Probing Pulsar Scattering between 120 and 280 MHz with the MWA

The interstellar medium (ISM) is a cold plasma that disperses and scatters pulsar radiation. Dispersion causes the broadband signal of a pulsar to be smeared out in time such that a particular pulse arrives at the observer later in time at lower frequencies than at higher frequencies. This is quantified in terms of the dispersion measure, $\mathrm{DM}={\int }_{0}^{D}{n}_{e}(l){dl}$, which is the integral over the electron density, n_e, along the line of sight to a pulsar at distance D. Dispersion is generally well understood, with the dispersion delay τ_dm scaling as the inverse of the observing frequency ν squared: τ_dm ∝ ν⁻². Thus, dispersion can be corrected for, modulo the time variability (e.g., You et al. 2007; Keith et al. 2013) and possible chromaticity (Cordes et al. 2016) of the DM. Both dependencies, DM(t) and the suggested DM(ν), originate in the inhomogeneous, turbulent ISM: the motion of a pulsar relative to the ISM and Earth causes the pulsar signal to probe different parts of the ISM over time. At the same time, density variations in the ISM cause radio waves to be scattered over a range of directions, resulting in an angular spectrum with a characteristic width θ_scatt, i.e., the scattering angle. The integrated effect is that an intrinsically narrow pulse becomes broadened in time as off-axis radiation is scattered back into the line of sight and arrives later at the observer due to the extra path length. The measurable effect of scatter broadening, the scattering delay τ, depends on frequency roughly as ν⁻⁴.

As such, scattering is the main hindrance to perform accurate pulsar timing at lower frequencies (ν ≲ 1 GHz) where pulsars tend to be brighter. At the same time this means that high sensitivity observations of pulsars at low radio frequencies are ideal to use both effects and their variation with time and frequency to study the properties of the ISM itself. The shape of a scattered profile and its evolution as a function of frequency give us insights into the underlying geometry, dynamics, and physics of the ISM. For example, the shape of the rising edge of a pulse depends on the distribution of the scattering medium along the line of sight, namely whether it is homogeneously distributed or organized in (possibly multiple) thin sheets. Similarly, the detailed structure of the scattering tail reveals the underlying structure function of density irregularities in the ISM (Lambert & Rickett 1999).

Since the impact of the ISM on pulsar radiation is strongest at low frequencies, observations of bright scattered pulsars at ν ≲ 300 MHz are well suited to differentiating between competing models of the ISM. In particular, it remains unclear whether Kolmogorov turbulence appropriately describes and quantifies variations in DM, the frequency scaling of pulsar scattering, and the scintillation properties of pulsars (e.g., Lam et al. 2016). The parabolic arcs in secondary spectra of scintillating pulsars (e.g., Stinebring et al. 2001; Brisken et al. 2010) disfavor a homogeneous, isotropically distributed scattering medium (e.g., Walker et al. 2004). Instead, they hint at localized, anisotropic scattering structures that could have a sheet-like geometry (e.g., Pen & Levin 2014; Braithwaite 2015).

The scattering delay induced by the ISM scales with frequency as ${\nu }^{-2\beta /(\beta -2)}$, where β is the power spectral index of density irregularities. For Kolmogorov turbulence we expect β = 11/3 (e.g., Rickett 1990). Most of the published measurements of α = −2β/(β − 2), however, deviate from the theoretically expected value α = −4.4. In fact, the majority of recently published measurements obtained from low-frequency observations of pulsars list spectral indices α > −4 (Lewandowski et al. 2015; Eftekhari et al. 2016; Geyer et al. 2017; Krishnakumar et al. 2017; Meyers et al. 2017). Evidence for shallower scaling also comes from a large sample of pulsar measurements by Bhat et al. (2004), who deduced a value of $\langle \alpha \rangle =-3.9\pm 0.2$ for the global scattering index. They also offer a possible explanation in terms of an inner scale and a crossover point that depends on this scale. Cordes & Lazio (2001) explain such "anomalous" scattering behavior by introducing filamentary structures along the line of sight; a scenario that can also give rise to shallow scattering. Xu & Zhang (2017) invoke supersonic turbulence and a frequency dependent volume filling factor of density irregularities to reconcile observations and theoretical models. Geyer & Karastergiou (2016) investigate the idea of anisotropic pulse broadening functions (PBFs) in the case of a thin screen, finding that the standard isotropic models can significantly underestimate the scattering time, especially at low frequencies. In a recent study, Geyer et al. (2017) compare spectral indices obtained by fitting isotropic and anisotropic models to multi-frequency scattering observations, finding that the anisotropic models yield values for α that are in better agreement with theoretical expectations.

