1. High Energy Astrophysics jlc@mssl.ucl.ac.uk http://www.mssl.ucl.ac.uk/
Pulsars
High Energy Astrophysics
References, -
1. High Energy Astrophysics, Volume 2, Second Edition, Longair
2. ‘Pulsar Astronomy’ by Lyne and Smith

2. 4. Pulsars: Pulsed emission; Rotation and energetics; Magnetic field; Neutron star structure; Magnetosphere and pulsar models; Radiation mechanisms; Age and population [3]

3. Introduction Pulsars - isolated neutron stars
Radiate energy via slowing down of rapid spinning motion (P usually ≤ 1sec, dP/dt > 0)
Neutron Stars – supported by degeneracy pressure; Fermi
exclusion principle restricts position hence Heisenberg uncertainty principle allows large momentum/high pressure
Pulsating X-ray sources / X-ray pulsators - compact objects (generally neutron stars) in binary systems
Accrete matter from normal star companion
(P ~ 10s, 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.
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 - 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 accretion and modulated with the neutron star spin period. X-ray pulsator period generally tens of seconds, some with ~ 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 X-ray pulsators are not aligned with the rotation axis.
See Smith, p12-13

4. Pulsars Discovered through their pulsed radio emission
Averaging over many pulses we see:
Period
1.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.
2. 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.
3. As more and more pulsars were discovered it became clear however that a new class of object had been discovered.
4. The original paper was published by Hewish et al. In Nature, 1968, volume 217, p709.
pulse
(~P/10)
interpulse

5. Pulse profiles Average pulse profile very uniform
Individual pulses/sub-pulses very different in shape, intensity and phase
Sub-pulses show high degree of polarization which changes throughout pulse envelope t
Generally pulse profile integrated over a sequence of a few hundred pulses is very uniform.
Individual component pulses or sub-pulses can be very different from one another, varying in intensity, shape and in the phase at which they occur within the integrated profile.
Figure shows a sequence of pulses in time compared to the average envelope for a few hundred pulses.
The most important characteristic of sub-pulses is their high degree of polarization. This also changes throughout the pulse envelope in its form and degree of polarization.
See Smith p91-93
average envelope

6. Pulsar period stability12
Period extremely stable: 1 part in indicates some mechanical clock mechanism
- this mechanism must be able to accommodate pulse-to-pulse variablity.
Pulsations of white dwarf ??? (but Crab pulsar period (P~1/30 sec) too short)
Rotation of neutron star ???
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.

7. Rotation of a neutron star
For structural stability:
Gravitational force > centrifugal force
where and P is the period
otherwise star would fly apart
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!).

8. Reducing: G = 6.67x10 m kg s ; P = 33x10 s => but so -3 -11 3 -1 -2
Stage 1: cancel m from both sides of the equation and substitute v=2pr/P.
Stage 2: simply rearranging the equation to put M/4pr^3 on left
Stage 3: substituting M/4pr^3 into density equation
Crab

9. Substituting numbers for Crab pulsar then:
so r > 1.3 x kg m
This is too high for a white dwarf (which has
a density of ~ 10 kg m ), so it must be a
neutron star. -3
kg m 14 -3 9 -3
1. 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=(2pr/P))

10. Pulsar energetics
Pulsars slow down => lose rotational energy - can this account for observed emission?
Rotational energy:
Pulsars are observed to slow down, ie dP/dt>0 thus they are losing rotational energy.
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. so

11. Energetics - Crab pulsar
Crab pulsar M ~ 1 M
- P = seconds R = 10 m
= x kg m 4 2
kg m
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. 38 2

12. and from observations: thus energy lost by the pulsar
From the equation on slide 9, dE/dt = ( 4.I.pi^2 / P^3 ) x dP/dt
Substituting for I from slide 10 - and of course energy is being lost so dE/dt is negative.

13. Rate of energy loss is greater than that inferred
from the observed keV emission, for which
the observed luminosity in the Crab Nebula is
~ 1.5 x watts.
Thus the pulsar can power the nebula.
Characteristic age for magnetic dipole energy loss
t = P/2 P
= /2 x s ~ 1300 years
Crab Nebula exploded in 1054 AD 30 •
It is interesting to note that the lifetime (characteristic age) of a pulsar, tau = P/2(dP/dt) – assumes magnetic dipole energy loss, for the Crab Nebula is about 1300 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 g-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.

