On R-W1 as A Diagnostic to Discover Obscured Active Galactic Nuclei in
  Wide-Area X-ray Surveys
  Capitalizing on the all-sky coverage of {\it WISE}, and the 35\% and 50\% sky
coverage from SDSS and Pan-STARRS, respectively, we explore the efficacy of
$m_{R}$ (optical) - $m_{3.4 \mu m}$ (mid-infrared), hereafter $R-W1$, as a
color diagnostic to identify obscured supermassive black hole accretion in
wide-area X-ray surveys. We use the $\sim$16.5 deg$^2$ Stripe 82 X-ray survey
data as a test-bed to compare $R-W1$ with $R-K$, an oft-used obscured AGN
selection criterion, and examine where different classes of objects lie in this
parameter space. Most stars follow a well-defined path in $R-K$ vs. $R-W1$
space. We demonstrate that optically normal galaxies hosting X-ray AGN at
redshifts $0.5<z<1$ can be recovered with an $R-W1>4$ color-cut, while they
typically are not selected as AGN based on their $W1-W2$ colors. Additionally,
different observed X-ray luminosity bins favor different regions in $R-W1$
parameter space: moderate luminosity AGN ($10^{43}$ erg s$^{-1} < L_{\rm 0.5-10
keV} < 10^{44}$ erg s$^{-1}$) tend to have red colors while the highest
luminosity AGN ($L_{\rm 0.5-10 keV} > 10^{45}$ erg s$^{-1}$) have bluer colors;
higher spectroscopic completeness of the Stripe 82X sample is needed to
determine whether this is a selection effect or an intrinsic property. Finally,
we parameterize X-ray obscuration of Stripe 82X AGN by calculating their
hardness ratios (HRs) and find no clear trends between HR and optical
reddening. Our results will help inform best-effort practices in following-up
obscured AGN candidates in current and future wide-area, shallow X-ray surveys,
including the all-sky {\it eROSITA} mission.
