Close binaries

In Section 1.1 we pointed out that the early-type components of close binaries rotate more slowly than the average of single stars of the same spectral type. In contrast, whereas the rotational velocities of single main-sequence stars of spectral type F5 and later are quite small (i.e., less than 10 km s-1), appreciable rotations are common among the late-type components of close binaries. It has long been recognized that the distribution of rotational velocities in the close binaries is caused mostly by tidal interaction between the components, although some other processes - such as stellar winds, gravitational radiation, and large-scale magnetic fields - may also play a definite role in some binaries. To be specific, all types of tidal interaction involve an exchange of kinetic energy and angular momentum between the orbital and rotational motions. If we neglect stellar winds, the total angular momentum will be conserved in the tidal process. However, due to tidal dissipation of energy in the outer layers of the components, the total kinetic energy will decrease monotonically. Accordingly, as a result of

Fig. 1.9. Histogram showing the frequency distribution of rotation periods of T Tauri stars. This figure combines the data for the Trapezium cluster, the Orion Nebula cluster, and other T associations. Source: Eaton, N. L., Herbst, W., and Hillenbrand, L. A., Astron. J., 110, 1735, 1995.

various dissipative processes, a close binary starting from a wide range of initial spin and orbital parameters might eventually reach a state of minimum kinetic energy. This equilibrium state is characterized by a circular orbit, where the stellar spins are aligned and synchronized with the orbital spin.

As we shall see in Sections 8.2-8.4, however, in detached binaries the synchronization of the components proceeds at a much faster pace than the circularization of their orbits. Accordingly, the rotation of each component will quickly synchronize with the instantaneous orbital angular velocity atperiastron,

where the tidal interaction is the most important during each orbital revolution. (As usual, e is the orbital eccentricity and is the mean orbital angular velocity.) Figure 1.10 illustrates this concept of pseudo-synchronism for a sample of selected eclipsing binaries with eccentric orbits for which we have accurate absolute dimensions. This figure compares the observed rotational velocities with the computed rotational velocities, assuming synchronization at periastron. We observe that most points scatter along the 45-degree line, indicating that pseudo-synchronization obtains in most close binaries of short orbital periods, either perfectly or approximately.

Observations show that an upper limit to the orbital period exists at which the observed rotational velocities begin to deviate very much from the synchronization (or

Fig. 1.10. Predicted versus observed rotational velocities assuming synchronization at peri-astron. The diagonal line is the locus of pseudo-synchronous rotation. Source: Claret, A., and Gimenez, A., Astron. Astrophys., 277, 487, 1993.

pseudo-synchronization) period. As was originally noted by Levato (1976), the orbital period below which main-sequence binary components are still rotating in synchronism depends on spectral type. Specifically, he found that the largest orbital period for full synchronism is about 4-8 days in the early B spectral range, decreases to a minimum value of about 2 days at mid A-type, and increases up to 10-14 days at mid F-type. Subsequent investigations have confirmed that the tendency toward synchronization between the axial rotation and orbital revolution is indeed stronger in the F-type and later types than in the hotter ones. However, these studies have also demonstrated that in the whole early spectral range synchronism (or pseudo-synchronism) extends up to binary separations substantially greater than previously held. For example, the rotational properties of a large sample of early-type double-lined spectroscopic binaries have been investigated by Giuricin, Mardirossian, and Mezzetti (1985). Their statistical study indicates that a considerable tendency toward pseudo-synchronization extends up to a distance ratio d/R ^ 20 in the early-type (from O to F5) close binaries. (Here d is the mean distance between the components and R is the radius.) In fact, only for d/R > 20 do pronounced deviations from synchronism at periastron become the rule in these binaries. In terms of orbital periods (for an easier comparison with Levato's underestimated upper limit

Fig. 1.11. Period-eccentricity distribution for a sample of spectroscopic binaries with Atype primaries. Single-lined binaries are shown as crosses; double-lined binaries are shown as filled circles. Source: Matthews, L. D., and Mathieu, R. D., in Complementary Approaches to Double and Multiple Star Research (McAlister, H., and Hartkopf, W. I., eds.), A.S.P. Conference Series, 32, 244, 1992. (Courtesy of the Astronomical Society of the Pacific.)

Fig. 1.11. Period-eccentricity distribution for a sample of spectroscopic binaries with Atype primaries. Single-lined binaries are shown as crosses; double-lined binaries are shown as filled circles. Source: Matthews, L. D., and Mathieu, R. D., in Complementary Approaches to Double and Multiple Star Research (McAlister, H., and Hartkopf, W. I., eds.), A.S.P. Conference Series, 32, 244, 1992. (Courtesy of the Astronomical Society of the Pacific.)

periods), a limiting value of d/R ^ 20 corresponds to orbital periods of about 26, 18, and 13 days at spectral types B2, A0, and A5, respectively.

