Figure 1. The orthogonal rotator which produces magnetic
dipole radiation in a vacuum. The physics here is covered
in most introductory electromagnetizm books The rotation
produces very large amplitude electromagnetic waves, with
the resultant energy loss slowing down the rotation.
It is essentially this theoretical expectation that gives the canonical pulsar
magnetic field of gauss, and to this day the magnetic field is defined to be
(Taylor) where is the period in seconds and is the dimensionless slowing down
rate. This model tells us little more about how pulsars generate pulses,
although pulses could be attributed to "crashing" of the electromagnetic waves
owing to plasma from the pulsar swept away by the waves. The complex pulse
shapes and their polarization properties make this mechanism an unlikely source
of the coherent radio emission.
B. The Aligned Rotator (Goldreich and Julian: GJ)
Here we have the other extreme where the rotation axis parallels the magnetic
field axis. The assumption of a pure magnetic dipole is obviously a
simplification, but a ubiquitous one. To my knowledge, no one has argued that a
pulsar would not function if it had a pure dipole magnetic field, although the
case of perfect alignment is problematical (read on) In this model, plasma about
the neutron star plays a central role in the dynamics, unlike the previous model
where it is included more or less incidentally. The source of the plasma was
originally assumed to be from the pulsar surface, owing to the huge
rotationally-induced electric field that would act on the surface if the
surroundings were vacuum. In the usual MHD approximation, one can directly
calculate what the required plasma density has to be through the assumption that
the magnetic field lines have to be equipotentials, the so-called
Goldreich-Julian density. In effect, the plasma is an extension of the
conducting neutron star and must rotate rigidly with the star. Consequently the
pulsar problem seemed to be defined self-consistently because centrifugal forces
would overpower magnetic trapping at or near the so-called light cylinder
distance where corotation would exceed the speed of light. The magnetic fields
would be forced open as shown in Fig. 2, plasma would flow out and have to be
replaced, and acceleration of plasma from the surface to do this would be the
natural place to look for radio emission. Indeed a number of subsequent models
simply concentrated on these magnetic polar caps as the site of radio emission,
such as the oft-cited Ruderman-Sutherland (RS) model .
Fig. 2 The Goldreich-Julian model where the magnetic field
lines are taken to be equi-potentials. It is
straightforward to show that the required charge
distribution corotates rigidly with the star and that the
space charge density is distributed as shown (the usual
choice is to have electrons over the magnetic poles,
although the opposite is possible).
This model had well-known short comings in that the charge distribution
over the polar caps was all of a single sign, as can be seen from the figure and
loss of such plasma would simply charge the star and halt the flow. However the
model was unable to "charge the star" because it was already fixed and had no
adjustable parameters (in fact the star is charged, but this charge cannot be
altered). Nevertheless people plugged ahead in hopes that this seeming
technicality would be patched up, and indeed this investigator spend much time
and energy on this effort, an effort that was, at the time, adjudged worthy of
support.
C. The Vacuum Gaps (Non neutral Plasmas)
The essential flaw was finally identified, although the resolution was implicit
in a number of suggestions by a number of workers. Ruderman and coworkers
proposed empty gaps in the outer magnetosphere (in order to accelerate particles
sufficiently to make the gamma-rays seen from the Crab pulsar). Holloway showed
by a simple Gedanken Experiment that regions of the GJ magnetosphere could have
empty regions (this work instantly invalidated the GJ model, but that was not so
quickly appreciated). I backed into this from trying to resolve the
inconsistencies in the above model. The bottom line when all was sorted out is
that the plasma around a pulsar is not approximated by the MHD equations but is
fundamentally different because it is entirely charge-separated and not, as the
textbooks generally assume for a plasma, quasi-neutral. Ironically at this same
time John Malmberg and collaborators at UCSD were trapping and observing the
properties of pure electron plasmas. The Malmberg device is quite simple, being
a solenoid that provides a magnetic field that confines the electrons axially
and having two end electrodes that provide an electric field that confine the
electrons along the axis, as shown in Fig. 3
Fig 3 Malmberg device. The axial magnetic field prevents
particles from moving to the cylindrical walls (B) while
charged rings (A & C) prevent the electrons from escaping
down the solenoid. The electrons are trapped in the form of
a rotating prolate spheroid and are extremely stable
(trapping for hours). The remaining details in the figure
concern injection and diagnostics.
Note that it would be impossible to confine a quasi-neutral plasma in such a
trap; the properties of the two plasmas are entirely different.
A mathematically simpler device having the same properties is the
Penning trap where it is a textbook exercise to show that electrons can be
trapped in the form of a sphere (in fact, as any axially symmetric spheroid).
