Cowling (1941) was the first to study the natural modes of oscillation in a centrally condensed star, suggesting that some of the gravity modes may enter into resonance with the periodic tidal potential in a close binary. As was originally noted by Zahn

(1975), in a binary component possessing a radiative envelope the resonances of these low frequency oscillations are heavily damped by radiative diffusion, which operates in a relatively thin layer below the star's surface. Owing to that dissipative process, the dynamical tide does not have the same symmetry properties as the forcing potential. Hence, a net torque is applied to the binary component, which tends to synchronize its axial rotation with the orbital motion.

To evaluate the characteristic time for synchronization in an early-type star, one must calculate the amplitude of the forced oscillation at the star's surface, taking radiative damping into account. Following Zahn's (1975) analysis, one can show that the bulge raised by the dynamical tide is much smaller than that produced by the equilibrium tide; however, unlike the latter, it can take any orientation with respect to the companion star, depending on the tidal frequency. Detailed calculations also show that the torque Td resulting from the dynamical tide is proportional to the product (R/a)6(Q, â€” a0)8/3, if the density gradient is assumed to be continuous across the core-envelope interface. (Compare with Eqs. [8.4] and [8.5].) It is therefore appropriate to introduce a new synchronization time tsyn defined as

Again letting Ii2 ^ â€”Td and making use of Eq. (8.17), one can estimate the time scale tsyn by tsyn = f,5,6 ( T ) . (8.27)

where the structural constant ES, which is the strict analog of the apsidal-motion constant k in Eq. (8.2), depends mainly on the size of the convective core. Along the main sequence, one has ES ^ 10â€”8 when M = 2M0 and ES ^ 10â€”6 when M = 1OM0. For moderately small eccentricities, one also has

with the secondary contributing a similar amount to the effective circularization time of the binary star. (Compare these two equations with Eqs. [8.20] and [8.22], respectively.)

Because the synchronization process involves a secular adjustment of the gaseous components in a binary star, Eqs. (8.17) and (8.27) clearly show that the degree of synchronism decreases as the distance ratio a/R and the orbital period P increase. Accordingly, if we consider a sample of binaries with a random distribution of ages, one may expect to find an increasingly mixed population of asynchronous and synchronous rotators as the orbital periods approach an upper period limit above which binaries are nearly all asynchronously rotating. Given any data set, it is therefore essential to check that the theory can indeed account for the whole period range for which there is still a significant tendency toward synchronization. The same remark can be made about orbital circularization.

Table 8.1. The critical values ac/R and Pc

Synchronization Circularization

Synchronization Circularization

Table 8.1. The critical values ac/R and Pc

1.6 |
6.11 |
1.21 |
4.44 |
0.75 |

2 |
7.05 |
1.59 |
4.99 |
0.95 |

3 |
6.81 |
1.92 |
4.85 |
1.10 |

5 |
6.52 |
2.19 |
4.68 |
1.33 |

7 |
6.72 |
2.69 |
4.80 |
1.62 |

10 |
6.67 |
3.30 |
4.77 |
2.00 |

15 |
7.04 |
3.98 |
4.99 |
2.38 |

Source: Zahn, J. P., Astron. Astrophys., 57, 383, 1977.

Source: Zahn, J. P., Astron. Astrophys., 57, 383, 1977.

Following Zahn (1977), we shall define the limiting separations for synchronization and orbital circularization as the distance ratios ac/R for which one has, respectively, tsyn/ta = 0.25 and tcir/ta = 0.25. (The time ta is the main-sequence lifetime of the binary star, which consists of two similar components.) Thence, by making use of Eqs. (8.17), (8.27), and (8.28), one can easily obtain the corresponding critical periods Pc. In Table 8.1 we list the numerical values of these limiting separations.

From Table 8.1 it is apparent that synchronism should be the rule up to a/R ^ 6-7 in the early-type, main-sequence stars. This is not in agreement with the observational results reported in Section 1.4, however, since they clearly show that the early-type (from O to F5) close binaries do exhibit a considerable tendency toward synchronization (or pseudo-synchronization) up to a/R ^ 20, with deviations from synchronism becoming the rule for a/R > 20 only. In fact, this mechanism is also much too weak to account for the high degree of orbital circularization that is observed in the early-type, main-sequence stars. Indeed, whereas Figure 1.11 clearly shows that some binaries with A-type primary stars have circular orbits with periods as long as 10 days, Table 8.1 indicates that the mechanism is effective only up to P ^ 1-2 days in these stars.

A similar result was obtained by Claret and Cunha (1997), who have integrated Eqs. (8.21) and (8.28) using a set of early-type main-sequence models. Again, unless the effects of radiative damping acting on the dynamical tide are artificially increased by several orders of magnitude, it is found that the resonance mechanism is unable to explain the longest-period circular orbits shown in Figure 1.11. The same result was obtained by Pan (1997), who found that this mechanism does not explain the observed degree of synchronism in early-type binaries with orbital periods P ^ 4-8 days. In other words, unless the orbital periods are shorter than a few days only, it is a most ineffective tidal process.

Attempts to patch up Zahn's (1975, 1977) calculations have been made. In particular, Goldreich and Nicholson (1989) have pointed out that the synchronization process caused by the tidal forcing of the gravity modes proceeds from the outside toward the inside of an early-type star. Thence, assuming that the tides induce differential rotation by synchronizing the outer layers of the star while leaving its interior roughly unaltered, Savonije and Papaloizou (1997) have shown that rotational effects could significantly influence the tidal response in the surface layers of a 20M0 main-sequence star. In particular, in contrast to subsynchronous stars, which tend to spin up toward corotation as a result of resonances with damped g-modes, it is found that supersynchronous stars spin down toward corotation due to resonances with damped r-modes, analogous to Rossby waves in the Earth's atmosphere. In my opinion, although these rotational effects might also improve the efficiency of the resonance mechanism in the less massive stars, they do not change the inescapable fact that this mechanism is a short-range one, since the corresponding times tsyn and tcir are proportional to (d/R)8 5 and (d/R)10'5, respectively. This is the reason why it is not likely to explain the largest circular orbits reported in Figure 1.11 and in p. 19n.

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