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Waves In Pair Plasma and Mechanism Of Radio Emission In Pulsars Thesis submitted in partial fulfillment of the requirements for the degree of ”DOCTOR OF PHILOSOPHY” by Ella Gruman Submitted to the senate of Ben-Gurion University of the Negev July 13, 2005 Beer-Sheva I Waves In Pair Plasma and Mechanism Of Radio Emission In Pulsars Thesis submitted in partial fulfillment of the requirements for the degree of ”DOCTOR OF PHILOSOPHY” by Ella Gruman Submitted to the senate of Ben-Gurion University of the Negev Approved by the advisor Prof. Michael Gedalin Approved by the Dean of the Kreitman School of Advanced Graduated Studies July 13, 2005 Beer-Sheva II This work was carried out under the supervision of Prof. Michael Gedalin In the Department of Physics Faculty of Natural Sciences III This work is dedicated to my parents, husband and children Shyrly, Eve and Haim, who deserve all my gratitude for their unconditional support, patience and love. IV Acknowledgments I would like to take this opportunity to express my gratitude to everybody who has contributed to this thesis in anyway. Especially I would like to point out the contribution of my supervisor, Prof. Michael Gedalin , for his great commitment, endless patience and support. His approach to scientific problems and his endurance in driving towards the target greatly inspired me in carrying out this work. A lot of thanks to a dear friend Dr. Doron Braunstain for his friendly assistance and support. Many thanks to my parents who helped me all along the way. To Kathy Hernic for her help in styling the text and for being such a great person. CONTENTS 1 INTRODUCTION TO PULSARS 1 1.1 Pulsar Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Observed Features of Pulsars Emission . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Pulse shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Frequency range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Spectrum of pulsar radio emission . . . . . . . . . . . . . . . . . . . . 11 1.2.5 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.6 The brightness temperature . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 The Plasma Creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Waves in Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5 Pulsars’ Radio Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA 23 2.1 25 Pulsars Conditions and Plasma Parameters . . . . . . . . . . . . . . . . . . . V CONTENTS 2.2 VI Dispersion relation and wave properties . . . . . . . . . . . . . . . . . . . . . 30 2.2.1 Infinite magnetic field approximation . . . . . . . . . . . . . . . . . . 37 2.2.2 Finite magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 Numerical Analysis: Waterbag Distribution . . . . . . . . . . . . . . . . . . 41 2.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3 THE EMISSION MECHANISM 49 3.1 The pulsar conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Hydrodynamic Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.1 Cold Plasma Approximation . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.2 Relative Spread in Momenta . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.3 Validity of Cold-plasma, Cold-beam Model . . . . . . . . . . . . . . . 70 3.2.4 Direct Refractive Conversion . . . . . . . . . . . . . . . . . . . . . . . 72 3.3 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Model for the Amplification Factor . . . . . . . . . . . . . . . . . . . 75 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.3.1 3.4 4 DATA ANALYSIS 83 4.1 Pulse Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Pulse Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3 Geometrical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 A VLASOV EQUATION 107 B PULSARS DATA 111 LIST OF FIGURES 1.1 The believed structure of pulsar. . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Chart record of individual pulses from P SR0329 + 54. . . . . . . . . . . . . . 5 1.3 Distribution of pulsar periods. . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 A sequence of 100 pulses and an average of 500 pulses from P SR1133 + 16. . 8 1.5 Integrated pulse profiles form different pulsars. . . . . . . . . . . . . . . . . . 9 1.6 Radio emission subpulses from P SR0329 + 54. . . . . . . . . . . . . . . . . . 10 1.7 Pulse spectra for six pulsars, illustrating the different forms observed. Power law spectral indexes α are given beside each curve. . . . . . . . . . . . . . . . 1.8 12 Characteristic polarization behavior of the integrated profiles of four pulsars. The dark-shaded area represents linearly polarized power and the light-shaded area circular polarization power. The pulsars are: a)P SRB0355+54, b)P SR0525+ 21, c) P SRB1800 − 21, d) P SR0144 + 59. . . . . . . . . . . . . . . . . . . . 13 1.9 Motion of particles in the open field lines region of the pulsars’ magnetosphere. 16 2.1 Transformation of the distribution function f (u) ∝ γ −3/2 from the plasma to the pulsar rest frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII 27 LIST OF FIGURES 2.2 VIII Ellipticity as a function of frequency for different angles of propagation, θ = 1◦ -crosses, θ = 2◦ -diamonds, θ = 5◦ -triangles. ∆ = 10−8 ,u0 = 2, ū = 0.01, η = 0.005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 43 Ellipticity as a function of frequency for different angles of propagation, θ = 1◦ -crosses, θ = 2◦ -diamonds, θ = 5◦ -triangles. ∆ = 10−6 ,u0 = 2, ū = 0.01, η = 0.005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 44 Ellipticity for various pairs of η and ū. η = 0.005 and ū = 0.01-crosses; η = 0.005 and ū = −0.01-diamonds; η = −0.005 and ū = 0.01-triangles; η = −0.005 and ū = −0.01-circles. Delta = 10−6 and θ = 1◦ . . . . . . . . . . 45 2.5 Ellipticity for different values of the pair η and ū. . . . . . . . . . . . . . . . 46 3.1 The growth rate for the cold beam cold plasma as a function of frequency for different angles of propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 The behavior of the normal modes in the transparency range for cold-cold case. 61 3.3 The double waterbag distribution. . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 K in the transparency range for the double waterbag case. . . . . . . . . . . . 64 3.5 Magnification of the instability range for the double waterbag case. . . . . . . 64 3.6 Growth rate vs frequency for the double waterbag distribution with γ1 = 10, γ2 = 50, γ3 = 500 and γ4 = 1000, and propagation angles θ = 0.01◦ , 0.05◦ , 0.1◦ and 0.5◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 65 Growth rate vs frequency for the double waterbag distribution with γ1 = 5, γ2 = 20, γ3 = 200 and γ4 = 500, and propagation angles θ = 0.05◦ , 0.1◦ , 0.5◦ and 1◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.8 The double overlapping soft bell distribution. . . . . . . . . . . . . . . . . . . 67 3.9 Transparency region for double overlapping soft bell distribution. . . . . . . . 68 LIST OF FIGURES IX 3.10 Phase velocity for double soft bell distribution. . . . . . . . . . . . . . . . . . 69 3.11 Growth rate for double soft bell distribution. . . . . . . . . . . . . . . . . . . 70 3.12 Matching at the critical point ωc = ω. . . . . . . . . . . . . . . . . . . . . . . 73 3.13 Gain as a function of frequency ξ − ω/ωr0 . . . . . . . . . . . . . . . . . . . . 78 3.14 Gain as a function of opening angle η = 3γb φ0 /8. . . . . . . . . . . . . . . . 79 3.15 Gain as a function of φ for different frequencies. . . . . . . . . . . . . . . . . 80 4.1 Pulse width at half-W0.5 and at tenth-W0.1 of the maximum height as a function of frequency for P SRB2110 + 27, P SRB1737 − 30, P SRB0950 + 08, P SRB0353 + 52, P SRB1706 − 16 and P SRB0950 + 08. . . . . . . . . . . . 4.2 86 log W0.5 as a function of log ω for P SRB2217+47, P SRB2303+30, P SRB0136+ 57 and P SRB0154 + 61. The width is measured in degrees and frequency in M Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 87 log W0.5 as a function of log ω for P SRB0820+02, P SRB1706−16, P SRB1737− 30, P SRB0611 + 22, P SRB0037 + 56, P SRB0353 + 52, P SRB0626 + 24 and P SRB0540 + 23. The width is measured in degrees and frequency in M Hz. . 4.4 88 log W0.5 as a function of log ω for P SRB0756−15, P SRB0628−28, P SRB0809+ 74, P SRB0823 + 26, P SRB0950 + 08, P SRB0919 + 06, P SRB2110 + 27 and P SRB1620 − 09. The width is measured in degrees and frequency in M Hz. . 89 4.5 Maximum intensity as a function of frequency. . . . . . . . . . . . . . . . . . 90 4.6 Maximum intensity as a function of frequency. . . . . . . . . . . . . . . . . . 91 4.7 Maximum intensity as a function of frequency. . . . . . . . . . . . . . . . . . 91 4.8 Maximum intensity as a function of frequency. . . . . . . . . . . . . . . . . . 92 4.9 Profiles for B0540+23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.10 Asymmetry parameter as a function of frequency for P SRB0628−28, P SRB0037+ 56, P SRB0136 + 57 and P SRB0626 + 24. . . . . . . . . . . . . . . . . . . . 95 LIST OF FIGURES X 4.11 Asymmetry parameter as a function of frequency for P SRB1737−30, P SRB2217+ 47, P SRB0540 + 23, P SRB0823 + 26, P SRB0809 + 74, P SRB2303 + 30, P SRB0611 + 22 and P SRB0154 + 61. . . . . . . . . . . . . . . . . . . . . . 96 4.12 Asymmetry parameter as a function of frequency for P SRB1706−16, P SRB1620− 09, P SRB0756 − 15, P SRB0353 + 52, P SRB0950 + 08, P SRB2110 + 27, P SRB0820 + 02 and P SRB0919 + 06. . . . . . . . . . . . . . . . . . . . . . 97 4.13 Plasma distribution function for different pulsars at different frequencies. . . 99 4.14 Plasma density as a function of radius, R, and angle ,φ, for the Lognormal distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.15 Gain factor for the Lognormal distributed plasma density. . . . . . . . . . . . 101 4.16 Matching between the observed and expected profiles. . . . . . . . . . . . . . . 102 ABSTRACT Pulsars are believed to be compact rotating neutron stars. Since their discovery pulsars are constantly observed. However the nonthermal (brightness temperature ∼ 1025 − 1030 K) radio emission coming from pulsars is still unexplained. A lot of suggestions have been made, however no one can point out the process or the processes that determine the observed radiation. In this research we propose a one stage mechanism based on the nonresonant electromagnetic beam instability as a generator of the observed spectrum. The pulsar radiation is created and propagates in a relativistic (γ ≈ 103 ) pair plasma. We believe that understanding the properties of the observed radiation in the radio range from pulsars lies first of all in the studies of low frequency waves, ω ∼ (109 − 1011 )Hz, in a typical to pulsars magnetosphere with a strong magnetic field B0 ≈ 1012 G. Those waves properties may explain some features of the observed radiation from pulsars (polarization and etc.). We XI LIST OF FIGURES XII examine the waves properties in the most general case of non-neutral, current-carrying pair plasma with arbitrary distribution function for electrons and positrons. We derive general dispersion relations including gyrotropic terms caused by the deviation from quasi-neutrality and relative streaming of electrons and positrons. We show that under conditions typical to the pulsar magnetosphere, waves with speed close to the speed of light can be substantially elliptically polarized. The dependence of the ellipticity on plasma parameters is shown. The analysis is illustrated by numerical example using the waterbag distribution that provides a semi-quantitative description of the wave features, except from their damping rate. We propose that the observed radiation is generated by the nonresonant electromagnetic beam instability. A low density high energy beam propagates through an ultrarelativistic pair plasma filling the pulsar magnetosphere. Quasi-transverse electromagnetic waves are generated due to the beam-plasma interaction. The instability causes a beam mode to grow over a broad range of lowe frequencies. The wave growth is not restricted to a narrow layer of the magnetosphere and can last over a large range of heights so that the net amplification caused by the nonresonant instability is much larger then by the resonant instability. As the wave propagates the growth continues until the wave frequency is equal to the resonant frequency. At this point the growth ceases and the beam mode joins the L − O mode. The waves can freely escape from the magnetosphere, no additional processes are needed. We analyze the instability in details for a cold plasma cold beam. We show that inclusion of the relativistic spread in momenta does not change the conclusions substantially. The derived local field line spectrum possesses some properties of the observed radiation from pulsars. LIST OF FIGURES XIII We compare our model predictions with observations. The observed radiation should give us some indication on the structure of the emission region. We propose a method of quantitative description of the pulse asymmetry. Based on the pulse structure we propose phenomenological functional form for plasma distribution across the emission region. CHAPTER 1 INTRODUCTION TO PULSARS For the last 37 years astrophysicists has tried to solve the mystery of radiation in the radio range from stars that are known as pulsars. The name, pulsars, was given them because of their most important feature. They emit incredibly regular pulses in the radio range. The first star of this type was detected on the 28th of November 1967 by Cambridge astronomers Jocelyn Bell and Antony Hewish. Hewish was awarded the Nobel Prize for this discovery in 1974. Today about 1000 pulsars are known, nevertheless the emission mechanism responsible for the pulsars’ detected radiation in the radio range is still a mystery. 