In the Penning trap one again has a uniform magnetic field but the electric
field is axisymmetric quadrupolar, formed by a ring and two end electrodes, a
"deformation" of the Malmberg configuration. The general properties of the pure
electron sphere have echoes of standard plasma physics: the oscillation
frequency of the sphere along the magnetic axis is essentially the plasma
frequency and the density in the sphere goes from a constant value inside to
zero outside in a distance of about a Debye length. The equipotentials in the
sphere are shown in Fig. 4.
Fig. 4 Sphere and equipotentials in a magnetic Penning trap.
Any of the equipotential lines could be taken to be
electrodes. The electron (or ion) sphere rotates as the
particles execute drift about the magnetic field axis
(vertical). Along the vertical axis, the self-repulsion of
the electrons is exactly balance by the quadrupolar
electrostatic field, and in fact this determines the charge
density, which vanishes at the surface. For zero
temperature the system freezed into a Wigner crystal.
The magnetized neutron star is essentially an inside-out Penning trap
with a gravitational vengeance. The Goldreich-Julian model lacked free
parameters because it was the special solution where the magnetosphere was
filled to capacity. Even then, the properties of non neutral plasmas could be
seen in hindsight. If the polar regions were filled with electrons, for
example, it would be impossible to add positive particles into this region. As
anyone who has studied ionospheric physics would immediately know, an electric
field is required in part to support these particles against gravity. If
particles of the opposite sign are introduced, this field AND gravity both
accelerate them to the surface. In the Penning trap the same thing would
eventually happen, but before then the test particle would drift along the
magnetic field lines to the surface of the sphere, pop out, and be accelerated
to the electrodes. Not only are these plasmas non neutral, they cannot be
neutralized. For that reason, they are entirely different. They support
density discontinuities with the vacuum; occupy limited volumes of space; cannot
be neutralized,;are extremely stable structures,;and break the apparent
self-consistency of the GJ model. All of this is detailed in Theory of Neutron
Star Magnetospheres.
D. The Klauss-Polstorff/Michel model
The obvious question is, "If GJ isn't right, what is?" The answer and the
resolution of Holloway's paradox together is simply that the negative and
positive regions of the GJ magnetosphere are limited in extent; the polar
regions are confined to domes and the equatorial regions are confined to a
torus. In between is a vacuum "gap" (actually the system is essentially all
vacuum with the torus and two domes of opposite charge). To drive this point
home, the entire configuration was numerically simulated with the result (for
the numerically tractable case of an aligned magnetosphere) as shown in Fig. 5.
Figure 5. Simulation of the aligned rotator. Dots are
charged rings removed from the surface. The dome and torus
about the neutron star (spherical quadrant at origin) are
detailed. The charge density discontinuities for both are
obvious. The system has to be positively charged to retain
polar electron domes. Solid lines are equipotentials,
dashed are magnetic field lines. The Holloway "gap" is
denoted H.
At first, this configuration was dismissed as a "dead pulsar."
Work continues to the present to somehow revive the GJ model in England (Mestel
& co.), in Japan (Shibata), and in Russia (Beskin, Istomin, and Gurevich). In
all of these models it is necessary to somehow circulate plasma currents inside
the magnetosphere. But the important point is that the vacuum gaps are unstable
to pair discharging and filling. I made the correct but misleading remark that
pair discharging would simply allow the system to approach the GJ configuration
but stop far short of where discharging could continue, at least in the case of
the slower pulsars; the magnetic fields drop to only a few gauss near the light
cylinder and pair-cascading is out of the question. But I was still thinking in
terms of the aligned rotator per se.
E. The Pair-Plasma Avalanche.
The idea of a pair/gamma-ray cascade was around at almost the beginning
of pulsar research and the basic paper predates it (Erber). The idea is that a
cascade can be driven by emission of gamma-rays by energetic electrons moving on
curved magnetic field lines followed by conversion of these quanta on magnetic
field "quanta" to produce electron-positron pairs, with these new leptons
radiating in turn., as illustrated in Fig. 6
Figure 6. Pair production cascade. Hard gamma-rays from
curvature radiation pair produce on the virtual magnetic
field "photons" to produce electron-positron pairs. In the
usual passive cascade, the particles attenuate in energy.
In the cascade, the electrostatic fields reaccelerate the
electrons and eject the positrons. The primary moves next
to where the pair is created, causing bunching.
The GJ model created some difficulty for this model in that the
space-charge-limited current of particles accelerated from the pulsar surface
was insufficient to drive such a cascade (which lead to the RS model's
assumption that particles could not be emitted from the surface because the
sense of rotation required them to be ions and the ions could arguably be
strongly bound). Ruderman & co. relocated the cascades to the vacuum gaps for
this reason. In the context of GJ model, such cascading was just another source
of current when the surface seemed a plentiful source to begin with. Thus
cascading seemed an interesting mechanism, but not an essential mechanism.
