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4 TV Cet

The detached eclipsing binary TV Cet (also HD 20173, BD  $+02^{\circ}$ 502, SAO 111068, HIP 15090, PPM 146367, AN 270.1934, FL 266; $\alpha_{2000} = 3^{\rm h} 14^{\rm m} 36.5^{\rm s}$, $\delta_{2000} = +2^{\circ}45'16.4''$, $V_{\max} = 8.6$ mag; Sp. F2+F5) is a rarely investigated binary with an eccentric orbit (e = 0.055) and relatively long orbital period of about 9.1 days. It was discovered to be a variable by Martynov (1951), who derived the first light elements

\begin{displaymath}{\rm Pri.~Min. = HJD~24~ 26692.494 ~+~ 9\hbox{$.\!\!^{\rm d}$ }1032 \cdot} E.
\end{displaymath}

Spectroscopically TV Ceti was studied by Popper (1967, 1968), who obtained the radial velocity curves with semiamplitudes K1 = 67.4 kms-1 and K2 = 73.8 kms-1. Four-color ubvy photometry was obtained by Jørgensen (1979) at ESO, Chile, between November 1972 and December 1974. He derived photometric elements and absolute dimensions of this binary ( $M_1 = 1.39 \pm 0.05~M_{\odot}$, $M_2 = 1.27 \pm 0.04~M_{\odot}$). He also presented three new times of minimum and refined light elements

\begin{displaymath}{\rm Pri.~ Min. = HJD~ 24~ 41685.6112~ +~ 9\hbox{$.\!\!^{\rm d}$ }103291 \cdot} E.
\end{displaymath}

From its spectral type and other known properties, TV Cet was listed by Giménez (1985) as a good candidate for the study of the contribution of general relativity to the secular displacement of the line of apsides, given that the relativistic effect is expected to be dominant in this particular case. More moments of minimum light obtained photoelectrically were published by Meyer (1972) and later by Caton & Hawkins (1987), Caton et al. (1989) and Agerer & Hübscher (1998). Aside from these occasional measurements of the times of eclipse, TV Ceti has remained a rather neglected system until recently. More than 20 years have elapsed since its last study, thus TV Cet was also included in our photometric program. From the Hipparcos photometry (Perryman 1997), we were able to determine one additional moment of minimum light. It is also given in Table 1, where epochs are calculated according to the light elements given by Jørgensen (1979).

All photoelectric times of minimum light published in Meyer (1972), Jørgensen (1979), Caton & Hawkins (1987), Caton et al. (1989) as well as Agerer & Hübscher (1998) were incorporated in our analysis. A total of 20 times of minimum light were used in our analysis, with 9 secondary eclipses among them.


  \begin{figure}
\par\includegraphics[width=8.8cm,clip]{TVCET.EPS} \end{figure} Figure 1: O-C graph for the times of minimum of TV Cet. The continuous and dashed lines represent predictions for primary and secondary eclipses, respectively. The individual primary and secondary minima are denoted by circles and triangles, respectively. Larger symbols correspond to the photoelectric measurements which were included in calculations with higher weight.

The eclipse timings listed partially in Table 1 allow the determination of linear ephemerides indepedently for primary and secondary eclipses with the following results(numbers between parentheses indicate errors in the last digits and E is the number of cycles):

\begin{displaymath}{\rm Pri.~Min. = HJD~ 24~ 41685.6122(2)~ + ~9\hbox{$.\!\!^{\rm d}$ }1032891(4) \cdot} E,
\end{displaymath}


\begin{displaymath}{\rm Sec.~Min. = HJD~ 24~ 41690.1038(2) ~+~ 9\hbox{$.\!\!^{\rm d}$ }1032860(4) \cdot} E.
\end{displaymath}

These linear ephemerides we propose also for current use. The difference in the apparent periods is of course a clear indication of the presence of significant apsidal motion. The method described by Giménez & García-Pelayo (1983), with equations revised by Giménez & Bastero (1995), was used for a more accurate calculation of the apsidal motion rate. The apsidal motion resulting from the final fit is $\dot{\omega}
= 0.000\,30 \pm 0.000\,08$ deg cycle-1, which is significant at the 3 $\sigma$ level.

Adopting the orbital inclination derived from the light curve solution of $i = 89.15^{\circ}$ (Jørgensen 1979), the apsidal motion elements can be computed. The parameters found and their internal errors of the least squares fit (in brackets) are given in Table 2. In this table $P_{\rm s}$ denotes the sidereal period, $P_{\rm a}$ the anomalistic period, e represents the eccentricity and $\dot{\omega}$ is the rate of periastron advance (in degrees per cycle or in degrees per year). The zero epoch is given by T0 and corresponding position of the periastron is represented by $\omega_0$.

