Astron. Astrophys. 361, 555-580 (2000)

3. Results and analysis

3.1. ISOCAM images

Out of the 24 dark cloud fields in our programme, 23 show absorption-like features in the mid-IR.

The fields presenting such features can be divided into two broad categories, those that show deep ( 10% - cf. Column 9 of Table 1), compact absorption features likely associated with dense cores, and those presenting weaker ( 10%) and more diffuse absorption whose nature is also more uncertain. Fig. 1 presents the ISOCAM images of 11 fields with strong absorption features, to which we added the widely studied core L1544 (e.g. Ward-Thompson et al. 1994; Tafalla et al. 1998; Williams et al. 1999; Ohashi et al. 1999), and from which we withdrew L1709C (the absorption features lying in the corners of the image, making the analysis difficult). The remaining fields are presented in Fig. 2. Fig. 3 shows examples of cuts through two of the deep absorption features (for L328 and L1689B). The morphologies encountered in the fields with compact absorption vary from spherical cores like L328 to more elongated, filamentary structures (e.g. L1689B, GF5).

Fig. 2a-f. ISOCAM images of the remaining cores. The fainter absorption features led us to enhance the contrast of the images, which resulted in poorer image appearance (the flatfield noise is clearly visible in some cases). Contour levels are: a for L1517B, 4.3 MJy/sr; b for L1512, 12.2 MJy/sr; c for L1672, 3.5, 4 MJy/sr; d for BHR78, 15, 16, 17; e for BHR111, 17, 18, 19 MJy/sr; f for R7, 20, 25, 30, 35 MJy/sr.

Fig. 2g-l. Contour levels: g for R60, 9.1, 9.3 MJy/sr; h for L1709C, 6.9, 7.4 MJy/sr; i for L204B, 12.2 MJy/sr; j for B68, 5.4, 5, 6 MJy/sr; k for BHR140, 9, 9.5 MJy/sr; l for B133, 7.4 MJy/sr.

Fig. 3. a North-South intensity cut through the  µm ISOCAM image of L328 (see Fig. 1), at b North-South intensity cut through the  µm ISOCAM image of L1689B at Both cloud cores clearly show up in absorption against the diffuse mid-infrared background.

3.2. Comparison with the millimeter continuum maps

All of the cores observed at 1.3 mm (Table 2) present extended, weak dust continuum emission which we were able to map. For all cores, the peak of the mid-IR absorption does correspond with a peak in the millimeter emission. This is illustrated in Fig. 4 which shows the mid-IR ISOCAM absorption contours superimposed on the 1.3 mm continuum maps of the eight cores positively detected at 1.3 mm (the maps of Oph D and L1544 are also shown in MAN98 and WMA99, respectively).

Fig. 4a-f. IRAM 1.3 mm continuum emission maps smoothed to 13" FWHM resolution (contours) superimposed on ISOCAM 7 µm absorption image (greyscale). Contour levels are: a 20, 40, 60 mJy/beam for L1544; b 30, 50, 70 mJy/beam for Oph D; c 20, 40, 60 mJy/beam for L1709A; d 10, 30, 50 mJy/beam for L1689B; e 20, 40, 60 mJy/beam for L310; f 20, 40 mJy/beam for L328.

Fig. 4g and h. Contour levels are: g 25, 50, 75, 100 mJy/beam for L429; h 10, 20, 30, 40 mJy/beam for GF5.

The good agreement between the respective images of these cores in both types of data confirms that the features we see with ISOCAM are due to absorption by the (rather large) column density of cold dust traced at 1.3 mm, and not by fluctuations in the mid-IR background. Furthermore, the similar morphologies seen in both wavebands suggest that the mid-IR absorption images trace the same dust as the millimeter emission maps, which justifies a detailed comparison of the column density profiles derived from both tracers.

