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PASJ: Publ. Astron. Soc. Japan 53, 85–92, 2001 February 25 c 2001. Astronomical Society of Japan. The Most Luminous Protostars in Molecular Clouds: A Hint to Understand the Stellar Initial Mass Function Kazuhito D OBASHI1 , Yoshinori YONEKURA2 , Tomoaki M ATSUMOTO3 , Munetake M OMOSE4 , Fumio S ATO1,5 , Jean-Philippe B ERNARD5 , Hideo O GAWA2 1 Department of Astronomy and Earth Sciences, Tokyo Gakugei University, Koganei, Tokyo 184-8501; [email protected] 2 Earth and Life Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531 3 Department of Humanity and Environment, Hosei University, Fujimi, Chiyoda-ku, Tokyo 102-8160 4 Institute of Astrophysics and Planetary Sciences, Ibaraki University, Mito 310-8512 5 IAS, bat 121, Campus d’Orsay, 91405 Orsay Cedex, France (Received 2000 March 15; accepted 2000 October 11) Abstract The maximum luminosity of protostars forming in molecular clouds has been investigated as a function of the parent cloud mass on the basis of a rich cloud sample searched for in the literature. In total, we gathered 499 molecular clouds among the published data, out of which 243 clouds were found to be associated with protostellar candidates selected from the IRAS point-source catalog. A diagram of the maximum stellar luminosity in each cloud and the parent cloud mass shows that the protostars in the clouds associated with H II regions are apparently more luminous than those in clouds away from H II regions over the entire cloud mass range investigated (1 < MCL /M < 106 ). In addition, we found that there are well-defined upper and lower limits in the maximum 1.5 stellar luminosity distribution with the lower limit having a steeper dependence on the cloud mass (LMAX ∝ MCL ) 0.8 than the upper one (LMAX ∝ MCL ). All of these features can be naturally accounted for if we assume that the luminosity function of protostars is controlled by the cloud mass and the external pressure imposed on the cloud surface. We introduce a simple model for the stellar luminosity function as a function of these quantities, which should be closely related to the stellar initial mass function. Key words: infrared: stars — ISM: clouds — ISM: molecules — stars: formation — stars: luminosity function, mass function 1. Introduction An interesting question has been how star-formation activity is related to the global physical properties of the parent molecular clouds. The cloud mass is apparently one of the most important parameters to characterize star formation, because massive stars are produced in giant molecular clouds (GMCs), while only low-mass stars are born in small dark clouds. It was Larson (1982) who first investigated the relation between the maximum stellar mass MSTAR-MAX and the parent cloud mass MCL . Among the literature, he collected data of cloud masses measured in 13 CO as well as the spectral type of the associated young stars, and found that the relation between the two quantities can be well fitted by a power-law, (MSTAR-MAX /M ) = 0.33(MCL /M )0.43 . Using an analogue of Larson’s method, we recently investigated the correlation between the luminosity of the most luminous protostar forming in a molecular cloud (LMAX ) and the parent cloud mass on the basis of the ∼ 120 star-forming clouds in the Cygnus, Cepheus, and Cassiopeia regions (Dobashi et al. 1994, 1996; Yonekura et al. 1997). Our studies were based on data obtained through a large scale 13 CO survey for molecular clouds carried out using the Nagoya 4 m telescopes, as well as a search for protostellar candidates based on the IRAS point-source catalog (1988). In these studies, we found that the LMAX –MCL relation can be 1.5 for the luminosity well fitted by the power-law LMAX ∝ MCL range 10 < LMAX /L < 105 . The luminosities of relatively massive protostars in this range may remain rather constant until they evolve into main-sequence stars, while those of lowmass protostars should largely change according to their evolutionary stages (e.g., Schaller et al. 1992). We have there3.8 , for fore adopted a stellar mass–luminosity relation, L ∝ MSTAR the main-sequence stars that we derived by fitting data summarized by Lang (1992) for the range 2 < MSTAR /M < 20. The LMAX –MCL relation which we found can then be converted into a relation between MSTAR-MAX and MCL , which is quite similar to Larson’s law, indicating that the two relations essentially trace the same general rule in star formation. In our previous studies, we also investigated the luminosity distribution of protostars (i.e., IRAS point sources with cold color) as a function of the parent cloud mass, which lead to an empirical luminosity function (LF) approximately expressed by the following formula for the ranges 10 < L/L < 104 and 102 < MCL /M < 105 : L −r MCL q dN = C0 , (1) dL L M where dN is the number of protostars found in the luminosity range between L and L + dL. The constant C0 and indices r and q were measured to be 5.8 × 10−3 , 1.6, and 0.9 for clouds apart from H II regions in the Cygnus region (Dobashi et al. 1996; also see Yonekura et al. 1997 for correction in 86 K. Dobashi et al. [Vol. 53, Table 1. Sample of molecular clouds. Sample number 1 2 3 4 5 1 2 3 4 5 Cloud sample Cloud mass range Nearby clouds Outer arm LMC (SEST) LMC (Columbia) Bright-rimmed clouds 1.2 < log(MCL /M ) < 5.1 4.1 < log(MCL /M ) < 5.4 4.6 < log(MCL /M ) < 