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\pubyear{1989}
\volume{226}
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\begintopmatter  %  start the two spanning material

\title{Dust envelopes around RV Tauri stars}
\author{A. V. Raveendran}
\affiliation{Indian Institute of Astrophysics, Bangalore 560034, India}

\shortauthor{A. V. Raveendran}
\shorttitle{Dust envelopes around RV Tauri stars}

% \acceptedline is to be defined at the Journals office and not
% by an author.

\acceptedline{Accepted 1988 December 15. Received 1988 December 14;
  in original form 1988 October 11}

\abstract {In the {\it IRAS\/} [12]--[25], [25]--[60]
colour--colour diagram, RV Tauri stars are found to populate
cooler temperature regions $(T<600\,{\rm K})$, 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.}

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

\maketitle  %  finish the two spanning material


\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 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).
%
\begintable*{1}
\caption{{\bf Table 1.} Data on the RV Tauri stars detected by {\it IRAS}.}
\halign{%
\rm#\hfil&\qquad\rm#\hfil&\qquad\rm\hfil#&\qquad\rm\hfil
#&\qquad\rm\hfil#&\qquad\rm\hfil#&\qquad\rm#\hfil
&\qquad\rm\hfil#&\qquad\rm#\hfil&\qquad\hfil\rm#\cr
Name&&\multispan4\hskip23pt\hss Flux density (Jy)$^a$\hss \cr
Variable&{\it IRAS}&12$\;\umu$m&25$\;\umu$m
&60$\;\umu$m&100$\;\umu$m&Sp.&Period&Light-&$T_0({\rm K})$\cr
&&&&&&group&(d)\hfill&curve\cr
&&&&&&&&type\cr
\noalign{\vskip 10pt}
TW Cam&04166$+$5719&8.27&5.62&1.82&$<$1.73&A&85.6&a&555\cr
RV Tau&04440$+$2605&22.53&18.08&6.40&2.52&A&78.9&b&460\cr
DY Ori&06034$+$1354&12.44&14.93&4.12&$<$11.22&B&60.3&&295\cr
CT Ori&06072$+$0953&6.16&5.57&1.22&$<$1.54&B&135.6&&330\cr
SU Gem&06108$+$2734&7.90&5.69&2.16&$<$11.66&A&50.1&b&575\cr
UY CMa&06160$-$1701&3.51&2.48&0.57&$<$1.00&B&113.9&a&420\cr
U Mon&07284$-$0940&124.30&88.43&26.28&9.24&A&92.3&b&480\cr
AR Pup&08011$-$3627&131.33&94.32&25.81&11.65&B&75.0&b&450\cr
IW Car&09256$-$6324&101/06&96.24&34.19&13.07&B&67.5&b&395\cr
GK Car&11118$-$5726&2.87&2.48&0.78&$<$12.13&B&55.6&&405\cr
RU Cen&12067$-$4508&5.36&11.02&5.57&2.01&B&64.7&&255\cr
SX Cen&12185$-$4856&5.95&3.62&1.09&$<$1.50&B&32.9&b&590\cr
AI Sco&17530$-$3348&17.68&11.46&2.88&$<$45.62&A&71.0&b&480\cr
AC Her&18281$+$2149&41.47&65.33&21.12&7.79&B&75.5&a&260\cr
R Sct&18448$-$0545&20.88&9.30&8.10&$<$138.78&A&140.2&a\cr
R Sge&20117$+$1634&10.63&7.57&2.10&$<$1.66&A&70.6&b&455\cr
V Vul&20343$+$2625&12.39&5.72&1.29&$<$6.96&A&75.7&a&690\cr
}
\tabletext{\noindent $^a$Observed by {\it IRAS}.}
\endtable


\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
$$
   L(\nu)=\mskip-25mu\int\limits_{\rm envelope}\mskip-25mu
   \rho(r)Q_{{\rm abs}}
   (\nu)B[\nu,T_{\rm g}(r)]\exp [-\tau(\nu,r)]\> {\rm d}V, \eqno\stepeq
$$
where
$Q_{{\rm abs}}(\nu)$ is the absorption efficiency at frequency $\nu$,
$\rho(r)$ is the dust grain density,
$T_{\rm g}(r)$ is the grain temperature,
$B[\nu,T_{\rm g}(r)]$ is the Planck function and
$\tau(\nu,r)$ is the optical depth at distance {\it r\/} from the
centre of the star.

The temperature $T_{\rm g}(r)$ 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:
\beginlist
\item (i) the attenuated and diluted stellar radiation;
\item (ii) scattered radiation, and
\item (iii) reradiation from other grains.
\endlist

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 $[\tau({\rm
UV})\gg 10]$, and at optical depths $\tau({\rm UV})<1$ 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 $Q_{{\rm abs}} (\nu)$ 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
$T_{\rm g}$ 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
$$
   L(\nu)=\!\!\int_{r_{1}}^{r_{2}}\!\!4\upi r^2\rho(r)\> Q_{{\rm abs}}
   (\nu)B[\nu,T_{\rm g}(r)]\> {\rm d}r, \eqno\stepeq
$$
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
$$
   L(\nu)\propto \nu^{2+\gamma-Q}
   \int_{X_0}^{\infty}{{x^Q}\over{{\rm e}^x-1}}{\rm d}x, \eqno\stepeq
$$
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 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.
%
\beginfigure{1}
\vskip 91mm
\caption{{\bf Figure 1.} 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 $\sim 0.05$.}
\endfigure

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 $Q_{{\rm abs}}(\nu)$ 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).