Obviously, the ISM cannot be fully described by an isotropic model in which the power spectrum of density irregularities follows a simple power law. In order to shed further light on the physics of the ISM, more detailed studies of the shape and the temporal and spectral evolution of scattered pulsar profiles at low frequencies (ν ≲ 300 MHz) are required. New instruments such as the Murchison Widefield Array (MWA; Tingay et al. 2013), the Low Frequency Array (LOFAR; van Haarlem et al. 2013), and the Long Wavelength Array (LWA; Taylor et al. 2012) can provide the sensitivity, the frequency coverage, and also the temporal resolution needed for making further progress in this area.

In this work we present a pilot study of pulsar scattering using the MWA. We selected three bright, moderate DM pulsars—J0534+2200, J0742−2822, and J0835−4510 (Table 1)—and observed them across the large frequency range of the MWA (80–300 MHz). We show that for bright pulsars the sensitivity of the MWA is sufficient to obtain high quality pulse profiles within a short integration time (∼5 minutes), allowing for a measurement of the frequency scaling index α for all three targets from a single observation.

In Section 2 we describe the MWA observations, and in Section 3 we present our results and an analysis of the scattering profiles. Our results are discussed in Section 4 and the conclusions are presented in Section 5.

We observed the three pulsars J0534+2200 (the Crab pulsar), J0742−2822, and J0835−4510 (the Vela pulsar) with the MWA using the Voltage Capture System (VCS; Tremblay et al. 2015). This observing mode records the channelized voltages from each of the 128 tiles,⁴ which can be summed coherently after performing calibration. In order to be able to characterize the scattering properties of each target we observed in the following manner: instead of observing over a contiguous band of maximal 30.72 MHz bandwidth, we spread the 24 available individual coarse channels across the accessible frequency range of 80–300 MHz. Each of these coarse channels has a bandwidth of 1.28 MHz which, before recording, is channelized into 128 fine channels (i.e., 10 kHz) in a polyphase filterbank. Multiple subbands can be constructed by either grouping consecutive coarse channels, or by using individual coarse channels as subbands.

We used different frequency setups for each target as indicated in Table 2. The number of subbands ranged from 4 to 20, hence the bandwidth varied between 1.28 and 7.68 MHz per subband. Observing duration varied from about 5 to 60 minutes depending on the target. For calibration purposes we also performed a short (∼2 minutes) observation of a strong nearby calibrator (nonpulsar continuum source) using the same frequency setup as for the respective pulsar prior to the target observations. The calibrator scans were not recorded with the VCS but instead with the MWA correlator (Ord et al. 2015), which produced visibilities at a cadence of 10 kHz and 2 s.

2.1. Calibration

2.2. Coherent Beamforming

We used the software packages PSRCHIVE (Hotan et al. 2004; van Straten et al. 2012) and DSPSR (van Straten & Bailes 2011) for further data processing and analysis. We processed the individual subbands for each pulsar separately, and the resultant, incoherently dedispersed pulse profiles are shown in Figures 1 and 3.

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One of our main goals is to quantify the physical nature of scattering along the sight lines to the three different pulsars. To that end, we explore different PBFs to fit the observed pulse profiles in each band and measure the frequency scaling of the scattering delay τ. However, for a least-squares fitting procedure to be easily applicable, the intrinsic pulse shape needs to be known with high accuracy and the characteristic scattering time needs to be small enough that the scattering tail does not extend beyond the pulsar period. In practice, this need not necessarily be the case; measured profiles can be composed of multiple components whose shapes and separations cannot be easily extrapolated to low frequencies from higher frequency observations. Moreover, long scattering tails result in an apparent merging of the components and render the determination of a baseline impossible. In the case of the Crab pulsar this is alleviated by analyzing individual giant pulses (e.g., Bhat et al. 2007; Meyers et al. 2017).

For the other two pulsars we employ a deconvolution technique based on the CLEAN algorithm (Högbom 1974), which was implemented for pulse profiles by Bhat et al. (2003). The method recovers an intrinsic pulse shape by deconvolving the observed profile with an assumed model of the PBF (Figure 2), nonetheless, trialing over a range of τ to determine the best-fit value. Two commonly adopted models are a thin screen model and a thick screen model.

