
\documentclass[journal]{rmaa}

\def\uv{$uvby-\beta$}


\title{Functional relationships for $T_{\rm \lowercase{eff}}$ and \lowercase{log}~$\lowercase{g}$ in \\ 
F-G supergiants from $\lowercase{uvby}-\beta$ photometry}

\author{A. Arellano Ferro\affil{Instituto de Astronom\'ia\\ Universidad Nacional Aut\'onoma de M\'exico, Mexico}}

\shortauthor{Arellano Ferro}

\shorttitle{Temperature and gravity from $\lowercase{uvby}-\beta$ in F-G supergiants}


\listofauthors{A.~Arellano~Ferro}

\indexauthor{Arellano Ferro, A.}


\fulladdresses{
\item A. Arellano Ferro: Instituto de Astronom\'ia, Universidad Nacional Aut\'onoma de M\'exico, Apdo. Postal 70-264, M\'exico D.F. 04510, 
Mexico (armando@astro.unam.mx).}


\ReceivedDate{2010 May 25}
\AcceptedDate{2010 July 2}
\SetYear{2010}


\resumen{A partir de datos fotoel\'ectricos en el sistema $uvby-\beta$ y de valores precisos sint\'eticos y espectrosc\'opicos de 
$T_{\rm eff}$ y log~$g$ en 50 supergigantes de tipos espectrales F-G, hemos calculado relaciones funcionales que permiten la estimaci\'on 
inicial de la temperatura efectiva y la gravedad en este tipo de estrellas. Se demuestra que aunque las calibraciones de $T_{\rm eff}$ 
fueron calculadas con los datos de estrellas supergigantes j\'ovenes y masivas, tambi\'en son v\'alidas para estrellas evolucionadas 
post-AGB y RV Tau de temperaturas similares. La gravedad superficial tambi\'en puede calcularse a partir del \'indice $\Delta [c_1]$ 
con una precisi\'on de 0.26 dex. Aunque se puede distinguir una correlaci\'on entre $M_V$ y $\Delta [c_1]$, no hemos encontrado una 
correlaci\'on que prediga $M_V$ de manera suficientemente precisa.}

\abstract{From photoelectric $uvby-\beta$ data and recent accurate synthetic and spectroscopic values of $T_{\rm eff}$ and log~$g$ for 
50 F-G supergiants, we have calculated functional relationships that lead to initial estimates of effective temperature and gravities 
for these types of stars. It is shown that while the $T_{\rm eff}$ relationships are calculated using the data on young massive supergiants, 
they are also valid for evolved stars of similar temperatures like post-AGB and RV Tau stars. The gravity can also be predicted from 
the $\Delta [c_1]$ index with an uncertainty of about 0.26 dex. Although a clear and significant trend between $M_V$ and $\Delta [c_1]$ 
is seen, no calibration is found that predicts accurate values of $M_V$.}

\addkeyword{stars: AGB and post-AGB}
\addkeyword{supergiants}
\addkeyword{techniques: photometric}


\begin{document}

\maketitle

\section{INTRODUCTION}

Back in the mid 1990's we calculated a set of functional relationships between several reddening-free indices in the Str\"omgren 
$uvby-\beta$ system and the effective temperature for 41 supergiants of luminosity classes I and II and with spectral types between 
A0 and K0. The main goal of those relationships was to provide initial values of the effective temperature for our own spectroscopic 
work aimed at calculating detailed atmospheric abundances for stars of intermediate temperatures. At that time we used the temperatures 
calculated by Bravo-Alfaro, Arellano Ferro, \& Schuster (1997) from 13-color photometry and the $uvby-\beta$ data from the catalogue 
of Arellano Ferro et al. (1998).

The relationships turned out to be quite useful for several problems related to the estimation of the temperature for these types of 
stars. However they remained unpublished mainly because the intrinsic scatter was rather large, probably due to the limited quality of 
the temperature data used.

Very recently a set of accurate temperatures, gravities and distances for 48 near ($d \leq 700$~pc) and 15 distant ($d \geq 700$~pc) A-G 
supergiants, were published (Lyubimkov et al. 2010). In the present paper functional relationships are worked out in view of these 
new values of physical parameters, with the aim of providing a tool to estimate initial values of $T_{\rm eff}$ and log~$g$ in yellow supergiants 
for subsequent spectroscopic work.

The paper is organized as follows: in  \S~\ref{data} the sources of the $uvby-\beta$ data are described, in \S~\ref{temp} the calibrations 
for the temperature as a function of the [$c_1$], [$m_1$] and H$\beta$ photometric indices are discussed, in \S~\ref{grav} the gravity 
calibration in terms of the $\Delta [c_1]$ index is calculated, in \S~\ref{pagb} we discuss the validity of the calibrations for post-AGB 
and RV Tau stars, in \S~\ref{MV} the attempts at a calibration of the absolute magnitude and its limitations are presented, and in \S~\ref{concl} 
we summarize our conclusions.


