Unraveling the Eclipse Mechanism of a Binary Millisecond Pulsar Using Broadband Radio Spectra

The observations were performed with the uGMRT (Gupta et al. 2017), which is a radio interferometric array consisting of 30 dishes. We performed observation on three different epochs. Observations on 2018 February 6 and 2018 April 17 were performed by splitting the total number of antennas into two sub-arrays at 300–500 and 650–850 MHz, whereas the observations on 2018 May 7 were performed using all antennas at 300–500 MHz. The observations were carried out in phased array mode where the spectral voltage signals from different antennas are coherently added together to form a single dish using the whole array. Coherent beam filterbank data at 48.28 kHz frequency resolution with 4096 frequency channels were recorded at every 81.92 μs. The observations were scheduled in such a way that the full eclipse phase (FEP) was covered.

We have used the GMRT pulsar tool gptool (A. Chowdhury et al. 2021, in preparation) to perform automated radio frequency interference (RFI) mitigation. The data were then corrected for interstellar dispersion with incoherent dedispersion (Lorimer & Kramer 2004) and then folded with the known ephemeris of the pulsar using dspsr (van Straten & Bailes 2010). We have used Tempo2 (Edwards et al. 2006; Hobbs et al. 2006) to calculate the difference between the observed and computed pulse time of arrivals (TOAs) using the known ephemeris of the pulsar derived from non-eclipse phase (NEP) TOAs. Then the excess electron column density in the eclipse medium is measured directly from the observed excess time delay of the pulsed emission (Lorimer & Kramer 2004) as

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{ex}}(\mathrm{pc}\,{\mathrm{cm}}^{-3})=2.4\times {10}^{-10}{t}_{\mathrm{ex}}(\mu {\rm{s}})f{\left(\mathrm{MHz}\right)}^{2},\end{eqnarray} \tag{ 1 }$$

where t_ex is the excess observed time delay, DM_ex is the excess dispersion measure (DM) calculated from the t_ex, and f is the observing frequency in MHz. The excess DM is then converted to excess electron column density as

$$\begin{eqnarray}&&{N}_{e}({\mathrm{cm}}^{-2})=3\times {10}^{18}\times \mathrm{DM}(\mathrm{pc}\,{\mathrm{cm}}^{-3}).\end{eqnarray} \tag{ 2 }$$

The observed data are flux calibrated using a flux calibrator 3C 286 (Appendix A.1). We used the flux calibrated pulse averaged flux density for the rest of the work (Appendix A.2).

The flux density reduced during the eclipse phase. In Figure 1(c), we show the variation of pulsed flux density in band 3 as a function of orbital phase. We define the range of the orbital phase 0.19–0.27 around superior conjunction (ϕ_b = 0.25) to be the FEP and 0.91–0.10 to be a part of the NEP. There is an approximate 60% reduction in the received pulse flux density in the FEP and a corresponding average delay in the arrival time of the pulses of 100 μs (Figures 1(a), (b)). In Figure 2(a), we show the variation in the delay in the pulse TOA in band 3 and the corresponding excess electron column density throughout the eclipse phase for three different epochs (Section 2). The variation in flux density, normalized with respect to the NEP flux density, is shown in Figure 2(b). We have marked three different regions in the eclipse phase; eclipse ingress phase (EIP), FEP, and eclipse egress phase (EEP) in Figure 2. In contrast to what was seen in band 3, in band 4, we detect no change in the observed flux density between the FEP and NEP greater than our noise limit of ∼100 μJy.

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To further study the frequency dependence of the flux density of the pulsar during the eclipse phase, we generated broadband spectra for PSR J1544+4937 at different orbital phases, namely, the NEP, FEP, EIP, and EEP, by splitting the observing band into smaller frequency slices. Different eclipse phases are shown in Figure 2. We divided the observing band into frequency slices of 10 MHz where the pulsar is still bright enough to potentially be detected in the NEP. In order to increase the signal-to-noise ratio during the brief ingress and egress phase, we averaged a 20 MHz bandwidth in both the EIP and EEP. The spectra at the NEP, FEP, EIP, and EEP are shown in Figure 3.