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In general, the exact form of the PBF associated with the ISM is unknown and depends on the distribution and the turbulence power spectrum of the scattering material along the line of sight to the pulsar. Assuming a square-law structure function for density inhomogeneities, the simplest PBF for a thin screen model is a one-sided exponential

$$\begin{eqnarray}&&{g}_{0}(t)=\left(\displaystyle \frac{1}{\tau }\right)\exp \left[-\displaystyle \frac{t}{\tau }\right]u(t),\end{eqnarray} \tag{ 1 }$$

where u(t < 0) = 0 and u(t ≥ 0) = 1.

More realistic models typically have a shallower rise time and a longer exponential tail, corresponding to different peak positions. Williamson (1972) showed that for a thick scattering screen, the PBF at small times t, i.e., during the rise time up until shortly after the peak of the emission, is given by

$$\begin{eqnarray}&&{g}_{1}(t)={\left(\displaystyle \frac{\pi \tau }{4{t}^{3}}\right)}^{1/2}\exp \left[-\displaystyle \frac{{\pi }^{2}\tau }{16t}\right].\end{eqnarray} \tag{ 2 }$$

The shape of the decaying tail can be described by an exponential, e^−t/τ, similar to g₀(t).

To evaluate the quality of the deconvolution procedure we make use of the figures of merit (FOM) defined in Bhat et al. (2003). In particular these are the positivity f_r (the PBF should not over-subtract the flux), the symmetry Γ (the deconvolved pulse should not show a residual scattering tail), and the consistency of the residuals with the off-pulse rms indicated by N_r (the relative number of bins that are within three standard deviations of the off-pulse region) and σ_r (the ratio of the rms in the CLEANed window to the rms in the off-pulse region). For convenience we summarize the definitions of each parameter in the Appendix. The deconvolution procedure should minimize f_r and Γ, while maximizing N_r and achieving σ_r ∼ 1. It is important to note that the absolute numbers of parameters Γ, N_r, and σ_r do not allow for a meaningful assessment of which scattering model provides a better fit. They merely serve to find the best-fit τ as illustrated in Figure 8.

We use f_r also to estimate the uncertainties of our measured τ. In a least-squares fitting routine, the minimum of the reduced chi-square statistic, ${\chi }_{\mathrm{red}}^{2}$, indicates the best-fit model. The parameter range for which ${\chi }_{\mathrm{red}}^{2}$ is unity above that minimum indicates the uncertainty of the measured parameter. Similarly, we use the range in τ for which f_r is unity above the best-fit model to estimate our uncertainties. In general the absolute number of this parameter carries little information. It is only in comparison to another model where a lower value of f_r indicates a better fit. Similarly, we also compute a reduced chi-square value, ${\chi }_{\mathrm{nCC}}^{2}$, from our best-fit model in order to have a relative measure for the goodness of fit between the thick screen model and the thin screen model. In this context we use the number of pulse phase bins in which CLEAN components were subtracted, nCC, as an estimate for the number of degrees of freedom.⁶ We emphasize that ${\chi }_{\mathrm{nCC}}^{2}$ computed in this way is merely another figure of merit that can be employed to prefer one model for scattering over another.

3.1. PSR J0742−2822

Using the above method, we analyzed the J0742−2822 data by adopting a PBF form as in Equation (1) and Equation (2). We have performed this for each subband separately; the resultant scattering delays τ and associated FOM are summarized in Table 3. Overall, the positivity parameter f_r and the parameter ${\chi }_{\mathrm{nCC}}^{2}$ are lower for the thin screen model than the thick screen one indicating that a thin screen geometry is a better representation of the data.

