The RMS Survey: The Luminosity Functions and Timescales of Massive Young Stellar Objects and Compact H ii Regions

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The RMS Survey: The Luminosity Functions and Timescales of
   Massive Young Stellar Objects and Compact H ii Regions

      Joseph C. Mottram1,2 , Melvin G. Hoare1 , Stuart L. Lumsden1 , Rene D. Oudmaijer1 ,
            James S. Urquhart3 , Ben Davies1 , Toby J. T. Moore4 , Joseph J. Stead1

                                         joe@astro.ex.ac.uk

                                            ABSTRACT

           We present the first determination of the luminosity functions of massive
        young stellar objects (MYSOs) and compact H ii regions within the Milky Way
        galaxy. These are determined from the large, well-selected sample of these sources
        identified by the Red MSX Source (RMS) survey. The MYSO luminosity function
        shows that there is not a significant population of these objects with a luminosity
        above ∼ 7×104 L⊙ , whilst the luminosity function for CH ii regions extends to
        L ∼ 6×105 L⊙ as expected. The lifetimes of these phases are also calculated as a
        function of luminosity by comparison with the luminosity function for local main
        sequence OB stars. These indicate that the MYSO phase has a lifetime just over
        105 yrs, whilst the CH ii region phase lasts of order 3×105 yrs or about 5% of
        the exciting stars main sequence lifetime. The lack of MYSOs above 105 L⊙ is
        consistent with the scenario in which objects around 10M⊙ are swollen and cool
        due to high accretion rates, but at higher masses contract rapidly to the main
        sequence allowing them to ionize their surroundings.

        Subject headings: stars: formation — stars: luminosity function, mass function
        — stars: early-type — HII regions — Galaxy: stellar content

  1
      School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK
  2
      Department of Physics and Astronomy, University of Exeter, Exeter, Devon, EX4 4QL, UK
  3
     Australia Telescope National Facility, CSIRO Astronomy and Space Science, Sydney, NSW 2052, Aus-
tralia
  4
  Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton
Wharf, Birkenhead, CH41 1LD, UK
–2–

                                     1.   Introduction

     Massive stars (M ≥ 8M⊙ ) dominate feedback processes in the plane of our galaxy due
to their strong ionising radiation, outflows, stellar winds and supernovae. In star formation
regions they have a strong influence on nearby star formation, with the potential to disrupt
their parent molecular cloud and/or trigger further star formation.
     Regions of massive star formation are usually more heavily embedded within their par-
ent molecular cloud, rarer and further away than sites where only low mass stars form. In
addition, massive stars form and evolve faster than low mass stars. These issues mean that
there are few large, well selected samples of young massive stars, restricting our understand-
ing of how massive stars form and evolve in the early parts of their lives. Original catalogues
of mid-infrared bright but radio quiet massive young stellar objects (MYSOs), where major
accretion is probably still ongoing but an H ii region has yet to form, contained only of order
50 sources (Wynn-Williams 1982; Henning et al. 1984) which were serendipitously discovered
and biased away from the galactic plane.
     MYSO samples based on IRAS colours have been constructed (Palla et al. 1991; Molinari
et al. 1996; Sridharan et al. 2002) but due to the low-resolution of the IRAS satellite (3′ by
5′ at 100 µm, Beichman et al. 1988) were chosen to avoid confused regions of high source
density. Large H ii regions were avoided by comparison with single dish radio surveys, but
more compact H ii regions were often found to contaminate these samples to some degree
when followed up with higher resolution radio observations (Molinari et al. 1998). These
criteria tended to biased these samples away from complex regions where many massive stars
form.
     The Red MSX Source (RMS) survey is based on a colour-selected sample of ∼ 2000 can-
didate MYSOs by Lumsden et al. (2002) from the 20′′ resolution MSX point source catalogue
(PSC, Egan et al. 1999, 2003). A series of complementary follow-up observations has been
carried out designed to identify contaminants (CH ii regions, low mass YSOs, evolved stars,
proto-planetary nebulae (proto-PNe) and PNe) and provide characteristics of the MYSOs.
These follow-up observations have included ∼ 1′′ resolution radio continuum observations
(Urquhart et al. 2007a, 2009), ∼ 1′′ resolution mid-IR observations (e.g. Mottram et al. 2007)
outside the SPITZER GLIMPSE survey regions (Benjamin et al. 2003), 13 CO molecular line
observations (Urquhart et al. 2007b, 2008) to obtain kinematic distances and low resolution
near-IR spectroscopy (e.g. Clarke et al. 2006).
     Each source within the RMS sample has been classified into one of several categories.
‘YSO’s are radio-quiet mid-IR point sources which are associated with 13 CO emission.
‘H ii regions’ are radio loud and/or resolved in the mid-IR indicating emission from Lα
–3–

