STAR FORMATION EFFICIENCIES AND LIFETIMES OF GIANT MOLECULAR CLOUDS IN THE MILKY WAY

Giant molecular clouds (GMCs) are stellar nurseries—most stars in most galaxies are born in GMCs. However, the fraction of a GMC's gas that ends up in stars before the cloud is disrupted is rather small, with typical estimates in the Milky Way ∼2%, e.g., Evans (1991). A related question regards the rate at which stars form in GMCs. Global measurements in both the Milky Way and external galaxies (Kennicutt 1998) establish that ∼2% of the molecular gas is turned into stars over the galactic dynamical time R/v_c, where R is the exponential scale length of the disk and v_c is the circular velocity. Because gas disks are in hydrostatic equilibrium, the galactic dynamical time is also the disk dynamical time H/v_T, where H is the disk scale height and v_T is the turbulent velocity of gas in the disk. Finally, the largest GMCs have diameters that are of order the molecular gas disk scale height (Malhotra 1994) and Table 1 below, and CO linewidths ∼5 km  s⁻¹, e.g., Solomon et al. (1987), so GMC dynamical times are again of the same order. This implies that only a few percent of the mass of a GMC is turned into stars in the GMC free-fall time.

The lifetimes of GMCs have been a matter of considerable and intense debate, with estimates ranging from a single dynamical or free-fall time τ_ff (Elmegreen 2000; Hartmann et al. 2001), to a few (Elmegreen 2007), up to 10 or more (Scoville & Hersh 1979; Scoville & Wilson 2004), although it is important to note that the last authors were careful to distinguish between the lifetime of a typical H₂ molecule (which is likely in excess of 10 GMC free-fall times) and that of the GMC of which it is a part. Recent work using the NANTEN telescope has determined that GMCs in the Large Magellanic Cloud have a lifetime of 20–30 Myr (Kawamura et al. 2009; Fukui & Kawamura 2010), about 3τ_ff.

If GMCs live much longer than 2–3 free-fall times, the question arises as to what supports them. As just noted, individual GMCs are observed to have CO linewidths (on large scales) of sufficiently high magnitude to support them against their own gravity. However, it is likely that such motions, often referred to as turbulence, decay on a dynamical time (Mac Low 1999; Ostriker et al. 2001). Hence, long-lived clouds require some energy source if their turbulence is to be maintained. Another possible way to prevent GMC collapse is to appeal to dynamically significant or dominant magnetic fields (e.g., Mouschovias 1976; Shu 1983); such fields (or weaker fields) could be the carriers of MHD turbulence (Arons & Max 1975), possibly from sources outside the cloud, which would obviate the need to maintain the turbulence from inside the cloud.

However, it is also possible that GMCs only live for 2–3 free-fall times, in which case neither magnetic fields nor turbulence are needed to support them; they could be in free fall, as suggested originally by Goldreich & Kwan (1974). These authors pointed out that the similarity between the optically thick CO emission lines and optically thin emission lines from other molecular species in the same clouds could be understood if the clouds had bulk motions with velocities in excess of the thermal linewidths. They argued that there was no clear way to support GMCs, so global collapse was to be expected, leading to large-scale bulk motions needed to explain the observed similarity in line profiles.

This suggestion was disputed by Zuckerman & Evans (1974). They give three arguments against free-fall collapse of GMCs. First, they argue that the resulting star formation rate would be too large. They arrive at this conclusion by assuming that upon collapse, all the gas in a GMC would be converted into stars. This ignores any feedback from the stars on the evolution of the cloud. As shown by Murray & Rahman (2010), Rahman & Murray (2010), and this paper, there is solid evidence that stellar feedback disrupts the host GMC when a fraction _GMC ≈ 0.08 of the GMC mass is converted into stars. This eviscerates the star formation rate argument of Zuckerman & Evans (1974), a point also made by Elmegreen (2007).

