An improved formulation for the N ( H _ { 2 } ) / I ( CO ) conversion factor or X-factor is proposed .
The statement that the velocity-integrated radiation temperature of the { } ^ { 12 } \kern - 0.8 ptCO J = 1 \rightarrow 0 line , I ( { } ^ { 12 } \kern - 0.8 ptCO ) , “ counts ” optically thick clumps is quantified using the formalism of ( ) for line emission in a clumpy cloud .
Adopting the simplifying assumptions of thermalized { } ^ { 12 } \kern - 0.8 ptCO J = 1 \rightarrow 0 line emission and isothermal gas , an effective optical depth , \tau _ { ef } , is defined as the product of the clump filling factor within each velocity interval and the clump effective optical depth as a function of the optical depth on the clump ’ s central sightline , \tau _ { 0 } .
The clump effective optical depth is well approximated as a power law in \tau _ { 0 } with power-law index , \epsilon , referred to here as the clump “ fluffiness , ” and has values between zero and unity .
While the { } ^ { 12 } \kern - 0.8 ptCO J = 1 \rightarrow 0 line is optically thick within each clump ( i.e. , high \tau _ { 0 } ) , it is optically thin “ to the clumps ” ( i.e. , low \tau _ { ef } ) .
Thus the dependence of I ( CO ) on \tau _ { ef } 
is linear , resulting in an X-factor that depends only on clump properties and not directly on the entire cloud .
Assuming virialization of the clumps yields an expression for the X-factor whose dependence on physical parameters like density and temperature is “ softened ” by power-law indices of less than unity that depend on the fluffiness parameter , \epsilon .
The X-factor provides estimates of gas column density because each sightline within the beam has optically thin gas within certain narrow velocity ranges .
Determining column density from the optically thin gas is straightforward and parameters like \epsilon then allow extrapolation of the column density of the optically thin gas to that of all the gas .
Implicit in this formulation is the assumption that fluffiness is , on average , constant from one beam to the next .
This is also required to some extent for density and temperature , but the dependence of the X-factor , X _ { f } , on these may be weaker .

One important suggestion of this formulation is that virialization of entire clouds is ir relevant . 
The densities required to give reasonable values of X _ { f } are consistent with those found in cloud clumps ( i.e . \sim 10 ^ { 3 } H _ { 2 } cm ^ { -3 } ) .
Thus virialization of clumps , rather than of entire clouds , is consistent with the observed values of X _ { f } .
And even virialization of clumps is not strictly required ; only a relationship between clump velocity width and column density similar to that of virialization can still yield reasonable values of the X-factor .
The underlying physics is now at the scale of cloud clumps , implying that the X-factor can probe sub-cloud structure .

The proposed formulation makes specific predictions of the dependence of X _ { f } on the CO abundance and of the interpretation of line ratios .
In particular , the { } ^ { 13 } \kern - 0.8 ptCO J = 1 \rightarrow 0 / { } ^ { 12 } \kern - 0.8 ptCO J = 1 \rightarrow 0 line ratio values observed in the Orion clouds suggest that \epsilon \simeq 0.3 \pm 0.1 .
If the majority of the { } ^ { 12 } \kern - 0.8 ptCO J = 1 \rightarrow 0 emission originates in structures with an r ^ { -2 } density variation , then the constraints on \epsilon also constrain the ratio of the outer-to-inner radii of the r ^ { -2 } region within the clumps .
Specifically , this ratio for spherical clumps must be 2 to 9 and for cylindrical clumps it must be 4 to 42 .
This is apparently consistent with observations , but higher spatial resolution is necessary to ensure that the observed ratios are not just lower limits .
This formulation also ties the narrow range of the observed values of the { } ^ { 13 } \kern - 0.8 ptCO J = 1 \rightarrow 0 / { } ^ { 12 } \kern - 0.8 ptCO J = 1 \rightarrow 0 line ratio to the relative constancy of the X-factor .

The properties of real clumps in real molecular clouds can be used to estimate the X-factor within these clouds and then be compared with the observationally determined X-factor .
This yields X-factor values that are within a factor of 2 of the observed values .
This is acceptable for the first attempt , but reducing this discrepancy will require improving the formulation .
While this formulation improves upon that of ( ) , it has shortcomings of its own .
These include uncertainties as to why \epsilon 
seems to be constant from cloud to cloud , uncertainties in defining the average clump density and neglecting certain complications , such as non-LTE effects , magnetic fields , turbulence , etc .

Despite these shortcomings , the proposed formulation represents the first major improvement in understanding the X-factor because it is the first formulation to include radiative transfer .