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\title{Dust envelopes around RV Tauri stars}
\author[A. V. Raveendran]
       {A. V. Raveendran \\
        Indian Institute of Astrophysics, Bangalore 560034, India}
\date{Accepted 1988 December 15.
      Received 1988 December 14;
      in original form 1988 October 11}

\pagerange{\pageref{firstpage}--\pageref{lastpage}}
\pubyear{1994}

\begin{document}

\maketitle

\label{firstpage}

\begin{abstract}
In the {\it IRAS\/} [12]--[25],  [25]--[60] colour--colour diagram, RV
Tauri stars are found to populate cooler temperature regions
$(T<600\,\rmn{K})$, distinctly different from those occupied by  the
oxygen and carbon Miras. The {\it IRAS\/} fluxes are  consistent with the
dust density in the envelope varying as the inverse square of the
radial distance, implying that the grain formation processes in these
objects are most probably continuous and not sporadic. It is found that
the spectroscopic subgroups A and B are well separated in the
far-infrared two-colour diagram, with group B objects having
systematically cooler dust envelopes. We interpret this as being due to
a difference in the nature of grains, including the chemical
composition, in the two cases.
\end{abstract}

\begin{keywords}
circumstellar matter -- infrared: stars.
\end{keywords}

\section{Introduction}

It has been well established that RV Tauri variables  possess infrared
emission far in excess of their expected  blackbody continuum, arising
from their extended cool dust envelopes (Gehrz \& Woolf 1970; Gehrz
1972; Gehrz  \& Ney 1972). Recently, Lloyd Evans (1985) and Goldsmith
et al.\ (1987) have given detailed descriptions of the
near-infrared properties of RV Tauri stars. In this paper we present an
analysis of the {\it IRAS\/} data of RV Tauri stars with the help of the
far-infrared two-colour diagram and a grid computed using a simple
model of the dust envelope. Such two-colour plots have already been
employed extensively by several investigators to study the
circumstellar envelopes around oxygen-rich and carbon-rich objects
which are in the late stages of stellar evolution (Hacking et al.\ 
1985; Zuckerman \& Dyck 1986; van der Veen \& Habing 1988; Willems \&
de Jong 1988).

Table 1 summarizes the basic data on the 17 objects  detected at
\hbox{60\,$\umu$m}. Apart from the {\it IRAS\/} identification and the flux
densities at 12-, 25-, 60-  and 100-$\umu$m wavebands, it gives the
spectroscopic groups of Preston et al.\ (1963), the  light-curve classes
of Kukarkin et al.\ (1969) and  the periods of light variation. The
list, which contains  about 20 per cent of all the known RV Tauri
stars, is  essentially the same as that given by Jura (1986). The 
spectroscopic subgroups are from either Preston et al.\ (1963) or Lloyd
Evans (1985).
\begin{table*}
 \centering
 \begin{minipage}{140mm}
  \caption{Data on the RV Tauri stars detected by {\it IRAS}.}
  \begin{tabular}{@{}llrrrrlrlr@{}}
   Name     &            & \multicolumn{4}{c}{Flux density (Jy)%
  \footnote{Observed by {\em IRAS}.}}\\
   Variable & {\it IRAS} & 12$\,\umu$m & 25$\,\umu$m & 60$\,\umu$m 
     & 100$\,\umu$m & Sp. & Period & Light- & $T_0\,(\rmn{K})$ \\
        &  &  &  &  &  & group & (d) & curve \\
        &  &  &  &  &  &       &     & type  \\[10pt]
 TW Cam & 04166$+$5719 & 8.27 & 5.62 & 1.82 & $<$1.73 & A & 85.6 & a & 555 \\
 RV Tau & 04440$+$2605 & 22.53 & 18.08 & 6.40 & 2.52 & A & 78.9 & b & 460 \\
 DY Ori & 06034$+$1354 & 12.44 & 14.93 & 4.12 & $<$11.22 & B & 60.3 &  & 295 \\
 CT Ori & 06072$+$0953 & 6.16 & 5.57 & 1.22 & $<$1.54 & B & 135.6 &  & 330 \\
 SU Gem & 06108$+$2734 & 7.90 & 5.69 & 2.16 & $<$11.66 & A & 50.1 & b & 575 \\
 UY CMa & 06160$-$1701 & 3.51 & 2.48 & 0.57 & $<$1.00 & B & 113.9 & a & 420 \\
 U Mon  & 07284$-$0940 & 124.30 & 88.43 & 26.28 & 9.24 & A & 92.3 & b & 480 \\
 AR Pup & 08011$-$3627 & 131.33 & 94.32 & 25.81 & 11.65 & B & 75.0 & b & 450 \\
 IW Car & 09256$-$6324 & 101/06 & 96.24 & 34.19 & 13.07 & B & 67.5 & b & 395 \\
 GK Car & 11118$-$5726 & 2.87 & 2.48 & 0.78 & $<$12.13 & B & 55.6 &  & 405 \\
 RU Cen & 12067$-$4508 & 5.36 & 11.02 & 5.57 & 2.01 & B & 64.7 &  & 255 \\
 SX Cen & 12185$-$4856 & 5.95 & 3.62 & 1.09 & $<$1.50 & B & 32.9 & b & 590 \\
 AI Sco & 17530$-$3348 & 17.68 & 11.46 & 2.88 & $<$45.62 & A & 71.0 & b & 480 \\
 AC Her & 18281$+$2149 & 41.47 & 65.33 & 21.12 & 7.79 & B & 75.5 & a & 260 \\
 R Sct  & 18448$-$0545 & 20.88 & 9.30 & 8.10 & $<$138.78 & A & 140.2 & a \\
 R Sge  & 20117$+$1634 & 10.63 & 7.57 & 2.10 & $<$1.66 & A & 70.6 & b & 455 \\
 V Vul  & 20343$+$2625 & 12.39 & 5.72 & 1.29 & $<$6.96 & A & 75.7 & a & 690 
\end{tabular}
\end{minipage}
\end{table*}

