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Slide 1
References, 1. High Energy Astrophysics, Volume 2,
Second Edition, Longair
2. ‘Pulsars’ by Smith
Pulsars
High Energy Astrophysics
[email protected]
http://www.mssl.ucl.ac.uk/
Slide 2
Introduction
• Pulsars - isolated neutron stars
- radiate energy via slowing down of rapid
spinning motion (P usually < 1sec, dP/dt>0)
• Pulsating X-ray sources / X-ray pulsators
compact objects (generally neutron stars) in
binary systems. Accrete matter from normal
star companion. (P ~ 10s secs, dP/dt<0)
Pulsars (ie objects which emit radio
pulses with very short periods) are
isolated, highly magnetic neutron stars
which radiate energy produced in the
slowing down of their rapid spinning
motion. Their rotational period is
generally less than 1 second and the
period is increasing with time.
Neutron stars are supported by
degeneracy pressure because their
internal densities are so high that
classical gas formulae are inappropriate
to describe conditions. The ‘pressure’ is a
result of Heisenberg’s Uncertainty
Principle and Fermi’s Exclusion
Principle, which state that particles
cannot occupy the same quantum state.
The resultant mechanical momenta of the
particles provides the pressure of the
degenerate gas.
Pulsating X-ray sources, which are also
known as X-ray pulsators, are compact
objects (generally neutron stars) in binary
systems which accrete matter from a
normal star companion. The energy
radiated is produced by the process of
accretion and is modulated with the spin
period of the neutron star. The period of
X-ray pulsators is generally tens of
seconds, although a few have periods of a
few seconds. Their period is decreasing
with time, ie. they are spinning up.
X-ray pulsators are effectively binary
pulsars.
The magnetic axes of both pulsars and Xray pulsators are not aligned with the
rotation axis.
See Smith, p12-13
Slide 3
Pulsars cont.
• Discovered in radio
Averaging over many pulses we see:
Period
pulse
~P/10
interpulse
Radio pulsars were discovered in 1967 by
Antony Hewish and his graduate student
Jocelyn Bell-Burnell using the
Cambridge radio telescope. Hewish
subsequently received the Nobel Prize for
this discovery. It was the first evidence of
strictly periodic pulses seen on a chart
recorder and it was speculated at the time
that these were signals from other
civilizations. As more and more pulsars
were discovered it became clear however
that a new class of object had been
discovered.
The original paper was published by
Hewish et al. In Nature, 1968, volume
217, p709.
Slide 4
• Measuring pulse complicated by Doppler
motion of Earth and frequency dispersion
in pulse arrival times.
• Individual pulses:
av. very constant, individual pulses variable
Slide 5
Pulsar period stability
• Period extremely stable: 1 part in 10 12
indicates some mechanical clock
mechanism - although this mechanism must
be able to accommodate pulse variablity.
• Pulsations of white dwarf??? (but Crab
pulsar period (P~1/30 sec) too short).
• Rotation of neutron star???
Identifying and measuring the periodic
signal emitted by a pulsar is complicated
by the Doppler motion of the Earth and
by the frequency dispersion in the pulse
arrival times.
Averaging over many pulses gives clear
pulse and interpulse profiles. However,
individual pulses are very variable and
variations in the flux can be observed on
timescales as short as 100 microseconds.
Let’s consider the stability of the period
of the pulsar. The period is observed to
be highly stable, up to 1 part in 1e12,
after allowing for the slowing down of
the spin. This suggests that some
‘mechanical’ clock exists, although this
mechanism must be able to
accommodate the pulse-to-pulse
variability of the pulse structure.
Originally, it was considered that these
could be analogous to the pulsations
observed in a white dwarf, as in the case
of longer period variables. However, this
is not a satisfactory explanation for the
Crab pulsar where the period P is about
1/30 second. (We will go through the
logic for this statement in the next few
viewgraphs.)
The rotation of the neutron star is thus the
most likely source of the pulsations.
Slide 6
We will now justify our statement that
the Crab pulsar cannot be a white dwarf it must be a neutron star.
We start with the statement that the
gravitational force of the neutron star
must be greater than its centrifugal force
(otherwise it would fly apart!).
Rotation of a neutron star
Gravitational force > centrifugal force
GMm mv 2

r2
r
where
v
2r
and P is the period
P
Slide 7
Stage 1: cancel m from both sides of the
equation and substitute v=2r/P.
Stage 2: simply rearranging the equation
to put M/4r^3 on left
Stage 3: substituting M/4r^3 into
density equation
Reducing:
M

