Circulating Subbeam Systems and the Physics of Pulsar Emission

Abstract

The purpose of this paper is to suggest how detailed single-pulse observations of ``slow'' radio pulsars may be utilized to construct an empirical model for their emission. It links the observational synthesis developed in a series of papers by Rankin starting in the 1980s to the more recent empirical feedback model of Wright (2003a) by regarding the entire pulsar magnetosphere as a non-steady, non-linear interactive system with a natural built-in delay. It is argued that the enhanced role of the outer gap in such a system indicates an evolutionary link to younger pulsars, in which this region is thought to be highly active, and that pulsar magnetospheres should no longer be seen as being ``driven'' by events on the neutron star's polar cap, but as having more in common with planetary magnetospheres and auroral phenomena. Comment: 15 pages, 3 figures

Full text

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arXiv:astro-ph/0306054v1 2 Jun 2003
Astronomy & Astrophysics manuscript no. strategy8
(DOI: will be inserted by hand later)
February 2, 2008
Circulating Subbeam Systems and the Physics of Pulsar Emission
Joanna M. Rankin1,3and Geoffrey A. E. Wright2
1Sterrenkundig Instituut, University of Amsterdam, Kruislaan 403, Amsterdam 1098 SJ, Netherlands e-mail:
jrankin@astro.uva.nl
2Astronomy Centre, University of Sussex, Falmer BN1 9QJ UK e-mail: gae@pact.cpes.susx.ac.uk
3On leave from: Physics Deptartment, University of Vermont, Burlington, VT 05405 USA e-mail:
Joanna.Rankin@uvm.edu
Received / Accepted
Abstract. The purpose of this paper is to suggest how detailed single-pulse observations of “slow” radio pulsars
may be utilized to construct an empirical model for their emission. It links the observational synthesis developed
in a series of papers by Rankin in the 1980’s and 90’s to the more recent empirical feedback model of Wright
(2003a) by regarding the entire pulsar magnetosphere as a non-steady, non-linear interactive system with a natural
built-in delay. It is argued that the enhanced role of the outer gap in such a system indicates an evolutionary link
to younger pulsars, in which this region is thought to be highly active, and that pulsar magnetospheres should no
longer be seen as being “driven” by events on the neutron star’s polar cap, but as having more in common with
planetary magnetospheres and auroral phenomena.
Key words. stars: pulsars: Polarisation – Radiation mechanisms: non-thermal
Introduction
A visitor to a pulsar observing session will see on the
oscillograph something quite unlike anything in the rest
of astrophysics: a never-ending dancing pattern of pulses:
sometimes bright, sometimes faint, sometimes in regular
patterns, sometimes disordered, sometimes switching off
entirely only to resurge with greater vigour. Variations
can be found on every time scale down to tiny fractions of
seconds.
Astrophysics is a field used to dealing with objects
which evolve over millions, over thousands of millions of
years, perhaps occasionally punctuated by dramatic cat-
aclysmic events, but generally affording no more than an
unvarying image through the telescope. How are we then
to deal with a phenomenon which is so alien to the com-
mon astrophysical experience?
It can be argued that the study of pulsars is more
than a study of complex physics: that it is a study of
complexity itself. Beyond the original insights, some 30
years ago now, that pulsars are rotating magnetised neu-
tron stars, emitting coherently in the radio band from a
roughly conical region above the magnetic polar caps, lit-
tle has been elicited from the welter of information gath-
ered over the decades to point us towards some fundamen-
tal understanding of the underlying mechanism by which
the pulsars emit.
Send offprint requests to: Joanna.Rankin@uvm.edu
This impasse has arisen partly because pulsars have
been treated primarily as steady-state astrophysical ob-
jects undergoing minor fluctuations which we detect in
subpulses, rather than as intrinsically non-steady, nonlin-
ear systems whose subpulses contain valuable information
about the nature of the system. Yet before any detailed
physics can be undertaken, it is essential to unravel the
embedded complexity and to discern the structure of the
underlying system. This point is well understood in many
branches of terrestrial physics where irregular time series
are commonplace. Why is it so difficult to predict the
weather? Why do animal populations dramatically rise
and fall in an apparently random manner? The point of
course is that although complexity may arise through the
operation of complex systems (as with the weather), it can
also do so through simple systems operating under simple
conditions—as in the classic population studies of Prof.
