What can we learn from phase alignment of gamma-ray and radio pulsar light curves?
aa r X i v : . [ a s t r o - ph . H E ] D ec What can we learn from phase alignment of γ -rayand radio pulsar light curves? C Venter , T J Johnson , and A K Harding Centre for Space Research, North-West University, Potchefstroom Campus, Private BagX6001, Potchefstroom 2520, South Africa Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771,USA Department of Physics, University of Maryland, College Park, MD 20742, USAE-mail:
Abstract.
The
Fermi
Large Area Telescope (LAT) has revolutionized high-energy (HE)astronomy, and is making enormous contributions particularly to γ -ray pulsar science.As a result of the many new pulsar discoveries, the γ -ray pulsar population is nowapproaching 100. Some very famous millisecond pulsars (MSPs) have also been detected:J1939+2134 (B1937+21), the first MSP ever discovered, as well as J1959+2048 (B1957+20),the first black widow pulsar system. These, along with other MSPs such as PSR J0034 − γ -ray light curves (LCs). Traditionally, pulsar LCs have been modelled usingstandard HE models in conjunction with low-altitude conal beam radio models. However, adifferent approach is needed to account for phase-aligned LCs. We explored two scenarios: onewhere both the radio and γ -ray emission originate in the outer magnetosphere, and one wherethe emission comes from near the polar caps (PCs) on the stellar surface. We find best-fit LCsusing a Markov chain Monte Carlo (MCMC) technique for the first class of models. The firstscenario seems to be somewhat preferred, as is also hinted at by the radio polarization data.This implies that the phase-aligned LCs are possibly of caustic origin produced in the outermagnetosphere, in contrast to the usual lower-altitude conal beam radio models. Lastly, weconstrain the emission altitudes with typical uncertainties of ∼
10% of the light cylinder radius.The modelled pulsars are members of a third γ -ray MSP subclass, in addition to two otherswith non-aligned radio and γ -ray LCs.
1. Introduction
For quite some time, there were only a handful of pulsars detected in the γ -ray waveband [1]. Thissmall sample already provided a starting point for spectral and light curve (LC) modelling aswell as preliminary population studies. However, the launch of the Fermi
Large Area Telescope(LAT) [2] heralded an exciting new era, particularly for high-energy (HE) pulsar physics. γ -ray pulsars galore Following the first
Fermi
Catalog which included about 1 500 γ -ray sources [3], the second Fermi
Catalog is in now production, containing nearly 1 900 γ -ray sources [4, 5] includingalmost 100 γ -ray pulsars [6, 7]. About a third of these are millisecond pulsars (MSPs), somehaving been found by observing non-variable, unassociated Fermi sources at high latitudes withadio telescopes and searching for pulsed radio signals [8, 9, 10]. Some of these MSPs exhibitthe unusual phenomenon that the peaks of their γ -ray and radio LCs occur at the same observerphase (‘longitude’ φ ), i.e., they are phase-aligned. Two classes of models have been invoked to explain HE pulsar radiation. The first class, low-altitude polar cap (PC) models [11, 12], assume that primary electrons are accelerated abovethe neutron star surface, and that magnetic pair production of curvature radiation or inverse-Compton-scattered γ -rays occurs in the intense B-fields close to the stellar surface. In thecase that pair creation is suppressed along the last open field lines, which mark the boundarybetween the corotating closed field line zone and the active open one, a slot gap (SG) [13]may form, corresponding to a two-pole caustic (TPC) geometry [14] which extends from thestellar surface up to near the light cylinder. The second class of models are the outer gap (OG)models [15, 16], which assume the production of HE radiation along the last open field linesabove the so-called null charge surface, where the Goldreich-Julian charge density becomes zero.In both classes of outer-magnetospheric models (SG and OG), the HE pulse profiles are theresult of the formation of caustics – the accumulation of photons in narrow phase bands due toa combination of special relativistic effects and magnetic field line curvature. Also, the narrowgaps in these models require abundant production of electron-positron pairs which will be ableto screen the accelerating E-field.For the case of MSPs, LCs and spectra have been modelled [17, 18, 19] using a pair-starvedpolar cap (PSPC) model [20]. The latter is similar to the traditional PC model, but in this casethe number of pairs is not sufficient to screen the accelerating E-field, such that the acceleratingregion above the star is pair-starved. MSP spectra and energetics have also been modelledusing an OG model [21, 22], while LC modelling using an annular gap model can furthermorereproduce the salient features of the γ -ray LCs [23].The outer magnetospheric models usually invoke relatively low-altitude ‘core’ and ‘conal’radio beams centred on the magnetic axis, in addition to the HE radiation being produced inextended regions reaching up to the light cylinder. The difference in the location of the γ -rayand radio emission regions thus implies that the corresponding pulse profiles will have non-zerolags (phase offsets) between the γ -ray and radio LCs. γ -ray light curves: a new MSP subclass The LCs of the first 8
Fermi -detected γ -ray MSPs [24] have been modelled [25], yielding twodistinct MSP subclasses: those whose LCs are well fit by a standard OG or TPC model, andthose whose LCs are well fit by a PSPC model. These fits are mutually exclusive. Importantly,this unexpectedly implied that there is copious pair production taking place even in MSPmagnetospheres with their characteristic low B-fields.A third class of MSPs emerged with the discovery of PSR J0034 − γ -ray LCs. Prior to this detection, such behaviour has only beenobserved for the Crab pulsar [27]. This phenomenon has now also been seen for PSR J1939+2134,PSR J1959+2048 [28], and PSR J2214+3000 [10]. Some other MSPs, including PSR B1821 − − γ -ray LCs, on the basis thattheir radio and X-ray profiles are phase-aligned [29, 30].
