,
due to the dust contained by the cloud. It is based on two assumptions:
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Sophie
Thoraval and
Patrick
Boissé
e-mail:
thoraval@ensapa.ens.fr
Laboratoire de Radioastronomie Millimétrique, Ecole normale supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, FRANCE
mag). First, we have
extended the classical ``star count'' method by using the full stellar
magnitude distributions as determined from CCD observations. We thus estimate
the amplitude of extinction variations and investigate the possible presence of
optically thick clumps (as suggested by CO observations). We also consider
extended background sources, like galaxies or HII regions, located behind
clouds which can be used to get some information on the amplitude of the
density fluctuations. We present here some preliminary results from the BVRI
photometric study of a stellar field and galaxies seen through translucent
clouds (
mag).
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:
, where 
is the number density of stars with magnitude less than
.
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:
(hereafter model ``H''), for which
reduces to 
: extinction then implies a horizontal shift
in the 
plane,
of the projected surface, the rest
of the surface being unextinguished. In this case, 
reduces to
and then, the curve for 
is that for
shifted vertically by an amount 
.
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?