From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
Date: Mon, 2 Sep 2019 13:46:59 +0900
Subject: Initial commit

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+%
+% v1.2 released 5th September 1994 (M. Reed)
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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.
+
+
+\begin{thebibliography}{99}
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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}
-- 
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