Using the QUARRY focal plane array at the FCRAO 14 m, we have mapped the central 630 x 75 pc (l x b = 4.dg3 x 0.dg5) of the Galaxy at ~50" (2 pc) resolution in both CO and HCN J = 1 --> 0 emission. Since HCN emission traces gas with densities >~100 times larger than that traced by CO, the HCN luminosity measures the mass of dense (n >~ 105 cm-3) molecular gas, and the HCN/CO intensity ratio measures gas density. The densest molecular gas is found in the molecular clouds associated with the l = 1.dg5 complex, Sgr A, Sgr B, and Sgr C. When compared with the Milky Way at the same spatial resolution, the HCN luminosities in the central kiloparsec of six nearby starburst galactic nuclei are typically larger (factors of ~0.5--20), but the CO luminosities are only moderately larger (factors of ~0.5--5). Averaged over the central 200 pc, the HCN/CO intensity ratio in the starburst galaxies is larger than in the Milky Way, but a few starburst nuclei have similar or smaller HCN/CO intensity ratios when averaged over larger regions. The spatial extent of elevated HCN/CO ratios (>~0.08) typically encompasses >~200 pc in the starburst galaxies but only ~50 pc in the Milky Way. These results suggest that starburst nuclei have larger masses of dense gas, higher average gas densities in the central 200 pc, and larger regions of enhanced gas density than the Galactic center. When averaged to 630 pc resolution, the HCN and CO spectra show two main velocity components: (1) a broad feature of width ~250 km s-1 centered on VLSR = 0 km s-1 and (2) a narrow feature of width ~90 km s-1 centered on VLSR = 50 km s-1. For the broad component, the HCN/CO J = 1 --> 0 intensity ratio is nearly constant over the entire velocity range, and its small value (R ~ 0.03) is typical of that found in normal galaxies at several kiloparsecs spatial resolution. The gas in the broad velocity component is spatially extended and may represent diffuse molecular gas bound not to individual clouds but rather directly to the Galactic center potential well in a central disk or bar. Gas in the narrow velocity component is associated with the dense giant molecular clouds associated with the l = 1.dg5 complex, Sgr A, B, and C.
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