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Astrophysical journal
A Preprint typ eset using L TEX style emulateap j v. 05/12/14

PSR B0329+54: STATISTICS OF SUBSTRUCTURE DISCOVERED WITHIN THE SCATTERING DISK ON RADIOASTRON BASELINES OF UP TO 235,000 KM
C. R. Gwinn1 , M. V. Popov2 , N. Bartel3 , A. S. Andrianov2 , M. D. Johnson4 , B. C. Joshi5 , N. S. Kardashev2 , R. Karuppusamy6 , Y. Y. Kovalev1,6 , M. Kramer6 , A. G. Rudnitskii2 , E. R. Safutdinov2 , V. I. Shishov7 , T. V. Smirnova7 , V. A. Soglasnov2 , S. F. Steinmassl8 , J. A. Zensus6 , V. I. Zhuravlev2 (Received January 18, 2015; Accepted March 15, 2016)

arXiv:1501.04449v3 [astro-ph.GA] 15 Mar 2016

ABSTRACT We discovered fine-scale structure within the scattering disk of PSR B0329+54 in observations with the RadioAstron ground-space radio interferometer. Here, we describe this phenomenon, characterize it with averages and correlation functions, and interpret it as the result of decorrelation of the impulseresponse function of interstellar scattering between the widely-separated antennas. This instrument included the 10-m Space Radio Telescope, the 110-m Green Bank Telescope, the 14 â 25-m Westerbork Synthesis Radio Telescope, and the 64-m Kalyazin Radio Telescope. The observations were performed at 324 MHz, on baselines of up to 235,000 km in November 2012 and January 2014. In the delay domain, on long baselines the interferometric visibility consists of many discrete spikes within a limited range of delays. On short baselines it consists of a sharp spike surrounded by lower spikes. The average envelope of correlations of the visibility function show two exponential scales, with characteristic delays of 1 = 4.1 ± 0.3 µs and 2 = 23 ± 3 µs, indicating the presence of two scales of scattering in the interstellar medium. These two scales are present in the pulse-broadening function. The longer scale contains 0.38 times the scattered power of the shorter one. We suggest that the longer tail arises from highly-scattered paths, possibly from anisotropic scattering or from substructure at large angles. Keywords: scattering -- pulsars: individual B0329+54 -- radio continuum: ISM -- techniques: high angular resolution
1. INTRODUCTION

All radio signals from cosmic sources are distorted by the plasma turbulence in the interstellar medium (ISM). Understanding of this turbulence is therefore essential for the proper interpretation of astronomical radio observations. The properties and characteristics of this turbulence can best be studied by observing point-like radio sources, where the results are not influenced by the extended structure of the source, but instead are directly attributable to the effect of the ISM itself. Pulsars are such sources. Dispersion and scattering affect radio emission from pulsars. Whereas dispersion in the plasma column introduces delays in arrival time that depend upon frequency and results in smearing of the pulse, scattering by density inhomogeneities causes angular broadening, pulse broadening, intensity modulation or scintillation, and distortion of radio spectra in the form of diffraction patterns. The scattering effects
1 University of California at Santa Barbara, Santa Barbara, CA 93106-4030, USA 2 Astro Space Center of Leb edev Physical Institute, Profsoyuznaya 84/32, Moscow 117997, Russia 3 York University, 4700 Keele St., Toronto, ON M3J 1P3, Canada 4 Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138, USA 5 National Centre for Radio Astrophysics, Post Bag 3, Ganeshkhind, Pune 411007, India 6 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel ¨ ¨ 69, Bonn 53121, Germany 7 Pushchino Radio Astronomy Observatory, Astro Space Center of Leb edev Physical Institute, Pushchino 142290, Moscow region, Russia 8 Physik-Department, Technische Universit¨ at Munchen, ¨ James Franck-Strasse 1, Garching bei Munchen 85748, Ger¨ many

