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3 Change in the period and explanation

The (O-C)1 residuals in Table 2 are calculated by using the ephemeris given by Cereda et al. (1988), i.e.

\begin{displaymath}{\rm Min.I} = {\rm HJD}~2443853.9096 + 0.3449094 E. \end{displaymath}

The (O-C) diagram of the change in the orbital period of V 781 Tau is shown in Fig. 1, where (O-C)1 residuals listed in Table 2 are used. It is clear that the period of this binary star decreased continually from 1949 to 1998. All epochs in Table 2 are used for a least square solution to obtain a quadratic ephemeris. The improved ephemeris is described as follows:


\begin{eqnarray*}{\rm Min.I}&=&{\rm HJD}~2443853.9110(15)+0.34490929(18)E \\
&& -2.5(2)~{10^{-11}}E^{2}
\end{eqnarray*}


which is used to calculate the (O-C)2 in Table 2. The period change rate of V781 Tau is $\delta{p}/p=-5.0~{10^{-11}}$.


  \begin{figure}\includegraphics[width=7.2cm,clip]{ds8864f1.eps}\end{figure} Figure 1: The O-C diagram of the period change for V 781 Tau. The solid circles indicate the photoelectric observations and the crosses express the photographic one

The secular instability of the secondary component of a W UMa type binary arises because they obtain the luminosity transferred from the primary component (Lucy & Wilson 1979; Hazlehurst 1985; Wang 1994). So it could be suggested that decrease in the period of V 781 Tau is caused by the contraction of the secondary component, then its shrinking velocity can be calculated from the decrease rate in the orbital period.

The Kepler's third law can be written as


A3 = 74.5p2M (1)

where A is the separation between the two components in solar radii, p represents the orbital period in days and M indicates the total mass of the two components in solar mass. From the definition of the relative radius of one of the two components, one may have

\begin{displaymath}A = \frac{R_{1} + R_{2}}{r_{1} + r_{2}}\cdot
\end{displaymath} (2)

According to Binnendijk (1970) and Lacy (1977), one may have

r1 + r2 =0.76 (3)

and

\begin{displaymath}\frac{R_{2}}{R_{1}} = q^{0.92}.
\end{displaymath} (4)

Inserting the Eqs. (4), (3) and (2) into the Eq. (1), one may acquire

\begin{displaymath}R_{2}^{3}\left(1+\frac{1}{q^{0.92}}\right)^{3} = 32.7p^{2}M.
\end{displaymath} (5)

Assuming conservation of total mass of the system, from the Eq. (5), one can obtain

\begin{displaymath}\frac{{\rm d}R_{2}}{{\rm d}t} = \frac{21.8pMq^{2.76}}{R_{2}^{2}(1 + q^{0.92})^{3}}\frac{{\rm d}p}{{\rm d}t}\cdot
\end{displaymath} (6)

Since the period of V 781 Tau decreases, from the Eq. (6) and the parameters published by Lu (1993), one may find a contracting velocity of the secondary component, ${\rm d}R_{2}/{\rm d}t =-6.77~{10^{-5}}~{\rm cm~s}^{-1}$, namely the contraction of 21.4 m/year, which is in agreement with the results shown in the study of the thermal relaxation oscillation of contact binaries by Wang (1994).


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