1.1 Pulsar Structure The most widely accepted model for pulsars is the rotating neutron star model [see, for example, Manchester and Taylor (1977)]. In there finale stage stars with mass 1.4MJ < M < 15MJ are expected to form a neutron star. After an ordinary star exhausts 1 CHAPTER 1. INTRODUCTION TO PULSARS 2 his stock of nuclear fuel, the star can no longer create outward pressure to balance the inward gravitational pull of its great mass. The stars’ core collapses releasing a huge amount of energy. This will cause a blast wave that ejects the stars’ envelope into interstellar space. This process is known as the supernova explosion. The envelope is the nebula. In some cases after the wind cleans the environment, a rotating neuron star can be observed. This star is most likely to be a pulsar. Neutron star is an object consisting primarily of neutrons. The mass of such a star nearly equals that of the sun, but its radius is only 10km. In the neutron stars the gravitational forces are brought to equilibrium by pressure of strongly compressed neutron matter. The principle argument identifying pulsars as neutron stars is based on the shortness of pulsar period [see Smith (1979)]. It is almost certain that the neutron stars we observe as pulsars possess extremely strong magnetic field of about Bo ≈ 1012 G. There have been suggested different models for the inner structure of the neutron stars. Most of them are based on the solution of the Tallman-Oppenheimer-Volkoff equation that describes hydrostatic equilibrium of spherically symmetric cold star [see Oppenheimer and Volkoff (1939)]. More realistic models taking into account the star rotation and the magnetic field were considered by Shapiro and Teukolsky (1983), Lindblom (1986) and Friedman et al. (1986). Despite the variety of models of the outer crust structure, most of them agree that the outer crust of the neutron star consists of iron nuclei and degenerate electrons. Because of the strong magnetic field, the surface layers are expected to form a dense lattice with high conductivity. Electric conductivity of the star is large enough to freeze the magnetic field in CHAPTER 1. INTRODUCTION TO PULSARS 3 the neutron star. The region surrounding the star is called the magnetosphere. The magnetic field varies in the pulsars’ magnetosphere as a magnetic dipole field, B = (R0 /R)−3 B0 , where R0 ≈ 106 cm is the radius of the neutron star and B0 is the magnetic field at the stars’ surface. Pulsars’ magnetosphere are assumed to consist of magnetically closed regions and two magnetically open regions. It is assumed that the closed zone is filled with plasma. The strong magnetic field of the pulsar, forces the charged particles to co-rotate with the star [Manchester and Taylor (1977), Smith (1979), Goldreich and Julian (1969)]. This plasma provides a co-rotating electric field, E = −(v × r) × B, where r is the position vector related to the center of the star, B is the magnetic field induction and v is the angular velocity of the rotating star. At some distance from the rotation axis where the co-rotation velocity equals to the speed of light RL v = c, the particles stop co rotate with the star. This distance is known as the light cylinder radius, RL = cP/2π, where P -denotes the pulsar period. The close region is defined by those field lines that are closed within the light cylinder. CHAPTER 1. INTRODUCTION TO PULSARS 4 Figure 1.1: The believed structure of pulsar. The polar caps are regions where the magnetic field lines are open in the sense that they cross the light cylinder. The observed radiation is produced by a particle beam, which is emitted along the open field lines. When the magnetic axis is oblique to the rotation axis, the radiation beam draws a circle in the sky. A pulse is observed when the line of sight crosses the radiation beam. The duration of the pulse is determined by the beam width. Figure 1.1 shows the believed structure of the pulsar. CHAPTER 1. INTRODUCTION TO PULSARS 1.2 5 Observed Features of Pulsars Emission The only source of energy available in pulsars is the rotation energy of the neutron star that is actually a magnetic dipole. However, the observed frequencies (107 − 1010 Hz) are different from those of the magnetodipole radiation (1 − 100Hz). The actual mechanism by which pulsars convert the rotation energy of the neutron star into the observed pulses is not completely understood yet. Observations, in the radio range, of the pulsars conclude that all of them share a number of common properties. Any proposed mechanism for pulsars’ radio emission should be able to explain the observed properties of the radio emission. 1.2.1 Period The most important property of pulsars is the emission of periodic sequence of pulses in the radio range. Chartered record of individual pulses is shown in Figure 1.2. Figure 1.2: Chart record of individual pulses from one of the first pulsars discovered, P SR0329 + 54. They were recorded at a frequency of 410M Hz. The pulses occur at regular intervals of about 0.714s. [After Manchester and Taylor (1977)]. The time interval between pulses is called the pulse period, P . Observed pulsar period lie between 1.553ms and 4.308s, the majority of them lying between 0.3s and 1.5s as shown in Figure 1.3 [Manchester and Taylor (1977)]. CHAPTER 1. INTRODUCTION TO PULSARS Morphology and Characteristics of Radio Pulsars 25 6 Figure 1.3: Distribution of pulsar periods. Fig. 20 The period distribution of 1300 pulsars Pulsars with very short period, P < 10ms, form a separate class of objects because their the majority of millisecond pulsars are members of binary star systems. Binary pulsars located between the two clusters will slowly drift toward the millisecond cluster in less than 108 years. magnetic field is significantly weaker. The mean period is remarkably constant up to six digits and more. However, two classes Assuming that the decay of pulsar periods is due to their dipole radiation, their characteristic age can be calculated from the very simple expresof period variations are observed: years. The “dash-dot” lines in Fig. 21 correspond to lines sion τchar = 21 P Ṗ of constant age. The of normal pulsars have is ofbeen the order a) Secular period variation: Incharacteristic all cases whereage accurate observations made, of pul8 10 years, whereas the age of millisecond pulsars is slightly above 109 years. 7 103 years line. The Crab pulsar is the isolated pulsar the sar’s period is found to increase in a steady way,closest P Ṗ −1 to ≈ 10 years, were Ṗ is the period Following classical electrodynamics theory, the surface p magnetic field of pulsars is given by the expression B0 = 3.3 × 10 × P Ṗ gauss. The “dash” lines in Fig. 21 correspond to lines of constant magnetic field. It b) Unpredictable irregularities: Statistic analysis show that there are small random fluctuais immediately noted that normal pulsars have a surface magnetic field of 12 aboutas10”glitches” gauss, inwhereas the period. surfaceThe magnetic of millisecond pulsarssee tions known the pulsars nature field of those glitches is unknown 9 is much lower, of the order of 10 gauss. derivative with respect to time. Faster pulsars tend to have larger 19 slowdown rates. [Cordes and Downsit(1985) and et al. (1988)]. unpredictable period fluctuations Finally should beCordes mentioned that theThe absence of pulsars in the lower right corner of the diagram is due to the existence of a “death line”, owing to the gradual decaying of the induced electrical potential of pulsars. Slow pulsars with low magnetic field cannot develop a large enough potential above their magnetic poles for discharges (and therefore radiation) to take place. The absence of millisecond pulsars below about 1010 years indicates CHAPTER 1. INTRODUCTION TO PULSARS 7 are small usually about several milliseconds. 1.2.2 Pulse shape Speaking about the pulse shape one has to distinguish between single pulse and integrated pulse. Singe pulse: The intensity of single pulse widely varies and at times pulses are completely missing as it can be seen in Figure 1.4. Integrated pulse The integrated pulse profile obtained by summing over large number of pulses. Contrary to the single pulse the average pulse energy over an interval of several hours is remarkably stable. confirmed. Pulsar radiation is highly polarized in a most complicated way. At low radio frequencies some pulsars are almost 100% linearly polarized. Others have very high and variable circular polarization. The development of polarization with frequency is radically different from all other radio sources. The polarization may be high at low frequencies while dropping rapidly to zero at high frequencies. Possibly this is a hint for a coherent (low frequencies) – incoherent (high frequencies) emission mechanism, an corroborated by high frequency pulsar spectra [72]. CHAPTER 1.effect INTRODUCTION TO PULSARS 8 3 (Top): A sequence of 100 single pulses from PSR 1133+16 plotted unFigure 1.4: Fig. A sequence of 100 pulses from P SR1133 + 16 recorded at 600M Hz. An derneath each other (Noise wings are suppressed). (Bottom): By adding the above average of 500single pulses is shown the bottom. pulses, we get theat Integrated profile Consecutive pulses are plotted vertically to show variations in individual pulse shapes and arrival times; the average pulse behavior is quite stable and periodic, however. [After Cordes (1979)]. The observed profiles are often rather complex having several components or identifiable peaks. The shape of integrated profiles is generally somewhat frequency dependent; however, the basic character of a pulsar’s profile usually remains the same at all frequencies [Manchester and Taylor (1977)]. Each pulsar has a unique profile as it can be seen in Figure 1.5. CHAPTER 1. INTRODUCTION TO PULSARS 9 Figure 1.5: Integrated pulse profiles for different pulsars. The profiles were recorded at frequencies between 400 and 650M Hz. The main profile of pulsar radio emission usually occupies only for a small part of the period. If total period corresponds to 360◦ , the radio emission pulse duration usually corresponds to 6◦ − 30◦ . Individual pulses usually consist of one or more subpulses as it can be seen in Figure 1.6. The subpulses typically have almost Gaussian shape and width 3◦ − 10◦ . When two or more subpulses are present in an individual pulse they are often overlap. A thorough study by Deshpande and Rankin (1999) of the subpulse structure provides indications on multi-beam structure of the emission zone, with each emitting beam apparently traced down to the polar cap zone. The subpulses correlate well at different frequencies. CHAPTER 1. INTRODUCTION TO PULSARS 10 Figure 1.6: Radio emission subpulses from P SR0329 + 54 at 1700M Hz. The time resolution is 70ms. 1.2.3 Frequency range All known pulsars emit broadband radio emission (16M Hz − 25GHz) in form of periodic pulses. Detailed investigation made by Izvekova et al. (1981) and Kuzmin et al. (1986) show that most pulsars have a low frequency and a high frequency cut off. The low frequency cut-off is νmin ' (50 − 300)M Hz and νmax ' (1.4 − 10)GHz is the high frequency cut off. CHAPTER 1. INTRODUCTION TO PULSARS 11 According to Malov and Malofeev (1981) the cut off frequencies are related to the period of the pulsar νmin ' 100P −(0.38±0.09) M Hz and νmax ' P −(0.62±0.19) GHz. 1.2.4 Spectrum of pulsar radio emission The average pulse energy over an interval of several hours is relatively stable. In general the radio frequency spectrum is quite steep with a cut-off in the low frequency limit as it can be seen in Figure 1.7. All know spectra show a fall of flux density S with frequency ν according to a power low S ∝ ν α where α is a spectral index. According to Manchester and Taylor (1977) and Malofeev et al. (1994) spectral indices of most pulsars are between α = −2 ± 1 with the peak in the distribution at about α = −1.5. In the case of spectra with a break in the slope, the spectral index below the break is close to α = −1. According to Smith (1979) the spectral indices are α = −3 ± 1. CHAPTER 1. INTRODUCTION TO PULSARS 12 Figure 1.7: Pulse spectra for six pulsars, illustrating the different forms observed. Power law spectral indexes α are given beside each curve.[After Sieber (1973)]. 1.2.5 Polarization Many pulsars show a high degree of linear polarization, in some cases up to 100% [Manchester et al. (1975)]. The amount and position angle of linear polarization frequently changes along the duration of the pulse. The degree of linear polarization of pulsars usually decreases with increasing frequency [Manchester and Taylor (1977)]. Resent studies on pulsars polarization conclude that many of them have a considerable circular polarization as well see [Han et al. (1998), Manchester et al. (1998), von Hoensbroech et al. (1998), von Hoensbroech and Lesh (1999)]. Figure 1.8 shows characteristic polarization behavior. CHAPTER 1. INTRODUCTION TO PULSARS Morphology and Characteristics of Radio Pulsars 13 19 Fig. 15 Characteristic polarization behaviour of pulsars. The dark-shaded area Figure 1.8: Characteristic polarization behavior of the integrated of polarized four pulsars. represents linearly polarized power and the light-shaded areaprofiles circularly The dark-shadedpower. area (a): represents linearly (b): polarized power and light-shaded areaPSR circular PSR B0355+54. PSR 0525+21. (c):the PSR B1800-21. (d): 0144+59. From [155] polarization power. The pulsars are: a)P SRB0355+54, b)P SR0525+21, c) P SRB1800−21, d) P SR0144 + 59.[After von Hoensbroech (1999).] been conducted [47] with the Lovell telescope at Jodrell Bank. The polarof a large sample ofwas pulsars was subsequently studied at the They highest The propertiesization of circular polarization summarized by Han et al. (1998). showed radio frequencies [156][155] up to the frequency of 32 GHz (Fig. 14). Some pulsar polarization properties can now be made. that the circulargeneralizations polarization about is common but diverse in nature. It is generally stronger in In Fig. 15 we show the polarization evolution with frequency for four characteristic types of pulsars. The pulsar B0355+54 (Fig. 15a) begins with CHAPTER 1. INTRODUCTION TO PULSARS 14 the center of the profile. The circular polarization often changes sense near the middle of the profile, but sign changes are occasionally observed at other longitudes. Relatively strong circular polarization is often observed in pulsars which also have high linear polarization. There are cases of both increasing and decreasing degree of circular polarization with frequency. Though von Hoensbroech et al. (1998) stated that the polarization is stronger at high frequencies. 1.2.6 The brightness temperature The brightness temperature is defined by Iν = Bν (Tb ) where Iν is the specific intensity and Bν is the Planck function. The brightness temperature is estimated to be Tb ∼ 1025 − 1030 K. The very high Tb implies that the radiation is not thermal and it is most likely that we are dealing with coherent emission mechanism. 1.3 The Plasma Creation It is believed that the radiation is formed in the region were the magnetic field lines are open therefore we are specifically interested in what happens there. The plasma which fills the pulsar magnetosphere is produced in the electromagnetic cascade [for review, see Arons (1981); Taylor and Stinebring (1986); Arons (1992); Melrose (1992, 1995); Usov (1996)]; see, however, Weatherall and Eilek (1997) regarding difficulties with the cascade picture), beginning most probably, with the charge particle (electrons) CHAPTER 1. INTRODUCTION TO PULSARS 15 acceleration along the magnetic field near the polar cap, in the so-called polar gap region, where absence of a dense plasma ensures the existence of a very strong Ek . These particles are accelerated to very high energies, γ ∼ 107 . While moving along the curved magnetic field lines, they emit high energy curvature photons. Those photons are radiated in the direction of particle motion, that is along magnetic field lines. Because of the curvilinearty of the magnetic field a photon starts crossing the field lines. These photons can produce pairs, γ + B → e+ + e− + B, when encountering a sufficiently strong perpendicular magnetic field, εγ (B⊥ /B) > 2me c2 . These secondary electron and positron are born with nonzero perpendicular momenta, therefore emit synchrotron photons. These last, in turn, produce more pairs and so on. The process develops in an explosive way and a relatively dense plasma is produce in a narrow layer (pair production front). A schematic description on the plasma creation is given in Figure 1.9. The produced pair plasma is believed to be dense enough, N >> NGJ = B/P ec, to shield out the parallel electric field, so that the plasma can freely stream outward along the magnetic field lines, but now this possibility is questionable [see Shibata et al. (1998)]. CHAPTER 1. INTRODUCTION TO PULSARS 16 Figure 1.9: Motion of particles in the open field lines region of the pulsars’ magnetosphere. The secondary plasma parameters depend on the details of the polar gap structure, in particular, on the electric field and height of the acceleration region [Oppenheimer and Volkoff (1939), Shapiro and Teukolsky (1983), Goldreich and Julian (1969), Shibata et al. (1998) and Hibschman and Arons (2001)]. The Lorentz factor for the secondary particles is estimated to be about γ ∼ 102 − 103 . The ratio between the number of secondaries and the number of primaries is estimated to be in the range M ∼ 102 − 103 [Manchester and Taylor (1977), Melrose (1995)], thought CHAPTER 1. INTRODUCTION TO PULSARS 17 Hibschman and Arons (2001) suggest a lower value. The efficiency of the pair production depends on the overall potential drop and the temperature of the polar cap. 1.4 Waves in Plasma The plasma in the pulsar magnetosphere is far from equilibrium, so that it is expected to be unstable and may give birth to modes. Understanding the properties of low frequency waves in the surrounding typical to pulsars magnetosphere is important since the observed radiation is generated in and propagates through the pair plasma. Early studies of the waves in plasma concentrated mainly on the relativistic plasma flow, assuming cold or only mildly relativistic distribution of electrons and positrons in the plasma rest frame [see Melrose (1996)]. Kinetic analysis of the highly relativistic plasma done by Kuz’menkov and Polyakov (1983) concentrated on longitudinal waves propagating along the magnetic field. Gedalin and Machabeli (1983a) derived a general expression for the dielectric tensor neglecting gyrotropic terms for oblique low frequency waves in one dimensional plasma (meaning that the particles move only along the magnetic field lines). Studies of waves in ultrarelativistic pair plasma with identical distribution function for electrons and positrons [see Gedalin and Machabeli (1982); Arons and Barnard (1986); Lyubarski (1995); Gedalin et al. (1998); Gruman (1998); ?] conclude that in this case the waves polarization is linear which leaves the circular polarization often present in the pulse difficult to explain. Many ways were proposed in order to explain the circular polarization. Most of the suggestions are referring to specific characteristics of the emission mechanism [see for example Radhakrishnan and Rankin (1990),Lyutikov et al. (1999)]. Lyubarski and Petrova (2000) considered the propa- CHAPTER 1. INTRODUCTION TO PULSARS 18 gation effects while Lyutikov (1999) turned to nonlinear interactions. Cheng and Ruderman (1977)and Arons and Barnard (1986) suggested that the polarization may be associated with the properties of the natural wave modes in plasma. 