THE CRAB PULSAR AT CENTIMETER WAVELENGTHS. I. ENSEMBLE CHARACTERISTICS

At low radio frequencies (below ∼5 GHz) and at high energies (optical, X-ray, γ-ray), the mean profile of the Crab pulsar shows two strong features, generally known as the Main Pulse and the Interpulse. The phase separation of the Main Pulse and Interpulse remains approximately constant, ∼140° of rotation phase (radio, Moffett & Hankins 1996; optical, Słowikowska et al. 2009; γ-rays, Abdo et al. 2010). The fact that both radio and high-energy components appear at approximately the same phase strongly suggests the regions emitting both components are located close to each other in the star's magnetosphere.

1.1.1. Radio Models

1.1.2. High-energy Models

1.1.3. Alternatives and Complications

This is not yet the entire story of radio emission from the Crab pulsar. Its mean radio profile is complex, with seven separate components identified so far. We list these in order of increasing phase in Table 1, following the nomenclature of Moffett & Hankins (1996).

The Crab pulsar is unusual in having such a large number of components spread throughout the rotation period. While there are many pulsars whose mean profiles contain multiple components (e.g., PSR B1237+25 which has five components, Hankins & Wright 1980) those components are typically bunched closely together in phase and are generally considered part of a single mean-profile component (for instance within the Main Pulse). By contrast, the seven components found in the Crab pulsar are widely spaced in phase, extending throughout the star's full rotation period. There are also a few pulsars, such as PSR B0826−34 and PSR B1929+10, which emit over most or all of their rotation period. While some authors (e.g., Biggs et al. 1985) have suggested the magnetic axis of such stars is oriented close to the line of sight (so-called aligned rotators), this picture is not necessarily consistent with the observed position angle behavior (e.g., Rankin & Rathnasree 1997).

While the mapping from magnetospheric position to rotation phase is far from linear, the wide phase spread of the Crab's components suggests that many separate radio emission sites exist throughout the star's magnetosphere. We have not seen much discussion in the literature of where in the magnetosphere these components may arise. It has occasionally been suggested that the Precursor is generated at low altitudes near the pulsar surface in the open-field-line region of the polar cap (e.g., Lyne et al. 2013). This suggestion agrees with caustic models which locate Main Pulse emission, both radio and high-energy, at high altitudes (e.g., Harding et al. 2011; low-altitude emission arising close to the polar cap will lead the high-altitude Main Pulse emission by several degrees of phase. Both Hankins & Eilek (2007) and Lyutikov (2008) speculated that the High-Frequency Interpulse might arise from high altitudes, close to the light cylinder, but these ideas have not been developed far enough to test against mean profiles. We are not aware of any suggestions for the spatial locations of the other components (the two High-Frequency Components and the Low-Frequency Component). While recent studies of pulsar magnetospheres (e.g., Li et al. 2012; Kalapotharakos et al. 2012; Contopoulos & Kalapotharakos 2010) do not specifically model the radio profile, the extended current sheets and dissipation regions revealed by their simulations seem very likely to be sources of coherent radio emission.

Our primary focus in this paper is the phenomenology of the mean radio profile of the Crab pulsar. We present continuously sampled data, recorded between 9 and 43 GHz at sub-microsecond time resolution. We use these data to derive and characterize mean profiles at these high radio frequencies, and also identify and characterize the statistics of bright single pulses within the data streams. In this paper we use the term "mean profile" to describe the intensity as a function of rotation phase. This quantity is derived by summing the received intensity synchronously with the star's rotation period. The same data product is also often called the "light curve," especially in the high-energy community.

We have also recorded strong individual pulses between 0.33 and 43 GHz, at time resolution down to a fraction of a nanosecond (Hankins & Eilek 2007; Crossley et al. 2010). These occasional strong pulses emitted by the Crab pulsar led to its discovery (Staelin & Reifenstein 1968) long before its periodicity was determined (Comella et al. 1969). These pulses have been called "giant pulses" because their intensities are far greater than the typical pulse. However, there is no standard criterion for labeling these single pulses as "giant," and in fact the distribution of pulse amplitudes in the Crab pulsar appears to be continuous from weak to strong (e.g., Karuppusamy et al. 2010). In this work we will therefore just refer to "single pulses," meaning those bright enough to be clearly discernible above the system noise. Such pulses are relatively rare; in most rotation periods the pulses are below the noise level. When we do detect strong, single pulses, they occur at the phase of the Main Pulse and the Interpulse, and very occasionally at the phases of the High-Frequency Components.