5.9 4.6 < log(MCL /M ) < 6.1 −0.3 < log(MCL /M ) < 3.8 Molecular line 13 CO (J = 1–0) CO (J = 1–0) 12 CO (J = 1–0) 12 CO (J = 1–0) Identified by eyes 12 Telescope Nagoya 4m NRAO 11m SEST 15m Columbia 1.2m POSS / ESO-R Total number of clouds 345 27 21 30 76 Number of clouds with protostars 118 17 12 20 76 Nearby molecular clouds collected from Dobashi et al. (1994; 1996) and Yonekura et al. (1997). The clouds are located in the Cygnus, Cepheus, and Cassiopeia regions whose distances are mostly ≤ 1 kpc from the sun. Two clouds in Cygnus whose association with IRAS point sources remain uncertain are excluded from the sample (Dobashi et al. 1995). Molecular clouds in the Outer Arm cataloged by Kutner, Mead (1981), Mead (1988), and Mead, Kutner (1988). Molecular clouds in the Large Magellanic Cloud observed with the SEST 15 m telescope. Data are taken from Kutner et al. (1997) and Caldwell, Kutner (1996). These clouds are located in the 30 Doradus complex and in the H II region N 11. Molecular clouds in the Large Magellanic Cloud mapped by the Columbia 1.2 m telescope (Cohen et al. 1988). Unresolved clouds in the 30 Doradus complex and a cloud toward N 11 are excluded. Cloud masses are rescaled using the conversion factor N(H2 )/WCO = 2.3 × 1020 cm−2 K−1 km−1 s (Johansson et al. 1998). Bright-rimmed clouds cataloged by Sugitani et al. (1991) and Sugitani, Ogura (1994) are collected as cloud sample showing definite association with H II regions. The clouds are originally identified by eyes on the Palomar Observatory Sky Survey (POSS) prints and the European Southern Observatory (ESO-R) atlas. Among 89 clouds listed in their original catalog, we excluded 13 clouds from our sample whose cloud mass estimate may suffer a significant error (see text). The excluded clouds are Nos. 1, 26, 27, 28, 58, 59, 60, 64, 75, 76, 77, 78, and 85 in the original catalog. the calibration), which may vary slightly from region to region (Yonekura et al. 1997). The relation described in equation (1) is quite important for research on star formation, because it should be closely related to the stellar initial-mass function (IMF). Interestingly, if we integrate equation (1) over L from L = LMAX to infinity with N = 1, +∞ dN dL = 1, (2) LMAX dL we obtain an LMAX –MCL relation with the same cloud mass de1.5 pendence as that we measured previously (LMAX ∝ MCL ). This suggests that the formation of the most luminous and massive stars might be a matter of probability controlled by the LF in equation (1). Besides the above, we found that compact clouds associated with H II regions have a tendency to produce more luminous protostars than those isolated from H II regions. This indicates that star formation in low-mass clouds may be strongly influenced by the high pressure in the H II regions (Dobashi et al. 1996), which was first pointed out by Sugitani et al. (1989) and was more recently confirmed by Yamaguchi et al. (1999). However, the cloud mass range for which we and other authors determined the LMAX –MCL relation was limited to 10 < MCL /M < 104 . It has been our particular interest to expand this study over a much wider cloud mass range and to probe into the general rules of star formation at the molecularcloud scale. The purpose of this paper is to present new results regarding the LMAX –MCL relation. We have collected a sample of about 500 clouds with various masses from the literature, which now allow us to study this relation over a much wider cloud mass range of 1 < MCL /M < 106 . We summarize our literaturesurvey for molecular clouds and a search for the associated protostellar candidates among the IRAS point sources in section 2. On the basis of the collected sample, we found that LMAX increases as a function of MCL , and also confirmed that clouds associated with H II regions are accompanied by more luminous protostars than clouds away from H II regions. These features identified in the newly obtained LMAX –MCL diagram are most likely to be valid over the entire cloud mass range investigated (1 < MCL /M < 106 ). The new findings in the above as well as an estimate for possible errors in determining the LMAX values due to the poor angular resolution of the IRAS observations are summarized in section 3. In section 4, we introduce a simple model for the luminosity function of protostars that can naturally account for the global features seen in the LMAX –MCL relation. Our conclusions are summarized in section 5. 2. Sample of Molecular Clouds In addition to the 345 nearby clouds evidenced in our earlier CO survey, which are mostly located within 1 kpc (Dobashi et al. 1994, 1995, 1996; Yonekura et al. 1997), we consider here 27 GMCs detected in the CO survey in the outer arm (Kutner, Mead 1981; Mead 1988; Mead, Kutner 1988). Moreover, 30 GMCs in the Large Magellanic Cloud (LMC) mapped in CO using the Columbia 1.2 m telescope were also collected by referring to the catalog summarized by Cohen et al. (1988) after excluding unresolved clouds located in the 30 Doradus complex as well as in the N 11 region. For these 2 complex regions in the LMC, we adopted CO data with a higher angular resolution taken using the SEST 15 m telescope to sample 21 GMCs (Caldwell, Kutner 1996; Kutner et al. 1997). The masses of all the GMCs collected here range from 104 M to 106 M . In order to evaluate the influence of H II regions on star formation, we further collected