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We modeled the FEP spectrum (F_ec(ν)) for free–free absorption, induced Compton scattering, and synchrotron absorption. We used the radiative transfer equation F_ec(ν) = F_nonec(ν)e^−τ(ν) to model the FEP spectrum, where F_nonec is the power-law spectrum fitted to the observed NEP spectrum, which is taken to be constant throughout the orbit. τ(ν) is the frequency dependent optical depth for different eclipse mechanisms proposed by Thompson et al. (1994). We used the nonlinear curve fitting function curve_fit of the SciPy package.

Analytical expressions of frequency dependent optical depths for the different mechanisms adopted from Thompson et al. (1994) are used in this study.

2.5.1. Free–Free Absorption

The radio emission from the pulsar can be absorbed by the free electrons of the eclipse medium by the free–free absorption. The free–free absorption optical depth is given by

$$\begin{eqnarray}&&{\tau }_{\mathrm{ff}}(\nu )\approx 3.8\times {10}^{-14}\displaystyle \frac{{f}_{\mathrm{cl}}}{{T}^{\tfrac{3}{2}}{\nu }^{2}L}{N}_{e}^{2}\mathrm{ln}\left(5\times {10}^{10}\displaystyle \frac{{T}^{\tfrac{3}{2}}}{\nu }\right),\end{eqnarray} \tag{ 3 }$$

where, T is the temperature of the eclipsing medium in Kelvin, N_e is the electron column density of the eclipse medium in cm⁻², ${f}_{\mathrm{cl}}=\tfrac{\langle {n}_{e}^{2}\rangle }{\langle {n}_{e}{\rangle }^{2}}$ is the clumping factor of the medium, and L is the absorption length in centimeters. Assuming a spherical distribution of eclipse medium around the companion, we consider the maximum absorption length; L_max = 2 R_E = 1.0 R_⊙ as twice the eclipse radius; R_E = 0.5 R_⊙. The observed FEP spectrum is fitted considering temperature, clumping factor, and absorption length as the free parameters using the expression for free–free optical depth in Equation (3). During fitting we put a physically motivated constraint that the temperature T > 0 and absorption length L ≤ L_max.

2.5.2. Induced Compton Scattering

The induced Compton scattering optical depth is given by

$$\begin{eqnarray}&&{\tau }_{\mathrm{ind}}(\nu )\approx 4\times {10}^{6}\displaystyle \frac{{N}_{e}{S}_{\nu }}{{\nu }^{2}}\langle f(\phi )\rangle | \alpha +1| {\left(\displaystyle \frac{{d}_{\mathrm{kpc}}}{a}\right)}^{2}M,\end{eqnarray} \tag{ 4 }$$

where S_ν is the mean flux density of the pulsar in millijansky at frequency ν MHz, α is the spectral index of the incident radiation, d_kpc is the distance to the scattering center from the observer in kiloparsecs, and a ∼ 1.2 R_⊙ (Bhattacharyya et al. 2013) is the distance between the companion and the pulsar in centimeters. f(ϕ) is an angular factor, which is averaged out over the scattering region (Thompson et al. 1994). Reflection off a plasma cloud of radius of curvature R_C will cause de-magnification $M\sim {\left(\tfrac{{R}_{C}}{2r}\right)}^{2}$, where r is the distance from the center of the curvature. The de-magnification factor varies between 0 and 1. We consider the NEP flux density as 6.5 ± 0.2 mJy. The distance to the binary system estimated from optical observations is 2–5 kpc (Tang et al. 2014) and we take the average value d_kpc = 3.5 kpc. The spectral index at the NEP is α_nonec = −2.8 ± 0.7 and the distance between the pulsar and the companion is a ∼ 1.2 R_⊙ (Bhattacharyya et al. 2013). Considering these values and maximum de-magnification factor M = 1, we calculated the induced Compton optical depth.

