 to 2, as in Fig. 2 by Lombardi et al. (2010).
In Fig. 10 we show the M − R diagram for four clouds with two power-law N -pdfs. Again, there are two cases with size of 10 pc, and other two with 50 pc. The indexes in the high extinction wing of the N -pdf are n = 3.5 and 6 (m = 2.5 and 5). In all cases, the index of the low extinction wing of the N -pdf is q = 0. We note that a very steep power-law N -pdf is needed, with an index of n = 6 (m = 5), in order to reproduce the slope of 1.6 reported by Lombardi et al. (2010) for the M − R diagram at intermediate radii.
In Fig. 11, the mass-size diagram for clouds with a lognormal function at low column densities, and a power-law regime at large extinctions N -pdf. The fiducial parameters are: Acut = 0.2, n = 4, and σ = 0.4. In the upper panel we vary the value where the transition between the lognormal and the power-law occurs: Acut = 0.2, 0.3 and 0.4. In Figure 11. M − R diagram for lognormal+power-law N -pdfs. In the upper panel, we vary the extinction value where the transition between the lognormal and power-law regime occurs, Acut. In the middle panel we vary the slope of the power-law regime, n. Finally, in the lower panel, we vary the width of the lognormal function, σ.
the middle panel we vary the slope of the power-law regime: n = 6, 5 and 4. Finally, in the lower panel we vary the width of the lognormal: σ =0.3, 0.4, 0.5, 0.6, 1 and 1.5.
What these figures show us is that the width of the lognormal function shifts the relation: the wider the N -pdf, the larger the mass at a given radius. Similarly, the slope n of the power-law in the N -pdf determines the slope a of the M − R relation: Assuming a power-law N -pdf, from eqs. (7) and (8), we can solve for R = S/π, and show that the Figure 12. n − a diagram, according to eq. (30), where a stands for the exponent of the M − R relation, while n stands for the exponent of the power-law of the N -pdf at large extinctions. We recall that the slope measured from the N -pdf when plotted in log-log scale is given by m = n − 1, as given in the upper x-axis. Notice that value a = 2 is forbidden, according to eq. 30.
index a in eq. (29) will be related to the index n of the high extinction wing of the N -pdf by Fig. 12 shows the n − a diagram, as given by eq. (30). We notice that a moves softly from ∼ 1.2 to 1.6 for realistic values of the power-law (n = 3.5 − 6, or m = 2.5 − 5). Note furthermore that value a = 2 is forbidden, according to eq. (30).
All together, our results and the observational results present the following puzzle: while most of the observed clouds exhibit a power-law distribution N -pdf for large extinctions, why such clouds exhibit an M ∝ R 2 relation at small radii, if the value a = 2 is forbidden for power-laws at large extinctions? In principle, this value will be allowed only for the few clouds that exhibit lognormal N -pdfs, as Coalsack, California, the Pipe, Lupus V, LDN 1719.
In order to explain this apparent contradiction, lets assume that we have a N -pdf with a shape that changes continuously, such that it can be approximated as a sequence of power-laws in small intervals of A: We now assume that the N -pdf decreases rapidly, such that the mass at a given threshold is dominated by the lowdensity region (this assumption is fulfilled by the observed clouds, since this is one of the conditions needed in order to obtain the third Larson relation, as we showed in the previous sections). In this case, we can integrate out eqs. (3) and (4) only for the small interval in which the power-law applies, i.e., and Solving these equations we obtain and Since we have defined S = πR 2, we can write the ratio Ai+1/Ai in terms of R from eq. (34): and insert this value in eq. (35) to obtain where δ < 1. Lets now analyze the limit cases. For large radii (low extinctions), πR 2 ∼ Stot. Note that the ratio (1−ni)/Aipi 25: observations by Kainulainen et al. (2009) show that the typical values of the parameters are: 3.5 ni 6, 0.1 Ai 2,(recall that AV ∼ 10A k, and our Ai are assumed to be in the K band) and pi ∼ 1 for Ai ∼ 0.1, and pi 10 −3 for Ai ∼ 2 (see N -pdfs in Kainulainen et al. 2009). Thus, the term in parenthesis becomes much larger than unity, and then, for large radii, recovering the solution for the single powerlaw case, eq. (30). Note that 2δ < 2, since δ < 1. As an example, lets take for instance the N -pdf of Lupus I cloud, shown in Fig. 4 by Kainulainen et al. (2009). The slope for AV 1 is approximately m = 3.3. Thus, n = 4.3 and 2δ = 1.13. This is the value of the slope that can be measured from Fig. 1 in Lombardi et al. (2010) for the black circles that denote Lupus 1.
Lets now analyze the opposite case. For small radii, and thus the expansion of eq. (37) in Taylor series gives the relation i.e., for small radii all the slope of the M − R curves should approach to two, the limit case. For intermediate radii, however, the solution must be given by eqs. (37) and (38).
In summary, as we can see from eqs. (37)-(41), the intracloud mass-size relation is completely defined by the shape of the N -pdf, and transitions from a slope of 2, at small radii, towards flatter slopes, at large radii.

The inter-cloud relation
Early discussions of the importance of the apparent constancy of the column density of MCs found by Larson (1981) emphasized the limited column densities probed by thenavailable 12 CO observations (Kegel 1989). Since that time, other tracers of column density such as near-infrared extinction with larger dynamic range, have been used to derive similar results. In particular, Lombardi et al. (2010) and Beaumont et al. (2012) have found M ∝ area at differing thresholds, though with differing normalization.
In the first sections of this work we showed how a wide variety of N -pdfs can reproduce the inter-cloud Larson's 3rd law, i.e., can produce M ∝ area relations for multiple clouds. Furthermore, we showed that if one could in principle threshold the volume density, one would get a M ∝ volume relation. This strongly suggests that these types of relations, rather than universal laws, are mostly an artifact of the thresholding process.
Operationally, cloud masses are obtained by integrating or summing column densities over the cloud area. It is thus not surprising that cloud masses are observed to correlate with areas, specially, if most of the mass is at low densities, i.e., occupying most of the area. If then the N -pdf is rapidly decreasing with increasing column density, this means that most of the column density and thus most of the mass is near the threshold. Thus one automatically gets a cloud mass linearly proportional to its area. Similarly, if one were able to measure densities directly, the same principle results in mass linearly proportional to volume.
The significance of the mass-area relationship then reduces to understanding why the mean or peak of the N -pdf seems not to vary much from region to region (Beaumont et al. 2012), and why the N -pdf has a rapid decrease with increasing column density in the thresholds of interest. We argue that there are good physical reasons for this. The first is that, at least in the Solar Neighborhood, an extinction of AV ∼ 1 − 2, equivalent to a hydrogen column density of N ∼10 21 cm −2, is approximately what is needed to shield CO from the dissociating photons of the interstellar radiation field (van Dishoeck & Black 1988;van Dishoeck & Blake 1998). It is difficult to set thresholds for molecular clouds below this average value (which corresponds to A ∼ 0.1 at K-band, the lowest level considered by Lombardi et al. 2010), since as it would not be a CO cloud and these studies are focused towards MCs.
Secondly, we argue that most of the mass of clouds is at low density, because the shielding value is roughly the column density at which one expects self-gravity to become important (Franco & Cox 1986;Hartmann et al. 2001). Numerical simulations (e.g., Ballesteros-Paredes et al. 2011b) indicate that the slope of the N