a sample of 76 clouds with relatively small masses ranging from 1 to 104 M from the catalogs of “bright-rimmed” clouds showing definite association with H II regions (Sugitani et al. 1991; Sugitani, Ogura 1994). In total, our sample includes 499 clouds. We divide the cloud sample into 5 groups according to the references, and summarize them in table 1. As we did in our previous studies, we searched for the most luminous protostars in each cloud using the IRAS point-source catalog (1988). The IRAS point sources regarded as protostars 13 No. 01] The Most Luminous Protostars in Molecular Clouds in this work are required to be located within the cloud extent given in the reference, and should be detected at least at 25 µm and 60 µm with a flux density at 60 µm greater than at 25 µm. These are the same selection criteria as used in Dobashi et al. (1996). For the IRAS sources in the LMC, we adopted the catalog produced by Schwering (1989) to avoid any confusion in the original IRAS point-source catalog. As a result, we found one or more protostars in 243 clouds among the 499 clouds of our sample. We calculated the luminosity of the IRAS sources following the method proposed by Myers et al. (1987), and then identified the most luminous source in each cloud. We basically adopted the cloud masses given in the references. The masses of the clouds taken from our 13 CO surveys were estimated assuming the local thermodynamic equilibrium (LTE) and using the 13 CO abundance ratio measured by Dickman (1978). The virial masses were adopted for GMCs in the outer arm as well as in the LMC mapped using the SEST 15 m telescope. In the case of the other GMCs in the LMC observed using the Columbia 1.2 m telescope, LTE masses listed by Cohen et al. (1988) were adopted after rescaling using more recent measurements of the N (H2 )/WCO ratio 2.3 × 1020 cm−2 K−1 km−1 s (e.g., Chin et al. 1997; Johansson et al. 1998). The masses of the bright-rimmed clouds were estimated from their apparent sizes on the optical images and from a typical density for the clouds (n[H2 ] = 3 × 104 cm−3 ), as suggested by the authors (Sugitani et al. 1991). In one of our earlier studies toward the Cygnus region (Dobashi et al. 1996), we found that the virial masses are generally in good agreement with the LTE masses for GMCs, especially in the case of star-forming clouds (see figure 4 in Dobashi et al. 1996), which was recently confirmed by Kawamura et al. (1998) for clouds in the Gemini and Auriga regions. Actually, the two methods (LTE and virial) yield consistent values mostly within a factor of < 2 for GMCs in the outer arm (Mead, Kutner 1988). We therefore conclude that the cloud masses which we gathered from the literature should be consistent within a small factor. However, errors in the cloud mass estimate could be larger for the bright-rimmed clouds. The authors of the original catalog measured the sizes of bright rims associated with the clouds through eye inspection and used them to estimate the cloud masses (Sugitani et al. 1991; Sugitani, Ogura 1994). Some of the bright rims in the catalog, however, are apparently tracing only a small fraction of more extended clouds, which may result in a significant error in determining the total cloud masses. Among 89 brightrimmed clouds contained in the original catalog, we identified 13 clouds suffering this problem. The masses of these clouds were calculated to be in the range 0.05–2500 M based on the sizes given in the catalog, but their true total masses may be much higher. Therefore, we excluded these clouds from our sample (see footnote in table 1). For some of the remaining bright-rimmed clouds, the authors of the original catalog made a comparison of their estimate based on the bright-rim sizes with the cloud masses already measured in 13 CO (Sugitani et al. 1989; 1991), and concluded that the cloud masses might be uncertain by a small factor. This is a major uncertainty concerning the cloud mass within our sample. 87 Fig. 1. Maximum stellar luminosity of protostars vs. parent cloud mass. Nearby molecular clouds, clouds in the outer arm, and bright-rimmed clouds are indicated by squares, diamonds, and circles, respectively. Upward and downward triangles denote clouds in the LMC, mapped using the Columbia 1.2 m telescope and the SEST 15 m telescope, respectively. Clouds associated with H II regions are denoted by open symbols, and the others are indicated by filled ones. Stellar luminosities corresponding to the stellar mass MSTAR = 6, 10, 20, and 120 M are indicated by the broken lines (Lang 1992). The upper and lower envelopes of the data distribution are denoted by the solid lines A and B, whose LMAX –MCL relations can be expressed by the power laws 0.8 1.5 LMAX ∝ MCL and LMAX ∝ MCL , respectively. 