\section {The photometric $\lowercase{uvby}-\beta$ data}
\label{data}

The $uvby-\beta$ photometric data were taken from the catalogue of Arellano Ferro et al. (1998). For about a dozen of stars photometric 
data were not available in that source and their data were obtained from the sources given in the Simbad database. When several 
measurements exist a simple average was calculated. In Table~\ref{T1} we present for the sample stars the reddening free indices 
[$c_1$] = $c_1-0.16(b-y)$ and [$m_1$] = $m_1+0.33(b-y)$, and values for their distance, effective temperature and gravity taken from Lyubimkov et al. (2010).


%%%%% TABLA 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption[]{Physical parameters and photometric indices for the sample F-G supergiants}
\label{T1}
\setlength{\tabnotewidth}{0.96\textwidth}
\tablecols{12}
\setlength{\tabcolsep}{1.25\tabcolsep}
\footnotesize
\begin{tabular}{lccccccccrcc}
\toprule
\multicolumn{1}{c}{Star} & D & $T_{\rm eff}$ & log~$g$ & [$c_1$] & [$m_1$] & H$\beta$ & $E(b-y)$ & $M_V$ & \multicolumn{1}{c}{$\Delta [c_1]$} & Sp. T. & Comment \\
  & (pc) & (K) & (dex) &  &  &  &  &  &  &  &  \\
\midrule
HR27    & ~380 & 6270  & 2.10   & 1.022 & 0.222 & 2.668 &  ~~0.100 & $-3.3$ & $-$0.236 & F2 II     &    \\
HR~157  & ~259 & 5130  & 2.15   & 0.293 & 0.504 & 2.585 & $-$0.031 & $-1.5$ & $-$0.130 & G2.5 IIa  & \\
HR~207  & ~935 & 5220  & 1.55   & 0.380 & 0.145 & 2.626 &  ~~0.553 & $-3.9$ &    0.000 & G0 Ib     & \\ 
HR~792  & ~397 & 5020  & 2.09   & 0.245 & 0.645 &       &  ~~0.034 & $-1.8$ & $-$0.062 & G5 II     & D   \\     
HR~849  & ~538 & 5020  & 1.73   & 0.209 & 0.707 & 2.605 &  ~~0.019 & $-8.6$ & $-$0.068 & G5 Iab:   & D   \\      
HR~1017 & ~156 & 6350  & 1.90   & 1.048 & 0.300 & 2.681 &  ~~0.097 & $-5.1$ &    0.114 & F5 Ib     & V,S   \\  
HR~1135 & ~170 & 6560  & 2.44   & 0.950 & 0.271 & 2.684 &  ~~0.046 & $-2.6$ & $-$0.093 & F5 II     & V,RRL \\
HR~1242 & ~629 & 6815  & 1.87   & 1.464 & 0.229 & 2.730 &  ~~0.310 & $-5.3$ &    0.236 & F0 II     &   \\      
HR~1270 & ~427 & 5060  & 1.91   & 0.260 & 0.635 & 2.580 &  ~~0.045 & $-2.1$ & $-$0.050 & G8 IIa    &    \\ 
HR~1303 & ~275 & 5380  & 1.73   & 0.451 & 0.479 & 2.609 &  ~~0.097 & $-3.5$ & $-$0.018 & G0 Ib     & D,SB \\   
HR~1327 & ~~98 & 5440  & 2.89   & 0.326 & 0.457 & 2.574 & $-$0.024 & $+0.4$ & $-$0.183 & G5 IIb    & S \\      
HR~1603 & ~265 & 5300  & 1.79   & 0.414 & 0.500 & 2.607 &  ~~0.002 & $-0.6$ & $-$0.009 & G1 Ib-IIa & D \\   
HR~1829 & ~~49 & 5450  & 2.60   & 0.366 & 0.423 & 2.576 &  ~~0.007 & $-0.6$ & $-$0.232 & G5 II     & S   \\      
HR~1865 & ~680 & 6850  & 1.34   & 1.451 & 0.202 & 2.730 &  ~~0.121 & $-7.1$ &    0.086 & F0 Ib     & V,S    \\ 
HR~2000 & ~224 & 5000  & 2.45   & 0.304 & 0.519 & 2.568 & $-$0.045 & $-0.4$ & $-$0.105 & G2 Ib-II  & S\\    
HR~2453 & ~633 & 4900  & 1.70   & 0.224 & 0.648 & 2.582 & $-$0.033 & $-2.4$ & $-$0.083 & G5 Ib     & D \\  
HR~2597 & ~935 & 6710  & 2.00   & 1.333 & 1.777 & 2.721 &  ~~0.193 & $-4.3$ & $-$0.071 & F2 Ib-II  & \\   
HR~2693 & ~495 & 5850  & 1.00   & 0.869 & 0.446 & 2.660 &  ~~0.013 & $-6.7$ &    0.337 & F8 Ia     & V,S    \\ 
HR~2786 & ~386 & 5260  & 1.90   & 0.367 & 0.556 & 2.619 &  ~~0.003 & $-2.7$ & $-$0.003 & G2 Ib     & D,S   \\    
HR~2833 & ~375 & 5380  & 2.21   & 0.391 & 0.535 & 2.586 & $-$0.010 & $-1.8$ & $-$0.005 & G3 Ib     &    \\  
HR~2881 & ~461 & 5300  & 1.66   & 0.413 & 0.557 & 2.622 & $-$0.029 & $-3.6$ &    0.042 & G3 Ib     & S    \\     