\section[]{Description of the Envelope\\* Model}

If we assume that the dust grains in the envelope are  predominantly of
the same kind and are in thermal  equilibrium, the luminosity at
frequency $\nu$ in the infrared is given by
\begin{equation}
   L(\nu)=\mskip-12mu\int\limits_{\rmn{envelope}}\mskip-12mu
   \rho(r)Q_{\rmn{abs}}(\nu)B[\nu,T_{\rmn{g}}(r)]\exp [-\tau(\nu,r)]\>
   \rmn{d}V,
\end{equation}
 where
 $Q_{\rmn{abs}}(\nu)$ is the absorption efficiency at frequency $\nu$,
 $\rho(r)$            is the dust grain density,
 $T_{\rmn{g}}(\nu)$    is the grain temperature,
 $B[\nu,T_{\rmn{g}}(r)]$  is the Planck function, and 
 $\tau(\nu,r)$        is the optical depth at distance {\it r\/} from the
                      centre of the star.

The temperature $T_{\rmn{g}}(r)$ is determined by the condition of energy
balance: amount of energy radiated = amount of energy absorbed. The
amount of energy absorbed at any point is proportional to the total
available energy at that point, which consists of:
\begin{enumerate}
  \item the attenuated and diluted stellar radiation;
  \item scattered radiation, and
  \item reradiation from other grains.
\end{enumerate}

Detailed solutions of radiative transfer in circumstellar  dust shells
by Rowan-Robinson \& Harris (1983a,b) indicate that the effect of
heating by other grains becomes significant only at large optical
depths at the absorbing frequencies $[\tau(\rmn{UV})\gg 10]$, and at
optical depths $\tau(\rmn{UV})<1$ the grains have approximately the
same temperature that they would have if they were seeing the starlight
unattenuated and no other radiation.

The Planck mean optical depths of circumstellar envelopes  around
several RV Tauri stars, derived from the ratios of the luminosities of
the dust shell (at infrared wavelengths) and the star, range from 0.07
to 0.63 (Goldsmith et al.\ 1987). There is much uncertainty in the
nature of the optical properties of dust grains in the envelope. The
carbon-rich RV Tauri stars are also reported to show the 10-$\umu$m
silicate emission feature typical of oxygen-rich objects (Gehrz \& Ney
1972; Olnon \& Raimond 1986). The pure terrestrial silicates or lunar
silicates are found to be completely unsuitable to account for the
infrared emission from circumstellar dust shells around M-type stars
(Rowan-Robinson \& Harris 1983a). We assume that the absorption
efficiency $Q_{\rmn{abs}} (\nu)$ in the infrared varies as
$\nu^{\gamma}$. ${\gamma}=1$ appears to provide a reasonable fit in a
variety of sources (Harvey, Thronson \& Gatley 1979; Jura 1986). Under
these circumstances the condition of energy balance implies that the
dust temperature $T_{\rmn{g}}$ will vary as $r^{\beta}$.

