In this review my aim is to mention very briefly the present state of knowledge over a wide area in order to provide some focus for the wealth of information that this symposium will bring forth. During 13 years a great deal has been discovered, but the jigsaw puzzle of the pulsar model is still in pieces. Some of these have intriguing shapes and seem to fit together, but they do not yet indicate what the picture will be.

Motivation for the study of the pulsar magnetosphere is multifold. First of all, energy-wise the magnetosphere is the dominant problem, for the energy emitted in the radio pulses is at most one per-cent of the energy loss inferred from the secular increase in the rotation period of the central neutron star. With the possible exception of gamma-ray emission in a few cases, the pulsing observed in all wavelengths can be regarded as a diagnostic of a more basic phenomenon. A successful model should in fact tell us the total rate of energy loss from the star and its distribution between a low-frequency wave, a possible wind, and emission in the radio, optical, X-ray and gamma-ray bands. As a necessary preliminary, the model should predict the strength of the currents leaving and returning to the star, the location of relativistic particles, and the conditions for transition to a quantum magnetosphere via pair production (Sturrock 1971). But even if magnetospheric theory did not offer such hopes of ultimate links with observation, the problem is so well-defined that it is a challenge to the theorist: it is just intolerable that we should not know how a rotating magnetized neutron star comes to terms with its environment.

A central question in magnetospheric theory of pulsars is: how does a rapidly rotating magnetized neutron star manage to short-circuit its huge unipolar induction voltage? This problem is best studied in an idealized geometry like the parallel rotator. The theory must at least include basic plasma physics with relativistic particle inertia. Even the simplest model will require a global self-consistent solution of both Maxwell's equations and relativistic MHD-equations of motion. Assuming time independence, axial symmetry and charge separation, we reduce this problem to three coupled, quasilinear second order partial differential equations for three dimensionless scalar quantities Φ, f and Γ (see Schmalz et al., 1980): Here x and z are cylindrical coordinates measured in units of the light cylinder radius. Φ is the stream function which is constant on stream lines, f = x · vφ/C is connected with the angular momentum (vφ = toroidal velocity), and Γ = 2∊γ = 2∊/(1 - v2/c2)1/2 where ∊ = mc2/(eBoa2ω) is a dimensionless parameter. Taking electrons (mass m, charge e) and typical neutron star parameters (radius a = 10 km, angular velocity ω = 30/s, polar field strength Bo = 108 T), we have ∊ ≈ −10−12.

The relatively slow progress in both analytical and numerical work on the structure of the pulsar magnetosphere calls for a reformulation of the equations governing the magnetosphere. Since the set of equations for the pulsar magnetosphere are nonlinear anyway, one should not hesitate to use nonlinear representations for the field intensities and the current from the very beginning. Those nonlinear representations can be chosen to fulfill certain equations automatically and are therefore more adapted to the problem. The aligned rotator was formulated to some extent in this way by Schmalz et al. (1979). We formulate here the general problem and use for this the calculus of differential forms and some theorems concerning the normal form of forms (= representation of a form with a minimal number of scalar functions and their gradients). In the language of forms Maxwell's equations read where F is the field intensity 2-form and j the current 3-form. The first equation states that F can be derived from a 1-form A: F = dA, the usual vector potential. A consequence of the second equation is dj = 0, the continuity equation for the current. Now since j is a simple closed 3-form, it can be represented by three scalar functions ⊘,χ,Ψ in the following way which obviously satisfies dj = 0. This is the normal form of the 3-form with the properties mentioned above.

The electric fields in the polar cap region of the magnetosphere of a pairless orthogonal rotator are being calculated using a modified Poisson's equation which takes explicit account of the effect of the displacement current on the potential Φ. This is the first step in a program to determine how return currents are formed in these models and the resulting global current flow patterns. The potential is being calculated from 0.1 RL to 3 RL, where RL is the light cylinder radius, using known results for ⊘ at 0.1 RL with an inner last closed magnetic field line corresponding to that of a point dipole. Considerable difficulty is being encountered because of the mixed nature of the equation and the irregular boundaries. Possible methods of overcoming these difficulties are indicated.

