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Astronomy&AstrophysicsReview
As seen, pseudo-synchronisation is in fact an excellent approximation for the great majority of the stars; most of the non-synchronous cases are found below relative radius 0.1, as expected from tidal evolution theory. Interesting exceptions are V459 Cas (no. 44 in Table \ref{tableMR}) in the bottom panel of Fig. \ref{vps} at relative radius \(\sim\)0.07, with stars rotating much faster than predicted, and the slightly eccentric RW Lac (no. 90) in the top panel of Fig. \ref{vps} at relative radius 0.04–0.05, with highly sub-synchronous rotation. These exceptions are consistent with the expected levels of synchronisation for such small relative radii, given that the overall tendency in binaries is to slow down radiative stars and spin up convective ones relative to their single counterparts.
The good quality of the data, as reflected in the near-invisible error bars, reveals a smaller dispersion among stars with convective rather than radiative envelopes due to the more efficient circularisation mechanism. The data also reveal a number of sub-synchronous radiative stars with relative radii above 0.1 that cannot yet be explained (e.g., V451 Oph or V1031 Ori). Detailed stellar models with integrated tidal evolution calculations will be needed to address this issue.
Using different symbols for stars in eccentric and circular orbits in Fig. \ref{vps} also reveals any effects of overcorrection for eccentricity in convective stars when adopting pseudo-synchronisation. EY Cep \citep{eycep} is an interesting case of a highly eccentric young binary where theory predicts a non-circular orbit as observed, but not that the two stars should rotate at the orbital rate. Presumably, synchronisation was achieved by deep convective envelopes, now or during the pre-main sequence phase, or the stars managed to slow down independently. In other cases, the two stars in the same system show clearly different behaviours. For example, the primary of V364 Lac rotates faster than expected, while the secondary rotates sub-synchronously; the opposite is the case in V396 Cas.
Overall, the normal pattern is that the larger primary stars are synchronised while the smaller secondaries are still on their way to the final state, either from faster rotation in radiative stars (\(\zeta\) Phe) or from slower speeds in stars with convective envelopes (BW Aqr). Nevertheless, cases exist that require special attention, including new observations and additional tidal modelling (e.g. V539 Ara or CV Vel). For clarity, the exceptionally fast-rotating secondary component of TZ For (spinning more than 15 times above the synchronous rate) has been excluded from Fig. \ref{vps}; the tidal history and special evolutionary configuration of this system have been studied in detail by \cite{1995A+A...296..180C}. Recent advances in tidal theory include the work of \cite{Kumar:96}, \cite{Witte:99a, Witte:99b, Witte:02}, and \cite{Willems:03}. An excellent summary of the topic and its applications can be found in the proceedings of the 3rd Granada Workshop on Stellar Structure \citep{Claret:05}.
Apsidal motion
Important additional information about stellar structure is available if the rate of apsidal motion in an eccentric binary system can also be measured. This is the case for 29 of the 44 eccentric systems in our sample, although for two of them the apsidal motion has not been measured with enough precision to allow for a significant comparison with theory. One of these systems is BP Vul \citep{2003AJ....126.1905L}; the other is the extremely interesting case of CM Dra, with the lowest stellar masses of the sample \citep{cmdra}.
The much-discussed system DI Her requires special mention \citep[see][]{Claret:98}. DI Her was excluded from our overall study because of the recent discovery, based on the Rossiter effect, that the spin axes of the stars are almost perpendicular rather than parallel to that of the orbit \citep{albrecht}. This configuration is, in fact, not unlikely in such a young binary with small relative radii, provided that the stars were initially formed with misaligned spin axes. In summary, when the observed misalignment, the Shakura effect \citep{1988ApJ...335..962C}, and the general relativistic contribution are accounted for in the tidal and rotational terms of the predicted apsidal motion, excellent agreement is obtained with the observed apsidal motion rate.
The observed apsidal motion in binary stars has two contributions due to the non-Keplerian dynamical behaviour of the component stars. The classical term is caused by the stellar distortions produced by rotation and tides, while the non-Newtonian term corresponds to the predictions of General Relativity. For a recent comparison between observed and predicted apsidal motion rates, including for the first time the effects of dynamical tides, see \cite{Claret:02}.