1. High Energy Astrophysics emp@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. ‘Pulsars’ by Smith

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 X-ray pulsators are not aligned with the rotation axis.
See Smith, p12-13

3. Pulsars cont. Discovered in radio Averaging over many pulses we see:
Period
pulse
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.
~P/10
interpulse

4. av. very constant, individual pulses variable
Measuring pulse complicated by Doppler motion of Earth and frequency dispersion in pulse arrival times.
Individual pulses:
av. very constant, individual pulses variable
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.

5. Pulsar period stability12
Period extremely stable: 1 part in 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???
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.

6. Rotation of a neutron star
Gravitational force > centrifugal force
where and P is the period
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!).

7. 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 -3
-11 3 -1 -2
Crab

8. Substituting numbers for Crab 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
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)) 9 -3

9. 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. 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^ then substituting, which leaves a factor of dP/dt. so

10. Energetics - Crab pulsar
Crab pulsar M = 1 solar mass 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

11. thus energy lost by the pulsar
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
and then substituting for I from slide 10 - and of course energy is being lost so dE/dt is negative.

12. Thus the pulsar can power the nebula.
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 x watts.
Thus the pulsar can power the nebula. 30
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 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.

13. Irregularities in pulsar emission
Short timescales - pulsar slow-down rate is remarkably uniform
Longer timescales - irregularities apparent - in particular, ‘glitches’
A glitch is a discontinuous change of period
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 P
glitch t

14. Glitches
Glitches are caused by stresses and fractures in the external layers, the so-called ‘crust’ of the neutron star.
For example,
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.

15. 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
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 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.
The 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 t
average envelope

16. Neutron Stars
General parameters: R ~ 10 km (10 m) r ~ kg m = 10 g cm M ~ solar masses surface gravity ~ 10 m s
We are going to find magnetic induction, B, of 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. 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).

17. Magnetic induction Magnetic flux,
Radius collapses from 7 x 10 m to 10 m
constant
surface
radius Sun 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

18. The general field of Sun is uncertain but should be ~ 0.01 Tesla.
Thus the field for the neutron star, B ~ 5 x 10 Tesla = 5 x Gauss
Next - how long does B last? 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 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). ns

19. Decay time of magnetic field
D - typical dimension over which field varies significantly (for n.s., D ~ 3 x 10 m)
s - conductivity
3000m
10km
Polar cap
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, s, can be found from the following:
Ohm’s Law states that j = (1/r) .E
j = current density
r = resistivity
E = electric field
and s = 1/r 3

20. Thus, t ~ (3 x 10 ) (10 ) (4p10 ) ~ 10 seconds ~ 3 x 10 years
But magnetic field Crab pulsar still intense after 1000 years => interior must be superconducting (s and t both very large)
Neutron stars very dense and zero-T energy supports star and prevents collapse. 3 10 -7 11 3
Substituting typical values into the relationship for t, 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

21. Neutron star structure
crust
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.
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. 17 -3
2x kg m
1km 14 -3
crystallization of neutron matter
4.3x kg m
9km 9 -3
10 kg m 18 -3
10km
kg m

22. 1. Between densities of 4.3 x kg m and 2 x10 kg m , the lowest energy state is reached when nuclei are embedded in an electron and neutron fluid.
2. Above 2x10 kg m , there is a continuous neutron fluid with electrons and protons as minor constituents. 14 -3 17 -3 17 -3
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

23. Observational evidence
From “Gravitationally-redshifted absorption lines in the X-ray burst spectra of a neutron star”, Cottam et al., 2002, Nature (in press)
Investigating the low-mass X-ray binary EXO
Neutron star primary accreting matter from an evolved red dwarf secondary star via Roche Lobe overflow through an accretion disc.
335ksec observation with XMM-Newton in

24. Low mass X-ray binary Neutron star primary Evolved red dwarf secondary
Roche point
Accretion disk

25. Outbursts Total exposure time = 335,000 sec in XMM-Newton RGS
28 X-ray bursts, typically lasting 100 sec
X-ray count rate in quiescence ~ 0.5 cts/sec
Burst count rate typically 9 ct/sec

26. X-ray absorption lines
cold bursts (<1.2keV)
hot bursts (>1.2keV)
quiescence
low-ionization
circumstellar
absorber
redshifted, highly ionized gas
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.
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.
z = M ~ solar masses R ~ 9-12 km

27. EXO
origin of X-ray bursts
circumstellar material

28. Pulsar Magnetosphere h First, defining scale height
The pressure difference supports the element of atmosphere h
See Smith, Chapter 6, Longair p107 h

29. The pressure difference is given by:
where r is the density
But
thus and
(where m is the mass of constituent particles)

30. Formula for scale height
Integrating:
=> pressure falls off exponentially with height in atmosphere with uniform temperature.
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.
has the dimensions of distance and is called the ‘scale height’.

