RadioAstron Studies of the Nearby, Turbulent Interstellar Plasma
With the Longest Space-Ground Interferometer Baseline
RadioAstron space-ground VLBI observations of the pulsar (catalog B0950+08), conducted with the 10-m space radio telescope in conjunction with the Arecibo 300-m telescope and Westerbork Synthesis Radio Telescope at a frequency of 324 MHz, were analyzed in order to investigate plasma inhomogeneities in the direction of this nearby pulsar. The observations were conducted at a spacecraft distance of 330,000 km, resulting in a projected baseline of 220,000 km, providing the greatest angular resolution ever achieved at meter wavelengths. Our analysis is based on fundamental behavior of structure and coherence functions. We find that the pulsar shows scintillation on two frequency scales, both much less than the observing frequency; but modulation is less than 100%. We infer that the scattering is weak, but a refracting wedge disperses the scintillation pattern. The refraction angle of this “cosmic prism” is measured as mas, with the refraction direction being approximately perpendicular to the observer velocity. We show that the observed parameters of scintillation effects indicate that two plasma layers lie along the line of sight to the pulsar, at distances of pc and pc, and traveling in different directions relative to the line of sight. Spectra of turbulence for the two layers are found to follow a power law with the indices , significantly different from the index expected for a Kolmogorov spectrum of turbulence, .
Small-scale fluctuations in the electron density of the interstellar medium (ISM) scatter radio waves from pulsars. We can study properties of both the ISM and the pulsar magnetosphere by measuring frequency-time characteristics of pulsar scintillation on the baseline of the cosmic interferometer resulting from scattering. Interferometry provides for comparison of scintillation at two different places at the same time.
The goal of the present study is to investigate the spatial distribution of scattering along the line of sight to pulsar (catalog B0950+08), one of the brightest and nearest pulsars. Its distance, measured by parallax, is pc (Brisken et al., 2002). The observations were conducted at a baseline projection of 220,000 km and with a spacecraft distance of 330,000 km providing angular resolution of 1 mas – the maximum ever achieved at meter wavelengths, on the longest baseline yet attained for VLBI observations, using the RadioAstron orbiting antenna.
1.1 Interstellar Scattering Material
Observations suggest the presence of three components of scattering material in our galaxy (Smirnova, Gwinn & Shishov, 2006; Smirnova & Shishov, 2008). The first is unevenly distributed material in the space between spiral arms (component A). The second corresponds to a cavity of depleted electron density extending as far as 300 pc from the sun in the direction perpendicular to the galactic plane and 50 to 100 pc in the Galactic plane (component B) (Snowden et al., 1990; Bhat at el., 1998). The power spectra of density fluctuations for components A and B are well described by a Kolmogorov spectrum with an index of 11/3 (Armstrong et al., 1995; Shishov & Smirnova, 2002). The third, component C, is located only about 10 pc from the sun and has an increased level of turbulence. This component is responsible for the intra-day variability of quasars at centimeter wavelengths (Dennett-Thorpe & de Bruyn, 2001; Rickett et al., 2002).
As discussed in earlier papers (Smirnova, Gwinn & Shishov, 2006; Smirnova & Shishov, 2008), the main contributor to the scintillation of the nearby pulsars (catalog J0437-4715) and (catalog B0950+08) is component C. The power spectrum of density fluctuations for these pulsars is flatter than the Kolmogorov spectrum. The index of the power-law is for (catalog PSR~B0950+08) and for (catalog PSR~J0437-4715). As shown by these studies, the level of turbulence in component C is a factor of 20 higher than that in the extended region responsible for scintillation of (catalog PSR~B0809+74) (Smirnova & Shishov, 2008). Phillips & Clegg (1992) assumed that the scattering material in the direction of B0950+08 was uniformly distributed, and found that the level of turbulence an order of magnitude lower than for any previously measured interstellar line of sight. Their results are also consistent with enhanced scattering in component C, at a distance of only 10 pc.
IDV sources show large-amplitude and rapid variability caused by scintillation in component C. The time scale of these variations is about the same as it would be for pulsars at centimeter wavelengths. Linsky, Rickett, & Redfield (2008) suggested that the partially-ionized surfaces of nearby interstellar clouds are responsible for intraday variability. They note that observed intraday variable sources lie behind such clouds; some where clouds collide. The nearby scattering screen for (catalog PSR~B0950+08) may have the same origin; indeed, its line of sight passes through one of the clouds they identify. The far edge of the Local Bubble may also scatter (catalog PSR~B0950+08). In this direction, the edge of the Local Bubble lies at a distance of 120 to 160 pc (Lallement et al., 2003). Smirnova & Shishov (2008) suggested the existence of strong angular refraction in the direction to (catalog PSR~B0950+08).
