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Toward An Empirical Theory of Pulsar Emission VIII: Subbeam

Circulation and the Polarization-Modal Structure of Conal Beams

Joanna M. Rankin1

Sterrenkundig Instituut ‘Anton Pannekoek’, Universiteit van Amsterdam, 1098 SJ Amsterdam NL

R. Ramachandran2

Stichting ASTRON, Postbus 2, 7990 AA Dwingeloo, The Netherlands

ABSTRACT

The average polarization properties of conal single and double profiles directly reflect thepolarization-modal structure of the emission beams which produce them. Conal componentpairs exhibit large fractional linear polarization on their inside edges and virtually completedepolarization on their outside edges; whereas profiles resulting from sightline encounters withthe outer conal edge are usually very depolarized. The polarization-modal character of subbeamcirculation produces conditions whereby both angular and temporal averaging contribute to thispolarization and depolarization.

These circumstances combine to require that the circulating subbeam systems which produceconal beams entail paired PPM and SPM emission elements which are offset from each otherin both magnetic azimuth and magnetic colatitude. Or, as rotating subbeam systems produce(on average) conal beams, one modal subcone has a little larger (or smaller) radius than theother. However, these PPM and SPM “beamlets” cannot be in azimuthal phase, because bothalternately dominate the emission on the extreme outer edge of the conal beam. While thisconfiguration can be deduced from the observations, simulation of this rotating, modal subbeamsystem reiterates these these conclusions. These circumstances are also probably responsible,along with the usual wavelength dependence of emission height, for the observed spectral declinein aggregate polarization.

A clear delineation of the modal polarization topology of the conal beam promises to addressfundamental questions about the nature and origin of this modal emission—and the modal parityat the outer beam edges is a fact of considerable significance. The different angular dependencesof the modal “beamlets” suggests that the polarization modes are generated via propagationeffects. This argument may prove much stronger if the modal emission is fundamentally onlypartially polarized. Several theories now promise quantitative comparison with the observations.

Subject headings: MHD — plasmas — pulsars: general, individual (B0301+19, 0329+54, 0525+21,0809+74, 0820+02, 0943+10, 1133+16, 1237+25, 1923+04, 2016+28, 2020+28 and 2303+30), radiationmechanism: nonthermal

1On leave from Physics Department, A405 Cook Bldg.,University of Vermont, Burlington, VT 05405 USA; email:rankin@physics.uvm.edu

2Current address: Astronomy Department, 601 Camp-bell Hall, University of California, Berkeley, CA 94720-3411USA; email: ramach@astron.berkely.edu

The Outer-Edge Depolarization Phenomenon

A humble fact about pulsar radio emission,which to our knowledge has attracted virtually nonotice or comment, is the following: The extremeouter edges of virtually all conal component pairsare prominently, and apparently accurately, depo-

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larized. Considerable comment has been made re-garding the obverse of this circumstance—that is,to the effect that the highest levels of fractionallinear polarization are usually found on the in-ner edges of conal components, indeed where itis sometimes nearly complete (Manchester 1971;Morris et al 1981).

Of course, longitudes corresponding to theouter edges of such conal component pairs arealso just where intervals of secondary polarization-mode dominance are seen in individual pulses,as we know from those well known stars whoseprofiles indicate a fairly central sightline traversethrough the conal beam: B0329+54 exhibits zonesof outer edge depolarization over a wide frequencyrange (Gould & Lyne 1998; hereafter GL98; Su-leymanova & Pugachev 1998, 2002) accompaniedby prominent outer edge “90◦ flips” in the positionangle (PA). Earlier average studies (Manchester1971; Morris et al 1981; Bartel et al 1982) togetherwith the more recent single-pulse analyses of Gil& Lyne (1995), von Hoensbroech & Xilouris et

al (1997), and Mitra (1999) provide an unusuallycomprehensive picture of this pulsar’s outer edgedepolarization. The phenomenon persists to thehighest frequencies, as can clearly be seen in the10.55-GHz profile of von Hoensbroech & Xilourisabove.

Other obvious exemplars are pulsars B0525+21and 1133+16, which clearly exhibit the outeredge depolarization phenomenon over the entirerange of frequencies that they can be observed[see the above papers as well as Blaskiewicz et al

(1991), von Hoensbroech (1999), and Weisberg et

al (1999)]. For 0525+21, which has a more centralsightline traverse (see Table 1), individual-pulsepolarization displays show that the weaker sec-ondary polarization mode dominates the primaryone (hereafter SPM and PPM, respectively) onlyon the extreme outer edges of its profiles; whereasfor 1133+16, which has a more oblique sightlinetraverse, SPM dominated samples can be seenover a larger longitude range.3

Reference to the now extensive body of pub-lished average polarimetry provides several hun-dred examples of pulsars whose conal component

3In this paper, the terms primary (PPM) and secondary(SPM) polarization mode denote little more than their rel-ative strength.

