\documentclass[journal]{rmaa}


\title{Orbital elements for eight binaries.\\  Study of the nature of wide components. I}


\author{  
F.~M. Rica Romero\altaffilmark{1,2}}

\altaffiltext{1}{Agrupaci\'on Astron\'omica de M\'erida, Spain.}
\altaffiltext{2}{Liga Iberoamericana de Astronom\'ia (LIADA), Argentina.}


\shortauthor{Rica}

\shorttitle{Orbital elements for eight binaries. I}


\fulladdresses{
\item Francisco Rica Romero: Agrupaci\'on Astron\'omica de M\'erida, C/Jos\'e Ruiz Azor\'in, 
14, 4 D, 06800, M\'erida, Spain and Liga Iberoamericana de Astronom\'ia (LIADA), Av. Almirante 
Guillermo Brown, 4998, 3000 Santa Fe, Argentina (frica0@gmail.com).}


\listofauthors{F.~M.~Rica~Romero}

\indexauthor{Rica Romero, F.~M.}


\ReceivedDate{2010 March 25}
\AcceptedDate{2010 June 5}
\SetYear{2010}
\SetVolume{46}


\resumen{Se calcularon nuevos par\'ametros orbitales y masas para ocho binarias. Las \'orbitas, en la mayor\'ia de estas
binarias, mostraron grandes residuos que aconsejaban obtener nuevos par\'ametros orbitales. Para determinar los par\'ametros
orbitales iniciales se us\'o el m\'etodo de Thiele-van den Bos modificado, para posteriormente usar el m\'etodo de correcci\'on
diferencial de Heintz para obtener los par\'ametros orbitales finales. El m\'etodo Thiele-van den Bos fue modificado para que
un valor aproximado de la constante areal permitiera obtener buenos resultados. La naturaleza de las componentes distantes fue 
estudiada mediante fotometr\'ia $BVIJHK$ y datos cinem\'aticos y astrom\'etricos. Se realiz\'o una b\'usqueda de nuevas 
compa\~neras alrededor de las binarias estudiadas.}

\abstract{New orbital parameters and masses for eight binaries were determined. Most of these were noted as being in need of 
correction due to large residuals to the previous best orbit. A modified Thiele-van den Bos method was used to obtain initial 
orbits which were improved using the differential correction method of Heintz. The Thiele-van den Bos method was modified so 
that an approximate value of the areal constant allowed good results to be obtained. The nature of wide components was studied 
using $BVIJHK$ photometry, historical astrometry and kinematical data. A search for new unreported companions around the binaries 
was carried out.\smallskip}


\addkeyword{binaries: general}
\addkeyword{binaries: visual}
\addkeyword{instrumentation: high angular resolution}
\addkeyword{techniques: high angular resolution}


\begin{document}

\maketitle

\section{INTRODUCTION}
\label{sec:intro}

As a result of part of the work on visual double stars carried out by the Double Star Section of Liga Iberoamericana de Astronom\'ia 
(LIADA) I present orbital parameters and other data (residuals, ephemerides, masses, parallaxes, apparent orbits, etc.) 
for eight visual binary systems: WDS03272+0944 (HDS 433), WDS 04070-1000  (HDS 521), WDS06345-1114 (HO 234), WDS09228-0950 AB (A 1342 AB), 
WDS12274-2843 (B 228), WDS18410+2450 (A 2988), WDS 21459+1153 AB (A 1223 AB) and WDS 23026+4245 AB (BU 1147 AB). For HDS 433 this is 
the first orbit published.

Initial orbits were calculated using the method of Thiele-van den Bos and were then improved using the differential correction method 
of Heintz (1978). Dynamical parallaxes were calculated using the Baize-Romani algorithm (1946). Bolometric corrections were taken from 
Allen's tables (1973). The nature of wide components was studied using $BVIJHK$ photometry, historical astrometry and kinematical data.

The astrophysical properties of the binaries were determined and discussed. All the orbits presented here, except that of HDS 521, have 
previously been announced in the Information Circulars of IAU Commission 26 (hereafter IAUDS). The orbital parameters listed in this paper 
are the final corrected versions and are slightly different from those announced in IAUDS.


%%%%% TABLA 1

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Orbital parameters, parallaxes and residuals}
\label{tab:Orbits}
\small
\begin{tabular}{lcccc}
\toprule
Name & HDS 433 & HDS 521 & HO 234 & A1342 \\
WDS  & 03272+0944 & 04070-1000 & 06345-1114 & 09228-0950 \\
ADS  & \nodata & \nodata & 5212 & 7334 \\
HIP  & 16083   & 19206 &  31356 & 45999\\
\midrule
Disc. date & 1983.7134 & 1991.25 & 1888.64 & 1905.99 \\
P [years] & $51.923\pm 15.054$ & $22.467\pm 1.812$ & $374.882\pm 153.887$ & $53.199\pm
0.193$ \\
T [BY] & $2008.282\pm 0.502$ & $1996.723\pm 0.230$ & $1912.738\pm 1.133$ &
$1957.346\pm 0.572$ \\
e & $0.568\pm 0.077$ & $0.678\pm 0.044$ & $0.345\pm 0.068$ & $0.038\pm 0.037$ \\
a [$''$] & $0.441\pm 0.027$ & $0.221\pm 0.017$ & $0.694\pm 0.048$ & $0.165\pm 0.011$ \\
i [$^{\circ}$] & $22.7\pm 5.1$ & $123.1\pm 5.1$ & $55.8\pm 3.0$ & $67.1\pm 4.1$  \\
$\omega$ [$^{\circ}$] & $3.0\pm 6.3$ & $255.0\pm 4.6$ & $179.3\pm 1.7$ & $173.6\pm 4.6$ \\
$\Omega$ [$^{\circ}$] & $112.1\pm 4.4$ & $37.0\pm 3.1$ & $35.9\pm 3.4$ & $23.7\pm 3.6$ \\
Residuals:   &   &   &   &    \\
RMS($\theta$)[$^{\circ}$] & 1.06 & 2.83 & 1.35 & 3.59 \\
RMS($\rho$) [$''$] & 0.007 & 0.006 & 0.013 & 0.009 \\
MA($\theta$)[$^{\circ}$] & 0.77 & 0.57 & 0.92 & 1.86 \\
MA($\rho$)[$''$] & 0.005 & 0.001 & 0.005 & 0.005 \\
a$^{3}$/P$^{2}$ & $7.21370 \times 10^{-5}$ & $10.59844 \times 10^{-5}$ & $4.42712 \times
10^{-6}$ & $9.61969 \times 10^{-6}$ \\
\noalign{\medskip}
\midrule
Name & B 228 & A 2988 & A 1223 AB & BU 1147 \\
WDS & 12274-2843 & 18410+2450 & 21459+1153 & 23026+4245 \\
ADS & 8555 & 11574 & 15300 & 16467 \\
HIP & 60775 & 91609 & 107461 & 113788 \\
\midrule
Disc. date & 1926.77 & 1916.43 & 1905.53 & 1889.54\\
P [years] & $44.483\pm 6.249$ & $57.172\pm 1.168$ & $44.046\pm 1.374$ & $73.997\pm 0.509$ \\
T [BY] & $1952.295\pm 1.251$ & $1941.464\pm 1.520$ & $1972.549\pm 1.175$ & $1870.280\pm 0.595$ \\
e & $0.666\pm 0.077$ & $0.688\pm 0.101$ & $0.324\pm 0.107$ & $0.800\pm 0.056$ \\
a [$''$] & $0.173\pm 0.107$ & $0.102\pm 0.019$ & $0.123\pm 0.018$ & $0.225\pm 0.011$ \\
i [$^{]circ}$] & $63.6\pm 12.0$ & $154.7\pm 19.0$ & $160.7\pm 40.4$ & $21.7\pm 46.0$  \\
$\omega$ [$^{\circ}$] & $164.9\pm 12.6$ & $306.8\pm 11.5$ & $154.1\pm 8.8$ & $356.4\pm 3.0$  \\
$\Omega$ [$^{\circ}$] & $137.8\pm 10.6$ & $329.6\pm 9.7$ & $14.8\pm 8.3$ & $159.5\pm 2.0$ \\
Residuals:  &  &  &  &   \\
RMS($\theta$)[$^{\circ}$] & 4.15 & 4.54 & 4.89 & 2.40 \\
RMS($\rho$) [$''$] & 0.011 & 0.018 & 0.023 & 0.016 \\
MA($\theta$)[$^{\circ}$] & 2.73 & 3.27 & 3.71 & 1.48 \\
MA($\rho$)[$''$] & 0.007 & 0.011 & 0.011 & 0.007 \\
a$^{3}$/P$^{2}$ & $1.51253 \times 10^{-5}$ & $3.18298 \times 10^{-6}$ & $7.78058 \times 10^{-6}$ & $9.24564 \times 10^{-6} $ \\
\bottomrule
\end{tabular}
\vspace{0.4cm}
\end{table*}

