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, $C(60\mathrm{/}100)$] 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_{\mathrm{C}\mathrm{O}(7-6)}$) to the total infrared luminosity ($L_{\mathrm{I}\mathrm{R}}$; 8$-$1000 $\mu$m) are fairly tightly distributed (to within $\sim$0.12 dex) and show little dependence on $C(60\mathrm{/}100)$. This makes $L_{\mathrm{C}\mathrm{O}(7-6)}$ a good SFR tracer, which is less contaminated by active galactic nuclei (AGN) than $L_{\rm IR}$andmayalsobemuchlesssensitivetometallicitythan$L_{\rm CO,(1-0)}$\mathrm{.}Furthermore,thelogarithmic[NII]205$\mu$mtoCO(7$-$6) luminosity ratio is fairly steeply (at a slope of $\sim$$-1.4$) correlated with $C(60\mathrm{/}100)$, with a modest scatter ($\sim$0.23 dex). This makes it a useful estimator on $C(60\mathrm{/}100)$ with an implied uncertainty of $\sim$0.15 [or $\lesssim$4 K in the dust temperature ($T_{\mathrm{d}\mathrm{u}\mathrm{s}\mathrm{t}}$) in the case of a graybody emission with $T_{\mathrm{d}\mathrm{u}\mathrm{s}\mathrm{t}}\gtrsim 30$ K and a dust emissivity index $\beta \ge 1$]. Our locally calibrated SFR and $C(60\mathrm{/}100)$ estimators are shown to be consistent with the published data of (U)LIRGs of $z$ up to $\sim$6.5.