The Scattering and Intrinsic Structure of Sagittarius A* at Radio Wavelengths

The basic properties of interstellar scattering have been summarized in several reviews (e.g., Rickett 1990; Narayan 1992; Thompson et al. 2017), and our specific scattering model is derived and discussed in detail in a companion paper (Psaltis et al. 2018). Here, we will only briefly summarize the key properties that are of immediate relevance for the remainder of this paper.

Interstellar scattering and scintillation at radio wavelengths is caused by density inhomogeneities in the ionized ISM. Neglecting a weak birefringence from the magnetic field (which is negligible for the observing wavelengths we consider), the local index of refraction of the ISM is given by $n\approx 1-\tfrac{1}{2}{\left(\tfrac{{\nu }_{{\rm{p}}}}{\nu }\right)}^{2}$, where ν is the wave frequency, ${\nu }_{{\rm{p}}}\approx 9.0\times \sqrt{\tfrac{{n}_{{\rm{e}}}}{1\,{\mathrm{cm}}^{-3}}}\,\mathrm{kHz}$ is the plasma frequency, and n_e is the electron density (see, e.g., Thompson et al. 2017). A density fluctuation δn_e along a path length dz then introduces a corresponding phase change $\delta \phi =-{r}_{{\rm{e}}}\lambda \times {dz}\times \delta {n}_{{\rm{e}}}$, where ${r}_{{\rm{e}}}\approx 2.8\times {10}^{-13}\,\mathrm{cm}$ is the classical electron radius and λ is the wavelength. Note that this dispersion relation is quite general and is independent of a specific ISM scattering model or geometry.

Along many lines of sight, the effects of scattering can be approximated via a single thin phase screen $\phi ({\boldsymbol{r}})$, where ${\boldsymbol{r}}$ is a two-dimensional vector transverse to the line of sight. Electron density fluctuations imprint their spectrum on the power spectrum $Q({\boldsymbol{q}})$ of phase fluctuations, which are typically characterized by a single, unbroken power law between some outer (r_out) and inner (r_in) scales: $Q({\boldsymbol{q}})\propto {| {\boldsymbol{q}}| }^{-\beta }$. This description is expected for a top-down turbulent cascade between an injection scale and a dissipation scale, and a Kolmogorov spectrum of density fluctuations gives β = 11/3 (Goldreich & Sridhar 1995).

The effects of scattering on interferometric measurements are conveniently characterized using the phase structure function of the scattering screen, ${D}_{\phi }({\boldsymbol{r}})\equiv \langle {[\phi ({\boldsymbol{r}}^{\prime} +{\boldsymbol{r}})-\phi ({\boldsymbol{r}}^{\prime} )]}^{2}\rangle \propto {\lambda }^{2}$. In the ensemble-average scattering limit (see Goodman & Narayan 1989; Narayan & Goodman 1989), the effects of scattering are to convolve an unscattered image with a scattering kernel or, equivalently, to multiply unscattered interferometric visibilities by the appropriate Fourier-conjugate kernel. This kernel is given by $\exp \left[-\tfrac{1}{2}{D}_{\phi }({\boldsymbol{b}}/(1+M))\right]$, where ${\boldsymbol{b}}$ is the vector baseline of the interferometer and M = D/R is the "magnification" of the scattering screen (D is the observer–screen distance; R is the source–screen distance). For spatial displacements smaller than r_in, the phase fluctuations will be smooth, $\phi ({\boldsymbol{r}}^{\prime} +{\boldsymbol{r}})\approx \phi ({\boldsymbol{r}}^{\prime} )+{\boldsymbol{r}}\cdot {\rm{\nabla }}\phi ({\boldsymbol{r}}^{\prime} )$. In this limit, ${D}_{\phi }({\boldsymbol{r}})\propto {r}^{2}{\lambda }^{2}$ (Tatarskii 1971). This expression—which makes no assumptions other than the cold plasma dispersion relation and smoothness of phase fluctuations below some scale—shows that ensemble-average scatter-broadening should act as a (possibly anisotropic) Gaussian blurring that scales as ${\theta }_{\mathrm{scatt}}\propto {\lambda }^{2}$ for baselines ${\boldsymbol{b}}\lesssim (1+M){r}_{\mathrm{in}}$ (i.e., on angular scales $\theta \gtrsim \lambda /((1+M){r}_{\mathrm{in}})$. Moreover, because the time-averaged scattering kernel from an ensemble of thin screens is determined by the cumulative convolution of all the individual screens, this generic asymptotic behavior is not limited to thin-screen scattering. At longer baselines, ${D}_{\phi }({\boldsymbol{r}})\propto {| {\boldsymbol{r}}| }^{\alpha }$, where $\alpha \equiv \beta -2$, and the corresponding image becomes non-Gaussian. In this regime, the angular broadening scales as ${\theta }_{\mathrm{scatt}}\propto {\lambda }^{1+\tfrac{2}{\alpha }}$ and the interferometric scattering kernel falls as ${e}^{-{| {\boldsymbol{b}}| }^{\alpha }}$. However, note that ${D}_{\phi }({\boldsymbol{r}})\propto {\lambda }^{2}$ regardless of α or the scattering model.

While scatter broadening is produced by phase fluctuations on the diffractive scale¹⁷ ${r}_{\mathrm{diff}}\sim \lambda /((1+M){\theta }_{\mathrm{scatt}})$, refractive scintillation is dominated by modes that are comparable to the refractive scale (i.e., the projected size of angular broadening on the scattering screen): ${r}_{\mathrm{ref}}\sim {\theta }_{\mathrm{scatt}}D$. The Fresnel scale ${r}_{{\rm{F}}}\equiv \sqrt{\tfrac{{DR}}{D+R}\tfrac{\lambda }{2\pi }}$, which is defined entirely by geometrical parameters of the scattering, corresponds to the geometric mean of the diffractive and refractive scales. When ${r}_{\mathrm{ref}}\gt {r}_{{\rm{F}}}\gt {r}_{\mathrm{diff}}$, the scattering is said to be "strong" (e.g., the scattering of Sgr A* is strong for all frequencies lower than a few THz). In this case, refractive effects are most naturally described using the power spectrum of phase fluctuations: $Q{({\boldsymbol{q}})\equiv (2\pi )}^{2}{\lambda }^{-2}\int {d}^{2}{\boldsymbol{r}}\langle \phi ({\boldsymbol{r}}^{\prime} +{\boldsymbol{r}})\phi ({\boldsymbol{r}})\rangle {e}^{-i{\boldsymbol{q}}\cdot {\boldsymbol{r}}}$. In this expression, the prefactor renders $Q({\boldsymbol{q}})$ independent of wavelength.

To describe a full scattering model then requires six parameters. Three are needed to characterize the long-wavelength behavior (Gaussian scatter broadening with a λ² dependence). As described before, we use the FWHM along the major and minor axes at a reference wavelength and the major axis position angle: θ_maj,0, θ_min,0, and ϕ_PA. In addition, the power-law index α, inner scale r_in, and outer scale r_out are needed. Psaltis et al. (2018) show how to compute the phase structure function, power spectrum, and scattering properties once these parameters are specified.

We caution that the exact specification of these parameters is not unique, and the radio scattering literature is particularly inconsistent in defining the inner scale. For example, Rickett (1990) and Smirnova & Shishov (2010) taper the power spectrum by ${e}^{-{q}^{2}{r}_{\mathrm{in}}^{2}}$, Coles et al. (1987) and Armstrong et al. (1995) use ${e}^{-{q}^{2}{r}_{\mathrm{in}}^{2}/2}$, Lambert & Rickett (1999) use ${e}^{-{q}^{2}{r}_{\mathrm{in}}^{2}/4}$, and Spangler & Gwinn (1990) use ${e}^{-{{qr}}_{\mathrm{in}}/(2\pi )}$. We use a power spectrum taper of the form ${e}^{-{q}^{2}{r}_{\mathrm{in}}^{2}}$.

Refractive scattering modes introduce many types of stochastic effects. They modulate the total flux density of an image, displace its centroid, and distort the overall image. They also introduce image substructure, even on scales for which the unscattered source was smooth. All of these effects introduce a new type of "noise" for interferometric measurements. This refractive noise has a fractional bandwidth of order unity and varies slowly, on the refractive timescale t_ref = r_ref/V_⊥, where V_⊥ is the characteristic relative transverse velocity of the observer, scattering, and source. At centimeter wavelengths, Sgr A* has ${r}_{\mathrm{ref}}\approx (2\times {10}^{13}\,\mathrm{cm})\times {\lambda }_{\mathrm{cm}}^{2}$. Taking V_⊥ ∼ 50 km s⁻¹ gives ${t}_{\mathrm{ref}}\sim (50\,\mathrm{days})\times {\lambda }_{\mathrm{cm}}^{2}$.

Johnson & Gwinn (2015) and Johnson & Narayan (2016) provide expressions for how to compute properties of refractive noise, including the variance of refractive fluctuations of a complex visibility measured on a vector baseline ${\boldsymbol{b}}$: ${\sigma }_{\mathrm{ref}}^{2}({\boldsymbol{b}})\,\equiv \langle {| {\rm{\Delta }}V({\boldsymbol{b}})| }^{2}\rangle$. However, for short baselines, this variance is not the correct quantity to apply to standard very long baseline interferometry (VLBI) analyses. Namely, part of the variance is due to variations in the total flux density, caused by refractive focusing, and part is due to image wander, caused by refractive deflections. Both of these effects would be eliminated in a typical VLBI analysis. The flux variations would be absorbed into the estimated total source flux density, and the image wander would be eliminated by centering the image (since VLBI has no concept of absolute position without absolute phase referencing).