have already been studied extensively theoretically (see, e.g., Prokhorov et al. 1975; Rickett 1977; Goodman & Narayan 1989; Narayan & Goodman 1989; Shishov et al. 2003) and observationally with ground VLBI of Sgr A (Gwinn et al. 2014) and pulsars (see, e.g., Bartel et al. 1985; Desai et al. 1992; Kondratiev et al. 2007), as well as with ground-space VLBI of PSR B0329+54 (Halca, Yangalov et al. 2001) and the quasar 3C 273 (RadioAstron, Johnson et al. 2016). Whereas the VSOP pulsar observations were done at a relatively high frequency of 1.7 GHz and with baselines of 25,000 km and less, ground-space VLBI with RadioAstron allows observations at one-fifth the frequency, where propagation effects are expected to be much stronger, and with baselines 10 times longer (Kardashev et al. 2013). Such observations can resolve the scatter-broadened image of a pulsar and reveal new information about the scattering medium (Smirnova et al. 2014). In this paper, we study the scattered image of the pulsar B0329+54 with RadioAstron. We demonstrate that the pulsar is detected on baselines that fully resolve the scattering disk. The interferometric visibility on these long baselines takes the form of random phase and amplitude variations that vary randomly with observing frequency and time. In the Fourier-conjugate domain of delay and fringe rate, the visibility forms a localized, extended region around the origin, composed of many random spikes. We characterize the shape of this region using averages and correlation functions. We argue theoretically that its extent in delay is given by the average envelope of the impulse-response function of interstellar scattering, sometimes called the pulse-broadening function. We find that the observed distribution is well-fit by


2

Gwinn et al.

Time Frequency

t

Fringe Rate

f

! V ( , t
1 F-

) )

F

-1 f t

! V ( , f
F


) )

as a function g (te ) of time te at the observer. Here, te is Fourier-conjugate to and varies at the Nyquist rate. The visibility is the product of Fourier transforms of g at the two antennas: ~ VAB = gA gB ~ ~ (2)

Delay

V ( , t

Ft

f

V ( , f

C ( , t ) = V ( , t )

2

K RL ( , t ) = C R ( , t ) C L ( + , t ) = VR ( , t ) VL ( + , t )
2 2



Figure 1. Relations among the interferometric visibility V in various domains, and functions derived from it. The fundamental observable is the visibility in the domain of frequency and time t, ~ V (, t); this is known as the cross-p ower spectrum, or cross sp ectrum. An inverse Fourier transform of to delay leads to the visibility V ( , t); this is the cross-correlation function of electric fields in the time domain (see Equation 21). A forward Fourier transform of t to fringe rate f leads to V ( , f ). A forward trans~ form of back to produces V (, f ), and an inverse Fourier trans~ form of f to t returns to V (, t) The square modulus of V ( , t) is C ( , t). The cross-correlation function in of CR for right- and CL for left-circular p olarization is KRL ( , t). We denote the Fourier transform by F, and quantities in the domain of frequency by the accent ~.

where g is the Fourier transform of g (te ). ~ We denote the typical duration of g (te ) as sc , the broadening time for a sharp pulse. Within this time span, g (te ) has a complicated amplitude and phase. The function g (te ) changes over longer times, as the line of sight shifts with motions of source, observer, and medium. This change takes place on a timescale tsc , and over a spatial scale Ssc . The shorter and longer timescales sc and tsc lead to our use of dual time variables: te , of up to a few times sc and Fourier-conjugate to ; and t, of a fraction of tsc or more and Fourier-conjugate to f . This duality is commonly expressed via the "dynamic spectrum" (see Section A.2). If the scattering material remains nearly at rest while the line of sight travels through it at velocity V , then one spatial dimension in the observer plane maps into time, and t
sc

= Ssc /V



(3)

a model that is derived from an impulse-response function that has two different exponential scales. We discuss possible origins of the two scales.
2. THEORETICAL BACKGROUND Our fundamental observable is the interferometric visibility V . In the domain of frequency , this is the product of electric fields at two antennas A and B : ~ ~ ~ VAB (, t) = EA (, t)EB (, t). (1)

