The GMRT High Resolution Southern Sky Survey for Pulsars and Transients. V. Localization of Two Millisecond Pulsars

Pulsars, the nature's best clocks, are excellent laboratories to test the laws of physics under extreme magnetic fields, density, and gravity. Millisecond pulsars (MSPs) are a subclass of pulsars with a few millisecond period (<30 ms) and remarkable rotational stability, allowing them to be used as probes for understanding pulsar intrinsic properties as well as probing the intervening interstellar medium. The long-term timing studies of these MSPs can precisely measure the unmodeled variations (called timing noise) in the time-of-arrival (ToA) of the pulses.

The pulsar timing array (PTA; e.g., Detweiler 1979) experiments use a set of MSPs with precise timing parameters to detect gravitational wave (GW) signals using the angular correlation between the residuals of the arrival times of pairs of pulsars (Hellings & Downs 1983). The GW frequency range to which PTA is sensitive is 1/baseline to 1/cadence (Stappers et al. 2018). Here, the cadence is the regularity of the observations (∼weekly), and the baseline is the overall timing observation span of pulsars (∼tens of years). This weekly cadence and years of timing baseline correspond to the frequency range 10⁻⁶–10⁻⁹ Hz, commonly known as the nanohertz range. The majority of the nHz GW spectrum is assumed to be contributed by an ensemble of merging supermassive black hole binaries, which forms an isotropic stochastic GW spectrum (Burke-Spolaor et al. 2019). The number of MSPs used in the experiment has the strongest influence on the detectability of nHz GW background signal in timing data (Siemens et al. 2013). This necessitates the identification of newly discovered MSPs suitable for inclusion in PTAs. However, such a high-precision timing follow-up of newly discovered MSPs is hindered by the large positional uncertainties associated with the discoveries from surveys with single dishes (e.g., GBNCC survey; Stovall et al. 2014; HTRU survey; Keith et al. 2010; PALFA survey; Cordes et al. 2006; etc.) or with incoherent array (IA) beam of interferometric array (e.g., the GMRT High Resolution Southern Sky, hereafter GHRSS survey; Bhattacharyya et al. 2016).

GHRSS survey (Bhattacharyya et al. 2016, 2019; and Singh et al. 2022) is a low-frequency survey for radio pulsars and transients. The survey uses an IA beam of the full GMRT array in band 3 (300−500 MHz) to target the sky away from the galactic plane (∣b∣ > 3°). This survey has so far detected 30 new neutron stars, including three millisecond pulsars, two weakly recycled pulsars, two rotating radio transients (RRATs), and 23 normal pulsars. ¹  The discovery positional error of pulsars in the GHRSS survey can be an order of a degree at 400 MHz. However the timing follow-up with sensitive phased-array (PA) beams of the GMRT using 70% of the array warrants $\leqslant 1^{\prime}$ positional accuracy. The time needed to determine the precise position from timing studies of newly discovered pulsars can take more than a year, with observations having regular cadence. Moreover, the covariance between position and pulsar period derivative (P) limits the convergence of the timing fit. In case the pulsar is in binary (specially with longer orbital period), the binary model can also have large covariance with position. The effects of such large covariance in timing fit can be minimized with a precise a priori astrometric position accelerating the convergence to an accurate timing model. Such precise arcsec localization can be achieved with interferometric imaging where pulsars are effectively seen as point sources. Roy & Bhattacharyya (2013) reported the arcsec localization of five MSPs discovered in a Fermi-directed survey (Bhattacharyya et al. 2013; Roy et al. 2015; Bhattacharyya et al. 2021; and Bhattacharyya et al. 2022) using gated imaging. Using these precise positions, Bhattacharyya et al. (2022) demonstrated phase coherent timing solutions for three of those MSPs (J1120−3618, J1646−2142, and J1828+0625) using the legacy GMRT. Along with a GHRSS MSP (J2144−5237), these three MSPs were followed by Sharma et al. (2022) using the upgraded GMRT (uGMRT). Sharma et al. (2022) reported ∼3.4 times better timing precision for these MSPs with the uGMRT wide-band receiver observations compared to the legacy GMRT narrowband observations reported in Bhattacharyya et al. (2022) and investigated the possibility of using them in the PTA experiment.