14. Neutron Stars
General parameters: R ~ 10 km (10 m) r ~ kg m = 10 g cm M ~ M
- surface gravity, g = GM/R2 ~ 10 m s
We are going to find magnetic induction, B, for a neutron star. 4 18 -3 15 -3
inner 12 -2
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. Particle positions are very tightly constrained so 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.

15. Magnetic induction Magnetic flux, constant
Radius collapses from 7 x 10 m to 10 m
constant
surface
RNS R 8 4
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.
Surface change gives

16. Thus the field for the neutron star:
The Sun has magnetic fields of several different spatial scales and strengths but its general polar field varies with solar cycle and is ≈ Tesla.
Thus the field for the neutron star:
B ~ 5 x 10 Tesla = 5 x Gauss
If the main energy loss from rotation is through magnetic dipole radiation then:
B ~ 3.3 x 1015 (P P) ½ Tesla
or ~ 106 to 109 Tesla for most pulsars 7 11 ns
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 pulsars typically lie in the range of to Tesla (but weaker for the millisecond pulsars, magnetic field B = 3e15 sqrt(P x dP/dt). •

17. Neutron star structure
crust
inner
outer
Heavy nuclei (Fe) find a minimum energy when arranged in a crystalline lattice
Neutron star segment
neutron liquid 1.
solid core?
Superfluid neutrons, superconducting p+ and e- 2. 17 -3
2x kg m
1km
The diagram shows a segment (ie a slice) through a neutron star.
Main Components: Crystalline solid crust (1) and neutron liquid interior (2)
Boundary at r = kg/m3 – the density of nuclear matter
Outer Crust: Solid; matter similar to that found in white dwarfs, ie heavy nuclei (mostly Fe) forming a Coulomb lattice embedded in a relativistic degenerate gas of electrons. Lattice is min energy configuration for heavy nuclei.
Inner Crust (1): Lattice of neutron-rich nuclei (electrons penetrate nuclei to combine with protons and form neutrons) with free degenerate neutrons and a degenerate relativistic electron gas. For r > kg/m3 – the neutron drip point, massive nuclei are unstable and release neutrons. Neutron fluid pressure increases as the density increases.
Neutron Fluid Interior (2): For 1 km < r < 9 km, ‘neutron fluid’ – superfluid of neutrons and superconducting protons and electrons. Enables B field maintenance. Density is < r < kg/m3. Near inner crust, some neutron fluid can penetrate into inner part of lattice and rotate at a different rate – glitches?
Core: Extends out to ~ 1 km and has a density of kg/m3. Its substance is not well known - it could be a neutron solid, quark matter or neutrons squeezed to form a pion concentrate.
On the very surface of the neutron star, densities fall below 109 kg/m3 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. 14 -3
crystallization of neutron matter
4.3x kg m
9km 9 -3
10 kg m 18 -3
10km
kg m

18. Regions of NS Interior Main Components: (1) Crystalline solid crust
(2) Neutron liquid interior
- Boundary at r = kg/m3 – density of nuclear matter
Outer Crust:
Solid; matter similar to that found in white dwarfs
Heavy nuclei (mostly Fe) forming a Coulomb lattice embedded in a relativistic
degenerate gas of electrons.
- Lattice is minimum energy configuration for heavy nuclei.
Inner Crust (1):
Lattice of neutron-rich nuclei (electrons penetrate nuclei to combine with protons and
form neutrons) with free degenerate neutrons and degenerate relativistic electron gas.
For r > kg/m3 – the neutron drip point, massive nuclei are unstable and
release neutrons.
Neutron fluid pressure increases with r
1.Zero temperature energy – the Fermi energy, supports the star and prevents further collapse. From Pauli principle, each allowed energy state can be occupied by no more than two electrons of opposite spin.
2. For electrons, even at 0o K, occupy a range of states of different Fermi energy states. In a collapsed star, they occupy a small volume and have well known positions. Hence by the uncertainty principle, they have a large momentum and generate a high temperature – independent pressure. The corresponding “classical” thermal KE would be ~ K. These electrons are called degenerate and so electron degeneracy pressure supports the star which in this case is a White Dwarf.
3. For collapse of high mass stars, the inert Fe core gives way to a neutron star where neutron degeneracy pressure supports the star against gravity and prevents further collapse. Since the available size is much smaller than for a WD, neutrons are forced to occupy states of even higher Fermi energy (E ~ 1 MeV) and so the resulting degeneracy pressure can support the neutron star.

