Astron. Astrophys. 327, 325-332 (1997)

2. Cloud properties

2.1. Individual cloud properties

Our sample consists of all the detected clouds located beyond 2 kpc from the Sun. Confusion to identify the individual clouds in the southern outer Galaxy occurs only at low velocities () which correspond to local clouds at distances within 1.5 kpc of the Sun. Also, because it is not possible to obtain accurate estimates of their distances kinematically, we have considered only those clouds located beyond 2 kpc from the Sun.

Since some clouds can be quite far from the Sun and subtend a small angular size, we included in our analysis only those whose angular extension was larger than 3 beam-widths. The angular extension of each cloud was measured within the 3 contour of the line intensity integrated over the velocity range of the cloud's CO emission. Also, clouds that were too close to the latitude boundaries of the survey were discarded to avoid errors in the determination of the sizes. Therefore, after applying to the deep CO survey such a selection criteria, we have ended up with a sample of 177 molecular clouds whose main physical properties are included in Table 2. Columns 1 and 2 correspond to the positions, in galactic coordinates, of the peak of the CO emission of each cloud. Columns 3 and 4 include and that correspond to the peak and FWHM of the gaussian fit, respectively. They have been determined from a gaussian fit to the cloud's composite spectrum, i.e., the sum of all spectra across the cloud's projected surface.

Table 2. Molecular clouds beyond 2 kpc from the Sun in the third galactic quadrant

Table 3. (continued)

2.1.1. Distance

Since outside the solar circle there is no distance ambiguity, heliocentric distances to all 177 clouds in Table 2 (column 5) were determined kinematically, using the rotation curve of Brand (1986), and Brand and collaborators (Wouterloot & Brand 1989; Brand & Blitz 1993) with the new galactic constants, and kpc (Kerr & Lynden-Bell 1986). Also the rotation curve of Alvarez et al. (1990), determined from CO emission in the inner Galaxy (fourth quadrant), was used for testing purposes. We have found that both rotation curves give very close results in the outer Galaxy indicating that the flat rotation curve at of Alvarez et al. (1990) can be extrapolated to the whole Galaxy.

2.1.2. Radius

We define the effective radius of a cloud (column 6 in Table 2) as , where A is the actual projected area obtained from the angular extent of each cloud, measured from the spatial maps, and the distance. The measurements of the angular area of a cloud might be subjected to significant errors mainly because the CO emission from the clouds in the outer Galaxy is rather weak and their angular extent is small as they are usually quite far away. The angular extent of each cloud was measured within the contour of the line intensity integrated over the velocity range of the cloud's CO emission.

2.1.3. Mass

The mass, , of each cloud was estimated directly from its luminosity on the empirically based assumption that the integrated CO line intensity is proportional to the column density of H₂ along the line of sight (e.g. Lebrun et al. 1983; Sanders et al. 1984; Bloemen et al. 1986). Thus, the masses were computed using the relation

where w is the mean molecular weight per H₂ molecule, X is the constant ratio of H₂ column density to integrated intensity, and is the luminosity given by

where is the line intensity integrated over all velocities and lines of sight within the boundaries of the cloud, and d is the heliocentric distance of the cloud. In practice, is computed by integrating the emission over the full velocity extent of the cloud, as exhibited in the galactic latitude-velocity or longitude-velocity diagrams, and over the face of the cloud, defined by its contour in the spatial maps.

Several authors (Mead & Kutner 1988; Digel et al. 1990; Sodroski 1991) have claimed that X in the outer Galaxy is larger than in the inner Galaxy varying from a factor of 2 (Mead & Kutner 1988), between 2 and 3 (Sodroski 1991), to (Digel 1991). We have adopted here a value for X equal to twice the value for the inner Galaxy because it is compatible with the three estimates used for the outer Galaxy. Strong et al. (1988), through an improved analysis of the work by Bloemen et al. (1986) together with new data, derived a value of molecules cm^-2 (K km s . However, this X has to be scaled down by 0.82 from the published value to account for the different calibration scale of the database from which it was derived (Bronfman et al. 1988), which gives molecules cm^-2 (K km s for the inner Galaxy (Murphy & May 1991; Mauersberger et al. 1996). Therefore, assuming a mean molecular weight per H₂ molecule of 2.72 times the mass of the H atom to account for the He content (Allen 1973), and adopting a value of molecules cm^-2 (K km s for our sample of clouds, we have

where is in and is in K km s^-1 kpc² deg². Note that denotes the total mass of molecular gas based on the integrated CO intensity. Table 2 (column 7) includes for the 177 clouds in our sample, in units of .

For comparison the virial mass of each cloud was also derived using the relation

where is in solar masses, is the effective radius of the cloud, in pc, and is the HPFW of the cloud's composite spectrum, in km s^-1. Eq. (4) assumes: 1) the cloud is in virial equilibrium, 2) the cloud is spherical with a density distribution, where r is the distance from its center, 3) the observed line width of the cloud is an accurate measure of the net velocity dispersion of its internal mass distribution, which is believed to be clumpy on many scales (e.g. Zuckerman & Evans 1974; Blitz & Stark 1986); or in other words, the cloud is free from magnetic or other non-gravitational forces of pressure (e.g. MacLaren et al. 1988). We are aware that magnetic forces and especially pressure terms may be important for clouds located in the galactic disk near spiral arms and regions of strong activity, like HII regions, supernovae, young stars, etc. (e.g. Myers & Goodman 1988; Elmegreen 1989; Mouschovias 1995), however, in the outer Galaxy the clouds are, in general, more isolated from their surroundings and the possible tidal effects that might take place are expected to be less important than those occurring inside the solar circle. In any case, for simplicity we have not considered in Eq. (4) the magnetic and pressure terms. We have adopted the density distribution considering the work of Fitzgerald et al. (1976), Dickman (1978), Snell (1981), Lorent et al. (1983), Arquilla & Goldsmith (1985) and Brand & Wouterloot (1995).