Fig. 1.— Log-linear plots of the polarized intensityprofiles of pulsar B1133+16, showing the full ex-tent of the edge depolarization—430 MHz (upper)and 1414 MHz (lower). The total intensity, Stokesparameter I, is given by the solid curve, the linearpolarization L by the dashed curve, and−V by thedotted one (only rh circular is observed in this pul-sar). The three respective curves were smoothedover five samples, normalized to the maximum inI, and the statistical bias in L was removed. The430-MHz and 21-cm sequences had lengths of 956and 2180 pulses, respectively.

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pairs have prominently depolarized outer edges.The effect is so widespread, indeed, that it isdifficult to identify completely convincing exam-ples to the contrary. The stars comprising fourof the five main profile classes (e.g., see PaperVI) of conal single (Sd), double (D), triple (T),and five-component (M) virtually all exhibit thephenomenon as do the few stars in the more re-stricted cT and cQ classes. It is worth notingthat good examples of outer edge depolarizationare found among stars with both inner and outerconal configurations; all three of the stars withinner cones discussed in the foregoing Paper VII[Mitra & Rankin (2002)] show the effect, thoughinterestingly, in each case it is more prominenton the trailing than on the leading edge.4 Thebest examples of stars with little or no edge de-polarization all either have (or probably have,given that some are yet not well observed) coresingle St or inner-cone triple T configurations;some are B0355+54, 0450+55, 0540+23, 0559–05, 0626+24, 0740–28, 0833–45, 0906–49 (mainpulse), 1055–52, 1322+83, and 1737–30, and notethat, overall, these stars have much shorter peri-ods than is typical for the normal pulsar popula-tion.

We then summarize the overall characteristicsof the edge-depolarization phenomenon:

1. The average linear polarizationL (=√

Q2 + U2)falls off much faster than the total power Ion the edges of the profile and decreasesasymptotically to near zero.

2. This phenomenon usually occurs over a verybroad band—essentially the entire range ofthe observations—and therefore, the edgedepolarization appears to be nearly indepen-dent of frequency.

3. This profile-edge depolarization is modalin origin, meaning that it largely occursthrough the incoherent addition of PPM andSPM power both within samples and frompulse to pulse.

4As discussed in Paper VII, many or most stars with inner-cone profile configurations also exhibit discernible emissionin the “baseline” region, far in advance of the leading com-ponent and sometimes after the trailing component, per-haps because a weak outer cone is also emitted.

4. The edge depolarization affects conal com-ponent pairs, and therefore must be re-garded as a roughly symmetrical, structuralfeature of conal emission beams.

5. Then, in terms of such beams:

• the outer edge depolarization requiresthat the modal power be about equal atlarge angles to the magnetic axis, and

• the proximity of the depolarized outeredges of conal component pairs to theirmore highly polarized inner edges re-quires that the weaker mode peak atslightly larger angles to the magneticaxis than the stronger one.

In the remainder of this paper we will ex-plore the causes and consequences of these cir-cumstances, drawing extensively on the earlier ar-ticles of this series, Papers I–VII (see References).We will show that these structural characteris-tics of conal emission beams—and therefore wellresolved conal component pairs—are almost cer-tainly the result of subbeam circulation as in pul-sar B0943+10 (see Deshpande & Rankin 2001).This circulation, in sweeping a series of polarizedsubbeams around the magnetic axis and past oursightline, is responsible for the outer-edge depo-larization and (sometimes periodic) modal fluc-tuations in pulsars where the sightline traversecuts the emission beam centrally (e.g., 0525+21);it is also largely responsible for the very differ-ent polarization effects observed in pulsars wherethe sightline traverse is oblique (i.e., B0809+74).Of course, only subbeams with particular angu-lar polarization patterns can produce the particu-lar sorts of depolarized profile forms that are ob-served, and in the remainder of this paper we en-deavor to understand what general features are re-quired of them. In the following sections, we firstbriefly describe our observations and then considerthe contrasting characteristics of stars, first withwell separated conal component pairs, and thenwith conal single Sd profiles. The penultimate sec-tion gives the results of modelling the polarizedemission beam, and we conclude with a summaryof our results and a discussion of their implica-tions.

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Table 1

Pulsar Parameters.

Pulsar P β/ρa f Source Date BW(B–) (s) (MHz) (MHz)

0301+19 1.388 0.45 430 AO 1974Jan5 10/320329+54 0.715 0.31 840 WSRT 2002Jan10 80/5120525+21 3.745 0.19 430 AO 1974Apr4 10/321133+16 1.188 0.78 430 AO 1992Oct19 10/32

1414 AO 1992Oct15 20/321237+25 1.382 ∼0 430 AO 1974Jan6 10/322020+28 0.343 0.49 430 AO 1992Oct16 10/32

0809+74 1.292 0.93 328 WSRT 2000Nov26 10/640820+02 0.865 0.98 430 AO 1992Oct19 10/320943+10 1.098 -1.01 430 AO 1992Oct19 10/321923+04 1.074 0.97 430 AO 1991Jan6 10/322016+28 0.558 0.96 430 AO 1992Oct15 10/322303+30 1.576 0.99 430 AO 1992Oct15 10/32

aThe sign of the magnetic impact angle β is specified only when itis known. ρ, the conal beam radius, is positive definite. The valuesrefer to 1 GHz and most are taken from Rankin (1993a,b; hereafter,Paper VI).

