Results of escape probability models from the RADEX program (Van der Tak et al. 2006) for four different kinetic temperatures, a range of H₂ densities and a range of CO molecular column density per km s^-1. The grey scale mapping and the black contour lines show the expected CO(2–1) emission T_R(co₂₁) [K] (see wedge to the right of each panel). Dashed blue lines show the CO(2–1)/CO(1–0) line temperature ratios 0.6, 0.9 and 1 as predicted by the RADEX program. Dash-dot red lines show the predicted CO(3–2)/CO(2–1) line temperature ratios 0.6, 0.9 and 1. The measured CO(2–1)/CO(1–0) line temperature ratio in r15 and r20 are close to one. So the possible solutions lie along the dashed-blue line of value one. We see that it is thus difficult to constrain the range of possible kinetic temperatures and H₂ densities. Nervertheless, even with a high T_kin of 500 K, the density must be  ≥ 10^2.5. Note also that if we consider a standard Milky Way M_gas/ conversion factor, then we expect N_CO/ΔV > 1    ×    10¹⁷ cm^-2 km s^-1 if f_2D ~ 10^-3 i.e. T_R ~ 10 K, which gives slightly higher densities (≥ 10^2.9). Finally, Bridges & Irwin (1998) reported a CO(3–2)/CO(2–1) line ratio close to one in the central  ~8 kpc region. If this is also true for the filaments, then the possible solutions would lie at the intersection of the dashed blue line (values  ~0.9–1) with the dash-dot red line (values  ~0.9–1), which means even higher densities. The high-J CO lines are thus important diagnostics to determine the gas properties.