where A(t) describes the acceleration of the common center of mass with respect to our frame of reference, W is the tidal potential, and F is the (turbulent) viscous force per unit volume.* Remaining symbols have their standard meanings (see Eqs. [2.27]). A dot designates a derivative with respect to time.

To make the problem tractable, we shall neglect the nonlinear terms u ■ grad u in Eq. (8.44). This implies that one has |u ■ gradu| ^ |2^01z x u|z and |u ■ gradu| ^ |grad W| (i.e., - < ^0 and - ^|2 < GM'/d3, where d is the mutual

* See, e.g., Landau, L. D., and Lifshitz, E. M.,Mechanics, Section 39, Oxford: Pergamon Press, 1959.

separation of the two components). For the sake of simplicity, we shall neglect the secular variations in time of the orbital angular velocity, and we shall assume also that the star is a barotrope (see Section 3.2.1). Given these simplifying approximations, by taking the curl of Eq. (8.44), we obtain d

which is quite similar to Eq. (8.32). The function F can be neglected in the bulk of the star; in the surface boundary layer, however, one has d ( d u \

where f¿V is the vertical (i.e., in the direction of gravity) coefficient of eddy viscosity. To ensure mass conservation we must also prescribe that div(p u) = 0. (8.47)

Boundary conditions (2.20) and (2.21) further imply that n • u = 0 and n x [n • T(u)] = 0 (8.48)

on the free surface of the tidally and rotationally distorted primary. As usual, n is the unit outer normal to the free surface, and T are the Reynolds stresses, which depend linearly on fzV and the first-order derivatives of u (see Section 3.6). Equations (8.45)-(8.48) specify the vector u completely.

Boundary-layer theory can be used to describe the general features of these time-dependent motions in the corotating frame. To be specific, one writes u = ^ u*(r, d, y)exp(-2^kt), (8.49)

k where the uk can be expanded in terms of radial functions and spherical harmonics (k = 0, 1, 2, ...). Thence, performing a boundary-layer analysis of Eqs. (8.45)-(8.47) and applying boundary conditions (8.48), one can obtain the permissible values for the j3k in Eq. (8.49). Obviously, the lowest eigenvalue j30 is the most important one since it defines the e-folding time t * of the transient motions, which is equal to (2^0^0)-1.

Detailed mathematical calculations show that there always exists a thin Ekman-type suction layer that induces a large-scale flow of matter within the almost frictionless interior of an asynchronous binary component. Figure 8.4 illustrates the streamlines of the tidally induced meridional flow in a model with constant density and constant eddy viscosity, when the mass ratio is equal to unity (M = M'). In the case of a spin-down (Q > Q0), these motions correspond to a quadrupolar circulation pattern that is weakly dependent on the longitude y, with the fluid entering the boundary layer in the equatorial belt and returning with decreased angular momentum to the poles. (The reverse phenomenon occurs in the case of a spin-up, when Q < Q0.) Given our simplifying approximations, the typical speed of the meridional flow is of the order of eT (5/R)(Qi — Q0)m while the e-folding time t* is approximately equal to (2Q0eT 5/R)—1, where 5/R is the relative boundary-layer thickness - which is of the order of (xV/2pQ,0R2)1/2 -and eT is the ratio of the tidal attraction to gravity at the equator. For moderately small

Fig. 8.4. Streamlines of the transient, tidally induced meridional circulation in an asynchronously rotating model with constant density and constant eddy viscosity, when > Q0. The rotation axis is vertical. The streamlines do not penetrate into the free boundary, with matter flowing from the equator to the poles in the outermost surface layers. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J.., 359, 155, 1990.

Fig. 8.4. Streamlines of the transient, tidally induced meridional circulation in an asynchronously rotating model with constant density and constant eddy viscosity, when > Q0. The rotation axis is vertical. The streamlines do not penetrate into the free boundary, with matter flowing from the equator to the poles in the outermost surface layers. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J.., 359, 155, 1990.

eccentricities and masses of comparable magnitude, it follows that

As usual, P (= 2n/ Q0) is the orbital period and a is the semimajor axis.