In view of the low value of the observed Planck mean  optical depth for
the stellar radiation and the nature of the assumed frequency
dependence of the absorption efficiency, the extinction of the infrared
radiation by the dust envelope can be neglected. If we consider the
envelope to be spherically symmetric, equation (1) reduces to
\begin{equation}
   L(\nu)=\!\!\int_{r_{1}}^{r_{2}}\!\!4\upi r^2\rho(r)\> Q_{\rmn{abs}}(\nu)B[\nu,T_{\rmn{g}}(r)]\> {\rmn{d}}r,
\end{equation}
where $r_1$ and $r_2$ are the inner and outer radii of the shell. For
a dusty density distribution $\rho(r)\propto r^{\alpha}$ and $r_2\gg
r_1$, equation (2) reduces to
\begin{equation}
   L(\nu)\propto \nu^{2+\gamma-Q}\int_{X_0}^{\infty}{{x^Q}\over
   {\rmn{e}^x-1}}\rmn{d}x ,
\end{equation}
where $Q=-(\alpha+\beta+3)/\beta$ and $X_0=(h\nu /kT_0)$. $T_0$
represents the temperature at the inner boundary of the dust shell
where grains start condensing. In a steady radiation pressure driven
mass outflow in the optically thin case, values of $\alpha$ lie near $-2$
(Gilman 1972). $\gamma$ and $\beta$ are related by
$\beta=-2/(\gamma+4)$.

In the {\it IRAS\/} Point Source Catalog (PSC, Beichman et al.\ 1985a),
the flux densities have been quoted at the effective wavelengths 12,
25, 60 and \hbox{100\,$\umu$m}, assuming a flat energy spectrum $[\nu F(\nu)=1]$
for the observed sources. For each model given by equation (3), using
the relative system response, the colour-correction factors (Beichman
et al.\ 1985b) in each of the {\it IRAS\/} passbands were
calculated and the fluxes were converted into flux densities expected
for a flat energy distribution, as assumed in the {\it IRAS\/} PSC, so
that the computed colours can be directly compared with the colours
determined from the catalogue quantities. Such a procedure is
more appropriate than correcting the {\it IRAS\/} colours for the energy
distribution given by a particular model and then comparing them with
those computed by the model.

\subsection{Colour--colour diagram}

The IR colour is defined as
\[
  [\nu_1]-[\nu_2]=-2.5\log [f(\nu_1)/f(\nu_2)],
\]
 where $\nu_1$ and $\nu_2$ are any two wavebands and $f(\nu_1)$
and $f(\nu_2)$ are the corresponding flux  densities assuming a flat
energy spectrum for the source.
In Fig.~1, we have plotted the [25]--[60] colours  of RV Tauri stars
against their corresponding [12]--[25]  colours derived from the {\it
IRAS\/} data. Filled circles  represent stars of group A and open circles
stars of group B. The two sets of near-parallel lines represent the
loci of constant inner shell temperature $T_0$ and the quantity $Q$
defined above. The models correspond to the case of absorption
efficiency $Q_{\rmn{abs}}(\nu)$ varying as $\nu$ (with $\gamma=1$ and
hence $\beta=-0.4$). We have omitted R Sct in Fig.~1 because it shows a
large deviation from the average relation shown by all the other
objects. R Sct has a comparatively large excess at 60$\,\umu$m, but the
extent of a possible contamination by the infrared cirrus (Low et
al. 1984) is unknown. Goldsmith et al.\ (1987) found no evidence
of the presence of a dust envelope at near-IR wavelengths and the
spectrum was consistent with a stellar continuum. This explains why R
Sct lies well below the mean relation shown by stars of groups A and C
between the [3.6]--[11.3] colour excess and the photometrically
determined (Fe/H) (Dawson 1979). R Sct has the longest period of
140$\,$d among the RV Tauri stars detected at far-infrared wavelengths
and does not have the 10-$\umu$m emission feature seen in other objects
(Gehrz 1972; Olnon \& Raimond 1986). R Sct is probably the most
irregular RV Tauri star known (McLaughlin 1932).