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


%%%%%% TABLA 2

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Ephemerides}
\label{tab:ephemerid}
\setlength{\tabnotewidth}{0.95\textwidth}
\setlength{\tabcolsep}{1.2\tabcolsep}
\tablecols{12}
\small
\begin{tabular}{cccccccccccc}
\toprule
& \multicolumn{2}{c}{03272+0944}  & & \multicolumn{2}{c}{04070-1000} & &
\multicolumn{2}{c}{06345-1114} & & \multicolumn{2}{c}{09228-0950} \\
\cmidrule{2-3}
\cmidrule{5-6}
\cmidrule{8-9}
\cmidrule{11-12}
& $\theta$ & $\rho$ & &$\theta$ &$\rho$ & &$\theta$ &$\rho$ & &$\theta$ &$\rho$ \\
DATE &(deg) &(arcsec) &  &(deg) & (arcsec) & &(deg) & (arcsec) & & (deg) & (arcsec) \\
\midrule
2010.0 & 160.9 & 0.207 & & 357.7 & 0.240 & & 0.8 & 0.597 & & 199.6 & 0.156\\
2011.0 & 182.5 & 0.231 & & 278.6 & 0.169 & & 1.4 & 0.604 & & 202.5 & 0.158\\
2012.0 & 199.5 & 0.262 & & ~~6.2 & 0.236 & & 2.1 & 0.611 & & 205.3 & 0.158\\
2013.0 & 212.7 & 0.296 & & 294.6 & 0.179 & & 2.8 & 0.617 & & 208.2 & 0.156\\
2014.0 & 223.1 & 0.332 & & ~15.5 & 0.220 & & 3.4 & 0.624 & & 211.2 & 0.152\\
2015.0 & 231.5 & 0.367 & & 308.8 & 0.191 & & 4.0 & 0.630 & & 214.5 & 0.146\\
2016.0 & 238.4 & 0.401 & & ~27.3 & 0.182 & & 4.6 & 0.637 & & 218.0 & 0.138\\
2017.0 & 244.3 & 0.434 & & 321.2 & 0.204 & & 5.2 & 0.643 & & 222.0 & 0.129\\
2018.0 & 249.4 & 0.464 & & ~52.4 & 0.098 & & 5.8 & 0.650 & & 226.7 & 0.118\\
\midrule
& \multicolumn{2}{c}{12274-2843} & & \multicolumn{2}{c}{18410+2450} & &
\multicolumn{2}{c}{21459+1153} & & \multicolumn{2}{c}{23026+4245}\\
\cmidrule{2-3}
\cmidrule{5-6}
\cmidrule{8-9}
\cmidrule{11-12}
& $\theta$ & $\rho$ & & $\theta$ & $\rho$ & & $\theta$ & $\rho$ & & $\theta$ & $\rho$ \\
DATE & (deg) & (arcsec) & & (deg) & (arcsec) & & (deg) & (arcsec) & & (deg) & (arcsec) \\
\midrule
2010.0 & 254.8 & 0.036 & & 236.7 & 0.113 & & 312.5 & 0.105 & & ~13.3 & 0.207\\
2011.0 & 108.1 & 0.146 & & 233.5 & 0.118 & & 301.9 & 0.099 & & ~16.8 & 0.190\\
2012.0 & 127.1 & 0.265 & & 230.5 & 0.123 & & 289.9 & 0.093 & & ~20.8 & 0.172\\
2013.0 & 137.5 & 0.271 & & 227.8 & 0.128 & & 276.4 & 0.088 & & ~25.6 & 0.154\\
2014.0 & 156.2 & 0.138 & & 225.2 & 0.132 & & 261.6 & 0.084 & & ~31.6 & 0.135\\
2015.0 & ~67.6 & 0.062 & & 222.8 & 0.136 & & 245.6 & 0.082 & & ~39.9 & 0.114\\
2016.0 & 121.1 & 0.226 & & 220.6 & 0.140 & & 229.2 & 0.082 & & ~53.8 & 0.089\\
2017.0 & 132.9 & 0.283 & & 218.4 & 0.143 & & 212.8 & 0.083 & & ~82.3 & 0.062\\
2018.0 & 144.8 & 0.220 & & 216.4 & 0.146 & & 197.1 & 0.085 & & 137.7 & 0.046\\
\bottomrule
\end{tabular}
\vspace{0.4cm}
\end{table*}

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


\section{METHOD OF ORBITAL CALCULATION}
\label{sec:method}

Before using any analytical method $\theta$ was corrected for precession and proper motion. $\theta$ and $\rho$ were plotted against time 
which allowed the detection of measures with important errors or quadrant problems. Measures with the largest errors were assigned zero weight.

Preliminary orbits were determined by the method of Thiele-van den Bos which was then improved by the author. Three base points and a crude 
value of the areal constant \textit{c} are needed. A set of Keplerian orbits changing the initial value of the areal constant in an appropiate 
range (i.e., 50 percent of the initial value of \textit{c}) was calculated. All the apparent orbits pass through the three given points. 
Only periodic orbits were considered. The step size and the search range for \textit{c} are customizable by the user.

The orbit with minimum RMS residuals was retained. In the next iteration the search range and the step size for the new value of 
\textit{c} were reduced. The process was repeated until the result for two iterations did not show a significant difference. Two or three 
iterations were sufficient for most of the binaries. The difference between the initial estimate of \textit{c} and the final value of \textit{c} 
was less than 15 percent for most binaries.

If a set of Keplerian orbits showed a flat gradient for the RMS residuals then those orbits were rejected by comparing of the mass calculated 
by dynamical parameters, with that determined using spectral types.

The three base points have to be chosen carefully where the observational data seem most reliable considering instrumentation, data density, 
or critical arc coverage. I also tried to cover as much of the observed arc as possible. This may allow the area around a single observation 
to represent a base point without additional observational coverage.

Initial orbits determined by the Thiele-van den Boss method were improved using the Heintz differential correction method (1978). The initial weights 
for measures were assigned using a data weighting scheme based on Hartkopf, McAlister, \& Franz (1989), Mason, Douglass, \& Hartkopf (1999a), 
Seymour et al. (2002), Docobo \& Ling (2003). The initial $\theta$ weights were 5 times larger than the $\rho$ weights (Heintz 1978) for visual measures. 
After several iterations in the differential correction process the measures with residuals larger than $3\sigma$  were assigned zero weight. 
Later, the non-zero weight measures were reassigned following the work of Irwin, Yang, \& Walker (1996).


