Nonlinear Phenomena in Rotating Astrophysical Objects

Contents

1 Introduction
1.1 Concept of Plasma
1.1.1 The Evolution of Plasma Physics
1.1.2 Plasma Characteristics
1.1.3 Classification of Plasmas
1.1.4 Occurrences of Plasmas
1.1.5 Applications of Plasmas
1.2 Electron-Positron Plasmas
1.2.1 Occurrences Electron-Positron Plasmas
1.3 Different Plasma Waves Phenomena
1.3.1 Short Notes on Alfvén Waves
1.3.2 Types of Alfvén Waves
1.4 Rotating Astrophysical Objects
1.5 Linearity and Nonlinearity of Waves
1.6 Existing Theoretical Nonlinear Approach
1.7 Nonlinear Wave Phenomena
1.8 Layout of The Book

2 Nonlinear Compressional Alfvén Waves in a Pair Plasma
2.1 Introduction
2.2 Governing Equations
2.3 Derivation of The K-dV Equation
2.4 Solution of The K-dV Equation
2.5 Numerical Analysis
2.6 Discussion

3 Nonlinear Solitary Waves in a Pair Plasma
3.1 Introduction
3.2 Governing Equations
3.3 Derivation of NLSE
3.4 Numerical Analysis
3.5 Modulational Instability
3.6 Discussion

4 Nonlinear Shear Alfvén Waves in a Pair Plasma
4.1 Introduction
4.2 The Mathematical Model
4.3 Derivation of The DNSE
4.4 Solution of The DNSE Equation
4.5 Numerical Analysis and Results
4.6 Conclusion

5 Summary

6 Appendix
6.1 Maxwell’s Equation in a Rotating Frame
6.2 Solution of the Korteweg-de Vries (K-dV) Equation

Bibliography

Dedication

To my Parents and Supervisors

(Dr. Sharmin Sultana and Prof. Dr. A. A. Mamun), for their fundamental motivation to all my work as well as their love and support

Acknowledgment

First, and foremost, I would like to acknowledge and thank my supervisor, Dr. S. Sultana [PhD (Queen ’ s University Belfast, UK)], whose patience and kindness, as well as her academic experience, had been invaluable to me. I wish to thank her for invaluable support, advice, encouragement, and excellent supervision during my research time. I have greatly benefited from her thorough deep knowledge and expertise in plasma physics research field in general and nonlinear waves in particular. The countless discussions we have had and the explanations as well as original ideas she gave me were an invaluable contribution to this thesis. Not only she has been invaluable for the development of my M.Sc. thesis, but also it has furthermore always been a great pleasure to work with her. I feel very lucky to have had such understanding and I am honored to be her student. I also appreciate and acknowledge her kind support as well as good advice for matters beyond my research work.

I would like to express my sincere and hearty appreciation to my joint su- pervisor Prof. A. A. Mamun [PhD (St. Andrews), AvH Research Awardee (Germany), AvH Fellow (Germany), Commonwealth Fellow (UK), Regular Associate (ICTP, Italy), Commonwealth Scholar (UK), Bangladesh Academy of Science Gold Medal Awadee] for his constant encouragement, unparallel stimulating influence, un-tiring efforts, and invaluable suggestions through- out the progress of my research work. I am very proud to do my research work with him, a very well known plasma physicist, who ’ s regular ambition is research, students progress, humanity, and discipline. I have never met such a person before.

My special thanks must go to the senior and junior members of our Plasma Physics Research Group for all the bad and good times we had together as well as for their continuous supports during my thesis work. I wish to extend my thanks to my lab-mates for their support and co-operation.

On a more personal basis, I would like to thank my parents, brothers, and sister-in-law for all the good they brought and are still bringing to my life as well as for their unwavering help, encouragement, and support.

I would like to acknowledge the Bangladesh Ministry of Science and Tech- nology for their financial support through the National Science and Technol- ogy (N.S.T) Fellowship. This fund has given me the freedom to focus on my studies.

Finally, I cannot thank everyone individually for the help they have given me in the production of this work, but I would like to finish with a final thank to all those people who have made my life so wonderfully enjoyable.