Appendix A shows how to compute a renormalized visibility variance, ${\hat{\sigma }}_{\mathrm{ref}}({\boldsymbol{b}})$, that eliminates these contributions. For example, the renormalized refractive noise is zero in the limit of zero baseline. On short baselines, it is dominated by changes in the overall image size from scattering—a property that we utilize in Section 5.2.2. We will include renormalized refractive noise in the error budget for our model fitting.

We will use a few supplementary measurements and assumptions about the scattering of Sgr A*. The first is for the scattering geometry of Sgr A*. Because the Galactic Center magnetar lies only 24 from Sgr A* (Bower et al. 2015a), its radio pulsations permit an estimate of temporal broadening associated with the scattering toward Sgr A*. This measurement can be combined with the angular broadening to estimate the location of the scattering material (Gwinn et al. 1993). For instance, if the scattering is isotropic, then the pulse-broadening function is exponential: g(t) = e^−t/τ. This expression follows by relating a radial distance r on the scattering screen to its corresponding geometric delay, $t(r)\approx \tfrac{{r}^{2}}{2c}\left(\tfrac{1}{D}+\tfrac{1}{R}\right)$, and then expressing the brightness distribution on the sky in terms of t. The 1/e scale of the temporal broadening, τ, is related to the FWHM angular size of the scattered image, θ_scatt, via (e.g., Cordes & Lazio 1997)

$$\begin{eqnarray}&&c\tau =\displaystyle \frac{{{Md}}_{\mathrm{src}}}{8\mathrm{ln}(2)}{\theta }_{\mathrm{scatt}}^{2},\end{eqnarray} \tag{ 1 }$$

where d_src = D + R is the distance from the observer to the source, and M = D/R. Because the magnetar shows angular broadening comparable to Sgr A*, the same scattering material is thought to dominate the angular broadening of each (Bower et al. 2014a, 2015a). The temporal broadening of the magnetar can then be combined with the angular broadening and distance to Sgr A* to estimate the location of the scattering material for both objects.

For anisotropic scattering, the pulse-broadening function is monotonically decreasing but not exponential. For a scattered image with FWHM θ_maj and θ_min along the major and minor axes, the pulse-broadening function takes the form¹⁸

$$\begin{eqnarray}\begin{array}{rcl}g(t) & = & {I}_{0}\left(\displaystyle \frac{4\mathrm{ln}(2)\mathrm{ct}}{{{Md}}_{\mathrm{src}}{\theta }_{-}^{2}}\right){e}^{-\tfrac{4\mathrm{ln}(2)\mathrm{ct}}{{{Md}}_{\mathrm{src}}{\theta }_{+}^{2}}},\\ {\theta }_{\pm } & \equiv & \displaystyle \frac{{\theta }_{\mathrm{maj}}{\theta }_{\min }}{\sqrt{{\theta }_{\mathrm{maj}}^{2}\pm {\theta }_{\min }^{2}}}.\end{array}\end{eqnarray} \tag{ 2 }$$

In this case, determining τ (i.e., solving g(τ) = g(0)/e) must be done numerically.

We will assume a distance to Sgr A* of 8.1 kpc (Gravity Collaboration et al. 2018). Using our estimated scattering kernel parameters (see, e.g., Table 2), we obtain ${\tau }_{1\mathrm{GHz}}/(1\,{\rm{s}})\,\approx 2.47M$. Using the measured value ${\tau }_{1\mathrm{GHz}}=1.3\pm 0.2\,{\rm{s}}$ (Spitler et al. 2014) then gives M = 0.53 ± 0.08. Note that this estimate differs slightly from the simpler approach of using the isotropic scattering result with the geometric mean of the major and minor scattering axes, which gives ${\tau }_{1\mathrm{GHz}}/(1\,{\rm{s}})\approx 2.77M$ (Bower et al. 2014a). Using the exact expression for anisotropic scattering, we obtain D = 2.7 kpc and R = 5.4 kpc.

There is now compelling evidence that at least some of the temporal broadening of the magnetar is local to the Galactic Center region (see, e.g., Dexter et al. 2017; Desvignes et al. 2018). Because angular broadening is more sensitive to scattering material closer to the observer, it is likely that the angular broadening and refractive effects are dominated by the scattering region that is distant from the Galactic Center. Because the temporal broadening caused by this material may be smaller than the value used above, the corresponding M for the scattering responsible for the angular broadening may be somewhat lower and the scattering material somewhat further from the Galactic center. However, our later results are insensitive to changes in M. Refractive noise scales as ${\sigma }_{\mathrm{ref}}\propto {M}^{-1+\tfrac{\alpha }{2}}$ (e.g., ${\sigma }_{\mathrm{ref}}\propto {M}^{-1/6}$ for a Kolmogorov spectrum), while our later inner scale constraint is proportional to (1 + M)⁻¹. Thus, even a change in our assumed temporal broadening by a factor of 2 would not strongly affect our conclusions, and so we will work within the single-screen framework for the remainder of this paper.

Our second assumption is that the outer scale for the scattering of Sgr A* is sufficiently large to be irrelevant for our calculations (specifically, we require ${r}_{\mathrm{out}}\gg 10\,\mathrm{au}$). We will discuss the validity of this assumption in Section 6.2.1.

One significant simplification in our model fitting approach is that we model the brightness distribution of the source on the sky as a wavelength-dependent anisotropic Gaussian. In Section 2, we demonstrated that this assumption is well motivated for the shape of the scatter broadening because it corresponds to universal scattering behavior in the limit of long wavelength. Moreover, our approach to estimate parameter uncertainties uses the full, non-Gaussian scattering model, so our final error budget accounts for limitations in the assumption of Gaussian scatter broadening. However, the intrinsic source may be non-Gaussian, especially when the emission region becomes optically thin. Nevertheless, even in this regime, the Gaussian source assumption is well motivated for model fitting and has a meaningful associated FWHM, as we will now demonstrate.

Specifically, the interferometric visibility $\tilde{I}({\boldsymbol{u}})$ on a short baseline ${\boldsymbol{u}}$ can be approximated as

$$\begin{eqnarray}\begin{array}{rcl}\tilde{I}({\boldsymbol{u}}) & = & \displaystyle \int {d}^{2}{\boldsymbol{x}}\,I({\boldsymbol{x}})\,{e}^{-2\pi i{\boldsymbol{u}}\cdot {\boldsymbol{x}}}\\ & \approx & \displaystyle \int {d}^{2}{\boldsymbol{x}}\,I({\boldsymbol{x}})[1-2\pi i{\boldsymbol{u}}\cdot {\boldsymbol{x}}-2{\pi }^{2}{({\boldsymbol{u}}\cdot {\boldsymbol{x}})}^{2}],\end{array}\end{eqnarray} \tag{ 3 }$$

where $I({\boldsymbol{x}})$ denotes the image, with ${\boldsymbol{x}}$ an angular coordinate on the sky (Thompson et al. 2017). The term that is linear in ${\boldsymbol{u}}$ reflects an interferometric phase that is proportional to the image centroid projected along the baseline direction (from the Fourier shift theorem). Standard VLBI observations (including all those used in this paper) do not have absolute phase information, so we can set this term to zero (i.e., we use the image centroid to define the origin of the sky coordinates). The remaining terms in Equation (3) show that the visibility amplitude decreases quadratically with baseline length for short baselines. The quadratic coefficient is proportional to the second moment of the image projected along the baseline direction. This second moment can then be used to define a characteristic image FWHM, using the relationship corresponding to a perfectly Gaussian image. For example, the major axis FWHM θ_maj is given by

$$\begin{eqnarray}&&{\theta }_{\mathrm{maj}}=\sqrt{-\displaystyle \frac{2\mathrm{ln}(2)}{{\pi }^{2}\tilde{I}(0)}{{{\rm{\nabla }}}_{{\hat{{\boldsymbol{u}}}}_{\mathrm{maj}}}^{2}\tilde{I}({\boldsymbol{u}})\rfloor }_{{\boldsymbol{u}}=0}},\end{eqnarray} \tag{ 4 }$$

where $\tilde{I}(0)$ is the total flux density of the image, and ${{\rm{\nabla }}}_{{\hat{{\boldsymbol{u}}}}_{\mathrm{maj}}}^{2}$ is the second directional derivative along the major axis direction. The three characteristic Gaussian parameters {θ_maj, θ_min, ϕ} can be determined by diagonalizing the image covariance matrix.

For this universal Gaussian behavior for the source visibility function to be applicable, the baselines must only marginally resolve the unscattered source. For Sgr A*, this assumption can be assessed post hoc using the inferred intrinsic size. Using the characteristic size ${\theta }_{\mathrm{src}}\sim (0.4\,\mathrm{mas})\times {\lambda }_{\mathrm{cm}}$ that we derive later (see Section 5.3), we estimate that projected baselines must have a length of approximately 3000 km for the normalized visibility function of the intrinsic source to fall to 1/e (this length is independent of wavelength because the source grows linearly with wavelength while angular resolution scales inversely with wavelength). For all observations we analyze other than 1.3 and 3.5 mm, the longest baselines are significantly shorter than this limit (because longer baselines heavily resolve the scattered source). Thus, for the wavelengths we analyze to estimate the scattering kernel (λ ≥ 7 mm), the quadratic expansion of Equation (3) and Gaussian approximation are well motivated for the intrinsic structure of Sgr A*.