Robert May [for a review see May (1986)]. And it is es-
sential to distinguish between them, and to know which
we are dealing with.
In the case of pulsars emphasis has certainly been laid
on the former of these assumptions. Theorists have ex-
plored the properties of time-independent magnetosphere
models (often axisymmetric about the rotation axis, so
they would not even pulse!) and assumed that the ob-
served radio phenomena are complex temporal or geo-
metrical “perturbations” of some underlying equilibrium.
Furthermore, many emission models have seen pulsar

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2Rankin & Wright: Rotating Subbeams & Pulsar Emission
“events” as being driven and determined by conditions
on the polar cap surface, reflecting the traditional view of
classical dynamics that systems have starting and ending
points, that causality has only one direction.
The problem of this approach is that detailed time-
structured observations have little to say in the construc-
tion and verification of these models. Perhaps it is possible
to take an alternative approach, well started in a series of
papers by one of us and her collaborators (“Towards An
Empirical Theory of Pulsar Emission”, I–VIII; hereafter
ETI–ETVIII), to use the observations to determine the
model—to ask the pulsars themselves how they work.
To do this we will adopt the view that, although appar-
ently complex, pulsar observations at both radio frequen-
cies and in the optical, x-ray and γ-ray regions may be the
by-products of a single simple underlying system. As far as
possible special pleading or exceptional circumstances will
not be introduced in order to explain difficult results. The
thesis explored in this review is that the simple picture of
a dipole rotating alone in vacuo, when inclined at different
angles and viewed from different angles, can give rise to
the myriad of beautiful complex phenomena observed in
pulsars at many wavelengths over the past decades. This
thesis will be put to the test.
Geometry is Pivotal
Let us assume that the only permanent features of any
pulsar are its underlying magnetic geometry and our par-
ticular view of it. Knowledge of these is the prerequisite to
establishing the degree of complexity (or simplicity) the
underlying flow of the emitting particles needs to possess
to account for the highly non-steady observations.
So what results, developed over the many years of pul-
sar research, can confidently be regarded as indicators of
a pulsar’s magnetic field geometry and thus give a start-
ing point in our quest? Below are listed the three most
influential ideas, all of which are closely associated with
a pulsar’s most fundamental observational property: its
remarkably stable and individual integrated profile.
– The most fundamental result—as fundamental today
as it was over 30 years ago for Radhakrishnan & Cook
(1969; hereafter R&C) and Komesaroff (1970)—is the
conal, single-vector-model (SVM) geometry implicit
in many profile forms and position-angle traverses.
Without question this is the most successful theoret-
ical idea yet articulated as it provides a fundamental
standpoint for explaining geometric aspects of the ob-
servations. Of course, it is probably a simplification or
abstraction of the actual physical environment. And
we must question whether its underlying assumptions
are entirely correct. But (as with the dipolar assump-
tion below) the best means of assessing its correctness
is to assume it true and then study any resulting dis-
crepancies.
– Second, the extension and development of the forego-
ing models (also Backer 1976) into a profile classifi-
cation system—the starting point of the “Empirical
Theory” noted above—and their subsequent evolution
into several broadly compatible means of estimating
the magnetic inclination and sightline impact angles α
and β (Lyne & Manchester 1988; ETVIa,b). This in
turn has led to the provisional conclusion that the the
integrated emission from most pulsars stems from one
or all of three different emission beams, the core and
the inner/outer cones, each roughly centered on the
magnetic axis.
– Third, it has emerged that pulsar emission beams are
nearly circular! While various workers have cogently
explored whether they might be latitudinally or lon-
gitudinally extended, no strong evidence has emerged
to the effect that they are non-circular (Biggs 1990;
McKinnon 1993). Indeed, probably they are somewhat
so, but their departures from circularity are evidently
small and less systematic than mere axial extension
(Arendt & Eilek 2003; Eilek & Arendt 2003).