May 2015 Tanvir I. Rajib

Abstract

The nonlinear propagation of electromagnetic (EM) waves in a pair plasma consisting of electron-positron (e-p) has been rigorously investigated in rotating astrophysical objects (e.g. pulsar magnetosphere). Two nonlinear approaches have been used to analyze the nonlinear wave phenomena in pulsar magnetosphere. The reductive perturbation method has been employed in order to derive the Korteweg-de Vries (K-dV) and derivative nonlinear Schrödinger equation (DNSE) as well as Wentzel, Kramers, and Brillouin (W.K.B) approximation technique has also been used to derive nonlinear Schrödinger equation (NLSE). The steady state solutions of these nonlinear equations have been obtained, and analyzed theoretically and numerically. The existence of solitary waves (SWs) in e-p plasma are found depending on the basic plasma parameters of considered e-p plasma. It has been seen that the basic features of SWs is significantly modified by the effects of rotational frequency (RF) and positron-to-electron thermal energy ratio (ER).

The results, which have been found in this dissertation are summarized as follows: (i). e-p plasma medium supports low phase speed, high frequency, compressional e-p Alfvén (CEPA) waves, and shear e-p Alfvén (SEPA) waves;

(ii). Both the amplitude and the width of CEPA, and SEPA SWs are seen to increase with the increase of ER. It is investigated that the RF does not have any influence on the amplitude and width of the K-dV and DNSE SWs observed in the e-p plasma of crab pulsar but the RF has significant effect on the basic properties of SWs obtained via NLSE; (iii). The frequency of EM waves propagated in the crab pulsar lies in the range of radio waves. It is also observed that the phase speed of the EM waves increases with the decreasing values of the angle between the magnetic moment and the rotational axis, and waves are modulationally unstable when W.K.B technique has been used. The findings of these investigations may use to understand the nonlinear EM waves phenomena in rotating astrophysical and laboratory plasmas.

List of Symbols

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Abbreviations

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Related Publications

1. T. I. Rajib, S. Sultana, and A. A. Mamun: Solitary Waves in Rotational Pulsar Magnetosphere: Astrophysics and Space Science (Netherlands), 357, 52 (2015).

2. T. I. Rajib, S. Sultana, and A. A. Mamun: Nonlinear Compressional Alfvén Waves in a Fully Relativistic Electron-Positron Plasma: IEEE Transactions on Plasma Science (USA), 45, 4 (2017).

3. T. I. Rajib, S. Sultana, and A. A. Mamun: Shear Alfvén Waves in a Magnetized Electron- Positron Plasma: IEEE Transactions on Plasma Science (USA), 99, 1 (2017).

Chapter 1

Introduction

1.1 Concept of Plasma

In the mid-19th century the Czech physiologist Jan Evangelista Purkinje introduced use of the Greek word plasma to denote the clear fluid which remains after removal of all the corpuscular material in blood. Half a century later, the American scientist Irving Langmuir proposed in 1922 that the electrons, ions, and neutrals in an ionized gas could similarly be considered as corpuscular material entrained in some kind of fluid medium and called this entraining medium plasma. However, it turned out that unlike blood where there really is a fluid medium carrying the corpuscular material, there actually is no fluid medium entraining the electrons, ions, and neutrals in an ionized gas [1].

Ionized gas is called plasma but any ionized gas can not be called plasma, of course; there is always some degree of ionization in any gas. There are some basic criteria that are required for being plasma formation. These criteria are necessary to distinguish between an ionized gas and plasma itself. The basic criteria, namely, charge-neutrality (“macroscopical neutrality”), collective behaviour, and collision condition.

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Figure 1.1: The transitions between the phases of matter. ‘Solid’ transforms into ‘liquid’ through melting. ‘Liquid’ transforms into ‘gas’ through ‘vaporization’ and ‘gas’ transforms into ‘plasmas’ through ionization [Source: powerlisting.wikia.com].