To estimate the scattering kernel of Sgr A*, we independently fit anisotropic Gaussian models to observations of Sgr A* at wavelengths from 1.3 mm to 30 cm. In principle, fitting a Gaussian to interferometric data is quite simple. In practice, the fitting is subtle and subject to many sources of noise and bias. These include thermal noise, station-based systematic errors in amplitude and phase, and refractive noise. We developed a simplified prescription for Gaussian model fitting that accounts for all these errors. Our prescription is motivated by tests on synthetic data sets (see Section 3.3); it sacrifices some exactness for the sake of computational efficiency. Nevertheless, our approach provides a reliable error budget despite shortcomings in the model fitting procedure.

For each observation, we began with complex visibilities that had a priori amplitude calibration applied but no self calibration. We first flagged all visibilities for which the elevation at either station was below 5°. Next, on a scan-by-scan basis, we flagged all stations that did not have a signal-to-noise ratio (S/N) of at least 12 on any baseline. Thus, at each time, a station was only included if it had at least one strong fringe detection. This station-based cut was chosen to avoid including visibilities with a noise bias from the fringe search; a baseline-based cut would also avoid spurious fringes but would potentially bias the set of unflagged, low-S/N visibility amplitudes upward. Next, we computed the expected renormalized refractive noise (see Section 2.2) for each point, and eliminated all visibilities for which the ensemble-average visibility function was less than four times the renormalized refractive noise. This cut eliminated visibilities that were dominated by refractive noise from the Gaussian model fits (we only performed this final cut for the Gaussian model fitting and included these visibilities for estimates of the long-baseline refractive noise).

Next, we jointly fit for complex, time-dependent station gains at every site and the Gaussian image parameters (i.e., self-calibration to a model), seeking the maximum a posteriori estimate of all parameters. For this estimate, we used flat priors on the station phases and Gaussian priors on the logarithm of the gain amplitude, centered on a gain of amplitude of unity and with wavelength- and array-dependent spread. We used 5% uncertainty for VLA data at 15–30 cm, 5% uncertainty at 3.6 cm for Very Long Baseline Array (VLBA) data, and 10% uncertainty at 1.3 cm (VLBA) and 7 mm (Korean VLBI Network and VERA Array (KaVA)). At 3.5 mm, the a priori calibration is sufficiently poor that we do not constrain the gain amplitudes (similar to an analysis using only closure quantities). These values can be validated after fitting the actual data and were guided by the expected performance for each wavelength/array combination. We assumed that the visibilities had complex Gaussian random noise, with a standard deviation that was the quadrature sum of the measured thermal noise and the renormalized refractive noise. In this way, we included additional tolerance for visibility errors from refractive noise.

To estimate uncertainties for the fitted parameters, we used a Monte Carlo approach. Namely, for each data set analyzed, we generated a representative ensemble of synthetic data sets and analyzed each using our procedure for the actual data. To create synthetic data sets, we scattered Gaussian source images using the stochastic-optics module of the EHT-imaging library (Chael et al. 2016; Johnson 2016). The source and scattering parameters were chosen to match the current best estimates in our iterative fitting procedure (see Section 3.4). We then sampled each scattered image on the observed (u, v) coordinates and added complex Gaussian noise that was equal to the measured thermal noise. Next, we injected two types of gain uncertainty to the measurements: (1) fluctuations of the station gains that were uncorrelated from scan to scan, and (2) an overall uncertainty in each station gain that was constant over the entire observation but different among the different synthetic data sets. Each of these gain errors was a Gaussian random variable with unit mean and wavelength-dependent uncertainty, matching the values given Section 3.2.

As a concrete example, a single realization of the simulated 1.3 cm VLBA data would have rapid jitter from thermal noise that was uncorrelated among all baselines and times, rapid station-based jitter from the gain errors (rms of $\sqrt{2}\times 10 \%$ of each visibility amplitude), and a constant station-based error (rms of $\sqrt{2}\times 10 \%$ of each visibility amplitude). For instance, all baselines to a particular antenna might be systematically underestimating the true flux density in one realization and overestimating in another. Each realization also produced an image with FWHM fluctuations from refractive image distortions and with additional noise on long baselines from refractive substructure.

To estimate our parameter uncertainties, we compute the rms of the parameter estimates from each simulated data set with respect to the true, ensemble-average parameter. Thus, our uncertainties account for thermal scatter in the model fitting, for systematic scatter from the refractive scattering, and for systematic errors and bias in the model fitting procedure.

Real data have additional imperfections that our simplified prescription does not capture, including bandpass errors, polarimetric leakage, and gain errors that are elevation dependent. However, the polarization of Sgr A* is negligible at cm wavelengths, and residual bandpass errors are small. As we will demonstrate, the dominant source of uncertainty for many of our measurements is refractive scattering, and our Monte Carlo approach fully accounts for this uncertainty.

As described in the previous sections, we independently fit Gaussian models to data at multiple frequencies. However, these fits used refractive noise corresponding to the scattering properties that are estimated using the full multi-frequency data set. Thus, our overall fitting procedure is iterative.

• 1.  
  We fit Gaussian models to the 1.3 and 3.6 cm data. These fits require estimates of r_in and α to determine the refractive noise to include in the model fitting procedure and in the Monte Carlo uncertainty estimation via synthetic data. We assume that the scattering dominates the intrinsic size at these wavelengths (as is supported by the λ² scaling), so we use these fits to estimate the three parameters that characterize the long-wavelength scattering behavior: θ_maj,0, θ_min,0, ϕ_PA.
• 2.  
  Keeping θ_min,0 and ϕ_PA fixed at the values obtained in step 1, we fit the 15–30 cm VLA data to obtain a tighter constraint on θ_maj,0.
• 3.  
  Having determined the three parameters of the long-wavelength ensemble-average image in steps 1 and 2, we use four additional pieces of evidence to constrain r_in and α.

  • (a)  
    The nearly perfect scaling of image size as λ² down to 1.3 cm and across the observing bandwidth at 1.3 cm, combined with constancy of position angle and image anisotropy over this frequency range. These properties suggest that scattering must dominate over intrinsic structure at all wavelengths longer than 1.3 cm and that the inner scale must exceed the diffractive scale at 1.3 cm.
  • (b)  
    The Gaussian scaling of visibility amplitude with baseline length at 1.3 cm and tentative non-Gaussian scaling of visibility amplitude with baseline length at 7 mm.
  • (c)  
    The magnitude of refractive visibility noise using long-baseline measurements at 1.3 cm and 3.6 cm, where the Gaussian image contribution is negligible.
  • (d)  
    An upper limit of 3% on image size fluctuations at 7 mm, as determined by historical data.
• 4.  
  We then repeat steps 1–3 using the full scattering model (θ_maj,0, θ_min,0, ϕ_PA, α, r_in) estimated in the previous pass to estimate a new set of scattering parameters.

The longest-wavelength observations we examine are with the Karl G. Jansky VLA. These observations span wavelengths from 15 to 29 cm. The recorded bandwidth was divided into 16 spectral windows, each 64 MHz. Of the 16 original windows, four were flagged by the VLA calibration pipeline in CASA. We analyzed each spectral window independently. For each, we averaged in frequency and in one minute intervals. Figure 1 shows representative baseline coverage for one spectral window.

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The long-wavelength data are subject to a challenge for Gaussian model fitting that does not affect our other observations. Namely, short baselines measure significant flux density that is not associated with Sgr A*; it is diffuse emission from the local Galactic Center environment. To eliminate contributions from this emission, we imposed a minimum baseline length u_min for visibilities used in the Gaussian fits. On baselines longer than ∼50 kλ, we did not see any indications of contaminating emission (e.g., via non-zero closure phases). Thus, we repeated Gaussian fits using u_min = 60 kλ, 80 kλ, and 100 kλ, and we then used the scatter of these solutions as our estimate for the measurement uncertainty.

Because the VLA baselines only modestly resolve Sgr A*, we did not find stable results among the different spectral windows when fitting all Gaussian parameters separately. In addition, the fitted position angle was highly degenerate with the major axis size; smaller position angles produce a smaller major axis size because of the anisotropic baseline coverage (see Figure 1). To mitigate this problem, we instead fit the Gaussian parameters holding the minor axis and position angle fixed to the values determined by 3.6 and 1.3 cm observations (i.e., we only fit for the total flux density and major axis size in each spectral window). With this reduction, we obtained self-consistent estimates of the major axis among all spectral windows (see Table 1).

Fitting a single λ² scattering law to our measured major axis sizes yields θ_maj,0 = 1.3799 ± 0.0067 mas. This uncertainty does not account for refractive scattering effects, and it underestimates thermal uncertainty because the measurements with varying u_min have identical thermal noise. The uncertainty in the assumed position angle, σ_PA ∼ 02, gives an additional uncertainty of $\approx (0.004\,\mathrm{mas})\times ({\sigma }_{\mathrm{PA}}/1^\circ )$, which is negligible. The uncertainty in the assumed minor axis is likewise negligible.

