\chapter{Results from the molecular radio line modelling}

\section{Gas expansion velocities}

The distribution of gas expansion velocities for the IRV/SRVs, derived
from the modelling of the CO circumstellar emission, has a median
value of 7 km s$^{-1}$, ranging from 2.2 to 14.4 km s$^{-1}$. Hence,
these stars sample the low end in gas expansion velocities of AGB
winds. The CO envelopes around the Miras were found to typically
expand at $\approx$18 km s$^{-1}$, though the Miras sample contained
three sources with intermediate expansion velocities ($\approx$7 km
s$^{-1}$). Thus, the stellar wind has a scaling of $v_{\rm e} \propto
\dot{M}^{0.7}$ at low to intermediate mass loss rates, gradually goes
over into a wind of about 20\,km\,s$^{-1}$, for the higher mass loss
rates.

There is no apparent difference in the gas kinematics between the SRVs
and the IRVs, but the latter group is less represented in the
sample. Modelling of C-rich SRVs and IRVs (Sch\"oier \& Olofsson 2001)
indicates that the gas expansion velocity in this type of stars is
larger (the median value is 9.5 km s$^{-1}$). Furthermore, the number
of stars with very low expansion velocities is larger in O-rich than
in C-rich variables. This might be indicating an interplay between the
chemistry and kinematics of the circumstellar envelopes.

In a few cases the derived SiO velocities are lower than the
velocities derived from the CO emission.Indeed, the SiO line profiles
are narrower in the sense that the main fraction of the emission comes
from a velocity range narrower than twice the expansion velocity
determined from the CO data.  On the other hand, the SiO line profiles
have weak wings so that the total velocity width of its emission is
very similar to that of the CO emission.  This effect becomes less
prominent among the high mass loss rate sources. These
``Gaussian''-like SiO line profiles were noted already by Bujarrabal
et al.  (1986) and they have been interpreted as due to the influence
of gas acceleration in the region which produces most of the SiO line
emission.  A high sensitivity multi-line study combined with
interferometric observations are required to fully tackle this
problem.


\section{Mass loss rates}

The distribution of the derived mass loss rates for the IRV/SRVs has
a median value of 2.0$\times$10$^{-7}$ M$_\odot$ yr$^{-1}$, extending
from 2.0$\times$10$^{-8}$ M$_\odot$ yr$^{-1}$ to 8.0$\times$10$^{-7}$
M$_\odot$ yr$^{-1}$. There is no significant difference between the
IRVs and SRVs, but the number of IRVs is quite low. The values are
similar to those found by Sch\"oier \& Olofsson (2001) for these types
of objects in their sample of C-rich stars, but this conclusion rests
on the uncertain CO fractional abundance assumed in both works.

There is a sharp cut-off at a mass loss rate slightly below 10$^{-6}$
M$_\odot$ yr$^{-1}$ which is very likely not a selection effect. This
upper limit is commonly exceeded by regular pulsators, and thus, it
might be indicating a connection between the mass loss mechanism and
the regularity and amplitude of the pulsation. However, a higher time
variability in the mass loss rates of SRVs and IRVs as compared to that of
Miras has been found by Marengo et al. (2001). This would lead to a,
on the average, lower mass loss rate estimate for the SRV/IRVs.

We found no correlation between mass loss rate and period or effective
temperature. This is also the case found for the C-type IRVs and SRVs
by Sch\"oier \& Olofsson (2001).  The two main characteristics of the
mass loss process, the mass loss rate and the gas expansion velocity,
are found to be correlated in our sample of M-type stars. Such
correlation ($\dot M \propto v_e^{1.4}$) is, however, weaker than for
the C-star IRVs and SRVs ($\dot M \propto v_e^{2.0}$). This result is
in line with theoretical predictions for an optically thin dust-driven
wind, but the outcome for an individual star also appears very
sensitive to the conditions in the region where these properties are
determined.

Among the Mira variables we found that 4 of the Miras have very high
mass loss rates ($>\approx$\,10$^{-5}$\,M$_\odot$/yr), 6 are high mass
loss rate objects ($>\approx$\,10$^{-6}$\,M$_\odot$/yr) and 2 are
intermediate mass loss rate sources
($>\approx$\,10$^{-7}$\,M$_\odot$/yr).

