Annu. Rev. Astron. Astrophys. 1999. 37:311-362 Copyright © 1999 by Annual Reviews. All rights reserved

5. CLUSTERED STAR FORMATION AND MASSIVE STARS

In this section, we will address the issues of clustered formation, regardless of mass, and high-mass star formation, which seems to occur exclusively in clusters. In this review, the term cluster refers to a group of forming stars, whether or not the eventual outcome is a bound cluster. Since high mass stars are rare, the nearest examples are more distant than is the case for low mass star formation. Together with the fact that they form in clusters, the greater distance makes it difficult to isolate individual events of star formation. On the other side of the balance, massive stars are more easily detectable at large distances because luminosity is such a strong function of mass. Heating of the surroundings makes them strong emitters in both dust and many spectral lines; the spectra of regions forming massive stars are often very rich, reflecting both a high excitation state and, in many cases, enhanced abundances, as complex chemistry is driven by elevated temperatures (van Dishoeck & Blake 1998). These features led early studies, when sensitivity was poor, to concentrate on regions forming massive stars. However, most of these advantages arise because the star is strongly influencing its surroundings; if we wish to know the preconditions, we will be misled. This problem is aggravated by the fast evolution of massive stars to the main sequence. Reviews of the topic of clustered star formation can be found in Elmegreen (1985), Lada & Lada (1991), and Lada (1999). Reviews focusing on the formation of massive stars include Churchwell (1993, 1999), Walmsley (1995), Stahler et al (2000), and Kurtz et al (2000).

5.1. Theoretical Issues

Some of the primary theoretical issues regarding the formation of massive stars have been reviewed by Stahler et al (2000). First, what is the relevant dividing line between low-mass and high-mass stars? For the star formation problem, the question is how far in mass the scenario for low-mass stars can be extended. Theoretically, the limit is probably about 10 M_⊙, where stars reach the main sequence before the surrounding envelope is dissipated. Observations of the physical conditions in regions forming intermediate mass stars (Herbig Ae/Be stars and their more embedded precursors) can reveal whether modifications are needed at even lower masses. Since accretion through disks plays a crucial role in the standard model, it is important to know the frequency and properties of disks around more massive stars.

Stars as massive as 100 M_⊙ seem to exist (Kudritzki et al 1992), but radiation pressure from the rapidly evolving stellar core should stop accretion before such masses can be built (e.g. Wolfire & Cassinelli 1987). In addition, massive stars produce very strong outflows (Shepherd & Churchwell 1996, Bachiller 1996). Since standard accretion theory cannot produce rates of mass accretion high enough to overwhelm these dispersive effects, some new effects must become important.

Related questions concern the formation of clusters. To what extent can the ideas of isolated core collapse be applied if there are competing centers of collapse nearby? If the collapse to form massive stars is supercritical, a whole region may collapse and fragment to form many stars.

The fact that the most massive stars are found near the centers of forming clusters has led to the suggestion that massive stars are built by collisional coalescence of stars or protostars (Bonnell et al 1997, 1998). This scheme requires high densities (n_⋆ ≥ 10⁴ stars pc⁻³); for a mean stellar mass of 1 M_⊙, this corresponds to n ≥ 2 × 10⁵ cm⁻³. Even higher stellar densities are seen in the core of the Orion Nebula Cluster (Hillenbrand & Hartmann 1998).

The special problems of making the most massive stars are a subset of the larger question of explaining the mass distribution of all stars. There may be variations in the IMF between clusters (Scalo 1998), which can test theories. Hipparcos observations of nearby OB associations have extended the membership to lower mass stars (de Zeeuw et al 1999), suggesting total masses of a few 10³ M_⊙.

The main questions these issues raise for observations are whether the mass and density of cores are sufficient for forming clusters and massive stars, and whether the mass distribution of clumps in a star forming region can be related to the mass distribution of stars ultimately formed.

