In order to relate the diversity of IR color indices observed among S stars to
the underlying physical parameters, synthetic IR color indices of a star embedded
in a circumstellar shell have been computed for various input parameters. In our
simple model, the star is assumed to radiate as a
black body at a temperature 
. The mass-losing star is surrounded by a
spherically-symmetric dust shell extending from 
to 
,
with density decreasing as r-2, r being the distance from the central
star. This is equivalent to assuming a steady mass-loss rate
at constant
outflow speed. The inner radius of the dust shell must be larger than or
equal to the radius where grains start condensing
(i.e., to the radius where the shell temperature drops below 1300 K for silicates,
or 1500 K for graphite and amorphous carbon).
The shell outer radius is chosen such that 
, which
ensures that the color indices of the shell have reached an asymptotic value.
A roughly logarithmic radial mesh is defined in the dust envelope so that each
shell is optically thin. In each shell, the grains are assumed to be in thermal
equilibrium, so that the energy absorbed by the grains exactly balances the energy
re-emitted. At the inner boundary, the radiation field is that
of a black body of temperature . The model IRAS flux densities are
calculated from the emergent spectrum by convolving it with the IRAS
filter bandpass (IRAS Explanatory Supplement, 1988).
The main shortcoming of the
code is the neglect of the scattering contribution, since only absorption
is taken into account.
Three types of dust grains have been considered: silicates (with a specific mass
of 3.5 g cm-3), graphite (with a specific mass of 2.25 g cm-3) and
amorphous carbon (with a specific mass of 1.85 g cm-3).
The grain radius is 0.2 
m in all cases. The absorption
coefficients as a function of wavelength are taken from
Draine &
Lee (1984) and Draine (1985) for silicates and graphite. For amorphous
carbon, the absorption coefficients have been generated with the
usual Mie formulae using the
optical constants provided by Rouleau & Martin
(1991).
At wavelengths 
m, the spectral index of the emissivity
coefficient has been taken equal to -2 for graphite grains, and to -1.5 for
amorphous-carbon and silicate grains (Ivezi
& Elitzur 1995).
The results of this model
are presented in Fig. 13 (click here) for dust shells made of either
silicate grains, graphite grains or amorphous carbon grains.
Constant dust mass loss rates
of 10-12, 10-10, 10-9, 10-8 and 10-7 
y-1 with a wind
velocity of 14 km s
have been adopted. The shell inner radius is set by the
grain-condensation temperature, so that the dust shells
computed in Fig. 13 (click here) are
not "detached'' (in the sense of Willems & de Jong 1988). The central
star has been assigned effective
temperatures of 4000 K
(solid line) and 3000 K (dashed line), and a luminosity L =
5000 
; these parameters turn out to have little impact on the
shell colors.
In the ([12] - [25], [25] - [60]) and (K - [12], [25] - [60]) diagrams, the silicate track is
quite distinct
from the graphite or amorphous-carbon tracks.
The graphite and amorphous-carbon tracks go directly from Region I to upper VII
(i.e., from A to D and upper C),
whereas the silicate track goes from I to II and IIIa (i.e., from A to
B and lower C).
These differences observed in Fig. 13 (click here) between carbon- and
oxygen-rich shells must be related to the different emissivities of
silicate and carbonaceous grains in
the IRAS bands, as discussed by Ivezi & Elitzur (1995).
The tracks for carbonaceous and silicate grains predicted by this
simple model outline the segregation observed in the
color-color diagrams between S stars with an oxygen-rich shell, as indicated
by the 9.7 m silicate feature (IRAS LRS class E),
and S stars with
featureless IR spectra (IRAS LRS class S - "stellar'' - or F
- "featureless''). It is important to note here that featureless
spectra are indeed predicted for carbonaceous grains
(Ivezi & Elitzur 1995).
The data on maser emission collected in Sect. 3 (click here) are compatible with this segregation. Those S stars which are SiO or OH masers (Table 3 (click here)) must have an oxygen-rich circumstellar environment, and lie indeed close to the silicate track (Fig. 12 (click here)). On the contrary, no SiO, OH or H2O maser emission has been detected for S stars in Regions D and upper C, along the tracks corresponding to carbonaceous grains. Those S stars might therefore possibly be surrounded by C-rich circumstellar shells, especially since they occupy the same region of the color-color diagram as the optical carbon stars (Chan & Kwok 1988).