As was already noted, in obtaining these results we have made use of several simplifying approximations. Since I want to avoid any misunderstanding about the applicability of the time scale t * to a sample of binary stars, I shall conclude this section by making a few practical remarks.

First, in view of application to actual binary stars, it is much more realistic to evaluate the ratio 8/R for a model in which p a (R — r)3 in the surface layers. To simulate the case of an eddy viscosity that is larger than the microscopic viscosity, we shall also assume that iV = 10N|rad, where |rad is the radiative viscosity and N is a constant (see Eq. [4.62]). To be specific, if we let p = pb(R — r)3 and |rad = ib(R — r), a straightforward dimensional analysis shows that 8/R = 10N/4(|b/pbQ0R4)1/4. For a

since, for unequal masses, one has

Cowling point-source model with electron-scattering opacity, this relation takes the form

Parenthetically note that the small exponent 1/4 considerably reduces the uncertainties on the coefficient of eddy viscosity in the surface layers.

Second, in deriving Eq. (8.50) we have neglected the secular variations in time of the orbital angular velocity For the sake of simplicity, we have also assumed that the free boundary of the primary is nearly coincident with the steady surface corresponding to synchronism. It is evident that these approximations impose a severe restriction on the mass ratio M'/M. Indeed, if the mass M was much larger than M', a small change in the rotational angular velocity ^ would lead to large variations in time of the quantities and a/R. Because such large changes were not permitted in our model calculations, it follows at once that Eq. (8.50) does not apply to binary systems having extreme mass ratios. Such a restriction is unimportant for binary-star systems, since the masses M and M' are in general of comparable magnitude. It is of paramount importance for planetary-satellite systems, however, for they generally have very small mass ratios.

Third, because we have neglected the nonlinear terms u ■ grad u in our analysis, the e-folding time defined in Eq. (8.50) is no more than a lower limit to the actual synchronization time, tsyn (say), in a real binary star. Indeed, as was properly shown in Section 8.4.1, nonlinearity increases the spin-up and spin-down times of theflow between parallel infinite plates (see Figure 8.2). Accordingly, because these laboratory problems are quite similar to ours, the synchronization time tsyn should also be much larger than the e-folding time t *. Finally, recall that we have considered barotropic models only; that is, we have explicitly assumed that Eq. (8.45) contains no term proportional to the vector grad p x grad p. General considerations in geophysics indicate that baroclinicity effects inhibit large-scale circulations. Hence, given the great similarities between the geophysical and astronomical problems, we conclude that in a more realistic stellar model the inherent departures from barotropy should also inhibit the tidally driven currents. Since the effects of nonlinearity and baroclinicity cannot be ascertained at this time, hereafter we shall make the reasonable assumption that tsyn = 100t *, where o is a constant of order unity (see Section 8.4.3).

Fourth, in an early-type binary component, the tidally driven currents are most probably confined to its radiative envelope, since the core-envelope interface can act as an effective barrier. This fact is of little concern to us, because it is only the surface rotation rates that can be measured. On theoretical grounds, a concomitant braking of the convective core by turbulent diffusion of linear momentum is quite plausible, however.

Fifth, since the tidally driven currents do not depend on eddy viscosity in the bulk of an asynchronous binary component, it follows at once that the Ekman-pumping process is also operative in stars possessing a deep convective envelope. In fact, the hydrodynamical mechanism should be more effective in late-type binary components than in early-type ones, because the ratio 8/R takes larger values in stars that have a larger eddy viscosity in their outermost surface layers.