This representation of the visibility is known as the cross spectrum, or cross-power spectrum. Because electric ~ fields at the antennas are complex and different, VAB is complex. Usually visibility is averaged over multiple accumulations of the spectrum, to reduce noise from background and the noiselike electric field of the source. The second argument t allows for the possibility that the visibility changes in time, as it does for a scintillating source, over times longer than the time to accumulate a single spectrum. Such a spectrum that changes in time is known as a "dynamic spectrum" (Bracewell 2000). The correlator used to analyze our data, as discussed in ~ Sections 3 and 4, calculates VAB (, t) (Andrianov et al. 2014). Hereafter, we omit the baseline subscript indicating baseline AB in this paper, except in sections of the Appendix where the baseline is important. Under the assumptions that the source is pointlike, and that we can ignore background and source noise, the impulse-response function of interstellar scattering g determines the visibility of the source. A single deltafunction impulse of electric field at the source is received

The averaged square modulus of g is the pulsebroadening function G = g (te )g (te ) S . Here, the subscripted angular brackets ... S indicate an average over realizations of the scattering. This function is the average observed intensity for a single sharp pulse emitted at the source. An average over time is usually assumed to approximate the desired average over an ensemble of statistically-identical realizations of scattering. We derive a number of representations of the visibility and quantities derived from it, and show that these provide straightforward means to extract the impulseresponse function. These functions are summarized in Figure 1, and discussed briefly here, and in detail in Section A of the Appendix. In particular, visibility in the domain of delay and time t is V ( , t). This is the correlation function of electric field at the two antennas A and B (Equation 21), and is the inverse Fourier trans~ form of V (, t) from to . We are also concerned with the square modulus of V ( , t) (see Section A.3.2): C ( , t) = |V ( , t)|
2

(4)

We calculate C for right- and left-circular polarizations separately, and then correlate them in delay to form KRL , the cross-correlation between polarizations: KRL ( , t) =
1 N

CR ( , t)CL ( + , t)

(5)

Here, KRL is the correlation of a single measurement of CR and CL , and N is the number of samples in CR and CL . When averaged over many realizations of the scattering material, KRL S is related to the statistics of the pulse-broadening function G. Most commonly, the average over many realizations of scattering material is approximated by averaging over a time much longer than tsc ; for this reason we omit the time argument for KRL ( ) S . Equivalently, evaluation of KRL ( , fmax ) = CR ( , fmax )CL ( + , fmax ) at the fringe rate fmax of the maximum magnitude of KRL


RadioAstron discovery of scaterring substructure in PSR 0329+54
Table 1 Diary of observations Ep och of Observations 2012 Nov 26 through 29 2014 Jan 1 and 2 Time Span 1 hr/day 12 hr Ground Telescopes GB WB, KL Polarizations RCP+LCP RCP Scan Length 570 s 1170 s

3

Table 2 Observations on Earth-Space Baselines Ep och Pro jected Baseline Length (103 km) 26 27 28 29 1 2 2 60 90 175 235 20 70 90 RA Observing Time (minutes) 60 60 60 60 60 100 120

2012 2012 2012 2012 2014 2014 2014

Nov Nov Nov Nov Jan Jan Jan

yields the same time average. For this theoretical discussion, fmax = 0; for practical observations, instrumental factors can offset the fringe rate from zero, so that fmax provides the most reliable time average. For a baseline that extends much further than the scale of scattering Ssc (see Equation 32): KRL ( )
S

= G( ) G- ( ) G( ) G- ( ) + 1 if = 0

(6)

Here, we introduce the symbol to indicate convolution, and denote the time-reverse of G as G- ( ) = G(- ). Our analysis method differs somewhat from Smirnova et al. (2014), who used structure functions of intensity, visibility, and visibility squared to study scattering of pulsar B0950+08 on an extremely long baseline to RadioAstron. The two methods are closely related theoretically. Structure functions are particularly valuable when the characteristic bandwidth approaches the instrumental bandwidth, and can be extended to cases where the signal-to-noise ratio is low, as they discuss.
3. OBSERVATIONS