On the basis of the first two points it may provision-
ally be concluded that pulsar emission appears to reflect
a magnetic field configuration which is nearly dipolar in
the emission region. While many of us have at times ap-
pealed to “non-dipolar effects” to explain sundry myster-
ies, no single instance yet exists where this explanation
can be clearly demonstrated. Indeed, although theory of
neutron stars and observations of them in other contexts
(e.g., x-ray binaries) suggest that pulsar surface magnetic
fields are probably not entirely dipolar—particularly in
the case of millisecond pulsars—our very failure to iden-
tify concrete instances of non-dipolar effects in ordinary
pulsars argues that the fields must be nearly dipolar at
the emission-region heights that the observations reflect.
Furthermore, clear evidence for non-dipolarity will prob-
ably come only by pushing the dipolar assumption so far
that counterexamples emerge. Many theorists have plau-
sibly argued that the magnetic field in the outer magne-
tosphere will be distorted by current flows and relativistic
effects (e.g., Michel 1991; Beskin et al. 1993; Mestel 1999;
Shibata 1995). But one must be beware of overlooking
more fundamental concepts by using multipole structures
close to the surface to explain difficult observations—i.e.,
one may fall into the trap of using complexity to explain
complexity.
Support for the third point, also consistent with the
dipolehypothesis, follows from the identification of cir-
culating subbeams systems in B0943+10 (Deshpande &
Rankin 1999) and B0809+74 (van Leeuwen et al. 2002):
it is then this subbeam circulation which produces the av-
erage conal form, and thus makes them roughly circular
in shape—i.e., symmetrical about the magnetic axis. The
subbeam circulation (identified observationally as sub-
pulse “drift”) may be provisionally regarded as a general
characteristic of conal beams—but the subbeams need not
be regularly spaced, nor steady over time; they can equally
well be formed in a sporadic or chaotic manner while still
retaining a circular symmetry about the magnetic axis.

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Rankin & Wright: Rotating Subbeams & Pulsar Emission3
For these reasons, the form of pulsar beams can best be
explained by assuming circularity and then assessing any
evidence for departures.
We can therefore adopt three assumptions, the SVM,
dipolarity and conal beam circularity, to jointly provide a
standpoint for constructing simple geometrical models for
most pulsars (e.g., Deshpande & Rankin 2001; hereafter
DR01). To these we can add three basic electrodynamic
concepts, also geometric in nature, which were established
in the early days of pulsar research. First, a light cylin-
der, at which corotating particles would attain the speed
of light. Second, a corotating zone whose bounding field
line would be the last to close within the light cylinder;
emission would thus be confined to the open field lines in a
region close to the polar cap and surrounding the magnetic
axis. Third, a surface on which the charge density would
be formally zero in a quasi-steady state, and which would
therefore be capable of forming an “outer gap” accelerator
(Holloway 1975). It is in this last region that γ- and x-ray
pulses are thought by many (Cheng et al. 1986; Romani
1996; Romani & Yadigaroglu 1995; Hirotani & Shibata
1999; Cheng et al. 2000) to be formed in young pulsars,
and it is not unreasonable to believe that it may continue
to play an important role even after its high-energy phase
is past (Chen & Ruderman 1993; Wright 2003a).
These are the geometric considerations which play a
central role in our approach, but attempts at “ab initio”
theorising will be eschewed: three decades of experience
and history have shown that general pulsar theories—
physical theories of pulsars attempting to deduce the be-
haviour of real pulsars from first principles—are inca-
pable of yielding significant, specific, falsifiable expecta-
tions about the observed emission of an actual individual
pulsar. Future more successful theories must be able to
do so, and simple semi-empirical models of the emission
geometry along the lines summarised here provide the es-
sential point of connection between our natural observa-
tions and the ramifications of physical theories. However
in this article, we stress again, the reader will find geom-
etry put not only to its traditional use of disentangling
the observer’s perspective of pulsar “events”, but given a
prominent role in determining their nature.
The Pulsar Family
Although the main focus of this article will be on “slow”
radio pulsars, it is important to stress that their properties
are likely to be closely related both to those of faster,
younger pulsars such as the Crab and Vela, which also
emit in the high-energy bands, and to the family of older
but rapidly spinning millisecond pulsars.
