Scintillation based search for off-pulse radio emission from pulsars

TL;DR

This paper introduces a scintillation-based method to detect off-pulse radio emission from pulsars by analyzing the correlation of intensity variations in dynamic spectra, offering advantages over previous techniques.

Contribution

The authors present a novel technique utilizing interstellar scintillation to identify off-pulse emission, demonstrating its effectiveness through simulations and real data, and highlighting its immunity to measurement non-idealities.

Findings

Applied method to PSR B0329+54, setting upper limits on off-pulse emission.

Simulations confirm the method's robustness against measurement non-idealities.

Technique is suitable for analyzing existing and future pulsar data.

Abstract

We propose a new method to detect off-pulse (unpulsed and/or continuous) emission from pulsars, using the intensity modulations associated with interstellar scintillation. Our technique involves obtaining the dynamic spectra, separately for on-pulse window and off-pulse region, with time and frequency resolutions to properly sample the intensity variations due to diffractive scintillation, and then estimating their mutual correlation as a measure of off-pulse emission, if any. We describe and illustrate the essential details of this technique with the help of simulations, as well as real data. We also discuss advantages of this method over earlier approaches to detect off-pulse emission. In particular, we point out how certain non-idealities inherent to measurement set-ups could potentially affect estimations in earlier approaches, and argue that the present technique is immune to such non-idealities. We verify both of the above situations with relevant simulations. We apply this method to observation of PSR B0329+54 at frequencies 730 and 810 MHz, made with the Green Bank Telescope and present upper limits for the off-pulse intensity at the two frequencies. We expect this technique to pave way for extensive investigations of off-pulse emission with the help of even existing dynamic spectral data on pulsars and of course with more sensitive long-duration data from new observations.

Tables

Table 1. Table 1 The estimated η 𝜂 \eta , for the 6 models (2 cases each).

AO	$N_{\mathrm{eff}}$	$\eta \pm {\sigma }_{\eta }$	$N_{\mathrm{eff}}$	$\eta \pm {\sigma }_{\eta }$
		$\left({\eta }_{\mathrm{true}}=0.01\right)$		$\left({\eta }_{\mathrm{true}}=0\right)$
1 A	628	$0.0099\pm 0.0007$	736	$0.0014\pm 0.0005$
1 B		$0.0111\pm 0.0006$		$0.0011\pm 0.0005$
2 A	609	$0.0139\pm 0.0009$	609	$0.0043\pm 0.0008$
2 B		$0.0106\pm 0.0008$		$0.0007\pm 0.0007$
3 A	517	$0.0137\pm 0.0012$	511	$0.0049\pm 0.001$
3 B		$0.0104\pm 0.001$		$0.0009\pm 0.0009$

Table 2. Table 2 Results from the dynamic spectral cross-correlation analysis of scintillation data for the on- and off-pulse regions. The estimates of N t subscript 𝑁 t N_{\mathrm{t}} and N ν subscript 𝑁 𝜈 N_{\nu} assume a conservative value of the filling factor κ = 0.2 𝜅 0.2 \kappa=0.2 .

$\nu$	730 MHz	810 MHz
$N_t$	$\sim 5$	$\sim 4$
$N_{\nu }$	$\sim 3$	$\sim 3$
$N_{\mathrm{eff}}$	14.4	12.7
${\sigma }_{\eta }$	0.0017	0.0021
$\eta \pm {\sigma }_{\eta }$	0.0002 $\pm$ 0.0017	0.0013 $\pm$ 0.0021
${\eta }_{\max }$	0.0045	0.0066

Equations

$$CC(0,0)=<\delta I_{\mathrm{on}}(\nu ,t)\delta I_{\mathrm{off}}(\nu ,t)>$$

$$\eta$$

$$\delta I_{\mathrm{off}}(\nu ,t)=\eta \delta I_{\mathrm{on}}(\nu ,t)+U(\nu ,t)$$

$${\sigma }_{\eta }=\frac{{\sigma }_{<\mathrm{U}>}}{\sqrt{<(\delta I_{\mathrm{on}}(\nu ,t){)}^2>}}$$

$${\sigma }_{<\mathrm{U}>}={\sigma }_U\sqrt{\frac{1}{N_{\mathrm{eff}}}}$$

$${\eta }_{\max }$$

$$S_{\mathrm{d}}={\left[8\pi r_{\mathrm{e}}^2{\lambda }^2{C}_{\mathrm{N}}^{2}zf(\alpha )/(\alpha +1)\right]}^{-1/\alpha }$$

$$S_{\mathrm{d}}=\sqrt{z{\lambda }^2{\nu }_{\mathrm{d}}/4c\pi }=r_{\mathrm{f}}\sqrt{{\nu }_{\mathrm{d}}/2\nu }$$

$$F(\nu)=\left\{\begin{array}[]{lc}1\,,&|\nu-\nu_{\mathrm{c}}|\leq 0.5BW\,,\\[3.0pt] 0.01\,,&0.5BW\leq|\nu-\nu_{\mathrm{c}}|\leq 3.5BW\,,\\[3.0pt] 0\,,&otherwise\\[3.0pt] \end{array}\right.$$

$$I(\nu ,t)=I_0\exp \{-((t-\tau (\nu ))-nP{)}^2/(2{\sigma }^2)\}$$

$$\tau (\nu )=4.15\times 10^{3}DM\left[\frac{1}{{\nu }^2}-\frac{1}{{\nu }_{\mathrm{ref}}^2}\right](s)$$

$$P_{3N}(q_x,q_y,q_z)=C_{\mathrm{N}}^2{q}^{-\beta };q=\sqrt{q_x^2+q_y^2+q_z^2}$$

$$P_{2\varphi }(q_x,q_y)$$

$$\varphi (x,y)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }g(q_x,q_y)\sqrt{P_{2\varphi }(q_x,q_y)}\exp \left[-j2\pi (x{q}_x+y{q}_y)\right]d{q}_{x}d{q}_y$$

$$\varphi =2\pi (2\pi {)}^{-\beta /2}(N\Delta r{)}^{-1+\beta /2}\sqrt{2\pi z(\lambda r_{\mathrm{e}}{)}^2{C}_{\mathrm{N}}^2}\left[{\mathcal{F}\mathcal{T}}^{-1}\left\{g{M}_0\right\}\right]$$

$$E_{\mathrm{s}}(x,y)=\exp (-j\varphi (x,y)).$$

$$E_{\mathrm{O}}(\xi ,\eta )=\frac{e^{-j\pi /2}{e}^{j2\pi z/\lambda }}{2\pi r_{\mathrm{f}}^2}\int \int \exp \left[j\varphi (x,y)+j\frac{(x-\xi {)}^2+(y-\eta {)}^2}{2r_{\mathrm{f}}^2}\right]dxdy$$

$$E_{\mathrm{O}}=\mathcal{I}\mathcal{F}\mathcal{T}\{\mathcal{F}\mathcal{T}\{h(x,y)\}\mathcal{F}\mathcal{T}\{\exp (-j\varphi )\}\}$$

$$N\ge \frac{(\nu /{\nu }_{\mathrm{d}})}{({\Delta \varphi }_{\min }{)}^{2/(\beta -2)}}$$

Full text

1. Introduction

It is the pulsed nature of the emission (as against continuous emission) that made the discovery of pulsars (Hewish et al. 1968) possible. Their average intensities, if were to manifest as continuous emission, are in most cases too weak to be detectable, in presence of possible confusion from other continuous sources. The pulsed emission has been studied in great detail, and has lead to our present understanding of the physical picture of pulsars. However, the question as to whether pulsar radiation indeed has any intrinsic continuous component, in addition to its distinguishing pulsed signature, or if the periodic emission extends well beyond the main/inter-pulse windows, have been issues of much interest since the early days of pulsar studies.