heated dust. H ii regions with spectral types later than B1 may have too low radio sur-
face brightness to be detected in our 1′′ radio observations, but would still exhibit strong
hydrogen recombination emission with Case-B (Baker & Menzel 1938) line ratios in our
near-IR spectroscopic follow-ups. For details on individual sources see the RMS database
(www.ast.leeds.ac.uk/RMS).
     In Mottram et al. (2010b, submitted) we used far-IR fluxes from Mottram et al. (2010)
and other data, along with the spectral energy distribution (SED) model fitter of Robitaille
et al. (2007) to obtain luminosity measurements of ∼ 800 of these sources. In this paper we
use the luminosity distributions obtained by Mottram et al. (2010b, in prep.), reproduced in
figure 1, to calculate the luminosity functions for MYSOs and CH ii regions in the galactic
plane of our Galaxy. These calculations will be discussed in §2, the results of which will be
presented in §3. We will then use these results to obtain estimates for the lifetime of these
phases as a function of luminosity using a method similar to that of Wood & Churchwell
(1989), which will be discussed in §4. Our conclusions are then presented in §5.

                             2.   The Luminosity Function

    We wish to derive a luminosity function defined as the number of objects per unit volume
per unit luminosity interval as a function of position in the Galaxy. In order to derive the
volume of the galactic plane within which sources of luminosity L are detected within the
RMS survey, a 3-D grid of cells was set up in cylindrical polar coordinates (i.e. r, θ and z),
centred on the galactic centre. The volume of each cell was calculated (r δr δz δθ), along
with the galactic coordinates (l & b) using the distance between the Earth and the galactic
centre of R0 = 8.5 kpc. The distance between each cell and the Earth was also calculated,
which was then used to obtain the total flux (FT otal ) that would be observed for a source of
luminosity L within the cell.
      Since the RMS sample is effectively a flux-limited sample in the MSX 21 µm band
we need to convert FT otal to FM SX 21 using the observed colours in the RMS sample. The
distribution of FT otal (Wm−2 ) / FM SX 21 (Wm−2 ) was obtained from SED model fits for
young RMS sources by Mottram et al. (2010b, submitted), and takes the form of a Gaussian
in terms of log ( FT otal / FM SX 21 )=1.40±0.26. There is no significant variation in this
ratio between YSOs and H ii regions. The mean and 1 σ values for FT otal / FM SX 21 were
then used to calculate the mean, maximum and minimum MSX 21 µm fluxes in Wm−2 from
FT otal for a source of luminosity L in the cell. These are then divided by the bandwidth of
the MSX 21 µm band (4.041 × 10−14 Hz, Cohen et al. 2000) in order to convert these fluxes
to Janskys. If FM SX 21 ≥ 2.75 Jy, the nominal flux limit of the MSX survey used when
–4–

Lumsden et al. (2002) defined the RMS sample, for the cell then the volume of the cell is
included within the total volume included in the RMS survey for that luminosity.
    The distribution of material and stars within our Galaxy is far from uniform, particularly
with regard to young sources. The distribution of young RMS sources with luminosities
L ≥ 1.3 × 103 L⊙ ( this will be discussed in §3) are shown in figure 1 in terms of galactocentric
radius (r) and distance from the galactic mid-plane (z). We fitted the distribution with the
functional form of a thin disk as modelled by Robin et al. (2003). This was modified here
with an offset from the galactic centre, r0 and has a different length-scale for the disk in order
to match the observed distribution in galactocentric radius. The distribution of sources in
terms of z is similar to that for H ii regions from Cox (2005) with scale-height zd .
     Given that RMS sources are not evenly distributed within the galaxy, a weighting func-
tion was used to weight the volume of each cell within the numerical integration by the
observed distribution. This function is a combination of the modified distributions discussed
above. It therefore has the form:

                                                         −(r − r0 )2                   −(r − r0 )2
                                                                                               
                                  −|z|
        weight(r, z) = exp                     exp                         − exp                          (1)
                                   zd                        rd2                           rh2