Young Pulsars
Through their capacity to produce optical, x-ray and γ-
ray emission, young pulsars have often been seen as a class
apart—not least because they are observed by a distinctly
different community of astronomers! Yet this is a danger-
ous view if we are to regard pulsars as exhibiting a con-
tinuum of behaviour which evolves as a pulsar ages. It
has seemed likely that the high-energy photons of young
pulsars are produced by a different mechanism—and prob-
ably in a different region of the magnetosphere—from the
coherent radio emission. It is then easy to believe that
those who study radio pulsars have little to learn from
the high-energy studies, and vice versa.
The stress we are laying on the role of geometric fea-
tures in determining phenomena should warn us against
this view. Indeed, it is largely through geometric argu-
ments that the outer gap has been identified by some
(Cheng et al. 1986) as a possible source of γ rays: and
the outer gap is directly linked by magnetic field lines to
what is certainly the site of the radio emission in slower
pulsars. Does outer-gap pair creation cease as soon as the
high-energy emission becomes undetectable? It is possible
to construct a viable emission model in which this pro-
cess plays a critical role (Wright 2003a), and if verified,
could provide a natural link between radio pulsars and
their high-energy siblings.
Millisecond Pulsars
These pulsars, thought to be older neutron stars which
have been “spun-up” through a history of accretion, have
relatively weak magnetic fields and often unusual pro-
files which do not conform to the pattern of slow pulsars
(Kramer et al. 1998, 1999). There are good theoretical ar-
guments for believing that their surface magnetic fields
are highly distorted (e.g., Ruderman et al. 1998), which
may cause profile distortion. However, virtually nothing
is known of their single pulse behaviour. For this reason
they lie outside much of the analysis here, but again we
would caution against rushing to multipole geometries as
quick explanations. At any large distance from the star the
dipole component will dominate, and, as we will strongly
suggest, dipole geometries are capable of creating great
intrinsic complexity.
Subbeam Circulation and Pulsar Phenomenology
Pulsar Profiles as Attractors
It is no coincidence that the three fundaments listed in
the opening section are all deductions based on the prop-
erties of integrated profiles. A pulsar’s profile is its indeli-
ble, individual and stable characteristic. This extraordi-
nary property has been recognized since the early days of
pulsar research. However, the invariance of profiles is prob-
ably responsible for seducing many theorists into taking
it as evidence of some underlying stability in the emis-
sion system, such that the ever changing behaviour of the
individual pulses can conveniently be ignored.
Yet they are nothing of the sort. Studies of non-linear
dynamical systems have repeatedly revealed the presence
of strange attractors, features which confine the highly
time-dependent variables of the system to a specific re-

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4Rankin & Wright: Rotating Subbeams & Pulsar Emission
gion of variable space, but in no way indicate conver-
gence to a steady state. A pulsar’s profile represents a
two-dimensional cross-section (Poincar´ e section) created
by our sightline intersecting an otherwise unseen three-
dimensional attractor. Nothing in the pulsar emits radi-
ation in the form of a profile. Profiles contain valuable
information about the quasi-chaotic system, but they are
not the system itself.
A powerful result of the 1980’s was the claim that pul-
sars have attractors in the form of nested cones (ETI,
ETVIa), and even that cones have approximately con-
sistent radii from pulsar to pulsar (relative to the size
of the polar cap) (ETVIa,b). Over the years there have
been associated claims that the true attractor structures
are less (Lyne & Manchester 1988) or more (Gangadhara
& Gupta 2001, 2003) ordered, but nonetheless the im-
plications of these findings remain profound. It has long
been assumed that pulsar emission emanated from parti-
cles closely bound to the magnetic field lines, so that the
emission components followed the contours of that field.
The consequence of any observations which suggest con-
sistent profile structure from pulsar to pulsar (such as the
“Empirical Theory”) is then that certain field lines are
preferentially selected by the particles—and very nearly
the same field lines in each pulsar. Explanations for this
in terms of the classic Ruderman & Sutherland (1975;
hereafter R&S) model then have to appeal to multipole
features in the surface magnetic field (Gil et al. 2002a,b;
Asseo & Khechinashvili 2002), yet this begs the obvious
question as to why each pulsar would have similar multi-
poles. Alternatively, it has been suggested that the cones
are formed by multiple refractions within the magneto-
sphere (e.g., Petrova 2000). But then, precisely because
profiles are only attractors and not the actual emission,
we would expect the subpulses in the inner and the outer
cones to have similar subpulse behaviour—and this seems
to be far from the case.