(1) Quasi-Neutrality: Quasi-neutrality (quasi, from the Latin, ‘as if’, ‘resembling’) describes the apparent charge neutrality of a plasma overall, while at smaller scales, the positive and negative charges making up the plasma, may give rise to charged regions and electric fields i.e. they are macroscopically neutral, where plasma dimension L >> Debye radius λ D. Since electrons are very mobile, plasmas are excellent conductors of electricity, and any charges that develop are readily neutralized, and in many cases, plasmas can be treated as being electrically neutral [2, 3].

(2) Collective Behavior: Collective behavior means that the behavior of a specific point of interest may affect by the state of distant regions from it (i.e. large number of particle in a Debye sphere). In a plasma system, any charged particles is acted upon by all other charged particles through the long-range electrostatic force. Thus, we can say that a plasma system is like a network of particles connected by mass-less springs⁴. All the charged particles (either electrons or ions) move together collectively to external disturbance. If a charged particle moves in a plasma (under the influence of any disturbance), it produces local concentration giving rise to electric fields. Motion of charged particles generates currents which produces magnetic field. These electric and magnetic fields affect the motion of other charged particles far away. This is called the collective behavior i.e. ND >> 1, where ND number of particles in the Debye sphere.

(3) Collision Condition: If ω is the frequency of typical plasma oscillations and τ is the mean time between collisions with neutral atoms, we require ωτ >> 1 for the gas to behave like a plasma rather than a neutral gas. Their motion is governed by electromagnetic forces rather than ordinary hydrodynamic forces i.e Plasma frequency ω p >> collision frequency ν.

A simple but complete definition of the plasma can be given as “ A plasma is defined as a microscopically neutral substance containing many interacting charged and neutral particles which exhibit collective behavior due to the long ranged electromagnetic forces ”.

Like gas, plasma does not have a definite shape or a definite volume unless enclosed in a container; unlike gas, under the influence of a magnetic field, it may form structures such as filaments, beams and double layer. It is a very interesting fact that 99 . 99 per cent of the visible matters in our universe is in the plasma state ⁴. In the Universe, plasma is the most common state of matter for ordinary matter both by mass and by volume ⁵, most of which is in the rarefied intergalactic plasma (particularly intracluster medium), and in stars. Some common plasmas are found in stars and neon signs. All the stars are made of plasma, and even the space between the stars is filled with a plasma, albeit a very sparse one. So, plasmas have been attracted a great deal of interest in recent days.

1.1.1 The Evolution of Plasma Physics

The field of plasma is indeed a large one and has grown in diversity since the early studies of Nobel Prize winning American chemist Irving Langmuir and his colleague Lewi Tonks [6, 7]. Some other engineers and scientists that contributed immensely to the field of plasma physics include:

(i). Edward Appleton and his colleague K.G. Budden - who basically developed the theory of electromagnetic wave propagation through non-uniform magnetized plasmas.

(ii). Hannes Alfvén who developed the magneto-hydrodynamics (MHD) theory; which basically treats plasma as a conducting fluid. (iii). James V. Allen who pioneered the exploration of the Earth’s magnetosphere and discovered the Van Allen radiation belts surrounding the earth.

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Figure 1.2: Graphical representation of different plasma systems for different plasma parameters [Courtesy of the contemporary physics education project].

1.1.2 Plasma Characteristics

There are three basic criteria that are required for plasma formation. The basic criteria, namely, charge-neutrality (“macroscopical neutrality”), collective behavior, and collision condition. Apart from these basic criteria there are also some other plasma characteristics such as Debye shielding, plasma frequency, cyclotron frequency, and collision frequency are discussed below:

(1) Debye Shielding: A fundamental characteristic of a plasma is its ability to shield the electric field of an individual charge particle or of a surface that is at some non zero potential. This characteristic provides a measure of the distance which is called Debye radius over which the influence of the electric field of an individual charged particle is felt by other charged particles inside plasma.

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Figure 1.3: Debye shielding [Ref. Chen].