8.4.3 The characteristic times

In Sections 8.2.2 and 8.3.1 we have expounded the main shortcomings of the two well-known mechanisms that are usually invoked to explain synchronism in the close binary stars. In Section 8.4.2 we have considered a third mechanism that tends to synchronize the axial and orbital motions in the components of a detached close binary. This spin-down (or spin-up) process involves large-scale meridional currents superposed on the azimuthal motion around the rotation axis of an asynchronous rotator. Figure 8.4 illustrates these transient motions, which vanish altogether when the tidally distorted body has reached a state of hydrostatic equilibrium in the frame corotating with the orbital angular velocity Q0, that is, when synchronization has been attained. They are quite similar, therefore, to the transient meridional currents that are responsible for the decay of various rotational motions near solid boundaries (see Figures 2.2, 2.3, and 8.1).

Although problems with solid boundaries have some obvious features in common with the double-star problem, it is evident that they differ in the manner by which the secondary meridional flow comes into existence. For example, in the case of a midlatitude cyclonic vortex in the Earth's atmosphere, turbulent friction acting on the ground slows down the azimuthal motion, thus producing a radial inflow of matter toward the rotation axis. By continuity, this horizontal transport is balanced by a small vertical flux of matter into the free atmosphere above the surface boundary layer. It is this upward motion that eventually produces the secondary flow illustrated in Figure 2.2 (as well as those illustrated in Figures 2.3 and 8.1). In contrast, in the double-star problem there are no solid boundaries that may spin down (or spin up) the azimuthal motion in an asynchronously rotating binary component. But then, as was explained in Section 8.4.2, it is the self-gravitational attraction of the star that acts as the "container," forcing the tidal bulges to remain almost aligned with the line joining the two centers of mass. Given this severe constraint on the free surface of a tidally distorted star, one can show that it is this forced lack of axial symmetry that prevents the azimuthal motion from being wholly one of pure rotation in an asynchronous rotator, thus leading to the formation of transient meridional currents, as illustrated in Figure 8.4.

Note also that the problems with solid and free boundaries differ in their respective time scales. In the former case, the spin-down time is proportional to the dynamical time scale (i.e., the final period) divided by the relative thickness of the Ekman pumping layer (see Eqs. [2.101] and [8.43]). In the case of a tidally distorted star, however, the e-folding time t* is proportional to the dynamical time scale (i.e., the orbital period) divided by the product eT (5/R), where eT is a measure of the small departure from axial symmetry (see Eq. [8.51]) and 5/R is the relative boundary-layer thickness (see Eq. [8.52]). Obviously, the presence of a free (rather than solid) boundary reduces the efficiency of the Ekman-pumping process. As we shall see in Section 8.4.4, however, even though 5/R lies in the range 10-5 to 10-3 and despite the fact that eT is also a small parameter, this is quite sufficient to provide the general trend of the observational data.

Making use of the results presented in Section 8.4.2, one finds that

tsyn(yr) = * (1 + * )3/8 {T) Uj Uj Uj or, by virtue of Kepler's third law, tsyn(yr) = 5.35 x 102+"-N/4 ■

where q is the mass ratio. As explained in Section 8.4.2, these formulae do not apply to binary systems having extreme mass ratios. The meaning of the adjustable factor 10o-N/4 is properly explained at the end of Section 8.4.2 also. Following Claret, Gimenez, and Cunha (1995), the most plausible values are o ^ 1.6, with N ^ 0 in a radiative envelope and N ^ 10 in a convective envelope. However, future discussions based on a larger sample of binaries could well lead to a smaller value for o and more refined values for N.

Simultaneously, because viscous dissipation retards the equilibrium tide, angular momentum is exchanged between the orbit and the rotation of each component, thus modifying the orbital eccentricity of the binary star. To a good degree of approximation, the ratio tsyn/ tcir is of the order of the ratio of rotational and orbital angular momentum. Hence, for moderately small eccentricities and masses of comparable magnitude, we can estimate the time to circularize the orbit by

Uyr) = 9.4 x ,03+. - <1^ ( ( 0"" ( [ P (day)]-,

where rg is the fractional gyration radius. As usual, the secondary makes a similar contribution to the effective circularization time of the binary (see Eq. [8.21]).

