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Narrow Fe K$\alpha$ fluorescent emission lines arising at $\sim$kpc-scaleseparations from the nucleus have only been detected in a few AGN. Thedetections require that the extended line emission be spatially resolved andsufficiently bright. Compared to narrow Fe K$\alpha$ lines arising closer tothe nucleus, they have much lower fluxes but show substantially largerequivalent widths, EW$_{\rm Fe K\alpha}$. We show that, in the optically-thinlimit, a purely analytical argument naturally predicts large, EW$_{\rmFeK\alpha}\sim$1 keV, values for such lines, regardless of the details ofequivalent hydrogen column density, $N_H$, or reprocessor geometry. Monte Carlosimulations corroborate this result and show that the simple analytic EW$_{\rmFeK\alpha}$ prescription holds up to higher $N_H$ approaching the Compton-thickregime. We compare to $Chandra$ observations from the literature and discussthat our results are consistent with the large EW$_{\rm FeK\alpha}$ valuesreported for local AGN, for which the line is detected in extended, up to$\sim$kpc-scale, regions. We argue that large EW$_{\rm FeK\alpha}$ fromkpc-scale regions in AGN should be ubiquitous, because they do not depend onthe absolute luminosity of the central X-ray source, and are measured onlyagainst the scattered continuum. We predict values to be of the order of$\sim$1 keV or larger, even for covering factors $\ll$1, and for arbitrarilysmall column densities. We propose that the large-scale molecular material thatis now routinely being detected with the Atacama Large Millimeter/SubmillimeterArray (ALMA) may act as an extended X-ray scattering reprocessor giving rise to$\sim$kpc-scale Fe K$\alpha$ emission.

The Ubiquity and Magnitude of Large FeK$\alpha$ Equivalent Widths in AGN Extended Regions