Measurements of pulse jitter and single-pulse variability in millisecond pulsars using MeerKAT
A. Parthasarathy, M. Bailes, R.M. Shannon, W. van Straten, S. Oslowski, S. Johnston, R. Spiewak, D.J. Reardon, M. Kramer, V. Venkatraman Krishnan, T.T. Pennucici, F. Abbate, S. Buchner, F. Camilo, D.J. Champion, M. Geyer, B. Hugo, A. Jameson, A. Karastergiou, M.J. Keith, M. Serylak
MMNRAS , 1–16 (2020) Preprint 22 January 2021 Compiled using MNRAS L A TEX style file v3.0
Measurements of pulse jitter and single-pulse variability inmillisecond pulsars using MeerKAT
A. Parthasarathy , , ★ , M. Bailes , , R.M. Shannon , , W. van Straten , S. Osłowski , ,S. Johnston , R. Spiewak , , , D. J. Reardon , , M. Kramer , V. Venkatraman Krishnan ,T. T. Pennucci , , F. Abbate , S. Buchner , F. Camilo , D. J. Champion , M. Geyer ,B. Hugo , , A. Jameson , A. Karastergiou , M. J. Keith , M. Serylak Centre for Astrophysics and Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany OzGrav: Australian Research Council Centre of Excellence for Gravitational Wave Discovery. Institute for Radio Astronomy & Space Research, Auckland University of Technology, Private Bag 92006, Auckland 1142, New Zealand Gravitational Wave Data Centre, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia CSIRO Astronomy and Space Science, Australia Telescope National Facility, PO Box 76, Epping NSW 1710, Australia Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA, Institute of Physics, Eötvös Loránd University, Pázmány P. s. 1/A, 1117 Budapest, Hungary South African Radio Astronomy Observatory, Cape Town, 7925, South Africa Department of Physics and Electronics, Rhodes University, Artillery Road, Grahamstown, South Africa, Oxford Astrophysics, Denys Wilkinson Building, Keble Road, OX1 3RH, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Using the state-of-the-art SKA precursor, the MeerKAT radio telescope, we explore the limitsto precision pulsar timing of millisecond pulsars achievable due to pulse stochasticity (jitter).We report new jitter measurements in 15 of the 29 pulsars in our sample and find that the levelsof jitter can vary dramatically between them. For some, like the 2.2 ms pulsar PSR J2241–5236,we measure an implied jitter of just ∼ ∼
100 ns/hr. While it is well known that jitter plays a central role to limitingthe precision measurements of arrival times for high signal-to-noise ratio observations, itsrole in the measurement of dispersion measure (DM) has not been reported, particularly inbroad-band observations. Using the exceptional sensitivity of MeerKAT, we explored this onthe bright millisecond pulsar PSR J0437–4715 by exploring the DM of literally every pulse.We found that the derived single pulse DMs vary by typically 0.0085 cm − pc from the mean,and that the best DM estimate is limited by the differential pulse jitter across the band. Wepostulate that all millisecond pulsars will have their own limit on DM precision which canonly be overcome with longer integrations. Using high-time resolution filterbank data of 9 𝜇 s, we also present a statistical analysis of single pulse phenomenology. Finally, we discussoptimization strategies for the MeerKAT pulsar timing program and its role in the context ofthe International Pulsar Timing Array (IPTA). Key words: stars: neutron; pulsars: general; methods: data analysis;
The precise monitoring of periodic radio pulses emitted from mil-lisecond pulsars (MSPs) has enabled some of the most stringenttests of fundamental physics. It has been used to test the theory ofgeneral relativity (Taylor & Weisberg 1982; Kramer et al. 2006b),alternative theories of gravity (Zhu et al. 2015; Voisin et al. 2020), ★ E-mail: [email protected] constrain neutron star equations of state (Demorest et al. 2010; An-toniadis et al. 2013; Cromartie et al. 2020), measure irregularities interrestrial time standards (Petit & Tavella 1996; Hobbs et al. 2020),detect planetary-mass companions (Wolszczan & Frail 1992) andcan potentially be used to detect and characterise nHz-frequencygravitational radiation (Hellings & Downs 1983; Foster & Backer1990). Precision long-term timing of an ensemble of the most sta-ble MSPs to <
100 ns root-mean-square (rms) residuals has ledto placing upper limits on the stochastic gravitational wave back- © a r X i v : . [ a s t r o - ph . H E ] J a n A. Parthasarathy et al. ground (Shannon et al. 2015; Lentati et al. 2016; Aggarwal et al.2019; Perera et al. 2019), allowing constraints on the formation andevolutionary scenarios of supermassive black holes and their hostgalaxies (Taylor et al. 2017).Pulsar timing residuals, which are the differences between theobserved pulse times-of-arrival (ToAs) and those predicted by a tim-ing model, are a fundamental diagnostic tool in assessing the qualityof the timing model. Numerous studies since the 1970s have shownthat the scatter in timing residuals are larger than that expectedfrom the formal uncertainties (i.e., the uncertainties reported froma match-filtered based arrival time determination algorithm) alone(Groth 1975; Cordes & Downs 1985; Osłowski et al. 2011; Shannonet al. 2014; Lam et al. 2019). This excess noise can be categorisedinto a time-correlated, red-noise component and an uncorrelatedwhite-noise component. One of the main contributing sources tothe red noise is caused by rotational irregularities in the pulsar’sspin period also known as spin noise or timing noise (Boynton et al.1972; Cordes 1980). Spin noise manifests as a low-frequency pro-cess in pulsar timing residuals over timescales of months to years.Many studies have attempted to characterise the strength and non-stationarity of spin noise across the pulsar population (Shannon& Cordes 2010; Parthasarathy et al. 2019) finding that although itis widespread in pulsars, it is weaker in millisecond pulsars (Lamet al. 2017). Additional contributions to the red-noise componentcan arise from quasi-periodic processes, due to magnetospherictorque variations (Kramer et al. 2006a; Lyne et al. 2010), unmod-elled planetary companions (Shannon et al. 2013; Kerr et al. 2015),unmodelled dispersion measure (DM) variations (Keith et al. 2013),turbulence in the interstellar medium (Cordes & Shannon 2010;Lam et al. 2017) and uncertainties in the solar system ephemeris(Champion et al. 2010; Caballero et al. 2018).On timescales of minutes to hours, the excess noise in thetiming residuals is typically dominated by the uncorrelated white-noise component. This excess white noise, in addition to radiometernoise, can arise from several sources, the most significant of which isfrom differences between the integrated pulse profile (the averagedphase-resolved light curve of the pulsar) and a template profile(average of a finite number of pulses). This difference contributesdirectly to the excess white noise which results in observed ToAuncertainties being higher than the predicted formal values. Theformal uncertainty in the arrival time is derived from the template-matching algorithm, which models the integrated pulse profile ( 𝑃 )as a scaled (by a factor 𝐴 ) and offset (by a constant 𝐵 ) version of thetemplate ( 𝑂 ) rotated by a phase shift 𝜙 with additional white noise 𝑁 ( 𝑡 ) , expressed as (Taylor 1992), 𝑃 ( 𝑡 ) = 𝐴𝑂 ( 𝑡 − 𝜙 ) + 𝐵 + 𝑁 ( 𝑡 ) . (1)Observations of single pulses from pulsars with high flux den-sities have exhibited variations in their pulse morphology along withcorrelated variations in their phase and amplitudes (Drake & Craft1968; Helfand et al. 1975; Jenet et al. 1998). Aside from the fact thatpulse profile variability adversely affects the attainable precision inpulsar timing experiments with MSPs, it is important to acknowl-edge that the general phenomenology has been extensively studiedacross the pulsar population on both short (seconds to hours) andlong (months to years) timescales. It has been known that emissionchanges can occur in pulsars on short timescales (Backer 1970b,c);where pulse profiles were observed to switch between two or moredistinct emission states (known as mode changing ) or where theprofile was either in a weak emission state or completely turned‘off’ (known as nulling ). Pulsars categorized as intermittent havebeen observed to cycle between quasi-periodic