31. Neutron star scale height
For a neutron star, g ~ m s T ~ 1 million K
thus h ~ 0.01m
Thus the atmosphere of a neutron star is only the order of 10cm! 12 -2

32. Forces exerted on particles
Particle distribution determined by gravity temperature electromagnetism e-
Gravity:
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).
Newton

33. Magnetic force: Newton13
This is a factor of larger than the gravitational force and thus dominates the particle distribution.

34. Neutron star magnetosphere
Neutron star rotating in vacuum:
Electric field induced immediately outside n.s. surface. w B
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.
pd on scale of neutron star radius:

35. Electron/proton expulsion
Neutron star particle emission w B
electrons
Cosmic rays
(From previous slide)
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.
protons

36. In reality...
In reality, the charged particles will distribute themselves around the star to neutralize the electric field.
=> extensive magnetosphere forms
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.

37. Pulsar models Magnetic and rotation axes co-aligned: e-
Co-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.
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 re-arranging, for the Crab pulsar, R_L, the radius of the light cylinder, is 1,600 km.
Radius of light cylinder must satisfy: p
light cylinder, R L

38. 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.

39. A better picture Radio beam 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

40. 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’ a
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,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 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.

41. 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.

42. Radiation mechanisms in pulsars
Emission mechanisms
Total radiation intensity
coherent
exceeds
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.
does not exceed
Summed intensity of spontaneous radiation of individual particles

43. Incoherent emission - example
eg. Radiating particles in thermodynamical equilibrium ie thermal emission.
blackbody => max emissivity
So is pulsar emission thermal?
consider radio: n~10 Hz; 100MHz; 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. 8

44. (1) Use Rayleigh-Jean approximation to find T: Watts m Hz ster-2 -1 -1
-25 -2 -1
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)

45. this is much higher than a radio blackbody temperature6
= 10 watts m Hz ster
From equation (1): -2 -1 -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.

46. 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 100 billion 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

47. Models of Coherent Emission
high-B sets up large pd => high-E particles e- e- p+
electron-positron
pair cascade
B=1e8Tesla
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.
1e16V
cascades results in bunches
of particles which can radiate
coherently in sheets

48. 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 filed lines are generally curved so that the transverse acceleration produces radiation. B
V. high mag fields; e- follow field lines very closely, pitch angle ~ 0

49. Curvature Radiation
This is similar to synchrotron radiation If v ~ c and r = radius of curvature, radiation v. 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:

50. Curvature vs Synchrotron
Synchrotron Curvature B B
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.

51. Spectrum of curvature radiation - similar to synchrotron radiation,
e- intensity c.r. << cyclotron or synchrotron => radio produced this way, need coherence
Flux n
1/3 -n e
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. n n m

52. X-rays from curvature radiation?18
At frequency 10 Hz luminosity ~ J/s requires g ~ and no. particles radiating nV ~ depending on density.
This is too many for such energetic particles => X-rays emitted by normal synchrotron 29 5 40 41

53. 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.

54. A better picture Radio beam 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

55. Magnetic poles Radiation source localized near mag poles.
(simple, axisymmetric case) b
Rad source localized near poles, narrow beam produced along mag field.
Polar caps defined by field lines tangential to light cylinder.
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.
light cylinder

56. 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.
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 pulsar-Earth 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.

57. Origin of subpulses subpulses
Brightening on boundaries between closed and open lines may produce subpulses
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.
boundary
co-rotating plasma

58. Light Cylinder Radiation source close to surface of light cylinder. P
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. P’
simplified case

59. Light cylinder – realistic but complex!
Top view
r=c/w B
aligned
w and B
Cross-section
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).
torus
aligned
w and B

60. What we see?

61. 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
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.

62. Long Period Pulsars
Not generally seen in optical or X-rays is this emission produced at light cylinder?
power radiated by synchrotron
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.
For dipole magnetic field :
Also :

63. and thus long period pulsars are weak emitters.
So if particles of the necessary energy E exist in all pulsars and emission occurs at R , we expect: radiated power
and thus long period pulsars are weak emitters. L

64. 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.

65. Magnetic energy & nebula
Neutron star slows down => energy sufficient to feed nebula
What about the magnetic energy?
Consider energy released at light cylinder.
Area = 4pR
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. 2R L R L 2 L

66. Magnetic field at R is stretched out to v~c. Magnetic energy density =
Mag energy crossing light cylinder per sec: L
But for mag dipole so

67. Substituting values for the Crab pulsar:
like rotational energy release, this is also comparable to observed emission from Crab Nebula

68. Age of Pulsars 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). 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. 4

69. 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!!!
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.