3. The LMAX –MCL Relation 3.1. Newly Obtained LMAX –MCL Diagram Figure 1 shows the LMAX –MCL diagram obtained using the newly collected 243 star-forming clouds. As can be seen in the figure, the LMAX –MCL relations obtained from all of the literature references surveyed are smoothly connected to each other over a large cloud mass range of 1 < MCL /M < 106 , suggesting that a general rule may control the population of the most luminous protostars regardless of the region sampled. It is noteworthy that the clouds associated with H II regions are systematically accompanied by more luminous IRAS sources than clouds away from H II regions over the entire cloud mass range investigated. In addition, there are well-defined upper and lower limits in the luminosity distribution (i.e., the lines A and B in figure 1) with the lower limit having a steeper slope 1.5 0.8 (LMAX ∝ MCL ) than the upper one (LMAX ∝ MCL ), and the vast majority of the data points lying between the two limits. Although the lower limit could be partially due to the detection limit of the IRAS survey, it should delineate a true limit, since the IRAS detection limit is lower than line B for most clouds. It is also interesting to note that, at least within the sample we collected here, there are only a few sources in figure 1 with luminosity exceeding 106 L corresponding to a stellar mass of ∼ 102 M (e.g., Lang 1992), which may indicate a very poor population of such very massive stars, or might represent the possible upper mass limit for a stable star (e.g., Maeder, Conti 1994). The LMAX –MCL relation shown in figure 1 implies that the parent cloud may need to have a huge mass of ∼ 106 M 88 K. Dobashi et al. in order to produce such a very massive star. In figure 1, the LMAX values apparently increase along with MCL , indicating that the parent cloud mass is a decisive parameter influencing the population of massive stars. Besides, the LMAX values in H II regions are generally higher than the others, strongly indicating that brighter and more massive stars can form in clouds surrounded by the high external pressure imposed by closeby H II regions. The external pressure may also play an important role in star formation by compressing molecular clouds, leading to more active star formation. The effect of high pressure from H II regions was already pointed out by Sugitani et al. (1989) for small bright-rimmed clouds. In this study, we confirmed its ubiquitousness over a large cloud mass range. It is also noteworthy that the dispersion of the LMAX values between clouds associated with and isolated from H II regions becomes smaller toward higher cloud masses, indicating that the external pressure is less crucial for star formation in more massive clouds. These trends indicate that star formation in molecular clouds may be controlled by the internal state of the cloud such as the density or the internal pressure, because these parameters should be determined mostly by the cloud mass, and are less affected by the external pressure in the case of massive clouds in virial equilibrium. In section 4, we introduce a model for the LF of protostars taking these two parameters (i.e., the cloud mass and the external pressure) into account. The upper and lower envelopes of the luminosity distribution (i.e., lines A and B in figure 1) indicate that there may be a physical mechanism limiting the LMAX values. As discussed in section 4, the upper envelope might be limited by the star-formation efficiency (SFE), i.e., the ratio of the total stellar mass to the mass of the entire cloud system including stars. If we assume that the LF given in equation (1) is scaled up by the external pressure, a very high SFE of 30–90% is expected for clouds located around the upper envelope in figure 1 where a large fraction of the cloud material should already be undergoing conversion into stars, and thus additional massive stars could not form. 3.2. Error in Determining the LMAX Values Because of the rather poor angular resolution of IRAS (i.e., a few arcminutes), some of the distant point sources selected here might not correspond to a single protostar, but include an unresolved star cluster. In such cases, some of the LMAX values plotted in figure 1 may represent the total luminosity of clusters or, in extreme cases, the total luminosity of protostars forming in an entire cloud, rather than the true maximum luminosities of single stars. Unlike the case of our earlier studies toward nearby molecular clouds, possible contamination due to the IRAS resolution can be considerable for much more distant clouds in the LMC and the outer arm. It is therefore important to estimate how much error can arise from this effect in order to interpret the global trend seen in figure 1. We consider here a “point source” which actually consists of a cluster of protostars, and express the total luminosity of the cluster and that of the truly most luminous source in it as LTOT and LMAX , respectively. If we assume a powerlaw stellar luminosity function with an index value of r for the cluster (i.e., dN/dL ∝ L−r ), the ratio of the maximum [Vol. 53, luminosity to the total luminosity, LMAX /LTOT , is then given by LMAX /LTOT = 2 − r for an index value of 1 < r < 2, which is a good approximation if we disregard the possible existence of the upper luminosity limit for stars (see appendices 1 and 2). In our previous studies, we found that the luminosity index r mostly ranges from 1.4 to 1.7 in nearby star-forming clouds (Dobashi et al. 1996; Yonekura et al. 1997). Using this range of r, roughly 30–60% of the total luminosity of a cluster is produced by the most luminous source. In other words, even if the “point sources” consist of unresolved clusters, their LMAX values plotted in figure 1 may need to be lowered only by 0.2–0.5 in logarithmic scale, which is unlikely to significantly change the global features seen in the LMAX –MCL diagram. The same argument holds for extremely distant cases where LMAX may actually represent