100 μas RESOLUTION VLBI IMAGING OF ANISOTROPIC INTERSTELLAR SCATTERING TOWARD PULSAR B0834+06

The SSs, A(τ, f_D), are squared amplitudes of the two-dimensional Fourier transforms, $\tilde{I}(\tau,f_{\rm D})$, of the dynamic spectra of intensity. Here, the transform variables conjugate to frequency, f, and time, t, are delay, τ, and Doppler frequency, f_D, respectively. The quantity $\tilde{V}(\tau,f_{\rm D},{\bf b})$ represents the complex two-dimensional Fourier transforms of the complex dynamic cross spectra. As some of the quantities we discuss in this work have not yet been introduced in the literature, we summarize in Table 1 the conventions we adopt throughout for the various products derived from the correlator auto and cross-correlations.

The dynamic spectra have a spectral resolution of 244 Hz over 8 MHz and temporal resolution of 6.25 s (about five pulsar rotation periods) over the observation duration of 6500 s, which are well suited to resolving the diffractive scintillation, whose timescale is ∼1 minute and frequency scale is ∼3 kHz. Thus, the SSs have a resolution of 125 ns in delay out to a maximum of 2.05 ms. They have resolution in Doppler frequency of 0.15 mHz over a width of ± 80 mHz. Custom software was used for this and all subsequent data processing.

For the first time, dynamic spectra of right minus left circular polarization and the corresponding SSs were computed to test for differential Faraday rotation in the ISM (Macquart & Melrose 2000). No detectable signal, significant at the 0.1% level, was found in any of these differenced spectra. In a simple model, this implies rotation measure differences across the image of less than ∼1.2 × 10⁻³ rad m⁻² on AU scales (Macquart & Melrose 2000).

The amplitude of the quantity $\tilde{V}(\tau,f_{\rm D},{\bf b})\,\tilde{V}(-\tau,-f_{\rm D},{\bf b})$ is plotted in Figure 1. This quantity is a generalization of the SS to the interferometric case and will be discussed later. The plot extends to a delay of 2 ms, which is considerably greater than has been published before, and is plotted with higher Doppler frequency and delay resolution than is usual. It thus reveals some interesting features. It is composed of a dense forest of fine arclets in which the apex of each arclet lies on or near the upward facing main arc and all arcs are parabolic with the same curvature, as shown by Hill et al. (2005). The disjoint group of arclets near 1 ms in delay with negative Doppler frequency is particularly striking and will be referred to as the 1 ms feature. Their apexes lie inside the main arc and, as we find below, they form a separate and distinct part of the scattered image. Some faint arclets on the positive Doppler frequency side can be identified with delays as high as 2 ms, although they are not readily visible in this figure.

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In single dish observations, information on the scattered image is limited to the power in the SS, obtained from $\tilde{I}(\tau,f_{\rm D})$. However, the addition of interferometric observations permits high-precision astrometry to be performed on each component of the secondary cross spectrum, which isolates the wavefield due to a single pair of interfering waves. By applying astrometry to each such pair individually, we can recover an apparent image of the scattered pulsar radiation.

For a particular pixel from $\tilde{V}(\tau,f_{\rm D},{\bf b})$ Equation (2) sums over all pairs of points that satisfy Equations (3). However, we now consider the case that only one pair of points satisfies the latter conditions, a condition that is valid for a linear or sufficiently anisotropic source. Then, the phase of $\tilde{V}$ is Φ_j,k = Φ_j(− b/2) − Φ_k(b/2) is

$$\begin{equation} \Phi _{j,k}= \phi _j-\phi _k + \frac{k D_{\rm s}}{2\beta } \bigg[ \theta _j^2-\theta _k^2 +\frac{\beta {\bf b}}{D_{\rm s}}\cdot ({ \mbox{$\btheta $} }_j+{ \mbox{$\btheta $} }_k) \bigg] \end{equation} \tag{ 4 }$$