2. Geometric models predicting phase-aligned light curves
The non-zero radio-to- γ lags of MSPs with phase-aligned LCs disqualify the application ofstandard models, since the phase alignment implies co-located emission regions. We considertwo model scenarios, one where the γ -rays and radio both occur in extended, high-altituderegions, and one where they originate at low altitudes. (For more details, see [31].) .1. High-altitude models We use the same framework as previously [25], except that the radio emission region is nowextended in altitude, and not assumed to be coming from a radio cone at a single altitiude. Wefurthermore limit the minimum and maximum radii of the radio and γ -ray emission regions,and assume constant emissivity as a function of altitude. These models are therefore calledaltitude-limited OG (alOG) and TPC (alTPC) models. Technically, the traditional OG modelis a specific instance of an alOG model, and similar for the traditional TPC and alTPC models. The so-called low-altitude Slot Gap (laSG) models represent an alternative possibility assuminga non-caustic origin of the emission. These are actually very-low-altitude geometric SG modelsresembling a hollow-cone beam close to the stellar surface. We modulate the emissivity asmotivated by detailed radiation models, where the emissivity rises and falls exponentially alongthe B-field lines, peaking at a distance of about one or two stellar radii. Additionally, we alsoinvestigated the case where the emissivity is constant.
3. Finding optimal LC solutions in multidimensional phase space
The models described above involve multiple free parameters describing the emission regionextent and width, as well as the pulsar geometry (inclination and observer angles α and ζ ). Inorder to pick statistically the best-fit parameters when comparing to the Fermi
LC data, wehave developed a Markov chain Monte Carlo (MCMC) maximum likelihood procedure [32] andapplied this to the alOG and alTPC models for three MSPs as indicated below (see Table 1).The γ -ray LCs are fit using Poisson likelihood while the radio LCs are fit using a χ statistic.These two values are then combined. Using this method, we can also derive uncertainties on theminimum and maximum altitudes of the emission regions.
4. Results
Increasing the lower limit of the emission region’s altitude while keeping the upper limit fixedhas the effect of growing the PC size (i.e., deleting a ring of radiation around the PC in thephaseplot, which indicates relative intensity per solid angle vs. ζ and φ ). This suppresseslow-level off-peak emission, especially for the alTPC case, so that the peaks become sharper.Conversely, decreasing the maximum radius while keeping the lower one fixed constrains theemission to a ring-like structure around the PC. As a result, the relative peak heights change inthe corresponding LCs, and low-level features may become more prominent.Figure 1 shows an example of best alOG and alTPC LC fits for PSR J1939+2134. Thebest-fit values for α and ζ , for the three MSPs we have modelled, are summarized in Table 1, Table 1.
Best-fit pulsar geometries for different models.PSR J0034 − α ζ α ζ α ζ alOG 12 ◦ ◦ ◦ ◦ ◦ ◦ alTPC 30 ◦ ◦ ◦ ◦ ◦ ◦ laSG1 10 ◦ ◦ ◦ ◦ ◦ ◦ laSG2 10 ◦ ◦ ◦ ◦ ◦ ◦ igure 1. LC fits for PSR J1939+2134 using alTPC and alOG models. Panel (a) shows the γ -ray data [28], while panel (b) shows the radio data.having typical errors of ∼ − ◦ . Best-fit emission altitudes and gap widths are presentedelsewhere [31]. It is interesting that in the laSG models the leading peak is wider while the trailing peak issharper, even for the low-altitude emission considered here. This means that the caustic effectsstart to appear, given the large corotation speed of MSPs already near their surface. By includingemission from higher altitudes, the peaks become broader. Also, the peak phase separation andwidth may be altered by choosing different values for the minimum and maximum emissionaltitudes. The constant-emissivity assumption leads to block-shaped LCs, and is therefore notconsidered viable. For the modulated-emissivity case, the peak widths may be fine-tuned usingfading parameters, while the peak separation depends sensitively on ζ . This class of modelsshows less variation in profile shape than the high-altitude ones, and there are also multiplecombinations of the free parameters that give very similar LCs, so a ‘best solution’ is probablynot unique. The different best-fit solutions usually are a trade-off between best fits for γ -ray vs.radio LCs. For this reason, we indicate two similar laSG solutions (labeled ‘laSG1’ and ‘laSG2’,differing only in fading parameters which set the emissivity fading properties) in Table 1.