A least-squares minimisation was used to obtain the parameters of this function from the
distribution of RMS sources, which resulted in the scale length of the disk and hole of
rd = 7.14 ± 0.04 kpc and rh = 1.32 ± 0.02 kpc respectively, the offset from the galactic
centre of r0 = 2.77 ± 0.02 kpc and a scale-height of the disk in z of zd = 44.2 ± 0.4 pc . This
is shown by the red solid lines in the bottom plots of figure 1. The asymmetry in z and the
slight increase in the number of sources at r = 10 kpc are probably due to features which
are near the location of the Sun (r0 = 8.5 kpc, z0 = 15 pc, Marshall et al. 2006).
      The total weighted volume for which a source with luminosity L should be included
within the RMS survey is therefore calculated by integrating the weighted cell volume over
all cells with FM SX 21 ≥ 2.75 Jy, -5◦ | b | ≤ 5◦ and 10◦ ≥ l ≥ 350◦ (see Lumsden et al. 2002).
This process was then repeated for logarithmic steps in luminosity. Increments of 0.1 kpc,
0.5◦ and 0.01 kpc were used in r, θ and z respectively, and all values were calculated for the
cell centre.
     The error in the luminosity bin is taken to be width of the bin and the error in the
luminosity function is calculated from the error in the number of sources. As the number
of sources is a discrete sampling of a continuous distribution, Poissonian statistics are used
to calculate this error and only bins with at least 3 sources in them are considered in the
following analysis.
–5–

Fig. 1.— Inputs to the luminosity function calculation. Top: the number of sources vs. lu-
minosity for RMS candidates with identifications of YSO and H ii reproduced from Mottram
et al. (2010b, submitted.). The blue arrows indicate the luminosity of a B0 type star, as
presented in Table 1. Bottom: the distribution of luminous (L & 1.3 × 103 L⊙ ) sources in
the RMS survey as a function of galactocentric radius (r, left) and distance from the galactic
mid-plane (z,right). The red lines indicate the least squares fit of the weighting function
(equation 1) to the data.
–6–

                                         3.    Results

     The volume of the galaxy included in the RMS survey as a function of luminosity, cal-
culated as discussed above, is shown in figure 2 and indicates that the RMS survey should be
& 50% complete above ∼ 9 × 103 L⊙ , which corresponds to a spectral type of ∼ B1V using
the calculations of Martins et al. (2005) corrected for the luminosity/temperature/mass rela-
tionship of Meynet & Maeder (2000), which is shown in Table 1. We restrict the luminosity
function and source distribution calculations to L ≥ 1.3 × 103 L⊙ as below this luminosity
the volume correction required is > 10. The calculated volume was then used to obtain
the luminosity functions for RMS survey YSOs and H ii regions. As the weighting function
causes the luminosity functions to be positionally dependent, the results were multiplied
by the weight for the galactic location of the Sun in order to obtain the local luminosity
functions for YSOs and H ii regions.
     The resulting luminosity functions are shown above the 50% completion limit in the
bottom plots of figure 2. At the higher luminosity end these luminosity functions simply
reflect the main features from the luminosity distribution; namely that there is no significant
population of MYSOs above 7 × 104 L⊙ or earlier than O8V. Over the luminosity range
9 × 103 L⊙ ≤ L ≤ 7 × 104 L⊙ , the MYSO luminosity function rises steadily such that
log(φM Y SO ) = (-1.0 ± 0.2) log(L) + (5.0 ± 0.8), where φM Y SO is in objects per kpc−3 per
dex of luminosity and and L is in solar luminosities.
     There is a turnover in the compact H ii region luminosity function around L = 4 × 104 L⊙ ,
09V). Above this the luminosity function is fit by log(φCHII ) = (-0.7 ± 0.1) log(L) + (4.0 ± 0.7).
Although the break in slope of the compact H ii region luminosity function coincides with
the 100% completeness limit, the fact that there is no similar break in the MYSO luminosity
function that continues to lower luminosities means that the break is likely to be real.
    The right-hand plot of figure 2 shows the local luminosity function for the combined
sample of all YSOs and compact H ii regions identified in the RMS survey. This shows the
overall function for the young mid-IR bright phase of massive star formation regardless of
whether the source is ionizing its surroundings or not. Above L = 4 × 104 L⊙ this combined
luminosity function is fit by log(φCombined ) = (-0.8 ± 0.1) log(L) + (4.8 ± 0.6).

