5 Variation in the orbital period of AC Tau

In order to study the changes in the orbital period of AC Tau, various times of light minimum are compiled. Many visual and photographic times of light minimum observed by many amateurs have been collected at Eclipsing Minimum Database and 7 visual timings from Cracow observatory were published by Kreiner (1976). These timings are listed in the first and sixth columns of Table 5. Timings with the same epoch numbers have been averaged and some times of light minimum listed in Table 5 are the mean values of these timings. With the following ephemeris given in the fourth edition of GCVS:

$${\rm MinI}=2445636.590+2\hbox{$.\!\!^{\rm d}$ }043356\times{E}$$ (14)

the residuals $({\rm O{-}C})_{1}$ of these timings are computed. These $({\rm O{-}C})_{1}$ valuse are listed in the fourth and ninth columns of Table 5 and are plotted in Fig. 6. One photographic minimum 2445057.522 shows large deviation from the general O-C trend formed by other points in Fig. 6. This point was not considered further in the discussion of the period variation.

As displayed in Fig. 6, we can conclude that the orbital period of the system is variable, and its variation is complex. Since the general trend of the $({\rm O{-}C})_{1}$ diagram may show a roughly parabolic distribution indicating a secular increase in the period, a least-squares solution of the $({\rm O{-}C})_{1}$values yields the following ephemeris:

$$% {\rm Min I}\!=\!2445636.6018\!+\!2.04337351\!\times\!{E} \!+\!1.99\ {10^{-9}}\!\times\!{E^{2}}$$ (15)

$\pm{14}$ $\pm{17}$ $\pm{2}$

where the coefficient of the square term represents the rate of change of the period. This ephemeris can be used for the estimation of future times of minima. A continuous period increase of ${\rm d}P/{\rm d}E=+3.98\ {10^{-9}}$days/cycle = $+7.11\ {10^{-7}}$ days/year is calculated which is equivalent to a period increase of 6.1 s/century. As in the case of RX Hya, the secular increase only indicates the general trend of the $({\rm O{-}C})_{1}$ diagram without describing any particular characteristic.

Figure 6: The O-C diagram of AC Tau. Also given in solid line is its description by a quadratic ephemeris

The $({\rm O{-}C})_{2}$ residuals from the quadratic ephemeris (15) are also listed in Table 5 and displayed in Fig. 7. The $({\rm O{-}C})_{2}$ values in Fig. 7 clearly suggest a periodic oscillation. With the least-squares method, the following periodic ephemeris is obtained:

$$({\rm O{-}C})_{2}\!=\!0.0069\!+\!0.0191\sin(0\hbox{$.\!\!^\circ$ }0677\!\times\!{E}\!+\!235.7^{\circ})$$ (16)

which can describe the general trend of the $({\rm O{-}C})_{2}$values wery well (solid line in Fig. 7). This ephemeris tell us a cyclic variation with a period of about T=29.8year and an amplitude of about $A=0\hbox{$.\!\!^{\rm d}$ }0191$. The cyclic variation of the O-C curve indicates that the orbital period varies in a periodic way.

Figure 7: Residuals of AC Tau from the quadratic ephemeris and their description by a periodic function