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–1– 28. HIGH-MASS STAR FORMATION: THEORY 28.1. The Effects of Radiation Pressure There are two critical differences between low and high-mass star formation: First, low-mass stars form in a time t∗f ∼ m∗ /ṁ∗ that is short compared to the Kelvin-Helmholtz time, tKH = Gm2∗ /R∗ L∗ , (1) whereas high-mass stars generally have tKH . t∗f (Kahn 1974). For example, for m∗ = 31M¯ , we have R∗ = 9.3R¯ and L∗ = 105.24 L¯ (Sternberg et al. 2003), corresponding to tKH = 18, 000 yr; as we shall see below, the formation time for such a star is of order 10 5 yr. As a result, whereas low-mass stars undergo extensive pre-main sequence evolution, high-mass stars accrete a significant amount of mass while on the main sequence. The second critical difference follows from the first: because high-mass stars are accreting while on the main sequence, they are hot and luminous, and this implies a radiation pressure large enough to overcome the force of gravity. The force per unit mass due to radiation pressure is Z ∞ κF F κF L κP (T )L κν Fν dν = = ' , (2) frad = 2c c c 4πr 4πr2 c 0 where κF is the flux-mean opacity and κP (T ) is the Planck-mean opacity. The approximation κF ' κP becomes exact as the frequency distribution approaches that of a blackbody. We can now define an Eddington limit for dust, LE, d , based on the condition that the radiative force match the gravitational force, κP (T )LE, d Gm∗ 4πcGm∗ = 2 ⇒ LE, d = . (3) 2 4πr c r κP Pollack et al. (1994) calculated κP (T ) for a dust model with an MRN distribution (dngr /da ∝ extending to 1 µm and a steeper distribution (∝ a−5.5 ) to 5 µm. The dominant constituents of the grains are olivine, which vaporizes at about 1200 K; refractory organic material, which vaporizes at about 600 K; and water ice, which vaporizes at about 140 K (all vaporization temperatures are evaluated at ρ = 10−12 g cm−3 ). The Planck opacity rises from 0 at T = 0 K to 3 cm2 g−1 at T = 100 K, and reaches a maximum of 8 cm2 g−1 at T ' 600 K. At higher temperatures, the opacity drops as the grains are vaporized. The effective Eddington limit is defined by the maximum opacity, which sets the minimum luminosity: ¶ µ m∗ 4 LE, d = 4.9 × 10 L¯ . (4) 30 M¯ a−3.5 ) By comparison, the standard Eddington limit is based on opacity due to electron scattering, κ es = 0.4 cm2 g−1 , which gives LE ' 106 L¯ at m∗ = 30 M¯ . The luminosity of a star increases rapidly with mass, and LE, d exceeds L∗ for m∗ & 15M¯ . How can stars more massive than this form? –2– Several mechanisms have been suggested to overcome the radiation pressure problem. The first is ram pressure. Wolfire & Cassinelli (1987) pointed out that the infalling gas could overcome the radiation pressure provided that the ram pressure exceeded the momentum flux in the radiation field. This is valid in the limit in which each photon transfers its momentum only once to the infalling gas; if the gas is sufficiently opaque, multiple scattering can occur and the energy rather than the momentum of the gas can be tapped. However, at the dust destruction front, UV radiation is absorbed at a high opacity (κP ∼ 200 cm2 g−1 ) and reradiated into the infrared, where the opacity is much lower. As a result, the approximation that the momentum transfer per emitted photon is hν/c is greatly improved. With this approximation, the ram pressure condition is 2 ρvin = ṁ∗ vin F > 4πr2 c ⇒ ṁ∗ > L , vin c (5) where vin = (Gm∗ /r)1/2 is the infall velocity. This condition must be evaluated at the dust destruction front at a radius r dd , which is given by the condition that the equilibrium temperature of a grain exposed to the stellar radiation field is just high enough to vaporize the grain, Tgr = Tdd . Recall that the grain absorption and emission cross sections are of the form Qπa2gr , where agr is the grain radius. Balancing absorption in the UV and emission in the IR, we have ¶ µ L∗ 2 4 πagr QUV = 4πa2gr QIR σSB Tdd . (6) 2 4πrdd Draine’s website has data on the values of Q and indicates that QUV ' 50QIR , so that rdd = µ QUV L∗ 4 QIR 16πσSB Tdd ¶1/2 1/2 = 1.8 × 1015 L∗5 cm, (7) where L∗5 = L∗ /(105 L¯ ). Inserting this result into equation (5), we find that the accretion rate needed to overcome radiation pressure is ṁ∗ > 1.4 × 10−4 Ã 5/4 L∗5 m∗ /30 M¯ !1/2 