1.2 Orbiting Antenna and Observations
Interferometric observation of pulsars with space and Earth antennas, for baselines of several Earth diameters, provides the possibility of localizing the scattering layers and also allows one to evaluate the influence of refraction on the received emission. Shishov (2010) presents a preliminary theoretical analysis of this approach. For the first such VLBI observations of the Crab pulsar with ground telescopes see Kondratiev et al. (2007). As we discuss in Section 2.2.8, the typical scale of the scattering pattern of a weakly-scattered pulsar is the Fresnel scale, larger than an Earth diameter. Space-Earth interferometry affords the possibility of measuring this scale directly, rather than allowing motions of pulsar, Earth and scattering material to carry it across a single antenna. We make use of both interferometric and time-lag analyses in this paper; the earlier work by Smirnova & Shishov (2008) used only single-dish observations and provides an interesting comparison.
Because scintillation is a stochastic process, observations must be compared with theoretically-predicted distributions. These distributions may not have analytical forms and can be quite difficult to compute (e.g., Gwinn, 2001; Johnson & Gwinn, 2013). Consequently, moments of the distributions, particularly in the form of structure functions, provide measures of the fundamental behavior of structure and coherence functions. Section 2 describes this approach in the present case.
In this paper, we present results obtained from observations of the pulsar (catalog PSR~B0950+08) carried out on 25 January 2012 at a frequency of 324 MHz with the RadioAstron space radio telescope in conjunction with the Arecibo 300-m telescope (AR) and the Westerbork synthesis array (WB). The RadioAstron project is an international collaborative mission involving a free-flying satellite, Spectr-R, carrying a 10-m space radio telescope (SRT) on an elliptical orbit around the Earth. This space telescope performs radio astronomical observations using very long baseline interferometry (VLBI) techniques in conjunction with ground-based VLBI networks. The orbit of the RadioAstron satellite evolves with time. It has an apogee between 280,000 and 350,000 km, a perigee between 7,000 and 80,000 km, a period of 8 to 9 days, and an initial inclination of . RadioAstron operates at the standard radio astronomical wavelengths of 1.19 – 1.63 cm (K-band), 6.2 cm (C-band), 18 cm (L-band), and 92 cm (P-band). Technical parameters of the on-board scientific equipment and measured parameters have been described in two main publications (Avdeev et al., 2012; Kardashev et al., 2013).
2 ISM Scattering Theory
2.1 Overview of Model
As we discuss in Section 4, pulsar (catalog B0950+08) shows scintillation with modulation , with typical bandwidth much less than the observing frequency . Typically is observed only in weak scattering, whereas is observed only in strong scattering (Cohen & Cronyn, 1974). These two regimes are distinct. In weak scattering the difference in phase among paths from source to observer is less than radians, but in strong scattering it is more.
We suggest that a strong gradient in the column density of refracting material – a prism – is responsible for the apparent paradox. Such a prism will disperse the wide-band scintillation pattern in the observer plane, so that maxima and minima appear at different places at different observing frequencies. Indications of strong refractive effects include the typically “refractive” scaling of scintillation bandwidth with frequency found by Smirnova & Shishov (2008), and the shift of the structure function of visibility with frequency, as a function of time lag, discussed in Section 4.3.4 and displayed in Figure 6. Smirnova & Shishov (2008, see their Figure 10) showed that the transition between weak and strong scattering for this pulsar takes place in the frequency range of 100 to 300 MHz, with of course strong scattering at lower frequencies.
We find that 2 screens at different distances are required to reproduce the observed properties of scintillation. In particular, we find that the structure function of visibility with observing frequency is composed of two components with different timescales and different behavior as a function of interferometer baseline. We discuss the structure function in Section 2.2, and present the structure functions formed from our observations in Section 4.2. Because the scintillation is weak, the observed scintillation pattern in the observer plane is the superposition of the patterns from the two screens.