GM 4 2 r

 2 =>
4r 3 P 2G
r2
P
but   M
4 3
r
3
so

3
P 2G
-3
G = 6.67x10 -11 m 3 kg -1 s -2 ; PCrab = 33x10 s
Slide 8
Substituting numbers for Crab then:

3
6.67  10 11  1100  10 6
kg m -3
so  > 1.3 x 10 14 kg m-3
This is too high for a white dwarf (which has
a density of ~ 10 9 kg m-3 ), so it must be a
neutron star.
Moreover, the radius of a white dwarf is
about 10,000 km, which would imply a
rotational velocity of 1.9e9 m/s…. And
since this is greater than the speed of
light, it is somewhat unlikely!
(calculation from v=(2r/P))
Slide 9
Pulsar energetics
• Pulsars slow down => lose rotational energy
- can this account for observed emission?
• Rotational energy:
E
so
1 2 I  4 2  2 I 2
I   2  
2
2 P 
P2
dE d  2 I 2 
4 I 2 dP
  2    3
dt dt  P 
P dt
Slide 10
Energetics - Crab pulsar
Crab pulsar
- M = 1 solar mass
- P = 0.033 seconds
- R = 10 4 m
I
Pulsars are observed to slow down, ie
dP/dt>0 thus they are losing rotational
energy. So can the energy lost via this
process account for the observed levels of
emission? Or is some other process
required to meet observed levels?
Rotational energy is given by half of the
moment of inertia (I) multiplied by the
square of the angular velocity (then just
substitute to put equation in terms of P).
Differentiating, d/dt = d/dP x dP/dt
So differentiating with respect to P;
d(P^-2)/dP
= -2.P^-3
then substituting, which leaves a factor of
dP/dt.
We are going to take the example of the
Crab nebula, calculate its rotational
energy losses and compare it to the
energy observed in its surrounding nebula
to see if the nebula can be powered by the
pulsar or requires an additional source.
2
2
MR 2   2  10 30  10 8 kg m 2
5
5
= 0.8 x 10 38 kg m 2
Slide 11
dE  4  0.8  10
 1 dP 

 10
 watts
dt
0.033 2
 P dt 
38
and
 1 dP 
 3  1042 
 watts
 P dt 
from observations: 1 dP
P dt
~ 10 11 s 1
thus energy lost
dE
31
by the pulsar  dt  3 10 watts
From the equation on slide 9,
dE/dt = ( 4.I.pi^2 / P^3 ) x dP/dt
and then substituting for I from slide 10 and of course energy is being lost so
dE/dt is negative.
Slide 12
This rate of energy loss is comparable to that
inferred from the observed emission, for
example in the 2-20keV range, the observed
luminosity in the Crab Nebula is approx.
1.5 x 10 30 watts.
Thus the pulsar can power the nebula.
Slide 13
Irregularities in pulsar emission
• Short timescales - pulsar slow-down rate is
remarkably uniform
• Longer timescales - irregularities apparent
- in particular, ‘glitches’
P
glitch
t
A glitch is a
discontinuous
change of
period
It is interesting to note that the lifetime of
a pulsar, tau = P/2(dP/dt) for the Crab
Nebula is about 1400 years - and this was
observed to explode in 1054. The
continuous supply of high energy
particles from the pulsar to the
surrounding nebula explains the observed
energy output levels from the nebula
since the original explosion.
The Crab pulsar (P=33 millisec) and the
Vela pulsar (P=89 millisec) have been
detected at radio, IR , optical, X-ray and
-ray wavelengths. Other pulsars tend to
be only radio emitters.
Over 500 pulsars are now known and all
of them are in our Galaxy except for one
in the Large Magellanic Cloud and one in
the Small Magellanic Cloud.
On short timescales, the order of a few
days or so, the slowing down of the
pulsar’s spin rate is remarkably uniform.
On longer timescales however,
irregularities of the pulse timing become
apparent. These are observed once every
few years in the Crab and Vela pulsars,
for example. The most studied of these
are the so-called ‘glitches’ in pulse
arrival time. These are discontinuous
changes of period, ie ‘steps’ in the pulse
arrival time distribution.
See Smith p60-61, Longair p104
Slide 14
Glitches
Glitches are caused by stresses and fractures
in the external layers, the so-called ‘crust’
of the neutron star.
For example,
P
~ 10 10
P
is the observed value for the Crab pulsar.
Glitches are believed to be caused by
‘starquakes’, analogous to earthquakes,
as the crust of the star establishes a new
equilibrium with respect to the superfluid
interior.
This model works for the Crab pulsar, but
not for the Vela pulsar because the
glitches occur too frequently.
An alternative model for the glitches is
the catastrophic unpinning of vortices in
the rotating neutron superfluid. These
vortices exist because, on a macroscopic
scale, a rotating superfluid must rotate
irrotationally. Thus a superfluid is made
up of an array of vortices whose axes lie
parallel to the rotation axis. The vortices
are either pinned to nuclei in the crust or
thread the spaces between them. As the
star slows down and angular momentum
is transferred outwards, the vortices
migrate. This is a jerky, not smooth
process and causes small glitches. The
giant glitches are believed to be caused
when the vortices are unpinned
catastrophically and a change in the
rotational speed results.
Slide 15
Pulse profiles
• Average pulse profile very uniform
• Individual pulses/sub-pulses very different
in shape, intensity and phase
t
Sub-pulses show high
degree of polarization
which changes throughout
pulse envelope
average envelope
Slide 16
Neutron Stars
• General parameters:
- R ~ 10 km (104 m)
- inner ~ 1018 kg m-3 = 1015g cm-3
- M ~ 0.2 - 3.2 solar masses
- surface gravity ~ 1012 m s -2
• We are going to find magnetic induction, B,
of a neutron star.
Generally, one studies the pulse profile
integrated over a sequence of a few
hundred pulses and this is very uniform.
But individual component pulses or subpulses can be very different from one
another, varying in intensity, shape and in
the phase at which they occur within the
integrated profile.
The figure shows a sequence of pulses in
time compared to the average envelope
for a few hundred pulses.
The most important characteristic of subpulses is their high degree of polarization.
This also changes throughout the pulse
envelope in its form and degree of
polarization.
See Smith p91-93
Neutron stars are supported by
degeneracy pressure because their
internal densities are so high that
classical gas formulae are inappropriate
to describe conditions. The ‘pressure’ is a
result of Heisenberg’s Uncertainty
Principle and Fermi’s Exclusion
Principle, which state that particles
cannot occupy the same quantum state.
The resultant mechanical momenta of the
particles provides the pressure of the
degenerate gas.
The surface gravity of a neutron star,
g(ns) is given by the equation
g(ns) = (GM)/R^2
= (6.67e-11 x 2e30) / (1e8) m/s^2
= 1e12 m / s^2
We are going to find the value for the
magnetic induction of a neutron star by
thinking of the Sun contracting with its
magnetic field to form a neutron star.
(Although is really not known where the
magnetic field in a neutron star comes
from).
Slide 17
Magnetic induction
Magnetic flux,
 BdS 
constant
surface
radius Sun
Radius collapses from 7 x 108m to 104 m
Surface
change
gives
2
Bns  7  108 
  5  109

