We present observations in CO(3-2) that, combined with previous observations in CO(2-1), constrain the physical properties of the filamentary molecular gas in the central $sim$6.5 kpc of NGC 1275, the central giant elliptical galaxy of the Perseus cluster. We find this molecular gas to have a temperature $gtrsim 20$ K and a density $sim$$10^2$-$10^4 {rm cm^{-3}}$, typically warmer and denser than the bulk of Giant Molecular Clouds (GMCs) in the Galaxy. Bathed in the harsh radiation and particle field of the surrounding intracluster X-ray gas, the molecular gas likely has a much higher ionization fraction than that of GMCs. For an ionization fraction of $sim$$10^{-4}$, similar to that of Galactic diffuse ($lesssim 250 {rm cm^{-3}}$) partially-molecular clouds that emit in HCN(1-0) and HCO$^+$(1-0), we show that the same gas traced in CO can produce the previously reported emissions in HCN(3-2), HCO$^+$(3-2), and CN(2-1) from NGC 1275; the dominant source of excitation for all the latter molecules is collisions with electrons. To prevent collapse, as evidenced by the lack of star formation in the molecular filaments, they must consist of thin strands that have cross-sectional radii $lesssim$0.2-2 pc if supported solely by thermal gas pressure; larger radii are permissible if turbulence or poloidal magnetic fields provide additional pressure support. We point out that the conditions required to relate CO luminosities to molecular gas masses in our Galaxy are unlikely to apply in cluster central elliptical galaxies. Rather than being virialized structures analogous to GMCs, we propose that the molecular gas in NGC 1275 comprises pressure-confined structures created by turbulent flows.
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