The Crab Pulsar at Centimeter Wavelengths II: Single Pulses
RRevised, 25 August 2016
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THE CRAB PULSAR AT CENTIMETER WAVELENGTHS II: SINGLE PULSES
T. H. Hankins , J. A. Eilek , G. Jones
Revised, 25 August 2016
ABSTRACTWe have carried out new, high-frequency, high-time-resolution observations of the Crab pulsar.Combining these with our previous data, we characterize bright single pulses associated with theMain Pulse, both the Low-Frequency and High-Frequency Interpulses, and the two High-FrequencyComponents. Our data include observations at frequencies ranging from 1 to 43 GHz with timeresolution down to a fraction of a nanosecond. We find at least two types of emission physics areoperating in this pulsar. Both Main Pulses and Low-Frequency Interpulses, up to ∼
10 GHz, arecharacterized by nanoshot emission – overlapping clumps of narrow-band nanoshots, each with itsown polarization signature. High-Frequency Interpulses, between 5 and 30 GHz, are characterized by spectral band emission – linearly polarized emission containing ∼
30 proportionately spaced spectralbands. We cannot say whether the longer-duration High-Frequency Component pulses are due to ascattering process, or if they come from yet another type of emission physics.
Subject headings: pulsars: general; pulsars: individual (Crab pulsar) INTRODUCTION
The pulsar in the Crab Nebula does not play by therules set down for “normal” pulsars. Most pulsars haveonly one radio pulse per rotation period, and a few havetwo, but seven components have been found so far in theCrab’s mean profile. In many pulsars the radio pulse ap-pears at a different rotation phase than the high-energypulses, but in the Crab the two main radio pulses appearat the same phases as their high-energy counterparts.To make matters more interesting, the seven radiocomponents do not behave as expected. The star’s meanradio profile is a strong function of frequency. Some com-ponents disappear, others appear, as one moves from lowto high radio frequencies. One might think one was look-ing at a totally different pulsar when observing belowor above a few GHz. Furthermore, the temporal andspectral characteristics of individual radio pulses changedramatically among components. This suggests that dif-ferent physical conditions are operating in the regionswhich emit some of the components – an idea that chal-lenges symmetries built into all current models of thepulsar magnetosphere.In previous work, our group studied the mean profileof the Crab pulsar at radio frequencies between ∼ ∼
43 GHz (Hankins et al.2015). We carried out high-time resolution observationsof individual bright pulses at frequencies ∼ [email protected] Physics Department, New Mexico Tech, Socorro, NM 87801 Adjunct Astronomer, National Radio Astronomy Observa-tory Jansky Fellow, National Radio Astronomy Observatory, So-corro, NM 87801 Columbia University, New York, NY 10027
TABLE 1Components of the Mean Profile
Component Acronym FrequencyRangePrecursor PC 0.3 – 0.6 GHzMain Pulse MP 0.3 – 4.9 GHzHigh-Frequency Interpulse HFIP 4.2 – 28.4 GHzLow-Frequency Interpulse LFIP 0.3 – 3.5 GHzHigh-Frequency Component 1 HFC1 1.4 – 28.0 GHzHigh-Frequency Component 2 HFC2 1.4 – 28.0 GHzLow-Frequency Component LFC 0.6 – 4.2 GHz
Note . — Frequency range over which component is de-tected in mean profiles. Occasional single pulses may be de-tected outside this range, but they are too rare to contributeto the mean profile. From Hankins et al. (2015).
Components of the mean profile
Each component in the mean profile of the Crab pul-sar evolves with observing frequency (Moffett & Hank-ins 1996; Hankins et al. 2015). Two bright componentsdominate the mean profile at low radio frequencies (be-low a few GHz): the Main Pulse and the Low-FrequencyInterpulse. The latter lags the Main Pulse by 145 ◦ ofphase. Two weaker components, also detectable below afew GHz, lead the Main Pulse by 36 ◦ (the Low-FrequencyComponent) and by 20 ◦ (the Precursor). At high radiofrequencies (between 9 and 30 GHz) the profile is dom-inated by three quite different components. Both theMain Pulse and the Low-Frequency Interpulse becomeweak and disappear altogether. A new component, theHigh-Frequency Interpulse, leads the Low-Frequency In-terpulse by ∼ ◦ . In addition, two new High-FrequencyComponents appear. At 5 GHz they follow the High-Frequency Interpulse by ∼ ◦ and ∼ ◦ of phase, andtheir phase lag increases at higher observing frequencies.We summarize these components in Table 1. Low or high emission altitudes?
Pulsar radio emission has long been held to come fromlow altitudes close to the star’s polar caps. In the Crab a r X i v : . [ a s t r o - ph . H E ] A ug Hankins, Eilek & Jonespulsar, however, the Main Pulse and the two Interpulsesappear at the same rotation phases as the two brightcomponents in the high-energy light curve (Abdo et al.2010). This is also the case for some millisecond pulsars(e.g., Johnson et al. 2016). Pulsed high-energy emis-sion is now understood to come from extended “caus-tic” regions at moderate to high altitudes in the mag-netosphere (e.g., Abdo et al. 2013). If the radio pulseswere to come from the surface, they would lead the high-energy pulses by a substantial fraction of the rotationperiod (e.g., Pierbattista et al. 2016). For the Crab, thephase agreement of radio and high-energy pulses sug-gests the relevant emission regions are in the same partsof the magnetosphere. In addition, our ∼ ◦ viewingangle relative to the star’s rotation axis (Ng & Romani2004) precludes our seeing more than one bright polar-cap pulse per rotation period. We therefore argue that radio emission from the Main Pulse and Interpulses ofthe Crab pulsar originates at high altitudes . In Eilek &Hankins (2016) we suggest that the emission zones maylie between (0 . − . R LC , where R LC is the radius ofthe light cylinder.The origin of the two High-Frequency Components isless clear. Their unusual rotation phases suggest to usthat they also come from high altitudes in the magne-tosphere, but not from the same regions that producethe Main Pulse or the Interpulses. The story of the Pre-cursor and Low-Frequency Component may be different,however. Because they appear earlier in rotation phasethan the Main Pulse, and they do not have clear high-energy counterparts, it may be that they come from lowaltitudes, above one of the star’s polar caps. If this is thecase, the star’s magnetic axis must be inclined ∼ ◦ toits rotation axis, in order for our sight line to intersect apolar cap once per rotation period. Single pulses and emission physics
In previous work we studied individual Main Pulsesand High-Frequency Interpulses between 1 and 10 GHz(Hankins et al. 2003; Hankins & Eilek 2007; Crossley etal. 2010; Hankins et al. 2015). We showed that MainPulses have very different characteristics from High-Frequency Interpulses when observed at sufficiently hightime resolution. Main Pulse emission is bursty on sub-microsecond timescales. The spectrum of each “mi-croburst” fills our observing band. Once in awhile aMain Pulse is resolved into well-separated “nanoshots,”each lasting no more than a few nanoseconds. High-Frequency Interpulse emission is also bursty, on few-microsecond timescales. Its spectrum, however, is verydifferent. Emission from the High-Frequency Interpulseis confined to proportionately spaced spectral emissionbands , with band separation 6% of the center frequency.We believe the very different characteristics of MainPulses and High-Frequency Interpulses show that differ-ent physics is operating in the parts of the magnetospherewhich lead to each component. We recall that three con-ditions must converge to create pulsar radio emission (seealso Eilek & Hankins 2016). (1) There must be a sourceof available energy which can be tapped for radio emis-sion. This may be relativistic plasma flow in the open In this paper we refer to optical, X-ray and γ -ray emission as“high-energy” emission. field line region; magnetic reconnection has also beensuggested. (2) There must be a mechanism by which theavailable energy is converted to coherent radio emission.This probably involves coherent charge motion and maystart with a plasma instability. (3) Finally, there mustbe a long-lived site in the magnetosphere within whichconditions are right for items (1) and (2) to work. Thepolar caps provide one such region; high-altitude causticsmay be another.We collectively refer to all of these requirements as the“emission physics” responsible for pulsed, coherent ra-dio emission. We believe our high-time-resolution obser-vations provide important clues to the emission physicsfor mean-profile components of the Crab pulsar. In thispaper we present new data for Main Pulses, both In-terpulses and the High-Frequency Components, and webriefly discuss possible emission physics for each of thesecomponents. In a companion paper, Eilek & Hankins(2016), we critique models of the radio emission mech-anism and use our data to constrain conditions in theemission zones for the bright components. What’s coming in this paper
