Download Waves In Pair Plasma and Mechanism Of Radio Emission In Pulsars

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?
B1620-09
1.27644574527
2.5798 · 10−15
-
B1706-16
0.65305471898199996
6.3088 · 10−15
St
B1737-30
0.60667854720000003
4.653 · 10−13
-
B2110+27
1.2028511498700001
2.6226 · 10−15
Sd
B2217+47
.53846924794850004
2.76503 · 10−15
St
B2303+30
1.5758847442699999
2.89567 · 10−15
Sd
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