%%%%%%% TABLA 3

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Stellar Data}
\label{tab:stellardata}
%\setlength{\tabnotewidth}{0.95\textwidth}
\setlength{\tabcolsep}{0.97\tabcolsep}
\tablecols{13}
\footnotesize
\begin{tabular}{ccccclccccccc}
\toprule
\multicolumn{2}{c}{Star}  &  & \multicolumn{3}{c}{WDS}  &  & Present Work  &  &
\multicolumn{4}{c}{Hipparcos} \\
\cmidrule{1-2}
\cmidrule{4-6}
\cmidrule{8-8}
\cmidrule{10-13}
   &     &  &               &              & \multicolumn{1}{c}{Spectral} &   & $\pi$ (Dynamical) &   &                &                
& $\pi$ (Trigonometric) & $\sum$ \\
WDS & HIP &  & $m_\mathrm{A}$ & $m_\mathrm{B}$ & \multicolumn{1}{c}{Type}     &   & (arcsec)          
&   & $H_\mathrm{pA}$ & $H_\mathrm{pB}$  & (arcsec)              & $M_{\odot}$ \\
\midrule
03272+0944  & ~16083 &  & 3.74 & 7.55 & B9Vn &  & \nodata &  & 3.74 & 7.55 & $0.01468\pm0.00101$ & $10.1~\pm6.4~$\\
04070-1000  & ~19206 &  & 7.29 & 8.60 & G0   &  & 0.0240  &  & 7.29 & 8.60 & $0.02400\pm0.00092$ & $~1.55\pm0.43$\\
06345-1114  & ~31356 &  & 7.73 & 9.05 & F0   &  & 0.0093  &  & 8.22 & 8.31 & $0.01046\pm0.00200$ & $~2.08\pm2.11$\\
09228-0950  & ~45999 &  & 6.99 & 7.84 & A3V  &  & 0.0069  &  & 6.99 & 7.84 & $0.00720\pm0.00080$ & $~4.25\pm1.66$\\
12274-2843  & ~60775 &  & 8.18 & 8.39 & A9V  &  & 0.0095  &  & 8.18 & 8.39 & $0.00988\pm0.00096$ & $~2.71\pm5.15$\\
18410+2450  & ~91609 &  & 8.08 & 9.08 & F0V  &  & 0.0042  &  & 8.08 & 9.08 & $0.00441\pm0.00085$ & $~3.79\pm3.07$\\
21459+1153  & 107461 &  & 9.77 & 9.60 & G0   &  & 0.0071  &  & 9.77 & 9.60 & $0.00723\pm0.00135$ & $~2.54\pm6.78$\\
23026+4245  & 113788 &  & 5.19 & 7.70 & A3Vn &  & 0.0070  &  & 5.26 & 7.43 & $0.00774\pm0.00051$ & $~4.49\pm1.11$\\
\bottomrule
\end{tabular}
\vspace{0.3cm}
\end{table*}

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


%%%%% TABLA 4

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Statistical Results}
\label{tab:statistical}
\setlength{\tabnotewidth}{0.95\textwidth}
\setlength{\tabcolsep}{1.2\tabcolsep}
\tablecols{10}
\small
\begin{tabular}{clclcccccc}
\toprule
\multicolumn{2}{c}{Star} &  & \multicolumn{1}{c}{Authors}  &  & \multicolumn{2}{c}{RMS} &  &
\multicolumn{2}{c}{MA} \\
\cmidrule{1-2}
%\cmidrule{4}
\cmidrule{6-7}
\cmidrule{9-10}
WDS & \multicolumn{1}{c}{Name}  &   &   &   & $\delta\theta(^{\circ})$ & $\delta\rho('') $  &   &
$\delta\theta(^{\circ})$ & $\delta\rho('')$ \\
\midrule
03272+0944 & HDS 433   &  & Rica (this paper)                  &  & ~~1.06 & 0.007 &  & ~~0.77 & 0.005\\
04070-1000 & HDS 521   &  & Rica (this paper)                  &  & ~~2.83 & 0.006 &  & ~~0.57 & 0.001\\
           &           &  & Balega et al. (2006)               &  & ~~4.29 & 0.007 &  & ~~0.97 & 0.002\\
06345-1114 & HO 234    &  & Rica (this paper)                  &  & ~~1.35 & 0.013 &  & ~~0.96 & 0.005\\
           &           &  & Heintz (1979)                      &  & ~~5.85 & 0.099 &  & ~~3.06 & 0.088\\
           &           &  & Baize (1958)                       &  & ~45.38 & 0.177 &  & ~39.44 & 0.170\\
09228-0950 & A 1342 AB &  & Rica (this paper)                  &  & ~~3.59 & 0.009 &  & ~~1.86 & 0.005\\
           &           &  & Starikova (1981)                   &  & ~~5.00 & 0.022 &  & ~~4.12 & 0.019\\
           &           &  & Finsen (1976)                      &  & ~45.52 & 0.018 &  & ~21.14 & 0.015\\
           &           &  & Ekenberg (1945)                    &  & 121.04 & 0.032 &  & 107.12 & 0.023\\
12274-2843 & B 228     &  & Rica (this paper)                  &  & ~~4.15 & 0.011 &  & ~~2.73 & 0.007\\
           &           &  & Heintz (1981)                      &  & ~42.89 & 0.069 &  & ~18.72 & 0.058\\
18410+2450 & A 2988    &  & Rica (this paper)                  &  & ~~4.54 & 0.018 &  & ~~3.27 & 0.011\\    
           &           &  & Starikova (1981)                   &  & ~~5.69 & 0.033 &  & ~~4.40 & 0.028\\
           &           &  & Baize (1955)                       &  & ~85.88 & 0.036 &  & ~61.84 & 0.032\\
21459+1153 & A 1223 AB &  & Rica (this paper)                  &  & ~~4.89 & 0.023 &  & ~~3.71 & 0.011\\
           &           &  & Baize (1955)                       &  & ~~6.17 & 0.043 &  & ~~3.89 & 0.018\\
           &           &  & Voronov (1934)                     &  & 103.49 & 0.077 &  & ~92.39 & 0.030\\
\bottomrule
\end{tabular}
\vspace{0.38cm}
\end{table*}

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


\vspace{0.2cm}

\section{RESULTS}
\label{sec:results}

Table~\ref{tab:Orbits} lists, for each binary, the date of its discovery, the final orbital elements, the root mean square (RMS) residuals 
and the mean absolute (MA) residuals. These are in both cases weighted averages calculated using the data-weighting scheme described in \S~\ref{sec:method}.

The $a^{3}/P^{2}$ values are also listed. Errors for the elements were calculated from the covariance matrix and the residuals of all observations.

Table~\ref{tab:ephemerid} presents ephemerides for the period  2010--2018. Table~\ref{tab:stellardata} lists stellar data for binaries: 
WDS magnitudes in Columns~(3) and (4) and WDS spectral types in Column~(5); in Column~(6), the dynamical parallaxes calculated using the 
orbital periods and semimajor axes obtained in this work; in Columns~(7) and (8) the apparent magnitudes published in the {\it Hipparcos} and 
{\it Tycho} catalogs (ESA 1997), in Column~(9) the corresponding trigonometric parallaxes (with their standard errors); and in Column~(10) the 
total mass of the binary (with an estimated standard error), as calculated from the {\it Hipparcos} parallax. I summarize in Table~\ref{tab:statistical} 
some statistical results concerning the orbits computed in this work and other orbits previously calculated. The RMS and MA residuals for 
$\theta$ and $\rho$ also are listed.

Figures~1--8 show the new apparent orbits drawn together with the observational data; the $x$ and $y$ scales are arcseconds. The solid 
ellipse represents the newly determined orbit, while the dashed ellipse represents previous orbit. The line passing through the origin 
is the line of nodes. Speckle measures are shown as filled squares, visual interferometric observations as open circles, visual measures 
as plus signs, and measures from the ESA {\it Hipparcos} instrument as filled diamonds. The rejected observations are shown as crosses. All 
measures were connected to their predicted positions on the new orbit by O-C lines.