phases in which the radio emission is either clearly present or invisible (Kramer et al.2006a; Camilo et al. 2012; Lyne et al. 2017) and in all cases theobserved emission changes have been correlated with the pulsar’srotational behaviour. Changes in pulse morphology have also beenattributed to precession of the pulsar’s spin axis, which causes dif-ferent regions of the emission beam to orient along our line-of-sight(Weisberg et al. 1989; Kramer 1998). In an important study, Cordes& Downs (1985) analysed 24 pulsars and concluded that pulse shapevariations or jitter were significant in a large fraction of their sampleand proposed that it is likely to occur in all pulsars with varyingdegrees of importance.Profile changes in MSPs have been reported in very few cases.Hotan et al. (2004) & van Straten (2013) showed that previousreports of profile instabilities in PSR J1022+1001 (Kramer et al.1999) are possibly due to instrumental polarimetric calibration er-rors while a recent study suggests that calibration alone cannotaccount for the observed profile variations (Padmanabh et al. 2021).Shannon et al. (2016) reported a broad-band profile change in PSRJ1643–1224 (also observed by Brook et al. 2016) which may havebeen due to a disturbance or a state change in the pulsar magneto-sphere. Very recently, the brightest and closest MSP, PSR J0437–4715 exhibited a significant change in its integrated pulse profile(Kerr et al. 2020), indicating that such abrupt profile changes may becommon among MSPs as well. In contrast to observations of youngpulsars, the profile changes in both MSPs were not accompaniedwith measurable changes in their spin down rate.A few studies have recently focused on studying jitter noisein MSPs and its effect on limiting the attainable precision throughpulsar timing (Osłowski et al. 2011; Liu et al. 2012; Shannon et al.2014; Lam et al. 2019). Jitter noise is a stochastic process com-mon to all pulsars, arising from intrinsic self-noise in the pulsaremission mechanism and is thought to be a wide-band phenomenon(Taylor et al. 1975; Rickett 1975). Since single pulse morphologychanges stochastically from pulse to pulse, and can be measurablein high signal-to-noise (S/N) observations, the averaged pulse pro-file ( 𝑃 ) will have a shape that is different from the template, thuscausing an excess scatter in the measured ToA uncertainty. Thisexcess scatter can be measured from its contribution to the rms ofthe ToAs, expressed as 𝜎 J ( 𝑁 p ) , where 𝑁 p is the number of aver-aged pulses or can also characterised with a dimensionless jitterparameter ( 𝑓 j ), as the ratio of 𝜎 J ( 𝑁 p ) to the pulse period ( 𝑃 ). Un-like the formal ToA uncertainty ( 𝜎 S / N ), 𝜎 J is independent of theS/N. Furthermore, additional scatter in the ToAs can also resultfrom narrow-band diffractive interstellar scintillation (DISS) alongwith the frequency dependence of the pulse profile, especially if theprofiles are averaged over large bandwidths (Demorest et al. 2013;Shannon & Cordes 2017). A second effect related to diffractive scin-tillation, is caused by stochasticity in the pulse broadening and hasbeen termed the finite-scintle effect (Cordes et al. 1990; Cordes &Shannon 2010). This effect can be is greater for pulsars which showlarger degrees of scatter broadening, so is especially important forhigh dispersion-measure pulse observed at lower frequencies.Osłowski et al. (2011) analysed 25 hours of high-precision tim-ing data of PSR J0437–4715 (using the Murriyang/64-m Parkes ra-dio telescope, at an observing frequency of ∼ 𝑁 p ∼ pulses), pulse jitter limits the attainabletiming precision to ∼
30 ns. Shannon & Cordes (2012) similarlyreported the jitter in PSR J1713+0747 to be ∼
20 ns in an hour.Jitter measurements for a sample of 22 MSPs as part of the ParkesPulsar Timing Array (PPTA; Manchester et al. 2013) project werereported by Shannon et al. (2014) with PSR J1909–3744 showingthe lowest levels of jitter noise of ∼
10 ns in an hour. They modelled
MNRAS000
MNRAS000 , 1–16 (2020) easurements of pulse jitter and single-pulse variability in millisecond pulsars using MeerKAT the contribution of jitter as a function of observing time ( 𝑇 ) as ≈ . 𝑊 eff √︁ 𝑃 / 𝑇 . More recently, Lam et al. (2019) detected jitter in 43MSPs as part of timing program of the North American NanohertzObservatory for Gravitational Waves (NANOGrav, Arzoumanianet al. 2018) and found significant frequency dependence of jitter in30 of them. These studies clearly show that pulse jitter is a genericproperty of MSPs, that it dominates the white noise budget whenobserving with increased S/N and that robust characterisation of jit-ter noise is vital in the search for stochastic nHz gravitational wavebackground using pulsar timing arrays.The MeerKAT radio telescope is several times more sensitivethan the Parkes radio telescope and offers the opportunity to detectand constrain jitter in many more MSPs at declinations < + ◦ . InSection 2 we describe the MeerTime MSP observing program, therelevant data processing by the observing backend and the variousprocessing steps implemented in the data reduction pipeline. Wediscuss the methodology used to estimate jitter, report jitter mea-surements for 29 MSPs and discuss its frequency dependence inSection 3. In Section 4 we discuss fundamental limits imposed onDM measurements in PSR J0437–4715. Statistical studies of singlepulses allow us to link timing variations and shape variations thusproviding further insights into characterising jitter noise. In Section5, we describe the statistical properties of MSP single pulses anddiscuss these results in the context of timing variations. Finally, inSection 6, we discuss the implications of our results and the role ofthe MeerTime Pulsar Timing Array (MPTA) programme in aidinghigh precision pulsar timing and PTA experiments. The MeerKAT radio telescope is the South African precursor for theSquare Kilometre Array (SKA) mid radio telescope located in theGreat Karoo region, and is capable of observing the large populationof known pulsars in the Southern and Northern hemispheres. TheMeerTime collaboration (Bailes et al. 2020) is one of the MeerKATLarge Survey Projects, which has been provisionally awarded manymonths of observing time. The MPTA is one of the four major sci-ence themes as part of MeerTime, which focuses on the precisiontiming of MSPs and is poised to contribute to an internationallycoordinated effort to detect the stochastic gravitational wave back-ground using pulsar timing arrays through the International PulsarTiming Array (IPTA, Hobbs et al. 2010; Verbiest et al. 2016; Pereraet al. 2019). The current observing strategy for the MPTA is to attainsub-microsecond formal timing precision on as many pulsars as pos-sible with integration times less than 2048 seconds. For each pulsar,the integration time is set to match the median expected time eitherbased on previous observations from the MeerKAT MSP censusprogram or previous monitoring campaigns. A maximum integra-tion time of 2048 seconds is imposed per epoch and a minimumintegration time of 256 seconds is used if sub-microsecond timingprecision is achievable in less than that time. The MPTA, since com-mencing observations from February 2019, attains timing precisionof < 𝜇𝑠 on ∼
70 MSPs in a total integration time of approximately11 hours. The Parkes Pulsar Timing Array, in comparison, achievessub-microsecond precision in 22 MSPs in 24 hours and NANOGravachieves the same precision on 47 MSPs (Alam et al. 2020a,b).For the jitter analysis presented here, we selected MSPs ob-served with MeerKAT that exceeded a S/N per pulse of unity follow-ing Shannon & Cordes (2012). These included ∼
350 observationsof 29 pulsars over a span of about a year (starting from February2019). Approximately 80% of these observations had an integration time of <
900 seconds. We used the L-band receiver with a systemtemperature of ∼