HR~3045 & ~370 & 4880  & 1.21   & 0.251 & 0.793 & 2.625 & $-$0.030 & $-4.4$ &    0.000 & G6 Ia     & D,S  \\   
HR~3073 & ~338 & 6670  & 2.61   & 1.145 & 0.271 & 2.700 &  ~~0.080 & $-2.2$ &    0.106 & F1 Ia     &   \\   
HR~3102 & ~161 & 5690  & 2.17   & 0.520 & 0.395 & 2.651 & $-$0.005 & $-1.8$ & $-$0.144 & F7 II     & S \\   
HR~3188 & ~325 & 5210  & 1.75   & 0.325 & 0.569 & 2.589 & $-$0.026 & $-3.1$ & $-$0.035 & G2 Ib     & D \\      
HR~3229 & ~267 & 5130  & 2.04   & 0.281 & 0.663 & 2.584 & $-$0.003 & $-2.1$ & $-$0.016 & G5 II     &  \\  
HR~3291 & ~900 & 6600  & 1.25   & 1.413 & 1.996 & 2.703 &  ~~0.224 & $-7.2$ &    0.165 & F3 Ib     & D \\ 
HR~3459 & ~236 & 5370  & 2.08   & 0.424 & 0.464 & 2.609 &  ~~0.005 & $-2.3$ & $-$0.072 & G1 Ib     & D  \\     
HR~4166 & ~177 & 5475  & 2.36   & 0.380 & 0.459 & 2.586 & $-$0.028 & $-1.4$ & $-$0.129 & G2.5 IIa  & \\     
HR~5143 & ~166 & 5190  & 2.75   & 0.277 & 0.540 & 2.543 & $-$0.023 & $+0.2$ & $-$0.113 & G5 II:    & \\      
HR~5165 & ~253 & 5430  & 2.37   & 0.480 & 0.416 & 2.591 &  ~~0.069 & $-1.7$ & $-$0.135 & G0 Ib-IIa &   \\
HR~6144 & ~330 & 7400  & 1.80   & 1.509 & 2.277 &       &  ~~0.169 & $-5.3$ &    0.000 & A7 Ib     & pAGB?\\  
HR~6536 & ~117 & 5160  & 1.86   & 0.323 & 0.524 & 2.598 &  ~~0.016 & $-2.6$ & $-$0.359 & G2 Ib-IIa & D \\   
HR~6978 & ~649 & 6000  & 1.70   & 0.780 & 0.383 & 2.636 &  ~~0.020 & $-4.4$ &    0.075 & F7 Ib     &    \\     
HR~7014 & 1000 & 6760  & 1.66   & 1.429 & 2.043 &       &  ~~0.226 & $-5.8$ &    0.195 & F2 Ib     & \\  
HR~7094 & ~855 & 6730  & 1.75   & 1.208 & 1.460 & 2.711 &  ~~0.227 & $-5.2$ & $-$0.026 & F2 Ib     & \\   
HR~7264 & ~156 & 6590  & 2.21   & 1.025 & 0.242 & 2.701 &  ~~0.042 & $-3.2$ & $-$0.140 & F2 II     & D,S \\   
HR~7387 & ~880 & 6700  & 1.43   & 1.406 & 1.976 & 2.715 &  ~~0.189 & $-6.7$ &    0.015 & F3 Ib     & D  \\      
HR~7456 & ~370 & 5550  & 2.06   & 0.459 & 0.421 & 2.611 &  ~~0.087 & $-2.2$ & $-$0.143 & G0 Ib     & D  \\      
HR~7542 & ~376 & 5750  & 2.15   & 0.605 & 0.426 & 2.633 &  ~~0.185 & $-2.2$ &    0.017 & F8 Ib-II  & pAGB?\\  
HR~7770 & ~960 & 6180  & 1.53   & 0.930 & 0.864 & 2.665 &  ~~0.329 & $-5.4$ &    0.080 & F5 Ib     & D \\     
HR~7796 & ~562 & 5790  & 1.02   & 0.818 & 0.425 & 2.645 &  ~~0.026 & $-6.6$ &    0.234 & F8 Ib     & V,S \\ 
HR~7823 & 1010 & 6760  & 1.92   & 1.294 & 1.674 & 2.729 &  ~~0.219 & $-4.7$ &    0.000 & F1 II     & \\      
HR~7834 & ~235 & 6570  & 2.32   & 1.022 & 0.259 & 2.691 &  ~~0.060 & $-3.1$ & $-$0.078 & F5 II     & V  \\   
HR~7847 & 1040 & 6290  & 1.44   & 1.022 & 1.044 & 2.684 &  ~~0.295 & $-5.9$ &    0.077 & F5 Iab    & \\  
HR~8232 & ~165 & 5490  & 1.86   & 0.537 & 0.481 & 2.600 &  ~~0.024 & $-3.3$ &    0.076 & G0 Ib     & D \\ 
HR~8313 & ~283 & 4910  & 1.58   & 0.203 & 0.721 & 2.593 & $-$0.032 & $-2.8$ & $-$0.069 & G5 Ib     & V \\
HR~8412 & ~284 & 5280  & 2.35   & 0.255 & 0.618 & 2.596 &  ~~0.024 & $-1.0$ & $-$0.066 & G5 Ia     & D \\      
HR~8414 & ~161 & 5210  & 1.76   & 0.386 & 0.562 & 2.596 &  ~~0.002 & $-3.1$ &    0.020 & G2 Ib     & D   \\      
HR~8692 & ~413 & 4960  & 1.90   & 0.210 & 0.662 & 2.597 & $-$0.008 & $-1.8$ & $-$0.087 & G4 Ib     & \\  
\bottomrule
\tabnotetext{}{Comments code: V-variable, D-Double, S-$uvby-\beta$ from Simbad, pAGB?- possible post-AGB star, RRL- RR Lyrae, SB-Spectroscopic Binary.}
\end{tabular}
%\vspace{0.2cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 1
  
