

330

210

300

240 270

Figure 2.1: Polar graph of the Archimedes spiral for a = −5.

ϕ(r) = ar,

a = const.

(2.23)

(see also Fig. 2.1). In this case, since ϕ00 = 0, from Eq.(2.18) it is easy to see that r¨ ∼ Ω, while Eq.(2.22) implies that |F| ∼ Ω as well. Therefore, in the case of the Archimedes spiral trajectory, we can predict that the asymptotic behavior of the functions Ω(t), v(t), v(t), ˙ and |F(t)| will be similar. RF treatment We see the LF treatment allows to solve the problem and to obtain the complete information about the dynamics of the bead motion along the fixed nonstraight (curved) trajectories. However it is quite instructive and much more convenient to consider the same problem in the frame of reference, rotating with the pipe-bead system (rotating frame - RF). In order to do this we, first, need to switch from Eq.(2.6) to the frame, rotating with the angular velocity ω. Employing the transformation of variables:


17

T = t,

(2.24)

X = rcosφ = rcos(ϕ + ωt),

(2.25)

Y = r sin φ = r sin(ϕ + ωt),

(2.26)

ds2 ≡ −(1 − ω 2 r2 )dt2 + 2ω 2 dtdϕ + r2 dϕ2 + dr2 .

(2.27)

we arrive to the metric:

For the straight pipe (ϕ = ϕ0 ) case Eq.(2.27) reduces to the metric ds2 = −(1−ω 2 r2 )dt2 + dr2 , which was basic metric for the (Machabeli & Rogava 1994) study. Now, for a curved pipe, defined by the equation Eqs.(2.1,2.2), Eq.(2.27) reduces to the following form: ds2 = −(1 − ω 2 r2 )dt2 + 2ωrΦdtdr + (1 + Φ2 )dr2 .

(2.28)

For the resulting metric tensor µ

kgαβ k ≡

¶

−(1 − ω 2 r2 ), ωrΦ , ωrΦ, 1 + Φ2

(2.29)

we can easily find out that: ∆ ≡ [− det(gαβ )]1/2 = (1 − ω 2 r2 + Φ2 )1/2 ,

(2.30)

and, apparently it is the same function ∆, defined previously by Eqs.(2.20,2.21). For this relatively simple, but nondiagonal, two-dimensional spacetime we can develop the “1 + 1” formalism. Doing so we follow as a blueprint the well-known “3+1 formalism, widely used in the physics of black holes (Thorne & MacDonald 1982; MacDonald & Thorne 1982; Thorne et al. 1986). Namely, we introduce definitions of the lapse function: ∆ α≡ √ = grr

s

1 − ω 2 r 2 + Φ2 , 1 + Φ2

(2.31)

and the one-dimensional vector β~ with its only component: βr ≡

ωrΦ gtr = . grr 1 + Φ2

(2.32)

Within this formalism Eq.(2.28) can be presented in the following way: ds2 = −α2 dt2 + grr (dr + β r dt)2 .

(2.33)


18

Note that for the metric tensor Eq.(2.29) t is the cyclic coordinate and, moreover, in the RF the motion of the bead inside the pipe is geodesic - there are no external forces acting on it. Hence the proper energy of the bead, E, must be a conserved quantity. Employing the definition of the four velocity U α ≡dxα /dτ we can write: E ≡ −Ut = −U t [gtt + gtr v] = const.

(2.34)

On the other hand, the basic four-velocity normalization condition gαβ U α U β = −1 requires h

U t = −gtt − 2gtr v − grr v 2

i−1/2

,

(2.35)

this equation, written explicitly, has the following form: h

U t = 1 − ω 2 r2 − 2ωrΦv − (1 + Φ2 )v 2

i−1/2

.

(2.36)

Recalling the expression Eq.(2.5) for the angular velocity of the bead Ω(t), measured in the LF, and the definition Eq.(2.7) of the Lorentz factor γ(t) in the same frame of reference we can easily see that: U t = [1 − r2 Ω2 − v 2 ]−1/2 = γ(t).

(2.37)

It is important to note that the conserved proper energy of the bead, E, defined by Eq.(2.34) may be written in terms of the Ω(t) and γ(t) functions simply as: E = γ(t)[1 − r2 (t)ωΩ(t)] = const.

(2.38)

Taking time derivative of this relation and rearranging the terms we will finally arrive to exactly the same Eq.(2.18) for the radial acceleration of the bead r¨ as in the LF treatment. Note also that Ψ = m0 E. But the convenience of the RF treatment goes much further. From Eq.(2.34) and Eq.(2.35) we can derive the explicit quadratic equation for the velocity: 2 (gtr + E 2 grr )v 2 + 2gtr (gtt + E 2 )v + gtt (gtt + E 2 ) = 0,

(2.39)

with the obvious solution: √

v = r˙ =

·

¸

q gtt + E 2 2 −g tr gtt + E ± E∆ . 2 (gtr + E 2 grr )

(2.40)

The “1 + 1” formalism helps to write equivalents of the same equations in a more elegant form. Namely, if we define the radial velocity


19

1 (v + β r ), α

(2.41)

γ˜ ≡ (1 − V 2 )−1/2 ,

(2.42)

Vr ≡ and corresponding Lorentz factor:

then, instead of Eq.(2.37), we will simply have: U t = γ˜ /α,

(2.43)

E = γ˜ [α − (β~ · V~ )].

(2.44)

while from the Eq.(2.34) we obtain:

Instead of Eq.(2.39) we will have [V 2 ≡ grr V r V r = Vr V r , β 2 ≡ grr β r β r = βr β r ]: ~ V~ ) + (α2 − E 2 ) = 0, (β 2 + E 2 )V 2 − 2α(β·

(2.45)

with the solution: "

s

#

E 2 + β 2 − α2 1 r V = 2 αβ ± E . β + E2 grr r

(2.46)

Note that the RF Lorentz factor defined by Eq.(2.42) and the LF Lorentz factor, specified by Eq.(2.34) do not equal each other γ˜ (t) 6= γ(t),

(2.47)

which is the manifestation of the obvious fact that Lorentz factor is not an invariant physical quantity. One can see that for the γ˜ (t) the following quadratic equation holds: (α2 − β 2 )˜ γ 2 − 2αE˜ γ + (E 2 + β 2 ) = 0,

(2.48)

with the following solution: ·

¸

q 1 αE ± |β| γ˜ (t) = 2 β 2 + E 2 − α2 . α − β2

(2.49)

Therefore, above developed theory allows us to look for the solution of the initial value problem for the bead moving along an arbitrarily curved pipe. The thorough consideration of different particular cases is beyond the scope of the present work. Instead, we


20

will give representative solutions for one of the simplest kinds of a spiral – Archimedes spiral – defined by Eq.(2.23). The scheme for the complete inspection of the problem for any given initial value problem is the following: First, one specifies the initial location of the bead (r0 ) and its initial radial velocity (v0 ). The values of the ϕ0 (r0 ) and Φ(r0 ) are fixed as soon as we specify the form of the pipe ϕ(r). The initial values for the Ω(0) and γ(0) are given by Eq.(2.5) and Eq.(2.7): Ω0 = ω + ϕ0 (r0 )v0 ,

(2.50)

γ0 = [1 − r02 Ω20 − v02 ]−1/2 ,

(2.51)

while the value of the bead proper energy, according to Eq.(2.38), is given as: E = γ0 [1 − r02 ωΩ0 ].

(2.52)

Working with the Eq.(2.40), as the first order ordinary differential equation for the radial position r(t) of the bead at any moment of time, we can subsequently calculate all other physical variables. On the Fig. 2.2 the set of solutions is given for the case of the Archimedes spiral with a = −5, rotating with the angular velocity ω = 2 for the bead, which initially was situated right over the pivot of the rotator (r0 = 0) and had initial radial velocity v0 = 0.1. These plots tell us that in the limit of large distance from the rotator the value of the radial velocity tends to the asymptotic value: lim inf v(t) = v∞ ≡ −ω/a,

(2.53)

which, in this case, is equal to v∞ = 0.4. The angular velocity of the bead in the LF tends to zero, as well as the absolute value of the pipe reaction force, implying that at infinity the bead asymptotically reaches the limit of the force-free motion. This limit is understandable also analytically, because from Eq.(2.5) and Eq.(2.40) we can see that: √ E a2 − ω 2 ω + , v→ (2.54) |a| ωa2 r2 √ E a2 − ω 2 . Ω→ (2.55) ωar2 From these expressions it is clear that this regime is accessible if the condition |a| > ω holds! Otherwise, the particle is not able to reach the infinity.


21

Radial distance

Radial velocity

Radial acceleration

1.4

0.25

1.2

0.4

0.2

1 0.15

0.3

r

a

r

Vr

0.8 0.6

0.1

0.4

0.2 0.05

0.2 0

0

2 t

0.1

4

0

Lorentz factor

2 t

0

4

0

Effective angular velocity

1.12

2 t

4

Reaction force

1.5

0.35 0.3

1.1 1.08

0.25

1

F

Ω

γ

0.2 1.06

0.15 1.04

0.5

0.1

1.02 1

0.05 0

2 t

4

0

0

2 t

4

0

0

2 t

4

Figure 2.2: Graphs for the radial distance r(t), velocity v(t), acceleration a(t), the Lorentz factor γ(t), angular velocity Ω(t) and the absolute value of the reaction force |F(t)| for the rotationally (ω = 2) driven bead, moving on the Archimedes spiral with a = −5. r0 = 0, v0 = 0.1.

Since the shape of the function r(t) is almost linear it is instructive to make plots for the functions v(r) for different values of the initial radial velocity v0 , but with all other parameters of the initial value problem being the same. On the Fig. 2.3 we plotted these functions for eight different values of the initial radial velocity. We see that when v0 = v∞ the movement of the particle is force-free (geodesic) and uniform during the whole course of the motion. Physically it means that for this particular value of the v0 the shape of the pipe follows the geodesic trajectory of the bead, in the RF, for the metric Eq.(2.27) on the rotating 2D disk, so the bead moves freely, without interacting with the walls of the pipe. When v0 < v∞ , the particle moves with positive acceleration and asymptotically reaches the force-free regime in the infinity. While, when v0 > v∞ the character of the motion is decelerative, but the force-free limit is reached, again, when the bead heads to infinity. One more example of the latter behavior, similar to the case shown on the Fig. 2.2, but plotted for the initial velocity v0 = 0.5 > v∞ = 0.4 is given on the Fig. 2.4. Here we see that, unlike the case given on the Fig. 2.2, the acceleration of the bead is negative all the time and it reaches zero “from below”, taking less and less negative values. While the angular velocity of the bead relative to the LF Ω(t) is also negative from the beginning but its absolute value decreases and reaches the zero as the particle tends to the infinity.


22

Radial Velocity 1

0.9

0.8

0.7

V

0.6

0.5

0.4

0.3

0.2

0.1

0

0

2

4

6

8

10 r

12

14

16

18

20

Figure 2.3: Graphs for the radial velocity v(r), when the initial value of the v(r) is taken to be: 0.01, 0.1, 0.3, 0.4, 0.5 (force-free value), 0.7, 0.9, 0.99. ω = 0.1, a = −0.2. Radial distance

Radial velocity

2

Radial acceleration

0.5

0 −0.02

0.48

1.5

−0.04 −0.06

1

ar

Vr

r

0.46

−0.08

0.44

−0.1

0.5

0

0.42

0

2 t

4

0.4

−0.12 0

Lorentz factor

2 t

4

−0.14

Effective angular velocity

1.16

4

0.7 0.6

−0.1

0.5 −0.2

0.4

1.12

F

Ω

γ

2 t Reaction force

0

1.14

0.3

−0.3

0.2

1.1

1.08

0

−0.4

0

2 t

4

−0.5

0.1 0

2 t

4

0

0

2 t

4

Figure 2.4: Graphs for the radial distance r(t), velocity v(t), acceleration a(t), the Lorentz factor γ(t), angular velocity Ω(t) and the absolute value of the reaction force |F(t)| for the rotationally (ω = 2) driven bead, moving on the Archimedes spiral with a = −5. r0 = 0, v0 = 0.5.


2.1.3

23

Conserved energy case

When the bead accelerates, it continuously takes energy from the rotator. So, if one needs to keep the rotation rate constant, one needs to supply the system with energy from outside. This is, certainly, less realistic setup than the assumption that the “rotatorpipe-bead” system is conservative, viz. its total energy Etot is constant. In this case, however, the bead acceleration can not be permanent, because asymptotically it extracts all energy from the rotator and reaches the regime: ω(t) → 0, EM → 0 and Em → Etot . Clearly, in this situation, it is more convenient to study the dynamics in the laboratory frame of reference (LF), in which the rotator and the pipe are rotating rigidly with the time-dependent angular velocity ω(t). As regards the bead, since the shape of the pipe is curved, its angular velocity relative to the LF is given by Eq.(2.5). Since the pipe is considered to be massless and absolutely rigid, it does not contribute any amount of energy and/or angular momentum to the total energy Etot and angular momentum Ltot of the system. The rotator for simplicity is assumed to be a sphere of the radius R and the mass M having the inertia moment 2 I = M R2 , 5

(2.56)

the energy EM =

I 2 ω (t), 2

(2.57)

and the angular momentum LM = Iω(t).

(2.58)

Note that Eqs.(2.56-2.58) are nonrelativistic expressions. If initially, at t = t0 , ω0 R ¿ 1,

(2.59)

then it will remain nonrelativistic during the whole course of the motion, because it is assumed that the bead constantly extracts energy from the rotator, while the latter slows down so that the angular velocity ω(t) is a monotonically decreasing function of time. The Eq.(2.59) condition seems to be valid for the known fastest rotators in the Nature — pulsars. For the Crab pulsar, for instance, R = 1.2 × 106 cm, ω0 = 190.4Hz and consequently ω0 R/c ' 7.6 × 10−3 . Even for the fastest millisecond pulsars ω0 R/c ≤ 0.25. This justifies the usage of nonrelativistic Eqs.(2.56-2.58) expressions in our analysis. The remaining part of the threefold system — the bead — is assumed to be of the rest mass m0 . Its angular velocity and radial velocity relative to the LF, at any given moment


24

of time, are Ω(t) and v(t) ≡ r, ˙ respectively. Even when the initial radial velocity of the bead is nonrelativistic (v0 ¿ 1), it is still necessary to write relativistic expressions for its energy and angular momentum, because the bead gains energy, accelerates and sooner or later its motion becomes relativistic. Therefore, its energy and angular momentum must be written as: Em = m(t) = m0 γ(t),

(2.60)

Lm = m0 γ(t)r2 (t)Ω(t) = m(t)r2 (t)Ω(t),

(2.61)

where the LF Lorentz factor γ(t) is defined by Eq.(2.34). The system “rotator-pipe-bead” is conservative, there is no energy inflow from outside. In this sense it principally differs from the one considered in the previous section, where either the external energy source was necessary to keep the rotation rate ω constant or the rotator was assumed to possess an infinite amount of energy. Now, since the system is conservative, its dynamics are governed by the conservation laws of its total energy Etot ≡ EM + Em and total angular momentum Ltot ≡ LM + Lm : I 2 ω (t) + m0 γ(t) = Etot , 2 Iω(t) + m0 γ(t)r2 Ω(t) = Ltot .

(2.62) (2.63)

And the solution of the problem reduces to the solution of these equations, linked with (2.13,2.14), for two unknown functions of time r(t) and ω(t) for an arbitrary initial value problem: initial location of the bead r0 = r(0) and the initial value of the rotation rate ω0 = ω(0) of the whole system. The detailed study of this problem is beyond the scope of this work and will be given in a separate publication.

2.1.4

Conclusion

The purpose of the present work was to study the dynamics of relativistic rotating particles with prescribed, curved trajectories of motion in the rotating frame of reference. The work is a natural generalization of the gedanken “pipe-bead” experiment considered (Machabeli & Rogava 1994). In that paper the authors considered the case of the straight rotating pipe and they found out that when the velocity of the bead, driven by the rotation of the whole device and sliding along/within the pipe, is high enough the character of the motion changes from the accelerated to the decelerated one. In particular, it was found that when the√bead starts moving from the pivot (r = 0) of the rotating pipe with initial velocity v0 > 2/2, the motion is decelerative from the very beginning.


25

In this work we consider the motion of rotationally driven particles along flat trajectories of arbitrarily curved shape. The practical motivation for this approach and its importance are related with the following two facts: 1. The ’pipe-bead’ (or the ‘bead-on-the-wire’) gedanken experiment is considered as a model for the study of dynamics of centrifugally driven relativistic particles in rotating magnetospheres, in various classes of astrophysical objects, like pulsars (Gold 1968; Gold 1969; Machabeli & Rogava 1994; Contopoulos et al. 1999) and AGNs (Blandford & Payne 1982; Cao 1997; Gangadhara & Lesch 1997; Rieger & Mannheim 2000). The role of ”pipes” is played by the magnetic field lines. 2. The shape of magnetic field lines is always curved. It implies that for the largescale, global dynamics of charged particles — driven by centrifugal forces and moving along curved field lines of rotating magnetospheres — it is important to know what qualitative changes occur when the form of the field lines is not linear but curved. In this work we studied this problem, on the level of the idealized gedanken experiment, both in the laboratory (LF) and in the rotating (RF) frames of reference. For the simple example of the Archimedes spiral we found that the dynamics of such particles may involve both accelerative and decelerative modes of motion. One important difference from the linear pipe case (Machabeli & Rogava 1994) is that for the case of a curved pipe the motion of the bead is not any more radially bounded: there exist regimes of motion when the bead may reach infinity. This result has simple physical explanation. For the case of the linear pipe, rotating with the constant angular velocity ω0 , the natural limit of the radial motion was given by the light cylinder radius, defined as RL ≡ ω0−1 . Now, in the case of the curved pipe, even when it rotates with the same constant rate, the bead can slide in the azimuthal direction, following the curvature of the pipe and having a variable angular velocity Ω(t). It means that now the role of the effective light cylinder is played by RL (t) = Ω(t)−1 , and, hence, all those radial distances become accessible, where r(t) < RL (t). Therefore, if both r(t) and RL (t) are monotonically increasing functions, but the former stays always smaller than the latter (evidently it was the case in above considered examples for the Archimedes spiral) then the bead can reach infinity. Moreover, we found that there are special solutions, which are force-free during the whole course of the motion. These are simply geodesics in the two-dimensional rotating metric Eq.(2.27). In the LF the motion of the bead in this case is radial, because its angular velocity Ω(t) stays zero all the time and, correspondingly, the light cylinder is at the


26

infinity from the very beginning. The form of the trajectory in the RF in this case, ϕ(r) = −(v0 /ω0 )r, is simply that trace, which a free bead could leave on the surface of the rotating disk during the course of its geodesic motion. Intuitively it is evident that if the pipe has this particular form the bead slides within it freely, without interaction with the walls of the pipe. We considered only one, simple, subclass of spiral trajectories as the representative example of the solutions, but the developed theory may be used for the study of the dynamics of particles moving along arbitrarily shaped flat trajectories. It means that this approach may find wide applications to different astrophysical situations where rotation impels plasma particles to move along curved magnetic field lines. We also gave basic equations and outlined the scheme for the solution of the more general version of the same problem, where the angular velocity of the rotating system is not assumed to be constant. Instead, it is assumed that the system rotator-pipe-bead is conservative and the rotator is allowed to exchange perceptible portions of energy with the bead. The formalism developed and the results found in this work suggest some important directions of the further research, which could be physically and astrophysically relevant. First, it seems worthy to consider the case of the charged bead, which naturally would radiate while performing its nonuniform motion along the curved pipe. The radiative energy losses and the change of the angular momentum of the bead due to radiation could affect the dynamics of the bead and the whole system, considered, again, to be conservative. In the astrophysical context it could be interesting to see how the radiation of the bead would appear for the distant observer. Second, it is quite natural to try to extend the analysis for the 3-D fixed trajectories of motion, considering a family of axisymmetric trajectory lines and imitating the structure of the pulsar dipole magnetic field. This could bring us at least one step closer to the understanding of the radiative processes in pulsar magnetospheres. Third, sooner or later, we should address the fluid (plasma) problem and try to see how a continuous stream of fluid particles would behave, moving along rotating curved trajectories. This will comprise one more step closer to the reality of the pulsar environment and could help to relate with each other total energy losses of a pulsar, estimated through its slowing rate, and its radiation losses and energy taken away by centrifugally driven plasma, forming eventually the pulsar wind. These efforts would, hopefully, bring us to the construction of the unified theory of the pulsar magnetosphere, where the inertial aspects of the particle dynamics would be taken into due account. One could then try to test the theory with the existing empirical (observational) data about the energy deposited by pulsars into their winds and the energy they lose via their radio emission. This way we could have a clue as of how

2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. 27 important inertial processes (often unfairly neglected) are in the dynamics of pulsars. Similar problems can be addressed also in the context of the centrifugal acceleration of particles in the jets in AGNs (Blandford & Payne 1982; Cao 1997; Gangadhara & Lesch 1997; Rieger & Mannheim 2000). Yet another promising field of application is the accretion of plasma on strongly magnetized neutron stars, which is believed to lead to the appearance of X-ray pulsars. The final stage of the accretion is dominated by the dipole magnetic field of the accreting neutron star. It is normally assumed that the motion of accreted plasma particles is guided to the magnetic poles of the star by the rotating family of polar magnetic field lines. So, in a certain sense, this problem is of the same kinematic nature as the one related with radio pulsars, except that this time plasma moves from the region outside of the light cylinder towards the star. Certainly here, again, radiative effects and collective plasma effects are essential, so the simple ’one-particle treatment can give only very approximate picture of the involved physical processes. But taking into account the plasma fluid effects and the role of the radiation on the dynamics of infalling plasma streams, one could try to show how important the rotational (inertial) processes are for the dynamics of the flows infalling on strongly magnetized neutron stars and what is the influence of these processes on the observational appearance of related X-ray sources.