\begin{figure}
 \vspace{302pt}
 \caption{Plot of [25]--[60] colours of RV  Tauri stars against their
  [12]--[25] colours after  normalizing as indicated in Beichman et al.\ 
  (1985b). Some of the objects are identified by their variable-star
  names. Typical error bars are shown in the bottom right-hand corner.
  The lines represent the loci for constant inner shell temperature and
  the quantity $Q$. Note the separation of group A and B stars at $T_0
  \sim$ 460$\,$\,K. Positions occupied by a sample of carbon and oxygen
  Miras are also shown. The $Q=1.0$ line differs from the blackbody line
  by a maximum of $\sim 0.05$.}
\end{figure}
The inner shell temperatures $(T_0)$ derived for the various objects
are also given in Table~1 and we find the majority of them to have
temperatures in the narrow range 400--600$\,$K. If the dependences of
$Q_{\rmn{abs}}(\nu)$ on $\nu$ and $\rho(r)$ on $r$ are similar in all
the objects considered, then in the colour--colour diagram they all
should lie along a line corresponding to different values of $T_0$ and
in Fig.~1 we find that this is essentially the  case. In view of the
quoted uncertainties in the flux measurements, we cannot attach much
significance to the scatter in Fig.~1.

At \hbox{100\,$\umu$m} the infrared sky is characterized by emission, called
infrared cirrus, from interstellar dust on all spatial scales (Low et
al. 1984), thereby impairing the measurements at far-infrared
wavelengths. In Fig.~2, we have plotted the [60]--[100] colours of the
six RV Tauri stars detected at \hbox{100\,$\umu$m} against their [25]--[60]
colours, along with the grid showing the regions of different values
for inner shell temperature $T_0$ and the quantity $Q$, as in Fig.~1.
The results indicated by Fig.~2 are consistent with those derived from
Fig.~1. AR Pup shows a large excess at \hbox{100\,$\umu$m} but, in view of the
large values for the cirrus flags given in the catalogue, the intrinsic
flux at \hbox{100\,$\umu$m} is uncertain.

\subsection{Radial distribution of dust}

\begin{figure*}
  \vspace*{174pt}
  \caption{Plot of the [60]--[100] colours of RV Tauri stars against
  their [25]--[60] colours after normalizing as indicated in Beichman
  et al.\ (1985b). The solid lines represent the loci for constant
  inner shell temperature and the quantity $Q$. The dashed line shows
  the locus for a blackbody distribution.}
\end{figure*}

From Fig.~1, it is evident that all RV Tauri stars lie between the
lines corresponding to $Q=1.5$ and 0.5. With
 \[
  \alpha=-(1+Q)\beta-3,
 \]
 these values suggest limits of $r^{-2.0}$ and $r^{-2.4}$ for the
dust density variation, indicating a near-constant mass-loss rate. Jura
(1986) has suggested that the density in the circumstellar envelope
around RV Tauri stars varies as $r^{-1}$, implying a mass-loss rate
that was greater in the past than it is currently. By fitting a power
law to the observed fluxes, such that $f(\nu)$ varies as $\nu^q$,
values of $q$ determined by him for the various objects given in Table~1
lie in the range  0.6--1.2, with a mean $\skew5\bar q=0.98$. The
assumption of a power law corresponds to the case of $X_0=0$ in
equation (3) and hence we get
 \[
  q=2+\gamma -Q.
 \]
Since we assume that $Q_{\rmn{abs}}(\nu)$ varies as $\nu$, the
resulting value for $Q$=2.0. None of the objects is found to lie in the
corresponding region in the colour--colour diagram. Even this extreme
value for $Q$ implies a density which varies as $r^{-1.8}$.