5.2. Overall Cloud and Core Properties

What do we know about the general properties of the galactic clouds? The broadest picture is provided by surveys of CO and ¹³CO. Surveys of significant areas of the Galaxy indicate that the power-law distribution in mass seen for small clouds ( dN(M) ∝ M^−α dM, Section 3.1) continues up to a cutoff at M ∼ 6 × 10⁶ M_⊙ (Williams & McKee 1997). Studies by Casoli et al (1984), Solomon & Rivolo (1989), and Brand & Wouterloot (1995) find 1.4 ≤ α ≤ 1.8 over both inner and outer Galaxy. Extinction surveys find flatter slopes (Scalo 1985), but Scalo & Lazarian (1996) suggest that cloud overlap affects the extinction surveys. The fact that α < 2 implies that most of the mass is in the largest structures, although there are issues of how to separate clouds at the largest scales. Since the star formation rate per unit mass, measured by CO, appears not to depend on cloud mass (Mead et al 1990, Evans 1991), the mass distribution supports the idea that most stars form in massive clouds (Elmegreen 1985). However, the enormous spread (> 10²) in star formation rate per unit mass at any given mass, together with the fact that most of the molecular gas is sterile, suggests that comparisons to overall cloud masses, measured by CO, are not particularly relevant.

Surveys in molecular lines indicative of denser gas have generally been biased towards signposts of star formation. Exceptions are the CS J = 2 → 1 survey of L1630 (Lada et al 1991) and the CS J = 1 → 0 and J = 2 → 1 surveys of L1641 (Tatematsu et al 1993, 1998). These two clouds, also called Orion B and Orion A, are adjacent. In both cases, the CS J = 2 → 1 maps showed more contrast than maps of ¹³CO (Bally et al 1987), with CS J = 1 → 0 being somewhat intermediate. Maps of higher-J transitions have been less complete, but show still less area covered by emission. The CS surveys detected less than 20% of the total mass in both clouds (Lada et al 1991, Tatematsu, personal communication). The J = 2 → 1 emission in L1641 is somewhat smoother than the J = 2 → 1 emission from L1630 (Tatematsu et al 1998), and this difference may be reflected in the distribution of star formation. Star formation in L1641 appears to include a distributed component (Strom et al 1993); in contrast, star formation in L1630 is tightly concentrated in clusters associated with massive cores of dense gas (Lada 1992, Li et al 1997).

Excitation analysis of CS lines with higher critical density in L1630 shows that the star-forming regions all contain gas with n ≥ 10⁵ cm⁻³ (Lada et al 1997). These results suggest that surveys in lines of high n_c(jk) are relevant for characterizing star formation regions. Of the possible tracers, CS and NH₃ have been most widely surveyed. In comparison to cores in Taurus, where only low-mass stars are forming, cores in the Orion clouds tend to be more massive and to have larger linewidths when observed with the same tracer (CS: Tatematsu et al 1993, NH₃: Harju et al 1993). The differences are factors of 2–4 for the majority of the cores, but larger for the cores near the Orion Nebula and those forming clusters in L1630 (Lada et al 1991).

CS transitions have been surveyed toward Ultra-Compact (UC) HII regions, H₂O masers, or luminous IRAS sources (see Kurtz et al 2000). Since the IRAS survey became available, most samples are drawn from the IRAS catalog with various color selection criteria applied. The most complete survey (Bronfman et al 1996) was toward IRAS sources with colors characteristic of UC HII regions (Wood and Churchwell 1989) over the entire Galactic plane. Bronfman et al found CS J = 2 → 1 emission (see Table 1 for density sensitivity) in 59% of 1427 IRAS sources, and the undetected sources were either weak in the far-infrared or had peculiar colors. Searches toward H₂O masers have used the catalogs of Braz and Epchtein (1983) and Cesaroni et al (1988). Surveys of CS J = 2 → 1 (Zinchenko et al 1995, Juvela 1996) toward southern H₂O masers found detection rates close to 100%, suggesting that dense, thermally excited gas surrounds the compact, ultra-dense regions needed to produce H₂O masers. The detection rate drops in higher J transitions of CS (Plume et al 1992, 1997) but is still 58% in the CS J = 7 → 6 line, which probes higher densities (Table 1). An LVG, multitransition study of CS lines found ⟨logn (cm⁻³)⟩ = 5.9 for 71 sources and a similar result from a smaller sample using C³⁴S data (Plume et al 1997). Densities derived assuming LVG fall between the average and maximum densities in clumpy models with a range of densities (Juvela 1997, 1998).