2.2

Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar.

One of the important stages in the development of pulsar radiation theory was the discovery that the rotation energy could transform into energy in the electrostatic field; this possibility stimulated the modeling of pulsar magnetospheres (Deutsh 1955; Goldreich & Julian 1969). In a typical model, the electrostatic field E0 , generated over the pulsar surface, has a finite projection along the dipole magnetic field B0 . The resulting electric force, eE0 exceeds the gravitational force near the neutron star surface, and ends up uprooting charged particles from the surface layer. These particles are accelerated in the electric field and radiate γ photons because of the curvature radiation in the dipolar field. For photon energies greater than twice the electron rest mass ( ²γ ≥ 2me c2 ), the electron-positron pair production (γ +B0 → e+ +e− +γ 0 ) in the ambience of the magnetic field becomes possible. The particles, so produced, repeat the cycle; they are accelerated, generate γ radiation which creates more electron-positron pairs. This cascading leads to the build-up of a large flux of relativistic e+ e− pairs ; the process continues till the pair

2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. 28 plasma screens the electric field E0 as shown in (Sturrok 1971; Tademaru 1973). The e+ , e− population in the pulsar atmosphere may be conveniently divided into three principal components: 1) the basic plasma mass (Bulk) with concentration npl , and a Lorentz factor γpl , 2) a tail with concentration nt , and Lorentz factor γt , and 3) the remnant of the primary electron or ion beam (most likely electrons). Note that cascading due to pair production is possible only in a highly relativistic scenario, because for nonrelativistic or even mildly relativistic velocities, the pair annihilation dominates pair production (Dirac 1930; Kompaneec D.A., 1978). The reader is referred to (Kazbegi et al. 1991; Machabeli & Usov 1979) for a guide to processes that define and control the behavior of radiation in the pulsar magnetosphere. The essence of this “standard model” is that the pulsar magnetosphere is permeated with a multi-component plasma. It is generally assumed that the energy contents of the three major components are of the same order: npl γpl ≈ nt γt ≈ nb γb . Theoretical estimates for the density and the energy of the primary component are: nb ≈ (1011 − 1012 )cm−3 ( also called the Goldreich Julian concentration) and γb ≈ 106 − 107 . The origin of the pulsar radiation is supposed to be in the magnetosphere, and hence the energy contained in the e+ e− plasma must be enough to account for the observable radiation. In these pulsar models, the measure of the region over which the electric field is nonzero is in great significance. The distance between the star surface and the region where it is screened out is called the vacuum gap, and may be estimated to be (103 − 104 )cm for E0 ≈ 107 G (Ruderman & Sutherland 1975). Unfortunately the particle energy inventory accumulated within the gap is not sufficient to explain the observed visible radiation. Several mechanisms have been invoked to increase the gap size. The intermediate formation of positronium (electron-positron bound state) which would hold back the decay, and lead to an increase in the gap was proposed in (Usov & Shabad 1985). According to (Arons & Sharleman 1979), the pulsar magnetic field lines, supposedly curved towards rotation, must be screwed even more; the twisted magnetic field lines will be rectified, and as a result will enlarge the gap. A different mode, the PFF (pair formation front) mechanism was introduced instead of the gap. Taking the heating of the stellar surface and the electron thermoemission into account leads to nonstationarity, and explains nonstationary behavior of pulsars (Kazbegi et al. 1996). In a series of papers, Muslimov and Tsigan have attempted to solve the gap problem via the general relativity route. Realizing that in the vicinity of the rotating neutron star, the space-time is slightly curved, they work out the creation of the electric field in Kerr metric (Muslimov & Tsygan 1992). As a result, the gap size is somewhat enlarged but not enough to make a difference. It seems that some additional source of energy or some new mechanism will be necessary to surmount the problem arising out of the “insufficiency” of the energy content

2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. 29 in the vacuum gap. For instance, it seems perfectly plausible and possible that one could draw upon the pulsar rotation energy to augment the energy content of the magnetospheric e+ e− plasma; the rotation energy, for example, could be transformed into energy associated with oscillations in the e+ e− plasma far from the pulsar surface. In this region the complicating effects of unremovable gravitation (having to use, the Kerr metric, for example) will not exist. This work is an attempt to formulate and investigate the problem associated with the parametric pumping of plasma oscillations in the pulsar magnetosphere taking into consideration the complicated nature of the multicomponent electron-positron substrate. Let us review a standard system to examine the plausibility of the proposed mechanism. For the Crab Nebula, the nebula radiation can be surely sustained by the rotational energy of the pulsar PSR 0531 located in the vicinity of the nebular center. The power of the nebula radiation exceeds the pulsar radiation power by two orders of magnitude (it is approximately 2 · 1038 erg/sec). The only source capable of providing such a prodigious ˙ ≈ IΩΩ, ˙ where I is pulsar’s moment of power is the slowdown of the pulsar rotation: W inertia with mass of order (1.5 − 2.5)M¯ (M¯ is solar mass), Ω and Ω˙ are star’s angular velocity and angular acceleration respectively. This power is equal to 5 · 1038 erg/sec. ˙ /W ' 2Ω/Ω ˙ The rotation energy loss rate is described by the ratio W = 2P˙ /P , where ˙ P = 2π/Ω is the neutron star’s rotation period, and P and P are measurable parameters for pulsars. Ratio P˙ /P for different pulsars ranges from 10−11 sec−1 (PSR 0531) to 10−18 sec−1 (PSR 1952+29). Thus, for ordinary plasma oscillations to be pumped by the pulsar rotation, their growth rate must not exceed P˙ /P , the measurable rate of rotation energy loss. A hydrodynamic approximation will be used to study the problem of rotation induced wave generation in plasmas. For relative simplicity one will assume that the e + e− plasma has two distinct energy ranges: the lower range (still highly relativistic) with npl , and γpl , and the beam with nb , and γb . These flows propagate along the rotating monopole like magnetic field lines.

2.2.1

The main consideration

It is supposed that for distances less than the radius of curvature of the field line, the magnetic field is monopole like (in this approximation the magnetic field lines may be supposed to be rectilinear). The problem of the motion of charged particles in the pulsar magnetosphere can be considered in the local inertial frame of the observers, who measure the physical quantities in their immediate vicinity. They are called the Zero Angular Momentum Observers (ZAMOs). Naturally, the proper time of the observer riding the particle, is different from the proper time of ZAMOs.

2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. 30 Transformation from the inertial frame to the frame connected with the pulsar magnetosphere can be done in the following way: t = t0 , ϕ = ϕ0 , r = r0 , z = 0,

(2.64)

then the interval in the corotating frame will have the form: ³

´

ds2 = − 1 − Ω2 r2 dt2 − dr2

(2.65)

where Ω is angular velocity of rotation. √ Note that the lapse function α = 1 − Ω2 r2 (c = 1) not only connects the proper time of ZAMOs with the universal time dτ = αdt, but also gives a gravitational potential: ∇α . α The equation of motion of the particle in the ZAMOs frame is expressed as: g=−

(2.66)

dp e = γg + (E + [VB]) (2.67) dτ m where γ = (1 − V2 )−1/2 is the Lorentz-factor and V = dr/dτ is the velocity of the particle determined in the so-called 1+1 formalism, and p → p/m is the dimensionless momentum. As it is shown (Chedia et al. 1996) the transition from the particle equation of motion to the Euler equation for fluid dynamics in the 1 + 1 formalism, may be easily fulfilled if one changes d/dτ in Eq. (2.67) by 1/(α∂t) + (V∇). The resulting equation describing the stream motion (neglecting the stream pressure) takes the following form: 1 ∂p ∇α e (2.68) + (V∇)p = −γ + (E + [VB]) α ∂t α m where V and p are now hydrodynamic velocity and momentum respectively. In order to rewrite this equation in the inertial frame, let us note that the ZAMOs momentum coincides with the momentum in the inertial frame. In fact, from the definitions p = γV, γ = αγ 0 , and V0 = dr/dτ (prime refers to quantities in the inertial frame), one can easily find that p = p0 . In the inertial frame, then Eq. (2.68) converts to (omitting primes for all quantities): ei ∂pi + (vi ∇)pi = −γα∇α + (E + [vi B]) , i = b, e, p, (2.69) ∂t m where b, e and p denote the beam, electron and positron components respectively. In Eq.(2.69), the force F = −γα∇α is the analog of the centrifugal force.

2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. 31 Adding the continuity and the Poisson equations: ∂ni + ∇(ni vi ) = 0, ∂t

(2.70)

∇E = 4πe(ne − np + nb )

(2.71)

As we are interested in the evolution of fluctuations, one can look for solutions of Eqs. (2.69-2.71) in the framework of a perturbation theory- an expansion in which terms like E 1 /mnγ (the small parameter in the approximation of weak turbulence for the plasma) are small: ∂pi ei + (vi ∇)pi = −γα∇α + (E + [vi B]) , i = b, e, p, ∂t m

(2.72)

E = E0 + E1 + ...

(2.73)

B = B0 + B1 + ...

(2.74)

pi = p0i + p1i + ...

(2.75)

where E0 , B0 and p0i are the leading terms, and E1 , B1 and p1i constitute the perturbations. In the zeroth approximation (taking into consideration the fact that the ejected particles not only move along the radius, but also corotate with the pulsar magnetosphere because of the frozen-in condition , E0 + [v0 B0 ] = 0) Eq. (2.69) will be reduced to the form (Machabeli & Rogava 1994): 

Ã

dr d2 r Ω2 r  = 1 − Ω2 r 2 − 2 2 2 2 dt 1−Ω r dt

!2  

(2.77)

where we have neglected the term (v0 ∇)p0 (taking this term into account is a separate problem, and is not examined within the framework of this work). One has to note that this equation is a special case of Eq. 2.18 for straight magnetic field lines. As is shown in (Machabeli & Rogava 1994), Eq. (2.77) allows an exact solution for particular initial conditions r(t0 = 0) = 0, V (t0 = 0) = V0 : r(t) =

f) V0 Sn(Ωt | m f) Ω dn(Ωt | m

(2.78)

where Sn and dn are Jacobian elliptical functions, the sine and the modulus respectively f = 1 − V02 . and m

2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. 32 Using the following properties of the mentioned Jacobian elliptical functions Sn(x | 0) = sin(x) and dn(x | 0) = 1 (Abramowitz & Stegun 1965), one may easily reduce Eq. (2.78) for the ultra-relativistic regime (V0 → 1) relevant to this work: V0 sin Ωt. Ω With this known asymptotic solution, the first order Eq. (2.69) reads: r(t) =

ei ∂p1i + (vi0 ∇)p1i = F1i + E1 , ∂t m 2 0 1 1 Fi = Ω rvi pi , vi0 = Vi0 cos Ωt.

(2.79)

(2.80) (2.81) (2.82)

where, in addition to Eq. q (2.79), we have used the Lorentz-factor expansion in the small parameter p1i /p0i : γ ≈ 1 + (p0i )2 (1 + p0i p1i /(1 + (p0i )2 )). In a similar fashion, the linearized Eq. (2.70) becomes: ∂n1i + div(n0i vi1 ) + div(n1i vi0 ) = 0. (2.84) ∂t The electron-positron continuity equations may be combined to obtain the evolution equation for the effective charge density npl = ne − np : ∂n1pl 1 0 + div(n0e vpl ) + div(n1pl vpl )=0 (2.85) ∂t 1 0 where vpl = ve1 − vp1 and it has been assumed that ve0 = vp0 ≡ vpl and n0e = n0p . We seek here a solution in which the density perturbations have no spatial dependence so that the last terms of Eqs. (2.84) and (2.85) are identically zero. Let us choose n1i (now i = pl, b) to have the form n1i = Ni e−

ikV0i Ω

sin Ωt

.

(2.86)

Then, if the rest of the perturbed quantities are allowed the spatial dependence p1i (r; t) = p1i (k; t)eikr , one obtains from Eqs. (2.84)-(2.85): ikp1i = −

3 γ0i ∂Ni −ikRi e 0 ni ∂t

(2.87)

3 where γ0i is the initial Lorentz factor, and Ri = VΩ0i sin Ωt, and the relation vi1 = p1i /γ0i (p1pl = p1e − p1p ) that is satisfied for Ωt ∼ Ω/ω ¿ 1 has been used.

2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. 33 Let us now go back to Eqs. (2.80,2.81,2.82) and write them separately for the two components- the plasma and the beam : e ∂p1b + (vb0 ∇)p1b = F1b + E1 , (2.88) ∂t m ∂p1pl e 0 (2.89) + (vpl ∇)p1pl = F1pl + 2 E1 . ∂t m Repeating the procedure applied to Eq. (2.88,2.89) and taking Eqs. (2.79) and (2.87) into consideration, one can find that the beam and the plasma components evolve as: ∂ 2 Nb en0b ikRb = −i e kE1 , 3 ∂t2 mγb0

(2.90)

en0pl ikRpl ∂ 2 Npl = −2i e kE1 . 3 ∂t2 mγp0

(2.91)

Combining Eqs. (2.90) and (2.91) eliminates the electric field to yield 3 2 n0b γp0 ∂ 2 Nb ik(Rpl −Rb ) ∂ Npl = e . 3 ∂t2 2n0pl γb0 ∂t2

(2.94)

Equation (2.94) is a rather complicated non-autonomous equation in time. To solve it, one can take its Fourier time transform (restoring the speed of light) along with that of the Poisson Eq.(2.71) [cf. Appendixes A,B Eqs. (A.2) and (B.2)] with the electric field eliminated to arrive at the coupled system 3 X n0b γ0pl ω Nb (ω) = 0 3 (ω + sΩ)2 Js (a)Npl (ω + sΩ), 2npl γ0b s 2

Ã

(2.95)

!

ω2 ω2 X ω − 3pl Np (ω) = 3pl Js (a)Nb (ω − sΩ), γ0pl γ0pl s 2

(2.96)

q

2 where a = kc/2Ωγ0pl , and ωpl = 8πn0pl e2 /m. Naturally one can see the appearance of convolution sums on the right hand sides of both equations. Substituting Nb from Eqs. (2.95) into (2.96), one can find:

Ã

!

Ã

2 ωpl ω2 X ω − (s − l)Ω 2 ω − 3 Npl (Ω) = 3b Js (a)Jl (a) γ0p γ0b sl ω − sΩ

!2

Npl (ω − (s − l)Ω),

(2.97)

2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. 34 q

where ωb = 8πn0b e2 /m. One could try to find the nature of the time evolution from Eq.(2.98), but it seems to be a little better to go back to npl given by Npl = n1pl eikRpl . Carrying out the algebra given in the appendix[cf. Appendix C, Eq. (C.2)], one may derive Npl =

+∞ X

(2.98)

Js (b)npl (ω − sΩ),

s=−∞

where b = kc/Ω. Substituting Eq.(2.97) into Eq. (2.98), one finally obtains the rather complicated dispersion: Ã

!

Ã

ω2 X ω2 X ω + (j + m)Ω Js (b)npl (ω−sΩ) = 3b Jl (a)Jj+l+m (a)Jm (b) ω − 3pl γ0p s γ0b jlm ω − lΩ 2

!2

npl (ω+jΩ).

(2.99) To extract some sense out of the above result, one can explore the dispersion relation 2 3 near the resonant condition, ω 2 ≈ ωpl /γ0pl . Near the resonance, the basic contribution P to the sum s Js (b)npl (ω − sΩ) comes from ω ≈ s0 Ω. Similarly the right hand side of (2.99) is reduced to a single term corresponding to ω ≈ −j0 Ω, ω ≈ l0 Ω and m = m0 = 3/2 3/2 2 3 −j0 = l0 = s0 . Rewriting ω 2 − ωpl /γ0pl as 2ωpl ∆/γ0pl (where ∆ = ω − ωpl /γ0pl ), then, reduces Eq. (2.99) the simple cubic equation: ∆3 ≈

ωb2 ωpl 3 3/2 2γ0b γ0pl

Js20 (a),

(2.100)

3/2

where s0 = [ωpl /Ωγ0pl ]. In addition to the real root, the dispersion relation of Eq. (2.100) allows the complex conjugate pair √ 1 i 3 ∆1,2 ≈ − M ± M, (2.101) 2 2 

1/3

ω 2 ωpl M =  b 3/2 Js20 (a) 3 2γ0b γ0pl

,

(2.102)

with comparable real and imaginary parts. The root with the positive imaginary part implies the instability that we were seeking.

2.2 Parametric mechanism of the rotation energy pumping by a relativistic plasma. Application to the Crab pulsar. 35

2.2.2

Estimates

The instability turns out to be rather strong. For typical pulsar parameters (that of the Crab nebula) n0b ≈ 107 cm−3 , γ0b ≈ 106 , γ0pl ≈ 10, Ω ≈ 190Hz, √ considering the longitudinal waves, one can plot the graph of the growth rate δ (δ = 3M/2 see Eq. (2.101,2.102)) as a function of the radial distance (normalized by the light cylinder radius). For simplicity let us examine longitudinal waves in a coupling point. In this case all three modes of a cold plasma (o, x and Alfvén modes) are indistinguishable, then one can estimate the 3/2 wave number according to the following approximate formula kc ≈ ωpl /γ0pl (Machabeli et al. 1999). Using mentioned pulsar parameters, from Eq. (2.101) the growth rate is estimated. In Fig. 2.5 one may see that δ is sensitive to the radial distance, in the range 0.75 ≤ R/Rcyl ≤ 1 it increases from ∼ 10−14 to ∼ 10−7 . (here Rcyl is the light cylinder radius, thus the radius, where the rotation velocity equals to the speed of light). It is clear that, δ becomes unreasonably large for R/Rcyl ≥ 0.85, which means that, the linear assumption will be grossly violated long before these distances are reached. The instability will have a linear growth over certain distances and then nonlinear saturation will set in after which other nonlinear mechanisms may take place in the energy pumping phenomenon. In estimating the growth one has invoked n0pl γ0pl ≈ n0b γ0b , and used the fact that the density goes as n ≈ n0 (r/R)3 (where r ≈ 106 cm is a radius of the neutron star) because the magnetic filed is monopole like.

2.2.3

Conclusions

The purpose of this work was to explore the possibility of pumping the rotational energy of the pulsar into the plasma. Considering a highly idealized system, the linear instability caused by rotation in a two component relativistic plasma embedded in a uniform magnetic field was examined. By using the hydrodynamic equations of motion, the continuity and the Poisson equation, it has been shown that the plasma waves can grow on the rotation energy with rather high growth rates. In fact the perturbation growth rate for distances R/Rcyl ≥ 0.85 is quite high in comparison with the supposed instability rates we were seeking. If this mechanism is, indeed, operational, then the only consistent scenario is that the linear stage with this growth rate is very short, and nonlinearities are turned in soon enough to considerably reduce the growth. Thus the need for a nonlinear theory is immediately and strongly indicated. Sooner or later we should consider this particular nonlinear effect, which will comprise one more step closer to the real scenario. There are two other shortcomings of this effort: a) the straight magnetic field lines (in the local frame of reference) has been examined, whereas real profiles are curved and b) only electrostatic waves have been considered. The preliminary results, however, unam-

2.3 On efficiency of particle acceleration by rotating magnetospheres in AGN. Application to Blazars.

36

−7

10

−8

10

−9

10

−10

δ

10

−11

10

−12

10

−13

10

−14

10

0.75

0.8

0.85

0.9

0.95

1

R / Rcyl

Figure 2.5: Graph of log(δ(R/Rcyl )). Set of parameters is following: n0b ≈ 107 cm−3 , γ0b ≈ 106 , γ0pl ≈ 10, Ω ≈ 190 Hz, r ≈ 106 cm.

biguously show that the energy content of the magneto spheric plasma can grow at the expense of the stellar rotational energy.

2.3

On efficiency of particle acceleration by rotating magnetospheres in AGN. Application to Blazars.

In this section we present another work which has been prepared and finished in UNITO. In the previous section the role of the centrifugal force in rotating systems has been investigated and applied for the Crab pulsar. Another class of astrophysical objects, where the same approach can be used, is AGN. A produced relativistic extragalactic jets have quite complex velocity configuration, usually they are characterized not only by the motion as a whole, but also are rotating. It is interesting to study the role of the centrifugal acceleration for AGN jets. Here we consider the problem of the origin of non thermal emission in AGNs. In order to demonstrate the effect of the centrifugal acceleration in producing of TeV energy photons in several blazars we consider relativistic electrons moving along straight rotating magnetic field lines and study the role of the centrifugal force in this process.