\begin{figure}[!t]
\centering
\includegraphics[width=0.98\columnwidth, height=0.98\columnwidth]{aarellanofig1.eps}
\caption{Temperature dependence of the reddening free [$c_1$] index. The open circles correspond to five A type 
stars which, along with the K1 star HR 461, are not included in the fit. The fit is valid for F-G type stars. Filled 
circles represent variable stars, empty triangles double stars and filled triangles non-variables. Asterisks 
correspond to two possible post-AGB stars in the sample of Lyubimkov et al. (2010). The error bars correspond to 
typical uncertainties of $\pm 200$ K and $\pm 0.019$ mag in $T_{\rm eff}$ and [$c_1$], respectively.}
\label{te_bc1}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 

\section {The $T_{\rm \lowercase {eff}}$ calibrations}
\label{temp}

Plots of $T_{\rm eff}$ as a function of [$c_1$] and [$m_1$] are shown in Figures~\ref{te_bc1} and \ref{te_bm1}, respectively. Variable, 
double and non-variable stars have been plotted with different symbols to check whether their nature contributes to the scatter or if 
they appear as outliers. For the variable stars we used averages from rather scarce photometry not covering the full variational cycles; 
nevertheless, they follow the trends as well as the non-variable and double stars.


%%%%% FIGURA 2

\begin{figure}[!t]
\centering
\includegraphics[width=0.98\columnwidth, height=0.98\columnwidth]{aarellanofig2.eps}
\caption{Temperature dependence of the reddening free [$m_1$] index. Symbols are as in Figure~\ref{te_bc1}. The error 
bars correspond to typical uncertainties of $\pm 200$ K and $\pm 0.021$ mag in $T_{\rm eff}$ and [$m_1$], respectively.}
\label{te_bm1}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The $T_{\rm eff}$-[$c_1$] correlation is good for F-G type stars. Five A-type stars included in the sample of Lyubimkov et al. (2010): 
HR~825, HR~1740, HR~2874, HR~3183 and HR~6081, identified with open circles, and the K1 star HR 461 (Yoss 1961) were not included in 
the fit. The line has the form:
\begin{equation}
\label{TeBc1} 
T_{\rm eff} = 1566.6 (\pm 50.7) [c_1] + 4704.0 (\pm 40.5)\,,
\end{equation}
with a standard deviation of 152~K and a correlation coefficient of $R=0.98$.