19. Regions of NS Interior (Cont.)
Neutron Fluid Interior (2):
For 1 km < r < 9 km, ‘neutron fluid’ – superfluid of neutrons and superconducting
protons and electrons.
- Enables B field maintenance.
- Density is < r < kg/m3.
Near inner crust, some neutron fluid can penetrate into inner part of lattice and
rotate at a different rate – glitches?
Core:
- Extends out to ~ 1 km and has a density of kg/m3.
- Its substance is not well known.
- Could be a neutron solid, quark matter or neutrons squeezed to form a pion
concentrate.

20. White Dwarfs and Neutron Stars
In both cases, zero temperature energy – the Fermi energy, supports the star and prevents further collapse
From exclusion principle, each allowed energy state can be occupied by no more than two particles of opposite spin
Electrons in a White Dwarf occupy a small volume and have very well defined positions – hence from uncertainty principle, they have large momentum/energy and generate a high pressure or electron degeneracy pressure
Corresponding “classical” thermal KE would have T ~ K and the related electron degeneracy pressure supports the star
For a high mass stellar collapse, inert Fe core gives way to a Neutron Star and neutron degeneracy pressure supports the star
NS has ~ 103 times smaller radius than WD so neutrons must occupy states of even higher Fermi energy (E ~ 1 MeV) and resulting degeneracy pressure supports NS

21. Low Mass X-ray Binary provides Observational Evidence of NS Structure
Neutron star primary
Evolved red dwarf secondary
Roche point
Accretion disk

22. Gravitationally Redshifted Neutron Star Absorption Lines
XMM-Newton found red-shifted X-ray absorption features
Cottam et al. (2002, Nature, 420, 51):
- observed 28 X-ray bursts from EXO
ISM
z = 0.35
Fe XXVI & Fe XXV
(n = 2 – 3) and O VIII
(n = 1 – 2) transitions
with z = 0.35
Red plot shows:
- source continuum
- absorption features
from circumstellar gas
1. If a photon starts out with wavelength lo at a radial distance r from a spherical gravitating mass M, then it will be red-shifted to a wavelength l where
z = (l-lo)/lo and l/lo = (1 – 2GM/c2r)-1/2. This is a General Relativistic formula.
2. To avoid the lines being broadened by differential gravitational redshifting if they come from a region that is extended in radius, we need the source to be localised in a shell whose thickness is << r.
3. This is certainly true for a neutron star where the surface layer where these lines are produced is extremely thin. Scale– height of a neutron star atmosphere is approx 10 cm!
Note: z = (l-lo)/lo and l/lo = (1 – 2GM/c2r)-1/2

23. X-ray absorption lines
Low T bursts
Fe XXV & O VIII
(T < 1.2 keV)
High T busts
Fe XXVI
(T > 1.2 keV)
quiescence
low-ionization
circumstellar
absorber
redshifted, highly ionized gas
z = 0.35 due to NS
gravity suggests:
M = 1.4 – 1.8 M
R = 9 – 12 km
For a neutron star with these parameters, the structure is consistent with being made up mostly of “normal” neutron star matter.
More exotic matter, such as strange quark matter or kaon condensates are unlikely, because, for this radius, the mass would have to be much lower (less than 1.1 solar masses) which is uncomfortably close to the lower limit for the formation of a neutron star.
3. The implied birth mass for this star is near the average for non-accreting stars, and, assuming accretion for about a billion years (remember that this is the primary component of an accreting binary system), its mass agrees with those estimated for other accreting neutron stars.

24. EXO
origin of X-ray bursts
circumstellar material

25. Forces exerted on particles
Pulsar Magnetospheres
Forces exerted on particles
Particle distribution determined by gravity electromagnetism e-
Gravity
1. Particle distribution around a neutron star is determined not only by gravity and temperature, but also by electromagnetic forces.
2. 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).
Newton

26. Magnetic force
RNS
Newton
PNS 13
This is a factor of larger than the gravitational force and thus dominates the particle distribution.

27. Neutron star magnetosphere
Neutron star rotating in vacuum:
Electric field induced immediately outside n.s. surface. w B
Potential difference on scale of neutron star radius is:
1. For a star rotating in a vacuum, the rotating magnetic field induces an electric field immediately outside the neutron star surface (E = (v x B)) and is given by the equation shown in the green box.
2. The potential difference on the scale of the neutron star radius is approx Volts.