1.5 Pulsars’ Radio Emission Since the pulsars’ discovery a lot of efforts were invested in explaining their radiation in the radio range. Many theoretical models have been proposed but not a single one is compelling [for review see Melrose (1991, 1992, 1995)]. The basic problems in building a model for radio emission is the uncertainty in the structure of the pulsars’ magnetosphere and the interpretation of the observed radio data. Though it is clear that the radio emission production should be a fast process, in most models suggested up today, the radiation formation involves two stages: a) generation of waves (not necessarily in the radio range) b) the conversion of these waves into the observed radiation. As previously mentioned the plasma in the pulsars’ magnetosphere is most probably unstable and can give birth to some modes. The second stage, the conversion of those modes into the observed radiation seems to bee the problem. There were suggested several ways in generating the observed spectrum. In models involving relativistic plasma emission Machabeli and Usov (1979); Lominadze et al. (1983); Gedalin and Machabeli (1983a,b); Asseo et al. (1990); Lyubarski (1996) initial waves should not be necessarily in the radio range. The waves may be excited in higher or CHAPTER 1. INTRODUCTION TO PULSARS 19 lower frequencies and be converted to the observed radiation by some nonlinear processes like three wave interaction and induced scattering. Early models indicate Langmuir wave instability or cyclotron instability as possible sources of primarily excited waves. In both cases waves are assumed to propagate along the external magnetic field. Our calculations Gruman (1998) in view of such model show that the time needed for spectra production is ,maybe, longer than the time that is needed for the pulse to propagate outward. The possibly formed spectrum seems to be flat and not consistent with observations. In some models, the coherent mechanism, related to bunches of particles radiate in phase with each other [(Sturrock, 1971; Ruderman and Sutherland, 1975; Buschauer and Benford , 1976, 1983; Benford , 1977; Benford and Buschauer , 1977)]. The volume of a bunch has to be less then a cubic wavelength. This model requires generation of more or less stable bunches. It is unclear what is the physical process that would create such long lasting bunches in an ultrarelativistic plasma. In a number of models it is suggested that electrostatic solitons can be formed that may play the role of such bunches. So far even the attempts to describe these bunches in a cold plasma are not especially successful. Another class of models is in favor of maser emission mechanism, for detailed discussion see (Manchester and Taylor , 1977; Melrose, 1993; Usov and Melrose, 1996; Kirk, 1980; Luo and Melrose, 1995). Maser emission is familiar in bound-state systems (atoms or molecules) in which the emission between two levels produces a narrow line. Negative absorption between two levels results from the higher-energy level being overpopulated relative to the lowerenergy level, called an inverted population. In plasma a maser emission can occur under CHAPTER 1. INTRODUCTION TO PULSARS 20 a variety of conditions where there is a continuum of states and a continuum of emission frequencies [Gedalin and Machabeli (1982)]. The particle distribution needs to have some feature that correspond to an inverted energy population that is, the distribution should be far from thermal equilibrium. In appropriate conditions, a single particle emitted emission may be greatly amplified when propagated through the non equilibrium medium with the inverse population Melrose (1993). Maser emission applies in random phase approximation so that the phase of the growing waves is irrelevant. Other models are based on a beam that penetrates the plasma [for details see Usov (1987), Lyubarski and Petrova (2000)]. Till now the main effort was concentrated on the resonant instability where energetic beam causes quasi-longitudinal subluminal waves to grow ( Asseo and Riazuelo , 2000; Lyubarski and Petrova, 2000; Magneville, 1990). In order to produce the observed radiation a certain mechanism converting those waves into quasi-transverse waves is needed . The beams’ properties are model dependent. In case the beam is highly relativistic γ ∼ 107 the growth, being to slow, is inefficient. Till now the main effort has been concentrated on resonant excitations of L-O mode propagating parallel to the magnetic field. Those waves are purely longitudinal and additional physical process is needed to convert those waves to escaping transverse waves. The beam can excite resonantly only subluminal waves resulting radiation at high frequencies. Since the pulsars emission is broadband conversion to lower frequencies is needed. In addition the growth is restricted only to a very narrow layer of the magnetosphere where the resonant condition is satisfied so that the effective growth is small. CHAPTER 1. INTRODUCTION TO PULSARS 1.6 21 Research Objectives The objective of this research is to build a quantitative model for pulsars radio spectrum formation. We propose one stage mechanism based on hydrodynamical nonresonant beam instability as a generator of the observed spectrum. This kind of mechanism has not been considered yet in the context of pulsars radio emission. The main idea of our model is based on the assumption that low energy, high density beam propagates throughout an ultrarelativistic pair plasma filling the pulsars’ magnetosphere. In the beam, quasi-transverse waves with frequencies below the resonant frequency are generated. Propagating outward, the waves grow nonresonantly until they reach the resonance point where the growth ceases and the wave escapes. The power spectrum is formed within the instability region. It is most likely that some features of the observed radiation, like polarization, are dictated by the surroundings through which the radiation propagates. In this case the observed polarization and the wave polarization at the point of spectrum formation maybe quite different, therefore understanding the impact of the surroundings on the waves features is quite important. On this purpose we will study the properties of low frequency waves in ultrarelativistic pair plasma in a strong magnetic field when the deviation from quasinutralety and relative velocity between electrons and positrons is taken into account. In order to emphasize some properties of those waves a numerical analysis will be performed. The dependence of the observed radiation on the plasma parameters will be investigated. CHAPTER 1. INTRODUCTION TO PULSARS 22 In order to check the validity of our model we will compare observations to predictions of the proposed mechanism for the radio emission generation. In order to get a better understanding about the structure of the emission region we will perform data analysis using data base of published pulse profiles maintained by the European Pulsar Network. The thesis is organized as follows: We start with a short review on the pulsars’ structure and observed properties. In the second chapter we will study the properties of low frequency waves in most general case taking into account relative velocity between the electrons and positrons distribution functions and deviation from quasineutrality for relativistic pair plasma. The dependence of the ellipticity on the plasma parameters and the propagation angle will be surveyed as well. Numerical analysis of the waves properties is done with the waterbag distribution. In the third chapter we analyze the nonresonant beam instability. On this base we propose a mechanism for radio spectrum formation in pulsars. We derive the local field line spectrum within a simple approximation model, taking into account the propagation along the curved field lines. In the forth chapter we compare some observed radiation features with the predictions. Phenomenological quantitative description of the pulse asymmetry and a possible density distribution function across the emission region are proposed. CHAPTER 2 LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA The features of low frequency waves in pulsars magnetosphere determine some properties of the observed radiation, polarization believed by us to be one of them. The observation of radio emission from pulsars show a high degree of linear polarization and in some cases there is a circular polarization as well. Waves in ultrarelativistic pair plasma with identical distribution function for electrons and positrons were studied comprehensively in cold and in hot regimes by [Gedalin and Machabeli (1982), Arons and Barnard (1986), Gedalin et al. (1998), Gruman (1998), Lyubarski (1995)]. In the case of symmetric plasma three types 23 CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA24 of waves are present the t-wave , Alf vén wave and L-O waves. All of them are linearly polarized. Those results live the circular polarization unexplained. Possible explanation for the observed circular polarization may be attributed to one of the following reasons: one suggestion is that the circular polarization is inherent to the emission process itself [see Radhakrishnan and Rankin (1990), Gil and Snakowski (1990a), Gil and Snakowski (1990b) Gangadhara (1997), Lyubarski and Petrova (1999), Lyutikov et al. (1999)]; the other may be the result of propagation effects. In this chapter we shall consider the second possibility. Linear polarization of the waves in the symmetric pair plasma results from the bi-diagonal form of the dielectric tensor. The off-diagonal terms (εxy , εyz ) vanish identically for the symmetric distribution functions. Asymmetry of the distribution functions (deviation from quasineutrality, relative velocity) would result in the appearance of the off-diagonal terms (or gyrotropic terms) thus making the polarization elliptic in general. The off-diagonal terms appears due to the nonzero charge density ρ ∼ eNGJ and parallel current density J ∼ eNGJ associated with the rotation of the magnetosphere [see Arons and Barnard (1986)] since NGJ = B · Ωrot /2πe. The off-diagonal terms are inversely proportional to the cyclotron frequency therefore they are small due to the conditions of the superstrong magnetic field of the pulsar magnetosphere. Nevertheless we will show that they can’t be neglected and that waves with the phase velocity close to the light speed have substantial elliptical polarization. The waves polarization is sensitive to the plasma parameters and the angles of propagation. The parameters of the secondary particles is poorly known. There is no general agreement on plasma parameters and estimates may differ by the order of magnitude or even more CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA25 [ Cordes (1992), Hibschman and Arons (2001)]. Therefore we will examine the influence of plasma parameters on the waves polarization. We will use the waterbag distribution to provide a semi-quantitative description of the wave features except their damping rate. For this distribution the dispersion relation can be calculated explicitly so that the dependence of polarization on the plasma parameters and propagation angles can be examined. This chapter is organized as follows. In the first section we briefly describe the plasma parameters used here. In the second section we drive the general dielectric tensor, dispersion relation for asymmetric plasma and general expression for polarization . We summarize the waves features in the infinite magnetic field limit. The effects of the finite magnetic field, including wave polarization, are considered. In the third section we use the waterbag distribution for a quantitative example. And finally in the fourth section we discuss the possible implications of our results for understanding of pulsar radiation. Results of this chapter were published in the following paper: Gedalin M., Melrose D.B., and Gruman E., Phys.Rev.E,. 57, 3399, (1998). 