In this paper we present the ensemble characteristics of the Crab pulsar's radio emission revealed by our data. We describe our observations in Section 2: we extend up to 43 GHz previous studies at lower radio frequencies, such as that of Cordes et al. (2004). In Section 3 we present the composite results: mean profiles as a function of frequency and the phases and widths of the components of the mean profile. In Section 4 we present ensemble characteristics of individual Mean Pulses and Interpulses: their arrival phase and width, their fluence distributions, and the "burstiness" of arrival times. In Section 5 we summarize our results and discuss the constraints they place on current and future models of the star's magnetosphere.

In a separate paper to follow (T. H. Hankins, et al. 2015, in preparation, Paper 2), we extend our previous single-pulse work (Hankins et al. 2003; Hankins & Eilek 2007; Crossley et al. 2010) to higher frequencies and more components, by presenting new observations of bright single pulses at the phases of the Main Pulse, the Low-Frequency and High-Frequency Interpulses, and the two High-Frequency Components.

We have extended the radio mean profiles of Moffett & Hankins (1996) up to 28 GHz in order to test for the frequency dependence of the spacing and width of the mean profile components. In Figure 1 we show our new profiles aligned with a selection of previously published profiles. As insufficient timing information was available, and because the observations were made at widely separated epochs, we have made two choices to align the profiles. For frequencies ≲5 GHz, we have set the Main Pulse phase at 0°. For frequencies ≳5 GHz we have assumed that the High-Frequency Interpulse phase is fixed at 13756 after the center of the Main Pulse. This phase separation is based on several data sets which show a detectable Main Pulse as well as a High-Frequency Interpulse at ν > 5 GHz. Although the mean profile of the Main Pulse is very weak above ∼5 GHz, we have detected occasional single pulses at the phase of the Main Pulse at all frequencies observed, up to 43 GHz (as in Section 4.1).

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Figure 1 shows that the nature of the mean profile changes dramatically between low and high radio frequencies, as follows.

• 1.  
  The mean profile shape changes abruptly at ∼5 GHz. The Main Pulse, which dominates the mean profile at low radio frequencies, nearly disappears above ∼5 GHz. The high-frequency mean profile is dominated by the Interpulse and the two High-Frequency Components.
• 2.  
  The Interpulse undergoes a sudden, discontinuous phase shift of ∼7° around 5 GHz. This transition coincides with the frequency above which the spectral and temporal characteristics of the Interpulse change dramatically (Hankins & Eilek 2007).
• 3.  
  The two High-Frequency Components can be identified in the mean profile from 1.4 to 28 GHz. Both components drift in phase, appearing at later phases at higher frequencies.
• 4.  
  The Precursor can be identified in mean profiles at 0.43 and 0.61 GHz, but is not seen at any higher frequency. Similarly, the Low-Frequency Component can be seen in mean profiles from 0.61 to 4.2 GHz, but is not seen at higher frequencies.

We simultaneously fitted single Gaussians to each of the profile components in Figure 1, ignoring components that are below the profile noise level. The phase of a component is defined as the peak of the fitted Gaussian. The phase uncertainties are the quadrature sums of the 1σ Gaussian centroid positions as obtained from the formal fitting procedure (Press et al. 1986). The sample point weighting in our least-squares fitting procedure is based on the signal-to-noise ratio (S/N), so that profiles with higher S/N dominate the fits.

In Figure 2 we show the phases of each of the fitted components. As in Section 3.1, we assume the Main Pulse has a phase of ϕ_MP = 0°, and the High-Frequency Interpulse has a phase of ϕ_HFIP = 13756. For the two High-Frequency Components we computed a fit of phase versus log(frequency), $\phi (\nu )=a+b{\rm log} \nu$, with ν measured in GHz. We tried fitting polynomials of the form $\phi ={{c}_{0}}+{{c}_{1}}\nu +{{c}_{2}}{{\nu }^{2}}$ but found that the χ² error of the polynomial fit to be much larger. We present the fitting coefficients a and b in Table 2, along with the extrapolated values of the phases at a reference frequency of 4 GHz. Before fitting for ${{\phi }_{{\rm LFIP}}}(\nu )$ and ${{\phi }_{{\rm LFC}}}(\nu )$ we extrapolated their observed epoch phases to MJD 53000 using frequency interpolated values of dϕ/dt from Lyne et al. (2013). The resulting secular phase changes are in all cases smaller than our fitting uncertainties.

