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The Event Horizon Telescope (EHT), a global 230 GHz VLBI array, achieves anangular resolution of ${\approx} 20\,\mu{\rm as}$, sufficient to resolve thesupermassive black hole Sagittarius A* (Sgr A*). This resolution may soonenable measurement of the black hole "shadow" size and asymmetry, predicted tobe ${\approx}50\,\mu$as and ${\lesssim} 3\,\mu$as, respectively. Measurementsthat depart from these values could indicate a violation of the "no-hairtheorem." However, refractive scattering by the turbulent ionized interstellarmedium distorts the image of Sgr A*, affecting its apparent size and asymmetry.In this paper, we present a general analytic approach to quantify the expectedimage wander, distortion, and asymmetry from refractive scattering. If theturbulence in the scattering material of Sgr A* is close to Kolmogorov, weestimate the mean refractive image wander, distortion, and asymmetry to be 0.53$\mu$as, 0.72 $\mu$as, and 0.52 $\mu$as at 230 GHz. However, alternativescattering models with flatter power spectra can yield larger values, up to 2.1$\mu$as, 6.3 $\mu$as, and 5.0 $\mu$as, respectively. We demonstrate that theseeffects can be reduced by averaging images over multiple observations. For asmall number of observations, the effects of scattering can be comparable to orgreater than those from black hole spin, and they determine a fundamental limitfor testing general relativity via images of Sgr A*.

Testing General Relativity with the Black Hole Shadow Size and Asymmetry of Sagittarius A*: Limitations from Interstellar Scattering