18 K, operating at a frequency range between 856MHz and 1712 MHz. The dual-polarization and channelized signalfrom the beamformer were processed in the Pulsar Timing UserSupplied Equipment (PTUSE) using the dspsr software library(van Straten & Bailes 2011). dspsr provides two signal processingpipelines and produces fold-mode and search-mode data products.Pulsar timing applications use the fold-mode pipeline to producefrequency- and phase-resolved averages of the polarised flux for thetarget pulsar. Search-mode applications (for single pulses) on theother hand, use digifits to produce high time resolution spectra(filterbanks) data products. A detailed description of the MeerKATpulsar timing infrastructure including polarisation calibration is pro-vided in Bailes et al. 2020. For the analysis presented here however,we only used the total intensity profiles.The coherently dedispersed folded archives and search-modedata products produced by PTUSE are automatically ingested by theMeerKAT kat-archive and transferred to the OzStar HPC facilityat Swinburne University of Technology through an authenticateddownload for post-processing. PTUSE produces folded archives of8 second integrations with 4 Stokes parameters, 1024 frequencychannels across the observing bandwidth with 1024 phase bins.Although the receiver has a bandwidth of 856 MHz, a portionof the bandwidth is ignored due to bandpass filter roll-off whichreduces the usable bandwidth to 775.75 MHz and thus the number offrequency channels to 928. These archives are further processed bythe fold-mode processing pipeline meerpipe , which generates RFI-excised full-frequency and time resolution psrfits based archives(van Straten et al. 2010) along with decimated products with user-specified frequency and time resolution and associated ToAs.The fragmented 8 second folded archives are integrated us-ing psrchive tool psradd and aligned using an up-to-date pulsarephemeris. The ephemerides are automatically checked using a se-ries of pre-defined standards which is then followed by a manualvetting process. These are automatically version controlled and usedby PTUSE , ensuring accountability. The noise-free templates usedfor timing are generated either manually (using the psrchive tool paas ) or automatically wavelet-smoothed using the psrchive tool psrsmooth . Frequency-dependent template profiles (or portraits) aregenerated for pulsars which show significant profile evolution acrossfrequency using PulsePortraiture (Pennucci 2019).To study single pulses, the filterbanks are recorded at a time-resolution of 9 𝜇𝑠 across 768 frequency channels and are stored inpsrfits format. For the analysis presented here, the post-processingof single pulses is done using dspsr.RFI excision is implemented using a modified version ofcoastguard (Lazarus et al. 2016) which uses a template profileto identify the phase-bins containing the pulsar signal and com-putes profile residuals by subtracting the observed profile from thetemplate. The folded data cube with full-frequency and time reso-lution after integrating the four Stokes parameters is used to exciseRFI. Using metrics as described in Lazarus et al. (2016), RFI miti-gation is performed on the Fourier transform of the profile residuals.For the analysis presented here, only frequency integrated templateprofiles are used for RFI excision. Furthermore, manual inspectionof the output archives is performed to ensure that no residual RFI ispresent in the data. http://dspsr.sourceforge.net/ https://bitbucket.org/meertime/meerpipe/src/master/ https://github.com/pennucci/PulsePortraitureMNRAS , 1–16 (2020) A. Parthasarathy et al.
The reference signal produced by the incoherent sum of thenoise diode signals in the MeerKAT array cannot be used to cal-ibrate the absolute gain, differential gain and differential phase ofthe system as it deviates from 100% linear polarization (Bailes et al.2020). However, the differential gain is calibrated to within 1% dur-ing the procedure that is used to phase up the tied array, leaving onlythe absolute gain and differential phase to be calibrated. The abso-lute gain can be calibrated using a separate set of flux calibrationobservations of a radio source known to have constant flux density,such as PKS B1934 −
638 and in a subset of MeerTime observa-tions, the differential phase also happens to be very close to zero.Therefore, for these observations, it is sufficient to perform onlythe feed hand (basis) and parallactic angle (projection) correctionswhich are implemented in the processing pipeline using pac .The cleaned and vetted archives are then decimated into vari-ous data products containing different number of frequency channelsand sub-integrations. The ToAs are then calculated by cross corre-lating these sub-banded observations with a frequency integratedtemplate profile in the Fourier domain. The formal uncertaintiesproduced by this method assumes that the only source of noise inthe measurement is white radiometer noise and thus, underestimatesthe true ToA uncertainty. The meta-data associated with each ToAfollows the IPTA convention (Verbiest et al. 2016). Wideband ToAsthat account for frequency-dependent profile evolution are also pro-duced (Pennucci et al. 2014). For the results presented here, we usedprofiles that are frequency-averaged to 32 channels with 8 secondsubintegrations.
In this section we first describe the methodology used to estimate jit-ter using frequency averaged ToAs followed by the use of widebandtemplates to examine the frequency dependence of jitter.
The sub-banded ToAs produced by meerpipe are fitted to curatedpulsar ephemerides to obtain sub-banded timing residuals. Sincethe observations typically have a 5 minute duration, only the spin-frequency ( 𝜈 ) and DM are fitted for and only ToAs derived from8 s profiles with S/N >
10 are retained in the analysis . To computethe rms uncertainty of jitter in 𝑇 sub ( = 𝜎 ( 𝑇 sub ) = 𝜎 ( 𝑇 sub ) − 𝜎 ( 𝑇 sub ) . (2)The uncertainties from the observed, frequency-averaged ToAsare induced in an idealised ToA data set to generate the simulateddata. Using the tempo2 fake tool (Hobbs et al. 2006), we simu-late ∼ 𝜎 ( 𝑇 sub ) . We assume that allexcess noise observed in the arrival time measurements are causeddue to jitter since other effects that lead to such short-timescale per-turbations vary more strongly with observing frequency and causeperturbations in the ToAs on longer ( ∼ hours) timescales than what We have cross-checked these measurements with a wideband template aswell and did not find any discrepancies. is reported here (Shannon & Cordes 2012). Distortions in the pulseprofile caused due to imperfect polarization calibration typicallytend to vary with the parallactic angle of the receiver and are causedon much longer timescales than a few minutes. The methodologypresented here is similar to that discussed in Shannon et al. (2014),except that we use simulated ToAs rather than simulated profiles.Since jitter is expected to scale proportionally to 1 / √︁ 𝑁 p , where 𝑁 p is the number of pulses, we can estimate the implied jitter in onehour to be, 𝜎 J ( ) = 𝜎 J ( 𝑇 sub ) / √︁ /( 𝑇 sub ) . (3)We also use the Bayesian pulsar timing package, temponest(Lentati et al. 2014) as a consistency check for our jitter mea-surements. To estimate jitter using temponest, we determine thestandard white noise parameter, EQUAD, used commonly in pulsartiming analyses . EQUAD represents a source of time-independentnoise which could arise from stochastic shape variations in the in-tegrated pulse profile.We claim to have detected jitter in a pulsar if 𝜎 ( 𝑇 sub ) isgreater than 95% of the simulated 𝜎 ( 𝑇 sub ) values, which sug-gests that the rms of the frequency-averaged residuals is higherdue to excess scatter in the observed ToAs than that caused due toradiometer noise. Table 1 reports the jitter measurements for 29 MSPs in our sampleusing frequency averaged ToAs. We constrain jitter in 13 pulsarsand report upper limits for the remaining. Pulsars with either newjitter measurements or new upper limits are highlighted in the table.PSR J2241–5236 has the lowest level of jitter hitherto reported.Our measurements are consistent within uncertainties to previouslypublished values which are also reported in the table.In Figure 1, we show a representative sample of frequencyaveraged timing residuals of eight pulsars with different levels ofjitter. To highlight the markedly different strengths of jitter noiseacross the population, the timing residuals are all plotted on thesame y-scale.