In this paper we present new single-pulse data between14 and 43 GHz, and combine it with our previous databelow 10 GHz to characterize the five bright componentsof the Crab pulsar’s mean profile.We begin with the Main Pulse and the Low-FrequencyInterpulse. After an overview of our observations, inSection 2, we discuss the short-lived microbursts whichcharacterize Main Pulses in Section 3. We show thatmicrobursts continue in Main Pulses up to 22 GHz, andthat Low-Frequency Interpulses have the same tempo-ral and spectral signature as Main Pulses. In Section 4we explore the polarization of Main Pulses down to few-nanosecond time resolution. We show that polarizationin Main Pulses can fluctuate on such short timescales,and argue that both Main Pulses and Low-FrequencyInterpulses are composites of overlapping nanoshots. InSection 5 we present one very interesting Main Pulsewe captured at 43 GHz, which suggests interesting newphysics at this high frequency.We then move to the High-Frequency Interpulse. InSection 6 we present new, high-frequency observations ofthe spectral emission bands. We show the proportionalfrequency spacing of the bands continues up to 30 GHz,and we critique some models which have been proposedto explain the bands. In Section 7 we show that in-dividual High-Frequency Interpulses show strong linearpolarization with constant position angle that is indepen-dent of the rotation phase at which the pulse appears. InSection 8 we show that the High-Frequency Interpulse ischaracterized by variable, intrinsic dispersion that mustarise locally to the emission region for that component.In Section 9 we turn to the two High-Frequency Com-ponents, and present single pulses in each componentthat we captured at 9 GHz. We show that individualHigh-Frequency-Component pulses are relatively weak,but last much longer, than typical Main Pulses or Inter-pulses. We consider possible origins of these two compo-nents, and ask whether they represent yet another typeof emission physics in this pulsar.Finally, in Section 10, we summarize our results, dis-cuss what we have learned about this pulsar, and pointingle Pulses from the Crab Pulsar 3out questions which still need answers. OBSERVATIONS
We recorded the data we present in this paper duringseveral observing sessions at the Karl G. Jansky VeryLarge Array, at the Arecibo Observatory, at the RobertC. Byrd Green Bank Telescope, and at the Goldstone-Apple Valley Radio Telescope. We used our Ultra High Time Resolution System(UHTRS), as described in Hankins et al. (2015) to cap-ture individual pulses from the Crab pulsar. In this sys-tem, a digital oscilloscope, triggered by pulses that ex-ceed six times the noise levels as tabulated in Table 2,sampled and recorded the received voltages from bothpolarizations. The data were processed off-line using co-herent dedispersion (Hankins 1971; Hankins & Rickett1975), which allowed time resolution down to the inverseof the receiver bandwidth.Because pulses bright enough for us to record with theUHTRS formally coincide with the high-flux tail of thenumber versus flux histogram for single pulses, as seenby Argyle & Gower (1972) and Lundgren et al. (1995) atlower frequencies, they might loosely be called “giant”pulses. However, it is not yet clear whether such high-flux pulses are physically similar to, or different from, themore common “weak” pulses. In fact, the distributionof pulse amplitudes in the Crab pulsar appears to becontinuous from weak to strong (see, e.g., Karuppusamyet al. 2010). In what follows – as also in Hankins &Eilek (2007); Crossley et al. (2010); Hankins et al. (2015)– we do not attempt to distinguish between weak andstrong pulse populations, but just discuss Main Pulses,Interpulses and High-Frequency-Component Pulses.We have included relevant telescope and observing pa-rameters in Appendix I. MICROBURSTS: VARIABLE ENERGY SUPPLY?
We previously found that Main Pulses, captured be-tween 1 and 10 GHz, contain microsecond-long radiobursts (Hankins & Eilek 2007; Crossley et al. 2010; Jess-ner et al. 2010). Microbursts typically last a few mi-croseconds, and are shorter at higher frequencies. Theycan occur anywhere in the probability envelope definedby the mean-profile component, which extends for sev-eral hundred microseconds (e.g., Hankins et al. 2015).Their spectrum is relatively broadband: it extends atleast across the 2.5 GHz observing band we used in Han-kins & Eilek (2007). In this section we extend our MainPulse study to larger bandwidths, higher radio frequen-cies and polarization studies. We also show that the Low- The Very Large Array (VLA) is an instrument of the NationalRadio Astronomy Observatory, a facility of the National ScienceFoundation operated under cooperative agreement by AssociatedUniversities, Inc. The Arecibo Observatory (AO) is operated by SRI Interna-tional under a cooperative agreement with the National ScienceFoundation (AST-1100968), and in alliance with Ana G. M´endez-Universidad Metropolitana, and the Universities Space ResearchAssociation. The Green Bank Telescope (GBT) is an instrument of theNational Radio Astronomy Observatory, a facility of the NationalScience Foundation operated under cooperative agreement by As-sociated Universities, Inc. The Goldstone-Apple Valley Radio Telescope (GAVRT) is op-erated by the Lewis Center for Educational Research with supportfrom the NASA Jet Propulsion Laboratory.
Fig. 1.—
Comparison of a Low-Frequency Interpulse and a MainPulse: both contain microbursts with spectra extending across theobserving band. Top, a typical Low-Frequency Interpulse seen at1.4 GHz, plotted with 320 ns time resolution and 1.5625 MHzspectral resolution, de-dispersed with dispersion measure (DM) of56.8270 pc-cm − . Contour levels are 0.5, 1, 2, 5, 10, and 20 kJy.Bottom, a typical Main Pulse at 1385 MHz, plotted at 320 ns timeresolution, and 1.5625 MHz spectral resolution, de-dispersed withDM of 56.79682 pc-cm − . Contour levels are 0.2, 0.5, 1, 2, 5 kJy.In this and later figures, the orange line to the left is the equalizedoff-pulse response function of the receiver. Frequency Interpulse has the same temporal and spectralcharacteristics as does the Main Pulse.
Microbursts in Low-Frequency Interpulses
Low-Frequency Interpulses and Main Pulses share sim-ilar temporal and spectral properties. In Figure 1 weshow one example of a Low-Frequency Interpulse anda comparable Main Pulse, both observed near 1.4 GHzat the Very Large Array. Although the Low-FrequencyInterpulse is not detectable in the mean profile muchabove 4 GHz (Hankins et al. 2015), we have captureda few single Low-Frequency Interpulses at higher fre-quencies. Jessner et al. (2010) also present one 8 GHzpulse which they captured at the rotation phase of theLow-Frequency Interpulse. These data show that thetemporal and spectral characteristics of Low-FrequencyInterpulses are the same as for Main Pulses. The emis-sion usually arrives in one or a few microsecond-long mi-crobursts, the spectrum of which is continuous acrossthe observing band. We show in Section 4.2 that anoccasional Low-Frequency Interpulse is found to contain Hankins, Eilek & Jonesseveral well-separated nanoshots. Based on these simi-larities to characteristics of the Main Pulse, we concludethat the same emission physics governs both components.
Microbursts in Main Pulses
To probe larger bandwidths, we used the GAVRT 34-mtelescope to record single Main Pulses with an observingband extending from 2 to 10 GHz, and 10 ns time res-olution. We show one example of a multi-burst MainPulse, captured by multiple CASPER iBob devices atGAVRT, in Figure 2. These results show that the mi-croburst spectrum extends at least over the full 8 GHzfrequency range in the pulses we captured with GAVRT. Fig. 2.—
A typical Main Pulse, with three microbursts, cap-tured between 2 and 10 GHz with 10 ns time resolution using theGoldstone-Apple Valley Radio Telescope. The spectrum resolutionis 8 MHz. The color bar at the top right describes the power in thefrequency-time plane. The left panel shows on-pulse (red) and off-pulse (blue) power as a function of frequency, integrated across thepulse. The high spikes in the on-pulse power correspond to strong,narrow-band spikes within each microburst. The right panel showsthe pulse shape at 8 frequencies within the pulse. All 8 pulses areover-plotted in the top figure, where the dotted lines separate theon-pulse and off-pulse regions used for the left panel. All powersgiven in terms of off-pulse noise (RMS).
To probe higher frequencies, we recorded single pulseswith our UHTRS between 14 and 43 GHz at the GBT.As shown in Figures 4 and 5 of Hankins et al. (2015)at frequencies above about 10 GHz the occurrence ofstrong Main Pulses is rare as compared with strong High-Frequency Interpulses. In fact, using the UHTRS we cap-tured only 6 Main Pulses above 10 GHz in several tens ofobserving hours. We show in Figure 3 two Main Pulses,one at 14 and one at 20 GHz. It is clear that the nature Collaboration for Astronomy Signal Processing and Electron-ics Research, https://casper.berkeley.edu of Main Pulses is the same above and below 10 GHz.The same microburst structure is still apparent at thesehigher frequencies, and the spectrum of each microburstcontinues across our full observing band. We concludethat Main Pulses are well described as broadband mi-crobursts across the entire 1 to 30 GHz frequency rangewhere we have studied them.
