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The shape of pulsar radio beams

Using all available multi-component radio pulse profiles for pulsars with medium to long periods and good polarisation data, we have constructed a two-dimensional image of the mean radio beamshape. This shows a peak near the centre of the beam but is otherwise relatively uniform with only mild enhancements in a few regions. This result supports the patchy-beam model for emission beams in which the mean beam shape represents the properties of the emission mechanism and observed pulse components result from emission sources distributed randomly across the beam.


INTRODUCTION
Pulsars are generally believed to be rotating neutron stars in which the observed pulses result from one or more emission beams sweeping across the Earth as the star rotates. Observations of radio polarisation (Radhakrishnan & Cooke 1969) led to the magnetic-pole model in which the emission beam was centred on the magnetic axis of a predominantly dipole magnetic field. On the assumption that the emission is directed radially, the observed pulse profile reflects the variations in emission intensity along a line of constant rotational latitude. Although more than 1000 pulsars are now known (Taylor, Manchester & Lyne 1993;Lyne et al. 2000) the shape of pulsar radio beams remains controversial.
There are two main areas of uncertainty. One concerns the outline shape of the radio beam. Originally assumed to be circular, some investigations (e.g. Narayan & Vivekanand 1983) argued for an elliptical beam extend in the latitude direction with a large axial ratio. In contrast, Biggs (1990) suggested that the beam was slightly compressed in the latitudinal direction. Most other investigations (e.g. Lyne & Manchester 1988;Björnsson 1998;Gil & Han 1996) have concluded that the emission beam is essentially circular, and we will assume this in the present investigation.
The second area of uncertainty concerns the form of the beam pattern. Early observations showed that there are two or more pulse components in many pulsars (e.g. Lyne, Smith & Graham 1971;Manchester 1971). Ruderman & Sutherland (1975) presented a detailed model for pulsar radio emission in which the beam had the shape of a hollow cone, more intense around the periphery, corresponding to the last open field lines emanating from the polar cap region. The radio emission was attributed to curvature radiation by positrons moving these field lines. Such radiation is linearly polarised in the plane of curvature of the magnetic field and hence the model naturally explained the smooth sweep of polarisation position angle seen in many pulsars, as well as the common occurrence of double-peaked pulse profiles. Backer (1976) first discussed the more-or-less central pulse component seen in many pulsars and suggested that it resulted from an axial beam. Rankin (1983) made the distinction between this central or 'core' emission and the outer or 'conal' emission, and showed that these two components of the profile had rather different properties. At least in short-period pulsars, core emission has a steeper spectrum than conal emission (Rankin 1983;Lyne & Manchester 1988) and the fluctuation properties of the two types of emission are very different (Rankin 1986). This model was subsequently extended to have two or more coaxial conal emission zones, either to account for multiple-component profiles (Gil & Krawczyk 1997;Qiao & Lin 1998) or for the appearance of components at different apparent radii from the conal axis (Rankin 1993a;Kramer et al. 1994;Mitra & Deshpande 1999). These latter analyses depend on a knowledge of the angle between the beam and rotation axes (α), usually computed from the observed width of a 'core' component. Unfortunately, the determination of this angle is very model dependent and this results in large uncertainties in the derived radii.
Based on a large sample of pulse shape and polarisation data, Lyne & Manchester (1988) found that pulse components in multiple-component profiles were not symmetrically located within the beam boundary. Furthermore, they were generally of very different intensities and, in some cases, missing altogether. Their results were consistent with components being randomly located within the beam boundary, leading to the 'patchy-beam' model. Gould (1994) and  Gould, Lyne & Smith (2000) confirmed these conclusions with an even larger data set. Manchester (1995) suggested an interpretation of these results in which the observed pulse profile is the product of a 'window function', which is a function of pulse period and radio frequency but common to all pulsars, and a 'source function' which determines the strength of the emission at a given point within the beam and is different for every pulsar. For example, the source function may be determined by the density or energy of the plasma beams along different field lines, whereas the window function is determined by the properties of the emission mechanism.