Recently, we discovered two new MSPs in the GHRSS survey pointing of J1243−47 and J2059−48. The two pulsars were identified in 10 minutes of IA beams pointing toward 12^h43^m00^s, −47^d00'00'' and 20^h59^m00^s, −48^d00'00''. Both pulsars have rotational periods less than 10 ms. The discovery positional error box associated with the two MSPs is of the order of a square degree. The precise positions of the pulsars are required to characterize their properties through follow-up timing experiments using PA beam.

In this paper, we demonstrate localization of the two GMRT-discovered MSPs within an arcsecond range using the recently upgraded 33 MHz offline gating correlator. We explain the observations in Section 2. The localization methodologies based on pulsar-gated imaging and multiple PA beam formation are described in Section 3. We show the localization results from the two techniques in Section 4. We report follow-up studies of the localized MSPs using uGMRT band 3 (300−500 MHz), band 4 (550−750 MHz), and band 5 (1060−1460 MHz) to estimate their ToA and dispersion measure (DM) uncertainties at various frequencies while discussing the possibility of including them in the PTA experiments. In Section 5, we summarize the paper.

The localization analysis using gated imaging and/or multiple PA beamformation uses baseband data recorded with the legacy GMRT system (GMRT Software Backend (GSB); Roy et al. 2010). The baseband observation details for GMRT MSPs are listed in Table 1. For continuum imaging of the GHRSS field J1243−47, we recorded 2.6 s visibilities with 4096 channels from the uGMRT (GMRT Wideband Backend (GWB); Reddy et al. 2017) in band 3. After localization we followed up on the two MSPs with sensitive PA beam in GWB band 3, band 4, and band 5. The details of the continuum observation as well as of the follow-up observations are given in Table 1.

We used the coherently dedispersed offline gated correlator to generate beams from 33.33 MHz baseband data, as discussed in next section. The baseband data was processed to form IA and/or PA beams with 1024 channels for J1242−4712 (in the field of J1243−47) having DM of 78.65 pc cm⁻³ and 256 channels for J2101−4802 (in the field of J2059−48) having DM of 25.05 pc cm⁻³. We used REALFFT+ACCELSEARCH of PRESTO (Ransom 2011) to find the best-fit topocentric period and period derivative(s) from the IA beam data sets. In order to localize an MSP using gated imaging, we generated 2 s gated visibilities folded at the best-fit topocentric period and period derivative(s). We used CASA software version 5.6.0–60 ²  (McMullin et al. 2007) to image both the continuum (from GWB) and the gated (from GSB) visibility.

After successful localization, in follow-up observations, the PA beams in band 3 and band 4 were recorded with online coherent dedispersion, in which the spectral voltages in each frequency subband are corrected for dispersive delays using the nominal value of DM. In band 5 observation, we obtained Stokes-I filterbank data sampled at 81.92 μs with 4096 channels for offline incoherent dedispersion. For J1242−4712 and J2101−4802, the intra-channel smearing values in band 5 are 32 and 10 μs, respectively. Both the coherent and incoherent dedispersion mode filterbanks were incoherently dedispersed to remove the inter-channel smearing.

We used REALFFT+ACCELSEARCH to find the best-fit period and period derivatives for each PA beam data set. The PA beam data was then folded in PRESTO using the PREPFOLD command at the best-fit rotational parameters. The beam data from band 3 and band 4 were folded into single time-integration, 512 subbands, and 512 profile bins. In band 5, we created a single subintegration, 64 subbands, and 64 bins. We converted PRESTO's folded PFD to PSRCHIVE (van Straten et al. 2012) format using the PAM command. The folded data cube was converted to FITS format using PSRCHIVE's PSRCONV command. The FITS file format is used for deriving ToA and DM uncertainty.

3.1. Coherently Dedispersed Gated Imaging

A continuum image covering an area more than ∼1° × 1° (similar to half-power-beamwidth at the lower edge of band 3) in the sky can contain hundreds of point sources. Identification of the pulsed emission is difficult when multiple continuum point sources are present in the same field. Gated imaging is a method that isolates the pulsating point source from continuum sources by binning the visibility signals in pulse longitude allowing to separate the ON emission from OFF-pulse region for each period of the pulsar. By subtracting the OFF emission from ON gate, this approach efficiently removes all background–foreground continuum sources from the ON−OFF gated image. In a single observation, it can result in an arcsecond unambiguous localization of the pulsar, which appears as a single point source in the gated image. Roy & Bhattacharyya (2013) provides a detailed discussion of this procedure. Here we briefly explain the steps involved in the processing.