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These results quantify the trends apparent in Figure 1. The Low-Frequency Interpulse shows very little phase drift with frequency. The phase separation between the Low-Frequency Interpulse and the Main Pulse, measured from our mean profiles, varies by only ∼11 between 430 MHz and 3.5 GHz. Similarly, inspection of our single-pulse data showed no evidence for phase drift with frequency between the Main Pulse and the High-Frequency Interpulse. Except for the ∼7° phase jump when the Low-Frequency Interpulse disappears and the High-Frequency Interpulse appears, each Interpulse remains at a nearly constant phase relative to the Main Pulse.

Hankins & Fowler (1986) measured the separation between the Main Pulse and the Interpulse from 0.196 to 2.695 GHz. Their phase data and frequency-dependent fits to the separations are consistent with the current work. They obtained ${{\phi }_{{\rm LFIP}}}(1\;{\rm GHz})=145\buildrel{\circ}\over{.} 9\pm 0\buildrel{\circ}\over{.} 6$ with a frequency exponent of b = −0.68 ± 0.40. This compares well with our ${{\phi }_{{\rm LFIP}}}(1\;{\rm GHz})=145\buildrel{\circ}\over{.} 36\pm 0\buildrel{\circ}\over{.} 06$ with a frequency exponent of b = −0.77 ± 0.12. The secular change of ${{\phi }_{{\rm LFIP}}}$ found by Lyne et al. (2013) over the approximately 20 yr from Hankins and Fowler's observations to Lyne's MJD 53000 reference date is far smaller than the quoted separation estimation error of Hankins & Fowler (1986).

In the frequency transition region between the Low-Frequency Interpulse and the High-Frequency Interpulse, we measured at 4.15 GHz, the Main Pulse to Interpulse separation as 1391 ± 05, intermediate between the phases of the Low-Frequency and High-Frequency Interpulses. Inspection of single-pulse data in this frequency range shows that both types of Interpulses can occur around ∼4 GHz; we thus interpret the intermediate phase of the mean-profile Interpulse at 4.15 GHz as the result of this mixture.

Similarly, neither the Precursor nor the Low-Frequency Component show any strong phase drift over the frequency range in which they appear in our mean profiles. However, the phases of the two High-Frequency Components increase with frequency; at 28 GHz both High-Frequency Components lag their 1.4 GHz counterparts by more than 20°.

The approximate constancy of the Main Pulse-Interpulse phase separation agrees with several other interpulse pulsars (e.g., Hankins & Fowler 1986). However, those authors found that the Main Pulse-Interpulse separation in PSR B1944+17 changes significantly, by ∼15° between 0.43 and 2.4 GHz, reminiscent of the phase drift we find in the High-Frequency Components of the Crab pulsar. Based on the phase shift in PSR B1944+17, in addition to polarization anomalies and synchronous nulling in the Main Pulse and Interpulse, Hankins & Fowler (1986) speculated the radio emission is either from a single magnetic pole or from high altitudes in that star. Might this also be the case for the Crab pulsar?

The Gaussians we fitted to the components in the mean profile for phase determination can also be used to characterize the width of each component. We show in Figure 3 the measured FWHM of each fitted Gaussian, as a function of frequency. To quantify the relation between width, w, and frequency, we fitted a linear function in log space to the data points: ${\rm log} w(\nu )=\alpha +\beta {\rm log} \nu$, where ν is the observing frequency in GHz. We present the fitting coefficients α and β in Table 3, along with the widths at 4 GHz, evaluated as $w(4\ {\rm GHz})={{10}^{\alpha }}{{\nu }^{\beta }}$.