which has terms that are symmetric and anti-symmetric in scattered points j, k. We cancel the anti-symmetric terms by summing the phase of $\tilde{V}$ at each point (f_D, τ) and its conjugate point at (−f_D, − τ). This eliminates the random screen phase leaving the symmetric part of the phase as

$$\begin{equation} \psi _{j,k}({\bf b}) = \Phi _{j,k}+\Phi _{k,j} = \frac{2\pi }{\lambda } {\bf b} \cdot ({{ \mbox{$\btheta $} }}_j + {{ \mbox{$\btheta $} }}_k) \,, \end{equation} \tag{ 5 }$$

which neatly encodes the average position of the two scattering points projected parallel to the baseline b. As we show below, the scattering is indeed highly anisotropic and the assumption of a single pair of interfering waves for each pixel is well justified.

In practice, the anti-symmetric phases are canceled by first forming the secondary cross spectrum, defined as

$$\begin{equation} C(f_{\rm D}, \tau,{\bf b}) = \tilde{V}(f_{\rm D}, \tau,{\bf b}) \tilde{V}(-f_{\rm D}, -\tau,{\bf b}) \,, \end{equation} \tag{ 6 }$$

for all non-negative values of τ. Subsequently, the resultant complex phasors are averaged over small regions (3 pixels in Doppler by 5 in delay) in order to reduce the effect of noise before the argument is determined:

$$\begin{equation} \psi ({\bf b}) = {\rm arg}(\langle C(f_{\rm D}, \tau,{\bf b}) \rangle) \,. \end{equation} \tag{ 7 }$$

Here, 〈〉 denotes the 3 × 5 averaging which properly weights the complex products before the phase is computed. The result is the geometric phase of the secondary cross spectrum sampled with a resolution of 0.63 μs in delay and 0.44 mHz in Doppler. As with any phase measurement the result is modulo 2π.

The right panel of Figure 1 shows ψ, the phase of the secondary cross spectrum, for the baseline from GBT to AO. In contrast to the sharply defined arclets in the amplitude plot, the phase is relatively smooth, appearing continuous between the arclets, as might be expected if the fine structure in amplitude comes from a continuous image with fine structure in brightness. Consequently, we can use astrometry from ψ to actually map out the scattered brightness distribution with remarkable precision. Furthermore, we can measure the position angle of the axis of elongation and estimate both the distance to the region of scattering and its velocity.

We can use Equation (5) to find $({ \mbox{$\btheta $} }_j + { \mbox{$\btheta $} }_k)$ for any point in the secondary cross spectrum if it is dominated by a single pair (${ \mbox{$\btheta $} }_j,{ \mbox{$\btheta $} }_k$) and there are at least two baselines with sufficient S/N. However, if we can identify the apex of an arclet we know ${ \mbox{$\btheta $} }_j$ or ${ \mbox{$\btheta $} }_k$ is zero, so we have a unique solution for the remaining angle. The amplitude plot in Figure 1 shows that the apexes of the identifiable arclets lie near the parabolic curve shown as a line. Hence, we sampled the secondary cross spectrum along that curve to approximate a set of apex positions even where the arclets are too densely spaced to be identified individually. We defined the main parabolic arc by the curvature a = 5.577 × 10¹⁶/f² (s³) for center frequency f Hz in each of our four sub-bands, as shown by the black line in Figure 1.

We selected data with a high enough S/N to estimate ${ \mbox{$\btheta $} }$ as follows. For each of the six baselines, we found the phasor 〈C(f_D, τ, b)〉 and defined S as the S/N relative to the rms value of |〈C〉| in a low power (noise dominated) region near the point being examined. Samples with delays within ±0.6 μs of the main parabola were selected if S>4 on the GBT–AO baseline and if S>3 on the WB–AO baseline.