The absolute magnitudes were calculated using the tables published by Zombeck (1990), which relate the spectral types to the absolute magnitudes. 
For red stars Henry et al.'s (1997) work was used, which relates optical-infrared colors to absolute magnitudes. I expect that most of these stars 
have near solar metallicity


%%%%% FIGURA 1

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth]{fricafig1.eps}
\caption{Apparent orbit for HDS 433.}
\label{fig:orbit1}
\vspace{0.2cm}
\end{figure}

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


%%%%% FIGURA 2

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth]{fricafig2.eps}
\caption{Apparent orbit for HDS 521.}
\label{fig:orbit2}
\vspace{0.5cm}
\end{figure}

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


%%%%%% FIGURA 3

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth]{fricafig3.eps}
\caption{Apparent orbit for HO 234.}
\label{fig:orbit3}
\vspace{0.4cm}
\end{figure}

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


%%%%%% FIGURA 4

\begin{figure}[!t]
\centering
\includegraphics[width=0.98\columnwidth]{fricafig4.eps}
\caption{Apparent orbit for A 1342 AB.}
\label{fig:orbit4}
\vspace{0.2cm}
\end{figure}

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


%%%%%% FIGURA 5

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth]{fricafig5.eps}
\caption{Apparent orbit for B 228.}
\label{fig:orbit5}
\vspace{0.4cm}
\end{figure}

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


%%%%%% FIGURA 6

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth]{fricafig6.eps}
\caption{Apparent orbit for A 2988.}
\label{fig:orbit6}
\vspace{0.4cm}
\end{figure}

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


%%%%% FIGURA 7

\begin{figure}[!t]
\centering
\includegraphics[width=0.98\columnwidth]{fricafig7.eps}
\caption{Apparent orbit for A 1223 AB.}
\label{fig:orbit7}
\vspace{0.3cm}
\end{figure}

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


\section{NOTES FOR INDIVIDUAL SYSTEMS}
\label{sec:Note}

Next, photometric, astrometric, spectroscopy and kinematical data were analysed. Spectral types and luminosity classes, 
masses and dynamical parallaxes were obtained. The dynamical parallaxes were calculated using the Baize-Romani method (1946).

The tables published in Montes et al. (2001) and Soderblom \& Mayor (1993) were consulted to check the membership of the binaries 
included in this paper to young kinematic groups.  

In the next paragraphs we adopt the grading scheme used in the Sixth Catalog of Orbits of Visual Binary Stars (Hartkopf \& Mason 2001).


\subsection{WDS 03272+0944 = HDS 433}
\label{sec:HDS433}

Since the first measure in 1983.7134 (Mason et al. 1999b) 8 additional measures for this binary have been obtained, which cover an 
arc of about  116 degrees. The orbit presented in this work is the first orbit published.

HDS 433 (2 Tau, $\xi$ Tau) is composed of stars of magnitudes 3.74 and 7.55 (ESA 1997). The main component is a short period triple 
spectroscopic system. There are several unresolved observations of this system. The high $\Delta m$ undoubtedly is the reason for the 
many unsuccessful attempts at resolution (Mason et al. 1999b). The {\it Hipparcos} catalog lists a proper motion of $+53.61\pm 1.02$~mas~yr$^{-1}$ 
in RA and $-38.12\pm 1.01$~mas~yr$^{-1}$ in DEC and a trigonometric parallax of $14.68\pm 1.01$~mas, which corresponds to a distance of 
$68.11_{-4.4}^{+5.1}$~pc.

In the astronomical literature HDS 433 has been classified as a B9V star (Jaschek 1978; Kennedy 1983; Buscombe \& Foster 1995). From the 
apparent magnitudes and the {\it Hipparcos} parallax, the absolute magnitude for the secondary is +3.4 which corresponds to a spectral 
type F5V.

Tokovinin (1997) obtained masses of $3.20~M_{\odot}$ for Aa, $3.10~M_{\odot}$ for Ab, $5.50~M_{\odot}$ for Ac and $1.24~M_{\odot}$ for B. The 
sum of the masses is $13.04~M_{\odot}$. A total mass of $10.1\pm 6.4~M_{\odot}$ was obtained, using the orbit of this work and the errors 
in the trigonometric parallax, $P$ and $a$.

\textit{Age and Stellar Population}. Using the {\it Hipparcos} proper motion, parallax and a radial velocity of $+22.0\pm 6.6$~km~s$^{-1}$ 
as given by Duflot, Figon, \& Meyssonnier (1995), in this work a peculiar velocity ($U$,$V$,$W$) = ($-$5, +4, $-$30)~km~s$^{-1}$ was 
calculated. According to Eggen's diagrams (1969a,b) this system is a member of the young galactic disk. A value of 0.19 for the 
kinematic age parameter of Grenon (1987), $fG$, was obtained in this work, corresponding to young-middle age thin disk stars of age 
$3-4$~Gyr. However, Strom, Wolff,  \& Dror (2005)  determined an age of only $12-15$~Myr.


%%%%% FIGURA 8

\begin{figure}[!t]
\centering
\includegraphics[width=0.98\columnwidth]{fricafig8.eps}
\caption{Apparent orbit for BU 1147 AB.}
\label{fig:orbit8}
\vspace{0.2cm}
\end{figure}

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


\subsection{WDS 04070-1000 = HDS 521}
\label{sec:HDS521}

The {\it Hipparcos} satellite discovered this system in 1991.25. HDS 521 has 18 measures which cover an arc of 295 degrees.

Balega et al. (2002) used the 6~m telescope at the Special Astrophysical Observatory in Celenchuk to obtain a measure in 1998. 
B. Mason et al. (in preparation) used a 4~m telescope to perform a measure in 2001. Balega et al. (2006) recently calculated the 
orbital parameters for this binary. Later, Horch et al. (2008) obtained 11 speckle measures in 2002 to 2004, which cover 12--13 degrees. 
HDS~521 is composed of stars of magnitudes 7.29 and 8.60 with a combined spectral type of G1/2V (Houk \& Swift 1999).

The {\it Hipparcos} satellite determined a high proper motion of $-127.56\pm1.02$~mas~yr$^{-1}$ in RA and $-141.56\pm1.05$~mas~yr$^{-1}$ 
in DEC, and a trigonometric parallax of $24.00\pm 0.92$~mas (which corresponds to a distance of $41.6_{-1.4}^{+1.7}$~pc).

Using the orbital parameters calculated in this work, masses were calculated taking into account the standard deviation for the {\it Hipparcos} 
trigonometric parallax, and for $P$ and $a$. A total mass of $1.55\pm 0.43~M_{\odot}$ was obtained. The dynamical parallax turns out to be 
24.00~mas, in excellent agree with the {\it Hipparcos} data.

Differential photometry from {\it Hipparcos} gives $\Delta m =+1.31$. The individual spectral types inferred from the differential 
photometry are F9V and G8V. In this work, individual spectral types of F7V and G6V were also obtained using $BVIJHK$ photometry and 
a combined proper motion.

From the spectral types, stellar masses of $1.14~M_{\odot}$ and $0.84~M_{\odot}$ were calculated using Allen's tables (1973). If the 
spectral types determined in this work are used, then stellar masses of $1.23~M_{\odot}$ and $0.90~M_{\odot}$ are obtained.  So the sum 
of the masses ranges from $1.98~M_{\odot}$ to $2.13~M_{\odot}$, consistent within the error margins.

Absolute magnitudes of +4.20 and +5.40 were determined using the apparent magnitudes and the {\it Hipparcos} trigonometric parallax. These values correspond 
to F9V and G7/8V stars.

\textit{Age and Stellar Population}. Nordstrom et al. (2004) obtained a metallicity $[\mathrm{Fe/H}]=-0.27$, an age of $8.0\pm 1.2$~Gyr, 
a radial velocity of $+10.2\pm 3.3$~km~s$^{-1}$ and a peculiar heliocentric velocity ($U$,$V$,$W$) = (+34, $-$2, $-$20)~km~s$^{-1}$. Nordstrom 
et al. (2004) determined that HDS 521 describes a galactic orbit with an excentricity of 0.13, a periastron of 7,700~pc and an apoastron 
of 9,360~pc.

According to Eggen's diagrams (1969a,b) HDS 521 is a member of the young Galactic disk. A value of 0.17 for Grenon's (1987) parameter $fG$ 
was obtained in this work, corresponding to young-middle age thin disk stars of age $3-4$~Gyr.


\vspace{0.35cm}

\subsection{WDS 06345-1114 = HO 234}
\label{sec:HO234}

\vspace{0.05cm}

Hough (1890) discovered this binary in 1888.64 and it has 33 measures which cover about 170 degrees. Baize (1958) calculated the first 
orbit with a period of 169 years ($a=0.38\arcsec$; $e=0.18$) more than twice shorter that the orbit calculated in this work. Heintz (1979) 
calculated the next  orbit which is a grade 4 orbit. At the same time of my publication in IAU Circ. Inf. of the orbit published in this 
work, Hartkopf, Mason, \& Rafferty (2008) published new orbital parameters with residuals slightly smaller than those for the orbit presented 
in this work. Hartkopf et al. used a new measure not used in this work. HO 234 is composed by stars of 8.22 and 8.31 magnitudes (ESA 1997). 
{\it Tycho}-2 catalog proper motions are $-27.4$~mas~yr$^{-1}$ in RA and $-1.5$~mas~yr$^{-1}$ in DEC. The {\it Hipparcos} trigonometric 
parallax of $10.46\pm2.00$~mas corresponds to a distance of $95.6_{-15.3}^{+22.6}$~pc.