37

One puzzling and interesting problem, related with active galactic nuclei (AGNs), is the origin of the ultra-high γ-ray nonthermal emission from Blazars. Some of them, such as Mrk 421, Mrk 501, PKS 2155-304, 1ES 2344-514, H1426+428, and 1ES 1959+650, are emitting TeV (1T eV ≡ 1012 eV ) photons and form a special class of ”TeV Blazars” (Dai et al. 2002; Catanese & Weeks 1999; Horns 2002). The standard blazar model implies a presence of a supermassive black hole, surrounded by an accretion disk and ejecting twin relativistic jets, one of which is seen almost end-on. The broadband emission spectrum of blazars is made of two components: the low-energy (from radio to optical/UV) part attributed to synchrotron radiation and the high-energy (from X-rays to γ-rays) component formed by the inverse Compton scattering (ICS) of softer photons (Kino et al. 2002; Kubo et al. 1998). The latter part of the spectrum is usually interpreted on the basis of the synchrotron self-Compton (SSC) model. However, the origin of accelerated and/or pre-accelerated electrons, the mechanism which is responsible for their efficient acceleration is still a matter of uncertainty. Proposed mechanisms such as the Fermi-type acceleration process (Catanese & Weeks 1999) and re-acceleration of electron-positron pairs as a feedback mechanism (Ghisellini et al. 1993) may account for the observed high energy emission up to 20 TeV. However, the Fermi-type acceleration in relativistic jets is efficient when the seed population of ”pre-accelerated” electrons possesses quite high (γmax ≥ 102 ) Lorentz factors (Rieger & Mannheim 2000). The origin of this “pre-acceleration” is still a matter of discussion. When one has to study a relativistic, rotating plasma flow in astrophysics, the concept of centrifugally driven outflows (CDOs) naturally comes into the spotlight. Often it turns out that centrifugally accelerated particles may acquire quite high energies. This concept, since the pioneering insights (Gold 1968; Gold 1969), has often been discussed (Machabeli & Rogava 1994; Chedia et al. 1996; 1996; Contopoulos et al. 1999) in the context of pulsar emission theory. Regarding the AGNs it was shown (Blandford & Payne 1982) that CDOs from accretion disks are realized if the poloidal magnetic field lines are inclined at an angle ≤ 600 to the equatorial plane of the disk. In the AGN context, the presence of the CDO would imply that despite the intense UV radiation (when via ICS soft photons are scattered against accelerated electrons and, as a result, electrons lose energy and photons gain energy), electrons may reach quite high, γmax ∼ 105 , Lorentz factors. If being efficient enough, this mechanism could be used not only for justifying the pre-acceleration of electrons but it could be considered also as an alternative way of the Blazar high-energy emission (up to ∼ 20T eV ) generation (Okumura et al. 2002). Using the approach suggested by Gangadhara (1996), where a test charge motion along a rapidly rotating field line was considered in the context of a millisecond pulsar, Gangadhara and Lesch examined the role of the centrifugal force on the dynamics of electrons


38

moving along straight magnetic field lines (in the local frame of reference), fixed in the equatorial plane and co-rotating with the spinning AGN (Gangadhara & Lesch 1997). They have shown that scattering of low-energy photons against accelerated electrons may lead to the generation of the nonthermal X-ray and γ-ray emission. This problem was re-examined by Rieger and Mannheim (Rieger & Mannheim 2000) who tried to specify whether the rotational energy gain of charged particles, moving along straight magnetic field lines (in the local frame of reference), is limited not only by ICS but also by the breakdown of the-bead-on-the-wire approximation (BBW). The latter would happen in the vicinity of the light cylinder when a Coriolis force acting on the particle and trying to ‘tear it off’ the field line would exceed the Lorentz force bounding the particle to the field line. According to their consideration, the maximum value of the Lorentz factor, being the subject of both limitations is γmax ∼ 1000, which is not enough for producing ultra-high energy photons emitted by TeV blazars. In real, three-dimensional astrophysical situations (for example jets) the magnetic field lines are not localized in the equatorial plane but are inclined with respect to it. In this work we re-examine the same problem considering the wide range of possible inclinations. We show that, for a wide range of AGN, the mechanism responsible for limiting the attainable maximum Lorentz factors is ICS, which under certain conditions can allow particles to reach quite high γmax ≥ 105 Lorentz factors. The BBW becomes important for the low luminosity (< 1041 erg/s) AGN, when γmax ∼ 108 . For higher luminosities (> 1041 erg/s) it can be dominant only for relatively small inclinations of the magnetic field lines with respect to the rotation axis (≤ 10◦ ). Therefore, contrary to the conclusions of Rieger and Mannheim (2000), we argue that CDOs could be efficient enough to account for the TeV Blazar emission. The work is arranged in the following way: In Sec. 2.3.1 we derive basic equations. In Sec. 2.3.2 we consider a case when magnetic field lines are located in the equatorial plane and show that the presence of the ICS allows a wide range of AGN to generate ultra-high photons emission. In Sec. 2.3.3 inclined magnetic field lines are considered. We show that in this case ICS still stays the dominant limiting mechanism for most of the AGN. In the final section we summarize the obtained results.

2.3.1

Main consideration

We consider a typical AGN with a central black hole mass MBH = 108 MJ (MJ denotes the solar mass) and an angular rate of rotation ω ∼ 3 × 10−5 s−1 (that is only 1% of the maximum possible rotation rate for black holes with the mentioned mass), which makes the light cylinder radius being located at rL ≈ 1015 cm. The values of the angular velocity and corresponding light cylinder radius are typical for AGN winds (Belvedere


39

et al. 1989; Rieger & Mannheim 2000). This spinning rate corresponds to the Keplerian radius ∼ 8×Rg (where Rg ≡ 2GMBH /c2 is the gravitational radius) which is about where softer photons are expected to originate (Peterson 1997), these photons are instrumental for the Inverse Compton mechanism. Rieger and Mannheim (2000) consider two limiting mechanisms for the maximum Lorentz factor: ICS, when soft photons are scattered against electrons gaining energy and suspending further electron acceleration and BBW when due to the violation of the Coriolis and Lorentz forces balance, acting on the electron, it comes off the magnetic field line. When the electron moves along the rotating magnetic field line it experiences the centrifugal force and as a result accelerates. The corresponding time scale describing the acceleration process can be defined as (Rieger & Mannheim 2000): tacc ≡

γ . dγ/dt

(2.103)

The acceleration lasts until the electron encounters a photon, which may limit the maximum Lorentz factor of the particle. This process may be characterized by the corresponding cooling time scale (e.g. Rybicki & Lightman 1979; Rieger & Mannheim 2000): tcool = 3 × 107

γ [s], (γ 2 − 1)Urad

(2.104)

where Urad = τ le ×LEdd /4πcR2 is the energy density of the radiation, γ-the Lorentz factor of the electron, τ ≤ 1, LEdd is the Eddington luminosity (for the mentioned mass of AGN LEdd ≈ 1046 erg/s), le ≡ L/LEdd (L is a disk luminosity), 10−7 ≤ le ≤ 1. In (Rieger & Mannheim 2000) the range 10−4 ≤ le ≤ 1 was used but in this work we consider the whole range of AGN luminosities (down to AGN with the lowest luminosities 1039 erg/s (le ≈ 10−7 ) (Jimenez-Bailon et al. 2003). Let us start by the general case when straight magnetic field lines are inclined by the angle θ to the rotation axis and supposing that magnetic field lines remain straight up to the light cylinder. Let us introduce a metric in the co-moving frame of reference Ã

ds2 = −c2

!

ω 2 R2 sin2 θ 1− dt2 + dR2 c2

(2.105)

derived from the Minkowskian metric (ds2 = ηαβ dxα dxβ , with ηαβ ≡ diag{−1, +1, +1, +1} and xα ≡ (ct; x, y, z)) after the following variable transformation: x = R sin θcosωt, y = R sin θ sin ωt and z = Rcosθ. One may easily check that defining Ω = ω sin θ one reduces 2.105 to:

2.3 On efficiency of particle acceleration by rotating magnetospheres in AGN. Application to Blazars. Ã 2

ds = −c

2

40

!

Ω2 R2 1− 2 dt2 + dR2 , c

(2.106)

which formally coincides with the metric in the co-moving frame of reference in the case of the inclination angle θ = 90◦ (Machabeli & Rogava 1994). This means that all kinematic expressions valid for the equatorial plane are also valid for the inclined trajectories of electrons if instead of ω we use Ω. For the equation of motion, from 2.106, we get: d ∂L ∂L , α = dτ ∂ x¯˙ ∂ x¯α

(2.107)

1 d¯ xα d¯ xβ L = − mc¯ gαβ 2 dλ dλ

(2.108)

(

g¯αβ

Ã

!

)

Ω2 R2 ≡ diag − 1 − 2 , +1 , c

(2.109)

d¯ xα . (2.110) dλ According to this approach all physical quantities are functions of a parameter λ. Then from Eqs. (2.107) for α = 0 using the four velocity identity g¯αβ (d¯ xα /dλ)(d¯ xβ /dλ) = −1 one can express the Lorentz factor of electrons as a function of the radial distance (Machabeli & Rogava 1994): α x¯α ≡ (ct; R), x¯˙ ≡

γ=√

³

1

f 1− m

Ω2 R 2 c2

´.

(2.111)

Combining Eqs. (2.103) and (2.111), for the acceleration time scale we have (Rieger & Mannheim 2000): q

tacc =

c 1−

r

Ω2 R 2 c2

³

f 1− 2Ω2 R 1 − m

Ω2 R2 c2

´.

(2.112)

f = (1 − Ω2 R02 /c2 − v02 /c2 )/(1 − Ω2 R02 /c2 )2 . R0 and v0 are initial position and where m initial radial velocity of the particle, respectively. For evaluating the efficiency of each of the limiting mechanisms one needs the expression for the maximum Lorentz factor attainable by an electron subject to ICS. Initially the electrons accelerate, this process lasts until the energy gain is balanced by the energy losses due to ICS. The latter happens when tacc ' tcool . From Eqs. (2.104), (2.111), and


41

(2.112) it follows that nearby the light cylinder, when ΩR/c → 1 (and γ → ∞) both time scales tend to zero, which means that ICS starts working efficiently in the vicinity of the light cylinder, hampering the subsequent acceleration of the electron. Using the corresponding condition for time scales: tcool ≈ tacc

(2.113)

and combining Eqs. (2.104,2.112,2.113) one may easily derive an approximate expression for the maximum Lorentz factor: ICS γmax

≈ 10

14

√

"

6Ω f m Urad (RL )

#2

,

(2.114)

where RL ≈ rL / sin θ. In order to derive the analogous estimation of the maximum Lorentz factor limited by the BBW mechanisms we can use the method developed by Gangadhara (1996). For this purpose one has to write down the force responsible for the BBW. The particle’s momentum in the laboratory frame of reference is: P = mvr er + mvz ez + mω × R,

(2.115)

where vr = sin θ(dR/dt) and vz = dz/dt. Differentiating this equation and taking into account dei /dt = ω × ei , i = r, z one may write down: dP = dt ³

´

Ã

dP dt

!

!

Ã

d(mr) − mω rer + ω mvr + eϕ , dt n 2

(2.116)

where dP denotes the time derivative of the momentum defined in the non-inertial dt n frame of reference. Finally for the inertial forces (taking into consideration the relationship: r = R sin θ) we have: !

Ã

Fin

d(mR) eϕ . = mω 2 R sin θer − Ω mv + dt

(2.117)

Generally the force responsible for BBW is the projection of Fin on the direction perpendicular to the magnetic field line: h

2 2 cos2 θ + Finϕ F⊥ = Finr

i1/2

,

(2.118)


42

where Finr = m0 γω 2 r sin θ,

(2.119)

Ã

Finϕ

!

d(γR) . = −m0 Ω γv + dt

(2.120)

Note that (2.118), in the limit θ = 900 , reduces to the expression (Rieger & Mannheim 2000): Ã

!

dR dγ F⊥ = m0 ω 2γ +R . dt dt

(2.121)

coinciding with the Coriolis force acting on the particle. While the electron moves along the magnetic field line, apart from the inertial forces, it also experiences the Lorentz force: q FL = vrel × B (2.122) c where vrel is the velocity of the electron relative to the magnetic field line and m0 and q are the electron’s rest mass and charge, respectively. By virtue of the Lorentz force, the electron gyrates around the magnetic field line and, during the course of motion, the Lorentz force changes from parallel to antiparallel to F⊥ . If the Lorentz force is greater than F⊥ the particle moves along the magnetic field line. The situation changes when F⊥ exceeds FL , in this case F⊥ forces the particle to come off the magnetic field line. Therefore, the “assumption that the particle sticks to the filed line is no longer valid and the BBW, as a limiting mechanism for the centrifugal acceleration becomes important. From the condition FL ≈ F⊥ (Rieger & Mannheim 2000) one can then derive an estimate of the maximum Lorentz factor limited by the BBW: BBW γmax ≈ A1 + [A2 + (A22 − A61 )1/2 ]1/3 + [A2 − (A22 − A61 )1/2 ]1/3 ,

(2.123)

where A1 = − A2 =

ctg 2 θ , f1/2 12m

q 2 le × LEdd + A31 , 2 f1/2 5 4m0 m c

which in the limit θ = 900 reduces to the form (Rieger & Mannheim 2000):

(2.124) (2.125)


43

8

6

x 10

5

γ

mas

4

3

2

1

0 3

4

5

6 7 L / 1040[erg s−1]

8

9

10 −6

x 10

ICS BBW Figure 2.6: Graphs of dependence of maximum Lorentz factors on L. γmax and γmax are presented

by the solid line and the dashed line respectively. Set of parameters is following: τ = 1, ωr0 /c = 0.4, v0 /c = 0.6 and ω = 3 × 10−5 s−1 .

BBW γmax

1 ≈ 1/6 f m

Ã

B(rL )q 2mωc

!2/3

,

(2.126)

where B(r) is the equipartition magnetic field strength (which means that magnetic field and radiation energy densities are equal) at the radius r and is given by the formula (Rieger & Mannheim 2000): B 2 (r) =

2.3.2

2le × LEdd . r2 c

(2.127)

Case of the magnetic field lines fixed in the equatorial plane

BBW (L) We start by considering the case when θ = 90◦ . From comparison of graphs of γmax 40 ICS and γmax (L) (see Fig. 2.6) one can see that for AGN with luminosities L > 8 × 10 erg/s the maximum Lorentz factor attainable by electrons via ICS (solid line) is less than the corresponding Lorentz factor attainable via BBW (dashed line), which means that


44

4

3.5

3

2.5

lg(t

)

cool

2

lg(t

)

acc

1.5

1

0.2

0.4

0.6

0.8

1

γ

1.2

1.4

1.6

1.8

2 5

x 10

Figure 2.7: Graphs of dependence of logarithms of time scales on the Lorentz factor. Set of parameters is the same as on Fig. 2.6, except l = 5 × 10−4 .

the latter one is unimportant and it becomes significant only for objects with L < 8 × 1040 erg/s. From Fig. 2.6 it is also clear that, for L > 8 × 1040 erg/s, increasing the luminosity power, the maximum attainable Lorentz factors decrease. From Eq. (2.114) ICS 2 we also can see that the maximum Lorentz factors scale as γmax ∼ 1/Urad , as a result of the dependence of tcool on Urad . Quite different is the conclusion made by Rieger and Mannheim (Rieger & Mannheim 2000). These authors estimated the upper limit of the maximum Lorentz factor attainable under the BBW for le < 10−3 (when according to Rieger and Mannheim (2000) ICS is unimportant) as γmax ∼ 1000 and even for the highest possible magnetic field strength - B(rL ) = 100G - as γmax ∼ 2500. The correct value of the corresponding Lorentz factor is γmax ∼ 4 × 108 . This circumstance changes the result of Rieger and Mannheim, because this value of γmax is, by many orders of magnitude, larger than the corresponding limit of γmax given by ICS (equal, in this case, to ∼ 3.6 × 103 ). This means that the latter is dominant and the lower limit of le at which the ICS still works must be shifted from 10−3 to 8×10−6 (see Fig. 2.6).The corresponding luminosity shifts from 1043 erg/s to 8 × 1040 erg/s. In particular, if one considers the case L = 2 × 1042 erg/s (when following the work (Rieger & Mannheim 2000) BBW must be important), using Eqs. (2.112,2.104,2.121,2.122) one can easily show that, during


45

−8 −8.5

lg(FL)

−9 −9.5 −10 lg(F⊥)

−10.5 −11 −11.5 −12 −12.5 −13

1

2

3

4

5

γ

6

7

8

9

10 4

x 10

Figure 2.8: Graphs of dependence of logarithms of the Lorents FL and Coriolis F⊥ forces on Lorentz factors. Set of parameters is the same as on Fig. 2.7.

the course of motion, the Lorentz force always exceeds Coriolis force and the maximum Lorentz factor is limited only by ICS (see Fig. 2.7, Fig. 2.8). Moreover, from Fig. 2.7 it is clear that, at the beginning, the electron acceleration time scale is smaller than the cooling time scale, which means that the electron will accelerate, but, as soon as the Lorentz factor of the particle becomes of the order of 105 , the electron energy gain is counter-balanced by the losses via ICS and further acceleration is impossible. Inverse Compton scattering of 0.1keV energy photons produces gamma rays with a maximum energy (Rieger & Mannheim 2000) 2 γmax T eV (2.128) 1010 Considering, as an example, a luminosity power of 2 × 1042 erg/s, we have seen that the maximum Lorentz factor is 105 , that gives a maximum gamma ray energy of ∼ 1T eV . Our approach is based on an assumption that electrons co-rotate with the spinning AGN, which is valid only inside the Alfvén zone, because in this region the magnetic field is very strong and, as a result, the flow follows it. On the other hand, particles reach their upper energy limit in a region very close to the light cylinder, which means that our approach is valid if the following condition is satisfied: (rL − rA )/rL ¿ 1 (where rA is

²ph ≈


46

the Alfvén radius). rA can be estimated by the expression: B(r)2 /(8π) ≈ m0 nc2 γ(r), where m0 and n are respectively the relativistic electron’s rest mass and the density. If one considers a range of nJ /nm = [10−4 ; 104 ], where nJ is the jet bulk density and nm the medium density (suppose that nm ≈ 1cm−3 ), by using Eqs.(2.111) and (2.127) one can show that for L = 2 × 1042 erg/s, (rL − rA )/rL ∼ [10−11 ; 10−3 ], which means that co-rotation is valid almost for the whole course of motion.

2.3.3

Case of the inclined magnetic field lines

In this section we examine how the situation changes by varying the inclination of magnetic field lines. For simplicity we consider again the case of straight magnetic field lines. The critical luminosity power (the luminosity power at which both mechanisms give the same maximum factors) depending on the inclination angle. From the ³ Lorentz ´ ³ changes ´ ICS BBW graphs of log γmax and log γmax [see Eqs. 2.114, 2.123 ] one can see that for a ³

´

ICS luminosity less than 8 × 1040 erg/s (see Fig. 2.6) the surface of log γmax is over the corresponding surface of the BBW, independently on the inclination angle (see Fig. 2.9). Thus, under a luminosity of 1041 erg/s, the maximum Lorentz factor attainable by the electron is limited only by the BBW. For higher luminosity AGNs the BBW also may be important, but for relatively smaller angles (see Fig. 2.9). For TeV Blazars, like Mrk421 and Mrk501 we have the ultra-high energy γ-ray emission in the range [1 − 20]T eV . The luminosity power of Mrk421 and Mrk501 is estimated as L ∼ 1044 erg/s (Bicknell et al. 2001). If we consider magnetic field lines inclined by the angle θ, then using Eqs. (2.114,2.128) one may express the mentioned angle in terms of ²ph :

µ

¶

²ph −1/4 4320c3 , θ ≈ × (2.129) ωL 1018 In Fig. 2.10 we show the graph of θ(²ph ). The set of parameters is following: ω = 3 × 10−5 s−1 and L ∼ 1044 erg/s. As it is seen from this figure, while considering magnetic field line’s inclination in the range ∼ [0.6◦ −1.2◦ ], one can explain the observed ultra-high energies from Mrk421 and Mrk501. As one can also see, by decreasing the inclination angle the corresponding energy of the photon increases. For understanding the behaviour shown in Fig. 2.10, one has to substitute the following expressions RL ≈ rL / sin θ and Ω = ω sin θ into Eqs. (2.104) and (2.112), then one gets tcool /tacc ∼ Ω−1 , which means that for smaller angles the ratio of time scales is higher. As a result, the particle will have more time for acceleration before the energy saturation and the energy gain will be o


47

14 13 12 11 lg(γmax)

10 9 8 7 6

5 80

0

60

20 40

40 L / 1040[erg/s]

60

20

80 0

¡

100

¢

θo

¡

¢

ICS BBW Figure 2.9: Three dimensional graphs of log γmax (θ, l) (bright surface) and log γmax (θ, l) (dark

surface). Set of parameters is the same as for Fig.1 except L and θ which vary respectively in ranges:[5 × 1040 , 7 × 1041 ]egr/s and θ ∈ [0.1o , 90o ].