The second argument of Zuckerman and Evans is that the line profiles of molecules such as H₂CO, which is seen in absorption toward H ii regions, should show relative velocity shifts compared to the CO emission lines; the H₂CO absorption will be blue shifted if the H ii regions are near the centers of the host GMC and the cloud is expanding, since the portion of the cloud between Earth and the H ii region is moving toward the observer, while the absorption will be red shifted if the cloud is collapsing. This argument makes the assumption that the absorption lines are formed by gas in the outer regions of the GMC, between the observer and the H ii regions, which provide the background continuum against which the H₂CO lines are seen. The observations described in Rahman & Murray (2010) establish the presence of outward velocities (in the expanding bubble walls) somewhat in excess of the turbulent velocities measured by CO observations of the GMCs we examine. Despite this, neither double-peaked CO emission lines, nor redshifted (relative to the CO lines) absorption lines have been reported toward these star-forming complexes (SFCs). High-resolution molecular line observations toward individual bright polycyclic aromatic hydrocarbon emission regions might reveal both red and blue shifted CO emission, and associated molecular absorption lines.

The third argument of Zuckerman & Evans (1974) against global infall of GMCs is that there appear to be many infall centers in a given GMC. However, there is no reason to believe that there will be a single infall center in a cloud that is undergoing global collapse: given the clumpy nature of the turbulent interstellar medium (ISM), any large clump will act as a local center for collapse of gas in its immediate vicinity.

In this paper, we estimate the star formation efficiency, and efficiency per free-fall time, of Milky Way GMCs selected by their free–free luminosity. The star formation efficiency _GMC is measured on a per-GMC basis; in principle it is defined as the total mass in stars, produced over the lifetime of the GMC, divided by the initial GMC mass. However, the observationally accessible quantities in the case of massive clusters are the stellar mass inferred from the ionizing flux (via free–free or Hα emission) and the current GMC mass, so that is what we use here. The star formation efficiency per free-fall time _ff is the fraction of a GMC (or other object) that is turned into stars per free-fall time of the GMC (Padoan 1995). Recent discussions provide both observational estimates of _ff (Krumholz & Tan 2007) and theoretical estimates (e.g., Krumholz & McKee 2005; Krumholz et al. 2009).

We show that both efficiencies vary widely, the latter by nearly three decades. Both have large upper limits of order ∼0.3 and ∼0.6, respectively. We also estimate the lifetimes of massive GMCs, finding that they live 1–3 free-fall times, consistent with the picture of Goldreich & Kwan (1974). In Section 2, we give our definition of the two efficiencies _GMC and _ff. In Section 3, we describe how we select our SFCs, and how we identify their host GMCs. We estimate the lifetimes of the host GMCs in Section 4. We outline the implications of our results and discuss its relation to previous work in Section 5; in particular, we compare our results with the predictions of turbulent star formation theories in Section 5.2. We present our conclusions in the final section.

In this section, we define and compute two types of star formation efficiency. The first is the efficiency _GMC of star formation in GMCs. This is nominally the fraction of a GMC that is converted into stars over the lifetime of a GMC. In fact what we actually measure in this paper is the fraction of a GMC's mass that currently resides in that GMC in the form of ionizing star clusters. This is because we find stars (and their parent GMCs) by using Wilkinson Microwave Anisotropy Probe (WMAP) to look for free–free radiation, which is powered almost exclusively by massive stars.

2.1. Star Formation Efficiency in a GMC

2.2. Star Formation Efficiency per Free-fall Time

In this section we measure the the efficiency, _GMC, of star formation in individual Milky Way GMCs, and the efficiency of star formation per free-fall time, _ff. We select the most rapidly star-forming GMCs in the Galaxy: the 32 GMCs we select are responsible for 31% of the star formation in the Galaxy.

We start with the WMAP free–free thermal radio fluxes of sources found by Murray & Rahman (2010). Examination of Spitzer and MSX images in the direction of the WMAP sources reveals one to several "star-forming complexes" in the direction of each of the WMAP sources (Rahman & Murray 2010). A SFC is identified on the basis of the morphology in the 8 μm Spitzer and MSX images, combined with measurements of radio recombination line (or molecular line) radial velocities taken from the literature. The radial velocity data are given in Table 4 of Rahman & Murray (2010). The result is a list of 40 SFCs with Galactic coordinates and radial velocities, (l, b, v_r). These complexes have characteristic sizes of ∼25 pc, ranging between 2 pc and 70 pc (see Table 1). The spread in v_r is ≲15–20 km  s⁻¹.

