Hyperbolic limit on the early arrival time of bright pulses from PSR J0835$-$4510 (Vela)

The Vela pulsar (PSR J0835-4510) is an excellent example of studying emission mechanisms due to its normal pulses and giant micro-pulses. This pulsar has been studied for long-term timing analyses (Dodson et al., 2007) as well as short-term single pulse studies of normal pulses (Johnston et al., 2001) and giant micro-pulses (Chen et al., 2020). This young, luminous, and close pulsar emits pulses across multiple wavelengths (Pellizzoni et al., 2010) and is highly linearly polarised ($>95\%$ at $1720$  MHz) with a period of approximately 89.3 ms (Radhakrishnan et al., 1969). Unlike the Crab pulsar, the Vela pulsar does not emit giant pulses exceeding the mean flux density by ten times (Knight, 2006; Cairns, 2004; Palfreyman et al., 2016). On the contrary, as reported by Johnston et al. (2001), the Vela pulsar emits giant micro-pulses with high peak-flux density and narrow pulse width at 660 MHz and 1413 MHz. Additionally, consecutive bright pulses with five times the mean flux density of the average pulse were detected, indicating that bright pulses may not be independent random events (Palfreyman et al., 2011).

We adopted the pulse-to-pulse analysis model introduced by Krishnamohan & Downs (1983) (hereafter referred to as KD83) that has provided valuable insights into the nature of the Vela pulsar. The KD83 model involves segmenting adjacent intensity intervals, or "gates", to encompass a wide range of peak pulse intensities. Pulses are then grouped and added based on peak flux density, resulting in gated pulse profiles. The gated profiles revealed that brighter pulses occur earlier in the pulse sequence. The emission region comprises four sub-components representing distinct emission zones at varying distances from the core, as found by KD83. In our study, we aim to extend this model by examining the behaviour of gated profiles at higher peak flux densities and applying it to a larger dataset to explore sub-components’ presence. The reasons behind the early arrival of bright pulses remain unexplained in the literature, and the KD83 model has yet to be applied to large datasets like ours. Hence, we examined regular, bright, and giant micro-pulses for six days using the KD83 model and then focused our analysis on a specific dataset. We test the peak flux density against the full width half maxima (FWHM) and phase of the peak to identify any correlations and attempt to discern sub-components in the higher gated profiles.

To establish continuity with today’s research while extending the gated profiles by a factor of 20. We have used the term "Fflux-density range(s)" instead of "gate(s)" and "flux-density integrated profile" instead of gated profiles. This paper describes our observations and data reduction process, including a critical comparison to the original analysis conducted by Krishnamohan & Downs (1983), in Section 2. We then present our results and discuss their implications in Sections 3 and 4, respectively.

The University of Tasmania’s 26 m radio telescope at Mt. Pleasant Observatory near Hobart, Tasmania, Australia, has been used to observe the Vela pulsar for long-term single-pulse studies since $2007$. In this work, we report on 111 hours of the Vela pulsar single pulses recorded in 2016 and 2020. The data for this analysis has a central frequency of $1376$ MHz with a bandwidth of $64$ MHz. Each raw file is $10$ s long, containing around $111$ single pulses. DSPSR (Digital Signal Processing Software for Pulsar Astronomy, (Van Straten & Bailes, 2011)) was used to fold the raw data into $8192$ timing bins with a $10.9~{}\mu$s time resolution. Each file was also integrated in polarisation and frequency using PSRCHIVE (Hotan et al., 2004).

Flux Density calibration was performed by pointing on and off at a known source calibrator and pulsing with a noise diode. Since the L-band setup of the radio telescope rarely changed, these were only performed sporadically.

