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

The Event Horizon Telescope (EHT), a global 230 GHz VLBI array, achieves an angular resolution of ${\approx} 20\,\mu{\rm as}$, sufficient to resolve the supermassive black hole Sagittarius A* (Sgr A*). This resolution may soon enable measurement of the black hole "shadow" size and asymmetry, predicted to be ${\approx}50\,\mu$as and ${\lesssim} 3\,\mu$as, respectively. Measurements that depart from these values could indicate a violation of the "no-hair theorem." However, refractive scattering by the turbulent ionized interstellar medium distorts the image of Sgr A*, affecting its apparent size and asymmetry. In this paper, we present a general analytic approach to quantify the expected image wander, distortion, and asymmetry from refractive scattering. If the turbulence in the scattering material of Sgr A* is close to Kolmogorov, we estimate 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, alternative scattering 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 these effects can be reduced by averaging images over multiple observations. For a small number of observations, the effects of scattering can be comparable to or greater than those from black hole spin, and they determine a fundamental limit for testing general relativity via images of Sgr A*.