Given instead magnetospheres with potentially huge gaps (Fig. 6), test particles
would not simply cascade in the vacuum regions but would become
radiation-reaction limited and convert all their electrostatic energy into
gamma-rays and pairs. At the same time, the new pairs would be winnowed by the
accelerating field, so an initial electron, say, would produce a pair and the
positron would be halted and accelerated backwards as shown in Fig. 6, leaving
two electrons. Since the system is extremely relativistic, the particles are
essentially co-moving and consequently the two effects cause a dense bunch to
form, with not only obvious but quantitatively promising implications for
coherent radio emission (Michel 1991). Owing to radiation reaction, on the
order of gammas are emitted before the first pair conversion, so the
exponentiation is virtually explosive. Emission from bunches has been a
standard street lamp in the search for the keys to coherent emission.
This avalanching is not to be confused with the passive cascading
investigated by Harding and co-workers, wherein the energetic primary has
already been accelerated elsewhere and the cascading takes place with no further
accelerating field. An equivalent view of the avalanching is shown in Fig. 7.
Figure 7. Bremmsstrahlung approximation for the avalanche.
The Lorentz-contracted field of the primary acts like a
virtual photon and is bent as the primary follows a curved
path, with the result that the upper portion is causally
disconnected from the primary (the curvature radiation) and
it is on this surface that the pairs are produced, which
emphasizes why bunching takes place.
F. The Oblique Rotator
Pulsar observers have long assumed that the pulsar magnetic field is in general
neither aligned nor orthogonal but some angle in between. So have the
theorists, except that they held out some hope that these simpler systems might
suffice in themselves and so offer a more tractable system for analysis. But
the above pair avalanching would do little in an aligned rotator as already
recognized: the magnetosphere near the star would fill up but the distant
magnetosphere would not and then the avalanching would cease; a dead pulsar. In
the case of an inclined rotator though, one has the complexity of no symmetry
axis but also the additional physics in the production of very large amplitude
electromagnetic waves by the orthogonal dipole component. If one imagines an
almost-aligned pulsar in which avalanching has attempted to fill the
magnetosphere, the essential thing to notice is that filling along the rotation
axis is much more extensive than filling toward the light cylinder as can even
be seen in Fig. 5. The reason is that the electrons on the axis are only
confined to the system by the net system charge. The effect of discharging is
to drive the system charge to zero since positrons can be ejected on field lines
leading beyond the light cylinder and be lost, while the electrons simply cause
the dome to grow. But given a slight inclination, there is additionally a wave
zone.
Thus if the dome were to extend into the wave zone, the electrons would be
driven off by the pondermotive force of the waves. Thus we have in the inclined
case a mechanism for driving off both the electrons and positrons, with the
potential of solving the current closure problem. Note in this respect that the
terminology "light cylinder" serves to focus attention on the old GJ model's
assupmtion that centrifugal effects were responsible for pulsar action, as if
particles could not be lost unless they reached this magical distance. As we
will argue below, the physically significant region is the spherical wave zone
about the neutron star, where particles are driven away by the huge pondermotive
forces of the EM waves, not be centrifugal forces per se.
G. The Acceleration and Deceleration Regions.
One problem with the pair avalanche is actually a virtue, namely that charged
particles flow both ways. In the early "particles-accelerated-from-the-surface"
models, particles only could flow away from the neutron star, which was the
fatal flaw in the GJ model: only particles of one sign were available. On the
other hand, with particles flowing toward the surface, one would expect them to
strike and heat the surface (Arons). In fact, small x-ray emitting spots have
apparently been observed on some nearby pulsars (Helfand). But such heating is
known not to be a big item in the pulsar energy budget. The possible resolution
to this problem was puzzling for some time but is fairly straightforward.
First, one must recognize that electrons are accelerated OFF the polar surface.
However, in the Polstorff model, the electric field in the vacuum gaps
accelerates electrons TOWARD the surface. The reason for this apparent paradox
is that the electron domes are bounded by surfaces of or so-called
Force-Free-Surfaces (FFS). By definition, the FFS is where the electric field
changes from accelerating in one direction to accelerating in the other
direction (the mathematical possibility that this zero could be of second order
is of no concern here). In the STATIC Polstorff model, there is no deceleration
zone because it happens to be filled with electrons out to the FFS. But in a
dynamic pulsar model the electrons from the acceleration zones beyond the FFS
are being precipitated toward the neutron star surface and to keep this process
from charging the neutron star negative, electrons are accelerated off the
surface (Fig. 8, below). Thus the system is much like having an electron beam
impinge on an isolated sphere; the sphere will charge up until either the beam
is deflected or field emission allows electrons to be emitted from the sphere.