We selected four calibrated observations that were longer than 18 hours and between the selected days in 2016 and 2020. These calibrated days showed that 1.0 on the Mt Pleasant relative scale was equivalent to 63.7 Jy (standard deviation of 0.72 Jy) as shown in Table 1. For this study, it is important to note that the relative flux densities of the days of observation are consistent. Figure 1 shows the signal-to-noise ratio of each 10 s file over the set of selected observations. The noteworthy observation is the apparent presence of cycles. This is most likely due to the differences in sensitivity of the two linear channels and low-noise amplifiers in the L-band receiver at Mt Pleasant. Vela is highly linearly polarised, so the signal moves from channel to channel as Vela passes across the sky. The signal-to-noise values are reasonably consistent across the observation days, even though they are years apart. This also implies that the system temperature was fairly stable throughout our analyses.

Palfreyman et al. (2016) showed the cyclic nature of the FWHM of the integrated pulse profile of a day, which could be related to the amount and type of pulses emitted by the Vela pulsar. $1.52$ ms was the median value of the extensive dataset (Palfreyman et al., 2016). Considering this possible relation, we selected a day, Modified Julian Date (MJD) $58851$, where the FWHM was greater than $1.52$ ms (hereafter 2020-data) and another day where it was less than $1.52$ ms, MJD $57661$ (hereafter 2016-data). The FWHM of the integrated pulse profiles of 2020-data and 2016-data were around $1.522$ and $1.499$ ms, respectively. This work assumes that MJD $58851$ (from 2020) could be a high-activity day, and MJD $57661$ (from 2016) could be a less active day, expecting fewer bright and giant micro-pulses. Bright and giant micro-pulses have higher peak flux density and smaller FWHM. The results of these two different activity phases of the Vela pulsar are discussed later in Section 3. Another influential factor was choosing days with minimum apparent radio frequency interference (RFI). Observed data of pulsar radio emission often suffer from various types of RFI, which could be caused by human communications technologies such as satellites, mobile base stations, or navigation radars. We did not select a day with an open day at the observatory.

We followed the model created by KD83 but with minor changes in the interval to further categorise and analyse our data. The intervals of peak flux for the KD83 gated profiles were not evenly spaced, but we distributed our flux-density range (hereafter FD-range) with smaller and equal intervals. We have created 80 peak flux intervals for the distribution of pulses, each of our FD-range being $1$ unit wide. For example, FD-range $1$ would include pulses of peak flux $1.00\leq S_{p}<1.99$, in arbitrary units (AU); i.e., $63.7\leq S_{p}<126.76$ Jy.

We integrated every pulse from each FD-range to create flux-density integrated (hereafter FD-integrated) profiles. In other words, we were averaging profiles within the same energy range. These FD-integrated profiles had signal-to-noise ratios varying from as high as 9075 (FD range 3 with 169942 pulses) to as low as 77 (FD range 76 with one pulse) because of a varying number of extracted pulses. We added pulses from the other two consecutive days to improve noisy FD-integrated profiles (mostly higher (brighter) profiles; > 20 for 2016-data and > 30 for 2020-data) with lower SNR values. We manually checked each pulse from brighter profiles to remove radio frequency interference (RFI). So all $16944$ pulses in these higher FD-range were manually checked and removed if RFI was predominant. In this analysis, we had $4,358,856$ pulses, and $1,348,242$ pulses were used to create our FD-integrated profiles. The pulse number difference is because the lower FD-ranges have pulses from just one day, and higher FD-ranges from three days. After comparing the number of pulses of 80 FD-ranges from both activity days, we focused on the phase of the peak of FD-integrated pulse profiles from high activity days to analyse the nature of the arrival time and FWHM. The sub-components are also analysed from a brighter profile of the 2020-data.

The peak flux density ($S_{p}$) of bright profiles was calculated using the radiometer equation,

where $S_{sys}=470~{}Jy$ is the system equivalent flux density ¹¹1https://ra-wiki.phys.utas.edu.au/, $n_{p}=2$ is the number of polarisation summed, $\Delta f=$ 64 MHz is the effective bandwidth and $T_{int}=10.9~{}\micro$s is the sampling interval (Lorimer & Kramer (2012)).

