3 Derived parameters of the clouds

3.1 Dust temperature

The 60 and 100 $\mu$m dust color temperature, $T_{\rm d}$, was calculated at each pixel in an image assuming that the dust in a single beam can be characterized by one single temperature ($T_{\rm d}$), and that the emission at 60 and 100 $\mu$m is due to blackbody radiation from dust grains at temperature $T_{\rm d}$, modified by a power-law emissivity. The flux density of optically thin emission from dust grains at wavelength $\lambda$ is given by (e.g. Arce & Goodman 1999)

$$% F_{\lambda} = B_{\lambda}(T_{\rm d}) N_{\rm d} \tau_{\lambd... ...ambda}(T_{\rm d}) N_{\rm d} \lambda^{-\beta} \Omega_{\lambda},$$ (1)

where $N_{\rm d}$ is the column density of dust grains, $B_{\lambda}(T_{\rm d})$ is the Planck function, $\tau_{\lambda}$ is the dust optical depth, $\Omega_{\lambda}$ is the solid angle at $\lambda$, and $\beta$ is the power-law index of the dust emissivity. If we calculate the ratio, R, of the flux densities at 100 $\mu$m and 60 $\mu$m (after smoothing the 60 $\mu$m images to the spatial resolution of the 100 $\mu$m images), we can construct a look-up table with the value of R calculated for a wide range of $T_{\rm d}$. For each pixel in the image, the table is then searched for the value of $T_{\rm d}$ that reproduces the observed 60 to 100 $\mu$m flux ratio at each pixel. Emissivities $\epsilon\propto\lambda^{-1}$ and $\epsilon\propto\lambda^{-2}$ were used to calculate two dust temperatures for each globular filament.

3.2 Dust optical depth and visual extinction

Assuming optically thin emission, we use the dust color temperature to calculate the dust optical depth at each pixel

$$% \tau_{100} = 10^{-17}\frac{F_{\lambda}(100~\mu{\rm m})}{B_{\lambda}(100~\mu{\rm m},T_{\rm d})},$$ (2)

where $B_{\lambda}(100~\mu{\rm m},T_{\rm d})$ is the Planck function and $F_{\lambda}(100~\mu{\rm m})$ is the observed 100 $\mu$m flux. We then use Eq. (5) of Wood et al. (1994) to convert from dust optical depth to visual extinction:

$$% A_{V} = 15.078\,(1-{\rm e}^{-\tau_{100}/641.3})~,$$ (3)

where $\tau _{100}$ is the optical depth given in units of 10^-6. This result is obtained by fitting the relation between $60~\mu$m optical depth ($\tau_{60}$) and A_V found by Jarrett et al. (1989) for the $\rho$ Ophiuchi region. Assuming optically thin emission, Wood et al. (1994) multiply the Jarrett et al. (1989) $\tau_{60}$ values by 100/60 to convert to $\tau _{100}$ and obtain Eq. (3) above. Thus, the derived extinction values are subject to the uncertainties in the conversion equation. However, Jarrett et al. (1989) show that there is a very tight correlation between $\tau_{60}$ and A_V for $A_{V}\leq5$ mag, implying very little uncertainty in the conversion of far-infrared optical depth to visual extinction, at least in the low ( $A_{V}\leq5$ mag) extinction regions of GF 17 and GF 20.

3.3 Errors and assumptions

In order to derive the dust temperature and optical depths, we have made the following assumptions: (1) the dust is optically thin at 60 and 100 $\mu$m, (2) dust emissivity is proportional to a power law $\lambda^{-\beta}$, with index $\beta=1$, and (3) the dust in the IRAS beam is at a single temperature. We discuss the validity of these assumptions below.

Draine & Lee (1984) have shown that the $100~\mu$m optical depth can be estimated in terms of the hydrogen column density along the line of sight as $\tau_{100}=3.6\times10^{-25}~N_{\rm H}$. Thus, $\tau_{100}\sim 1$ only for $N_{\rm H}$ in excess of $2.7\times10^{24}$ cm^-2, which is well above typical Galactic plane values. In addition, the largest $\tau _{100}$ we find in our images is ${\sim}3\times10^{-5}$. Thus, assumption (1) is valid.

The errors introduced by assuming a constant $\beta$ along the line of sight are hard to estimate, since we do not have any way to measure how much $\beta$ changes in our regions of study. There is a general agreement that the emissivity index depends on the grain size, composition, and physical structure (Weintraub et al. 1991), and the general consensus in recent years has been that $\beta$ has a value most likely between 1 and 2, that in the general ISM $\beta$ is close to 2, and in denser regions with bigger grains $\beta$ is closer to 1 (Beckwith & Sargent 1991; Mannings & Emerson 1994; Pollack et al. 1994). We have performed tests with $\beta=2$ and find that our results are not significantly affected: for both clouds, the dust color temperature calculations with $\beta=2$ and $\beta=1$ are consistent within $20\%$. Thus, we conclude that assumption (2) is acceptable.