ICS higher. That is why γmax ∼ 1/Ω2 [see Eq.(2.114)], thus for smaller angles the result of acceleration is more effective than for higher inclinations, where acceleration is stronger but saturation appears very soon, hampering subsequent energy gain for particles. As a ICS result of the angle dependence of γmax (θ), energies of produced photons become higher for smaller inclinations [see Fig. 2.10 and Eq. (2.129)]. The dependence on the angle is very useful also for studying different spinning rates. In our analysis, we considered a certain value of the angular velocity, but, on the basis of the above consideration one can predict what may happen for other values of ω. As we have already seen, ω and θ are equivalent, in a sense that dynamics of particles depends on Ω, but not on ω or θ independently. By applying (8) and (15) one may see ²ph ∼ (ω sin θ)−4 , which means that for smaller spinning rates the corresponding energies of produced photons are higher. This happens because for smaller angular velocities particles have more time for acceleration and their resulting energy may be higher (see the corresponding discussion about the inclination angles).


48

1.3

1.2

1.1

1

θo

0.9

0.8

0.7

0.6

0.5

2

4

6

8

10

12

14

16

18

20

εph

Figure 2.10: Graph of dependence of θ(²ph ) on ²ph . Set of parameters is following: ω = 3 × 10−5 s−1 and L ∼ 1044 erg/s.

2.3.4

Conclusion

In the present work our main attention was focused on the ultra-high emission from AGN, the role of the centrifugal acceleration in this process and the efficiency of different mechanisms limiting this acceleration. For this purpose we re-examined and generalized the considered problem (Rieger & Mannheim 2000), where it has been argued that together with the ICS, also BBW is an efficient limiting mechanism for the electron acceleration. They argued that the BBW limits γmax to a few thousands, which is much less than ∼ 105 (the minimum value of the Lorentz factor needed to produce TeV photons). Revisiting the same problem we showed that for AGN’s considered by Rieger and Mannheim (2000) as the acceleration limiting mechanism only ICS is important and consequently the upper limit of the maximum Lorentz factor may be much more than several thousands. In particular, we have shown that when the magnetic field lines are straight and inclined to the axis of rotation by the angle θ = 90◦ , for AGN with luminosity L > 8 × 1040 erg/s, ICS is dominant and BBW becomes significant only for AGN with lower luminosities L < 8 × 1040 erg/s. Therefore, on an example of the AGN with the luminosity power 2 × 1042 erg/s, when as it was shown only ICS is important (for θ = 90◦ ), we calculated the approximate value of the maximum Lorentz factor 105 and argued that by virtue of


49

the ICS electrons with such Lorentz factors can produce 1T eV energy γ-rays. We also have considered the dependence of validity of the mentioned mechanisms on the inclination angle of the magnetic field lines with respect to the rotation axis and have shown that also in this case, for the wide range of possible inclination angles, ICS is the most important mechanism for limiting the Lorentz factor. The latter may reach values up to ∼ 108 . BBW becomes dominant mostly for relatively lower luminosities with relatively higher angles, or with higher luminosities and smaller angles. Exception is a case when the luminosity is less than 8 × 1040 erg/s, when independently on the inclination angle the BBW is dominant. It is easy to estimate by Eq. 2.123 that the BBW minimum value of γmax also is very high ∼ 107 . Letting magnetic field lines to have inclination angles in the range ∼ [0.6◦ − 1.2◦ ] we have shown that the corresponding energies of photons due to ICS, are of the order of [1 − 20]T eV which is sufficient to account for the emission of TeV blazars. Finally, we can summarize our results by following: for the wide range of AGN from two considered mechanisms ICS still remains the dominant factor for limiting the maximum Lorentz factor of accelerated particles, letting them to produce very high energy photons [1 − 20]T eV . For low luminosity power AGN, at the other hand, BBW may be significant but it also may provide with high maximum Lorentz factors of accelerated electrons. An important restriction of our approach is that we studied only straight magnetic field lines, whereas in realistic astrophysical situations the magnetic field lines are curved. Therefore, it is interesting to generalize our approach and specify how the considered problem changes, when the curvature of the magnetic field lines is taken into account. It might be especially important if we consider particle dynamics on a longer time span, over large length-scales, when the curvature of the field lines can not be neglected. Mathematical formalism for such study does exist (Rogava et al. 2003) and it can be applied to the problem under consideration. The next limitation is related with the fact that the magnetic field is not influenced by a particle motion up to the region in the immediate vicinity of the light cylinder, where the magnetic field is affected by the motion of the plasma. Besides our model up to now was based on the study of the dynamics of a single particle, neglecting complex processes in real relativistic plasmas. It is reasonable and necessary to generalize our approach and examine a more realistic model.

Chapter 3 Instabilities in astrophysical outflows In this chapter we present instabilities induced by a velocity shear. Recently it was fully realized that collective phenomena in shear flows are characterized by, so called ”nonmodal” processes, related with ”non-normality” (Trefethen et al. 1993) (non-selfadjointness) of linear dynamics of perturbations in some shear flows. According to this approach, instead of considering the normal modes and eliminating the time dependence, one succeeds in finding a transformation annihilating the spatial dependence and solving the linear analysis problem with respect to time. In the astrophysical context the original method of nonmodal approach, stemming from the classic paper (Lord Kelvin 1887) by Lord Kelvin, has been revived by Goldreich and Lynnden-Bell (1965). The presence of velocity shear can strongly modify the plasma response to perturbations. Expected changes include: strong transient growth, new modes of energy transfer, the conversion of the point spectrum to a continuous spectrum, and even a possible disappearance of eigenmodes. This consideration can be applied for several examples from astrophysical plasmas. Chagelishvili et al. (1994), considering a compressible Couette flow demonstrated a new mechanism of energy transfer from the background flow to the sound-type perturbations induced by the velocity shear flow. In the context of MHD waves, the problem has been studied by Chagelishvili et al. (1996) and it has been argued that for two-dimensional waves in unbounded, parallel hydromagnetic flow with uniform velocity shear the Alfvén, fast and slow magnetosonic waves are coupled. A problem considered from the same point of view was studied by Rogava and Mahajan (1997), where it was shown that the shear flow can induce a coupling between the sound waves and the internal gravity waves. Considering the energy transfer problem in the escaping radiation from pulsars (Mahajan et al. 1997), it has been demonstrated that the shear flow efficiently converts the Langmuir waves into propagating electromagnetic waves. In all previous works the velocity

3.1 Amplification of MHD waves in astrophysical flows.

51

configuration was very simple. It is interesting to consider more complicate structure of the shear flow, studying the nonmodal instabilities. In the first section we demonstrate a linear analysis of shear driven non modal instabilities in helical flows, studying transformations of MHD waves into each other and possible energy transfer from the mean flow into excited waves. Flows having velocity shears are common in many astrophysical situations and their analysis might be a key for understanding phenomena in related objects. This work has been done in Georgian National Astrophysical Observatory. The second part, finished in Università degli Studi di Torino, is dedicated to the consideration of a linear analysis of the Kelvin-Helmholtz instability in relativistic magnetized plane parallel flows. The KHI can be used to predict the onset of instability and transition to turbulent flow (Matsumoto & Hoshino 2004), can be applied in a wide range of problems: in Solar physics, planetary physics and Geophysics, for example in order to understand: processes in the solar photosphere, surrounded by strong downflows, which come into swirling motion (Kolesnikov et al. 2004), the structure of the Saturn atmosphere (Pu & Kivelson 1984) and a physical nature of a behaviour of the global vortex configuration in the Earth’s atmosphere (Hasegawa et al. 2006). A special subclass of problems where the Kelvin-Helmholtz instability is supposed to be fundamental is problems related to astrophysical jets. It is strongly believed that a role of the KHI in a formation process of a complicate morphology of astrophysical jets can be very important. By Massaglia et al. (1997) the problem of the non-linear Kelvin-Helmholtz instability has been discussed for explaining the origin of the emission knots in HerbigHaro jets. Unlike the young stellar objects, the extragalactic jets are relativistic and in order to implement the KHI for them, it is necessary to take the relativistic effects into account, generalize the approach and study the KHI. In the second section we consider the linear analysis of the Kelvin-Helmholtz instability for planar relativistic magnetized flows and study corresponding regimes versus physical parameters. For understanding a non linear behaviour of the problem, in the third section of the chapter, we study the KHI for the shearing layer case in a magnetized and non magnetized flows, investigating specific features of non linearity: creation, multiplication and merging processes of vortices.

3.1

Amplification of MHD waves in astrophysical flows.

Lately it became quite commonly presumed and hoped that swirling three-dimensional motion occurs in different kinds of space plasma flows. The presumption is grounded


52

on the mounting evidence from different branches of observational astronomy, while the hope is related to the long-standing aspiration of theoreticians to find in astrophysical flows natural laboratories for testing of their expectations and ideas. One of the most remarkable observational indications came from Solar and Heliospheric Observatory—Coronal Diagnostic Spectrometer (SOHO-CDS) data that led to the identification (Pike & Mason 1998) of macrospicules having both rotational and jet-like features. Pike and Mason presented evidence for the existence of blue- and red-shifted emission on either side of an axis of a macrospicule stretching above the limb from a footpoint region on the disk. They interpreted these observations as indicating the presence of a rotation within these tall squalls of hot and magnetized, swirling plasma flows — christened as solar tornados — observed both on the limb and the disk of the Sun. This discovery allows to argue (for earlier arguments in the same vein see (Pneuman & Orrall 1986; Shibata & Uchida 1986)) that the wide range of dynamic events in the solar atmosphere, including micro flares, jets, plumes, surges, spicules and macrospicules, may exhibit complicated, kinematically nontrivial, three-dimensional sheared plasma motions. Generally speaking it looks quite credible that rotation plays a major role in the dynamics of chromospheric and transition region features. It seems plausible to admit that the presence of these complicated patterns of plasma motion deeply influence the dynamics of MHD waves in the solar atmosphere, contribute to the coronal heating, and to the acceleration of the solar wind. Another message of observational evidence for swirling flows recently came from the totally different class of astronomical objects: Herbig-Haro (HH) jets. Two different observations of young stellar objects in HH 212 (Davis et al. 2000) and DG Tau (Bacciotti et al. 2002), contain serious indications for the presence of rotation in these jets. These results are in accordance with predictions of the popular magnetocentrifugal jet launching model (Blandford & Payne 1982). The possible impact of these complicated, helical motions on physical processes occurring within these flows is yet to be understood. The growing evidence in favour of astrophysical helical flows poses a twofold challenge. At the one hand it naturally solicits for the further theoretical study of general aspects — generation, equilibrium, stability and internal dynamics — of helical flows in the framework of plasma astrophysics. At the other hand, after acquiring better understanding of basic physics, it suggests to build concrete prototype models, closely adjusted to specific examples of observed swirling flows, which might appear to be useful for shedding some light on the puzzling observational appearance of related astronomical objects. The basic theory of shear flows tells us that dynamics of waves and vortices, sustained by these flows, are substantially affected by the differential character of motion. On the basic mathematical level these phenomena, somewhat misleadingly christened as ”nonmodal processes”, are related to the non-self-adjointness of linear dynamics of perturbations in


53

SF (Trefethen et al. 1993). The variety of these processes, that could be labeled as ”shear-induced nonmodal processes” (SINP), is quite well-understood for simple, plane parallel flows and thoroughly described in the recent literature (for the latest review see, e.g., (Bodo et al. 2001) and references therein). It is known that SINP lead to the generation of new modes of plasma collective behaviour, to new forms of flow-wave, wavewave and vortex-wave interactions provoked and fueled by ‘parent’ shear flows. These phenomena, originally disclosed in hydrodynamics, take place in various kinds of plasmas and might have a number of astrophysical applications, including pulsar magnetospheric plasmas, solar atmospheric phenomena and galactic gaseous disk dynamics. The interest towards possible applications was strengthened by the recent numerical evidence that real-space appearance of SINP is easily recognizable and robust even in the presence of tangible dissipation (Bodo et al. 2001). Real astrophysical shear flows are almost always involved in motions with complex kinematics/geometry and kinematic complexity is known to bring an additional variety to SINP. When geometry and kinematics of flows are complex a whole bundle of new effects arises, like ”echoing” (repetitive) transient pulsations and different kinds of shearinduced (including parametric) instabilities (Mahajan & Rogava 1999). The study of kinematically complex SF was initiated in hydrodynamics (Lagnado et al. 1984; Craik & Criminale 1986; Criminale & Drazin 1990) and this approach is still largely unknown for the plasma and astrophysics community. Therefore, it is an intriguing challenge and a task of a big practical importance to study these processes in MHD flows of nontrivial geometry and kinematics. The first step in this direction was recently made ((Rogava et al. 2003), hereafter referred as Paper I). In this study the simplifying assumption of incompressibility was adopted, which cuts off modes of acoustic origin (slow magnetosonic waves, SMW, and fast magnetosonic waves FMW) and allows to concentrate on the investigation of the dynamics of Alfvén waves (AW). The subject of the interaction between flows and AW is interesting in a number of astrophysical applications (Balbus & Hawley 1991; Tagger et al. 1992; Tagger & Pellat 1999; Ryutova et al. 2001; Varnièr & Tagger 2002). In Paper I it was found that helical shear flows are efficient amplifiers of AW. In purely ejectional flows (i.e., when no rotation is present) AW are amplified transiently via algebraic, shearinduced instability. In a swirling flow AW are exponentially unstable: depending on the mode of differential motion both usual and parametric instabilities appear. These results were discussed in the context of their possible (observable) manifestations. It was argued that they might account for the generation of the large-amplitude Alfvén waves – e.g., within ‘tornado-like’ patterns existing in the solar atmosphere. It was suggested that they could lead to the efficient self-heating of flows: the kinetic energy of the flow, being extracted by amplified Alfvén waves, returns back to the flow in the form of


54

the thermal energy because AW are eventually damped via magnetic diffusion. Finally, it was argued that these instabilities might serve as an initial (linear and nonmodal) phase in the ultimate subcritical transition to MHD Alfvénic turbulence in various kinds of astrophysical shear flows. It is known that the incompressibility condition, used in the Paper I, is quite restrictive. Often the usage of this condition introduces its own imprints on the dynamics of perturbations and it is not trivial to distinguish the genuine SINP from the phenomenological effect imposed by the incompressibility approximation. Therefore it is quite important to consider MHD waves in helical flows without the usage of the incompressibility condition and to study the SINP for the full spectra of MHD waves containing together with the AW also the SMW and the FMW. This task is undertaken in this paper. We find that the range of processes, sustained by helical flows is extremely rich. It encompasses different kinds of wave transformations, wave beatings. Most importantly, usual and parametric shear instabilities, which has been found in the Paper I, are found to appear for the full spectra of MHD waves containing AW, SMW and FMW. We see that flows of such a high degree of complexity efficiently intertwine all three MHD wave modes and efficiently exchange energy with them. The relevance of these results to the physics of helical flows and to the understanding of possible observational appearance of related astronomical objects are pointed out and critically discussed.

3.1.1

Theory

Our aim is to study linear collective MHD modes in helical flows. For this purpose we need to write equations of the ideal linearized MHD for the evolution of perturbations within the flow. In Paper I we studied only incompressible perturbations, while now we consider fully compressible case, so our starting equations are [Dt ≡∂t + (V·∇)]: Dt ρ + ρ(∇·V) = 0,

(3.1)

1 B Dt V = − ∇P − ×(∇×B), ρ 4πρ

(3.2)

Dt B = (B·∇)V − B(∇·V),

(3.3)

∇·B = 0.

(3.4)

Our equilibrium model, used in Paper I, assumes a homogeneous MHD plasma (ρ0 = const), embedded in a homogeneous, vertical magnetic field (B0 ≡[0, 0, B0 = const]). We consider instantaneous values of all physical variables as sums of their mean (equilibrium)


55

and perturbational components: B≡B0 +B0 , ρ≡ρ0 +ρ0 , etc. Applying this decomposition we convert Eqs.(3.1-3.4) into the following set for perturbation variables [Dt ≡∂t +(U0 ·∇)]: Dt d + ∇·u = 0,

(3.5)

Dt u + (u·∇)U0 = −Cs2 ∇d + CA2 [∂z b − ∇bz ],

(3.6)

Dt b = (b·∇)U0 + ∂z u + ez (∇·u),

(3.7)

∇·b = 0,

(3.8)

with d≡ρ0 /ρ0 and b≡B0 /B0 . Note also that for compressible perturbations p0 = Cs2 ρ0 with Cs the homogeneous speed of sound. If we introduce a new vector quantity h≡b − ez d then the above system can effectively be reduced to the following set of second-order equations: Dt2 h − [(h·∇)U0 ·∇]U0 − (Cs2 + CA2 )∆h = CA2 [∂z2 h − ∇(∂z hz ) − ez ∂z (∇·h)],

(3.9)

which describes evolution of all three MHD modes — SMW, AW and FMW — influenced by the presence of the equilibrium flow with an arbitrary U0 . In paper I we worked with the velocity field specified by U(r)≡[0, rΩ(r), U (r)],

(3.10)

with Ω(r) = A/rn , where r = (x2 + y 2 )1/2 is a distance from the rotation axis, while A and n are some constants. In particular, n = 0 and A = Ω0 for rigidly rotating plasmas, while n = 3/2 and A = (GM )1/2 , for the Keplerian rotation. In the framework of the method of the nonmodal approach, for studying the nonexponentially evolving disturbances we consider a background velocity field of the most general form: U(x, y, z) = Ux (x, y, z)ex + Uy (x, y, z)ey + Uz (x, y, z)ez ,

(3.11)

which we expand in the neighborhood of the point: A(x0 , y0 , z0 ), (|x − x0 |/|x0 | ¿ 1, etc): Ui (A) = Ui,x (A)(x − x0 ) + Ui,y (A)(y − y0 ) + Ui,z (A)(z − z0 ), so one can introduce the shear matrix:

(3.12)








56



Ux,x Ux,y Ux,z a1 a2 a3    S=  Uy,x Uy,y Uy,z  ≡  b1 b2 b3  Uz,x Uz,y Uz,z c1 c2 c3

(3.13)

On has to note that this matrix should be traceless due to the continuity equation for a homogeneous equilibrium density flow: ∇ · U = 0, which means that a1 + b2 + c3 = 0. Then one may easily show that Eq. (3.11) can be reduced by following: Ux (x, y, z) ' U0x (x, y, z) + a1 x + a2 y + a3 z,

(3.14)

Ux (x, y, z) ' U0y (x, y, z) + b1 x + b2 y + b3 z,

(3.15)

Ux (x, y, z) ' U0z (x, y, z) + c1 x + c2 y + c3 z.

(3.16)

Correspondingly the convective derivative, for the velocity perturbation V ≡ U + u can be presented by three components: Qux + a1 x + a2 y + a3 z,

(3.17)

Quy + b1 x + b2 y + b3 z,

(3.18)

Quz + c1 x + c2 y + c3 z,

(3.19)

where Q is the following linear operator: ∂t +(U0x +a1 x+a2 y +a3 z)∂x +(U0y +b1 x+b2 y +b3 z)∂y +(U0z +c1 x+c2 y +c3 z)∂z . (3.20) So the problem is to eliminate the spatial dependence in Eq. (3.20). One may directly check out that the following ansatz: Ψ(x, y, z; t) ≡ Ψ(kx (t), ky (t), kz (t); t)ei(Ξ1 −Ξ2 ) , Ξ1 (x, y, z; t) ≡ kx (t)x + ky (t)y + kz (t)z, Z

Ξ2 (kx (t), ky (t), kz (t); t) ≡ U0x

Z

kx (t)dt + U0y

(3.21) (3.22)

Z

ky (t)dt + U0z

kz (t)dt

(3.23)

will satisfy the required condition, if k(t) satisfies the following one: ∂t k + S T · k = 0.

(3.24)

In our case one can show that the linear shear matrix for this has the following form (Rogava et al. 2003):




57



σ A1 0  S=  A2 −σ 0  , C1 C2 0

(3.25)

where we denote the matrix elements in a more convenient form. In this case Eq. (3.24) transcribes to the following set of equations1 : kx(1) + σkx + A2 ky + C1 kz = 0,

(3.26)

ky(1) + A1 kx − σky + C2 kz = 0,

(3.27)

while kz = const. These equations imply that , while kx (t) and ky (t) may have algebraic, exponential or periodic time dependence. For swirling flows the differential rotation parameter n plays decisive role in determining the evolution scenario for the wave number vector |k(t)|: when n < 1 (including the rigid rotation case) the time evolution of the |k(t)| is periodic, while when n > 1 (including the Keplerian rotation regime) |k(t)| evolves exponentially. Note that the following nonlinear combination of kx (t) and ky (t): ∆≡kx ky(1) −ky kx(1) +kz (C1 ky −C2 kx ) = A2 Ky2 −A1 kx2 +2σkx ky +2kz (C1 ky −C2 kx ) = const. (3.28) is a conserved quantity. In the case of pure, two-dimensional rotation (kz = C1 = C2 = 0) it reduces to the constant ∆r ≡kx ky(1) − ky kx(1) (Mahajan & Rogava 1999), while for the case of a pure outflow (A1 = A2 = σ = 0) it reduces to the conservation of the quantity ∆e ≡C1 ky − C2 kx (Rogava et al. 2000). Therefore, we see that the wave-number invariant ∆ in the case of the helical flow is the sum of the ∆r and the ∆e functions. Using the same method as in Paper I (Mahajan & Rogava 1999) we can effectively convert the system Eqs. (3.5-3.8) to the set of first oder ordinary differential equations [H≡ih, D≡id]:

1

D(1) = k·u,

(3.29)

u(1) + S·u = −(Cs2 + CA2 )kD + CA2 [kz H − kHz + ez kz D],

(3.30)

H(1) = S·H − kz u,

(3.31)

k·H = −kz D,

(3.32)

Hereafter F (n) will denote n-th order time derivative of a function F .