the total luminosity of protostars forming in one entire cloud. To summarize this section, we found the following three features in the newly obtained LMAX –MCL diagram, which are most likely to be valid over the wide range of the parent cloud mass, 1 < MCL /M < 106 : First, the maximum luminosity of protostars LMAX increases as a function of the parent cloud mass MCL . Second, it is likely that there are upper and lower limits in the LMAX distribution. Third, LMAX values in clouds associated with H II regions are systematically higher than in clouds away from H II regions. Some IRAS point sources selected as candidates for protostars may in fact be unresolved star clusters. For such sources, we estimate that the LMAX values in figure 1 would need to be lowered by 0.2–0.5 in logarithmic scale. However, this should not significantly affect the major features mentioned above. 4. A Model of the Luminosity Function Here we propose a simple model for the LF of protostars which can account for the global features seen in figure 1. We assume the following three physical processes or conditions: (1) The luminosity function of protostars in a cloud, dN/dL, is a function of the stellar luminosity L, the parent cloud mass MCL , and the external pressure imposed on the cloud surface PEXT , and can be written in the form, dN = f (L, MCL , PEXT ) = φ(L)g(MCL , PEXT ), (3) dL where φ(L) is the luminosity function of a cloud with unit cloud mass located in vacuum, i.e., dN/dL = φ(L) for MCL = 1 M and PEXT = 0. (2) We assume that star-forming clouds are globally in, or close to, virial equilibrium. Although this need not be true on the small scale for the material actually involved in forming stars, this assumption is plausible on a large scale even for star-forming clouds, as shown by observations (e.g., Dobashi et al. 1996). Some sources keeping the star-forming clouds in the virial equilibrium are suggested by Nakano (1998). (3) We assume that the gaseous velocity dispersion ∆V in starforming clouds can be uniquely determined as a function of MCL and PEXT , and the factor g(MCL , PEXT ) in equation (3) to scale up φ(L) is related to ∆V as g ∝ ∆V 3 . This is because the number of protostars N observed at the present epoch is No. 01] The Most Luminous Protostars in Molecular Clouds 89 probably proportional to the total cloud mass and should be inversely proportional to the time scale to form stars. If we take the total cloud mass to be the Jeans mass MJ measured from ∆V (i.e., MJ ∝ ∆V 3 n−1/2 , where n is the gas density), and assume that the star-forming time scale is proportional to the free-fall time of the cloud τff (∝ n−1/2 ), N is then proportional to MJ /τff ∝ ∆V 3 . In the case of a uniform cloud with radius R, the virial theorem is written as, 4π R 3 PEXT = 2 3GMCL 3MCL kTEQ −a , µ 5R (4) where k, G, µ, and TEQ are the Boltzmann constant, the gravitational constant, the mean molecular weight, and the cloud temperature respectively (e.g., Spitzer 1978). The factor a is defined according to the cloud shape and should be close to unity for realistic clouds. We adopt a = 1 in the following. For TEQ , we adopt the Doppler temperature, which is related to the gaseous velocity dispersion in the cloud (∆V ) as µ∆V 2 , (5) TEQ = 8k ln 2 if we define ∆V at the FWHM. Using equations (4) and (5), assumptions 1–3 result in the following stellar luminosity function: dN q S ∝ φ(L)∆V 3 ∝ φ(L)MCL (C1 C2 PEXT MCL + 1)3/2 , (6) dL where C1 = 20π/(3G). The values of the indices q , S, and the constant C2 are related to the cloud shape factor ξ and the gas density n [i.e., (MCL /M ) = n(R/pc)ξ ] and are expressed as q = (3/2)(1 − 1/ξ ), S = −2 + 4/ξ , and C2 = n−4/ξ . The values of n and ξ are determined using the cloud sample in Cygnus (Dobashi et al. 1994) to be 107.3 and 2.4 respectively. A complete measurement or theoretical prediction of the LF per unit cloud mass φ(L) is not easy to establish at the moment. We therefore adopt an approximation by substituting MCL = 1 M in the empirical luminosity function obtained in our earlier studies [equation (1)], which results in φ(L) = 5.8 × 10−3 (L/L )−1.6 . Using the parameters mentioned above, equation (6) becomes dN L −1.6 MCL 0.87 −3 = 5.8 × 10 dL L M 1.5 PEXT /k MCL −0.33 × 3.94 +1 . (7) 104 K cm−3 M Note that the cloud mass dependence under PEXT = 0 (q = 0.87) agrees well with the empirical LF in equation (1) (i.e., q = 0.9). Integrating equation (7) over L, and replacing L with LMAX giving N = 1, we obtain the following model-based LMAX –MCL relation: LMAX MCL 1.5 = 4.4 × 10−4 L M 2.5 MCL −0.33 PEXT /k × 3.94 +1 . (8) 104 K cm−3 M Fig. 2. The LMAX –MCL relation compared with the model calculation. Solid lines denote the LMAX values calculated using equation (8) leaving the external pressure PEXT /k as a free parameter. The open circles denote clouds associated with H II regions, and the other clouds are indicated by plus signs. The LMAX values calculated from the above equation is compared with the data in figure 2, leaving the external pressure PEXT /k as a free parameter. As can be seen in the figure, the model predicts the actual LMAX –MCL relation well, tracing the lower and upper limits of the luminosity distribution at PEXT /k = 0 and ∼ 105.5 K cm−3 , respectively. Finally, we note that equation (7) can be converted into an IMF which is useful to estimate the SFE of molecular clouds, if