2.5.3. Cyclotron Absorption

Since the companion has a magnetic field, cyclotron absorption is another possible eclipse mechanism. The cyclotron frequency is ${\nu }_{B}=\tfrac{{eB}}{2\pi {m}_{e}c}$ and the corresponding cyclotron harmonic at frequency ν is $m=\tfrac{\nu }{{\nu }_{B}}$. There are two components of cyclotron optical depths for two polarization components of the incident radiation. One component optical depth (τ_∥) is parallel to ( k × B ) × k and another component (τ_⊥) is perpendicular to ( k × B ) × k , where k is the wavevector and B is the magnetic field vector. Considering the LOS angle with the magnetic field is 90°, τ_∥ is very small compared to the τ_⊥ at higher cyclotron harmonics. Thus, considering the LOS angle with the magnetic field is 90°, the cyclotron absorption optical depth for perpendicular polarization component corresponding to cyclotron harmonic m is given by Thompson et al. (1994) as

$$\begin{eqnarray}&&\tau {{}_{\mathrm{abs}}}_{\perp ,\tfrac{\pi }{2}}^{(m)}=\displaystyle \frac{\pi }{2}\displaystyle \frac{{m}^{m+1}}{m!}{\left(\displaystyle \frac{{{mk}}_{B}T}{2{m}_{e}{c}^{2}}\right)}^{(m-1)}\displaystyle \frac{{n}_{e}{e}^{2}{L}_{B}}{{m}_{e}c\nu },\end{eqnarray} \tag{ 5 }$$

where ${L}_{B}=| \tfrac{{ds}}{d\mathrm{ln}B}|$ is the scale length of magnetic field variation. For simplicity, we consider the scale length of electron density variations and magnetic field variations as similar. Thus, the quantity n_e L_B is taken to be equal to the average electron column density N_e (Thompson et al. 1994). The cyclotron approximation is valid if the temperature is $T\leqslant \tfrac{{m}_{e}{c}^{2}}{2{{km}}^{3}}$ (Thompson et al. 1994).

2.5.4. Synchrotron Absorption

In the trans-relativistic case, thermal electrons dominate the absorption if the cyclotron harmonic is $m\lt \tfrac{1}{2}(p+1)$ (Thompson et al. 1994). Thus, thermal electrons dominate the absorption at lower cyclotron harmonics. At higher harmonics, we consider only the synchrotron absorption by nonthermal electrons with energy density distribution $n(E)={n}_{0}{E}^{-p};{E}_{\min }\lt E\lt {E}_{\max }$. The optical depth is given by

$$\begin{eqnarray}&&{\tau }_{\mathrm{syn}}=\left(\displaystyle \frac{{3}^{\tfrac{(p+1)}{2}}{\rm{\Gamma }}\left(\tfrac{3p+2}{12}\right){\rm{\Gamma }}\left(\tfrac{3p+22}{12}\right)}{4}\right){\left(\displaystyle \frac{\sin \theta }{m}\right)}^{\tfrac{p+2}{2}}\displaystyle \frac{{n}_{0}{e}^{2}}{{m}_{e}c\nu }L,\end{eqnarray} \tag{ 6 }$$

where L is the absorption length and p is the power-law index. Here we have considered an average magnetic field strength along the LOS and homogeneous distribution nonthermal electron. We considered a tiny fraction (∼1%) of total electron density (n_e ) as a typical value of nonthermal electron density (Thompson et al. 1994). For a chosen value of LOS angle (θ) we fitted the observed spectrum with B and p as free parameters.

