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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.
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