We can confirm from the models of Lester, Gray, \& Kurucz (1986) that the $c_1$ index is temperature sensitive for $T_{\rm eff} \leq 7000$~K 
and that for hotter stars the sensitivity of the index changes and becomes more gravity dependent. This explains why the A-type stars do not 
follow the smooth correlation displayed by the F-G stars.

The $T_{\rm eff}$-[$m_1$] correlation shows a non-linear form that can be represented by:
$$T_{\rm eff} = 6081.3 (\pm 817.0) [m_1]^2 -9294.9 (\pm 731.8) [m_1]$$
\begin{equation}
\label{TeBm1}
+~8486.7 (\pm 149.5)\,.~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{equation}
\noindent As was done for equation~(1) the five A-type stars and the K1 star HR 461 were not included. The standard deviation is 152~K, the correlation 
coefficient $R=0.98$.

In order to explore a possible color dependence of the above calibrations, we have selected the dereddened color $(b-y)_0$. To calculate 
the reddenings for the F-G supergiants we used the calibration given by equation~(5) of Arellano Ferro \& Parrao (1990) which provides 
$E(b-y)$ from ($b-y)$, $c_1$ and $m_1$ indices with an accuracy of about $\pm 0.03$ mag. The color excesses so calculated are given in 
Column~8 of Table~\ref{T1}. Since most of these supergiants have low reddening, and given the scatter in the reddening calibration, some 
small negative values are expected. These negative values should be taken as indicating zero reddening. The resulting equations with the 
color term included have slightly smaller standard deviations and higher correlation coefficients. They are:
$$T_{\rm eff} = 501.6 (\pm 255.1) [c_1] -1980.1 (\pm 467.3) (b-y)_0$$
\begin{equation}
\label{Tec1by}
+~6252.0 (\pm 367.0)\,,~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
\end{equation}
\noindent with the standard deviation being 129~K and the correlation coefficient $R=0.98$, and
$$T_{\rm eff} = 2244.0 (\pm 958.6) [m_1]^2 -2350.0 (\pm 1407.1) [m_1]$$
\begin{equation}
\label{Tem1by}
-~2244.0 (\pm 454.7) (b-y)_0 + 7375.8 (\pm 236.8)\,.
\end{equation}
\noindent with the standard deviation being 119~K and the correlation coefficient $R=0.99$.

At this point it is important to remark that since Lyubimkov et al. (2010) used [$c_1$], among other indices, to estimate $T_{\rm eff}$, the 
correlations in Figures~\ref{te_bc1} and \ref{te_bm1} are expected. Naturally, the calibrations of equations~(1) to (4) are expressed in 
terms of the temperature scale of Lyubimkov et al. (2010). What it is offered here are functional relations that can be comfortably used to 
estimate $T_{\rm eff}$ in F-G supergiants from the $c_1$ photometry, with comparable accuracies. 

The H$\beta$ index is also linearly correlated with the temperature. For the linear fit between H$\beta$ and $T_\mathrm{eff}$, using the 
47 F-G stars with H$\beta$ in Table~\ref{T1}, the standard deviation is 234 K and the correlation coefficient is $R=0.93$. Since this 
calibration is of less quality than the four represented by equations~(\ref{TeBc1}) to (\ref{Tem1by}), we have not illustrated it. 


%%%%%% FIGURA 3

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth, height=0.99\columnwidth]{aarellanofig3.eps}
\caption{Dependence of gravity on the index $\Delta [c_1]$. Symbols are as in Figure~\ref{te_bc1}. The segmented line 
represents the calibration of Arellano Ferro \& Mendoza (1993), shifted by 0.3--0.5 dex towards lower gravities. See 
text for discussion. The error bars correspond to typical uncertainties of $\pm 0.25$~dex and $\pm 0.019$~mag in log$~g$ 
and $\Delta [c_1]$, respectively.}
\label{te_Dc1}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{The \lowercase {log}~$\lowercase {g}$ calibration}
\label{grav}

Calibrations of the surface gravity in yellow supergiants and bright giants have been performed in the past by Gray (1991) and Arellano Ferro 
\& Mendoza (1993). The strategy was to use the [$m_1$]-[$c_1$] plane, on which the locus of a ``standard line'', determined from F-G Ib 
supergiants exclusively, was defined by Gray (1991). Then, a vertical parameter; $\Delta [c_1] = [c_1]-[c_1](standard~line)$, can be measured. 
It has been shown that $\Delta [c_1]$ is correlated with log~$g$. Arellano Ferro \& Mendoza (1993) used the spectroscopic gravities for 27 F-G 
supergiants given by Luck \& Bond (1989) to calibrate the correlation. In this work we attempt a new calibration using the new values 
of log~$g$ given by Lyubimkov et al. (2010). The value of $\Delta [c_1]$ for each star in the sample is given in Column~10 of Table~\ref{T1}. 
Figure~\ref{te_Dc1} shows the $\Delta [c_1]$ dependence on log~$g$. As before, the five A type and the K1 star in the sample of Lyubimkov et al. 
(2010) have not been included. Among the F-G stars three outliers were noted in the $\Delta [c_1]$-log~$g$ plane; HR~1242 (F0~II) and HR~3073 
(F1~Ia), both of which are non-variables, and the double star HR~6536. We have no ready explanation for their discordant position, and they were 
ignored. The remaining 47 stars display a clear relationship and the straight line has the form:
\begin{equation}
\label{gc1}
{\rm log}~g = -2.836 (\pm 0.349) \Delta [c_1] + 1.841 (\pm 0.040),
\end{equation}  
\noindent with the standard deviation being 0.267 dex and the correlation coefficient $R=0.77$.