There have been several attempts to detect off-pulse emission from pulsars (as summarized in Table 3 and discussed in Section 5 of Basu et al. 2011). Most attempts were primarily aimed at detection of unpulsed emission component of magnetospheric origin (for example, Hugunein et al. 1971; Bartel et al. 1984; Perry & Lyne 1985; Hankins et al. 1993; Basu et al. 2011,2012), which is indeed the focus of this paper. In contrast, some were prompted by, and were aimed to test, the proposition of Blandford et al. (1973) - “existence of ghost supernova remnants around old pulsars”. Detection of unpulsed emission of magnetospheric origin is indeed challenging, when based on apparent intensity in the off-pulse region, particularly in presence of a variety of unresolved astronomical sources and the resulting confusion. Such contaminants could include pulsar companions, if any, nearby galactic/extragalactic radio sources, and diffuse background emission, in addition to the following sources associated with the pulsar. They may include (e.g. as discussed by Hankins et al. 1993) weak halos (Blandford et al. 1973), remnants of the progenitor supernova, shock structures or synchrotron nebulae, and detectable bow shock. All these contaminations are unavoidable because of finite beam-width of single-dish telescopes and non-negligible side-lobes of interferometers.

After several non-detections and some reports of detections that were refuted subsequently, the off-pulse emission has attracted renewed attention with Basu et al. (2011, 2012) reporting detection of off-pulse emission from B0525+21 and B2045-16 based on their GMRT observations. It is worth noting, that in their study of 20 pulsars, including B0329+54, B0525+21 and B2045-16, at 2.7 and 8.1 GHz using the NRAO 3-element interferometer, Huguenin et al. (1971) found no significant unpulsed emission, implying an upper limit of 20 mJy within 10 arcsec of the pulsar directions. Much later, Bartel et al. (1984) made observations of pulsars B0329+54 and B1133+16 at 2.3 GHz using Mark III VLBI, and also ruled out continuous emission above their detection limit (2.5 mJy). Soon after, Perry & Lyne (1985), reported their interferometric observations at 408 MHz, on 25 pulsar including B0329+54 and B0525+21, made using 76m MK 1A telescope at Jodrell Bank and the 25m telescope at Defford, with baseline of 127 km. They claimed detection of unpulsed emission from 4 pulsars B1541+09, B1929+10, B1604-00 and B2016+28. However, later it became clear that B1541+09 and B1929+10 are aligned rotators (Hankins et al. 1993; Rathnasree & Rankin 1995), and the unpulsed emission from B1604-00 and B2016+28 were shown to be from unrelated background sources (Strom & Van Someren Greve 1990; Hankins et al. 1993).

The recently reported detections of off-pulse emission (Basu et al. 2011; 2012) from two long period pulsars B0525+21 (3.75 s) and B2045-16 (1.96 s) are based on the imaging mode of GMRT, and also at two frequencies (325 and 610 MHz). Although the authors have discussed some effects that could potentially contaminate off-pulse region with leakage from the emission that is otherwise confined to the main-pulse window, and have attempted some tests based on which they claim absence of such leakage. We consider these tests inadequate to rule out “leakage”, since there are a few different aspects, associated with commonly employed receiver setups, that have noticeable potential for undesirably spilling the main pulse contribution across off-pulse region.

Ideally, we need a method that is immune to such contamination, as far as possible, while making reliable estimation of possible off-pulse or unpulsed emission intrinsic to pulsar.

Owing to their compact size and pulsed emission, pulsars have been an excellent probe of the ISM since their discovery. Primarily, they have revealed the distribution of free electron in the Galaxy, through direct measures of column density (from the observed dispersion) and spatial distribution of electron density irregularities (from scintillations and angular/temporal broadening as a result of scattering). The highly polarized nature of their radiation also allows Faraday Rotation measurements, sampling the magneto-ionic component of the intervening medium, and their pulsed nature facilitates some of the clearest measurements of HI absorption along their sight-lines. Of these, the diffractive scintillation effects are readily observable in pulsar directions thanks to their tiny angular sizes, and become apparent only in the cases of some extra-galactic sources having the required compact angular size, such as in early phases of $\gamma$-ray burst (GRB) afterglow sources (see for example, Frail et al. 1997; Macquart & de Bruyn 2006).

The diffraction induced chromatic modulation of intensity, when combined with relative motions, translates to intensity variations across time and frequency. Similarly it is only the pulsed nature of the radio pulsars which makes the dispersion effect measurable, and also to reveal the temporal broadening due to scattering. However, the latter can be probed indirectly via other manifestation of scattering (such as decorrelation scales in frequency and/or angular broadening), even in the case of continuous sources. The camaraderie between the pulsars and the interstellar medium is indeed reciprocal. For example, ultra-high angular resolution probe of pulsar emission is made possible by the ISM acting as a lens. This was first pointed out by Lovelace (1970) and has been followed-up by many (e.g. Cordes & Wolszczan 1988; Pen et al. 2014; and references therein). Here, the diffractive/refractive effects due to large scale irregularities are considered as providing interstellar interferometric measurements capable of resolving even magnetospheric emission regions of pulsars. The refractive effects leading to multiple imaging manifest themselves as fine-scale corrugations or drift patterns within scintles in the dynamic spectra resulting from diffractive scintillations (e.g. Wolszczan and Cordes 1987, Gupta et al. 1994,1999).

In this paper, we present a technique which advantageously uses such interstellar-scale telescope for search and detection of unpulsed emission, if any, from pulsars. Our technique (described in Section 2) is based on diffractive interstellar scintillation (DISS) and its correlated imprint on the pulse intensity and any off-pulse emission intrinsic to the pulsar, and has the potential for providing more reliable measurement of intrinsic off-pulse/unpulsed emission, without needing conventional interferometric measurements, i.e. which are possible even with single-dish observations.

In Section 3, we demonstrate sensitivity of our technique using simulated dynamic spectra over wide band, and assess its immunity to various known sources of contamination in the off-pulse region. Discussion of one such potential contaminant is given in Appendix A. The details of the DISS simulation are presented in the Appendix B. In Section 4, we illustrate application of our technique to real data, using observations on B0329+54 at two radio frequencies. We summarize the main conclusions of our paper in Section 5.

2. Scintillation-based technique for search/detection of unpulsed emission from pulsars

In this section, we present a new technique based on diffractive interstellar scintillation (DISS) and assessment of correlation between dynamic spectra for the pulse and off-pulse intensities. It effectively renders measurements with fine angular resolution offered by interstellar diffraction to distinguish pulsar emission region from sources of confusion, even in close proximity to the pulsar. This DISS correlation criterion effectively and readily discriminates against all discrete and diffuse radio emissions on angular scales larger than that of the pulsar magnetosphere, since they will be devoid of DISS imprint in their dynamic spectra, let alone show any correlation with pulse intensity variations. Any confusing compact source, unresolved by the observing telescopes, and compact enough to show DISS, will show a dynamic spectral signature, i.e., the scintillation pattern, significantly different from that associated with the pulsar emission. In fact, differences between the scintillation patterns associated with even the different components within the pulse profile have been probed to assess spatial separation, if any, between the apparent sites of emission (Cordes et al. 1983).

If indeed, a pulsar has a component of intrinsic emission that is unpulsed/continuous, we expect its intensity modulation due to interstellar scintillation to be closely related to, if not matching, that of the pulsed component. For the desired correlation to exist between the diffractive scintillation spectra of intensities in the two longitude regions, the spatial transverse separation between the associated emission regions should ideally be well within the equivalent spatial resolution of the interstellar aperture/interferometer at work. As Cordes et al. (1983) have already noted, the spatial scale $S_{\mathrm{d}}$ of the diffraction pattern in the observer plane also (reciprocally) defines the associated spatial resolution at the source distance.

A suitable data set for implementation of our technique is, in general, an appropriately sampled data cube of intensity $I(\nu,t,\phi)$, as a function of rotational longitude $\phi$, radio frequency $\nu$ and time $t$, and over wide frequency and time spans of, say, $\Delta\nu_{\mathrm{BW}}$ and $\Delta t_{\mathrm{obs}}$, respectively.