Maps of the cores provide size and mass information. Based on the sizes and masses of 28 cores mapped in the CS J = 2 → 1 line (Juvela 1996), one can compute a mean size, ⟨ l ⟩ = 1.2 ± 0.5 pc, mean virial mass, ⟨M_v⟩ ∼ 5500 M_⊙, and ⟨M_n⟩ ∼ 4900 M_⊙. While the two mass estimates agree on average, there can be large differences in individual cases. Remarkably, cloud structure does not introduce a big uncertainty into the cloud masses: using a clumpy cloud model, (see Section 5.4), Juvela (1998) found that M_n increased by a factor of 2 on average compared to homogeneous models and agreed with M_v to within a factor of 2. Plume et al (1997) obtained similar results from strip maps of CS J = 5 → 4: ⟨ l ⟩ = 1.0 ± 0.7 pc (average over 25 cores); ⟨M_v⟩ = 3800 M_⊙ (16 cores). As usual mean values must be regarded with caution; there is a size distribution. As cores with weaker emission are mapped, the mean size decreases; an average of 30 cores with full maps of J = 5 → 4 emission gives ⟨ l ⟩ = 0.74 ± 0.56 pc, with a range of 0.2 to 2.8 pc (Y Shirley, unpublished results).

Churchwell et al (1992) surveyed 11 UC HII regions for CS J = 2 → 1 and J = 5 → 4 emission, leading to estimates of n ≥ 10⁵ cm⁻³. Cesaroni et al (1991) surveyed 8 UC HII in three transitions of CS and C³⁴S and estimated typical sizes of 0.4 pc, masses of 2000 M_⊙, and densities of 10⁶ cm⁻³. More extensive surveys have been made in NH₃; Churchwell et al (1990) found NH₃ (J,K) = (1,1) and (2,2) emission from 70% of a sample of 84 UC HII regions and IRAS sources with similar colors. They derived T_K, finding a peak in the distribution around 20 K, but a significant tail to higher values. Further studies in the (4,4) and (5,5) lines toward 16 UC HII regions with strong (2,2) emission (Cesaroni et al 1992) detected a high fraction. Estimates for T_K ranged from 64 to 136 K and sizes of 0.5 pc. Two sources indicated much higher densities and NH₃ abundances. Follow-up studies with the VLA (Cesaroni et al 1994, 1998) revealed small (∼ 0.1 pc), hot (T_K ∼ 50 −200 K), dense (n = 10⁷ cm⁻³) regions with enhanced NH₃ abundances. These hot cores (discussed below) are slightly displaced from the UC HII, but coincide with H₂O masers.

Magnetic fields strengths have been measured with the Zeeman effect toward about 10 regions of massive star formation. The fields are substantially stronger than the fields seen in isolated, low-mass cores, but the masses are also much higher. In most cases the mass to flux ratio is comparable to the critical ratio, once geometrical effects are considered (Crutcher 1999b). Given the uncertainties and sample size, it is too early to decide if regions forming massive stars are more likely to be supercritical than regions forming only low-mass stars.

