# Harvard Astronomy 201b

## ARTICLE: The Galactic Distribution of OB Associations in Molecular Clouds

In Uncategorized on March 4, 2011 at 5:19 am

### Read the Paper by J.P. Williams and C.F. McKee (1997)

Summary by Vicente Rodriguez Gomez

### Abstract

Molecular clouds account for half of the mass of the interstellar medium interior to the solar circle and for all current star formation. Using cloud catalogs of two CO surveys of the first quadrant, we have fitted the mass distribution of molecular clouds to a truncated power law in a similar manner as the luminosity function of OB associations in the companion paper to this work. After extrapolating from the first quadrant to the entire inner Galaxy, we find that the mass of cataloged clouds amounts to only 40% of current estimates of the total Galactic molecular mass. Following Solomon & Rivolo, we have assumed that the remaining molecular gas is in cold clouds, and we normalize the distribution accordingly. The predicted total number of clouds is then shown to be consistent with that observed in the solar neighborhood where cloud catalogs should be more complete. Within the solar circle, the cumulative form of the distribution is c(>M)=105[(Mu/M)0.6-1], where c is the number of clouds, and Mu = 6 × 106 M is the upper mass limit. The large number of clouds near the upper cutoff to the distribution indicates an underlying physical limit to cloud formation or destruction processes. The slope of the distribution corresponds to dc/dMM−1.6, implying that although numerically most clouds are of low mass, most of the molecular gas is contained within the most massive clouds.

The distribution of cloud masses is then compared to the Galactic distribution of OB association luminosities to obtain statistical estimates of the number of massive stars expected in any given cloud. The likelihood of massive star formation in a cloud is determined, and it is found that the median cloud mass that contains at least one O star is ~105 M. The average star formation efficiency over the lifetime of an association is about 5% but varies by more than 2 orders of magnitude from cloud to cloud and is predicted to increase with cloud mass. O stars photoevaporate their surrounding molecular gas, and even with low rates of formation, they are the principal agents of cloud destruction. Using an improved estimate of the timescale for photoevaporation and our statistics on the expected numbers of stars per cloud, we find that 106 M giant molecular clouds (GMCs) are expected to survive for about 3 × 107 yr. Smaller clouds are disrupted, rather than photoionized, by photoevaporation. The porosity of H II regions in large GMCs is shown to be of order unity, which is consistent with self-regulation of massive star formation in GMCs. On average, 10% of the mass of a GMC is converted to stars by the time it is destroyed by photoevaporation.

### Introduction

This article by Jonathan P. Williams and Christopher F. McKee (1997) was motivated by the question “can one determine, a priori, how many stars are likely to form in a molecular cloud?”  It turns out that this is possible for low-mass stars (McKee 1989), but not for more massive ones, such as O and B types, because of the enormous amounts of ionizing and dissociating radiation they produce. This radiation can quickly destroy their molecular environment and form large HII regions.

In view of the complex interaction between massive stars and their natal molecular clouds, Williams and McKee adopt an empirical approach to infer the number of massive stars that have already formed in molecular clouds. In particular, they focus on giant molecular clouds (GMCs), since they are large enough to generate OB associations.

The first half of the publication deals with estimating the mass spectrum of molecular clouds in the Galaxy, using existing data from CO surveys. In the second half, this mass spectrum is combined with a luminosity distribution of OB associations (McKee & Williams, 1997) to determine the distribution of OB associations within a given cloud. Finally, this result is used to calculate many other quantities and distributions of interest, such as (1) the number of cloud-association pairs of a given mass-luminosity in the Galaxy, (2) the probability that a cloud does not form any massive stars, (3) the most likely brightest association in a cloud, (4) the distribution of star formation efficiencies for a given association within a cloud, (5) the average star formation efficiency for all the associations within a cloud, (6) the filling factor of HII regions in GMCs, and (7) the rate at which the HII regions destroy the clouds.