The two dynamic spectra, $I_{\mathrm{on}}(\nu,t)$ and $I_{\mathrm{off}}(\nu,t)$, to be tested for mutual correlation, are to be constructed for the apparent average intensity across (i) an appropriate number of bins spanning or within the pulse window, and (ii) a chosen set of bins or longitude range in the off-pulse region which is well-separated from the pulse window. All of the (dynamic) spectra here are assumed to be already corrected for any non-uniformity in spectral response of the observing system within the observed band.111An estimate of the required normalized spectral gain response ($G(\nu)$), to be used for dividing all the observed spectra, can be made by averaging the observed off-pulse spectra over the entire time span of observation to first obtain a mean uncalibrated spectrum $<S_{\mathrm{off}}>(\nu)$, and then normalizing it with band-averaged intensity $\bar{S}_{\mathrm{off}}$, such that $G(\nu)=<S_{\mathrm{off}}>(\nu)/{\bar{S}_{\mathrm{off}}}$.

Sensitivity in the estimation of correlation depends on the signal-to-noise ratio in estimation of the two dynamic spectra, and the degrees of freedom provided by the richness in the dynamic spectra, quantifiable to the first order in terms of number of scintles. Naturally, scintillation dynamic spectra obtained from longer duration observations with wide spectral coverage are desired, if not essential. The dynamic spectral resolutions in time and frequency, say, $\delta t_{\mathrm{res}}$ and $\delta\nu_{\mathrm{res}}$, respectively, need to be adequately finer than the respective decorrelation scales ($t_{\mathrm{s}}$ and $\nu_{\mathrm{d}}$), which together characterize the average size of scintles. The dynamic spectra, therefore, are to be smoothed optimally to reduce the uncertainty in estimation of the intensity variations due to scintillation, without washing out details in the ISM induced diffractive variation of interest.

In practice, the dynamic spectra are not free of (additive) random noise in estimating intensity at each pixel in the time-frequency plane, but the magnitude of this noise is expected to be largely consistent with the system temperature and the integration employed (quantified by the relevant time-bandwidth product). Thus, in general, $I_{\mathrm{on}}(\nu,t)=I^{p}_{\mathrm{on}}(\nu,t)+U_{\mathrm{on}}(\nu,t)$, and $I_{\mathrm{off}}(\nu,t)=I^{p}_{\mathrm{off}}(\nu,t)+U_{\mathrm{off}}(\nu,t)$, where the $U_{\mathrm{on}}$ and $U_{\mathrm{off}}$ represent random noise (with zero mean and standard deviations $\sigma_{\mathrm{on}}$ and $\sigma_{\mathrm{off}}$ respectively), which is uncorrelated from pixel to pixel, and contaminates the respective underlying pulsar dynamic spectra $I^{p}_{\mathrm{on}}$ and $I^{p}_{\mathrm{off}}$. These delta-correlated noise contributions, of magnitude $\sigma^{2}_{\mathrm{on}}$ and $\sigma^{2}_{\mathrm{off}}$, will be clearly noticeable as such at zero-lag in the respective auto-correlation functions, $ACF_{\mathrm{on}}$ and $ACF_{\mathrm{off}}$, of the dynamic spectra, on top of the the otherwise smoothly varying auto-correlations of $I^{p}_{\mathrm{on}}$ and $I^{p}_{\mathrm{off}}$, respectively. Hence, the zero-lag auto-correlation of the underlying intensity variation is estimated routinely by interpolation from correlations at adjacent lags.

The average cross-correlation between the intensity variations in two dynamic spectra, $I_{\mathrm{on}}(\nu,t)$ and $I_{\mathrm{off}}(\nu,t)$, defined as

[TABLE]

at zero lags, is to be assessed for significance against uncertainties, where $\delta I_{\mathrm{yy}}(\nu,t)=I_{\mathrm{yy}}(\nu,t)-<I_{\mathrm{yy}}>$ for yy state (on or off), and $<x>$ indicates ensemble average of $x$ across the span of ($\nu,t$). The uncertainty in the estimated correlation, in the best case (i.e. dynamic spectra free of noise and other undue contaminants), will be dominated finally by the finiteness of available scintle statistics. In case of detection of significant correlation, the off-pulse emission intensity as fraction $\eta$ of on-pulse intensity can be estimated as

[TABLE]

where $AC_{\mathrm{on}}(0,0)$ is the average zero-lag auto-correlation of (on-)pulse intensity variations, which includes the variance $\sigma^{2}_{\mathrm{on}}$ of the delta-correlated noise $U_{\mathrm{on}}(\nu,t)$.

In the discussion so far, the apparent intensity fluctuations across the dynamic spectrum for the on-pulse region are, in an ideal case, assumed to be primarily a manifestation of the interstellar scintillations across the observing band. However, a finite but small part of these may be due to 1) variations in the system noise, including the sky noise (other than that from the pulsar), in addition to 2) the contribution from aliased spectral range, if any. The former additive contributions equally affect the dynamic spectrum for the off-pulse region, and may undesirably contribute to the apparent correlation between the dynamic spectra. It is therefore important that the on-pulse dynamic spectrum $I_{\mathrm{on}}(\nu,t)$ is obtained after subtraction of $I_{\mathrm{off}}(\nu,t)$ from the corresponding spectrum for the on-pulse region. The version of $I_{\mathrm{off}}(\nu,t)$, to be used for subtraction here, should be for intensity averaged over the entire off-pulse region, as far as possible. In case of any genuine unpulsed intensity with correlated variations with those for on-pulse region, the suggested subtraction would result in an under-estimation of $\eta$ by an amount ${\eta}^{2}$. On the other hand, even intrinsic variations in the off-pulse region that are uncorrelated between the two dynamic spectra would be unduly subtracted from the on-pulse dynamic spectrum, and would introduce a negative bias in $\eta$ estimate. The magnitude of such bias is given by the ratio of variance ${\sigma}^{2}_{U}$ of these uncorrelated variations in the off-pulse spectrum to that for variations in the on-pulse spectrum (i.e. ${\sigma}^{2}_{U}/({AC_{\mathrm{on}}(0,0)-{\sigma}^{2}_{\mathrm{on}}})$). In any case, the negative bias will be limited to ${\eta}^{2}_{\mathrm{max}}$, where $\eta_{\mathrm{max}}$ is as defined later in the Equation 6. The advantage of thus removing any common unpulsed intensity variations, either due to sky or system, from $I_{\mathrm{on}}(\nu,t)$, in terms of obtaining a more reliable estimate of $\eta$, overwhelms the undesirability of the the mentioned bias, which is expected to be insignificant any way.

Of course, any intrinsic variability in the pulsar intensity would leave an unavoidable (multiplicative) imprint in the dynamic spectrum. The spectral scales of intrinsic variability are expected to be much wider than those associated with interstellar scintillation. There is no a priory basis yet for expecting the possible unpulsed component, if any, to have correlated intrinsic variability. Hence, in general, any independent intrinsic variability of intensities in the two regions would reduce the net cross-correlation, and in any case, increase the uncertainty in the estimation of the unpulsed intensity. Fortunately, any pulse-to-pulse variations in intrinsic intensity are expected to average out, with suitable temporal smoothing of the dynamic spectrum ($P\ll\Delta t_{\mathrm{res}}<t_{\mathrm{s}}$). Any residual variation, on time scales shorter than $\Delta t_{\mathrm{res}}$, would be indistinguishable from the random uncertainty in estimation of the dynamic spectral elements. The combined magnitude of these fluctuations would be readily apparent in the auto-correlation function across the first few time-lags, as the delta-correlated contribution. In comparison, the auto-correlation due to scintillation-induced intensity variations is expected to decorrelate on a relatively longer time-scales ($t_{\mathrm{s}}$).

The expected implicit linear inter-relationship between the patterns (after removing the respective mean values), assessed through formal cross-correlation, can be modeled explicitly as follows

[TABLE]

where, the first term on the right-side is the best-fit model, and $U(\nu,t)$ is the apparent deviation or the part of observed off-pulse dynamic spectrum that is uncorrelated in time and frequency with $\delta I_{\mathrm{on}}(\nu,t)$, with its nominal mean $\left<U(\nu,t)\right>=0$, and other quantities as defined earlier. The uncorrelated part $U(\nu,t)$ includes also any measurement uncertainties in $I_{\mathrm{off}}$ and also the model $\eta\,I_{\mathrm{on}}$. In the above formulation, as in the Equation 2, $\eta$ is a measure of the ratio $\delta I_{\mathrm{off}}/\delta I_{\mathrm{on}}$.