The relation between the sidereal and the anomalistic period, $P_{\rm s}$ and $P_{\rm a}$, is given by

\begin{displaymath}P_{\rm s} = P_{\rm a} \,(1 - \dot{\omega}/360^\circ)
\end{displaymath}

and resulting apsidal motion period U, directly given by  $\dot{\omega}$ is,

\begin{displaymath}U = 360^\circ P_{\rm a}/\dot{\omega}.
\end{displaymath}

The O-C residuals for all times of minimum with respect to the linear part of the apsidal motion equation are shown in Fig. 1. The quasi-linear predictions, corresponding to the fitted parameters, are plotted as continuous and dashed lines for primary and secondary eclipses, respectively.
  \begin{figure}
\par\includegraphics[width=8.8cm,clip]{tv3.eps} \end{figure} Figure 2: O-C2 diagram for the times of minimum of TV Cet after substraction the terms of apsidal motion. The curve represents a light-time effect for the third body orbit with a period of 28.5 years and an amplitude of about 0.003 days. The individual primary and secondary minima are denoted by circles and triangles, respectively.

Subtracting the influence of apsidal motion, the O-C2 diagram in Fig. 2 can be plotted. The sinusoidal variation of these values are remarkable and could be caused by a light-time effect. A preliminary analysis of the possible third body orbit gives the following parameters:

P3 (period) = $10\,415 \pm 80$ days
  = 28.5 years
T3 (time of periastron) = JD $24\,50505 \pm 40$
A (semiamplitude) = $ 0\hbox{$.\!\!^{\rm d}$ }0027 \pm 0\hbox{$.\!\!^{\rm d}$ }0004 $
e3 (eccentricity) = $ 0.25 \pm 0.03 $
$\omega_3$ (lenght of periastron) = $ 148\hbox{$.\!\!^\circ$ }1 \pm 1\hbox{$.\!\!^\circ$ }4 .$

These values were obtained together with the new mean linear ephemeris

\begin{displaymath}%
{\rm Pri.~Min.} ={\rm HJD~ 24 41685.5839} ~+~ 9\hbox{$.\!\!^{\rm d}$ }10328692 \cdot E,
\end{displaymath}


\begin{displaymath}~~~~~~~~~~~~~~~~~~~~~~~~~~~~\,\pm0.0004~~~\pm0.00000032
\end{displaymath}

by the least squares method. Assuming a coplanar orbit ( $i_3 = 90^{\circ}$) and a total mass of the eclipsing pair $M_1 + M_2 = 2.66~M_{\odot}$(Jørgensen 1979), we can obtain a lower limit for the mass of the third component $M_{3, \min}$. The value of the mass function is $f(M) = 0.000\,13~M_{\odot}$, from which the minimum mass of the third body follows as $0.10~M_{\odot}$. A possible third component of spectral type M8 with the bolometric magnitude about +12 mag could be practically invisible in the system with a F2 primary ( $M_{\rm bol} = +3.2$ mag, Harmanec 1988). Therefore, new high-accuracy timings of this eclipsing binary are necessary in order to confirm the light-time effect in this system.

The acceleration of the rate of apsidal motion caused by the presence of the third body $\dot{\omega}$(Martynov 1973) is

\begin{displaymath}\dot{\omega} = \frac{3}{4} \lambda \, m^2
+ \frac{225}{32} \lambda^2 \, m^3 + \dots ,
\end{displaymath} (1)

where

\begin{displaymath}\lambda = \frac{M_3}{M_1 + M_2 + M_3} , {\rm ~~~~~and~~~~~}
m = \frac{P_{\rm s}}{P_3}\cdot
\end{displaymath} (2)

This correction for the apsidal motion is negligible in this system due to the relatively long period P3 of the third body orbit.

More precise non-linear light elements of the eclipsing pair including the term of the light-time effect with the circular orbit are

\begin{displaymath}{\rm Pri.~ Min. = HJD~ 24 41685\hbox{$.\!\!^{\rm d}$ }6122~ +~ 9\hbox{$.\!\!^{\rm d}$ }1032891 \cdot} E
\end{displaymath}


\begin{displaymath}\hspace{2.7cm}+~ 0\hbox{$.\!\!^{\rm d}$ }0027 \sin (M + 148\hbox{$.\!\!^\circ$ }1 ),
\end{displaymath}


\begin{displaymath}{\rm Sec.~ Min. = HJD~ 24 41690\hbox{$.\!\!^{\rm d}$ }1038 ~+~ 9\hbox{$.\!\!^{\rm d}$ }1032860 \cdot} E
\end{displaymath}


\begin{displaymath}\hspace{2.7cm}+ 0\hbox{$.\!\!^{\rm d}$ }0027\sin (M + 148\hbox{$.\!\!^\circ$ }1 ),
\end{displaymath}

where

\begin{displaymath}M = 2 \pi (T_0 + P_{\rm s} E - T_3)/P_3,
\end{displaymath}

is a mean anomaly of the third body.


 

 
Table 2: Apsidal motion elements of TV Cet and V451 Oph.
Element TV Cet V451 Oph
T0 [HJD] 24 41685.5839 (3) 24 45886.5310 (2)
$P_{\rm s}$ [days] 9.10328692 (8) 2.19659703 (12)
$P_{\rm a}$ [days] 9.10329452 (8) 2.19667490 (12)
e 0.0545 (4) 0.0120 (5)
$\dot{\omega}$ [deg $\:\rm {cycle^{-1}}$] 0.00030 (8) 0.0128 (4)
$\dot{\omega}$ [deg $\:\rm {yr^{-1}}$] 0.012 (3) 2.12 (6)
$\omega_0$ [deg] 100.9 (0.2) 253.4 (0.5)
U [years] 30000 (8000) 170 (5)



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