2.1 Pulsars Conditions and Plasma Parameters The analysis is carried out in the plasma rest frame in order to remove the streaming motion of plasma. Variables measured in the pulsar rest frame will be denoted by prime, and variables in the plasma rest fame will be unprimed (plasma moving strictly along magnetic CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA26 field lines with Lorentz factor γp ). The transformation rules between the frames are: u0 = γp u + up γ, γ 0 = γp γ + up u, (2.1) 0 0 u = γp u − up γ , γ = γp γ 0 − up u0 , where u = γv/c and γ = √ 1 + u2 . The invariance of the distribution function implies f 0 (u0 ) = f (u). The production mechanism for the secondary pair plasma implies a distribution function in the pulsar frame that is limited by some u0min & 1 from below and by some u0max from above. The 4-speed u0max is thought to be greater than the average hu0 i by about an order of magnitude, where it is assumed that the distribution decreases rapidly with energy and that up is high. The corresponding limits of umin and umax in the plasma rest frame depend on γp so that umax increases with increasing Lorentz-factor of the flow. In Figure 2.1 we illustrate transformation of the distribution function f (u) ∝ γ −3/2 , |u| < umax from the plasma rest frame into the pulsar frame. CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA27 a 1.1 1 f 0.9 0.8 0.7 0.6 -1 -0.5 0 0.5 u b 1.1 1 c 1.1 1 1 0.9 f' f' 0.9 0.8 0.8 0.7 0.7 0.6 0 10 20 30 40 0.6 50 0 u' d 1.2 1 104 2 104 u' 3000 4000 5000 3 104 4 104 5 104 1 0.8 0.8 0.6 f f 2000 e 1.2 1 1000 0.4 0.6 0.4 0.2 0.2 0 -10 -5 0 u 5 10 0 0 u' Figure 2.1: Transformation of the distribution function f (u) ∝ γ −3/2 from the plasma (a and d) to the pulsar rest frame b ,c and e. The parameters are as follows: for a and b umax = −umin = 1 and up = 20; for c umax = −umin = 1 and up = 2000; for d and e umax = −umin = 10 and up = 2000. For umax = −umin = 1 and up = 20 the distribution in the pulsar rest frame is limited by u0min ≈ 8 and u0max ≈ 48. For umax = −umin = 1 and up = 2000 the limits are u0min ≈ 800 and u0max ≈ 4800. In the case of umax = −umin = 10 and up = 2000 the limitations are u0min ≈ 100 and u0max ≈ 40000. For large multiplicities of M ∼ 104 − 105 , one has up ∼ 102 CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA28 so that in the plasma frame umax ∼ 1 − 10. The electromagnetic cascade begins most probably with the charge particle acceleration along the magnetic field lines in the polar gap up to very high energies γ ∼ 107 . Secondary plasma parameters depend on the details of the polar gap structure, particularly, on the electric field and high of acceleration region. In our calculations we adapted values that correspond to a pulsar with a period of P = 0.2sec, radii of R0 ≈ 106 cm and magnetic field B0 ≈ 1012 G at the surface of the neutron star. For those values the Goldreich-Julian density at the surface in the polar gap region is NGJ = B0 /P ec ≈ 3.5×1011 cm−3 . The ration between the secondary and the primary particles is vague [see Lyutikov (1998), Shibata et al. (1998)] and make take the values M ∼ 102 − 106 . For numerical calculations we will adopt the dense case, M = 105 . For this value the plasma density in the pulsar frame is N 0 = M NGJ = 3.5 × 1016 cm−3 . Being highly relativistic the characteristic Lorentz factor is considered to be γ ∼ γpr M −1 ∼ 102 and the spread in Lorentz factor is about hγi ∼ γp ∼ 102 . Then the plasma density in the plasma rest frame is N ∼ N 0 /γp ≈ 3.5 × 1014 cm−3 so that the plasma rest frame frequency is ωp = p 4πN e2 /m ≈ 1012 s−1 . The gyrofrequency is Ω = qB/mc. The ratio between the plasma frequency and the gyrofrequency is (ωp /Ω)2 ≈ 10−14 (R/R0 ). The plasma frequency, the gyrofrequency and the ratio between them are given in Table 2.1. In most models the emission zone is believed to be inside the light cylinder whose radius is RL = cP/2π ≈ 1010 cm. The plasma density in these region varies as N ∼ R−3 . We are interested in the frequencies observed within the radio range ω ∼ 109 − 1011 sec−1 which CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA29 Distance Plasma frequency Gyrofrequency R0 ωp [sec−1 ] Ω[sec−1 ] ωp2 /Ω R0 1 × 1012 2 × 1019 4 × 10−15 10R0 3 × 1010 2 × 1016 4 × 10−12 102 R0 1 × 109 2 × 1013 4 × 10−9 103 R0 3 × 107 2 × 1010 4 × 10−6 104 R0 1 × 106 2 × 107 4 × 10−3 Table 2.1: Plasma frequency, gyrofrequency and the ratio between them at different distances from the surface of the neutron star. becomes ω ∼ 107 − 109 sec−1 in the plasma rest frame for γ = 102 . The Goldreich-Julian density states NGJ = B · Ωrot /2πe ∼ N± /M , so that the deviation from quasi-neutrality and the current density in the plasma rest frame are ρ0 /N 0 e ∼ 1/M Manchester and Taylor (1977) and j 0 /N 0 ec ∼ 1/M Lyubarski (1992). The charge density and current density in the plasma rest frame may be obtained from the following transformation rules: ρ0 = γp ρ + up j/c, j 0 /c = γp j/c + up ρ, (2.2) 0 0 ρ = γp ρ − up j /c, j/c = γp j 0 /c − up ρ0 . CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA30 2.2 Dispersion relation and wave properties In order to study the properties of low frequency waves in the pulsars magnetosphere we start by deriving the dispersion relation. We begin with Maxwell equations (c = 1) ∇ · E = 4πρ, (2.3) ∇ · B = 0, (2.4) ∇×E=− ∇×B= ∂B , ∂t ∂E + 4πJ. ∂t (2.5) (2.6) By introducing small perturbations δB , δE ∼ ek·r−ωt in the electric and magnetic field B = B0 + δB, (2.7) E = δE, (2.8) substituting this into equations (2.6) and (2.5), combining them together we get: −k × k × δE = ω 2 δE + 4πiωJ, (2.9) 1 4πi k2 J = 0, 1 − 2 δE + 2 k(k · δE) + ω ω ω (2.10) and subsequently CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA31 ji = σij δEj , δij + (2.11) 4πi k2 1 σij − 2 δij + 2 ki kj δEj = 0, ω ω ω where σij is the conductivity tensor and εij = δij + 4π σ ω ij (2.12) is the dielectric tensor. The only nontrivial solution will give us the dispersion relation, which is equal to: det k n2 δij − ni nj − εij k= 0, (2.13) where n =| ~n | is the reflective index. In order to find the conductivity tensor from (2.11) distribution function is required since j = P s qs R d3 vvfs . The subscript s denotes the type of species (electrons or positrons) and fs is the distribution function to be determined. The distribution function is found by using the Vlasov equation that describes ultrarelativistic electron positron plasma: ∂ ∂ qs ∂ fs + v fs + (E + v × B )fs = 0, ∂t ∂r ms ∂u (2.14) where u = p/m, v = u/γ, γ 2 = 1 + u2 and c = 1. The external magnetic field is taken to be B0 = (0, 0, B0 ). The current is equal to: jz = X Z qs vz fs,0 u⊥ du⊥ duz , (2.15) s jx = X1X s 2 σ Z qs v⊥ fs,σ u⊥ du⊥ duz , (2.16) CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA32 jy = X1X 2 s Z iσqs v⊥ fs,σ u⊥ du⊥ duz , (2.17) σ in which the distribution function is represented as: n=∞ X fs = fs,n (u⊥ , uz )exp(−inφ). (2.18) n=−∞ The only components of fs that need to be obtained are fs,0 and fs,σ where σ = ±1. The complete solution of Vlasov equation is given in Appendix A. Here we will give the final result only. fs,o = Fs,o (u⊥ , uz ) − iαk⊥ v⊥ αk⊥ v⊥ Ey µo Fs,o + Ex µo Fs,o + 2Ω̃ζ 2Ω̃2 ω (2.19) + fs,σ 2 v⊥ vz iαk⊥ iαk⊥ v⊥ µo Fs,o Ez + µo Fs,o Ex . 2Ω̃2 ζ 2Ω̃2 2 2 iαζ k⊥ v z αk⊥ v⊥ Ez k⊥ vz α Ey µo Fs,o + =− Eσ + Ez µo Fs,o + Eσ + µo Fs,o . (2.20) ζ ζ 2iσ Ω̃ 4σ Ω̃2 ζ 2Ω̃2 The notations are: α = q/m, Ω̃ = qBo /mγ, ζ = ω−kz vz , µo = [ζ (∂/∂u⊥ ) + kz v⊥ (∂/∂uz )] /ω and Fs,0 is the initial distribution function. The dielectric tensor then becomes: εzz 2 Z 2 2 Z X ωp,s X ωp,s k⊥ v z u⊥ ∂ =1+ Fs,o du⊥ duz − u2⊥ u2z γ −1 ζ −1 µo Fs,o du⊥ duz , (2.21) 2 ω ζ ∂uz 2ωΩs s s CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA33 εyz = −εzy = i Z 2 X ωp,s k⊥ 2ωΩs s εxz = εzx = − (2.22) u2⊥ uz µo Fs,o du⊥ duz , (2.23) Z 2 X ωp,s k⊥ s εxy = −εyx u⊥ uz ζ −1 µo Fs,o du⊥ duz , 2ωΩ2s Z 2 X ωp,s u2⊥ µo Fs,o du⊥ duz , =i 2ωΩ s s (2.24) εxx Z 2 X ωp,s =1− u2⊥ γζµo Fs,o du⊥ duz , 2 2ωΩ s s (2.25) εyy 2 2 ωp,s k⊥ = εxx + 2ωΩ2s (2.26) Z u4⊥ γ −1 µo Fs,o du⊥ duz . The wave vector k is taken to be, k = (k⊥ , 0, kk ). The gyrofrequency is Ωs = qs B0 /ms 2 = 4πNs qs2 /ms . The distribution functions Fs,0 (u⊥ , uz ) are and the plasma frequency is ωps arbitrary. Normalizing the distribution function R Fs,0 u⊥ du⊥ duz = 1 the above expressions can be partially integrated and then the dispersion relation takes the following form: εzz = 1 + 2 2 2 2 2 2 X ωp,s X ωps X ωps X ωps k⊥ kz u2z u2⊥ ∂ k⊥ u2z u2⊥ uz ∂ u2z h i+ h i − h i − h 2 is , s 2 Ω2 γ s 2 Ω2 3 2 Ω2 ω γζ ∂u ω 2ω γ 2ω γ ζ ∂u z z s s s s s s s (2.27) εyz = −εzy = i X s εxz = εzx = − 2 X ωp,s k⊥ εxy = −εyx 2ω 2 Ωs uz u2⊥ ∂ is ], γζ ∂uz (2.28) 2u2z − u2⊥ is ], γ (2.29) + kz h [−2ωhuz is + kz h 2 s εxx 2 k⊥ ωp,s uz [−2ωh is 2 2ω Ωs γ 2 X ωp,s uz =i [−2ω + kz h is ], 2 2ω Ωs γ s 2 2 2 X ωp,s 2 2 u⊥ 2 2uz − u⊥ =1− [−2ω hγis − ω h is + 4ωkz huz is − kz h is ], 2ω 2 Ω2s γ γ s (2.30) (2.31) CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA34 εyy = εxx + 2 2 X ωp,s k⊥ 3 Here the averaging is h...i = R u4⊥ u2⊥ u4⊥ ∂ is ]. [−4h is + h 3 is + kz h 2 4ωΩ2s γ γ γ ζ ∂uz (2.32) (...)f0 (uz )duz where f0 (uz ) is the distribution function in equilibrium. We assume that the distribution function of the pair plasma in the pulsar magnetosphere is one dimensional Fs,0 ∝ (u⊥ )u⊥ . This is justified since the perpendicular energy of the particles is radiated away. As well we shall assume Fs,0 = F̃s,0 (uz )(u⊥ )u⊥ and normalized to R F̃s,0 duz = 1. Applying all that mentioned above, the dielectric components become: εxx = εyy ≡ ε⊥ = 1 + 2 X ωps Ω2s s εzz = εk = 1 − (hγis ) − 2nk huz is + n2k hu2z γ − 1is ), 2 X ωps ω s Ws (nk ) + 2 s εxy = −εyx = −i 2 X ωps n⊥ s 2 X ωps n⊥ s Ω2s hu2z γ −1 is , 2 X ωps (1 − nk huz γ −1 is ), ωΩ s s εyz = −εzy = −i εxz = εzx = 2 2 X ωps n⊥ Ω2s ωΩs huz γ −1 i, (huz is − nk hu2z γ −1 is ), (2.33) (2.34) (2.35) (2.36) (2.37) where n = k/ω, nk = n cos θ, n⊥ = n sin θ and θ is the angle between the external magnetic field. The Ws is equal to: 1 Ws (nk ) = − nk Z 1 df0s duz , 1 − nk vz + iτ duz (2.38) CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA35 with τ → 0. Let us introduce the following notations: N+ = N0 (1 + η), N− = N0 (1 − η), X m1 = (2.39) (1 + sη)hγis , (2.40) (1 + sη)huz is , (2.41) u2z (1 + sη)h is , γ (2.42) s m2 = X s m3 = X s W = X (1 + sη)Ws , (2.43) (1 + sη) = 2η, (2.44) uz is . γ2 (2.45) s δ1 = X s δ2 = X (1 + sη)h s Substituting those into the dielectric tensor we get: ε⊥ = 1 + ∆m1 − ∆m2 n cos θ + ∆m3 n2 cos2 θ, εk = 1 + ∆m3 n2 sin2 θ − W , Z2 √ i ∆δ2 cos θ − δ1 εxy = , Z √ i ∆δ2 n sin θ εyz = − , Z (2.46) (2.47) (2.48) (2.49) CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA36 εxz = ∆m2 n sin θ − ∆m3 n2 sin θ cos θ, (2.50) where ∆ = ωp2 /Ω2 , ωp2 = 4πN0 e2 /m, Z = ω/ωp , δ1 = ρ/N0 e and δ2 = j/N0 ec. The general dispersion relation can be written as: A2 Z 2 + A1 Z + A0 = 0, (2.51) A2 = d22 (d11 d33 − d213 ), (2.52) A1 = d11 d22 W − d11 d223 − d33 d212 + 2d12 d23 d13 , (2.53) A0 = −W d212 , (2.54) d11 = (1 − ∆m3 )n2 cos2 θ + 2∆m2 n cos θ − (1 + ∆m1 ), (2.55) d22 = n2 (1 − ∆m3 cos2 θ) + 2∆m2 n cos θ − (1 + ∆m1 ), (2.56) d33 = (1 − ∆m3 )n2 sin2 θ − 1, (2.57) where √ d12 = ∆(δ2 n cos θ − δ1 ), d13 = −(1 − ∆m3 )n2 sin θ cos θ − ∆m2 n sin θ, (2.58) (2.59) √ d23 = ∆δ2 n sin θ. (2.60) CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA37 The polarization rations are given then by: rx ≡ Ex d12 d23 − Z 2 d22 d13 = , |E| D (2.61) ry ≡ Z(d11 d23 − d12 d13 ) Ey = , |E| D (2.62) Ez Z 2 d11 d22 − d212 = , |E| D (2.63) rz ≡ where D= q (d12 d23 − Z 2 d22 d13 )2 + Z 2 (d11 d23 − d12 d13 )2 + (Z 2 d11 d22 − d212 )2 . (2.64) Now we shall examine the dispersion relation in two different cases: the infinite magnetic field and the finite magnetic field. 2.2.1 Infinite magnetic field approximation The infinite magnetic field approximation corresponds to the region of the pulsar magnetosphere close to the pulsar surface. In this approximation Ω → ∞ that is ∆ → 0. This case was studied by [Gedalin and Machabeli (1982), Arons and Barnard (1986),Lyubarski (1995), Gedalin et al. (1998),Lyubarski and Petrova (2000)]. In the infinite magnetic field approximation our equation became: d23 = d12 = 0, (2.65) CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA38 d11 = n2 cos2 θ − 1, (2.66) d22 = n2 − 1, (2.67) d33 = n2 sin2 θ − 1, (2.68) d13 = −n2 sin θ cos θ, (2.69) A0 = 0, (2.70) A1 = d11 d22 W = (n2 cos2 θ − 1)(n2 − 1)W, (2.71) A2 = d22 (d11 d33 −d213 ) = (n2 −1)[(n2 cos2 θ−1)(n2 sin2 θ−1)−(−n2 sin θ cos θ)2 ] = −(n2 −1)2 . (2.72) We reestablished that in the case of infinite magnetic field the plasma modes are: a) a magnetosonic t-wave with dispersion relation n = 1. b) mixed Langmure-Alfvén mode with dispersion relation n2k − 1 ω2 = 2 W (nk ). ωp2 n −1 (2.73) The plasma is transparent when (n2k − 1)(n2 − 1)(W (nk )) > 0. Since W (nk < 1) > 0 only the superluminal L mode with n2 < 1 exists. We are interested in the oblique wave with |n2 − 1| << 1. In the approximation of nk << 1 ⇒ W = hγ −3 i so that ω 2 = ωp2 hγ −3 i at n = 0 and at high frequency limit n2 = 1 − ωp2 sin2 θhγ −3 i/ω 2 except for θ ≈ 0. CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA39 In the case of compact distribution, that is if there exists um such that F = 0 for | uz |> um , 2 the Alfvén wave is undamped in the range 1 < n2k < vm . In the case of infinite magnetic field the all modes have linear polarization. For the t-mode Ex = Ez = 0 and Ey 6= 0. For the L − A mode Ey = 0 , Ex 6= 0 and Ez 6= 0. 2.2.2 Finite magnetic field In the case of finite magnetic field the dispersion equation dose not split into two separate equations as before and its general form is quit complicated. We shall analyze the dispersion relation in the pulsar frame being interested in waves that don’t damp on the Cerenkov resonance. In order to satisfy the weak cyclotron damping condition for a flat distribution [proposed by Arons and Barnard (1986) and used by us for the numerical representation] nk vmax < 1 or nk vmin > 1 is required [Gedalin et al. (1998)]. The limit of low frequency long wavelength is: γ|1 − n cos θv| << Ω/ω. According to Cheng and Ruderman (1977) the polarization is formed at r ∼ 103 R0 where Ω ∼ 1010 −1011 s−1 , while the observed frequencies are ω109 − 1011 s−1 , so that ω/Ω ≤ 1. Bringing those conditions together we fined that either 2 2 −1/γmax << nk − 1 . 1/2γmax or 1/2γmin . nk − 1 << 1/2γmax . For γmin ≈ 10 and γmax ≈ 103 the second condition cannot be satisfied so that we shall work in the range of 2 −1/γmax << nk − 1 . 1/2γmax . It is most likely that the Alfvén waves don’t participate in the formation of the observed spectrum from pulsars for two reasons [see Gedalin et al. (1998), Gruman (1998)]. First the Alfvén wave frequency is limited from above (nk ) and second it’s dumping. The damping of CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA40 Alfvén wave results from the continuity of the distribution function at uz = um . The Alfvén the damping condition holds for R ≥ 103 R0 , limiting the existence of the Alfvén wave to the range R < 103 R0 . So that in what follows we shall not consider it. We are interested in the range of small θ and n ≈ 1. We will define n cos θ = 1 + χ where |χ| << 1. Using this approximation for equations (2.55)-(2.60) and neglecting high order terms in χ and θ we get: d11 = 2χ − ∆m3 + 2∆m2 − ∆m1 , (2.74) d22 = 2χ + θ2 − ∆m3 + 2∆m2 − ∆m1 , (2.75) d33 = θ2 − 1, (2.76) √ d12 = ∆(δ2 − δ1 ), d13 = θ(∆m3 − ∆m2 − 1), (2.77) (2.78) √ d23 = ∆δ2 , (2.79) (m1 + m3 − 2m2 ) , 2 (2.80) where we define: χ1 = ∆ and χ2 = χ1 − θ2 . 2 (2.81) CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA41 The refraction index under this approximation is n = 1 + χ + θ2 /2, where we used the expansion of the cos function. The complete dispersion relation (2.51) in this case will take the following form: 2Z 2 (W − Z 2 )(χ − χ2 )2 − W Z 2 θ2 (χ − χ2 ) − ∆(Z 2 − W )(δ2 − δ1 )2 = 0. (2.82) The solutions are: χ − χ2 = W Z 2 θ2 ± p W 2 Z 4 θ4 + 8Z 2 ∆(Z 2 − W )2 (δ2 − δ1 ) . 4Z 2 (W − Z 2 ) (2.83) The equation solution for (χ − χ2 ) will actually give us the refraction index n. The electric field relation under this approximation is: | Ey Z|χ − χ2 | |= √ . Ez ∆|δ2 − δ1 | (2.84) The last two equations describe the waves properties in the desired approximation. In order to examine waves properties one has to consider specific distribution function since there 2 is no universal approximation for W in the range −1/2γmax < χ < 1/2γmax . In the next section we will use the waterbag distribution in order to examine the waves features. 2.3 Numerical Analysis: Waterbag Distribution In order to emphasize the waves properties we will perform numerical analysis with waterbag distribution function. Taking into account the deviations from quasi-neutrality and CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA42 current the following waterbag distributions for electrons and positrons shifted one relative to the other are considered: f+ = N+ Θ(u2 − uz )Θ(u1 + uz ), u1 + u2 (2.85) f− = N− Θ(u2 + uz )Θ(u1 − uz ), u1 + u2 (2.86) This kind of distribution was first proposed as an approximation by Arons and Barnard (1986). In our previous studies we found [ see, Gedalin et al. (1998); Gruman (1998) and Melrose et al. (1999)] that a more realistic distribution functions add dissipation of waves with phase ω/kk < u− . One the other hand in the high phase velocity range n < nm the waves’ features do not depend significantly on the exact form of the distribution function. p The ratio ry = iEy / Ex2 + Ey2 + Ez2 will be used as a signature of elliptical polarization [Gedalin et al. (2001)]. In case of the waterbag distribution equations (2.40)-(2.45) turn to: m1 = u1 γ1 + u2 γ2 + ln[(γ1 + u1 )(γ2 + u2 )] , u1 + u 2 m2 = η(u2 − u1 ), m3 = 1 γ1 − u1 [(u1 γ1 + u2 γ2 ) − ln ], 2(u1 + u2 ) γ2 − u2 δ1 = 2η, (2.87) (2.88) (2.89) (2.90) CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA43 δ2 = W = 2(γ1 − γ2 ) , u 1 + u2 (2.91) v2 ηn cos θ(v22 − v12 ) 2 v1 . + ) + ( u1 + u2 1 − v12 n2 cos2 θ 1 − v22 n2 cos2 θ (u1 + u2 )(1 − v12 n2 cos2 θ)(1 − v22 n2 cos2 θ) (2.92) We shell define: u0 = u 1 + u2 , 2 (2.93) ū = u 1 − u2 . 