The flux density drops to 85% of the NEP phase flux density at orbital phase ϕ_i = 0.16 and ϕ_e = 0.30, where ϕ_i and ϕ_e are defined as the start and end of the EIP and EEP, respectively (Figure 2(b)). The orbital phase range ϕ_e − ϕ_i corresponds to the eclipse radius R_E ∼ 0.5 R_⊙. We observed the excess electron column density at the ingress and egress as N_e ∼ 1.0 × 10¹⁶ cm⁻² (Figure 2(a)). Considering the constant electron density along the LOS, we estimate an average electron density; n_e ≈ 1.5 × 10⁵ cm⁻³ at the eclipse boundaries (Polzin et al. 2018).

Using the 10 MHz frequency slices, we find that in the FEP the pulsar is not detected above 5σ at frequencies below an eclipse onset frequency of ν_c = 345 ± 5 MHz where the pulsar has a flux density of S_c = 1.1 ± 0.2 mJy (Figure 3(a)). The error on the eclipse onset frequency is defined as the half width of the frequency slice. The estimated eclipse onset is an order of magnitude more precise ν_c compared to previous estimates for any eclipsing MSP (Bhattacharyya et al. 2013; Polzin et al. 2019).

Due to the large uncertainty of ν_c , previous studies could only put a limit on the physical parameters of the eclipse medium. With an order of magnitude of a more precise ν_c we can calculate the observed optical depth at the ν_c . We use an average value of the observed electron column density in the FEP of N_e = 1.5 × 10¹⁶ cm⁻² (Figure 2(a)). We calculate the optical depth τ_ν,c ≈ 1.7 at ν_c since the flux density of the pulsar decreases from 6.5 ± 0.2 mJy in the NEP to 1.1 ± 0.2 mJy in the FEP (Figure 3(a)). This direct estimation of the optical depth at the FEP allows us to provide a strong constraint on the physical parameters of the eclipse medium, which was not possible before.

At ν_c the eclipse medium is transitioning from an optically thick to thin regime which is hitherto unexplored for any other eclipsing MSP systems, since the majority of previous studies were performed at frequencies corresponding to the optically thick regime. In Figure 3(a), we show the spectrum at the FEP and NEP for the observation on 2018 February 6. Fitting a power law, ${S}_{\mathrm{nonec}}(\nu )\propto {\nu }^{{\alpha }_{\mathrm{nonec}}}$, where α_nonec = −2.8 ± 0.7 is the spectral index of the NEP spectrum. By comparison, the spectrum in the FEP has a break in the spectrum. We fit a broken power law to the observed FEP spectrum given as

$$\begin{eqnarray}S(\nu )=\left\{\begin{array}{ll}{S}_{t}{\left(\displaystyle \frac{\nu }{{\nu }_{t}}\right)}^{{\alpha }_{\mathrm{low}}}, & \nu \lt {\nu }_{t},\\ {S}_{t}{\left(\displaystyle \frac{\nu }{{\nu }_{t}}\right)}^{{\alpha }_{\mathrm{high}}}, & \nu \gt {\nu }_{t}\end{array}\right.,\end{eqnarray} \tag{ 7 }$$

where ν_t is the turnover frequency, S_t is the peak flux density at the turnover frequency ν_t , α_low is the spectral index for frequencies below ν_t , and α_high is the spectral index for frequencies above ν_t . The fitted power-law spectrum for the NEP is shown by the magenta solid line and the fitted broken power law for the FEP is shown by the green solid line in Figure 3(a). The errors on ν_t , α_low, α_high, and S_t are the errors from fitting. Above a turnover frequency of ν_t = 416 ± 33 MHz the spectral index α_high = −1.9 ± 0.2 and below ν_t , the spectrum has a positive spectral index, α_low = 3.1 ± 0.1. The peak flux density at ν_t obtained from broken power-law fit is S_t = 2.4 ± 0.3 mJy. This difference between the NEP and the FEP spectral shape has not been reported before in any other eclipsing binary MSP. A comparison between the NEP spectrum with those in the EIP and the EEP are shown in the lower panels of Figure 3, and we can see that there are differences compared to the FEP spectrum suggesting that there may be different eclipse mechanisms in place in the different regions, or the physical properties of the material are sufficiently different.