The orbit calculated in this work has a period more than two times larger and a major semiaxis about 60 percent greater than the previous 
orbit. A total mass of $2.08\pm2.11~M_{\odot}$ was obtained. In the astronomical literature HO 234 has been classified as a F3V star (Buscombe 
1998) and F2V (Houk \& Swift 1999). In this work a combined spectral type of F4V (F4V+F5V) was obtained using $BVIJHK$ photometry and the 
combined proper motion. From the apparent magnitudes and the {\it Hipparcos} parallax, the absolute magnitudes for the components were 
determined, which correspond to spectral types F4/5V. Using the historical measures, the photometric difference is 0.13 magnitudes. Stellar 
masses of 1.39 and $1.35~M_{\odot}$ were calculated using Allen's tables (1973). The sum of the masses is $2.74~M_{\odot}$.

The dynamic parallax obtained, 9.3~mas, is in agreement with the {\it Hipparcos} parallax. Nordstrom et al. (2004) obtained a metallicity 
$[\mathrm{Fe/H}]=-0.30$, and an age of $1.9_{-0.3}^{+0.4}$~Gyr. The projected rotational velocity is 20~km~s$^{-1}$ (Glebocki, Gnacinski, 
\& Stawikowski 2000). However these data were not useful to estimate the age for the stellar system. The orbital elements have 
previously been announced in the Information IAUDS No. 162 (Rica 2007).


\subsection{WDS 09228-0950 = A 1342 AB}
\label{sec:A1342AB}

This system is KU Hya, a magnetic $\alpha\,CVn$ variable star in which brightness changes of 0.05 magnitudes in a period of 33.97 days have 
been observed (Hensberge et al. 1981). Aitken (1929) discovered this binary in 1905.99 and it has 72 measures which cover two revolutions.

Ekenberg (1945) calculated the first orbit ($P=15.0$~yr; $e=0.10$; $a=0.166\arcsec$). Finsen (1976) calculated another orbit with a period 
of 25.4 years ($e=0.962$; $a=0.28\arcsec$). Starikova (1981) also calculated an orbit, graded 2. Since then, 15 new measures have been made, 
covering about 161 degrees.

Using the historical measures listed in the WDS database, the photometric difference is 0.04 magnitudes. So the {\it Hipparcos} apparent magnitudes 
were not used for this system. A 1342 AB is composed of stars of 7.3 and 7.3 magnitudes (this work). According to the {\it Tycho}-2 catalog 
the proper motion is $-28.8$~mas~yr$^{-1}$  in RA and $-16.0$~mas~yr$^{-1}$ in DEC. The {\it Hipparcos} trigonometric parallax of $7.20\pm0.80$~mas 
corresponds to a distance of $138.9_{-13.9}^{+17.4}$~pc.

Using the orbital parameters, a total mass of $4.25\pm1.66~M_{\odot}$ was obtained. In the astronomical literature A 1342 AB has been classified 
as an A5 star (Cowley et al. 1969; Buscombe \& Foster 1995), an A3V star (Abt 1981) and an F1 star (Abt \& Morrell 1995). In this work, a combined 
spectral type of A5V was obtained using $BVIJHK$ photometry and a combined proper motion. From the apparent magnitudes and {\it Hipparcos} 
parallaxes the absolute magnitudes for the components were determined, which correspond to spectral types A3/4V.

Grenier et al. (1999) calculated a radial velocity of $+19.6\pm5.0$~km~s$^{-1}$.

Stellar masses of 2.07 and $2.05~M_{\odot}$ were calculated from Allen's tables (1973) using the absolute magnitudes. The sum of the masses is 
$4.12~M_{\odot}$. The dynamic parallax obtained is 6.9~mas, in agreement with the {\it Hipparcos} parallax.

\textit{Age and Stellar Population}. In this work a peculiar velocity of ($U$,$V$,$W$) = ($-15, -22, -6$)~km~s$^{-1}$ was calculated. According 
to Eggen's diagrams (1969a,b) A 1342 AB is a member of the young galactic disk. A value of 0.13 for Grenon's $fG$ was obtained in this work, 
corresponding to young-middle age thin disk stars of age $3-4$~Gyr.


\subsection{WDS 12274-2843 = B 228}
\label{sec:B228}

Van den Bos (1928) discovered this double in 1926.77. B 228 has 28 measures which cover about two revolutions. Van den Bos and Finsen obtained 
more than half of the historical measures.

The only existing orbit for B 228 was calculated by Heintz (1981) and it is of grade 4. Since then, 11 new measures have been made (9 of 
them were speckle measures, one measure is micrometrical, and the other measure was made by {\it Hipparcos}) covering about 326 degrees. 
B~228 is composed of stars of 8.18 and 8.39 magnitudes (ESA 1997). {\it Tycho}-2 determined a proper motion of $-53.8$~mas~yr$^{-1}$ in RA and 
$-5.7$~mas~yr$^{-1}$ in DEC. The {\it Hipparcos} trigonometric parallax of $9.88\pm0.96$~mas corresponds to a distance of $101.2_{-9.0}^{+10.9}$~pc.

The orbit calculated in this work is less excentric and more inclined than the previous orbit. The semimajor axis is larger by about 14 percent. 
A total mass of $2.71\pm5.15~M_{\odot}$ was obtained.

Cannon \& Pickering (1918-1924) classified B 228 as an F0 star and Houk (1982) as an A9V star. In this work a combined spectral type of F3V was 
obtained using $BVIJHK$ photometry and a combined proper motion.

From {\it Tycho}-2 apparent magnitudes and {\it Hipparcos} parallaxes, the absolute magnitudes for the components were determined, which 
correspond to individual spectral types F3V and F4V.

Stellar masses of 1.45 and $1.38~M_{\odot}$ were calculated using Allen's tables (1973). The sum of the masses is $2.83~M_{\odot}$ is good agreement 
with the sum of masses obtained by the use of dynamical parameters.

The dynamic parallax obtained is 9.5~mas, in good agreement with the {\it Hipparcos} parallax.


\subsection{WDS 18410+2450 = A 2988}
\label{sec:A2988}

Aitken (1918) discovered this system in 1916.43. A 2988 has 60 measures which cover nearly two revolutions.

Baize (1955) calculated an orbit with a dynamic parallax of $0.0074\arcsec$. Starikova (1981) also calculated an orbit with a dynamic 
parallax of $0.006\arcsec$. Since then, 27 new measures have been obtained (nearly half of the historic measures).

A 2988 is composed of stars of 8.08 and 9.08 magnitudes (ESA 1997). {\it Tycho}-2 lists a proper motion of $+1.9$~mas~yr$^{-1}$ in RA and 
$+8.5$~mas~yr$^{-1}$ in DEC. The {\it Hipparcos} trigonometric parallax of $4.41\pm0.85$~mas, corresponds to a distance of $226.8_{-36.6}^{+54.1}$~pc.

The orbit calculated in this work is more excentric and inclined than the previous one. The semimajor axis is smaller by about 22 percent. 
A total mass of $3.79\pm3.07~M_{\odot}$ was obtained.

In the astronomical literature A 2988 has been classified as an A5 star (Cannon \& Pickering 1918--1924), an A7 star (Grenier et al. 1999), 
an A8V (Christy \& Walker 1969) and an F0V star (Cowley \& Cowley 1965; Kennedy 1983). In this work a combined spectral type of F1V was obtained 
using $BVIJHK$ photometry and a combined proper motion. From the apparent magnitudes and {\it Hipparcos} parallaxes the absolute magnitudes 
for the components were determined, which correspond to spectral types F0V.

Rufener (1981) determined that A 2988 is a Delta Scuti variable star. It is catalogued as NSV 24547. Grenier et al. (1999) calculated a 
radial velocity of $-36.3\pm7$~km~s$^{-1}$.