The ionization fraction in the somewhat more massive Orion cores appears to be very similar to that in low-mass cores: −6.9 < logx_e < −7.3 (Bergin et al 1999). The most massive cores in their sample have x_e ≤ 10⁻⁸, as do some of the massive cores studied by de Boisanger et al (1996). Expressed in terms of column density, the decline in x_e appears around N ∼ 3 × 10²² cm⁻². At x_e = 10⁻⁸, t_AD = 7 × 10⁵ yr, about 0.1 t_AD in isolated, low-mass cores. Even if the massive cores are subcritical, their evolution should be faster than that of low-mass cores.

To summarize, the existing surveys show ample evidence that massive star formation usually takes place in massive (M > 10³ M_⊙), dense (n ∼ 10⁶ cm⁻³) cores, consistent with the requirements inferred from the study of young clusters and associations, and with conditions needed to form the most massive stars by mergers. Cores with measured B_z seem to be near the boundary between subcritical and supercritical cores.

5.3. Evolutionary Scenarios and Detailed Theories

To what extent can an evolutionary scenario analogous to the class system be constructed for massive star formation? Explicit attempts to fit massive cores into the class system have relied on surveys of IRAS sources. Candidates for massive Class 0 objects, with L>10³ L_⊙, have been found (Wilner et al 1995, Molinari et al 1998). One difficulty with using the shape of the spectral energy distribution for massive star formation is that dense, obscuring material usually surrounds objects even after they have formed, and a single star may be quite evolved but still have enough dust in the vicinity to have the same spectral energy distribution as a much younger object. The basic problem is the difficulty in isolating single objects. Also, the role of disks in massive regions is less clear, and they are unlikely to dominate the spectrum, as they do in low mass Class II sources. Other markers, such as the detection of radio continuum emission, must be used as age indicators. Hot cores provide obvious candidiates to be precursors of UC HII regions, but some have embedded UC HII regions and may be transitional (Kurtz et al 2000). The chemical state of massive cores may also provide an evolutionary sequence; Helmich et al (1994) suggested an evolutionary ordering of three sources in the W3 region based on their molecular spectra and chemical models (see van Dishoeck & Blake 1998).

While theories for clustered and massive star formation are much less developed, some steps have been taken (e.g. Bonnell et al 1998, Myers 1998). The larger Δ v in regions forming massive stars imply that turbulence must be incorporated into the models. Myers & Fuller (1992) suggested a “thermal-non-thermal” (TNT) model, in which n(r) is represented by the sum of two power-laws and the term with p = 1 dominates outside the radius (r_TNT) at which turbulent motions dominate thermal motions. McLaughlin & Pudritz (1997) develop the theory of a logatropic sphere, which has a density distribution approximated by p = 1. Collapse in such a configuration leads to power laws in the collapsing region with similar form to those in the collapsing isothermal sphere, but with higher densities and lower velocities. These ideas lead to accretion rates that increase with time and timescales for forming massive stars that are much weaker functions of the final stellar mass than is the case for the isothermal sphere (Section 4.5). Recent simulations of unmagnetized fragmentation that follow the interaction of clumps find that the mass spectrum of fragments steepens from α = 1.5 to a lognormal distribution of the objects likely to form stars (e.g. Klessen et al 1998). To avoid an excessive global star formation rate (Section 2) and distortion of the clump mass spectrum in the bulk of the cloud, this process must be confined to star-forming regions in clouds.

5.4. Filaments, Clumps, Gradients, and Disks

An important issue is whether the dense cores have overall density gradients or internal structures (clumps) that are likely to form individual stars and whether the mass distribution is like that of stars. Unfortunately, the terms “clumps” and “cores” have no standard usage; I will generally use “cores” to refer to regions that appear in maps of high-excitation lines and “clumps” to mean structures inside cores, except where a different usage is well established (e.g. “hot cores”). Myers (1998) has suggested the term “kernels” to describe clumps within cores. Cores themselves are usually embedded in structures traced by lines with lower critical density, and these are also called clumps by those who map in these lines. Many of these low-density structures are quite filamentary in appearance: examples include the ¹³CO maps of L1641 (Orion B) of Bally et al (1987). In some cases, this filamentary structure is seen on smaller scales in tracers of high density or column density (e.g. Johnstone & Bally 1999, Figure 2).

