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 [ N ii ] at 205 \mu { m } ( [ N ii ] _ { 205 \mu { m } } ) .
For local ( U ) LIRGs , the ratios of the CO ( 7 - 6 ) luminosity ( L _ { CO ( 7 - 6 ) } ) to the total infrared luminosity ( L _ { 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 _ { CO ( 7 - 6 ) } a good SFR tracer , which is less contaminated by active galactic nuclei ( AGN ) than L _ { IR } and may also be much less sensitive to metallicity than L _ { CO ( 1 - 0 ) } .
Furthermore , the logarithmic [ N ii ] _ { 205 \mu { m } } /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 _ { dust } ) in the case of a graybody emission with T _ { dust } \gtrsim 30 K and a dust emissivity index \beta \geq 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 .