BSun  10 4 
Slide 18
The general field of Sun is uncertain but
should be ~ 0.01 Tesla.
Thus the field for the neutron star,
Bns ~ 5 x 107 Tesla = 5 x 1011 Gauss
Next - how long does B ns last?
Slide 19
Decay time of magnetic field
Decay time of
magnetic field:
3000m
  D 2 0
10km
Polar cap
D - typical dimension over which field varies
significantly (for n.s., D ~ 3 x 10 3 m)
 - conductivity
The integral of the magnetic field around
the surface of the star must be a constant
and the radius collapses from 7e8 m to
1e4m. Thus the ratio of the magnetic
fields of the neutron star to the Sun is
equivalent to the ratio of their radii
squared… and this is approximately 5e9.
So the magnetic field of a 1 solar mass
neutron star is 5 billion times that of the
Sun.
If we can assume that magnetic braking is
responsible for the slowing-down of
pulsars, then we can calculate the
magnetic field strengths at the surface.
From observed dP/dt, we find that
magnetic field strengths of most pulars
typically lie in the range of 2 million to 2
billion Tesla (but weaker for the
millisecond pulsars, magnetic field B =
3e15 sqrt(P x dP/dt).
The decay time of the magnetic field is
given by the equation shown. The
‘dimension’ in the case of a neutron star
is assumed to be the radius of the polar
cap where the magnetic field lines are
concentrated. This is ~2% of the total
surface of the neutron star thus for a
radius of 10,000m is about 3000m.
The conductivity, , can be found from
the following:
Ohm’s Law states that j = (1/) .E
j = current density
 = resistivity
E = electric field
and = 1/
Slide 20
Thus,
t ~ (3 x 103 ) (1010 ) (410 -7)
~ 1011seconds ~ 3 x 10 3 years
But magnetic field Crab pulsar still intense
after 1000 years => interior must be
superconducting ( and  both very large)
Neutron stars very dense and zero-T energy
supports star and prevents collapse.
Slide 21
Neutron star structure
crust
Neutron star segment
neutron
1.
liquid
solid
Superfluid
core?
neutrons, 2.
superconducting
p+ and e1km
crystallization
of neutron
9km
matter
10km
1018 kg m -3
Heavy nuclei (Fe)
find a minimum
energy when
arranged in a
crystalline lattice
2x1017 kg m -3
4.3x1014 kg m -3
109 kg m -3
Substituting typical values into the
relationship for we find a value for the
decay time of a neutron star of 3000
years. However, since the magnetic field
is still very intense after 1000 years, we
must conclude that the interior of the
neutron star is superconducting, ie the
conductivity and thus the lifetime are
both very large.
It is interesting at this point to consider
the internal structure of neutron stars.
Neutron stars are extremely dense stars,
where it is the zero-temperature energy
(the Fermi energy) of the particles, rather
than their thermal energy, which supports
the star and prevents further collapse.
See Smith p36, Longair p84
The diagram shows a segment (ie a slice)
through a neutron star. The core extends
out to about 1km and has a density of
1e18 kg/m^3. Its substance is not well
known however, - it could be a neutron
solid, quark matter or a pion concentrate.
It may not even exist.
2. From 1km out to 9km, there is a
‘neutron fluid’, a superfluid made up of
neutrons and superconducting protons
and electrons. The density in this region
is between 2e17 and 1e18 kg/m^3.
1. In the inner crust, there is a lattice of
neutron-rich nuclei with free degenerate
neutrons and a degenerate relativistic
electron gas. The neutron fluid pressure
increases as the density increases.
The outer crust is solid and its matter is
similar to that found in white dwarfs, ie
heavy nuclei (mostly Fe) forming a
Coulomb lattice embedded in a
relativistic degenerate gas of electrons.
The density falls away relatively quickly
in this region, down to a billion kg/m^3
in the outer crust.
On the very surface of the neutron star,
densities dall below a billion kg/m^3 and
matter consists of atomic polymers of
56Fe in the form of a close packed solid.
The atoms become cylindrical, due to the
effects of the strong magnetic fields.
Stresses and fractures in the crust cause
the glitches in the pulse period.
Slide 22
1. Between densities of 4.3 x 10 14 kg m -3 and
2 x10 17 kg m -3, the lowest energy state is
reached when nuclei are embedded in an
electron and neutron fluid.
2. Above 2x1017 kg m -3, there is a continuous
neutron fluid with electrons and protons as
minor constituents.
Slide 23
The degenerate neutron fluid is a
superfluid, ie its viscosity is zero. The
degenerate electron gas has a high
conductivity and the proton fluid may be
superconducting. This is sufficient to
maintain the magnetic field for most of
the life of the pulsar.
See Smith p39-40, Longair p 94-95
See Smith, Chapter 6, Longair p107
Pulsar Magnetosphere
First, defining scale height
p
h
h
p
p
h
h
h
The pressure
difference
supports the
element of
atmosphere
Slide 24
The pressure difference is given by:
p
h  h    g
h
where  is the density