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Observations

The source and character of our observationsare summarized in Table 1. The Arecibo Obser-vatory (AO) recordings were made under two po-larimetry programs, the first in the early 1970sand the second in 1992, and both are describedin Rankin & Rathnasree (1997). The 328- and840-MHz sequences were made using the Wester-bork Synthesis Radio Telescope (WSRT) with itspulsar machine PuMa, and these are described inRamachandran et al (2002).

The Depolarization Pattern of Conal Com-

ponent Pairs

Let us now look in more detail at the mannerin which the outer edges of conal component pairsare depolarized. Turning first to pulsar 1133+16,Figure 1 shows the relative behaviors of the log-arithms of Stokes parameters I, L, and −V asa function of longitude for a 430-MHz sequence(top) and a 21-cm sequence (bottom). Here wecan follow the behavior of the fractional polariza-tion far out into the “wings” of the star’s profile.We see not only that the depolarization persiststo very low intensity levels, but also that its linearand circular polarization generally decrease withor faster than the total power down to the pointwhere the noise fluctuations begin to dominate at2–4×10−4 (note that only the absolute value ofthe noisy quantities can be plotted).

Probably, this behavior is typical of many pul-sars, but only for a few, such as 1133+16, canpolarized profiles with such a large dynamic rangebe computed. Even for 1133+16 it would be in-teresting to compute a more sensitive such display.These observations from Arecibo were only some40 minutes long, so with care it should be possibleto reduce the relative noise level much further. If,then, it is generally true that the outer edges ofconal profiles—and thus the outer edges of conalbeams—are accurately depolarized on average, itprovides a strong constraint on the angular beam-ing characteristics of the modal emission.

We can look at this outer-edge depolarization inmore detail by conducting an appropriate mode-segregation analysis on selected sequences. Twosuch algorithms were described in Deshpande &Rankin’s (2001) Appendix, and we use here thethree-way mode segregation method, because it

provides the greatest flexibility. It produces twofully polarized PPM and SPM pulse sequences anda fully depolarized UP sequence, while making norestrictive assumptions about the origin of the de-polarization. Briefly, the I and L of each sampleare compared with a noise threshold, and its re-spective L and I–L portions accumulated in threepartial sequences depending on whether the sam-ple is PPM dominated, SPM dominated (relativeto a model PA traverse which defines the former),or essentially unpolarized (UP).

The results of these analyses for pulsars withprominent conal component pairs are given in Fig-ure 2, where the heavier curves give the usualtotal-power (I) and total linear (L) profiles, whilethe lighter curves show the PPM (dashed curve),SPM (dotted curve) and UP (solid curve) pro-files. A similar (but more primitive) analysis forB1737+13 (Rankin et al 1986) can also be com-pared, as can the excellent modal polarizationstudies of B2020+28 and 0525+21 by McKinnon& Stinebring (1998, and 2000, respectively; here-after MS98 and MS00). The sources of the varioussequences are given in Table 1, where we also tab-ulate β/ρ. For the stars considered in this section,|β|/ρ < 0.8, a sightline geometry which produceswell resolved conal component pairs. Note, by con-trast, that the conal single Sd pulsars consideredin the next section all have |β|/ρ > 0.9.

For most of the stars (all but B2020+28), wesee a fairly consistent picture. The weaker modeonly rarely has sufficient intensity to dominate asample, so the aggregate SPM power is typicallyonly some 10% that of the PPM—and almost all ofthis SPM power is found on the “wings” of the pro-files. Often, the SPM power peaks slightly furtherout than the PPM and exhibits a narrower angu-lar width. Note, further, that the UP distributionbehaves very similarly to both the PPM and SPMcurves, so we may view some portion of the UPpower as the accumulation of samples which weredepolarized by equal contributions of PPM andSPM power—and indeed, the UP curves alwaysasymptotically approach the overall I curves atvery low power levels. This behavior could also bedemonstrated by applying the two-way modal “re-polarization” technique in Deshpande & Rankin,which proceeds under the assumption that the de-polarized samples contain equal PPM and SPMlevels of power.

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Fig. 2.— Three-way, mode-segregated average profiles for pulsars with prominent conal component pairs,B0301+19, 0329+54, 0525+21, 1133+16, 1237+25, and 2020+28. The heavier solid and dashed curves givethe total power (Stokes I) and total linear L; whereas, the lighter dashed, dotted and solid curves give thePPM, SPM and UP power, computed according to the algorithm in Deshpande & Rankin’s (2001) Appendix(see text).

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Fig. 3.— Three-way, mode-segregated average profiles for six conal single Sd pulsars, B0809+74, 0820+02,0943+10, 1923+04, 2016+28, and 2303+30 as in Fig. 2.

