A single pulse study of a millisecond pulsar PSR J0621+1002

The polarization profile for PSR J0621+1002 is shown in Figure 1. The overall pulse width is about 175 degrees, nearly half of the pulse period. The profile features are consistent with the EPTA observation at 1.4 GHz  (Desvignes et al., 2016). The pulse profile has three components and the fractional linear polarization for the second component is higher than the first and third components.

The position angles (PAs) vary significantly across the first and the third components and it is relatively smooth across the second component. Generally, for pulse profile with three components, the first and third pulse components are related to the cone emissions and the second pulse component is associated with the core emission (Backer, 1976; Rankin, 1983) and the gradient of PA swing is large near the pulse centre (Lyne & Manchester, 1988). The second component of PSR J0621+1002 exhibits a relatively smooth PA swing which does not agree with the core emission. This component also shows high fractional linear polarization which is characterized by the cone emission. Therefore, we suggested that all the three components of PSR J0621+1002 are related to the cone emissions. Our measured flux density at 1250 MHz and RM are 1.90 mJy and $53.0\pm 0.1\,{\rm rad\,m^{-2}}$, respectively, which are consistent with the previously published results (Sobey et al., 2019).

Single-pulse stack of 300 pulses is shown in Figure 2. To investigate the pulse modulation behavior, longitude-resolved fluctuation spectrum (LRFS, Backer 1970b) and two-dimensional fluctuation spectrum (2DFS, Edwards & Stappers 2002) was carried out for each pulse component. 2DFS is a useful tool to determine whether subpulses are drifting in pulse longitude (Weltevrede, 2016) and LRFS can be used to detect periodicity of subpulse modulation. Using PSRSALSA package with FFT-size of 1024, the LRFS and 2DFS of all the data are shown in the upper and bottom panels of Figure 3, respectively. There is a clear periodic pulse intensity modulation for both the first and the third pulse components. The fluctuation spectrum of the first pulse component has a clear peak of $3.0\pm 0.1\,P$ with the pulse period $P$, which means that the vertical separation of drift bands of this component is coherent with $P_{3}=3.0\pm 0.1\,P$. However, the fluctuation spectrum of the third pulse component is diffused at the peaks of $3.0\pm 0.1\,P$ and 200$\pm$1 $P$ which suggests that the vertical separation of drift bands is not fully coherent. The asymmetric feature (along X=0) in the 2DFS of the first component suggests the existence of pulse drifting across the pulse longitude. However, our resolution is not enough to fully resolve the feature and yield a measurement of $P_{2}$.

The $P_{3}$ for the first and third components seem identical. Correlation analysis between them was carried out to check whether the intensity variation following this period is phase-locked (more details see Kou et al. 2020). We found no evidence that the intensity variations for the first and third pulse components are phase-locked.

Pulse energy distributions of the off-pulse and on-pulse regions for the first and the third pulse components are presented in Figure 4. All the energies are normalized by the mean on-pulse energy. On-pulse region is determined as the longitude range where the pulse intensity significantly exceeds (larger than 3 $\sigma$) the baseline noise and the off-pulse energy is measured from a region with the same duration as the on-pulse region in the baseline.

Pulse energy of the off-pulse region follows a narrow normal distribution, while that of the on-pulse region can be described as log-normal distribution with a high-energy power-law tail. The energy cutoff is at 1.7E/$\langle E\rangle$ for the power-law tail (the vertical black line in Figure 4) and the spectral index is $-8.0\pm 0.1$. Pulse energy distribution of the first pulse component can be fitted well by a log-normal function. Pulse energy of the third pulse component also follows a log-normal distribution with a power-law tail. Generally, pulse energy for MSPs can be well modeled using log-normal or Gaussian distribution (Shannon et al., 2014). However, log-normal energy distribution with an excess of high-energy pulses has been detected in many normal pulsars (Mickaliger et al., 2018).

ToA uncertainties on short time-scales can be described as (Cordes & Shannon, 2010; Liu et al., 2012; Lam et al., 2016):

where $\sigma_{\rm rm}$, $\sigma_{\rm J}$, $\sigma_{\rm scint}$ and $\sigma_{\rm 0}$ are the uncertainties induced by radiometer noise, jitter noise, instability of short-term diffractive scintillation and all other possible contributions. $\sigma_{\rm rm}$/$\sigma_{\rm J}$ is proportion to the signal to noise ratio (S/N) of single pulse. For highly sensitive radio telescopes, such as FAST (Liu et al., 2011, 2012; Hobbs et al., 2019), $\sigma_{\rm rm}$ is expected to be less dominant especially for bright pulsars.