2.2 Characteristics of the ISM structure function
This paper considers interferometric observations of scattering. The interferometric visiblity is as a function of frequency, time and baseline is the fundamental observable. We are concerned with the fluctuations of . We analyze these fluctuations using the fundamental behavior of structure and coherence functions. As we describe in this section, we relate the modulus of the visibility, the product of electric fields at two positions, to the product of the intensities at those positions. The resulting expressions are excellent when the noise level is low. In the case of a high level of noise, it is advantageous to measure fluctuations in the squared modulus of the interferometer cross-power spectrum . In Appendix A, we derive the analogous relations that account for noise. In particular, we show that for statistics of the contributions of noise and signal are simply additive, whereas for statistics of or the contributions of noise and signal are much more difficult to separate. We therefore deal with the statistics of in this paper.
2.2.1 Field and propagation
We define as the electric field of pulsar emission, in the absence of a turbulent plasma, where is the offset of the observing frequency from the band center , and is time. This electric field also includes modulation by the receiver bandpass. The field after propagation through the ISM can be represented as (Shishov, 2010)
where scattering in the ISM results in the factor , given that is the spatial coordinate in the observer plane perpendicular to the line of sight. Here and throughout the paper, we use boldface type to denote vector quantities, such as . To obtain the cross-power spectrum – i.e., the response of an interferometer with baseline averaged over a fixed realization of the scattering – we multiply by and average over statistics of the source electric field:
Here, the angular brackets with subscript indicate an average over the noiselike statistics of the electric field of the source. The flux density of the source, corrected for bandpass, is ; and the effects of scattering are expressed by . Note that depends on frequency within the observed band and on time , but we omit these arguments for clarity in the equations below.
2.2.2 Visibility and fluctuations
Consider fluctuations of the modulus of the interferometer response (the dynamic cross-power spectrum):
where the intensity is the square modulus of electric field at a single position:
Again, note that , , and depend on and , but we omit these arguments for simplicity. We normalize the average flux density of the source, so that the average over realizations of scintillations is unity: . We are concerned with the fluctuations of intensity about this average: .
In the regime of weak scintillation, , so can be approximated as
Thus, fluctuations in the interferometric visibility are given by
2.2.3 Screens and statistics
We suppose that the scattering material lies in two phase-changing screens located at distances and from the observer. The distance of the source form the observer is . Each screen produces some variation of phase as a function of position. We can characterize the statistics of the phase screen by two spatial structure functions of phase fluctuations and , where and are the differences of the spatial coordinates in the plane of the phase-changing screen:
where identifies the screeen, and is the screen phase at . Here, the angular brackets with subscript indicate an average over an ensemble of statistically-identical scattering media.
Smirnova & Shishov (2008) showed that the spectrum of turbulence of the interstellar medium in the direction of (catalog PSR~B0950+08) has a power-law form, , where is spatial frequency. Consequently, the spatial structure functions of phase fluctuations exhibit a power-law form as well. We describe the structure functions of the two phase-changing screens with the expressions
where is the wavenumber, is the wavelength, and represents the angle of scattering at phase-changing screen . We assume a power-law form for the structure functions, with indices , . Note that this equation introduces the assumption that scattering is isotropic. We make this assumption for the rest of the paper. In principle, effects of anisotropy could be detected by comparing results on several baselines of comparable lengths and different orientations. Thus, ongoing and future observations should be able to refine the results presented here.
As was pointed out in Smirnova & Shishov (2008, see figure 10), the transition from strong to weak scintillation takes place in the frequency range 100 - 300 MHz. We will show in section 4.1 that modulation index at frequency 324 MHz is about 0.35, so the scintillation is weak Martin & Flatte (1988). In other words, the fluctuations in phase introduced from propagation are small: . The modulation indices resulting from scattering at the screens are and , defined formally in Equations 28 and 30 below. We can apply weak scattering theory because for a power-law spectrum of turbulence, the difference between approximate and exact values of doesn’t exeed of (Martin & Flatte, 1988).
For a nearby screen at distance , we can use the plane wave approximation, so that
For a phase-changing screen at distance , the structure function of phase fluctuations at the screen is related to that at the observer by:
Note that these equations describe the effects of the screen as a “shadowgraph”, where effects of the screen are projected directly onto the observer plane. This is characteristic of weak scattering.