3.3. Modelling of the mid-infrared absorption

In order to derive information about the column density structure from the mid-IR intensity, we have used a simple geometrical model: each dense core is embedded within a lower-density parent molecular cloud. On the line of sight to the core, we measure i) a background intensity arising from the rear side of the parent cloud that is attenuated by the absorption from the core, and ii) a foreground intensity arising from the front side of the parent cloud. We can thus express the mid-IR intensity along the line of sight as:

where and are the background and foreground intensities respectively, and is the dust opacity at a projected radius from core centre and at observed wavelength . The foreground emission, , includes the contribution from the zodiacal light emission, .

The mid-infrared background emission, , is thought to arise from very small grains presenting aromatic emission features (Léger & Puget 1984; Allamandola et al. 1989; Boulanger et al. 1996). These grains/molecules are excited by the interstellar far-UV (FUV) radiation field and re-emit this energy as a group of spectral lines/bands in the mid-infrared, the Unidentified Infrared Bands (UIBs). The ISOCAM LW2=[5.0, 8.5 µm] and LW6=[7.0, 8.5 µm] filters that we used for our observations include the 6.2 µm and 7.7 µm emission features of these UIBs.

According to recent studies, the small particles responsible for the UIBs are distributed throughout molecular clouds but because of high interstellar extinction in the FUV band, they are ususally only excited in the outer regions where (Hollenbach & Tielens 1997; Bernard et al. 1993). For this reason, the interiors of (quiescent) clouds should be relatively free of UIB emission, and only their outer envelopes should significantly contribute to the mid-IR background and foreground (e.g. Bernard et al. 1993; Boulanger et al. 1998).

As a first step, we neglected the spatial fluctuations of the background and foreground, and , and thus assumed and to be equal to their mean values across the field, and (see 3.5.2 below for a critical discussion of this assumption).

For a given gas-to-dust ratio, the H₂ column density is related to the dust opacity by , where is the dust extinction cross section ⁴ at the wavelength of observation . We assume to follow the Draine & Lee (1984) dust model in the mid-IR as a first approximation, but the value is rather uncertain (see Sect. 3.5). We adopted cm² for the LW2 filter, cm² for the LW6 filter, and cm² for the LW9 filter, i.e., we calculated at the central wavelength of each filter. A more accurate value could be obtained by integrating the extinction curve over the response of the ISOCAM filters, but as we will see in the following, the exact value of has only little influence on the derived column density profiles.

The quantities and are unknown, but they can be constrained using independent measurements of the H₂ column density at two different radii. These constraints enable the ISOCAM absorption profile to be converted into a column density profile.

3.3.1. Central column density from millimeter continuum data

We used our 1.3 mm continuum maps to estimate the column density averaged over the flat central region of each core. As millimeter dust continuum emission is optically thin, the column density averaged over the flat inner part (of radius ) of the core (see WMA99) may be derived from the 1.3 mm flux density integrated over the same area, (cf. AWM96, MAN98):

where is the solid angle of the flat inner region, is the dust opacity per unit mass column density at  = 1.3 mm, is the mean molecular weight, and is the Planck function at  = 1.3 mm, for a dust temperature T. We assumed a single, representative dust temperature  K for all the cores. Based on recent ISOPHOT measurements of L1544, Oph-D, L1689B, and other similar starless cores (e.g. Ward-Thompson & André 1999; Lehtinen et al. 1998), we believe that this value of is likely to be within  K of the true dust temperature in most cases. Considering the additional uncertainty on the millimeter dust mass opacity in dense cores, which we took equal to  cm²g^-1 (see, e.g., Preibisch et al. 1993; AWM96; and Kramer et al. 1999), the estimates of listed in Table 2 are uncertain by a factor of on either side of their nominal values.