(a) Antenna-based coherent dedispersion and offline beamforming. We record Nyquist-sampled baseband data from individual antennas using the GSB. The baseband data for each antenna is then coherently dedispersed to remove the dispersive delays caused by the line-of-sight electron density. The geometric delay between the antennas is corrected during offline correlation and beamforming of coherently dedispersed voltages. Spectral visibility and an IA beam are generated from the correlator and beamformer respectively.

(b) Best-fit topocentric model. The rotational and orbital parameters (for binary systems) are not precisely known for the newly discovered MSPs. We search for the best-fit topocentric period and period derivative(s) using the IA beam. The IA beam is then folded at the derived parameters to locate the ON and OFF bins in the pulse longitudes.

(c) Gating ON- and OFF-pulse visibilities. Coherently dedispersed baseband data are used to construct correlated visibility, which are binned and folded individually over the pulse longitude (say n_bin) using the derived topocentric rotational parameters. The gating correlator produces n_bin visibility files corresponding to the bins on the pulse profile. Each bin file contains the complex visibility products as a function of time, frequency, and baselines.

(d) Imaging. We generate the optimal number of bins along pulse longitude so that ON-pulse emission is contained inside a single bin or in a few adjacent bins (n_ON). The OFF-pulse emission is contained within the remaining n_bin − n_ON bins. The OFF-pulse visibility file(s) are chosen from a set of n_bin − n_ON files having no contributions from ON-pulse emission (even in case of wide profiles or profile with inter-pulse). In case of multiple ON and OFF bins, the corresponding visibility files are added to form a ON and OFF bin visibility data set. The OFF-pulse visibilities are then subtracted from the ON-pulse visibilities. Following this subtraction, all continuum sources will be eliminated from the ON–OFF visibility file, leaving just the pulsar. The gated visibilities of the pulsar field are flagged for bad time and frequency chunks using flagcal (Prasad & Chengalur 2012). flagcal is a software that flags and calibrates GMRT UV data. The same software is used to calibrate the target field using the data of an external calibrator field observed before the target. The position of the pulsar is determined by imaging ON, OFF, and ON−OFF gated visibility data (after calibration) using CASA. CASA is used for deconvolution and self-calibration.

3.1.1. Limitations of the Technique

The gated correlator uses narrowband data with a bandwidth of 16.67 MHz. The availability of lower observational bandwidth for gating makes it difficult to localize the fainter MSPs discovered from GHRSS survey over 200 MHz bandwidth. For low signal-to-noise ratio (S/N) in the image domain, the noise fluctuations in the ON and OFF images might cause false detection in the ON−OFF image, which can imitate pulsars. Also, due to the gating of ON- and OFF-pulses on a restricted number of bins (∼20), we lose sensitivity for narrow duty-cycle pulsars. Since the gating correlator simultaneously writes n_bin visibility files to a disk each at a rate of >2 MB s⁻¹, the number of bins is limited by the available disk writing speed. The time required to generate gated visibilities from 1 hr of baseband data is of a few days for MSPs, depending on pulse period and the number of phase bins. Moreover, the pulsars with ∼mJy (or less) flux density and located near the beam's edge are sometime difficult to spot on the ON image.

3.1.2. Upgrade in Bandwidth of Correlator

We implemented a baseband recording capability with 33.33 MHz bandwidth (twice of the previously supported bandwidth) with 2 bits per Nyquist sample. The corresponding upgrade was made to the offline correlator, resulting in improved sensitivity for imaging fainter pulsars.

In order to validate this newly developed mode, we observed pulsar B1929+10 with 33.33 MHz gating correlator. We formed 14 gated visibility files, corresponding to 14 bins of the pulse longitudes, which were imaged for the test pulsar B1929+10. Each visibility file yields an estimate of the pulsar's flux density at its location. We compared the flux density of the test pulsar from each visibility file with the pulse profile from simultaneously generated IA beam to validate the functionality of binning and folding during the gating process. Figure 1 depicts the normalized flux density at the pulsar location in individual gated images from the upgraded gating correlator. In the same plot, the pulsar IA beam profile produced from a simultaneous beamforming operation over the same 33.33 MHz bandwidth is overplotted. The similarity of the two curves validates the proper functioning of the 33.33 MHz offline gating correlator without any contamination of OFF-pulsed emission by the ON-pulse energy.

