We present correlations between 9 CO transition ($J=4-3$ to $12-11$) and beam-matched far-infrared (Far-IR) luminosities ($L_{mathrm{FIR},,b}$) among 167 local galaxies, using {it{Herschel}} Spectral and Photometric Imaging Receiver Fourier Transform
Spectrometer (SPIRE; FTS) spectroscopic data and Photoconductor Array Camera and Spectrometer (PACS) photometry data. We adopt entire-galaxy FIR luminosities ($L_{mathrm{FIR},,e}$) from the {it{IRAS}} Revised Bright Galaxy Sample and correct to $L_{mathrm{FIR},,b}$ using PACS images to match the varying FTS beam sizes. All 9 correlations between $L_{mathrm{CO}}$ and $L_{mathrm{FIR},,b}$ are essentially linear and tight ($sigma$=0.2-0.3 dex dispersion), even for the highest transition, $J=12-11$. This supports the notion that the star formation rate (SFR) is linearly correlated with the dense molecular gas ($n_{mathrm{H}_2}gtrsim10^{4-6},cm^{-3}$). We divide the entire sample into three subsamples and find that smaller sample sizes can induce large differences in the correlation slopes. We also derive an average CO spectral line energy distribution (SLED) for the entire sample and discuss the implied average molecular gas properties for these local galaxies. We further extend our sample to high-{it{z}} galaxies with CO $J=5-4$ data from the literature as an example, including submillimeter galaxies (SMGs) and normal star-forming BzKs. BzKs have similar FIR/CO(5-4) ratios as that of local galaxies, an follow well the locally-determined correlation, whereas SMG ratios fall around or slightly above the local correlation with large uncertainties. Finally, by including Galactic CO($J=10-9$) data as well as very limited high-{it{z}} CO $J=10-9$ data, we verify that the CO(10-9) -- FIR correlation successfully extends to Galactic young stellar objects, suggesting that linear correlations are valid over 15 orders of magnitude.
We study the global SF law - the relation between gas and SFRs in a sample of 181 local galaxies with L_IR spanning almost five orders of magnitude, which includes 115 normal galaxies and 66 (U)LIRGs. We derive their atomic, molecular gas and dense m
olecular gas masses using newly available HI, CO and HCN data from the literature, and SFRs are determined both from total IR and 1.4 GHz radio continuum (RC) luminosities. In order to derive the disk-averaged surface densities of gas and SFRs, we have used high-resolution RC observations to measure the radio sizes for all galaxies. We find that dense molecular gas (as traced by HCN) has the tightest correlation with that of SFRs, and is linear in (N=1.01 +/- 0.02) across the full galaxy sample. The correlation between densities of molecular gas (traced by CO) and SFRs is sensitive to the adopted value of the alpha_CO used to infer molecular gas masses from CO luminosities. For a fixed value of alpha_CO, a slope of 1.14+/-0.02 is found. If instead we adopt values of 4.6 and 0.8 for disk galaxies and (U)LIRGs, respectively, we find the two distinct relations. If applying a continuously varying alpha_CO to our sample, we recover a single relation with slope of 1.60+/-0.03. The SFRs is a steeper function of total gas than that of molecular gas, and is tighter among low-luminosity galaxies. We find no correlation between SFRs and atomic gas.
We study the global star formation law - the relation between the gas and star formation rate (SFR) in a sample of 130 local galaxies with infrared (IR) luminosities spanning over three orders of magnitude (10^9-10^12 Lsun), which includes 91 normal
spiral galaxies and 39 (ultra)luminous IR galaxies [(U)LIRGs]. We derive their total (atomic and molecular) gas and dense molecular gas masses using newly available HI, CO and HCN data from the literature. The SFR of galaxies is determined from total IR (8-1000 um) and 1.4 GHz radio continuum (RC) luminosities. The galaxy disk sizes are defined by the de-convolved elliptical Gaussian FWHM of the RC maps. We derive the galaxy disk-averaged SFRs and various gas surface densities, and investigate their relationship. We find that the galaxy disk-averaged surface densities of dense molecular gas mass has the tightest correlation with that of SFR (scatter ~ 0.26 dex), and is linear in log-log space (power-law slope of N=1.03 +/- 0.02) across the full galaxy sample. The correlation between the total gas and SFR surface densities for the full sample has a somewhat larger scatter (~ 0.48 dex), and is best fit by a power-law with slope 1.45 +/- 0.02. However, the slope changes from ~ 1 when only normal spirals are considered, to ~ 1.5 when more and more (U)LIRGs are included in the fitting. When different CO-to-H2 conversion factors are used to infer molecular gas masses for normal galaxies and (U)LIRGs, the bi-modal relations claimed recently in CO observations of high-redshift galaxies appear to also exist in local populations of star-forming galaxies.