Using the historical measures listed in the WDS, the photometric difference is 0.0. {\it Hipparcos} obtained a difference in magnitude of 
1.0. A more accurate value for the differential photometry is needed.

Assuming a zero photometric difference, stellar masses of $1.64~M_{\odot}$ and $1.64~M_{\odot}$ were calculated using Allen's tables (1973) 
and spectral types F1V for both components.  The sum of the masses is $3.28~M_{\odot}$.

The dynamic parallax obtained is 4.2~mas, in excellent agreement with the {\it Hipparcos} parallax. The orbital elements have previously been 
announced in the Information IAUDS No. 160 (Rica 2006b).

\textit{Age and Stellar Population}. Bartkevicius \& Gudas (2002) calculated a peculiar velocity ($U$,$V$,$W$) = ($-19.9, -16.9, -0.8$)~km~s$^{-1}$ 
and classified it as a intermediate age thin disk star.


\subsection{WDS 21459+1153 = A 1223 AB}
\label{sec:A1223}

Since Aitken (1906) discovered it in 1905.53, A 1223 AB has been measured on 32 occasions. These measures cover more than two revolutions. 
Tokovinin (1985) performed an interferometric measure and two speckle measures were made (Hartkopf et al. 2000; Scardia et al. 2000). 
{\it Hipparcos} obtained a measure in 1991.

Voronov (1934) calculated the first orbit ($P=23.9$~yrs; $e=0.21$; $a=0.12\arcsec$). Another orbit was calculated by Baize (1955) and 
graded 4. Since then 13 measures have been made covering nearly a revolution.

A 1223 AB is composed of stars of 9.77 and 9.60 magnitudes (ESA 1997) with a combined spectral type of G0V (Kennedy 1983).

{\it Tycho}-2 (H\o g et al. 2000) determined a proper motion of $-83.4$~mas~yr$^{-1}$ in RA and $-85.6$~mas~yr$^{-1}$ in DEC. The {\it Hipparcos} 
trigonometric parallax of $7.23\pm1.35$~mas corresponds to a distance of $138.3_{-21.8}^{+31.8}$~pc.

Masses were calculated taking into account the standard deviation for the {\it Hipparcos} trigonometric parallax and the errors of $P$ and $a$. 
A total mass of $2.54\pm6.78~M_{\odot}$ was obtained.

Eggen (1969c) obtained a spectral type of G2V while Christy \& Walker (1969) obtained G1V with individual spectral types of G0V and G2V. In this 
work, a spectral type of G3V was obtained using $BVIJHK$ photometry and a combined proper motion. From the apparent magnitudes and {\it Hipparcos} 
parallaxes the absolute magnitudes for the components were determined, which correspond to spectral types of F7/8V and F8/9V.

Differential photometry from {\it Hipparcos} yields $-0.17$. A mean differential photometry using the five last historical results from WDS 
catalog yields +0.14. From the spectral types of the literature, stellar masses of 1.09 and $1.09~M_{\odot}$ were calculated using Allen's 
tables (1973). The sum of the masses is $2.18~M_{\odot}$ in agreement, within errors, with the total mass obtained using orbital parameters.

The dynamic parallax is 7.1~mas in excellent agreement with the {\it Hipparcos} parallax. Individual masses of 1.13 and $1.12~M_{\odot}$ were 
determined using the Baize-Romani method (1946).

The orbital elements have previously been announced in the Information IAUDS No. 159 (Rica 2006a).

\textit{Age and Stellar Population}. Bartkevicius \& Gudas (2002) obtained a peculiar velocity of ($U$,$V$,$W$) = (+73, $-45$, +24)~km~s$^{-1}$.

According to Eggen's diagrams (1969a,b) A 1223 AB is a member of the old Galactic disk. A value of 0.37 for $fG$ was obtained in this work 
corresponding to old age thin disk stars, or to thick disk stars of age about 10~Gyr.


\subsection{WDS 23026+4245 = BU 1147 AB}
\label{sec:BU1147}

Since Burnham (1894) discovered it in 1889.54,  BU 1147 AB (= 2 And = HD 217782 = BD+41 4665) it has collected 66 measures which cover 
nearly two revolutions. A previous orbit was calculated by Baize (1974) and graded 4. Since the previous orbit, 39 measures, many of 
which are speckle measurements, have been made, covering 26 degrees.

BU 1147 AB is composed of stars of 5.26 and 7.43 magnitudes (ESA 1997) with a combined spectral type of A0V (Kennedy 1983).

The {\it Tycho}-2 catalog (H\o g et al. 2000) proper motion is $+58.9\pm1.0$~mas~yr$^{-1}$ in RA and $+6.3\pm1.0$~mas~yr$^{-1}$ in 
DEC. The {\it Hipparcos} trigonometric parallax (van Leeuwen 2007) of $7.74\pm0.51$~mas corresponds to a distance of $129.2_{-8.0}^{+9.1}$~pc.

The total stellar mass was calculated taking into account the standard deviation for the {\it Hipparcos} trigonometric parallax and the errors 
of the orbital parameters. A total mass of $4.49\pm1.11~M_{\odot}$ was obtained.

According to different references, the combined spectral type is A0V (Palmer et al. 1968; Kennedy 1983; Bartkevicius \& Lazauskaite 1997), 
A1V (Abt \& Morrell 1995) or A3Vn (Cowley et al. 1969). Since in the literature the individual spectral types were not found, I determined 
them from the differential magnitude (+2.17 from {\it Hipparcos}) and the combined spectral type of Cowley et al. (1969) using the tables 
published by Christy \& Walker (1969) and Edwards (1976). Spectral types of B8V/A2V and F1V/F4V were determined for the primary and secondary.

From {\it Hipparcos} apparent magnitudes and parallaxes, the absolute magnitudes, corrected for the reddening calculated in this paper ($E(B-V)=0.06$) 
were determined ($-0.48\pm0.14$ and $+1.49\pm0.14$). The primary component could be a subgiant star of an age of about 500~Myr and it has been 
suspected to be a spectroscopic binary but not confirmed. The absolute magnitude of the secondary is about 1.5 magnitudes brighter than that 
of an F1V/F4V star.

From the spectral types of the literature, stellar masses of 2.61 and $1.40~M_{\odot}$   were calculated using Allen's tables (1973). The sum of 
the masses is $4.01~M_{\odot}$ in agreement, within errors, with the total mass calculated using orbital parameters and the {\it Hipparcos} parallax.

The dynamic parallax is 7.0~mas. Individual masses of 3.32 and $1.87~M_{\odot}$ were determined using the Baize-Romani method (1946).

The secondary component is catalogued as the suspected variable star NSV14396. Frolov (1970) included it in a list of Delta Scuti stars, 
although is not clear if the variable is the main or the secondary component.

\textit{Age and Stellar Population}. Bartkevicius \& Gudas (2002) obtained a peculiar velocity of ($U$,$V$,$W$) = ($-13.94$, $-1.26$, $-7.44$)~km~s$^{-1}$. 
According to Eggen's diagrams (1969a,b) BU 1147 AB is a member of the young Galactic disk. A value of 0.07 for $fG$ corresponds to young-medium 
age thin disk stars about $3-4$~Gyr old.

\vspace{0.2cm}

\section{STUDY OF WIDE COMPANIONS}
\label{sec:wide}

The astronomical literature was consulted in order to obtain photometric, astrometric and kinematic data. VizieR, Simbad (Wenger et al. 2003) 
and the ``services abstract'' tools were used from the website of Centre De Donn\'ees Astronomiques de Strasbourg. Photometry in the $B$, $V$ 
and $I$ bands came from the {\it Hipparcos} (ESA 1997) and {\it Tycho}-2 catalogs (H\o g et al. 2000). Infrared $J$, $H$ and $K$ photometry came 
from the Two Micron All Sky Catalogue (Cutri et al. 2000), hereafter 2MASS. The proper motions were taken from the {\it Tycho}-2 Catalogue. 
This catalog was chosen because the {\it Hipparcos} proper motions could be affected by Keplerian motion due to its smaller baseline. Historical 
astrometric data were kindly supplied by Mason. Spectral types and other astrophysical data were taken from other sources.