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The measured widths of the components are, of course, affected by data acquisition systematics, which vary from observation to observation. These include the instrumental time resolution, the dedispersion technique and the value of dispersion measure used, the accuracy of the pulsar ephemeris used to calculate the pulse periods used to form the mean profiles, and, particularly below 1 GHz, the variable scattering broadening in the Crab Nebula and the interstellar medium. An upper estimate of the component width broadened by scattering can be obtained by scaling the values for the scattering decay times, τ_D, measured by Kuzmin et al. (2008) at 111 MHz over 2.5 years, as ν⁻⁴, and multiplying the result by 2.46 to convert τ_D to w, the component FWHM (Although some authors assume Kolmogorov turbulence for the general ISM, which would imply ν^−4.4, the low-frequency pulse broadening of the Crab pulsar is more consistent with Gaussian turbulence; e.g., Crossley et al. 2010). Using the largest value of their observed range, $8\lt {{\tau }_{{\rm D}}}\lt 26$ ms, we get for our lowest observation frequency $w(430\ {\rm MHz})\approx 2.46*26*{{(430/111)}^{-4}}=0.28$ ms, or about 31 of pulse phase, which is slightly larger than the size of the plot symbols in Figure 3. Since none of our observations were made during a known scattering event, we conclude that our width measurements are not strongly biased by scattering broadening.

Despite these uncertainties, we find there is very little change of the component widths over the observed frequency range. The β coefficients we find are significantly non-zero only for the Low-Frequency Interpulse and the High-Frequency Component 2, and we note that the formal fit to the Low-Frequency Interpulse is strongly influenced by two very high S/N points at 0.43 and 1.4 GHz. Thus, only High-Frequency Component 2 shows clear evidence for width changing with frequency.

For many pulsars the mean profile components are broader at low frequencies. For instance, Rankin (1983) fitted the half-power component widths below ∼1 GHz in several pulsars as $w(\nu )\propto {{\nu }^{\beta }}$, with β ranging from −0.5 to nearly zero. This "radius-to-frequency mapping" is often interpreted as lower-frequency radio emission coming from higher altitudes within the open field line region (but still close to the star's surface). At higher frequencies, however, component widths tend to stabilize (e.g., Rankin 1983; Thorsett 1992). Our finding of frequency-independent widths for most mean-profile components of the Crab pulsar agree with this general trend. We are not aware of any other pulsar with a mean-profile component that broadens at higher frequencies, as High-Frequency Component 2 does in the Crab.

In addition to the mean intensity profile, we used the GUPPI system to track the occurrence rate of pulses strong enough to be detected as single pulses (above the system noise, typically 6σ). In Figure 4 we combine our new data (above 5 GHz) with data from Cordes et al. (2004) to show the frequency evolution of pulse-count histograms of the arrival phase of individual bright pulses. This data product highlights strong, infrequent single pulses which may be hidden in, or distributed differently from, the mean intensity profile (Figure 1).

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The histograms in Figure 4 demonstrate that Main Pulses occasionally occur well above 5 GHz, even though they are too infrequent to be detected in the intensity-weighted mean profiles in Figure 1. In particular, Main Pulses are evident at 9 and 14 GHz, and a few were detected at 28 GHz. Interpulses occur up to 43 GHz, the highest frequency we used. In Hankins & Eilek (2007) are shown individual examples of Main Pulses captured with our UHTRS at 9 GHz; in Paper 2 we show examples of single Main Pulses and Interpulses detected above 10 GHz, including one Main Pulse recorded at 43 GHz. We note that both Main Pulses and Interpulses are so rare and weak at 43 GHz that the mean profile does not show either component; thus we did not include 43 GHz data in Figure 1.

The histograms in Figure 4 show no significant detections at the phases of any of the other pulse components. The non-zero histogram values outside the nominal phases of the Main Pulse and Interpulses result from noise spikes or interference that exceeded the S/N threshold.

The pulse-count histograms in Figure 4 clearly show the striking difference between the count rates of the Main Pulse and those of the two Interpulses. At low frequencies the Main Pulse dominates, but above ∼5 GHz—the frequency range where the Interpulse changes nature—the Interpulse dominates the count rates, just as it does the intensity-weighted mean profile in Figure 1.

This can also be shown in terms of count rates, expressed as pulses/second. In Figure 5 we combine lower-frequency results from the literature with our current high-frequency data, to show the measured rates of the Main Pulse and two Interpulses, N_MP and N_IP. Although the measured count rates depend strongly on the observational systematics, such as telescope sensitivity and the Crab Nebula background level, the ratio ${{N}_{{\rm IP}}}/{{N}_{{\rm MP}}}$ is much less sensitive to these effects.