At this juncture, it is worth noting that the two mechanisms presented in Sections 8.2 and 8.3 are mutually exclusive, in the sense that one of them applies to stars having a deep convective envelope whereas the other one applies to stars having a radiative envelope. Because the third mechanism can be operative in both groups of stars, however, the resonance mechanism and the hydrodynamical mechanism both produce secular changes in the spin and orbital parameters of the early-type stars. Similarly, because the tidal-torque mechanism and the hydrodynamical mechanism are not mutually exclusive, both of them can be operative in the late-type binaries. As we shall see in Section 8.4.4, in the early-type binaries it is always the hydrodynamical mechanism that is the most effective; Eqs. (8.53) and (8.55) thus provide the dominant contributions to the times tsyn and tcir. In contrast, in stars having a deep convective envelope both the tidal-torque mechanism and the hydrodynamical mechanism can be operative, albeit for different values of the parameters R, L, and P. Accordingly, for these binaries Eqs. (8.19) and (8.21) can be used to discuss jointly the synchronization and orbital circularization caused by both mechanisms, provided that one inserts

and, for each component,

in the corresponding equations.

To conclude this section, it may not be inappropriate to stress again the importance of solving Eqs. (8.19) and (8.21) rather than making comparison between time scales. As was already noted in Section 8.2.2, the latter approach is probably adequate, in most cases, for binary components evolving without mass loss on the main sequence. For rapidly contracting or expanding components, however, it might prove quite misleading. It is also worth noting that any meaningful comparison between theory and observation can be made on a statistical basis only, preferably with a large sample of binaries. In other words, one should not evaluate the merits of a theory by making use of a few short-period binaries that display a very eccentric orbit and/or a large degree of asynchronism. These binaries may have had a large initial eccentricity, with a large initial departure from synchronism. Other evolutionary processes, which were not included in the theory, might also significantly modify both axial rotation and the strength of tidal interaction. Recall also that the tidal-torque and hydrodynamical mechanisms both depend on two adjustable parameters: the viscous time scale T in Eqs. (8.20) and (8.22) and the factor 10CT-N/4 in Eqs. (8.53)-(8.56). One should not expect these two quantities to be universal constants, however. As a matter of fact, the mathematical difficulties of the problem are such that there is little or no hope of calculating their values from first principles alone. It would thus seem that progress can be made only through an efficient cooperation between theory and observation, using - as I said - a very large sample of binaries.

8.4.4 Pseudo-synchronization and orbital circularization

Let us first discuss the early-type main-sequence binaries. As was pointed out in Section 8.3.1, the resonance mechanism is unable to account for the observed degree of orbital circularization among these stars. However, by making use of Eqs. (8.55) and (8.56), one can show that the hydrodynamical mechanism is quite effective in inducing orbital circularization during the main-sequence lifetime ta of these stars. Yet, if one considers a model consisting of two A-type components with little or no turbulence in their outer layers (i.e., with N = 0), a comparison between the times tcir and ta shows that this mechanism can explain circular orbits up to P ^ 6 days only. This is somewhat shorter than the 10-day cutoff shown in Figure 1.11. This finding strongly suggests either that turbulence in the outer layers might play a somewhat greater role in these stars (with N ^ 4, say, instead of N = 0) or that their observed eccentricity distribution might result from main-sequence and pre-main-sequence circularization. Accordingly, a mere comparison of the time scales tcir and ta is probably insufficient in this case; the problem requires a direct integration of Eqs. (8.21) and (8.55).

As far as synchronization is concerned, the hydrodynamical mechanism is also a much more efficient process than the resonance mechanism, since the time tsyn depends primarily on the factor (a/ R)4'125 instead of (a/R)85 (see Eqs. [8.27] and [8.53]). The presence of a smaller exponent makes it a long-range mechanism that can enforce synchronization up to a/R ^ 20 in the early-type main-sequence binaries. This is in agreement with the observations reported in Section 1.4. Note also that the effects of the hydrodynamical mechanism can still be felt for larger separations, up to P k 100 days (say) - without bringing complete synchronization beyond P k 15-25 days, however. This is in agreement with the finding that most late A-type dwarfs in binaries with P < 100 days are Am stars, with rotational velocities smaller than 100 km s-1 (see Section 6.3.5). There is thus no need to invoke pre-main-sequence braking to explain the paucity of normal A-type dwarfs from binaries with orbital periods smaller than 100 days.

