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A CS survey of small molecular cores

We are addressing the sites of isolated low mass star formation in the solar neighbourhood, i.e. small cloud cores within one kiloparsec. We aim at determining the physical parameters of the cores, i.e., temperature, volume density, column density and (radial) velocity fields, and the status of star formation, i.e., whether embedded objects are present within the cores. Surveying small dark clouds in both celestial hemispheres we study the physical conditions of low-mass star formation for detectable core masses M>0.01Msun. The target list is drawn from catalogues of optically selected dark cloud cores, where the visual extinction exceeds 5 magnitudes. The selected probe is the CS molecule that needs high densities for excitation of its rotational levels. To gauge the state of excitation, the cores were observed in two transitions. In a limited number of cases, optical depths were derived from complementing lines of the rarer isotopologue C34S for the (2-1) and (3-2) transitions. Making small (3arcmin by 3arcmin) maps, the 471 optically selected cores were searched for CS(2-1) and 315 (67%) were detected (T_A*>3sigma). In general, the position of peak CS emission does not coincide with the optically determined centre of the cores. The cores appear cold (T<10K) and, in the majority of cases, the CS emission is optically thin (tau<1). On the arcminute scales of the observations, the median column density of carbon monosulfide is N(CS)=7.E12/cm2. For an average abundance of N(CS)/N(H2)=1.E-8, the median mass of the detected cores is 1.0Msun. The line shapes are most often Gaussian with widths exceeding that due to thermal broadening of<0.1km/s. The observed median FWHM=0.7km/s, i.e. non-thermal turbulence contributes dominantly to the line widths


Introduction
Many cold and dense cloud cores in the interstellar medium will eventually evolve towards dynamical instability and form stars. For the cloud contraction to proceed, the generated compressional heat needs to be channeled away. This can be achieved by means of radiation losses through collisionally excited spectral lines. The first major computation of the expected cooling rates in molecular cloud conditions were presented by Goldsmith & Langer (1978). Water vapour (H 2 O) was found to be the dominant cooling agent at densities exceeding 10 5 cm −3 , irrespectively of the kinetic temperature of the gas. These early models did not consider the effects of molecular depletion from the gas phase due to freeze-out onto dust grains: during more recent years, observations from space indicated much lower abundances than what had been assumed in the earlier works.
From an observational point of view, such studies require trustworthy statistics and, hence, large homogeneous samples. To identify the dense regions where the water cooling could be expected to become efficient, we selected molecular probes Lynds' highest opacity classes = 5 and 6). The core size was initially set not to exceed 10 , but this effective radius was redefined to be at most 4 (see: Clemens & Barvainis 1988). Here, the size is defined as r eff = √ a × b with the semiaxes of the source ellipse in arcminutes. This procedure could be expected to ensure a volume limited sample, since the number of foreground stars N star is small and the distance D to dark clouds is proportional to N star , e.g. d = 320 N 0.57 star pc over a 5 sky area (Herbst & Sawyer 1981). The resulting 'southern' catalogue contained 193 cores.
All in all, the initial target list ('northern' + 'southern') contained 538 objects. The majority of observed dense cores has a distinctly non-spherical geometry. Typical aspect ratios are of the order of 2:1 (see, e.g., Hartley et al. 1996, Clemens & Barvainis 1988, Benson & Myers 1989).

Distribution of the cores in the sky
Not entirely unexpected, the majority of cores is found in, or relatively close to, the galactic plane ( Fig. 1), where the high density of background stars provides a high contrast for the dark globules. Comparing the LSR-velocities of the core gas with models of the galactic rotation indicates that most of the cores are within 1 kpc from the Sun, i.e. that they indeed belong to the Solar Neighbourhood (| υ lsr | < ∼ 10 km s −1 ).