28. Electron/proton expulsion
Neutron star particle emission w B
electrons
Cosmic rays?
protons
(From previous slide)
1. The potential difference on the scale of the neutron star radius is approx Volts.
2. 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 as in cosmic rays.

29. In reality... => extensive magnetosphere forms
Charged particles will distribute themselves around the star to neutralize the electric field.
=> extensive magnetosphere forms
Induced electric field cancelled by static field
arising from distributed charges or -
E + 1/c (W x r) x B = 0
where E and B are electric and magnetic fields and
W is the vector angular velocity of the neutron star
1. 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.
See Smith p48 and Manchester & Taylor p178.]

30. Magnetosphere Charge Distribution
Rotation and magnetic polar axes shown co-aligned
Induced E field removes charge from the surface so charge and
currents must exist above the surface – the Magnetosphere
Light cylinder is at the radial distance at which rotational velocity of
co-rotating particles equals velocity of light
Open field lines pass through the
light cylinder and particles stream
out along them
Feet of the critical field lines are at
the same electric potential as the
Interstellar Medium
Critical field lines divide regions of
+ ve and – ve current flows from
Neutron Star magnetosphere

31. Pulsar models Here magnetic and rotation axes co-aligned: e-
Co-rotating plasma is on magnetic field lines that are closed inside light cylinder
Radius of light cylinder must satisfy:
Particles are able to move along, but not across, the magnetic field lines.
2. 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.
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 rc defines the ‘light cylinder’ and satisfies the condition (2*pi*R_L)=c.
Substituting (P=0.033 secs) and re-arranging, for the Crab pulsar, rc, the radius of the light cylinder, is 1,600 km.
light cylinder, r c

32. A more realistic model... For pulses, magnetic and rotation axes
cannot be co- aligned.
Plasma distribution and magnetic field
configuration complex for Neutron Star
Radio
Emission
Velocity- of -
Light Cylinder
For r < rc, a charge-separated co-rotating magnetosphere
Particles move only along field lines;
closed field region exists within field-lines
that touch the velocity-of-light cylinder
Particles on open field lines can flow out of
the magnetosphere
Radio emission confined to these open-field
polar cap regions
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.

33. A better picture Radio beam r=c/w Open magnetosphere Light cylinder B
Closed magnetosphere
Neutron star
mass = 1.4 M
radius = 10 km
B = 10 to 10 Tesla
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. 4 9

34. The dipole aerial Even if a plasma is absent, a spinning neutron
star will radiate – and loose energy, if the
magnetic and rotation axes do not coincide.
This is the case of a ‘dipole aerial’ – magnetic analogue of the varying electric dipole a
1. 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.
2. The wave radiated from the magnetic poles has an angular frequency,w. A field of 1e8 Tesla can in fact explain the energy losses observed… ie. magnetic dipole radiation is the principal way in which pulsars loose energy.
3. 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.
4. The magnetic dipole is offset from the rotation axis so that it displays a varying dipole moment at large distances (see the previous slide).
5. Thus electromagnetic energy is radiated from the star and this is extracted from the rotational energy, ie the spin period decreases.

35. Quick revision of pulsar structure
Pulsar can be thought of as a non-aligned rotating magnet.
Electromagnetic forces dominate over gravitational in magnetosphere.
Field lines which extend beyond the light cylinder are open.
Particles escape along open field lines, accelerated by strong electric fields.

36. Radiation Mechanisms in Pulsars
Emission mechanisms
Total radiation intensity
coherent
exceeds
incoherent
does not exceed
Recall that TB = K for radio emission. 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.
1. Remember throughout that the outflow of particles results in an interaction with the magnetic field.
2. 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.
3. Coherent light sources emit waves of the same frequency and the same phase, ie waves combine crest-to-crest with each other.
4. Total amplitude of the summed wave is proportional to the number of sources
5. Total intensity is proportional to the square of the number of sources.
Summed intensity of spontaneous radiation of individual particles

37. Incoherent emission - example
For radiating particles in thermodynamic
equilibrium i.e. thermal emission.
Blackbody => max emissivity
So is pulsar emission thermal?
Consider radio: n~108 Hz or 100MHz; l~3m
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.

38. (1) Use Rayleigh-Jeans approximation to find T: Watts m Hz ster-2 -1 -1
-25 -2 -1
Crab flux density at Earth, F~10 watts m Hz
Source radius, R~10km at distance D~1kpc
then:
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).
(1)

39. this is much higher than a radio blackbody temperature!
So - 6 -2 -1 -1
In = 10 watts m Hz ster
From equation (1):
this is much higher than a radio blackbody temperature!
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.