\section{SiO depletion onto dust grains} 

Previous work strongly suggests that the SiO abundance in the CSE is
markedly lower than that in the stellar atmosphere. The decrease in
the SiO abundance with radius is very likely linked to two different
processes taking place in the CSE. Photodissociation due to
interstellar UV radiation is a well-known mechanism which reduces the
abundances of molecules in the extended CSE, but for SiO we must also
take into account the depletion of SiO onto grains closer to the star.

Most likely the radial distribution of the SiO molecule is determined
by a combination of the photodissociation and condensation processes.
Thus, one can imagine an initial SiO abundance determined by the
stellar atmosphere chemistry.  For low mass loss rates, the abundance
decreases only slowly beyond the condensation radius until the
photodissociation effectively destroys all remaining SiO molecules.
For high mass loss rates, the abundance declines exponentially beyond
the condensation radius with an e-folding radius which depends on the
amount of condensation onto dust grains.

The distribution of SiO abundances for the IRV/SRV sample has a median
value of 8$\times$10$^{-6}$, and a minimum of 1$\times$10$^{-6}$ and a
maximum of 5$\times$10$^{-5}$. For the IRVs and SRVs the median
results were 8$\times$10$^{-6}$ and 6$\times$10$^{-6}$, respectively.
For the high mass loss rate ($\dot{M} > 5 \times
10^{-6}$\,M$_{\odot}$\,yr$^{-1}$) Miras we found a distinct division
into a low abundance group (on average 4$\times$10$^{-7}$) and a high
abundance group (on average 7$\times$10$^{-6}$), while the
intermediate mass loss rate ones seemed to follow the trend of the
IRV/SRVs. We could not identify any reason for this bifurcation. In
general, there was a trend of decreasing SiO abundance with increasing
mass loss rate which continues towards high mass loss rates.  This was
interpreted as an effect of increased adsorption of SiO into dust
grains the higher the mass loss rate, i.e., the density of the
envelope. The derived SiO abundances are at least a factor of a few
lower than the maximum value of 7$\times$10$^{-5}$ allowed by solar
abundances.


\section{Peculiar objects}

Four objects in our sample have gas expansion velocities that are
lower than 3 km s$^{-1}$. Such low velocities are surprising since
they translate for a 1M$_\odot$ star into an accelerating region five
times larger than normally accepted. However, Winters et al. (2000)
concluded that for low radiative acceleration efficiencies only low
mass loss raters and low gas expansion velocities are produced. A
scenario involving more complicated geometries/kinematics is indeed
suggested by the broad line wings seen in their SiO line profiles.

The double-component line profiles seen in the CO circumstellar
emission towards four stars in our sample could represent examples of
such complicate scenarios. Such double-component line profiles are
also seen in the SiO circumstellar emission of two of the stars. This
type of sources have been found both among M- and C-type star. In the
modelling we have simply decomposed the emission into two components
for each source, assuming that the emissions are additive. The mass
loss rates derived are higher for the broader component by, on
average, an order of magnitude. The fits to the narrow components
result in very low gas expansion velocities ($\approx$1 km
s$^{-1}$). One possible interpretation is outlined in Kahane \& Jura
(1996) where they report on a bipolar outflow around one of our
sources, X Her. They were able to spatially resolve the blue- and
red-shifted sides of the broad component, and inferred a (weakly
collimated) bipolar outflow, while the low-velocity emission, i.e.,
the narrow component, appears to outline a symmetric envelope. Knapp
et al. (1998), on the other hand, suggested a scenario where episodic
mass-loss plays a role. A similar narrow spike in the middle of a
broader line could also be interpreted in terms of emission from a
long-lived circumstellar disk (Kahane et al. 1998, Bergman et
al. 2000).



\section{Achievements in Papers III \& IV}

The modelling of CO circumstellar radio line emission performed in
{\bf Paper III} yielded terminal gas expansion velocities for the
stars in our sample. These expansion velocities are more accurate than
those derived from direct line profile fits since the effects of,
e.g., the turbulent velocity are taken into account. The model values
are in general lower than those found in the literature.