Once all the SFCs are identified, we divide the free–free flux in each WMAP source between the SFCs contained in that source. In two cases, G283 and G327, there is only a single SFC in the WMAP source. We assign the entire free–free flux of the WMAP source to that SFC; in both cases the size of the SFC (determined by the extent of the region outlined by the radio recombination line velocity measurements) is similar to that of the WMAP source, supporting the view that only that source is responsible for the free–free emission seen by WMAP in that direction.

In four of the WMAP sources there are exactly two SFCs; we split the flux between the SFCs based on the radio recombination line fluxes associated with each source. Such a division is a rather crude approximation; ground-based radio continuum fluxes toward each of these complexes are systematically lower than the continuum fluxes measured by WMAP, but the flux ratios vary unpredictably from object to object. In its favor, this recombination line ratio split was consistent with the relative amount of 8 μm flux associated with the SFCs (accounting for the fact that the total 8 μm flux appears to scale as the square of the free–free flux, Rahman & Murray (2010)).

Rahman & Murray (2010) identified four SFCs in G10, three in G34 and G311, and five in G337. In all these cases the SFCs are well separated in (l, b, v_r) space, as was the 8 μm emission in (l, b). The division of the free–free emission amongst the sources was made in the same manner as for the previous cases.

The two WMAP sources G24 and G30 are very confused, with eight and six SFCs, respectively; the corresponding catalog numbers are in the range 5–18 inclusive. We again used the recombination line ratios to assign the free–free flux to each source, but we have less faith in the results; to indicate this, we plot these points using open polygons, as opposed to the filled polygons plotted for the less confused WMAP sources.

Having assigned fluxes to each SFC, we then use the kinematic distance D to that region to calculate the free–free luminosity L_ν = 4πD²f_ν emitted by that region, and the rate Q = 1.33 × 10²⁶L_ν s⁻¹ (Murray & Rahman 2010) of ionizing photons emitted by each source per second required to power the observed free–free luminosity. In cases where the distances listed in Rahman & Murray (2010) and the relevant GMC catalog do not agree, we use the distance of the former. In the case of Rahman & Murray sources 5, 7, 9, 12, 15, and 35, where those authors do not resolve the distance ambiguity, we use the distance that is closer to that of the relevant GMC catalog distance.

Following McKee & Williams (1997) we then increase the rate of ionizing photons by a factor 1.37 to account for the fact that some of those photons are absorbed by dust, and hence do not contribute to the free–free emission detected by WMAP.

The live stellar mass was then calculated using Equation (3); a given GMC almost certainly has given birth to stars older than 〈t_ms〉, so M_* as defined here is a lower limit on the total mass of stars formed in the GMCs we examine.

The next step in the calculation of _GMC and _ff is to identify the host GMCs of our SFCs. We search the GMC catalogs of Solomon et al. (1987), Grabelsky et al. (1988), Bronfman et al. (1989), and Heyer et al. (2009) for objects having centers at the same Galactic longitude within ±0.4 deg, Galactic latitude within ±0.4 deg (∼50  pc at D = 10  kpc, comparable to the radius of a typical cloud), and the same heliocentric radial velocity within ±20 km  s⁻¹.

We find matches for 32 out of our 40 SFCs; results are given in Table 1. A number of GMCs are listed in both Solomon et al. (1987) and Heyer et al. (2009), in which case we use the values in the latter. We corrected the GMC radii to account for the difference between our assumed value for the distance to the Galactic center R₀ = 8.5  kpc and that used in the GMC surveys. We have followed Williams & McKee (1997, their Table 1) in correcting the masses of the GMCs in the first three surveys. Heyer et al. use the same distance to the Galactic center we do (8.5  kpc); we use M_GMC = 2 ×  M_LTE, as those authors recommend (corresponding to $X_{\rm CO}=1.9\times 10^{20}\,{\rm cm }^{-2} (\rm{K }\,{\rm km \,\, s}^{-1})^{-1}$).

Two of the matches are to objects in Grabelsky et al. (1988) that those authors identify as "cloud complexes," assigning them catalog numbers 26 and 35 (corresponding to our numbers 29 and 32). The complexes contain more than one GMC, but Grabelsky et al. (1988) do not give separate masses or radii for the component GMCs. The smaller of the two complexes has a mass nearly twice that of the most massive GMC, and is well separated from the GMCs in the plots below. In the figures we plot these complexes as open circles, but connect them to filled squares; the latter represents an attempt to break up the complexes, assuming that the most massive GMC contains half the mass of the entire complex. However, in calculating the various averages presented below, we have simply used the total mass contained in the complex.