The other short term noise source that can contribute to excesswhite noise is related to propagation in the interstellar medium.Stochasticity in the pulse-broadening function, referred to as thefinite-scintle effect (Cordes et al. 1990; Cordes & Shannon 2010;Lam et al. 2016) will cause arrival time variations that can besignificant for highly scattered pulsars. The strength of this effectdepends on the pulse broadening time 𝜏 , and the number of scin-tles in the observation 𝑁 𝑠 , 𝜎 FS = 𝜏 /√ 𝑁 𝑠 . The numbers of scintlesis 𝑁 𝑠 = ( + 𝜂 Δ 𝜈 / 𝜈 𝑑 )( + 𝜂 Δ 𝑇 / 𝑡 𝑑 ) , where Δ 𝑇 and Δ 𝜈 are theobserving time and bandwidth while 𝑡 𝑑 and 𝜈 𝑑 are the diffractivescintillation time and bandwidth, and 𝜂 ≈ . − Since we use frequency-averaged ToAs, EQUAD and ECORR resultin similar estimates of jitter noise. The ECORR parameter models short-timescale noise processes that result in correlated sub-banded TOAswithin an epoch/observation, but which are otherwise uncorrelated betweenepochs/observations. (NANOGrav Collaboration et al. 2015).MNRAS000
The other short term noise source that can contribute to excesswhite noise is related to propagation in the interstellar medium.Stochasticity in the pulse-broadening function, referred to as thefinite-scintle effect (Cordes et al. 1990; Cordes & Shannon 2010;Lam et al. 2016) will cause arrival time variations that can besignificant for highly scattered pulsars. The strength of this effectdepends on the pulse broadening time 𝜏 , and the number of scin-tles in the observation 𝑁 𝑠 , 𝜎 FS = 𝜏 /√ 𝑁 𝑠 . The numbers of scintlesis 𝑁 𝑠 = ( + 𝜂 Δ 𝜈 / 𝜈 𝑑 )( + 𝜂 Δ 𝑇 / 𝑡 𝑑 ) , where Δ 𝑇 and Δ 𝜈 are theobserving time and bandwidth while 𝑡 𝑑 and 𝜈 𝑑 are the diffractivescintillation time and bandwidth, and 𝜂 ≈ . − Since we use frequency-averaged ToAs, EQUAD and ECORR resultin similar estimates of jitter noise. The ECORR parameter models short-timescale noise processes that result in correlated sub-banded TOAswithin an epoch/observation, but which are otherwise uncorrelated betweenepochs/observations. (NANOGrav Collaboration et al. 2015).MNRAS000 , 1–16 (2020) easurements of pulse jitter and single-pulse variability in millisecond pulsars using MeerKAT Table 1.
Jitter measurements and upper limits for 29 MSPs in our sample. For each pulsar, the parameters reported here are corresponding to the brightestobservation, i.e, with the highest average S/N per pulse. For reference, the median S/N per pulse computed from all selected observations per pulsar is alsoreported. Columns two and three report the period and the DM while columns six to eight report the integration time, the mean ToA error and the weightedRMS values in 8 s sub-integrations. The last two columns report the implied jitter in one hour and a reference to a previously published jitter measurementwhere available. S+2014 refers to Shannon et al. (2014) and L+2019 refers to Lam et al. (2019) (values are scaled from single-pulse rms values). Pulsar namesin bold represent either a new detection or a new upper limit.PSR P DM S/N pulse , max S/N pulse , med T obs 𝜎 ToA
WRMS Implied 𝜎 J ( hr ) Previous 𝜎 J ( hr ) (ms) (pc cm − ) (s) ( 𝜇 s) ( 𝜇 s) (ns) (ns)J0030+0451 4.9 4.4 1.9 1.4 370 0.298 1.765 <
60 60 ± J0125–2327 ±
13 -J0437–4715 5.8 2.6 139 112 180 0.019 0.868 50 ±
10 48.0 ± J0636–3044 ±
30 -J0711–6830 5.5 18.4 2.1 1.6 256 0.619 1.284 60 ± <
90 (S+2014)
J0900–3144 <
130 -J1017–7156 2.3 94.2 2.4 2.2 256 0.149 0.259 < <
100 (S+2014)J1022+1001 16.5 10.3 23.3 5.6 256 0.167 2.054 120 ±
20 280 ±
140 (S+2014), 265 ±
20 (L+2019)J1024–0719 5.2 6.5 2.1 1.8 256 0.204 0.886 <
30 18 ±
10 (L+2019)
J1045–4509 ± <
900 (S+2014)
J1157–5112 <
690 -J1600–3053 3.6 52.3 1.9 1.4 256 0.118 0.711 < <
200 (S+2014)J1603–7202 14.8 38.0 8.1 3.8 512 0.297 3.947 180 ±
40 300 ±
56 (S+2014)
J1622–6617 <
300 -
J1629–6902 <
60 -J1643–1224 4.6 62.4 2.4 2.0 256 0.303 1.647 < <
500 (S+2014), 31 ±
12 (L+2019)
J1730–2304 ± <
400 (S+2014)J1744–1134 4.1 3.1 6.6 2.6 512 0.026 0.686 30 ± ± ± J1756–2251 <
500 -
J1757–5322 ±
45 -
J1802–2124 <
80 -J1909–3744 2.9 10.4 9.6 3.2 256 0.021 0.199 9 ± ± ± J1918–0642 <
55 -
J1946–5403 < <
80 59 ± J2039–3616 <
25 -J2129–5721 3.7 31.9 3.3 3.3 360 0.285 0.493 < <
400 (S+2014)J2145–0750 16.1 9.0 11.1 3.6 256 0.331 3.883 200 ±
20 192 ± ± J2241–5236 ± <
50 (S+2014) a dispersion measure above 50 pc cm − . Based upon its diffractivescintillation bandwidth and time scale we expect that the contribu-tion of the finite-scintle effect to be 40 ns, which is lower than themeasured level of jitter noise. The high dispersion measure pulsarPSR J1017 − 𝜎 J <
10 ns.This pulsar however has an under-turbulent line of sight with ascintillation bandwidth of Δ 𝜈 ≈ ∼ By utilizing the large fractional bandwidth of MeerKAT, we areable to study the frequency dependence of jitter in our sample ofMSPs. Based on the S/N of the observation we generate ToAs persub-band to compute 𝜎 J as a function of frequency channel. We donot detect significant frequency dependence of jitter in any otherpulsars except in PSR J0437–4715. Higher S/N observations ofthese pulsars, especially during scintillation maxima might enablesuch studies. For PSR J0437–4715, we find that jitter decreases with in-creasing observing frequency. In the lower part of the band at ∼ 𝜎 J to be 63 ±
25 ns, at ∼ 𝜎 J to be 50 ±
15 ns, while at ∼ 𝜎 J to beonly 24 ±
20 ns. This is consistent with Shannon et al. (2014), whoreported jitter to be modestly greater at lower frequencies. How-ever, it is important to note that PSR J0437–4715 shows significantfrequency-dependent profile evolution which could likely bias theuncertainties on the jitter measurements at lower and higher bandsdue to template fitting errors. To account for these limitations, wegenerate a wideband template that models the frequency evolutionof the profile.Using PulsePortraiture and the methodology described inPennucci (2019), we create a frequency-dependent smoothed tem-plate for PSR J0437–4715. Using this, we compute sub-bandedToAs with uncertainties that account for the frequency-dependentprofile evolution. Figure 2 shows the frequency dependence of jit-ter using frequency-averaged and wideband templates. Using theseToAs, at ∼
910 MHz, we measure 𝜎 J to be 64 ±
20 ns, at ∼ 𝜎 J to be 50 ±
13 ns, while at ∼ 𝜎 J to be 42 ±
12 ns. The deviations in the measurements of 𝜎 J using frequency-averaged and sub-banded ToAs are consistentwithin uncertainties. It must be noted that as the template deviates MNRAS , 1–16 (2020)
A. Parthasarathy et al.
Figure 1.