2.2.4 Cosmic prism
We suppose that a cosmic prism, or gradient of refracting material, is located between the pulsar and the phase-changing screens. This prism deflects radiation from the pulsar at a frequency-dependent refractive angle. We parametrize this refraction using , the apparent displacement of the source location as observed from the observer plane at frequency . Similar strong angular refraction has been detected in the direction to (catalog PSR~B0329+54) from analysis of multi-frequency observations (Shishov et al., 2003). Smirnova & Shishov (2008) showed that strong angular refraction exists in the direction to (catalog PSR~B0950+08), so we assume that the refractive angle is significantly greater than the scattering angle at either screen:
Thus, our model of turbulent interstellar plasma in the direction of the pulsar is characterized by the following parameters: , , , , , , .
2.2.5 Shifts in frequency, position, and time
The cosmic prism dominates the angular deflection of the source, and in particular is greater than that produced by the screens. In the presence of angular refraction, with the dispersion produced by the interstellar plasma, a change in frequency of from the fiducial frequency leads to an apparent displacement of the source position by an angle
where is the frequency at the center of the observing band.
The combination of a cosmic prism and a phase-changing screen leads to a shift in the scintillation pattern. For a phase-changing screen at distance from the observer, and the cosmic prism beyond the screen, the apparent displacement of the source leads to a displacement of the scintillation pattern in the observer plane by a distance , given by (Little & Hewish, 1966; Shishov, 2007)
The further away the screen, the greater the dispersion of the scintillation pattern at the Earth produced by the prism beyond the screen.
On the other hand, if the observer travels at velocity perpendicular to the line of sight, and if screen 1 moves at speed , then observer’s spatial displacement relative to the scintillation pattern increases with the change of time at velocity :
If this displacement is parallel to the dispersion of the cosmic prism, then the observer will notice a shift of the scintillation pattern in frequency, as a function of time.
For a more distant screen, at distance from the observer, the spherical form of the waves must be taken into account:
where is the velocity of screen 2, and is the velocity of the pulsar.
Thus, a change in frequency will cause a change in the diffraction pattern analogous to a change in position in the direction of ; and a delay in time is equivalent to a change in position in the direction of a linear combination of , , and , for a moving observer, screen, or pulsar. These equivalences arise because the intensity variations from weak scintillation have wide intrinsic bandwidth: they are dispersed only by the cosmic prism. We apply these equivalences further in Section 2.2.7 below.
2.2.6 Formation of structure functions
We construct the structure function of intensity in the observer plane:
In weak scattering, the structure functions of the two screens add to produce the observed structure function, as we indicate here. We wish to relate this to the structure function for interferometric visibility.
Consider observations of intensity at two locations in the observer plane separated by , and separated in frequency by and in time by . The structure function characterizing the fluctuations of intensities is:
where or 2 identifies the screen. This equation relates the structure functions of intensity to those of interferometric visiblity. A similar equation provides the reverse relation:
Also note that in the short-baseline limit, the structure function for the interferometric visibility is that of intensity, with the correction for bandpass.
2.2.7 Equivalence of time, frequency and baseline
We can use the results of the previous section to infer dependence of the structure function on time and frequency as well as baseline, by noting that the cosmic prism renders a change in time or frequency equivalent to a change in position. We can therefore define a generalized position variable , which includes effects of dispersion on time and frequency behavior of the scintillation pattern for each screen:
Because we assume that the scattering in the screens is isotropic, the structure function for each screen depends only on the magnitude of its argument:
The direction of refractive dispersion divides the components of the other vectors into those parallel and perpendicular to the direction of that dispersion. The components of the vectors may be different because the conversion of and to and differs for the screens.
Here, is the angle between the direction of dispersion , and the velocity of the ray relative to screen ; and is the angle between and the baseline .
Physically, this equation states that the structure function for variations in intensity is that set by optics of a static screen without refraction ;
however, refraction and motion introduce correlation in frequency set by the scale and in time set by the scale .
Note that the effects of delays in time or changes in frequency depend sensitively on the angles .
Note slso that is a symmetrical function of ,
as Equation (22) shows. Therefore is a symmetrical function of and is a symmetrical function of .
2.2.8 Fresnel scales
The Fresnel scale for a nearby screen at distance is given by
For a nearby screen, the structure function of intensity variations is given by the simple relations (Prokhorov et al., 1975):
where is the modulation index of the nearby layer.
Consequently, for a distant scattering screen,
where is the modulation index of a distant scattering screen. Physically, Equations (28) and (30) represent the fact that the diffraction pattern from a static, nonrefracting screen in weak scattering has a scale equal to the Fresnel scale. Together with Equation (22), these equations describe the statistics of the diffraction pattern from a screen in the presence of a cosmic prism, and with motions of source, screen, or observer.