We consider different possible eclipse mechanisms (Thompson et al. 1994) to explain our observations (Section 2.5). Radiation at a frequency below the plasma frequency, ν_p = 8.5 (n_e /cm⁻³) kHz, of the eclipsing material will not be able to propagate through it. If we consider that the observed ν_c corresponds to ν_p in the FEP we find that the corresponding electron density is n_e,p = 1.6 × 10⁹ cm⁻³. This is in stark contrast to the electron density in the eclipse region, n_e ∼ 1.5 × 10⁵ cm⁻³, estimated by assuming a spherically distributed eclipse medium and the electron densities derived from the delayed pulse arrival times (Figure 2(a)). This suggests that the eclipse onset at ν_c cannot be explained by the plasma frequency cutoff. On the other hand, radiation at frequencies higher than ν_p undergo refraction due to the intervening medium. If the angle of refraction is sufficiently large, the pulsed emission will be refracted out of the LOS resulting in an eclipse. If refraction is the cause of the observed eclipse at ν_c , the delay of the pulse arrival time at the eclipse boundaries is ∼10–100 ms (Thompson et al. 1994), which is much larger than the observed delay in pulse arrival time at the eclipse boundaries ∼100 μs (Figure 2(a)). This suggests that the refraction at the frequencies below ν_c is not sufficient to explain the observed eclipse. Pulse broadening due to excess DM and scattering in the FEP is ∼45 μs at ν_c (Appendix A.3), which is less than the time resolution of the data, 81.92 μs, and so it cannot reduce the observed pulse detection.

We have then modeled the spectrum in the FEP using the radiative transfer equation (Section 2.5) for three mechanisms: free–free absorption, induced Compton scattering, and synchrotron absorption. We have assumed that the spectra at the NEP does not change for other reasons. We consider a given mechanism as the most plausible when the observed optical depth (τ_ν,c ) and the observed FEP spectrum can be reproduced with physically acceptable parameters. We have modeled the FEP spectra for three different epochs to see whether eclipse mechanism or the eclipse medium properties varies significantly between epochs.

3.6.1. Free–Free Absorption

The free–free absorption optical depth (τ_ff) (Equation (3)) due to the free electrons in the eclipse medium depends on three physical quantities: temperature (T), absorption length (L), and clumping factor (${f}_{\mathrm{cl}}=\tfrac{\langle {n}_{e}^{2}\rangle }{\langle {n}_{e}{\rangle }^{2}}$). Clumping factor is a measure of inhomogeneity of the medium. To obtain the observed value of τ_{ν, c} from free–free absorption the temperature of the medium would need to be T = 130 ${({f}_{\mathrm{cl}})}^{\tfrac{2}{3}}$. Thus, either a very low temperature (T ∼ 1000 K) or high clumping factor (f_cl ∼ 10⁹) is required to create an eclipse at ν_c due to free–free absorption. We fit (brown solid line in Figure 4(a)) the observed spectrum in the FEP (green circles in Figure 4(a)) using the frequency dependent optical depths of free–free absorption but did not obtain a good fit (${\chi }_{\mathrm{red}}^{2}\sim 2$). The mismatch between the observed and modeled FEP spectrum suggests that the free–free absorption cannot produce the observed frequency dependent eclipses of PSR J1544+4937.