Fig. 3.—
Two Main Pulses recorded above 10 GHz. In agree-ment with Main Pulses seen at lower frequencies (Hankins & Eilek2007), these two examples contain a few microbursts, and have aspectrum extending across our observing band. Top, a Main Pulseat 12 −
16 GHz, shown with 12.8 ns time resolution, 78 MHz spec-tral resolution, and de-dispersed with DM of 56.78900 pc-cm − .The contour levels are 0.1, 0.2, 0.5 and 1 kJy. Bottom: a MainPulse at 18 . − . − . The contour levels are 0.2, 0.5, 1 and 5 kJy. The mi-crobursts here are narrower than those in Figure 1, consistent withthe width-frequency trend found in Hankins et al. (2015). A note on dispersion measure
At our high time resolution, details of the pulses aresensitive to the exact value of the dispersion measure(DM) used for the coherent dedispersion operation. Todetermine an “optimum DM” for the figures presented inthis paper we generally started with the DM value givenby the Jodrell Bank Crab Pulsar Monthly Ephemeris for each observing epoch. From there, we followed one ∼ pulsar/crab.html ingle Pulses from the Crab Pulsar 5of two paths. For sharp pulses, such as bursts withina Main Pulse, we calculated the intensity variance afterdedispersion with a range of DM values around the Jo-drell value and used the DM which maximized the vari-ance. Alternatively, for broad pulses such as the High-Frequency Components and the High-Frequency Inter-pulse, we experimented with small DM changes to deter-mine “by eye” the DM value that best aligned the start ofthe pulse in the dynamic spectrum. With both methods,we typically found it necessary to refine our “optimum”DM values up to ∼ − pc-cm − relative to the Jodrellvalue.In Section 8 we report on systematic methods to deter-mine the DM of single pulses that are more appropriatefor analyzing a large number of pulses. Our results inthat section verify the size of the difference between theDM of individual pulses and the Jodrell value. Sporadic energy release in the emission zones
Microbursts show that the Main Pulse and Low-Frequency Interpulse emission zones are neither uniformnor homogeneous. They must be variable in space andtime. We envision isolated, short-lived regions where en-ergy builds up until it some or all of it is released in aburst of coherent radio emission. After a period of qui-escence, perhaps the same region is re-energized, or per-haps the process moves to a different part of the emissionzone. One possibility, which we discuss in Eilek & Han-kins (2016), is that this sporadic energization involves a“sparking” cycle (Ruderman & Sutherland 1975) in anunsteady pair cascade. The details are not yet clear,however, because it is not known how such a pair cas-cade cycle operates (if at all) at the high altitudes wherethe Main Pulse and Low-Frequency Interpulse originate. MAIN PULSES: POLARIZATION AND NANOSHOTS
The Main Pulse and Low-Frequency Interpulse areonly weakly polarized in the mean radio profile (Mof-fett & Hankins 1999; S(cid:32)lowikoska et al. 2015). S(cid:32)lowikoskaet al. (2009) also found low fractional polarization forthe optical
Main Pulse and Interpulse, and suggestedthe cause is depolarization from extended emission intwo-pole caustics (Dyks et al. 2004). Their suggestion issupported by rapid position angle swings in the two opti-cal pulses, and is consistent with a high-altitude causticorigin for optical emission.However, observations of single radio pulses tell a dif-ferent story. Weak polarization in those pulses is morecomplicated than simple caustic effects. Radio observa-tions of single pulses by Hankins et al. (2003) with 2 nstime resolution, Soglasnov (2007) with ∼
30 ns time reso-lution, and Jessner et al. (2010) with 2 ns time resolution,show rapid changes of linear and circular polarization ontime scales of a few nanoseconds. Such rapid variabilitydepolarizes individual pulses observed at low time resolu-tion, as well as the associated mean-profile components.
Polarization suggests nanoshots
Our larger sample of Main Pulses confirms and extendsthese results. In particular, we find that some – but notall – Main Pulses show rapid polarization fluctuations ona timescale of a few nanoseconds. To simplify the dis-cussion in this section, we use “low time resolution” to mean smoothing to ∼ −
100 ns, and “high time resolu-tion” to mean smoothing to only a few nanoseconds. Ourbest possible time resolution, of course, is the inverse ofour observing bandwidth, i.e., a fraction of a nanosecondfor most of the observations presented here. To reduceuncertainty in the polarized flux, we always smooth ourdata by at least several times that limit.
Fig. 4.—
Polarization of a Main Pulse, captured between 8 and10.5 GHz, de-dispersed with DM of 56.75252 pc-cm − and dis-played with 51.2 ns time resolution. Top panel shows total ( I ;black) and linearly polarized ( L ; green) intensity. Middle panelshows total and circularly polarized ( V ; red) intensity. Bottompanel shows position angle of linear polarization. Orange lines –indistinguishable from zero-intensity axis in this example but ap-parent in later figures – indicate 4 times the off-pulse noise level;only polarization above this threshold is shown. In this pulse, theposition angles are ordered in the first three microbursts, but be-come disordered in the strongest microburst and the lagging tail.Compare the polarization behavior to that of the pulses in Figures5 and 6. Low time resolution: ordered polarization is rare
Some Main Pules observed at low time resolution showstrong linear polarization and ordered position angle be-havior. Jessner et al. (2010) showed two examples. InFigure 4 we show another, at 51.2 ns time resolution.The leading part of this pulse shows strong polarizationand ordered position angle rotation. However, the po-sition angle behavior appears to change with each mi-croburst – clearly inconsistent with the simple rotating-vector model of Radhakrishnan & Cooke (1969). Inter-estingly, the position angle behavior of this pulse changesagain in the last two microseconds, when it becomes ap-parently random.Such disordered behavior is typical of most of the MainPulses we have captured, where well-ordered position an-gle behavior is the exception rather than the rule. At Hankins, Eilek & Jones
Fig. 5.—
Polarization of two microbursts in another Main Pulse,captured between 8 and 10.5 GHz, de-dispersed with DM of56.76080 pc-cm − , and displayed with 3.2 ns time resolution. Lay-out is the same as in Figure 4. Both the amount and the positionangle of the linear polarization change rapidly in this pulse, as doesthe magnitude and sign of the circular polarization. Compare thepolarization behavior to that of the pulses in Figures 4 and 6. low time resolution, most Main Pulses have disorderedposition angle behavior, and linear polarization a smallfraction of the total intensity.The story is similar for circular polarization. SomeMain Pulses show significant circular polarization at lowtime resolution, such as the pulse in Figure 4. In otherexamples circular polarization is significant, but changessign between microbursts. Still other pulses show littleor no circular polarization when smoothed to low timeresolution. High time resolution: polarization fluctuates rapidly
The story becomes more interesting when pulses arestudied at high time resolution. Main Pulses we havecaptured often show significant linear and circular polar-ization which fluctuates on timescales no longer than afew nanoseconds.In Figures 5 and 6 we show two typical examples dis-played at 3.2 ns time resolution. In both examples, theposition angle of the linear polarization fluctuates ontimescales of a few nanoseconds. These pulses wouldbe only weakly polarized if studied at lower time res-olution. Circular polarization in the pulse in Figure 5changes sign on similar timescales. This pulse loses cir-cular polarization when studied at lower time resolution.By contrast, circular polarization of the pulse in Figure6 is predominantly negative, but rapid sign changes canbe seen in the two leading microbursts.Figures 5 and 6 also show strong, rapid fluctuationsin total intensity. Because these fluctuations are much larger than off-pulse noise, we believe they are physi-cal. We have argued before that microbursts in theMain Pulse and the Low-Frequency Interpulse are notthe smallest unit of emission. Our previous high timeresolution observations show that Main Pulses can occa-sionally be resolved into shorter-lived structures, whichwe have called “nanoshots” (Hankins et al. 2003; Hankins& Eilek 2007, see also Jessner et al. 2010). Such rapidtotal intensity fluctuations are very suggestive of blendednanoshots, in agreement with the amplitude-modulatednoise model of Rickett (1975). Our polarization datasuggest that each nanoshot has its own polarization iden-tity – its own linear and circular polarization signature –which may or may not change from one nanoshot to thenext.
Fig. 6.—
Polarization of several microbursts in a third MainPulse, captured between 8 and 10.5 GHz, de-dispersed with DMof 56.75900 pc-cm − , and displayed with 3.2 ns time resolution.Layout is the same as in Figure 4. Both the intensity and theposition angle of the linear polarization change rapidly in this pulse,similar to the pulse in Figure 5. Circular polarization in this pulsefluctuates rapidly, but maintains the same sign throughout mostof the pulse; this behavior differs from the pulse in Figure 5, butresembles the pulse in Figure 4. Nanoshots observed directly
Very occasionally the nanoshots in a Main Pulse aresufficiently sparse to be studied directly. Figure 7 showssuch examples, a Main Pulse and a Low-Frequency In-terpulse. Both are shown at relatively low time reso-lution, to improve the clarity of the dynamic spectra.(See Hankins et al. 2003; Hankins & Eilek 2007; Eilek &Hankins 2016, for other examples). The dynamic spec-trum in Figure 7 shows that well-isolated nanoshots havea relatively narrowband spectrum, with emission band-width small compared to our observing bandwidth. Ex-ingle Pulses from the Crab Pulsar 7amples are the nanoshots at 9, 22.5 and 24 microsecondsin the top panel of Figure 7. The spectrum of othernanoshots extends across our observing band, suggestingthey contain substructure which is not resolved at thistime resolution. The existence of nanoshots in a Low-Frequency Interpulse is consistent with our other evi-dence that these two components have similar temporaland spectral characteristics.