M¯ yr−1 . (8) Note that this is considerably greater than the typical accretion rates for low-mass stars. Other mechanisms that have been suggested to overcome the effects of radiation pressure are - Rotation: Nakano (1989) pointed out that since the infalling gas is rotating, it would fall into a disk, where it is shielded from the stellar radiation. Jijina & Adams (1996) worked out the dynamics of rotating, dusty gas falling towards a massive protostar, and concluded that this effect was adequate to permit the formation of stars up to about 100 M ¯ . Their calculations assumed that only the momentum of the emitted photons could reverse the infall. –3– - Beaming: Krumholz et al. (2005) showed that the cavities produced by protostellar outflows would allow the reprocessed radiation to escape, reducing the radiation pressure in the infalling gas and permitting accretion over a substantial range of solid angle. Their calculations were based on the solution of the radiative transfer equation and therefore allowed for tapping the energy of the emitted photons. - 3D effects: Krumholz et al. (PPV poster and in prep.) have shown that in 3D numerical simulations, the radiation is able to escape through low-column regions while the gas can accrete in high-column regions. - Stellar coalescence: Bonnell et al. (1998) made the radical suggestion that the effects of radiation pressure could be overcome by building up massive stars through the direct collision of low-mass stars (see below). 28.2. High-Mass Star Formation as Scaled-Up Low-Mass Star Formation The conventional view is that high-mass star formation is a scaled up version of low-mass star formation: The accretion rate is ṁ∗ ' c3 /G, where the effective sound speed c includes the effects of thermal gas pressure, magnetic pressure, and turbulence (Stahler et al. 1980, although they did not address the issue of high-mass star formation). Wolfire & Cassinelli (1987) found that accretion rates of order 10−3 M¯ yr−1 are needed to overcome the effects of radiation pressure for the highest stellar masses, and attributed this to the high values of c in high-mass star forming regions. By modeling the spectral energy distributions (SEDs) of high-mass protostars, Osorio et al. (1999) inferred that high-mass stars form in somewhat less than 10 5 yr. Behrend & Maeder (2001) assumed that the accretion rates are proportional to the observed protostellar outflows, and inferred a rapidly accelerating accretion (i.e., m∗ reaches infinity in a finite time in the absence of other effects) that produced a massive star in ∼ 3 × 105 yr; this phenomenological model has an accretion rate that can exceed the value ṁ∗ ∝ c3 that is expected on dynamical grounds, however. The most detailed model of high-mass star formation is the turbulent core model (McKee & Tan 2002; 2003; hereafter MT03). They assumed that high-mass stars form in turbulent, gravitationally bound cores (virial parameter αvir ∼ 1). They assumed that the turbulence is self-similar on all scales above the Bonnor-Ebert scale, where thermal pressure dominates. They also assumed that the star-forming clump and the protostellar cores within it are centrally concentrated. The pressure and density therefore have a power-law dependence on radius, p ∝ r −kρ , ρ ∝ r−kρ . It follows that the cores are polytropes with ¶ µ 1 γp . (9) p = Kρ , γp = 2 1 − kρ Since the Bonnor-Ebert scale is small, the core is approximately singular. –4– The regions of high-mass star formation studied by Plume et al. (1997) have surface densities Σcl ∼ 1 g cm−3 , corresponding to visual extinctions AV ∼ 200 mag. By contrast, regions of low-mass star formation have Σ ∼ 0.03 g cm−2 , corresponding to AV ∼ 7 mag (Onishi et al. 1996). MT03 showed that observed star clusters in the Galaxy have surface densities comparable to those of high-mass star forming regions, with values ranging from about 0.2 g cm −3 in the Orion Nebula Cluster to about 4 g cm−2 in the Arches Cluster. The typical Galactic globular cluster has Σ ' 0.8 g cm−2 . We can express the properties of the star-forming clump and core in terms of the surface density of the clump by recalling that in a self-gravitating gas p ∼ GΣ2 . Since the pressure in the core is comparable to that in the clump in which it is embedded, it follows