Measuring Star-formation Rate and Far-Infrared Color in High-redshift
  Galaxies Using the CO (7-6) and [NII] 205 micron Lines
  To better characterize the global star formation (SF) activity in a galaxy,
one needs to know not only the star formation rate (SFR) but also the
rest-frame, far-infrared (FIR) color (e.g., the 60-to-100 $\mu$m color,
$C(60/100)$] of the dust emission. The latter probes the average intensity of
the dust heating radiation field and scales statistically with the effective
SFR surface density in star-forming galaxies including (ultra-)luminous
infrared galaxies [(U)LIRGs]. To this end, we exploit here a new spectroscopic
approach involving only two emission lines: CO\,(7$-$6) at 372 $\mu$m and [NII]
at 205 $\mu$m. For local (U)LIRGs, the ratios of the CO (7$-$6) luminosity
($L_{\rm CO\,(7-6)}$) to the total infrared luminosity ($L_{\rm IR}$; 8$-$1000
$\mu$m) are fairly tightly distributed (to within $\sim$0.12 dex) and show
little dependence on $C(60/100)$. This makes $L_{\rm CO\,(7-6)}$ a good SFR
tracer, which is less contaminated by active galactic nuclei (AGN) than $L_{\rm
IR}$ and may also be much less sensitive to metallicity than $L_{\rm
CO\,(1-0)}$. Furthermore, the logarithmic [NII] 205 $\mu$m to CO (7$-$6)
luminosity ratio is fairly steeply (at a slope of $\sim$$-1.4$) correlated with
$C(60/100)$, with a modest scatter ($\sim$0.23 dex). This makes it a useful
estimator on $C(60/100)$ with an implied uncertainty of $\sim$0.15 [or
$\lesssim$4 K in the dust temperature ($T_{\rm dust}$) in the case of a
graybody emission with $T_{\rm dust} \gtrsim 30$ K and a dust emissivity index
$\beta \ge 1$]. Our locally calibrated SFR and $C(60/100)$ estimators are shown
to be consistent with the published data of (U)LIRGs of $z$ up to $\sim$6.5.
