Sub-galactic scaling relations between X-ray luminosity, star-formation
  rate, and stellar mass
  X-ray luminosity ($L_X$) originating from high-mass X-ray binaries (HMXBs) is
tightly correlated with the host galaxy's star-formation rate (SFR). We explore
this connection at sub-galactic scales spanning ${\sim}$7 dex in SFR and
${\sim}$8 dex in specific SFR (sSFR). There is good agreement with established
relations down to ${\rm SFR {\simeq} 10^{-3}\,M_\odot \, yr^{-1}}$, below which
an excess of X-ray luminosity emerges. This excess likely arises from low mass
X-ray binaries. The intrinsic scatter of the $L_X$-SFR relation is constant,
not correlated with SFR. Different star formation indicators scale with $L_X$
in different ways, and we attribute the differences to the effect of star
formation history. The SFR derived from H$\alpha$ shows the tightest
correlation with X-ray luminosity because H$\alpha$ emission probes stellar
populations with ages similar to HMXB formation timescales, but the
H$\alpha$-based SFR is reliable only for $\rm sSFR{>}10^{-12}\,M_\odot \,
yr^{-1}/M_\odot$.