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Figure 5 shows that bright Main Pulses are somewhat more frequent than bright Low-Frequency Interpulses below ∼5 GHz, but the arrival rates differ only by a factor of a few. However, above ∼5 GHz, the story changes. Bright Main Pulses become increasingly rarer; we typically caught only one every couple of minutes. However, High-Frequency Interpulses are much more frequent; we typically detected them 30–1000 times more often than Main Pulses in our continuously sampled observations between 9 and 43 GHz. This statistic, ${{N}_{{\rm IP}}}/{{N}_{{\rm MP}}}$, is somewhat sensitive to the amplitude distributions of the single pulses (discussed below in Section 4.4); but when the rare Main Pulses above 10 GHz are detected, they are certainly no weaker than the High-Frequency Interpulses and sometimes much stronger.

Pulse-count histograms (Figure 4) and mean profiles (Figure 1) do not tell the full story of the emission physics in the Crab pulsar, because individual pulses emitted at the phase of a given component are generally much narrower than the ensemble width of the component. Our UHTRS observations, obtained at time resolutions from 100 μs down to 0.25 ns, allow us to resolve individual bright pulses, and thus characterize their duration, over a wide frequency range. We have recorded single pulses at the VLA at 0.33, 1.4, 4.8, and 8.4 GHz with 10 to 100 ns time resolution (partial results reported in Crossley et al. 2010); at the Arecibo telescope with frequency bands 4–6, 6–8, and 8–10.5 GHz and 0.4 ns time resolution (partial results reported in Hankins & Eilek 2007); and at the GBT with frequency bands 8–10, 12–15.6, 18–22.4, 22–26.5, 28–32, and 41–46 GHz, with 0.25 ns time resolution (as reported in this paper, and in more detail in Paper 2 to follow).

Because the time signature of individual Main Pulses is quite different from that of individual High-Frequency Interpulses (Hankins & Eilek 2007), we need a measure of width that is not overly sensitive to the detailed pulse structure. We choose to calculate the equivalent width (EW) of the pulse intensity. After experimentation, we found that the EW of the intensity and the EW of the intensity autocorrelation function (ACF) give similar results, differing only by a small multiplicative factor. (The intensity ACF shows a "zero-lag spike" due to the noisy structure of the pulse and the contribution of the receiver and sky noise. Therefore, the ACF zero-lag amplitude is strongly dependent upon the S/N (Rickett 1975). If used in the calculation of ACF EW, the zero-lag value would strongly bias the width estimate to narrower values. Therefore, for our analysis we truncated the ACF value at zero lag and replaced it by fitting a parabola to adjacent lags before computing the ACF EW.) In Table 4 we present the number of pulses captured with the UHTRS at each frequency and the mean intensity EW at each frequency. In Figure 6 we show the the EW of the pulse intensity against frequency, averaged over all pulses recorded at each frequency. Below 2 GHz our mean EWs for the Main Pulse are consistent with the range of single pulse widths presented in Crossley et al. (2010) and Majid et al. (2011).

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Comparing these data to the durations of the mean-profile components (Table 3 and Figure 3) shows a key result: individual bright pulses are much shorter than the duration of their associated mean-profile component. From Section 3.3 we know the width of the Gaussian fit to the mean profile of the Main Pulse, at a few GHz, is ∼370 μs. For comparison, single bright Main Pulses last at most only ∼20 μs, and usually less, over the same frequency range. Similarly, the width of the High-Frequency Interpulse in the mean profile is ∼730 μs between 5 and 30 GHz, but the width of individual bright Interpulses is no more than ∼2 μs, and usually less, in this frequency range. This result makes it clear that the mean-profile components do not arise from spatially extended regions with uniform radio emissivity. Rather, the mean profile components are an envelope inside of which short-lived bursts occur.

Figure 6 also shows that the bursts which constitute single pulses become shorter at high frequencies. This is true for Main Pulses, Low-Frequency Interpulses and High-Frequency Interpulses. To quantify this behavior, we carried out least-squares fits of the form ${\rm EW}(\nu )\propto {{\nu }^{\alpha }}$. For Main Pulses between 1.24 and 9 GHz, we find α = −1.76 ± 0.13. This fit is approximately consistent with the value of α ∼ −2 given by Crossley et al. (2010) over a small frequency range (see their Figure 6). For Low-Frequency Interpulses in the range 1.4 to 4.8 GHz we find α = −1.38. For High-Frequency Interpulses in the range 6.5 to 24.25 GHz we find α = −1.18 ± 0.06. This fit is approximately consistent with the widths reported for single High-Frequency Interpulses by Hankins & Eilek (2007) over a much narrower frequency range.