However, if we abandon the unwritten assumption
of these models that pulsar magnetospheres are systems
driven from the polar cap—that the tiny tail wags the sub-
stantial dog—then we are forced to postulate that some-
how the outer magnetosphere selects the critical fieldlines.
The natural choice for these fieldlines, on both geomet-
ric and physical grounds, would be the cones which con-
nect the outer gap’s upper and lower extrema to the polar
cap (as exemplified in the model of Fitzpatrick & Mestel
1988a,b). There is anyway strong evidence that the outer
gap plays a critical role in the production of γ rays in
young pulsars ( Romani & Yadigaroglu1995), and it would
be natural that it might continue to play an important, if
not directly detectable, role in slower pulsars. The open-
ing angles of these critical field lines seem, on reasonable
assumptions about the emission heights, to have the right
proportions to account for the attractor cones of ET (Gil
et al. 1993; Wright 2003a), and at these heights the mag-
netic field is almost purely dipolar. It is not impossible
that the precise fieldlines preferred in any given pulsar
may be at some intermediate value, especially in more
Fig.1. A carousel depicting the structure and themes of
this paper: the individual topics are linked to the under-
lying principles via their geometric interpretations
inclined pulsars—and may vary in time, resulting in mul-
ticonal attractors.
We are consequently led to understand that it is the
downward-moving particles which determine the emission
site. These particles must be accelerated over the vast
distances from the outer gap towards the pole (Mestel
1985; Beskin et al. 1993), and particles of opposite sign
must be accelerated back to the gap. This concept thus
shares many features with the free acceleration models
of Arons & Scharlemann (1979), Mestel (1999), Mestel
& Shibata (1994) and Jessner et al. (2001), although the
scale of operation is greater than envisaged by these au-
thors. More recently, by invoking inverse-Compton scat-
tering as the principle emission mechanism for producing
pairs in older pulsars, promising models have begun to ap-
pear (Hibschmann & Arons 2001a,b; Harding et al. 2002;
Harding & Muslimov 2002a,b) in which the acceleration
zone is extended further up into the magnetosphere, and
in which pair creation may fail to quench the local elec-
tric field in slow pulsars, thus leaving a residual potential
difference extending to “infinity”—a feature which could
naturally correspond to the magnetosphere-wide scale re-
quirements of the empirical model. However, in all these
models the implied so-called “return flow” should in the
present view be seen as the primary flow, and none have
explored the possibility of azimuthally-dependent emis-
sion implied by both observations and the feedback system
of Wright (2003a) (see Figure 2).
The new model may therefore theoretically reproduce
the system attractors—the double cone. But to develop
it further on the empirical basis we have promised above,

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Rankin & Wright: Rotating Subbeams & Pulsar Emission5
we must focus our attention on the pulse-sequence be-
haviours, and deduce the model’s properties from them.
The behaviours can be conveniently discussed under four
headings which summarize four basic emission phenom-
ena: “drift”, core emission, mode-changing, and nulling.
These headings are largely suggested by the manner of
their detection and observation. However, it is essential to
bear in mind that some or all are often present in a single
pulsar (e.g., B0031–07, B1237+25), and may well spring
from different aspects of the same physical mechanism. A
fifth heading, “emission cycles”, is therefore added, un-
der which we discuss the apparent “rules” or “memories”
which may link these phenomena. The principle headings
of our discussion are gathered together graphically in the
carousel of Figure 1.
“Drift”/Non-“drift”
Subpulse “drift” is a crucial clue towards solving the pul-
sar puzzle, as it exhibits the stunningly beautiful capac-
ity for order in pulsar radio emission. It is a feature
found only in conal regions of the profile—and indeed only
then when our sightline passes obliquely along the outer
edge of the emission cone (thus producing a conal sin-
gle, or Sd profile). And this drift can range from being
gradual—with subpulses moving slowly across the pulse
window over up to 20 rotation periods—to being rapid—
presenting an on-off effect to the observer. Its intermit-
tent presence in the emission of predominantly “slow”
pulsars is powerful evidence of the unpredictable regu-
larity characterisitic of quasi-chaotic systems. The emis-
sion of some pulsars varies systematically, although not
periodically or even predictably, between discreet drift-
ing patterns (e.g., B0031–07, B1944+17, or B2319+60),
but many/most stars usually exhibit much less order in
their pulse sequences (PSs). No pulsar is known which
permanently emits with one single drifting pattern. On
the other hand, few pulsars have conal emission which is
fully chaotic. Most at least occasionally exhibit sequences
which, however brief, are more or less orderly.