Fig. 7.—
Two examples of nanoshots. Top: a single Main Pulsebetween 8 and 10.5 GHz. Shown with time resolution 51.2 ns,spectral resolution 19.5 MHz, and de-dispersed at 56.73736 pc-cm − . The contour levels are 0.05, 0.1, 0.2, 0.5 and 1 kJy. Bottom:a single Low-Frequency Interpulse seen between 6.4 and 7.6 MHz,shown at time resolution 64.0 ns, spectral resolution 39 MHz, andde-dispersed at DM of 56.74001 pc-cm − . The contour levels are0.1 and 0.2 kJy. Note, the low-level emission between nanoshotsis real, well above the 5.9 Jy level of the off-pulse noise. Bothexamples illustrate the characteristics of resolved nanoshots: theyare extremely short-lived and have frequency spread less than ourobserved bandwidth. Compare Figure 8 which shows some of thenanoshots at higher time resolution. In Figure 8 we explore the nanoshot polarization byzooming into a high-time-resolution display of some ofthe nanoshots in the Main Pulse of Figure 7. At thistime resolution, 4.0 ns, substructure can be seen in mostof the nanoshots. The nanoshots in this example aredominated by linear polarization, although weak circularpolarization can also be seen. Resolved nanoshots wehave captured in other pulses show different polarizationsignatures. For instance, in Eilek & Hankins (2016) we show a Main Pulse in which some nanoshots are linearlypolarized and others are circularly polarized.Interestingly, each nanoshot in Figure 8 has its own po-sition angle “sweep”; there is no sign of ordered positionangle behavior through the entire pulse. This is also thecase for other polarized nanoshots we have captured. Wetake this as another indication that the Radhakrishnan& Cooke (1969) model does not apply here.
Fig. 8.—
A zoom into the nanoshot clumps seen between 5 and16 µ s in upper Figure 7, to display their polarization. Layout isthe same as in Figure 4, except that polarization above 3 times thebackground noise is shown here. At the displayed time resolution,4.0 ns, substructure can now be seen in most of the nanoshots,consistent with their blended dynamic spectrum in Figure 7. Inthis example, the nanoshots are mostly linearly polarized, withsome weak circular polarization. Note that each nanoshot clumphas its own position angle behavior. Emission physics of the nanoshots
We believe nanoshots in the Main Pulse and the Low-Frequency Interpulse provide an important test of ra-dio emission models. A successful model must explainthree key characteristics. (1) The basic units of the radiobursts are nanosecond-long “shots” of coherent emission.(2) The product of the center frequency and duration ofsuch a nanoshot obeys νδt ∼ O (0 . AN UNUSUAL MAIN PULSE AT 43 GHZ
Although the Crab pulsar becomes fainter at high fre-quencies, we managed to capture one Main Pulse at 43GHz. This is the highest frequency at which a singlepulse has been captured from this pulsar; we spent 12hours of observing time to catch it. We show this un-usual pulse in Figure 9. Unlike Main Pulses we havecaught at lower frequencies, this pulse lasted 2 µ s, withno sign of internal sub- µ s microbursts. More strikingly,the upward sweep of the narrow emission band in its dy-namic spectrum is unique among all the Main Pulses wehave observed. Fig. 9.—
The total intensity of a Main Pulse recorded at 43.25GHz and de-dispersed using DM of 56.794 pc-cm − (taken fromJodrell Bank monitoring for our observing date) is shown with atime resolution of 44.8 ns. The frequency resolution of the dynamicspectrum is 78 MHz. The Intensity contour levels in the dynamicspectrum are 0.2, 0.5, 1, and 2 kJy. The off-pulse noise level forthe total intensity is 15.2 Jy, and for the dynamic spectrum, 110Jy. We checked whether the unusual dynamic spectrum ofthis pulse could be due to dedispersion processing withthe wrong DM. We used the DM determined from Jodrell Bank monthly Crab monitoring at 1.4 GHz appropriatefor our observing date. Dedispersion with a much lowerDM ( ∼
41 pc-cm − , well below the 56 .
794 pc-cm − Jo-drell value) would be necessary to “straighten up” theemission spectrum. We cannot believe that such a “hole”in the electron density would have appeared in the CrabNebula, then vanished again, within only a few days. The upward sweep in frequency is also very unlikely toarise from a frequency error in the local oscillator setupor in our processing. In the dispersion removal parame-ters, we would have to use a center frequency of 60 GHzto eliminate the frequency sweep. We conclude that theupwards sweep in Figure 9 is real – and completely un-expected.We also checked whether our identification of this as aMain Pulse could be wrong. Using TEMPO to convertthe arrival time of this single 43 GHz pulse to rotationphase, we found the pulse arrived within 6 µ s of theMain Pulse phase predicted by TEMPO. Furthermore,although the dynamic spectrum of this pulse is sugges-tive of a single emission band from a High-FrequencyInterpulse, the bandwidth separation ∆ ν (cid:39) . ν whichwe measured at lower frequencies (Hankins & Eilek 2007,also Section 6) would put adjacent bands at ∆ ν = ± . THE HIGH-FREQUENCY INTERPULSE: SPECTRALEMISSION BANDS
In Hankins & Eilek (2007) we discovered, to our sur-prise, that the High-Frequency Interpulse is strikinglydifferent from its Low-Frequency counterpart. In ad-dition to the 7 ◦ phase shift, the two components havedifferent temporal and spectral characteristics. High-Frequency Interpulses contain one broad burst of emis-sion, typically several microseconds long, rather thanseveral shorter microbursts. We have never found sparsenanoshots in a High-Frequency Interpulse. Furthermore,we found a totally unexpected spectrum between 5 and10 GHz in the High-Frequency Interpulse. Rather thancontinuous, broadband emission, its dynamic spectrumcontains a set of discrete emission bands . These bandsare not uniformly spaced, as one might expect from har-monic emission. Instead, we found they are proportion-ally spaced , with frequency separation ∆ ν ∼ . ν be-tween 5 and 10 GHz. In this section we show that the The DM monitored by Jodrell Bank varied by ∼ .
20 pc-cm − over ∼
12 years; our 20 and 23 GHz observations, made onlyseveral days earlier, are consistent with the Jodrell Bank value. see http://tempo.sourceforge.net ingle Pulses from the Crab Pulsar 9emission bands continue up to at least 30 GHz. Fig. 10.—
Two High-Frequency Interpulses observed above 10GHz. Top, a pulse captured between 12 and 16 GHz, de-dispersedwith DM of 56.801254 pc-cm − . The contour levels are 0.2, 0.5,1, and 2 kJy. Bottom, a pulse captured between 25.9 and 30.3GHz, de-dispersed with DM of 56.799770 pc-cm − . The contourlevels are 0.5, 1, and 2 kJy. Both pulses are displayed with timeresolution 25.6 ns, and spectral resolution 78.125 MHz. In thetop pulse, five spectral bands can be seen, with three band “sets”within each band. In the bottom pulse, three spectral bands canbe seen. Note the larger frequency spacing between bands in thelower example, consistent with the fit in equation 1, also shown inFigure 11. Emission bands at higher frequencies
High-Frequency Interpulses are more abundant thanMain Pulses above 9 GHz (Hankins et al. 2015); we wereable to catch 240 new examples between 12 and 30 GHz.Figure 10 shows two typical examples, at 14 and 28 GHz.These new data show that the character of the High-Frequency Interpulse continues unchanged at these highfrequencies. The spectral bands are still there, and theirspacing increases with frequency, as expected from ourprevious results.To explore the band spacing at these new frequen-cies, we measured the band center frequencies of manyof the High-Frequency Interpulses we captured above 10GHz. In Figure 11 we combine these results with ourearlier measurements (Hankins & Eilek 2007). It is clearthat our previous linear relation between band spacing(∆ ν ) and band-center frequency ( ν ) continues to 30 GHz. Specifically, we fit the band spacings with a linear func-tion,∆ ν = − . ± . . ± . ν (1)finding a reduced chi-squared value of 1.3949 (all fre-quencies measured in GHz). We also tested quadraticfits. We found no evidence of a quadratic relationshipbetween ∆ ν and ν , nor did we find any evidence of anoffset from the origin. Our band spacing measurementsare are consistent with a straight line passing throughthe origin and having a slope of about 6%.Although the data in Figure 11 extend from 5 to 30GHz, we have no direct evidence that the bands them-selves exist throughout that range in any one pulse. Ourlargest bandwidth is 4 GHz, too narrow to sample morethan a few of the bands at any one time. However, wehave never seen a High-Frequency Interpulse that didnot show emission bands throughout our observing band-width. This fact, plus the clear linear relation in Figure11, suggests to us that every High-Frequency Interpulsecontains continuous spectral bands from (at least) 5 to30 GHz. If this is the case, there must be about 30 suchbands over that full range. Fig. 11.—
The Interpulse emission band spacing is plottedagainst the band center frequency, using data from this paper andfrom Hankins & Eilek (2007). The plot is extended down to (0,0)to emphasize that the first term of the linear fit is consistent withzero. The band widths, measured by Gaussian fits, are shown onthe same scale: the bands are much narrower than their spacing.
We also measured the spectral widths of the Inter-pulse bands, by fitting Gaussians to individual bands.Although some of the fits were contaminated or biasedbecause of overlapping band sets, the general behavior ofthe bandwidths can be seen in Figure 11. We find thatthe spectral width of an emission band does not increaseas fast as the band separation does.We occasionally see emission bands in the High-Frequency Interpulse that appear to drift upwards in fre-quency, such as in the lower panel of Figure 10. However,upon close examination of many pulses, we find the ap-pearance of upward drift is usually caused by the super-position of new band sets that begin slightly later in timeat slightly higher band frequencies. This can be seen inthe upper panel of Figure 10; another example is shownin Figure 7 of Hankins & Eilek (2007). We have never0 Hankins, Eilek & Jonesseen any emission bands in a High-Frequency Interpulsewith the strong upward frequency drift – about 5 percentof the band center – that we found in the pulse shown inFigure 9.