The details about the process to estimate spectral types and luminosity classes for the companions, the calculation of interstellar reddening, 
determination of the nature for the stellar systems, the expected semiaxis major and the orbital period were explained in detail in Rica (2008) 
and Benavides et al. (2010).


\subsection{The criterion of Halbwachs}
\label{sec:HALBWACHS}

Several criteria were used to determine if the wide components are gravitationally bound. These criteria were commented in detail in 
Rica (2008) and Benavides et al. (2010). In this paper we use also the criterion of Halbwachs (1986) who presented a search of common 
proper motion systems among AGK 3 star. The selection of physical systems was based upon the ratio between the angular separation 
($\rho$), and the proper motion ($\mu$), which he called $T$. $T$ is the time after which a system has covered on the sky a distance 
equal to $\rho$. Because of the inaccuracy in proper motion of the AGK 3 catalog, Halbwachs studied pairs with  $\mu \geq +50$~mas~yr$^{-1}$ 
and listed all the pairs with $T < 1000$ years in a table, where optical pairs were estimated to be 1 percent.


\subsection{A 1223 C}
\label{sec:A1223C}

The component C is a star with a 13.5 magnitude (Rapaport et al. 2001) which was measured for the first time by Aitken (1906) in 1905 
when the angular separation was $1.60\arcsec$ (in direction of $349\arcdeg$) from the AB close pair. Since 1905 it has been measured 
on 10 occasions, last in 2000 from the 2MASS catalog when the angular separation was $11.87\arcsec$ (in direction of $39.4\arcdeg$).

Using magnitude $r$ from the Carlsberg Meridian Catalog Number 14 (CMC 2006, hereafter CMC14) and $JHK$ from 2MASS, I determined the $V$ 
magnitude for C using the following relation (John Greaves, private communication):
\begin{equation}
V=0.6 \times (J-K) + r'_\mathrm{CMT}-0.03\,.
\end{equation}

Greaves took a large number of $V$ magnitudes from the catalog ``$UBVRI$ photometry of faint field stars'' (Skiff 
2007),\footnote{\url{http://cdsarc.u-strasbg.fr/viz-bin/Cat?II/277}.} and matched them against CMC14 plotting $V-r'_\mathrm{CMT}$ and $J-K_\mathrm{s}$. 
The expression (1) is only valid for $-0.2 < J-K < 1.2$ and $9 < r'_\mathrm{CMT} < 14$. The standard deviation for $V$ calculated in 
this way is 0.06 magnitudes. A $V$ magnitude of 13.32 was calculated for component C in agreement with Rapaport's value.

The Catalog of Rectilinear Elements (Hartkopf et al. 2006) lists a relative motion of C with respect to AB of $\Delta x=+87.9\pm0.7$~mas~yr$^{-1}$ 
and $\Delta y=+79.5\pm1.0$~mas~yr$^{-1}$. The proper motion of C is not listed in the literature so I calculated it using the proper motion 
of AB and the relative motion of C: $\mu(\alpha)=+4.5\pm1.4$~mas~yr$^{-1}$ and $\mu(\delta)=-6.1\pm1.6$~mas~yr$^{-1}$.

In this work a spectral type of F2 was obtained based on $VJHK$ photometry. No spectral type information was founded in literature.

According to the very different proper motions (much greater than $3\sigma$) of C and AB, this component is not gravitationally bound 
to AB system and so it is an optical companion. This pair must be flagged with ``U'' code (``\textit{Proper motion or other technique 
indicates that this pair is non-physical}'') in the Note column of the WDS Index Catalog.


\subsection{RST 5568 C}
\label{sec:RST5568C}

This is a star with 11.66 magnitude (average of historical photometries from WDS catalog) at about $2.0\,\arcsec$ from A 1342 AB. It was 
measured first in 1938 by van den Bos (1938) when the angular separation was $1.88\arcsec$ from the AB close pair. Since then, it has been 
measured 12 times, last in 1976 by Worley (1978) when the angular separation was $1.96\arcsec$.

I calculated the relative motion of C with respect to AB: $\Delta x=-2.8\pm1.3$~mas~yr$^{-1}$ and $\Delta y=-0.8\pm1.3$~mas~yr$^{-1}$. The 
baseline of the 12 measures used was only 38 years. The proper motion of C was not listed in the literature so we calculated it using the 
proper motion of AB and the relative motion: $\mu(\alpha)=-31.6\pm2.4$~mas~yr$^{-1}$ and $\mu(\delta)=-16.8\pm2.5$~mas~yr$^{-1}$.

No spectral type was found in the literature and I did not estimate it due to the lack of photometric and astrometric data. If this wide 
companion were gravitationally bound to AB then it would be a K2V star.

In order to determine the nature of the wide C component, two tests, based on probability theory were used. Using the work of Ciardullo 
et al. (1999) the probability P that a star of 11.66 magnitude could be located by chance at a distance of $1.9\arcsec$ from another star 
is 0.003 percent. In a list of optical pairs studied by LIADA, no optical double was found with $P < 0.09$, so component C is likely bound 
to the AB pair.

Grocheva \& Kiselev (1998) propose to use the real distribution of proper motions for estimating $P$. In this work if $P < 0.01$ the pair is 
considered physical. The value of $P$ for RST 5568 AB-C is 0.002, and so the component C is surely bound to the AB close pair.

Based on a normal distribution there is a probability of 55 percent that RST 5568 AB-C is a common proper motion pair. The parameter $T$ of 
Halbwachs is 56~years and so this system is likely be physical according to this criterion.

I calculated a hypotetical parallax, $\pi_\mathrm{hyp}$, for the stellar system of 0.0062~mas (160.2~pc) in good agreement with the {\it Hipparcos} 
parallax  of $0.0072\pm0.0008$~mas ($138.9_{-13.9}^{+17.4}$~pc). So the C component is likely bound to the AB close binary.

The expected semimajor axis would be 260~UA and the orbital period $\sim 1,900$~yrs (assuming a circular and face-on orbit). The expected 
semimajor axis was calculated using the expression obtained by Fischer \& Marcy (1992) The arc covered by this component is only 2.7 degrees. 
If we take into account this angular motion this binary completes a orbit in about $\sim 5,000$~yrs.


\subsection{BU 1147 C}
\label{sec:BU1147C}

The component C was added to the system BU 1147 AB in 1911 by Burnham (1913) when he observed a star of 13.6 magnitude at $90.55\arcsec$ 
from the close pair, in direction $191.6\arcdeg$. Since 1911 it has been measured only 4 times, last in 1998 ($195.1\arcdeg$ and 
$91.53\arcsec$, from 2MASS).

In order to obtain a modern estimate for the magnitude of C, photographic catalogs were used. The photometric data of the Guide Star Catalogue 
1.2 (hereafter GSC) was calibrated, using {\it Tycho}-2 stars for the region of the sky where the component C is located. In this part of the sky, 
GSC photometry is 0.15 magnitude brighter than the $V$ band. A magnitude $V=13.17\pm0.16$ was calculated. From $B$ and $R$ photometric data of 
the USNO-B1.0 catalog, a value of $V=13.35\pm0.30$ was obtained using the expression
\begin{equation}
V=B_{j}-0.46 \times (B_{j}-R_{f})\,,
\end{equation}
cited in L\'epine,  Rich, \& Shara (2005). From $B1$ and $R1$ photometry of USNO-B1.0 a magnitude $V=13.0\pm0.5$ was obtained. In this case I used 
the expression determined by Greaves (2008),\footnote{\url{http://www.aerith.net/astro/color_conversion.html}.}
\begin{equation}
V=0.444 \times B1 + 0.556 \times R1\,.
\end{equation}

A weighted mean gave a $V=13.19\pm0.21$.

No spectral type information was found in the literature. From the $V$ magnitude and the $J$, $H$ and $K$ magnitudes listed in 2MASS, I 
determined a spectral type of K2-4. A giant nature was rejected because then the tangential velocity would be about 280~km~s$^{-1}$ 
and this is an unlikely value. So I conclude that the C component is a K3V dwarf at about 175~pc. I estimated an interstellar reddening of 
0.08 for the $B-V$ color. The spectral type of K3V is corrected for this effect.