To estimate the total uncertainty, we repeated the Gaussian fits using multi-frequency sets of synthetic data generated from 10 simulated realizations of the scattering. For each realization, we estimated a scattering law from the fitted Gaussian parameters. Note that while these data do not include diffuse emission, we used the same procedure with cutoffs of ${u}_{\min }=60\,k\lambda$, $80\,k\lambda$, and $100\,k\lambda$. These estimates of ${\theta }_{\mathrm{maj},0}$ had a scatter of 0.011 mas relative to the ensemble-average value for the simulations. This scatter accounts for refractive noise, thermal noise, and systematic noise from gain errors, but it does not account for systematic uncertainty from diffuse structure. Adding our two estimated uncertainties at quadrature yields our final estimate: ${\theta }_{\mathrm{maj},0}=1.380\pm 0.013\,\mathrm{mas}$.

We also reanalyzed the VLA observations reported in Bower et al. (2006) using the same procedure as for the VLA observations. These observations included the single VLBA antenna at Pie Town in addition to the VLA, thereby extending the longest baselines by a factor of ≈2. However, they had the disadvantage of a radio transient located only 27 south of Sgr A*, with a flux density that was ∼5% of Sgr A* (Bower et al. 2005). For our analysis, we adopted an image-domain approach to remove the transient. Namely, we performed maximum entropy imaging independently for each sub-band. For each image, we then computed the interferometric visibilities for the transient by windowing the image on a region of radius 18 centered on the transient. We subtracted these from the measured visibilities (because the Fourier relationship is linear) and used the remainder for Gaussian model fitting to Sgr A*. With this procedure, we found ${\theta }_{\mathrm{maj},0}=1.4082\pm 0.0075$. This value is at modest tension (1.9σ) with the VLA-only results, especially because refractive effects are likely correlated between the two epochs (at these wavelengths, the refractive timescale is ∼100 yr), so each of the two measurements would be similarly biased. Because uncorrected contamination from the transient may bias the measured Gaussian values for Sgr A*, we adopted the measurement and uncertainty of the VLA-only results.

We analyzed observations at λ = 3.6 cm taken with the VLBA in 2014. These observations also included the Green Bank Telescope (GBT), but we did not detect fringes between the GBT and the inner VLBA, so we only used the inner six VLBA stations (Brewster: BR; Fort Davis: FD; Kitt Peak: KP; Los Alamos: LA; Owens Valley: OV; Pie Town: PT) for our analysis. These observations recorded four contiguous 128 MHz channels and spanned approximately 3.5 hr. They used NRAO 530 as a calibration source. After a global fringe search in AIPS (Greisen 2003), we averaged the data in frequency and in 30 s intervals before Gaussian fitting.

Even without detailed analysis, the effects of refractive substructure are evident in the closure phases of these data. On triangles that resolve the source, the closure phases are markedly non-zero, demonstrating that the underlying image is inconsistent with any smooth, scatter-broadened structure (see Figure 2). Our Gaussian fitting procedure gives the major and minor axis sizes to within an estimated uncertainty of less than 2%, even when including refractive effects in the error budget (see Table 1).

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After the Gaussian fits, we self-calibrated the full data set to the Gaussian model. However, this procedure must be done with care because the longest baselines are dominated by refractive noise and are inconsistent with the pure Gaussian model. While our approximate prescription for model fitting with substructure (i.e., simply inflating the thermal noise with the rms renormalized refractive noise; see Section 3.2) gives reliable results for Gaussian model fitting, we found that it could downward bias long-baseline visibilities. To perform the self-calibration without biasing long-baseline visibility amplitudes, we first derived time-dependent gain solutions using only "short" baselines, for which the Gaussian model visibility was four times the renormalized refractive noise. We then applied these self-calibration solutions to all baselines. We dropped any visibilities that did not have simultaneous self-calibration solutions for both stations. In this way, long baselines that are dominated by refractive noise still obtain reliable self-calibration to the Gaussian model. Figure 3 shows our 3.6 cm data after self-calibration in this way. However, these data only have eight baselines with consistently strong detections (and there are six stations to self-calibrate), so we cannot reliably synthesize an image.

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We analyzed observations at λ = 1.3 cm taken with the VLBA+GBT in 2014. These observations were analyzed in Gwinn et al. (2014), who reported the initial discovery of refractive substructure in Sgr A*. As with the 3.6 cm data, these observations recorded four contiguous 128 MHz channels, spanned approximately 3.5 hr, and used NRAO 530 as a calibration source. They include strong detections to the VLBA antennas at North Liberty (NL) and Hancock (HN) in addition to the sites noted in Section 4.2. After a global fringe search in AIPS (Greisen 2003), we averaged the data in frequency and in 30 s intervals before Gaussian fitting. However, we averaged the data to four 128 MHz sub-bands and separately analyzed each.

For these data, the overall baseline coverage is well matched to the scattered image, and we can tightly constrain the Gaussian parameters separately within each sub-band. For each, the thermal uncertainty on the major axis size is less than 0.1%, and we clearly identify the λ² scaling of image size across the four sub-bands (see Figure 4). However, the uncertainty from refractive distortion is an order of magnitude larger, so we can only constrain the ensemble-average FWHM to within approximately 1%.

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For these data, there is sufficient baseline coverage to reliably synthesize an image. Figure 5 shows a maximum entropy image reconstruction using the EHT-imaging library (Chael et al. 2016). The effects of refractive substructure are evident in the substructure of the image. However, the effects of substructure are most striking in the visibility domain, where long baselines from GBT to the inner VLBA give strong detections on baselines for which the Gaussian visibility contribution is negligible. Figure 6 shows the final self-calibrated visibilities, following the procedure described in Section 4.2.

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The KaVA array has been conducting regular monthly monitoring of Sgr A* at 7 mm since 2014 September as part of the KaVA AGN large program (Kino et al. 2015; Zhao et al. 2017). The KaVA baselines range from 300 to 2300 km and provide excellent (u, v) coverage for Sgr A* observations (Figure 7; see also Akiyama et al. 2014). In particular, the KaVA coverage along the north–south direction is significantly better than VLBA coverage at this frequency, so the KaVA data are better suited to estimate the minor axis size.

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We analyzed data from the experiment r14308a, which were obtained in 2014 November. The data were recorded with 256 MHz total bandwidth, spanned 5.5 hr, and had an on-source time for Sgr A* of 220 min. NRAO 530 and two nearby SiO masers (OH 0.55–0.06, VX Sgr) were observed as calibrators (Cho et al. 2017). The correlated data were analyzed with AIPS in a standard pipeline. Most stations had good fringe detections in this experiment. After a global fringe search, the data were averaged in 30 s intervals and across the entire bandwidth for Gaussian model fitting. See G.-Y. Zhao et al. (2018, in preparation) for more details of the monitoring and data analysis.

Figure 7 shows the data from this observation, after our Gaussian model fitting and self-calibration. For these observations, the contribution of renormalized refractive noise is insignificant and is only comparable to the thermal noise on the longest baselines.

For λ = 3.5 mm, we analyzed data from the first VLBI observations using the Large Millimeter Telescope Alfonso Serrano (LMT) in concert with the VLBA. These observations recorded 480 MHz of bandwidth and spanned 7.5 hr (with approximately 3.4 hr on Sgr A*). We averaged the data in 10 s intervals and across the full bandwidth before Gaussian model fitting. For additional details about these observations, see Ortiz-León et al. (2016), who originally reported and analyzed them.

Using our Gaussian fitting procedure, we found values and uncertainties very close to those reported in Ortiz-León et al. (2016) using self-calibration. This agreement is expected because the only significant adaptation in our current approach is to include refractive noise, and the renormalized refractive noise is less than the thermal noise for all points. More recent data, reported by Brinkerink et al. (2016), also include the GBT and shows marked non-Gaussianity in the closure phases. For these data, which achieve significantly better sensitivity, including refractive noise in the error budget for model fitting may be significant.

Since 2007, the EHT has observed Sgr A* using a 1.3 mm VLBI array with stations in California, Arizona, and Hawaii. Recently, Lu et al. (2018) reported observations that included a fourth station (in Chile). With only three or four stations, there is insufficient baseline coverage to create an image. In addition, the measured visibilities are markedly non-Gaussian (Johnson et al. 2015; Fish et al. 2016; Lu et al. 2018), as expected because of complex intrinsic structure from optically thin emission near the black hole. Nevertheless, even under these circumstances, the image FWHM is still a meaningful quantity that is reliably constrained by sparse coverage because it represents universal behavior of the interferometric visibility function on short baselines (see Section 3.1). Thus, we will now estimate this characteristic FWHM and its uncertainty at 1.3 mm using previously published EHT data.

Because the EHT baseline joining CARMA-SMT (California–Arizona) does not significantly resolve Sgr A* at 1.3 mm, current EHT measurements primarily constrain the source size in the direction of the California–Hawaii and Arizona–Hawaii baselines, close to east–west (i.e., roughly along the major axis of the scattering kernel). Early detections were consistent with a Gaussian image having a FWHM of approximately 40 μas (Doeleman et al. 2008; Fish et al. 2011). However, more recent measurements with improved sensitivity and calibration find visibility amplitudes on the shortest Hawaii baselines (California–Hawaii) that are strongly inconsistent with the Gaussian model (Johnson et al. 2015; Lu et al. 2018). Thus, the appropriate FWHM is not that of the Gaussian fits, which are incompatible with the data, but can instead be estimated by computing the characteristic FWHM of models that do fit the short- and intermediate-baseline visibility amplitudes.

