8 Discussions and conclusions

In the previous sections, the changes in the orbital periods of six Algol-type binaries, UW Cyg, RX Hya, AK Ser, AC Tau, UW Vir and VV Vul, are analyzed. It is found that the periods of AK Ser and VV Vul may vary in secular increase, the period of UW Cyg shows a cyclic change. However, the periods of the other three systems, RX Hya, AC Tau and UW Vir vary in complex ways. Periodic variations are found to be superimposed on the long-time increase components.

Algol-type binaries are classified as semi-detached systems whose less massive components are filling their critical Roche Lobe. Assuming the conservation of angular momentum, the orbital periods of Algols should increase during their evolution. The secular increase of the periods in the five system, RX Hya, AK Ser, AC Tau, UW Vir and VV Vul can be explained in terms of the mass transfer from the less massive secondary to the more massive primary. This is consistent with the semi-detached configuration of these systems. If such period changes are due to conservative mass transfer, then by using the parameters listed in Table 1 in the well-known equation:

$$\Delta{P}/P=3(M_{1}/M_{2}-1)\Delta{M_{1}}/M_{1}$$ (23)

the mass transfer rates for theses binaries are computed and listed in Table 8. Since no absolute parameters of VV Vul have been published, its parameters are estimated during the calculation. With the spectral type A2/3V given by Halbedel (1984), the mass of the primary component is estimated to be $M_{1}=2.5\,M_{\odot}$, and considering a typical value of mass ratio q=0.3 for Algols, the mass of the secondary is about $0.75\,M_{\odot}$.

 

 

Table 8: Mass transfer rates for five semi-detached binaries

Stars	P (days)	dP/dt(days/year)	dM/dt($M_{\odot}$/year)
RX Hya	2.281645	$+1.04\ {10^{-6}}$	$1.32\ {10^{-7}}$
AK Ser	1.922580	$+7.94\ {10^{-7}}$	$2.41\ {10^{-7}}$
AC Tau	2.043356	$+7.11\ {10^{-7}}$	$3.62\ {10^{-7}}$
UW Vir	1.8107755	$+1.73\ {10^{-6}}$	$1.81\ {10^{-7}}$
VV Vul	3.411361	$+3.43\ {10^{-6}}$	$3.59\ {10^{-7}}$

 

 

Table 9: The values of the masses and the orbit radii of the third bodies in UW Cyg, RX Hya, AC Tau and UW Vir

	UW Cyg	RX Hya	AC Tau	UW Vir	Units
A	0.0383	0.0431	0.0191	0.0306	days
T	49.3	54.3	29.8	45.9	years
$a_{12}^{\prime}\sin i^{\prime}$	6.64	7.47	3.31	5.30	AU
f(m)	$1.21\ {10^{-1}}$	$1.41\ {10^{-1}}$	$4.09\ {10^{-2}}$	$7.07\ {10^{-2}}$	$M_{\odot}$
$m_{3}(i^{\prime}=90^{\circ})$	1.12	1.50	0.75	0.87	$M_{\odot}$
$m_{3}(i^{\prime}=70^{\circ})$	1.21	1.62	0.80	0.94	$M_{\odot}$
$m_{3}(i^{\prime}=50^{\circ})$	1.60	2.12	1.03	1.23	$M_{\odot}$
$m_{3}(i^{\prime}=30^{\circ})$	3.00	3.92	1.81	2.23	$M_{\odot}$
$m_{3}(i^{\prime}=10^{\circ})$	27.0	32.8	11.5	17.2	$M_{\odot}$
$a_{3}(i^{\prime}=90^{\circ})$	13.64	16.83	10.77	13.40	AU
$a_{3}(i^{\prime}=70^{\circ})$	13.43	16.59	10.74	13.20	AU
$a_{3}(i^{\prime}=50^{\circ})$	12.46	15.55	10.24	12.37	AU
$a_{3}(i^{\prime}=30^{\circ})$	10.18	12.88	8.924	10.46	AU
$a_{3}(i^{\prime}=10^{\circ})$	3.257	4.433	4.044	3.904	AU