Frequency averaged timing residuals for eight pulsars from our sample showing varying levels of jitter. All y-axes are plotted on the same scale forease of comparison. from an accurate description of the profile, we get an artificial lowervalue of jitter. The varying levels of jitter in higher and lower fre-quency bands in PSR J0437–4715 can most likely be attributed tothe narrowing of the pulse profile at higher frequencies.Owing to our ability to measure jitter in individual sub-bandsand also due to the high flux density of the pulsar, we can esti-mate the degree of correlation of jitter between the bands. We see a high degree of correlation ( ∼ ∼ ∼
910 MHz, we compute the correlation strength ( 𝑟 j ) asa function of increasing channel separation as shown in Figure 3.The total observing bandwidth is divided into 32 channels and foreach frequency channel we compute a mean Spearman correlation MNRAS000
910 MHz, we compute the correlation strength ( 𝑟 j ) asa function of increasing channel separation as shown in Figure 3.The total observing bandwidth is divided into 32 channels and foreach frequency channel we compute a mean Spearman correlation MNRAS000 , 1–16 (2020) easurements of pulse jitter and single-pulse variability in millisecond pulsars using MeerKAT Figure 2.
Jitter as a function of observing frequency for PSR J0437–4715,using ToAs generated from frequency-averaged (blue) and wideband tem-plates (purple). The frequency-averaged points are offset by 5 MHz forclarity. The observing bandwidth is averaged to 32 frequency channels.There is a moderate dependence of jitter on observing frequency.
Figure 3.
ToA correlation as a function of observing frequency (in MHz)for PSR J0437–4715 showing the increasing decorrelation of jitter withincreasing channel separation. The lowest frequency channel at a frequencyof ∼
900 MHz is considered as the reference frequency. coefficient by bootstrapping the corresponding 8 s sub-integrations.It is clear that with increasing channel separation, the ToAs begin todecorrelate, implying that the bandwidth of the process that causesjitter is comparable to the bandwidth of our observations.
The emergence of decorrelation of jitter with observing frequencyin PSR J0437–4715, implies that at the lower and higher frequencybands, the emission statistics are increasingly independent of eachother. In the top panel of Figure 4, we show the post-fit timing resid-uals (generated using a wideband template) from each frequencychannel plotted serially in time across a 256 second observation.Each cluster of ToAs is 8 s long with the color representative ofthe frequency band of observation. A striking feature is the vary-ing frequency dependence of the arrival times on a timescale of ∼ − pc with a standard deviation of 2.7 × − cm − pc as shown in the left panel of Figure 4. We also investigatethis effect by analysing the single pulses from this pulsar. For eachpulse with full frequency resolution, we compute the arrival timesusing a wideband template and estimate the DM. The right panelof Figure 4 shows the distribution of estimated DMs with a medianvalue of 2.643 cm − pc and a standard deviation of 8.5 × − cm − pc, which is consistent with what is expected from extrapolating the8-s subintegrations.The left panel of Figure 5 shows a single 8 s integration ofwideband timing residuals selected from Figure 4 but after fitting forDM, and the right panel shows the corresponding profile residuals.The clear presence of structures in the post-fit timing residualssuggests that there may be other (possibly intrinsic) processes thatproduce a spectral dependence on the ToAs. It must be noted that thefit for DM also absorbs the 1 / 𝜈 contributions from such processes,thereby introducing a large scatter in the DM estimates. Analysingconsecutive single pulses, we find that the characteristics of thespectral structures changes from pulse to pulse, potentially causingthe varying frequency dependence as shown in Figure 4. This islikely to be “spectral jitter", a phenomena where the amount ofjitter varies stochastically across the observing band.To understand why such a spectral structure arises, we anal-ysed phase-resolved modulation index of the pulsar using its singlepulses. We find that the modulation index of the main component(C1; see Figure 6) is much higher than the other (wings) parts of theprofile. The shape of the 8-second integrations is thus dependenton the instantaneous modulation of each profile components, whichcauses the profile shape to significantly deviate from the averageprofile.An alternative, more speculative perspective is that the ob-served apparent DM variations could arise due to changes in theplasma density inside or nearby the pulsar magnetosphere. We notethat the magnitude of the observed DM variations, 8.5 × − cm − pc, would in principle allow for such a possibility. Previous work,for instance, by Wu & Chian (1995) or Luo (1998), have proposedthat DM variations arising from non-linearities in the pulsar magne-tosphere may lead to a strong dependence between radio luminosityand DM fluctuations. Since we do not find any correlation betweensingle-pulse DM estimates and S/N and that the observed arrivaltime variations do not show 𝜈 − scaling, the argument for a disper-sive origin for the frequency dependent variations is not stronglysupported.In summary, it is evident that in PSR J0437–4715, spectraljitter places a fundamental limit on the precision of DM estimateson short timescales and it is likely that this phenomena can beobserved in other bright nearby pulsars.A similar effect has been observed in PSR J1713 + ∼ . 𝜇 s MHz − , which extrapolates to 0 . 𝜇 s MHz − for8 s subintegrations. In J0437 − MNRAS , 1–16 (2020)
A. Parthasarathy et al.
Figure 4. Top panel:
Post-fit wideband timing residuals of PSR J0437–4715 estimated using 8 s integrated profiles containing 32 frequency channels overa 256 s observation. Each set of ToAs are colored based on the frequency channel as indicated in the plot. The timing residuals are plotted serially (as ToAnumbers) to showcase the varying frequency dependence for each ToA set.