One such model is an annulus. The fitted annulus in Johnson et al. (2015) gives a characteristic FWHM of 58.5 μas for the intrinsic source (as defined in Equation (4)). For comparison, the annulus model from Doeleman et al. (2008) gives 51.5 μas. Two-Gaussian model fits that also include closure phase measurements and baselines to APEX give FWHMs of 55.2 μas and 60.4 μas along the east–west direction or 62.5 μas and 60.5 μas along the major axis of the scattering (for Models A and B of Lu et al. 2018). Because the east–west scatter-broadening is ≲20 μas at this frequency, our revisions to the scattering model and remaining uncertainties have little effect on the estimated intrinsic FWHM. The uncertainties are instead dominated by the sparse baseline coverage, and we estimate a plausible range of 51–63 μas for the FWHM of the intrinsic source along the major axis of the scattering based on the span of these fitted models. Note that this range extends beyond the expected diameter of the black hole shadow (51 ± 3 μas), so it does not necessitate that the accretion flow be viewed at large inclination. Accounting for both source and scattering uncertainties, we adopt a plausible range of 53–66 μas for the FWHM of the scattered image of Sgr A* at 1.3 mm along the major axis of the scattering kernel.

The north–south FWHM at 1.3 mm is comparatively poorly constrained. Krichbaum et al. (1998) reported detections of Sgr A* at λ = 1.4 mm on the baseline joining Pico Veleta and an antenna of the IRAM interferometer at Plateau de Bure. These observations had a baseline length $| {\boldsymbol{u}}| \approx 0.7\times {10}^{9}$, but the baseline was aligned close to east–west (position angle approximately 70° east of north). Lu et al. (2018) have recently reported detections at λ = 1.3 mm on baselines from APEX to California and Arizona, which are oriented close to north–south, but these heavily resolve the source. Thus, they are unreliable for estimating a FWHM using Equation (4) or for computing the second moment of a fitted model. Instead, we estimate a maximum size of the source along the scattering minor axis by requiring that the CARMA-SMT baseline amplitude be at least 80% of the zero-baseline flux density over the Greenwich Sidereal Time (GST) range from −0.5 to 4.0 hr, as is supported by both a priori calibration (Lu et al. 2018) and polarization arguments (Johnson et al. 2015). For a major axis FWHM of ∼60 μas, this requirement gives an upper limit to the minor axis FWHM of approximately 90 μas. To obtain a corresponding lower limit, we require that the correlated flux density on the SMT-APEX baselines for the scattered image never exceed 10% of the zero-baseline flux density over the GST range from 0.0 to 2.5 hr (otherwise it would exceed measurements on this baseline; Lu et al. 2018). This constraint only requires that the scattered source have a minor axis FWHM that exceeds 25 μas. Combining these limits, we obtain a plausible range for the FWHM along the scattering minor axis direction of 25–90 μas (of course, the scattering position angle need not correspond to that of the scattered or unscattered image at 1.3 mm).

Finally, we note that the EHT has detected persistent non-zero closure phases of Sgr A* on the California–Arizona–Hawaii triangle, demonstrating that the scattered image structure is not point symmetric (Fish et al. 2016). However, these results do not imply that the intrinsic or scattered FWHM is asymmetric because the non-zero closure phases may be produced by image substructure. For instance, Model B in Lu et al. (2018) fits both the visibility amplitudes and closure phases but has little asymmetry in the FWHM, with major and minor axes FWHMs of 60.5 μas and 60.3 μas, respectively.

The three asymptotic parameters of our scattering model can be estimated directly from the Gaussian fits to long-wavelength data. These parameters can also be directly compared with the results of previous studies.

For the major axis normalization, our analysis of the VLA data from 15–30 cm gives ${\theta }_{\mathrm{maj},0}=1.380\pm 0.013\,\mathrm{mas}$. For comparison, our fits to the $3.6\,\mathrm{cm}$ VLBA observation gives ${\theta }_{\mathrm{maj},0}=1.412\pm 0.024\,\mathrm{mas}$. Thus, the two estimates are consistent to within their stated uncertainties. We will adopt the VLA estimate and uncertainty for our constraint on ${\theta }_{\mathrm{maj},0}$.

Because we could not reliably fit the minor axis and position angle using the VLA or VLA+PT data, we use VLBI measurements at shorter wavelengths to estimate these parameters. The minor axis of the scattering is small enough at 1.3 cm that intrinsic structure may be significant. Taking only the 3.6 cm measurement and full uncertainty gives ${\theta }_{\min ,0}=0.703\pm 0.013\,\mathrm{mas}$. This estimate represents an upper limit to the scattering size because we have not included a contribution from intrinsic structure. However, our representative intrinsic source size derived below using the full set of shorter-wavelength data (see Section 5.3) would bias this upward by only ≲0.01 mas, which is within our measurement uncertainty.

Despite the relatively complete baseline coverage at 3.6 cm (see Figure 3), the position angle is rather poorly constrained at this wavelength. The reason for the poor constraint is that there are only eight baselines that are dominated by the Gaussian structure, and these baselines must constrain the (time-dependent) self-calibration solutions for the six participating stations. For comparison, among those same six stations, the 1.3 cm data have fifteen baselines that are dominated by the Gaussian structure. Thus, the self-calibration at 1.3 cm is heavily over-constrained, and the measured position angle has small uncertainties despite the more limited baseline coverage. Because we find a position angle that is consistent with a constant value over wavelengths from 3.5 mm to 3.6 cm, it is unlikely that intrinsic structure changes the position angle appreciably at wavelengths of 1.3 or 3.6 cm. In addition, for the scattering model of Psaltis et al. (2018), the position angle of the scattering kernel is independent of wavelength. Thus, we estimate the scattering position angle by combining the measured position angles at 1.3 cm and 3.6 cm, giving ϕ_PA = 81.9 ± 0.2.

Table 2 compares our newly derived Gaussian parameters with previously reported estimates. Note that the three observations used to derive our parameter estimates (2015 VLA observations, and VLBA observations at 3.6 and 1.3 cm) were not used by any of these previous studies. Relative to past work, the major and minor axes are consistent with the values found by Shen et al. (2005), but our major axis normalization is 4.7σ larger than the estimate of Bower et al. (2006) and 3σ larger than that of Bower et al. (2015a) (both relied on the same VLA+PT image-domain analysis at long wavelengths). Our major axis uncertainty is similar to these previous results, largely because of the increased error budget to accommodate refractive fluctuations, while our minor axis uncertainty is significantly smaller than all past work. While our position angle is somewhat larger than most previous studies, it is close to the value and uncertainty estimated by Bower et al. (2015a).

Table 2.  Estimated Asymptotic Gaussian Scattering Parameters

Reference	${\theta }_{\mathrm{maj},0}$ (mas)	${\theta }_{\min ,0}$ (mas)	P.A. (deg)
Lo et al. (1998)	1.430 ± 0.020	0.760 ± 0.050	80 ± 3
Shen et al. (2005)	1.390 ± 0.020	0.690 ± 0.060	80
Bower et al. (2006)	1.309 ± 0.015	${0.640}_{-0.050}^{+0.040}$	${78}_{-1.0}^{+0.8}$
Lu et al. (2011)	1.335 ± 0.014	0.817 ± 0.042	⋯
Psaltis et al. (2015b) a	1.320 ± 0.040	0.820 ± 0.210	77.8 ± 9.7
Bower et al. (2015a)	1.320 ± 0.020	0.670 ± 0.020	81.8 ± 0.2
This Work	1.380 ± 0.013	0.703 ± 0.013	81.9 ± 0.2

Notes. These parameters give the scattering kernel at the reference wavelength λ₀ ≡ 1 cm.

^aUnlike the other entries in this table, Psaltis et al. (2015b) reanalyzed a sample of published Gaussian parameter fits rather than analyzing new or archival observations directly.

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The remaining two parameters of our scattering model, α and r_in, can be constrained in two ways: through a change in the scatter-broadening law from its asymptotic behavior at long wavelengths and through stochastic signatures of refractive scattering. For both types of constraints, the effects of α and r_in must be considered jointly; α will determine the asymptotic behavior at short wavelengths, but r_in determines the scale on which the scattering transitions between the two asymptotic regimes. Many previous efforts have constrained α by fitting the wavelength dependence of scatter-broadening to a power law λ^β or by quantifying Gaussianity of the scattered image (e.g., Lo et al. 1998; Bower et al. 2004; Lu et al. 2011). However, these studies have implicitly assumed the limit ${r}_{\mathrm{in}}\to 0$, effectively fitting centimeter data to the properties of the scattering expected for the asymptotic regime $\lambda \to 0$. As we will demonstrate, jointly fitting the two parameters is imperative to derive meaningful parameter constraints for α and r_in.

We will now derive a series of constraints α and r_in. In Section 5.2.1, we derive constraints from the refractive noise on long baselines at 3.6 and 1.3 cm. In Section 5.2.2, we determine constraints from the stringent limits on refractive fluctuations of the image size at 7 mm. In Section 5.2.3, we derive constraints based on the λ² scaling of scatter broadening at centimeter wavelengths. In Section 5.2.4, we establish constraints based on Gaussianity of the scattered image at 1.3 cm. While any of these individual constraints can only weakly constrain the parameter pair (α, r_in), the cumulative constraints are quite restrictive, summarized in Figure 9. We discuss these constraints and give our recommended characteristic values in Section 5.2.5.