2 (2.94) In all the following figures we will plot only one mode. 0.2 0.18 parameters 0.16 0.14 ry 0.12 0.1 0.08 0.06 0.04 0.02 0 0 200 400 600 800 1000 omega 1200 1400 1600 1800 2000 Figure 2.2: Ellipticity as a function of frequency for different angles of propagation, θ = 1◦ crosses, θ = 2◦ -diamonds, θ = 5◦ -triangles. ∆ = 10−8 ,u0 = 2, ū = 0.01, η = 0.005. CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA44 We start by examining the dependence of the ellipticity on the propagation angle. Figure 2.2 shows ellipticity as a function of frequency for different angles of propagation θ = 1◦ ,θ = 2◦ and θ = 5◦ . The plasma parameters are: ∆ = 10−8 , u0 = 2, ū = 0.01 and η = 0.005. The ellipticity rapidly decreases to nearly zero when the propagation angle increases. At small angles the ellipticity is not negligible. Ellipticity increases with the increase of frequency. Since we assumed that |ω − kz vz | << ωhγi we shall consider only the range ω/ωp << ∆−1/2 . 0.2 0.18 parameters 0.16 0.14 ry 0.12 0.1 0.08 0.06 0.04 0.02 0 0 50 100 150 omega 200 250 300 Figure 2.3: Ellipticity as a function of frequency for different angles of propagation, θ = 1◦ crosses, θ = 2◦ -diamonds, θ = 5◦ -triangles. ∆ = 10−6 ,u0 = 2, ū = 0.01, η = 0.005. CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA45 Figure 2.3 shows the dependence of polarization as before except that now ∆ = 10−6 . The ellipticity is substantially stronger. 1 0.8 0.6 0.4 ry 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 50 100 150 omega 200 250 300 Figure 2.4: Ellipticity for various pairs of η and ū. η = 0.005 and ū = 0.01-crosses; η = 0.005 and ū = −0.01-diamonds; η = −0.005 and ū = 0.01-triangles; η = −0.005 and ū = −0.01-circles. ∆ = 10−6 and θ = 1◦ . We examine the ellipticity for different signs of η and ū as well, η = 0.005 and ū = 0.01crosses; η = 0.005 and ū = −0.01-diamonds; η = −0.005 and ū = 0.01-triangles; η = −0.005 and ū = −0.01-circles. For Figure 2.4 we choose ∆ = 10−6 and θ = 1◦ . The sense of polarization depends on the sign of the parameters. CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA46 1 0.8 0.6 0.4 ry 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 50 100 150 omega 200 250 300 Figure 2.5: Ellipticity for different values of the pair η and ū.η = 0.005, ū = 0.01 -crosses; η = 0.01, ū = 0.01 -diamonds; and η = 0.005, ū = 0.02 -triangles. In Figure 2.5 we show the dependence of ellipticity on the values of η and ū, η = 0.005, ū = 0.01 -crosses; η = 0.01, ū = 0.01 -diamonds; and η = 0.005, ū = 0.02 -triangles. We left ∆ = 10−6 and θ = 1◦ . In general, the sense of the polarization is determined by the sign of d11 d22 − d212 ,which depends on the refraction index or frequency. For |n2 − 1| << sin2 θ << 1, the sense of the polarization is determined by δ2 − δ1 , that is, by the gyrotropic term ε12 . CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA47 2.4 Discussion and conclusions In this part of the research we have studied the properties of low frequency waves in nonneutral, current carrying pair plasma. The goal was to scrutinize the origin of the circular polarization present in the pulsar radiation. The observed diverse circular polarization properties may be related to the pulsar emission mechanism or to the propagation effects occurring in the pulsar magnetosphere [Melrose (1995)]. We concentrated on the possibility of explaining the circular polarization as a consequent of the propagation effects. We have shown that relative streaming of electrons and positrons changes the polarization of the normal modes in the pulsar plasma. In the electromagnetic range | n−1 | 1 the waves are elliptically polarized and the ellipticity depends on the charge and current densities. The higher the current in the plasma the stronger the effect. The ellipticity is stronger in the outer layers of the pulsar magnetosphere and is enhanced with the increase of ωp /Ω. It disappears when this ratio becomes too small. Whenever the radio spectrum is formed the normal plasma modes generated close to the surface of the neutron star will have linear polarization till they reach the distance of ≈ 103 R0 and then entering the region where elliptical polarized waves are allowed to propagate there polarization may transform (partly) into elliptical. The effect of polarization change towards elliptical polarization is stronger at higher frequencies in agreement with observations [von Hoensbroech et al. (1998)]. The polarization is stronger for small angles of propagation as well in agreement with observations [see Han et al. (1998), Rankin (1983a), Lyne and Manchester (1988)]. Wide range of variations of the polarization properties from pulsar to pulsar and with phase can be related to the high sensitivity of the CHAPTER 2. LOW FREQUENCY WAVES IN ASYMMETRIC RELATIVISTIC PAIR PLASMA48 polarization to the plasma parameters (relative velocity, density). The change of the normal mode features has to be taken into account in every model and calculations of the pulse properties. However this explanation of elliptical polarization does not exclude the possibility that other processes may contribute to the ellipticity of the observed radiation. CHAPTER 3 THE EMISSION MECHANISM In this chapter we will build a model for pulsar radiation based on the nonresonant hydrodynamic beam instability. We believe that the radio emission from pulsars is generated by beam instability. The mean idea of the model is that low energy dense beam penetrates the plasma. Such beam is the result of nonstationary avalanche pair generation [for details see Usov (1987); Ursov and Usov (1988); Asseo and Melikidze (1998)] or in the steady avalanche regime where the high energy tail of pair plasma distribution transforms into a dense beam due to the inverse Compton scattering [detailed description see in Daughtery and Harding (1982); Lyubarski and Petrova (2000)]. The resulting distribution consists of a plasma body having γp ∼ 10 − 100 and a beam having γb ∼ 103 . Quasi-transverse waves in the beam are generated. The instability causes a beam mode to grow over a broad range of lower frequencies. The wave growth is not restricted to a narrow layer of the magnetosphere and can last over a large range of heights. As the wave is propagating the growth continues 49 CHAPTER 3. THE EMISSION MECHANISM 50 till the wave frequency is equal to the resonant frequency. At this point the growth ceases and the spectrum is formed. From this point the generated radiation, consisting of electromagnetic waves in radio range, can escape directly into the universe. No additional processes are needed Gedalin et al. (2002) and Gedalin et al. (2002). As we believe that the radiation is formed well inside the pulsars magnetosphere, the infinite magnetic field approximation is used [Cordes (1992); Lyubarski and Petrova (2000)]. This chapter is organized as follows: we start by defining the pulsars conditions. Afterwards in the second section we will review the hydrodynamic beam instability and apply it to the case of cold plasma-cold beam. We show that the qualitative features of the instability are not sensitive to the details of the distribution function. In the third section we will drive the local field line spectrum within a simple approximate model taking into account the propagation along the curved field line. The results of this chapter were published in the following papers: 1. Gedalin M., Gruman E., and Melrose D.B., Phys.Rve.Lett. 88, (2002). 2. Gedalin M., Gruman E., and Melrose D.B., Mon. Not. R. Astron. Soc. 337, 442, (2002). 3.1 The pulsar conditions The geometry of the instability region is very important for the model of radio emission generation. We shall consider the geometry of perpendicular rotation. In this case the magnetic axis is along the x-axis and the rotation is along the z-axis in the pulsar rest CHAPTER 3. THE EMISSION MECHANISM 51 frame. We do not distinguish between the rotating and non-rotating frames Lyubarski and Petrova (2000); Gedalin and Melrose (2001). The magnetic field geometry is of a dipole. Taking x = r cos φ and y = r sin φ the magnetic field lines are given by: sin φ √ = const. r (3.1) The angle between the magnetic field and the magnetic axis is then: tan ψ = 3 sin φ cos φ . 3 cos2 φ − 1 (3.2) The last open magnetic field line is the line that passes through the point located at the light cylinder distance with coordinates r = RL and φ = 90◦ where RL is the light cylinder radius. The equation for the last open field line is: r sin φmax = r . RL (3.3) The radio waves are generated in the magnetic tube which is limited by the last field line. For the generation that occurs at some RE the maximum angle width of the emission zone is sin φE = p RE /RL . If the generation occurs in the deep magnetosphere, R0 << RE << RL then φE << 1 and the maximum opening angle of the field line is ψE ≈ 3φE /2. The polar cap zone is the region of open field lines and is limited by the last open field line. The angular size of the polar cap at the pulsar surface is sin φ0 = p R0 /RL . CHAPTER 3. THE EMISSION MECHANISM 52 The parallel inhomogeneity scale in the emission region is Lk ∼ RE . The transverse inhomogeneity scale is L⊥ . RE φE depending on the pair production. Since the radio waves may be treated as plane waves with frequency ω and wave vector k = (k⊥ , kk ) (⊥ and k refer to the local magnetic field direction), the geometrical optics’ condition should be satisfied, that is, kk Lk >> 1 and kk Lk >> 1. The condition kk Lk >> 1 is always satisfied because we are only interested in waves with θ = arctan(k⊥ /kk ) << 1. The transverse condition limits the propagation θ from below: θ >> θc ∼ 1/kL⊥ & 1/kRE φE . Thus, strictly parallel propagation is prohibited by the plasma inhomogeneity. Whether waves with θ > θc can be considered as nearly parallel depends on what extent the wave dispersion changes over the rang θ < θc . 3.2 Hydrodynamic Instability We will use the infinite magnetic field approximation since we believe that the radiation is formed well inside the pulsars magnetosphere [Cordes (1992)]. As a first step, in order to emphasis the model properties, we will use the cold beam approximation, neglecting kinetic effects and thermal spread that is justified in a case of hydrodynamic instability [ Mikhailovskii (1974)]. Later we will show that including the spread in velocity does not affect our conclusions significantly. We assume that the distribution function is f (u) = np fp (u) + nb fb (u) where p stands for pair and b for beam components. The distribution functions are normalized R fb du = 1 and CHAPTER 3. THE EMISSION MECHANISM R 53 fp du = 1. The dispersion relation for this plasma beam system in the limit of B → ∞ is: εk = tan2 θ , z2 − 1 (3.4) where z = ω/kk and ωp2 εk = 1 − 2 W (z) + εb . kk The dispersion function for the pairs is: W = R∞ −∞ (3.5) (v − z − iτ )(dfp /du)du, τ → +0, and the beam contribution is: εb = −ωb2 γb−3 (ω − kk vb )−2 . For the cold beam approximation the dispersion relation (3.4) becomes: ωb2 z 2 − z02 ωp2 W (z) = 2 − ≡ K(z), γb3 (ω − kk vb )2 z −1 kk2 (3.6) where z0 = 1/ cos θ. In the absence of the beam K(z) = 0 the dispersion relation can be written in the parametric form as: kk2 = ωp2 W (z 2 − 1) , z 2 − z02 ω2 = ωp2 W z 2 (z 2 − 1) . z 2 − z02 (3.7) Adding the beam will give additional solutions, the beam modes, and hydrodynamic instabilities may be attributed to a beam mode becoming intrinsically growing. The instability is said to be nonresonant when the beam mode does not coincide with a natural mode of the pair plasma, and resonant when it does. The contribution of the beam is significant only when the dominator in the left hand side of (3.6) is small, near z = vb . In this case we can CHAPTER 3. THE EMISSION MECHANISM 54 writ ω = kk vb + δω where |δω| << ω. The solution is: ω h vb2 − z0 ωp2 W (vb )vb2 i−1/2 ωb pb = − . ω2 γb3 K(vb ) γb3 vb2 − 1 δω = p (3.8) The imaginary part of the oscillation frequency is as a rule called the growth rate if positive and decay rate if negative. In order that δω should be imaginary and positive K(vb ) < 0, that is, ω 2 < ωp W (vb )vb2 (vb2 − 1)/(vb2 − z0 ) and this may occur only if W (vb ) > 0. This is the nonresonant part of instability which sets on the ”beam mode” ω ≈ kk vb . The instability becomes resonant when the ”beam mode” is also a normal mode of the plasma without beams. In this case the solution of (3.8) is invalid. For the resonant case the right hand side of (3.6) have to be Taylor expanded up to the first nonzero term which gives: √ 2 3 ωb 1/3 Γr = Imδω = , 2 γb3 K 0 (3.9) where K0 ≡ ∂K ∂ω = res 2vb2 γb4 tan2 θ ωp2 vb3 0 − W (vb ), ω ω (3.10) and W 0 (z) = dW/dz. The derived expressions give the growth rate for arbitrary propagation and beam parameters. We are particularly interested in the behavior of the instability for small angels θ << 1 and highly relativistic beam vb ≈ 1 − 1/2γb2 , γb >> 1. CHAPTER 3. THE EMISSION MECHANISM 55 The nonresonant instability growth rate will take the form: i−1/2 ωb h ωp2 W (vb ) 2 2 Γn = √ − 1 − γ θ , b γb ω2 (3.11) is slightly increasing function of the propagation angle. In the range γb−2 . θ . γb−1 (electromagnetic mode) the dependence on the angle is negligible. The maximum beam frequency that is also the frequency at which the resonant instability occurs is: 2 ωmax = ωp2 W (vb ) . 1 + γb2 θ2 (3.12) The frequency is almost independent of the propagation angle. The resonant growth dependence on the propagation angle is determined by the relative importance of the two term in K 0 . For a wide class of distributions the approximation W 0 (vb ) ∼ γp2 W (vb ) holds. This means that the angle dependence of the resonant growth rate is insignificant if θ . γp /γb2 and decreases slowly with θ & γp /γb2 . Assuming linear growth the equation for the wave amplitude is: daω = Γ(ω, R)aω . dt (3.13) The growth rate Γ(ω) is a function of plasma parameters np , nb , γp and γb by which it depends on the radius R. The plasma is streaming outward at a speed close to the speed of light and CHAPTER 3. THE EMISSION MECHANISM 56 the solution of (3.13) is: Z 0 R aω (R) = aω (R ) exp Γ(ω, R)dR, (3.14) R0 where R0 is the radius at which the instability sets on. The measured radiation intensity is proportional to the square of this amplitude. The efficiency of the wave generation determined by the gain factor G that is equal to: Z G(ω, θ) = exp 2 Γ(ω, θ, R)dR . c (3.15) The frequency of a wave propagating outwards changes along the path because the plasma in pulsar magnetosphere is inhomogeneous. The plasma density drops according to the distance from the neutron star as n ∝ R−3 . Hence the resonant conditions can be satisfied only for a short distance. For a given frequency ω the resonant condition |ω − kk vb | < Γr (R) will be satisfied only for ∆R/R ∼ Γr /ωr . Still a wave with a given ω << ωr can grow nonresonantly while propagating until ω ≈ ωr . At the resonant point the beam mode joins the plasma L − O mode. Beyond this point the amplification stops and the wave escapes as L − O mode. The nonresonant growth occurs throughout a much longer path in the pulsar magnetosphere then the resonant growth. As a result the nonresonant gain factor is much bigger. Beyond the resonant frequency the instability ceases and the L − O waves are in the transparency range. CHAPTER 3. THE EMISSION MECHANISM 57 The unstable mode polarization is: Ex tan θ = = γb2 tan θ. Ez 1 − z2 (3.16) For θ . γb−2 the wave is longitudinal (E k B0 ) and transversal (E ⊥ B0 ) otherwise. Polarization of the growing waves is unimportant since the polarization evolves as the wave propagates outward and observed polarization may be quite different from that at the point of emission. 