Apart from 6 photoelectric timings for RX Hya, other times of light minimum analyzed in this paper are visual or photographic, and all times of light minimum for the six systems measured for the primary minimum. This may be caused by the fact that the secondary is more difficult to observe by such methods. The cyclic variations of the O-C residuals in the four systems, UW Cyg, RX Hya, AC Tau and UW Vir, are only formed by the times of primary minimum. We do not know whether the secondary timings follow the same trend of oscillation. However, for RX Hya and UW Vir, Lucy & Sweeney (1980) have pointed out that their orbits are circular; for the other two, UW Cyg and AC Tau, since their orbital periods are small (P<4days), a strong mutual tidal interaction between the components may have made their orbits circular. This indicates that the observed cyclic oscillations in O-C residuals of the four systems are not caused by apsidal motion. The oscillatory character of the (O-C) variations in these systems may be result of light time effects due to the presence of additional bodies. As displayed in Figs. 2, 7 and 9, the sine-like O-C variations for UW Cyg, AC Tau and UW Vir suggest that their third bodies are moving in circular orbits around the common centre of the gravity of the three bodies. For RX Hya a small eccentricity ( $e^{\prime}=0.34$) of the orbit of the third body is suggested by the observations. Other orbital parameters are also calculated in Sect. 3.

With the semi-amplitudes of O-C oscillations, the value $a_{12}^{\prime} \sin{i}$ can be computed. Then using the following equation:

$$f(m)\!=\!\frac{(M_{3}\sin{i^{\prime}})^{3}} {(M_{1}+M_{2}+M_{... ...c{4\pi^{2}}{GT^{2}}\times(a_{12}^{\prime}\sin{i^{\prime}})^{3}$$ (24)

we can obtain the mass functions f(m) for the additional bodies. Taking the absolute parameters listed in Table 1, the values of the masses and the orbital dimensions of the third bodies for several different values of $i^{\prime}$ are computed and listed in Table 9. If we assume that the orbital inclination is perpendicular to the visual line (i.e., $i^{\prime}=90^{\circ}$), the values of the masses of the additional bodies are computed to be m₃=1.12, 1.50, 0.75and 0.87 $M_{\odot}$ for UW Cyg, RX Hya, AC Tau and UW Vir. At this case the orbital radii of the third bodies should be: a₃=13.64, 16.83, 10.77, 13.40AU respectively.

As listed in Table 9, for RX Hya, the minimum mass of the third body is $m_{3}=1.50~M_{\odot}$ which is smaller than the corresponding value $2.25~M_{\odot}$ derived by Vyas & Abhyankar (1989) from their orbital period analysis. If the third component is main sequence star as the components of RX Hya, the computed mass will correspond to a spectral type G1-3 which is nearly close to the spectral type of this system (F1). This indicates that the spectral line of the third body should be visible in the spectrum and the third star should contribute to the total luminosity of the system. In the absence of such evidences, the presence of the third body needs further study.

For UW Cyg, the minimum mass of the third body ( $m_{3}=1.12~ M_{\odot}$) corresponds to a spectral type G5-7 which is also close to the spectral type of this system (F0). From the spectroscopic and photoelectric observations, we should be able to find the presence of the third body. In the cases of AC Tau and UW Vir, since the values $0.75~M_{\odot}$ and $0.87~M_{\odot}$are also the minimum masses of the third bodies, there is a possibility to see their spectral lines in the spectra. However, to the best of my knowledge, no spectroscopic and photometric studies of the three systems, UW Cyg, AC Tau and UW Vir, have been published. This emphasizes the urgent need of future photometric and particularly spectroscopic observations to decide on the hypothetical third bodies.

Acknowledgements

Many thanks are given to Prof. F. van't Veer for his useful comments and improvements of the author's English writing. The author also thanks the supports from the Chinese Natural Science Foundation and from the 973 scheme.