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5.2.1. Constraints from Refractive Noise at 3.6 and 1.3 cm

5.2.2. Constraints from Refractive Fluctuations of the Image Size

Refractive scattering causes variations in the observed angular size of a source (Blandford & Narayan 1985). For observations that span many refractive timescales, the observed level of variability can then be used to constrain the scattering model. Because the intrinsic source may also be time-variable, measurements of the image size variability can only give an upper limit for the variations attributable to scattering. As with other refractive effects, fluctuations of image size will increase with increasing α and r_in.

For Sgr A*, the most stringent constraints on image size fluctuations come from observations at λ = 7 mm. Even without accounting for refractive noise, observations over the past 25 years or so consistently find a major axis size in the range of $680\mbox{--}750\,\mu \mathrm{as}$ (e.g., Backer et al. 1993; Krichbaum et al. 1993; Lo et al. 1998; Bower et al. 2004, 2014b, 2015a; Lu et al. 2011; Akiyama et al. 2014; Zhao et al. 2017). A uniform distribution over the entire range $680\mbox{--}750\,\mu \mathrm{as}$ has a standard deviation of $20.2\,\mu \mathrm{as}$, or fractional variations of 2.8%. Note that this range is inflated by measurement uncertainties in the reported sizes (in addition to scattering and intrinsic variability). Thus, we estimate that the fractional scatter of the major axis size at $\lambda =7\,\mathrm{mm}$ from refractive distortion is certainly less than 3%.

We can compare this limit to the expected refractive fluctuations, which can be computed semi-analytically via the framework for renormalized refractive noise developed in Appendix A. Namely, on short baselines, the renormalized refractive noise is dominated by refractive fluctuations in image size. For a short baseline ${\boldsymbol{u}}$, the renormalized visibility (i.e., the visibility after normalizing the total flux density and centering the image) is given by

$$\begin{eqnarray}&&\hat{V}({\boldsymbol{u}})=1-\displaystyle \frac{{\pi }^{2}}{4\mathrm{ln}2}{\theta }_{\parallel }^{2}{u}^{2}+{ \mathcal O }({\theta }_{\parallel }^{3}{u}^{3}),\end{eqnarray} \tag{ 5 }$$

where ${\theta }_{\parallel }$ is the source size projected along the baseline direction (see Equations (3) and (4)). Because of scattering, the instantaneous source size will not match the ensemble-average value, $\langle {\theta }_{\parallel }\rangle ;$ this discrepancy is what produces renormalized refractive noise ${\hat{\sigma }}_{\mathrm{ref}}({\boldsymbol{u}})$ on short baselines. Explicitly,

$$\begin{eqnarray}\begin{array}{rcl}{\hat{\sigma }}_{\mathrm{ref}}({\boldsymbol{u}}) & = & \sqrt{\langle {\rm{\Delta }}\hat{V}{({\boldsymbol{u}})}^{2}\rangle }\\ & \approx & \displaystyle \frac{{\pi }^{2}}{4\mathrm{ln}2}{u}^{2}\sqrt{{({\theta }_{\parallel }^{2}-{\langle {\theta }_{\parallel }\rangle }^{2})}^{2}}\\ & \approx & \displaystyle \frac{{\pi }^{2}}{2\mathrm{ln}2}{u}^{2}\langle {\theta }_{\parallel }\rangle \sqrt{\langle {\rm{\Delta }}{\theta }_{\parallel }^{2}\rangle }\\ \Rightarrow \,\displaystyle \frac{\sqrt{\langle {\rm{\Delta }}{\theta }_{\parallel }^{2}\rangle }}{\langle {\theta }_{\parallel }\rangle } & \approx & \displaystyle \frac{2\mathrm{ln}2}{{\pi }^{2}}\displaystyle \frac{{\hat{\sigma }}_{\mathrm{ref}}({\boldsymbol{u}})}{{u}^{2}{\langle {\theta }_{\parallel }\rangle }^{2}}.\end{array}\end{eqnarray} \tag{ 6 }$$

The red lines in the right panel of Figure 9 show contours for the values of α and ${r}_{\mathrm{in}}$ that would produce 1%, 3%, and 5% fractional fluctuations of the major axis size at $\lambda =7\,\mathrm{mm}$. For these calculations, we hold the ensemble-average image size fixed (approximating it by our measured size), so the intrinsic size is also a function of these parameters because the scattering kernel depends on them. The requirement that the fluctuations are smaller than 3% then gives an α-dependent upper-limit on ${r}_{\mathrm{in}}$.

Observe that the shapes of the image fluctuation contours are very similar to those of the refractive noise at 1.3 cm on the fixed baseline ${\boldsymbol{u}}=(207.6,-30.4)\times {10}^{6}$. This similarity is expected because both effects are dominated by scattering modes on the same angular scale. Specifically, at 7 mm, the dominant modes for image distortion are on the scale of the image size, ${\theta }_{\mathrm{maj}}\approx 0.7\,\mathrm{mas}$. At 1.3 cm, the dominant modes for refractive noise are those matched to the angular resolution of the long baselines, $1/| {\boldsymbol{u}}| \sim 1\,\mathrm{mas}$.

5.2.3. Constraints from the ${\lambda }^{2}$ Scaling of Scattered Size

5.2.4. Constraints from the Image Gaussianity

5.2.5. Recommended Characteristic Values for α and ${r}_{\mathrm{in}}$

Using the scattering model determined in Sections 5.1 and 5.2, we now explore the expected scattering properties for Sgr A* and estimate its wavelength-dependent intrinsic size. Table 3 summarizes our estimates for the parameters of this scattering model and provides additional derived quantities.

Table 3.  Estimated Scattering Model for Sgr A*

Parameter	Estimate
Geometrical parameters
Scattering screen magnification	$M=D/R=0.53\pm 0.08$
Earth-scattering distance	$D=2.7\pm 0.3\,\mathrm{kpc}$
Sgr A*-scattering distance	$R=5.4\pm 0.3\,\mathrm{kpc}$
Scattering parameters
Reference wavelength	${\lambda }_{0}\equiv 1\,\mathrm{cm}$
Gaussian major axis FWHM	${\theta }_{\mathrm{maj},0}=1.380\pm 0.013\ \mathrm{mas}$
Gaussian minor axis FWHM	${\theta }_{\min ,0}=0.703\pm 0.013\ \mathrm{mas}$
Gaussian position angle	${\phi }_{\mathrm{PA}}=81\buildrel{\circ}\over{.} 9\pm 0\buildrel{\circ}\over{.} 2$
Power-law index of ${D}_{\phi }({\boldsymbol{r}})$	$\alpha ={1.38}_{-0.04}^{+0.08}$
Power-law index of $Q({\boldsymbol{q}})$ and ${P}_{{n}_{{\rm{e}}}}({\boldsymbol{q}})$	$\beta =\alpha +2={3.38}_{-0.04}^{+0.08}$
Inner scale	${r}_{\mathrm{in}}=800\pm 200\,\mathrm{km}$
Scattering transitions
Gaussian–inertial kernel transition	$\lambda =5\,\mathrm{mm}$
Weak–strong scattering transition	$\lambda =0.2\,\mathrm{mm}$

Note. Values and uncertainties for α, β, and ${r}_{\mathrm{in}}$ use our tentative upper-limit from non-Gaussianity at 7 mm (see Figure 9). We define the Gaussian-inertial kernel transition as the wavelength for which the diffractive scale and inner scale are equal. At significantly longer wavelengths, the scattering kernel will be Gaussian (determined by the three asymptotic Gaussian parameters); at significantly shorter wavelengths, the shape will be determined by α. The weak–strong transition is the wavelength for which the diffractive scale and refractive scale are equal. For Kolmogorov turbulence, α = 5/3 and β = 11/3.

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Figures 10 and 11 show our estimated major axis FWHM, minor axis FWHM, and position angle as a function of wavelength (these values are given in Table 1). After normalizing by ${\lambda }^{-2}$, these sizes show a significant increase with decreasing wavelength for $\lambda \lesssim 1.3\,\mathrm{cm}$. Because $\alpha \lt 2$, the scattering law can only become steeper than ${\lambda }^{2}$ at short wavelengths. Thus, this increase in normalized size robustly indicates intrinsic structure at millimeter wavelengths.

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Figures 10 and 11 also show the estimated FWHM of the scattering kernel, including the uncertainty spanned by the plausible range of α and ${r}_{\mathrm{in}}$ (see Figure 9). For these estimates, we do not define the FWHM using the second derivative of the visibility amplitude on a zero baseline (as in Section 3.1). We instead identify the baseline length at which the scattering kernel falls to half, $\exp \left[-\tfrac{1}{2}{D}_{\phi }({{\boldsymbol{u}}}_{1/2}/(1+M))\right]\equiv 1/2$, and then derive a representative image FWHM based on the relationship for a Gaussian image: ${\theta }_{\mathrm{FWHM}}=\tfrac{2\mathrm{ln}2}{\pi {u}_{1/2}}$. The kernel uncertainties become larger at shorter wavelengths because of our limited constraints on α and ${r}_{\mathrm{in}}$. At 1.3 mm, the uncertainty is ∼20% on both the major and minor axis FWHM.

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Figure 12 shows our estimates of the visibility domain scattering kernel and its uncertainties at six representative wavelengths. This kernel is required to "deblur" measurements of Sgr A*, so it is fundamental to scattering mitigation strategies (Fish et al. 2014; Johnson 2016). At millimeter wavelengths, the kernel differs significantly from the prediction of the simple Gaussian/${\lambda }^{2}$ model, and the remaining uncertainty in the scattering kernel on long baselines is significant, primarily because of uncertainties on α and ${r}_{\mathrm{in}}$. Thus, we expect that the blurring effects of scattering are significantly weaker than have been assumed for Sgr A*.