40. Incoherent X-ray emission?
In some pulsars, eg. Crab, there are also pulses at IR, optical, X-rays and g-rays.
- Are these also coherent?
Probably not – brightness temperature of X-rays is about 1011 K, equivalent to electron energies 10MeV, so consistent with incoherent emission.
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 X-rays.
However it is not well known whereabouts in the magnetosphere the processes which produce the higher energy pulses actually take place.
radio
coherent
IR, optical, X-rays, g-rays
incoherent

41. Models of Coherent Emission
high-B sets up large pd => high-E particles e- e- e+
electron-positron
pair cascade
B = 1.108Tesla
R = 104 m
1. The actual mechanism by which coherent radio emission is produced is not well understood.
2. 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 1e18V).
3. As they stream away they emit curvature radiation and the high energy photons produced interact with low energy photons to produce electron-positron pairs.
4. These cascades produce bunches of particles, of dimension < a radio wavelength (~ 1 – 100 m), which can emit coherently in sheets.
5. For N electrons in a bunch, radiation intensity is N2 times that of an individual charge.
6. This model requires short rotation periods (about a second) and high magnetic fields (100 million Tesla) or the cascades cannot take place.
7. Theory shows that B.P2 ≥ 107 Tesla/second is required to produce electron-positron cascades..
1.1018V
cascades results in bunches
of particles which can radiate
coherently in sheets

42. Emission processes in pulsars
Important processes in magnetic fields : cyclotron synchrotron
Curvature radiation => Radio emission
Optical & X-ray emission in pulsars
=>
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 field lines are generally curved so that the transverse acceleration produces radiation. B
High magnetic fields; electrons follow field lines very closely, pitch angle ~ 0o

43. Curvature Radiation
This is similar to synchrotron radiation If v ~ c and r = radius of curvature, the radiation very similar to e- in circular orbit with: e-
where n is the gyrofrequency L
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.
‘effective frequency’ of emission is given by:

44. Curvature vs Synchrotron
Synchrotron Curvature B B
1. 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.
2. 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.

45. n exp(-n) n Spectrum of curvature radiation (c.r.)
- similar to synchrotron radiation,
For electrons:
intensity from curvature radiation << cyclotron or synchrotron
If radio emission produced this way, need coherence
Flux n
1/3
exp(-n) n n m
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.

46. Beaming of pulsar radiation
Beaming => radiation highly directional
Take into account
- radio coherent, X-rays and Optical incoherent
- location of radiation source depends on frequency
- radiation is directed along the magnetic field lines
- pulses only observed when beam points at Earth
Model:
- radio emission from magnetic poles
- X-ray and optical emission 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.
3. 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 :
Stability of the pulses indicates that there is little opportunity for the emission region to wander about its mean position: and
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.

47. Observational Evidence for Pulsar Emission Sites
Radio pulses come from particles streaming away from the NS in the magnetic polar regions:
Radio beam widths
Polarized radio emission
Intensity variability
Optical and X-ray brightening occurs at the light cylinder
Radiation at higher energies only observed from young pulsars with short periods
Only eight pulsar-SNR associations from more than 500 known pulsars
Optical and X-radiation source located inside the light cylinder
Pulse stability shows radiation comes from a region where emission position does not vary
High directionality suggests that emission is from a region where field lines are not dispersed in direction i.e. last closed field lines near light cylinder
Regions near cylinder have low particle density so particles are accelerated to high energies between collisions

48. The better picture - again
Radio beam
The better picture - again
r=c/w
Open
magnetosphere
Light cylinder B
Closed magnetosphere
Neutron star
mass = 1.4 solar masses
radius = 10 km
B = 10 to 10 Tesla
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. 4 9

49. Light Cylinder Radiation sources close to surface of light cylinder P
Outer gap region
- Incoherent emission
X-ray and
Optical beam
Radio
Beam
Polar cap region
- Coherent emission
Light Cylinder
Outer gap region
- Incoherent emission
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.
Gaps of low particle density exist between the last closed field line and the light cylinder. Particles can be accelerated to very high energies.
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.
Simplified case – rotation and magnetic axes
orthogonal

50. Relativistic beaming may be caused by motion of source with v ~ c near the light cylinder
- radiation concentrated into beam width
Also effect due to time compression (2g2), so beam sweeps across observer in time:
g ~ 2 – 3 needed to explain individual pulse widths
(the Lorentz factor)
1. 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 concentrated in a beam along the direction of motion, the beam width being approximately 1/gamma where gamma is the Lorentz factor.
2. 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).
3. The observed widths of typical individual pulses can be explained with gammas of 2 or 3.