Left panel:
Estimated values of DM for every 8 s subintegration from fits to thetiming residuals shown in the top panel. The horizontal dashed line represents the median estimated DM value of 2.6419 cm − pc. Right panel:
Distributionsof measured DM values from ∼ typical drift of ≈ 𝜇 s GHz − . The similarity in the values canbe explained by the pulsars having comparable levels of jitter noiseand pulse profile evolution. The phenomenon of jitter can be better understood by examiningshape variations of single pulses. Single pulses from MSPs havebeen studied in relatively few cases compared to more slower, nor-mal pulsars, owing to their low S/N per pulse. Out of the 29 pulsarslisted in Table 1, we present a statistical analysis of single pulses foreight of them with detected jitter measurements. Some pulsars withinferred jitter measurements (from Section 3) did not have associ-ated search mode observations to enable single pulse analysis andothers with upper limits on jitter had very low S/N single pulses.For each pulsar, a set of statistical properties, derived fromfrequency-averaged pulse profiles were determined to study andcompare their emission properties. These are described below: The drifts in the MeerKAT observations of PSR J0437 − Characterising variations in single pulse morphology:
Ex-amining the brightest pulses and their phase-resolved modulationcan provide insights into the state of plasma emission. We exam-ine a number of statistical properties to characterise single pulseamplitude and shape variability.We compute the phase-resolved modulation index as, 𝑚 𝐼 ( 𝜙 ) = √︃ 𝜎 𝐼 ( 𝜙 ) − 𝜎 𝐼 ( 𝜙 ) , (4)where 𝜎 𝐼 ( 𝜙 ) and 𝐼 ( 𝜙 ) are the rms and the mean intensity computedat phase 𝜙 and 𝜎 off is the rms intensity computed from an off-pulsewindow.We also investigate temporal correlations in intensities of sin-gle pulse emission, in particular to check whether emission showsthe pulse-pulse correlation observed in many young pulsars (e.g.,Backer 1970a). We implemented this through analysis of the timeseries of maximum intensity of the individual pulsars. In particularwe calculated the auto-correlation function (ACF) of the time series. Pulse energy distributions:
Analysis of single pulse energydistributions provide a measurement of energy contained in a pulseor sub-component(s) of a pulse and the type of distribution allows usto probe the pulse emission mechanism. We measure the integratedS/N by defining windows around the main component and various
MNRAS000
MNRAS000 , 1–16 (2020) easurements of pulse jitter and single-pulse variability in millisecond pulsars using MeerKAT Figure 5. Left panel:
Sub-banded timing residuals of a single 8 s integrated profile of PSR J0437-4715 plotted serially in time, after fitting for DM.
Rightpanel:
Pulse profile residuals as a function of observing frequency computed by subtracting an average pulse frequency evolution model from the same 8 sintegrated profile. sub-components of the main pulse and interpulse depending onthe profile morphology. In cases where multiple sub-componentswere present, the size of the windows were chosen to be similar(wherever reasonable) to enable a more direct statistical comparison.The instantaneous S/N over a selected window is computed as, 𝑆 / 𝑁 = (cid:205) 𝑁 window 𝑖 = ( 𝐴 𝑖 − 𝐵 )√ 𝑁 window 𝜎 off , (5)where 𝐴 𝑖 is the pulse flux density at the 𝑖 -th bin, 𝐵 is the meanoff-pulse flux density, 𝑁 window is the number of phase bins in theselected component window and 𝜎 off is the off-pulse rms flux den-sity. RFI excision on single pulses is implemented using standardpsrchive tools and coastguard, whenever necessary. Residual RFIwas manually inspected and removed from the analysis. A least-squares minimisation of the data, fitted by a model for the pulseenergy distribution was performed using log-normal and Gaussiandistributions. The log-normal model was defined as, 𝑃 ( 𝑆 ) = 𝑆𝜎 ℓ √ 𝜋 exp (cid:34) − (cid:0) log ( 𝑆 ) − 𝜇 ℓ (cid:1) 𝜎 ℓ (cid:35) , (6)where 𝑆 is the S/N , and 𝜇 ℓ and 𝜎 ℓ parameterise the distribution. Forthe Gaussian model, we fit for the mean ( 𝜇 𝑔 ) and standard deviation( 𝜎 𝑔 ) of the pulse energy distribution. A 𝜒 statistic was used forquantifying the goodness of fit of the preferred distribution. Timing properties of single pulses:
By computing frequency-averaged single pulse ToAs and investigating correlations betweensingle pulse properties and corresponding ToAs, we study the timingproperties of single pulses. We also measure jitter as a function ofthe number of pulses integrated ( 𝑁 p ) and show that the arrival timeuncertainties obtained from averaged profiles due to template fittingare consistent with the measured pulse-to-pulse variations and thatjitter typically scales as 1 / √︁ 𝑁 p . The single pulses of PSR J0437–4715 have been studied exten-sively and owing to its high flux density, pulse shape variationscause excess timing uncertainty, at least four times greater than thatpredicted from radiometer noise alone (Osłowski et al. 2014). Weanalysed ∼ ∼ . We computed the ACF from the peak flux intensities of sin-gle pulses and find no evidence of temporal correlations amongst thepulses. The level of jitter noise estimated by integrating increasingnumber of pulses from 10 to ∼ / √︁ N p and is consistent within uncertainties overmultiple epochs. Our statistical analysis of single pulses from PSRJ0437–4715 show that they are consistent with previously publishedanalyses (Jenet et al. 2001, Osłowski et al. 2014, and Shannon et al.2014). The off-pulse variance was not subtracted to unbias the measured modu-lation index in Osłowski et al. (2014)MNRAS , 1–16 (2020) A. Parthasarathy et al.
Figure 6.
Single pulse S/N histograms for six MSPs. The various histograms shown for each pulsar correspond to selected windows across the pulse profile.The windows used for each pulsar are shown in the respective sub-plot containing the integrated profile (solid line) and the mean profile which is derived fromaveraging the brightest 100 or 1000 single pulses (dashed line). The bottom axis for these sub-plots represent the number of phase bins used while the top axisshows the corresponding phase in turns. The S/N across a selected window is computed using Equation 5.
PSR J0125–2327 is a binary pulsar with a spin-period of ∼ ∼ 𝜎 J to be 48 ±
13 ns in anhour. Statistical properties of single pulses from PSR J0125–2327have not been previously reported. A total of ∼ ∼
15 per pulse. Inthe observed band, the pulse profile has multiple components witha main strong component towards the leading edge of the profileand multiple weaker components towards the trailing edge. The dis-tribution of S/N computed from single pulses follows a log-normaldistribution as shown in Figure 6 for two selected windows overthe full on-pulse region and the peak component (C1). Unlike PSR
MNRAS000
MNRAS000 , 1–16 (2020) easurements of pulse jitter and single-pulse variability in millisecond pulsars using MeerKAT Figure 7.
Phase-resolved modulation index for PSRs J1022+1001 and J2145–0750 shown across the on-pulse region in the upper panel. The middle panelshows the integrated profile from 15000 single pulses and the lower panel shows the mean profile from averaging the brightest 100 pulses. The vertical greydashed line represents the phase of maximum S/N for the profile shown in the lower panel.