3.3.2. Outer column density from millimeter line data

The second value of the column density we can use to calibrate the ISOCAM column density profile is given by our C¹⁸O(1-0) line observations of the outer parts of the cores. Assuming optically thin emission and Local Thermodynamical Equilibrium (LTE), we can determine the value of the C¹⁸O column density from the following equation:

where is the measured antenna temperature of the C¹⁸O(1-0) line, is the gas kinetic temperature in the cloud, and dv is in km s^-1 (e.g. Rohlfs & Wilson 1996). We used Eq. (2) with  K to estimate the C¹⁸O column density in the outer parts of the core. This method would not be reliable towards the centre of the core where C¹⁸O(1-0) may be optically thick. Moreover at the centre, the possibility of depletion of gas molecules onto grains makes the estimation of the column density unreliable (e.g. Kramer et al. 1999). We adopted the following relation between H₂ and C¹⁸O column densities (Frerking et al. 1982):  cm^-2. The values of the column density derived in this way, as well as the distance from the core centre at which it was evaluated, are given in Table 2. The exact position where was measured is indicated by a triangle-symbol on the images of Fig. 1. The uncertainty on the H₂ column density derived from molecular line data has two major origins: first, the excitation state of the C¹⁸O(1-0) molecules is not accurately known and the LTE hypothesis is only an approximation (cf. White et al. 1995), and second, the C¹⁸O abundance ratio with respect to H₂ is also uncertain in our cores. Based on comparisons with more sophisticated calculations performed with a Monte-Carlo radiative transfer code (Blinder 1997) for a realistic pre-stellar core model, we judge that our LTE estimate of the column density in the outer parts of each core, , is good to better than a factor of .

3.3.3. Estimates of the background and foreground intensities

Given the values of and derived from millimeter observations, it is easy to estimate and using Eq (1) for and , and then deduce the column density from the mid-IR intensity at any position in the ISOCAM image. The values of and that we calculated in this way, using the independent constraints that and , are listed in Table 2. In addition to `best' values, we give permitted ranges for and that were determined using the maximum and minimum values of and (given the factor uncertainty on both these quantities).

Another approach consists in estimating the values of and directly from the ISOCAM images. The sum is constrained by the mean mid-IR intensity measured in the map outside the dense core (cf. column 7 of Table 1; this also corresponds to the "baseline level" measured on intensity cuts such as those of Fig. 3). Furthermore, one can set straightforward lower and upper limits on the foreground intensity since has to be less than the value of the minimum intensity in the image, , and greater than the zodiacal emission, . Hence, and, since  ,  .

This second method assumes that can be measured outside the dense core, which is not necessarily the case if, e.g., the core extends beyond the limits of the ISOCAM map. Moreover, according to Eq. (1), the minimum intensity in the image is equal to the intensity at the maximum of the absorption (), which in practice can greatly overestimate the value of since one typically has (see below). For these reasons, we preferred to adopt the first approach based on millimeter measurements of the central and outer column densities (see above).

In the case of the Oph D dense core, however, it is possible to determine accurate values of and at more than one wavelength using both methods, since large-scale () ISOCAM images of the Oph main cloud are available (cf. Abergel et al. 1996). This provides an important consistency check. At 6.75 µm, we obtain  MJy/sr and  MJy/sr using millimeter observations (see Table 2), while we find 4.6 MJy/sr  MJy/sr and 6.3 MJy/sr  MJy/sr using the second method. The agreement is thus very good. The background and foreground intensities can also be estimated at  µm (cf. Table 2). As can be seen in Fig. 5c) and Fig. 5d) below, they yield a column density profile for Oph D which is virtually identical to that derived at 6.75 µm. Furthermore, our estimated values ( MJy/sr and  MJy/sr) are consistent with the idea that the mid-IR background is primarily due to the emission of UIB carriers while the foreground is dominated by the zodiacal light. Indeed, the emission spectrum of aromatic particles is such that the ratio of the LW2-filter to LW3-filter intensities is (Boulanger et al. 1996), which is precisely what we find for the background emission in Oph D. By contrast, the 7-15 µm spectrum of the foreground emission is rising and compatible with that of the zodiacal light ( MJy/sr at  µm and  MJy/sr at  µm).