If we consider an electric field is applied into a plasma by putting two charged metal surfaces connected to a battery. The charged surfaces would attract potentials of opposite charges and almost immediately a cloud of ions will surround the negatively charged surfaces, and a cloud of electrons will surround the positively charged surface. If the plasma is cold and there is no thermal motions, there would be just as many charges in the cloud as on the surfaces, and the shielding would be perfect. On the other hand, if the temperature is finite, there would have enough thermal energy to escape from the electrostatic potential well. The edge of the cloud occurs at the radius where the potential energy is approximately equal to the thermal energy kBTs, where kB is the Boltzmann’s constant and Ts is the temperature of the species s, and the shielding is not complete. Potentials of the order kB Ts/qs can leak into the plasma and cause a finite electric field to exit ⁴.

A measure of the plasma shielding is called the Debye length λ Ds which is given by mathematically in CGS system

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where qs is the charge of the species s and nso is the equilibrium plasma density of the species s.

Here, describing the plasma as quasi-neutral means that its neutral enough to assume nio = neo = no for normal electron-ion plasma (here, nio and neo are equilibrium number density of ion and electron, respectively). An effective Debye length λ D is related to the electron Debye length λ De and ion Debye length λ Di, which can be mathematically defined as in CGS system

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(2) Characteristics Frequency: When a plasma is instantaneously disturbed from its equilibrium, the resulting internal space charged field gives rise to collective particle motions which tend to restore the original charge neutrality. These collective motions are characterized by a neutral frequency of oscillation known as plasma frequency ω p. There are two different types of characteristics frequency, namely plasma frequency and collision frequency.

(a) Plasma Frequency ω p: When the plasma particles (i.e. ions, electrons, or positrons) are displaced from their equilibrium position, a space charge field will be built up in such a direction as to restore the neutrality of the plasma by pulling the particles back to their original positions. But because of their inertia they will overshoot and will be again pulled back to their original position by the space charge field of the opposite polarity. Again because of their inertia they will overshoot and thus continuously oscillate about their equilibrium positions, i.e. when a plasma is instantaneously disturbed from its equilibrium positions, the resulting internal space charge field gives rise to collective particle motions which tend to restore the original charge neutrality. These collective motions are characterized by a natural frequency of oscillation known as plasma frequency ω p. This plasma frequency will not the same for electrons, ions, and dust grains, but will depend on the mass and charge of the plasma particles. The plasma frequency ω p is defined as in CGS system

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where nso, qs, and ms are the un-perturbed number density, charge, and mass of the plasma species s, respectively.

(b) Collision Frequency ν sn: The collision frequencies are associated with the collision of plasma particles (electrons, ions, and positrons) with stationary neutral. These are electron-neutral collision frequency ν en, ion-neutral collision frequency ν in, and the positron-neutral collision frequency ν pn. The collision frequency ν sn for the scattering of the plasma species s by the neutral is ν s = nn σ n s V T s, where nn is the neutral number density, σ n s isthescatteringcrosssection,and V T s = (kB Ts/ms)¹ / ² is the thermal speed of the species s. The oscillations will be slightly damped only when ν s < ω p.

(3) Cyclotron Frequency: The number of rotations of plasma species per second around the axis of magnetic field is called the plasma cyclotron frequency. Mathematically, it can be defined as in CGS system

Abbildung in dieser leseprobe nicht enthalten

where B 0 is the external magnetic field and c is speed of light in vacuum.

1.1.3 Classification of Plasmas

Based on the relative temperatures of the electrons, ions, and neutrals plasmas are classified as either Thermal or Non-Thermal plasma. Thermal plasmas are plasmas in which the plasma is said to approach a state of local thermodynamic equilibrium (LTE). LTE occurs when the temperatures of the electrons and the relatively heavier particles (ions and neutrals) are equal or in other words, the particles are in thermal equilib-rium with each other. These thermal plasmas are generally produced by atmospheric arcs, sparks, and flames.