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3.6.2. Induced Compton Scattering

3.6.3. Possibility of Cyclotron Absorption

3.6.4. Synchrotron Absorption

While modeling the observed FEP spectrum with synchrotron absorption discussed in Section 3.6.4, we assumed that the nonthermal electron distribution in the eclipse medium is homogeneous and the average LOS angle to the magnetic field is 90°. Best-fit values of B and p for the three epochs are given in Table 1, which are broadly consistent. We obtain a mean value of magnetic field strength, B ∼ 13.7 ± 1.2 G, across the three epochs, which is similar to the characteristic magnetic field strength B_E . We estimated the nonthermal electron energy density, U_nth ≈ 10⁻⁴ erg cm⁻³, which is only a tiny fraction of the total pulsar wind energy density, U_E = 4.37 erg cm⁻³. The estimated LOS averaged magnetic field strength using the synchrotron absorption model depends on the LOS angle (θ) with the magnetic field (Equation (6)). To explore further we fit the FEP spectra for different values of θ between the range of 5°–90° in 10° steps and the estimated values of B are shown in Figure 5. The upper and lower limits of the estimated LOS averaged magnetic field strengths are 162 ± 14 and 14.0 ± 1.1 G, respectively, which are also a similar order of magnitude to B_E .

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When considering the influence of the different eclipse mechanisms we assumed that the spectrum in the NEP is representative of that seen throughout the orbit, if there were no eclipse. Scintillation could change the flux density of the pulsar; however, the decorrelation bandwidth for diffractive scintillation is ∼1.1 kHz (Appendix A.3) given the DM of the pulsar and thus will be averaged out over the 10 MHz frequency slices. Moreover, we would not expect scintillation to have this sort of orbital phase dependence. Although we have ruled out free–free absorption and induced Compton scattering as the main eclipse mechanism, we now consider if they may make a significant fractional contribution to the eclipses. Considering the average electron column density N_e ∼ 1.5 × 10¹⁶ cm⁻² at the FEP, and a typical temperature of the stellar wind (Thompson et al. 1994) T ∼ 10⁸−10⁹ K and clumping factor f_cl ∼ 1, we have estimated the free–free optical depth; τ_ff ∼ 10⁻⁹–10⁻¹⁰ and the induced Compton scattering optical depth is ∼4 × 10⁻² at the ν_c . Thus, both free–free absorption and induced Compton scattering do not contribute significantly to the observed optical depth. Synchrotron absorption is therefore solely responsible for the observed frequency dependent eclipsing and the estimated physical parameters are not affected by a contribution to the optical depth by other mechanisms.

Most previous studies (Fruchter et al. 1988, 1990; Stappers et al. 1998; Polzin et al. 2018, 2019, 2020) have concluded that the low frequency eclipses may be caused by cyclotron/synchrotron absorption. The magnetic field strength, B, is a crucial parameter for cyclotron/synchrotron absorption. Recent studies of PSR B1957+20 (Li et al. 2019) and PSR J2256-1024 (Crowter et al. 2020) estimated B of a few milligauss at the eclipse boundaries, which is significantly different from B_E and insufficient to cause cyclotron absorption at the observing frequency. However, recent polarization observations of PSR J2051-0827 (Polzin et al. 2019) using the Parkes UWL (Hobbs et al. 2020) at the FEP provides a limit on the parallel component B_∥ = 20 ± 120 G of the magnetic field, which is close to the characteristic magnetic field strength for this pulsar. Our estimate of the magnetic field strength, using a completely different method for PSR J1544+4937, is consistent with B_E . Thus, the measurement of a very low magnetic field strength at the eclipse boundaries were possibly due to the observations at the eclipse boundaries where the eclipse medium properties may be significantly different than in the FEP (Khechinashvili et al. 2000).

Our new method can be utilized to understand the eclipse mechanisms and the evolution of other eclipsing MSPs in compact binaries, including the transitional MSP J1227-4853 (Roy et al. 2015; Kudale et al. 2020). In the future more sensitive observations will allow us to perform the spectral modeling with higher orbital phase resolution covering the eclipse region. This method can also be used for a diverse range of binaries, with different companion types and orbital properties, for which spectral modulation and DM variation with orbital phase is, or expected to be, observed. For example, this method can probe the eclipse properties of PSR B1259-63 (Johnston et al. 2005), which is orbiting a massive Be star and the observed eclipses are thought to be caused by the circumstellar disk of the Be star and related systems, such as PSR J1740-3052 (Madsen et al. 2012). For the double pulsar binary, PSR J0737-3039, where the observed flux density of one pulsar is modulated by the second pulsar (Lyne et al. 2004; McLaughlin et al. 2004; Breton et al. 2012), an adoption of our simultaneous broadband observing approach may be used to probe the magnetosphere of the second one. Thus, even though primarily developed for the study of the eclipse properties of the MSPs in compact orbits, this technique has the potential to probe the stellar environment for a wide range of binaries via the modeling of broadband radio spectra.

