Modelling CO emission - II. The physical characteristics that determine the X factor in Galactic molecular clouds

We investigate how the X factor, the ratio of the molecular hydrogen column density (N-H2) to velocity-integrated CO intensity (W), is determined by the physical properties of gas in model molecular clouds (MCs). The synthetic MCs are results of magnetohydrodynamic simulations, including a treatment of chemistry. We perform radiative transfer calculations to determine the emergent CO intensity, using the large velocity gradient approximation for estimating the CO population levels. In order to understand why observations generally find cloud-averaged values of X = X-Gal similar to 2 x 10(20) cm(-2) K-1 km(-1) s, we focus on a model representing a typical Milky Way MC. Using globally integrated N-H2 and W reproduces the limited range in X found in observations and a mean value X = X-Gal = 2.2 x 10(20) cm(-2) K-1 km(-1) s. However, we show that when considering limited velocity intervals, X can take on a much larger range of values due to CO line saturation. Thus, the X factor strongly depends on both the range in gas velocities and the volume densities. The temperature variations within individual MCs do not strongly affect X, as dense gas contributes most to setting the X factor. For fixed velocity and density structure, gas with higher temperatures T has higher W, yielding X proportional to T-1/2 for T similar to 20-100 K. We demonstrate that the linewidth-size scaling relationship does not influence the X factor - only the range in velocities is important. Clouds with larger linewidths sigma, regardless of the linewidth-size relationship, have a higher W, corresponding to a lower value of X, scaling roughly as X proportional to sigma(-1/2). The 'mist' model, often invoked to explain a constant X-Gal consisting of optically thick cloudlets with well-separated velocities, does not accurately reflect the conditions in a turbulent MC. We propose that the observed cloud-averaged values of X similar to X-Gal are simply a result of the limited range in N-H2, temperatures and velocities found in Galactic MCs - a nearly constant value of X therefore does not require any linewidth-size relationship, or that MCs are virialized objects. Since gas properties likely differ (albeit even slightly) from cloud to cloud, masses derived through a standard value of the X factor should only be considered as a rough first estimate. For temperatures T similar to 10-20 K, velocity dispersions sigma similar to 1-6 km s(-1)and N-H2 similar to 2-20 x 10(21) cm(-2), we find cloud-averaged values X similar to 2-4 x 10(20) cm(-2) K-1 km(-1) s for solar-metallicity models.