For each sample of 〈C〉, we first formed a "dirty image" from the six baselines in the same sense as for traditional synthesis imaging (Taylor et al. 1999). At the dirty image maximum a two-dimensional Gaussian was fitted to determine a position, and both S and the shape of the Gaussian were used to assign positional uncertainties. However, since C is a product of two visibilities this method gave too much relative weight to the high S/N baselines and led to apparent positional errors much smaller than the scatter of the points from neighboring samples along the main arc. In addition, there are lobe ambiguities in the synthesized beam, since two of four antennas (JB and WB) are relatively close to each other and so probe nearly the same spatial frequencies. The result is a dirty beam that is nearly doubly periodic with lattice vectors a₁ = (59.0, − 61.9) mas and a₂ = (34.2, 17.8) mas at 322.5 MHz (scaling linearly with wavelength).

After some investigation, we adopted a simpler astrometric method based on a weighted linear least-squares fit of the observed phases, ψ, using Equation (5) (with one of the angles set to zero). Under conditions of high S/N, one can show that the phase error is $\propto 1/\sqrt{S}$ and so we used weights $\propto \sqrt{S}$. Because the phase is observed modulo 2π there are ambiguities in position, corresponding to the lobe ambiguities. Consequently, we used a first guess position as an initial model and wrapped the phase (e.g., on the transatlantic baselines) to be within ±π of that solution before applying the least-squares fit. For the points from the main arc, our first guess was at the origin and the results are plotted as error bars (in a pale color) in Figure 2. Points from the main arc form a continuous elongated image close enough to the origin that the phase ambiguity is not a problem for them.

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An empirically determined line passing through arclet apexes was used to sample the points in the SS associated with the 1 ms feature. The astrometry of these points, however, forms a distinct group offset from the origin in celestial coordinates, making it difficult to resolve the phase ambiguities. Accordingly, we used first guess positions from the two closest synthesized beam lobes which are roughly equidistant from the origin. The cyan points on the left show the fitted positions starting from the central lobe; the cyan points on the right show them starting from a position lobe-shifted by the lattice vector −a₂, which corresponds to the addition of 2π to the phase on the transatlantic baselines. In Section 4, we discuss the resolution of this ambiguity in favor of the right-hand position.

The major conclusion from Figure 2 is that the scattered image from the main arc is remarkably extended along a straight line with very few points more than 2σ from that line. As noted above highly anisotropic scattering was already inferred by the emptiness of the SS (particularly the absence of power at f_D ∼ 0, τ>0) and the many discrete arclets. The apparent flaring of the points away from the origin is due to the increasing size of the error bars because the SS amplitude decreases with angle of scattering. We define the straight line fitted to the points as the "scattering axis." Note that the plot is not a proper image of the scattered brightness, since the amplitude |C| must be transformed by a Jacobian into the brightness. Hence, the axial ratio is not easily estimated and we postpone the discussion of it to a later section.

The astrometric positions discussed in the preceding section and plotted in Figure 2 are based on the assumption that samples along the main arc correspond to the apexes of an underlying arclet. However, this is not always true, as is obvious in the 1 ms feature. So we have made an effort to identify all the discrete arclets, estimate the apex of each in an optimal way, and then map the phases at each apex to an angular position. At 322.5 MHz, we found 62 well-defined arclets (with apex delays τ>0.1 ms) of which seven are in the 1 ms feature. At smaller delays, the arclets are too crowded to identify separately.

We formed a template by averaging all the arclet amplitudes, centered by their apex, and we fitted a straight line in phase versus f_D to the template to obtain the phase gradient versus f_D in the template. We then fitted the amplitude and phase templates to each arclet using a weighted least-squares fit. Hence, we obtained estimates for the delay, Doppler, phase, and amplitude at the apex and errors in all four parameters. We then estimated ${ \mbox{$\btheta $} }$ and its error at each apex and repeated the procedure for all four sub-bands.