that Σcore ' Σcl . Under the assumption that half the mass of the core is incorporated in the final star of mass m ∗f , MT03 found that the radius of the core is µ ¶ m∗f 1/2 1/2 Rcore = 0.06 Σcl pc, (10) 30M¯ where Σcl is the surface density (in g cm−2 ) of the several thousand M¯ clump in which the star is forming. As discussed in Lecture 24, such a core accretes onto the central protostar at a rate ṁ∗ = φ∗ m∗ , tff (11) where tff is the initial free-fall time of the gas that is just accreting onto the star and φ∗ ' 1.62 − 0.48kρ = O(1). (12) MT03 adopted kρ = 23 , comparable to the values inferred for the star-forming clumps studied by Plume et al. Recall from Lecture 24 that a freely falling core with this density gradient has φ∗ = 2[3(3 − kρ )]1/2 /kρ = 2.8, so the turbulent pressure slows the collapse by a factor of about 3. Larson-Penston accretion can have much larger values of φ∗ , but this is unlikely to occur in a turbulent medium since such a flow is highly organized on the scale of the entire core. Numerically, the typical accretion rate and the corresponding time to form a star of mass m ∗f is µ ¶ ¶ m∗f 3/4 3/4 m∗ 0.5 M¯ Σcl ṁ∗ ' 0.5 × 10 , 30M¯ m∗f yr ¶1/4 µ m∗f −3/4 5 Σcl yr. t∗f ' 1.3 × 10 30 M¯ −3 µ For typical values of Σcl ∼ 1 g cm−2 , the star formation time is of order 105 yr and the accretion rate is of order 10−3 M¯ yr−1 . This accretion rate is large enough to overcome the effects of radiation pressure at the dust destruction front, thereby addressing one of the key theoretical difficulties for models of high-mass star formation. –5– There are several theoretical difficulties with this model. First, it treats the turbulence in the core and clump as microturbulence, which acts as a local pressure. As we discussed in connection with the virial theorem, this is approximately valid when averaging over an ensemble, but it can be in significant error when looking at a particular core at a particular time. Numerical simulations will test the validity of this approximation. Second, why doesn’t the core simply fragment into smaller stars? Note that such fragmentation is allowed by the observations only if the mass distribution of cores does not determine the IMF, contrary to current observations. Dobbs et al. (2005) simulated the collapse of a high-mass core similar to that considered by MT03. In the isothermal case, they found that the core fragmented into many pieces, which is inconsistent with the formation of a massive star. With a more realistic equation of state, however, only a few fragments formed. Krumholz (2006) has shown that heating due to the central protostar almost completely suppresses fragmentation and concludes that a high-mass core will form one or two high-mass stars. Finally, the model neglects feedback due to the ionizing radiation of the star itself and due to other stars in the cluster. Nonetheless, the turbulent core model is consistent with several set of observations: (1) Existing observations are consistent with having the IMF determined by the core mass distribution, as assumed in the model. (2) The model is scale free for masses above the Bonnor-Ebert mass and distances greater than stellar radii, so there is no difficulty with having star formation occur in the crowded environment observed in regions of high-mass star formation. (3) Since it is an extrapolation of theories of low-mass star formation, it includes accretion disks and outflows. As discussed in the PPV review by Cesaroni et al. (2006), accretion disks are observed around protostars with m∗ ' 10 − 20 M¯ (note that Jim Moran does not believe that these observations are conclusive, however); to date such disks have not been seen around O4-O8 protostars. Outflows from massive protostars are common, although they are less collimated for L & 10 5 L¯ (Arce et al. 2006; Beuther et al. 2006). 