The clumpy structure of molecular clouds measured in low-excitation lines suggests that dense cores will be clumpy as well. Suggestions of clumpy structure came from early work comparing densities derived from excitation analysis in different tracers, but smooth density gradients provided an alternative (e.g. Evans 1980). Multitransition studies of three cores in CS (Snell et al 1984), C³⁴S (Mundy et al 1986) and H₂CO (Mundy et al 1987) found no evidence for overall density gradients; the same high densities were derived over the face of the core, while the strength of the emission varied substantially. This was explained in a clumpy model with clump filling factors of the dense gas f_v ∼ 0.03 to 0.3, based on a comparison of M_v with M_n (Snell et al 1984). This comparison forms the basis for most claims of unresolved clumps.

Observations with higher resolution support the idea of clumps postulated by Snell et al (1984). For example, Stutzki and Güsten (1990) deconvolved 179 clumps from a map of C¹⁸O J = 2 → 1 emission near M17. Because of overlap, far fewer clumps are apparent to the eye; assumptions about the clump shape and structure may affect the deconvolution. Maps of the same source in several CS and C³⁴S lines (Wang et al 1993) could be reproduced with the clump catalog of Stutzki & Güsten, but only with densities about 5 times higher than they found. Thus the clumps themselves must have structure, either a continuation of clumpiness or smooth gradients. Since the inferred clumps are now similar in size to the cores forming low mass stars, a natural question is whether massive cores are fragmented into many clumps which can be modeled as if they were isolated cores. In favor of this view, Stutzki and Güsten noted that the Jeans length was similar to the size of their clumps.

A significant constraint on this picture is provided by the close confinement of the clumps; unlike the picture of isolated core formation, the sphere of influence of each clump will be limited by its neighbors. A striking example is provided by the dust continuum maps of the ρ Ophiuchi cloud (Figure 1), our nearest example of cluster formation, albeit with no very massive stars. Within about six cores of size 0.2 pc, Motte et al (1998) find about 100 structures with sizes of 1000-4000 AU. They deduce a fragmentation scale of 6000 AU, five times smaller than isolated cores in Taurus. Thus the “feeding zone” of an individual clump is considerably less and the evolution must be more dynamic, with frequent clump-clump interactions, than is the case for isolated star formation. This picture probably applies even more strongly to the more massive cores. In ρ Ophiuchi, the clump mass spectrum above 0.5 M_⊙ steepens to α = 2.5, close to the value for stars (Motte et al 1998), in agreement with predictions of Klessen et al (1998). A similar result (α = 2.1) is found in Serpens, using millimeter interferometry (Testi & Sargent 1998). While more such studies are needed, these results are suggesting that dust continuum maps do trace structures that are likely precursors of stars, opening the study of the origin of the IMF to direct observational study.

Some of the less massive, relatively isolated cores, such as NGC 2071, S140, and GL2591, have been modeled with smooth density and temperature gradients (Zhou et al 1991, 1994b, Carr et al 1995, van der Tak et al 1999). Models with gradients can match the relative strengths of a range of transitions with different excitation requirements, improving on homogeneous models. Zhou et al (1994) summarized attempts to deduce gradients and found preliminary evidence that, as core mass increases, the tendency is first toward smaller values of p. The most massive cores showed little evidence for any overall gradient and more tendency toward clumpy substructure. This trend needs further testing, but it is sensible if more massive cores form clusters. Lada et al (1997) found that the L1630 cores forming rich embedded clusters with high efficiency tended to have larger masses of dense (n > 10⁵ cm⁻³) gas, but a lower volume filling factor of such gas, indicating more fragmentation.