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Fig. 4.— Color polarization display of a 200-pulse portion of the 430-MHz observation in Figs. 2 and 5. Thefirst column gives the total intensity (Stokes I), with the vertical axis representing the pulse number andthe horizontal axis pulse longitude, colour-coded according to the left-hand scale of the top bar to the leftof the displays. The second and third columns give the corresponding fractional linear polarization (L/I)and its angle (χ = 1

2tan−1 U/Q), according to the top-right and bottom-left scales. The last column gives

the fractional circular polarization (V/I), according to the bottom-right scale. Plotted values have met athreshold corresponding to 2 standard deviations of the off-pulse noise level. Note the 2.63 c/P1 modulationassociated with the outer conal component pair—and that this modulation has a strikingly modal characteras can be seen particularly clearly in the orthogonal chartreuse and magenta PAs.8

Though pulsar 2020+28’s modal behavior ap-pears more complex (e.g., Cordes et al 1978;MS98), we see many of the same features—forinstance, that the UP power approaches the totalpower on the extreme edges of profile. Indeed,MS98’s analysis based on “superposed modes”suggests similar conclusions. The well measuredprofile demonstrates that its “two” componentseach have a good deal of structure—seen as“breaks” in the total-power curves—but the PPM,SPM and UP curves demonstrate, in addition,that much of the complexity is modal in origin.The complex modal behavior of this pulsar de-serves much fuller study, and a well measuredpolarimetric pulse sequence in the 100–200-MHzrange would add much to our knowledge.

Overall, we see that the conal component pairsdepicted in Figure 2 all have moderate to high lev-els of fractional linear polarization—that is, typ-ically some 50%—though most have narrow, in-terior regions of longitude where the linear polar-ization is higher. We shall see that this stands insharp contrast to the Sd pulsars considered in thefollowing section. Our point is that when |β|/ρis relatively small—producing well resolved conalcomponent pairs—the mode mixing depolarizesthe outer edges, but not the profile interior. Thisthen reflects properties—both dynamic in termsof modulation phenomena and polarizational—ofthe conal emission beam, and we must reflect onjust how this is possible.

The Depolarization Patterns of Conal Sin-

gle Stars

We have just considered a group of stars inwhich our sightline makes a fairly central traversethrough their emission cone(s), and we now turn tomembers of the conal single Sd group, all of whichare configured by a tangential traverse along theaverage emission cone. Here we have the opportu-nity both to explore the conal depolarization phe-nomena in a very different geometrical context andthen to investigate how the modulation and de-polarization phenomena are connected. Figure 3gives mode-segregated polarization plots (similarto those in Fig. 2) for six Sd stars. Here, it isimportant to keep in mind that each of these pul-sars has prominent “drifting” subpulses, so thatthe profiles give only a static average of the sub-

pulse polarization. The displays of Fig. 3 showthat the UP (perhaps, mode-mixed) power is typ-ically 50% of the total, so that the overall modalcontributions are comparable and the aggregatelinear polarization is often small. While all of thetotal-power profiles are roughly unimodal (onlyB0820+02 is really symmetrical), the modal pro-files are more complex; the two peak at differentlongitudes in B0809+74; the SPM has a doubleform in B2016+28; and we have already noted thepeculiar “triple” form of the aggregate linear inpulsar 0820+02.

As a class, the Sd stars exhibit conspicuouslydepolarized profiles at metre wavelengths. Indeed,this has been one of the great obstacles to under-standing their characteristics, because, for many(i.e., 0809+74), the modal complexity and lowfractional linear polarization make it difficult toaccurately determine even such a simple param-eter as the PA sweep rate (e.g., Ramachandranet al 2002). Paradoxically, some also have nearlycomplete linear polarization at certain longitudesand frequencies (i.e., as does 0809+74’s leadingedge at higher frequencies) suggesting that mode-mixing is not always operative.

The Sd pulsars are also the profile class mostclosely associated with the problematic phe-nomenon of “absorption”. It was in 0809+74that the effect was first identified (Bartel et al

1981; Bartel 1981)—that is, evidence that partsof the profiles were “missing”—and strong evi-dence to this effect through subbeam-mappingmethods have also been adduced for 0943+10(Deshpande & Rankin 2001). Surely one couldimagine from 0943+10’s asymmetric profile thata part of its trailing-edge emission is “absorbed”,though 0809+74’s more symmetric leading edge atmeter wavelengths gives little clue that emissionappears to be missing here as well. In short, thecircumstances defining the profile edges appear tobe more complicated for conal single stars thanfor the other species, and their modal polarizationcharacteristics are an aspect of this complexity.

What is the Relative Polarization-Modal

Phase in Conal Component Pairs?

We have learned in the foregoing two sectionsthat well resolved conal component pairs are mostdepolarized on their extreme outer edges, while

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the polarized modal emission in conal single starsaccrues to the depolarization essentially over theentire width of the profile. These circumstancesbegin to illuminate the polarization configurationof the subbeams; and, indeed, we saw in Desh-pande & Rankin (2001; fig. 19) that for 0943+10the discernible SPM emission was found in be-tween the 20 PPM subbeams. We have found asimilar configuration for pulsar 0809+74, wherethe PPM and SPM power centers are displacedfrom each other systematically in both magneticazimuth and colatitude by perhaps 20% of the sub-beam spacing (Rankin et al 2002).