28.3. High-Mass Star Formation as Different from Low-Mass Star Formation Several models of high-mass star formation have been proposed that are qualitatively different from the canonical inside-out collapse theory for low-mass star formation. Keto (2002, 2003) modeled the growth of massive stars as being due to Bondi accretion, so that the accretion rate ṁ∗ ∝ m2∗ under his assumption that the ambient medium has a constant density and temperature. As with the Behrend-Maeder model, the fact that the accretion rate grows faster than linearly implies that the stellar mass would reach an infinite value in a finite time in the absence of other effects. As Keto points out, the Bondi accretion model assumes that the self-gravity of the gas is negligible. The condition that the mass within the Bondi radius Gm∗ /c2 be much less than the stellar mass can be shown to be equivalent to requiring ṁ∗ . c3 /G; for the value of c ' 0.5 km –6– s−1 considered by Keto, this restricts the accretion rate to ṁ∗ . 3 × 10−5 M¯ yr−1 , smaller than the values he considers. (One can show that when one generalizes the Bondi accretion model to approximately include the self gravity of the gas, the accretion rate is indeed about c 3 /G.) Bonnell and collaborators have proposed two alternative models. In the competitive accretion model (Lecture 24), small stars (m∗ ∼ 0.1M¯ ) form via gravitational collapse, but then grow by gravitational accretion of gas that was initially unbound to the star–i.e., by Bondi-Hoyle accretion. This model naturally results in segregating high-mass stars toward the center of the cluster, as observed. Furthermore, it gives a two-power law IMF that is qualitatively consistent with observation (Bonnell et al. 2001). Simulations by Bonnell et al. (2004) are consistent with this model. However, there are two significant difficulties: First, radiation pressure disrupts Bondi-Hoyle accretion once the stellar mass exceeds ∼ 10M¯ (Edgar & Clarke 2004), so it is unlikely that competitive accretion can operate at masses above this. There is no evidence for a change in the IMF in this mass range, however, which suggests that competitive accretion does not determine the IMF at lower masses either. Second, competitive accretion is effective only if the virial parameter is much less than observed, as discussed in Lecture 24. The most radical and imaginative model for the formation of high-mass stars is that they form via stellar collisions during a brief epoch in which the stellar density reaches ∼ 10 8 stars pc−3 (Bonnell et al. 1998; Bonnell & Bate 2002), far greater than observed in any Galactic star cluster. This model also results in an IMF that is in qualitative agreement with observation, although it must be borne in mind that the simulations to date have not included feedback. In their PPIV review, Stahler et al. (2000) supported the merger model, emphasizing that gas associated with protostars could increase the effective collision cross section and permit merging to occur at lower stellar densities. More recently, Bonnell & Bate (2005) have suggested that binaries in clusters will evolve to smaller separations due to accretion, resulting in mergers. However, a key assumption in this model is that there is no net angular momentum in the accreted gas, which makes sense in the competitive accretion model but not the gravitational collapse model. Stellar dynamical calculations by Portegies Zwart et al. (2004), which did not include any gas, show that at densities & 108 stars pc−3 it is possible to have runaway stellar mergers at the center of a star cluster, which they suggest results in the formation of an intermediate mass black hole (m BH À 102 M¯ ). It should be noted that they inferred that this could have occurred based on the currently observed properties of the star cluster (although with the assumption that the tidal radius is greater than 100 times the core radius), not on a hypothetical ultra-dense state of the cluster. Bally & Zinnecker (2005) discuss observational approaches to testing the merger scenario, and suggest that the wideangle outflow from OMC-1 in the Orion molecular cloud could be due to a protostellar merger that released 1048 − 1049 erg. While it is quite possible that some stellar mergers occur near the centers of some star clusters, the hypothesis that stellar mergers are responsible for a significant fraction of high-mass stars faces several major hurdles: (1) the hypothesized ultra-dense state would be quite luminous, yet has never been observed; (2) the mass loss hypothesized to be responsible for reducing the cluster density from ∼ 108 stars pc−3 to observed values must be finely tuned in order –7– to decrease the magnitude of the binding energy by a large factor; and (3) it is difficult to see how this model could account for the observations of disks and collimated outflows. 