The average star-forming GMC mass in our ionizing luminosity-selected sample is 1.5 × 10⁶ M_☉. The ionizing luminosity-weighted GMC mass is 2.3 × 10⁶ M_☉. The average and Q-weighted GMC radii are 61  pc and 82  pc, respectively. Similarly, the average and Q-weighted free-fall times are 9  Myr and 10  Myr.

These averages include the confused GMCs associated with SFCs 5 to 18. These GMCs have been investigated by both Solomon et al. (1987) and Heyer et al. (2009); the masses of these GMCs differ between the two studies (as noted above, we use the results of Heyer et al. 2009 where possible). While the average properties of the confused sources (and their host GMCs) are similar to those of the unconfused sources, the measured properties of individual confused sources may be unreliable.

If we confine our attention to unconfused GMCs, the average and Q-weighted radii are 85  pc and 110  pc. Similarly, the average and Q-weighted free-fall times for the unconfused GMCs are 13  Myr and 15  Myr, while the corresponding masses are 2.1 × 10⁶ M_☉ and 2.9 × 10⁶ M_☉.

Having found the mass of the host GMCs, we can estimate the star formation efficiency. The results are plotted in Figure 2.

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The sample average 〈_GMC〉 = 0.08 for our 32 SFCs. We stress again that _GMC as defined here is a lower limit to the total star formation efficiency of the host GMC. Any stars in clusters older than 〈τ_ms〉 will not be detected by WMAP and hence are not included in our estimate of M_*.

This average is a bit higher than that usually quoted (typical estimates are more like 0.02), but we have selected the most active SFCs in the Galaxy, rather than a Galaxy wide average.

The Q-weighted efficiency is defined as 〈_GMC〉_Q ≡ Σ_iQ_iG,i/(Σ_iQ_i); we find 〈_GMC〉_Q = 0.07. Finally, if we weight by the mass of the host GMC, we find $\langle \epsilon _{\rm GMC }\rangle _{M_{\rm GMC}}\equiv \Sigma _i M_{\rm GMC,i} \epsilon _{G,i}/(\Sigma _i M_{\rm GMC,i})=0.03$.

3.1. The Star Formation Rate per Free-fall Time

The result that ∼30% of the star formation in the Milky Way occurs in 32 GMCs is truly remarkable. This becomes apparent when we compare the total molecular gas mass in the Milky Way, M_tot ≈ 10⁹ M_☉ (Dame 1993), to the mass in the 32 star-forming GMCs, with total gas mass M ≈ 5.9 × 10⁷ M_☉; 30% of the star formation takes place in clouds that contain only 6% of the molecular gas mass. For context, the number of GMCs with masses above 10⁵ M_☉ ≈ 1100, while the number with masses above 10⁶ M_☉ ≈ 250. Thus, the bulk of massive GMCs in the Milky Way do not house WMAP sources.

All the SFCs we examine contain expanding bubbles, with typical expansion velocities of ∼10 km  s⁻¹ and radii R_b ranging from 3  pc to 100  pc (Rahman & Murray 2010). In the more vigorously star-forming GMCs, the outward force exerted by radiation from the stars exceeds the inward force of gravity acting on the GMC; we use

for the force due to radiation, where ξ = 8 × 10⁻¹¹ erg  s⁻¹ ·  s, appropriate for our choice of IMF. The outward force due to the pressure of the ionized gas is estimated as F_gas = 4πR²_bP, where P is the gas pressure. We use the expression P = nkT, where k is Boltzmann's constant, the temperature of the ionized gas is T = 7000 K, and

where α_rec is the recombination coefficient.

For the force of gravity we estimate

The ratio F_rad/F_grav is given in Column 9 of Table 2, and the ratio (F_rad + F_gas)/F_grav is plotted in Figure 5. More than half of the unconfused sources have a total outward force larger than the force of self-gravity. This is consistent with our interpretation of the radial velocity spreads as being due to expansion of the bubble walls, and strongly suggests that the star clusters are disrupting their host GMCs. We note that in almost all our sources, F_rad>F_gas, as expected for such massive clusters (Murray et al. 2010).