%%%%% TABLA 2

\begin{table*}[!t]
\vspace{0.1cm}\
\centering
\caption[]{Physical parameters and photometric indices for\\ a sample of \lowercase{post}-AGB and RV Tau stars}
\label{T2}
\small
\begin{tabular}{lrccrccl}
\toprule
\multicolumn{1}{c}{Star} & $T_{\rm eff}$ & log $g$ & [$c_1$] & [$m_1$] & \multicolumn{1}{c}{$\Delta [c_1]$} & Sp. T. & Comment \\
                         & (K)         & (dex)   &         &         &                                    &        &  \\
\midrule  
HD~46703  & 6000 & 0.40 & 1.069 & 0.165 & $-$0.463 & F7~IVw    & pAGB\\      
HD~56126  & 7167 & 0.67 & 1.375 & 0.393 &    0.700 & F5~Iab:   & pAGB\\      
HD~95767  & 7300 & 1.30 & 1.111 & 0.266 &    0.049 & F3~II     & pAGB\\    
HD~107369 & 7600 & 1.50 & 1.575 & 0.119 & $-$0.213 & A2~II/III & pAGB\\       
HD~108015 & 6800 & 1.25 & 1.187 & 0.237 &    0.000 & F4~Ib/II  & pAGB\\      
HD~112374 & 6600 & 0.80 & 1.057 & 0.272 &    0.020 & F3~Ia     & pAGB\\         
HD~148743 & 7200 & 0.50 & 1.509 & 0.169 &    0.000 & A7~Ib     & pAGB\\    
HD~161796 & 6850 & 0.37 & 1.447 & 0.251 &    0.312 & F3~Ib     & pAGB\\
HD~163506 & 6550 & 0.60 & 1.309 & 0.193 & $-$0.085 & F2~Ibe    & pAGB\\      
HD~172481 & 7250 & 1.50 & 1.297 & 0.164 & $-$0.257 & F2~Ia0    & pAGB\\
HD~179821 & 6775 & 0.90 & 1.003 & 0.601 &    0.675 & G5~Ia     & pAGB\\        
HD~190390 & 6600 & 1.60 & 1.071 & 0.190 & $-$0.338 & F1~III    & pAGB\\
\midrule    
HD~170756 & 5900 & 1.13 & 0.679 & 0.534 &    0.281 & K0~III      & RV Tau\\     
AR~Pup    & 6300 & 1.50 & 0.879 & 0.328 &    0.025 & F0~Iab      & RV Tau\\   
IW~Car    & 6700 & 2.00 & 1.211 & 0.270 &    0.160 & A4~Ib/II    & RV Tau\\      
RU~Cen    & 6000 & 1.00 & 0.741 & 0.457 &    0.234 & F7/F8;A4~Ib & RV Tau\\     
EN~TrA    & 6150 & 1.25 & 0.968 & 0.312 &    0.074 & F2~Ib       & RV Tau\\
\bottomrule
\end{tabular}
\vspace{0.4cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Since there are evidences that log~$g$ depends on $T_{\rm eff}$ for a given luminosity class (Straizys \& Kuriliene 1981) we attempted to 
incorporate the color term ($b-y)_0$ but found it to be insignificant.

Equation~(\ref{gc1}), when compared with equation~(5) of Arellano Ferro \& Mendoza (1993), has smaller dispersion and a slightly higher 
correlation coefficient. However, we should note that the present calibration indicates lower gravities by 0.3--0.5 dex for a given 
$\Delta [c_1]$. This invited a comparison of the gravities used by Arellano Ferro \& Mendoza (1993), taken from Luck \& Bond (1989), with 
the gravities used in the present work from Lyubimkov et al. (2010) which are listed in Table~\ref{T1}. There are ten F and early G type 
stars in common; Figure~\ref{LGvsLG} shows that the gravities of Luck \& Bond (1989) are systematically smaller. Luck \& Bond (1989) found 
that their spectroscopic gravities, obtained mostly from the ionization equilibrium condition on Fe~I and Fe~II lines, are systematically 
smaller by 0.3 dex than gravities derived via the membership of stars in clusters and their photometry. Similar discrepancies were found by 
Luck \& Lambert (1985) for classical cepheids when spectroscopic gravities were compared with gravities obtained via PLC relationship. These 
authors forward a possible explanation as due to the fact that models built under the assumption of hydrostatic equilibrium may not be a 
good representation for the extended atmospheres of supergiant stars. The reader is referred to the detailed discussions on this point given 
in the above papers, and similar arguments might be invoked to explain the discrepancies displayed in Figure~\ref{LGvsLG}. 