A related question which has had no investi-gation at all is the following: what is the mod-ulation phase relationship between the PPM andSPM power on the outside edges of pulsars withconal component pairs? Such a question is nottrivial to answer because only a few of such starshave modal modulation which is strongly periodic(while virtually all of the Sd stars, for instance, inFig. 2 exhibit a good deal of regularity). Two pul-sars which do have periodic modulation featuresare B1237+25 and 2020+28.

Figure 4 exhibits the character of this subpulsemodulation in pulsar B1237+25. This sequencewas chosen for its brightness and relative freedomfrom nulls, and in consequence its outer compo-nents show a particularly sharp feature at 2.63c/P1. This modulation can be seen very clearlyin the first column which gives the total power I.The modal character of the modulation, however,is most obvious in the third column depicting thePA, where the alternating magneta and chartreusecolors represent orthogonal PAs. Further effectsof this modal modulation can be seen in the vary-ing levels of associated depolarization (second col-umn) and the correlated variations of circular po-larization (forth column). Note that virtually allof these modal modulation effects are confined tothe outermost pair of conal components, usuallyreferred to as components I and V.

Figure 5 provides a more quantitative analy-sis of this modal modulation under B1237+25’souter conal component pair. The top display givesthe PPM profile after a three-way segregation ofthe modal power along with a curve showing thefraction of this power which is modulated at thefeature frequency of 2.63 c/P1; whereas the lowerpanel gives the phase of this modulation. As noted

Fig. 5.— Modulation amplitude and phase ofthe three-way segregated PPM (top) and SPM(bottom) power in pulsar B1237+25 at 2.63 c/P1.Note that 40–60% of the fluctuation power underthe outer cone is modulated at this frequency andthat the modal sequences have roughly oppositephases. The relationship at other longitudes is dif-ficult to interpret, because the mode segregationis less definitive and the fluctuating power smallor negligible. The sequence here is a superset ofthat in Fig. 2.

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above, we chose a part of the pulse sequence withfew nulls, which also had a particularly “pure”modulation feature. Clearly, the phase is onlyreliable under the outside conal component pair,where the modulation represents a large fractionof the total modal power. The lower display givessimilar information for the SPM-segregated par-tial sequence. Results for the UP partial sequenceare irrelevant here and thus not shown.

Remarkably, we see here that the PPM andSPM power are roughly out of phase under theouter conal component pair. The error in thisphase difference is relatively small as evidencedby the stable SPM phase under the outer compo-nent pair. Thus, when computed over the 256-pulse sequence, we have strong evidence that themodal power is emitted in a manner which is farfrom “in phase”. This in turn indicates that themodal power is systematically modulated, just asis the total power. Furthermore, that there is SPMpower to segregate implies (as can also be seen inFig. 4) that, at times, the weaker SPM dominatesthe PPM.

This behavior can be understood if both modesare, in general, present in every sample and com-bine incoherently—which is just the situation of“superposed modes” favored by MS00.

Geometry of Conal Beam Depolarization

As discussed earlier, conal component pairsexhibit large fractional linear polarization ontheir inner edges and pronounced (often nearlycomplete) depolarization on their outer edges.The three-way mode-segregation method pro-vides some vital clues to understanding this phe-nomenon. The power corresponding to the weakerSPM is sufficient to dominate the PPM only onthe outer “wings” of the profile.

The mode-segregation analyses above revealtwo important characteristics of the emissionbeam configuration. First, the SPM emissionis generally shifted further outward, away fromthe magnetic axis, than the PPM emission. Ifthis modal radiation is emitted (in some averagesense) by conal beams, then the emission conalregion corresponding to the SPM beam must havea little larger radius than that of the PPM.

Second, as we saw in Fig. 5 the PPM and SPMpower is substantially out of phase. Given the

Fig. 6.— A gray scale representation of ourrotating-subbeam model. Note the respective setsof PPM (‖ polarized) and SPM (⊥ polarized)“beamlets”, which in turn comprise the PPM (in-ner, solid) and SPM (outer, dashed) subcones.There is also an unpolarized, non-drifting centralcore component.

small |β|/ρ for 1237+25—such that the sightlinecuts the conal beams close to the magnetic axis—the phase difference suggests that emission ele-ments within the respective modal beams are off-set in magnetic azimuth! And, indeed, this is justthe polarized-beam configuration observed in therotating subbeam systems of conal single Sd pul-sars B0809+74 (Rankin et al 2002) and 0943+10,where systematic longitude offsets between themodes (at |β|/ρ ∼ 1) also indicate offsets in mag-netic azimuth. In summary, the modal conal emis-sion patterns are offset in both magnetic colati-tude and azimuth.

We can begin to conceive, given the aboveobservational indications, how complex are themodal depolarization dynamics of conal beams.The familiar polarization properties of conal com-ponent pairs are produced by central sightline tra-jectories (small |β|/ρ) and represent an angular av-erage over the modal “beamlets”. For conal single(Sd) stars, however, the impact angle |β| is veryclose to the radius of the emission cone ρ, and the

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Fig. 7.— Simulated linear polarization his-tograms: (a) a conal double profile modeled onB0525+21, and (b) a conal single (“drifter”) pro-file modeled on P0820+02. See text for de-tails. No effort has been made to model StokesV (dotted line). Pulsar names have been given as“0000+00”, just to indicate that the profiles are‘simulated’.

observed average polarization will depend first onjust how the sightline cuts the modal cones, andsecond on how this modal power is both angularlyand temporally averaged.

