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4 Discussion

The decrease in the orbital period of V 781 Tau has presented the evidence to test the interesting results presented by Wang (1994), who suggested that the two subtypes of W UMa binaries should be in two different thermal relaxation oscillation states: the secondary components of the W-subtypes of W UMa type binaries are shrinking whereas the ones of A-subtypes are swelling. According to Wang (1994), a W-subtype W UMa binary has a contracting secondary component, which releases some gravitational energy to support its surface effective temperature higher than that of the primary component. The luminosity radiated by the secondary component consists of the three parts, the first is from the nuclear reaction in the secondary, the second is from the transferred luminosity from the primary and the third is from the released gravitational luminosity by the secondary component. Suppose the secondary component with a mass of m2 and a radius of R2 spherically symmetric, then its gravitational energy $E_{{\rm g}}$ is written (Kippenhahn & Weigert 1990) as follows:

\begin{displaymath}E_{{\rm g}} = \frac{3Gm_{2}^{2}}{(n - 5)R_{2}},
\end{displaymath} (7)

where n = 3 for the main-sequence stars. For the low mass secondary components of W-subtype W UMa binaries, n = 1.5 adopted by Wang (1994). From the Eq. (7), the following equation can be derived

\begin{displaymath}\frac{{\rm d}E_{{\rm g}}}{{\rm d}t} = \frac{3Gm_{2}^{2}}{(5 - n)R_{2}^{2}}\frac{{\rm d}R_{2}}{{\rm d}t}\cdot
\end{displaymath} (8)

Lu (1993) gave the physical parameters of V 781 Tau and pointed out that the components of this system are main-sequence stars. A luminosity generated from the nuclear reactions of the secondary component with a mass of $0.501~m_{\odot}$ is about $0.048~L_{\odot}$. Assuming that the transferred luminosity from the primary component could only make the surface effective temperature of the secondary component the same as that of the primary component, then a maximum value of the transferred luminosity was about 0.65 $L_{\odot}$. The observed luminosity of the secondary component is $0.71~L_{\odot}$ and it can be supposed that the over-luminosity of $0.012~L_{\odot}$ was the gravitational luminosity released by the secondary component. From the Eq. (8), one may find a minimum contracting velocity of the secondary component, ${\rm d}R_{2}/{\rm d}t = -2.45~10^{-6}~{\rm cm~s}^{-1}$. This minimum shrinking velocity is less than the contracting velocity of $6.77~10^{-5}~{\rm cm~s}^{-1}$ calculated from the decrease in the period, because the superior limit of the transferred luminosity from the primary component is adopted. In fact, adopting the contracting velocity of $6.77~10^{-5}~{\rm cm~s}^{-1}$ calculated from the decrease in the period and from the Eq. (8), one may find the gravitational luminosity released by the secondary component, ${\rm d}E_{{\rm g}}/{\rm d}t = 0.330~L_{\odot}$. Thus, the transferred luminosity from the primary component is only about $0.33~L_{\odot}$ rather than $0.65~L_{\odot}$.

The paradox of the over-luminous secondary component of the W UMa binary was explained by the introduction of large-scale energy transfer from the primary to the secondary (Lucy 1968). In the present discussion for V 781 Tau, the secondary component of the system is continuously shrinking at the velocity of $6.77~10^{-5}~{\rm cm~s}^{-1}$ and releasing the gravitational luminosity of $0.330~L_{\odot}$, while the orbital period of the system was continuously decreasing with the change ratio of $\delta{p}/p=-5.0~{10^{-11}}$. Then the transferred luminosity from the primary component may be only a part of the over-luminosity, the other

part may be from the gravitational luminosity released by the secondary component when it shrinks. According to Mochnacki (1981), the ratio of energy transfer of V 781 Tau should be of $\Delta{L}/L_{1} = 0.45$, but in the present discussion, that is only of $\Delta{L}/L_{1} = 0.22$.

It is interesting to notice that the W-phenomena of W UMa binaries arise also because of the contracting secondary components rather than only because the energy transfer from the primary to the secondary. The transferred luminosity from the primary should never cause the temperature of the secondary component to be higher than that of the primary if there were non-gravitational luminosity released by the secondary component.

In order to test the contracting model of the secondary components of the W-subtype W UMa Binaries, observations and analyses in the orbital period of more systems will be needed.

Acknowledgements
The authors would like to thank Dr. F. Li for his assistance in the observation and Dr. P.G. Niarchos for his useful advice. The authors would also like to express their gratitude for the support from the Chinese National Science Foundation Committee and the Chinese Academy of Sciences.


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