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The largest bubbles appear likely to disrupt the host GMC; the bubble walls have enough momentum to sweep up the rest of the gas in the host cloud, driving it to r ∼ 200 pc, at which point tidal shear will complete the disruption. We see these clouds in their death throes.

We can estimate the lifetimes of massive (M_GMC ≳ 10⁶ M_☉) Milky Way GMCs against disruption by the effects of the star clusters that form in them. To do so we assume that the massive GMCs harboring the massive clusters we examine here are drawn from the massive GMC population as a whole. This assumption implies that all massive GMCs will eventually form massive star clusters, but leaves open the possibility that less massive GMCs (say with masses below ∼10⁵ M_☉) do not necessarily form many stars. For example, cloud fragments from the objects we have found, which may well have masses as large as 10⁵ M_☉, may not be self-gravitating. If they are not, they may avoid any substantial star formation until the next time they find themselves inside a (larger) gravitationally bound object.

To proceed with our estimate of massive GMC lifetimes, we need to estimate the number of GMCs in the parent population of our star-forming objects. We do so in two different ways. First, we estimate the number of GMCs in the Milky Way that are required to produce a fraction f_* ≈ 0.31 of the observed total star formation rate, $\hbox{\it dM}_*/dt=1.3\,M_\odot \,\,{\rm yr }^{-1}$. This requires an estimate of the GMC distribution function, which is known to follow a power-law dN_GMC/dm ∼ m^−α truncated at an upper mass M_U (see the Appendix). The most massive GMC known in the Milky Way is object number 24 in Grabelsky et al. (1988, which does not correspond to any of the bright WMAP sources considered in this paper). We follow McKee & Williams (1997) and adopt M_U = 6 × 10⁶ M_☉, but we also report the result of using M_U = 1.1 × 10⁷ M_☉, which we take as a very firm upper limit. The power-law index α ≈ 1.5–1.6 (Casoli et al. 1984; Dame et al. 1986; Digel et al. 1996; Rathborne et al. 2009). We truncate the power law at the lower end as well, denoting the lower limit to the GMC mass distribution by M_L; our results depend only weakly on M_L.

The second method of estimating GMC lifetimes uses the mass function of GMCs integrated from M_U down to a critical GMC mass M_crit. This critical GMC mass can be defined in multiple ways. We define it as that mass which results in a mean GMC mass equal to the Q-weighted mean GMC mass in our sample.

We proceed to the first estimate. To find the star formation rate in clouds with masses greater than some mass M_GMC, we use the Galaxy-wide average star formation rate per free-fall time, and integrate from M_U down toward the least massive clouds, with mass M_L. Let f_* be the fraction of the Galactic star formation rate that occurs in clouds with mass M_f* or larger. We show in the Appendix that the critical mass (denoted by $M_{f_*}$ here since it is defined by the fraction of the total star formation rate produced by clouds at or above that mass) is

where δ = (7/4) − α. Figure 6 shows the cumulative fraction of the star formation rate produced in GMCs with masses greater than M, for two values of M_U, our best estimate 6 × 10⁶ M_☉ (Williams & McKee 1997), and a firm upper limit M_U = 1.1 × 10⁷ M_☉, and two values of α (1.5 and 1.6). Taking the two extreme cases, the value of $M_{f_*=0.31}=1.05\times 10^6\,M_\odot$ (M_U = 6 × 10⁶ M_☉, α = 1.6) and $M_{f_*=0.31}=2.9\times 10^6\,M_\odot$ (M_U = 1.1 × 10⁷ M_☉, α = 1.5).

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Figure 7 shows the cumulative number N(>M) of GMCs with masses M_GMC>M. The number of Milky Way GMCs with masses larger than M(f_* = 0.31) = 1.05 × 10⁶ M_☉ is 87 (left panel, dashed vertical line, and for M(f_* = 0.31) = 2.9 × 10⁶ M_☉ there are 210 GMCs (right panel, dashed vertical line). We use the relation

where we assume that the average lifetime of the star clusters in our sample is 〈τ_ms〉 = 3.9  Myr. Note that the luminosity of a cluster decays much more slowly after 3.9  Myr; the luminosity decays to half the luminosity at 〈τ_ms〉 only when the cluster age is ≈8  Myr (about a dynamical time for the massive GMCs we consider here). Thus, the radiation pressure from the cluster acts for about 2〈τ_ms〉 (while the H ii gas pressure essentially vanishes after 〈τ_ms〉).