a mass–luminosity relation for protostars is assumed. As mentioned in section 1, it may be plausible to assume that the 3.8 mass–luminosity relation of main-sequence stars, L ∝ MSTAR , obtained in the range 2 < MSTAR /M < 20 (Lang 1992) is also valid for protostars for the luminosity range of equation (1) (10 < L/L < 104 ). If we adopt this mass–luminosity relation, a power-law IMF with a stellar-mass dependence close to that measured among field stars (e.g., Scalo 1986; Rana 1987) can be derived from equation (7). Given a lower stellar mass cutoff in the resulting power-law IMF, one can estimate the SFE of molecular clouds. For a fixed value of the minimum stellar mass of 0.25 M (Scalo 1986), SFE of a few percent is predicted for clouds in vacuum (i.e., the line with PEXT /k = 0 in figure 2), while a SFE as high as 30–90% is predicted for clouds influenced by a high external pressure PEXT /k = 105.5 K cm−3 giving the line close to the upper limit of the LMAX –MCL distribution. Although the assumption of a constant minimum stellar mass for all clouds may not reflect the truth, the above estimate indicates that the upper envelope of the LMAX –MCL distribution may be determined by the full SFE; in short, cloud-scale star formation may follow the probability defined by equation (7) under the condition SFE < 100%. 90 5. K. Dobashi et al. Conclusions On the basis of a rich cloud sample searched for in the literature and protostellar candidates selected from the IRAS point source catalog, we have investigated the maximum luminosity of protostars as a function of the parent cloud mass over a large cloud mass range, 1 < MCL /M < 106 . The main findings of this work are summarized in the following points: (1) We found that the maximum luminosity increases along with the parent cloud mass in the cloud mass range 1 < MCL /M < 106 . (2) Protostars forming in clouds associated with H II regions are apparently more luminous than those in clouds away from H II regions over the entire cloud mass range investigated, indicating that the increased pressure of the H II regions influences star-formation in molecular clouds. (3) There are well-defined upper and lower limits in the maximum stellar luminosity distribution with the lower limit having 1.5 ) than a steeper dependence on the cloud mass (LMAX ∝ MCL 0.8 the upper one (LMAX ∝ MCL ). (4) We propose a model for the stellar luminosity function as a function of the parent cloud mass and the external pressure imposed on the cloud surface, which can naturally account for the global features identified in the LMAX –MCL diagram. K. D. is very grateful to T. Umemoto, N. Hirano, and N. Ohashi for fruitful discussions. Thanks are due to the anonymous referee for useful comments on errors affecting the LMAX values. This work was financially supported by the Grant-in-Aid for Scientific Research by the Japanese Ministry of Education, Science, Sports and Culture (Nos. 10147204, 10147208, 11134209, 12021203, and 12740123) as well as by the Japan Society for the Promotion of Science (Nos. 11740122 and 11740125). This work was also supported by Sasagawa Scientific Research Grant from the Japan Science Society (No. 10-082). Appendix 1. Estimate for the LMAX /LTOT Ratio In the following, we estimate the ratio of the total luminosity of a star cluster to the luminosity of the most luminous source in it. Here, we assume a cluster in which the stellar luminosities distribute following the power-law luminosity function, dN/dL = C0 L−r , where r and C0 are constant. The relation between the maximum luminosity LMAX and the luminosity function is given by Lup cut dN dL = 1, (A1) dL LMAX where Lup cut is the upper luminosity cutoff. The total luminosity of the cluster LTOT can be regarded as the sum of the luminosity from the most luminous source and that from all other members in the cluster; thus, LTOT can be expressed as LMAX dN L (A2) LTOT = dL + LMAX , dL Llow cut where Llow cut is the lower luminosity cutoff. If we set the two luminosity cutoffs as Llow cut → 0 and Lup cut → ∞, equations [Vol. 53, (A1) and (A2) with dN/dL = C0 L−r yield a LMAX /LTOT ratio of LMAX = 2−r (A3) LTOT for the luminosity index range 1 < r < 2. The above equation should give a good estimate for the LMAX /LTOT ratio when the LMAX value satisfies the condition Llow cut LMAX Lup cut . The true Llow cut value may come from a turn-over in IMF at ∼ 0.25 M (e.g., Scalo 1986). The corresponding luminosity cannot be determined uniquely, because the luminosities of low-mass young stars should vary greatly along with the stellar evolution, while massive stars with a luminosity > 10 L may remain rather constant (e.g., Schaller et al. 1992). However, we regard the condition Llow cut LMAX as realistic for most of our sample. If we select IRAS point sources in the nearby Taurus cloud complex using the same criteria as in this work, we find that the luminosity distribution of the IRAS sources follows a powerlaw luminosity function down to a few times 0.1 L . The Llow cut can therefore be assumed to be lower than this on average, while most of the sources used in this work are much more luminous. The upper luminosity cutoff, Lup cut , has not been definitely determined up to now. The most luminous stars in surveys toward massive star-forming regions including the galactic center (e.g., Nagata et al. 1993) are found to be as luminous as ∼ 106.5 