2.2. Statistical cloud properties

We have investigated some power-law relations found by several authors (e.g. Larson 1981; Myers 1983; Dame et al. 1986; Solomon et al. 1987; Sodroski 1991). Fig. 1 shows the relation between line-width and radius, , for the clouds in our sample. The straight line is a least-squares fit (corr. coeff. 0.61) given by

with and . The exponent is in agreement with previous determinations which range between 0.4 and 0.5 (e.g. Larson 1981; Dame et al. 1986; Solomon et al. 1987; Mead & Kutner 1988; Sodroski 1991).

Fig. 1. Logarithm of the observed line width (FWHM) of each cloud in Table 2 versus the logarithm of its radius . The straight line shows the least-squares fit to the data given by the equation

Fig. 2 shows the relationship between the mean H₂ number density and radius for the clouds in our sample. The mean H₂ number density, was computed directly from the values of and given in Table 2, assuming a mean molecular weight per H₂ molecules of 2.72 times the mass of the H atom. A least-squares fit of

to the data in Fig. 2 (corr. coeff. -0.82) gives and , where is in cm^-3 and is in pc. The power-law exponent is similar to the value found by Sodroski (1991) for his sample of outer Galaxy clouds.

Fig. 2. Logarithm of the mean H2 number density of each cloud in Table 2 versus the logarithm of its radius . The straight line represents the least-squares fit to the data given by the equation

For clouds in virial equilibrium the following condition has to be fulfilled

For our clouds , so we will assume they are in approximate virial equilibrium. It should be noted that in recent years several authors (e.g. Maloney 1990; Issa et al. 1990; Combes 1991; Adler &Roberts 1992) have questioned whether the size-line width relationship indicates that the clouds are gravitationally bound objects.

To compare the virial masses with those computed from luminosities, following Dame et al. (1986) and Sodroski (1991), we have plotted the ratio of the observed composite line widths, , to the values expected in virial equilibrium, , as a function of the effective radius, (Fig. 3). Because of the assumptions made (3.1.3) represents the net FWHM velocity dispersion of the internal mass distribution of a virialized spherical cloud with density distribution, mass and radius , and is given by

where is in km s^-1, is in and is in pc.

Fig. 3. Ratio of the observed line width (FWHM) to the virial theorem velocity width (FWHM) versus cloud radius. The straight line represents the least-squares fit to the data given by the equation - . Since this line is not perfectly horizontal at the clouds in our sample are in approximate virial equilibrium only

From Fig. 3 we can see that the line fitted to the data is not perfectly horizontal at , indicating some dependence of the ratio on the size of the clouds. This effect, that can also be seen in Fig. 3 of Sodroski (1991) and Fig. 4 of Leisawitz (1990), could be interpreted as a dependence of X on cloud size, something already commented by Scoville et al. (1987). Also Fig. 3 shows that the dispersion in the data decreases with increasing size indicating perhaps that only the larger clouds are self gravitating, that is, closer to virial equilibrium than the smaller ones. Furthermore, since the fit line is nearly horizontal at 1, the conversion factor X adopted here appears to be adequate since it yields masses which are compatible, in a statistical sense, with the virial masses derived under the assumptions described above.

Fig. 4. a Size distribution and b line width distribution of the 177 outer Galaxy clouds for which physical properties have been derived. Our sample shows that these clouds are smaller than 50 pc with spectral lines narrower than (FWHM)

The cloud size, line width and mass distribution of the clouds in our sample are shown in Figs. 4a, 4b and 5, respectively. When comparing the distribution of our clouds with those in the inner Galaxy (e.g. Dame et al. 1986; Solomon et al. 1987), it becomes evident that the clouds in the outer Galaxy are smaller ( pc), less massive () and with narrower lines ( km s^-1). We believe this effect to be real, and not due to an insufficient sample, as argued by Wouterloot & Brand (1995) analyzing their outer Galaxy clouds, because our ensemble of 177 clouds is large enough to provide a statistically significant result. Our results also differ from those of Sodroski (1991) who, using a sample of 32 outer Galaxy clouds, found no difference between the inner and outer Galaxy clouds. We believe our results are more reliable because Sodroski (1991) uses low resolution data () which increases the apparent cloud's size (because of the blending of clouds produced by the large synthesized low-resolution beam of the Columbia telescopes).

Fig. 5. Mass distribution of the 177 outer Galaxy clouds for which physical properties have been derived. The distribution shows that these clouds are less massive than and that our sample is incomplete for clouds with

Fig. 6 represents the derived mass spectrum d d for our sample of clouds, where N is the number of clouds of a given mass. In Fig. 6 we have taken bins of for to , and for to . The solid line corresponds to a weighted least-squares fit to the data points for (corr. coeff. -0.95), below which the data are incomplete (Fig. 5), and is given by the equation

Fig. 6. Mass spectrum d dM versus for our sample of outer Galaxy clouds. The straight line represents the weighted least-squares fit to the data points with and is given by the equation

where the weights are based on the number of clouds in each bin. The slope of this power-law (-1.45) is less steep than that found by Brand & Wouterloot (1995) (-1.62) for their extended sample of 112 outer Galaxy clouds taken from various authors, but it is closer to that found by Solomon & Rivolo (1989) (-1.5) for 440 clouds in the first galactic quadrant. Therefore, this similarity in the mass spectrum for clouds located inside and outside the solar circle suggests that the cloud formation mechanism is apparently independent of the galactocentric distance.

© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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