The peak flux of the integrated pulse profile of $\sim 18$ hr from 2020-data, at $1376$ MHz is $106.29~{}(\pm 0.097)$ Jy arriving at phase $-0.04~{}(\pm 0.01)$ ms and is similar for 2016-data. The uncertainty associated with phase value is from bin width. As mentioned before, the FWHM of $\sim 18$ hr integrated profile from 2020-data is $1.522~{}(\pm 2.0\times 10^{-04})$ ms whereas, it is $1.499~{}(\pm 2.156\times 10^{-04})$ ms for 2016-data. The mean flux density (MFD) of integrated pulse profiles from both datasets is $2.3596~{}(\pm 2.6\times 10^{-04})$ Jy. According to the relationship described in Palfreyman et al. (2016) between the width and Vela’s activity, the data from 2020 corresponds to high activity days of the Vela pulsar, while the data from 2016 corresponds to low activity days.

According to Kramer et al. (2002) and Johnston & Romani (2002), the Vela pulsar has been identified as an excellent candidate for emission studies. It is due to its unique characteristics, including the close correlation between giant micro-pulses and classical giant pulses and the emission of normal pulses. The Vela pulsar serves as an ideal case for investigating the properties and mechanisms of these different types of pulsar emissions. The Vela pulsar emits pulses in a wide frequency range exhibiting various structures of individual pulses that vary from the average integrated pulses profiles  (Cairns et al., 2001, 2003; Shannon et al., 2014). These individual pulses also vary in peak flux density; hence, different nomenclature is used to differentiate them (Cordes et al., 1988; Johnston et al., 2001; Edwards & Stappers, 2002; Palfreyman et al., 2011). We found minimal mention of analyses on the variability of the pulse shape for the same frequency for an extensive period. In this work, the observational period lets us study and compare the Vela pulsar for the different years and peak flux density for a fixed observed frequency. The FD-integrated profile model (KD83) would reduce uncertainties arising from different individual pulse shapes because the pulse properties vary for pulses even with similar frequencies and peak flux densities.

Our fainter profiles have normal pulses. In contrast, the pulses from brighter profiles are giant micropulses that exhibit properties similar to giant pulses of the Crab pulsar and two millisecond pulsars, B1821$-$24 and B1937+21 (Staelin & Reifenstein III, 1968; Cognard et al., 1996; Romani & Johnston, 2001). The Vela pulsar and two other pulsars (B1706$-$44 and B0950$+$08) have been recognised as giant micro-pulse emitters (Johnston et al., 2001; Kramer et al., 2002; Johnston & Romani, 2002; Cairns, 2004). These are examples of high-emission pulsars. Another example of such sporadic, bright emission would be PSR J0901-4624 (Raithel et al., 2015). From our analyses, we have found that the type of pulses Vela emits is connected with the FWHM of the integrated pulse profiles and is twice as many for 2020-data when the Vela pulsar is in considerably high activity mode. The two brightest pulses from 2020-data have FWHM of $0.250\,$ms and $0.305\,$ms and peak flux densities of $4989.62\pm 0.86$ Jy and $4969.87\pm 0.73$ Jy, making them exceptional as per the definition of Johnston et al. (2001). The brightest pulse we found for 2016-data is from FD-range 79, with a peak flux density of $5076.25\pm 0.09$ Jy and FWHM $0.265\,$ms.