We also computed the intensity EWs of Main Pulses we captured at 0.33 GHz, finding the average width to be 400 ± 50 μs. At this frequency the pulse shape is dominated by interstellar and Nebular scattering broadening (e.g., Crossley et al. 2010). We can, however, estimate the intrinsic single-pulse width at this frequency by extrapolating our power-law fit (upper panel of Figure 6) down to 0.33 GHz, which gives ∼200 μs. Scaling the typical scattering broadening time of τ_D(1 GHz) ≃ 2 μs at 1 GHz (Crossley et al. 2010, Figure 6) by ν⁻⁴ down to 0.33 GHz we obtain ${{\tau }_{{\rm D}}}(0.33\ {\rm GHz})\simeq 170\ \mu {\rm s}$. If the 200 μs intrinsic pulse is convolved with a 170 μs scattering function then the expected observed pulse width should be on the order of $\sqrt{{{200}^{2}}+{{170}^{2}}}\approx 260\ \mu {\rm s}$, which is not inconsistent with our width measurement.

We used the continuously sampled GUPPI data to determine the fluence distribution of High-Frequency Interpulses at each frequency we observe between 9 and 43 GHz, averaging over two-second time bins. (There were not enough Main Pulses in our data to provide reliable statistics). As is conventional, we fitted our fluence distributions with power laws. In Figure 7, also in Table 5, we show the distribution of power-law indices of our data from 9 to 30 GHz along with power-law indices from lower frequency observations from the literature.

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There are, of course, many effects that can bias the indices. In addition to different systematic effects which can change with different observations, and the difficulty of establishing true power-law behavior when only a small range of fluence is sampled, the fluence distribution can vary with frequency and between the Main Pulse and the two Interpulses. For example, Popov & Stappers (2007), working at 1.197 GHz, found that when the giant pulses are sorted according to their width, the Main Pulse distributions are best fit by two power-law indices, and the Interpulse by one. Also Karuppusamy et al. (2010), working at 1.38 GHz, found the amplitude distributions are power-law only at the higher intensity tail of the distributions. Therefore, we do not interpret these power-law indices as proof of true power-law behavior of the fluence distribution, but rather as a general indication of the relative abundance of stronger or weaker pulses in that distribution.

Nonetheless, despite these uncertainties, Figure 7 shows that the Main Pulse and both Interpulses share a modest trend. As also noted by Cordes et al. (2004) and Crossley et al. (2010), there seem to be relatively fewer "super-giant" pulses at higher frequencies in each of these components.

We also used the continuously sampled GUPPI data to study the arrival statistics of bright single pulses, at each observing frequency between 9 and 28 GHz. The different frequencies were not simultaneous, because they were observed on different days, but each Main Pulse/Interpulse time series at a given frequency was extracted from data taken on one observing run. In Figure 8 we present three metrics to characterize the arrival statistics on timescales from 2 to 10,000 s.

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4.5.1. The Data: Fluctuating Arrival Rates and Fluences

4.5.2. The Fluctuations: Intrinsic or Interstellar?

Rapid fluctuations such as seen in the left panel of Figure 8 are often ascribed to diffractive interstellar scintillation (DISS) in the strong scattering regime (e.g., Cordes & Rickett 1998). Because we have the intensity time series over a wide range of wavelengths, we can determine whether DISS is the cause of the burstiness we see between 9 and 28 GHz in the Crab pulsar.

In Figure 9 we show the ACFs of each time series shown in the left panel of Figure 8. The DISS timescale, ${{\tau }_{{\rm DISS}}}$, is typically defined as the half-power point of this ACF (e.g., Cordes et al. 2004). Direct inspection of the ACFs in Figure 9 shows that ${{\tau }_{{\rm DISS}}}$ fluctuates around ∼50 to 100 s over this range, but does not show any systematic increase with observing frequency ν. This behavior is inconsistent with standard DISS theory (e.g., Rickett 1975), which predicts that ${{\tau }_{{\rm DISS}}}(\nu )\propto {{\nu }^{x}}$, where x = 1.0 for Gaussian turbulence. Thus, ${{\tau }_{{\rm DISS}}}$ should increase by a factor of ∼3 over our 9 to 28 GHz observed frequency range—which is not the case. This suggests that the burstiness we observe is intrinsic to the pulsar.