L (e.g., Najarro et al. 1997; Massey, Hunter 1998). Even higher luminosity 106.6 –107.2 L is inferred in the case of the Pistol star (Figer et al. 1999a). Although the existence of a definite maximum stellar luminosity has not been evidenced, Lup cut should have a value around the highest luminosities reported to date (106.5 –107 L ). In fact, as shown in appendix 2, equation (A3) can be erroneous in the case of OB clusters with a very high LMAX value exceeding ∼ 106.5 L , while it mostly gives an accurate LMAX /LTOT ratio for cases with lower LMAX values. The luminosity of the most luminous IRAS sources in our sample is 106.4 L , and the vast majority of other sources are less luminous than 106 L , as can be seen in figure 1. Therefore, the condition LMAX Lup cut is valid for most of our samples, except for a few sources with the highest luminosities. Appendix 2. Measured LMAX /LTOT Ratio We now examine how well equation (A3) can reproduce the observed LMAX /LTOT ratio toward compact OB clusters unresolved by IRAS. Okumura et al. (2000) recently made near-infrared observations toward W 51 at 7 kpc, and identified more than 20 H II regions as well as the exciting OB stars (see table 7 in Okumura et al. 2000). Using their list for the spectral types of the exciting stars, we estimated the stellar luminosities following Lang (1992), and found that in most cases the most luminous star dominates the total luminosity from all of the exciting stars in each H II region. The minimum LMAX /LTOT ratio among them (∼ 36%) is obtained toward the H II region “d” in their list where LMAX and LTOT are estimated to be 9.8 × 105 L and 2.8 × 106 L , respectively. Although we cannot derive the luminosity function of the individual H II regions due to rather poor statistics, the high No. 01] The Most Luminous Protostars in Molecular Clouds 91 Table 2. LMAX /LTOT ratios. Object No. 1 2 3 4 5 6 7 Cluster or Cloud name OB Clusters in W 51 Trapezium Quintuplet R 136 Cyg OB 7 ρ Oph Taurus Distance (kpc) 7 0.45 8 50 0.8 0.16 0.14 Luminosity index r — 1.6 1.3 1.5 1.6 1.4 1.8 Measured LMAX /LTOT > 36% 39% 27 ± 12% 10% 44% 66% 16% Predicated LMAX /LTOT — 40% 70% 50% 40% 60% 20% Total luminosity log(LTOT /L ) 6.5 ∼5 7.4–7.6 7.6 3.9 2.4 1.6 Note: Measured and predicted values of LMAX /LTOT ratios are listed (see appendix 2). The measured values are derived based on the references below, and the predicted values are derived from equation (A3) using the luminosity index r in the list. Objects Nos. 1–4 are star clusters, and the others are nearby molecular clouds. References for each object. — (1) Okumura et al. 2000; (2) McCaughrean, Stauffer 1994; (3) Figer et al. 1999a, b; (4) Massey, Hunter 1998; (5) Dobashi et al. 1994; (6) Loren 1989; (7) Mizuno et al. 1995. LMAX /LTOT ratios estimated here imply that a large fraction of the total luminosity of the clusters may be attributed to the most luminous sources. Another example can be found in Orion. In the central 0.2 pc × 0.2 pc region of the Trapezium cluster, McCaughrean and Stauffer (1994) detected 123 stars through their near-infrared observations, and provided the K magnitude for individual stars whose bolometric luminosity may be around at ∼ 105 L in total (e.g., Figer et al. 1999b). Using the list given by McCaughrean and Stauffer (1994; see their table 1), one can find that the K band flux distribution of the detected sources follows well a power-law luminosity function with an index value of r = 1.6 in the range mK < 11.5 mag. The brightest source has a magnitude of mK = 4.41 mag, while the total flux summed over the 123 sources amounts to mK = 3.39 mag. This leads to a ratio of the maximum to total flux as high as 39%, which is consistent with what can be expected from equation (A3) with r = 1.6. However, in the case of the much brighter “Quintuplet” cluster located near the galactic center (e.g., Nagata et al. 1990; Okuda et al. 1990), a rather low LMAX /LTOT ratio is obtained in comparison with those in W 51 or Trapezium: Figer et al. (1999b) recently made near-infrared photometry for the massive members therein, and listed their luminosities (table 3 in Figer et al. 1999b). Based on their list, we derived an LMAX /LTOT ratio of ∼ 27 ± 12%. The large uncertainty (i.e., ±12%) is due to the ambiguity in the luminosity estimate of the most luminous member (i.e., the Pistol star) whose luminosity may be in the range 106.6 –107.2 L (Figer et al. 1999a). It is interesting to note that this ratio is much lower than what can be predicted by equation (A3) (roughly 70%) with the luminosity index of r ∼ 1.3 which one can derive in the stellar luminosity range > 105.5 L using the data from Figer et al. (1999b). This mismatch may be caused by the existence of an upper luminosity cutoff around 106.5 –107 L which should break the condition LMAX Lup cut assumed to derive equation (A3). A more apparent discrepancy can be found in the case of R 136, a very bright cluster exciting the 30 Doradus H II region in the LMC. Using the data of the cluster members identified by Massey and Hunter (1998; see their table 3), we derived a LMAX /LTOT ratio of ∼ 10% with a high LMAX value of ∼ 106.6 L , while the luminosity index r ∼ 1.5 can be derived in the range > 106.0 L . We further investigate the validity of equation (A3) in predicting the LMAX /LTOT ratio for distant clouds by summing up the luminosity contributions