Table 3.  Measured Scattering Delays for J0742−2822

Frequency	Thin Screen							Thick Screen
(MHz)	τ (ms)	fr	Nr	σr	Γ	${\chi }_{\mathrm{nCC}}^{2}$	nCC	τ (ms)	fr	Nr	σr	Γ	${\chi }_{\mathrm{nCC}}^{2}$	nCC
230.40	4.4(4)	0.77	0.19	1.95	0.22	42	17	3.8(7)	0.55	0.19	1.25	0.31	112	6
220.16	3.7(16)	0.12	0.18	1.23	0.55	29	21	3.7(14)	0.63	0.18	1.13	0.54	89	7
211.20	5.4(14)	0.12	0.18	0.89	0.31	32	18	5.6(9)	0.20	0.19	0.89	0.19	91	6
209.92	5.7(9)	0.30	0.19	1.23	0.23	32	17	5.6(8)	0.62	0.19	0.97	0.08	110	5
199.68	5.5(13)	0.11	0.19	1.57	0.02	45	18	6.1(6)	0.70	0.19	1.13	−0.19	129	6
189.44	7.8(15)	0.05	0.20	2.03	0.29	40	14	7.8(18)	0.14	0.19	1.00	0.29	172	3
179.20	7.8(25)	0.02	0.19	0.90	0.50	36	14	9.8(42)	0.37	0.20	0.91	−0.29	147	3
168.96	8.3(30)	0.03	0.19	1.16	0.22	46	15	8.1(56)	0.05	0.19	1.03	0.63	129	5
α	−2.5(6)							−2.9(7)

Note. Numbers in brackets indicate the uncertainty in the last digit.

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The middle and right panels of Figure 3 depict the measured scattering delays as a function of frequency for both the thin and thick screen models, respectively. Also shown are power-law fits of the form τ ∝ ν^α. For the thin screen model we obtain α = −2.5 ± 0.6, which is in agreement with the measurements of Lewandowski et al. (2015, α = −2.52 ± 0.3) but lower than those of Geyer et al. (2017) who measure α = −3.8 ± 0.4. The use of the thick screen model, yields a slightly higher α = −2.9 ± 0.7, which agrees with the thin screen estimate within the uncertainties.

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3.2. The Vela Pulsar

In the lowest band of our observations, the effect of scatter broadening becomes too large to enable a detection of the pulsar, and the detection in the two bands centered at 164.5 and 179.8 MHz is only marginal (Figure 1). Therefore, we restrict our analysis to the highest two bands, centered on 210.56 and 256.64 MHz. The frequency scalings as indicated by the measured delays (Table 4) are α = −4.0 ± 1.5 (thin screen model) and α = −5.0 ± 1.5 (thick screen model). The FOM also listed in Table 4 do not indicate a clear trend for which model reproduces the observations better. The post-CLEANing residuals are similar in their noise characteristics (indicated by σ_r and N_r) and the skewness Γ is also very similar between the models. Both the f_r—parameter and ${\chi }_{\mathrm{nCC}}^{2}$ indicate that the thick screen model is a better representation of the observations at 256 MHz, while at 210 MHz the thin screen model provides a better fit.

Table 4.  Measured Scattering Delays for the Vela Pulsar

Frequency	Thin Screen							Thick Screen
(MHz)	τ (ms)	fr	Nr	σr	Γ	${\chi }_{\mathrm{nCC}}^{2}$	nCC	τ (ms)	fr	Nr	σr	Γ	${\chi }_{\mathrm{nCC}}^{2}$	nCC
256	14.1 ± 1.7	1.6	0.26	1.66	0.33	21	5	11.4 ± 2.5	0.53	0.28	1.65	0.36	21	5
210	31.2 ± 8.1	0.2	0.31	1.24	0.86	22	5	32.6 ± 4.7	0.28	0.31	1.46	1.13	38	3
α	−4.0 ± 1.5							−5.0 ± 1.5

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3.3. The Crab Pulsar

The complexity of the pulse profile of the Crab pulsar and the degree of scattering at MWA frequencies render the analysis of average profiles unsuitable for the estimation of its scattering delays. The scattering tail extends well beyond the pulsar period at the lower frequencies.

The Crab pulsar is known for giant pulses exceeding the flux density of its regular pulses by orders of magnitude. Therefore, we can use a single giant pulse to study the effects of the ISM on the Crab pulsar's emission. Intrinsically, giant pulses are of ∼μs time duration (e.g., Popov & Stappers 2007; Bhat et al. 2008), so can be treated as impulsive signals given the native resolution of our VCS recordings (100 μs). Hence, the problem of an unknown intrinsic pulse shape does not apply to giant pulses and a normal least-squares fitting routine can be applied when fitting scattering models. We tried both PBF forms in Equations (1) and (2) and, additionally, also fit a modified exponential function defined as

$$\begin{eqnarray}&&{g}_{2}(t)={t}^{\gamma }\exp \left[-\displaystyle \frac{t}{\tau }\right]u(t),\end{eqnarray} \tag{ 3 }$$

where γ is a free parameter in the range 0–1 describing the rise time of the pulse; u(t) is as defined for Equation (1). Both Karuppusamy et al. (2012) and Ellingson et al. (2013) have made use of this PBF and found that this functional form fits their data better at low frequencies. We restrict our analysis to the lower three bands where the signal-to-noise ratio (S/N) is highest (Figure 1). The data and associated fits are shown in the left column of Figure 5 and the measured scattering delays along with the reduced ${\chi }_{\mathrm{red}}^{2}$ values are summarized in Table 5.