But p   kT
m
thus

(where m is the mass of
constituent particles)
p
mg
pm
 p
and
h
kT
kT
Slide 25
Formula for scale height
Integrating:
 mg 
p  p0 exp  
h
 kT 
=> pressure falls off exponentially with height
in atmosphere with uniform temperature.
 kT  has the dimensions of distance

h0  
 mg  and is called the ‘scale height’.
This is the expression for pressure in a
non-ionized atmosphere. If the
atmosphere is ionized, then the same
number of ions and electrons must be
present to maintain neutrality. If
hydrogen is the only constituent of the
gas, then each proton and electron act as
independent particles of mass:
(m(p)+m(e-))/2 ~ m(p)/2.
Thus p = p_0 exp (- (m(p)gh) / 2kT )
The formula shows that pressure falls off
exponentially with height in an
atmosphere with a uniform temperature.
Slide 26
Neutron star scale height
For a neutron star,
g ~ 1012 m s-2
T ~ 1 million K
thus h0 ~ 0.01m
Thus the atmosphere of a neutron star is
only the order of 10cm!
Slide 27
Forces exerted on particles
Particle distribution determined by
gravity
etemperature
Fg ns
electromagnetism
FB
Gravity:
Fg ns  me g ns  9  10 31  1012  10 18
Newton
In fact, the particle distribution around a
neutron star is determined not only by
gravity and temperature, but also by
electromagnetic forces.
Taking the example of the gravitational
force on an electron, for a typical neutron
star this is approx 1e-18 Newtons (on a
proton, the gravitational force is 2e-15
Newtons).
Slide 28
Magnetic force:
FB  evB  1.6  10 19


2 10 4 m
10 8 T
33  10 3 s

5
 3  10 Newton
This is a factor of 1013 larger than the
gravitational force and thus dominates
the particle distribution.
Slide 29
Neutron star magnetosphere
Neutron star rotating in vacuum:

B
Electric field induced
immediately outside n.s. surface.
E  Bv  10 8  2  10 6 Vm 1
 2  1014 Vm 1
pd on scale of neutron star radius:
  ER  1018 V
Slide 30
For a star rotating in a vacuum, the
rotating magnetic field induces an electric
field immediately outside the neutron star
surface and is given by the equation
shown in the green box.
The potential difference on the scale of
the neutron star radius is approx. 1e18
Volts. Electric potentials generated in this
simple way are sufficient to overcome
electrostatic binding forces, thus
electrons and protons are expelled from
the neutron star surface and are
accelerated almost to the highest
energies, ie in cosmic rays.
(From previous slide)
Electron/proton expulsion
Neutron star particle emission