The uncertainty $\sigma_{\mathrm{\eta}}$ in its estimate can be expressed as

[TABLE]

where $\sigma_{\mathrm{<U>}}$ is the reduced uncertainty in the mean of $U(\nu,t)$, and is related to standard deviation in $U(\nu,t)$ as

[TABLE]

where $N_{\mathrm{eff}}$ is the effective size of the ensemble. The $<U>$ and ${\sigma}_{U}$ are, in practice, computed using all of the $N$ samples available in the dynamic spectral array, including $U(\nu,t)$. The total number of points $N$ in these arrays is equal to $N_{\mathrm{\nu 0}}N_{\mathrm{t0}}$, where $N_{\mathrm{\nu 0}}$ is the number of spectral channels and $N_{\mathrm{t0}}$ number of time bins/sections in the dynamic spectrum. However, since all the points/pixels in the dynamic spectrum are not independent, particularly when the random measurement noise is much smaller than the intensity variations due to scintillation. Hence, in such cases, $N_{\mathrm{eff}}$ is often much smaller than $N$, and represents rather the number of independent samples in the dynamic spectrum. We have used the number of scintles as defining $N_{\mathrm{eff}}$, so that our uncertainty estimate $\sigma_{\mathrm{\eta}}$ corresponds to worst-case error. The definition of number of scintles, as given in Cordes & Lazio (1991), is $N_{\mathrm{eff}}=N_{\mathrm{t}}\times N_{\nu}$, where $N_{\mathrm{t}}=1+\kappa({\Delta t_{\mathrm{obs}}}/{t_{\mathrm{s}}})$, $N_{\nu}=1+\kappa({\Delta\nu_{\mathrm{BW}}}/{\nu_{\mathrm{d}}})$, where $\kappa$ is an empirically obtained number (we can call it filling factor), lies in range $0.1-0.5$. If $N_{\mathrm{eff}}$ for one spectrum is different from that for the other, we use the geometric mean of the two $N_{\mathrm{eff}}$ values.

For dynamic spectra spanning long durations, explicit attention would be needed to examine if they are affected by possible slow variations in pulse intensity within the span, due to intrinsic variations and/or originating from extrinsic reasons, including refractive scintillations and any instrumental gain variations that remain to be corrected. The correlation scales across frequency for these are expected to be generally wide. Hence, any contamination in the off-pulse region, as mentioned above, is likely to be modulated the same way, resulting in spurious correlation corrupting the correlation of interest. It may become necessary therefore to either estimate slow modulation, and correct at least the on-pulse dynamic spectra accordingly, or estimating the correlation or $\eta$ using dynamic spectra of shorter spans at a time, repeating the analysis for each of such sections separately, and then computing a weighted average of $\eta$, combining independent estimates made using subsets of data.

Before proceeding further, we wish to draw attention to a particular ready utility of the dynamic spectra of the apparent intensity variations in the on-pulse and off-pulse regions. We argue that, regardless of the details of contamination, and the presence or the lack of correlation between the two dynamic spectra, it is possible to define a hard upper-limit for the unpulsed intensity, as

[TABLE]

where $AC_{\mathrm{off}}(0,0)$ is the average zero-lag auto-correlation of observed intensity variations in the off-pulse region, which includes the variance $\sigma^{2}_{\mathrm{off}}$ of the delta-correlated noise $U_{\mathrm{off}}(\nu,t)$. When $AC_{\mathrm{off}}(0,0)\gg\sigma^{2}_{\mathrm{off}}$, fractional uncertainty ($1\sigma$) in $\eta_{\mathrm{max}}$ would be $1/\sqrt{2N_{\mathrm{eff}}}$. However, even when $\delta I_{\mathrm{off}}(\nu,t)$ appears to consist of only delta-correlated noise, i.e. $\eta_{\mathrm{max}}\approx 0$, the uncertainly would at best be limited to $\sigma_{\mathrm{off}}/\sqrt{N<(\delta I_{\mathrm{on}}(\nu,t))^{2}>}$.

2.1. Implications of relative location of possible off-pulse emission region

In general, the apparent emission in the off-pulse region would be a combination of the intrinsic and confusing sources of continuous emission, and the discussed correlation would be correspondingly partial, but providing a measure of the intrinsic component (spatially confined within the transverse scale $S_{\mathrm{d}}$).

Given the form of spatial distribution of electron density irregularities in the ISM as detailed in Appendix B, this spatial resolution scale $S_{\mathrm{d}}$, same as the diffraction pattern scale, is given by the following relation (Armstrong et al. 1995).

[TABLE]

where, $r_{\mathrm{e}}$ is the classical radius of electron, $\lambda$ is the radio wavelength, $\alpha$ = $\beta-2$, and $z$ is the effective propagation distance through the ISM.222 For uniformly distributed scattering, $z$ would correspond to the distance to the pulsar. For Kolmogorov turbulence, $\beta=11/3$ ($\alpha=5/3$) and the numerical value of the function $f(\alpha)$ is $\approx 1.12$.

The ISM parameters in the above equation are not directly measurable, although can be estimated.333The diffractive scintillation time-scale $t_{\mathrm{s}}$ (decorrelation time) is directly related to $S_{\mathrm{d}}$, where $t_{\mathrm{s}}=S_{\mathrm{d}}/V$, but the equivalent velocity $V$ of the medium relative to the pulsar sight-line is not independently known in most cases. On the other hand, the associated angular scatter broadening $\theta$, which ultimately limits the resolution in imaging observations, also relates to the above spatial scale, as $\theta=\lambda/2\pi S_{\mathrm{d}}$. A related and more readily measurable quantity is the scintillation decorrelation bandwidth $\nu_{\mathrm{d}}$, or alternatively the temporal scatter broadening $\tau$ of the pulse, where $\tau=z{\theta}^{2}/2c\approx 1/2\pi\nu_{\mathrm{d}}$, and $c$ is the speed of light. Thus, $S_{\mathrm{d}}$ can be estimated from $\nu_{\mathrm{d}}$ measurement, as444The form of this expression is consistent with that in Equation 13 of Cordes et al. (1983).

[TABLE]

where $r_{\mathrm{f}}$(=$\sqrt{z\lambda/2\pi}$) is the Fresnel scale, and $\nu$ is the observing radio frequency.

A positive result in our proposed correlation test would not only conclusively confirm the claimed detections, but would constrain the apparent size and spatial separation, at the so-called “retarded emission time” (Cordes et al. 1983), between the corresponding emission regions, and the level of correlation would provide clues on the relative spatial separation. A negative result, on the contrary, would not necessarily imply absence of unpulsed emission, unless the resolution scale $S_{\mathrm{d}}$ is large enough to cover the entire spatial extent within which emission can be considered as intrinsic to the pulsar. Considering the maximum separation between relevant emission regions to be the so-called light-cylinder radius $r_{\mathrm{L}}$ ($=cP/2\pi$), the above requirement implies that $r_{\mathrm{L}}\lesssim S_{\mathrm{d}}$, where $P$ is the pulsar period555 For the spin periods in the range 1.4 ms$-$11.8 s, the range of light-cylinder radius corresponds to $\sim 10^{4}-10^{8}$m..

This condition can also be expressed as $cP/2\pi\lesssim r_{\mathrm{f}}\sqrt{\nu_{\mathrm{d}}/2\nu}$, assessment of which would require an estimate of $z$, in addition to that of the decorrelation bandwidth $\nu_{\mathrm{d}}$. Although independent distance measurement is desired, $z$ estimated from dispersion measure would also render useful for the present purpose. It is worth emphasizing that the above stated condition is not model dependent, i.e. independent of $\beta$.

For a given pulsar, i.e. given $P$, $z$ and $C_{\mathrm{N}}^{2}$, the observing frequency $\nu$ can be chosen suitably, to see if the above condition can be met. The condition is more likely to be satisfied in cases of higher frequency probe of scintillation patterns for relatively nearby short-period pulsars.666 The underlying basic dependencies, as in the Equation 7, imply that the spatial resolution scale $S_{\mathrm{d}}$ broadens with increasing radio frequency ($\nu$) and with decreasing integrated scattering measure $C_{\mathrm{N}}^{2}z$. This diffraction pattern scale, in the weak scattering regime at adequately high frequencies, would of course saturate to its upper limit, that is the Fresnel scale $r_{\mathrm{f}}$, equal then to the refractive scale at its lower limit.