In non-thermal plasmas, the thermal motion of the ions can be ignored. As a result, there is no pressure force, the magnetic force can be ignored, and only the electric force is considered to act on the particles. Furthermore, the electrons are not in thermal equilibrium with the heavier particles (for example ions). The temperature of the ions and neutrals are generally at a much lower temperature sometimes around room temperature, whereas the electrons are at a much higher temperature. This is sometimes referred to as non-LTE (NLTE). Examples of non-thermal plasmas include the Earth’s ionosphere, and the flow discharge in a fluorescent tube.

1.1.4 Occurrences of Plasmas

Plasmas are common in nature and found nearly everywhere. For instance, stars are predominantly plasma as are most space and astrophysical objects. Plasmas are also found on Earth where they find a wide range of uses. Plasmas are indeed complex and exist in differing situations by many orders of magnitude. Most importantly, plasmas do not normally exist in ordinary human experience and as a result, people do not have the sort of intuition for plasma behavior that they have for solids, liquids or gases.

Plasmas occur naturally and can also be man made, and comprise the majority of the universe encompassing among other phenomena, the solar corona, solar wind, nebula and the earth’s ionosphere. In the earth’s atmosphere, plasma is often observed as a transient event in the phenomenon of lightning strikes. Since air is normally non-conducting, large potential differences can be generated between clouds and earth during storms and these lightning discharges occur to neutralize the accumulated charge in the clouds. As one progresses further into near-space altitudes, the earth’s magnetic field interacts with the charged particles streaming from the sun. These particles are diverted and often become trapped by the earth’s magnetic field. The trapped particles are most dense near the magnetic poles.

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Figure 1.4: Lightning is the example of naturally occurred plasma [Source: http://plasmauniverse.info].

1.1.5 Applications of Plasmas

The physics of plasma is a topic of growing importance. Different types of plasmas regulate different applications and different natural phenomena. However, many fundamental considerations span the broad parameter ranges that characterize the many natural and man-made plasmas that are important in our lives. Plasma technologies are numerous and involve many industries. As a result, the progress in plasma research has led to a wide range of plasma applications. The number of industrial applications of plasma technologies is extensive and involves many industries, especially electronics, lightning, coatings, treatment and processing of materials, metallurgy and energy systems. Such a situation takes place, for example, in thermal plasma deposition of protective coatings, in plasma stabilization of flames, in plasma conversion of fuels ⁸.

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Figure 1.5: Thermonuclear fusion reaction [Source:

www.fusion.kit.edu/english/79.php].

The main issue for practical use of any chemical process in a particular plasma system is to find the proper regime and optimal plasma parameters among the numerous configurations available. The list below shows some of the more important practical applications of plasma physics:

(i). Plasma etching;

(ii). Material analysis; (iii). Plasma propulsion;

(iv). Controlled thermonuclear fusion;

(v). The magneto-hydrodynamic generator;

(vi). Nanotechnology: agglomeration and coagulation; (vii). Microbiology: electrostatic Disruption of bacteria.

1.2 Electron-Positron Plasmas

An electron-positron (e-p) plasma is a fully ionized gas composed of electrons and positrons having equal masses and charges with opposite polarity, is considered not only as a building block of our early universe ⁹, but also as an omnipresent ingredient of a number of astrophysical objects, such as active galactic nuclei ¹⁰, pulsar magnetospheres ¹¹, solar flares ¹², fireballs producing gamma ray bursts ¹³, etc. It is believed that after the Big Bang, e-p plasmas occurred in the MeV-epoch of the early Universe. Electron-positron plasmas are also observed in laboratory experiments in which the positrons can be used to probe the particle transport in the tokamak plasmas. Processes of electron- positron pair production can occur during intense short laser pulse propagation in plasmas ¹⁴. However, because of the rather long lifetime of positrons, most of the astrophysical and laboratory plasmas [9, 12, 13] becomes an admixture of electrons, positrons, and ions. It is found that the e-p plasmas behave differently from typical electron-ion (e-i) plasmas. The wave propagation in such a two component e-p plasma [15-33] has attracted much interest in the study of space plasmas.