Is there frequency memory in the emission bands?
Searching for more clues on the origin of the emis-sion bands, we also explored whether bands in separateHigh-Frequency Interpulses have any “frequency mem-ory.” Are the band frequencies, as well as their fractionalspacing, steady over time?
Fig. 12.—
Two examples of individual dynamic spectra for allpulses recorded during one observing day, with the average spec-trum for the day shown in the top panel. The example on the topappears to show preferred frequencies over the entire day; the ex-ample on the bottom does not. In each example, gaps in the dataare due to system, pointing and calibration checks of the telescope.
One way to check this uses multiple pulses within onerotation period. We know the phase windows associatedwith components of the mean profile are much wider thanthe duration of a single pulse (Hankins et al. 2015). Forinstance, at > ∼ ∼ µ s, much less than the ∼ µ swidth of the associated component of the mean profile.Our sampling window is several times wider than thecomponent width, and we find that about 1/4 of thedata records show multiple High-Frequency Interpulsesseparated up to several hundred microseconds. If the bands have frequency memory, we would expect each ofthe multiple High-Frequency Interpulses in one rotationperiod to have similar band frequencies. However, whenwe inspected examples of multiple pulses, we did not findany consistent frequency memory.We also checked whether the emission bands within theInterpulse have preferred frequencies within a few-hourobserving run. Here the result was mixed. Figure 12shows two examples from days where a sufficient numberof pulses was recorded. In the upper panel it is clear thatboth the single-pulse and daily average spectra show fa-vored band frequencies at about 19, 20.2 and 21.4 GHz.The lower panel shows what appears to be a randomset of single-pulse band frequencies with no preferredfrequencies in the day average spectrum. Overall, weconclude there is no strong evidence for band frequencymemory. Emission physics for the High-Frequency Interpulse
Neither of the models we discussed in Section 4.3 canaccount for the emission bands in the High-Frequency In-terpulse. This is also true of the other usual suspects forpulsar radio emission (e.g., Melrose 1979; Eilek & Han-kins 2016). However, since the bands were discovered,several ad hoc models have been suggested to explainthe spectral bands.
Direct emission models
Resonant cyclotron emission.
Lyutikov (2007) sug-gested the bands are harmonic emission of plasma wavesexcited by the anomalous cyclotron instability. With hisspecific parameter choices, which require high plasmadensities close to the light cylinder, the spacing of thefirst few harmonics approximately agrees with the 6–10GHz results we presented in Hankins & Eilek (2007).Unfortunately, higher-order harmonics of the the wavemode he proposes do not obey the simple 6% propor-tionality we observe (even if 30 harmonics could some-how be excited in the first place, which may be unlikely;M. Lyutikov, private communication 2015).
Beamed superluminal emission.
Ardavan (1994) pro-posed that beamed emission from a superluminal polar-ization current pattern, outside the light cylinder, ac-counts for pulsar radio emission. Ardavan et al. (2008)applied this to the Crab pulsar, suggesting the spectralbands are due to monochromatic oscillations of the po-larization current. With their parameter choices, thefirst few bands they predict do approximately match our6–10 GHz data in Hankins & Eilek (2007). However,their model also predicts band spacing increasing withfrequency, as ∆ ν ∝ ν / . This disagrees with the simple∆ ν (cid:39) . ν proportionality we observe. Double plasma resonance.
The emission bands we findin the High-Frequency Interpulse are reminiscent of so-called zebra bands found in type IV solar flares (e.g.,Chernov et al. 2005). The most successful model for ze-bra bands is a double plasma resonance, which requiresthe upper hybrid frequency be an integer multiple of thecyclotron frequency (e.g., Chen et al. 2011; Eilek & Han-kins 2016). If the spatial structure of the local densityand magnetic field allows this resonance to be satisfiedat more than one location, a set of double-resonant har-monics can be excited. Zheleznyakov et al. (2012) sug-gested this model may also explain the High-Frequencyingle Pulses from the Crab Pulsar 11Interpulse. They note that low magnetic fields and highplasma density are needed, and speculate that such con-ditions may exist close to the light cylinder. They didnot, however, discuss the specific field and density struc-tures that would be needed to create the uniformly pro-portional spacing we observe.
Propagation models
Stimulated Compton Scattering.
Petrova has suggestedthat the High-Frequency Interpulse results from stimu-lated Compton scattering of the Main Pulse. In Petrova(2008) she argues that induced Compton scattering ofthe Main Pulse creates the High-Frequency Interpulse.Alternatively, in Petrova (2009) she proposes that Comp-ton scattering of the Main Pulse creates the Precursor,and the Precursor itself is then scattered to create theHigh-Frequency Interpulse. Both of these models arechallenged by the emission bands we observe in the High-Frequency Interpulse. Petrova (2008) suggests that scat-tering of individual nanoshots in the Main Pulse, whichconcentrates their power toward the edges of the shot’semission band, can lead to bands in the dynamic spectra.However, she presents no explanation of how this modelcan lead to the regular, proportional spacing we observeover a factor of six in frequency.
Interference models.
In Hankins & Eilek (2007) wespeculated that the emission bands may be a propaga-tion effect. If a radiation beam could be split coherently,perhaps by reflection, it may interfere with itself. Alter-natively, if low-density cavities exist in the plasma, theymight impose a discrete frequency structure on escapingradiation. Similar models have been proposed for zebrabands (e.g., Ledenev et al. 2001; Labelle et al. 2003). Wealso noted that the incoming radiation must be broad-band – extending at least from 5 to 30 GHz – in order forthese models to work. This requirement appears to ex-clude many standard pulsar radiation mechanisms whichpeak around the local plasma frequency. We suggestedthat linear acceleration emission, for instance in a doublelayers within gap regions (e.g., Kuijpers 1990), may bea possible alternative. Unfortunately, although we findsuch models attractive, we have not come up with a wayto make the small, long-lived plasma structures in themagnetosphere that such models require. POLARIZATION OF THE HIGH-FREQUENCYINTERPULSE
Observations by Moffett & Hankins (1999) showed thatthe High-Frequency Interpulse is strongly polarized inmean profiles. At 5 and 8 GHz, they found that com-ponent shows 50–70% linear polarization. Jessner et al.(2010) found this also true for a few individual High-Frequency Interpulses at 15 GHz.
Data: uniform polarization position angle
Our data support these results. High-Frequency Inter-pulses that we have captured between 5 and 24 GHz aresimilarly polarized, typically with 80–90% linear polar-ization. Circular polarization is occasionally seen, butis generally weak (10–20%) or undetectable. Figure 13shows one such example; in Eilek & Hankins (2016) weshow two other examples. The polarization position an-gle in these examples remains nearly constant throughout the pulse, in agreement with the results of Jessner et al.(2010) for High-Frequency Interpulses at 15 GHz.
Fig. 13.—
Polarization of a typical High-Frequency Interpulse,captured between 17.8 and 22.8 GHz, de-dispersed at DM of56.79476 pc-cm − , and displayed at 51.2 ns time resolution. Lay-out the same as in Figure 4, except that polarization greater than3 times off-pulse rms noise level is shown. This example is typicalof the majority of High-Frequency Interpulses we have observed: itshows strong linear polarization, only weak circular polarization,and a nearly constant position angle across the pulse. Not only do we see no significant position angle evolu-tion within most individual High-Frequency Interpulses,but the position angle is also independent of rotationphase at which the pulse occurred. Figure 14 illustratesthis for 18 pulses captured in one observing day: the po-sition angle remains approximately constant across thefull range of rotation phase at which we captured pulses.The upper panel of this Figure shows both the pulses andtheir arrival phase on the same timescale. Because indi-vidual pulses are very short-lived compared to the 0.03window of rotation phase within which the pulses arrivedon this observing day, no detail can be seen within anyof the pulses in this display. Therefore, in the two lowerpanels of this Figure we expand the timescale of eachpulse by factor of 50, while maintaining their relativearrival phases, so that structural details can be seen.Occasionally, however, we captured some High-Frequency Interpulses with significant position angle ro-tation across the pulse. Figure 15 shows one such exam-ple. Such pulses often have weaker fractional polarizationthan pulses with constant position angles.
Constraints on the emission region
Although we do not know the magnetospheric locationof the High-Frequency Interpulse emission region, we can2 Hankins, Eilek & Jones
Fig. 14.—
Polarization of a sequence of High-Frequency Inter-pulses, captured within 13 minutes on one observing run and dis-played with 51.2 ns time resolution. Upper panel shows the pulsesplotted as a function of the rotation phase at which they appeared(a 0.03 span of phase corresponds to ∼ − .
005 pulsephase, in this figure. make two general statements based on our polarizationdata. (1) The emission region must be spatially local-ized. If it were extended along the full caustics, thoughtto be the origin of pulsed high-energy emission, the High-Frequency Interpulse would be significantly depolarized(e.g. Dyks et al. 2004), which is not the case. (2) Themagnetic field direction must be approximately constantwithin the emission region, and must remain stable dur-ing the duration of each days observing run. Signifi-cant variations in the field direction would be reflectedin variable polarization position angles for single High-Frequency Interposes, which is not the case. DISPERSION OF THE HIGH-FREQUENCY INTERPULSE
The 9 GHz example pulses we presented in Hankins& Eilek (2007) showed that single High-Frequency In-terpulses can have higher dispersion measure than singleMain Pulses. In this section we explore this result in ourfull data set.