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With the estimated size of the scattering kernel in place, we can deconvolve the scattering from the observed Gaussian size to estimate an intrinsic FWHM at each wavelength. Table 4 and Figure 13 show the estimated intrinsic size along the major and minor axes of the scattering kernel as a function of wavelength. The plotted uncertainties account for uncertainties in our estimates of the scattered size (from thermal noise, systematic effects, and refractive distortion) and in the estimated scattering kernel. The inferred intrinsic size is nearly isotropic and scales approximately as θ ∝ λ. The largest intrinsic anisotropies are at 3.5 mm (up to ∼1.3:1) and 1.3 mm (up to ∼2:1, but poorly constrained). Because the total flux density I₀ of Sgr A* rises with frequency over this range (see, e.g., Lu et al. 2011; Bower et al. 2015b), the brightness temperature ${T}_{{\rm{b}}}\propto {\lambda }^{2}{\theta }_{\mathrm{src}}^{-2}{I}_{0}$ also rises. For instance, using ${\theta }_{\mathrm{src}}\sim (40\,\mu \mathrm{as})\times {\lambda }_{\mathrm{mm}}$, we estimate ${T}_{{\rm{b}}}\sim 1.1\times {10}^{10}\,{\rm{K}}$ at 1.3 cm and ${T}_{{\rm{b}}}\sim 3.1\times {10}^{10}\,{\rm{K}}$ at 1.3 mm.

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Table 4.  Estimated Intrinsic Size of Sgr A*

λ (cm)	${\theta }_{\mathrm{maj}}$ ($\mu \mathrm{as}$)		${\theta }_{\min }$ ($\mu \mathrm{as}$)
	Char.	Plausible	Char.	Plausible
3.598	${4000}_{-1900}^{+1200}$	${4000}_{-4000}^{+1800}$	⋯	⋯
1.261	${550}_{-370}^{+210}$	${550}_{-550}^{+280}$	${560}_{-90}^{+80}$	${560}_{-160}^{+120}$
0.698	${327}_{-46}^{+41}$	${327}_{-69}^{+58}$	${274}_{-13}^{+12}$	${274}_{-25}^{+23}$
0.348	${143}_{-6}^{+6}$	${143}_{-12}^{+11}$	${114}_{-5}^{+5}$	${114}_{-8}^{+7}$
0.131	${56}_{-6}^{+6}$	${56}_{-7}^{+7}$	${59}_{-31}^{+30}$	${59}_{-31}^{+30}$

Note. These estimates of intrinsic FWHM correspond to directions along the major and minor axes of the scattering kernel. We give uncertainties for both the assumed characteristic scattering parameters (which only account for $\pm 1\sigma$ measurement uncertainties) and for the full range of plausible scattering parameters (which account for remaining uncertainties in the scattering kernel). We omit the estimated intrinsic minor axis at 3.6 cm because the minor axis scattering was estimated from this measurement (see Section 5.1).

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Our estimates for the approximately linear wavelength dependence of intrinsic size are typical for stratified emission in synchrotron self-absorbed systems (e.g., Blandford & Königl 1979; Falcke & Markoff 2000; Davelaar et al. 2018), and they are plausible for both disk- and jet-dominated models for the radio emission of Sgr A*. However, the lack of asymmetry in the inferred intrinsic size and the stable position angle of the scattered image both argue against intrinsic structure that is highly asymmetric for $3.5\,\mathrm{mm}\lesssim \lambda \lesssim 1.3\,\mathrm{cm}$. For instance, the model of Falcke & Markoff (2000) predicts an image asymmetry of roughly 4:1, and recent general relativistic magnetohydrodynamic (GRMHD) simulations of jets show asymmetry of ∼2:1 at 7 mm (Davelaar et al. 2018). The intrinsic size of Sgr A* we find is qualitatively consistent with radiatively inefficient accretion flow (RIAF) models (e.g., Özel et al. 2000; Yuan et al. 2003; Yuan & Narayan 2014), which are more plausible for producing a nearly isotropic image at wavelengths as long as 1.3 cm. Recent general relativistic radiation magnetohydrodynamic (GRRMHD) simulations show good agreement with our estimated size at 1.3 mm and also show a similar size trend, but they generally underpredict the size at 7 mm and 1.3 cm (Chael et al. 2018), perhaps highlighting the contribution from a non-thermal population of electrons.

Note that there has not been consistency in how image FWHM from simulations is defined. For comparison with Gaussian image sizes reported here and elsewhere, simulations should compute the image FWHM from the second moment along principal axes of the image brightness distribution (see Section 3.1). While some simulation papers have adopted this convention for comparison (e.g., Mościbrodzka et al. 2009, 2012; Chael et al. 2018; Davelaar et al. 2018), others have developed ad hoc definitions for the reported image size (e.g., Falcke & Markoff 2000; Özel et al. 2000; Chan et al. 2015; Psaltis et al. 2015a) or do not state their procedure for estimating the size. An alternative is to fit or compare simulations directly to measured interferometric visibilities (e.g., Broderick et al. 2009, 2016; Dexter et al. 2010; Kim et al. 2016; Pu et al. 2016; Gold et al. 2017).

We now re-evaluate our assumptions for the scattering of Sgr A*, and discuss implications of our findings.

6.2.1. The Outer Scale of Turbulence

All of our calculations and model fits have assumed that the outer scale of turbulence is effectively infinite. We now evaluate this assumption a posteriori. In particular, Goldreich & Sridhar (2006) estimated that ${r}_{\mathrm{out}}\lesssim {10}^{11}\,\mathrm{cm}\times {\left(\tfrac{R}{130\mathrm{pc}}\right)}^{5/2}\,\times {\left(\tfrac{T}{{10}^{4}{\rm{K}}}\right)}^{3/4}$. For the previously assumed value of $R=130\,\mathrm{pc}$ (Lazio & Cordes 1998), they noted that this scale is unacceptably small, as it produces too much heating and it does not correspond to a reasonable astronomical scale for nonlinear density fluctuations. In addition to these objections, our measurements of refractive noise give a lower bound for the outer scale because refractive noise will be suppressed on angular scales larger than $\sim \,{r}_{\mathrm{out}}/D$. Thus, our measurements of refractive noise on baselines with $| {\boldsymbol{u}}| \sim {10}^{7}$ at 3.6 cm show that ${r}_{\mathrm{out}}\gtrsim {(2\pi )}^{-1}D/{10}^{7}\sim {10}^{14}\,\mathrm{cm}$. With the modified distance to the scattering (see Section 2.3 and Bower et al. 2014a), the problems identified by Goldreich & Sridhar (2006) are mitigated, as we now discuss in detail.

Specifically, an upper limit on the outer scale can be estimated as the scale on which the scattering power spectrum requires density fluctuations of order unity. Suppose that the scattering material is statistically homogeneous over a region of length z along the line of sight. Electron density fluctuations $\delta {n}_{{\rm{e}}}({\ell })$ on a scale ℓ then introduce corresponding screen phase fluctuations of $\delta \phi ({\ell })\sim {r}_{{\rm{e}}}\lambda \sqrt{{\ell }z}\delta {n}_{{\rm{e}}}({\ell })$ (because of the random walk through z/ℓ regions; see Section 2.1). Taking $\delta {n}_{{\rm{e}}}{({\ell })/{n}_{{\rm{e}}}\sim ({\ell }/{r}_{\mathrm{out}})}^{(\alpha -1)/2}$, we obtain,

$$\begin{eqnarray}\begin{array}{rcl}\delta \phi ({\ell }) & \sim & {n}_{{\rm{e}}}\lambda {r}_{{\rm{e}}}\sqrt{z{\ell }}{\left(\displaystyle \frac{{\ell }}{{r}_{\mathrm{out}}}\right)}^{(\alpha -1)/2},\\ \Rightarrow \,{r}_{\mathrm{out}} & \lesssim & {({n}_{{\rm{e}}}\lambda {r}_{{\rm{e}}}\sqrt{{{zr}}_{\mathrm{diff}}})}^{2/(\alpha -1)}{r}_{\mathrm{diff}},\end{array}\end{eqnarray} \tag{ 7 }$$

where ${r}_{\mathrm{diff}}\approx \lambda /((1+M){\theta }_{\mathrm{scatt}})$ is the diffractive scale (i.e., $\delta \phi ({r}_{\mathrm{diff}})\sim 1$). For Sgr A*, ${r}_{\mathrm{diff}}\approx {10}^{8}\,\mathrm{cm}$ at $\lambda =1\,\mathrm{cm}$. With our characteristic value $\alpha =1.38$, we then find

$$\begin{eqnarray}&&{r}_{\mathrm{out}}\lesssim (10\,\mathrm{pc})\times {\left(\displaystyle \frac{{n}_{{\rm{e}}}}{10{\mathrm{cm}}^{-3}}\right)}^{5.26}{\left(\displaystyle \frac{z}{10\mathrm{pc}}\right)}^{2.63}.\end{eqnarray} \tag{ 8 }$$

For comparison, Armstrong et al. (1995) estimate ${r}_{\mathrm{out}}\gtrsim 30\,\mathrm{pc}$ for the scattering material within 1 kpc. For Equation (8) to violate our measured lower limit for the 3.6 cm refractive noise would require much lower electron densities ${n}_{{\rm{e}}}\lesssim 0.1$ or larger values of α (approximately $\alpha \gtrsim 5/3$, for the characteristic values of z and ${n}_{{\rm{e}}}$ given in Equation (8)), although these results are highly sensitive to the screen thickness, z. From the dispersion measure of the Galactic Center magnetar (e.g., Eatough et al. 2013; Kravchenko et al. 2016), we can only estimate an upper bound on the plasma density, ${n}_{{\rm{e}}}\,\lt (180\,{\mathrm{cm}}^{-3})/(z/10\,\mathrm{pc})$. Regardless, the outer scale required by our scattering model is not implausible.