The astrometric positions and errors of the 62 arclets are plotted in black in Figure 2. The error bars are smaller than from the main arc because the phases were determined from a weighted average of the phases over the entire arclet. In addition, the scatter should be smaller because they are known to correspond to interference with the bright centroid component at the origin. However, the lobe uncertainty in the location of the seven arclets from the 1 ms feature remains. As we discuss below, we resolve the ambiguity in favor of the lower right lobe position marked by the black error bars derived from the seven apexes. The apexes near the main arc have positions extending out to 20 mas from the origin and show good linear alignment with the inner points from the main arc. A straight line fitted to the inner main arc has a position angle of −252 ± 05 (east of the declination axis).⁶ We use this axis to define θ_∥ and define θ_⊥ by a 90° clockwise rotation from it.

We have used three techniques to resolve the lobe ambiguity. The first relies on the frequency dependence of the positions obtained from each arclet apex. As discussed below in Section 3.3, we find that the angular positions of the scattered features that cause the arclets are essentially independent of frequency. Thus, we can use the fact that the correct lobe choice should give a position that is independent of frequency. We computed the two-dimensional mean for the 1 ms apexes from a weighted average position of its imaged points separately for each of the four sub-bands. For the lobe position at a negative R.A. offset in Figure 2, the mean position was independent of frequency within the errors of estimation (∼ 0.1 mas). All other lobe choices show systematic frequency dependence in their positions in agreement with that expected for a lobe error.

Confirmation comes from two further tests. The second technique relies on Equation (3b) relating delay to the angular offset of an arclet apex and is discussed in Section 4.1. Similarly, the third technique relies on Equation (3a) for the Doppler frequency and is discussed in Section 4.2.

Hill et al. (2005) found the wavelength scaling of the apex Doppler frequencies of the four arclets which they identified in pulsar B0834+06 to scale ∝λ⁻¹. Hence, using Equation (3a) they concluded that the scattering angle responsible for the arclet was independent of wavelength. We have done a similar analysis on the apex Doppler frequencies of all the identified arclets in our four sub-bands. In addition, we analyzed the wavelength scaling of the apex delay in the four sub-bands, which is ∝θ² as in Equation (3b). We combined the fits to estimate a single scaling parameter γ where θ ∝ λ^γ. We estimated γ separately for three groups of apexes and tabulate the values in Table 2.

The results show that the scattering responsible for each arclet originates at a location that is essentially independent of wavelength across the 10% range spanned by the four sub-bands. This is entirely incompatible with the value γ = 2, to be expected if the individual arclets came from separate ray paths caused by refraction due to differing gradients in the column density of electrons. The conclusion applies to all arclets in the SS, with the possible exception for the 1 ms feature whose position scales weakly λ^0.06. Such a weak scaling might apply if the arclet were caused by a lens-like concentration of plasma, such as that invoked by Romani et al. (1987) to explain ESEs, but with a transverse extent very much smaller than its distance from the direct path.

Equation (3b) relates delay τ to the angular offset θ for an arclet apex, for which the second angular component is at the origin. Thus,

$$\begin{equation} \hspace{.6pc}\tau = (D_{\rm eff}/2c) \theta ^2, \end{equation} \tag{ 8 }$$

where

$$\begin{equation} D_{\rm eff}=D_{\rm p} (1-\beta)/\beta. \end{equation} \tag{ 9 }$$

In Figure 3, we plot on logarithmic scales θ² from the astrometry against delay with the various groups of data from Figure 2. Note that there is a bias to θ² due to the uncertainties in right ascension and declination. Since we have estimates of those uncertainties we subtracted the bias before fitting.

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Two separate straight lines through the origin were fitted to the (blue) points sampled along the main parabola and from the 1 ms feature. The apex points were also fitted separately for three regions: for apex delays less than 0.4 ms, delays bigger than 0.4 ms with positive Doppler and with negative Doppler (i.e., the 1 ms feature). The fits were weighted proportional to the reciprocal of the standard deviation in θ². The derived effective distances are given in Table 3 with their formal 1σ errors for all four sub-bands.