%%%%% FIGURA 4

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth, height=0.99\columnwidth]{aarellanofig4.eps}
\caption{Comparison of log~$g$ values for ten stars in common between Lyubimkov et al. (2010) and Luck \& Bond 
(1989). See text for discussion.}
\label{LGvsLG}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{The case of \lowercase{post}-AGB stars}
\label{pagb}

We often have wondered if the calibrations of physical parameters that have been worked out for young yellow supergiants would be equally valid 
for more evolved and less massive post-AGB stars of similar temperature and gravity. We found a few well known post-AGB and RV Tau stars in 
the catalogue of Stasi\'nska et al. (2006) with reported values of effective temperature and gravity estimated by spectroscopic techniques, and 
with Str\"omgren photometry available in the Simbad database. These stars are listed in Table~\ref{T2} and plotted in Figure~\ref{pAGB} with 
blue and red symbols on the [$c_1$]-$T_{\rm eff}$ and [$m_1$]-$T_{\rm eff}$ planes of Figures~\ref{te_bc1} and \ref{te_bm1}. The stars are rather faint, 
and their spectral types may be difficult to determine with high accuracy. For this reason, and to avoid too small a sample we included several 
late A stars and one K0 star. For the [$c_1$] index the post-AGB and RV Tau stars follow very well the trend defined by the F-G 
supergiants and, therefore, it seems that the calibration is also valid for these types of stars stars. We have refrained from fitting the 
straight line with all points since the difference with equation~(1) would be negligible. For the [$m_1$] index the post-AGB's and RV Tau also 
follow the parabola of the F-G supergiants, except for three stars HD~46703 (F7~IVw), HD~56126 (F5~Iab:) and HD~179821 (G5~Ia). HD~46703 has the 
spectral type of a subgiant, which may explain its peculiar position. However, for HD~56126 and HD~179821 we have no explanation at hand.


%%%%% FIGURA 5

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth, height=0.99\columnwidth]{aarellanofigura5.eps}
\caption{Inclusion of post-AGB and RV Tau stars in the $T_{\rm eff}$ calibrations. The black symbols are as in Figure~1. 
The red circles are post-AGB stars and the blue circles are RV Tau stars. The red triangles are the two post-AGB 
candidate stars HR~6144 and HR~7542 plotted as asterisks in Figures~1,~2~and~3.}
\label{pAGB}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Let us now explore, in a similar way, the role of post-AGB stars on the $\Delta [c_1]$-log~$g$ plane. As for the case of the young supergiants 
we started by plotting the stars on the [$m_1$]-[$c_1$] diagram from which one can estimate $\Delta [c_1]$. Figure~\ref{pAGBlg} illustrates both 
these planes. Except for the already noted outliers HD~46703, HD~56126 and HD~179821, both the post-AGB and the RV Tau stars follow the 
distribution on the [$m_1$]-[$c_1$] plane. However, on the $\Delta [c_1]$-log~$g$ plane the post-AGB stars are at odds with the distribution 
of the supergiant stars. The five RV Tau stars (blue circles) seem to follow the supergiant trend rather well. With the available information 
we cannot say whether the post-AGB stars follow a completely different relationship, or if this is a result of the gravities reported by 
Stasi\'nska et al. (2006) being anomalously too small. In either case it is clear that the relationship in equation~(5) should be used only for 
the young supergiant stars.


\section{Comments on the determination\\ of $M_V$}
\label{MV}

It would be desirable to posses a calibration that could predict the luminosity of young yellow supergiants given an observational parameter, 
such as for example a color index. Since these stars are luminous they could be used as distance indicators. We have attempted before to 
calculate such calibration (Arellano Ferro \& Parrao 1990; Arellano Ferro \& Mendoza 1993). However, an accurate result has been rather elusive. 
On the other hand a neat calibration of $M_V$ from the intensity of the OI 7774 triplet, valid for a vast range of luminosities, was obtained 
by Arellano Ferro, Giridhar, \& Rojo Arellano (2003), which predicts $M_V$ with an uncertainty of 0.38~mag.