Data: excess dispersion measure
Fig. 15.—
Polarization of a High-Frequency Interpulse whichshows significant position angle rotation. Captured between 17.7and 22.4 GHz, de-dispersed at DM of 56.80600, and displayed at51.2 ns time resolution. Layout the same as in Figure 4, exceptthat polarization greater than 3 times off-pulse rms noise level isshown. The linear polarization in this pulse is weaker than thatin Figure 13, which is typical of High-Frequency Interpulses whichshow position angle rotation.
Because inspecting individual pulses (as in Section 3.3)is daunting for large numbers of pulses, we used the fol-lowing methods to analyze our full data set. We first de-dispersed each pulse using the monthly DM values givenby Jodrell Bank (“JB”; Section 3.3), spline interpolatedto each observing epoch.We split the Fourier Transform of each dedispersedpulse into the top and bottom halves of our observingband, then transformed each half band back to the timedomain and obtained the unsmoothed intensity. We thencarried out a series of correlations to determine the timedelay between the two band halves. We formed the cross-correlation function (CCF) of the upper and lower bands,and found the time at which that CCF peaked by fit-ting a parabola around the CCF maximum. We call this“method 1.” We also formed the autocorrelation func-tions (ACFs) of the intensity profiles in the upper andlower bands, and took their geometric mean. To findthe CCF peak we then cross correlated the CCF andthe mean ACF, and found the time lag at which thatcross-correlation peaked. We call this “method 2.” InFigures 16 and 17 we use method 2, which reduces thenoise in the final correlation. In Figure 18 we compareboth methods and verify that they give similar results.Once the time delay between the upper and lower bandhalves is determined, we convert it to an excess DM foreach pulse, relative to the JB value: DM tot = DM JB + δ (DM). Here we use the usual relation between pulseingle Pulses from the Crab Pulsar 13arrival time, t p , and center frequency, ν : Aν t p ( ν ) = DM tot (2)where the constant A = 2 . × − s-pc-cm − , andDM ≡ (cid:82) n e dL , measured in pc-cm − , is defined as theintegral of the election density n e over the path length L .We note that this expression for t p ( ν ) invokes the ν − functional form which holds for cold plasma, such asthe interstellar medium. There is no a priori reason toassume this dispersion law also holds for the magneto-spheric plasma. However, because the magnetosphericdispersion law is unknown, we cannot specify a betterform. We might expect a different frequency dependencein the magnetosphere to appear as a frequency depen-dence in our δ (DM) calculations, but in practice thatturned out not to be the case. Example: one illustrative day
Fig. 16.—
Excess dispersion measures, δ (DM), for individualMain Pulses (top) and High-Frequency Interpulses (bottom) cap-tured between 8.0 and 10.5 GHz, within 100 minutes on one ob-serving day, given relative to the Jodrell Bank value. Dotted linesshow the mean δ (DM) value for each pulse type; solid lines cor-respond to the JB value ( δ (DM) = 0). The δ (DM) values for theMain Pulses are approximately consistent with the JB value, butthe δ (DM) values for the High-Frequency Interpulses are generallylarger than the JB value. The scatter in the High-Frequency Inter-pulse DM values is real. The formal error bars, showing the uncer-tainty of the peak of a parabolic fit to the maximum of the CCF –are smaller than the plotted points. Outliers with | δ (DM) | > . − for the Main Pulse, | δ (DM) | > .
04 pc-cm − for the High-Frequency Interpulse, or fit uncertainty σ DM > .
01 pc-cm − foreither pulse have been deleted. These outliers result from unusualfrequency-dependent differences between the pulse shapes in thetwo bandpass halves. To illustrate the excess dispersion measures we findfor single pulses, in Figure 16 we show the δ (DM) val-ues for one observing day which contained a good num-ber of both Main Pulses and High-Frequency Interpulses.We see two important trends. (1) The values for MainPulses are approximately consistent with the JB value.(2) The dispersion measures for the High-Frequency In-terpulses can be larger or smaller than the JB values,and show significant scatter. The δ (DM) scatter for theHigh-Frequency Interpulse appears random. We find nocorrelation between δ (DM) and pulse time of arrival, ei-ther as function of rotation phase (shown in Figure 16)or clock time during the 1-hour observing run.A few points in Figure 16 have δ (DM) <
0. We do notinterpret this as inverted dispersion behavior ( dt p /dν >
0) in the pulsar. Rather, it seems likely that the JBvalue, which is based on mean profiles, may contain asmall contribution from the pulsar itself.
Systematics: all of the pulses
The trends illustrated in Figure 16 hold true in general,for nearly 740 pulses we captured between 2 and 30 GHz,in more than 50 observing runs spaced over 9 years. InFigure 17 we collect δ (DM) values for all of these pulses,separated by pulse type but combining all frequenciestogether. This figure shows that the High-Frequency In-terpulses are, on average, more dispersed than the MainPulses and are also more dispersed than the Jodrell Bankmean value. We find the mean δ (DM) ∼ .
010 pc-cm − for the High-Frequency Interpulse, with formal standarddeviation of ∼ .
016 pc-cm − . Furthermore, there is sig-nificant scatter about this mean. Individual pulses canhave δ (DM) as large as ∼ .
04 pc-cm − , or they can haveDM less than the Jodrell Bank value ( δ (DM) < δ (DM) values for theMain Pulse are consistent with the Jodrell Bank value.To explore the frequency dependence of the δ (DM) in-trinsic to the pulsar, In Figure 18 we separate the mean δ (DM) values by pulse type and by frequency. Pulsesat 2–4 GHz are Main Pulses and Low-Frequency Inter-pulses. Pulses at higher frequencies are Main Pulsesand High-Frequency Interpulses. This figure shows threetrends. (1) There is no DM difference between MainPulses and Low-Frequency Interpulses. We find no evi-dence of intrinsic DM associated with Low-Frequency In-terpulses. (2) High-Frequency Interpulses are on averagemore dispersed than Main Pulses, and we find significantscatter in their excess DM. (3) We find no evidence of fre-quency dependence in the excess DM of High-FrequencyInterpulses. The excess dispersion of the High-FrequencyInterpulse can plausibly be described by the 1 /ν behav-ior we assumed in equation (2). Origin of the intrinsic dispersion
High-Frequency Interpulses can be significantly moredispersed than either Main Pulses or Low-Frequency In-terpulses. We find the excess dispersion can fluctuatefrom zero to ∼ .
04 pc-cm − on timescales of no morethan a few minutes. Our results constrain the natureand/or location of the emission zone for this component.The δ (DM) scatter of the High-Frequency Interpulsecannot be caused by the Crab Nebula or the interstel-4 Hankins, Eilek & Jones Fig. 17.—
Excess dispersion measures, relative to the JodrellBank value, for the Interpulse (top) and the Main Pulse (bottom).We included 356 Main Pulses and 386 Interpulses, all but 3 ofwhich are High-Frequency Interpulses. The pulses were capturedat frequencies from 2 to 30 GHz, and are distributed over frequencyas described in the caption of Figure 18. Outliers are excluded,as described in Figure 16. The spread in δ (DM) for Interpulsesis clearly larger than the measurement uncertainty, and must beintrinsic to the pulsar. lar medium between us and the Crab pulsar. In nei-ther of these can the dispersion have changed in lessthan a minute (the typical time between subsequentbright pulses we captured). Furthermore, there is noreason why dispersion from the Nebula or the interstel-lar medium changed in phase with the pulsars rotation(which would be needed to enhance dispersion of only theHigh-Frequency Interpulse). We conclude the dispersionmeasure fluctuations are intrinsic to the magnetosphericemission region for the High-Frequency Interpulse .Furthermore, the excess DM cannot be from somedense region that just happens to sit above the emis-sion region for the High-Frequency Interpulse, nor can itbe due to a longer propagation path through the mag-netosphere for that component, In either case we wouldsee large δ (DM) in every High-Frequency Interpulse –which is not the case. Instead, the excess DM must ariselocally in the emission zone, perhaps due to turbulentfluctuations in the plasma density around the emissionregion. We discuss this further in Eilek & Hankins (2016)where we present one possible example of the dispersingmedium. HIGH-FREQUENCY COMPONENTS
Although High-Frequency Components (HFCs) areeasy to detect in mean profiles between 3.5 and 28 GHz
Fig. 18.—
Differences in mean DM, for Main Pulses and In-terpulses, binned by observing frequency and calculated by twodifferent methods. Squares use our method 1, while triangles usemethod 2, as described in Section 8.1. Outliers have been excludedas described in Figure 16. Bin at 2–4 GHz contains 48 MP and 3LFIP; bin at 4–8 GHz contains 219(226) MP and 64(66) HFIP; binat 8–16 GHz contains 80(81) MP and 217(218) HFIP; bin at 16–32 GHz contains 3(2) MP and 106(102) HFIP. First numbers arefor pulses used in method 2; numbers in parentheses are for pulsesused in method 1. Note slight differences in outlier exclusion forthe two methods. We find no strong evidence for any frequencydependence of the excess DM for HFIPs, nor do we find any excessDM for the LFIPs. Acronyms are given in Table 1. (see Figure 1 in Hankins et al. 2015), the associated sin-gle pulses are generally fainter than Main Pulses or Inter-pulses, which makes them hard to observe individually.Jessner et al. (2005) captured a few dozen at 8.35 GHz,and Mickaliger et al. (2012) reported “a few” HFC singlepulses at 8.9 GHz, but neither paper discussed the char-acteristics of single High-Frequency Component pulses.In one exceptional observing run at the Arecibo Obser-vatory we were able to catch 30 bright pulses between 8and 10.5 GHz. These were evenly split between the firstand second High-Frequency Components (which we re-fer to as HFC1 and HFC2, when needed, to shorten thenotation).