B
electrons
protons
Cosmic
rays
The potential difference on the scale of
the neutron star radius is approx. 1e18
Volts. Electric potentials generated in this
simple way are sufficient to overcome
electrostatic binding forces, thus
electrons and protons are expelled from
the neutron star surface and are
accelerated almost to the highest
energies, ie in cosmic rays.
Slide 31
In reality...
• In reality, the charged particles will
distribute themselves around the star to
neutralize the electric field.
=> extensive magnetosphere forms
• Number difference +ve and -ve charges:
n  n ~ 7  10 8
B
m 3
Pperiod
(B in
Tesla)
The previous scenario only applies to a
star rotating in a vacuum, but in reality,
the charged particles will distribute
themselves around to neutralize the
electric field, ie. They will form an
extensive magnetosphere around the
neutron star.
It can be shown that the number
difference of positive and negative
charges is given by the equation above.
See Smith p48 and Manchester & Taylor
p178.
Slide 32
Crab pulsar particle density
• This relationship gives an indication of the
particle density n:
• take, for example, the Crab pulsar -
n  7  108
108
m 3  1018 m 3
3  10  2
Slide 33
Pulsar models
Magnetic and rotation axes co-aligned:
eCo-rotating plasma,
mag field lines are
closed inside light
cylinder
Particles are able to move along, but not
across, the magnetic field lines. In this
model, the plasma is ‘carried along’ with
the neutron star in the equatorial regions
(ie. It co-rotates with the star). Streams of
charged particles leave the star at high
latitudes where the field lines are open.
Radius of light cylinder
must satisfy:
p
light cylinder, R L
2RL
c
P
Of course the plasma can co-rotate with
the neutron star only out to the radius at
which v=c (at larger radii, v exceeds c
which is impossible). This radius R_L
defines the ‘light cylinder’ and satisfies
the condition (2*pi*R_L)=c.
Substituting (P=0.033 secs) and rearranging, for the Crab pulsar, R_L, the
radius of the light cylinder, is 1,600 km.
Slide 34
A more realistic model...
• Note that if radiation pulses are to be
predicted, magnetic axis and rotation axis
cannot be co-aligned.
• => plasma distribution and magnetic field
configuration around a neutron star is much
more complicated.
The model presented on the previous
slide assumes that the magnetic axis and
rotation axis are co-aligned. However,
this cannot be the case in this type of
model if we are to be able to see pulsed
radiation from a spinning neutron star.
Such a misalignment implies that the
plasma distribution and magnetic field
configuration around a neutron star is
much more complicate than this simple
picture suggests.
Slide 35
This illustrates the model for a pulsar –
where the axis of the neutron star’s
magnetic field is offset from the rotation
axis. The parameters shown are typical
for a pulsar.
Slide 36
Note that even if there is no plasma
surrounding the neutron star, the star will
radiate if the magnetic and rotation axes
do not coincide due to ‘magnetic
braking’. This is what is known as a
dipole aerial.
The wave radiated from the magnetic
poles has an angular frequency,. A field
of 1e8 Tesla can in fact explain the
energy losses observed… ie. magnetic
dipole radiation is the principal way in
which pulsars emit.
Magnetic braking is one of the most
important processes which leads to the
slowing down of the rotation of the
neutron star, thus the loss of energy and
the decay of the magnetic field. The
magnetic dipole is offset from the
rotation axis so that it displays a varying
dipole moment at large distances (see the
previous slide). Thus electromagnetic
energy is radiated from the star and this is
extracted from the rotational energy, ie
the spin period decreases.
The dipole aerial
Even if a plasma is absent, a spinning neutron
star will radiate if the magnetic and rotation
axes do not coincide.
This is the case of a

‘dipole aerial’
dE
  4 R 6 B 2 sin 2 
dt
Slide 37
Quick revision of pulsar structure
1. Pulsar can be thought of as a non-aligned
rotating magnet.
2. Electromagnetic forces dominate over
gravitational in magnetosphere.
3. Field lines which extend beyond the light
cylinder are open.
4. Particles escape along open field lines,
accelerated by strong electric fields.
Slide 38
Radiation mechanisms in pulsars
Emission mechanisms
Total radiation
intensity
exceeds
does not
exceed
Summed intensity of
spontaneous radiation
of individual particles
coherent
incoherent
It is still not well understood which
physical mechanisms cause the actual
pulsed emission in pulsars, although we
have a nice model for the geometry and
structure of pulsars.Therefore, in this
section, we will be considering radiation
mechanisms involving particles to
investigate what is actually causing the
pulses.
Remember throughout that the outflow of
particles results in an interaction with the
magnetic field.
In astrophysical conditions, emission
mechanisms can be divided into coherent
and incoherent, depending on whether or
not the total radiation intensity can
exceed the summed intensity of the
spontaneous radiation of individual
particles.
Coherent light sources emit waves of the
same frequency and the same phase, ie
waves combine crest-to-crest with each
other. The total amplitude of the summed
wave is proportional to the number of
sources and the total intensity is
proportional to the square of the number
of sources.
Slide 39
Incoherent emission - example
eg. Radiating particles in thermodynamical
equilibrium ie thermal emission.
blackbody => max emissivity
So is pulsar emission thermal?
consider radio: ~10 8 Hz; 100MHz; 3m
Slide 40
Use Rayleigh-Jean approximation to find T:
I   
2kT 2
Watts m-2 Hz -1ster -1
c2
Flux density at Earth, F~10-25 watts m -2 Hz -1
Source radius, R~10km at distance D~1kpc
then:
I   
(1)


 
 D 2  10  25 3  1019
F
 F  2  
2

10 4
R 
2
We will consider an example of
incoherent emission, that of radiating
particles in thermodynamical
equilibrium, ie thermal emission. We
know that blackbody radiation gives the
maximum emissivity. So can pulsar
emission be thermal?
We will consider the radio emission from
pulsars (1e8Hz; 100MHz; 3m) and use
the Rayleigh-Jeans approximation on the
long-wavelength side of the peak.
We are going to calculate the expected
blackbody emission from a pulsar and
compare this with the observed emission.
Remember that since blackbody emission
is the maximum emissivity that any
(incoherent) source can have, if the
observed emission is higher than the
blackbody flux, then we know that the
emission must be coherent.
The temperature on the surface of a
neutron star is about a million K and we
want to know if the radio pulses are due
to a coherent mechanism. Therefore we
assume a blackbody spectrum so that the
radio emission is on the long-wavelength
side of the blackbody peak (which must
be in X-rays). Then we are going to
calculate the brightness temperature
appropriate for the flux actually observed
in the radio, and then see if this is a
reasonable temperature.
For the long-wavelength side of the radio
peak, we use the Rayleigh-Jeans
approximation for a blackbody (top
equation). Then assuming a radius for the
neutron star of 10km and a linear distance
to the source of about 1kpc, the intensity
at the surface of the neutron star (ie if it
were a blackbody source) would be that
given by equation (1).
Slide 41
= 106 watts m -2 Hz -1 ster -1
From equation (1):
T