The results of these inter-comparisons on Oph D confirm the validity of the millimeter method we used to determine and for the other cores.

We note that for many sources in Table 2, suggesting that the zodiacal emission is often the main contributor to the foreground. This situation is expected if most of the target clouds are illuminated from behind, which is indeed believed to be the case for the  Oph cloud, located in front of the Sco OB2 association (e.g. de Geus et al. 1989; de Geus 1992). As most of our fields were chosen to be nearby and devoid of strong (e.g. foreground) infrared sources, the fact that the front sides of the clouds appear to contribute negligible foreground emission is likely the result of a selection effect.

3.4. Radial column density profiles

3.4.1. Derivation of the profiles

In order to characterize the radial structure of each core and reduce the influence of spatial fluctuations in the background and foreground, we first derived mean radial intensity profiles by averaging the mid-IR emission in the ISOCAM images according to the apparent morphology of the core. For L1689B, we averaged the intensity over elliptical annuli of increasing radii separately for a 40^o sector in the South and a 40^o sector in the Western side of the core. For L328, Oph D and L429, the intensity was averaged over circular annuli of increasing radius. Only the South-Western sector of the profile was considered for Oph D and only the Western half for L328. For very elongated, filamentary structures (L310, L1709A, L1544, GF5), we averaged cuts perpendicular to the main axis of the cores, but only kept the Southern half of the absorption intensity profile (except GF5 for which the North side was kept) ⁵. The sectors over which the profiles were averaged (for the elliptically/spherically symmetric cores L1689B, L328 and L429) are shown as dotted lines on Fig. 1 as are the ranges of cuts which were averaged for the filamentary cores (L1544, L1709A, L310, GF5).

The radial column density profiles that were derived in this way using our `best' values for and (see above) are displayed in Fig. 5 as crosses. The vertical cross sizes represent statistical error bars on the mean radial profile, estimated from the dispersion of the values averaged at each radius, divided by the square root of the number of values. In addition to these random errors at each radius, there is also a global uncertainty on the column density profiles resulting from our uncertain knowledge of and . The dashed curves lying above and below the `best' profiles limit the range of column density profiles that are consistent with the mid-IR absorption data, given the permitted range of values for and in each case (see Table 2). The upper dashed curve corresponds to the profile for the smallest value of and the greatest value of , while the lower dashed curve corresponds to the greatest value of and the smallest value of . It is important to stress that there is a simple, global relation between each permitted profile, , and the `best' profile, , of the type:

where A and B are two parameters depending only on and , and is again the extinction cross-section at the mid-IR wavelength .

Fig. 5a-f. Column density profiles of the target pre-stellar cores (crosses), as well as global error bars (dashed curves), derived from the ISOCAM images and adjusted so as to match the values obtained in the flat central region with 1.3 mm continuum data and in the outer region with C18O(1-0) line observations (see Sect. 3.3). Theoretical models are plotted for comparison (see Sect. 4).

Fig. 5g-j. The dotted line is an power-law (corresponding to the profile of a singular isothermal sphere at T=10 K), convolved with the ISOCAM psf. The thin solid line corresponds to the column density profile of a finite-sized sphere model with uniform density for , for , and surrounded by a medium of uniform column density.

For comparison with the profiles derived from the ISOCAM images, the thin solid curves in Fig. 5 represent the column density profiles of finite-sized sphere models with uniform density for , for , and surrounded by a medium of uniform column density. The dotted curves plotted in Fig. 5 are power laws (which correspond to the column density profile of a singular isothermal sphere at 10 K). These models as well as other theoretical models will be discussed in Sect. 4. (Note that the convolution with the ISOCAM psf has no effect on the models beyond an angular radius of  6".)

