5 Discussion

Our new observations of S 1082 confirm the eclipses reported by Goranskij et al. (1992), the small radial-velocity variations of the narrow lines in the spectrum (Mathieu et al. 1986) and the time-variation of a broad-lined component seen in high-resolution spectra (van den Berg et al. 1999; Shetrone & Sandquist 2000). In addition, we now have found that the radial-velocity shifts of the broad-lined component and of a third component in the spectra vary on the photometric period. This clearly demonstrates that the narrow-lined star, which is a blue straggler on its own, is not part of the eclipsing binary - this solves the seeming contradiction in the properties of this system. The broad variable component of the H$\alpha$ line discussed by van den Berg et al. (1999) is the Stark-broadened H$\alpha$ line of the hot component Aa of the inner binary. The ultraviolet flux measured by Landsman et al. (1998), higher than expected for a star at the B-V colour of S 1082, is only partly explained as a consequence of the bluer colour, i.e. higher temperature of the third star.

Comparison with Figs. 2 and 3 of Dempsey et al. (1993) shows that the X-ray luminosity of S 1082 (Belloni et al. 1998) is typical for a subgiant in an RS CVn system. The X-ray emission could therefore be caused by magnetic activity in the rapidly rotating subgiant Ab. Thus, our model can explain the X-rays of S 1082.

Our solution to the photometry is symmetric (see Fig. 8) and does not explain the asymmetry between phases 0.25 and 0.75; nor the variability in the form of the secondary eclipse between our first two observation runs. This indicates that variable spots are present and accordingly that the best-fit parameters of the system are subject to some additional uncertainty. However, we do not think our main conclusions are affected.

We think that the eclipsing binary forms a bound triple with the third star, for two reasons. First, the radial-velocity measurements of the third star B - the narrow-lined system in S 1082 - indicate that it is in a $\sim$1000 day orbit around a companion. The systemic velocities of both the 1000 day and the 1.07 day binaries are compatible with the radial velocity of the cluster. A chance alignment of two such bright cluster members is unlikely. Second, if we relax the constraint that the binary is at the cluster distance in solving the photometry, we find that the best solutions tend to be an Algol binary at higher mass ($\gtrsim$4 $M_{\odot }$) and larger distance (about twice the cluster distance). A high-mass binary at such a large distance from the galactic disk is unlikely.

Assuming then that the binary forms a hierarchical triple with the third star, in M 67, we note that the 1-$\sigma$ lower bound to the mass of the binary is at about three times the turnoff mass of the cluster. The formation of the binary must thus have involved at least three stars. The third star is a blue straggler on its own account, and thus according to most current models its formation involves two stars. We have to conclude that the formation of S 1082 required the interaction of no less than five stars!

This interaction may have started with a binary-binary encounter, in which two stars collided directly and merged; one of the remaining two stars ended in a close orbit around the merger, the fourth star in a wide orbit around the inner binary. The fourth star must be a blue straggler, which either was already present in the original encounter or was later exchanged into the system. That triples are formed easily by binary-binary encounters, and that they live long enough to undergo subsequent exchange encounters with a binary, is shown for example by the computations of Aarseth & Mardling (2001). It is less obvious that mergers are common in such encounters, as most binary-binary encounters are between relatively wide systems. Merger products tend to be subluminous (Sills et al. 2001), in accordance with the properties of the more massive star Aa in the inner binary. The main problem with this scenario for the formation of the S 1082 system is that a merger product is only subluminous for a very limited period; comparable to its thermal time scale. This reduces the available time within which the outer star of the first encounter is exchanged with a binary or blue straggler in a subsequent encounter, and indicates that the blue straggler was present in the initial binary-binary encounter. Alternatively, an initial encounter between two triple systems can have led immediately to the currently observed configuration. In any case, the probability of catching the merger product while it is strongly subluminous is disconcertingly small.

Aarseth & Mardling (2001) note that the inclination of the outer orbit in a hierarchical triple with respect to the inner orbit can induce a large eccentricity in the inner orbit; subsequently, tidal forces in the inner orbit cause it to shrink. Thus, the inner orbit may be eccentric, and smaller than in the past. Our solution for the inner binary implies that the cool star has a radius which is a sizeable fraction of the binary separation $R_{\rm Ab}/a_{\rm A} \approx 0.3$. According to Eq. (2) of Verbunt & Phinney (1995) the circularisation time scale of the inner binary A is $\sim$10³ year; the eccentricity of A and asynchronicity of Ab thus will be determined by the competing effects of the perturbation by the outer star B and the tidal forces in the inner binary. Only if the latter are substantially reduced with respect to the Zahn (1977) formulation - as suggested by e.g. Goodman & Oh (1997) - do we expect measurable eccentricity and asynchronicity.

Whereas our new observations have allowed us to resolve the apparent contradictions in the earlier data and to determine the system parameters, we conclude that these parameters - in particular those of the primary Aa of the inner binary - are hard to understand in terms of standard stellar and binary evolution, even when stellar encounters and mergers are taken into account. This situation is remarkably similar to that in the study of two other members of M 67, located below the subgiant branch in the colour-magnitude diagram, both of which turn out to be binaries (Mathieu et al. 2001, in preparation); and indeed to our lack of understanding of the blue straggler population in the cluster. Many of these stars may be mergers, which raises the question whether the object that results when two stars merge may follow an evolutionary track which is very different from that of an ordinary star of the same mass; and whether they can do so for a period of time which significantly exceeds the thermal time scale.

Further studies of the remarkable triple system S 1082 should provide a better-sampled radial-velocity curve, necessary to determine whether the inner binary is eccentric, and to measure its systemic velocity more accurately to establish cluster membership. Better-quality light curves may also serve to improve the accuracy of the inner orbit, e.g. if the inner orbit is eccentric the separation of the eclipses is different of 0.5. A long-term sampling of the light curves will be required to improve the interpretation of the O-C in the timings of the primary eclipse.

Acknowledgements

The authors wish to thank M. Janson, R. Dijkstra, G. Geertsema, R. Cornelisse and G. Nelemans for doing part of the observations. We also want to thank D. Latham for discussions. M.vdB. is supported by the Netherlands Organization for Scientific Research (NWO).