The diagnostic potential of the weak field approximation for investigating the quiet Sun magnetism: the Si I 10 827 Å line

Aims. We aim to investigate the validity of the weak field approximation (WFA) for determining magnetic fields in quiet regions of the solar photosphere using the polarization caused by the Zeeman effect in the Si I 10 827 angstrom line. Methods. We solved the NLTE line formation problem by means of multilevel radiative transfer calculations in a three-dimensional (3D) snapshot model taken from a state-of-the-art magneto-convection simulation of the small-scale magnetic activity in the quiet solar photosphere. The 3D model used is characterized by a surface mean magnetic field strength of about 170 G. The calculated Stokes profiles were degraded because of the atmospheric turbulence of Earth and light diffraction by the telescope aperture. We apply the WFA to the Stokes I, Q, U, V profiles calculated for different seeing conditions and for the apertures of the VTT, GREGOR, EST and DKIST telescopes. We compare the inferred longitudinal and transverse components of the magnetic field with the original vertical and horizontal fields of the 3D model. Results. We find that with a spatial resolution significantly better than 0.5 '' the surface maps of the magnetic field inferred from the Stokes profiles of the Si I 10 827 angstrom line applying the WFA are close to the magnetic field of the model on the corrugated surface, corresponding to line optical depth unity at Delta lambda approximate to 0.1 angstrom for a disk-center line of sight. The correlation between them is relatively high, except that the inferred longitudinal and transverse components of the magnetic field turn out to be lower than in the 3D model. Conclusions. The use of the WFA for interpreting high-spatial-resolution spectropolarimetric observations of the Si I 10 827 angstrom line obtained with telescopes like GREGOR, EST, and DKIST allows the longitudinal and transverse components of the magnetic field to be retrieved with reasonable precision over the whole quiet solar photosphere, the result being worse for telescopes of lower aperture.