However, the line profiles of optically thick lines predicted by models with gradients are usually self-absorbed, while the observations rarely show this feature in massive cores. Clumps within the overall gradients are a likely solution. The current state of the art in modeling line profiles in massive cores is the work of Juvela (1997, 1998), who has constructed clumpy clouds from both structure tree and fractal models, performed 3-D radiative transport and excitation, and compared the model line profiles to observations of multiple CS and C³⁴S transitions in massive cores. He finds that the clumpy models match the line profiles much better than non-clumpy models, especially if macroturbulence dominates microturbulence. Structure trees (Houlahan & Scalo 1992) match the data better than fractal models, but overall density and/or temperature gradients with p + q ≈ 2 are needed in addition to clumps.

The study of gradients versus clumps in regions forming intermediate mass stars could help to determine whether conditions change qualitatively for star formation above some particular mass and how this change is related to the outcome. Using near-infrared observations of regions with Herbig Ae/Be stars, Testi et al (1997) found that the cluster mode of star formation becomes dominant when the most massive star has a spectral type earlier than B7. Studies of the far-infrared emission from dust remaining in envelopes around Herbig Ae/Be stars found values of p ranging from 0.5 to 2 (Natta et al 1993). Maps of dust continuum emission illustrate the difficulties: the emission may not peak on the visible star, but on nearby, more embedded objects (Henning et al 1998, Di Francesco et al 1998). Detailed models of several suitable sources yield p = 0.75 to 1.5 (Henning et al 1998, Colomé et al 1996). Further work is needed to determine whether a change in physical conditions can be tied to the change to cluster mode.

Many Herbig Ae stars have direct evidence for disks from interferometric studies of dust emission (Mannings & Sargent 1997), though fewer Herbig Be stars have such direct evidence (Di Francesco et al 1997). For a review, see Natta et al (2000). Disks may be more common during more embedded phases of B star formation; Shepherd and Kurtz (1999) have found a large (1000 AU) disk around an embedded B2 star. The statistics of UC HII regions may provide indirect evidence for disks around more massive stars. Since such regions should expand rapidly unless confined, the large number of such regions posed a puzzle (Wood & Churchwell 1989). Photoevaporating disks have been suggested as a solution (Hollenbach et al 1994). Such disks have also been used to explain very broad recombination lines (Jaffe & Martín-Pintado 1999). Kinematic evidence for disks will be discussed in Section 5.5.

A particular group of clumps (or cores) deserve special mention: hot cores (e.g. Ohishi 1997). First identified in the Orion cloud (Genzel & Stutzki 1989), about 20 are now known (see Kurtz et al 2000). They are small regions (l ∼ 0.1 pc), characterized by T_K > 100 K, n > 10⁷ cm⁻³, and rich spectra, probably reflecting enhanced abundances, as well as high excitation (van Dishoeck & Blake 1998). Theoretical issues have been reviewed by Millar (1997) and Kaufman et al (1998), who argue that they are likely to be heated internally, but they often lack radio continuum emission. Since dynamical timescales for gas at such densities are short, these may plausibly be precursors to the UC HII regions.

The evidence for flatter density distribution in regions of intermediate mass support the relevance of models like the TNT or logatropic sphere models in massive regions (Section 5.3), but it will be important to study this trend with the same methods now being applied to regions forming low mass stars, with due regard for the greater distance to most regions forming massive stars. The increasingly fragmented structure in more massive cores and the increased frequency of clusters above a certain mass are consistent with a switch to a qualitatively different mode of star formation, for which different theories are needed. Finally, the common appearance of filaments may support a continuing role for turbulence in dense regions, since simulations of turbulence often produce filamentary structure (e.g. Scalo et al 1998).