1.2.1 Occurrences Electron-Positron Plasmas

Two component e-p plasmas occur naturally in astrophysical backgrounds [9-12, 23, 34-49] and have been encountered in laboratory experiments. The occurrences of e-p plasmas are given below:

I. Space and Astrophysical Plasmas:

- Magnetars;
- Microquasars;
- Neutron Stars;
- Early Universe;
- Solar Atmosphere;
- Center of Our Galaxy;
- Active Galactic Nuclei;
- Pulsars Magnetosphere;
- Ultra Intense Laser Fields;
- Cosmological Gamma Ray Fireballs.

II. Laboratory Plasmas:

- Tokamaks;

- Semiconductor Plasma.

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Figure 1.6: Schematic diagram of a tokamak [Source: physics.stackexchange.com].

1.3 Different Plasma Waves Phenomena

Waves means periodic oscillations which carry energy from one place to another place. A wave is a disturbance that moves through a medium in a manner that at any point the displacement is a function of time. In general, there are three most common types of wave known in the literature, such as : longitudinal/electrostatic (e.g. ordinary sound wave), transverse/electromagnetic (e.g. light waves), and mixed of electrostatic and electromagnetic (e.g. extraordinary waves). There are six terms that are used to describe relations among the four quantities as external magnetic field B0, wave number k, perturbed electric field E1, and perturbed magnetic field B1. Mention that for parallel propagation of waves angle θ between wave vector k and electric field E1 or magnetic field (B0 or B1) is zero i.e. θ = 0 ◦. But in case of perpendicular propagation θ will be 90 ◦.

If k is along B0 (i.e. k · B0 = 1), we call parallel wave. If k · B0 = 0, the wave is perpendicular. If k · E1 = 1, the wave is longitudinal, while if k · E1 = 0 the wave is transverse. When B1 = 0 the wave is electrostatic, while if B1 = 0, the wave is electromagnetic. The waves can travel

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Figure 1.7: Propagation of electromagnetic waves along +x direction [Source: miniphysics.com].

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Figure 1.8: Transverse and Longitudinal waves [Source: physics.stackexchange.com].

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Figure 1.9: The electric field vectors circularly polarized electromagnetic waves [Source: Wikipedia].

across B0 or along B0. It’s also true that waves can travel at any angle to B0. If so happens, there will be two modes for any angle of propagation, and their properties will be some combination of the properties of ordinary, extra-ordinary, Right hand circularly polarized (RHCP), and Left hand circularly polarized (LHCP) waves [4, 50].

1.3.1 Short Notes on Alfvén Waves

In 1942, after the discovery of new solar phenomena, the Swedish physicist Hannes Alfvén unified the mutual interactions between ionized gases and magnetic fields by writing the equations describing the motion of electromagnetic fluids. The mathematical solution of this new fluid consists of waves of electrons and ions that were found not only in laboratory plasma experiments but also in the plasma of our atmosphere and sun. These waves were later called Alfvén waves.⁵¹.

An Alfvén wave in a plasma is a traveling oscillation of the plasma species s (=electrons, positrons, ions, and dust particles, etc.) and the magnetic field. The heavy plasma species mass density provides the inertia and the magnetic field line tension provides the restoring force. The wave propagates in the direction of the magnetic field and can be excited in any electrically conducting fluid permeated by a magnetic field. The motion of the heavy plasma species and the perturbation of the magnetic field are in the same direction

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Figure 1.10: An Alfvén wave propagating along external magnetic field [Ref. Chen].

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Figure 1.11: Magnetic field perturbation associated with a shear-Alfvén wave [Source: Farside.ph.utexas.edu].

and transverse to the direction of propagation. Alfvén waves are known to be a important mechanism for transporting energy and momentum in many geophysical and astrophysical hydrodynamic systems.

Mathematically, the phase speed of Alfvén’s wave or simply Alfvén speed can be defined as in CGS system

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It is clear from above equation that the magnetic pressure B 02 / 4 π) gives rise to the restoring force, and the mass density (nsoms) of the plasma density provides the inertia.

1.3.2 Types of Alfvén Waves

Basically, there are two types of Alfvén waves in the literature as compression Alfvén waves and shear Alfvén waves.

i. Compressional Alfvén Waves

The compressional Alfvén waves propagate across the magnetic field B0. It is also known as the first Alfvén waves.

