3 Variation in the orbital period of RX Hya

The orbital period of RX Hya has been studied by Vyas & Abhyankar (1989). They collected 6 photoelectric and 55 visual times of light minimum and pointed out that the O-C curve showed a cyclic variation with a period of 72.46 years and an amplitude of $0\hbox{$.\!\!^{\rm d}$ }069$. By omitting a few visual timings of earlier epoch, they obtained a rather large increase rate ( ${\rm d}P/{\rm d}t=1.02\ {10^{-5}}$ days/year) for the change of the orbital period. After Vyas and Abhyankar's study, many times of light minimum have been collected at Eclipsing Minimum Database. With these timings, the changes in the orbital period of the system are analyzed. In order to study the variations in the period of the system, the residuals $({\rm O{-}C})_{1}$ based on the following ephemeris:

$${\rm MinI}=2443447.700+2\hbox{$.\!\!^{\rm d}$ }2816450\times{E}$$ (4)

given in GCVS, are calculated. These $({\rm O{-}C})_{1}$ values are listed in the fifth and ninth columns of Table 3 and shown graphically against epoch number in Fig. 3. During the calculation, timings with the same epoch have been averaged and some times of light minimum listed in Table 3 are the mean values.

Figure 3: O-C diagram of RX Hydrae. Circles refer to visual and photographic, and dots to photoelectric observations. Also given in solid line is the description of the general trend of the O-C curve

As displayed in Fig. 3, the orbital period of the system varies in some complex ways. Since the general trend of the $({\rm O{-}C})_{1}$ diagram may show a roughly parabolic distribution indicating a long-time increase in the orbital period, with the weight 3 for visual and photographic, and 8 for photoelectric times of light minimum, a second-order least-squares solution of the $({\rm O{-}C})_{1}$ values yields the following ephemeris:

$${\rm Min I}\!=\!2443447.7165\!+\!2.28162647\!\times\!{E} \!+\!3.25\ {10^{-9}}\!\times\!{E^{2}}$$ (5)

$\pm{82}$ $\pm{15}$ $\pm{2}$

where the coefficient of the square term represents the rate of change of the period and a continuous period increase of ${\rm d}P/{\rm d}E=+6.5\ {10^{-9}}$days/cycle = $+1.04\ {10^{-6}}$ days/year is calculated which is equivalent to a period increase of 9.0s/century. The secular increase only indicates the general trend of the $({\rm O{-}C})_{1}$ diagram without describing any particular characteristics.

The $({\rm O{-}C})_{2}$ residuals from the quadratic ephemeris (5) are also listed in Table 3 and displayed in Fig. 4. The $({\rm O{-}C})_{2}$ values in Fig. 4 clearly show a periodic oscillation. Most of the observations are photographic or visual and all the times of light minimum are primary, this may be caused by the fact that the secondary is more difficult to observe by such methods. We do not know whether secondary times of light minimum follow the same trend of the O-C variation. The circular orbit of RX Hya (Lucy & Sweeney 1980) suggests that the periodic oscillation of the $({\rm O{-}C})_{2}$ residuals is not caused by apsidal motion and that the light-time effect caused by the presence of a third body may be the real reason. The not strictly sine-like variation of the $({\rm O{-}C})_{2}$ curve suggests that the third body is moving in an elliptical orbit. The following formula:

$$({\rm O{-}C})_{2}=a_{0} +\sum_{i=1}^{2}{[a_{i}\sin(i{\omega}{E})+b_{i}\sin(i{\omega}{E})]}$$ (6)

has been used to express these $({\rm O{-}C})_{2}$ values which are the result of various numerical trials with different expressions and give the best fitting with the observations (solid line in Fig. 4). By least-squares solution of equation (6), the following values are obtained: a₀=0.0031, a₁=0.0369, b₁=-0.0224, a₂=0.0021, b₂=0.0073 and $\omega=0\hbox{$.\!\!^\circ$ }0414$. With $\omega=360^{\circ}P_{\rm e}/T$, the orbital period of the eclipsing binary around the centre of mass of the triple system is determined to be: T=54.3years.

Figure 4: O-C residuals of RX Hydrae based on the quadratic ephemeris (5). Solid line is the fit of the third body solution

From a visual inspection of the trend of these $({\rm O{-}C})_{2}$residuals, it is seen that the orbital eccentricity of the eclipsing binary around the common centre of the gravity of the three bodies is small, and the orbital parameters can be directly determined from the folowing formular (Kopal 1959):

$$a_{12}^{\prime}\sin{i}^{\prime}=c\sqrt{a_{1}^{2}+b_{1}^{2}}$$ (7)

$$e^{\prime}=2\sqrt{\frac{a_{2}^{2}+b_{2}^{2}}{a_{1}^{2}+b_{1}^{2}}}$$ (8)

$${\omega}^{\prime}=\arctan\frac{(b_{1}^{2}-a_{1}^{2})b_{2}+2a_{1}a_{2}b_{1}} {(a_{1}^{2}-b_{1}^{2})a_{2}+2a_{1}b_{1}b_{2}}$$ (9)

$${\tau}^{\prime}=t_{0}-\frac{P^{\prime}}{2\pi} \arctan\frac{a_{1}b_{2}-b_{1}a_{2}}{a_{1}a_{2}+b_{1}b_{2}}$$ (10)

where c is the speed of light, $a_{12}^{\prime}$ is the semi-major axis of the eclipsing pair around the common centre of the gravity of the triple system, $i^{\prime}$, $e^{\prime}$, $w^{\prime}$, ${\tau}^{\prime}$ and $P^{\prime}$ are the usual elements of the third body orbit. The results are: $a_{12}^{\prime}\sin i^{\prime}=7.47$AU, $e^{\prime}=0.34$, $w^{\prime}=43.5^{\circ}$ and ${\tau}^{\prime}=2447568\hbox{$.\!\!^{\rm d}$ }5$.