\journalname
Astronomy&AstrophysicsReview
The general relativistic term.
The relativistic term of the apsidal motion was examined, e.g., by \cite{Claret:97}, and reviewed more recently by \cite{2007IAUS..240..290G}, who analysed 16 systems. He found good agreement between predicted and observed rates for 12 of them, but the sample was not of the same quality as that presented here. For our comparison between predicted and observed apsidal motions, we have considered all the systems in Table \ref{tableMR} with eccentric orbits and well-determined apsidal motions, and with a predicted relativistic contribution of at least 40% of the observed total rate. This allows us to minimize the influence of possible errors in the models used to compute the tidal contribution.
Only six systems fulfil these conditions, excluding the special case of DI Her, discussed above. For these systems, the expected apsidal motion rates were computed including the general relativistic contribution, using theoretical models of internal structure as described below, and the observed rotational velocities (see Table \ref{tableMRsup}). The predicted and observed apsidal motion rates, in degrees per cycle, are compared in Fig. \ref{apsidal1}. The good agreement seen there, together with the resolution of the ‘DI Her enigma’, seems to indicate that apsidal motion as an argument in favour of alternative theories of gravitation is a closed case.
The tidal terms.
Applying the standard correction for the general relativistic contribution to the observed apsidal motion for the rest of the systems in Table \ref{tableaps}, we can compute the average internal structure constant, \(\log k_{2}\). In order to ensure that the resulting values are of good accuracy, we consider only systems where the tidal and rotational effects contribute at least 40% of the observed total apsidal motion. Moreover, pinpointing individual main-sequence systems between the ZAMS and the TAMS (terminal-age main sequence) requires a precision of at least 0.1 in \(\log k_{2}\). Only 18 binaries satisfy these quality criteria, and we list the derived values of \(\log k_{2}\) for these stars in Table \ref{tableaps}.
Fig. \ref{apsidal2}a shows \(\log k_{2}\) for these systems as a function of the mean mass of the stars (using the same weighting procedure as implicit in the observed structure constant). Theoretical values from ZAMS models for the solar chemical composition from \cite{1995A+AS..109..441C} are also shown. It is clear that the precision of the data allows us to follow the stars as they evolve beyond the ZAMS, towards smaller values of \(\log k_{2}\) (greater central concentration); note that the lower-mass stars are generally less evolved. No correction for variation in metal content has been made in this general plot.
That the downward shift of the points in Fig. \ref{apsidal2}a is primarily due to evolution is clearly seen in Fig. \ref{apsidal2}b, which shows the difference between the observed and theoretical (ZAMS) \(\log k_{2}\) values as a function of the difference in the observed and ZAMS values of the mean surface gravity, \(\Delta\log g\). The nearly linear correlation and increasing dispersion in \(\Delta\log k_{2}\) with increasing \(\Delta\log g\) were suggested already by the evolutionary models of \cite{1995A+AS..109..441C} (his Fig. 7). Corrections for mild degrees of evolution, derived from this relation, were applied to the computed tidal contributions when assessing the general relativistic terms above.
Given that differences in metal content or rotation of the stars were not taken into account, the observed tight correlation is encouraging. Computing specific models for the observed mass, chemical composition and degree of evolution of each of the component stars would no doubt provide useful constraints on the adopted input physics when compared with these observations. The next few years should see a significant increase in the number of systems with reliably determined internal structure constants.