3.4.2. Results

It can be seen from Fig. 5 that all the column density profiles present a flattening in their inner regions - though this flattening is only marginal in the case of L429. On the other hand, the slopes and shapes of the outer parts of the profiles appear to vary from core to core and may be separated into two broad categories. For four of the cores (GF5, L429, L1709A and the Western part of L1689B), the profile beyond the flat inner part follows a near power law with projected radius , i.e. , and does not show any clear steepening up to the boundary of the image. By contrast, for the other cores (L328, Oph D, L310, L1544 and the Southern side of L1689B), the profile steepens beyond some radius , until it merges with the ambient molecular cloud. Three cores in our sample present "edges", defined as regions where the column density profile drops significantly steeper than a power law. Globally, we may divide each column density profile into four parts: a flat inner region of radius , followed by a near power-law portion of index up to a radius , a region with a steep slope from to that marks the core "edge", and finally a relatively flat plateau once the column density has reached the average value it has in the ambient molecular cloud. Depending on the cases, one or two parts may be absent.

The values of the core opacities at 7 µm, the column densities, as well as the masses, average slopes, and density contrast deduced from the ISOCAM profiles of Fig. 5 are given in Table 3. The peak column density, , was read on the profiles shown in Fig. 5 (in most cases, because the central plateau is not strictly flat). The average slopes and of the column density profile between and , and between and , respectively, were derived by fitting segments of straight lines to the data. The maximum values of and were obtained by assuming the maximum possible value of together with the minimum possible value of (taking into account the factor of  2 uncertainty), while the minimum values of and were obtained by assuming the minimum possible value of together with the maximum possible value of .

Table 3. Core parameters derived from the ISOCAM absorption analysis.
Notes:
(1) Column density averaged over the flat inner part of the core of radius .
(2) Peak column density read from the ISOCAM profiles of Fig. 5.
(3) Opacity at the centre of the core at  7 µm.
(4) is the radius of the flat inner part of the ISOCAM profiles of Fig. 5 and is slightly different from (cf. Table 2) for L1544, Oph D and L1689B.
(6) Radius of the core edge (when there is one).
(7) Average density over the flat inner core region, estimated from a piecewise power-law model fitting the profile (see text).
(8) Integrated core mass within a circular area of radius .
(9) Integrated core mass within a circular area of radius .
(10) Index of the power-law fitting the column density profile between and (Column 4).
(11) Index of the power-law fitting the column density profile between (Column 4) and .
(12) Density contrast between the centre and the edge of the core. For Oph D, L1689B and L1709A, the value of was taken from the large scale ¹³CO observations of Loren (1989a), and for L1544, it comes from Butner et al. (1995). As no estimate of is available for the remaining cores in the literature, a lower limit for the density contrast was estimated from the model described in Sect. 4.1.

3.5. Uncertainties, difficulties, limitations

3.5.1. Dust extinction curve in the mid-IR

Among the parameters involved in the relationship between the mid-IR intensity and the H₂ column density is the dust extinction coefficient . This paragraph focuses on the influence of on the derived column density profiles. The uncertainty on stems from two main factors. First, the ISOCAM filters encompass a certain bandwidth over which the value of varies. In our study, we used for the value for the wavelength at the centre of the filter (ie at 6.75 µm for the LW2 filter and 7.75 µm for the LW6 filter). Considering the rapid variations (steep increase) of towards 8 µm (eg. Draine & Lee 1984), the values we adopted in Sect. 3.3 above could be underestimated by up to a factor of  2. Second, the dust properties in dense cores are known to depart from those calculated by Draine & Lee (1984) for the diffuse interstellar medium (Krügel & Siebenmorgen 1994, see in particular their Fig. 12). In environments like dense cores, in which temperatures can be as low as 10 K and densities as high as - cm^-3, coagulation of particles, fluffiness and the presence of ice mantles on the grains can change the value of the dust extinction coefficient, depending on the nature of the grains (amorphous carbon or coated silicate grains). By using the Draine & Lee value, we could thus be underestimating the value of the extinction coefficient by a factor of  2.