(RDR). This type of recorder was also used at the KL, while the Mk5B recording system was used at the GB and WB. Table 1 summarizes the observations. The data were transferred via internet to the Astro Space Center (ASC) in Moscow and then processed with the ASC correlator with gating and dedispersion applied (Andrianov et al. 2014). To determine the phase of the gate in the pulsar period, the average pulse profile was computed for every station by integrating the autocorrelation spectra obtained from the ASC correlator. The autocorrelation spectra VAA (, t) are the square modulus of electric field at a single antenna. In November 2012 the pro jected baselines to the space radio telescope were about 60, 90, 175, and 235 thousand kilometers for the four consecutive days, respectively. Data were recorded in 570-second scans, with 30-second gaps between scans. In January 2014 the projected baselines were about 20, 70, and 90 thousand kilometers during the 12-hour session. Data were recorded in 1170-second scans. The RA operated only during three sets of scans of 60, 100 and 120 min each, with large gaps in between caused by thermal constraints on the spacecraft. The auto-level (AGC), phase cal, and noise diode were turned off during our observations to avoid interference with pulses from the pulsar. Table 2 gives parameters of the Earth-space baselines observed.
4. DATA REDUCTION

The observations were made in two sessions: the first for one hour each on the four successive days November 26 to 29, 2012, and the second for a total of 12 hours on the two days January 1 and 2, 2014. The first session used the 10-m RadioAstron Space Radio Telescope (RA) together with the 110-m Robert C. Byrd Green Bank Telescope (GB). The second session used the RA together with the 14 â 25-m Westerbork Synthesis Radio Telescope (WB), and the 64-m Kalyazin Radio Telescopes (KL). Both right (RCP) and left circular polarizations (LCP) were recorded in November 2012, and only one polarization channel (RCP) was recorded in January 2014. Because of an RA peculiarity at 324 MHz, the 316­332 MHz observing band was recorded as a single upper sideband, with one-bit digitization at the RA and with two-bit digitization at the GB, WB, and KL. Science data from the RA were transmitted in real time to the telemetry station in Pushchino (Kardashev et al. 2013) and then recorded with the RadioAstron data recorder

4.1. Correlation All of the recorded data were correlated with the ASC correlator using 4096 channels for the November 2012 session and 2048 channels for the January 2014 session, with gating and dedispersion activated. The ON-pulse window was centered on the main component of the average profile, with a width of 5 ms in the November 2012 session and 8 ms in the January 2014 session. These compare with a 7-ms pulse width at 50% of the peak flux density (Lorimer et al. 1995). The OFF-pulse window was offset from the main pulse by half a period and had the same width as the ON-pulse window. The correlator output was always sampled synchronously with the pulsar period of 0.714 s (single pulse mode). We used ephemerides computed with the program TEMPO for the Earth center (Edwards et al. 2006). The results of the correlation were tabulated as cross power spectra, ~ V (, t), written in standard FITS format. 4.2. Single-Dish Data Reduction Using autocorrelation spectra at GB, KL, and WB, we measured the scintillation time tsc and bandwidth sc = 1/2 sc . The results are given in Table 3. Our analysis using interferometric data, for which the noise baseline is absent and the spectral resolution was higher, is more accurate for the constants 1 and 2 as discussed


4

Gwinn et al.
Table 3 Measured Scattering Parameters of PSR B0329+54 Ep och (1) Nov 2012 Jan 2014 tsc (s) (2) 114 ± 2 102 ± 2 sc (kHz) (3) 15 ± 2 7±2 wn (ns) (4) 50 ± 5 43 ± 3 wnf (mHz) (5) 20 ± 2 25 ± 3 1 = 1/k (µs) (6)
1

2 = 1/k (µs) (7) 23 ± 3 ­

2

4.1 ± 0.3 7.5 ± 0.3

Note. -- Columns are as follows: (1) Date of observations, (2) Scintillation time from autocorrelation spectra as the half width at 1/e of maximum, (3) Scintillation bandwidth from single-dish autocorrelation spectra as the half-width at half maximum (HWHM), (4) HWHM of a sinc function fit to the central spike of the visibility distribution along the delay axis, (5) HWHM of a sinc function fit to the central spike of the visibility distribution along the fringe rate axis, (6) Scale of the narrow component of |KRL ( )| (Section 5.2.3), (7) Scale of the broad component of |KRL ( )| (Section 5.2.3).