In any case, if any intrinsic unpulsed emission were to originate within the angular scale $S_{\mathrm{d}}/z$ around the pulsar, we expect to find the expected correlation signature. Although such cross-correlation (at zero-lag) is expected to fall significantly and rapidly, as $\exp[-(\Delta S/S_{\mathrm{d}})^{(\beta-2)}]$ (Cordes et al. 1983), with increasing separation $\Delta S$. However, if the separation, even if large (i.e. many times $S_{\mathrm{d}}$), happens luckily to be near parallel (within angle $(S_{\mathrm{d}}/\Delta S)^{c}$, for $S_{\mathrm{d}}\leq\Delta S$), then again significant cross-correlation would be expected, but now at time-lag $\Delta t\sim t_{\mathrm{s}}\Delta S/S_{\mathrm{d}})$, if the scattering transfer function can be considered as essentially frozen over those time scales. The above considerations necessitate exploration of the discussed correlation over a range of lags in both time and frequency, as we do in our tests and analysis to follow.

3. Tests with Simulated Dynamic Spectra: Assessment of sensitivity and immunity

Here, we illustrate application of our technique to simulated scintillation dynamic spectra, and assess its performance, in terms of ability to reliably estimate off-pulse/unpulsed intensity intrinsic to the pulsar, and immunity to potential contaminants in the off-pulse region.

As mentioned in Section 2, and illustrated in Appendix A, one of the subtle contamination of the off-pulse region could come from genuine main-pulse signal itself, if it is not adequately filtered out from the spectral regions beyond the observing band. These aliased contributions (from possible image bands relevant to heterodyning, and regions inadequately attenuated by band-defining filter before digitization) appear at longitudes that are, in general, offset from the main-pulse window (see Figure A1), depending on dispersion measure and frequency separation.

Fortunately, the scintillation-induced intensity pattern would significantly differ for spectral separations larger than the decorrelation scales, $\nu_{\mathrm{d}}$, particularly when $\nu_{\mathrm{d}}\ll\Delta\nu_{\mathrm{BW}}$, and even the overall shape and sizes of the scintles (characterized by the decorrelation scales, $\nu_{\mathrm{d}}$ and $t_{\mathrm{s}}$), themselves vary systematically with $\nu$, more rapidly with decreasing frequency. Any aliased contribution from other bands will have their own different scintillation-induced imprint, and hence, is not expected to contribute to any significant net correlation. This forms the basis of our expectation for potential immunity of our scintillation correlation method against aliasing-induced contribution which disguises as off-pulse emission, and we assess it by using simulated dynamic spectrum over a spectral range several (7) times the nominal bandwidth of observation.

A detailed description of our simulation of diffractive scintillation is presented in Appendix B, and resultant dynamic spectrum spans 115.5 MHz (7x16.5 MHz) centered at 270 MHz, and covers a duration of about $1000t_{\mathrm{s}}$ seconds. The time and spectral sampling here is $\sim t_{\mathrm{s}}/4$ seconds and 64.45 kHz ($\sim\nu_{\mathrm{d}}/20$), respectively.

This simulated dynamic spectrum, is treated as directly corresponding to an on-pulse intensity pattern. A small section ($\sim 180t_{\mathrm{s}}$; or 3 hr, if $t_{\mathrm{s}}=60s$) of this pattern is shown in Figure 1, sampled across 1792 spectral channels and 700 time bins (out of the simulated duration spanning 4000 time bins).

The central spectral region, of 16.5 MHz width, is treated as the observing band, and the associated scintillation pattern is assumed to directly simulate an observed on-pulse dynamic spectrum. As an example, Figure 2 presents a zoomed portion over a short duration ($\sim 25t_{\mathrm{s}}$, or say, 25 minutes), where the scintle scales in both the dimensions are clearly discernible.

Dynamic spectrum corresponding to the off-pulse region, on the other hand, is constructed by appropriately superposing the intensity variation simulated across the seven bands, following different assumed levels of genuine unpulsed (off-pulse) emission, and those of aliasing from contaminating bands, if any. For completeness, enabling a range of assessments, we consider the following three kinds of off-pulse dynamic spectra, as having (a) only genuine unpulsed emission, (b) genuine unpulsed emission plus contamination from aliasing, and (c) no genuine unpulsed component, but with only aliased contributions.

In (a) and (b), the dynamic spectral contribution as due to a genuine unpulsed emission is readily obtained from the on-pulse dynamic spectrum, suitably scaled by an assumed factor $\eta$. Unless mentioned otherwise, $\eta$ is assumed to be 0.01. We assume, for simplicity, that the aliased bands, contributing in (b) and (c), are attenuated by also the same factor (i.e. 0.01 or -20 dB), The off-pulse dynamic spectra, across the same span (16.5 MHz, with 256 channels), wherein any aliased contribution from other bands is added together, with or without band-flips, as appropriate. An off-pulse dynamic spectrum thus simulated for the case (c), only with aliased contribution from only two bands adjacent to the observed band (i.e. immediate upper and lower band) on either side, is shown in Figure 3 (depicting a similarly zoomed section as in Figure 2).

The Figure 4 presents the auto-correlations, and the cross-correlation maps computed using the the respective dynamic spectra shown in Figure 2 and Figure 3.

The results in Table 1 are for the following two distinct cases of simulated off-pulse dynamic spectrum; namely, for $\eta_{\mathrm{true}}$ = 0.01 and 0. In each case, the aliasing-induced contamination from adjacent spectral bands is explicitly included (with chosen attenuation), for aliasing-order (AO) ranging for 1 to 3, making a total of six versions of simulated off-pulse spectra. These are separately used along with a common on-pulse dynamic spectrum to estimate $\eta$ in each case. For example, an aliasing-order “k” corresponds to a spectral span of k.$\Delta\nu_{\mathrm{BW}}$ on either side of the observing band as being the source of contamination.

The estimates of uncertainty in $\eta$ depend of the $N_{\mathrm{eff}}$, which is computed following the same procedure as described in Section 2. In each case, having dynamic spectra that are same for the on-pulse, but differently constructed for the off-pulse region, the $N_{\mathrm{eff}}$ is computed based on the decorrelation scales seen in the latter (i.e. off-pulse) spectra. The decorrelation scales as seen to effectively broaden with the number of independent spectral patterns contributing to the constructed off-pulse dynamic spectra, as would be expected. The simulated intensities in the dynamic spectra are essentially exponentially distributed. These distributions would approach to Gaussian, when additive measurement uncertainties are significant. The overall positive bias in the estimates of $\eta$ computed from the entire span is understood as due to the slow modulation of pulse intensity (owing to refractive scintillations) which is shared by the contaminants of the off-pulse dynamic spectrum. When the suggestion made in an earlier section is followed, i.e. $\eta$ is estimated separately for each of the shorter spans, and such estimates combined appropriately, the average $\eta$ estimate is largely free of such bias, without loss of sensitivity. This can be appreciated from the comparison of the results presented in Table 1.777 The 6 models correspond to 2 sets, with and without genuine unpulsed emission component, in each of the three aliasing-orders. Two estimates of $\eta$ are presented for each model; A: using the entire span together, and B: using eight sub-spans separately for $\eta$ estimation, and the weighted average of such estimates computed. The latter is largely free of the bias corrupting the former estimates. See the main text for details.

The $\eta$ estimates in all considered cases are consistent with their respective model/assumed values $\eta_{\mathrm{true}}$ within the mentioned uncertainty. The $N_{\mathrm{eff}}$ changes systematically with aliasing-order, indicating possible increase in the decorrelation scales ($\nu_{\mathrm{d}}$ and $t_{\mathrm{s}}$).

The correlation maps in Figure 5 are presented to illustrate the implication of the relative location of the region corresponding to the intrinsic unpulsed/off-pulse emission, for a location offset of $0.1R_{LC}$. These are a result of our extended simulations, to directly obtain special versions of off-pulse dynamic spectra (see the text at the end of Appendix B), and enable us to examine modification of cross-correlation signature for different magnitudes and specific orientations ($||$, $\perp$ and 45o to $V$) of location offsets (for unpulsed emission source) within the light cylinder. The corresponding correlation maps (such as in Figure 5), for different magnitude and orientation of the location offset, indeed show the expected qualitative correspondence (in terms of shift and/or reduction of the correlation peak, as discussed in Section 2.1) in all of the specific cases we simulated.

