Relation between photospheric magnetic field and chromospheric emission

We simultaneously observed the Stokes parameters of the photospheric iron line pair at 630.2 nm and the intensity profile of the chromospheric Ca II H line at 396.8 nm in a quiet Sun region at a heliocentric angle of 53 deg. We perform a statistical analysis of network and inter-network properties.The H-index is the integrated emission in a 0.1 nm band around the Ca core. We separate a non-magnetically, H_non, and a magnetically, H_mag, heated component from a non-heated component, H_co in the H-index. The average network and inter-network H-indices are equal to 12 and 10 pm, respectively. The emission in the network is correlated with the magnetic flux density, approaching a value of H 10 pm for vanishing flux. The inter-network magnetic field distribution peaks around 200 G and its mean absolute flux density is 11 Mx cm$^{-2}$. We find that a dominant fraction of the calcium emission caused by the heated atmosphere in the magnetic network, has non-magnetic origin (H_mag 2pm, H_non 3pm). Considering the effect of straylight, the contribution from an atmosphere with no temperature rise to the H-index (H_co 6pm) is about half of the observed H-index in the inter-network. The H-index in the inter-network is not correlated to any property of the photospheric magnetic field, suggesting that magnetic flux concentrations have a negligible role in the chromospheric heating in this region. The height range of the thermal coupling between the photosphere and low/mid chromosphere increases in presence of magnetic field. In addition, we demonstrate that a poor signal-to-noise level in the Stokes profiles leads to a significant over-estimation of the magnetic field strength