IRC +10 216 revisited II: the circumstellar CO shell

12CO and 13CO J = 6-5 observations of IRC +10 216 with the JCMT are presented. A spherically symmetric radiative transfer code is used to model these and other CO observations of the carbon star IRC~+10~216/CW Leo. Compared to previously published model calculations a much larger set of observational data is used as constraints. A grid of models is calculated with the following parameters: luminosity, mass loss rate, dust-to-gas ratio, dust opacity and CO abundance. A comparison with CO J = 1-0 data obtained from the literature points towards a preferred luminosity of 10000-15000 \lsol. Notwithstanding the overall good agreement, there remain discrepancies. The best fitting models (for each of the considered luminosities) cannot accommodate the more extended emission seen in the mapping data. To fit the data for radial offsets >50\arcsec the mass loss rate must be a factor of 5 higher in the outer envelope. A comparison is made between the mass loss rates and dust-to-gas ratios derived from the CO modelling and those derived from our previous dust modelling. The best agreement is obtained for the model with 15000 \lsol. We conclude that the likely luminosity of IRC +10 216 is between 10000 and 15000 \lsol and that its distance is between 110 and 135 pc. The present-day mass loss rate is (1.5 $\pm$ 0.3) $\times 10^{-5}$ \msolyr and the gas-to-dust ratio is a 700 $\pm$ 100. The dust opacity at 60 $\mu$m is found to be of order of 250 cm$^2$gr$^{-1}$. The CO abundance is 1.1 $\times$ 10$^{-3}$ relative to H$_2$.