58

Note that Eqs.(3.29-3.32) contain two first order ordinary differential equations with time-dependent coefficients. Time-dependence of these coefficients is completely determined by the temporal evolution of the wave number vector k(t) and, therefore, governed by Eq. (3.24). We can reduce this set to the set of second order equations for the components of the vector H, which is the nonmodal form of the Eq.3.9. It can be written in the following (vector) form: H(2) + Cs2 k(k·H) − S 2 ·H + CA2 [(k − kz ez )(k·H) + (kz2 H − kkz Hz )] = 0.

(3.33)

Note that in the helical flow, specified by the shear matrix 3.25, the square of the matrix is equal to (Rogava et al. 2003): 



Γ2 0 0  2 ||S || =  0 −Γ2 0  . ε1 ε2 0

(3.34)

where we use notation: Γ≡(σ 2 + A1 A2 )1/2 , ε1 ≡A2 C2 + σC1 and ε2 ≡A1 C1 − σC2 . Note that ||S 3 || = Γ2 ||S||. Splitting both H and k vectors into their longitudinal and transverse components — H≡(H⊥ , Hz ), k≡(k⊥ , kz ) — we can write more explicit form of (3.9) revealing the nature of coupling between the MHD modes, imposed by the presence of the velocity shear: Hz(2) + Cs2 kz2 Hz + Cs2 kz (k⊥ ·H⊥ ) − (ε1 Hx + ε2 Hy ) = 0, (2)

H⊥ + [CA2 kz2 − Γ2 ]H⊥ + (Cs2 + CA2 )k⊥ (k⊥ ·H⊥ ) + Cs2 k⊥ kz Hz = 0.

(3.35) (3.36)

It is easy to see that in the absence of the shear flow Eqs. (3.35-3.36) give standard expressions for the dispersion properties of all these modes. The much simpler version of this system was analyzed before for plane-parallel SF of standard MHD plasmas (Chagelishvili et al. 1996). In the astrophysical context the similar kind of coupled ODE’s were studied for MHD waves in the solar wind plasmas (Poedts et al. 1998), in galactic gaseous discs (Rogava et al. 1999) and in cylindrical rotationless flux tubes (Rogava et al. 2000). The actual ‘route’ of the wave number vector k(t) temporal ”drift” (caused, in its turn, by the differential character of the plasma motion) plays the crucial role in the time evolution of physical perturbations. In parallel flows k⊥ (t) exhibits linear time dependence. However, even in this relatively simple case we have the whole set of SINP (Rogava et al. 2000): waves exchange energy with the flow and the velocity shear makes waves coupled with one another, making possible their reciprocal transformations.


59

Helical flows are expected to exhibit evolutionary regimes characteristic to both purely ejectional and purely rotational flow patterns. However, we expect that shear-induced wave transformations (SIT) are more distinctly exhibited by parallel flows without rotation in the transverse plane, because these flows are stable and SIT is the only major kind of SINP occurring in the flow. In helical flows, presumably featuring different sorts of shear-induced instabilities, SIT could be less- or not-pronounced. It should be noted that the parameters ε1 and ε2 exist only when both rotation and outflow are present and they are nonzero only when the forces that determine the kinematic portrait of the flow are non-conservative (Craik & Criminale 1986). Thus their role in the dynamics of MHD waves sustained by swirling flows can be quite significant.

3.1.2

Discussion

The evolution of linear MHD modes in kinematically complex helical flows might be quite complicated. Before considering any special and/or particular cases we have to recall the following three levels of complexity arising within this problem. First, the medium itself is complex enough, because it sustains three different linear modes of oscillations: SMW, AW, and FMW. Although in the absence of shear these modes are decoupled, still an arbitrary perturbation excited within this medium is normally a superposition of these three normal modes (Sturrock 1994). Second, in the presence of a simple, plane-parallel flow with a linear velocity profile: (a) one mode - FMW - becomes able to draw energy out of the background flow; (b) depending on the plasma-β different waves become coupled and are able to transform into each other; (c) the system starts exhibiting beat wave phenomena. The resulting picture of the MHD wave dynamics becomes considerably complex (Rogava et al. 2000). Third, when incompressibility condition is used and the presence of the rotation and stretching of flow lines in the transverse cross section of the flow is ”allowed” one finds that new kinds of shear instabilities emerge (Rogava et al. 2003). This is expected to happen also for compressible (acoustic) wave modes, because even in the simplest twodimensional, nonmagnetized flow with kinematic complexity (Mahajan & Rogava 1999) usual and parametric instabilities do appear together with asymptotically persistent, ”echoing” solutions. Therefore it seems reasonable to suppose that helical MHD flows, possessing all these levels (or degrees) of complexity plus the complexity of the specifically helical nature, related with the existence of the ε1 and ε2 ”helical” parameters, must exhibit highly complicated collective processes, dominated by different regimes of shear-induced variability and instability.


60

For numerical purposes it is more suitable to deal with the dimensionless version of the (3.29-3.32) set: %(1) = Kx (τ )vx + Ky (τ )vy + vz ,

(3.37)

vx(1) + Σvx + a1 vy = −Kx (τ )²2 % + bx − Kx (τ )bz ,

(3.38)

vy(1) + a2 vx − Σvy = −Ky (τ )²2 % + by − Ky (τ )bz ,

(3.39)

vz(1) + R1 vx + R2 vy = −²2 %,

(3.40)

b(1) x = Σbx + a1 by − vx ,

(3.41)

b(1) y = a2 bx − Σby − vy ,

(3.42)

Kx (τ )bx + Ky (τ )by + bz = 0,

(3.43)

derived from Eqs.(3.29-3.32) with the usage of dimensionless notation: τ ≡CA kz t, Kx (τ )≡kx (t)/kz , Ky (τ )≡ky (t)/kz , vi ≡ui /CA , Σ ≡ σ/CA kz , a1,2 ≡A1,2 /CA kz , R1,2 ≡(C1,2 /kz CA ), ²≡Cs /CA . It is instructive to calculate how the total energy of perturbations being, in this case, the sum of kinetic, compressional and magnetic energies: Etot ≡Ekin + Ec + Em ,

(3.44)

Ekin ≡(vx2 + vy2 + vz2 )/2,

(3.45)

Ec ≡²2 %2 /2,

(3.46)

Em ≡(b2x + b2y + b2z )/2,

(3.47)

changes in time. We can easily see that the evolutionary equation for the total energy is: E (1) = (a1 + a2 )(bx by − vx vy ) + Σ[(vy2 − vx2 ) + (b2x − b2y )] + R1 (bx bz − vx vz ) + R2 (by bz − vy vz ), (3.48) The scope of this work is not full investigation of all possible regimes of evolution. Instead, our purpose is to see whether shear instabilities, disclosed in the incompressibility limit for AW, appear also in the compressible case, for the blend of SMW, AW, and FMW. It is instructive and convenient to unfold these phenomena by following the scheme used in our previous studies of cylindrical flows ((Rogava et al. 2000) and the Paper I) and to see how two, major regimes of k(t)-dynamics (periodic and exponential) affect the course of collective phenomena in flows with different values of the plasma-β.


61

High-β plasmas In parallel flows, when ²2 À1, FMW are decoupled from the AW and SMW, while the latter two are coupled (Rogava et al. 2000) and may transform into each other. Swirling flows show more complicated behaviour. When γ 2 ≡(Γ/CA kz )2 < 0, i.e., when temporal evolution of wavenumber vectors k(t) is periodic, we encounter with two new effects: 1. The appearance of ”echoing” waves, consisting of repetitive, modulated bumps of AW and SMW, exchanging energy with the mean flow. Fig. 3.1 displays an example of such process. From the figure we can surmise that mutual AW* )SMW transformations still happen and the resulting mixture of waves ”pulsates” taking and giving energy from/to the background helical flow! This process has wellpronounced quasiperiodic nature. 2. Since coefficients in Eqs.(3.37-3.43) vary periodically it is plausible to expect that for certain cases the system must also exhibit some kind of self-parametric instability. The term ”self-parametric” seems appropriate, because it is the consequence of the velocity shear inherent to the system and forcing on itself (Argentina et al. 1999). This kind of instability was first discovered for plain acoustic waves in 2-D flow patterns of neutral flows (Mahajan & Rogava 1999). Similar sort of instability was found for Alfvén waves in the Paper I. Numerical examination of the Eqs.(3.37-3.43) allowed us to find similar instabilities for the compressible case. One example is given on the Fig. 3.2. The values of all parameters (except R1 ) here are the same as on the Fig. 3.1, but R1 = 0.4. The figure shows that in this case hydromagnetic oscillations amplify exponentially, extracting energy from the flow. Note that for the existence of this kind of parametric instability it is not necessary to have any periodicity in the background flow, but it is essential to have periodic time variation of the wave number vector. Let us turn our attention, now, to flows with ”sharper” rate of differential rotation, with n > 1. In this case γ 2 > 0 and the temporal evolution of k(t)’s becomes exponential. In the absence of dissipation these flows host rather robust shear instabilities. The similar kind of instability was found for acoustic waves in a 2-D flow (Mahajan & Rogava 1999) and in the Paper I it was detected for Alfvén waves as well. The example of such instability for the mixture of AW and SMW is shown on the Fig. 3.3. The parameters are the same as for the Fig. 3.1, only a1 = 0.1 (it reverses the sign of the Λ2 ). We see that in this case the mixture of AW and SMW undergoes rather strong, exponential enhancement of its amplitude and energy.


x10 -3

2

62

Density

D

1 0 -1 -2 0

500

1000

1500

2000

2500

3000

3500

4000

2500

3000

3500

4000

t Total energy

25 20

E

15 10 5 0

0

500

1000

1500

2000 t

Figure 3.1: The temporal evolution of the density D(τ ) and the normalized total energy of perturbations Etot (τ )/Etot (0), which exhibits quasiperiodic and pulsational behavior. The set of parameters is: ² = 10, Kx (0) = 10, Ky (0) = 8, R1 = 0.8, R2 = 2, Σ = 0, a1 = −0.1, a2 = 0.1.


63

Density

0.1

D

0.05 0 -0.05 -0.1 0

500

1000

1500

2000

2500

3000

3500

4000

2500

3000

3500

4000

t 1.5

x10 5

Total energy

E

1

0.5

0

0

500

1000

1500

2000 t

Figure 3.2: The evolution of parametrically unstable blend of AW and SMW. The set of parameters is the same as on Fig. 3.1, except R1 = 0.4.


x10 -3

2

64

Density

D

1 0 -1 -2 0

5

10

15

20

25

30

35

40

45

50

30

35

40

45

50

t Total energy

40

E

30 20 10 0

0

5

10

15

20

25 t

Figure 3.3: Shear instability of low-frequency MHD oscillations. The plot features temporal evolution of the same kind of initial value problem as on Fig. 3.1. The only difference is that The evolution of parametrically unstable blend of AW and SMW. The set of parameters is the same as on Fig. 3.1, except that a1 = 0.1 now, making the sign of γ 2 positive.


65

Vx

0.15 0.1 0.05 0 -0.05 -0.1 0

50

100

150

200

250

300

350

400

450

500

300

350

400

450

500

t Total energy

15

E

10

5

0

0

50

100

150

200

250 t

Figure 3.4: The temporal evolution of initially excited AW, which became partially transformed into FMW and exhibits quasiperiodic, pulsational behavior in low-β plasma flow. The set of parameters is: ² = 0.1, a1 = −0.1, a2 = 0.1, Σ = 0, R1 = 0.01 and R2 = 0.9. Fig. 3.4 shows time evolution of the velocity perturbation vx and the total energy normalized on its initial value Etot (τ )/Etot (0).


66

Vx

1 0.5 0 -0.5 -1 0

500

1000

1500

2000

2500

3000

3500

4000

2500

3000

3500

4000

t Total energy

1000 800

E

600 400 200 0

0

500

1000

1500

2000 t

Figure 3.5: The parametrically unstable solution. The values of the parameters are: ² = 0.1, a1 = −0.1 a2 = 0.1, Σ = 0, R1 = 0.01 and R2 = 0.83.


67

Vx

0.15 0.1 0.05 0 -0.05 -0.1 0

10

20

30

40

50

60

70

50

60

70

t Total energy

25 20

E

15 10 5 0

0

10

20

30

40 t

Figure 3.6: The shear instability in the γ 2 > 0 case. The values of the parameters are: ² = 0.1, a1 = 0.1 a2 = 0.1, Σ = 0, R1 = 1 and R2 = 1.


68

Low-β plasmas In this case hydromagnetic oscillations offer quite a different picture. Namely, the SMW mode is decoupled from the other two MHD modes—its dispersion curve runs well below the Alfvén dispersion ”horizontal” and it is not coupled with other two (AW and FMW) modes. The FMW, on the other hand is coupled with the AW. When rotation is absent and there is only parallel outflow the coupling ensures linear transformation of the AW into the FMW (see as an example Fig. 3.6a-d in (Rogava et al. 2000)). In the helical flow, getting certain ”input” of initial AW and/or FMW oscillations, these waves will keep transforming into each other. Besides the mixture of waves might exhibit the same kind of ”echoing” and unstable behavior as it was seen in high-β plasmas. Numerical simulations support this expectation. Making values of a1 and a2 nonzero (a1 = −0.1 and a2 = 0.1, so that γ 2 < 0) and using following parameters: R1 = 0.01 and R2 = 0.9 we see (Fig. 3.4) the appearance of interesting patterns of ”echoing” solutions with quasiperiodic variability of the perturbation total energy1 . We find that parametric instabilities are also characteristic to low-β plasmas. The example is given on Fig. 3.5. The parametric nature of the instability is apparent from the remarkable narrowness of the range of parameter values, where the instability is present. Note that the set of parameters used for Fig. 3.5 is the same as for Fig. 3.4 except R2 = 0.83. If one takes the value of R2 less by 0.01, then the instability disappears and the system again displays the ”echoing” behaviour. When the differential rotation is characterized by n > 1 profile (γ 2 > 0) the system shows strong exponential shear instability. The corresponding plots are given on Fig. 3.6.

3.1.3

The case of β ' 1

From the studies of parallel flows we know that in terms of wave couplings and mutual transformations this is the most complex case: all MHD wave modes are coupled and may transform into each other. In helical flows the presence of wave transformations may be less visible, overshadowed (when γ 2 < 0) by quasiperiodic modulation of waves appearing as repetitive, (”echoing”) bundles of mixed AW, SMW and FMW modes (see Fig. 3.7); or by self-parametric instability (see Fig. 3.8). While when γ 2 > 0 the waves are unstable in the similar way (see Fig. 3.9) as in previously considered high-β and low-β cases. 1

Note that here, as elsewhere on the plots of this work, the total energy is normalized on its initial value.


69

Bz

0.2 0.1 0 -0.1 -0.2 0

500

1000

1500

2000

2500

3000

3500

4000

2500

3000

3500

4000

t Total energy

20

E

15 10 5 0

0

500

1000

1500

2000 t

Figure 3.7: The temporal evolution of the wave blend that exhibits quasiperiodic, pulsating behavior in ² = 1 plasma flows. The set of parameters is: a1 = −0.1 a2 = 0.1, Σ = 0, R1 = 0.1 and R2 = 0.85. The plots show time evolution of the magnetic field Bx perturbations and the normalized energy Etot (τ )/Etot (0).


70

Bz

0.5

0

-0.5 0

500

1000

1500

2000

2500

3000

3500

4000

2500

3000

3500

4000

t Total energy

600

E

400

200

0

0

500

1000

1500

2000 t

Figure 3.8: The evolution of parametrically unstable blend of waves. The set of parameters is: a1 = −0.1 a2 = 0.1, Σ = 0, R1 = 0.1 and R2 = 0.8005. Note that the range of R2 when the evolution of the wave mixture is parametrically unstable is very narrow: [0.8001; 0.8011].


71

Bz

0.2 0.1 0 -0.1 -0.2 0

10

20

30

40

50

60

70

50

60

70

t Total energy

80

E

60 40 20 0

0

10

20

30

40 t

Figure 3.9: The shear instability in the λ2 > 0 case. The set of parameters is: a1 = 0.1 a2 = 0.1, Σ = 0, R1 = 1 and R2 = 1.


3.1.4

72

Conclusion

The goal of this work was to find out whether the exotic SINP originally found for 2-D kinematically complex velocity patterns of neutral flows (Mahajan & Rogava 1999) and for 3-D helical flows of magnetized conducting flows in the incompressible limit, appear also for the full spectra of MHD waves sustained by compressible MHD medium. We found that this is indeed the case! Therefore we can now firmly claim that the range of the SINP, typical for the flows of complicated, helical nature, is broad. These SINP persist to show up both in the incompressible and compressible cases. They are present in flows with arbitrary values of plasma-β. In relatively mildly sheared flows (with n < 1, i.e., including rigidly rotating systems) MHD modes appear to be rather stable exhibiting either ”echoing” pulsational behaviour or relatively long-time-scaled parametric instabilities. One can expect that in helical flows of this nature, especially in well-beamed, or well-collimated flows such as jets, shear flow effects are not likely to lead to disruptive instabilities. Instead, through quasiperiodic interchange of energy with the mean flow, they might tend to exhibit certain modes of quasi stable and quasiperiodic structuring both in the space and in time. In more strongly sheared systems (n > 1 including Keplerian rotation) waves within swirling flows become subject to potentially very fast-growing shear instabilities, which would either lead to the disruption of ”parent” flow patterns or to the development of the MHD turbulence with subsequent phase transition to turbulent rotational flow systems. We have to bear in mind that in this case the exponential growth of the |k(t)| inevitably makes the spatial length-scales of perturbations smaller and smaller. It implies that the effects of the viscous decay and/or magnetic diffusion, neglected while we consider the MHD flow as an ”ideal” one, must sooner or later become important and lead to the dissipation of the energy gained by the exponentially increasing waves into the heat. It can be argued that in accretion-ejection systems, where the rotational law seems to be quasikeplerian, these instabilities may account for the transition to turbulence in accreted plasma flows. Alternatively, this process might lead to effective ”self-heating” of these flows, when the energy acquired by waves from the flows through the agency of the shear instability would eventually transform into heat via diffusion. Speaking about swirling astrophysical flows we are keen to use the term ”cosmic tornado” for this class of flows, because they are reminiscent of powerful and dangerous tornados in the Earth’s atmosphere. Recently such structures, called solar tornados, were identified in the polar regions of the solar atmosphere (both on the limb and the disk) by SOHOCDS observations (Pike & Mason 1998). Another class of cosmic tornados are, probably, stellar jets, because recent observational results (Bacciotti et al. 2002; Davis et al. 2000) seem to confirm the predictions of various magnetocentrifugal jet acceleration models

3.2 Linear analysis of the Kelvin-Helmholtz instability for relativistic magnetohydrodynamics flows.

73

(Shu et al. 1995; Ferreira & Pelletier 1995; Camenzind 1997; Lery et al. 1999; K¨ onigl & Pudritz 2000) about the presence of a swirling motion within the jets. The third class of astrophysical flows with possible presence of tornado-like motion are accretion columns – magnetically channeled shear flows of plasma to a neutron star’s (or a white dwarf’s) magnetic pole. The infalling matter is decelerated approaching the star surface, but it can also form a shock high above the star’s surface (Hujeirat & Papaloizou 1998). Originally the formation of rotational accretion columns was considered in the astrophysical flow dynamics context (Cassen 1978), while now accretion columns associated with X-ray pulsars and cataclysmic binaries are modeled either as thin slabs or tall columns of infalling matter. It seems reasonable to surmise that accretion columns comprise 3-D swirling plasma flows. Bearing in mind these perspectives we should stress that the results of this work are quite general and (deliberately) not adjusted to either of three kinds of swirling astrophysical flows. For all these classes of ‘cosmic tornados’ the level of our factual knowledge about their basic kinematic features is still inadequate for building of any credible concrete models. However we hope that these future models, based on the gained data and implying real-space sophisticated simulations, will show how the generic processes disclosed in this work and the Paper I might influence the overall dynamics of real ‘cosmic tornados’.

3.2

Linear analysis of the Kelvin-Helmholtz instability for relativistic magnetohydrodynamics flows.