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We find that the lifetime of the parent GMCs is ∼11  Myr and ∼26  Myr for the two extreme cases.

The Q-weighted average free-fall time of these GMCs is ≈9  Myr, and 10  Myr for unconfused GMCs, so our first estimate is that massive GMCs live 1.1–2.9 free-fall times before they are disrupted by expanding bubbles produced by the star clusters they contain.

Now for the second estimate for the lifetimes of massive GMCs, which uses the Q-weighted mean GMC mass for the parent GMCs in our ionizing luminosity-selected sample of SFCs. We use the GMC mass function, $\hbox{\it dN}_{\rm GMC}/dm$, to calculate the mean GMC mass as a function of the lower mass cutoff (denoted M_crit). We then adjust M_crit until the mean GMC mass is equal to the Q-weighted mean GMC mass in our sample. Having found the appropriate M_crit, we then calculate the number of GMCs having masses above M_crit, and compare that to the number of GMCs in our sample.

The Q-weighted mean GMC mass of our sample is 2.3 × 10⁶ M_☉. The M_crit that gives the same mean mass, assuming M_U = 6 × 10⁶ M_☉ and α = 1.6, is 9.5 × 10⁵ M_☉, corresponding to N(>9.5 × 10⁵ M_☉) = 231; see Figure 7. The estimated GMC lifetime is ∼28  Myr, or ∼3τ_ff.

If we use M_U = 1.1 × 10⁷ M_☉ and α = 1.5, we find N(>4.9 × 10⁵ M_☉) = 343, a lifetime of 42  Myr, or ∼4.2τ_ff. These lifetimes are somewhat larger than our first estimate; averaging we take the lifetime of massive GMCs in the Milky Way to be 27 ± 12  Myr, or ∼2.7 ± 1τ_ff.

The ionizing luminosity of some of our SFCs is rather low, while our original WMAP sample was chosen for its high luminosity. The confusion of the WMAP sources results in the inclusion of low luminosity SFCs, which in turn leads to our overcounting the number of GMCs in a true luminosity-selected sample. We can try to correct for this by making a cut in Q, at a value we will denote Q_cut. When we do so, we find that the average GMC mass increases as a function of Q_cut. So while it is true that increasing Q_cut decreases the number of star-forming GMCs in our sample, which tends to increase the estimated lifetimes of the GMCs, at the same time 〈M_GMC〉(Q_cut) also increases, which tends to decrease the estimated lifetimes of the GMCs. The net result is that the ratio N_GMC/N_cut increases slowly, from 4.6 to 6.1, and the estimated lifetime increases by 30% by the time Q_cut = 5 × 10⁵¹  s⁻¹; at this value of Q_cut, there are only seven sources left in the sample, and the mean mass is no longer well defined, i.e., the mean GMC mass begins to drop with increasing Q_cut beyond this value.

A number of authors, e.g., Mooney & Solomon (1988), Mead et al. (1990), and Evans (1991), have examined the ratio L_FIR/L_CO (a proxy for _GMC) in Milky Way GMCs. They found that this ratio varies by nearly three orders of magnitude; this is consistent with the findings in this paper, and is illustrated in Figure 3.

Similar result have been found recently by Schruba et al. (2010), who examine the ratio of H_α to CO luminosity in nearby galaxies. They find that selecting for H ii regions leads to very high ratios on small scales, whereas selecting for CO peaks leads to very low ratios on small scales. Again, this suggests large variations in _GMC on small scales (although their smallest scales tend to be larger than individual GMCs in the relevant host galaxies).

Along the same lines, Lada et al. (2010) examine nearby SFCs in the Milky Way and argue that the star formation rate per free-fall time varies strongly from cloud to cloud.

The massive clusters in the sample examined here are capable of ionizing a substantial fraction of the mass in their host GMCs. For a cluster emitting Q ionizing photons per second, all of which are absorbed by hydrogen in the host GMC, we have

where α_rec ≈ 3.57 × 10⁻¹³ cm³ s⁻¹ is the recombination coefficient.