The above tests with simulated data demonstrate the sensitivity of our technique in reliably searching/estimating possible intrinsic unpulsed emission using pulsar dynamic spectra, and confirm its desirable immunity to possible contaminants of off-pulse dynamic spectra.

4. Illustration of our technique: A case study with data on B0329+54

We now apply our technique to data from observation on pulsar B0329+54, made using multi-band receiver system (MBR; Mann et al. 2013) with the Robert C. Byrd Green Bank Telescope (GBT), on July 25, 2009.

From among many pulsars observed in ten well separated bands simultaneously, we have chosen B0329+54 based on sensitivity considerations, given that it is one of the brightest pulsars known. Width of each band $\Delta\nu_{\mathrm{BW}}$ is 16.5 MHz, and time-span of the observation $\Delta t_{\mathrm{obs}}$ is 1 hour. Other considerations include choice of the frequency band, as well as the resolutions in time and frequency, with which we can expect to see scintillation features in the the on-pulse dynamic spectrum. These choices depend largely on the decorrelation bandwidth and timescale, which are given by $\nu_{\mathrm{d}}(\text{kHz})=59C_{\mathrm{-4}}^{-1.2}\lambda^{-4.4}D^{-2.2}$ and $t_{\mathrm{s}}(s)=149C_{\mathrm{-4}}^{-0.6}\lambda^{-1.2}D^{-0.6}v_{7}^{-1}$, respectively (Romani et al 1985888Romani et al. (1986) has some typographical errors in these expressions; a wrong exponent of $C_{\mathrm{-4}}$ in the expression for $\nu_{\mathrm{d}}$, and $v$ instead of $v_{7}$ the expression for $t_{\mathrm{s}}$(s). ). Adequately fine sampling of the scintles, and ensuring as large a number of scintles as possible within the spectro-temporal span, require that $\delta\nu_{\mathrm{res}}\ll\nu_{\mathrm{d}}\ll\Delta\nu_{\mathrm{BW}}$ and $\delta t_{\mathrm{res}}\ll t_{\mathrm{s}}\ll\Delta t_{\mathrm{obs}}$. Based on these criteria, the data at 730 MHz and 810 MHz are found suitable, while other data at lower and higher frequencies did not meet these criteria. For B0329+54, at 730 MHz, the estimated $\nu_{\mathrm{d}}$ and $t_{\mathrm{s}}$ are $\sim 100$ kHz to $\sim 1800$ kHz and 970 s to 240 s, respectively, at 810 MHz, the corresponding $\nu_{\mathrm{d}}$ and $t_{\mathrm{s}}$ are $\sim 200$ kHz to $\sim 2900$ kHz and 1100 s to 270 s, respectively, for $C_{\mathrm{N}}^{2}\sim 3\times 10^{-5}$ and $\sim 3\times 10^{-4}\ m^{-20/3}$. Available measurements by Stinebring et al. (1996) and Wang et al. (2008) at 610 and 1540 MHz, respectively, imply $\nu_{\mathrm{d}}$ and $t_{\mathrm{s}}$ to be about 750 kHz and 450 s, respectively, at our lower frequency. For comparison, our estimated decorrelation scales (from correlation analysis such as shown in Figure 8) of about 1 MHz, 360 s and 1.3 MHz, 400 s, at 730 and 810 MHz, respectively, are largely consistent with the above mentioned values, within the uncertainties. Our dynamic spectra have frequency resolution of 64.45 kHz, and time-resolution of 18 s, with 200 time bins across 3600 s.

The recorded raw voltage time sequences corresponding to a bandwidth of 16.5 MHz were analyzed to obtain dynamic spectra with frequency resolution of 64.45 kHz (i.e., across 256 channels). After suitable corrections for dispersion and gain compression,999 When signal levels are even slightly larger than the limit within which a radio receiver has linear response, output power becomes less than that expected from its linear response, amounting to reduction in the gain. “Gain compression” refers to this reduction in gain, or non-linear response. In the context of dispersed pulsar signals, if not corrected, such a situation can cause a dip proportional to the pulse intensity, with a spread in longitude corresponding to the dispersion delay across the observed bandwidth. This effect can contaminate off-pulse region significantly, in cases of bright and high dispersion measure pulsars. if any, dynamic spectra for various choices of ranges in the longitude were obtained separately. Figures 6 and 7 show these pairs of dynamic spectra, in which the spectral or time ranges affected by radio frequency interference have been removed.

For our data, after removing bad time-sections and RFI channels, we have performed the dynamic spectral correlations, to estimate various quantities mentioned above, including the measure of correlation $\eta$. More specifically, we estimate $N_{\mathrm{t}},\ N_{\nu},\ N_{\mathrm{eff}},\ \eta$ and $\sigma_{\mathrm{\eta}}$, along with ${\eta}_{\mathrm{max}}$, as listed in Table 2.

As is apparent from these estimates, the off-pulse emission can be said to be less than about 0.5% of the main pulse flux density (corresponding a 3-$\sigma$ limit). The present reduction in the uncertainties in $\eta$ is moderate, in comparison with the hard limit $\eta_{\mathrm{max}}$, and is consistent with the relatively small statistics (i.e. $N_{\mathrm{eff}}$).

At this stage, our presentation of these results is mainly as an illustration, but with improved spectro-temporal span of the data (i.e. large $N_{\mathrm{eff}}$), we can expect a significant refinement in these estimations and their uncertainties.

As can be seen readily from the 1-d plots corresponding to the auto-correlations shown in Figure 8, the random measurement noise de-correlates at non-zero lags, and the sharp drop in the auto-correlation with respect to its value at zero-lag provides a ready estimate of its relative contribution (i.e. $\sigma^{2}_{\mathrm{on}}$ or $\sigma^{2}_{\mathrm{off}}$), which is to be discounted while estimating decorrelation bandwidth or time-scale.

From the cross-correlation map (the bottom plot in Figure 8), and in general, the level of cross-correlation, or the lack of it, can be assessed across the respective lags, and interpreted either in terms of upper limits on the unpulsed emission intensity, or possible separation of the associated emission region from that of the main-pulse emission.

5. Discussion and Conclusions

The new technique proposed here, for searching for intrinsic off-pulse/unpulsed radiation for pulsars, is inspired by the expectation that such emission originating from apparent location(s) matching to or in the vicinity of that of the pulsed component (compared at their respective retarded emission times) would also carry a scintillation imprint similar to that measurable for the pulse intensity.

Needless to say that a systematic search for unpulsed emission at a range of frequencies, with appropriate spectro-temporal resolution and spans, will naturally be rewarding. On the other hand, for data sets at sufficiently nearby frequencies, a combined estimate (or upper limit) for $\eta$ can be obtained. For example, such an upper limit (3$\sigma$) for unpulsed intensity of B0329+54 would be 0.4%, based on the data at the two frequencies.

Since our method, although a truly high angular resolution probe, is based on longitude-resolved dynamic spectral information, it can be expected to be applied to most of the archived observations on pulsars, made from single-dish and synthesis telescopes, in addition to future observations, as long as the scintles, or the refractive fringing when present, are at least Nyquist sampled. In fact, it would not be surprising to see a massive initiative to search of the continuous/unpulsed emission in the near future, using existing data and new observations.

