Spatially Resolved Water Emission from Gravitationally Lensed Dusty Star
  Forming Galaxies at z $\sim$ 3
  Water ($\rm H_{2}O$), one of the most ubiquitous molecules in the universe,
has bright millimeter-wave emission lines easily observed at high-redshift with
the current generation of instruments. The low excitation transition of $\rm
H_{2}O$, p$-$$\rm H_{2}O$(202 $-$ 111) ($\nu_{rest}$ = 987.927 GHz) is known to
trace the far-infrared (FIR) radiation field independent of the presence of
active galactic nuclei (AGN) over many orders-of-magnitude in FIR luminosity
(L$_{\rm FIR}$). This indicates that this transition arises mainly due to star
formation. In this paper, we present spatially ($\sim$0.5 arcsec corresponding
to $\sim$1 kiloparsec) and spectrally resolved ($\sim$100 kms$^{-1}$)
observations of p$-$$\rm H_{2}O$(202 $-$ 111) in a sample of four strong
gravitationally lensed high-redshift galaxies with the Atacama Large
Millimeter/submillimeter Array (ALMA). In addition to increasing the sample of
luminous ($ > $ $10^{12}$L$_{\odot}$) galaxies observed with $\rm H_{2}O$, this
paper examines the L$_{\rm H_{2}O}$/L$_{\rm FIR}$ relation on resolved scales
for the first time at high-redshift. We find that L$_{\rm H_{2}O}$ is
correlated with L$_{\rm FIR}$ on both global and resolved kiloparsec scales
within the galaxy in starbursts and AGN with average L$_{\rm H_{2}O}$/L$_{\rm
FIR}$ =$2.76^{+2.15}_{-1.21}\times10^{-5}$. We find that the scatter in the
observed L$_{\rm H_{2}O}$/L$_{\rm FIR}$ relation does not obviously correlate
with the effective temperature of the dust spectral energy distribution (SED)
or the molecular gas surface density. This is a first step in developing
p$-$$\rm H_{2}O$(202 $-$ 111) as a resolved star formation rate (SFR)
calibrator.