To our knowledge, no existing literature extensively investigates these pulses together. The resultant hyperbolic nature of the FD-integrated pulse profile phase with the asymptotic value of $800$ micro-seconds could support the theory of different origins (Cairns, 2004) for both types of pulses. The plausible hypothesis for the origin of giant pulses is the collapse of plasma-turbulent wave packets (Hankins et al., 2003) or emission from a rotating carousel of discharges that circulates the magnetic axis (Gil & Melikidze, 2003). Large flux densities, smaller-restricted phase windows and power-law distribution put giant and giant micro-pulses in similar categories (Cairns, 2004). We have also found power-law distribution for the analysed bright pulses. The comparable properties could indicate a similar emission process. The mean flux density of FD-integrated profiles exceeding a threshold of 20 was five times greater than that of the integrated profile of $\sim 18$ hr. Notably, these profiles exhibited bright pulses, which can be characterized as conforming to a power-law distribution. Furthermore, these pulses demonstrated a narrowed FWHM, measuring less than $0.5366$ ms, and their peak fluxes ranged from $1275$ Jy to $3000$ Jy.

The $89.3$ ms period of Vela pulsar may occasionally fluctuate due to activities such as glitches (Palfreyman et al., 2018; Manchester et al., 2005), bright pulses (Palfreyman et al., 2011), consecutive bright pulses (Palfreyman et al., 2011), nulling (Ashton et al., 2019). We analysed around 4425693 pulses but have not found nulling. Among these, pulses with peak flux greater than $1400$ Jy were found in the window of $0.3$ ms phase, ranging from approximately $-0.6$  to $-0.9$ ms. From the $89.3$ ms period and $8$ ms observed radio emission region, our 59 FD-integrated profiles (FD-range 20 onwards) fall in just the $0.3$ ms phase range. This small phase region for higher FD-integrated profiles would support the theory of separate origins for normal pulses and giant micropulses (Cairns, 2004; Hankins et al., 2003; Johnston & Romani, 2002; Romani & Johnston, 2001; Gil & Melikidze, 2003).

According to the research conducted by Krishnamohan & Downs (1983), the emission of the Vela pulsar comprises four distinct emission zones, all of which we could identify in our study. Nevertheless, only two components have been found for higher FD-integrated profile 70. The smaller two emission zones were missing, which might indicate that they are too small for detection or not present. However, the other present components had different phase and peak values, which might show the emission region is moving. This difference is another point supporting the theory put forward by Cairns (2004). The Vela pulsar has classical properties such as microstructure, high degree of polarisation, S-like position angle swing, and features related to high-energy emissions, such as giant micro-pulses and different profile components (Johnston et al., 2001). Since we can now relate giant and giant-micro pulses, the emission process can also be connected to the radio and high-energy emissions.

This study aims to extend the KD83 gated profiles of the Vela pulsar by a factor of approximately 20 in flux density and to examine the characteristics of the brighter profiles. We have also investigated the correlation between the width of the integrated pulse profiles and the emission of bright and giant micro-pulses of the Vela pulsar and found a positive correlation between them.

Our analysis indicates that the arrival times of these FD-integrated profiles have a hyperbolic relationship with their peak flux density, with an asymptote of $-0.85$ ms compared to the peak of the integrated profile at $1376$ MHz. This behaviour could suggest a connection between the disappearance of emission sub-components and their phase delay at higher peak intensities, potentially indicating that the emission region of brighter pulses is near the outermost part of the emission cone beyond which the magnetic field lines are closed. While normal pulses had four emission components, we observed that giant micro-pulses had only two components, indicating different emission regions for these two types. Moreover, we found that the emission of bright pulses follows a power-law distribution similar to that of giant micro-pulses.

We thank the referee for the extremely helpful and positive comments, which improved the paper. The authors would like to thank Emeritus Professor John Dickey from the University of Tasmania for his feedback on the early drafts of this paper. Most of this work was based on AM’s MAppSci thesis from the University of Tasmania. A.M. would like to acknowledge the financial support from the OzGrav scholarships, which made much of this work possible. S.S. acknowledges the support from Australian Government Research Training Program Scholarship. We acknowledge the use of the ATNF Pulsar Catalogue, Manchester et al. (2005) ²²2https://www.atnf.csiro.au/research/pulsar/psrcat/.

The article’s data will be shared by the corresponding author upon reasonable request.