The simplest model puts all of the scattering in a single relatively thin region at the same effective distance. Comparing D_eff for the differing regions of the SS, we find D_eff in the range 1100–1400 pc. Since we found no significant scaling of the apex θ with frequency (see Section 3.3), we combine data from all four sub-bands to create a combined estimate shown as a fifth row in the table. We note that there is a possible contribution to inconsistencies among the sub-bands from a partial narrowing of the two outer sub-bands at AO by a front-end bandpass filter.

The average of the estimates in Column 2 from delays less than 0.4 ms from both the main arc and from the arclet apexes is 1171 ± 23 pc. In Column 4, the 1 ms cluster gives 1301 ± 40 pc, which overlaps at ±2σ with the value 1121 ± 59 pc obtained from the 1 ms apexes. This small discrepancy appears in Figure 2 as an offset in the centroid of the blue and black points. Since the arclet apexes have been specifically identified so that one of the ${ \mbox{$\btheta $} }$ parameters can confidently be set to zero and since D_eff for the 1 ms apexes is within 1σ of those from the main arc, we conclude that the scattering region responsible the 1 ms feature is located at the same distance as that causing the main arc. Without the lobe-shift, a fit for D_eff gives values from 1700 to 2600 pc. The large discrepancy in this distance provides the second confirmation of the lobe-shift for the 1 ms points.

A possibly significant difference is seen between the apexes with positive Doppler and delays >0.4 ms in Column 3, which give a significantly larger D_eff = 1378 ± 60 pc. Whereas all of the sub-bands give a larger distance there are only a few identified arclets in each sub-band and they are of relatively low S/N in all four sub-bands so that accurate correction for the noise bias to θ² becomes more critical. Thus, the evidence is relatively weak for a different distance for these points at large delay and positive Doppler.

We now take D_eff = 1171 pc, combined with the pulsar distance of 640 pc in Equation (9) to obtain the fractional scattering distance β = 0.353 ± 0.005. This is consistent within 2σ of the value of 0.29 ±  0.04 obtained by Hill et al. (2005) using the same pulsar distance but without the advantage of knowing the full geometry of the scattering screen. However, its true uncertainty is dominated by uncertainty in the pulsar distance which is estimated from the dispersion measure. Dispersion measure distances are notoriously unreliable, especially for nearby pulsars and thus are only accurate to ∼40% (Brisken et al. 2002); this gives a screen distance range of 250–580 pc from the Earth.

An independent analysis is provided by Equation (3a), relating the Doppler frequency for the arclet apexes to their astrometric position. At each apex, the Doppler frequency can be written as

$$\begin{eqnarray} f_{\rm D} &=& { \mbox{$\btheta $} }\,{\bm{\cdot}}\, {\bf V}_{\rm eff}/\lambda \nonumber \\ &=& (\theta _{\parallel } V_{\rm eff\parallel } + \theta _{\perp } V_{\rm eff\perp })/\lambda, \end{eqnarray} \tag{ 10 }$$

where

$$\begin{equation} {\bf V}_{\rm eff} = {\bf V}_{\rm ISS}/\beta \approx {\bf V_{\rm p}}(1-\beta)/\beta. \end{equation} \tag{ 11 }$$

The final approximation ignores the velocities of the Earth and of the screen relative to that of the pulsar. Then, the ratio V_eff/D_eff estimates the pulsar proper motion independent of its distance and of β. However, here we do not make that approximation but use the measured proper motion to estimate the velocity of the scattering screen.

In Figure 4, we plot θ_∥ against f_D which would give a straight line whose slope gives an estimate of λ/V_eff∥, providing that the value of θ_⊥V_eff⊥ were the same or negligible for all points. As shown in Figure 2, the points fitted by the straight line have small θ_⊥ scattered about zero. Hence, we were able to estimate V_eff∥ from the various groups of points and tabulate the results in Table 3 excluding all the points with delays >0.4 ms.

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If we combine inner apex fits with values from the main arc, we obtain V_eff∥ = 305 ± 3 km s⁻¹. We now assume the same V_eff∥ to apply to the 1 ms apexes and since these points have a significant measured θ_⊥ we can fit for the perpendicular velocity also, obtaining V_eff⊥ = −145 ± 9 km s⁻¹.