It is possible that higher orders of regularity are
present, even in apparently chaotic emission, which defy
detection by current methods. It may be that we are lim-
ited by current analytical tools, designed to identify spe-
cific correlations rather than to measure the underlying
complexity. Power spectra and cross-correlations pick up
strong periodicities at specific phases of the pulse window
and are powerful tools when the emission is highly regu-
lar. But how, for example, could a systematically decaying
or oscillating drift rate be detected? Near-chaotic systems
can exhibit great subtlety in their behaviour.
How do the differing geometrical circumstances found
within the pulsar population produce the immense va-
riety of patterns—both in the emission of a single star
and among those with ostensibly similar characteristics?
It is suspected that slow systematic drift over many pe-
riods may be a characteristic of pulsars with small mag-
netic inclination angles (well known in this category are
B0809+74, B0031–07 and B0818–13—all thought to be
aligned within about 15◦), a result which would suggest
that the entire magnetosphere—and not just conditions
near the surface—plays a role in fixing the subpulse be-
haviour. However, it is no less important to understand
an unusual Sd pulsar with no drift, such as B0628–28,
as it is to understand the regularities of B0943+10 or
B0809+74, and to account for the more irregular patterns
found in those pulsars with larger magnetic inclinations.
Also a puzzle are the properties of the conal doubles (type
D) stars, where our sightline cuts the emission cone more
centrally (e.g., B0525+21 and B1133+16); here some sub-
pulse regularity is observed but apparently far less than
in their close kin, the Sdstars.
Nonetheless, from both an observational and theoreti-
cal standpoint the natural starting point of any study of
“drift” is to examine those pulsars with the most regu-
larly behaved drifting subpulses, and by far the best and
brightest known exemplars are B0943+10 and B0809+74.
Observations of these have given us the telling image of
a circular “carousel” of emitting subbeams (Deshpande &
Rankin 1999; DR01). B0943+10 in particular, when emit-
ting in its highly regular “B” mode, exhibits precisely 20
subbeams which circulate around the magnetic axis about
every 37 rotation periods (or about 41 s). This star has
provided us our first opportunity to count the number
of subbeams and to confirm the geometric aspects of the
R&S model. Yet it is now known that even this “B” mode
adopts slightly varying circulation speeds on largely un-
predictable timespans (Rankin et al. 2003). And the well-
known pulsar B0809+74, after being thought for decades
to have a near-clockwork regularity in its drifting pattern,
has recently been found to drift on occasions at a consis-
tently slower rate (van Leeuwen et al. 2002).
The task of accounting for drifting subpulses has only
made limited progress over the years since the publi-
cation of the 1975 R&S polar gap model. Recently Gil
and coworkers have described multipole models in which
“sparks” on the polar cap can be made to adequately
mimic the observed drift of certain pulsars (e.g., Gil &
Sendyk 2000), but this inevitably involves some arbitrari-
ness in the choice of the magnetic field structure. However,
it is possible to produce drifting subbeams naturally, and
without invoking multipoles, through the operations of
the feedback model sketched in the previous subsection
(Wright 2003a): one can suppose the formation of pair-
creation “nodes” in regions both around the polar cap
close to the surface, and in the outer gap, which “fire”
particles at each other and thus create a self-sustaining
system. The nodes will appear to precess in tandem both
about the magnetic axis and around the outer gap. This
system, although still owing much in its physical processes
to the R&S model (i.e., pair creation and the E×B par-
ticle drift), depends on interactions between widely sepa-
rated regions of the magnetosphere. Thus a natural time
delay is built into the system, and hence leads to the pos-
sibility of chaotic or quasi-chaotic behaviour. The system