Table 5.  Measured Scattering Delays for the Crab Pulsar and Implied Frequency Scalings

	Square Law Structure Function						Fully Diffractive Kolmogorov Turbulence
Frequency	Thin		Thick		Modified thin		Thin		Thick		Double thin
(MHz)	τ (ms)	${\chi }_{\mathrm{red}}^{2}$	τ (ms)	${\chi }_{\mathrm{red}}^{2}$	τ (ms)	${\chi }_{\mathrm{red}}^{2}$	τ (ms)	${\chi }_{\mathrm{red}}^{2}$	τ (ms)	${\chi }_{\mathrm{red}}^{2}$	τ (ms)	${\chi }_{\mathrm{red}}^{2}$
121	58(1)	0.97	44.9(9)	0.85	40(2)	0.95	47(13)	1.26	46(2)	1.16	47(13)	1.26
165	18.3(2)	1.06	12.8(1)	1.08	11.9(2)	1.01	14(2)	1.44	13.2(3)	1.47	14(2)	1.52
210	8.4(1)	1.16	5.72(6)	1.01	4.66(9)	1.08	6.4(9)	1.58	6.75(7)	1.40	6.3(9)	1.84
α	−3.5(1)		−3.8(2)		−3.9(1)		−3.7(2)		−3.6(3)		−3.7(2)

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Figure 5 (right column) also contains the results of numerical fits where we employed PBFs that are based on a fully diffractive Kolmogorov model of turbulence (e.g., Lambert & Rickett 1999). This work is motivated by the fact that none of the above three models (Equations (1)–(3)), which are based on the assumption of a square-law structure function of the ISM, fits the data well across all three bands. Moreover, all three models assume that the scattering occurs in only one location along the path. For the Crab pulsar it has been suggested that there may be two distinct contributing scattering locations, namely material within the Crab Nebula itself and the ISM between Earth and the pulsar (Vandenberg 1976). Therefore, we are motivated to explore a set of models in which the scattering occurs at multiple locations along the ray path. In particular, we investigate the case in which the scattering occurs on multiple thin screens. It may be shown that the multiple thin screen geometry gives rise to a scattering kernel whose shape is the convolution of the scattering kernels of the two screens individually.

The scattering delays that we measure for each of the six models indicate a frequency scaling −4.0 < α < −3.4 (Table 5). This is slightly steeper than but consistent with the the results of, e.g., Meyers et al. (2017) and Eftekhari et al. (2016), and in excellent agreement with a global scattering index $\langle \alpha \rangle =-3.9\pm 0.2$ determined by Bhat et al. (2004). The parameter γ in Equation (3) evaluates to 0.18 ± 0.02 at all frequencies. This is in good agreement with the values found by Karuppusamy et al. (2012) and lower but still within three standard deviations of the results found by Ellingson et al. (2013).

4.1. PSR J0742−2822

The work of Johnston et al. (1998) suggested that the interstellar scattering along this line of sight is dominated by a thin screen located within the Gum Nebula. Our measurements are well in line with this notion as the FOM that we compute (Table 3) favor the thin screen model over the thick screen one. For all eight frequency bands considered in the analysis, both f_r and ${\chi }_{\mathrm{nCC}}^{2}$ are lower for the thin screen model than the thick screen one, indicating a better fit to the data. The thin screen model yields a scaling index α = −2.5 ± 0.6 that is in agreement with previous work (e.g., Lewandowski et al. 2015). However, the measured scattering delays τ and the frequency scaling α = −2.9 ± 0.7 as obtained from assuming a thick screen model agree with the thin screen estimates within one standard deviation.

One can also compare the recovered deconvolved pulse shapes between models with those obtained at higher frequencies. We show our profiles in Figure 4 (left), which hint at a double-peaked pulse regardless of the assumed underlying model for scattering. Similar double-peaked profiles with matching component separations are evident at higher frequencies for this pulsar (e.g., Kramer 1994; Zhao et al. 2017, and references therein). However, as the S/N in the individual subbands is low, these reconstructed pulse shapes should be considered as first-order estimates only. The measured scattered profiles (Figure 3, left), on the other hand, are very reliable as they were constructed from about 1800 individual pulses.