The Kelvin-Helmholtz instability (KHI henceforth) applies to two flows in relative motion. It plays a dynamically important role in a number of different astrophysical scenarios, such as accretion disks, planet magnetospheres, stellar and extragalactic jets and so forth. The problem has been extensively studied, beginning with the pioneering work of Von Helmholtz and Monats (1868) and Lord Kelvin (1871), successively generalized by Gerwin (1968) for compressible flows, where it has been shown that the flow becomes stable for Mach numbers which are larger than a certain critical value. Magnetic field effects were investigated by (Choudhury & Lovelace 1986), who studied the instability versus Alfvén velocity and the inclination of the wave number projection onto the interface and found unstable regimes corresponding to standing and traveling waves. It has been emphasized that, in the linear approximation, the contribution of the Lorentz force has overall stabilizing effects. A relativistic non magnetized extension was considered by Turland and Scheuer (1976) and Blandford and Pringle (1976a) and, more recently, by Bodo et al. (2004), who showed that an analytical expression to the dispersion relation


74

can be found in a frame where the two flows have equal and opposite velocities. Using the definition of relativistic Mach number it has been shown that in the laboratory frame the stability criteria are the same as those in the classical case. Earlier attempts to generalize the problem as to include both magnetic and relativistic effects were carried on by Ferrari et al. (1980), where the authors have discussed results in connection with models of extragalactic jets. Their model, however, neglected the effects of the displacement current and considered only non-relativistic Alfvèn velocities. They have shown, that in a cold gas for non relativistic flow velocities, in order to kill the instability, one needs the Alfvèn velocity equal to the speed of sound. For a hot gas with strong magnetic fields their approximation does not work at all, because the Alfvèn velocity becomes relativistic and correspondingly this case is out of the approximation. A numerical simulation of the KHI for a relativistic magnetized hot plasma was considered by Bucciantini and Del Zanna (2006) where the authors examined the KHI problem, applying it for the synchrotron emission in the Crab pulsar wind nebulae. In the present work we re-examine the KHI for a (special) relativistic magnetized flow by considering the full system of equations without neglecting the displacement current and studying as non relativistic as relativistic Alfvén velocities. It is shown that the stability issue can be parameterized in terms of four parameters which we conveniently choose to be the relativistic Mach number, the flow velocity, the Alfvén speed and the inclination of the projection of the wave vector onto the plane of the interface. The work is organized as follows. In §3.2.1 we derive the dispersion relation, in §3.2.2 the corresponding results are present and in §3.2.3 we summarize the results.

3.2.1

Dispersion Relation

The evolution of a relativistic magnetized flow is governed by the equations of particle number and energy-momentum conservation (Anile 1989): 3 ~ X ∂U ∂ f~i + = 0, i ∂t i=1 ∂x

(3.49)

complemented by the induction equation: ~ ∂B ~ × ~v × B ~ = 0, −∇ ∂t

(3.50)

~ and f~i are the vector of conserved variables and correwhere ~v is the flow velocity, U sponding fluxes given by:


75

~ = [ργ, wt γ 2 v j − b0 bj , wt γ 2 − b0 b0 − pt ]T , U

(3.51)

f~i = [ργv i , wt γ 2 v i v j − bi bj + pt δ, wt γ 2 v i − b0 bi ]T ,

(3.52)

where we have assumed, for simplicity, a flat Minkowskian metric. In Eq. (3.49) we have introduced the covariant magnetic field h

i

~ B/γ ~ + γ(~v · B)~ ~ v , bα = γ(~v · B),

(3.53)

wt = ρh + |b|2 ,

(3.54)

1 pt = p + |b|2 . 2

(3.55)

the total enthalpy with |b|2 = bα bα , and total pressure

Here γ is the Lorentz factor, p is the thermal (gas) pressure, and h is the specific enthalpy. Equations are written in units of c = 1, where c is the speed of light. Rest mass density and velocities are labeled with ρ and ~v . Proper closure is provided by specifying an equation of state which we take as the constant Γ-law: h=1+

Γ p , Γ−1ρ

(3.56)

where Γ is the polytropic index of the gas. Our setup consists of a planar interface lying in the xz plane at y = 0 separating two fluids moving in the x direction with opposite velocities ~v (y > 0) = β~i and ~v (y < 0) = −β~i. The fluids have the same density and pressure and are threaded by a uniform longitudinal ~ = B0~i. magnetic field B We start our analysis by introducing small deviations around the equilibrium state. This is better achieved by working in the rest-frame of one of the two fluids, where perturbations are sought in the form: 0

~e = B ~e 0 + B ~e + ... , B 0

(3.57)

~ve = vf ~0 + ... ,

(3.58)

ρe = ρe0 + ρe0 + ... ,

(3.59)

pe = pe0 + pe0 + ... ,

(3.60)


76

where tilde denotes quantities in the rest frame. The perturbed terms of the first order can be expressed by the following: h ³

í

e x e e+m e 0 ∝ exp i k f± ze − ω e ± te Ψ ± e + l± y ±

,

(3.61)

0

e 0 = (B ~e , ~ve0 , ρe0 , pe0 ). Subscripts + and − correspond to the regions y > 0 and where Ψ ± ± ± ± ± f± and ω e ± are wave numbers and the frequency respectively. y < 0, respectively. ke± , le± , m Proper linearization of the equations leads to the dispersion relation, the solutions of which are given by the the Alfvén mode (Komissarov 1999a): 2 e± ω = VA2 , 2 e k±

(3.62)

q

where VA = B0 / ρ0 h0 + B02 is the Alfvén velocity, and a pair of magneto-acoustic waves: Ã

!2

i 1h 2 µe ± − νe±2 , 4

(3.63)

ke 2 + le2 µe ± = Cs2 + VA2 − Cs2 VA2 e 2 ±e2 ± 2 f± k± + l± + m

(3.64)

ke 2 νe± = Cs2 VA2 e 2 e2± f2± k± + l± + m

(3.65)

2 e± ω 1 − µe ± 2 2 2 + le± ke±

=

where

s

Cs =

Γp0 ρ 0 h0

(3.66)

Here Cs is the sound speed. Notice that both Cs and VA are invariant under Lorentz boosts in the x-direction. Notice that only Eq. (3.63) gives rise to unstable modes, since the roots of Eq.(3.62) are always real. In order to obtain the dispersion relation in the laboratory frame, one has to transform all the quantities by mean of a Lorentz transformation: e ± = γ(ω ∓ kβ) , ω

(3.67)

ke± = γ(k ∓ ωβ) ,

(3.68)

f± = m , le± = l, m

(3.69)


77

Ψ0± ∝ exp [i (kx + l± y + mz − ωt)] .

(3.70)

Direct substitution into the induction equation (3.50) and into the y and z components of the momentum equation in (3.49) yields B0

h i ke∓ 0 0 0 e ∓ vy± By± + ωB02 + γρ0 h0 ω − l± (p0± + B0 Bx± ) = 0, γ

(3.71)

B0

h i ke∓ 0 0 0 e ∓ vz± Bz± + ωB02 + γρ0 h0 ω − m(p0± + B0 Bx± ) = 0, γ

(3.72)

0 0 0 (ω ∓ kβ)Bx± − B0 (l± vy± + mvz± ) = 0,

(3.73)

0 0 + kB0 vy± = 0, (ω ∓ kβ)By±

(3.74)

0 0 (ω ∓ kβ)Bz± + kB0 vz± = 0,

(3.75)

e ∓ are given by Eqs.(3.67,3.68) and for compactness we do not show their where ke∓ and ω expressions in the Eq.(3.71,3.72). By using Eq.(3.55) one may express the total pressure perturbation as:

pt± = p± +

0 l± B02 vy± . ω ∓ kβ

(3.76)

and imposing total pressure continuity (pt+ = pt− ) at the interface one obtains (together with Eq. 3.71) ρ0 h0 γ 2 (ω − kβ)2 + (ω 2 − k 2 )B02 l+ = , (3.77) l− ρ0 h0 γ 2 (ω + kβ)2 + (ω 2 − k 2 )B02 where we took into account the condition of matching displacement at the interface: 0 0 vy+ vy− = . ω − kβ ω + kβ

(3.78)

A second equation can be derived directly from (3.63): h

2 = −m2 + l±

i

4 2 2 2 e∓ e∓ − Cs2 VA2 ke∓ −ω (Cs2 + VA2 )ke∓ ω 2 2 2 e∓ e∓ +ω ) − (Cs2 + VA2 )ω Cs2 VA2 (ke∓

,

(3.79)

which, together with (3.77), gives the desired dispersion relation. A number of considerations follow.


78

• Positive imaginary roots of ω identify unstable modes. • In the non-relativistic limit (i.e. β → 0, ρh >> p, B02 and h → 1) gas our dispersion relation reduces to the form given by (Choudhury & Lovelace 1986) (see Eqs.(7,A2a)); • In the limit of vanishing magnetic field Eqs.(3.77,3.79) considerably simplify and the dispersion relation reduces to the one derived by Bodo et al. (2004)(see Eqs.(3.78,3.79)). • For the general case the system of equations Eqs.(3.77,3.79) has to be solved numerically.

3.2.2

Discussion

Our configuration depends on 4 parameters: the fluid velocity β, the relativistic Mach number Mr = γβ/γs (where γs = (1 − Cs2 )−1/2 ) the Alfvénic Mach number ζ = VA /Cs and the ratio f (= m/k) giving the angle between the wave number projection on the xz plane and the propagation direction. In general, relativistic effects come into play whenever one of β, Mr or ζ (or a combination of them) describes situations of high fluid velocities, hot gas or strong magnetic field, respectively. Non Relativistic case Let us now consider the case of a non-relativistic fluid with β = 0.001 (f = 0, Γ = 5/3). In Fig. 3.10 we plot Im(φ) as a function of Mr , for several values of the Alfvén Mach number in the range 0 ≤ √ ζ ≤ 0.9. For vanishing magnetic field (ζ = 0) the instability disappears when Mr > 2. Higher values of ζ induce a stabilizing effect by raising the pressure and forcing the fluid to be channeled along the field lines. This has two effects: one is a decrease in the maximum value of the growth rate and the other is narrowing down the instability Mach range ∆ ≡ Mrmax − Mrmin , where Mrmax (Mrmin ) is the smallest (largest) value of the Mach number below (above) which the fluids are stable. Notice that the values of the sound speed at the growth rate maxima are non relativistic. In Fig. 3.11 we show the dependence of the critical range of the relativistic Mach number on the magnetic field. As it is clear from the plot, when ζ → 1, the relativistic Mach number range tends to zero and the instability is completely killed. For ζ > 1 the interface is always stable. Our results are thus in full agreement with those of Ferrari et al. (1980) for the same parameter range (β = 0.001, f = 0).


79

Figure 3.10: Plots of dependence of Im(φ) on the relativistic Mach number, labels on each curve indicate the values of ζ. Set of parameters is: β = 0.001, ζ = [0, 0.2, 0.5, 0.8, 0.9], f = 0 and Γ = 5/3.

Figure 3.11: Plot of dependence of the range of critical values of the relativistic Mach number ∆ on the magnetic field for β = 0.001, (f = 0, Γ = 5/3).


80

Figure 3.12: Effects of magnetic field on growth rate Im(φ) for β = 0.7, labels on each curve indicate the value of ζ. The set of parameters is: β = 0.7, ζ = [0, 0.2, 0.3, 0.34], f = 0 and Γ = 4/3. When the magnetic field corresponds to ζ = 0.35, the flow is already stable.

Relativistic case In the previous subsection all the relevant quantities were describing essentially non-relativistic flows. We now consider situations in which the fluids exhibits relativistic behavior at least for some parameter range. In Fig. 3.12 we show the behaviour of the growth rate versus Mr for β = 0.7, f = 0, Γ = 4/3. As in the previous case, higher values of the relativistic Mach number correspond √ to lower sound speeds. Since the latter cannot exceed the upper limiting value of 1/ 3 (which is a result of the fact that √ the enthalpy is always more than 1, see Eqs. (3.56,3.66)), Mr is lower-bounded by 2βγ ' 1.386. When the magnetic field increases (i.e. higher ζ), both the growth rate peak and the Mach number range ∆ (for which the instability exists) decrease. This behaviour has been already discussed in the non-relativistic case (§3.2.2), where the flow was shown to become stable for values of the Alfvén speed close to the speed of sound. For the present case (β = 0.7), however, stability is approached when the Alfvén speed VA ' 0.35Cs , see Fig. 3.12. This pattern is confirmed by the profile of ∆ (the critical Mach number instability range) as function of ζ (shown in Fig. 3.13). As anticipated, the instability disappears for ζ ' 0.35, an effect which can also be justified by the increased kinematic inertia. In Fig. 3.14, indeed, we show the critical value of ζ (above which the instability is suppressed) as a function of the flow velocity β. Thus ζcr monotonically decrease from ∼ 1 (at β = 0.6) to 0 at β ' 0.71. Higher value of β lead to a stable interface, even


81

Figure 3.13: Plot of dependence of critical values of the relativistic Mach number for β = 0.7 (f = 0, Γ = 4/3).

without magnetic field, which is a major difference from the classical MHD case. The last result was already introduced (Bodo et al. 2004), where it was shown that positive growth rates are subject to the condition "

β < Cs

2 1 + Cs2

#1 2

.

(3.80)

Since the right hand side is√a monotonically increasing function of Cs , we conclude that for β ≥ 0.7071 (Cs = 1/ 3) the instability is suppressed by kinematic effects only, regardless the value of the magnetic field. We remind the reader that the previous considerations have been obtained for f = 0, that is, when the transverse component of the wave vector parallel to the interface was vanishing (i.e. m = 0). For the general case in which f 6= 0, the growth rate becomes sensitive to the inclination of the wave vector projection in the xz plane by monotonically increasing with f . This is best shown in Fig. 3.15, where we plot the dependence of the growth rate on f for a fixed relativistic Mach number Mr = 1.4 and different values of ζ. The same behavior is observed in the classical (1968; Choudhury & Lovelace 1986), relativistic non magnetic (Bodo et al. 2004)√cases, and it can be ascribed to the reduced effective Mach number proportional to k/ k 2 + m2 + l2 . In terms of the Mr this has the net effect of broadening the instability range by effectively increasing the uppermost


82

Figure 3.14: Effect of the kinematic factor on the instability. As it is clear, for β ' 0.71 the instability is already forbidden by the relativistic effects independently on Mr , which means, that for any values of the Mach number, the instability does not exist.

limit Mrmax . Eventually, for very large f ’s, one can show from Eqs.(3.77,3.79) that the asymptotic value of the growth rate can be expressed by following: "

1 β 2 − ζ 2 Cs2 lim Im(φ) = 2 f →∞ Cs 1 − β 2 ζ 2 Cs2

#1 2

.

(3.81)

Equation (3.81) also demonstrates that the saturated value decreases with ζ, ultimately hampering the instability when ζCs = β, i.e., VA = β. Thus when the inclination angle increases the magnetic field has a less stabilizing effect and, as a result, the growth rate increases (see Fig. 3.15). Unlike the case f = 0, the critical value of ζ is greater than 1. In Fig. 3.16 we show contours for different levels for the same Lorentz factor and for the fixed Alfvén Mach number equal to 1.6. As it is clear from the plots, there are regions in the space f − Mr where the Kelvin Helmholtz instability appears. As one can see, for small inclinations (f < 0.2, see the Fig. 3.16) there is no instability range, but increasing the angle, the flow may become unstable. In order to understand why these instabilities appear let us consider Fig. 3.17, where we show as imaginary (dotted dashed lines) as real (solid lines) roots of the equation governing the dispersion relation corresponding to a cut of the contour plots in Fig. 3.16 for f = 2. As it is clear from the figure, initially, when there is no instability, one has two real roots (with their counterparts also, see


83

Figure 3.15: Effect of the inclination factor f on the growth rate Im(φ) for β = 0.7, labels on each curve indicate the value of ζ. The set of parameters is: β = 0.7, Mr = 1.4 ζ = [0, 0.6, 1, 1.15, 1.21, 1.22] and Γ = 4/3. It is clear that when the magnetic field corresponds to ζ = 1.23, the flow is already stable.

Figure 3.16: Contour plots for constant φ lines. The set of parameters is: β = 0.7, ζ = 1.6 and Γ = 4/3.


84

Figure 3.17: Plots of imaginary (dotted dashed) and real (solid) roots. Set of parameters is: β = 0.7, ζ = 1.6, Γ = 4/3 and f = 2.

A and B in Fig 3.17) and the instability appears at the point a, when two (four with counterparts) real roots merge and as a result an imaginary root appears, corresponding to the instability (see a − b in the Fig. 3.17). Increasing the Mach number, already merged real root corresponding to C, at the point b is divided again by two branches, thus appears one more real root (see D and E) and as a result of it the imaginary root that was responsible for the instability disappears (see the point b). The same happens for the second instability range, when two real roots merge (see E and E 0 ) one more imaginary root appears, making the system unstable (see the point c ). One can see from Eq. 3.63 that each of the regimes correspond to slow (a − b) and fast (c) waves.

3.2.3

Summary

1. A linear analysis of the relativistic magnetized Kelvin-Helmholtz instability problem in the vortex sheet approximation has been presented. Solutions to the the dispersion relation have been sought in terms of four parameters giving the strength of the magnetic field, the relativistic Mach number, the flow velocity and the spatial orientation of the wave vector. 2. Similar to the non-magnetic counterpart, we have found that the flow becomes linearly stable when the relativistic Mach number exceeds a critical value.


85

3. In the √limit of vanishing Lorentz force, this threshold reaches the maximum value of 1/ 2, in a frame where the fluids have equal and opposite velocities. 4. For increasing magntic field strength, the maximum unstable growth rate decreases and the instability exists for a narrow range of Mach number values. This range shrinks with higher Alfvén velocity, but it broadens for increasing relative orientation (α) between the flow velocity and the projection of the wave vector in the plane of the interface separating the two fluids. We have shown that for the Alfvénic Mach number more than 1, one has two regimes of instabilities which are linked to the slow/fast magnetosonic waves. 5. In the parallel case (α = 0) and for small relative velocities, the instability disappears (independently of the Mach number) when the the Alfvén velocity approaches the speed of sound. 6. The main difference from the non relativistic MHD is that, at higher flow velocities for the same case (α = 0), kinematical effects stabilize the flow even for smaller values of the Alfvén velocity. Furthermore, the computed growth rates attain lower values than √ their classical counterparts. When the flow velocity becomes higher than 1/ 2, the flow does not need any more the magnetic field for stabilization. 7. When α → π/2, on the other hand, the maximum growth rate increases and corresponding value of the Alfvén speed required for stability is given by the flow bulk velocity. For supersonic flows, this value will clearly exceeds the speed of sound. The Kelvin-Helmholtz instability may be applied for the physics of extragalactic jets. Two major problems can be emphasized. The first one is the origin of jet morphologies, which can be associated with the KHI of a jet surface. For this particular case a rich observational data providing morphologies of extragalactic jets may be extremely important (Pagels et al., 2004). Another problem concerns the deceleration process of jets. According to this approach the evolution of Kelvin-Helmholtz instabilities induces the turbulent entrainment of an ambient matter, which leads to the deceleration process of the jet (Bodo et al., 2003). For this reason it is natural to investigate the KHI for a more realistic geometry and study the stability problem of a cylindrical magnetized relativistic flow.

3.3 The Kelvin-Helmholtz instability for relativistic Fluid, for a planar velocity shear layer approximation. Non linear analysis. 86

3.3

The Kelvin-Helmholtz instability for relativistic Fluid, for a planar velocity shear layer approximation. Non linear analysis.

The previous work was dedicated to the linear analysis of the KH instability for a planar case in a vortex sheet approximation and the instability was analyzed in terms of four physical parameters: flow velocity, relativistic Much number, the Alfvén Mach number and the inclination factor. For understanding a physical behaviour of the KHI one should consider it for relatively large time ranges, but the linear approach has a limitation of the time, up to which the linear analysis is valid. For this reason it is natural to examine also a non linear behaviour of the KHI and see how the system evolves in time. In this case another problem arises, the system of equations can not in general be reduced into a compact form, solvable analytically. In order to study the non linear behaviour of the KHI we implement the PLUTO code (Mignone et al. 2006) and apply it for the system of equations: Eqs.(3.49,3.50). Numerical study of the KHI is not new, several papers were dedicated to the KHI problem. A numerical investigation of the KHI was considered in the context of the accreting neutron stars (Wang & Robertson 1984), in Downes and Ray (1998) authors presented results of simulation of the development of the instability in a cooled slab symmetric system applying it for the young stellar objects (YSOs). For hydrodynamic case the relativistic consideration of the KHI problem was offered in series of works (Perucho et al. 2004a; Perucho et al. 2004b; Perucho et al. 2005), where corresponding results were classified for a wide range of the Lorentz factors. The KHI problem for the magnetorelativistic hot fluid was considered by Bucciantini and Del Zanna (2006) in the context of synchrotron modulation in pulsar wind nebulae. In this work we examine as non-magnetized as magnetized flows and analyze what changes in the instability behaviour when magnetic field is taken into account.

3.3.1

Physical problem

We investigate the stability of two flows in relative motion, for a more realistic situation, with a velocity shear layer. The equations governing the mentioned physical system is the same as in the previous section, (see Eqs(3.49,3.50)). As for the linear analysis we describe the system from the frame, where both flows have the same velocities, but opposite directions (along x-direction). The velocity shear is along the y-axis:

3.3 The Kelvin-Helmholtz instability for relativistic Fluid, for a planar velocity shear layer approximation. Non linear analysis. 87 vx = v tanh(y)

(3.82)

Hereafter we label the flow in the region y > 0 (the region where vx > 0) and y < 0 (region with vx < 0) by subscripts ”+” and ”−” respectively. Initially the problem depends on three parameters: the sound speed Cs ≡ [Γp/(ρh)]1/2 defined for a perfect gas, the relativistic Mach number: Mr = γM/γs , where γs is the Lorentz factor corresponding to the speed of sound and ζ - the Alfvén Mach number.