Now, in Section 8.2.2 we have shown that the tidal-torque mechanism is quite ineffective in inducing orbital circularization in the late-type main-sequence binaries (see Eqs. [8.23]-[8.25]). To show that the hydrodynamical mechanism can be operative in these stars, let us apply Eq. (8.56) to a typical solar-type binary component. For reasonable values of the parameters, one finds that tcir(yr) = 3 X 107-N/4[P(day)]49/12 , (8.59)

where N is a fitting constant. In fact, because we have let iV = 10N |rad, the factor 10N is some mean value of the Reynolds number Re in the surface layers (see Eq. [2.51]). Since N is a free quantity in Eq. (8.59), we shall thus prescribe that for each cluster listed in Table 1.2 one has tcir = ta at P = Pcut. For the three oldest clusters one obtains N k 9.3-9.7 or Re k 109-1010. For the Pleiades one must let N k 12, whereas the pre-main-sequence cluster requires the value N k 14. These two values may not be quite reliable, however, because Eq. (8.59) is directly applicable only to static stars. Anyhow, these crude evaluations of N are quite reasonable for late-type main-sequence binaries because, owing to the extreme smallness of the microscopic viscosity, the outer layers of aconvective envelope can easily sustain Reynolds numbers of the order 109-1010. Hence, we conclude that the hydrodynamical mechanism can be responsible for orbital circu-larization on the main sequence, even though it may not be equally efficient during the pre-main-sequence contraction. Since the two relevant mechanisms are not mutually exclusive, this result strongly suggests that the tidal-torque mechanism is the dominant one during the pre-main-sequence phase up to 8 days (as reported in Section 8.2.2), whereas the hydrodynamical mechanism becomes fully responsible for orbital circularization during the main-sequence phase, at a much slower pace, beyond ta = 1 Gyr (say). If so, then, the inefficiency of the tidal-torque mechanism on the main sequence is no longer an issue. Obviously, an integration of Eqs. (8.21) and (8.58) would be most welcome.

In Section 8.2.2 we also pointed out that the tidal-torque mechanism does not wholly account for pseudo-synchronization in the late-type main-sequence binaries. Making use of Eq. (8.54), one can show that the hydrodynamical mechanism is operative in these stars, although it is difficult to quantify with any certainty the counter-effects of magnetically driven winds on the synchronization process. In this connection, let us mention the work of van't Veer and Maceroni (1992), who have shown that the hydrodynamical mechanism is much more effective than the tidal-torque mechanism in the angular-momentum losing, G-type binaries belonging to the main-sequence group.

A very interesting case of asynchronism is that of the double-lined eclipsing binary TZ Fornacis. It is a system with an orbital period of 75.7 days and a circular orbit. Its components have nearly equal masses (M = 2.05M0 and M' = 1.95M0) but unequal radii (R = 8.32R0 and R' = 3.96R0). The more massive component is rotating synchronously with the orbital motion while the companion is spinning 16 times faster than the orbital period rate. This puzzling binary star has been recently investigated by Claret and coworkers, who integrated Eqs. (8.19) and (8.21) along the evolutionary path of each binary component. Within the theoretical and observational error bars, these calculations describe how the binary components may pass through a stage characterized by a synchronous primary and a supersynchronous companion in circular orbits about their common center of mass. However, two independent sets of calculations strongly suggest that the case of TZ Fornacis can be explained either by the hydrodynamical mechanism alone or by a combination of the tidal-torque and resonance mechanisms. Since all three mechanisms can operate in atidally distorted star, one may therefore argue that they become almost equally efficient during some periods of post-main-sequence evolution, so that all of them must be taken into account during this expanding phase.

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