It is a well-known fact that circular (or nearly circular) orbits greatly predominate in short-period binaries. Since tidal interaction between the components of close binaries will tend to circularize their orbits, the precise determination of the cutoff period above which binaries display eccentric orbits appears to be a valuable test for the tidal theories. Giuricin, Mardirossian, and Mezzetti (1984) have studied the period-eccentricity distribution for a large sample of early-type detached binaries, excluding systems believed to have undergone (or to be undergoing) mass exchange between the components. They found that almost all binaries have circular or nearly circular orbits for orbital periods P smaller than 2 days. However, a mixed population of circular and eccentric orbits was found in the period range 2-10 days. Beyond P = 10 days all orbits are eccentric. A similar result was obtained by Matthews and Mathieu (1992), who investigated the period-eccentricity distribution of a sample of spectroscopic binaries with A-type primary stars. Figure 1.11 clearly shows that all binaries with orbital periods less than P ^ 3 days have circular or almost circular orbits (i.e., e < 0.05). Binaries with periods between 3 and 10 days are found with either circular or eccentric orbits, with the maximum eccentricity increasing with period. The longest-period circular orbit is at P = 9.9 days. This is exactly the kind of distribution one may expect to find for a sample of detached binaries with a random distribution of ages, where the population of circular and eccentric orbits becomes increasingly mixed as the P s tend toward an upper limit period above which all orbits become eccentric.* For comparison, Figure 1.12 illustrates

* More recently, Mermilliod (1996, Fig. 2) has shown that this upper limit period was actually close to 25 days for a sample of 39 late-B and A-type binary stars belonging to open clusters. Note also that most of the O-type binaries with periods less than 30 days have circular orbits, whereas the long-period systems have eccentric orbits (Massey, 1982, p. 258).

Fig. 1.12. Period-eccentricity distribution for a sample of spectroscopic binaries with red giant primaries. Source: Mermilliod, J. C., Mayor, M., Mazeh, T., and Mermilliod, J. C., in Binaries as Tracers of Stellar Formation (Duquennoy, A., and Mayor, M., eds.), p. 183, Cambridge: Cambridge University Press, 1992.

Fig. 1.12. Period-eccentricity distribution for a sample of spectroscopic binaries with red giant primaries. Source: Mermilliod, J. C., Mayor, M., Mazeh, T., and Mermilliod, J. C., in Binaries as Tracers of Stellar Formation (Duquennoy, A., and Mayor, M., eds.), p. 183, Cambridge: Cambridge University Press, 1992.

the period-eccentricity distribution of spectroscopic binaries with red giant primaries. Not unexpectedly, because red giants reach larger radii than main sequence stars, circular orbits are found for larger orbital periods. Note also the mixed population of circular and eccentric orbits in the period range 80-300 days. Again, this is caused by the mixing of all red giants, since the sample contains a range in age and mass.

It will be shown in Sections 8.2-8.4 that the degree of circularity of an orbit depends on how long the tidal forces have been acting on the components of a close binary. The study of binaries belonging to clusters is of particular interest, therefore, since these are the only stars for which one has some information about their ages. Mayor and Mermilliod (1984) were the first to study the orbital eccentricities for a coeval sample of late-type binaries in open clusters (33 red-dwarf binaries in the Hyades, Pleiades, Praesepe, and Coma Berenices open clusters). They found that all binaries with periods shorter than 5.7 days display circular orbits whereas all orbits with longer periods have significant eccentricities. More recently, it has been found that other coeval samples with different evolutionary ages exhibit transitions between circular and eccentric orbits at distinct cutoff periods. It is immediately apparent from Table 1.2 that the transition period Pcut increases monotonically with the sample age ta. Accordingly, the observed ta-Pcut relation strongly suggests that the circularization mechanism is operative during the main-sequence lifetime of the stars - pre-main-sequence tidal circularization is permitted but not required by present observations. This provides a very important test for the tidal mechanisms since the theoretical circularization time cannot exceed the sample age at cutoff period.

Tidal interaction in the RS CVn stars poses quite a challenging problem also. In fact, in these chromospherically active binaries there is still a tendency toward synchronization in the period range 30-70 days, up to P = 100 days. However, asynchronous rotators are present in all period groups, even among binaries with orbital periods of 30 days or less. In these systems one also finds that the rotation periods are either shorter or longer than the orbital periods, independent of the orbital eccentricities. As was shown by Tan, Wang, and Pan (1991), however, asynchronous RS CVn stars have orbital eccentricities that are larger, on the average, than the eccentricities of pseudo-synchronously rotating systems.

Table 1.2. The observed ta-Pcut relation.

Cutoff Period

Age

Binary Sample

(day)

(Gyr)

Pre-main-sequence

4.3

0.003

Pleiades

7.05

0.1

Hyades/Praesepe

8.5

0.8

M67

12.4

4.0

Halo

18.7

17.6

Source: Mathieu, R. D., Duquennoy, A., Latham, D. W., Mayor, M., Mazeh, T., and Mermilliod, J. C., in Binaries as Tracers of Stellar Formation (Duquennoy, A., and Mayor, M., eds.), p. 278, Cambridge: Cambridge University Press, 1992.

Source: Mathieu, R. D., Duquennoy, A., Latham, D. W., Mayor, M., Mazeh, T., and Mermilliod, J. C., in Binaries as Tracers of Stellar Formation (Duquennoy, A., and Mayor, M., eds.), p. 278, Cambridge: Cambridge University Press, 1992.

These authors also found that the chromospheric activity in their sample of asynchronous binaries is lower, on the average, than in synchronous RS CVn stars. If so, then, other braking mechanisms (e.g., magnetically driven winds) must be interfering with tidal interaction in these giant binary stars. To make the problem even more complex, let us note that Stawikowski and Glebocki (1994) have found another basic difference between the synchronous and asynchronous long-period RS CVn stars, when their primary component is a late-type giant or subgiant: Whereas for synchronously rotating stars the assumption about coplanarity of their equatorial and orbital planes isjustified, in most asynchronous binaries the rotation axis of the primary is not perpendicular to the orbital plane. A similar result was obtained by Glebocki and Stawikowski (1995, 1997) for late-type main-sequence binaries and short-period RS CVn stars with orbital periods shorter than about 10 days. Pseudo-synchronism and coplanarity will be further discussed in Section 8.2.1.

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