### Determination of the mass spectrum of molecular clouds

Williams and McKee used the data from four cloud catalogs: three of the first quadrant (DECT, SRBY and SYCSW), which contain a vast amount of GMCs, and one of the solar neighborhood (Dame et al. 1986), essentially for calibration purposes. There were four steps involved in the process:

1. Cloud masses from the different catalogs, which are by no means absolute, were adjusted to a uniform set of parameters: (1) X, the conversion factor of CO to $H_2,$ given by $X=N_{H_2}/W_{CO},$ and (2) $\alpha_{vir},$ a parameter in the formula for the virial mass, $M_{vir} = 5 R \sigma^2/\alpha_{vir}G.$ Clouds for which the distance was ambiguous were removed from the analysis.
2. Clouds were binned by mass and the resulting distribution was modeled with a truncated power law. The upper limit was the largest GMC in the catalogue, of about $6 \times 10^6 M_{\odot}.$
3. The fit was extrapolated to lower masses, which cannot be observed, and the distribution was integrated. The total cloud mass was found to be 2.5 times less than the total mass of molecular gas measured in the inner Galaxy (Bronfman et al. 1988). Therefore, the surveys do not represent the true cloud distribution and the model had to be modified in one of two ways: performing a uniform scaling of the distribution (model A), or adopting a steeper distribution (model B).
4. A comparison with the number of clouds in the solar neighborhood favored model A over model B.

Figure 1. Distribution of the number of clouds with respect to mass for the solar neighborhood (d < 1 kpc).

The resulting distribution of clouds in the Galaxy was found to be

$\frac{d\mathcal{N}_c}{d\ln M} = 63\left(\frac{6\times 10^6 M_{\odot}}{M}\right)^{0.6},$

for $M \le 6\times 10^6 M_{\odot}$ and galactocentric distances between 1.7 and 8.5 kpc.

### Comparison with the distribution of OB association luminosities

Having determined the number of OB associations in McKee & Williams (1997), the next step was to obtain the distribution of associations of luminosity S in a given cloud of mass M. The first obvious constraint was that the overall number of associations of luminosity S summed over all clouds must equal the total number of associations of luminosity S in the Galaxy. But even with this constraint, the allocation of associations in clouds is not uniquely determined. To solve this, Williams & McKee introduced two additional assumptions:

1. The luminosity of the brightest association in a cloud is less than some maximum value, $S \le S_{max}(M).$
2. The number of associations of each luminosity S in a cloud of mass M is half of that expected in a cloud of mass 2M.

With these conditions, Williams and McKee determined the average number of OB associations for clouds of a given mass, and then assumed Poisson statistics for the distribution of OB associations about the mean. Many interesting quantities can be determined from these results, such as the distribution for the brightest association per cloud, the total number of cloud-association pairs in the Galaxy, the probability that a cloud be devoid of O stars (figure 2), and the star formation efficiencies (SFE), i.e. what fraction of the cloud mass is transformed into stars (figure 3).

Figure 2. Probability that a cloud of mass M does not contain an O9.5 star.

Figure 4. Destruction timescale for cloud photoevaporation. The solid line includes the effect of overlapping associations, while the dotted line does not.

Using estimates of the destructive effect of an OB association on a cloud, Williams and McKee also estimated the timescale over which OB associations ionize and disrupt their molecular surroundings (figure 4). The results show that high-mass clouds, with $M \begin{smallmatrix} > \\ \sim \end{smallmatrix} 3 \times 10^5 M_{\odot},$ are destroyed by large numbers of small associations over a timescale of 30-40 Myr, while low-mass clouds are disrupted by O stars, rather than photoionized.

Finally, comparing the results on cloud photoevaporation with the previously obtained efficiencies shows something unexpected: the average SFE over the life of an association is 5%, while the average SFE over the life of a cloud is 10%. If this discrepancy is real and not an artifact of the model, it can be explained by assuming that only half of the GMCs are actively forming stars. This way the lifetime of a star-forming cloud becomes comparable to the observed lifetime of associations, about 20 Myr. Nevertheless, Williams and McKee confirm once again that star formation is a fairly inefficient process: there are clouds with a considerable amount of molecular material and hardly any OB associations.