The proper motion of C listed in the UCAC-2 catalog is $\mu(\alpha)=-5.3\pm5.6$~mas~yr$^{-1}$ and $\mu(\delta)=-5.1\pm5.6$~mas~yr$^{-1}$. 
The USNO-B1.0 catalog lists $\mu(\alpha)=-6\pm1$~mas~yr$^{-1}$ and $\mu(\delta)=-4\pm2$~mas~yr$^{-1}$.

Using historical measures, a weighted linear fit was performed to calculate a relative motion of C with respect to AB of 
$\Delta x=-70.7\pm2.7$~mas~yr$^{-1}$ and $\Delta y=-2.9\pm2.3$~mas~yr$^{-1}$. The baseline of the 4 measures used was 87 years. 
From the proper motion of AB and relative motion of C, I calculated a proper motion $\mu(\alpha)=-11.8\pm2.9$~mas~yr$^{-1}$ 
and $\mu(\delta)=-9.2\pm2.5$~mas~yr$^{-1}$ for C.

In view of the very different proper motions (much greater than $3\sigma$) of C and AB, this component is not gravitationally bound to the AB 
system and so it is an optical companion. This pair must be flagged with ``U'' code (\textit{``Proper motion or other technique indicates that 
this pair is non-physical.''}) in the Note column of the WDS Index Catalog.


\subsection{LYS 25 D}
\label{sec:LYS25ABD}

The component D was added to the system BU 1147 AB in 1984 by Louys (1987) when he took a photograph with an astrograph and observed a star 
of 12.1 magnitude (inferred from the differential magnitude obtained by Louys) at $142.79\arcsec$ from the close pair, in direction $282.7\arcdeg$. 
Three more measures were added to WDS from the Washington Fundamental Catalog (WFC) and 2MASS. The most recent measure has an epoch of 1998.78 
($282.9\arcdeg$ and $142.48\arcsec$).

In order to obtain a more accurate estimate for the magnitude of D, the photometric data from the GSC catalog were calibrated using {\it Tycho}-2 
stars, for the region of the sky where component D is located. In this part of the sky, GSC photometry is 0.15 magnitude brighter than $V$ band 
and thus a magnitude $V=10.68\pm0.16$ was calculated.

From the magnitude $r$ from CMC14 and $JHK$ from 2MASS, I determined the $V$ magnitude for D using the relation of John Greaves (private communication) 
commented in this work for A 1223 C. A $V$ magnitude of $10.87\pm0.06$ was calculated for component D.

The \textit{TASS Mark IV Photometric Survey of the Northern Sky} (Droege et al. 2006) lists 30 photometric points in the $V$ and $I$ bands for 
component D with mean magnitudes $V=10.97\pm0.21$ and $I=9.57\pm0.26$. From the three values for $V$, a weighted mean gave $V=10.85\pm0.08$.

No spectral type information was found in the literature. From the $V$ and $I$ magnitudes from TASS and the $J$, $H$ and $K$ magnitudes listed 
in 2MASS (in addition to kinematic data), I determined that component D is a K1III giant located at a distance of about 950~pc. This spectral 
type was corrected for the reddening ($E(B-V)=0.19$) which was calculated in this work.

The TASS catalog lists a Welch-Stetson (Welch \& Stetson 1993, hereafter WS) variability index of 40.28. The WS index is a normalized measure 
of the degree of correlation of $V$-band and $I$-band variations from their mean values. The variability index will be close to zero for stars 
with random variations which are independent in each passband, or stars which vary by less than the uncertainty of each measurement. It will be 
a large (greater than 2 or 3) positive value for stars which really do change significantly in brightness. Nothing is listed in \textit{The 
International Variable Star Index (VSX)}  so component D is likely to be an uncatalogued variable star.

The proper motion of D listed in the UCAC-2 catalog is $\mu(\alpha)=+6.8\pm1.0$~mas~yr$^{-1}$ and $\mu(\delta)=-1.4\pm0.8$~mas~yr$^{-1}$.

{\it Tycho}-2 lists for the close AB pair a proper motion of $\mu(\alpha)=+58.9\pm1.0$~mas~yr$^{-1}$ and $\mu(\delta)=-6.3\pm1.0$~mas~yr$^{-1}$. 
In view of the very different proper motions (much greater than $3\sigma$) D is not gravitationally bound to the AB system and so it is an optical 
companion. This pair must be flagged with ``U'' code (\textit{``Proper motion or other technique indicates that this pair is non-physical.''}) 
in the Note column of the WDS Index Catalog.


\section{SEARCH FOR NEW BOUND COMPANIONS}
\label{sec:NEW_COMPANION}

SIMBAD and VizieR tools were used to find uncatalogued bound companions consulting astrometric catalogs and photographics plates. First I searched 
for common proper motion companions. For nearby stars I searched for bound companions by plotting the neighbour stars in a color-magnitude diagram 
where apparent 2MASS $J-K$ and $K$ data are plotted. A dwarf sequence at the distance of the target was also plotted. So dwarf companion candidates 
must be located on or near this dwarf sequence. Subdwarf or white dwarf companions will not be detected unless we use reduced proper motion diagrams. 
Giant companions will be bright enough to be detected and they will be located well above the dwarf sequence.

The expression of Abt (1988) which relates stellar masses of the primary with the maximum projected separation (in AU) was used to define the sky 
region to search. I selected companion candidates consulting 2MASS where only stars located within the search region and with a $S/N > 10$ 
were selected.


\subsection{Two Stars in the Vicinity of HDS 433}
\label{sec:HDS433_NEWCOMPANION}

In the search for uncatalogued companions I found two red dwarfs with high proper motion. Although these two stars are not bound to 
HDS 433 AB I studied their astrophysical properties. At coordinates AR = $03^{h}\,26^{m}\,59.66^{s}$ and DEC = $+9\arcdeg\,49\arcmin\,04\arcsec$ 
I found an M1.5V dwarf of magnitude $V=15.36$ (from the catalog GSC-2.3) located at 125~pc. Using astrometry from USNO-A2.0 and 2MASS for the 
epochs 1949.882 and 1999.935 I calculated a proper motion $\mu(\alpha)=+16$~mas~yr$^{-1}$ and $\mu(\delta)=+94$~mas~yr$^{-1}$. At coordinates 
AR = $03^h\,27^m\,40.17^s$ and DEC = $+09\arcdeg\,44\arcmin\,58\arcsec$ I found a M3V dwarf of magnitude $V=14.84$ (from the catalog GSC-2.3) 
located at 46~pc. The USNO-B1.0 catalog lists a proper motion $\mu(\alpha)=+44\pm1$~mas~yr$^{-1}$ and $\mu(\delta)=-112\pm2$~mas~yr$^{-1}$.

\vspace{0.2cm}

\section{CONCLUSIONS}
\label{sec:conclusions}

I have presented the first orbital solution for HDS 433 (8 measures covering 116 degrees) and updated orbital solutions for 7 binaries in addition 
to the masses.

Data from the literature were consulted and an astrophysical study was carried out. The B9V primary component of HDS 433 is a short period triple 
spectroscopic system. The sum of the masses for this young stellar system is $13.4~M_{\odot}$. Strom et al. (2005) determined an age of only 
$12-15$~Myr.

A 1342 AB is the magnetic $\alpha$ CVn variable star cataloged as KU Hya while A 2988 is a Delta Scuti cataloged as NSV 24547.

Wide components were studied astrophysically. For A 1223 C, a value for the V magnitude of 13.32 was obtained. The proper  motion and spectral type 
(F2) of C were determined for the first time. Based in the large difference in the proper motions, the component C is not bound to AB close binary.

RST 5568 C, the wide component of A 1342 AB, has not been resolved since 1976. The relative motion of C with respect AB was calculated. The proper 
motion of C was  determined for the first time. According to the same tests, I concluded that C is likely to be bound to the AB close binary. The orbital period 
ranges from $1900-5000$ years.

I searched for new bound companions to the close binaries. No new companion was detected. However, two uncatalogued red dwarfs with high proper motion 
were discovered.

\vspace{0.2cm}

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\end{document} 