Fe XII and Fe XIII Line Widths in a Southern Coronal Hole up to 1.5 Solar Radii

The non-thermal broadening of spectral lines formed in the solar corona is often used to seek the evidence of Alfv\'en waves propagating in the corona. To have a better understanding of the variation of line widths at different altitudes, we measured the line widths of the strong Fe \textsc{xii} 192.4 \mbox{\AA}, 193.5 \mbox{\AA}, 195.1 \mbox{\AA} and Fe \textsc{xiii} 202.0 \mbox{\AA} in an off-limb southern coronal hole up to 1.5 $R_\odot$ observed by the \textit{Extreme Ultraviolet Spectrometer} (EIS) on board the \textit{Hinode} satellite. We compared our measurements to the predictions from the Alfv\'en Wave Solar Model (AWSoM) and the SPECTRUM module. We found the Fe \textsc{xii} and Fe \textsc{xiii} line widths first increase monotonically below 1.1 $R_\odot$, then keep fluctuating between 1.1 - 1.5 $R_\odot$. The synthetic line widths of Fe \textsc{xii} and Fe \textsc{xiii} below 1.3 $R_\odot$ are notably lower than the observed ones. We discuss several possible sources of this discrepancy and suggest that some other heating mechanisms besides the dissipation of the Alfv\'en waves are required to understand the coronal heating and solar wind acceleration from coronal holes below 1.3 $R_\odot$