 
I  c 2
106 3  108
K
2
2k
2  1.4  10  23 108
 3 10 29 K
2
2
K
this is much higher
than a radio blackbody
temperature
Slide 42
Incoherent X-ray emission?
• In some pulsars, eg. Crab, there are also
pulses at IR, optical, X-rays and -rays.
• - Are these also coherent?
• Probably not – brightness temperature of Xrays is about 100 billion K, equivalent to
electron energies 10MeV, so consistent with
incoherent emission.
radio
coherent
IR, optical, X-rays, g-rays
incoherent
Slide 43
Models of Coherent Emission
high-B sets up large pd => high-E particles
e-
ep+
electron-positron
pair cascade
B=1e8Tesla
1e16V
cascades results in bunches
of particles which can radiate
coherently in sheets
Now we have the intensity at the surface
of the neutron star, re-arranging (!) and
inserting numerical values, we find that
the temperature, if this were radiating like
a blackbody, would be about 3e29K
which is far, far too high to be a radio
emission temperature. This corresponds
to particle energies of kT~3e25 eV!!
Since particles of such high energies are
not observed, there cannot be incoherent
radiation of this type in the radio band.
Coherent emission can ‘raise’ the flux up
to levels observed without the need for
extremely high particle energies.
So we have deduced that the emission
which produces the radio pulses must be
coherent to match observed flux levels.
But what about in the X-rays?
In fact IR, optical, X-ray and gamma-ray
pulse emission can be attributed to an
incoherent mechanism in a neutron star.
Thus different mechanisms are at work in
different wavelengths - coherent in the
radio but incoherent in the optical and Xrays.
However it is not well known
whereabouts in the magnetosphere the
processes which produce the higher
energy pulses actually take place.
The actual mechanism by which coherent
radio emission is produced is not well
understood.
The enormous induced electric fields can
drag electrons away from the neutron star
surface accelerating the particles to a
million times their rest mass energy
(assuming a magnetic field of 1e8 Tesla,
which sets up a potential difference
between pole and equator of 1e16V). As
they stream away they emit curvature
radiation. The high energy photons
produced interact with low energy
photons to produce electron-positron
pairs.
These cascades produce bunches of
particles which can emit coherently in
sheets. This model requires short rotation
periods (about a second) and high
magnetic fields (100 million Tesla) or the
cascades cannot take place. BP^2 must be
1e7 Tesla per second or more.
Slide 44
Emission processes in pulsars
• Important processes in magnetic fields :
- cyclotron
Optical & X-ray
emission in pulsars
- synchrotron
• Curvature radiation => radio emission
B
V. high mag fields; efollow field lines very
closely, pitch angle ~ 0
Slide 45
Curvature Radiation
• This is similar to synchrotron radiation.
If ve- ~ c and  = radius of curvature,
radiation v. similar to e- in circular orbit
with:
c
where L is the
L 
2
gyrofrequency
‘effective frequency’ of
emission is given by:
 m   L 3
Slide 46
Curvature vs Synchrotron
Synchrotron
Curvature
B
B
In a strong magnetic field, the processes
which are likely to be most important are
cyclotron (for non-relativistic particles)
and synchrotron (relativistic particles).
These can produce the optical and X-ray
emission of pulsars.
Radio emission is produced by curvature
radiation. This is caused by very high
magnetic fields when the electrons follow
field lines very closely, with a pitch angle
close to zero. The filed lines are generally
curved so that the transverse acceleration
produces radiation.
Curvature radiation is very similar to
synchrotron radiation. If the velocity of
the electrons is close to the speed of light
and ro is the radius of curvature, the
radiation is then very similar to that of an
electron in a circular orbit with a
gyrofrequency given by the expression
shown.
In synchrotron radiation, the pitch angle
of the particle is relatively large and most
of the synchrotron power comes from the
shade region shown, ie an annular region.
In curvature radiation where the magnetic
field is unusually strong, particles follow
the field lines very closely so that the
pitch angle is very much smaller and
most power is emitted over a much
smaller radius.
Slide 47
• Spectrum of curvature radiation
- similar to synchrotron radiation,
Flux
 1/3
e -