The distances provided by Lyubimkov et al. (2010) could in principle serve to explore a calibration of $M_V$ in terms of a photometric index. 
The absolute magnitudes listed in Column~9 of Table~\ref{T1} were calculated from the distances (Column~2) and the reddenings (Column~8). Since 
$\Delta [c_1]$ is correlated with log~$g$ one would expect a relationship between $M_V$ and $\Delta [c_1]$. In Figure~\ref {MVDc1} the plot 
of $\Delta [c_1]$ vs $M_V$ is shown. Although a trend is clearly seen, the scatter is large ($\sigma$ = 1.3 mag). Inclusion of the color term, 
$(b-y)_0$ does not improve the relationship ($\sigma$ = 1.1 mag).


%%%%% FIGURA 6

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth, height=0.99\columnwidth]{aarellanofigura6.eps}
\caption{Inclusion of post-AGB and RV Tau stars in the log~$g$ calibrations. The top panel shows the [$m_1$]-[$c_1$] 
plane and the fiducial line definded by Gray (1991), relative to which the $\Delta [c_1]$ parameter is measured. The 
lower panel shows the distribution of the different groups of stars in the $\Delta [c_1]$- log~$g$ plane. Symbols are 
as in Figure~\ref{pAGB}. See text for discussion.}
\label{pAGBlg}
\vspace{0.35cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We find of interest to consider a sample of F-G supergiants and 5 well documented post-AGB stars listed by Arellano Ferro et al. (2003) (their 
Table 3) for which values of $M_V$ have been calculated from the O~I~7774 calibration. $uvby-\beta$ photometry for all these stars is available 
in the {\it Simbad} data base and thus we can calculate their $\Delta [c_1]$ parameters. They are plotted in Figure~\ref {MVDc1} (blue and red 
symbols) along with the F-G supergiants of Table~\ref{T1}. We see that both groups of stars follow the same trend and display a similar scatter. 
One can conclude from this that while the distances in Lyubimkov et al. (2010) are consistent with the absolute magnitudes predicted from the 
strength of the O~I~7774 feature in F-G supergiants, a reliable calibration of $M_V$ in terms of a photometric index, for example $\Delta [c_1]$, 
is not foreseen.


%%%%% FIGURA 7

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth, height=0.99\columnwidth]{aarellanofigura7.eps}
\caption{Trend of $M_V$ with $\Delta [c_1]$. The black symbols are as in Figure~\ref{te_bc1}. The blue circles correspond 
to a sample of F-G supergiants and the red circles to a well documented post-AGB stars taken from Table~3 of Arellano Ferro 
et al. (2003), where values of $M_V$ were calculated from the O~I~7774 feature.}
\label{MVDc1}
\vspace{0.4cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Conclusions}
\label{concl}

New accurate values of $T_{\rm eff}$ and log~$g$ (Lyubimkov et al. 2010) lead to reliable functional relationships for the Str\"omgren reddening free 
indices [$c1$] and [$m1$], which allow to predict the effective temperature and gravity of F-G supergiants. The temperature calibrations are given 
in equations~(\ref{TeBc1}) and (\ref{TeBm1}) which predict the effective temperature with an standard deviation of 152~K. If the reddening is 
available, alternative calibrations, given by equations~(\ref{Tec1by}) and (\ref{Tem1by}) provide slightly smaller standard deviations (129 and 
119~K respectively). These calibrations are valid for the more evolved post-AGB stars of similar temperature and gravity. Given the fact that RV Tau 
stars are variables of large amplitude, it is rewarding to see that the five RV Tau stars included follow the temperature trends very closely. 
Perhaps their mean temperature and photometric indices are not too different from the temperatures reported by Stasi\'nska et al. (2006) and the 
mean photometric indices calculated here.

The stellar gravity can also be predicted from the  $\Delta [c_1]$ index (Gray 1991) from the calibration given by equation~(\ref{gc1}), with an 
accuracy 0.28 dex. The color term was found to be insignificant. This equation is valid for F-G yellow supergiants and very likely for RV Tau stars. 
However, well known post-AGB stars of similar temperature and gravities (see Table~2) do not follow the calibration.

We have not been able to determine a reliable calibration that can predict the absolute magnitude $M_V$ for F-G supergiant stars from photometric 
indices as accurately as from the strength of the OI 7774 feature.


\acknowledgments

Numerous comments and suggestions made by Prof. Sunetra Giridhar and an anonymous referee are thankfully acknowledged. This work was 
supported by DGAPA-Universidad Nacional Aut\'onoma de M\'exico grant through project IN114309. This research has made extensive use of 
the {\it Simbad} data base operated at CDS, Strasbourg, France and of the NASA ADS Astronomy Query Form.


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\end{document}