ToA uncertainties induced by short-term diffractive scintillation can be described as (see Cordes & Shannon (2010)): $\sigma_{\rm scint}^{2}=\tau^{2}/N_{\rm scint}$ with the pulse-broadening time-scale $\tau$ and the number of scintles $N_{\rm scint}$ in the observation. The number of scintles $N_{\rm scint}=(1+\eta\Delta\nu/\nu_{\rm d})(1+\eta\Delta T/t_{\rm d})$, where $\Delta\nu$ and $\Delta T$ are the observing bandwidth and time, $\nu_{\rm d}$ and $t_{\rm d}$ are the diffractive scintillation bandwidth and time, and $\eta\approx 0.3$ is the scintillation filling factor (Cordes & Shannon, 2010). For PSR J0621+1002, the diffractive scintillation bandwidth, diffractive scintillation time and pulse-broadening time are 1.19 MHz, 323.4 s and 0.16 $\mu$s (Cordes & Lazio, 2002), respectively. In our observation, the observing bandwidth and time are 400 MHz and 1560 s, respectively; the $\sigma_{\rm scint}$ is about 10 ns which is negligible.

Intrinsic single pulse shape and phase variations introduce jitter noise. We carry out single pulse timing analysis for PSR J0621+1002 and the ephemeris is provided by Desvignes et al. (2016). psrchive program paas was used to fit the integrated profile of the entire observation and formed a noise-free template. The ToAs are obtained by cross-correlating single pulse profiles with the standard template. The timing residuals of single pulses are shown in Figure 5 and the post-fit timing residuals are divided into two classes. Single pulses with timing residuals in the range of -5$-$4 ms and -10$-$-5 ms are defined as class A and class B, respectively. The number of pulses in class A and class B are 52429 and 1637, respectively.

The average pulse profiles of class A and class B are shown in Figure 6. It is expected that the average profiles of the two classes are different. The pulse profile of class A has a stronger third pulse component, and that of class B has a stronger first pulse component. PAs of both classes show complicated variations. However, there is no clear evidence that the PA swings of two classes of pulses are different because of the limited signal to noise ratio (S/N).

As mentioned above, $\sigma_{\rm scint}$ in Equation 1 is negligible. We can assume that all of the excess error in the arrival time measurements are attributed to jitter noise. Jitter noise can then be obtained by calculating the quadrature difference between the observed rms timing residual and the ToA uncertainty (Shannon et al., 2014): $\sigma_{\rm J}^{2}{(N_{\rm p})}=\sigma_{\rm obs}^{2}{(N_{\rm p})}-\sigma_{\rm ToA}^{2}{(N_{\rm p})}$ where ${N_{\rm p}}$ is the number of averaged pulses. Estimates of jitter noise of PSR J0621+1002 are shown in Figure 7. The red solid line in Figure 7 is the fitted result for the jitter noise scaling $\sigma_{\rm J}{\rm(N_{p})}\propto N_{\rm p}^{-1/2}$. The expected jitter noise for one-hour-long observation is 0.51 $\mu$s.

The expected timing precision for a fraction $f$ of $N$ pulses: $\sigma_{\rm J}(f,N)=\sigma_{\rm J}(f,1)/\sqrt{fN}$ (Shannon et al., 2014), where $\sigma_{\rm J}(f,N)$ is the jitter noise in the $fN$ selected single pulses. While using a fraction of pulses, the timing precision improvement is achieved if and only if $\sigma_{\rm J}(f,1)<\sigma_{\rm J}(1,1)\sqrt{f}$.

We examined the achievable timing precision using only the pulses in class A or class B. The pulses in class A and class B and all the single pulses formed three noise-free templates, respectively. ToAs for each class are obtained by cross-correlating the single pulse profile with the corresponding standard template. As shown in Figure 8, the $\sigma_{\rm J}$ does decrease if only the pulses in one class are selected, but it does not decrease to where the timing precision improvement can be achieved (the filled grey area in Figure 8).

We also examined the achievable timing precision for PSR J0621+1002 using only bright pulses. All the single pulses are divided into five classes with the S/N less than 10, in the range of 10 to 20, 20 to 30, 30 to 50, and larger than 50. Note that the S/N is determined by dividing the peak flux density of a single pulse by the rms of the off pulse region. The pulse numbers of these five classes are 37185, 12050, 3022, 1476, and 333, respectively. We summed all the single pulses in each class and formed five noise-free templates, respectively. ToAs for each pulse class are obtained by cross-correlating the single pulse profile with the corresponding standard template. As shown in Figure 8, The $\sigma_{\rm J}$ does decrease for brighter pulses since they originate from a narrower region of the pulse phase. However, timing precision improvement is not achieved.