m
• e- intensity c.r. << cyclotron or synchrotron
=> radio produced this way, need coherence
The spectrum of curvature radiation has
the same form as that of synchrotron
radiation, varying with the cube root of
frequency at low frequencies and falling
exponentially above the peak frequency.
For likely values of the parameters, it
turns out that the intensity produced by
an electron by curvature radiation is
much smaller than that radiated by the
cyclotron or synchrotron processes. If
radio emission is produced in this way,
then coherence must play a big part and
could solve the problem of energetics.
Coherence may indicate that bunches of
particles are travelling and emitting
together.
Slide 48
X-rays from curvature radiation?
• At frequency 1018Hz
luminosity ~ 10 29 J/s
requires  ~ 105
and no. particles radiating nV ~ 10 40- 1041
depending on density.
• This is too many for such energetic particles
=> X-rays emitted by normal synchrotron
Slide 49
Beaming of pulsar radiation
• Beaming => radiation highly directional
• Take into account
- radio coherent, X-rays incoherent
- location radiation source dep on frequency
• Model:
- radio from magnetic poles
- X-rays from light cylinder
For periodic pulses to appear from a
rotating neutron star, it is necessary that
the radiation is highly directional.
We also have to take into account the
coherence of radio emission (at least) and
possibly differences in the location of the
source of radiation at different
frequencies (eg radio and X-ray emission
may not be emitted from the same
regions).
These pieces of observational evidence
point towards the following structure for
a pulsar:
1. Radio pulses due to particles streaming
away from the star at the magnetic poles.
Supporting evidence from this model
comes from radio beam widths and
polarization of the emission. Another
possibility is magnetic braking where
waves are emitted from the polar caps.
2. X-ray brightenings occuring at the
light cylinder. The evidence for this
theory is from the fact that high energy
radiation is only observed from young
pulsars with short periods. For example
there are only 8 pulsars associated with
supernova remnants out of over 500
known pulsars.
Slide 50
This illustrates the model for a pulsar –
where the axis of the neutron star’s
magnetic field is offset from the rotation
axis. The parameters shown are typical
for a pulsar.
Slide 51
For the sake of simplicity, the diagram
illustrates the axisymmetric case for this
model.
The source of the radio emission is
localized near the magnetic poles and the
emission mechanism must be such as to
produce a narrow beam along the
magnetic field. Note that the polar caps
are defined by the magnetic field lines
which lie tangential to the light cylinder.
Magnetic poles
Radiation source localized near mag poles.

(simple, axisymmetric case)
Rad source localized near
poles, narrow beam
produced along mag field.
Polar caps defined by
field lines tangential to
light cylinder.
2  ~ 1  10 
light cylinder
Slide 52
Important observed properties
• Pulses observed only when beam points at
Earth.
• Rad source probably localized within light
cylinder close to neutron star surface…
- no ‘wandering’ and directionality
• Problem: ALL radiation mechanisms at
different frequencies (coherent or not) must
have same orientation along magnetic field.
Slide 53
Origin of subpulses
subpulses
Brightening on
boundaries between
closed and open
lines may produce
subpulses
boundary
co-rotating plasma
Pulses from the rotating neutron star are
only observed when the beam points at
the Earth, ie for those pulsars whose
magnetic axis lies close to the pulsarEarth line of sight.
The source of radiation is probably
localized inside the light cylinder, rather
close to the surface of the neutron star
and this is for two reasons : a) the
stability of the pulses indicates that there
is little opportunity for the emission
region to wander about its mean position:
and b) the high degree of directionality of
the radiation suggests that it is produced
in a region where the field lines are not
greatly dispersed in direction and this is
true near the surface.
One difficulty with this idea is that
despite differences in character (ie
whether coherent or not) of radiation
mechanisms at different frequencies, all
of them must provide the same
orientation of radiation along the
magnetic field.
Brightening on the boundaries between
closed and open field lines could give rise
to the subpulses. As the star rotates, the
brightening moves across our line of
sight.
The brightening may be caused by the
presence of current sheets between the
closed and open lines. Such lines do not
lie in the same plane because the open
ones are distorted by the rotation of the
neutron star. Current sheets originate
where the lines change direction.
Slide 54
Light Cylinder
Radiation source close to surface of light
cylinder.
P
P’
Slide 55
simplified case
The source of the pulsar radiation at high
energies may be close to the surface of
the light cylinder. We will consider a
simplified case, illustrated above, which
shows the equatorial field pattern of a
rotating dipole where the rotation axis is
perpendicular to the dipole moment.
P is on the tangent to the observer and P’
is the point where an open field line
crosses the light cylinder. Both points are
possible locations of emission given the
likely presence of high energy electrons
around a pulsar. The restricted range in
longitudes where this occurs (about
10degrees) when the magnetic axis is
offset from the rotation axis causes
radiation pulses to be observed and
matches the observed pulse widths of
pulsars.
The actual structure of a pulsar is likely
to be very complex.
See from above, the magnetic field lines
do not rotate rigidly as in a plane, but are
likely to follow a spiral pattern, caused
by the rotation of the star. The example is
shown for the magnetic field and rotation
axis co-aligned – offset them and the
geometry becomes even more complex.
In cross-section, the region of closed
magnetic field lines forms a torus around
the neutron star (again for the magnetic
field and rotation axis co-aligned). The
green line marks the critical boundary ie
the boundary between closed and open
field lines. It is the crossing of this field
line with the light cylinder which may
form the subpulses. The blue line shows
an open field line.
The right hand box shows the composite
view although, again – this is a simple
view (the lines will follow a spiral
pattern, for example).
Slide 56
What we see?
Slide 57
• Relativistic beaming may be caused by ~c
motion of source near light cylinder radiation concentrated into beam width :
  1 ,