In summary, the scattering model provides three observable parameters which we list in Table 4. We can express the velocities in R.A., decl. and relate them to the measured pulsar proper motion velocity V_p,α = 6.1 km s⁻¹, V_p,δ = 156 km s⁻¹. The derived model parameters are also listed in the table. Equation (11) relates the measured effective velocity to the scintillation velocity V_ISS which also depends on the transverse projected velocity of the Earth and the scattering screen. Using the pulsar velocity cited above gives an estimate of the ISM screen velocity relative to the Sun (∼16 km s⁻¹), which is comparable in magnitude to other measured interstellar velocities.

The ISM velocity analysis provides the third confirmation of the lobe-shift applied to the 1 ms astrometry, since in the alternate position V_eff⊥ is reversed and could only be reconciled with the pulsar velocity if there were an implausibly high velocity (>200 km s⁻¹) for the scattering screen relative to the Sun.

In the discussion so far we have assumed that the astrometry is correctly centered on the emission centroid. However, since the scattered image is highly anisotropic, the centering done by self-calibration in the primary analysis would have been more precise in the perpendicular direction than in the parallel direction. Close examination of the initial versions of the plots as in Figures 3 and 4 revealed small but significant asymmetries, which allowed us to improve the centering in the parallel direction.

In the Doppler plot of Figure 4 a straight line fit did not pass exactly through the origin and this fitted offset in θ_∥ was found to be on the order of 0.5 mas differing slightly between sub-bands. In the delay plot of Figure 3, the θ² values at a given delay with positive Doppler frequencies were systematically shifted relative to those with negative Doppler frequencies. We were able to minimize the χ² for the straight line by fitting for a positional offset in the parallel direction. We found that in each sub-band the optimum parallel offsets were consistent between the Doppler and delay estimation methods within their errors. An average of these two offsets was consequently applied to the astrometry in each sub-band before the analysis described in the two foregoing sections.

The synthesized beam size of our VLBI array (its basic angular resolution) was about 35 mas at 327 MHz. In the astrometry shown in Figure 2, we obtain positional accuracies as small as ±0.5 mas near the center of the image due to the high S/N and much larger errors away from the center. However, we can do even better from the apex positions of the arclets in the SS using Equations (8) and (10), which we will refer to as "back-mapping." This is possible since we used the VLBI astrometry to obtain accurate values for the model parameters D_eff, V_eff, and α. The equations give the magnitude of θ from the delay τ and the component θ_V parallel to V_ISS from f_D. There is an inherent ambiguity in such a mapping since it does not determine the sign of the angular coordinate orthogonal to the velocity. However, we can use the VLBI image to resolve that ambiguity and show the positions of each apex mapped back into in R.A., decl. in Figure 5 overplotted for the four sub-bands. The mapped astrometry points are plotted as red error boxes on a background of the error bars of the VBLI positions. The apex points mapped back in this fashion extend along the same axis as the VLBI positions but with a narrower spread in θ_⊥ due to their even smaller errors.

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With errors δτ and δf_D in the apex positions, we get positional errors δθ ∼ c δτ/(D_effθ) and δθ_V ∼ δf_Dλ/V_eff. With δτ ∼ 1 μs, we obtain δθ ∼ θ⁻¹_mas, i.e., 0.05 mas at θ_mas = 20 mas, which improves on the 0.5 mas from astrometry, particularly at large scattering angles where the S/N worsens; and with δf_D ∼ 1 mHz the resolution δθ_V ∼ 0.6 mas. The resulting confidence region is shown by the red box which is narrow in radial θ coordinate and wide in the transverse direction.

The map emphasizes the remarkable anisotropy of the scattering disk and also shows evidence that the 1 ms feature is also highly extended along a roughly parallel direction. This is an important aspect that will constrain any physical model for the scattering.