28.4. High-Mass Star Formation in Clusters High-mass stars form in clusters. Observations of other galaxies by Kennicutt, Edgar & Hodge (1989) show that typically dNa ∝ L−1 , (13) d ln L where Na is the number of OB associations and L is the Hα luminosity. From an analysis of observations of large H II regions in the Galaxy, McKee & Williams (1997) showed that dN a /d ln S ∝ S −1 , where S is the ionizing photon luminosity, consistent with the extragalactic observations of Kennicutt et al. This is valid only at large values of S, where the number of high-mass stars, N∗h (defined as stars more massive than 8M¯ ) is large enough to fairly sample the IMF. MW97 −1 assumed that, since S ∝ N∗h at large S, the relation dNa /d ln N∗h ∝ N∗h would extrapolate down to low values of the ionizing luminosity. Furthermore, since OB associations are observed via their ionizing luminosity, they are counted only when they are young; the count is thus proportional to their birthrate. They found that in the Galaxy, dṄa (N∗h ) 2400 = d ln N∗h N∗h Myr−1 . (14) Extending this down to N∗h ' 1 and assuming that each association lives for about 20 Myr (corresponding to several generations of active star formation), this birthrate is equivalent to the total star formation rate in the Galaxy. In other words, it is consistent to assume that most stars form in associations that have at least one high-mass star. 28.4.1. Mass Segregation High-mass stars are observed to be concentrated in the centers of clusters. In the Orion Nebula Cluster, Hillenbrand & Hartmann (1998) find that, for a cluster radius of 2 pc, half the 0.3 − 1 M¯ stars lie within a projected radius of about 0.8 pc of the center. On the other hand, half the stars more massive than 5 M¯ lie within 0.25 pc. A more dramatic example of mass segregation occurs in NGC 3603: Sung & Bessell (2004) find that the slope of the high-mass IMF, Γ ≡ −dN∗ /d ln m∗ , ranges from 0.5 inside 6 00 to 1.2 ± 0.2 beyond 12 00 . Stolte et al. (2006) find that the ratio N∗ (m∗ > 4 M¯ )/N∗ (< 4 M¯ ) drops by almost a factor 5 between 700 − 2000 and 2700 − 3300 . The issue raised by these observations is whether this mass segregation primordial or is it due to dynamical effects? A star of mass ms moving through a cluster containing stars of mass hm∗ i ¿ ms experiences –8– a drag force due to dynamical friction F = −0.43 ln Λ µ Gm2s r2 ¶ , (15) where r is the distance to the center of the cluster and where the numerical coefficient has been evaluated for a singular isothermal sphere (Binney & Tremaine 1987, p. 427). Here ln Λ is the Coulomb logarithm and Rvc2 M bmax v 2 ' = , (16) Λ= G(ms + hm∗ i) Gms ms where bmax ' R is the maximum impact parameter, M is the mass of the cluster, and where we have assumed that the velocity of the star v is about equal to the circular velocity given by vc2 = GM (r)/r. Note that the drag force depends on the mass of the cluster only through the Coulomb logarithm. Following Binney & Tremaine, we estimate the time for the star to sink to the center by assuming that it spirals in with a velocity ' vc , which is constant in an SIS. The specific angular momentum is L = rvc , and the rate of change of the specific angular momentum is dL dr Fr = vc = dt dt ms ⇒ r dr Gms = −0.43 ln Λ. dt vc (17) Integrating this, we find that the time for the star to sink to the center of the cluster is tsink = 1.17 2 v rpc r 2 vc c5 = 2.68 × 108 ln Λ Gms (ms /M¯ ) ln Λ yr → 7 Myr, (18) where the final numerical evaluation is for a 30 M¯ star at an initial radius of 1 pc in the ONC, which has a mass of 1800 M¯ and a 1D velocity dispersion of 2.24 km s−1 for 1 − 3 M¯ stars √ (Hillenbrand & Hartmann 1998; note that vc = 2σ). This is significantly greater than the age of the ONC, which is . 3 Myr, suggesting that the mass segregation is primordial. Bonnell & Davies (1998) carried out a number of numerical simulations of star cluster evolution and concluded that it was not possible for stellar-dynamical processes to produce the observed degree of mass segregation. The effects of gas on the dynamical friction were not included, however, and this can increase the rate of dynamical friction somewhat (Ostriker 1999).