1
1   
2
(the
Lorentz
factor)
• Also effect due to time compression (2 2 ),
so beam sweeps across observer in time:
P
 P  1
 2 
3
 2  2  4
 
Slide 58
Long Period Pulsars
• Not generally seen in optical or X-rays
- is this emission produced at light cylinder?
power radiated by
synchrotron
 E 2B2
For dipole magnetic field :
Also :
RL 
cP
2
B  r 3
In addition, the relativistic motion of the
source near the light cylinder may cause
relativistic beaming of the radiation. In
simple terms, the radiation is concentrate
in a beam along the direction of motion,
the beam width being approximately
1/gamma where gamma is the Lorentz
factor.
For a pulsar there is a further effect due
to a time compression by a factor of
2xgamma^2 when the source is travelling
towards the observer so that the beam
sweeps across the observer in the time
given by the relationship shown (P/(2pi)
is the time taken to cover 1 radian).
The observed widths of typical individual
pulses can be explained with gammas of
2 or 3.
Long period pulsars are not generally
seen at optical or X-ray wavelengths exceptions are the Crab and Vela pulsars.
If the optical and X-ray pulses are
produced at the light cylinder this would
explain why these are not observed.
Slide 59
• So if particles of the necessary energy E
exist in all pulsars and emission occurs at
RL , we expect:
- radiated power  P 6
• and thus long period pulsars are weak
emitters.
Slide 60
In summary...
• Radio emission
- coherent
- curvature radiation at polar caps
• X-ray emission
- incoherent
- synchrotron radiation at light cylinder
Slide 61
Magnetic energy & nebula
• Neutron star slows down
=> energy sufficient to feed nebula
• What about the magnetic energy?
Consider energy
released at light
2RL
cylinder.
RL
2
Area = 4RL
To summarize:
The radio emission from pulsars is
probably by coherent curvature radiation
from the polar caps
The X-ray emission is probably due to
incoherent synchrotron radiation at the
light cylinder.
We have seen a simple calculation which
demonstrated that the slowing down of
the neutron star’s rotation produces
enough energy to explain the pulsar
emission and feed the surrounding
nebula.
We will now consider the release of
magnetic energy from the pulsar at the
light cylinder and see how this compares.
Slide 62
• Magnetic field at R L is stretched out to v~c.
B2
• Magnetic energy density = 2
0
• Mag energy crossing light cylinder per sec:
B2
R3
PB 
4RL2 c But for BR  B0 03
L
2 0
RL
mag dipole
6
so
 R  2cRL2
PB  B02  0 
 RL   0
Slide 63
Substituting values for the Crab pulsar:
6

 
4
8
6
2  10  2 3  10 10
PB  10 8  6 
7
4 10
 10 
 
2
Js 1
 10 31 Js 1
like rotational energy release, this is also
comparable to observed emission from Crab
Nebula
Slide 64
Age of Pulsars
.
Ratio P / P (time) is known as ‘age’ of pulsar
In reality, may be longer than the real age.
Pulsar characteristic lifetime ~ 10 7 years
Total no observable pulsars ~ 5 x 10 4
The ratio P/Pdot has the dimension of
time and is often referred to as the age of
the pulsar (assuming it was formed with a
rotation period of a few millisec). For the
Crab and Vela pulsars, the determination
of the age of the pulsars was instrumental
in associating the pulsar with the SNR.
In reality, P/Pdot may be significantly
longer than the real age because pulsars
are not perfect rotating magnetic dipoles,
their magnetic fields decay and Pdot was
larger in the past. The Crab Nebula is
about 3000 years old.
Most pulsars are thought to have a
characteristic lifetime of about 10 million
years.
Integrating the distributions of pulsars
observed in relatively small regions of
the sky over the whole of the Galaxy, we
find that there are about 50,000
observable pulsars (in the Galaxy). This
is a minimum number: there may be
many more at fainter luminosities and we
also have to correct for beaming factors
(which brings this number up to about
200,000.
Slide 65
Pulsar Population
• To sustain this population then, 1 pulsar
must form every 50 years.
• cf SN rate of 1 every 50-100 years
• only 8 pulsars associated with visible SNRs
(pulsar lifetime 1-10million years, SNRs
10-100 thousand... so consistent)
• but not all SN may produce pulsars!!!
To sustain a population of 200,000
pulsars currently observed, 1 pulsar must
form every 50 years (characteristic
lifetime / no. pulsars).
This is consistent with the estimated rate
of 1 SN every 100 years in the Galaxy.
Moreover only 8 pulsars have been
associated with visible SNRs (eg. Crab,
Vela and PSR1509-58) out of more than
500 pulsars. This is entirely consistent
with the difference in lifetimes of typical
pulsars and SNRs.
Note however that not all SN may
produce pulsars… we need another origin
for the production of pulsars, eg accretion
induced collapse.