2.2.9 Fresnel scales for frequency and time
The Fresnel scales given by Equations (27) and (29) give rise to corresponding scales in frequency and time through the action of the cosmic prism, which relates frequency shifts and time lags with changes in position via Equations (14) through (19). Because the expressions are important in comparing theory with observation, we present expressions for these scales here. The Fresnel frequency scales are:
for a nearby and more distant screen, respectively. Note that the frequency scale is largest for a nearby screen, and decreases with increasing screen distance. The Fresnel time scales are simply:
Here, we assume that the velocities of the screens are small: and .
2.3 Temporal Coherence Function
The inverse Fourier transform of gives the temporal coherence function, averaged over statistics of the source electric field:
where is a time lag of coherence function. Then, the value of averaged over the statistics of the turbulent medium is
Here, is the temporal coherence function defined by the source and is the spatial coherence function of the scattered field.
Given these functions, we can characterize fluctuations of by its second moment:
Then, we can write the mean squared modulus of , averaged over statistics of turbulent medium, as
where corresponds to unscattered emission , and consists of two parts that give the contributions of the nearby and distant screens:
where is the flux density of the source integrated over frequency, , is a power phase spectrum, and and are components of the spatial frequency parallel and perpendicular to the direction of refractive angle. The power spectrum can also be written as
For the second component, , we find that
Note that the function is a random function of . Hence, if , the distribution of is approximately normal for fixed . Then, for a normally distributed random complex value, we have the relationship
If , then will be the primary contributor to for , while will affect predominantly for .
3 Observations and initial data reduction
We observed the pulsar (catalog B0950+08) for one hour on January 25, 2012 using the RadioAstron 10-m space radio telescope in concert with the Arecibo 300-m telescope and the Westerbork synthesis array. We observed dual polarizations across a 16 MHz band centered on 324 MHz. Data were recorded continuously for 5 min scans with a 30 s interval after each scan to write the data to disk.
Using the Astro Space Center correlator, we performed the first steps of data reduction, which involved removing the dispersion from the pulsar signal, calculating the complex spectrum for each telescope, and calculating the cross-spectra for all pairs of telescopes. The signal was correlated in a 15 ms gate around the maximum of the average pulse, and the noise was evaluated in a gate separated by 52 ms from the maximum. For most of our analysis, we averaged the correlator output over four pulsar periods ( s) and employed a frequency resolution of 125 kHz (128 channels); although we also used single-pulse spectra with the same frequency resolution in some cases, as noted below.
4 Analysis and Results
4.1 Spectra and Correlation Analysis
Pulsar (catalog B0950+08) has a high level of intrinsic variability, and even exhibits giant pulses with more than 100 times the mean flux density (Smirnova, 2012). Because this intrinsic variability is much more rapid than that of scintillation, we normalized each spectrum by its mean value. We also corrected the spectra for the receiver passband, which we estimated by averaging the off-pulse spectra over the entire observation (3570 s).
The passbands show narrow-scale interference (only one frequency channel in bandwidth) throughout the entire observation. This interference increases signal in individual channels of single-dish spectra, but reduces gain in the corresponding channels of cross-power spectra. Hence, for each spectrum, we replaced the intensities in contaminated channels with the mean value of two neighboring channels. Figure 1 shows several individual pulsar cross-power spectra at different times for the Arecibo-Westerbork baseline, prior to normalization by the mean intensity. Two frequency scales of variability are evident: a small scale, 30 - 40 channels (3.75 - 5) MHz, as well as a large scale, exceeding the 16 MHz receiver band. The small-scale structure changes only slowly with time, and remains similar over time periods of 200 s (pulses 749 and 949) and 540 s (pulses 1576 and 2032), whereas the character of the wider structure changes little over the 1-hour observation.
Figure 2 shows the variability of individual pulses over the experiment. The lower panel of the figure shows the mean intensity of individual pulses as a function of time. Here, the subscripted angular brackets indicate an average over frequency channels of one single-pulse integration. Fast variability reflects intrinsic pulse-to-pulse varaiations, wheras the slow modulation reflects scintillation. The middle panel of Figure 2 shows the standard deviation of flux density in individual single-pulse spectral as a function of time. For each spectrum, we used the autocorrelation function at a lag of one channel (to exclude the contribution of noise) to find . In the upper panel of the figure we show the modulation index as a function of time; null sequences correspond to 30 s gaps in data. The modulation index is equal to 0.35 to 0.4, indicating weak scintillation at MHz for (catalog PSR~B0950+08). The fast fluctuations of are due to noise or weak pulses. The slow variations of are caused by the wide-bandwidth component of scintillation; the narrow-bandwidth component of scintillation averages out over the observing band.