Single High-Frequency-Component Pulses
The temporal and spectral characteristics of High-Frequency-Component pulses are reminiscent of MainPulses and Low-Frequency Interpulses. To illustrate,in Figure 19 we show the total intensity and dynamicspectrum of two example pulses. Single High-Frequency-Component pulses – both HFC1 and HFC2 – can haveone or a few components. The dynamic spectrum ofa High-Frequency-Component pulse is relatively broad-band, extending across our 2.5 GHz observing band. Inboth of these aspects they resemble Main Pulses andLow-Frequency Interpulses.There are differences, however. Unlike the Main Pulseand the Low-Frequency Interpulse, the High-FrequencyComponents show strong linear polarization. This is ev-ident in the pulsar’s mean profile (Moffett & Hankins1999), and is consistent with the single pulses we cap-tured. While many of those pulses were too weak tomeasure significant polarization, the stronger pulses typ-ically showed ∼
50% linear polarization (at 3 . µ s timeresolution), with approximately constant position angles.We did not catch any pulses with significant circular po-ingle Pulses from the Crab Pulsar 15 Fig. 19.—
Two examples of High-Frequency Component pulsesrecorded between 8 and 10.5 GHz at the Arecibo Observatory.Above, a pulse from HFC1; below, a pulse from HFC2. Both areplotted with a spectral resolution of 39 MHz, and de-dispersed atDM of 56.76080 pc-cm − . The time resolution of the upper pulseis 7.8 µ s, and the lower pulse, 28.7 µ s. These pulses last muchlonger than the Main Pulses shown in Figure 3 and the Interpulsesshown in Figure 10, but are also much weaker. larization or significant position angle rotation.High-Frequency-Component pulses are fainter, andlast longer, than Main Pulses and Low-Frequency Inter-pulses. The peak flux of the High-Frequency-Componentpulses we captured was no more than a few Jy, muchfainter than the 100 − ± µ s forHFC1 pulses and 283 ± µ s for HFC2 pulses. Forcomparison, the equivalent widths of single Main Pulsesand High-Frequency Interpulses are typically ∼ − µ sat the same frequency (Hankins et al. 2015). Origin of High-Frequency Components?
We have found little discussion in the literature on thepossible origin of the High-Frequency Components. Per-haps they come from scattering of another component,or perhaps they are separate emission phenomena.
Scattering
It is tempting to consider the High-Frequency Com-ponents as scattered versions of a different profile com-ponent. Their broader width is reminiscent of pulsebroadening in propagation through turbulent media, andtheir total fluence at 9 GHz is similar to that of theMain Pulses and High-Frequency Interpulse we see atthe same frequency. This picture is challenged, how-ever, by the differences between single pulses belonging toeach component. Main Pulses are not strongly polarized,but High-Frequency-Component pulses (probably) showstrong polarization. Main Pulses are rare above 10 GHz,where the High-Frequency Component is strong. The dy-namic spectra of High-Frequency-Component pulses doesnot show the distinctive spectral bands characteristic ofHigh-Frequency Interpulses. It is not clear that simpleturbulent scattering could account for these differences.Petrova (2008, 2009) also proposed a scattering model,based on stimulated magnetized Compton scattering asa pulse passes through the relativistic pair plasma in themagnetosphere. For the Crab pulsar, she argues thatthe Main Pulse originates at low, polar-cap altitudes,then is scattered twice, both scatterings happening athigh altitude. The first scattering produces the Low-Frequency Component, which is itself then scattered tohigher frequencies to produce the two High-FrequencyComponents. Petrova points out that, if the details areright, the scattered radiation can be strongly polarizedand can emerge at higher frequencies than the incomingradiation.In addition to explaining the different characteristics ofthe High-Frequency Components and the pulses whichare scattered to make them, we note that any scatter-ing model must explain why two discrete High-FrequencyComponents exist – rather than a broad range of suchemission – and why the phases of those two componentschange with frequency. More work is needed here beforesuch models can fully explain the Crab pulsar.
Separate emission phenomena
If the High-Frequency Components are not scatteredversions of some other component, they must arise fromseparate emission regions somewhere in the magneto-sphere. Their unusual rotation phases suggest they arenot low-altitude, polar cap components, but – as withthe Main Pulse and both Interpulses – come from highmagnetospheric altitudes.Most models of caustic emission assume the causticsconnected to the two polar caps are symmetric, withuniform emission throughout the extended gap region.These models usually produce double-peaked mean pro-files, but with the right choices of emissivity distributionand viewing angle they can predict more complex meanprofiles (e.g., Bai & Spitkovsky 2010; Breed et al. 2015).It may be that the High-Frequency Components comefrom a “just right” mix of geometry and emissivity dis-tribution within a high-altitude caustic. As a specificexample, Romani & Watters (2010) noted that a subsetof the last closed field lines can extend for a very long dis-tance before crossing the light cylinder or the null chargesurface. They speculated that these field lines may de-fine a disjoint, high-altitude gap surface which could bethe source of the High-Frequency Components.6 Hankins, Eilek & JonesIf the High-Frequency Components arise from phys-ically distinct emission regions, must a distinct emis-sion physics also be operating in those regions? Clearlywe cannot invoke the same emission mechanism as forthe High-Frequency Interpulse, because we do not seespectral emission bands. Because we have not capturedenough strong pulses to comment on the existence ofnanoshots, we cannot easily rule out the microscale emis-sion mechanism that operates in the Main Pulse andthe Low-Frequency Interpulse. The longer duration ofHigh-Frequency-Component bursts, compared to MainPulses and Low-Frequency Interpulses, could be due todifferences in the energy storage and/or release mech-anisms in the emission zones for each component. Thehigher polarization of High-Frequency-Component pulsescould, perhaps, reflect a stronger magnetic field in theiremission zones. We therefore conclude that a third typeof emission mechanism may be operating in the High-Frequency-Component source regions, but one is not re-quired.
SUMMARY AND MORE QUESTIONS
In this paper we presented our new single-pulse ob-servations of the Crab pulsar between 10 and 43 GHz,and combined these results with our previous work tocharacterize each of the five emission components whichdominate the pulsar’s mean radio profile above 1 GHz.Our results clarify the properties of single pulses fromeach of these components, but also raise new questionsabout radio emission sites in the Crab’s magnetosphere.
Summary by component
We begin by summarizing the results of our single-pulse observations of the Crab pulsar, organized bymean-profile component.
Mean Pulse and Low-Frequency Interpulse
Main Pulse emission between 1 and 30 GHz comes inone or a few microbursts, each burst on the order ofmicroseconds long. The spectrum of a microburst isrelatively broadband; it is continuous across our few-GHz observing bandwidth. Microbursts show com-plex substructure; both intensity and polarization varyon timescales as short as several nanoseconds. Veryrarely the nanoshots within a Main Pulse are sparseenough to be studied individually. The spectra of thesenanosecond-long flashes of coherent radio emission is rel-atively narrowband, smaller than our observing band-width. We argue that every Main Pulse is a clump ofsuch nanoshots, with varying central frequencies, whichare usually not resolved because they overlap in time.Low-Frequency Interpulses show the same temporaland spectral characteristics as Main Pulses. Even thoughthese two components likely come from different partsof the pulsar’s magnetosphere – probably high-altitudecaustics associated with the star’s two magnetic poles –we infer they involve the same emission physics.Both Main Pulses and Low-Frequency Interpulses be-come rare above ∼
10 GHz, but we did capture one MainPulse at 43 GHz. Its spectrum and polarization are verydifferent from the spectra of Main Pulses at lower fre-quencies. A sample of one hardly justifies further discus-sion, but we find this one unusual pulse intriguing.
High-Frequency Interpulse
High-Frequency Interpulses are very different fromMain Pulses and Low-Frequency Interpulses. Al-though they do contain microbursts, we have neverseen nanoshots in any High-Frequency Interpulse. High-Frequency Interpulses are not seen below ∼ ν = 0 . ν , over this full frequency range. Althoughwe have not observed a single pulse over the full 25 GHzrange, we hold it likely that the spectrum of a singleHigh-Frequency Interpulse contains at least ∼
30 emis-sion bands. Because these differences from the MainPulse and Low-Frequency Interpulse are so striking, weinfer that different emission physics is operating in thiscomponent.High-Frequency Interpulses are partially dispersedwithin the pulsar magnetosphere. The intrinsic disper-sion varies rapidly, changing by ∼ .