Despite the uncertainty inherent to the modelling (estimated to be
about $\pm$20\% for those sources with three or more observed
transitions, and $\pm$50\% for the rest), it is expected that the
model work presented in {\bf Paper III} produces the most accurate
mass loss rates for these types of objects so far. However, on an
absolute scale they are uncertain by at least a factor of a few for an
individual object. A comparison with the mass loss rates estimated by
Knapp et al. (1998) for the six stars in common shows differences by
less than a factor of two, except for one source. Kastner (1992)
presented self-consistent model calculations for CO circumstellar
emission that allow them to derive a formula for estimating the mass
loss rate from the CO($J$=1$\rightarrow$0) antenna temperature and the
expansion velocity of the envelope. The simpler analysis of similar
objects performed in Kerschbaum et al. (1996) and Kerschbaum \&
Olofsson (1998) based on this formula results in mass loss rates that
are on average a factor of ten lower for the same distances and CO
abundance. Thus, the results of this work confirm that the formulae by
Knapp \& Morris (1985) and Kastner (1992) relating the gas mass loss
rate with easily determined observables are not appropriate for low
mass loss rate objects. This conclusion was also drawn by Sch\"oier et
al. (2001) from their modelling of CO circumstellar emission in a
sample of C-rich stars. It is not surprising since both formulae were
calibrated against the high mass loss rate star IRC+10216.

An extensive radiative transfer analysis of circumstellar SiO
`thermal' radio line emission from a large sample of M-type AGB
variable stars was performed in {\bf Paper~IV}. Through the modelling
SiO fractional abundances in the CSEs of a large sample of M-type AGB
stars were determined. The analysis of the circumstellar SiO radio
line emission presented in {\bf Paper~IV} is considerably more
uncertain than that of the CO radio line emission presented in {\bf
Paper~III}. This is partly because the SiO line emission predominantly
comes from the inner regions where the observational constraints are
poor, but also partly because the behaviour of the SiO molecule is
more complex, e.g., adsorption onto grains. Furthermore, the size of
the SiO envelope is crucial to the modelling but it could only be
estimated at a first approximation.  Multi-line SiO modelling and
interferometeric SiO line brightness size estimates of a limited
number of sources yielded a relation between the size
of the SiO envelope and the density measure $\dot{M}/v_{\rm e}$.

The derived SiO abundances are in all cases below the abundance
expected from stellar atmosphere chemistry, the median for the total
sample is down by a factor of ten.  This is interpreted in terms of
depletion of SiO molecules by the formation of silicate grains in the
CSEs, with an efficiency which increases with the mass loss rate.  In
addition, there is a trend of decrasing SiO abundance with increasing
mass loss rate. It is also found a difference in shape between the SiO
and CO radio line profiles of an object. In all, the results in {\bf
Paper~IV} strongly suggest that SiO radio line emission can be used as
a sensitive probe of circumstellar dust formation and dynamics.

\section{Outlook}

From the observational point of view, interferometric maps taken in a
small sample of semiregular and irregular M-type variables would help
to better constrain the envelope sizes. In a larger sample multi-line
spectra are desirable in order to be able to uniquely determine the
free parameters in the modelling, i.e., mass loss rates and the dust
properties of the CSEs in the modelling of the CO circumstellar radio
line emission, and the SiO abundances and envelope sizes in the
modelling of the SiO circumstellar radio line emission. Furthermore,
the range in mass loss rates of the stars observed in SiO is
relatively narrow. Only a few high mass loss rate Miras have been
included in the study. In order to make a more conclusive analysis of
the depletion of SiO molecules onto dust grains, a larger sample of
stars with mass loss rates between 10$^{-6}$--10$^{-5}$ M$_\odot$
yr$^{-1}$ is needed.

From the modelling point of view, a few improvements can be done in
order to better reproduce the conditions prevailing in the CSEs. In
the modelling of CO circumstellar emission, more accurate
photodissociation models are necessary. Also, a more rigorous handling
of the turbulent velocities in the CSEs is desirable. In the modelling
of SiO circumstellar emission, a full treatment of the dust in the
envelopes, a more accurate approach to the cooling in the inner
envelope by molecules like, e.g., H$_2$O, to study the effects of gas
acceleration in the line shapes and intensities and a better
constrained SiO envelope size determination are a step further to more
realistic models.