Figure 3 shows the mean cross-correlation of the interferometric visibility on the Arecibo-Westerbork baseline, after the visibility is averaged over frequency:
The correlation is shown for nonzero temporal lags of s, where = 1, 2, …, 35. Intrinsic pulsar fluctuations are uncorrelated at these large lags, so the correlation arises from scintillation. The characteristic timescale of scintillation, defined as the lag at half maximum, is s. Because the correlation is averaged over frequency, this reflects the effect of the broadband variation in Figure 2.
4.2 Estimated structure function
We used our observations to estimate the structure function of the interferometric visibility, and compared this estimate with the theoretical results of Section 2. To form our estimate, we normalized each spectrum by its mean in frequency, and also corrected for the receiver bandpass:
where is the receiver bandpass and is its value at the center frequency. As mentioned in subsection 2.2 and discuss further in Appendix A, we used the squared intensity when the noise level was comparable to the signal (i.e., on the space baselines). So we calculated a mean structure function for both baselines as:
where we include an additional shift in time s for a significant reduction of the effects of noise for the structure function at zero frequency lag. As we show in Appendix A, in weak scintillation the structure function for the squared modulus of the visibility is proportional to the structure function for the modulus of visibility. Consequently, for our calculated normalized structure function we can use all theoretical relations from section 2. Structure functions for Arecibo-Westerbork and RadioAstron-Arecibo baselines normalized by are plotted with frequency lag in Figure 4, for different time lags . We calculated the structure function for both positive and negative frequency lags, .
4.3 Comparison with theory
4.3.1 Simple model
The structure functions shown in the two panels of Figure 4 have qualitatively different forms. Comparison of the two shows that the structure function on the shorter Arecibo-Westerbork baseline comprises a narrow-bandwidth component and a broader-bandwidth component. For the long RadioAstron-Arecibo space baseline, the narrow-bandwidth component is absent; we see only the broad-bandwidth structure. (The sharp detail at lag s is caused by noise.) The narrower component also appears only at small time lags, whereas the broader component appears at both large and small time lags. The two frequency scales correspond to two effective layers of turbulent plasma, separated in space, where scattering of pulsar emission take place.
As a simple model for the structure function, we adopt a piecewise-linear form, displayed in Figure 5. Formally, we take in Equation (9); this is adequate for the determination of characteristic scales. This leads to the form
where we have combined results of Sections 2.2.7 through 2.2.9. The index runs over the two screens. Note that the displacements are added vectorially to form the arguments , but the arguments for different screens are completely independent.
Although the dependence of the structure function on is linear, interplay among the arguments can lead to different dependences on . For example, if dispersion by the cosmic prism is perpendicular to the velocity of the ray through the screen so that , and both frequency and time offsets contribute with and , then the variation of with becomes approximately quadratic, as Equation (2.2.7) shows. Likewise, for and , the dependence is approximately quadratic.
On the space-Earth RadioAstron-Arecibo baseline, the long baseline suppresses most of the narrowband structure and some of the wideband structure, as Figure 4 (bottom) shows. The shape of the structure function becomes more quadratic with increasing . As we argue in more detail below, this suggests that the interferometer baseline is perpendicular to the velocity of the ray relative to the more distant screen, screen 2.
The two components of the structure function on the shorter baseline represent the structure functions from the two screens:
Consequently, for the shorter Arecibo-Westerbork baseline, we expect the structure function to take the form:
The increase of the structure function up to the maximum frequency difference that is reliably sampled for our 16-MHz observing bandwidth, MHz, indicates that is equal to or greater than . This increase appears for both long and short baselines. Consequently, the present data do not allow us to explore the third possibility, , in Equation (51). From inspection of Figure 4 (top), the structure function on the shorter baseline has a change of slope at the frequency corresponding to the Fresnel scale for screen 2, at MHz (25 channels). This is the transition from the first possibility to the second in Equation (51).