Although the relevant correlation would reduce exponentially with increasing apparent angular offset $\Delta S/z$ (Cordes et al. 1983) of the source to be searched, opportunity to detect significant peak in the correlation map (at a non-zero lag in time) is to be expected when the orientation of the offset is along the pattern velocity (or within an alignment margin of $(S_{\mathrm{d}}/\Delta S)^{c}$), as illustrated in our simulations. Interestingly, given the reported pulsar spin-velocity alignment, for young pulsars in particular (Johnston et al. 2005), the location offset along latitudinal direction is likely to be along the pattern velocity, when the scintillation speed is dominated by pulsar motion. Regardless of the possible apparent location of the region responsible for unpulsed emission, if any, within the light-cylinder, it is unlikely that the associated key morphology (say, w.r.t. rotation axis) would differ significantly from pulsar to pulsar. This combined with the expected variety in the orientation of the apparent velocity of the diffractive scintillation pattern (again viewed with respect to the pulsar spin-axis) and in $S_{\mathrm{d}}$ (depending on sight-line and frequency), suggests that there would be adequate number of known pulsars offering conducive situation for the proposed probe to be rewarding, in either detecting their elusive continuous emission, or ruling it out to the extent possible.

In summary, the existence of unpulsed emission from pulsars is yet to be fully established, let alone be understood. However, if detected unambiguously, any unpulsed radio radiation intrinsic to pulsars would indeed be a precious token of the mysteries of emission mechanism in radio pulsars that are yet to be unraveled, and could provide important missing clue to further our understanding of the key processes at work. Our proposed detection method, exploiting the resolving power of the interstellar telescope, as a powerful tool for reliable and sensitive search/detection of unpulsed/off-pulse emission, should open a new window for this promising exploration.

We gratefully acknowledge contribution from Karishma Bansal in the very early phase of this work (during her Visiting Studentship at the Raman Research Institute). We thank our anonymous referee for constructive comments and suggestions.

Appendix A A: Aliasing in dynamic spectra,

and possible contamination in the off-pulse region

In this section, we discuss and illustrate how the observed data, and the off-pulse region in particular, be affected if the spectral filtering in the receiver chain were to be imperfect, allowing a non-zero fraction of sky signal out side the band of interest to pass through, even though attenuated significantly. Although, in practice, a variety of non-idealities in spectral filtering are possible101010 The key filtering stages include, a) image-band rejection before heterodyne or mixing stage, and b) band-defining before digitization (at Nyquist rate) of signal either at baseband or that located around a chosen center frequency (where harmonic or band-pass sampling at Nyquist rate is employed)., for illustration purpose, we consider a simple case of the band-defining filter having a non-zero response in adjacent spectral ranges on either side of the intended band of observation.

It is easy to see that the contribution from a dispersed pulse in these out-of-band spectral regions would be aliased in the Nyquist sampled band of interest. On dedispersion, these aliased contributions from the folded bands would not only spill out side the pulse window, but can systematically spread across a large part of the off-pulse longitudes, and could disguise as off-pulse emission. For an illustrative example of this effect, lets assume that the filter function over the desired band (of width = $BW$) is flat, as corresponding to a perfect rectangular filter, but this non-ideal filter offers finite, though high, attenuation in other spectral regions. In our simulation, we consider the out-of-band region on either side to be 3 times wider than $BW$, and relative attenuation to be 20 dB, such that the filter response $F(\nu)$ as function of frequency $\nu$ is given by

[TABLE]

where, $\nu_{\mathrm{c}}$ is the center frequency of observation.

Assuming a train of Gaussian pulses from a pulsar, the intensity pattern across time and frequency can be expressed as

[TABLE]

where, $P$ is the pulsar period, $\sigma$ is the standard width of the Gaussian, $n$ is number of periods defining the time span of the simulated sequence. The dispersion delay at frequency $\nu$, with respect to that at a reference frequency $\nu_{\mathrm{ref}}$, is given by

[TABLE]

where $DM$ is the dispersion measure in $pc\ cm^{-3}$ and the frequencies are in MHz. A dynamic spectrum containing dispersed pulses was simulated assuming $BW$= 16 MHz centered at $\nu_{\mathrm{c}}$= 300 MHz, $\sigma$ = 0.04 s, and for pulsar parameters similar to that of B0329+54 ($DM$ = 26.77 $pc\ cm^{-3}$, $P$= 0.714472578 s).

The top panel of Figure A1 (or 8) shows the dedispersed pulse sequence when no aliasing of sky signal occurs from outside of the desired band of bandwidth $BW$ (a case of ideal filtering). The middle panel shows a similarly obtained sequence, but now including aliasing (of remaining small level of sky signal) from the adjacent bands as described above. Note that the dispersed pulse contribution from the aliased adjacent bands would be at a much lower level, but will make its appearance in the off-pulse region. These contributions will occupy different longitude spreads after dedispersion, depending on the alias order, in addition to the $DM$ and $P$. Significant contamination in both the on-pulse and the off-pulse regions, as result of aliasing, is apparent from the difference between the sequences in the top two panels, as shown in the bottom panel. Here, we have deliberately used a lower $\nu_{\mathrm{c}}$ (=300 MHz), than those for the data we present (i.e. 730 MHz or 810 MHz), so that the mentioned contamination is more pronounced.

Appendix B B: Simulation of Diffractive Interstellar Scintillation (DISS) and Dynamic Spectra

The two main steps in simulation of DISS dynamic spectra, using a thin-screen approximation, are: (i) generation of a random phase screen following an assumed spatial distribution of electron density irregularities in the intervening ISM, which modifies the emerging wave-front, and (ii) calculation of resultant intensity, of the received signal at the observer’s location, as a function frequency and time.

As mentioned already, the most accepted 3D spatial power spectral description of the turbulent ISM is a power-law spectrum across spatial frequency $q$, ranging from $q_{\mathrm{min}}$ to $q_{\mathrm{max}}$, (Armstrong et al. 1995)

[TABLE]

where, $C_{\mathrm{N}}^{2}$ is the level of turbulence, and the power-law index $\beta=11/3$ for the Kolmogorov turbulence. Armstrong et al. (1995) have given evidence of the validity of Equation B1 for $q_{\mathrm{min}}=10^{-12}\ m^{-1}\ <\ q\ <\ q_{\mathrm{max}}=10^{-6}\ m^{-1}$, and have derived the typical turbulence strength $C_{\mathrm{N}}^{2}\sim 10^{-3}\ m^{-20/3}$, but the $C_{\mathrm{N}}^{2}$ can deviate significantly from this typical value, depending on direction and distance to the source.

A convenient way to study propagation effects due to this 3-D distribution of refractive index irregularities in the ISM is to model the modification of the incident wave-front by an equivalent thin phase-changing screen, located between the source and the observer. We use this thin screen approximation (see Lovelace 1970; Romani et al. 1986) for our present simulations, wherein the equivalent 2-D spatial power spectrum ($P_{\mathrm{2\phi}}$) of the phase deviation $\phi$ is given by

[TABLE]

where, $z$ is the distance to the source, $r_{\mathrm{e}}=2.82\times 10^{-15}$ m is the classical radius of the electron, and $\lambda$ is the wavelength of the propagating radiation.

B.1. Generation of Equivalent Thin Phase-Screen using FFT-Based Technique

The spatial power spectrum of the equivalent 2-D thin screen and the associated distribution of the random phase deviation $\phi(x,y)$ across that screen in transverse plane $(x,y)$ have the following Fourier relationship

[TABLE]

i.e., $\phi(x,y)$ is the (inverse) Fourier transform of the product $g(q_{x},q_{y})\sqrt{P_{\mathrm{2\phi}}(q_{x},q_{y})}$, where $g(q_{x},q_{y})$ is a Hermitian-symmetric complex Gaussian variable representing zero-mean white noise process, with unity variance (Johansson & Gavel 1994). We obtain $\phi(x,y)$ distribution using the above relation, employing FFT technique for computational ease.

The discrete form needed for simulation, of the Equation B3, for a square screen, $\phi$, made-up of $N\times N$ grid points, is

[TABLE]

where, $\Delta r$ is spatial sampling interval, $g$ is a $N\times N$ matrix (the procedure to obtain it is explained in the following) and $M_{0}$ is also a $N\times N$ matrix whose elements are $M_{0}(i,j)=[(i-N_{\mathrm{c}})^{2}+(j-N_{\mathrm{c}})^{2}]^{-\beta/4}$, where the origin is defined at $N_{\mathrm{c}}=N/2+1$, and the contribution at zero spatial frequency is set to zero, i.e. $M(N_{\mathrm{c}},N_{\mathrm{c}})=0$. The recipe for getting the Gaussian random matrix $g$ is as follows:

1. (i)

generate a complex matrix (say) $M$ of size $N\times N$, whose elements are $a+j\,b$, where, $j=\sqrt{-1}$, $a$ and $b$ are independent Gaussian random numbers with zero mean and unity variance. 2. (ii)

obtain the 2-D discrete Fourier transform of ‘$M$’, which will be the required matrix $g$ ($g=\mathcal{FT}^{-1}\left\{M\right\}$).