3.2.1 Cold Plasma Approximation To illustrate the results in the simplest possible way we shall use the cold plasma approximation, fp = δ(u − up ) so that: W (z) = γp3 (z 1 . − vp )2 (3.17) In the absence of the beam dispersion relation takes the following parametric form: kk2 = ωp2 (z 2 − 1) , γp3 (z 2 − z02 )(z − vp )2 ω = kk z. (3.18) In the ultrarelativistic limit γb >> γp >> 1 the plasma-beam dispersion function is: W = 4γp + ωb2 , ωp2 γb3 (δω)2 (3.19) CHAPTER 3. THE EMISSION MECHANISM 58 For small θ the nonresonant instability growth rate becomes: Γn = −1/2 ωb 4ωp2 γp 2 2 − 1 − γ θ . b 3/2 ω2 γb (3.20) p 1/2 The instability becomes resonant when the frequency is ωr = 2ωp γp / 1 + γb2 θ2 (the resonant frequency) and the resonant growth rate is: √ 1/2 3 ωb2 ωp γp Γr = . 2 γb3 (γb4 θ2 + γp2 )(1 + γb2 θ2 )1/2 (3.21) For small propagation angles θ . 1/γb the growth rates are almost independent of θ so that the expressions using the parallel propagation approximation are: −3/2 Γn = ωb γb 1/2 the resonant frequency is ωr = 2ωp γp (4ωp2 γp /ω 2 − 1)−1/2 , and the resonant growth rate is: √ 3 (ωp ωb2 )1/3 . Γr = √ 2 3 2 γb γp1/2 1/2 For low frequencies ω << 2ωp γp (3.22) (3.23) the growth rate of the nonresonant instability becomes: Γn = ωb 3/2 1/2 2ωp γb γp ω. (3.24) CHAPTER 3. THE EMISSION MECHANISM 59 It is worth mentioning that Γn (ω = ωr )/Γr ∼ (γp /γb )1/2 (np /nb )1/3 implies that for moderate np γp /nb γb the ratio of the two growth rates approximates a unity. Approximately the growth rate is: ω Γ = p 3 H(ω − ωr ), 2 γb γp (3.25) where H(x > 0) = 1 and H(x < 0) = 0. This approximation is valid for the whole range, except the narrow frequency range around the resonant. 1 Γ ←θ ω Figure 3.1: The growth rate for the cold beam cold plasma as a function of frequency for different angles of propagation. The instability range squeezes as the propagation angle increases: higher growth rates and larger regions of instability correspond to smaller propagation angels. In Figure 3.1 we plotted the growth rate as a function of frequency for different angles of propagation in the cold beam cold plasma approximation. This graph is qualitative and is not to scale. The instability range squeezes as the propagation angle increases. CHAPTER 3. THE EMISSION MECHANISM 60 Now we shall estimate the gain factor assuming that γp and γb do not change during the 1/2 outflow and ωr ∝ np ∝ R−3/2 . For a given ω the gain factor is: Z ∞ G = exp 2 R1 h ω R i Γ(R, ω) b0 1 dR = exp 2 3/2 x(x−2/3 − 1) , c cγb (3.26) where ωp0 is the plasma frequency at r = R1 and x = ω/2ωp0 γ 1/2 . The maximum of the gain factor is at x ≈ 0.2 that is: Gmax ≈ exp h 0.25ω R i b0 0 3/2 . (3.27) cγb Beyond the resonant frequency the instability ceases and the L − O waves are in the transparency range. The behavior of the normal modes in the transparency region is shown in Figure 3.2. The graph is qualitative and actually the distance between −1/2γb2 and −1/2γp2 is much bigger since γb >> γp . The superluminal mode with z − 1 > θ2 /2 is the L − O mode. It corresponds to ω = kc in the high frequency limit, ω/ωr >> 1. Four other modes have ω = kk vp or z − 1 = 1/2γp2 (two) and ω = kk vb or z − 1 = 1/2γb2 ( other two) asymptotically, and stand in the plasma and beam frame ,respectively. We should mention that the beam mode is electromagnetic and it polarization found from the dispersion relation is : E⊥ θ ≈ . Ek 1/2γb2 + θ2 The ratio is large for 1/γb2 << θ (3.28) CHAPTER 3. THE EMISSION MECHANISM 61 kll -1/2γ 2 p -1/2γ 2 b 2 θ /2 z-1 Figure 3.2: The behavior of the normal modes in the transparency range for cold beam-cold plasma case. The instability occurs in the range −1/2γp2 < z − 1 < −1/2γb2 where there is a gap between the dispersion curve and the z-axis. The instability ceases in the lowest point of this curve, where two complex conjugate solutions of the dispersion relation become a double real solution. The unstable mode should proceed further along one of this bifurcation branches. 3.2.2 Relative Spread in Momenta To justify our neglect of the spread in velocity or momentum in the beam we shall consider a hot plasma distribution and show that it does not affect our results significantly. We consider ultrarelativistic distributions γ >> 1. In the ultrarelativistic limit v = p 1 − 1/γ 2 = 1 − 1/2γ 2 . We define y = 1/2γ 2 and use it as an independent variable. Under this approximation the dielectric function takes the form: ε = 1 + εp + εb , (3.29) CHAPTER 3. THE EMISSION MECHANISM 62 where: ω2 εs = 2s Ws , k Z Ws = dfs , y+z−1 (3.30) fs is a function of y and df = (df /dy)dy. Normalization condition is: Z f dy = 1. (2y)3/2 (3.31) z2 − 1 k2 W, = ωp2 z 2 − z02 (3.32) Then the dispersion relation is equal to: K2 = where z0 = 1/ cos θ , W = Wp + αWb and α = nb /np . We are interested only in the forward propagation that is K > 0, so that W is well defined for Imz > 0 and should be analytically continued to Rez ≤ 0. Waves with Rez < 0 are subluminal ω/k < 1, while Rez > 0 corresponds to superluminal waves. Waterbag Distribution We will consider a double waterbag distribution of the form f = np Ap H(y1 − y)H(y − y2 ) + nb Ab H(y3 − y)H(y − y4 ), (3.33) where H(y) is the step function H(y > 0) = 1 , H(y < 0) = 0. From the normalization condition we shall obtain: 1 1 −1 Ap = √ −√ , 2y2 2y1 (3.34) CHAPTER 3. THE EMISSION MECHANISM 63 and 1 1 −1 Ab = √ −√ , 2y4 2y3 (3.35) respectively. A qualitative graph of the waterbag distribution shown in Figure 3.3. f y 1 y 2 y 3 y y 4 Figure 3.3: The double waterbag distribution .The high of the distribution depends on the values of np and nb respectively and the width depends on the chose of y1 , y2 , y3 , y4 . The function W is: W = ωp2 Ap (y1 − y2 ) ω 2 Ab (y3 − y4 ) + b . (x + y1 )(x + y2 ) (x + y3 )(x + y4 ) (3.36) In Figure 3.4 we plot K(x) in the transparency range. We used γ1 = 10, γ2 = 50, γ3 = 500, γ4 = 1000, α = 0.1 and θ = 0.1◦ . In Figure 3.5 we magnified the instability region. CHAPTER 3. THE EMISSION MECHANISM 100 64 K 80 60 40 20 0 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 x 0.01 Figure 3.4: K as a function of x for the double waterbag distribution . The parameters are α = 0.1, θ = 0.1◦ , γ1 = 10, γ2 = 50, γ3 = 500 and γ4 = 1000. 100 K 80 60 40 20 0 -0.0002 -0.00015 -0.0001 -5 10 -5 0 x Figure 3.5: Magnification of the instability range. The growth rate in this case is plotted in the Figures 3.6- 3.7. As it can bee seen the growth rate is similar to the one we have found for the cold case. That is, the unstable mode behaves in the same way as in the cold case. CHAPTER 3. THE EMISSION MECHANISM 65 1 Γ 0.00008 0.00006 0.00004 0.00002 4 2 6 8 10 12 K Figure 3.6: Growth rate vs frequency for the double waterbag distribution with γ1 = 10, γ2 = 50, γ3 = 500 and γ4 = 1000, and propagation angles θ = 0.01◦ , 0.05◦ , 0.1◦ and 0.5◦ . 1 Γ 0.0002 0.00015 0.0001 0.00005 2 4 6 8 10 12 14 K Figure 3.7: Growth rate vs frequency for the double waterbag distribution with γ1 = 5, γ2 = 20, γ3 = 200 and γ4 = 500, and propagation angles θ = 0.05◦ , 0.1◦ , 0.5◦ and 1◦ . CHAPTER 3. THE EMISSION MECHANISM 66 Soft Bell Distribution The double waterbag distribution discontinuous at the edges. It has been shown by us Gedalin et al. (1998),Gruman (1998) that the dumping of the Alfvèn wave results from the continuity of the distribution function. The dumping condition holds for R ≥ 103 R0 and depends on the details of the distribution function behavior. Only the L−O mode is expected to propagate freely beyond the resonance point. To illustrate what happens in this case we consider ”double soft bell” distribution f = np Ap (y − y1 )2 (y − y2)2 H(y1 − y)H(y − y2 ) + nb Ab (y − y3 )2 (y − y4)2 H(y3 − y)H(y − y4 ). (3.37) From the normalization condition we obtain: 105 , Ap = √ √ √ 5 √ 4 2( y1 − y2 ) (y1 + y2 + 5 y1 y2 ) (3.38) 105 Ap = √ √ . √ 5 √ 4 2( y3 − y4 ) (y3 + y4 + 5 y3 y4 ) (3.39) The expression for function W is long and we won’t present it here. The distribution function is shown in Figure 3.8. CHAPTER 3. THE EMISSION MECHANISM 10 67 f 8 6 4 2 0 y 2 y y 3 y 1 Figure 3.8: The double overlapping soft bell distribution with γ1 = 5, γ2 = 60, γ3 = 20, and γ4 = 80. This time the distribution is continuous with the first derivative, and there are no gaps (γ3 < γ2 ). In Figure 3.9 we plotted K = √ ReF and ImF . CHAPTER 3. THE EMISSION MECHANISM 68 K 300 150 250 K 200 150 100 50 0 100 -y x -y -y 0 3 2 4 50 0 y x 0 1 6 104 Im F 4 104 2 104 Im F 0 4000 -2 104 -4 104 -6 104 2000 -y2 -y4 X 0 0 -2000 -4000 -y 1 -y x 3 0 Figure 3.9: Transparency region for overlapping soft bell distribution with γ1 = 5, γ2 = 60, γ3 = 20, γ4 = 80. It is evident that above a certain frequency only L−O mode with z > 1/ cos θ can propagate, that is, the plasma is not transparent for other modes. Figure 3.10 and Figure 3.11 show CHAPTER 3. THE EMISSION MECHANISM 69 the phase velocity and the growth rate of the unstable mode respectively. It can be seen √ that the maximum growth rate occurs at the frequency ω = Kωp < ω γ2 and below this maximum the growth rate Γ/ω = Imz is almost independent of ω. Thus, the effective resonance frequency ωr decreases relative to the frequency predicted by the cold plasma√ beam analysis. The instability ceases at ω = ωc ≈ 2ωp γ2 , which also corresponds to the frequency beyond which no other modes can propagate except L − O mode. 1 Re(z − 1) -y4 2.5 5 7.5 10 12.5 15 17.5 K -y2 -y3 Figure 3.10: Phase velocity for double soft bell distribution. CHAPTER 3. THE EMISSION MECHANISM 70 1 Im(z − 1) y2 y4 2.5 5 7.5 10 12.5 15 17.5 K Figure 3.11: Growth rate for double soft bell distribution. 3.2.3 Validity of Cold-plasma, Cold-beam Model In this section we show that the instability growth rate may be taken from cold-cold approximation even if the true distribution is far from being cold. To simplify the analysis we shall assume that the plasma and beam distributions are well separated in the velocity space. In other words we assume that the plasma body is contained within γ1 < γ < γ2 and the beam is contained within γ3 < γ < γ4 , with γ3 >> γ2 . This does not necessarily mean that there are no particles in the regions γ < γ1 , γ2 < γ < γ3 and γ > γ4 . However, we do assume that number of these particles and their contribution are negligible. We also assume that the distribution function is steep at the low-energy and high-energy ends. We do not make any prior assumption about the behavior of the distribution inside the defined intervals. The behavior of the unstable mode is determined by the function W (z) when Imz > 0. As we have seen above, in the nonresonant instability regime z = vn + iσ, where CHAPTER 3. THE EMISSION MECHANISM 71 vn . v3 and 1/σ22 & σ & 1/γ32 . In this case W ≈ hγip − where h...i = R αhγ−3ib , σ2 (3.40) f (...)dγ. Equation (3.40) with the expressions for the cold case (3.19) one can see that the only change which has to be done is the substitution γp → hγip and γb−3 → hγ −3 ib ∼ 1/γ33 . By these substitutions the dispersion relation is reduced to that of the cold case. To this, one has to add that the only non-damping waves are superluminal L − O mode z − 1 > x0 . From the above we conclude that the beam instability develops qualitatively in the similar way in the relativistic pair plasma, rather independently of the precise shape or the distribution function. The basic features of the instability are: a)the oblique instability growth rate does not differ much from the growth rate of the parallel propagation; b)oblique unstable modes are electromagnetic, in contrast with the electrostatic parallel waves; c)resonant instability is most important in homogeneous plasma but less important than the non-resonant counterpart for the outflowing pulsar plasma; e)at the point where the instability ceases the unstable mode directly converts by refraction into L − O mode. CHAPTER 3. THE EMISSION MECHANISM 72 It is worth mentioning that for the distributions found in numerical [Daughtery and Harding (1982) and Hibschman and Arons (2001)] (f ∝ γ −3/2 or f ∝ γ −2 ) the resonance frequency 1/2 depends only weakly on the maximum Lorentz factor of the plasma (hγi ∝ γ2 3.2.4 or hγi ∝ ln γ2 Direct Refractive Conversion In an inhomogeneous pulsar plasma the point ω = ωc would correspond to the point of refractive conversion of the unstable beam mode into L−O mode. The propagating unstable mode growth until ω ≈ ωr and then propagates almost without changing up to the critical radius where ω = ωc . This mode cannot propagate any further, yet kk remains finite. As we have seen only one mode is allowed to propagate forward beyond this point: this is the L − O mode. Just before this point the only backward propagating mode is also the L − O mode. Thus, at the point ωc = ω the incident beam mode (ω, kix , kiz ) and backward L − O mode (ω, krx , krz ) should match the forward L − O mode (ω, kf x , kf z )and evanescent mode by putting kez = −κ and further κ → ∞. The system is shown in Figure 3.12. CHAPTER 3. THE EMISSION MECHANISM 73 x Transmitted L-O Reflective L-O Evanescent Incident beam B z Figure 3.12: Matching at the critical point ωc = ω. At the matching point the electric and magnetic fields should be continuous. The only components that present are Ex , Ez , By and one also has kix = krx = kf x = kex . For all this modes kx kz Ex = 2 , Ez kz − ω 2 By = zEx , (3.41) where z = ω/kz . As to the evanescent mode |kz | → ∞, in this mode By = Ex = 0, so that it should not be taken into account in the By and Ex continuity condition, while continuity of Ez always can be satisfied. So that: Eix + Erx = Ef x , zi Eix − |zr |Erx = zf Ef x , (3.42) CHAPTER 3. THE EMISSION MECHANISM 74 where we have taken into account that for the backward wave zb < 0. The above equations give: Ef x zi + |zr | = . Eix zf + |zr | (3.43) In our case zi ≈ zf ≈ |zr | ≈ 1 so that Ef x ≈ Eix , that is, the conversion of the unstable mode into forward propagating L − O mode is almost lossless. In the refraction point ω and k⊥ do not change. The unstable mode has ω ≈ kk while the L − O mode has ω = k. Thus, the propagation angle of the LO mode is related to the √ propagation angle of the unstable mode by the relation θ0 = θ/ 1 − θ2 where we assumed θ << 1. Therefore the refraction is accompanied by some increase of the propagation angle, although this increase is negligible for small angles of propagation. 3.3 Mechanism The proposed mechanism for the radio radiation pulsars include three stages: a)radio noise from the background or from incoherent emission, is amplified in the nonresonant, oblique-angle instability, with amplification occurring over a broad frequency range; b)while the wave propagates outward the ratio of the wave frequency to the characteristic frequency increases, and when this ratio reaches unity the amplification ceases; c)at the point where the ratio reaches unity the waves, which grow in the beam mode, evolve into non-dumping electromagnetic L − O mode waves which then freely escape, at least until they reach the region where the wave frequency is equal to the cyclotron frequency of the ambient particles [Luo and Melrose (2001) . CHAPTER 3. THE EMISSION MECHANISM 75 The spectrum is formed locally, that is, each magnetic field is an emitter by itself. The observed pulse and spectrum are produced by the extended zone of elementary emitters. In almost all previous models Lyubarski and Petrova (2000) (and references therein) it is implicity assumed that the beam instability is most efficient when parallel propagating waves, θ = 0, are excited. We have shown Gedalin et al. (2002) that the efficiency of oblique wave generation does not change much for small angles of propagation. For the same growth rate the efficiency of the instability is proportional to the available phase space. As seen from the above calculations, the beam instability can be considered as (quasi) parallel if θ . 1/γb2 , while in the range 1/γb2 < θ . 1/γb the instability is of oblique character. The ratio of the available phase spaces is ∼ γb in favor of the oblique regime. Moreover, strictly parallel propagation might not be realized at all, because of the inhomogeneity of the plasma tube in the direction perpendicular to the magnetic field. 