Flux calibration of the pulsar is performed using the on source scan of a standard flux calibrator source 3C 286 and off source scan at 5° away from the flux calibrator to convert the data into physical flux density unit Jansky. We used the following relation to calculate the sensitivity of the phased array for each 1 MHz frequency chunk for the 200 MHz band:

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{\mathrm{on}}-{P}_{\mathrm{off}}}{{P}_{\mathrm{off}}}={S}_{\mathrm{cal}}\displaystyle \frac{G}{{T}_{\mathrm{sys}}},\end{eqnarray} \tag{ A1 }$$

where P_on is the mean power for the scan on 3C 286 and P_off is the mean power while pointing the antennas at the cold sky 5° off from 3C 286, T_sys is the overall system temperature in Kelvin, S_cal is the flux density of calibrator source in jansky and G is the antenna gain in Kelvins per janksy. We then multiply each 1 MHz frequency bin of baseline subtracted incoherently dedispersed and folded pulsar data set with $\tfrac{{T}_{\mathrm{sys}}}{G}$ estimated using Equation (A1) to obtain a flux calibrated data set.

The mean flux density is obtained as the area under the pulse profile divided by the number of pulse phase bin (n_bin) in the pulse profile. We have accounted for the variable pulse width across frequency while calculating the mean flux density. The total error on the mean flux density is calculated as a quadratic sum of the off pulse rms noise (σ_off) scaled to the number of pulse phase bins in the profile ($\sigma \,=\,\tfrac{{\sigma }_{\mathrm{off}}}{\sqrt{{n}_{\mathrm{bin}}}}$) and a 10% (f_err) flux scale uncertainty (Chandra & Kanekar 2017). Thus, the total error on the mean flux density is

$$\begin{eqnarray}&&{\sigma }_{\mathrm{total}}=\sqrt{{\sigma }^{2}+{\left({f}_{\mathrm{err}}\times {S}_{\mathrm{psr}}\right)}^{2}},\end{eqnarray} \tag{ A2 }$$

where S_psr is the mean flux density of the pulsar.

The intrinsic pulse width of the pulsar radio emission gets broadened due to the dispersion and scattering. The effective pulse width is the quadratic sum of the intrinsic pulse width (t_intrinsic), pulse broadening due to dispersion measure (t_DM), and scattering (t_s ). The effective pulse width can also be expressed in terms of the observed pulse profile as

$$\begin{eqnarray}&&{t}_{\mathrm{eff}}=\sqrt{{t}_{\mathrm{intrinsic}}^{2}+{t}_{\mathrm{DM}}^{2}+{t}_{s}^{2}}=\displaystyle \frac{{S}_{\mathrm{mean}}}{{S}_{\mathrm{peak}}}P,\end{eqnarray} \tag{ A3 }$$

where S_mean is the mean flux density, S_peak is the peak flux density, and P is the pulsar rotation period. We have estimated the effective pulse width at 345 MHz is 345 μs. To obtain the scattering timescale we have assumed that the intrinsic pulse width is 10% of the pulse period, t_intrinsic = 215.9 μs. Pulse broadening due intra-channel dispersion for a finite channel width of 48.28 kHz is t_DM = 228 μs. Thus, using Equation (A3), the estimated value of the scattering timescale is t_s ≈ 141 μs and corresponding decorrelation bandwidth is ${\rm{\Delta }}f=\tfrac{1}{2\pi {t}_{s}}\sim 1.1$ kHz.