Measuring line formation depths by cross-spectral analysis

Abstract

Context. Numerical three-dimensional simulations of the solar photosphere have progressed greatly in the last 15 years. Observational tests are needed to independently verify the realism of these simulations. Aims. We aim to measure the perspective shift between monochromatic images at different wavelengths taken away from disk center. We investigate the feasibility of our method by applying it to simulated spectra of the Fe i line pair at 630.15 and 630.25 nm calculated from several snapshots of a three-dimensional simulation of solar magneto-convection. Methods. We present a method to determine line formation depths from spectroscopic observations without relying on assumptions about an atmospheric model. Our method is based on the measurement of a perspective shift, which is detected as a linear phase term in the cross-spectrum of the images. In principle this detection is independent of the spatial resolution of the observations, and provides a valuable test for numerical simulations of the solar photosphere. Results. To obtain accurate formation heights we need to correct spectra for convective Doppler shifts, and we need to accumulate successive phase shifts between images in nearby wavelengths, rather than compare images from the continuum and core directly. The comparison of images provides large dissimilarities, which result from the temperature contrast inversion in the granulation with height. We verify that the cross-spectrum phase of the simulated images shows the expected linear behavior with spatial frequency when considering two close enough wavelengths in a spectral line profile. This linear behavior is however only obtained at small spatial frequencies, i.e. for large granular structures. Derived line formation heights of the two lines range from 239 and 287 km above the continuum formation height for the 630.15 nm line, and from 138 to 201 km for the 630.25 nm line, with significant variation between snapshots. Formation height estimates from optical depth unity give on average 319 km and 244 km respectively. Conclusions. Our numerical tests validate measurements of line formation depths from cross-spectra between images at different wavelengths and stress the value of measuring the phase of the cross-spectra as an important test for numerical simulations.