Mass sensitivity of the core survey
We adopted a limiting antenna temperature of 20 mK (averaged over the 9-point map, cf. Sect. 3) or 60 mK (for any individual position) in the CS (2-1) line. This sensitivity corresponds roughly to a limiting LTE column density 2 N(CS) = (3 − 5) × 10 11 cm −2 . For a CS abundance relative to H 2 , X(CS) = 1 × 10 −8 (Irvine et al. 1987), this would correspond to a hydrogen column N(H 2 ) = a few × 10 19 cm −2 . This limit is much smaller than the expected column densities of the molecular gas associated with the high dust extinction (several A V ) through the centres of the cores, since (on the average) N(H 2 ) = 0.94×10 21 A V cm −2 mag −1 (2).
At the distance of 500 pc, an object 1 in size (the approximate size of the telescope beam), i.e. with diameter of 0.145 pc, the minimum detectable mass would thus be of the order of 10 −2 M . Likely effects due to subthermal excitation, line opacity, low beam filling and/or convolutedness of the object could increase this number to a higher value. However, the survey should be sensitive to the standard scale of star formation, viz. the Jeans-mass (Jeans 1927). For a of a cloud of ideal gas, M Jeans = 0.57 T 10 K 3 2 n(H 2 ) 10 5 cm −3 − 1 2 M . The numerical constant assumes the mean molecular weight µ = 2.4 for metal rich, fully molecular gas. A cloud of mass larger than M Jeans will be prone to gravitational collapse on a free-fall time scale t ff = 1.0 × 10 5 n(H 2 ) 10 5 cm −3 − 1 2 yr. In real, rotating clouds, various pressure gradient forces, including magnetic field stresses, will potentially prolong this time scale by factors of several (e.g., Ciolek & Mouschovias 1995). The matter will fall freely toward the centre of mass with a velocity υ ff = 0.20 M

Observations and data reduction
During several observing runs in 1996 and 1997, we exploited three different observatories, viz. the National Radio Astronomy Observatory (NRAO), Kitt Peak, Arizona, USA, the Onsala Space Observatory (OSO), Onsala, Sweden, and the European Southern Observatory, La Silla, Chile. There, we used the single-dish antennae of 12 m, 20 m and 15 m diameter, respectively. The La Silla telescope is a Swedish-ESO joint facility, i.e. the Swedish ESO Submillimetre Telescope (SEST).
The pointing accuracy of the SEST and OSO 20 m was regularly checked using SiO maser stars as references and was found to be about 4 (rms). The planets Jupiter and Saturn were used for the NRAO 12 m telescope, resulting in an accuracy of 8 (rms). From the observation of overlapping sources we determined the 3-telescope inter-calibration for the various transitions to be within 15%-20%.
The CS (2-1) and , i.e. short-hand for 12 C 32 S (J → J − 1), transitions (and their isotopic lines, see Table 1) could be observed simultaneously at the SEST, whereas for the northern sources these observations were split between OSO and Kitt Peak, cf. Table 2. SIS-mixer receivers were used as front-ends at all telescopes, whereas as back-ends we used Autocorrelators, Hybrid Autocorrelators (AC, HAC) and Acousto-optical spectrometers (AOS).
All observations were done in frequency switching mode, with switch frequencies of 4 MHz and 6 MHz (about 8 to 18 km s −1 ). Standard reductions included the removal of artefacts (interference spikes), the folding of the spectra and the polynomial fitting of the baseline and its removal. Finally, the properly weighted data were averaged to generate one spectrum per line and per position.
Integration times were typically of the order of 60 s per position. For each source, and for each line, a 3×3 map was obtained, aligned with the equatorial coordinate system and with 60 spacings. This was a trade-off between an acceptable sampling rate and a sufficient sky coverage within the available observing time. These small maps should give us a hint at the degree of compactness of the CS emission.

CS (2-1) and CS (3-2) detection rates
Of the originally selected sources, 471 (88%) were observed in CS (2-1) and 431 (80%) in CS . Table 5 summarises the observational results. Detections at the ≥ 3σ level of the antenna temperature T A were two thirds (315) in the (2-1) and one third (141) in the (3-2) transition. These relatively high detection rates clearly support the view that the selected dark dust patches in the sky are sites of high molecular gas density, approaching those of the critical values of the CS transitions.

Line intensities and line widths
Limiting the sample to cores with |υ LSR |< 20 km s −1 results in the distributions 3 for 283 CS (2-1) sources shown in Fig. 2. The median line intensity is T mb dυ = 0.6 K km s −1 and the observed The spatial LSR-velocity distribution of these detected cores. A simple Oort-type model is shown by the dashed lines to guide the eye, corresponding to distances of 0.5 kpc and 1.0 kpc, respectively. Formally, these loci are given by υ lsr = 13 d sin 2 , where d is the distance in kpc and is the galactic longitude. median line width is FWHM = 0.7 km s −1 . That this result is not limited by the resolution of the spectrometers is verified in Fig. 3, where similar distributions are shown for objects for which both valid (T A > 3σ) CS (2-1) and CS  data are available (132 error of that average and σ N is the square root of the number of cores in that bin. sources), since the higher frequency lines provide a better velocity resolution (cf. Table 2).
These observed line widths are much larger than those from purely thermal broadening, viz. υ th = 0.06 √ T/(10 K) km s −1 , and are thus dominated by other types of motion, including nonthermal turbulence.