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We can also make some simple scaling estimates, following methods in Cordes & Rickett (1998) also B. Rickett (2014, private communication). The exponential decay constant at 330 MHz, ${{\tau }_{{\rm D}}}\sim 200\ \mu {\rm s}$, is thought to be due to interstellar scintillation (e.g., Crossley et al. 2010). If this is the case, we can estimate the interstellar scintillation bandwidth at 330 MHz as $\delta {{\nu }_{{\rm ISS}}}(330\;{\rm MHz})=1/2\pi {{\tau }_{{\rm D}}}\sim 800$ Hz. This quantity is predicted to increase with frequency, as $\delta {{\nu }_{{\rm ISS}}}\propto {{\nu }^{s}}$, where s = 2(x + 2), thus s = 4.0 for Gaussian turbulence. For instance, we expect $\delta {{\nu }_{{\rm DISS}}}\sim 0.34$ MHz at 1.5 GHz, 670 MHz at 10 GHz, and 55 GHz at an observing frequency of 30 GHz. At observing frequencies where our receiver bandwidth, BW, is comparable to or less than the scintillation bandwidth (e. g., $\delta {{\nu }_{{\rm DISS}}}\gt 0.2(BW)$, Cordes & Rickett 1998), burstiness such as we observe could be due to interstellar scintillation. For our 800 MHz GUPPI bandwidth, frequencies ≳7 GHz could be affected by scintillation. However, in addition to the fact that our observed ${{\tau }_{{\rm DISS}}}$ does not show the expected linear increase with frequency, our scaling estimates also suggest the observed burstiness is intrinsic, as follows.

The scintillation bandwidth allows us to predict the fluctuation timescale expected from DISS, as ${{\tau }_{{\rm DISS}}}(\nu )\simeq ({{r}_{{\rm F}}}/v){{(\delta {{\nu }_{{\rm ISS}}}/\nu )}^{1/2}},$ where ${{r}_{{\rm F}}}={{(cz/2\pi \nu )}^{1/2}}$ is the Fresnel scale, z is the distance to the scattering screen, and v is the speed of the pulsar relative to that screen (e.g., Cordes & Rickett 1998). We estimate $v\sim 120$ km/s, from a Hubble Space Telescope measurement of the pulsar proper motion (Kaplan et al. 2008), predicting ${{\tau }_{{\rm DISS}}}\sim 27$ s at 330 MHz, and ∼80 s at 1 GHz. For a sanity check, we can compare our expected fluctuation timescale at 1.5 GHz to the results of Cordes et al. (2004). These authors formed a time series of giant pulses detected from the Crab pulsa, and measured the $1/e$ width of its ACF as ∼25 s. Identifying this timescale as ${{\tau }_{{\rm DISS}}}$ (which they argued should be detectable at 1.5 GHz with their observing bandwidth), they noted their result is a factor of ∼3 shorter than expected from interstellar scintillation. They suggested the discrepancy is due to unusually high speeds of filaments within the Crab Nebula; but intrinsic variability from the pulsar on shorter timescales than ${{\tau }_{{\rm DISS}}}$, may be another possibility.

With these scaling arguments, we predict ${{\tau }_{{\rm DISS}}}\sim 800$ s at 10 GHz, and ∼2500 s at 30 GHz. Because these timescales are significantly longer than the burstiness we observed, we believe the burstiness we observe is not due to interstellar scintillation but rather is intrinsic to the radio emission mechanism in the Crab pulsar.

Our main result is the dramatic change of the radio emission around frequencies of a few GHz as follows.

5.1.1. Low-frequency Radio Profile

5.1.2. High-frequency Radio Profile

5.1.3. Constraints on Future Models

5.1.4. How Sudden is the Change?

5.1.5. Relation of Radio and High-energy Components

5.1.6. Unsteady Radio Emission

5.1.7. Mean-profile Components Contain Short-lived Bursts

5.1.8. Single Pulses at High Time Resolution