The tidal-torque mechanism was originally discussed by Darwin (1879) with reference to a planet-satellite system. As was shown by Zahn (1966), it is also effective in binary-star components possessing an extended outer convection zone. In this model each component possesses tides lagging in phase behind the external field of force on account of eddy viscosity in its convective envelope. Accordingly, this misalignment of the tidal bulges with respect to the line joining the two centers of mass will introduce a net torque between the components. This torque will cause, in turn, a secular change in spin angular momentum of the individual components, the effects of which will be reflected in secular changes in the orbital elements of the binary.
8.2.1 Darwin's weak-friction model
For the sake of clarity, I shall first derive the synchronization time for a system of two rotating stars in circular orbits about their common center of mass; the rotation axes are assumed to be perpendicular to the orbital plane. We take as the origin of our system of coordinates the center of mass of the primary (of mass M and radius R). We shall also assume that the radii of the components are smaller than their mutual distance d, so that the secondary may be treated as a point mass when studying the tides raised on the primary. To lowest order in the ratio r /d, the tidal potential W due to the secondary (of mass M') is
where r is the distance from the primary's center, & is the angle between the direction to the field point and the line joining the two centers of mass, and P2 is the Legendre polynomial of degree two. The dynamical tide will be neglected altogether.
If viscous dissipation is negligible, the equilibrium tide raised by the secondary can be described by an effective potential We whose value at the primary's surface is given by
where k is the apsidal-motion constant, which depends on the density stratification in the tidally distorted star. Outside the primary, the potential We will be the external solution of Laplace's equation, with Eq. (8.1) defining its boundary value at r = R. One obtains
whenever the tidal bulges are symmetrical about the line joining the two centers of mass.
Turbulent friction introduces a small time lag At, however; the tidal bulges lag (or lead) by a small angle S if the rotational angular velocity Q of the primary is smaller (or greater) than the orbital angular velocity Q0. This produces a torque component in the gravitational attraction of the two stars. The tidal torque T felt by the secondary is equal to M' fed = —M'd We/d&, where the angular derivative is evaluated at r = d and & = S. One readily sees that
The tidal torque acting upon the primary is exactly opposite of this torque.
Now, in the so-called weak-friction approximation one assumes that the small angle S is linearly proportional to the departure from synchronism, with this angle being also proportional to the strength of viscous dissipation. We shall thus write t2
where tff = (GM/R3)-1/2 is the free-fall time and T is a typical time scale on which significant changes in the orbit take place through tidal evolution. Since the latter is inversely proportional to the efficiency of viscous dissipation, we shall further let T = R2/vt, where vt is the coefficient of eddy viscosity (see Section 2.4). We shall also assume that the primary goes through a succession of rigidly rotating states, during which the tidal torque causes a slow but inexorable change in the spatially uniform angular velocity Q. One thus has IQ ^ —T, where I is the moment of inertia of the primary about its rotation axis. (A dot designates a derivative with respect to time.) Thence, we can estimate the characteristic time for synchronization, tsyn, by
where r2 = (I/MR2)1/2 is the fractional gyration radius and q = M'/M is the mass ratio. Simultaneously, via the torque T, angular momentum is transferred from the primary's spin to the secondary's orbit. This results in a secular change in the distance ratio d/R and, hence, in the orbital angular velocity Q0.
The weak-friction model is ideally suited for a detailed study of tidal interaction in detached close binaries that have significant eccentricities. Following Hut (1981), we shall assume that the deviations from coplanarity are small enough to be treated linearly.
To simplify the discussion, the secondary is also assumed to be point like so that only on the primary tides will be raised. If so, then, it can be shown that the resulting tidal evolution equations for the primary are as follows:
dm _ ^ k q2 /R\ m Q0 ~dt =— T~ri\a) (i — e2)6 Q
where a is the semimajor axis, e is the eccentricity, ^ is the rotational angular velocity, and rn is the angle between the orbital plane and the equatorial plane of the primary. For brevity, we have defined the following quantities:
which is the ratio of rotational to orbital angular momentum (see Eq. [8.18]). Finally, one can write Kepler's third law in the form
where is the mean orbital angular velocity.
By making use of Eqs. (8.7)-(8.9), one easily verifies that the total angular momentum of the system is conserved. Here we have
Jti1 Q + MMMm' [G (M + M)a(i — e2)]i/2} = 0, (8.i8)
a since the rotational contribution from a point-mass companion can be neglected. In the case of two extended deformable bodies, Eqs. (8.7) and (8.8) can be applied to each binary component, interchanging the role of primary and secondary and adding both contributions to the orbital parameters a and e.