Positional coincidences and source sizes
As described in the previous section, 9-point maps (3 × 3 ) were obtained for each core. Inspection of these small maps (Table 5) reveals that the CS (2-1) peak emission most often does not coincide with the optically determined centre position. The fraction of sources for which positional agreement exists amounts to 38%. One explanation could be that the molecular line observations better probe the regions of maximum density (core centre) than the optical data which saturate at A V ∼ 5 mag, i.e., becoming indiscriminant for higher opacity regions. In Fig. 4, the optically determined core sizes and those found from the CS (2-1) observations are shown. The selection is biased and the CS mapping incomplete. However, from the figure it is evident that the sample is peaked towards small CS cores with an average value of 3.1 arcmin 2 . Even though the observed map is small and under-sampled, one can still get a rough estimate of the core extent in CS, simply by adding the number of map positions, each 1 arcmin 2 in size, where the detected signal is larger than half the peak intensity.
As already mentioned, the peak intensity of the CS line is not always found at the centre position. In addition, several cores show evidence of being extended on scales larger than our limited maps. We will therefore potentially underestimate the sizes of a number of cores. By assuming that the core is symmetric Fig. 5. Examples of mirroring the observed maps of extended sources (large dots symbolise peak integrated line emission, whereas the smaller symbols refer to those positions where the emission is above 50% of the peak value). Left: The peak is at a map edge. Right: The peak is in a map corner. around its peak and by mirroring the map with respect to this position, a somewhat better estimate of the source size can be found (see Fig. 5 for examples).
Finally, the resulting size distribution is also shown in Fig. 4. The average core size is about (4 ± 1) arcmin 2 (optical: 3.5 ± 0.2, CS (2-1): 3.1 ± 0.2, extrapolated CS: 4.6 ± 0.3). These independent size estimates (optical and CS) are in good agreement, indicative of a tight relation between the dust and the molecular gas (the average size ratio log [Optical/CS (2 − 1)] = −0.085, with a variance of 0.092). This indicates that, on the average, the beam should to a large extent be filled around the peak position.

The volume limited sample
The number of cores N subtending a given solid angle Ω can be written as dN dΩ = dN dD dD dΩ , where D is the distance to the core. For the case that the cores are of the same size and are evenly distributed in space, the following applies dN dD ∝ Ω −1 and that dD dΩ ∝ Ω −3/2 . This integrates to N(Ω) ∝ Ω −3/2 , so that the slope of this distribution is d log N d log Ω = −1.5. The core size distributions of Fig. 4 all have slopes of about −1.5, consistent with the result of Eq. 12 (optical: −1.62 ± 0.4, CS (2-1): −1.26 ± 0.28, extrapolated CS: −1.28 ± 0.16). We take this to support our view that the selected core sample is reasonably homogeneous and volume limited.

Optically thin emission: C 34 S (2-1) and C 34 S (3-2)
The solution to the equation of radiative transfer of molecular radio lines reads e h ν/kT −1 and the symbols have their usual meaning. f beam is the beam filling factor, T ex is the excitation temperature of the transition (the line source function), T bg represents a background radiation field, which at mm-wavelengths is the cosmic microwave radiation (T bg = 2.725 K), and τ is the line optical depth.
Assuming that the lines of C 34 S are formed in the same layers as the corresponding CS lines (= C 32 S) and that the line opacities scale as their relative abundances, i.e. x = τ(C 32 S)/τ(C 34 S) = X( 32 S)/X( 34 S) ( = 22.5), means that the radiative transfer equation for the ratio of these lines reduces to . This transcendental equation can be solved iteratively for the optical depth in the C 34 S lines, τ(34), and these are presented for a number of cores in Table 3. Quite in accordance with expectation, these values are all smaller than unity. For the optically thin C 34 S lines we derive the excitation temperature and column density of the molecule from degenerated rotation diagrams (see, e.g., Goldsmith & Langer 1999). For these two transitions, i.e. the C 34 S (2-1) and (3-2) lines, the excitation temperatures, T ex , and column densities, N(C 34 S), are listed in Table 3. Having determined the opacities from the C 34 S lines, the same analysis can be done for the CS lines, which results in the values presented in Table 4 4 . From this it is evident that the limited number of cores with detectable C 34 S emission has only moderately high optical depths in the lines of the main isotope of sulfur. The majority of cores can be expected therefore to exhibit τ < ∼ 1. From Tables 3 and 4 it is evident that the derived excitation temperatures are remarkably low. If these were meant to represent true gas kinetic temperatures, we have to conclude that assuming LTE (Local Thermodynamic Equilibrium) would likely be erroneous. Else, for unit beam filling, it would seem that the volume densities of cores, averaged over the beam, are generally lower than the critical densities of the transitions.