3.3.2

Setup

We consider a uniform medium with the density ρ = 1, the shear velocity, given by Eq.(3.82) with the following pressure: p=

ρCs2 Γ Cs2 Γ − Γ−1

(3.83)

and the velocity perturbation along the y-axis which has a form with maximum effect on the interface and diminishing for | y |→ ∞. This behavior may be provided by following expression: vy = µV0y

sin(kx ) cosh(y)

(3.84)

µ = 1 for y > 0 and µ = −1 for y ≤ 0, V0y is the perturbation amplitude, which in our case is 0.05v, kx = 0.2 is the wave number, that corresponds to the wave length λ = 10π. The numerical domain corresponds to a rectangular: 0 ≤ x ≤ 2λ, 0 ≤ y ≤ λ with a uniform mesh ∆x = 2λ/Nx , ∆y = λ/Ny where Nx = 256 and Ny = 128. In order to avoid reflection effects from the upper boundary at y = λ one can extend the domain along this axis up to y = 800 with a stretched grid with 128 points. Along x-axis on boundaries (x = 0 and x = λ) we use the periodic conditions, whereas at the upper boundary (y = 800) - the outflow condition. At the lower boundary y = 0 we use a symmetry with a reflection condition with respect to the central point (λ, 0): Ψ(x, 0) = ±Ψ(2λ − x, 0)

(3.85)

where Ψ is a physical quantity (ρ, vx , vy , Bx , By , p), plus sign corresponds to scalar quantities and the minus sign corresponds to vector quantities.


3.3.3

Numerical simulations

As we have mentioned, our initial magnetohydrodynamic system depends on three characteristic velocities, the speed of sound, the Alfvén speed and the flow velocity. In relativistic thermodynamics in a high temperature gas the speed of sound is limited from 1/2 the causality principle, √ as we have noticed the speed of sound Cs ≡ [Γp/(ρh)] can not exceed a value: 1/ 3. It is reasonable to consider two extreme cases of thermodynamics: the cold √ gas, with a speed of sound Cs = 0.1 and the maximum possible sound speed: Cs = 1/ 3. For each of them we study three different cases for the relativistic Mach number: √ 0.5, 1.4 (the reason of considering this case is that it is very close to the limiting value 2 (see Fig. 3.10) over which in linear approximation there is no instability) and 5. Cold plasma Let us consider the cold plasma with the speed of sound Cs = 0.1 neglecting the magnetic field. From the linear analysis in the vortex sheet approximation (see the previous section and also (Bodo et al. 2004)) it was shown that the instability √ exists only up to some critical value of the relativistic Mach number (Mr = 2), over which the system becomes stable. It shows the tendency of the stability effect, and it is clear that even for a case of the velocity shear layer, increase of Mr may cause the corresponding decrease of the KHI due to the compressibility and causality effects. A non linear regime leads to the formation of vortices which show many interesting features, like merging and multiplication phenomena of vortices. For this reason one should introduce the vorticity in relativistic hydrodynamics. Defining the enthalpy flow four-vector by following: qα = hvα

(3.86)

the set of equations (3.49-3.50) can be written down in a more compact form: vβ

∂qα ∂h − =0 ∂xβ ∂xα

(3.87)

∂qβ ∂qα − ∂xβ ∂xα

(3.88)

Introducing a tensor: Ωαβ = one can reduce Eq.(3.87) into a form: v β Ωαβ = 0

(3.89)


Figure 3.18: Color scale of the vorticity (upper) and natural logarithm of pressure (below) distributions for four different times: t = 1250 corresponding to the creation of vortices, t = 14500 corresponding to the starting time of the merging, t = 15000 - the moment when the vortices are already merged and t = 50000- the final state of the system. Set of parameters is: Cs = 0.1, Mr = 0.5 and ζ = 0.

Defining Ω = (Ω23 , −Ω13 , Ω12 ) = ∇ × q

(3.90)

one can show from the spatial part of Eq.(3.89) (after taking the curl of it), that Ω in analogy of the Kelvin circulation in the non relativistic case satisfies the equation: ∂Ω + ∇ × (v × Ω) = 0 (3.91) ∂t On the Fig. 3.18 we show the color scale of the vorticity and the pressure distribution for t = 1250. Time is measured in units of Lsh /c, where Lsh is the initial shear length, and c is the speed of light. One can see that two vortexes are created, that is reasonable

3.3 The Kelvin-Helmholtz instability for relativistic Fluid, for a planar velocity shear layer approximation. Non linear analysis. 90 because initially we had two wave lengths in a region 0 ≤ x ≤ 2λ. On the next picture (Fig. 3.18, t = 14500) we show the process of merging, the vortex which is on the left is starting to go to the left boundary and merge with its mirror counterpart, whereas the vortex which is on the right is merging on the right boundary with its mirror counterpart (see Fig. 3.18, t = 15000 ), from this moment, already merged vortices always stay at boundaries. The merging process is a result of fully non-linear behavior of the system. From the Fig. 3.18 (see pressure distribution for t = 14500) one can see that when the merging process starts, the pressure between two vortices is higher than the pressure on the left hand side from the left vortex and on the right hand side from the right vortex respectively, this difference in pressure provides the merging of vortices. During the instability evolution the interaction between two flows can lead to momentum exchange and mixing process as well. The average velocity profile changes reflect the momentum exchange process, and the analysis of the the velocity shear profile width can measure the efficiency of the mixing. For this purpose one may define the following quantity: δv =

Z +∞ −∞



Ã

V x (y) 1 − V0

!2   dy

(3.92)

where

1ZL V x (y) ≡ (3.93) Vx (x, y)dx L 0 is longitudinally averaged value of Vx (here L is a length of the domain along x-axis). This quantity can measure the width of the velocity shear layer. On the other hand the momentum exchange may happen not only by the waves that transport momentum, but also due to the direct mixing of flows. For this reason it is natural to introduce a passively advected tracer (color): ∂(T ρ) + ∇ (T ρv) = 0 (3.94) ∂t having the value 1 for one flow and -1 for another. Defining the tracer width δT in an analogy with δv δT =

Z +∞ −∞



Ã

T (y) 1 − T0

!2   dy

(3.95)

one can distinguish the efficiency of the direct mixing and wave mechanism in the context of momentum exchange. On Fig. 3.19 we show the behaviour of δv (t) and δT (t). As it is clear from the graphs, initially they show the linear regime of the instability, but in time


Figure 3.19: Two plots of δv (solid line) and δT (dotted line) versus time. Set of parameters is the same as for previous graphs.

the linear approximation comes to the end and brings into the non linear regime that is reflected also on shapes of the curves. As it is shown, the curves have a step like form, which is a direct consequence of the non linear behaviour. Comparing the corresponding time scales from Fig. 3.18 and Fig. 3.19 one can argue that the first step corresponds to the formation of vortices and the second step corresponds to the merging process. From the Fig. 3.19 it is clear that δT reaches higher saturation value than δv , this means that in the momentum exchange, the role of direct mixing is very efficient, on the other hand if the wave mechanism were dominant, the shape would be more continuous unlike the case shown in Fig. 3.19. Another example is the flow with the relativistic Mach number Mr = 1.4. In Fig. 3.20 we show the color scale of vorticity and pressure for four different times, corresponding to the vortex formation, one intermediate state corresponding to the multiplication process of vortices, the merging process and the final state of the system. As it is clear from Fig. 3.20, the formation time is larger with respect to the same time scale in the previous case, which is a natural consequence of the compressibility, that amplifies stability effects. Another difference from the first case is a more elongated shape of vortices due to the increased asymmetry along the flow motion. As one can see from the picture for t = 23350, the pressure distribution forces vortices to merge. Between vortices pressure is less than on the other sides and vortices move to each other, merge and finally instead


Figure 3.20: Color scale of the vorticity (upper) and natural logarithm of pressure (below) distributions for four different times t = 2700 corresponding to the creation of vortices, t = 15100 corresponding to the multiplication of vortices, t = 23350 corresponding to merging processes and t = 38000 the final state of the system. Set of parameters is: Cs = 0.1, Mr = 1.4 and ζ = 0.

of having four, at the very and of the process remain only two vortices (see Fig. 3.20, t = 38000). On Fig. 3.21 one can see that δv and δT do not have the step like behaviour any more as it was in the case of Mr = 0.5, now it is more continues, which shows, that the wave mechanism is more efficient for the momentum exchange than the direct mixing. This is a reason why a value of the tracer layer width is less than a value of the velocity layer width. The last example we consider for the cold plasma is Mr = 5. In the Fig. 3.22 one can see color scales of the vorticity and the pressure distribution. As it is shown on the first picture the instability appears very late (t = 51150) in comparison with the formation time in previous cases. Correspondingly a shape becomes dependent on the Mach number, the higher the Mach number the more elongated the vortex, up to the


Figure 3.21: Two plots of δv (solid line) and δT (dotted line) versus time. Set of parameters is: Cs = 0.1, Mr = 1.4 and ζ = 0.

last example, when a region with maximum vorticity at t = 51150 is so elongated that it is even difficult to call this structure a vortex. In due course the longitudinal size becomes less, and the vortex tends to be fixed nearby the vertical boundary (see Fig. 3.22, t = 100000). In the next figure, we show velocity and tracer shear widths Fig. 3.23, it is clear that, momentum exchange is mostly provided by the wave mechanism, confirmed by a continuous shape of the curve and by the difference between δT and δv . For understanding the effect of the magnetic field on dynamics of the instability, in Fig. 3.24 for Mr = 1.4 we show color scales for the vorticity and magnetic field distributions (natural logarithm of the magnetic field). Initially we have the uniform magnetic field having only x component and corresponding to ζ = 0.2. As one can see from this picture, due to the frozen-in condition, the flow follows the magnetic field, this is the reason why the magnetic field and the vorticity have similar configurations, (see Fig. 3.24, t = 2700). The magnetic field makes the flow more complicate. One can see that after creating two vortices, they split into several small ones, initially four vortices arise (see Fig. 3.24, t = 4250) which again split into eight even smaller vortices (see Fig. 3.24, t = 5300), but in due course due to the additional magnetic pressure the growth rate is decreasing finally coming into the disappearance of vorticities (see Fig. 3.24, t = 20000). Also in this case, the wave mechanism is more efficient than the direct mixing, as one can see


Figure 3.22: Color scale of the vorticity and the natural logarithm of pressure distributions for four different times t = 51150, t = 51300, t = 53000 and t = 100000. Set of parameters is: Cs = 0.1, Mr = 5 and ζ = 0.

from the Fig. 3.25, asymptotic values of δv and δT are less than corresponding values for the same sound speed and the Mach number but for ζ = 0 (see Fig. 3.21), which again is the influence of the magnetic field. We have considered three cases of flow velocities, and it was shown that for low values of the relativistic Mach number, the produced vortex is almost circular unlike the high value Mach numbers, when the vortex becomes more prolated up to the case of Mr = 5 when the relativistic effects are not any more negligible and a shape of the region with maximum vorticity does not look like a typical vortex like structure. From the point of view of shear widths, one may argue that the higher the Mach number, the less the role of material mixing and the higher the momentum exchange mechanism. Initially for the low Mach number the value attained by δT was higher than the value attained by δv : (see Fig. 3.19), situation drastically changed for Mr = 1.4, when unlike the previous example


Figure 3.23: Plots of δv (solid line) and δT (dotted line) versus time. Set of parameters is: Cs = 0.1, Mr = 5 and ζ = 0.

the value reached by the velocity width became even more than the corresponding value for the tracer (see Fig. 3.21), showing that the exchange of momentum trough waves was efficient. We have seen also that for the same case, but taking into account the magnetic field, the instability was decreasing due to the magnetic pressure. Increasing again the Mach number, we found that an efficiency of the wave mechanism became higher (see Fig. 3.23). Comparing three different cases of velocities from the context of characteristic times of vortex formation and merging and time of dynamical saturation it was shown that the system became more stable for higher Mach numbers, which can be explained as an increase of the role of compressibility and causality effects, that we have also seen in the previous section. Warm plasma After examining a cold plasma, with non relativistic √ sound speed, it is reasonable now to consider the maximum possible sound speed (1/ 3), which means to study the warm plasma and see, how results change for this extreme thermodynamic case. √ In the Fig. 3.26 for Mr = 0.5 and Cs = 1/ 3 we how the distribution of the vorticity. The vortex formation time in this case is less than in the corresponding case for the cold gas (see Fig. 3.18), also comparing the velocity and tracer widths (see Fig. 3.19 and Fig. 3.27) one can see that growth rate in the case of the warm plasma is higher. The same


Figure 3.24: Color scale of the vorticity (upper) and natural logarithm of pressure (below) distributions for four different moments: t = 2700 corresponding to the creation of vortices, t = 4250 and t = 5300 corresponding to two intermediate states (multiplication of vortices) and t = 20000 - the last state of the system after damping of the instability due to the strong magnetic field. Set of parameters is: Cs = 0.1, Mr = 1.4 and ζ = 0.2.

relation is valid also for merging time scales (see Fig. 3.18, Fig. 3.22), because as it is clear from these figures, in a cold gas the merging begins at t ∼ 14500 whereas for the hot plasma the merging process starts at t ∼ 2500. As one can see, the mixing effect is quite efficient (δv < δT ), which causes the fact, that the layer widths have a precise step like behaviour. Increasing the Mach number, the situation changes: in Fig. 3.28 for Mr = 1.4 we show again the vorticity distribution for the formation, multiplication and merging processes (two states, corresponding to the beginning and almost ending of the merging process). The vortex formation time in this case is larger than for Mr = 0.5, the system is more stable due to the compressibility. It is seen in Fig. 3.29 that unlike the previous case (Mr = 0.5) the value attained by the velocity mixing layer is higher than the tracer


Figure 3.25: Plots of δv (solid line) and δT (dotted line) versus time. Set of parameters is: Cs = 0.1, Mr = 1.4 and ζ = 0.2.

mixing layer, and their behaviours are more continues with respect to the first case, confirming that the wave mechanism is responsible for the momentum exchange process. As we have already seen, the non linear dynamics may lead to a complicate picture, when apart from the formation and merging, one has also multiplication of the vortices. In Fig. 3.30 one can see that when the instability arises, instead of two vortices (being a consequence of the initial perturbation with two wave lengths in a domain) four vortices are formed (see Fig. 3.30, t = 33800), which also become unstable and convert into two vortices via the merging (see Fig. 3.30,t = 34800). This process is not predictable analytically due to the non linear character of the phenomenon. The Kelvin-Helmholtz instability in this case is not as efficient as for Mr = 0.5 or Mr = 1.4, because of very low growth rate. From Fig. 3.31 one may see that for t ∼ 104 in the system there was no significant growth, whereas for the previous two cases, for the same time scale the vortices were formed and already merged. Also in a case of the hot √ gas, it is interesting to consider a role of the magnetic field. In Fig. 3.32 for Cs = 1/ 3 Mr = 1.4 and ζ = 0.2 we show distributions of the vorticity and magnetic field (natural logarithm of the magnetic field). As in the previous case of the cold gas, we see that the flow follows the magnetic field, this is the reason why the magnetic field and the vorticity have similar configurations, (see Fig. 3.32, t = 800). As it is clear from the pictures, instead of two vortices, which had to arise due to the


Figure 3.26: Color scale of the vorticity (upper) and natural logarithm of pressure (below) distributions for four different moments: t = 200 corresponding to the creation of vortices, t = 2200 corresponding to the starting process of merging, t = 2650 corresponding to the merging itself and t √ = 50000 - the final state of the system when remains only one vortex. Set of parameters is: Cs = 1/ 3, Mr = 0.5 and ζ = 0.

initial configuration, four vortices appear (see Fig. 3.32, t = 1450), in due course one can observe that two effects are working together, the merging of vortices and the damping process due to the magnetic field. This is why, we do not see clearly the process of merging. The magnetic field plays a crucial role for killing the instability and as a result, finally we do not see any more an unstable region in the flow (see Fig. 3.32, t = 4750), the distributions are smooth. The Fig. 3.25 shows the same behaviour as in a case of the cold gas: a mechanism that is responsible for momentum exchange, is the wave mechanism, because as we see from the graphs: δvasy > δTasy .


Figure √3.27: Two plots of δv (solid line) and δT (dotted line) versus time. Set of parameters is: Cs = 1/ 3 and Mr = 0.5.

3.3.4

Conclusion

We have studied the non-linear stability problem of the two dimensional relativistic flow dynamics. The three different cases for the relativistic Mach number (Mr = √ {0.5, 1.4, 5}) were considered for the cold (Cs = 0.1) and extremely hot gas (Cs = 1/ 3). We have studied the dependence of characteristic time scales of the vortex formation, multiplication and merging processes versus the Mach number, sound speed and the Alfvén Mach number. We have shown, that due to the causality principle, the higher the Mach number, the more stable the system. For analyzing the role of direct mixing in the momentum exchange, the velocity and tracer layer widths were introduced. By comparing the corresponding values attained by δT and δv we have shown that for low velocities, the direct mixing was more efficient, than the wave mechanism, whereas increasing the Mach number both layer widths became more continues and unlike low Mach numbers, values attained by δv exceeded the corresponding value of δT , confirming the wave character of the momentum exchange mechanism. Next interesting sequence of the non-linear behaviour was the merging of the vortices. It was shown that the velocity and tracer mixing layer’s step like behaviour was associated with the merging process, which as we have seen, took place for a case when the direct mixing was dominant, and for a case when the wave mechanism was efficient. Examining the extreme thermodynamics, thus √ considering the maximum speed of sound (1/ 3), we found that by virtue of the high

3.3 The Kelvin-Helmholtz instability for relativistic Fluid, for a planar velocity shear layer approximation. Non linear analysis.100

Figure 3.28: Color scale of the vorticity (upper) and natural logarithm of pressure (below) distributions for four different moments: t = 800 corresponding to the creation of vortices, t = 3000 corresponding to the multiplication of vortices, t = 6450 corresponding to the merging process and t =√20000 - the final state of the system when one one has only two vortices. Set of parameters is: Cs = 1/ 3, Mr = 1.4 and ζ = 0.

sound speed the instability rate was higher than in the parallel cases for the cold gas (see corresponding tame scales in cold and hot plasmas). As it has been shown there was no principal difference among vortex shapes. On the other hand a strong dependence on the flow velocity has been indicated, as it was demonstrated, the higher the velocity, the more elongate a corresponding shape of the vortex. Another factor leading to the asymmetry of a vortex configuration, is the magnetic field. It has been shown that due to the frozen-in condition, the flow followed the magnetic field and as a result the latter influenced the configuration of vortices, elongating them. As it was also predicted analytically, we have seen that the magnetic field has an overall stabilizing effect, leading to decreasing of the growth rate and for very strong fields to killing the instability.


Figure √3.29: Two plots of δv (solid line) and δT (dotted line) versus time. Set of parameters is: Cs = 1/ 3, Mr = 1.4 and ζ = 0.


Figure 3.30: Color scale of the vorticity distribution for six different times t = 32500, t = 32800 (beginning of the formation), t = 33800 (almost formed vortices), t = 34800 (beginning of merging), t√= 32850(merging) and t = 50000 (state corresponding to the saturation) Set of parameters is: Cs = 1/ 3 and Mr = 5 and ζ = 0.


Figure √3.31: Two plots of δv (solid line) and δT (dotted line) versus time. Set of parameters is: Cs = 1/ 3 and Mr = 5 and ζ = 0.


Figure 3.32: Color scale of the vorticity (upper) and natural logarithm of pressure (below) distributions for four different moments: t = 800 corresponding to the creation of vortices, t = 3000 corresponding to the multiplication of vortices, t = 6450 corresponding to the merging process and t =√20000 - the final state of the system when one one has only two vortices. Set of parameters is: Cs = 1/ 3, Mr = 1.4 and ζ = 0.2.


Figure √3.33: Two plots of δv (solid line) and δT (dotted line) versus time. Set of parameters is: Cs = 1/ 3, Mr = 1.4 and ζ = 0.