The behavior of the solutions of Eqs. (8.7)-(8.10) around a state of equilibrium has been thoroughly investigated by Hut (1981). In particular, for moderately small eccentricities, he was able to derive the time scales for the exponential relaxation of the relevant parameters. In detached close binaries for which the orbital angular momentum is much larger than the sum of the rotational angular momenta, Hut found that the characteristic time for orbital circularization, tcir, is much larger than the other three, which are of comparable magnitude.* Making use of Eq. (8.9), one easily obtains the linearized equation
which confirms the order-of-magnitude estimate given in Eq. (8.6). For two extended bodies in almost circular orbits about their common center of mass, Eq. (8.8) further implies that
where the figures 1 and 2 refer to the primary and secondary, respectively. For the primary one has
the secondary (of mass M' and radius R', say) makes a similar contribution to the effective circularization time, which is the harmonic mean of the circularization times obtained for the individual components.
One readily sees from Eqs. (8.20) and (8.22) that the ratio tsyn/tcir is of the order of the parameter n evaluated at equilibrium (see Eq. [8.16]). Since this quantity is much smaller than one in a detached close binary, we perceive at once that the synchronization of the components proceeds at a much faster pace than the circularization of the orbit. To the best of my knowledge, Hut (1981) was the first to point out that the rotation of each component in a detached close binary will synchronize with the instantaneous angular velocity at periastron, since during each revolution the tidal interaction will be the most important around that position (see Eq. [1.8]). Recall also that the inclination m decreases rather quickly while at the same time rotation tends to synchronize with revolution, whereas the eccentricity of the orbit decreases at a much
* This ordering ofthe time scales was originally noticed by Alexander (1973, Figs. 7-10), who integrated numerically the tidal-friction equations for the close binary system AG Persei.
slower pace. This property of a detached close binary, which is probably independent of the exact nature of the underlying dissipative process, is a most likely explanation for the correlation between synchronism and coplanarity, as reported at the end of Section 1.4.
In deriving Eqs. (8.9) and (8.20) we have explicitly assumed that the tidally distorted star remains in a state of uniform rotation throughout its tidal evolution. As was shown by Scharlemann (1982), however, a tidally distorted star with an extended, differentially rotating convective envelope can be synchronized on the average, at a specific latitude on the surface of the star. Of course, because the tidal torque is applied mainly to the outer convective regions, the radiative core might rotate at a quite different speed - unless there is a strong coupling between the inner core and the outer envelope. In fact, even though such a coupling might exist in the late-type stars, once a star has evolved away from the main sequence it develops a helium-rich core whose rotation becomes decoupled from that of the envelope. This is particularly relevant to the case of a close binary star that has achieved synchronism and orbital circularity on the main sequence, since post-main-sequence expansion will desynchronize the components while maintaining a circular orbit as they move up to the giant branch. As we shall see in Section 8.4.1, in that case one must integrate Eqs. (8.19) and (8.21) along the evolutionary paths of the binary components, retaining the time dependence of the radii R and R' in the functions fan and far.
For tidally distorted stars possessing a deep convective envelope, it is generally believed that turbulent friction operating over the whole of that envelope is responsible for the tidal torque T, so that the characteristic time T must be a convective friction time scale derived from stellar envelope parameters (see Eqs. [8.4] and [8.5]). Lacking any better theory of turbulent convection in a star, we shall thus let T = R2/vt and vt = LcVc, where Lc is the typical size of the largest eddies and Vc is a typical convection velocity. Mixing-length theory provides crude estimates of these quantities. Following Zahn (1966), we have that the convective friction time scale T is of the order of (MR2/L)1/3, where L is the total luminosity of the star. (One has T ^ 160 days for a solar-type star.) By virtue of Eqs. (8.17) and (8.20), this particular prescription for the eddy viscosity implies that the synchronization time tsyn is proportional to the fourth power of the orbital period P (= 2n/ ^0) - that is, tsyn a (a/R)6 or tsyn a P4. Similarly, by making use of Eqs. (8.17) and (8.22), one readily sees that tcir a (a/R)8 or tcir a P16/3. These rather high exponents make the characteristic times tsyn and tcir strongly dependent on the separation of the components or, equivalently, on the orbital period.
The results obtained in Section 8.2.1 clearly show that the degree of synchronism and orbital circularization depends on how long the tidal torque has been operative on the binary components. Accordingly, because main-sequence stars evolve with almost constant radius, much information can be gained by merely comparing the evolutionary age of a main-sequence star with the corresponding time scales tsyn and tcir. The study of main-sequence close binaries belonging to open clusters is of particular interest, therefore, since stellar evolution theory provides an estimate of their ages from isochrone fitting. Here we shall mainly discuss the problem of orbital circularization, making use of the observational data summarized in Table 1.2 and Figure 1.12.
As was noted in Section 1.4, in a sample of low-mass main-sequence binaries belonging to the same cluster, all binaries with periods shorter than a cutoff period - Pcut, say -have circular orbits, whereas binaries with longer periods have orbits with a distribution of eccentricities. From Table 1.2 it is also apparent that an older sample of binaries has a longer transition period, with Pcut increasing monotonically with the sample age ta. This is in perfect agreement with the fact that in a coeval sample of binaries the cutoff period is time dependent, because the tidal interaction has more time to extend its influence in an old cluster than in a young one. The crucial test for Darwin's tidal-torque theory is to check whether the circulation time defined in Eq. (8.22) is indeed shorter than (or equal to) the age ta at P = Pcut.