Goldsmith et al.\ (1987) have reported that the  simultaneous
optical and near-IR data of AC Her can be fitted by a combination of
two blackbodies at 5680 and 1800\,K, representing, respectively, the
stellar and dust shell temperatures, and suggested that in RV Tauri
stars the grain formation is a sporadic phenomenon and not a continuous
process. Apparently, they have been influenced by the remark by Gehrz
\& Woolf (1970) that their data in the 3.5--11$\,\umu$m region of AC Her
indicated a dust temperature of $\sim$300\,K. We find that the {\it K--L\/}
colours given by Gehrz (1972), Lloyd Evans (1985) and Goldsmith et
al. (1987) are all consistent with each other. Surely, hot dust ($\sim
1800\,$K), if present at the time of observations by Goldsmith et
al. (1987), would have affected the {\it K--L\/} colour significantly.
AC Her, like other members of its class, is found to execute elongated
loops in the ({\it U--B\/}), ({\it B--V\/}) plane (Preston et al.\ 
1963), indicating that significant departure of the stellar continuum
from the blackbody is to be expected. Further, their data show only a
marginal excess at the near-IR wavelengths. We feel that the case for
the existence of hot dust around AC Her and hence for the sporadic
grain formation around RV Tauri stars is not strong. In Fig.~3 we find that AC Her and RU Cen lie very close to R Sct
which, according to Goldsmith et al.\ (1987), shows no evidence
for the presence of a hot dust envelope.

\subsubsection{Comparison with oxygen and carbon Miras}

In Fig.~1 we have also shown the positions of a sample of oxygen-rich
and carbon-rich Miras. At the low temperatures characteristic of the
Miras, a part of the emission at 12$\,\umu$m comes from the photosphere.
For a blackbody at 2000$\,$K, the ratio of fluxes at wavelengths of 12
and 2$\,\umu$m $(f_{12}/f_{2})\sim 0.18$. The Miras shown in Fig.~1 have
$(f_{12}/f_{2})$ ratios larger than twice the above value. It is
clear that the three groups of objects populate three different regions
of the diagram. Hacking et al.\ (1985) have already noticed that
there are distinct differences between the {\it IRAS\/} colours of
oxygen-rich and carbon-rich objects. On the basis of an analysis, using
a bigger sample of bright giant stars in the {\it IRAS\/} catalogue, this
has been interpreted by Zuckerman \& Dyck (1986) as being due to a
systematic difference in the dust grain emissivity index. U Mon shows
the 10-$\umu$m silicate emission convincingly and, in most of the other
objects for which low-resolution spectra in the near-infrared have been
reported (Gehrz 1972; Olnon \& Raimond 1986), the 10-$\umu$m emission
may be partly attributed to silicates. Hence it is reasonable to expect
that, in the envelopes around at least some of the RV Tauri stars, the
dust grains are predominantly of silicates, as in the case of oxygen
Miras (Rowan-Robinson \& Harris 1983a). The fact that none of the RV
Tauri stars is found in the region of the two-colour diagram occupied
by the oxygen Miras indicates that the emissivity indices of the
silicate grains in the two cases are different. Because of the higher
temperatures and luminosities, the environment of grain formation will
be different in RV Tauri stars.

\subsubsection{Correlation with subgroups}

Preston et al.\ (1963) have identified three  spectroscopic
subgroups, which are designated as groups A, B and C. Objects of group
A are metal-rich; group C are metal-poor; group B objects are also
metal-poor, but  show carbon enhancements (Preston et al.\ 1963; Lloyd
Evans 1974; Dawson 1979; Baird 1981). It is interesting to see that
Table~1 contains no group C objects and that in Fig.~1 there is a clear
separation of the two spectroscopic subgroups A and B, with the
demarcation  occurring at an inner shell temperature of about 450$\,$K,
group B stars having lower temperatures than group A. SX Cen is the
only exception. Lloyd Evans (1974) has reported that metal lines are
stronger in SX Cen than in other group B objects. It may be worth
noting that SX Cen has the shortest period among the 100 or so objects
with the RV Tauri classification. RU Cen has the coolest inner  shell
temperature, as already suggested by the near-infrared spectrum (Gehrz
\& Ney 1972).
\begin{figure}
  \vspace*{174pt}
  \caption{Plot of ({\it K--L\/}) colours of RV Tauri stars detected by
  {\it IRAS\/} against their corresponding ({\it J--K\/}) colours. The
  position of AR Pup is indicated. The three objects lying close to the
  blackbody line are AC Her, RU Cen and R Sct.}
\end{figure}