In the latter case, the potential floats to whatever is required for the two
currents to be equal,
Figure 8. Sketch of a slightly (not illustrated) oblique rotator whose orthogonal dipole creates a wave zone at beyond which charged particles are driven from the system. The avalanching takes place beyond the natural force-free-surface, with positrons being ejected into the wave zone and the electrons being precipitated into a deceleration zone which de-energizes them while at the same time pulling electrons from the surface and ejecting them into the wave zone to close the current loop. and the incident beam is largely decelerated before striking the sphere. Of course some current must pass through the sphere and through the neutron star in the case of the pulsar. This is partially how energy is extracted from the system (along with radiation reaction from the large amplitude electromagnetic waves produced by the orthogonal component of the magnetic dipole). The fact that these two torques on the star, conduction currents through the polar caps and radiation reaction, are coupled is in itself interesting because pulsars seem mainly to be oblique rotators, neither totally aligned nor totally orthogonal but at some intermediate angle. In contrast, simple theory predicts that they should align quickly. The fact that the two are interconnected is perhaps a clue as to why neither extreme is realized. 2. SO WHAT'S NEW? Let me summarize what I regard as the essential developments that now make pulsar theory promising. A. The discovery that the GJ model was not unique and that the magnetosphere can support vacuum gaps. In initially this seems to have been regarded as something of a disaster to be overcome, myself included. In fact, it seems an essential step forward. B. The realization that acceleration in vacuum gaps can account for RADIO emission. Originally the vacuum gaps were postulated as sites for gamma-ray emission. The possibility of cascading was well-known, but the cascading was generally thought to take place following particle acceleration, not during it. Given the possibility (indeed, requirement) of cascading in vacuum gaps, one also has a possible mechanism of forming dense bunches that would radiate coherently at long wavelengths (radio frequencies). There are other possibilities too, given that both upward and downward electron currents flow, which also opens the possibility of two-stream instability as original proposed (Sturrock). C. Pulsar magnetospheres are complicated systems. A standard puzzle has been why pulsar pulses are all so different, like finger prints. If all you had was an inclined magnetic dipole and rotation, the system seemed too simple. Consequently, complexity was attributed to magnetic surface inhomogeneities (which may still play a role). But with pair-avalanching attempting to fill the pulsar magnetosphere, the system suddenly becomes rather complicated. It is no longer clear that the radiation comes from bunches moving up (as was the case when the discharging was thought to be all from the surface) but may be dominated by bunches coming down, in which case the radiation does not simply exit the system but must pass close to the neutron star and through the surrounding magnetosphere. D. The pulsar wind is very likely dominated by electron-positron pairs. This issue is an old one that has paralized wind theory for some time. Given some progress with resolving the radio emission problem, we should become more confident in the pair plasma nature of pulsar winds. E. Solution of the current closure problem The inability to remove plasma but of a single sign in the GJ model came to be known as the "current closure problem." This problem can be resolved in the inclined rotator case by the large amplitude electromagnetic waves of the orthogonal component of the magnetic dipole driving off the electron dome, required by the aligned component, which in turn acts to empty the magnetosphere, which then permits the pair avalanching, etc. 3. IMPLICATIONS The theoretical discussion outlined above leads even without further analysis to some intertesting implications. Whether these implications are "correct" depends on how well the model stands up to detailed analysis. For example, it was originally criticized on the basis that polar cap heating would be too severe and cause all pulsars to be bright x-ray objects. Given the existence of a natural decelleration region over the polar caps, this concern seems less obvious. Indeed a few pulsars have been observed to have x-ray emitting spots (or at least the data can be so interpreted), consistent with some heating which seems inevitable because some current must pass through the pulsar to extract rotational energy from it. A. There may not be "two" kinds of pulsar depending on the rotation sense. In the GJ model one sign of rotation produced electron polar caps which made a natural candidate for polar cap emission. Reversing the sign (plus insisting that the particles come from the surface) seemed much less promising because ion, even protons, make very weak synchrotron sources. If instead on has discharging over the polar caps, the direction of the discharge makes less difference particularly if the radiation is caused by avalance bunching. There might still be an interesting issue of positron transport at the polar caps since they have to move to more poleward magnetic field lines to be ejected from the star, and ions might indeed substitute for them. B. The wind from pulsars is probably an electron/positron plasma Regardless of the possible injection of ions from the polar caps, the avalanche discharge can potentially inject a large number of positrons per primary current carrier. Thus a pair plasma, to first approximation, should describe the wind (as inferred theoretically from other arguments by Kennel and Coroniti 1981). 4. SUMMARY It would apprear that we once again have a physically sensible, straightfoward theoretical model to analyze. The "loss" of the GJ model, which may not have been widely noticed, was a step forward because it made (some of) us realize the importance of non neutral plasmas. If the model suggested here is to fail, it will have to be because there is some definite flaw which will invalidate it, but that flaw will also shed more light on how pulsars work and point us toward a final model. Or maybe we are already there.