Goldreich & Sridhar (2006) have also provided an alternative model for interstellar scattering from folded magnetic field structures. Their proposed model reproduces the ${\lambda }^{2}$ scaling and Gaussian scatter-broadening for Sgr A*, but it also predicts significantly suppressed refractive scintillation. Furthermore, intrinsic structure would be blurred out on small angular scales from the scattering in this model. Thus, our measurements of image substructure at 1.3 cm conclusively reject the folded field model for the scattering of Sgr A* in its simplest form. However, we will demonstrate later that this model is compatible with our measurements if the inner scale (corresponding to the thickness of current sheets in this model) is significantly larger than expected: ${r}_{\mathrm{in}}\sim 2\times {10}^{6}\,\mathrm{km}$. In this case, the Goldreich & Sridhar (2006) spectrum would instead produce significantly enhanced refractive effects at millimeter wavelengths.

6.2.2. The Inner Scale of Turbulence

6.2.3. The Power-law Index of Turbulence

Figure 14 shows our constraints on the power spectrum of phase fluctuations, $Q({\boldsymbol{q}})$, along the direction of the scattering major axis. Refractive noise on a long baseline ${\boldsymbol{u}}$ is dominated by refractive modes with ${\boldsymbol{q}}\sim 2\pi {\boldsymbol{u}}/D$. Thus, our measurements of refractive noise at 3.6 and 1.3 cm constrain the power in wavenumbers ${q}^{-1}\,\sim \,{10}^{13}\mbox{--}{10}^{14}\,\mathrm{cm}$. In addition, our measurements of the asymptotic Gaussian angular broadening constrain the power in wavenumbers ${q}^{-1}\,\sim \,{r}_{\mathrm{in}}$, with the exact constraint also weakly dependent on α: $Q({r}_{\mathrm{in}}^{-1})\propto {r}_{\mathrm{in}}^{4}/{\rm{\Gamma }}(1-\alpha /2)$.

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As is evident from Figure 14, larger values of the inner scale require a flatter power spectrum for the measured refractive noise to be compatible with the measured angular broadening (see also Figure 9). Allowing arbitrarily small inner scales, we find $\alpha \lesssim 1.6$, while including our derived constraints on the inner scale, we obtain $\alpha \lt 1.47$. Thus, a Kolmogorov spectrum (α = 5/3) is incompatible with our measurements, as is an α = 3/2 spectrum (see, e.g., Iroshnikov 1964; Kraichnan 1965; Sridhar & Goldreich 1994; Goldreich & Sridhar 1995). Our results are at odds with measurements for the local ISM (e.g., Armstrong et al. 1995), the wavelength dependence of pulsar temporal broadening (e.g., Löhmer et al. 2001; Bhat et al. 2004; Lewandowski et al. 2013), and VLBI of heavily scattered sources (Spangler & Gwinn 1990), all of which tend to infer somewhat larger values of α. We now outline possible generalizations to our scattering model that might render higher values of α, including a Kolmogorov spectrum, feasible.

The first possibility is an outer scale of turbulence that is similar to the scales probed at 3.6 cm, thereby reducing the 3.6 cm refractive noise but perhaps not the 1.3 cm refractive noise (which probes smaller scales). The 3.6 cm refractive noise corresponds to scattering modes with a transverse scale of ∼4 au on the scattering screen, so the required outer scale is ${r}_{\mathrm{out}}\sim 1\,\mathrm{au}$. This value is somewhat smaller than the lower limit estimated by Armstrong et al. (1995). Moreover, this is a finely tuned constraint, requiring the outer scale to be precisely matched to our observing parameters—a smaller outer scale would be inconsistent with the observed 1.3 cm refractive noise, and a larger outer scale would not affect the 3.6 cm noise.

A second possibility is that there is extra power located near the dissipation scale. Spectral flattening near the inner scale has been seen in the solar wind (e.g., Neugebauer 1975; Celnikier et al. 1983; Coles et al. 1991) and possibly also in the ISM (Smirnova & Shishov 2010). A pile-up in power by a factor of ∼15 would reconcile our measurements with a Kolmogorov spectrum (see Figure 14); a factor of ≈2 would be needed for an $\alpha =3/2$ spectrum.

A more radical possibility, which is not excluded by our data, is that the spectrum is extremely shallow and the inner scale is correspondingly large. In particular, the model of Goldreich & Sridhar (2006) produces a power spectrum with α = 0, which would be consistent with all our measurements if ${r}_{\mathrm{in}}\sim 2\times {10}^{6}\,\mathrm{km}$. This inner scale is a factor of 20 larger than the characteristic value used by Goldreich & Sridhar (2006) and requires an outer scale that is 400 times larger than expected, or a few kpc. Nevertheless, our measurements are insufficient to rule out this type of power spectrum, and it would produce refractive signatures for the EHT that are approximately 10 times stronger than those predicted by our recommended characteristic model. Thus, we expect continued studies at 1.3 mm (and possibly 3.5 mm) will be able to conclusively confirm or reject this scattering model for Sgr A*.

We have analyzed all our data in the context of a single scattering model. The anisotropy in this model is determined by the magnetic field wander along the line of sight relative to its preferred orientation (which determines the minor axis of the scattering ellipse). Psaltis et al. (2018) give three representative models for the field wander: "von Mises," "Dipole," and "Boxcar." The von Mises model represents the angular field wander using a generalized Gaussian distribution for circular quantities, the Dipole model uses a change of variables to rescale the principal axes of the power spectrum, and the Boxcar model has a power spectrum that is isotropic across a restricted range of angles and is zero elsewhere. Because of the efficient computational tools developed in Appendix B, all of our results have used the Dipole model. We now evaluate how sensitive our conclusions are to this choice.

Psaltis et al. (2018) show that the shape of the scattering kernel is almost independent of the choice of scattering model. However, the refractive noise along the minor axis is sensitive to the scattering model. Because our measurements of refractive noise are predominantly along the major axis, our results are not strongly affected by the choice of scattering model.

For example, the mean refractive noise on our long baselines at 3.6 cm changes by less than ±2% among the three scattering models (well within our uncertainty from sampling only a few elements of the refractive noise). Likewise, the 95% confidence intervals are almost identical for the three models. For the average refractive noise on long baselines at 1.3 cm, the von Mises and Dipole models agree to within 1%, but the Boxcar model differs by 5%. Again, these differences are negligible within our error budget.

Thus, we conclude that our specific choice of scattering model is irrelevant for our results. Equivalently, our current measurements provide no firm guidance for discriminating among these models for the magnetic field wander. Future measurements of refractive noise on long baselines along the minor axis could immediately rule out the Boxcar model and would be sensitive to differences between the von Mises and Dipole models (see, e.g., Figures 9 and 13 in Psaltis et al. 2018).

For the scattering parameters that we have identified, a Gaussian scattering kernel is likely a good approximation for Sgr A* for centimeter wavelengths, although the full non-Gaussian kernel shape should be used for continued studies at millimeter wavelengths. The full kernel shape is especially important for scattering mitigation in imaging with EHT data (Fish et al. 2014; Johnson 2016).

We have shown that refractive noise is a critical component of the error budget when model fitting to observations of Sgr A*. In addition, we caution that inferences of intrinsic size should account for refractive image distortion. At long wavelengths, stochastic changes from refractive distortion can exceed the contribution of intrinsic structure (see Figure 15). Our results suggest that intrinsic structure can only be securely decoupled from refractive distortion for $\lambda \lesssim 3.6\,\mathrm{cm}$ (minor axis) or $\lambda \lesssim 1.3\,\mathrm{cm}$ (major axis). Note that these are fundamental limitations; they would apply even if the sensitivity and baseline coverage of the observations were perfect.

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Our work has two significant implications for imaging Sgr A* with the EHT. First, we have shown that the scattering kernel may be much smaller than has been estimated, so the blurring effects of scattering may be less severe than have been assumed (see Figure 12). Second, we find $\alpha \lesssim 1.47$, which produces significantly less refractive noise than the standard Kolmogorov picture; it predicts that the renormalized refractive noise is at most ∼1% of the zero-baseline flux density (see Figure 16). This estimate reinforces the conclusions of Fish et al. (2016) and Lu et al. (2018) that refractive noise is unlikely to be a significant component of the error budget for past EHT observations. Both these implications improve the prospects for horizon-scale imaging at 1.3 mm. However, continued observations must also account for the possibility of strong refractive effects from shallow spectra, such as from the model with α = 0 and ${r}_{\mathrm{in}}=2\times {10}^{6}\,\mathrm{km}$. While this model remains speculative and lacks support from other lines of sight, it would produce striking differences from our characteristic model for EHT observations (an increase in refractive noise by a factor of 10) and should be tested further with long-baseline measurements at 1.3 mm and 3.5 mm.

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