The scattered brightness distribution can be recovered from the SS if the intensity scintillations are weak, or if they are highly anisotropic (Cordes et al. 2006). In this case, the scintillations are highly anisotropic and we can make one-dimensional, strip integrated brightness distributions, B₁(θ_∥), by strip integrating the two-dimensional distribution over θ_⊥. Then, we have

$$\begin{eqnarray} &&B_1 (\theta _1) B_1 (\theta _2) = A(\tau,f_{\rm D}) |J|, \end{eqnarray} \tag{ 12 }$$

where τ = (θ²₁ − θ²₂)D_eff/2c, and f_D = (θ₁ − θ₂)V_effcos α/λ and the Jacobian |J| = f_DD_eff/c. Then, by sampling A(τ, f_D) along the main arc where θ₂ = 0, we can estimate B₁(θ). This estimate is plotted in Figure 6 without the 1 ms feature, which cannot be represented on this plot since it is not on the main parabola.

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If the scattering comes from homogeneous anisotropic Kolmogorov turbulence then the strip integrated brightness distribution is the Fourier transform of the one-dimensional correlation function of the electric field, B₁(θ_∥) = FT   [exp(−(s_∥/s₀)^5/3)]. We have plotted three such models over the observations in Figure 6, each with a different value of s₀, the coherence scale of the electric field. The middle model fits the data near the origin (s₀ ∼ 10⁴ km) and the other two give a rough estimate of the range of s₀ necessary to match the measurements. They correspond to changing the rms electron density by a factor of 2 (wider curve) or 0.5 (narrower curve). One can see in the expanded view on the right that near θ_∥ = −10 mas there is a change in the rms electron density by a factor of 4 in a very small distance.

The finest structure has an angular scale of about 0.1 mas, which corresponds to 0.05 AU. This plot can also be made using the VLBI astrometric θ_∥. In this case, the large-scale structure is the same but the small-scale structures seen clearly in Figure 6 are smoothed out by the larger error bars in the VLBI astrometry.

While we have clear evidence for highly anisotropic scattering, it is difficult to estimate the axial ratio R, which is usually defined by a contour at, say, half-power in the two-dimensional scattered brightness distribution. Estimates of the width in θ_∥ can be made from the one-dimensional brightness in Figure 6. However, the width at the same level in θ_⊥ is harder to estimate, since we have no information on brightness versus θ_⊥. As an alternative, we can define an apparent axial ratio R_ap from the scatter of the points in Figure 2 independent of their brightness. Here, we define R_ap as the ratio of the rms width in θ_⊥ to that in θ_∥. The observed scatter in θ_⊥ will of course be broadened by the astrometric errors, so that only an upper bound can be found on θ_⊥ due to ISS.

We apply these ideas to Figure 7, which shows the astrometry from the main arc as a scatter plot of θ_⊥ against θ_∥ superimposed from all four sub-bands. The black error bars show θ_⊥ averaged into 0.5 mas bins in θ_∥ where the length of the bar is the standard deviation in each average. Taken as a group the black points from the inner ±5 mas are consistent with a Gaussian distribution with zero mean and 0.38 mas rms; there are no points more than ±2σ from zero and their error bars, which have not been corrected for the measurement error, have a mean value of 0.3 mas. Thus, the true scatter in θ_⊥ is smaller than these errors, implying an intrinsic perpendicular rms width less than ∼0.3 mas. Taking the half-width at half-power in θ_∥ to be 3 mas from Figure 6 gives a lower bound on the axial ratio R ≳ 10. Similar estimates can be made from the ratios of the parallel to the perpendicular rms widths of the apex astrometry from both the VLBI and back-mapping methods. These yield R_ap ≳ 27 and R_ap ≳ 20, respectively.

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The scattering axis is neither parallel nor perpendicular to the Galactic plane. However, according to the Viriginia Tech Spectral-Line Survey⁷ the pulsar lies ∼1° from a 5° long bright Hα filament at a position angle within about 10° of our scattering axis. Although there is a rough agreement in position angle we have no other evidence to support an association with our 30 mas long filament.