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3.2. Multiple PA Beam Formation

4.1. Localization

The continuum visibility collected for the field of J1243−47 is flagged and calibrated (using 1154−350 calibrator) with flagcal . We performed three self-calibration cycles, the first two using only phase calibration mode and the third using amplitude and phase calibration mode. The Pybdsm based source extraction routine reports 245 point sources above the 5σ level with σ = 0.1 mJy within a 1.3° × 1.3° GWB continuum image of J1243−47 field. The MSP J1242−4712 was found among 245 sources by creating multiple PA beams using baseband data. It was located 18' away from the phase center of the GWB continuum image, which was the GHRSS survey pointing location. The pulsating point source, identified in the GWB continuum image, is shown on the left panel of Figure 2. Figures 3(a) and (c) show the IA and PA beams generated using baseband data at the location of this point source, where PA beam pointed on the pulsar gives ∼3.9 times more S/N than the IA beam.

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We followed a similar strategy to locate J2101−4802. The GWB continuum image of this field, with size of 2° × 2°, generated after calibrating (using 2225−049 calibrator) and flagging, contained 188 point sources. However, none of the PA beams formed using 70% array gave expected detection significance compared to the IA beam. One of the PA beams showed very faint detection (even weaker than the IA beam) and provides a possible location of the pulsar with few arcminutes error. We used the gating imaging technique to generate visibilities in 19 bins over the pulse longitude taking the possible location as the new phase center. The ON and OFF bins were identified from the simultaneously formed IA beam profile. Imaging the ON−OFF visibility bin, we found the pulsar 51' away from the original phase center of the GWB image (GHRSS discovery location), and the nearest point source (in GWB image) where we get the faint detection in PA beam was at a distance of 14 away from the location of the pulsar. The distance of 51' from phase center is outside the FWHM of the primary beam in GMRT band 3 (75' at 400 MHz). The GWB continuum image of this field contained no point sources above 5σ (where σ = 0.1 mJy), at the gated location. Thus a pulsar with flux density of ∼1 mJy at 400 MHz may not be detected in the continuum image if it is outside the FWHM of primary beam of GMRT. The gated image of the pulsar is shown on the right side of Figure 2. Figures 3(b) and (d) show the IA and PA beams generated using baseband data at the gated location, where detection significance in PA beam is ∼2.5 times more than that of the IA beam. The J2000 positions, as well as other parameters for the two MSPs, are listed in Table 2.

Table 2. The Table Shows the Basic Parameters of the Two MSPs

Pulsar Name	Type	P	Frequency Band	DM	Flux	J2000 Position	Offset from Survey Pointing Center
		(ms)		(pc cm−3)	(mJy)	(hh:mm:ss.s dd:mm:ss.s)	(')
J1242−4712	binary	5.31	Band 3	78.6472 (6)	2.00 (8)	12:42:12.80 (1)	18
			Band 4	78.643 (3)	1.22 (7)	−47:12:17.3 (3)
			Band 5	78.71 (8)	0.31 (6)
J2101−4802	binary	9.48	Band 3	25.055 (1)	10.2 (6)	21:01:55.115 (5)	51
			Band 4	25.095 (5)	4.0 (1)	−48:02:02.9 (5)
			Band 5	25.0 (1)	1.23 (9)

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4.2. Follow-up Investigations

4.2.1. Integrated Profiles

4.2.1.1. J1242−4712

J1242−4712 has a single component profile in band 3 (Figure 4(a)), with W10 and W50 (width at one-tenth and half of peak intensity) of 1010 ± 10.38 and 432 ± 10.38 μs, respectively. It has three profile components in band 4 (Figure 4(b)). The W50 for the middle (strongest) component is 356 ± 10.38 μs. This pulsar's band 5 profile (Figure 4(c)) also has three central components, with the middle one being the strongest. The W50 value for the middle component is 456 ± 83.02 μs.

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The number of profile components increased from one to three between band 3 and band 4. From band 4 to band 5, the ratio of the third component (from left to right in Figures 4(b) and (c)) to the first component changes from 1.24 to 1.33, and the ratio of the second component to the third component changes from 6.16 to 1.53. This pulsar's profile clearly shows significant evolution from band 3 to band 4 to band 5.