In order to understand this situation more fully,we have attempted to simulate the depolarizationprocesses in conal single and double pulsars. To doso, we generated an artificial pulsar signal such aswould be detected by a pulsar backend connectedto a radio telescope (e.g., WSRT with its PuMa

processor). We computed this (partially) polar-ized signal using the recipe given in our Appendix,together with a rotating subbeam model inter-acting with a specific observer’s sightline. Thissubbeam system, with pairs of modal “beamlets”which could be offset in both magnetic colatitudeρ and azimuth, flexibly modeled properties seenin both the Sd and D stars; and model pulses se-quence were computed using relations very muchlike the inverse cartographic transform in Desh-pande & Rankin (2001). Further, a low-level,non-drifting and unpolarized component, with aGaussian-shaped pattern peaked along the mag-netic axis, could be added to simulate weak coreemission. Here, we have so far ignored the natureof the circular polarization but hope to address itin future work.

The modal “beamlet” pairs rotate rigidly witha period P3 around the magnetic axis, with theirrotation phase “locked” to each other. Their re-spective trajectories have different radii (offset inmagnetic azimuth), and the “beamlets” also havesomewhat different radial widths. These charac-teristics are required in order both to permit highpolarization on the inner edges of conal compo-nent pairs and to ensure that their outer edges arefully depolarized. In order to specify the radialillumination pattern of the modal “beamlets”, wehave used a hybrid function with ranges of bothGaussian-like and exponential behavior,

P (θ) =exp[−θ/2σ2] + exp[−θ2/2σ2]

1 + exp[−θ2/2σ2](1)

where θ is the radial distance from the centreof the “beamlet”, and σ is its Gaussian-like rms

scale. This functional form was chosen to providea smoothly falling function near the “beamlet”peak and an exponential-like behavior on its edges.Although there is no physical basis for this choice,it seems to reproduce rather nicely the outer edges

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of the profiles shown in Fig. 1.

A schematic picture of our simulation model isthen shown in Figure 6. Othogonally polarizedsets of modal “beamlets” are shown in greyscale,which slowly rotate so as to form the two modalsubcones. The peaks of the respective PPM (‖polarization) and SPM (⊥ polarization) subconesare indicated by solid and dashed curves. A weak,non-drifting and unpolarized core beam is also in-cluded.

Figure 7 then shows some results from our sim-ulations. The top panel represents an attemptto model a conal double (D) pulsar with prop-erties similar to the canonical pulsar B0525+21.So, we have taken α, β, and P some 21◦, 1.5◦ and3.75 s, respectively. Further, in order to modelits 430-MHz profile, we took the mean radii of itstwo modal subbeam systems to be 3.0 and 3.6◦.We also assumed that its two orthogonal modesare fully linearly polarised. The rotating-subbeamsystem corresponding to the PPM and SPM eachhave 8 subbeams, with σ scales of 1.3◦ and 0.88◦

each, and the peak amplitudes of the SPM “beam-lets” are about 60% of their PPM counterparts.

We also modeled the central core component[which for B0525+21 should have an observedwidth of 1.77◦ (see Paper IV, eq. 5)] as a non-drifting, unpolarized pencil beam with a Gaussianprofile centered along the magnetic axis. However,since our sightline intersects this weak emissionfar off on its beam edge (β ∼ 1.5 degrees), it con-tributes little to the model sequence and profile.

As can then be seen, the fractional linear po-larization of the model profile reaches a maximumon the inner edges of the two components anddrops sharply on their outer edges, just as is ob-served [c.f., Fig. 2 and, for instance, Blaskiewiczet al (1991)]. Note the ‘S’-shaped PA traverseand the parallel modal PA stripes on their outeredges, which correspond to those samples wherethe SPM sometimes dominates the PPM. Thismodal display is also usefully compared directlywith the corresponding 430-MHz PA histogramof B0525+21 in Hankins, Rankin & Eilek (2002).Clearly, we have made no attempt to model thecircular polarization.

The bottom panel of Fig. 7 then depicts our ef-fort to simulate the polarized emission-beam con-figuration of a conal single (Sd) star, and here we

have taken pulsar B0820+02 as an example. Inthis case we took α, β, and P some 19◦, 5.5◦

and 0.865 s, and the radii of the subcones cor-responding to the PPM and SPM were 4.5◦ and5.1◦, respectively—nearly equal to |β| as expected.In this case, the weak unpolarized core beam hasa computed with of 4.05◦, and again contributeslittle to the model sequence and profile. Of course,we cannot know for Sd stars just how far out thesightline crosses the conal beam, so we can adjustthis point slightly to match particular polarizationcharacteristics.