Group B objects follow a different mean relationship from those of group
A, having systematically larger 11-$\umu$m excess for a given excess at
3$\,\umu$m (Lloyd Evans 1985). For a general sample of RV Tauri stars,
the distinction between the oxygen-rich and carbon-rich objects is not
that apparent in the {\it JHKL\/} bands. In Fig.~3 we have plotted the
near-IR magnitudes of the objects given in Table~1 (except V Vul which
has no available measurements) in the {\it J--K, K--L\/} plane. The
colours,  taken from Lloyd Evans (1985) and Goldsmith et al.\ 
(1987), are averaged if more than one observation exists, because the
internal agreements are found to be often of the order of observational
uncertainties, in accordance with the earlier finding by Gehrz (1972)
that variability has relatively little effect on colours. Barring RU
Cen and AC Her, it is evident that stars belonging to group B show
systematically larger excesses at {\it L\/} band for a given excess at
{\it K}. The low excesses at near-IR wavelengths for AC Her and RU Cen
are consistent with the very low dust temperatures indicated by the
far-infrared colours.
%
\begin{figure*}
\vbox to 220mm{\vfil
Landscape figure to go here. This figure was not part of the original 
paper and is inserted here for illustrative purposes.\\ See the author
guide for details on how to handle landscape figures or tables, and
\verb|mnland.tex|.
\caption{}
\vfil}
\label{landfig}
\end{figure*}

It is already well established that from {\it UBV\/} photometry one can
distinguish between groups A and B,  members of group A being
significantly redder than those of group B (Preston et al.\ 1963).
Similarly, Dawson (1979) has found that the two spectroscopic groups
are well separated in the DDO colour--colour diagrams when mean colours
are used for the individual objects.

The clear separation of the spectroscopic subgroups A and  B in the IR
two-colour diagram suggests that the natures of dust grains in the
envelopes in the two cases are not  identical. This is to be expected
because of the differences in the physical properties of the stars
themselves. The average colours of group B stars are bluer than group
A, but the envelope dust temperatures of B are cooler than those of A.
The near-IR spectra of AC Her and RU Cen are extremely similar (Gehrz
\& Ney 1972). The striking similarities in the optical spectra of AC
Her and RU Cen have been pointed out by Bidelman (O'Connell 1961). We
feel that the physical properties, including the chemical composition,
of the grains  formed in the circumstellar envelope strongly depend on
those of the embedded star. This, probably, explains the diversity of
the energy distributions of RV Tauri stars in the near-infrared found
by Gehrz \& Ney (1972). On the basis of the observed differences in
chemical abundances and space distribution of RV Tauri stars, Lloyd
Evans (1985) has already pointed out that there is no direct
evolutionary connection between group A and group B objects, thus
ruling out the possibility that group B objects are the evolutionary
successors of group A, in which grain formation has stopped and the
cooler temperatures for the former are caused by an envelope expansion.

Kukarkin et al.\ (1969) have subdivided RV Tauri stars 
into two classes, RVa and RVb, on the basis of their light curves; the 
former shows a constant mean  brightness, whereas the latter shows a
cyclically varying  mean brightness. Extensive observations in the
near-infrared show that, on average, RVb stars are redder than RVa
stars, and Lloyd Evans (1985) has suggested that in RVb stars dust
shells are denser in the inner regions and hence radiate strongly in
the 1--3$\,\umu$m region. Fig.~3 confirms this; RVb objects show
systematically larger ({\it J--K\/}) and ({\it K--L\/}) colours than RVa
objects. Apparently, there is no distinction between objects of the two
light-curve types at far-infrared wavelengths (Fig.~1).

\section{Conclusions}

In the [12]--[25], [25]--[60] colour diagram, RV Tauri stars populate
cooler temperature regions $(T<600 \,\rmn{K})$, distinctly different from
those occupied by the oxygen and carbon Miras. Using a simple model
in which
\begin{enumerate}
  \item the envelope is spherically symmetric,
  \item the IR-emitting grains are predominantly of the same kind, and
  \item in the infrared the absorption efficiency $Q_{\rmn{abs}}
        (\nu)\propto\nu$,
\end{enumerate}
we find that the {\it IRAS\/} fluxes are
consistent with the density in the envelope $\rho(r)\propto r^{-2}$,
where {\it r\/} is the radial distance. Such a dependence for the dust
density implies that the mass-loss rates in RV Tauri stars have not
reduced considerably during the recent past, contrary to the suggestion
by Jura (1986). In the two-colour diagram, the blackbody line and the
line corresponding to $\rho(r)\propto r^{-2.2}$ nearly overlap and the
present data are insufficient to resolve between the two cases. The
latter case is more physically reasonable, however.