4.2.1.2. J2101−4802

J2101−4802 has three strong components near its center and two weak components at the beginning and end of the profile in band 3 (Figure 4(a)). The W10, which corresponds to the first and second central components, is 1690 ± 18.52 μs. The W50 value for the second central component (the strongest) is 629 ± 18.52 μs. The number of profile components in band 4 is the same as in band 3 (Figure 4(b)). The W50 for the first central component (the strongest) in band 4 is 482 ± 18.52 μs. The profile in band 5 has three central components, the middle of which is almost nonexistent (Figure 4(c)). The W50 for the first (the strongest) component is 448 ± 148.13 μs.

The sizes of the three central components vary across frequency bands. The ratio of the first to third central component (from left to right in Figures 4(a), (b), (c)) varies from 1.30 to 2.37 to 2.41 from band 3 to band 4 to band 5, respectively. From band 3 to band 4, the ratio of the second to first central component decreases from 3.54 to 0.62, and the second component almost disappears in band 5. This MSP also shows significant profile evolution from one band to the other.

4.2.2. ToA and DM Precision

We used PulsePortraiture ³  software (Pennucci et al. 2016; Pennucci et al. 2014; Pennucci 2019) for the timing analysis of the two pulsars, which can estimate ToA and DM from each epoch's observation simultaneously. The wide-band timing technique is illustrated by Pennucci et al. (2014), Liu et al. (2014), and it extracts ToAs and DMs using a two-dimensional (2D) template that is a function of rotational phase bin and frequency. This 2D template is created in PulsePortraiture using principal-component-decomposition (Pennucci 2019). To extract the wide-band ToA and DM from the observational data, we used the same strategy described in Sharma et al. (2022).

In band 3, we have used only a single epoch of observation for both MSPs. The S/N is 220 for J1242−4712 and 470 for J2101−4802. The time-averaged, frequency-resolved FITS data are subjected to a principal-component-analysis (PCA). PCA returns one eigenvector with mean profile for both of the MSPs. A linear combination of the smoothed eigenvector and the mean profile (as described in Equation (10) of Pennucci 2019) is used to create the frequency-dependent template. This template is used to extract a single ToA and DM from the epoch FITS (using Equations (7) and (8) of Pennucci et al. 2014). Table 3 shows the ToA and DM uncertainty of the observations in band 3.

Table 3. The Table Shows the Observation Duration, S/N of the GWB PA beam, ToA, and DM Uncertainty for the Two MSPs in Bands 3, 4, and 5

Pulsar Name	Frequency Band	Duration	S/N	σToA	σDM
		(minutes)		(μs)	(10−4 pc cm−3)
	Band 3	45	203	0.744	0.90
J1242−4712	Band 4	47	175	0.444	3.64
	Band 5	82	43	4.366	1.10 × 102
	Band 3	45	470	0.539	0.63
J2101−4802	Band 4	60	181	1.348	1.18 × 101
	Band 5	60	37	5.417	1.14 × 102

Note. ToAs are calculated from a single epoch FITS in bands 3, 4, and 5.

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In band 4 and band 5, a single FITS was available for both MSPs. The FITS file is subjected to PCA, which produces a mean profile with no eigenvectors. The smoothed mean profiles are used to extract the ToA and DM from band 4 and band 5 FITS. Note that different frequency band observations are analyzed using their corresponding templates.

We obtained less than 1 μs ToA uncertainty for both MSPs in band 3 observations. The DM precision for the two MSPs in this band is of the order of 10⁻⁵ pc cm⁻³. The ToA precision for J1242−4712 increases from 0.744 to 0.444 μs, while the DM precision decreases by 4 times from band 3 to band 4. For J2101−4802, the ToA precision decreases from 0.539 to 1.348 μs, and the DM precision shifts by more than 1 order (6.3 × 10⁻⁵ to 1.18 × 10⁻³ pc cm⁻³) from band 3 to band 4. In band 5, J1242−4712 and J2101−4802 have ToA uncertainty of 4.366 and 5.417 μs, respectively, with DM uncertainty of the order of 10⁻² pc cm⁻³. For J1242−4712, we found that the DM values in band 3 and band 4 (Table 2) are similar within ±2σ_DM of band 4. The DM values from band 4 and band 5 are similar within ±1σ_DM of band 5. For J2101−4802, we find ±8σ_DM change in DM value from band 3 to band 4 with respect to the DM uncertainty of band 4. The DM values from band 4 and band 5 are similar within ±1σ_DM of band 5. Note that the absolute DM values described here are derived from single epoch observations (using PDMP command of PSRCHIVE) in different frequency bands of the uGMRT.