This paper briefly discusses the salient features of quasi-steady electrodynamics; that is to say it considers certain general properties of Maxwell's equations and their solutions when these equations are subjected to the constraint where ω is a constant angular frequency, ⊘ and t are the azimuthal angle and time, and the operators are applied only to quantities which are scalars or the cylindrical components of a vector (Mestel 1971).

The possible existence of electron-positron plasmas in the magnetosphere of neutron stars gives rise to the question in which way the e+e− annihilation process is modified by the strong magnetic fields prevailing there. On the basis of a correct quantum mechanical treatment of the two-body problem in an external magnetic field we have calculated lifetimes of the ground state as well as of excited states of bound e+e− pairs (positronium). We find that the 1γ decay remains strictly forbidden, whereas the 2γ process is strongly enhanced with respect to the field-free case.

We argue that the aligned rotator model originally introduced by Goldreich and Julian possesses “static” solutions wherein particles of both sign are trapped within the magnetosphere. Simple models of such distributions are given. For such models to have dynamic properties (constant particle emission, relativistic stellar wind, etc.) one must suppose that the equatorial particles are transported away by (e.g.) flux tube interchange, rather than by flow along field lines as originally proposed.

An electric field near a rotating magnetized conducting sphere is found in the cases when the sphere (a) rotates in vacuum and (b) is surrounded by a magnetosphere. The accretion of the interstellar neutral hydrogen onto the cold neutron star is discussed.

We review efforts to construct a self-consistent model of pulsar magnetospheres that links the particle source near the pulsar to the outflowing relativistic wind and couples the wind to the surrounding nebula. Pair production source models (Sturrock, 1971; Ruderman and Sutherland, 1975; Arons, 1979) produce outflowing positronic plasmas which are too dense to support a plasma wave of relativistic amplitude (Kennel et al., 1973). Therefore an appropriate description seems to be a relativistic magnetohydrodynamic (MHD) wind. Radiation from a relativistic wind will be collimated forward and can only be observed along the line of sight to the wind source. This makes relativistic winds essentially unobservable and provides an almost invisible method of energy transport. Photons can be emitted in all directions and the existence of the wind inferred only after the wind has been decelerated by a confining shock (causing the flow to become subsonic and non-relativistic).

Large amplitude waves (hereafter l.a.w.) have been studied mainly in connection with pulsars. A rotating neutron star, with an intense non-aligned magnetic dipole field, surrounded by a vacuum, radiates beyond the light cylinder distance, large amplitude electromagnetic vacuum waves of very low frequency (Ostriker and Gunn, 1969). In such a rotating configuration electromagnetic effects completely dominate and imply the existence of a relativistic plasma in the pulsar magnetosphere (Goldreich and Julian, 1969). Because the oblique vacuum model predicts a value of 3 for the braking index of the pulsar whereas the observed or computed values are different, a realistic model has to include both the relativistic plasma outflow and the electromagnetic wave emission. The vacuum wave model has been changed to include self-consistent plasma effects (Asséo et al., 1975; Asséo et al., 1978) in plane geometry and inhomogeneities linked to spherical geometry (Asséo et al., 1981). This results in very restrictive conditions for the possibility of propagation of the l.a.w.

It is argued that pulsar windzones are pulsed γ-ray emitters, and X-ray emitters. The wind consists of a strong subluminal wave, in approximate equipartition with a relativistic electron-positron plasma; it also contains a weak frozen-in magnetic flux. Trailing filaments may be responsible for large-angle particle scattering, giving rise to one-sided X-ray appearance.

A general picture of pulsar magnetospheres is outlined which suggests polar current flow should proceed within a magnetic flux tube with conducting boundaries. The dynamics of this current flow is considered, including the effect of formation of pair plasma on the potential. A slot gap surrounding the stream of pair plasma is described, and the electric fields and J·E work done in the gap are calculated. The inapplicability of curvature emission to the formation of γ-ray pulses is discussed. The dynamics of pair plasma flow is described, including electrostatic trapping of low energy positrons, polar cap bombardment and heating, and the emission of soft X-rays from the cap. The relation of this current flow and polar cap heating to the formation of fluctuations on the radio subpulse time scale is also described. The importance of the boundary layer between the pair plasma and the slot gap is illustrated by calculations of the boundary layer structure and microscopic instability, which show that this region may be the source of pulsar emission.