The ratio of the two subcone radii in the respec-tive examples chosen above are different. In thefirst case (B0525+21), it is 0.83 (3◦/3.6◦), while inthe second case (B0820+02) 0.88. Although thesetwo ratios are quite close in their values, it is un-clear what might cause this ratio to vary from starto star. By contrast, within the dynamical picturewe present here, the aggregate polarization prop-erties must be independent of parameters such asP2 (the subpulse separation in longitude), P3 (thetime for subpulse to drift through a longitudinalinterval of P2), and P3 (the subbeam circulationperiod). It is also important to note that the ag-gregate profile characteristics are completely in-dependent of the total number of circulating sub-beams.

In two particulars, our simulations depart sig-nificantly from what is observed: First, as a con-sequence of assuming that the modal emission isfully linearly polarized, we generally obtain higherlevels of aggregate linear polarization than is seenin the profiles we are attempting to model. Thissuggests, as yet inconclusively, that the modalbeams are not fully polarized. Second, we findmuch less scatter in the model PAs around the ge-ometrically determined PA traverse. While thebest observations have for some time suggestedthat this excessive scatter could not be the resultof the system noise, more quantitative statementshave not been easy to make. However, McKinnon& Stinebring (1998; 2000) have developed statisti-cal analysis tools which should make a more mean-ingful assessment practical, and we plan to pursuethis question in a future paper (Ramachandran &Rankin 2002).

13

Summary and Discussion

The results of this paper can be summarizedsuccinctly: Conal beams have a rotating subbeamstructure which also entails displacements betweenthe PPM and SPM radiation in both magnetic lat-itude and azimuth. This results in the outer-edgedepolarization seen in conal component pairs aswell as the complex (and often nearly complete)depolarization found in pulsar profiles that repre-sent an oblique sightline trajectory along the outeredge of the conal beam. It also provides a newand fundamental reason why the modal emissionis so often statistically “disjoint” (see Cordes et al1978). These characteristics of conal emission canbe identified in a variety of ways, and the conclu-sions verified by detailed models and simulations.

It is also likely that these effects largely explainthe frequency dependence of the fractional linearpolarization in the classic cases of conal doubleprofiles (i.e., B1133+16) first problematized byManchester, Taylor & Huguenin (1973). Manymore recent studies have pointed to both the thesecular decline at high frequencies and the mid-band “break” point below which the aggregatefractional linear increases no further (e.g., McK-innon 1997). And closely associated with theseprofile effects are pulse-sequence phenomena rang-ing from the purported “randomizing” of the PAat high frequencies to distributions of polarizationcharacteristics in subpulses. If we understand thatthe PPM and SPM “cones” have a significant dis-placement in magnetic colatitude at meter wave-lengths, then radius-to-frequency mapping (seePaper VII) almost certainly tends to reduce thisdisplacement at higher frequencies. Perhaps thecharacteristic depolarization of conal beams atvery high frequencies (as well as the “random”PAs) is simply the result of modal beam overlap.Perhaps the “breaks” mark the frequency at whichthe modal beams diverge to the point that no fur-ther depolarization occurs. It will be satisfying totest these ideas in future detailed studies.

The origin of “orthogonal mode” emission hasbeen a topic of debate for decades. Numer-ous models have been suggested wherein the twomodes are intrinsic to the emission mechanism it-self (e.g., Gangadhara 1997) and, lacking strongcontrary evidence, some bias has developed in fa-vor of this assumption. However, direct produc-

tion implies that the modes be fully (elliptically)polarized and associates them with a basic emis-sion mechanism which is itself still unknown [for areview, see Melrose (1995)].

The possibility that disjoint orthogonal modescan arise from propagation effects was also ex-plored very early by several authors (Melrose 1979;Allan & Melrose 1982). The central idea here isthat the natural wave modes, being linearly po-larized in two orthogonal planes, have different re-fractive indices, and become separated in spaceand angle during their propagation. This phe-nomenon of refraction in the magnetosphere wasexplored rigorously by Barnard & Arons (1986).

A recent work of Petrova (2001) has addressedthese issues in greater detail. According to hermodel, the primary pulsar radiation is comprisedof only one (ordinary) mode, which is later par-tially converted into extraordinary-mode emission.It is in this conversion that the orthogonal po-larization modes arise. Therefore, the transitionfrom one mode to the other, as observed in pulsaremission, can be understood as due to switchingbetween a “significant” and “insignificant” conver-sion. At any given time and pulse longitude, thedisjoint mode is the sum of two incoherently su-perposed modes. This nicely explains the partialpolarization observed in the pulsar radiation.

Conversion to the extraordinary mode, inPetrova’s model, is easiest for those rays whichare refracted outward, away from the magneticaxis, and such emission apparently comprises theconal beam—though her work yet gives no under-standing about why there should be two distincttypes of conal beams that are both present in somecases. It is further unclear how the ordinary orextraordinary mode would be polarized, thus howit then could be identified as a specific PPM orSPM in a given pulsar, and why one or the othershould experience a greater angular offset in mag-netic colatitude. Finally, this model appears tobe fully symmetric in azimuth, so that it is againhard to see how the wave-propagation effects canexplain the observed angular offsets in magneticazimuth.