In this paper, we determine the two-dimensional shape of the window function at frequencies around 1 GHz by averaging the observed pulse profiles of multi-component pulsars having reliable polarisation data. This analysis depends only on the observed width of the pulse profile and the 'impact parameter', that is the minimum angle between the line of sight and the beam axis, usually given the symbol β. In contrast to the beam inclination angle α, the impact parameter is normally well determined from the maximum rate of change of position angle at the profile centre (Lyne & Manchester 1988). We assume that emission is seen across the entire polar cap and choose only pulsars with profiles where this appears to be the case. In Section 2 we describe the data set and our methods of determining the two-dimensional beam pattern. Results are presented and discussed in Section 3 and Section 4 summarises our conclusions.

THE DATA SET AND ANALYSIS METHOD
In order to reliably determine the two-dimensional beam shape we choose pulsars which have pulse profiles with two or more pulse components, good signal/noise ratio and good quality polarisation data.
For most pulsars with two or more pulse components, we see emission from right across the polar cap. The polarisation information allows a measurement of the normalised impact angle, βn, that is, the impact parameter β expressed as a fraction of the beam radius. This parameter is computed assuming α = 90 • , but it is not very sensitive to the actual value of α (Lyne & Manchester 1988). The polarisation data also allow a check on the assumption that we are seeing emission right to both edges of the beam. Only those pulsars for which the maximum rate of change of position angle is close to the pulse centre were included in the sample. Single component pulsars are not included in the sample as it is often difficult to reliably determine if they are predominantly core emission (low βn) or from the outer edge of the beam (high βn).
Millisecond pulsars were also excluded from the sample as they generally have very wide and complex profiles and polarisation variations which do not fit the simple rotatingvector model (e.g. Navarro et al. 1997, Stairs et al. 1999. It is therefore difficult or impossible to obtain reliable values of βn for these pulsars. Very short-period pulsars also commonly have wide or interpulse profiles and are excluded from the sample for similar reasons.
Pulse profiles in a given pulsar usually evolve with frequency, with some components becoming stronger and other becoming weaker or even disappearing at higher or lower frequencies. At low frequencies, the core component tends to dominate the profile, whereas at high frequencies the conal emission is generally dominant. To ensure relative uniformity in the data set, we restricted our attention to profiles at frequencies between 600 and 1600 MHz. The results we obtain therefore represent pulsar emission at frequencies around 1 GHz.
We checked the profiles available on the pulsar profile database ⋆ of European Pulsar Network maintained at Max-Planck-Institut für Radioastronomie (Lorimer et al. 1998), which contains pulsar profiles from more than 50 papers, including the large datasets from Gould & Lyne (1998), Manchester, Han & Qiao (1998) and Hoensbroech & Xilouris (1997. Normalised impact parameters were obtained from Lyne & Manchester (1988), Rankin (1993), Gould (1994) and Manchester et al. (1998). The final data set of 87 pulsars which satisfy the above criteria is listed in Table 1.
To determine the average beamshape, we first defined the duration of each pulse profile by taking the points at ⋆ See http://www.mpifr-bonn.mpg.de/div/pulsar/data/  which the signal rose above 3σ, where σ is the rms deviation of the off-pulse noise. Next, the mean intensity of the observed emission within the pulse was normalised to unity. The normalised impact parameter, βn, gives the offset of the locus of line of sight across the polar cap in the latitude direction as a fraction of the beam radius. We assumed a circular beam of unity radius and a straight trajectory of the emission point across the polar cap. The beamshape is assumed to be symmetric about the equator, i.e., the sign of βn is ignored.