The spectroscopic subgroups A and B are well separated in  the {\it
IRAS\/} two-colour diagram, with group B objects  having systematically
cooler dust envelopes. If we consider only the objects detected by {\it
IRAS}, we find that stars belonging to group B show systematically
larger excess at {\it L\/} band for a given excess at {\it K}.
Apparently, there is no correlation between the light-curve types (RVa
and RVb) and the far-infrared behaviour of these objects. It is fairly
certain that the physical properties, including the chemical
composition, of the embedded stars are directly reflected by those of
the dust grains. Most probably, the grain formation process in RV Tauri
stars is continuous and not sporadic as suggested by Goldsmith et
al.\ (1987).

\section*{Acknowledgments}

I thank Professor N. Kameswara Rao for some helpful suggestions, 
Dr H. C. Bhatt for a critical reading of the original version of the 
paper and an anonymous referee for very useful comments that improved 
the presentation of the paper.


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\appendix

\section[]{Large gaps in L\lowercase{y}${\balpha}$ forests\\* due to fluctuations in line distribution}

(This appendix was not part of the original paper by A.V.~Raveendran 
and is included here just for illustrative purposes.)

Spectroscopic observations of bright quasars show that the mean number
density of Ly$\alpha$ forest lines, which satisfy certain criteria, 
evolves like $\rmn{d}N/\rmn{d}z=A(1+z)^\gamma$, where $A$ and~$\gamma$ are two 
constants.  Given the above intrinsic line distribution we examine the 
probability of finding large gaps in the Ly$\alpha$ forests.  We 
concentrate here only on the statistics and neglect all observational 
complications such as the line blending effect (see Ostriker, Bajtlik 
\&~Duncan 1988).

Suppose we have observed a Ly$\alpha$ forest between redshifts $z_1$ 
and~$z_2$ and found $N-1$ lines.  For high-redshift quasars $z_2$~is 
usually the emission redshift $z_{\rmn{em}}$ and $z_1$ is set to 
$(\lambda_{\rmn{Ly}\beta}/\lambda_{\rmn{Ly}\alpha})(1+z_{\rmn{em}})=0.844
(1+z_{\rmn{em}})$ to avoid contamination by Ly$\beta$ lines.  We
want to know whether the largest gaps observed in the forest are 
significantly inconsistent with the above line distribution.  To do 
this we introduce a new variable~$x$:
%
\begin{equation}
x={(1+z)^{\gamma+1}-(1+z_1)^{\gamma+1} \over
     (1+z_2)^{\gamma+1}-(1+z_1)^{\gamma+1}}.
\end{equation}
%
$x$ varies from 0 to 1.  We then have $\rmn{d}N/\rmn{d}x=\lambda$, where $\lambda$ 
is the mean number of lines between $z_1$ and $z_2$ and is given by
%
\begin{equation}
\lambda\equiv{A[(1+z_2)^{\gamma+1}-(1+z_1)^{\gamma+1}]\over\gamma+1}.
\end{equation}
%
This means that the Ly$\alpha$ forest lines are uniformly
distributed in~$x$. The probability of finding $N-1$ lines between $z_1$
and~$z_2$, $P_{N-1}$, is assumed to be the Poisson distribution.
%
\newpage
%
\begin{figure}
\vspace{11pc}
\caption{$P(>x_{\rmn{gap}})$ as a function of $x_{\rmn{gap}}$ for, 
 from left to right, $N=160$, 150, 140, 110, 100, 90, 50, 45 and~40.}
\label{appenfig}
\end{figure}

\subsection{Subsection title}

We plot in Fig.~\ref{appenfig} $P(>x_{\rmn{gap}})$ for several $N$ values. 
We see that, for $N=100$ and $x_{\rmn{gap}}=0.06$, $P(>0.06)\approx20$ 
per cent.  This means that the probability of finding a gap with a size 
larger than six times the mean separation is not significantly small.  
When the mean number of lines is large, $\lambda\sim N>>1$, our 
$P(>x_{\rmn{gap}})$ approaches the result obtained by Ostriker et al.\ 
(1988) for small (but still very large if measured in units of the mean 
separation) $x_{\rmn{gap}}$, i.e., $P(>x_{\rmn{gap}})\sim N(1-
x_{\rmn{gap}})^{N-1}\sim N {\rmn{exp}}(-\lambda x_{\rmn{gap}})$.

\bsp

\label{lastpage}

\end{document}