of all known sources in nearby clouds, as if the cloud was observed at a much larger distance with IRAS. This estimate is useful when determining the LMAX value for a very distant cloud where the maximum luminosity of the IRAS sources might represent the total luminosity of all associated protostars. In the Cyg OB 7 cloud complex (800 pc, cloud No. 71 in the list of Dobashi et al. 1994), there are 74 IRAS point sources satisfying our selection criteria for protostars whose luminosity distribution is well fitted by a power-law with an index value of r = 1.6. Their total luminosity is calculated to be 7870 L , while the luminosity of the brightest source is 3460 L (LMAX /LTOT = 44%). In the case of the ρ Oph cloud complex (160 pc), 16 IRAS point sources can be selected under the same selection criteria within the 13 CO emitting region evidenced by Loren (1989). The best-fitting power-law index value for the luminosity distribution of the IRAS sources is r = 1.4, and their total and maximum luminosities are 234 L and 155 L respectively, yielding a high LMAX /LTOT ratio of 66%. Although a large number of faint young stellar objects have been identified in the ρ Oph region through more recent and sensitive observations (e.g., Wilking et al. 1989; Greene, Young 1992), the fraction of these faint sources accounts for only a few percent of the total luminosity, LTOT , if the faint sources follow the same powerlaw luminosity function as the bright IRAS sources. A rather low LMAX /LTOT value is found in the Taurus cloud complex. Within the 13 CO emitting region in Taurus observed by Mizuno et al. (1995), we identified 49 IRAS point sources as candidates for protostars whose LTOT and LMAX values are 42.3 L and 6.9 L , respectively. Using the best-fitting index r = 1.8 for the 49 IRAS point sources, the resulting ratio LMAX /LTOT = 16% is consistent with the value expected from equation (A3). The above measurements of the LMAX /LTOT ratios are summarized in table 2. Equation (A3) gives a good approximation of the LMAX /LTOT ratios in most cases, except for the brightest clusters with a total luminosity > 107 L in which the assumed condition, LMAX Lup cut , to derive equation (A3) should not hold. 92 K. Dobashi et al. References Caldwell, D. A., & Kutner, M. L. 1996, ApJ, 472, 611 Chin, Y. N., Henkel, C., Whiteoak, J. B., Millar, T. J., Hunt, M. R., & Lemme, C. 1997, A&A, 317, 548 Cohen, R. S., Dame, T. M., Garay, G., Montani, J., Rubio, M., & Thaddeus, P. 1988, ApJ, 331, L95 Dickman, R. L. 1978, ApJS, 37, 407 Dobashi, K., Bernard, J.-P., & Fukui, Y. 1996, ApJ, 466, 282 Dobashi, K., Bernard, J.-P., Yonekura, Y., & Fukui, Y. 1994, ApJS, 95, 419 Dobashi, K., Bernard, J.-P., Yonekura, Y., Nozawa, S., Morimoto, S., Abraham, P., Kumai, Y., Hayashi, Y., & Fukui Y. 1995, PASJ, 47, 837 Figer, D. F., McLean, I. S., & Morris, M. 1999b, ApJ, 514, 202 Figer, D. F., Najarro, F., Morris, M., McLean, I. S., Geballe, T. R., Ghez, A. M., & Langer, N. 1999a, ApJ, 506, 384 Greene, T. P., & Young E. T. 1992, ApJ, 395, 516 IRAS Point Source Catalog 1988, Joint IRAS Science Working Group (Washington, DC: US GPO) Johansson, L. E. B., Greve, A., Booth, R. S., Boulanger, F., Garay, G., de Graauw, Th., Israel, F. P., Kutner, M. L., et al. 1998, A&A, 331, 857 Kawamura, A., Onishi, T., Yonekura, Y., Dobashi, K., Mizuno, A., Ogawa, H., & Fukui, Y. 1998, ApJS, 117, 387 Kutner, M. L., & Mead, K. N. 1981, ApJ, 249, L15 Kutner, M. L., Rubio, M., Booth, R. S., Boulanger, F., de Graauw, Th., Garay, G., Israel, F. P., Johansson, L. E. B., et al. 1997, A&AS, 122, 255 Lang, K. R. 1992, Astrophysical Data: Planets and Stars (New York: Springer Verlag) Larson, R. B. 1982, MNRAS, 200, 159 Loren, R. B. 1989, ApJ, 338, 902 Maeder, A., & Conti, P. S. 1994, ARA&A, 32, 227 Massey, P., & Hunter, D. A. 1998, ApJ, 493, 180 McCaughrean, M. J., & Stauffer, J. R. 1994, AJ, 108, 1382 Mead, K. N. 1988, ApJS, 67, 149 Mead, K. N., & Kutner, M. L. 1988, ApJ, 330, 399 Mizuno, A., Onishi, T., Yonekura, Y., Nagahama, T., Ogawa, H., & Fukui, Y. 1995, ApJ, 445, L161 Myers, P. C., Fuller, G. A., Mathieu, R. D., Beichman, C. A., Benson, P. J., Schild, R. E., & Emerson, J. P. 1987, ApJ, 319, 340 Nagata, T., Hyland, A. R., Straw, S. M., Sato, S., & Kawara, K. 1993, ApJ, 406, 501 Nagata, T., Woodward, C. E., Shure, M., Pipher, J. L., & Okuda, H. 1990, ApJ, 351, 83 Najarro, F., Krabbe, A., Genzel, R., Lutz, D., Kudritzki, R. P., & Hillier, D. J. 1997, A&A, 325, 700 Nakano, T. 1998, ApJ, 494, 587 Okuda, H., Shibai, H., Nakagawa, T., Matsuhara, H., Kobayashi, Y., Kaifu, N., Nagata, T., Gatley, I., & Geballe, T. R. 1990, ApJ, 351, 89 Okumura, S., Mori, A., Nishihara, E., Watanabe, E., & Yamashita, T. 2000, ApJ, 543, 799 Rana, N. C. 1987, A&A, 184, 104 Scalo, J. M. 1986, Fund. Cosmic Phys., 11, 1 Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, A&AS, 96, 269 Schwering, P. B. 1989, A&AS, 79, 105 Spitzer, L., Jr. 1978, Physical Processes in the Interstellar Medium (New York: Wiley) Sugitani, K., Fukui, Y., Mizuno, A., & Ohashi, N. 1989, ApJ, 342, L87 Sugitani, K., Fukui, Y., & Ogura, K. 1991, ApJS, 77, 59 Sugitani, K., & Ogura, K. 1994, ApJS, 92, 163 Wilking, B. A., Lada, C. J., & Young, E. T. 1989, ApJ, 340, 823 Yamaguchi, R., Saito, H., Mizuno, N., Mine, Y., Mizuno, A., Ogawa, H., & Fukui, Y. 1999, PASJ, 51, 791 Yonekura, Y., Dobashi, K., Mizuno, A., Ogawa, H., & Fukui, Y. 1997, ApJS, 110, 21