3.3.1 Model for the Amplification Factor We will consider a narrow magnetic tube in which the electromagnetic waves are amplified. Let I(ω, θ, R) be the wave intensity in the unstable region. Then dI/dt = 2Γ(ω, θ, R)I where frequency ω plays the role of a parameter, since a stationary plasma is considered. We neglect for the time being the effects related to the pulsar rotation. This radiation transfer equation can be rewritten as follows: dθ dI dI + = 2ΓI, dR dR dθ (3.44) CHAPTER 3. THE EMISSION MECHANISM 76 where the group velocity of the excited waves is vb ≈ 1, and the evolution of the angle θ between the wave vector and the magnetic field direction is described by the following equation Barnard and Arons (1986) dθ 3φ 3θ = − . dR 4R 2R (3.45) Equation (3.44) valid only in the instability region where ω < ωc (R, θ). The beam modes intensity will grow till ω = ωr at R = Rr (θ, ω). Away from this radius the mode ω ≈ kk vb propagates up to radius Rc where ω = ωc , so that the angle θ continues to grow, while the intensity does not change. Let say that the lowest radius at which the instability sets in is R0 then the wave intensity at Rr is: I(ω, θ, Rr ) = GI(ω, θ0 , R0 ), (3.46) where G is the amplification factor that is equal to: Z Rr G = exp 2 R0 Γ(ω, θ0 (R), R)dR , c (3.47) where θ0 (R) is the solution of (3.45) with the boundary condition θ0 = θ at R = Rr . As we already saw from Figure 3.1 the highest unstable frequency ωr remains almost the same for θ < θc ∼ 1/γb , and decreases approximately linearly with the angle increase for CHAPTER 3. THE EMISSION MECHANISM 77 θ > θc . Let us consider a simplified model where the growth rate is: Γ(R, θ0 ) = ω n 1/2 1 3/2 b H(ωr (R) − ω)H(θc − θ0 ). np γp γb (3.48) In this approximation either θ = θc and ωr (Rc ) > ω or θ < θc and ωr (Rc ) = ω. If the instability onset occur at r = R0 on field line with φ(R = R0 ) = φ0 and let the initial wave parameters be ω, θ0 . The amplification factor is : ln G = 2ω b 1 np γp γb n 1/2 3/2 (Rc − R0 ), (3.49) where Rc is the minimum of the two radii found from the condition ωr (Rc ) = ω or θ(Rc ) = θc . In the first case the eventual propagation angle is 3φ0 R0 3/2 3φ0 Rc 1/2 θ = θ0 − + < θc . 8 Rc 8 R0 (3.50) In the latter θ = θc . Neglecting the rapidly decreasing term ∝ R−3/2 in equation 3.50 we get ξ 1/3 − ξ, if η < ξ 1/3 ; n 1/2 ω R b p0 0 ln G = 4 × 3/2 np cγb η −2 − 1, if η > ξ 1/3 , (3.51) where ξ = ω/ωr0 and η = 3φ0 /8θc . Here we use ωr ∝ R( − 3/2). Figure 3.13 shows the dependence of the gain G on the frequency ξ (dimensionless) and the dependence on the CHAPTER 3. THE EMISSION MECHANISM 78 field line opening angle η (normalized) is shown in Figure 3.13. 1 ln G 0.3 0.2 0.1 0.2 0.4 0.6 0.8 1 ξ Figure 3.13: Gain as a function of frequency ξ − ω/ωr0 . The propagation angle of the waves is θ + 3φc /2 . 5θc to the magnetic axis at the point where the instability ceases. The maximum beam half-width at the decoupling site (beyond which the waves propagate along straight lines is & 5θc , which is ∼ 5◦ for γb ∼ 50. The simplified model used here neglects the wings θ > θc , so we expect the resulting width to be somewhat larger. The pulse half-peak intensity width should be determined by ωr = ω, so we expect R ∝ ω −2/3 and width ∝ R1/2 ∝ ω −1/3 . CHAPTER 3. THE EMISSION MECHANISM 79 1 ln G 0.3 0.2 0.1 0.2 0.4 0.6 0.8 1 η Figure 3.14: Gain as a function of opening angle η = 3γb φ0 /8. In order to check how the gain factor changes as a function of φ (angle between magnetic axis and a field line) we considered a particle density distribution of the following form: np = n0 3sin2 φ R0 (1 − ). R 8 (3.52) The behavior of gain factor across the opening magnetic field lines is plotted in Figure 3.15, the parameters are np0 = 1020 m−3 , nb /np = 30, γp = 100 , γb = 50. CHAPTER 3. THE EMISSION MECHANISM 80 7 ω=10 Hz 43.5 8 ω=10 Hz 13 12.5 12 43 Ln G Ln G 11.5 42.5 11 10.5 10 9.5 42 9 -0.6 -0.4 -0.2 0 φ[rad] 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 φ[rad] 0.2 0.4 0.6 Figure 3.15: Gain as a function of opening angle φ (between the magnetic axis and a field line) for different frequencies. The parameters are np0 = 1020 m−3 , nb /np = 30, γp = 100 , γb = 50. 3.4 Discussion and conclusions In the above analysis we showed that the nonresonant oblique beam instability may be quite efficient in generation of low frequency electromagnetic waves for wide class of particle distributions. In the plasma which flows outwards in the expanding magnetic field the nonresonant instability becomes more important than the resonant one. Two difficulties with the resonant instability, too small growth factor and too high frequency, may be solved with the non-resonant instability. The unstable waves are in a beam mode below the resonant frequency, as the waves propagates outward the beam mode joins on to the L − O mode, so that the waves can freely leave the pulsar magnetosphere. One of the most important features is the high bright temperature. Roughly estimating the initial effective temperature of the electromagnetic wave spectrum is Ti ∼ γp me c2. For γp ∼ 10 this temperature is Ti ∼ 1011 K. In order to achieve brightness temperature of CHAPTER 3. THE EMISSION MECHANISM 81 3/2 1023 − 1026 K the amplification factor should be G ∼ 1012 − 1015 that is ωp0 R0 /cγb & 100 at the site of the instability onset. This estimate constraints any beam instability scenario. From observations one can conclude that the maximum intensity should be observed at √ the frequency v ωpo γp . Together with the previous estimate one finds that a frequency 3/2 ω can be efficiently generated if ωR0 /cγb & 100. For γb ∼ 102 and ω ∼ 109 s−1 one finds R0 & 3 × 106 which is within the expected region. The spectrum, derived in the previous section, possesses some of the features of the observed spectrum. It fall towards higher and lower frequencies, and the farther the frequency from the maximum the steeper the slop. The derived shape is not a power-law spectrum. However this is a single field line spectrum, while power law is found for spectra integrated the whole pulse. The integrated spectrum is the sum of the emissions coming from many single spectra from a number of field lines. The plasma parameters may change across the emission region. So may the maximum frequency and the amplification factor. Hence the resulting integrated spectrum may substantially differ from the single field line spectrum. Another assumption made in the course of the calculations is that the instability onset occurs at some radius where the beam-plasma system is already finally formed. However instability may begin even before the beam takes its final shape (beam instability and inverse Compton scattering may coexist). This would result in the broadening of the frequency range of the instability towards higher frequencies (the instability stars at smaller radii) and alter the shape of the spectrum. CHAPTER 3. THE EMISSION MECHANISM 82 The proposed mechanism predicts efficient generation of electromagnetic waves propagating at angles θ . 1γb which, for most accepted estimates is within several degrees and √ typically less than the observed pulse widths. The high-frequency ω >> ωp γp , L-O mode propagates along straight lines, so that additional refraction zones are needed to increase the pulse width, unless the effective γ participating in the instability are sufficiently low. The waves that are excited by the described beam instability, are completely linearly polarized. This results from our neglect of gyrotropy that is the distribution of electrons and positrons is identical. As we saw in chapter 2 inclusion of gyrotropy is essential when considering circular polarization. Moreover it is not quite clear where the eventual polarization is formed (although we tend to believe that the limiting polarization region is farther from the pulsar surface). CHAPTER 4 DATA ANALYSIS Radiation from pulsars has been constantly measured ever since their discovery. Integrated profiles, at a single frequency, were published by Manchester (1971), Hamilton et al. (1977), McCulloch et al. (1978),Manchester et al. (1980), Rankin and Benson (1981), Rankin et al. (1989),Xilouris et al. (1989) and Wu et al. (1993). Multifrequency studies were done by Lyne, Smith and Graham (1971),Morris et al. (1981),Kuzmin et al. (1986) Lyne and Manchester (1988),Malofeev et al. (1994), Qiao et al. (1995), and Gould and Lyne (1998). Integrated profiles have provided the basis for studies of the emitting region location within the light cylinder Radhakrishnan and Rankin (1990). The dependence of pulse width on the frequency was studied by Kuzmin et al. (1986). Adding their measurements to the observations of Manchester and Taylor (1981), Morris et al. (1981) and Sieber et al. (1975) they examined the pulse width behavior as a function 83 CHAPTER 4. DATA ANALYSIS 84 of frequency range for 13 pulsars. They analyzed the pulse width at half maximum intensity W0.5 concluding that there is a decrease in the width with frequency in the interval (100 − 400)M Hz. At higher frequencies the pulse width decreases slower and it might even reverse the trend (3 cases). Their attempt to find the pulse width at a tenth of the maximum height W0.1 was unsuccessful since at this level the profiles of pulses were influenced by noise. Gould and Lyne (1998) studied about 300 pulsars over a period of several years. They calculated both W0.5 and W0.1 . For W0.1 it was found that, generally, the pulse width increases at lower frequencies. About 10%−15% showed a significant increase in width at the higher frequency. Rankin (1983b, 1990) found that core component width is related to the polar cap geometry, Wcore = 2◦ .45P −0.5 / sin α, where P is the pulsar period and α denotes the angle between the rotation and magnetic axes of the star, but no correlation between the pulse width and frequency was found. In this chapter we perform data analysis with the following objectives: first, to make a preliminary comparison with our model, and second, to achieve better description of the emission zone structure in the pulsar magnetosphere. We used the data base of published pulse profiles maintained by the European Pulsar Network, available at : http://www.mpifrbon.mpg.de/pulsar/data. This analysis was made possible due to the kind help of Lyne A.G. who updated the pulse profiles according to our request. We selected 20 pulsars with only one peak in the integrated profiles. The pulsars’ details are given in Appendix B. This chapter is organized as follows: we start by examining the pulse width as a function of frequency. The dependence of maximum intensity upon frequency is examined in the second CHAPTER 4. DATA ANALYSIS 85 section. In the third section we present a geometrical analysis of average pulse profiles in an attempt to achieve a more quantitative description of the pulse and hopefully a better understanding of the emission geometry. In particular, we suggest to use the pulse moments in order to get a better understanding on the symmetry of the emission region. Consequently a possible distribution function across the emission region is proposed. The results of this chapter will be submitted for publication soon. 4.1 Pulse Width In our attempt to understand the mechanism responsible for the generation of the observed radiation we examine the pulse width at half intensity. The width W should be determined by the geometry of the magnetic field. There is some controversy as to which of the pulse width is more appropriate for the emission analysis, W0.5 or W0.1 . Kuzmin et al. (1986) claimed that W0.1 would give a more reliable results than W0.5 . Having better data we decided to examine if their is any difference between the behavior of W0.1 and W0.5 as a function of frequency. In Figure 4.1 the widths W0.1 and W0.5 are plotted as functions of frequency for six pulsars. No difference was found in the behavior of the pulse width between W0.1 and W0.5 . All pulsars in our sample show the same behavior. CHAPTER 4. DATA ANALYSIS 86 B2110+27 B1737-30 7 30 25 W [deg] W [deg] 0.5 5 0.1 Pulse width [deg] Pulse width [deg] 6 4 3 W 0.5 20 0.1 15 10 5 2 0 400 600 800 1000 1200 Frequency [MHz] 1400 1600 1800 400 600 800 B0919+06 1400 1600 1800 35 W [deg] W [deg] 0.5 0.1 30 Pulse width [deg] 20 Pulse width [deg] 1000 1200 Frequency [MHz] B0950+08 25 15 10 5 25 W [deg] 0.5 W [deg] 0.1 20 15 0 10 400 600 800 1000 1200 Frequency [MHz] 1400 1600 1800 400 600 800 B0353+52 1400 1600 1800 14 40 12 W [deg] Pulse width [deg] 0.5 W [deg] 0.1 30 20 10 0 400 1000 1200 Frequency [MHz] B1706-16 50 Pulse width [deg] [deg] W [deg] 10 W [deg] 8 0.5 W [deg] 0.1 6 4 600 800 1000 1200 Frequency [MHz] 1400 1600 1800 400 600 800 1000 1200 Frequency [MHz] 1400 1600 1800 Figure 4.1: Pulse width at half-W0.5 and at tenth-W0.1 of the maximum height as a function of frequency for P SRB2110 + 27, P SRB1737 − 30, P SRB0950 + 08, P SRB0353 + 52, P SRB1706 − 16 and P SRB0950 + 08. The pulse width should be determined by the local emission mechanism and probably even more important by structure of the emission region (geometry, plasma distribution, etc). In our model the resonant condition between the beam and the plasma depends on the plasma density and therefore is a function of distance from the neutron star. The resonant conditions for higher frequencies are fulfilled closer to the neutron star than for lower frequencies. Since the magnetic field line diverges slowly from the surface of the star outwards, we expect that CHAPTER 4. DATA ANALYSIS 87 the pulse width will be larger for low frequencies. According to our model R ∼ ω −2/3 , for the dipole magnetic field the angle between the field line and the magnetic axis θ ∼ R1/2 that is θ ∼ ω −1/3 . We expect that W is roughly proportional to the opening angle, that is W ∼ θ ∼ ω −1/3 . B2303+30 0.8 0.66 0.78 0.64 0.5 Log(W ) 0.68 0.5 Log(W ) B2217+47 0.82 0.76 0.62 0.74 0.6 0.72 0.58 0.7 0.56 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 2.6 2.7 2.8 B0136+57 2.9 3 Log(ω) 3.1 3.2 3.3 3.1 3.2 3.3 B0154+61 0.84 0.8 0.83 0.78 0.5 Log(W ) 0.5 Log(W ) 0.82 0.81 0.8 0.76 0.74 0.79 0.72 0.78 0.77 0.7 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 2.6 2.7 2.8 2.9 3 Log(ω) Figure 4.2: log W0.5 as a function of log ω for P SRB2217 + 47, P SRB2303 + 30, P SRB0136 + 57 and P SRB0154 + 61. The width is measured in degrees and frequency in M Hz. CHAPTER 4. DATA ANALYSIS 88 B0820+02 B1706-16 0.954 0.78 0.952 0.76 0.5 Log(W ) 0.5 Log(W ) 0.95 0.948 0.946 0.74 0.72 0.944 0.7 0.942 0.94 0.68 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 2.6 2.7 2.8 0.89 1.2 0.88 0.8 0.85 0.4 0.84 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 3.1 3.2 3.3 3.1 3.2 3.3 B0353+52 1.4 0.58 1.3 0.56 1.2 0.5 Log(W ) 0.6 0.5 Log(W ) B0037+56 0.54 1.1 0.52 1 0.5 0.9 0.48 0.8 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 2.6 3.3 2.7 2.8 B0626+24 2.9 3 Log(ω) B0540+23 1.05 0.96 1 0.94 0.95 0.92 0.5 Log(W ) 0.5 3.3 0.86 0.6 Log(W ) 3.2 0.87 0.5 1 0.5 3.1 B0611+22 1.4 Log(W ) Log(W ) B1737-30 2.9 3 Log(ω) 0.9 0.85 0.9 0.88 0.8 0.86 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 2.6 2.7 2.8 2.9 3 Log(ω) Figure 4.3: log W0.5 as a function of log ω forP SRB0820 + 02, P SRB1706 − 16, P SRB1737 − 30, P SRB0611 + 22, P SRB0037 + 56, P SRB0353 + 52, P SRB0626 + 24 and P SRB0540 + 23. The width is measured in degrees and frequency in M Hz. CHAPTER 4. DATA ANALYSIS 89 B0628-28 0.64 1.25 0.63 1.245 0.5 Log(W ) 1.255 0.5 Log(W ) B0756-15 0.65 0.62 1.24 0.61 1.235 0.6 1.23 0.59 1.225 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 2.6 2.7 2.8 B0809+74 2.9 3 Log(ω) 3.1 3.2 3.3 B0823+26 1.2 0.6 0.58 1.15 0.5 Log(W ) 0.5 Log(W ) 0.56 1.1 0.54 0.52 0.5 1.05 0.48 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 2.6 2.7 2.8 B0950+08 0.95 1.125 0.9 1.12 0.5 1.11 1.105 3.3 3.1 3.2 3.3 3.1 3.2 3.3 0.8 0.75 0.7 1.1 0.65 1.095 1.09 0.6 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 2.6 2.7 2.8 B2110+27 2.9 3 Log(ω) B1620-09 0.6 0.62 0.58 0.6 0.56 0.58 0.5 Log(W ) 0.5 3.2 0.85 1.115 Log(W ) 0.5 Log(W ) 3.1 B0919+06 1.13 Log(W ) 2.9 3 Log(ω) 0.54 0.52 0.56 0.54 0.5 0.52 0.48 0.5 0.46 0.48 2.6 2.7 2.8 2.9 3 Log(ω) 3.1 3.2 3.3 2.6 2.7 2.8 2.9 3 Log(ω) Figure 4.4: log W0.5 as a function of log ω for P SRB0756 − 15, P SRB0628 − 28, P SRB0809 + 74, P SRB0823 + 26, P SRB0950 + 08, P SRB0919 + 06, P SRB2110 + 27 and P SRB1620 − 09. The width is measured in degrees and frequency in M Hz. CHAPTER 4. DATA ANALYSIS 90 In Figures 4.2- 4.4 we plotted the pulse width at half intensity, W0.5 , as a function of frequency. The width is measured in degrees and frequency in M Hz. 4.2 Pulse Intensity Without involving geometrical factors we expect the pulse intensity to decrease with the increase of frequency. However, the observed pulses come from extended sources so that the shape of the emission zone and plasma distribution should substantially affect the pulse shape. We plotted the maximum intensity as a function of frequency in Figures 4.5- 4.8. B0353+52 B0756-15 1200 B0626+24 B1620-09 1000 B0037+56 Max Intensity B0136+57 B0154+61 800 B0611+22 600 400 200 0 400 600 800 1000 1200 1400 1600 Frequency [MHz] Figure 4.5: Maximum intensity as a function of frequency. CHAPTER 4. DATA ANALYSIS 91 B1706-16 B0919+06 4000 B0611+22 3500 B2303+30 B0820+02 Max Intensity 3000 2500 2000 1500 1000 500 0 400 600 800 1000 1200 1400 1600 Frequency [MHz] Figure 4.6: Maximum intensity as a function of frequency. 