To summarise, the important conclusions ofthis work are as follows:

• The average profiles of pulsars with conalcomponent pairs exhibit low fractional po-

14

larization on their outer edges, and oftenhigh fractional linear polarization on theirinner edges.

• This very general behavior can be under-stood in terms of the dynamic averaging,along the observer’s sightline, of emissionfrom a rotating system of subbeams withsystematic modal offsets.

• The “beamlet” pairs corresponding to thePPM and SPM emission are offset not onlyin magnetic latitude, but also in magneticlongitude. In other words, the respective av-erage modal beams can be visualized as dis-tinct emission cones with somewhat differentangular radii. Dynamically, the “beamlet”pairs maintain a fixed relation to each otheras they circulate about the magnetic axis.

• The outer-edge depolarization requires thatthe PPM and SPM subcones have nearlyidentical specific intensity and angular de-pendence in this region. This would appearto place strong constraints on their physicalorigin.

• The causes of these remarkable angular off-sets between the PPM and SPM emission isunclear. Propagation effects can more easilyexplain the shifts in magnetic latitude thanlongitude.

We thank Avinash Deshpande for importantanalytical assistance and Mark McKinnon, Rus-sell Edwards, and Ben Stappers for discussionsand critical comments. We also thank the lattertwo for help with the WSRT observations. Oneof us (JMR) also gratefully acknowledges grantsfrom the Netherlands Organisatie voor Weten-schappelijk Onderzoek and the US National Sci-ence Foundation (Grant 99-86754). Arecibo Ob-servatory is operated by Cornell University undercontract to the US NSF.

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Appendix: Simulation of Partially Polar-

ized Radiation

Let us consider the two complex signals,X(t) = [XR(t) + j XI(t)] and Y (t) = [YR(t) +j YI(t)], which represent the Nyquist-sampledbaseband voltages from the two orthogonal lin-ear dipoles (X & Y) of a radio telescope. Thesubscripts R and I indicate the real and the imag-inary parts of the complex signal, and j =

√−1.

The Stokes parameters are defined as

I = 〈X X⋆ + Y Y ⋆〉Q = 〈X X⋆ − Y Y ⋆〉U = 〈Re[X Y ⋆ + X⋆ Y ]〉

= 〈2 |X | |Y | cos θ(t)〉V = 〈Im[X Y ⋆ + X⋆ Y ]〉

= 〈2 |X | |Y | sin θ(t)〉 (2)

The 〈 〉 symbols indicate time-averaging, Re

and Im the real and imaginary parts, and su-perscript “⋆” the complex conjugate. Q andU describe the linear and V the circular po-larization, obeying the well known inequality,I ≥

√

Q2 + U2 + V 2. The angle θ is the phasebetween X(t) and Y (t), which is given by θ =[tan−1(XI/XR) − tan−1(YI/YR)].

From the sampling theorem, we know that asignal varies at a rate given by the reciprocal of

This 2-column preprint was prepared with the AAS LATEXmacros v5.0.

16

the bandwidth ∆ν; samples having this resolutionare fully polarized and can be represented by apoint on the Poincare sphere. Polarimetry, then,always entails averaging over a time sufficient toadequately reduce the statistical errors.

To generate a realistic partially polarizedNyquist-sampled baseband signal, we adopted thefollowing procedure. A randomly polarized voltagesample in the X-dipole was defined as

X iRu =

√

1/2 Pr Gix cos(φ)

X iIu =

√

1/2 Pr Gix sin(φ)

where φ = 2π U ix , (3)

where Pr is its amplitude, Gix is a Gaussian-

distributed random variable with zero mean andunity rms amplitude, and U i

x a uniform randomvariable with equal density between 0 and 1. Sim-ilarly, for the Y-dipole,

Y iRu =

√

1/2 Pr Giy cos(φ)

Y iIu =

√

1/2 Pr Giy sin(φ)

where φ = 2π U iy , (4)

and Giy, U i

y are different random variables asabove.

For linear polarization, the signal voltages are

X iRl = cosχ Pl G

il cos(φ)

X iRl = cosχ Pl G

il sin(φ)

Y iRl = sinχ Pl G

il cos(φ)

Y iRl = sinχ Pl G

il sin(φ)

where φ = 2π U il , (5)

and for circular polarization they are

X iRc =

√

1/2 Pc Gic cos(φ)

X iIc =

√

1/2 Pc Gic sin(φ)

Y iRc = −

√

1/2 Pc Gic sin(φ)

Y iIc = +

√

1/2 Pc Gic cos(φ)

where φ = 2π U il , (6)

and, Gil , U

il , G

ic, U

ic are other random variables as

above.

The partially polarized observed voltage corre-sponding to a given sample is then

X iR =

[

X iRr + X i

Rl + X iRc

]

X iI =

[

X iIr + X i

Il + X iIc

]

Y iR =

[

Y iRr + Y i

Rl + Y iRc

]

Y iI =

[

Y iIr + Y i

Il + Y iIc

]

, (7)

and the Stokes parameters corresponding to thissample are computed according to Eqn. 2. TheseStokes-parameters samples are averaged over Nsamples to achieve a desired resolution and statis-tical significance in the simulated time series.

17