5.5. Kinematics

Lada et al (1991) found only a weak correlation between linewidth and size, disappearing entirely for a different clump definition, in the L1630 cores (Goodman Type 2 relation, see Section 4.3). Caselli & Myers (1995) also found that the Type 1 linewidth-size relation (with only non-thermal motions included, Δ v_NT ∝ R^γ) is flatter in massive cloud cores (γ = 0.21 ± 0.03) than in low mass cores (γ = 0.53 ± 0.07); in addition, the correlation is poor (correlation coefficient of 0.56) though the correlation is better for individual cores (Type 3 relations). They also noted that non-thermal (turbulent) motions are much more dominant in more massive cores and find good agreement with predictions of the TNT model. The “massive” cores in the Caselli & Myers study are mostly the cores in Orion with masses between 10 and 100 M_⊙. The much more massive (⟨M_v⟩ = 3800 M_⊙) cores studied by Plume et al (1997) exhibit no statistically significant linewidth-size relation at all (correlation coefficient is 0.26) and the linewidths are systematically higher (by factors of 4–5) for a given size than would be predicted by the relationships derived for low and intermediate mass cores (Caselli & Myers 1995). In addition, the densities of these cores exceed by factors of 100 the predictions of density-size relations found for less massive cores (Myers 1985). The regions forming truly massive stars are much more dynamic, as well as much denser, than would be expected from scaling relations found in less massive regions. The typical linewidth in massive cores is 6–8 km s⁻¹, corresponding to a 1-D velocity dispersion of 2.5–3.4 km s⁻¹, similar to that of the stars in the Orion Nebula Cluster (Hillenbrand & Hartmann 1998). Larson (1981) noted that regions of massive star formation, like Orion and M17, did not follow his original linewidth-size relation, suggesting that gravitational contraction would decrease size while keeping Δ v roughly constant or increasing it.

Searching for collapse in massive cores is complicated by the turbulent, clumpy structure, with many possible centers of collapse, outflow, etc. A collapse signature may indicate an overall collapse of the core, with accompanying fragmentation. In fact, self-absorbed line profiles from regions forming massive stars are rather rare (Plume et al 1997). A possible collapse signature has been seen in CS emission toward NGC 2264 IRS, a Class I source with L ∼ 2000 L_⊙ (Wolf-Chase & Gregersen 1997). If an HII region lies at the center of a collapsing core, absorption lines should trace only the gas in front and should be redshifted relative to the emission lines. Failure to see this effect, comparing H₂CO absorption to CO emission, supported an early argument against the idea that all clouds were collapsing (Zuckerman & Evans 1974). A more recent application of this technique to dense cores, using high-excitation lines of NH₃, showed no preference for red-shifted absorption (Olmi et al 1993) overall, but a few dense cores do show this kind of effect. These sources include W49 (Welch et al 1988, Dickel & Auer 1994), G10.6–0.4 (Keto et al 1988), and W51 (Zhang & Ho 1997, Zhang et al 1998a).

Dickel & Auer (1994) tested different collapse scenarios against observations of HCO⁺ and favored free-fall collapse, with n ∝ r^−1.5 and v ∝ r^−0.5 throughout W49A North; they noted that more complex motions are present on small scales. Keto et al (1988) used NH₃ observations with 0.3″ resolution to separate infall from rotational motions toward the UC HII region, G10.6–0.4. Zhang & Ho (1997) used NH₃ absorption and Zhang et al (1998a) added CS J = 3 → 2 and CH₃CN observations to identify collapse onto two UC HII regions, W51e2 and W51e8, inferring infall velocities of about 3.5 km s⁻¹ on scales of 0.06 pc. Young et al (1998) have tested various collapse models against the data on W51e2 and favor a nearly constant collapse velocity (v ∼ 5 km s⁻¹) and n(r) ∝ r⁻². Mass infall rates of about 6 × 10⁻³ (Zhang et al 1998a) to 5 × 10⁻² (Young et al 1998) M_⊙ yr⁻¹ were inferred for W51e2. Similar results were found in G10.6–0.4 (Keto et al 1988), and even more extreme mass infall rates (10⁻² to 1 M_⊙ yr⁻¹) have been suggested for W49A, a distant source with enormous mass (∼ 10⁶ M_⊙) and luminosity (L ∼ 10⁷ L_⊙) (Welch et al 1988). These high infall rates may facilitate formation of very massive stars (Section 5.1) and help confine UC HII regions (Walmsley 1995).