3.5 10 4 4 3 10 B0809+74 Max Intensity 2.5 10 B0950+08 4 B0628-28 B2217+47 4 2 10 1.5 10 4 4 1 10 5000 0 400 600 800 1000 1200 1400 1600 1800 Frequency [MHz] Figure 4.7: Maximum intensity as a function of frequency. CHAPTER 4. DATA ANALYSIS 92 1400 B0540+23 B2110+27 B1737-30 Max Intensity 1200 1000 800 600 400 200 0 400 600 800 1000 1200 1400 1600 Frequency [MHz] Figure 4.8: Maximum intensity as a function of frequency. As it can be seen, generally there is a decrease in intensity as the frequency increases (except for three pulsars) in agreement with our model. Those three pulsars seem to show irregular intensity behavior. From the profiles as they appear in the EPN Archive one can not see any deviation. In figure 4.9 we have attached five profiles of one of those pulsars B0540+23. Four Stokes parameters are plotted: I-intensity and V, Q, U -denote the state of polarization parameters. CHAPTER 4. DATA ANALYSIS 93 Figure 4.9: Profiles for B0540+23 as a function of the pulse phase at different frequencies as it appears in the EPN data archive. Four Stokes parameters are plotted: I-red, Q-blue,U indigo, V -rouge CHAPTER 4. DATA ANALYSIS 4.3 94 Geometrical Analysis So far there was no phenomenological quantitative description on the pulse profile asymmetry. Here we suggest using the moments of the observed pulse as follows: R I(ϕ)(ϕ − ϕ0 )n dϕ R , I(ϕ)dϕ mn = (4.1) where n is the order of the moment, I is the intensity and at ϕ0 the intensity is maximal. The asymmetry parameter is defined by: M3 = where σ = √ m3 , σ3 (4.2) m2 which is the measure of the pulse width. In a case of perfect symmetry the third moment will be equal to zero. The greater M3 the bigger is the lack of symmetry. Third moment and the asymmetry parameter have been calculated for each pulsar. In Figures 4.10- 4.12 the asymmetry parameter as a function of frequency is plotted. CHAPTER 4. DATA ANALYSIS 95 B0628-28 B0037+56 1.2 2 1 1.5 0.8 1 m3 m3 0.6 0.4 0.5 0.2 0 0 -0.2 -0.4 400 600 800 1000 1200 Frequency [MHz] 1400 1600 1800 -0.5 400 600 800 B0136+57 1000 1200 Frequency [MHz] 1400 1600 1800 1400 1600 1800 B0626+24 1.1 2 1 1.9 0.9 1.8 m3 m3 0.8 0.7 0.6 1.7 1.6 0.5 1.5 0.4 0.3 400 600 800 1000 1200 Frequency [MHz] 1400 1600 1800 1.4 400 600 800 1000 1200 Frequency [MHz] Figure 4.10: Asymmetry parameter as a function of frequency for P SRB0628 − 28, P SRB0037 + 56, P SRB0136 + 57 and P SRB0626 + 24. CHAPTER 4. DATA ANALYSIS 96 B1737-30 B2217+47 0 2 -0.2 1.5 -0.4 m3 m3 1 0.5 -0.6 -0.8 0 -0.5 400 -1 600 800 1000 1200 Frequency [MHz] 1400 1600 -1.2 400 1800 600 800 B0540+23 1000 1200 Frequency [MHz] 1400 1600 1800 1400 1600 1800 1400 1600 1800 1400 1600 1800 B0823+26 2 1.5 1.8 1 0.5 m3 m3 1.6 0 1.4 -0.5 1.2 1 400 -1 600 800 1000 1200 Frequency [MHz] 1400 1600 -1.5 400 1800 600 800 B0809+74 B2303+30 0.4 1 0.2 0.5 0 m3 m3 0 -0.5 -0.2 -0.4 -1 -1.5 400 -0.6 600 800 1000 1200 Frequency [MHz] 1400 1600 -0.8 400 1800 600 800 B0611+22 1.2 0.7 1 0.6 0.8 0.5 0.6 m3 m3 1000 1200 Frequency [MHz] B0154+61 0.8 0.4 0.3 0.4 0.2 0.2 0 0.1 -0.2 0 400 1000 1200 Frequency [MHz] 600 800 1000 1200 Frequency [MHz] 1400 1600 1800 -0.4 400 600 800 1000 1200 Frequency [MHz] Figure 4.11: Asymmetry parameter as a function of frequency for P SRB1737 − 30, P SRB2217 + 47, P SRB0540 + 23, P SRB0823 + 26, P SRB0809 + 74, P SRB2303 + 30, P SRB0611 + 22 and P SRB0154 + 61. CHAPTER 4. DATA ANALYSIS 97 B1706-16 B1620-09 -0.8 1.2 -1 1 0.8 m3 m3 -1.2 -1.4 0.6 -1.6 0.4 -1.8 -2 400 600 800 1000 1200 Frequency [MHz] 1400 1600 0.2 400 1800 600 800 B0756-15 1600 1800 1400 1600 1800 1400 1600 1800 m3 m3 1400 1.6 0.2 1.5 0 1.4 -0.2 600 800 1000 1200 Frequency [MHz] 1400 1600 1.3 400 1800 600 800 B0950+08 1000 1200 Frequency [MHz] B2110+27 -1.9 1 -2 0.8 -2.1 m3 0.6 -2.2 0.4 -2.3 0.2 -2.4 600 800 1000 1200 Frequency [MHz] 1400 1600 0 400 1800 600 800 B0820+02 -0.6 1 -0.8 0.8 -1 0.6 -1.2 0.4 -1.4 0.2 -1.6 600 800 1000 1200 Frequency [MHz] 1000 1200 Frequency [MHz] B0919+06 1.2 m3 m3 1800 1.7 0.4 0 400 1600 1.8 0.6 -2.5 400 1400 B0353+52 0.8 -0.4 400 1000 1200 Frequency [MHz] 1400 1600 1800 -1.8 400 600 800 1000 1200 Frequency [MHz] Figure 4.12: Asymmetry parameter as a function of frequency for pulsars. CHAPTER 4. DATA ANALYSIS 98 It is evident that the pulses are asymmetric. We expect the observed asymmetry to be related to the distribution of plasma density across the emission region. So far most fits for observed pulse profiles were done with Gaussian distribution Kramer (1994) which obviously can not describe the pulse asymmetry. Cairns et al. (2001) suggested that the intensity distribution of single pulses in Vela is lognormal. We can get information about the plasma distribution directly from the pulse profiles. Let us examine the particle distribution as a function of phase. According to the resonant 1/2 condition ωr = 2ωp γp that is: R 3 0 Rr ∼ ω2 . no (4.3) The measured intensity I is equal to : ln I ∼ ln G ∼ ω(Rr − R0 ). (4.4) Let us assume that the instability sets on at the same R0 for all field lines, then n0 ∼ (ln I)3 . ω (4.5) The calculated density as a function of angle is plotted in Figure 4.13 for 4 pulsars with core emission. CHAPTER 4. DATA ANALYSIS 99 B2216+ 47 925MHz 0.35 B1706-16 1642MHz 0.1 0.3 0.08 0.2 Ln3I/ω Ln3I/ω 0.25 0.15 0.06 0.04 0.1 0.02 0.05 0 170 175 180 Pulse Phase[degree] 185 0 170 190 B0540+23 1408MHz 0.14 1.2 0.12 185 190 B0823+26 610MHz 0.8 0.08 Ln3I/ω Ln3I/ω 180 Pulse Phase[degree] 1 0.1 0.06 0.6 0.4 0.04 0.2 0.02 0 150 175 0 160 170 180 190 Pulse Phase[degree] 200 210 80 85 90 95 Pulse Phase[degree] 100 105 Figure 4.13: Plasma distribution function for different pulsars at different frequencies. From Figure 4.13 it can be seen that the particle distribution is most likely to be asymmetric. The only symmetric distribution here is of B0823 + 26 at 610M Hz, as can be seen from the Figure 4.10 its third moment is almost zero. We suggest to approximate plasma density distribution with the Lognormal function : n∼ 1 −[ln ϕ/ϕc ]2 exp , w|ϕ| 2w2 (4.6) where w is the half width and at ϕc the distribution takes its maximum value. For convenience we take ϕ > 0. The Lognormal distribution function for the particles is plotted in CHAPTER 4. DATA ANALYSIS 100 Figure 4.14and the corresponding gain factor calculated according to our model is plotted in Figure 4.15. Figure 4.14: Plasma density as a function of radius, R, and angle ,φ, for the Lognormal distribution. CHAPTER 4. DATA ANALYSIS 101 Ln G 0 0.2 0.4 φ[rad] 0.6 0.8 1 1.2 Figure 4.15: Gain factor for the Lognormal distributed plasma density. Figure 4.15 shows the general shape and is not to scale. In Figure 4.16 we plotted the observed and the expected profiles. CHAPTER 4. DATA ANALYSIS B1737-30 102 610MHZ PSR B0626+24 70 1642MHZ Intensity [arbitrary units] Intensity [arbitrary units] 800 600 400 60 50 40 30 20 200 10 0 164 168 172 176 180 184 188 192 196 0 160 200 180 Pulse Phase [degree] Pulse Phase[degrees] B1620-09 1642MHz Intensity [arbitrary units] 100 80 60 40 20 0 178 180 182 Pulse Phase (degree) Figure 4.16: Matching between the observed and expected profiles. Black is the observed and red is the calculated profile. It can be seen that the calculated expected pulse profile is rather similar to the observed profiles. Proper matching of the distribution parameters along with the analysis of the frequency dependence requires further studies and is beyond the scope of the present work. CHAPTER 4. DATA ANALYSIS 4.4 103 Discussion and conclusions We have performed an analysis of the pulse profiles for a number of pulsars. For most pulsars the pulse intensity behaves as could be expected from a simple model which probably indicates that the geometry of the emission region is not too complicated. Similarity of the pulse width at one half and one tenth of the maximum intensity implies that emission originates in the deep magnetosphere. We have proposed a phenomenological quantitative description of the pulse asymmetry and a possible approximate functional form for plasma density distribution across the emission region. SUMMARY We studied properties of low frequency waves in surrounding typical to the pulsar magnetosphere. Dispersion relation for low frequency waves in relativistic pair plasma with non-zero charge and current density in the plasma was derived. The inclusion of non-zero charge and current densities imply that the natural wave modes are elliptically polarized. The polarization depends on the propagation angle, being stronger at smaller angles, and increases with frequency in agreement with observations. The effect is negligible in the inner magnetosphere but becomes more strong as the waves propagate outwards to the regions of weaker magnetic field where ωp /Ω increases. The effect is significantly sensitive to the plasma parameters (relative deviation from quasineutrality and current density). This sensitivity may be responsible for the large variety of polarization features of different pulsars and across a pulse. We proposed a mechanism based on the nonresonant hydrodynamical beam instability which is capable of direct generation of electromagnetic waves in the radio range. In the 104 CHAPTER 4. DATA ANALYSIS 105 pulsar magnetosphere conditions the nonresonant instability becomes more efficient then the resonant one. The unstable waves in the beam mode grow continuously in a nonresonant way over a large range of heights. At the point where the instability ceases the unstable mode directly converts by refraction into L−O mode and can freely leave the pulsar magnetosphere. The mechanism predicts efficient generation of electromagnetic waves propagating at angles θ . 1γb . The derived spectrum falls towards higher and lower frequencies in agreement with observations. The gain factor is frequency dependent and is influenced by plasma parameters. The mechanism given above describes the formation of spectra by one field line. The spectrum that is formed by individual field line is not a power spectrum, however one has to remember that the integrated spectrum is sum of emissions coming from many single spectra from a number of field lines. We performed data analysis using a sample of 20 pulsars. We compared our model predictions with observations. For most pulsars the pulse intensity decreases as a function of frequents in agreement with our model. The pulse width behavior indicates that the radiation is formed in deeper magnetosphere. We propose a method of quantitative description of the pulse asymmetry. We propose that the pulses asymmetry may be a result from asymmetric distribution function. Based on the pulse structure we propose the lognormal function for plasma distribution across the emission region. The results of this research were published in the following papers: 1. Gedalin M., Gruman E., and Melrose D.B., Mon. Not. R. Astron. Soc. 325, 715, (2001). CHAPTER 4. DATA ANALYSIS 106 2. Gedalin M., Gruman E., and Melrose D.B., Phys.Rve.Lett. 88, (2002). 3. Gedalin M., Gruman E., and Melrose D.B., Mon. Not. R. Astron. Soc. 337, 442, (2002). APPENDIX A VLASOV EQUATION Linear solution of Vlasov equation ∂fs ∂fs qs ∂fs +v + (E + v × B) = 0, ∂t ∂r m ∂u (A.1) where: u= u p , v = , γ 2 = 1 + u2 , c = 1, m γ (A.2) We introduce: fs = Fs + δfs , (A.3) B = B0 + B, (A.4) E = E0 + E, (A.5) 107 APPENDIX A. VLASOV EQUATION 108 where B0 = (0, 0, B0 ), E0 and Fs are solutions of the Vlasov equation and B, E and δfs are small perturbations ∝ ε << 1 After arranging and assuming that E0 = 0 equation (A.1) becomes: ∂ ∂ qs ∂ qs ∂Fs,0 +v + v × B0 δfs = − (E1 + v × B0 ) , ∂t ∂r ms ∂u ms ∂u (A.6) we will solve this equation by the variables’ separation method [see Gedalin (1994)]. Let us assume that δfs contains two parts, one that changes fast and the other that changes slowly. δfs = δ f˜s (f ast) + δ f¯s (slow). (A.7) qs ∂ (δE + v × δB1 ) Fs,o = Q, ms ∂u (A.8) Next we denote In the same way will be separated into two parts: fast and slow. Q = Q̃(f ast) + Q̄(slow), (A.9) The two separable equations of (A.6) can be written as follows: ∂ ∂ qs ∂ +v + v × B0 ∂t ∂r ms ∂u ∂ ∂ qs ∂ +v + v × B0 ∂t ∂r ms ∂u δ f˜s = Q̃, (A.10) δ f¯s = Q̄, (A.11) APPENDIX A. VLASOV EQUATION We are interested in (A.10) only. 109 We shall work in cylindrical coordinates with u = (v⊥ cos ϕ, v⊥ sin ϕ, vz ) and k = (k⊥ , 0, kz ). As well, let us assume that all perturbations are proportional to exp i(k · r − ωt). After substitution equation (A.10) becomes: es B0 1 ∂ ˜ −i (ω − v · k) δ f˜s − δ fs = Q̃, ms γ ∂ϕ Defining: ζ = ω − kz vz , Ω = qs B 0 ms , Ω̃ = ∂ ˜ δ fs + i ∂ϕ Ω γ (A.12) and arranging the equation. ( A.12) we get: ζ k⊥ v⊥ cos ϕ − Ω̃ Ω̃ Q̃ δ f˜s = − , Ω̃ (A.13) the solution of this equation is: ∞ X δ f˜s = i ei(n−l−σ) h l,σ,n=−∞ ζ + (n − σ) Ω̃ i Jl k⊥ v ⊥ Ω̃ Jn k⊥ v ⊥ Ω̃ Q̃σ , (A.14) where we used: exp[ −ikΩ⊥ v⊥ sin ϕ] = ∞ X exp[−ilϕ]Jl k⊥ v ⊥ Ω , (A.15) l=−∞ and expanded Q̃ = ∞ P Q̃σ exp[−iσϕ]. To determine Qσ . σ=−∞ ∂ qs (δE + v × δB1 ) F̃s,o = Q̃, ms ∂u (A.16) using: ∇×E=− ∂B ∂t ⇒ B= 1 ω k × E, (A.17) APPENDIX A. VLASOV EQUATION 110 we obtain : Q̃ = − mqss E + = − mess P σ=0,±1 1 ω ∂ (k (v · E) − E (v · k)) ∂u F̃s,o = n h exp (−iσϕ) Ez (1 − σ 2 ) ∂u∂ z + σ 2 k⊥ vz ∂ 2ω ∂uz i 2 + Eσ σ2 h kz v⊥ ∂ ω ∂uz + ζ ∂ ω ∂u⊥ io F̃s,o , (A.18) and the final solution is: δ f˜s = e im ∞ P l,σ,n=−∞ ei(n−l−σ) J ζ+(n−σ) Ω̃] l [ k⊥ v ⊥ Ω̃ Jn k⊥ v⊥ Ω̃ ·, (A.19) n h · Ez (1 − σ 2 ) ∂u∂ z + σ 2 k⊥ v z ∂ 2ω ∂u⊥ i 2 + Eσ σ2 h kz v ⊥ ∂ ω ∂uz + ζ ∂ ω ∂u⊥ io F̃s,o . APPENDIX B PULSARS DATA The pulsars’ data and classification. The classification is taken from Rankin (1990). Ststands for isolated core components of stars with core-single profiles, Sd-conal single stars, T-triple stars 111 APPENDIX B. PULSARS DATA 112 Pulsar Period [sec] dP/dt Classification B0037+56 1.11822446071 2.879 · 10−15 - B0136+57 0.27244979157400001 1.07003 · 10−14 St B0154+61 2.3517238322199998 1.88841 · 10−13 St B0353+52 0.1970300350276 4.7666 · 10−16 - B0540+23 0.2459740892957 1.542378 · 10−14 St B0611+22 0.33492505401 5.963 · 10−14 St B0626+24 0.47662265393800002 1.99705 · 10−15 St B0628-28 1.2444170725999999 7.107 · 10−15 Sd B0756-15 0.682264336409999950 1.617 · 10−15 - APPENDIX B. PULSARS DATA Pulsar Period [sec] 113 dP/dt Classification B0809+74 1.292241435530999900 1.683 · 10−16 Sd B0820+02 0.86487275188000001 1.039 · 10−16 Sd B0823+26 0.53066079757999995 1.7094 · 10−15 St B0919+06 0.43061967258 1.37202 · 10−14 T B0950+08 0.25306506819000002 2.2915 · 10−16 Sd? 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