                                       4.     Timescales

     Wood & Churchwell (1989) used the number of main sequence stars to obtain an estimate
of the lifetime of CH ii regions from the number of such sources they observed. In terms of the
luminosity functions of YSOs and H ii regions (φY SOs (L) and φH II (L)), the same concept can
–7–

Fig. 2.— Results of the luminosity function calculations. Top: Volume of the galaxy within
which sources should be detected in the RMS survey as a function of luminosity. The solid
blue and red horizontal lines indicate 50% and 10% completion respectively. Bottom: The
local luminosity functions for YSOs (black) and H ii regions (red)(left) and the combined
local luminosity functions for sources identified as either YSOs or H ii regions (right) in the
RMS survey. The results of the total least squares fits to the luminosity functions over
limited ranges of L are indicated by red, green and blue solid lines and are discussed in §3.
The spectral type scale is presented in Table 1
–8–

be applied here using the luminosity function of main sequence stars (φM Sstars (L)) to obtain
the lifetimes of YSOs and H ii regions (tY SO (L) and tH II (L)) as a function of luminosity.
This was calculated using:

                                                                    
                                                        φY SOs (L)
                             tY SO (L) = tM S (L)                                          (2)
                                                         φM S (L)

where the main sequence lifetime (tM S ) of an OB star is given by log(tM S ) = (9.57 ± 0.15) -
(0.74 ± 0.06) log(L) + (0.040 ± 0.006) log2 (L) calculated from a least squares fit to the
results of the models including stellar rotation by Meynet & Maeder (2000) with an initial
velocity of vini = 300 kms−1 . This starting velocity is chosen as these models predict the
correct ratio of red to blue supergiants, and it is the only set of models which extend beyond
60 M⊙ .
     The local main-sequence OB star luminosity function in terms of MV was obtained from
re-binning the results on dwarf stars of Reed (2005) resulting in a least-squares fit of the
form log(φ(MV )OB ) = (4.98 ± 0.18) + (0.69 ± 0.10) MV , where φ(MV )OB is in kpc−3 per
magnitude bin. This is shown in figure 3.
      This was then converted from a function of MV to a function of luminosity using
MV = (4.2 ± 0.5) - (2.5 ± 0.2) log(L) + (0.14 ± 0.02) log2 (L) which is a least squares
fit to the data in Table 1.
     The YSO and H ii region lifetimes obtained as functions of luminosity are shown in
figure 4. Fits to the data give log(τM Y SO ) = (-0.46 ± 0.21) log(L) + (7.6 ± 0.9) and
log(τCHII ) = (-0.23 ± 0.12) log(L) + (6.9 ± 0.6). Given the mostly linear main sequence
luminosity function then these lifetime plots show mostly the same features as the luminosity
functions for MYSOs and CHIIs themselves. They are fairly shallow showing that the lifetime
of these phases is not a strong function of luminosity and is of order a few times 105 years.
Wood & Churchwell (1989) concluded that the lifetimes of H ii regions powered by O stars
are about 15% if their main sequence lifetimes. Here we find the compact H ii region lifetimes
are about 5% of their main sequence lifetimes, although the main sequence lifetimes we are
using are about twice as long as those used by Wood & Churchwell.

                             5.   Discussion & Conclusions

     We have obtained, for the first time, the luminosity functions of YSOs and compact
H ii regions in the galactic plane. These clearly show that there is no significant population
of the mid-IR bright, radio quiet MYSOs above ∼ 7 × 104 L⊙ . This lack of high luminosity
–9–

Fig. 3.— OB star luminosity function, calculated from the results of Reed (2005) (solid
circles). This compares well with that for main sequence stars from McCuskey (1966) (open
circles) from his Tables 8 and 12.
– 10 –

Table 1: Properties of main-sequence stars from the calculations of Martins et al. (2005)
corrected for the luminosity/temperature/mass relationship of Meynet & Maeder (2000).
The main-sequence lifetimes were calculated from the quadratic fit given in the text to the
results of the models including stellar rotation by Meynet & Maeder (2000) with an initial
velocity of vini = 300 kms−1 .