To illustrate the problem, we shall assume that M = M0, R = R0, and q = 1. Although k probably lies within the range 0.01-0.02,1 shall give an edge to the theory and let k = 0.05. By virtue of Kepler's third law, Eq. (8.22) can be recast in the practical form tcir(yr) = 6 X 105 T(yr) [P(day)]16/3. (8.23)
Because we are considering similar binary components, the effective circularization time may differ from this by a factor of two, which is unimportant for our purpose. Since T is a free parameter in Eq. (8.23), let us prescribe that for each cluster the characteristic time tcir is equal to its age ta at P = Pcut. Letting Pcut = 18.7 days at ta = 17.6 Gyr, one easily verifies that ta (yr) = 3 x 103 [PCut(day)]16/3 (8.24)
gives a moderately good fit for the other coeval samples listed in Table 1.2. Since Eqs. (8.23) and (8.24) must be equivalent at P = Pcut, one readily sees that the mechanism is operative on the main sequence provided one has
which is much shorter than T ^ 160 days. Now, with R ^ R0, Lc ^ R0/10, and T ^ 2 days, the formula T = R2/vt implies that the typical convection velocity Vc should be of the order of 40 km s-1. Obviously, this independent evaluation of T is also too large by about two orders of magnitude. *
At this juncture it is appropriate to mention the work of Claret and Cunha (1997), who have integrated Eqs. (8.21) and (8.22) using a set of low-mass stellar models that are slowly evolving on the main sequence. Unless turbulent dissipation is artificially
* Zahn (1989) has argued that one has vt a P in the short-period binaries, thus implying that one should let tsyn a P3 and tcir a P13/3 in these stars. According to Goldman and Mazeh (1991), however, one has vt a P2 in the short-period binaries, so that one should let tsyn a P2 and tcir a P10/3. Unfortunately, although these modified versions of the standard theory can provide a somewhat better fit to the slope of the observed log 4-log Pcut relation, they are still unable to resolve the basic weakness of the tidal-torque mechanism: Given a reasonable theoretical value for the convective friction time scale T, the circularization times are much too long at P = Pcut during the main-sequence phase. This inadequacy has been also confirmed by the independent analysis of Goodman and Oh (1997), who concluded that some mechanism other than turbulent convection circularizes solar-type binaries.
enhanced by a factor around 100-200, their calculations show that the tidal-torque mechanism is most ineffective in inducing orbit circularization on the lower main sequence. This result thus brings confirmation to the foregoing order-of-magnitude calculation.
Of direct relevance to the present discussion are the results of Zahn and Bouchet (1989), who have studied the pre-main-sequence evolution of solar-type binary stars. During this contraction phase, because a star undergoes great changes in size and structure, it is necessary to follow in time the dynamical state of the binary star along the evolutionary paths of its components (see Eqs. [8.21] and [8.22]). Their calculations strongly suggest that most of the orbital circularization takes place during the Hayashi phase, with the subsequent decrease in eccentricity on the main sequence being quite negligible. They found that the cutoff period of any sample should lie between 7.2 and 8.5 days, independent of the sample age. Unfortunately, this conclusion is not at all supported by the observational data reported in Table 1.2, which strongly suggest that the circularization mechanism is operative during the main-sequence lifetimes of the stars - pre-main-sequence tidal circularization is permitted but not required by present observations.
Let us also note that the tidal-torque mechanism, which is quite ineffective on the main sequence, may become operative again during the post-main-sequence phases. This is quite apparent from the work of Verbunt and Phinney (1995), who have shown that turbulent friction acting on the equilibrium tide can generate circular orbits up to P ^ 200 days in binaries containing giant stars. Yet, Figure 1.12 clearly shows that there exists a mixed population of circular and eccentric orbits in the whole period range 80-300 days. Obviously, independent calculations based on Eqs. (8.21) and (8.22) are needed to ascertain whether this mechanism can remain operative up to P = 300 days, or whether an additional circulation mechanism becomes operative during the expanding phases of stellar evolution.
For completeness, let us briefly discuss the problem of pseudo-synchronism in late-type main-sequence binaries. This is a much more difficult exercise, however, because the relevant observations are still very scarce. Letting T ^ 160 days in Eq. (8.22), one finds that the tidal-torque mechanism does contribute to the synchronization process up to P ^ 20 days in these binary stars. However, there exist a few binaries, with orbital periods in the range 40-50 days, that exhibit a definite tendency toward pseudo-synchronization in their solar-type components. Accordingly, we are led to conclude that this mechanism might not wholly account for the presence of pseudo-synchronous rotators in these binaries. A similar comment was made by Maceroni and van't Veer (1991) in their study of the dynamical evolution of G-type main-sequence binaries.
These results present quite a dilemma if one assumes that Darwin's tidal-torque theory alone can explain the whole set of observational data for the late-type binaries. As we shall see in Section 8.4.4, there is no longer any problem when one relaxes the assumption of strict uniform rotation, thus making allowance for tidally driven meridional currents in the asynchronously rotating components of a detached close binary.
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