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 $Q_{{\rm abs}}(\nu)$ 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.
%
\beginfigure*{2}
\vskip 5.9cm
\caption{{\bf Figure 2.} 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.}
\endfigure

At 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 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 100$\,\umu$m but, in
view of the large values for the cirrus flags given in the
catalogue, the intrinsic flux at 100$\,\umu$m is uncertain.


\subsection{Radial distribution of dust}

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 $\bar q=0.98$. 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 $Q_{{\rm abs}}(\nu)$ 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 $\sim300\,$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).
%
\beginfigure{3}
\vskip 59mm
\caption{{\bf Figure 3.} Plot of $(K$--$L)$ colours of
RV Tauri stars detected by {\it IRAS\/} against their corresponding
$(J$--$K)$ 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.}
\endfigure

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.
%
\ifsinglecol
  \pageinsert
    \vfil
    \centerline{Landscape figure to go here. This figure was not
      part of the original paper and is inserted here for illustrative
      purposes.}
    \centerline{See the author guide for details on how to handle landscape
      figures or tables, and {\tt mnland.tex}.}
    \centerline{{\bf Figure 4.}}
    \vfil
  \endinsert
\else
  \beginfigure*{4}
    \vbox to 646pt{\vfil
    \centerline{Landscape figure to go here. This figure was not
      part of the original paper and is inserted here for illustrative
      purposes.}
    \centerline{See the author guide for details on how to handle landscape
      figures or tables, and {\tt mnland.tex}.}
    \caption{{\bf Figure 4.}}
    \vfil}
  \endfigure
\fi

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 $(T<600\,\rm {K})$,
distinctly different from those occupied by the oxygen and
carbon Miras. Using a simple model in which
\beginlist
\item (i) the envelope is spherically symmetric,
\item (ii) the IR-emitting grains are predominantly of the same kind, and
\item (iii) in the IR the absorption efficiency $Q_{{\rm abs}} (\nu)\propto\nu$,
\endlist
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 {$\bf L\lowercase{y}\balpha$} 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\looseness=1\ observations of bright quasars show that the mean number
density of ${\rm Ly}\alpha$ forest lines, which satisfy certain criteria,
evolves
like ${\rm d}N/{\rm d}z=A(1+z)^\gamma$, where $A$ and~$\gamma$ are two
constants. Given
the above intrinsic line distribution we examine the probability of finding
large gaps in the ${\rm Ly}\alpha$ 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 ${\rm Ly}\alpha$ forest between redshifts $z_1$
and~$z_2$
and found $N-1$ lines.  For high-redshift quasars $z_2$~is usually the emission
redshift $z_{\rm em}$ and $z_1$ is set to $(\lambda_{\rm Ly\beta}/\lambda_{\rm
Ly\alpha})(1+z_{\rm em})=0.844(1+z_{\rm em})$ 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~$x$:
$$
   x={(1+z)^{\gamma+1}-(1+z_1)^{\gamma+1} \over
     (1+z_2)^{\gamma+1}-(1+z_1)^{\gamma+1}}. \eqno\stepeq
$$
$x$ varies from 0 to 1.  We then have ${\rm d}N/{\rm d}x=\lambda$,
where $\lambda$ is the mean number of lines between $z_1$ and $z_2$ and is
given by
$$
   \lambda\equiv{A[(1+z_2)^{\gamma+1}-(1+z_1)^{\gamma+1}]\over\gamma+1}.
     \eqno\stepeq
$$
This means that the ${\rm Ly}\alpha$ 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.
%
\beginfigure{5}
\vskip 11pc
\caption{{\bf Figure A1.} $P(>x_{\rm gap})$ as a function of
$x_{\rm gap}$ for, from left to right, $N=160$, 150, 140, 110, 100, 90, 50,
45 and~40.}
\endfigure

\subsection{Subsection title}

We plot in Fig.~A1 $P(>x_{\rm gap})$ for several $N$ values.
We see that, for $N=100$ and $x_{\rm gap}=0.06$, $P(>0.06)\approx 20$
per cent. This means that the
\ifsinglecol\else \vadjust{\vfill\eject}\fi
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
$P(>x_{\rm gap})$ approaches the result obtained by Ostriker et~al.\ (1988)
for small (but still very large if measured in units of the mean separation)
$x_{\rm gap}$, i.e., $P(>x_{\rm gap})\sim N(1-x_{\rm gap})^{N-1}\sim N
{\rm exp}(-\lambda x_{\rm gap})$.


\bye

% end of mnsample.tex
