Please Note: the e-mail address(es) and any external links in this paper were correct when it was written in 1995, but may no longer be valid.
Laboratoire de Radioastronomie Millimétrique, Ecole normale supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, FRANCE
The classical star count method consists of comparing the number of stars per unit area counted towards the cloud and the number in a nearby reference field free from extinction (see e.g. Dickman (1978)) to obtain the extinction, , due to the dust contained by the cloud. It is based on two assumptions:
The extinction is then provided by the expression:
where and are the surface densities of stars counted towards the cloud and the reference field respectively. This quantity, , is a good estimate of the extinction in the case of an homogeneous cloud, but if is not distributed uniformly over the cell it is no longer related in a simple way to the amount of dust (in particular, it can be very different from the average over the cell depending on the degree of clumpiness).
As shown in the following this method can be generalized to investigate the structure of the dust distribution by using the full magnitude distribution of the stars seen through the cloud.
In order to illustrate the information concerning the density structure of the foreground cloud contained in the magnitude distribution, let us compute the magnitude distribution of stars observed towards the cloud (). If the obscuring cloud is close to us, the number of stars in front of it can be neglected and is related to , the distribution of reference stars, through
where is the probability distribution of values over the cloud. For the above relation to be valid, (which directly involves the structure of the cloud) has to be the same over the region studied (i.e. the assumption of a constant in the classical analysis is replaced by an assumption of statistical uniformity). For the sake of simplicity, two extreme models are considered:
Figure 1: Simulated magnitude distributions seen in a reference field of 1766 stars towards (, ) (full line) and towards: 1) a clumpy cloud (model C), consisting of opaque clumps covering 76% of the cloud surface (filled triangles) and: 2) a homogeneous cloud, (model H) of extinction (filled squares).
Note that for both cases, images of the two stellar fields corresponding to each of these two models would be indistinguishable. If a small magnitude range is considered, their magnitude distributions would have the same form in the plane. However, real distributions are curved when considered over a broad enough magnitude range. As a consequence, magnitude distributions can be used to discriminate between models H and C, as shown in Figure 1, where results are represented from a simulation based on the galactic model of Robin & Crézé (1987).
Figure 2: Observed magnitude distributions in the -band towards: 1) the cloud (, ) (filled squares) and 2) a reference field (full line). The distribution for a clumpy model, consisting of opaque clumps covering 76% of the cloud surface is also represented (filled triangles).
To apply the above method to real clouds, we have performed photometry of a low galactic latitude cloud (, ) and also of a reference field (at ) using the 1.2m telescope at the Observatoire de Haute Provence (OHP) and the 2m telescope at the Pic du Midi observatory. The surface area covered per field is about . In Figure 2, the observed magnitude distributions for the -band are displayed. One can easily check that the distribution obtained for the cloud field can be derived from the reference distribution by just an horizontal shift (this corresponds to a uniform visual extinction of 1.7 mag). On the other hand, one can easily verify that for any value of the surface filling factor , model C described above cannot provide an acceptable fit to the observed distribution (see Figure 2, where the distribution for model C is represented for only one value of ; we suggest the reader makes a copy of Figure 2 on a transparency and superimposes the shifted reference field distribution on the cloud distribution).
This result shows that the studied cloud is more close to an homogeneous slab of dust than to a clumpy cloud made of small optically thick clumps. Furthermore, this test was performed in three other bands () and the same behaviour of the magnitude distribution was found. The extinctions obtained in the four bands are consistent with the average interstellar extinction curve ( in the visible range, cf Savage & Mathis (1979)): this is an additional test which implies that the dust extinction properties are also uniform. In summary, these first results indicate that the dust distribution is rather homogeneous contrary to that expected from a uniform dust to gas ratio assumption. Of course these results are preliminary and several points still have to be studied:
Finally, we have recently complemented this study by spectroscopic observations of a few stars seen located behind this cloud in order to measure the extinction towards these stars, and also to search for small scale variations of interstellar features like diffuse interstellar bands or NaI lines.
As discussed above, background stars can be used efficiently to probe the global statistical properties of the dust distribution. However, local information is lost in this method. To compare the spatial distributions of the dust and gas, one really needs to map the extinction at a resolution similar to that of CO or HI observations. This can be achieved if an extended source is present behind the cloud studied, like HII regions or galaxies, which provide a smooth background. One example is provided by the dark fragments seen in front of HII regions, such as the elephant trunk globules or tear drops in the Rosette nebula (Schneps et al. (1980)).
Figure 3: -band image of a high latitude cloud in ursa major with an anonymous background galaxy. The field size is about and the major axis of the galaxy is about . Note that on this image also appears some extended emission which is due to stellar radiation from the Galactic plane backscattered by the HLC.
Figure 4: Brightness profile (-band) along the major axis () of an edge-on galaxy seen through a CO rich cirrus. Within the uncertainties (photon noise on the original CCD images: vertical bars), the curve is perfectly symmetrical.
Figure 5: Cut in the emission line along the major axis of the galaxy (beam size ).
Individual galaxies can also be used although they provide a background which obviously cannot be considered as uniform. In favourable conditions, one may however be able from broad band imaging to model the intrinsic spatial variations of the background. This can be achieved for some E or SO galaxies which provide a smooth light distribution devoid of small scale variations. Another interesting case is edge-on spiral galaxies which are so distant (angular size of about one arcminute) that their internal structure is washed out at the arcsecond resolution. It is true that it may be difficult to determine whether observed features in the light distribution are intrinsic or due to the foreground cloud structure, but at least, it is easy to set upper limits on the amplitude of small scale extinction variations. Indeed, if the brightness profile is seen to be symmetric around the galaxy center, this is certainly an intrinsic property of the galaxy (the chance that for instance an intrinsic brightness excess - due to spiral arms for instance - is just compensated by a coincident excess in the extinction being extremely unlikely). We present here results about an anonymous galaxy (, , B1950.0) found by chance towards a high latitude cloud in Ursa Major. For this galaxy, photometry (Figure 3) at OHP and CO line observations with the IRAM 30m telescope have been performed. The luminosity profile in the -band along the major axis, displayed in Figure 4 shows no deviations from symmetry to a very high accuracy, for which we derive an upper limit mag. On the other hand in Figure 5, the cut in the line along the same axis shows a large variation of the line strengths of about a factor 2. This indicates that the amount of molecular gas is not uniform over this area. In this sort of transluscent cloud mag, so CO is very sensitive to small variations of physical conditions (see Stark (1994)), but it is not clear whether such effects are strong enough to account for such large variations. Then, there is no evidence for fluctuations in the dust distribution in front of the galaxy at scales between 2 and 40 arcsec, while CO observations reveal a clear gradient in the amount of molecular gas. If CO is a good tracer of molecular gas in this cloud (for which from de Vries et al. (1987)), the dust-to-gas ratio should then not be uniform at these small scales.
The dust distribution inside the studied molecular clouds looks quite homogeneous: we could find no evidence for opaque clumps contrary to what is observed in the gas distribution. These results give rise to new questions: why are the spatial distribution different for the gas and the dust, and which processes are responsible for the structure in the gas and dust distributions?