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Furthermore, we can consider the scintillation properties at higher frequencies implied by a certain frequency scaling. For diffractive scintillation, the decorrelation bandwidth Δν_d is related to the scatter broadening time τ via the relation Δν_d = C/(2πτ), where C is of order unity (Lambert & Rickett 1999, C ranges from 0.65 to 1.2 depending on assumed geometry and underlying density spectrum). For PSR J0742−2822 an α = −2.5 (or α = −2.9 for the thick screen) would predict a decorrelation bandwidth Δν_d ≈ 0.1 MHz (Δν_d ≈ 0.2 MHz, thick screen) at 4.8 GHz that is only about an order of magnitude lower than what has been measured by Johnston et al. (1998) at that frequency (Δν_d = 8.83 MHz). Considering the time variability of scintillation and the uncertainties of our measurements our results are well in line with these earlier findings.

4.2. The Vela Pulsar

4.3. The Crab

None of the models that we fitted to the brightest giant pulse in our data set reproduce the observed pulse shapes well across all three frequency bands (Figure 5). They generally fail to capture the rise times correctly, with the model for a thick scattering screen describing the leading edge best, i.e., the residuals show nearly no deviation from flatness in this particular time range of the pulse. However, this model, like all others, does not reproduce the shape of the scattering tail in a way that the residuals show no systematic offset from zero. At a central frequency of 165 MHz the model for a thick scattering screen decays too steeply, i.e., it does not account for all the power still present right after the peak of the pulse, while at 210 MHz this model predicts more power at later times in the tail than is actually present. The latter might indicate the presence of an inner scale of turbulence as observed by, e.g., Rickett et al. (2009).

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Similar to the analytic models, none of the numerical models we employed appropriately characterize the immediate rise time of the pulse. Compared to the analytic models, the residuals of the single thin and double thin models are, however, somewhat flatter both right after the peak of the pulse and at later times in the scattering tail. We interpret this as an indication that, compared to the analytical models, the numerical models for thin screens better account for both the small-scale scattering structures as well as the larger-scale structures.

The fitting results of both the analytic and the numerical approach are strongly influenced by the fact that the scattering tail extends beyond a single pulse period. This becomes evident in Figure 6 where we plot the raw data in all three subbands alongside a five millisecond running average. In all three subbands one can identify an increase in power in the scattering tail at intervals equal to the pulse period. We interpret these peaks as pulses succeeding the giant pulse, ultimately leading to a measurement of the scatter broadening time that is larger than that induced by the ISM.

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Similarly, the influence of subsequent pulses makes it virtually impossible to distinguish between a simple exponentially decaying scattering tail and one that decays slower as is expected for Kolmogorov turbulence (Lambert & Rickett 1999).

We observed the three pulsars J0534+2200 (the Crab pulsar), J0742−2822, and J0835−4510 (the Vela pulsar) across the entire frequency range available with the MWA (80–300 MHz). We employed both analytical and numerical models for interstellar scattering to measure the pulse scatter broadening times induced by the ISM between frequencies of 148 and 256 MHz. Based on these measurements we were able to estimate the frequency scaling of interstellar scattering implied by the different models, which generally agree within their uncertainties despite the fact that the measured τ differ significantly between models. We find that both a thin and a thick screen model are consistent with our data, albeit a preference for a thin screen geometry for both J0742−2822 and J0835−4510. The thin screen models imply frequency scalings α = −2.5 ± 0.6 and α = −4.0 ± 1.5 for J0742−2822 and the Vela pulsar, respectively. These numbers are in agreement with previous publications (e.g., Lewandowski et al. 2015). In the case of the Crab pulsar our data indicate that none of our employed models represent the data accurately across all observed frequencies, with the model for a thick scattering screen fitting the rise times of the analyzed giant pulse best. The frequency scaling implied by this model is α = −3.8 ± 0.2. Overall, the spectral scaling indices we measure are all shallower than what is expected for pure Kolmogorov turbulence (α = −4.4). The existence of an inner scale of turbulence could at least partially explain this discrepancy (e.g., Bhat et al. 2004; Rickett et al. 2009).