Large transverse velocity gradients have been seen in some hot cores, including G10.6–0.4 (Keto et al 1988), W51 (Zhang & Ho 1997), and IRAS 20126+4104 (Cesaroni et al 1997, Zhang et al 1998b). For example, gradients reach 80 km s⁻¹ pc⁻¹ in the NH₃ (J,K) = (4,4) line and 400 km s⁻¹pc⁻¹ in the CH₃CN J = 6 → 5 line toward G29.96–0.02 and G31.41+0.31 (Cesaroni et al 1994b, 1998). Cesaroni et al (1998) interpret the gradients in terms of rotating disks extending to about 10⁴ AU. Using the dust continuum emission at 3 mm, they deduce disk masses up to 4200 M_⊙, in the case of G31.41+0.31.

5.6. Implications for Larger Scales

Since most stars form in the massive cores discussed in this section (Elmegreen 1985), they are most relevant to issues of star formation on a galactic scale. If the luminosity is used to trace star formation rate (e.g. Rowan-Robinson et al 1997, Kennicutt 1998), the star formation rate per unit mass is proportional to L/M. Considering clouds as a whole, ⟨L/M⟩ ∼ 4 in solar units (e.g. Mooney & Solomon 1988), with a spread exceeding a factor of 10² (Evans 1991). Using CS J = 5 → 4 emission to measure M in the dense cores, Plume et al (1997) found ⟨L/M⟩ = 190 with a spread of a factor of 15. The star formation rate per unit mass is much higher and less variable if one avoids confusion by the sterile gas. The average L/M seen in dense cores in our Galaxy is similar to the highest value seen in ultra-luminous infrared galaxies, where M is measured by CO (Sanders et al 1991, Sanders & Mirabel 1996). The most dramatic starburst galaxies behave as if their entire interstellar medium has conditions like those in the most active, massive dense cores in our Galaxy. Some studies (e.g. Mauersberger & Henkel 1989) indeed found strongly enhanced CS J = 5 → 4 emission from starburst galaxies. It will be interesting to observe high-J CS lines in the most luminous galaxies to compare to the conditions in massive cores in our Galaxy. Perhaps the large scatter in L/M seen in galaxies will be reduced if CS, rather than CO, is used as a measure of the gas, ultimately leading to a better understanding of what controls galactic star formation rates (see Kennicutt 1998).

5.7. Summary of Clustered Star Formation

The cloud mass distribution found for lower mass objects continues to massive clouds, but less is known about the distribution for dense cores. Very massive cores clearly exist, with sufficient mass (M > 10³ M_⊙) to make the most massive clusters and associations. These cores are denser and much more dynamic than cores involved in isolated star formation, with typical n ∼ 10⁶ cm⁻³ and linewidths about 4–5 times larger than predicted from the linewidth-size relation. Pressures in massive cores (both thermal and turbulent) are substantially higher than in lower mass cores. The densities match those needed to form the densest clusters and the most massive stars by coalescence. There are some cores with evidence of overall collapse, but most do not show a clear pattern. There is some evidence that more massive regions have flatter density profiles, and that fragmentation increases with mass, but more studies are needed. High resolution studies of nearby regions of cluster formation are finding many clumps, limiting the feeding zone of a particular star-forming event to l ∼ 6000 AU, much smaller than the reservoirs available in the isolated mode. In some cases, the clump mass distribution approaches the slope of the IMF, suggesting that the units of star formation have been identified. Studies of intermediate mass stars indicate that a transition to clustered mode occurs at least by a spectral type of B7.