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Figure 1.12: Compressional Alfvén waves [Source: Google Sites].

In case of obliquely propagating Alfvén wave (θ = 90 ◦), which compress either the magnetic field or the plasma density, but does not bend (shear) the magnetic field lines and that, for finite θ, this mode compresses of both the plasma density and magnetic field lines. The nonlinear analysis for obliquely propagating Alfvén waves in which case one can derive the Korteweg-de Vries (K-dV) equation to examine the properties of these Alfvén solitons ⁵².

ii. Shear Alfvén Waves

The waves with finite Larmor radius effects are known as the shear Alfvén waves or slow Alfvéns. The shear Alfvén wave propagates with the wave magnetic field vector perpendicular to the background field. The dispersion relation for this wave can be obtained by combining the two Maxwell equations for the curl of the electric and magnetic fields. The propagation vector of the waves make a very small angle with the steady magnetic field B0. The perturbed part of the magnetic field B(¹ ) is perpendicular to the steady part B0 ⊥ B(¹ ), where B0 ≫ B(¹ ).

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Figure 1.13: Shear Alfvén Waves [Source: Google Sites].

For parallel propagation (θ = 0 ◦), the Shear Alfvén wave which does not compress either the magnetic field or the plasma density, but bends (shears) the magnetic field lines and that, for finite θ, this mode reduces to the compression of both the plasma density and magnetic field lines. The nonlinear analysis for wave propagating exactly along with external magnetic field in which case one can derive cubic nonlinear Schrödinger equation (NLSE) or derivative nonlinear Schrödinger equation (DNSE) and examine the properties of the Alfvén solitons.

1.4 Rotating Astrophysical Objects

There are many rotating astrophysical objects in our universe such as neutron star, rotating black hole, and pulsar, etc. As our main concern in this dissertation is to analyze the characteristics of solitary waves in pulsar magnetosphere, so we will only discuss about neutron star and pulsar here. A neutron star ⁵³ is a type of stellar remnant that can result from the gravitational collapse of a massive star during supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with slightly larger mass than protons. Neutron stars are very hot and are supported against further collapse by quantum degenerate pressure due to the Pauli exclusion principle. This principle states that no two neutrons (or any other fermionic particles) can occupy the same place and quantum state simultaneously. A typical neutron star has a mass between about 1 and 2 solar masses [54, 55], with a corresponding radius of about 12 km if the Akmal-Pandharipande-Ravenhall (APR) equation of state (EOS) is used ⁵⁶.

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Figure 1.14: Neutron star [Source: www.aoi.com.au].

Many astronomers refer to them as pulsars because when observed they appear to pulse because of their very fast periods. A neutron star’s gravita- tional field is so immense that the escape velocity is roughly 1 / 3 the speed of light or 10⁵ km/s ⁵⁷. A neutron star is formed in two stages. First, within a second after nuclear fusion on the star’s surface ceases, gravity crushes the star’s atoms. This forces protons (positively charged particles) and electrons (negatively charged particles) together to form neutrons (uncharged particles) and expels high-energy subatomic particles called neutrinos. The star’s core, which started out about the size of Earth, is compacted into a sphere less than 60 miles (97 kilometers) across. In the second stage, the star undergoes a gravitational collapse and then, becoming energized by the neutrino burst, explodes in a brilliant supernova. All that remains is an extremely dense neutron core, about 12 miles (19 kilometers) in diameter with a mass nearly equal to that of the sun. A sugar-cube-sized piece of neutron star would weigh billions of tons. Neutron stars have been observed to “pulse” radio and x-ray emissions believed to be caused by particle acceleration near the magnetic poles, which need not be aligned with the rotation axis of the star. External viewers see these beams as pulses of radiation whenever the magnetic pole sweeps past the line of sight. The pulses come at the same rate as the rotation of the neutron star, and thus, appear periodic. Neutron stars which emit such pulses are called pulsars.