J0437 − PSR J1022+1001 is a relatively bright MSP with a rotation period of ∼
16 ms. Owing to its rotational stability and low mean arrival timeerrors, it is a part of the Parkes Pulsar Timing Array program (PPTA)(Manchester et al. 2013). Previous measurements of pulse shapevariations have shown that there is excess scatter in ToAs, largerthan that expected from only radiometer noise (Liu et al. 2015). Thepulse profile at 20 cm wavelengths has a double peaked structureand each component has a different spectral index, resulting instrong evolution of the pulse profile across the observing band. Weanalysed ∼ ∼
30. Inthe left panel of Figure 7, we show the phase-resolved modulationindex computed across the on-pulse region. The modulation index isa measure of intensity variations from pulse to pulse. In this pulsar,the modulation index grows increasingly stronger across the pulseprofile and becomes strongest towards the trailing edge of the secondpeak component suggesting high levels of amplitude modulation inthe pulses originating from these phase ranges. The modulationindex of the first component is ∼ two times smaller than the secondcomponent. The brightest pulses are dominated by emission fromthe second peak component and coincide in phase with the trailingedge of the second component as shown by the vertical dottedlines. From the ACF of pulse peak flux intensities, we find noevidence of temporal correlations amongst pulses. The single pulseS/N distributions of the pulse profile and its peak components are shown in Figure 6 and follow a log-normal distribution. While theintegrated pulse profile formed from 100 brightest pulses showsevidence of both the components, the second component (C2) ismuch more prominent than the first (C1) which is also supported bythe S/N distributions which show that the brightest pulses originatefrom C2. We estimate 𝜎 J to be 130 ±
20 ns in an hour for PSRJ1022+1001 and find that our measurements of jitter scale as 1 / √︁ 𝑁 p which are consistent over multiple epochs. PSR J1603–7202 has a spin period of ∼
15 ms and at 20 cm wave-lengths has two sub-components connected by a more dominantbridge of emission. We analysed ∼ ∼
17. Unlike in PSR J1022+1001, the modu-lation index of this pulsar is strongest towards the leading edge offirst component and is much weaker towards the second component.The S/N distribution of both the components follow a log-normaldistribution. We estimate 𝜎 𝐽 to be 180 ±
40 ns in an hour and findthat it scales as 1 / √︁ 𝑁 p similar to other pulsars. PSR J1744–1134 has a spin period of ∼ ∼ ∼
12. Unlike pulsars discussed so far, the S/N distributionsfor both the peak pulse component and the interpulse tends towardsa Gaussian distribution consistent with low amplitude modulation
MNRAS , 1–16 (2020) A. Parthasarathy et al. as shown in Figure 6. In contrast to PSR J1022+1001, where themodulation index peaks towards the trailing edge of the secondcomponent, in PSR J1744–1134, it gets weaker towards the peakcomponent implying low levels of amplitude modulation. We do notdetect single pulse emission from the interpulse (C1). We estimate 𝜎 J to be 30 ± / √︁ 𝑁 p . PSR J1909–3744 is one of the most precisely timed pulsars owingto its extremely stable rotational behaviour, narrow pulse profile andhigh flux density. At 20 cm wavelengths, the pulse profile consistsof a narrow main component with an interpulse. We analysed ∼ ∼
20 but do notdetect single pulses from the much fainter interpulse. In the leftpanel of Figure 8, we show ∼
200 consecutive pulses in which thepulse emission appears to null occasionally as highlighted by twohorizontal dashed lines at ∼ − >
18) having later times of arrival than pulses with averageS/N (5 < S/N <
18) as shown in Figure 9. We estimate the value of 𝜎 J to be 9 ± / √︁ 𝑁 p . PSR J2145–0750 has a spin period of ∼
15 ms and is a relativelybright pulsar at 20 cm wavelengths. Its pulse profile is dominatedby two main components connected by a bridge of emission and aprecursor. We detect single pulses from both the main componentsand none from the precursor. We analysed ∼ ∼
50. The modulation index shows complexstructures, with high levels of modulation towards the leading edgeof the first component and the trailing edge of the second componentwith the bridge also exhibiting a high modulation index as shownin the right panel of Figure 7. The S/N histograms of the twocomponents (C1 and C2) and the full on-pulse window show log-normal distributions as shown in Figure 6. The wide variability inphase and amplitude exhibited by the single pulses results in largelevels of jitter noise. We estimate a 𝜎 J value of 200 ±
20 ns in anhour and find that jitter scales as 1 / √︁ 𝑁 p . PSR J2241–5236 has a spin period of ∼ ∼ ∼
15. We did not detect any single pulses from the interpulse.The S/N histograms of the single pulses follow an approximatelyGaussian distribution as shown in Figure 6. Its modulation index isweakest towards the peak of the pulse profile, similar to PSR J1909–3744 indicative of low levels of amplitude modulation. Accordingly,this pulsar exhibits the lowest levels of jitter hitherto reported ofjust 3.8 ± >
7) largelyresemble the integrated profile implying that their ToAs are expectedto have similar arrival times as pulses with average S/N (5 < S/N <
7) as shown in Figure 9.The S/N of single pulses were not sufficient to estimate thescaling relation of jitter. The observation that was used to measurejitter from the folded archives (in Section 3) had an estimated me-dian single pulse S/N of ∼
10. However, there was no associatedsingle pulse data for that particular observation. In comparison, themedian single pulse S/N that are presented here are only ∼
4. Anal-ysis of single pulses during scintillation maxima would prove veryinteresting for precision timing analysis.
We have presented the first short-term high-precision pulsar timingresults using the MeerKAT radio telescope including a study ofsingle pulse phenomenology for a selection of MSPs. We foundthat in the highest S/N observations, stochastic pulse shape andamplitude variations cause excess scatter in the ToAs, which arehigher than that expected from radiometer noise alone. Out of the29 MSPs in our sample, we reported new jitter measurements for15 pulsars, out of which six pulsars had constraint measurementswhile upper limits were reported for the remaining. In the remaining14 pulsars, we found that our measurements are either consistentwith previously reported values or have tighter upper limits. PSRJ2241–5236 has the lowest levels of jitter reported in any pulsarof just ∼ ∼ <
11 ns in an hour,which promise to be excellent pulsars for high-precision timingexperiments. New detections of jitter in MSPs clearly confirm thatjitter is a generic property in all pulsars and our ability to detect jitterin many more pulsars will grow with increasing sensitivity of radiotelescopes. It also shows that the levels of jitter vary markedly acrossthe population, even between pulsars with similar spin periods andpulse profiles.We find evidence for the frequency dependence of jitter in PSRJ0437–4715 and showed that jitter decreases only moderately withincreasing observing frequency. We also found that jitter decorre-lates over the observing bandwidth of MeerKAT implying that thepulse emission statistics will become increasingly independent toeach other with wider bandwidths. This is consistent with analysisof the pulsar obtained with the 64-m Parkes radio telescope, whichshowed no correlation in jitter in observations obtained at bandscentred at 730 MHz and 3100 MHz (Shannon et al. 2014). Thedecorrelated nature of emission also results in individual pulseshaving spectral structures that vary from pulse-to-pulse and ow-ing to the very high S/N of each pulse across the observing band,we detect this effect in the pulsar timing residuals as time-varying
MNRAS000
MNRAS000 , 1–16 (2020) easurements of pulse jitter and single-pulse variability in millisecond pulsars using MeerKAT Figure 8. Left panel:
Consecutive single pulses of PSR J1909–3744 shown across the on-pulse region indicating potential pulse nulling behaviour as depictedby the two horizontal dashed lines.
Right top panel:
Single pulse S/N histograms for PSR J1909 − Right bottompanel:
Profile formed from integrating the amplitudes of eight consecutive pulses over the region highlighted in the left panel is shown in purple relative to anintegrated profile formed from adding eight pulses in the emission region (black dashed line) for reference. frequency dependence. Measurements of DMs on short-timescaleswill thus have a large scatter, placing a fundamental limit on theprecision with which pulsar DMs can be measured. In addition tothe limitations posed to the measurement of the gravitational wavebackground signal, this effect can also lead to limiting the precisionof orbital parameters in highly relativistic systems, especially dueto covariances between the DM and timing model parameters whencombining observation from multiple orbits.We speculate that with more sensitive radio telescopes withlarger bandwidths, pulse jitter will dominate the error budget.
We reported the first single pulse study of eight MSPs usingMeerKAT. Single pulse statistics for PSRs J0125-2327 and J2241–5236 have not been previously studied. We also reported the firstobservation of pulse nulling in an MSP, seen in PSR J1909–3744.Most pulsars in our sample with detected single pulses showed log-normal distributions consistent with previous results with MSPs(Shannon et al. 2014) and slow spinning pulsars (Burke-Spolaoret al. 2012). However, pulsars with the lowest levels of jitter noise,PSRs J2241–5236, J1909–3744 and J1744–1134 showed approxi-mately Gaussian distributions.Studying energy distributions enables us to distinguish be-
MNRAS , 1–16 (2020) A. Parthasarathy et al.