However, the above picture is not totally satisfactory, since
(i) the S stars with silicate emission are actually located
in between
the silicate and carbonaceous tracks, (ii) many stars in Regions D
and E have large 60 m excesses that cannot be reproduced by the
tracks displayed in Fig. 13 (click here), (iii) at least one star
(R Gem) moves from the region of silicate dust (lower C) into the
region of carbonaceous dust (upper C) during its variability cycle,
and (iv) the photosphere of S stars is oxygen-rich, so that their
circumstellar shell may be expected to be oxygen-rich as well.
The first two mismatches are in fact not specific to S stars, and
possible solutions were already suggested by
Ivezi & Elitzur (1995). They include either (a)
invoking a mixture of
silicate and carbonaceous grains (see however the discussion of
Sect. 3 (click here) on maser emission), (b) decreasing the
spectral index of the
emissivity at long wavelengths to values smaller than -1.5,
or (c) considering detached envelopes in the sense advocated by
Willems & de Jong (1988).
Figure 13:
Left panel: The ([12] - [25], [25] - [60]) colors predicted for
circumstellar shells (with ) made
of silicate grains, graphite grains or amorphous-carbon grains,
surrounding a star with L = 5000 and = 4000 K (solid lines) or
3000 K (dashed lines).
The diamonds along the curves correspond to dust mass loss rates of
10-12, 10-10, 10-9, 10-8 and
10-7 y-1 (from left to right), with a wind velocity of 14 km s.
Black bodies fall along the central straight line labelled BB.
Right panel: same as left for (K - [12], [25] - [60])
As discussed in Sect. 2.4 (click here),
the S stars located in Regions D and E have very specific
properties that may help to identify the origin of their large 60 m
excess. First, the
prototypical SC stars GP Ori and FU Mon located in Region E
have a C/O ratio equal to unity to within 1% (Dominy et al. 1986).
Similarly, TT Cen is a rare CS star which
exhibits ZrO bands at some times and C2 bands at other times
(Stephenson 1973). According to Stephenson (1973),
these variations are probably
caused by temperature changes in an atmosphere with a C/O ratio very
close to unity, or perhaps even to a secular change in the atmospheric C/O
ratio.
Most of the stars in Regions D and E are actually SC
stars or at least S stars with a large C/O spectral index.
Finally, several stars in Region E have very
extended shells that are resolved at 60 m and, in one case
(RZ Sgr), visible in the optical (Whitelock 1994).
All these properties point towards these stars being in a very
rare and short-lived evolutionary phase.
Based on this evidence, we suggest that the large 60 m excess of
SC stars populating Regions D and E
finds a natural explanation in the much debated concept of
interrupted mass loss first proposed by Willems & de Jong (1988).
The cessation of mass loss when C/O gets close
to unity (because all C and O atoms are then locked into the CO molecule
instead of being involved in dust-forming molecules)
causes the dust shell to detach from its parent star, and to cool down
as it expands into the interstellar medium without being replenished
at its inner side. As shown by Willems & de Jong (1988) and
Chan &
Kwok (1988), the colors of the dust shell then describe a
counter-clockwise loop in the color-color diagram, starting from the
region of stars with silicate emission and ending close to the photospheric
point (Region A) when the shell has dissolved into the interstellar
medium, after passing through Region E. The mass loss resumes when
C/O reaches values above
unity, and the star then enters the region of heavily-obscured
infrared carbon stars (Chan & Kwok 1988; lower VII in
Fig. 13 (click here)). The SC stars found in that region (RZ Peg, UY
Cen and BH Cru, a sister case of TT Cen uncovered by Lloyd Evans 1985)
may actually be on the lower part of that loop.
The application of the Willems & de Jong scenario (implying that the
loop described in the color-color diagram corresponds to a brief
evolutionary phase) to the
rare, supposedly short-lived SC phase would thus not face
the difficulty of inconsistent time scales
generally used to argue against it (e.g., Zuckerman & Maddalena 1989).
Further support for this idea comes from
the peculiar CO line profiles observed for many of the stars
populating Region E (FU Mon: see
Fig. 15 (click here) below; DK Vul, RZ Sgr, TT Cen and UY Cen:
Sahai
& Liechti 1995), as discussed in
Sect. 6.3 (click here).