3 Period analysis

Using the data gathered over a period of nine years, Kilkenny et al. (1994) showed that the orbital period of HW Vir was decreasing. Although they reviewed the possible mechanisms that would cause period decrease, they could not reach a conclusion as to which one was the most likely candidate. Among the probable mechanisms, the gravitational radiation effect, apsidal motion and mass transfer were ruled out. The remaining causes like the existence of a third body, mass loss through stellar winds and angular momentum loss through magnetic braking have been thoroughly studied. Of all these mechanisms, a period decrease through angular momentum loss has been given the highest probability. As stated previously, the final decision would be reached only after new observations.

In the present study, we obtained two primary minimum times and one secondary minimum time. Minimum times that have been obtained ever since the system was found to be an eclipsing binary are given in Table 1.

  

Table 1: Times of minima of HW Vir. References: (1) Menzies & Marang (unpublished); (2) Kilkenny et al. (1994); (3) Marang & Kilkenny (1989); (4) Wood et al. (1993); (5) Selam et al. (IBVS4109); (6) This study

O-C (I) deviations are obtained by using the light elements given by Menzies & Marang (1986),

$${\rm Min~I~=~JD~Hel.~2445~730.55607~+~0\hbox{$.\!\!^{\rm d}$}1167196311}~E.$$

O-C (I) deviations are plotted in Fig. 2.

  

Figure 2: The deviations between the observed and calculated times of minima

As it is seen from the figure, there is no significant deviation until $E=23\ 000$. For $E\gt 23\ 000$ it is obvious that O-C (I) deviations decrease continuously. Assuming that O-C (I) varies linearly with E we recalculated the light elements. The new elements are,

$${\rm Min~I~=~JD~Hel.~2445~730.556503~+~0\hbox{$.\!\!^{\rm d}$}1167195820}~E$$

$$\hspace{4.7cm} \pm 13 \hspace{2cm} \pm 42.$$

O-C deviations show a sine-like scattering around the linear fit. This sine-like curve, although it has not shown a complete cycle yet, seems to be nearing its one cycle completion.

Such a variation could be accounted for in three different ways:

1. Mass transfer. Components of HW Vir are sdB and low mass main sequence stars. It is known that in such a system, where mass accretion occurs from a low mass star to a high mass one, the orbital period increases. Nevertheless, it is also known from the light curve analysis that in the HW Vir system, the low mass star has not filled its Roche lobe yet (see Sect. 4). For this reason it is not reasonable to account for the O-C deviations in terms of mass transfer.

2. Magnetic braking. It was stated by Patterson (1984) that no matter how small the rate of mass-loss is, magnetic braking would eventually cause a decrease in period. However, in the HW Vir system, since the low mass star is very much smaller than its own Roche lobe it becomes unrealistic to expect that the low mass star will lose mass through stellar wind. Besides, the variation of the orbital period of HW Vir shows an uniform decreasing pattern which would have shown increasing and decreasing

portions otherwise. Therefore, neither magnetic braking is the likely mechanism that causes O-C deviations in the system.

3. A third body. Sinusoidal variations in O-C may be attributable either to an orbital motion around a third body or to apsidal motion. But the observational facts such as circular orbit inferred from the radial velocities (Hilditch et al. 1996) are the strongest evidence that would compel one to rule out the apsidal motion as a possible mechanism for period change. What remains behind is the possibility of orbital motion around a third body.

The additional time delay of any observed eclipse due to orbiting around a third-body can be represented by,

$$\Delta {T} = \frac{a\sin{i}}{c}[({1-e^{2}})\sin(\upsilon+\omega)/(1+e\cos\upsilon)+e\sin\omega]\,,$$

c = 2.590 10¹⁰ km d^-1 is the velocity of light. In this case the resulting eclipse ephemeris is given by, $t_{\rm ec} = t_0+EP^{'}+\Delta {T}$where t₀ is the starting epoch, E is the integer eclipse cycle number. We have chosen Irwin's (1952, 1959) definition for the zero of the light-time effect. The linear least squares solution was applied in order to obtain the best fit and the standard deviations of the parameters calculated. The results are given in Table 2.

 

Table 2: The parameters of the third-body orbit

 

The final parameters, given in Table 2, were used to obtain the calculated light-time values and the computed light-times were plotted in Fig. 3 along with the observed values.

  

Figure 3: O-C diagram for HW Vir

The fit seems to be quite good. The present analysis see to cover the 69 per cent of the whole cycle of the orbital motion. Further observations will justify or falsify the third body hypothesis.

Using the asini value given in Table 2, the mass function has been computed to be $0.000023~M_\odot$. Assuming a combined mass of $0.5~M_\odot + 0.14~M_\odot$ for the eclipsing pair the mass of the third body can be evaluated from the mass function depending on the inclination of the third-body orbit with respect to the plane of the sky. The mass of the third component was found to be $0.022~M_\odot$ for inclination 90$^\circ$, and $0.025~M_\odot$ for the inclination 60$^\circ$. For the inclination above 17$^\circ$ the mass computed by us would lead to a sub-stellar mass. Therefore, such a low mass of tertiary component would make it too faint for direct detection. If we assume the inclination is about 17$^\circ$ the mass of the third body would be $0.008~M_\odot$. The visual magnitude of such a body would be $20^{\rm m}$ for the distance of 141 pc.