Figure 9.
ToAs computed from a randomly chosen subset of ∼ 𝜇𝑠 . tween different models of plasma behaviour based upon theoreticalpredictions. For example, self-organised criticality (Bak 1996; Baket al. 1988) describes self-consistent systems that interact withoutany preferred distance or timescales. It predicts a power-law dis-tribution of pulse intensities. However, models based on stochasticgrowth theory (Robinson et al. 1992; Robinson 1995) describe in-teractions in an independent homogeneous medium with preferreddistances and timescales. This model predicts a log-normal distribu-tion of pulse intensities. Cairns et al. (2004) found Gaussian energydistributions at the edges of the pulse profile in slow spinning pul-sars. They attribute this to either irregularities in the density of theplasma or a superposition of multiple weak emission components.Following from this, it can perhaps be speculated that for the threepulsars in our sample showing Gaussian energy distributions, ourline of sight traverses the edge of the emission region. It is alsointeresting to note that these three pulsars have single componentGaussian profiles with an interpulse.In pulsars that show high levels of jitter noise, we found thatthe phase-resolved modulation index increased towards the mainemission component, such as in PSRs J0437–4715, J1022+1001,J1603–7202 which have wide profiles with multiple components; whereas in pulsars with the lowest levels of jitter noise, the mod-ulation index showed the opposite trend and approached a minimatowards the main emission component, such as in PSRs J1744–1134, J1909–3744 and J2241–5236 which have narrow profileswith single components and a weak interpulse.The phase-resolved modulation index can be used to distin-guish between pulsar emission models and previous theoreticalworks have suggested that the modulation index depends upon somefunction of the pulsar period and its period derivative. It is however,somewhat arbitrary to define a modulation index from the phase-resolved modulation index profile because it is highly dependenton the pulse phase. In this case, we chose a value that is represen-tative of the modulation index near the main emission componentof the integrated pulse profile. We used a Spearman correlationcoefficient to determine the correlations between the modulationindex and other pulsar parameters and found moderate correlationsof 0.62 ± ± MNRAS000
ToAs computed from a randomly chosen subset of ∼ 𝜇𝑠 . tween different models of plasma behaviour based upon theoreticalpredictions. For example, self-organised criticality (Bak 1996; Baket al. 1988) describes self-consistent systems that interact withoutany preferred distance or timescales. It predicts a power-law dis-tribution of pulse intensities. However, models based on stochasticgrowth theory (Robinson et al. 1992; Robinson 1995) describe in-teractions in an independent homogeneous medium with preferreddistances and timescales. This model predicts a log-normal distribu-tion of pulse intensities. Cairns et al. (2004) found Gaussian energydistributions at the edges of the pulse profile in slow spinning pul-sars. They attribute this to either irregularities in the density of theplasma or a superposition of multiple weak emission components.Following from this, it can perhaps be speculated that for the threepulsars in our sample showing Gaussian energy distributions, ourline of sight traverses the edge of the emission region. It is alsointeresting to note that these three pulsars have single componentGaussian profiles with an interpulse.In pulsars that show high levels of jitter noise, we found thatthe phase-resolved modulation index increased towards the mainemission component, such as in PSRs J0437–4715, J1022+1001,J1603–7202 which have wide profiles with multiple components; whereas in pulsars with the lowest levels of jitter noise, the mod-ulation index showed the opposite trend and approached a minimatowards the main emission component, such as in PSRs J1744–1134, J1909–3744 and J2241–5236 which have narrow profileswith single components and a weak interpulse.The phase-resolved modulation index can be used to distin-guish between pulsar emission models and previous theoreticalworks have suggested that the modulation index depends upon somefunction of the pulsar period and its period derivative. It is however,somewhat arbitrary to define a modulation index from the phase-resolved modulation index profile because it is highly dependenton the pulse phase. In this case, we chose a value that is represen-tative of the modulation index near the main emission componentof the integrated pulse profile. We used a Spearman correlationcoefficient to determine the correlations between the modulationindex and other pulsar parameters and found moderate correlationsof 0.62 ± ± MNRAS000 , 1–16 (2020) easurements of pulse jitter and single-pulse variability in millisecond pulsars using MeerKAT ship is a consequence of the ‘sparking gap’ model first theorized byRuderman & Sutherland (1975).Future observations of more pulsars will enable us to iden-tify correlations with the various ‘complexity parameters’ (Gil &Sendyk 2000; Lou 2001) that aid in distinguishing between differ-ent emission models which will prove powerful in probing deeperinto understanding pulsar emission models. We also measure thecorrelation of 𝜎 J (measured in 1 hr) with various estimates of thepulse widths at 50% ( 𝑊 ), 10% ( 𝑊 ) and the effective pulse width( 𝑊 eff ). We estimate the effective pulse width following the definitionin Downs & Reichley (1983) and Cordes & Shannon (2010), 𝑊 eff = Δ 𝜙 (cid:205) 𝑖 [ 𝑃 ( 𝜙 𝑖 + ) − 𝑃 ( 𝜙 𝑖 )] , (7)where Δ 𝜙 is the phase resolution of the pulse profile in units of timeand 𝑃 is the pulse profile which is normalised to a peak intensity ofunity. We find a moderate correlation of 0.64 ± 𝑊 and nosignificant relationship with 𝑊 eff . These results provide excellentconfirmation to those reported in Lam et al. (2019) and suggestthat the level of pulse jitter does not significantly depend on the‘sharpness’ of the pulse profile. Although the MeerKAT MSP data set is not yet sensitive to grav-itational wave radiation due to its short timing baselines, there area number of ways in which it can contribute to high precision pul-sar timing and PTA experiments. Limitations in precision timingexperiments are typically due to an incomplete understanding ofsystematics in both the instrument and analysis methodologies ordue to lack of sensitivity. Precise estimates of DM variations arecrucial for PTA experiments, which can be improved by conduct-ing near simultaneous global observing campaigns of a few MSPswith telescopes like MeerKAT, GBT, Parkes, Effelsberg and theFive Hundred Meter Aperture Spherical Radio Telescope. Suchcampaigns, as was conducted previously on PSR J1713 + ∼ The MeerKAT telescope is operated by the South African Radio As-tronomy Observatory, which is a facility of the National ResearchFoundation, an agency of the Department of Science and Inno-vation. This work made use of the gSTAR and OzSTAR nationalHPC facilities. gSTAR is funded by Swinburne and the AustralianGovernment Education Investment Fund. OzSTAR is funded bySwinburne and the National Collaborative Research InfrastructureStrategy (NCRIS). This work is supported through Australian Re-search Council (ARC) Centre of Excellence CE170100004. A.P.acknowledges support from CSIRO Astronomy and Space Science.R.M.S. acknowledges support through ARC grant CE170100004.M.B, S.O, and R.M.S. acknowledge support through ARC grantFL150100148. R.M.S. also acknowledges funding support throughAustralian Research Council Future Fellowship FT190100155. FAgratefully acknowledge support from ERC Synergy Grant “Black-HoleCam” Grant Agreement Number 610058. T.T.P. is supportedthrough a NANOGrav Physics Frontiers Center Postdoctoral Fel-lowship from the National Science Foundation Physics FrontiersCenter Award Number 1430284. This work also made use of stan-dard Python packages (Oliphant 2006, Jones et al. 2001, McKin-ney 2010, Hunter 2007), Chainconsumer (Hinton 2016) and Bokeh(Bokeh Development Team 2018).
DATA AVAILABILITY
The data underlying this article will be shared on reasonable requestto the corresponding author.
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