To check how the column density profile was affected by the value of , we calculated it for different values of . As the innermost parts of the core are denser and more shielded from the interstellar radiation field (and thus colder), they are more likely to be affected by grain coagulation and forming of ice mantles, resulting in a higher extinction coefficient than in the low-density outer parts. We therefore determined the column density profiles supposing a linear increase of from 1.35cm² on the edges of the core to 2.7cm² (which is twice the previous value) at the centre of the core. We also varied the value of by factors of 2 and 3 to check the influence of a high value of on the shapes of the profiles.

The results of this study show that if varies within a factor of  2-3 the column density profiles change only marginally. The same holds true if we assume a linear gradient in or in Log() from the centre to the edge of the cloud (so that in the end varies within a factor of 2). To summarize, our conclusions regarding the slopes of the column density profiles are not sensitive to the existing uncertainties on .

3.5.2. Simulations of non uniform background and foreground intensities

To check the influence of our assumption that the foreground and background intensities are spatially uniform, we carried out simulations of the total infrared intensity measured by the camera (and of the column density inferred), if the absorbing dense core is embedded in an ambient cloud of uniform column density, itself bounded by a mid-IR emitting shell (cf. Fig. 6 - left panel). We considered a spherically symmetric core model with uniform density for , for (corresponding to a column density profile), and uniform temperature  K throughout. (This is similar to the simple model compared to the data in Fig. 5 and discussed in Sect. 4.1 below.) The parent cloud (corresponding to radii ) is also assumed to be at  K. By contrast, the outer shell (beyond ) is taken to be much warmer at  K, so that its maximum emission is in the mid-IR. Thus, we have a thin outer region emitting mid-IR radiation and a cold, dense inner cloud/core only absorbing in the mid-IR. Such an idealized model of emission by an outer cloud in thermal equilibrium at  K is clearly not realistic and only meant to be illustrative of a background medium emitting in the mid-IR. Since it is the thin spherical shell beyond that produces the background and foreground, and are not uniform in this model but present `limb-brightened' peaks at (see Fig. 6 - right panel).

Fig. 6. Schematic representation of the simple model cloud/core system discussed in the text (left panel). The right panel shows the simulated mid-IR intensity profile (solid curve) emerging from such a model. The dashed curve represents what the intensity profile would be without the absorbing dense core/cloud. The illustrative case shown has  pc,  pc, and  pc. Note some similarity with the observed profiles of Fig. 3.

When is significantly smaller than , our simulations show that and vary only slowly over the spatial extent of the absorbing dense core (see the example of Fig. 6). In such a situation, the column density profile derived from the simulated mid-IR intensity of the model in the same way as for the ISOCAM data is almost indistinguishable from the original model profile. On the other hand, if so that the peaks in and occur immediately outside the core, then the profile derived under the assumption that and are spatially uniform departs markedly from the true column density profile as the temperature discontinuity is approached for . In particular, the derived profile drops sharply near , i.e., displays a sharp edge, even when the warm outer shell has the same density profile as the cold inner cloud core (so that there is no physical edge in the density structure but only a discontinuity in temperature). This shows that our method of converting intensities into column densities enables us to detect the existence of a discontinuity, but does not necessarily allow us to conclude about its nature.

In practice, however, it seems reasonable to assume that we are in the situation of Fig. 6 and that and have nearly uniform values over distance scales of several tenths of parsecs. The  Ophiuchi cloud complex may be used as an example. In this case, there is convincing evidence from the IRAS study of Bernard et al. (1993) that the mid-IR background emission originates from photo-excited PAH-like molecules distributed in a sort of shell corresponding to the outer layers of the large-scale ¹³CO molecular cloud mapped by Loren (1989a). This geometry is very reminiscent of Fig. 6 with on the order of a few pc , i.e., much bigger than the rather small ( pc) cores we are studying here (see, e.g., Table 3). This means that the cores are embedded deep inside the layers responsible for the mid-IR emission. Another piece of evidence comes from the ISOCAM images themselves which do not show strong intensity fluctuations on small scales.

© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000
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