This expression assumes that all the ionizing photons emitted by the stars in a GMC are absorbed in that GMC. However, the WMAP results show that a substantial fraction, and in many cases a majority, of the ionizing photons escape the host GMC. For example, source 29, at l = 298°, has a radius in excess of a degree (about twice the WMAP beam). This corresponds to R ≈ 200 pc, compared to R_GMC = 137 pc; the bulk of the free–free emission comes from outside the host GMC in this object. We estimate that on average, half or more of the ionizing photons escape from the host GMC. In addition, it is estimated that about 35% of the ionizing photons emitted by hot stars are absorbed by dust (McKee & Williams 1997).

Finally, if all the ionizing photons are absorbed before they reach R_GMC, the ionized mass will again be smaller than the estimate in Equation (19).

If we account only for the presence of dust, we find that, on average, ∼50% of the GMC gas could be ionized. This is consistent with the fact that the CO-inferred masses correlate well with the virial masses in the GMC catalogs we use; the average ratio of CO-inferred mass to virial mass is also 0.5. (This paper uses the virial GMC masses throughout.) However, it is likely that much of the non-molecular gas contributing to the viral mass is atomic rather than ionized. Indeed, if the sizes of the WMAP sources are any indication, only a small fraction of the non-molecular gas can be ionized; most of the photons leak out of the host GMC, and so cannot ionize as much gas inside R_GMC.

We note that we are using the currently measured virial mass, along with the current, less than 4  Myr old (ionizing) stellar mass. The mass of the cloud cannot change on a timescale much less than the dynamical time (the bubble wall expansion velocity and the ionized gas sound speed are both comparable to the reported GMC linewidths), either by feedback driven loss of mass or by accretion. Since the star clusters we find are less than half a GMC free-fall time old, their masses reflect the current GMC mass rather than any previous (larger or smaller) mass associated with their host GMC. It follows that the high values of _GMC and _ff are not the result of recent mass loss from the host GMCs.

Figure 5 shows that the sum of radiation pressure and gas pressure forces exceeds the self gravity in 19 of our 32 GMCs, indicating that these GMCs should be in the process of being disrupted. The presence of expanding bubbles in these systems is consistent with this notion, and with the predictions of Murray et al. (2010). This strongly suggests that these GMCs are at the end of their lives, either as converging flows or as gravitationally bound objects.

Both the observations and the theory of Murray et al. (2010) suggest that the process of disruption takes about a free-fall time, since the expansion velocity of the bubbles is v ∼ 15  km  s⁻¹, similar to the observed CO linewidths. The time for turbulence to decay (assuming the clouds are initially turbulence supported) is about one free-fall time, so the time for an unsupported GMC to collapse is ∼1–2τ_ff. The total life time of an unsupported GMC, from formation through collapse and star formation to disruption, would then be 1.5–2.5τ_ff, similar to the value we measure. In other words, the fact that massive GMCs in the Milky Way appear to live only about two free-fall times is consistent with the suggestion of Goldreich & Kwan (1974) that all GMCs are in the process of collapse; the short lifetimes of massive GMCs obviate the need for any means of support, including driven turbulence or magnetic support.

As noted in the introduction, the argument of Zuckerman & Evans (1974), that rapid collapse of GMCs would lead to a star formation rate in the Milky Way that was much higher than the observed rate, does not apply if the stars that are formed disrupt the host GMC. We have seen that the GMCs we examine which have _GMC ∼ 0.08 appear to be in the process of disruption. This suggests that the star formation rate is much less than M_GMC/τ_ff because only a small fraction of M_GMC is converted into stars before the cloud is disrupted by radiation pressure.

While our results are consistent with free-fall collapse of GMCs, they do not require such a collapse. It may be, for example, that the bulk of the material in GMCs is held up by magnetic fields, but that over one dynamical time (∼10 Myr), some 10%–15% of the GMC accretes onto one or two massive clumps. Roughly half of this material will form stars, with the rest dispersed back in to the ISM, resulting in the observed 〈_GMC〉_Q ∼ 0.08. It may also be that the GMCs are not gravitationally bound to begin with; measurements of the virial parameters of massive clouds are near unity.

The last statement is consistent with the short lifetimes we have inferred for massive GMCs—dissipating sufficient energy to make a strongly bound GMC would require more than a single free-fall time. One could also imagine that some supply of energy could maintain a cloud with a virial parameter near unity. However, since the clouds only live for one or two free-fall times, it is difficult to envision a steady state arising, in which decay of turbulent energy is balanced by some source of energy.