Chapter 4 Summary This thesis is concentrated on various aspects of centrifugal acceleration and instabilities in different astrophysical scenarios. The Chapter 2 was dedicated to the role of centrifugal force in different astrophysical scenarios. In 2.1 we considered dynamics of particles moving along rotating curved trajectories fixed in the equatorial plane. Analyzing the solutions we have shown, that particles can avoid the light cylinder problem in a case of the Archimedes spiral. In section 2.2 we considered a possibility of exciting the electrostatic waves in a pulsar magnetosphere, driven by the centrifugal force. Presenting a linear analysis of the kinematics of a rotating plasma flow we have shown that at the very initial stage of the instability, it becomes so strong, that a non-linear regime is strongly needed. In section 2.3 we have studied the problem of the non thermal emission in AGNs. We have shown that for a case of the straight magnetic field lines, the centrifugal acceleration is very important in producing of the TeV energy photons via the inverse Compton scattering and applied our model for the TeV energy Blazars. In the chapter 3 we study so called non modal instabilities in swirling astrophysical flows and relativistic aspects of the Kelvin-Helmholtz instability. The section 3.1 was dedicated to the non-modal instabilities in swirling astrophysical flows. Considering compressible spectrum of Alfvén, slow magnetosonic and fast magnetosonic waves, we have shown that the swirling character of the flow makes the MHD waves unstable, provoking the energy transfer from the mean flow to the excited waves. In the section 3.2 we studied the linear Kelvin-Helmholtz instability for a magnetized relativistic perfect gas in the plane parallel case for the vortex sheet approximation. We have examined the instability versus physical parameters. In the section 3.3, for extending the study of the Kelvin-Helmholtz instability for longer

Summary

107

time scales, we have considered the nonlinear regime for a planar velocity shear layer case by implementing the numerical code PLUTO. Depending on the physical parameters, we have studied the creation, multiplication and merging processes of vortices. Overall, the main frame of investigation throughout this thesis lies on the analysis of the role of centrifugal acceleration in the physics of high energy radiation astrophysical sources: AGNs and pulsars and the velocity shear flow driven KHI and non-modal instabilities. We hope that our efforts contribute to a better understanding of different aspects of acceleration and instabilities in related astrophysical objects.

Bibliography Abramowicz M.A. & Lasota J. P., 1974, Acta Physica Polonica B5, 327. Abramowicz M.A. & Prasanna A.R., 1990, MNRAS, 245, 720 Abramowicz M.A. & Miller J.C., 1990, MNRAS, 245, 729 Abramowicz M.A., 1990, MNRAS, 245, 733 Abramowicz M.A., 1992, MNRAS, 256, 710 Abramowitz M. & Stegun I.A., 1965, Handbook of Mathematical Functions, Natl. Bur. Stand. Appl. Math. Ser. No. 55, (U.S. GPO, Washington, D.C., 1965) Anile A. M., 1989, Relativistic fluids and Magneto-Fluids. Cambridge Univ. Press, Cambridge Anzer U. & Boerner G., 1983, A&A, 122, 73 Argentina M., Coullet P., & Risler E. 1999, Phys. Rev. Lett. 86, 807 Arons J. & Sharleman E.T., 1979, ApJ, 231, 854 Aurière, M. 1982, A&A, 109, 301 Bacciotti F., Ray T.P., Mundt R., Eisloffel J., & Solf J. 2002, ApJ, 576, 222 Balbus S. A., & Hawley J. F. 1991, ApJ, 376, 214 Belvedere G., Paternó L. & Pidatella R.M., 1989, MNRAS, 237, 827 Bicknell G., Wagner S. & Groves B., 2001, AIPC, 558, 261B Bini D., De Felice F. & Jantzen R.T., 1999, Class. Quantum, Grav., 16, 2105

BIBLIOGRAPHY

109

Birkinshaw M., 1991, MNRAS, 252, 505 Blandford R.D. 1999, in Astrophysical Disks, proc. ASP, 160 Blandford R.D., & Payne D.G. 1982, MNRAS, 199, 883 Blandford R. & Pringle J., 1976, MNRAS, 176, 443 Bloom S.D. & Marscher A.P., 1996, ApJ, 461, 657 Bodo G., Poedts S., Rogava A.D., & Rossi P. 2001, A&A, 374, 337 Bodo G., Mignone A. & R. Rosner, 2004, Phys.Rev.E, 70,036304 Bodo G., Rossi P., Mignone A., Massaglia S., Ferrari A., 2003, NewAR, 47, 577 Bucciantini N. & Del Zanna L., 2006, A&A, 454, 393 Camenzind M. 1997, in IAU Symp. 182, Harbig-Haro Flows and the Birth of Low Mass Stars, ed. B. Reipurth & C. Bertout (Dordrecht: Kluwer), 241. Canizares, C. R., Grindlay, J. E., Hiltner, W. A., Liller, W., & McClintock, J. E. 1978, ApJ, 224, 39 Cao X., 1997, MNRAS, 291, 145 Cassen P. 1978, ApJ, 219, 336 Catanese, M. & Weeks, T. C. 1999, PASP, 111, 1193 Chagelishvili G.D., Rogava A.D., Segal I.N., 1994, Phys. Rev. E, 50, 49 Chagelishvili G.D., Rogava A.D., Tsiklauri D.G., 1996, Phys. Rev. E, 53, 6028 Chedia, O. V., Kahniashvili, T. A., Machabeli, G. Z., Nanobashvili, I. S. 1996, ApSS, 239, 57 Choudhury S. Roy, & Lovelace R.V., 1986, ApJ, 302, 188 Contopoulos, I., Kazanas, D., & Fendt, C. 1999, ApJ, 511, 351 Craik A. D. D., & Criminale W. O. 1986, Proc. R. Soc. Lond. A, 406, 13 Criminale W. O., & Drazin P.G. 1990, Stud. Appl. Math., 83, 123

BIBLIOGRAPHY

110

Dai, Z. G., Zhang, B., Gou, L. J., Meszaros, P. & Waxman, E. 2002, ApJ, 580, L7 Davis C.J., Berndsen A., Smith M.D., Chrysostomou A., & Hobson J. 2000, MNRAS, 314, 241 De Felice F. & Usseglio-Tomasset S., 1991, Class. Quantum Grav., 8, 1817 De Felice F. & Usseglio-Tomasset S., 1993, Class. Quantum Grav., 10, 353 De Felice F., 1990, Rendiconti di Matematica, Serie VII, 10, 59 De Felice F., 1991, MNRAS, 252, 197 De Felice F., 1995, Phys.Rev. A , 52 Del Zanna L., Bucciantini N., & Londrillo P., 2003, A&A, 400, 397 Dermer C.D., Schlickeiser R. & Mastichiadis A., 1992, A&A, 256, L27 Deutsh A., 1955, Ann. Astrophys., 18, 1 Dirac P.A.M., 1930, Proc. Cambredge Phil. Soc., 26, 361 Djorgovski, S., & King, I. R. 1984, ApJ, 277, L49 Downes T.P., Ray T.P., 1998, A&A, 331, 1130 Rybicki G.B., Lightman A.P., 1979, Radiative Processes in Astrophysics. Wiley, New York Ferrari A., 1998, Ann. Rev. Astron. Astrophys., 38, 539 Ferrari A., Trussoni E., & Zaninetti L., 1980, MNRAS, 193, 469 Ferreira J., & Pelletier G. 1995, A&A, 295, 807 Gangadhara R.T., Lesch H., 1997, A&A 323, L45 Gangadhara, R. T. 1996, A&A, 314, 853 Gerwin R.A., 1968, Rev. Mod. Phys. 40, 652 Ghisellini, G., Haardt, F., & Fabian A.C. 1993, MNRAS, 263, L9 Gold, T. 1968, Nature, 218, 731

BIBLIOGRAPHY

111

Gold, T. 1969, Nature, 221, 25 Goldreich P. & Julian W.H., 1969, ApJ., 157, 869 Goldreich P. & Lynnden-Bell D., 1965, MNRAS, 130, 125 Hénon, M. 1961, Ann.d’Ap., 24, 369 Hagiwara, K., & Zeppenfeld, D. 1986, Nucl.Phys., 274, 1 Hardee, P.E., Rosen, A., Hughes, P.A., Duncan, G.C., 1998, ApJ, 500, 559 Harris, W. E., & van den Bergh, S. 1984, AJ, 89, 1816 Hartman R.C., et al., 1999, ApJS, 123, 79 Hasegawa H., Fujimoto M., Takagi K., Saito Y., Mukai T. & Réme H., 2006, JGRA, 11109203H Hauser, M. G., & Dwek, E. 2001, ARA&A, 39, 249 Heiles, C. & Troland, T. H., 2003, ApJS, 145, 329H Heyl J.S., 2000, Ap.J., 542, L45 Horns, D., 2002, preprint (astro-ph/0209454) Hujeirat A., Papaloizou J.C.B. 1998, A&A, 340, 593 Jimenez-Bailon E., Santos-Lleo M., Mas-Hesse J.M., Guainazzi M., Colina L. et al. 2003, ApJ, 591, 791 K¨ onigl A., & Pudritz R.E. 2000, in Protostars and Planets IV, ed. V. MAnnings, A.P. Boss, & S.S. Russell (Tucson: Univ. Arizona Press), 759. Königl A, 1981, ApJ, 243, 700 Kazbegi A., Machabeli G. & Melikidze G., 1991, MNRAS, 253, 377 Kazbegi A., Machabeli G., Melikidze G. & Shukre C., 1996, A&A, 309, 515 Kim, W.-T., Ostriker, E., & Stone, J. M., 2003, ApJ, 599, 1157 King, I. R. 1966, AJ, 71, 276

BIBLIOGRAPHY

112

King, I. R. 1975, Dynamics of Stellar Systems, A. Hayli, Dordrecht: Reidel, 1975, 99 King, I. R., Hedemann, E., Hodge, S. M., & White, R. E. 1968, AJ, 73, 456 Kino, M., Takahara, F. & Kusunose M. 2002, ApJ, 564, 97 Kolesnikov F., B¨ unte M., Schmitt D., & Sch¨ ussler M., 2004, A& A, 420, 737 Komissarov S.S., 1999, MNRAS, 303, 343 Kompaneec D.A., 1978, Pisma v AJ, 4, 304 Kron, G. E., Hewitt, A. V., & Wasserman, L. H. 1984, PASP, 96, 198 Kubo, H., et al. 1998, ApJ, 504, 693 Lagnado R. R., Phan-Thien N., & Leal L. G. 1984, Phys. Fluids, 27, 1094 Lery T., Henriksen R.N., & Fiege J.D. 1999, A&A, 350, 254 Lord Kelvin, 1871, Philos. Mag. 42, 362 Lord Kelvin, 1887, Philos. Mag. 24, 188 Lynden-Bell, D., & Wood, R. 1968, MNRAS, 138, 495 MacDonald D.A. & Thorne K.S., 1982, MNRAS, 198, 345 Machabeli, G.Z., Rogava, A.D. 1994, Phys. Rev. A, 50, 98 Machabeli G. & Usov V.V., 1979, Soviet Astron. Lett., 5, 238 Machabeli G.Z., Nanobashvili I.S. & Rogava A. D., 1996, Izv. Vuz. Radiofizika, 39, 39 Machabeli G.Z., Vladimirov S. V. & Melrose D.B., 1999, Phys. Rev. E, 59, 4552 Mahajan S. M., & Rogava A. D. 1999, ApJ, 518, 814 Mahajan S.M., Machabeli G., & Rogava A.D., 1997, ApJ, 479, L129 Mannheim K., 1993, A&A, 269, 67 Maraschi L., Ghisellini G. & Gelotti A., 1992, ApJ, 397, L5

BIBLIOGRAPHY

113

Marti J.Ma, Muller, E., Font, J.A., Ibanez, J.Ma, and Marquina, A., 1997, ApJ, 479, 151 Massaglia S., Micono M., Ferrari A., Bodo G., Rossi P., 1997, IAUS, 182, 335 Matsumoto Y. & Hoshino M., 2004, AGUFMSM, 13B, 1214M Mignone A., Bodo G., Matsakos T., Tesileanu O., Zanni C., Ferrari A., ApJ, (accepted) Muslimov A.G. & Tsygan A.I., 1992, MNRAS, 255, 61 Newell, E. B., & O’Neil, E. J. 1978, ApJS, 37, 27 Okumura K., Asahara A., Bicknell G.V. et al. 2002, ApJ, 579, L9-L12 Ortolani, S., Rosino, L., & Sandage, A. 1985, AJ, 90, 473 Osmanov Z.N., Machabeli G. & Rogava A.D., 2002, Phys. Rev. A, 66, 042103 Pagels A., Krichbaum T.P., Graham D.A., Alef W.,Kadler M., Kraus A., Klare J., Witzel A., Zensus J.A., Greve A., Grewing M., Booth R., Conway J., 2004, astroph/0409328 Perucho M., Hanasz M., Mart J.M., Sol H.,2004, A&A, 427, 415 Perucho M., Mart J.M.,Hanasz M., 2004, A&A, 427, 431 Perucho M., Hanasz M., Mart J.M., Miralles J.A., 2005, A&A, 801, 214 Perucho M., Lobanov A., Mart J.M., Hardee P.E., 2006, A&A, 456, 493 Perucho M., Lobanov A., 2006, astro-ph/0607386 Peterson, C. J. 1976, AJ, 81, 617 Peterson, B. M. 1997, An introduction to active galactic nuclei, (Cambridge University Press, Cambridge), 34 Pike C. D., & Mason H. E. 1998, Solar Phys., 182, 333 Pneuman G. W., & Orrall F. Q. 1986, in Physics of the Sun, Vol. II, ed. P. A. Sturrock (Dordrecht: Reidel), 71 Poedts S., Rogava A. D., & Mahajan S. M. 1998, ApJ, 505, 369

BIBLIOGRAPHY

114

Prasanna A.R., 1991, Phys. Rev. D, 43, 1418 Pu Z.-Y. & Kivelson M.G., 1984, ChJSS, 4, 105 Ray T., 1982, P&SS, 30, 245 Rieger, F. M., & Duffy, P. 2004, ApJ, 617, 155 Rieger, F. M., & Mannheim, K. 2000, A&A, 353, 473 Rieger, F. M. 2005, preprint (astro-ph/050294) Rindler W., 1960, Special Relativity, (Oliver and Boyd, Edinburgh and London 1960) Rogava A.D. & Mahajan S.M., 1997, Phys. Rev. E, 55, 1185 Rogava A. D., Poedts S., & Heirman S. 1999, MNRAS, 307, L31 Rogava A. D., Poedts S., & Mahajan S. M. 2000, A&A, 354, 749 Rogava, A. D., Dalakishvili, G., Osmanov Z.N., 2003, Gen. Rel. and Grav. 35, 1133 Rogava A.D., Mahajan S.M., Bodo G., & Massaglia S. 2003, A&A, 399, 421 Rosen, A., Hughes, P.A., Duncan, G.C., Hardee, P.E., ApJ, 1999, 516, 729 Ruderman M.A. & Sutherland P.G., 1975, ApJ, 196, 51 Rudnick, G. et al., 2003, ApJ, 599, 847 Ryutova M., Habbal S., Woo R., & Tarbell T. 2001, Sol. Phys., 200, 213 Shibata K., & Uchida Y. 1986, Solar Phys., 103, 299 Shu F.H., Najita J.R., Ostriker E., & Shang H. 1995, ApJ, 455, L155 Sikora M. Begelman M.C. & Rees M.J., 1994, A&A, 421, 153 Spitzer, L. 1985, Dynamics of Star Clusters, J. Goodman & P. Hut, Dordrecht: Reidel, 109 Sturrock P. A., 1994, Plasma Physics, Cambridge University Press, Cambridge Sturrok P., 1971, ApJ., 164, 529

BIBLIOGRAPHY

115

Swanenburg B.N., et al., 1990, Nature, 275, 298 Tademaru, 1973, ApJ., 183, 625 Tagger M., & Pellat R. 1999, A&A, 349, 1003 Tagger M., Pellat R., & Coroniti F. V. 1992, ApJ, 393, 708 Thomas R. M. C. & Gangadhara R.T., 2005, A&A 437, 537 Thorne K.S. & MacDonald D.A., 1982, MNRAS, 198, 339 Thorne K.S., Price R.H. & MacDonald D.A., 1986, eds. Black Holes: The Membrane Paradigm (Yale University Press, New Haven 1986) Trefethen L.N., Trefethen A.E., Reddy S.C., Driscoll T.A. 1993, Science, 261, 578 Treu, T. et al., 2003, ApJ, 591, 53 Turland B.D. & Scheuer P.A.G., 1976, MNRAS, 176, 421 Uberoi C., 1984, JRG, 89, 56652 Usov V.V. & Shabad A., 1985, ApSS, 117, 309 Varnièr P., & Tagger M. 2002, A&A, 394, 329 Von Helmholtz H., Monats K., 1868, Preuss. Akad. Wiss. Berlin, 23, 215 Wang Y.-M., Robertson J.A., 1984, A&A, 139, 93 Wilson A.S. & Colbert E.J.M. 1995, ApJ, 438, 62 Zanni C., Massaglia S., Bodo G., Rossi P., Capetti A., and Ferrari A., Mem. S.A.It. Suppl. Vol. 1, 155, 2003, Zanni C., Murante G., Bodo G., Massaglia S., Rossi P., Ferrari A., 2004, preprint (astroph/0103200)

Appendix A 4.1

Derivation of Eq. (2.95)

Note that:

Z

0

dω 0 e−iω t Nf (ω 0 )

Nf (t) =

(A.1a)

where f = b, pl then

Z

dte

iω 0 t

Z Z

=−

Z Z ∂2 Z 0 0 −iω 0 t 0 dω e Nf (ω ) = dtdω 0 eiωt (−iω 0 )2 e−iω t Nf (ω 0 ) = 2 ∂t Z

0

dtdω 0 eit(ω−ω ) (ω 0 )2 Nf (ω 0 ) = −2π

dω 0 (ω 0 )2 δ(ω − ω 0 )Nf (ω 0 ) =

= −2πω 2 Nf (ω)

(A.1b)

where the following representation of the delta function has been examined: 1 Z δ(x) = dkeikx , 2π c V0pl − V0b sin Ωt ≈ Rpl − Rb = Ω Ω

Ã

(A.1c)

!

1 1 c − 2 sin Ωt ≈ − 2 sin Ωt, 2 2γ0b 2γ0pl 2γ0pl Ω

(A.1d)

then e−ik(Rpl −Rb ) ≈ eia sin Ωt , a=

kc . 2 2γ0pl Ω

(A.1e) (A.1f )

2 , and the following We have considered the approximate expression: V0i ≈ 1 − 1/2γ0i observable fact: γ0b À γ0pl . As one can see, the speed of light has been restored again.

4.2 Derivation of Eq. (2.96)

117

By using the following identity: e±ix sin Ωt =

X

Js (x)e±isΩt ,

(A.1g)

s

one may easily transform e−ik(Rp −Rb ) into its Fourier mode: Z

dte

iωt

X

Js (a)e

s

=−

X

∂2 Z 0 dω 0 e−iω t Np (ω 0 ) = 2 ∂t

Z Z

X

0

dtdω 0 (ω 0 )2 Np (ω 0 )eit(ω+sΩ−ω ) =

Js (a)

s

= −2π

isΩt

Z

dω 0 (ω 0 )2 Np (ω 0 )δ(ω + sΩ − ω 0 ) =

Js (a)

s

= −2π

X

Js (a)(ω + sΩ)2 Np (ω + sΩ).

(A.1h)

s

Combining Eqs. (A.1b) and (A.1h), taking into consideration Eq. (2.94), one finally will obtain: n0 γ 3 X ω 2 Nb (ω) = b0 0pl3 (ω + sΩ)2 Js (a)Npl (ω + sΩ). (A.2) 2npl γ0b s

4.2


One can easily transform Eq. (2.71) into the following form: ´

³

kE1 = 4πe Nb (t)e−ikRb + Npl (t)e−ikRpl .

(B.1a)

Taking Eq. (2.91) into account, one can have: ´ 8πen0pl ³ ∂ 2 Npl (t) ik(Rpl −Rb ) = − N (t) + N (t)e . pl b 3 ∂t2 mγ0pl

(B.1b)

For Fourier expansion of the left-hand side of Eq. (B.1b) one can analogously go to Eq. (A.1b) write down: Z

dte

iω 0 t

∂2 Z 0 dω 0 e−iω t Npl (ω 0 ) = −2πω 2 Npl (ω) 2 ∂t

(B.1c)

while, for the second term in a bracket, one finds: X s

Z Z

Js (a)

0

dtdω 0 eit(ω−sΩ−ω ) Nb (ω 0 ) = −2π

X s

Z

Js (a)

dω 0 Nb (ω 0 )δ(ω − sΩ − ω 0 ) =

4.3 Derivation of Eq. (2.98)

118

= −2π

X

Js (a)Nb (ω − sΩ).

(B.1d)

s

Combining Eqs. (B.1c), and (B.1d) we will obtain Eq. (2.96): Ã

!

2 2 X ωpl ωpl ω − 3 Npl (ω) = 3 Js (a)Nb (ω − sΩ) γ0pl γ0pl s 2

where

s

ωpl =

4.3

8πn0pl e2 . m

(B.2a)

(B.2b)


From Eq. (2.86) we have: Nf (t) = n1f (t)eikRf .

(C.1a)

Taking into consideration Eq. (A.1g), and a condition γa À 1 one finds: Nf (t) =

X s

where

Js (Af )eisΩt n1f (t)

(C.1b)

kc Ω kc Apl = . 2 2Ωγ0pl Ab =

(C.1c) (C.1d)

Fourier analysis of the right-hand side of the Eq. (C.1b) gives following: X

Z Z

Js (Af )

0 it(ω+sΩ−ω 0 )

dtdω e

0

nf (ω ) = −2π

s

X

Z

Js (Af )

dω 0 nf (ω 0 )δ(ω + sΩ − ω 0 ) =

s

= −2π

X s

Hence we will have: Nf (ω) =

Js (Af )n1f (ω − sΩ)

X s

Js (Af )n1f (ω − sΩ)

(C.1e) (C.2)