The profile intensity was interpolated on to an x − y array representing a semi-circular beam of unity radius along a line at y = βn between x values of ± 1 − β 2 n using a fivepoint polynomial interpolation routine (Press et al. 1992). To allow for uncertainties in the value of βn, possible nonlinearities in the beam trajectory across the polar cap, and to represent the finite width of a subpulse beam, the pulse profile is broadened in latitude (y) assuming a guassian form, exp[−(β −βn) 2 /0.1], between ∆β = ±0.3. Examples of computed beam components for two pulsars are shown in Fig.  1.

RESULTS AND DISCUSSION
Integrating all pulse profiles in the sample gives the beam shape shown in Fig. 2. This figure represents the probabilty of observing beam components at each location. However, it is strongly affected by the sample distribution of βn. As shown in Fig. 3, this has a strong peak at βn ∼ 0.25, indicating that multi-component pulse profiles are most likely to be detected at about this impact parameter. The deficit at βn ∼ 0 is an observational selection effect which results from the smearing of rapid position angle changes near the profile centre due to finite subpulse beamwidths and instrumental broadening of the profile. Few multi-component profiles are observed at high βn.
The beam distribution shown in Fig. 2 was normalised to correct for the non-uniform distribution of βn using the gaussian-smoothed distribution shown in Fig. 3, giving the final average beam shape shown in Fig. 4. Since there are only one or two pulsars in bins with βn > 0.8 (Fig. 3), there is large statistical uncertainty in the average profile shape for these bins. Therefore we do not plot this region of the normalised beam pattern.
The overall impression given by Fig. 4 is that of a 'patchy' beam. There are some systematic features, but in general the intensity is relatively uniform across the whole beam. The central or 'core' feature is significantly displaced toward later longitudes. The next most prominent property of the average beam after the core component is the presence of a few patches of slightly enhanced emission. Most of these are located within an irregular and rather broad annular maximum at a normalised radius of about 0.7 which is somewhat stronger on the leading side of the beam. Underlying this is more-or-less uniform emission over the whole of the emission beam. The decline at the beam edge is not very sharp.
Although there is no evidence for double or multiple cones in Fig. 4, our present data set is dominated by twoand three-component pulsars. A larger sample of pulsars with more than three components is needed to reliably distinguish multiple-cone models from a patchy beam models. However, our results suggest that, if multiple cones exist, they are at different radii relative to the beam radius in different pulsars. They also show that the conal emission is not confined to a single annular region at the beam boundary.
These results are consistent with the idea that components in pulse profiles are largely determined by the distribution of sources across the polar cap -the 'source function'. The rather smooth distribution distribution of intensity in the mean beam shape suggests that these source regions are randomly distributed for different pulsars as suggested by Lyne & Manchester (1988) and Manchester (1995) the 'patchy-beam' model. The number of identifiable source components across the profile is mostly limited by the finite width of subpulse beams. Except in a few cases with very high signal/noise ratio, e.g. PSR B0740−28 (Kramer 1994), it is generally not possible to identify more than four or five peaks or components, and often only one or two. Some very wide components identified by gaussian fitting, e.g. PSRs B2319+60, B2021+51 (Kramer 1994), are most probably regions of distributed emission producing overlapping subpulse beams.
We believe Fig. 4 is a good representation of the mean radio beam (for βn < 0.8) emitted at frequencies around 1 GHz by 'typical' pulsars, that is, the pulsars of medium or long period which dominate the sample of known radio pulsars. It represents the 'window function' in the model of Manchester (1995) which is determined by the effective gain or efficiency of the radio emission process.
We have computed the average radio beamshape at frequencies about 1 GHz of pulsars with medium to long periods. This beamshape has a peak near its centre and a mild, broad and rather irregular enhancement at a normalised beam radius of about 0.7, but is otherwise rather uniform. The decline at the beam edge is gradual.
These results suggest that the presence and location of profile components are determined by a 'source function' which varies randomly from pulsar to pulsar. The summing of these randomly distributed components results in a relatively uniform average beam profile which we interpret as the 'window function' representing the properties of the emission process common to all longer-period pulsars.