4.2.3. Comparison of ToA and DM Precision with Other PTA MSPs

The North American Nanohertz Observatory for GWs (NANOGrav; Jenet et al. 2009) aims to detect stochastic GW background using a set of 47 MSPs. Using PulsePortraiture software, Alam et al. (2021) reported the wide-band timing of all of these MSPs observed with the Arecibo and GBT over a frequency range of 315−2380 MHz. The median scaled ToA uncertainty and median raw DM uncertainty of each of these MSPs in various frequency bands are given in Table 4 and Figure 2 of Alam et al. (2021). We chose the most precise median ToA and DM uncertainty among these different frequency bands for each MSP and plotted the histograms as seen in Figure 5. In the same histograms, we have added the best ToA and DM precision obtained for J1242−4712 and J2101−4802. This shows that the scaled ToA precision for both MSPs are well within the NANOGrav samples, and certainly they have very high DM precision (within top 10% of NANOGrav measurements). This illustrates the prospect of both these newly discovered pulsars for inclusion in the PTAs.

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The offline gated correlator at GMRT can unambiguously localize pulsars within arcsecond positional accuracy in a single observation. An accurate initial position facilitates follow-up with sensitive PA beam at multiple frequency bands and also accelerates the convergence of a timing model by removing covariance between long-term varying parameters, such as position and period derivative. We report an increase in the bandwidth of the offline gated correlator from 16.67 to 33.33 MHz and validate its functionality. This significant increase in correlator bandwidth also opens up new aspects, such as investigating the OFF-pulse emission of pulsars including MSPs.

Using gated imaging and multiple PA beam search, we successfully localized J1242−4712 and J2101−4802 with positional accuracy within 1 arcsecond at distances of 51' and 18' from the GHRSS survey pointing centers. J1242−4712 and J2101−4802 show more than 200 and 450 detection significance, respectively, in PA beam observations of band 3 for an hour of integration. The detection S/N of J1242−4712 does not differ significantly from band 3 to band 4. However, more than an hour of observation is required to achieve S/N of more than 200 for J2101−4802 in band 4 indicating its steep spectral nature between 300 and 750 MHz. In band 5, both pulsars show weaker detection significance of ∼40.

J1242−4712 and J2101−4802 have the most precise ToAs in band 4 and band 3, respectively. The minimum ToA uncertainty for J1242−4712 is 444 ns, and for J2101−4802, it is 539 ns. Both pulsars have the best DM precision in GMRT band 3, specifically 9 × 10⁻⁵ pc cm⁻³ (for J1242−4712) and 6.3 × 10⁻⁵ pc cm⁻³ (for J2101−4802). We compared the ToA–DM precision of the two MSPs (in this work) to the best median ToA–DM precision of 47 NANOGrav PTA MSPs. This comparison shows that the ToA–DM precision of the two MSPs is comparable to that of the 47 PTA MSPs. The long-term follow-up timing of these two MSPs with the PA beam in band 3 of the GMRT is currently ongoing. The results from the timing studies will be reported in the future GHRSS publications.

Sharma et al. (2022) recently reported the long-term wide-band timing of four GMRT-discovered MSPs. Considering band 3 detection S/N, those four MSPs are roughly 4 times weaker than the two MSPs reported in this paper. The timing of those four MSPs resulted in ∼μs order timing precision from 2−4 yr of timing baseline. The arcsecond localization of J1242−4712 and J2100−4802 will facilitate in the quicker convergence of their model parameters and the achievement of a phase-coherent solution from the follow-up observations. We expect to get even better timing precision from the follow-up timing for the two newly localized MSPs, which will be reported in future publications. An increase in the number of well-timed MSPs in the PTA experiments may aid to the detection significance toward the stochastic GW signals hidden in timing noise.

We gratefully acknowledge the Department of Atomic Energy, Government of India, for its assistance under project No. 12-R&D-TFR-5.02-0700. The GMRT is run by the National Centre for Radio Astrophysics—Tata Institute of Fundamental Research. We thank the GMRT telescope operators for their assistance with the observations. We also thank Timothy T. Pennucci and Scott M. Ransom for providing the data files for Figure 2 of Alam et al. (2021) allowing us to compare these two new MSPs with the NANOGrav samples.