51. In summary...
Radio emission coherent curvature radiation at polar caps
X-ray emission incoherent synchrotron radiation at light cylinder
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.

52. Age of Pulsars Ratio (time) is known as ‘age’ of pulsar•
Ratio (time) is known as ‘age’ of pulsar
In reality, may be longer than the real age.
Pulsar characteristic lifetime ~ 10 years
Total no observable pulsars ~ 5 x 10 7
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).
1. For the Crab and Vela pulsars, the determination of the age of the pulsars was instrumental in associating the pulsar with the SNR.
2. 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.
3. Most pulsars are thought to have a characteristic lifetime of about 10 million years.
4. 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. 4

53. Pulsar Population
To sustain this population then, 1 pulsar must form every 50 years.
cf SN rate of 1 every years
only 8 pulsars associated with visible SNRs (pulsar lifetime 1-10million years, SNRs thousand... so consistent)
but not all SN may produce pulsars!!!
1. 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 PSR ) 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.

54. PULSARS
END OF TOPIC

55. crystallization of neutron matter
crust
inner
outer
Heavy nuclei (Fe) find a minimum energy when arranged in a crystalline lattice
Neutron star segment
neutron
liquid
solid core?
Superfluid neutrons, superconducting
p+ and e- 17 -3
2x kg m
1km
The diagram shows a segment (ie a slice) through a neutron star.
Main Components: Crystalline solid crust (1) and neutron liquid interior (2)
Boundary at r = kg/m3 – the density of nuclear matter
Outer Crust: Solid; matter similar to that found in white dwarfs, ie heavy nuclei (mostly Fe) forming a Coulomb lattice embedded in a relativistic degenerate gas of electrons. Lattice is min energy configuration for heavy nuclei.
Inner Crust (1): Lattice of neutron-rich nuclei (electrons penetrate nuclei to combine with protons and form neutrons) with free degenerate neutrons and a degenerate relativistic electron gas. For r > kg/m3 – the neutron drip point, massive nuclei are unstable and release neutrons. Neutron fluid pressure increases as the density increases.
Neutron Fluid Interior (2): For 1 km < r < 9 km, ‘neutron fluid’ – superfluid of neutrons and superconducting protons and electrons. Enables B field maintenance. Density is < r < kg/m3. Near inner crust, some neutron fluid can penetrate into inner part of lattice and rotate at a different rate – glitches?
Core: Extends out to ~ 1 km and has a density of kg/m3. Its substance is not well known - it could be a neutron solid, quark matter or neutrons squeezed to form a pion concentrate.
On the very surface of the neutron star, densities fall below 109 kg/m3 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. 14 -3
crystallization of neutron matter
4.3x kg m
1018 kg m-3
9km
10 kg m 9 -3
10km

56. Relativistic beaming may be caused by ~ c motion of source near light cylinder - radiation concentrated into beam width :
Also effect due to time compression (2g ), so beam sweeps across observer in time:
(the Lorentz factor) 2
1. 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 concentrated in a beam along the direction of motion, the beam width being approximately 1/gamma where gamma is the Lorentz factor.
2. 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).
3. The observed widths of typical individual pulses can be explained with gammas of 2 or 3.

57. Pulsar Model Radio emission from magnetic poles
Radio pulses due to particles streaming away from the neutron star in polar regions along open field lines
Observed radio beam widths and polarized emission support this model
X-ray and optical emission from light cylinder
Radiation only seen from young short period pulsars

58. Pulsars Period interpulse (~P/10) pulse
1.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.
2. 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.
3. As more and more pulsars were discovered it became clear however that a new class of object had been discovered.
4. The original paper was published by Hewish et al. In Nature, 1968, volume 217, p709.

59. Pulse profiles t average envelope
Generally pulse profile integrated over a sequence of a few hundred pulses is very uniform.
Individual component pulses or sub-pulses can be very different from one another, varying in intensity, shape and in the phase at which they occur within the integrated profile.
Figure shows a sequence of pulses in time compared to the average envelope for a few hundred pulses.
The most important characteristic of sub-pulses is their high degree of polarization. This also changes throughout the pulse envelope in its form and degree of polarization.
See Smith p91-93