A pulsar (short for pulsating radio star) is a highly magnetized, rotating neutron star that emits a beam of electromagnetic radiation. This radiation can only be observed when the beam of emission is pointing toward the Earth, much the way a lighthouse can only be seen when the light is pointed in the direction of an observer, and is responsible for the pulsed appearance of emission. The formation of a pulsar begin when the core of a massive star is compressed during a supernova, which collapses into a neutron star. The neutron star retains most of its angular momentum, and since it has only a tiny fraction of its progenitor’s radius (and therefore its moment of inertia is sharply reduced), it is formed with very high rotation speed. A beam of radiation is emitted along the magnetic axis of the pulsar, which spins along with the rotation of the neutron star. The rotation slows down over time as electromagnetic power is emitted. When a pulsar’s spin period slows down sufficiently, the radio pulsar mechanism is believed to turn off (the so-called “death line”). This turn-off seems to take place after about 10-100 million years, which means of all the neutron stars in the 13.6 billion year age of the universe, around 99 percent no longer pulsate. The longest known pulsar period is 8.51 seconds ⁵⁸.

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Figure 1.15: X-ray picture of Crab Pulsar [Source: commons.wikimedia.org].

The Crab Pulsar (PSR B0531+21) is a relatively young neutron star. The star is the central star in the Crab Nebula, a remnant of the supernova SN 1054, which was widely observed on Earth in the year 1939 [59-61] Discovered in 1968, the pulsar was the first to be connected with a supernova remnant ⁶². The Crab Pulsar is one of very few pulsars to be identified optically. The optical pulsar is roughly 20 km in diameter and the pulsar “beams” rotate once every 33 milliseconds, or 30 times each second.

1.5 Linearity and Nonlinearity of Waves

We will first discuss the difference between linear and nonlinear system before 21 going to introduce linear and nonlinear wave structures. One may describe a linear system in very simple way saying “a linear system is a system in which the output is linearly proportional to input ”. For an example, we consider an amplifier (or any other system) to an input signal y = y 0 cos ω t, where ω is the signal frequency resulting in an output which can be express as output ∝ input (see Fig.1.16) in case of small amplitude i.e the response of the system is linear to its input. For more than one input signal, one may expect the output as a combined effect (i.e. a linear superposition) of all input, that is,

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where a 1, a 2, and a 3 are constant quantities.

The nonlinear (not linear) system, on the other hand, is a system which does not satisfy the superposition principal, that is, the output is not proportional to its input; which is shown in Fig.1.16. In other words, the variables which are to be solved for, can not be written as a linear combination of the independent components. If the amplitude (viz. the velocity, density, electrostatic potential etc.) of the input signal become very large distortion might occur because of the harmonic generation of the input signal, and the out is no longer proportional to the input. There is no generic to describe nonlinear system. For weak nonlinearity, one may write the output as

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where ej (j = 1 , 2 , 3 ····) is assumed to be constant quantity, and e 1 >> e 2 >> e 3 ····· For comparable e 1 ,e 2 ,e 3, etc. the system will be strongly nonlinear and may produce a response of chaotic nature. However, we shall focus on the case where e 1 >> e 2 >> e 3, as we are interested in analyzing theoretically nonlinear wave phenomena in a weakly nonlinear plasma medium via the fluid dynamical model.

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Figure 1.16: General representation of a linear and nonlinear system [Source: Authors own work].

We assume a wave y = y 0 ei (kx −ω t) propagating in a dispersive and weakly nonlinear medium. The wave will be subject to the dispersion and nonlinear properties of the medium. The phase speed of the wave changes because of the dispersive property of the medium, and the wave spreads out while propagating. On the other hand, the crest of the wave moves faster than the rest due to nonlinearity and wave steepening leads the wave to break down. The balanced between the two properties (the wave broadening due to the dispersive effect and the wave steepening due to nonlinear effect) leads the wave to acquire a stable structure . The waves originated by following this mechanism are known as solitary waves or solitons in physics.

In linear theory, the wave amplitude is assumed to be sufficiently small to ignore contributions of terms of second order and higher (i.e. nonlinear terms) in wave amplitude.

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