 Sp        Tef f   log(g)     R     log(L)     M       MV     log(Q0)     tKH         tM S
 Type    (103 K)            (R⊙ )    (L⊙ )   (M⊙ )    (mag)    (ph/s)   (104 yrs)   (106 yrs)
 O3         45      4.0     20.6     6.18     103      -5.8     49.98      1.1         3.1
 O4         43              19.6     6.06      84      -5.5     49.85      1.0         3.3
 O5         41              13.5     5.65      52      -5.2     49.40      1.4         4.3
 O6         39      4.05    10.4     5.34     37.3     -4.9     49.00      1.9         5.4
 O7         37               8.8     5.10     30.2     -4.7     48.63      2.6         6.5
 O8         35               7.6     4.88     24.1     -4.5     48.27      3.2         7.7
 O9         33               8.4     4.68     20.2     -4.3     47.86      3.2         9.1
 O9.5      31.5     4.1      6.2     4.52     18.1     -4.1     47.46      5.1        10.5
 B0        29.5              5.6     4.32     15.3     -4.0     47.07      6.3        12.6
 B0.5       28               5.2     4.16     13.7     -3.7     46.48      7.9        14.6
 B1         26               4.7     3.94     11.7     -3.3     45.98     10.6        18.1
 B1.5       24      4.15     4.2     3.71     10.0     -2.8     45.19     14.7        22.8
 B2         21               3.3     3.27     7.0      -2.4     44.88     25.3        36.6
 B2.5       19      4.2      3.0     3.01     6.0      -2.0     44.28     37.1        49.1
 B3        17.5              4.1     2.74      5.2     -1.5     43.69     38.0        67.7
– 11 –

Fig. 4.— The lifetime of MYSOs (black) and H ii regions (red) as a function of luminosity.
The red and black dashed lines indicate the results of total least squares fits to the functions
over limited ranges of L and are discussed in §4. The green solid line indicates the Kelvin-
Helmholtz timescale, while the blue line indicates the main-sequence lifetime. The spectral
type scale is presented in Table 1
– 12 –

MYSOs could mean that the immediate progenitors of the most luminous CH ii regions are
mid-IR faint or dark. Luminous far-IR cores and methanol masers that are too faint to be
in our MSX-selected sample are being found (e.g. Chambers et al. 2009; Ellingsen 2006).
     However, it is also important to consider the luminosity function in terms of the mech-
anism that must inhibits the formation of an H ii region around MYSOs, which are enough
to ionize their surroundings if they were on the main sequence. Previously it has been
suggested that MYSOs are on the main sequence, but that the accretion rate and/or high
density of the surrounding material quench or trap the H ii region close to the star (e.g.
Walmsley 1995; Keto 2003; Tan & McKee 2003). In this case the H ii region would have
a high enough emission measure (EM) to be optically thick at radio wavelengths, e.g.
τ1 cm = 3 × 10−10 EM (cm−6 pc)>>1 (Osterbrock 1989). However, such regions are un-
likely to still be optically thick in the near-IR recombination lines (e.g. Brα, Brγ) which
have line centre optical depths of order τBrα ∼ 10−5 τ1 cm (Hummer & Storey 1987). This
would imply that MYSOs should exhibit very strong near-IR recombination line emission,
whereas they actually only show weak, broad lines consistent with emission from stellar
winds (Bunn et al. 1995).
      Alternatively, as first suggested by Hoare & Franco (2007) the MYSO could be swollen
up due to high accretion rates. The recent stellar evolution calculations including high accre-
tion rates by Yorke & Bodenheimer (2008) and Hosokawa & Omukai (2009) have suggested
that the radius of stars with masses between ∼ 8M⊙ and ∼ 30M⊙ increase dramatically (i.e.
1-2 orders of magnitude with respect to the main sequence) due to high accretion rates which
deliver high entropy material onto the stellar surface. An accreting star would therefore only
begin forming an H ii region when its mass is high enough that the Kelvin-Helmholtz time
becomes so short that it shrinks to its MS size even whilst accreting. Given the lack of
a significant MYSO population above 7×104 L⊙ equivalent to ∼ 25M⊙ this would suggest
that accretion rates around this mass range are several times 10−4 M⊙ yr−1 by inspection
of figure 14 from Hosokawa & Omukai (2009). If an MYSO stops accreting significantly for
some reason it will also contract onto the main sequence on the Kelvin-Helmholtz timescale
and once there would form a lower luminosity H ii region as seen in the RMS sample.
     The luminosity functions of MYSOs and H ii regions presented in this paper can be used
to test models of the time-evolution of the mass accretion rate and stellar properties. Such
modelling will be discussed in a future publication (Davies et al. 2010, in prep.).

    JCM is partially funded by a Postgraduate Studentship and by a Postdoctoral Research
Associate grant from the Science and Technologies Research Council of the United King-
dom (STFC). The authors would like to that Cameron Reed for his help with the OB star
– 13 –

luminosity function.

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