02 pc-cm − , ontimescales of a few minutes. High-Frequency Interpulsesshow strong linear polarization, with a position anglethat is generally constant across the pulse and indepen-dent of the rotation phase within the “window” definedby the component in the mean profile. These resultssuggest that the emission region which creates the High-Frequency Interpulse is highly dynamic, probably turbu-lent, with a magnetic field that has a nearly constantdirection throughout the region. High-Frequency Components
Because High-Frequency-Component pulses are fainterthan their Main Pulse and Interpulse counterparts, theyare hard to observe. We managed to catch a modest num-ber between 8 and 10.5 GHz. These pulses are faint butlong-lived, lasting several hundred microseconds. Theirspectra are broadband, extending across our 2.5 GHz ob-serving bandwidth. They tend to be linearly polarized,with approximately constant position angles that are in-dependent of the rotation phase of the pulse.If the High-Frequency Components come from sepa-rate emission regions in the magnetosphere, their longerduration and high polarization may reflect different dy-namics in their natal regions. Alternatively, it may bethat they are secondary phenomena, produced by scat-tering or re-direction of other mean-profile components.
What have we learned?
We continue by summarizing what our results revealabout radio emission physics, and radio loud regions, inthe magnetosphere of the Crab pulsar. • At least two types of radio emission physics operatein the magnetosphere of the Crab pulsar.
The Main Pulse and the Low-Frequency Interpulseare characterized by nanoshot emission : narrow-band,nanosecond-long flashes of emission, usually clumped to-gether in microbursts. This emission is strong at low ra-dio frequencies, but dies out above several GHz. Becauseeach nanoshot can have its own polarization signature,this emission type can be depolarized in mean profiles.The High-Frequency Interpulse is characterized byingle Pulses from the Crab Pulsar 17 spectral band emission: longer bursts of strongly polar-ized emission containing of distinctive spectral emissionbands. This emission is strong at high radio frequen-cies but dies out below 4 GHz. The steady position an-gle of the linear polarization means this emission type isstrongly polarized in mean profiles.We suspect the High-Frequency Components may bea third type of radio emission, but we do not haveenough data to characterize them well. We have no high-time-resolution data on individual Precursor or Low-Frequency Component pulses, so cannot comment ontheir radio emission physics. • There are several radio emission sites in the magne-tosphere of the Crab pulsar.
The similar radio characteristics of the Main Pulse andthe Low-Frequency Interpulse, and their phase coinci-dence with high-energy pulses, suggests the emission re-gions for these two components lie somewhere along thehigh-altitude caustics above each of the star’s two mag-netic poles.The rotation phases of the Precursor and the Low-Frequency Component suggest those components comefrom low altitudes. Perhaps these two components ariseclose to one of the pulsar’s polar caps, in accordance withstandard models of rotation-powered pulsars.The different radio characteristics of the High-Frequency Interpulse suggest its emission region is phys-ically separate from that for the Low-Frequency Inter-pulse, despite the similar rotation phases of those com-ponents. The strong polarization and fluctuating disper-sion of the High-Frequency Interpulse suggest its emis-sion zone is a spatially localized, turbulent region.The unusual phases of the two High-Frequency Compo-nents suggest yet more radio emission zones – or localizedscattering sites – exist somewhere in the magnetosphere.The variable phases of these components suggests thatthe location of these emission or scattering sites changeswith observed radio frequency.Because the structure of the extended caustics is stillpoorly understood, we are not able to constrain wherein the magnetosphere the emission zones for the varioushigh-altitude radio components might be. Perhaps ra-dio data such as ours can guide future modeling of thedensity and magnetic field structures within the caustics.
What mysteries remain?
To conclude, we note that our observations have raisedas many questions as they have answered. We suggest afew such questions here; no doubt the reader can thinkof others. • Is there an unseen, High-Frequency Main Pulse inthe Crab pulsar? If the High-Frequency Interpulsecomes from a separate part of a caustic that givesrise to the Low-Frequency Interpulse, perhaps thecaustic that gives rise to the Main Pulse also hassuch a region, which we have not yet detected. • Do the main radio emission mechanisms in theCrab pulsar – nanoshot emission and spectral-band emission – occur in other pulsars? How doeslow-altitude, polar cap emission compare to high-altitude emission in the Crab pulsar? • What physical conditions trigger a particular typeof radio emission? For instance, is the triggerrelated to density, magnetic field, rotation rate,and/or particle acceleration in gap regions? Or isthe trigger due to something totally different? • Why do only a few pulsars show complex, high-altitude radio emission that is phase-aligned withhigh-energy pulsed emission? Is there a fundamen-tal difference in the high-altitude magnetospheresof these objects, or do many pulsars have high-altitude radio emission – perhaps at higher radiofrequencies – which has not yet been found?We hope that our detailed study of the Crab pulsar caninspire similar observations of other pulsars, and futuremodeling of likely magnetospheric emission zones, thatmay answer some of these questions.We thank the technical, operations, and computerstaffs at the Green Bank Telescope, the Very Large Ar-ray and the Arecibo Observatory for their help with thedata acquisition equipment, observing, and for providingsome of the computing environment we used for the ob-servations. We thank Jared Crossley, Jeff Kern, DavidMoffett, and James Sheckard for help with the Areciboobservations. We thank an anonymous referee for helpfulcomments which have strengthened the paper. This workwas partially supported by NSF grant AST-0607492.
APPENDIX
NOISE AND TRIGGER LEVELS
We summarize our observing parameters in Table 2. Intensity calibrations were performed using standard calibrationsources. Differences in noise levels depend upon total receiver bandwidths, spectral resolution, detector smoothingtime constant, telescopes, and at the higher frequencies, weather. The combination of the Crab Nebula spectral indexand frequency dependent antenna beam widths causes the contribution from the Crab Nebula to vary greatly over thefrequency range we observed. The narrow synthesized beam of the VLA resolves most of the Nebula and hence hasfar lower on-source system temperatures at low frequencies than the Arecibo telescope and the Green Bank Telescopewhere the system temperatures are completely dominated by the Nebular contribution, even at the highest frequencieswe used.The integration time constant for the square-law detected trigger channel was chosen to match the dispersion sweeptime across the trigger channel bandwidth. In most cases the Trigger Level (column 10) is lower than the Intensity NoiseLevel (column 11) because the Trigger Detector Time Constant (column 9) is much longer than the time resolution ofthe dedispersed intensity shown in the figures. For figures in which we show both the total intensity and the dynamicspectrum the noise levels in columns 11 and 12 of Table 2 reflect the full-band noise level and the noise level for8 Hankins, Eilek & Jones
TABLE 2Off-Pulse Noise and Trigger Levels
Figure Freq. Observation Telescope On Sensi- Data Trigger Trigger Trigger Intensity Spectrumin (GHz) Date Source tivity Channel Channel Detector Level Noise Noisetext T sys (K/Jy) Band- Band- Time- (Jy) Level Level(K) width width constant (Jy) (Jy)(MHz) (MHz) ( µ s)1a 1.38 1998/12/27 VLA a
140 1.4 d
50 0.25 40 270 38 1301b 1.42 1994/02/23 VLA 140 2.5 50 0.25 40 150 20 663a 14.00 2010/12/02 GBT b
100 1.5 4000 450 78 3.0 14 733b 20.25 2009/04/13 GBT 56 1.0 4260 450 25 4.5 7.2 374 9.25 2005/01/05 AO c
102 4.0 2220 140 83 2.0 3.45, 6 9.25 2007/01/01 AO 102 4.0 2200 140 83 2.1 137a 9.25 2005/08/08 AO 65 4.0 2200 140 83 1.3 2.1 157b 7.00 2006/03/11 AO 168 5.0 1070 50 70 4.8 5.9 228 9.25 2005/08/08 AO 65 4.0 2200 140 83 1.3 7.39 43.25 2009/04/18 GBT 70 0.46 4060 1000 20 7.9 9.6 9210a 14.00 2010/12/02 GBT 100 1.5 4000 450 77 3.0 9.9 7510b 28.00 2010/12/05 GBT 82 1.0 4450 1000 22 7.0 15 11012a 20.25 2009/04/13 GBT 56 1.0 4250 450 25 4.512b 14.00 2010/12/02 GBT 100 1.5 4000 450 77 3.013, 14 20.25 2009/04/12 GBT 56 1.0 4250 450 25 4.5 3.615 20.00 2010/12/02 GBT 101 1.0 4650 450 26 7.9 9.619a 9.25 2008/04/13 AO 101 4.0 2070 450 270 0.6 0.3 119b 9.25 2008/04/13 AO 101 4.0 2070 450 270 0.6 0.14 1
Note . — a. Very Large Array. b. Green Bank Telescope. c. Arecibo Observatory. d. Only half of the VLA antennas were used for thisobservation. an individual spectral channel. The noise levels and sensitivity of the GAVRT telescope over the 2.5 to 10.5 GHzfrequency range (Figure 2) are more complex. The on-source T sys is ∼
140 K at 2 GHz, 60 K at 7 GHz, and 45 K at10 GHz (see Figure 4.11, Jones 2009). The GAVRT telescope sensitivity ranges between 0.65 and 0.83 K/Jy (Jones,et al. 2010).
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