below, so we quote those values in Table 3. 4.3. VLBI Data Reduction The ASC correlator calculates the cross-power spec~ trum, V (, t), as discussed in Sections 2 and A.3.1. The resolution of the resulting cross-power spectra is 3.906 kHz for the 2012 observations and 7.812 kHz for the 2014 observations. Because the scintillation bandwidth was comparable to the channel bandwidth for the 2014 observations, as shown in Table 3, and because the single recorded polarization at that epoch prevented us from correlating polarizations to form KRL , as discussed in Section 5.2.3, we focus our analysis and interpretation on the 2012 observations.
5. ANALYSIS OF INTERFEROMETRIC VISIBILITY

for a useful analysis. However, we decided to analyze the data from the time series of multiple pulses. ~ Fourier transform of the cross spectrum, V (, t), to the delay/fringe-rate domain yields V (, f ) and concentrates the signal into a central region, and thus provides a high signal-to-noise ratio. The sampling rate of individual cross spectra in the time series was the pulse period of 0.714 s, as noted in Section 4.1. The time span of cross spectra used to form V ( , f ) varied, ranging from 71.4 s to 570 s, depending upon the application. 5.1. Distribution of visibility In Figure 2 we display the magnitude of the visibility in the delay/fringe-rate domain, |V ( , f )|, for a 500-s time span. The data were obtained on 29 November 2012 in the RCP channel for a pro jected 200M GB-RA base~ line. The cross spectra, V (, t), from which we obtained |V ( , f )| were sampled with 4096 spectral channels across the 16-MHz band, at the pulsar period of 0.714 s; consequently, the resolution was 0.03125 µs in delay, and 2 mHz in fringe rate. As Figure 2 shows, no dominant central spike is visible at zero delay and fringe rate, as would be expected for an unresolved source. Our long baseline interferometer completely resolves the scattering disk. Instead we see a distribution of spikes around zero delay and fringe rate that is concentrated in a relatively limited region of the delay-fringe rate domain. The locations of the various spikes appear to be random. Because the scattering disk is completely resolved on our long baseline, we conclude that the spikes are a consequence of random reinforcement or cancellation of paths to the different locations of the two telescopes, and hence interferometer phase. In Figure 2, the distribution of the magnitude of visibility is relatively broad along the delay axis and relatively narrow along the fringe rate axis. The extent is limited in delay to about the inverse of the scintillation bandwidth, sc = 1/2 sc ; and in fringe rate to about the inverse of the diffractive timescale tdiff . Within this region, the visibility shows many narrow, discrete spikes. If statistics of the random phase and amplitude of scintillation are Gaussian, and the phases of the Fourier transform randomize the different sums that comprise the visibility in the delay-fringe rate domain, then the square modulus

We investigated the scattering of the pulsar from the visibility in the delay-fringe-rate domain, V ( , f ). We studied the statistics of visibility V ( , f ) as a function of delay, fringe rate, and baseline length. If there were no scattering material between the pulsar and the observer, we would expect for |V ( , f )| one spike at zero delay and fringe rate with magnitude that remains constant as a function of baseline length, and with width equal to the inverse of the observed bandwidth in delay, and the inverse of the scan length in fringe rate. Scattering material in between changes this picture. First we expect the spike at zero delay and fringe rate to decrease in magnitude with increasing baseline length, perhaps to the point where it would become invisible. Second, we expect additional spikes to appear around the spike at zero delay and fringe rate. The distribution of these spikes give us invaluable information about the statistics of the scattering material. As we discuss in this section, we fitted models to the distribution of visibility, as measured by the correlation function KRL , and thus derived scintillation parameters that describe the impulse-response function for propagation along the line of sight from the pulsar. We also computed the maximum visibility as a function of pro jected baseline length, as we discuss in detail in a separate paper (Paper II: Popov et al., in preparation). For strong single pulses the visibility in the cross spec~ tra, V (, t), had signal-to-noise ratios sufficiently large


RadioAstron discovery of scaterring substructure in PSR 0329+54

5

Figure 2. Magnitude of visibility in the delay-fringe rate domain |V ( , f )|, for a 500-s time span on 29 November 2012 in the RCP channel, on the RA-GB baseline. Visibility is normalized for autocorrelation: |V (0, 0)| = 1. The axes show instrumental offsets, including about 6 µs in delay. Top: three-dimentional representation; bottom: two-dimentional representation.