I will describe a number of quantitative results and qualitative aspects of work in progress on the “slot gap” model for the emission regions of a pulsar, proposed by Arons and Scharlemann (1979, hereafter AS) and by Arons (1979b, hereafter I). The results and ideas concern the dynamics and emission physics of relativistic plasma and current flow along the polar field lines of an isolated, rotating magnetized neutron star with the axis of its dipole moment oblique to the rotation axis - a pulsar. The quantitative modelling is confined to the physics within a polar flux tube and is therefore “local”. Full detail will be given in papers now being written. However, development of this type of model requires the use of boundary conditions which follow from a qualitative view of the whole magnetosphere. Because this view involves some qualitative ideas not widely discussed in the literature, I describe it briefly.

Positively charged particle emission from pulsar polar caps evolves through many stages as the cap temperature cools. In “infant” pulsars, stripped Fe ions interact with secondary e± from a 1012 V discharge maintained just above the cap to give coherent radio emission at very high frequencies in a very broad beam. In “adolescent” pulsars such as the Crab and Vela, lower energy Fe ions interact with streams of protons produced in the surface by photonuclear reactions to give lower frequency highly linearly polarized radiation. There is no cap e± discharge. In mature “adult” pulsars proton and/or Fe ion streams from discharge heated patches on the polar cap interact with a relativistic e± plasma to give coherent radio subpulse beams which can drift, have strong submillisecond modulations, and orthogonal polarization mode switching. Young pulsars, and some older ones, can support additional more energetic e± discharges on some open field lines in the outer magnetosphere. These all give strong double beams of GeV γ-rays and weaker γ-ray beams above ~ 1012 eV. With Crab pulsar parameters an e± plasma associated with such a discharge also can give optical and X-ray double beams. If illuminated by the normal pulsar beam (precursor) coherent inverse Compton scattering also contributes a double beam of harder radio frequencies.

Possible observational consequences are outlined for pulsar models with positive ion outflow at the polar caps together with e+-e− pair production discharge there. A characteristic thermal x-ray luminosity is maintained by discharge heating in regions of positive current outflow. A decrease in polar cap thermal x-ray emission may occur during radio nulls. Two mechanisms are identified which can yield temporal modulation of the outflowing ion and e+-e− plasmas, and which may lead to modulation of coherent radio emission on observed microstructure timescales. These are: (1) polar cap temperature oscillations which occur preferentially in pulsars of low surface magnetic field, and (2) the tendency of sparks to migrate toward the convex side of the magnetic field lines.

Triggering of the spark discharges in the Ruderman and Sutherland model by background gamma rays is shown to be effective only within a narrow range of neutron star magnetic fields centred on 2.5 × 1012 gauss. This calculated range of field strengths is in good agreement with ‘observed’ values, suggesting that (a) such triggering is operative, and (b) that neutron stars with much stronger fields do not function as pulsars.

In the rotating “lighthouse” interpretation of pulsars, the average radio pulse profile is generally assumed to be a section through the average beam shape radiated by a spinning neutron star. The radio emission received in any one particular period usually differs markedly from the average profile. Such variations have been classified as subpulses (~ 3 ms) and micropulses (~ .3 ms); see Cordes (1979) for examples. However, there is not general agreement as to whether these variations are caused by the rotation of a steady narrow beam or by temporal variations of a wider beam. We first discuss these questions independently of the emitting particle location, and then apply them to a particular model.

Galactic pulsars have been discovered by their radio emission. The emission has the common feature of being periodic, polarized, weakly frequency dependent (in the frequency range 100 MHz 10 GHz). Their energy is large for radio emission but still negligible as compared with the observed emission in the γ-ray range. They are also characterized by a very high temperature of brightness (1027–1031 Kelvin) which implies a coherent emission mechanism (this temperature is much larger than the possible particle energy). The only radiation mechanism which may produce such emission seems to be some kind of relativistic plasma instability. Unfortunately, available plasma models for pulsar magnetospheres are up to now too preliminary to make any definite statement. Nevertheless we will adopt the most popular scenario which relates the radio emission to the field aligned currents (relativistic electrons, positrons, ions) produced in the vicinity of the neutron star. These currents may result from a spark gap mechanism with pair creation (as proposed by Ruderman and Sutherland (1975) and others) or implied by the star's neutrality (as a return current system).