4.3.2 Evaluation of model parameters
From inspection of Figure 4 we observe that approximately doubles between and :
If we substitute for into this expression from Equation (51), using second possibility because , we find:
Note also that the modulation index found for the entire scan, , as illustrated in Figure 2, is approximately equal to , because the bandwidth of the observations limits the modulation index. Thus:
We can set bounds on from the facts that , as observed above; and from the fact that Smirnova & Shishov (2008, see Figure 10) find , at our observing frequency. We then find
The smaller value corresponds to our lower limit on , and the larger to the upper limit. Consequently, our model comprises two screens, both weakly scattering. Screen 1 has the greater modulation index , and a larger frequency scale corresponding to the Fresnel scale . This suggests that screen 1 is closer, as can be seen from Equations (31 and 32).
4.3.3 Scales and distances
We can use the behavior of the structure function with time, as well as frequency, to estimate the Fresnel scales and distances of the scattering screens. The amplitude of the narrower component of the structure function decreases with increasing of time shift, and falls to zero at time lag, s. The cross-correlation coefficient of spectra decreases with time lag and falls to half-maximum as discussed in Section 4.1 above, and as shown in Figure 3. This suggests that the typical timescale for the narrow component, produced by screen 2, is less than 1000 sec.
We can evaluate the Fresnel scale for screen 1 by comparing the structure function at and , at . At this time lag, the structure function at is 0.42 times that at as seen in Figure 4.
From this we obtain an expression for :
We eliminate in favor of and using Equation 55 and find:
where in the last line we have used our observational limits for , and for used the speed of the Earth, relative to the Local Standard of Rest, at the date of observation . Using Equation (27) we find for the distance of screen 1, ,
Thus, screen 1 is quite close to the Earth.
Using Equation (31), we obtain for the refractive angle
where the larger value for arises from the smaller value for , leading to a closer screen (smaller value of ).
Using this value for refractive angle and MHz in the expression for the frequency scale corresponding to refraction by the nearby screen, Equation (32), we find for the value of
where the range of screen distances arises from the range of possible values for , with the closer distance for screen 2 associated with a closer distance for screen 1, and both arising from the smaller value for . Screen 2 is always more distant, and may lie at a significant fraction of the pulsar distance of .
The Fresnel scale for screen 2 is given by Equation (29):
where the smaller value corresponds to the lower limit on , and the larger to the upper limit.
We note that our reconstruction of Fresnel scales, screen distances, and refraction angle is roughly consistent with the observation that the narrow component of the structure function is supressed on the long RadioAstron-Arecibo baseline, as shown in Figure 4 (bottom). We expect this component to become decorrelated over a distance of , or 140,000 to 270,000 km, whereas the projected baseline length is 220,000 km. Our interferometer results favor the greater distances in this range from screen 1.
4.3.4 Asymmetry of the structure function
The cosmic prism disperses the scintillation pattern across the observer plane, so that particular intensity maxima and minima appear at different positions at different frequencies. If the screen moves parallel to the direction of dispersion, then the observer notes a shift in the frequency of the scintillation pattern with time, as given by and in Equations (13-19). If multiple screens are present, as we argue above, and if they have different velocities, then the observer will notice different rates for the different resulting patterns.
The shift in frequency of the scintillation pattern with time or position leads to an asymmetry in frequency of the structure function: , for nonzero time lag or finite baseline length . This asymmetry increases proportionately with or . For two screens with different velocities, displacements of the structure functions and increase with different rates. If the screens are moving with velocities with opposite directions of projection onto , the resulting displacements have opposite sign.
As an example of this asymmetry, Figure 6 shows the mean structure functions for the long RadioAstron-Arecibo baseline at large time lags. The line shows the best-fitting parabola at s. The minimum is shifted by 750 kHz, or 6 channels, toward . As Equation (24) suggests, a minimum of the structure function lies where
We can estimate directions of refractive dispersion and screen velocity from the measured asymmetry of the structure function. For our observations on the short Arecibo-Westerbork baseline, the weak asymmetry of the structure function, along with a relatively rapid decorrelation of the narrow component at s as seen in Figure 4 (top), suggests that the angle between the vectors and is close to .
We quantify the asymmetry of the normalized difference of structure functions for positive and negative frequency lags by the ratio:
For particular values of , the extrema as a function of lie at
For weak-scattering models such as the simple model discussed in Section 4.3.1, where the structure function for each screen has a single minimum and becomes constant for large displacements, extrema of tend to lie near or at the minima of the structure functions of the individual screens.
We display the ratio