Assumption (3) is certainly not valid near local heat sources. Langer et al. (1989) and Draine (1990) have considered this problem by examining a simple two-component model where the two regions have different dust temperatures and optical depths. They have shown that the calculated dust temperature is dominated by the emission from the hot dust component, even if this component represents only a few percent of the total mass of dust. Essentially our temperature determination, from which we calculate the $100~\mu$m and $60~\mu$m emission, is an emissivity weighted rather than a mass weighted dust temperature. Thus, our single-temperature assumption is violated in the immediate vicinity of stars embedded in the cloud. If a star heats the dust in its vicinity, the calculated T_60/100 will be dominated by the hot dust, the derived $\tau _{100}$ will be too low and consequently A_V will be underestimated. Hence, the masses of the clouds that have embedded stars with unresolved dust temperature gradients will also be underestimated. Even away from the point sources, complex temperature structure may be present and must be kept under consideration when interpreting the derived dust optical depths and temperatures. Without higher spatial resolution observations or modeling the dust temperature distribution close to embedded stars we cannot remove this effect.

3.4 Gas column densities

In estimating the physical parameters of the molecular gas in GF 17 and GF 20, we have assumed that the main beam temperature, $T_{\rm mb}$, is a good approximation of the source brightness temperature, since the molecular structures observed are extended with respect to the telescope beam. We assume a plane-parallel, LTE, isothermal, and optically thin radiative transfer model for the ¹³CO transition line.

The excitation temperature of each line of sight in each cloud was computed from the peak CO intensity under the assumptions of high optical depth of the CO line and a beam-filling factor of unity, using the equation

$$% \frac{T_{0}}{T_{\rm ex}}={\rm ln}\left[1+\frac{T_{0}}{J(T_{\rm bg})+T_{\rm mb}}\right],$$ (4)

where $T_{0}=h\nu/k$, $T_{\rm bg}$ is the background temperature, which we take to be 2.73 K, and J(T) is given by $J(T)=(h\nu/k)[{\rm exp}(h\nu/kT)-1]^{-1}$. For densities typical of these dark clouds, at least in the dense cores, the CO is expected to be thermalized by collisions at the gas kinetic temperature, so that the excitation temperature should be close to the kinetic temperature of the gas, $T_{\rm ex} \simeq T_{\rm k}$. By averaging all positions in each cloud, we find mean excitation temperatures of 16.8 K and 17.7 K for GF 17 and GF 20, respectively.

From the ¹³CO observations, we can obtain the column densities of the cloud with the usual assumptions that (1) CO is optically thick, (2) ¹³CO is optically thin, (3) the CO and ¹³CO transitions have the same excitation temperatures, and (4) the same beam filling factor. The optical depth of the ¹³CO line can be obtained by using the measured temperatures of CO and ¹³CO at each position toward the cloud:

$$% \frac{T_{\rm mb}({\rm CO})}{T_{\rm mb}(^{13}{\rm CO})}\approx \frac{[1-{\rm exp}(-X\tau_{13})]}{[1-{\rm exp}(-\tau_{13})]},$$ (5)

where $\tau_{13}$ is the optical depth of the ¹³CO transition, and X is the isotopic abundance ratio of the two species, which we assume to be about 89 (Gahm et al. 1993). We find that CO is optically very thick across the entire mapped regions, with $\tau_{12} \sim 90$ near the densest regions of GF 17 and GF 20.

Combining $\tau_{13}$ and $T_{\rm ex}$, the LTE column density of ¹³CO can be estimated as (e.g. Bourke et al. 1997)

$$% N(^{13}{\rm CO})=2.42\times10^{14}\frac{T_{\rm ex}+0.88}{1-{\rm exp}(-5.29/T_{\rm ex})}\int{\tau_{13}{\rm d}v},$$ (6)

where v is measured in kms^-1. From this, we obtain H₂ column densities in GF 17 and GF 20 by assuming an [H₂/¹³CO] ratio of $7\times10^{5}$ (Frerking et al. 1982).

We derive an estimate of the cloud mass from the ¹³CO observations by integrating Eq. (6) over the solid angle subtended by the source. For the areas covered by our ¹³CO maps, we find gas masses of $36~M_{\odot}$ and $31~M_{\odot}$ for GF 17 and GF 20, respectively, assuming a distance of 150 pc to the Lupus complex. In GF 17, our ¹³CO maps are limited to a region which is much smaller than the full extent of the cloud. Assuming similar conditions outside the areas covered in this study, the total mass of GF 17 may amount to ${\sim}200~M_{\odot}$, in good agreement with the estimate by Andreazza & Vilas-Boas (1996), and we regard this as an upper limit since the density is seen to drop significantly towards the cloud boundaries. On the other hand, GF 20 (which was almost completely mapped) has very low mass, a factor of ${\sim} 3$ smaller than the previous estimate by Tachihara et al. (1996).