6

Gwinn et al. of the distribution of magnitude for a range of baseline lengths, as a function of delay. The maxima lie near zero fringe rate, as expected. Under the plausible and usual assumption that the correct fringe rate lies at the fringe rate, fmax , where the distribution peaks, the crosssection represents the visibility averaged over the time span of the sample: V ( , f
max

) = V ( , t)

t

(7)

Figure 3. Examples of the fine structure of the magnitude of visibility, |V ( , fmax )|, as a function of delay , with fringe rate fixed at the maximum of the delay-fringe rate visibility near zero fringe rate, fmax . From lowermost to upp ermost, the curves corresp ond to progressively longer baselines, with the telescop es indicated and the approximate baseline pro jections given in M in parentheses. Curves are offset vertically, and the upp er 2 magnified as the vertical scale indicates, for ease of viewing. All curves show 71.4 s of data. Upp ermost curve is from 2012 Novemb er 29; the rest are from 2014 January, when multiple ground telescop es provided shorter baselines. Note variation in scattering time b etween epo chs as given in Table 3. The uppermost panel is the cross-section of the data shown in Figure 2, but for 71.4 s integration. Visibility is normalized as in Figure 2. The b est estimate of instrumental delay has b een removed for each curve.

of V ( , f ) should be drawn from an exponential distribution, multiplied by the envelope defined by the deterministic part of the impulse-response function, as discussed in the Appendix. Along the delay axis, |V ( , f )| takes the general form suggested by Figures 2 and 3: a narrow spike surrounded by a broad distribution. We found that the central spike takes the form of a sinc function in both delay and fringe rate coordinates, as expected for uniform visibility across a square passband (Thompson et al. 2007). The widths are somewhat larger than values expected from observing bandwidth of 16 MHz and time span of 71.4 s, of wn = 31.25 ns and wnf = 14 mHz respectively, probably because of the non-uniformity of receiver bandpasses and pulse-to-pulse intensity variations, respectively. The broader part of the distribution takes an exponential form along the fringe-rate axis in this case; more generally, the form can be complicated, particularly over times longer than 600 s. Traveling ionospheric disturbances may affect the time behavior of our 92-cm observations; in particular, they may be responsible for the 20 to 25 mHz width of the narrow component in fringe rate, as noted in Table 3. We do not analyze the broader distribution in fringe rate further in this paper; we will discuss this distribution, and the influence of traveling ionospheric disturbances, in a separate publication (Paper III, Popov et al. in preparation). Because of the relatively small optical path length of the ionosphere, even at = 92 cm, they cannot affect the cross spectrum (Hagfors 1976). The distribution of the magnitude of the visibility in delay/fringe-rate domain changes with baseline length. Figure 3 displays cross-sections through the maximum

