Interferometric Fringe Visibility Null as a Function of Spatial Frequency: a Probe of Stellar Atmospheres

We introduce an observational tool based on visibility nulls in optical spectro-interferometry fringe data to probe the structure of stellar atmospheres. In a preliminary demonstration, we use both Navy Precision Optical Interferometer (NPOI) data and stellar atmosphere models to show that this tool can be used, for example, to investigate limb darkening. Using bootstrapping with either multiple linked baselines or multiple wavelengths in optical and infrared spectro-interferometric observations of stars makes it possible to measure the spatial frequency $u_0$ at which the real part of the fringe visibility ${\rm Re}(V)$ vanishes. That spatial frequency is determined by $u_0 = B_\perp/\lambda_0$, where $B_\perp$ is the projected baseline length, and $\lambda_0$ is the wavelength at which the null is observed. Since $B_\perp$ changes with the Earth's rotation, $\lambda_0$ also changes. If $u_0$ is constant with wavelength, $\lambda_0$ varies in direct proportion to $B_\perp$. Any departure from that proportionality indicates that the brightness distribution across the stellar disk varies with wavelength via variations in limb darkening, in the angular size of the disk, or both. In this paper, we introduce the use of variations of $u_0$ with $\lambda$ as a means of probing the structure of stellar atmospheres. Using the equivalent uniform disk diameter $\theta_{\rm UD, 0}(\lambda_0)$, given by $\theta_{\rm UD, 0} = 1.22/u_0(\lambda_0)$, as a convenient and intuitive parameterization of $u_0(\lambda_0)$, we demonstrate this concept by using model atmospheres to calculate the brightness distribution for $\nu$ Ophiuchi and predict $\theta_{\rm UD, 0}(\lambda_0)$, and then comparing the predictions to coherently averaged data from observations taken with the NPOI.