Adding up the star formation from all the GMCs, ∑_GMCM_GMC/(2τ_ff) (the factor two comes from our estimate of the GMC lifetime) the star formation rate is only a factor of ∼2 larger than the observed Milky Way star formation rate. This implies that the fragments of the disrupted GMCs must reassemble in ∼2τ_ff; if they did not, the star formation rate would be lower than observed. In addition, if the fragments of the disrupted 10⁶ M_☉ GMCs do not reassemble after a short time, more of the molecular gas would reside in smaller clouds, contrary to observations. Both these observations strongly suggest that the cloud fragments reassemble in less than an orbital period.

We note that the lifetimes of the majority of massive GMCs cannot be much larger than a few τ_dyn, for two reasons: first, we would see massive GMCs between spiral arms, something for which there is little observational evidence. Second, the star formation rate would be much less than that observed. In other words, the majority of the most massive GMCs in the Milky Way cannot be supported against collapse by any mechanism that suppresses star formation for more than two or three GMC free-fall times.

5.1. Dynamical Star Formation

5.2. Turbulence and Star Formation

We have presented four major observational results in this paper. First, we have estimated a lower limit to the fraction of gas in a massive Milky Way GMC that will be converted into stars over the lifetime of that GMC, finding _GMC ≈ 0.08. Second, we have measured the star formation rate per free-fall time, _ff ≈ 0.16, for our ionizing luminosity-selected sample of GMCs. Third, we have estimated the lifetime of massive (M_GMC ≈ 10⁶ M_☉ Milky Way GMCs, finding that they live two to three free-fall times. Finally, we have shown that many if not most of these clouds are in the process of being disrupted by the radiation pressure of the star clusters they have given birth to.

The strong implication of these results is that the rate of star formation in Milky Way GMCs, and by extension, in the Galaxy as a whole, is not set by the properties of supersonic turbulence.

This research has made use of NASA's Astrophysics Data System. The author is supported in part by the Canada Research Chair program and by NSERC of Canada.

The GMC mass distribution function is generally fit by a simple power law

with α ≈ 1.5 (Casoli et al. 1984; Dame et al. 1986; Digel et al. 1996; Rathborne et al. 2009). These papers show that Milky Way GMCs have masses M_L < m < M_U with M_L ≈ 10³ M_☉ and M_U = 6 × 10⁶ M_☉; we adopt this value as our best estimate following Williams & McKee (1997). There are two objects in Table 2 that appear to have GMC masses above this limit, numbers 29 and 32 (Grabelsky et al. catalog numbers 26 and 35) but both are described by those authors as "cloud complexes." They list three other objects with masses in excess of 6 ×  10⁶ M_☉, none of which they consider to be a single GMC. We take M_U = 1.1 × 10⁷ M_☉ as a firm upper limit on the mass of the most massive GMC in the Galaxy.

The total mass in molecular gas is M_g = 10⁹ M_☉ (Dame 1993); we assume that it is all in GMCs.

It is useful to employ the natural logarithm of the mass as the independent variable

where β = α − 1. The quantity N_U is approximately the number of clouds in the mass range from M_U/2 to M_U.

Using this notation, the total mass in GMCs is related to the upper and lower mass cutoff, and the power-law index, by

The observed quantities are M_Tot, M_U, M_L, and the power-law slope α, so we solve for N_U in terms of these four:

where γ ≡ (2 − α). Using the numerical values given above, we find

The number of GMCs with mass larger than m is

Performing the integral we find

From Equation (A5),

The total number of clouds is $N_{\rm Tot}\approx \rm{13{,}000}$.

Next we calculate the star formation rate of the Galaxy under the assumption that

with a fixed _ff, i.e., the same value of _ff is used for all GMCs. The total star formation rate is

To evaluate τ_ff we will use the fact that the surface density Σ₀ ≡ M_GMC/(πR²_GMC) of GMCs is constant, independent of M_GMC, e.g., Solomon et al. (1987). Then τ_ff(M_GMC) = τ_ff(M_U)(M_GMC/M_U)^1/4 and

Letting δ = (7/4) − α, the fraction of star formation that takes place in GMCs with masses greater than m is

It follows that the GMC mass $M_{f_*}$ at which the cumulative star formation rate reaches a fraction f_* of the total star formation rate is