The top panel of Figure 3 shows this cross-section through Figure 2. The next lower panel shows the crosssection for the slightly shorter KL-RA baseline. The three lower plots give the equivalent cross-sections for 10 times and 100 times shorter pro jected baselines. These three short-baseline cross-sections are qualitatively different from the long baseline cross sections: the visibility has a central spike resulting from the component of the cross-spectrum that has a constant phase over frequency, as well as the broad distribution from the component that has a varying phase over frequency. The central spike is strongest for the shortest baseline and weaker for the next longer baselines, as expected based on the results of Sections A.3.1 and A.3.2. At very long baselines the central spike is absent even after averaging the visibility over the whole observing period, and only the broad component is present. As expected from Figure 2, in the delay/time domain the broad component appears as spikes distributed over a range of about 10 µs in delay. These spikes keep their position in delay for the scintillation time of about 100 to 115 s, as listed in Table 3. The character of the broad component changes with baseline length as well: mean and mean square visibility are the same for short and long baselines; but excursions to small and large visibilities are more common for a long baseline (Gwinn 2001, Eq. 12). 5.2. Averages and Correlation Functions Averages of the visibility, and averages of the correlation function of visibility, extract the parameters of the broad and narrow components of visibility. Such averages approximate the statistical averages discussed in Sections 2 and A. They seek to reduce noise from the observing system and emission of the source, as well as variations from the finite number of scintillations sampled, while preserving the statistics of scintillation. The averages and correlation functions allow the inference of parameters of the impulse-response function of propagation from the statistics of visibility. 5.2.1. Square Modulus of Visibility C The mean square modulus of visibility, C ( ) S = |V ( )|2 S , provides useful and simple characterization of visibility. To approximate the average over realizations of scattering ... S , we average over many samples in time t and over bins in delay . We realize the average over time by evaluating V ( , f ) at the fringe rate of maximum amplitude fmax , as discussed in Section 2. We also average over 16 lags in delay . The resulting average shows a broad component surrounding the origin; on shorter baselines, it shows a spike at the origin. The broad distribution samples the properties of the fine structure seen in Figures 2 and 3, and the spike


RadioAstron discovery of scaterring substructure in PSR 0329+54

7

Figure 4. Cross-section of the mean square visibility in the delay/fringe-rate domain C ( ) S = |V ( , fmax )|2 along the delay axis, at the fringe rate fmax where the magnitude of visibility p eaks, close to zero mHz. The visibilities for the GB-RA baseline on 2012 Nov 28 at 21:40 UT are shown as op en circles. The visibilities were computed by an inverse Fourier transform of the sp ectra, ~ V (, t), over 71.4 s time spans, and then by averaging over 6 observing scans, each 570 s long. They were then further averaged in delay, over 16 points or 0.5 µs, to smo oth fluctuations. The dashed horizontal line shows the offset contributed by background noise. The solid gray line shows the reconstructed form given by Equation 34, offset by the noise level, with parameters taken from the fit shown in Figure 5. The light dashed curve shows only the narrow comp onent of the two-exp onential model. Units of visibility are correlator units.

Figure 5. An example of the correlation function K ( , fmax ) t on 2012 Nov 28, averaged over 570 s starting at 21:40:00 UT. The data were normalized by the square ro ot of KRR and KRR at = 0. The best-fitting parameters for a 2-exp onential fit of the form of Equation 35 are as indicated.

to those seen on the shorter baselines in Figure 3. We argue in Section A that the spike in C ( ) S is related to the average visibility, and the broad component to the impulse-response function. Figure 4 shows an example of the broad component of C ( ) S . This is estimated as |V ( , fmax )|2 , by selecting the peak fringe rate fmax to average in time for each of 6 scans, averaging the results for the scans, and averaging over 16 lags of delay to smooth the data. These averaging procedures serve to approximate the average over an ensemble of realizations of scattering. Background noise adds complex, zero-mean noise to V ( , f ), with uniform variance at all lags; this adds a constant offset to the average C ( ) S = |V ( , fmax )|2 . 5.2.2. Correlation Function K Using Equation 5, we estimated KRL ( ) S , the averaged cross-correlation function between the square modulus of right-circular polarized (RCP) and of left-circular polarized (LCP) of visibility in the delay domain. (Note that KRL ( ) S is not the correlation function of the average C S , but rather the average of the correlation function CR CL S .) Because the background noise in the two circular polarizations is uncorrelated, they do not contribute an offset to KRL ( ) S . This allows us to follow the effects of the impulse-response function to much lower levels than for C S . The correlation function KRL ( ) S is thus less sub ject to effects of noise, and is more sensitive to the broad component of the distribution, than C S . To compute an estimate of KRL ( ) S , we calculated the squared sum of real and imaginary components of V ( , t), the inverse Fourier transform of the cross-power spectrum. We formed these for each strong pulse, and

no