5. Discussion
Pulsars with radio and γ -ray LCs which are non-aligned in phase may be modelled by γ -rayemission regions extended over a large range of altitudes (OG / TPC / PSPC models) inconjunction with a conal radio beam at relatively lower emission altitudes. However, phase-aligned LCs require co-located γ -ray and radio emission regions. We studied the viability ofreproducing such phase-aligned LCs using both high-altitude and low-altitude geometric models.We found that both classes of models could produce LC fits that can capture the most prominentfeatures of the pulse profiles. .1. Model constraints We have found best fits for the two angles describing the pulsar geometry, α and ζ , as indicatedin Table 1. These have typical errors of ∼ − ◦ . In addition, we were able to limit the emissiongeometry for the high-altitude models, and found that the typical errors on the best-fit emissionaltitudes are ∼
10% of the light cylinder radius. Furthermore, the maximum radio emissionaltitude seems to be better constrained than the γ -ray one. Since the low-altitude models canproduce very similar LCs for similar values of its free parameters, we regard the inferred valuesof these free parameters when applying the model to the data as less robust than the case ofthe altitude-limited models. While we used the MCMC technique to search for the optimal set of parameters that producedthe closest match to the LC data for the case of the altitude-limited models, we have not yetimplemented this technique for the laSG models. Since the parameter space has not been fullyexplored for the latter class of models, we cannot really claim to have found the best fit forthe alSG models. It is therefore difficult to quantitatively favour one class of models abovethe other. However, we did calculate the likelihood of the manually-selected best-fit laSG LCsas well as the MCMC-selected best-fit LCs of the altitude-limited models, and found that thealtitude-limited models give better fits than the laSG ones, while the alTPC models provideslightly better LC fits than the alOG models. This implies that the phase-aligned γ -ray andradio LCs are most probably of caustic origin, produced in the outer magnetosphere, and theradio emission is most likely originating near the light cylinder (see also [33]). By studying the visibility of two samples of γ -ray and radio pulsars, it was concluded [34] thatradio and γ -ray beams must have comparable sky coverage, especially for pulsars with large spin-down luminosities. This implies that the radio emission should originate in very wide beamsat a significant fraction of the light cylinder, motivating studies of high-altitude caustic radioemission. However, this scenario is limited in application to pulsars having nearly phase-alignedLCs (with phase differences of up to ∼ . γ -ray one. The bulk of the γ -ray pulsar population exhibits quite large ( ∼ . − . γ -ray phase lags, so that caustic radio emission is probably is not ubiquitous in youngpulsars. Furthermore, the γ -ray profiles are usually double-peaked, while the radio ones aresingle-peaked. Unless the radio emissivity has a strong altitudinal or azimuthal dependence,both radio and γ -ray LCs would be mostly double-peaked if they result from caustic emission,which is not observed. On the other hand, radio caustics may be more common for short-periodMSPs, as there are many more examples of MSPs with phase-aligned LCs. Detailed populationstudies involving both young and old pulsars will provide more quantitative answers to thisquestion. The rotating vector model [35] generally provides a good description of radio polarization data(position angle as function of phase). However, this model assumes a static dipole B-field with norotation effects included, and is probably not valid for fast-spinning MSPs which are expected tohave significant B-field distortion. Caustic emission models predict rapid position angle swingswith phase, coupled with depolarization [36], since the emission from a large range of altitudesand B-field orientations is compressed into a narrow phase interval to form the peaks. Rapidchanges in the polarization angle and low levels of linear and circular polarization near the peakphases have indeed been seen for the MSPs modelled in this paper, and may be indicative ofcaustic effects. Polarization signatures will therefore be important to help discriminate betweenCs produced by caustic emission (such as occurs in alOG / alTPC models) or not (e.g., in thelaSG model).
6. Conclusions
Future studies should include development of full radiation models which will be able toreproduce both the multiwavelength LC shapes, polarization properties, as well as the energy-dependent behaviour of the spectra of the γ -ray pulsars. Ways in which to induce abundantpair creation in low-B MSP magnetospheres, such as using offset-dipole [37] or multipole [21]B-fields, will need to be investigated. Acknowledgments
CV is supported by the South African National Research Foundation. AKH acknowledgessupport from the NASA Astrophysics Theory Program. CV, TJJ, and AKH acknowledgesupport from the
Fermi
Guest Investigator Program as well as fruitful discussions with DickManchester and Matthew Kerr.
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