B.2. Electric Field Distributions and Resultant Intensity in the Observer’s Plane

Having generated $\phi(x,y)$ (i.e., the discrete $\phi$, Equation B4), the electric field distribution at the thin screen can be given by

[TABLE]

In the thin screen approximation, the ray optics is applicable and so the electric field received at any point $(\xi,\eta)$ on the observer plane, can be represented by the Fresnel-Kirchhoff integral (Born & Wolf 1980)

[TABLE]

where, $r_{\mathrm{f}}=\sqrt{\lambda z/2\pi}$. This integral can be either calculated from 2-D numerical integration or methods using Fourier transform. We have used angular spectrum method to calculate the electric field, i.e., via following relation (in discrete form)

[TABLE]

where, $h(x,y)$ is called transfer function. By use of the above method to get $E_{\mathrm{O}}$, the spatial sampling interval at the observer’s plane will be the same $\Delta r$ as that of phase distribution of the thin phase screen. So corresponding to each value of input frequency/wavelength, we will have phase distribution $\phi$ of the thin screen (in discrete form a matrix of size say $N\times N$) and the electric field $E_{\mathrm{O}}$ at the observer’s plane (again a matrix of size $N\times N$). From this matrix $E_{\mathrm{O}}$, we select a spatial 1D cut, say $E_{\mathrm{O}}(r)$, which may be an arbitrarily chosen row or column, and obtain $E_{\mathrm{O}}(\nu,r)$ by varying only $\nu$ in uniform steps over the range of interest. The spatio-spectral description of observed intensity is trivially obtained as $I_{\mathrm{O}}(\nu,r)$ = $|E_{\mathrm{O}}(\nu,r)|^{2}$. This $I_{\mathrm{O}}(\nu,r)$ can be translated into $I_{\mathrm{O}}(\nu,t)$, i.e. the dynamic spectrum, by assuming a velocity $V_{\mathrm{trans}}$ along $r$ for the intensity pattern in the observer’s plane, which depends on the relative transverse velocities of the pulsar, the scattering medium and the observer.

B.2.1 Our Simulation Parameters and Results

Diffractive effects correspond to spatial frequency range $q\sim 10^{-8}\ m^{-1}$ to $10^{-6}\ m^{-1}$ of the ISM irregularities (Stinebring 1996; Rickett 1988; Narayan 1988; Wang, Manchester 2008 ). We have used a square (scattering) screen so sampling interval, say $\Delta r$, in x and y directions are equal, i.e. $\Delta x=\Delta y=\Delta r$. The Nyquist sampling criteria demands $\Delta r=1/(2q_{\mathrm{max}})$. What size of the phase screen will be suffice for simulation of DISS ? The observer receives radiation from a cone of half-angle $\theta_{\mathrm{S}}\approx r_{\mathrm{ref}}/z\approx r_{\mathrm{mp}}/z\approx\lambda/(2\pi S_{\mathrm{d}})$, (Cordes 1986) where, $r_{\mathrm{ref}}$ is refractive length scale, $r_{\mathrm{mp}}$ is multi-path propagation length scale (strong scintillation), $S_{\mathrm{d}}$ is diffractive length scale, $z$ is distance from the observer to the thin screen and $\lambda$ is the wavelength of the radiation from pulsar. So to properly simulate DISS, the phase screen size $r_{\mathrm{max}}$ should be at least $\sim r_{\mathrm{mp}}\approx r_{\mathrm{f}}^{2}/S_{\mathrm{d}}$ in each of the two dimensions. Hence the required number of grid points $N$ across $r_{\mathrm{mp}}$, for a $N\times N$ matrix describing the screen, would be $N\geq r_{\mathrm{max}}/\Delta r=2q_{\mathrm{max}}r_{\mathrm{f}}^{2}/S_{\mathrm{d}}$ Thus for the case of our data on B0329+54, where $\nu$=810 MHz and $z=1.44$ kpc, $r_{\mathrm{mp}}\sim 10^{11}$ m. To satisfy the criteria $\Delta r\leq 1/(2q_{\mathrm{max}})$, the required $N\sim 2^{18}$ [$N=10^{11}/(5\times 10^{5})\sim 2\times 10^{5}\sim 2^{18}$] ! To generate a phase screen of this overwhelmingly large dimension, of order $2^{18}\times 2^{18}$, and the subsequent Fourier analysis involving bigger dimension (i.e., $2^{19}\times 2^{19}$) is not only computationally intensive (even with use of FFTs), but well beyond the readily available computing resources.

However, we note the red nature of the underlying spatial spectrum of phase variation $P_{\mathrm{2\phi}}\propto q^{-\beta}$ ($\beta$ is +ve). The associated structure function for phase $\phi$, at a given scale $r$ can be expressed as $D_{\mathrm{\phi}}(r)=(1^{c})^{2}(r/S_{\mathrm{d}})^{(\beta-2)}$, given that for the diffractive (or the coherence) scale $S_{\mathrm{d}}$, the phase structure function is 1 $radian^{2}$ (Armstrong et al. 1995). Since contribution to phase fluctuations from smaller spatial scales is expected to decrease rapidly, for the relevant values of $\beta$, we consider revision of the sampling scale $\Delta r$, such that $D_{\mathrm{\phi}}(\Delta r)\leq{\Delta\phi}^{2}_{\mathrm{min}}$, for the desired small phase variation ${\Delta\phi}_{\mathrm{min}}$ that is to be sampled duly. With this criterion, and recalling Equation 8, we express the required grid dimension as

[TABLE]

For example, with ${\Delta\phi}_{\mathrm{min}}$ = 0.1 radian, and $\beta$ = 11/3, $N\geq 16(\nu/\nu_{\mathrm{d}})$. Choosing a suitable ratio $(\nu/\nu_{\mathrm{d}})$, say 200, the requirement of $N\geq 3200$ appears feasible with computational constraints, i.e. without needing supercomputers, and more importantly, without compromising significantly on the details of the phase screen.

In our present simulations, we have used $\nu_{\mathrm{d}}\sim 1.35$ MHz, to keep reasonable correspondence with the discussed observations, but use a relatively smaller $\nu$ of 270 MHz. We use $N=4096$, so that the screen and the diffraction patterns are sampled with adequate details (corresponding to $\Delta r=5\times 10^{5}$ m and $r_{\mathrm{max}}\sim 2\times 10^{9}$)111111It is worth noting that now the Fresnel scale $r_{\mathrm{f}}$ and $z$ are artificially small and $C_{\mathrm{N}}^{2}$ is correspondingly large, as a result of the spatial dynamic range we have chosen., and the resultant dynamic spectrum suffices for demonstrating the key aspects of our technique. It is worth pointing out that now the Fresnel scale $r_{\mathrm{f}}$ and $z$ are artificially small and $C_{\mathrm{N}}^{2}$ is correspondingly large, as a result of the spatial dynamic range we have chosen. However, the scale of direct relevance to us, that is the diffraction pattern scale $S_{\mathrm{d}}$ corresponds to typical 4 samples across $r$, implying $(t_{\mathrm{s}}/4)$ as the sampling interval in the dynamic spectrum. The overall time scale, and $t_{\mathrm{s}}$, can be defined by choice of the velocity $V_{\mathrm{trans}}$, if required. In any case, the simulated time span corresponds to $\sim 1000t_{\mathrm{s}}$.

For completeness, to examine the sensitivity of the correlation technique to the relative location off-pulse emission region, we extended our simulations to obtain the off-pulse dynamic spectra separately from that for the pulsed component. Using the common description for the phase pattern, we added a suitable extra phase-gradient to it for simulating new phase screen, corresponding to the location offset, and used the resultant intensity pattern for constructing off-pulse dynamic spectra. The magnitude and direction of the location offset were varied, and the resultant cross-correlation maps were examined (see Figure 5).