The scattering delays that we measured in these short, multiband, single-epoch observations with the MWA fit very well into what has been measured previously (Figure 7). In these previous works, the large scatter of the measured delays at similar frequencies is most likely caused by combining data from (i) separate observations spanning long time ranges, (ii) multiple dissimilar telescopes, and (iii) varying techniques to measure the scattering delay. Effectively, this will not only affect the measured delays but also the implied frequency scaling index of pulsar scattering, α. In this work, we accomplished these measurements from relatively short, single-epoch observations with the MWA alleviating all of the above difficulties. This is a testimony to the sensitivity and flexibility of the MWA and it underscores the importance of pulsar studies at low radio frequencies.

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We thank the two anonymous referees for insightful reviews and for several comments that helped to improve the content and presentation in this paper. F.K. acknowledges funding through the Australian Research Council grant DP140104114 and from the Swedish Research Council. N.D.R.B. acknowledges the support from a Curtin Research Fellowship (CRF12228). The authors acknowledge the contribution of an Australian Government Research Training Program Scholarship for supporting this research. This scientific work makes use of the Murchison Radio-astronomy Observatory, operated by CSIRO. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site. Support for the operation of the MWA is provided by the Australian Government (NCRIS), under a contract to Curtin University administered by Astronomy Australia Limited. We acknowledge the Pawsey Supercomputing Centre, which is supported by the Western Australian and Australian Governments. Parts of this research were conducted by the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project No. CE110001020.

Facility: MWA. -

Software: PSRCHIVE, DSPSR, PYTHON, MATLAB.

Here we summarize the definitions of the FoM as defined in Bhat et al. (2003). An example of how we chose the best-fit model is shown in Figure 8.

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The parameter f_r is defined as

$$\begin{eqnarray*}&&{f}_{r}=\displaystyle \frac{1}{N{\sigma }_{\mathrm{off}}^{2}}\displaystyle \sum _{i=1}^{N}{\left[{\rm{\Delta }}y({t}_{i})\right]}^{2}{U}_{{\rm{\Delta }}y},\end{eqnarray*}$$

where U_Δy is the unit step function defined such that

$$\begin{eqnarray*}{U}_{{\rm{\Delta }}y}=\left\{\begin{array}{ll}1 & \mathrm{if}\,({\rm{\Delta }}y({t}_{i})+1.5{\sigma }_{\mathrm{off}})\lt 0,\\ 0 & \mathrm{otherwise},\end{array}\right.\end{eqnarray*}$$

and N is the total number of time bins t_i during which the pulse is "on." σ_off denotes the off-pulse rms noise while the parameter Δy(t_i) denotes the value of time bin t_i in the CLEAN residual (i.e., the residual profile after subtraction of all CLEAN components). Effectively, f_r is a measure for positivity, i.e., how much oversubtraction occurs given the CLEANed pulse of a certain model.

N_r = N_f/N_tot denotes the relative number of points N_f that satisfy $| {y}_{i}-\langle {y}_{\mathrm{off}}\rangle | \leqslant 3{\sigma }_{\mathrm{off}}$, where $\langle {y}_{\mathrm{off}}\rangle$ denotes the off-pulse mean signal and N_tot are the total number of points in the profile.

σ_r = σ_offc/σ_off is the ratio between the rms of the residual after deconvolution and the off-pulse rms in the scattered profile. It is a relative measure for how noise-like the residual of the deconvolved pulse is.

The skewness Γ is a measure for the symmetry of the CLEANed profile and is computed as

$$\begin{eqnarray*}\begin{array}{rcl}{\rm{\Gamma }} & = & \displaystyle \frac{\langle {t}^{3}\rangle }{\langle {t}^{2}{\rangle }^{3/2}},\mathrm{with}\\ \langle {t}^{n}\rangle & = & \displaystyle \frac{{\displaystyle \sum }_{i=1}^{{n}_{c}}{\left({t}_{i}-\bar{t}\right)}^{n}{C}_{i}}{{\displaystyle \sum }_{i=1}^{{n}_{c}}{C}_{i}},\\ \bar{t} & = & \displaystyle \frac{{\displaystyle \sum }_{i=1}^{{n}_{c}}{t}_{i}{C}_{i}}{{\displaystyle \sum }_{i=1}^{{n}_{c}}{C}_{i}}.\end{array}\end{eqnarray*}$$

Here, the summation parameter i = 1, ..., n_c runs through all CLEAN components of amplitudes C_i and associated times t_i.