CS column densities and core masses
For the present sample, the analysis of the previous section, including maps of two transitions and of their weak isotopic lines, could be done only for a limited number of cores. The size of the analysable sample can be increased by applying the C 34 S method to the set of CS data for which two transitions have been observed. Assuming optically thin emission in the (2-1) and  lines of the main isotope (justified shortly), results in the T ex distribution shown in Fig. 6 for 90 cores (T ex > 3σ). These temperatures are, again, generally quite low, 6 K on the average. For 69 cores (N > 3σ), the data quality permits the determination of the CS column densities, which are of the order of 10 13 cm −2 and thus consistent with the values of Table 4.
These column densities are not extremely sensitive to the actual excitation temperature (within reasonable limits, of course), so that the adoption of a reasonable average value would permit us to use the entire data set. Consequently, we used (somewhat arbitrarily) the value of 10 K (see, e.g., 1). However, as will be demonstrated a posteriori, this choice is not critical to our final results (see below). Once T ex is known, the line optical depth τ can be determined from the observations. This is shown in Fig. 7, where the distributions of both the CS (2-1) (291 cores) and (3-2) (131 cores) line depths are depicted. As expected, the optical depth in the (3-2) transition is larger than that in the (2-1) line. Also evident from the figure is the fact that the majority of cores is optically thin in these lines.   From these optically thin transitions we derive the CS column densities. These are shown in Fig. 8, separately for the CS (2-1) and  observations. Obviously, these distributions are very similar, the (2-1) data being less noisy. Both yield practically identical results, i.e. the median CS column density of nearly 200 cores is N(CS) = 7 × 10 12 cm −2 . This is comparable with the values of Table 4 which were based on a more detailed and accurate analysis of the strong line cores.
Assuming a 'universal' CS abundance relative to H 2 of X(CS) = 10 −8 and a common distance of 500 pc, the column density distribution of Fig. 8 can be transformed into a core mass distribution (Fig. 9). As before (Sect. 4.3), this procedure is based on (for the area) adding the pixels with valid data (> 3σ) above the 50% peak value. For the sample of nearly 200 cores, the observed apparent mass range is from 10 −2 M to 10 2 M , where the low mass cut-off is set by the lack of observational sensitivity. The median mass is 1 M , indicating that perhaps up to about 10 2 cores are Jeans-unstable.
The corresponding distribution of volume densities n(H 2 ) can be obtained assuming that the extent of the core along the line of sight is similar to that in the plane of the sky, i.e. n ≈ N/ √ π × area (Fig. 10). We find that the median density is n(H 2 ) = 10 3 Table 3, but for the corresponding C 32 S (2-1) and (3-2) observations  1976). The results of Fig. 9 could then suggest that less than half of the cores are potentially unstable. However, from the results shown in Fig. 10, it is also evident that the (averaged) densities are much lower than the critical density of the CS (2-1) transition. This is clearly inconsistent with the assumption of LTE, since thermalisation of the level populations requires densities at least one order of magnitude above n crit .
The actual value of the CS abundance could be lower than that assumed above. This is also suggested by the large relative offset of the virial mass distribution shown in Fig. 9, which is based on the observed line widths of the CS (2-1) line and the

Line profiles
In order to avoid spurious effects due to the noise in the observed spectra we selected those observations the line intensity of which is above the 7σ level of the rms-noise (this particular value T mb, 0 > 7σ was found empirically). Ideally, the velocity of the object with respect to the telescope υ 0 should be determined from optically thin lines, that are known to be symmetric. However, since we have only a limited amount of C 34 S data, the central (= systemic) velocities of the CS (2-1) lines were determined through the application of the high order moment υ T 3 mb (to damp out the noise), i.e. υ 0 =  Fig. 11.
The result for those 158 cores for which T mb, 0 > 7σ is shown in Fig. 12

Conclusions
Based on extensive mm-wave observations of a large sample of small dark cores in both celestial hemispheres in rotational lines of the CS molecule we conclude the following: • The rate of CS detections is high. Specifically, out of 471 cores observed in CS (2-1), 315 were detected above 3σ. For CS , these numbers are 431 and 141, respectively. • For 38% of the detected cores does the optical centre coincide with the position of maximum CS emission. • Core temperatures are found to be low ( < ∼ 10 K). • On the scales of these observations, the majority of cores is optically thin in the CS emission, with the median column density N(CS) = 7 × 10 12 cm −2 . • The lines are generally symmetric and narrow, but wider than due to thermal broadening alone. The median line width is FWHM = 0.7 km s −1 .