Astron. Astrophys. 327, 11-21 (1997)

4. Comparison with the observational data and conclusions

The analysis performed previously (Figs. 2-12) allowed us to perform a direct comparison between the observational data for and those predicted theoretically. Before doing that, has to be corrected by the relativistic contribution (or that by Moffat) following

or

where the symbol clas refers to the classical contribution, obs the observed value, GR the prediction by GR and NST refers to the non-symmetrical theory.

The resulting value of , or the equivalent , will be compared with theoretical predictions.

4.1. Using the relativistic correction

Eq. 13 gives us the rate of advance of periastron in degree per cycle. The contribution due to the GR is given in column 8 of Table 1. In order to compute the observed we use Eqs. 8 and 13 while for the corresponding theoretical value Eq. 14 was used. Stellar models computed taking into account rotation are more mass concentrated than the standard ones (for details, see Claret & Giménez 1993a). The theoretical values were corrected by the effect of the stellar rotation. Ho ve found that the correction is not too large and it is about 0.02-0.05 in log and depends on the value of the rotation rate at the surface of the star.

The results can be seen in Fig. 14. The old effect - real stars seemed to be more centrally condensed than predicted by the models - is not detected. Indeed the agreement is very satisfactory. The system BW Aqr presents the maximum discrepancy but in the sense that the components seemed to be less mass concentrated than predicted by the models. Fig. 15 shows that the differences do not depend on the masses. There is also n ence of this parameter with log as one can see in Fig. 16. This means that the stellar models are able to predict the apsidal motion in good agreement with the observations. In addition, the correction given by GR is also supported. The last point is very important in the light of the discussions on the validity of the predictions of the advance of the periastron given by the GR. These results are not substantialy changed if one assumes that all the stars of the sample are synchronized at t eriastron instead of using the observed rotational velocities.

Fig. 14. Theoretical and observed apsidal motion constants. The shift in the periastron was computed correcting by the prediction of the General Relativity (Eqs. 8, 13 and 15).

Fig. 15. Discrepancies between the observed and theoretical values of the apsidal motion constant (see text) as a function of . This latter value was computed using a similar definition as in Eq. 14. The shift in the periastron was computed correcting by the prediction of the General Relativity (Eqs. 8, 13 and 15).

Fig. 16. Differences between the observed and theoretical values of the apsidal motion constant (see text) as a function of observed log g weighted as in Eq. 14. The shift in the periastron was computed correcting by the prediction of the General Relativity (Eqs. 8, 13 and 15).

4.2. The non-symmetrical theory of gravitation (1984)

In his paper of 1984 Moffat derived an alternative formulation for the shift of periastron. Such formulation is based on a non-symmetrical tensor and on a non-symmetrical affine connection . This theory predicts a slower relativistic apsidal motion rate than that given by the GR. In some cases, this rate can even be reversed. As commented before the old comparisons between observations and theoretical predictions for the apsidal motion indicated that real stars seemed to be more concentrated in mass than predicted by the stellar models. As the equations by Moffat predicted a slower apsidal motion (less concentrated stars) this theory seemed to be adequate to be applied to the "problematic" systems. In fact Moffat (1984) pre-calibrated his equation using systems whose disagreements with respect to the theory were notorious (DI Her, AS Cam, Y Cyg, etc). His equation to describe the shift of the periastron is

with

Note that for some situations - small l - the prediction by Moffat is the same as that given by the GR. The parameter l is a constant of integration and can be calculated using the following relation:

where the masses are given in solar units.

The non-symmetrical theory of gravitation should also be able to fit the data for the other systems and not only for the systems used in the pre-calibration. We have performed such a test. In Fig. 17 we show the results using the predictions by Moffat (1984). One can conclude that the relativistic corrections given by GR are in better agreement with the observations than those by Moffat. The later predicts too slow apsidal motions and a systematic deviation with respect to was detected.

Fig. 17. Theoretical and observed apsidal motion constants. The shift in the periastron was computed correcting by the prediction of the the non-symmetrical theory (Eqs. 8, 16 and 17). Note the systematic deviation in the predictions given by this theory.

4.3. The non-symmetrical theory of gravitation (1989) - NST89

In 1989, Moffat revised his theory. The basic difference concerning apsidal motion with respect to Eq. 17 is that in the parameter the variable l was changed to K where

with

and is given by

where M is the mass of the star, Y is the mass fraction of helium, is the mass of the proton, Z is number of protons, N is the number of neutrons, is the baryon number, is number of cosmions and . The parameters , and are universal coupling constants for protons, neutrons and cosmions respectively.

This formulation depends on parameters which are not well established and also present some inconsistencies as we comment below:

1. Let us examine Eq. 22. If one gives typical numerical values (see Moffat 1989, Table II) one can see that the main contribution to is due to the last term. This term is around 10⁶ larger than the other factors and the calibration by Moffat depends strongly on the stellar cosmion number . Although its importance in the NST89 calculations this factor is not well constrained.
2. Another important restriction to the NST89 calculations is the large number of free parameters present in the formulation.
3. In addition, the differences in the chemical compositions attributed to the components of the systems used in the calibration are a little confuse since the primaries always present a helium content different from the secondaries. This goes against the current theory that in a binary system both stars have the same age and chemical composition. Moreover, NST89 are not able to predict the shift in the periastron position a priori.
4. There is also a dependence on the adopted systems used in the calibration. In fact, the astrophysical parameters used in the 1989 calibration have been changed being substituted by new observations and new advances in the stellar models. In this way, the discrepancies, which are very important in order to to establish the calibration, have decreased with time. Another additional problem was also detected: the quality of the photometric and spectroscopic data used by Moffat. Let us examine with more detail the four systems used in the calibration. Giménez & Clausen (1994) have determined the masses for AG Per as 5.36 and 4.90 for the primary and secondary respectively. This means that the recent masses are about 20% larger than previous ones, used by Moffat. The data for AS Cam and mainly for Vir (see Claret & Giménez 1993a) are not sufficiently accurate to be used in this kind of investigation. Only the astrophysical data for DI Her are accurate enough to test apsidal motion. Of course, the new determinations of have also an influence in the calibration.
5. On the other hand, the primaries and secondaries used by Moffat in his calibration follow two different relationship between the mass and the ratio of cosmions to baryons. For example, the mass of the secondary of DI Her is only 0.2% smaller than the mass of the primary of AG Per (given in that paper). However, the corresponding ratio for the two stars is about 65%.
6. Eqs. 20 and 21 predict that for a mass ratio of 1.0 the correction due to NST89 is the same as that given by GR. In order to test his theory, Moffat (1989) only used systems with q =1.0 (V1143 Cyg, V889 Aql and V541 Cyg) after the calibration. However, among this three systems only one fulfilled the basic requirements for apsidal motion test (V1143 Cyg). Whatever the quality of the data, it is clear that the predictions by NST89 were not really compared with the apsidal motion data: of a total seven systems used, four were used in the calibration while three presented a mass ratio of 1.0. The later give, by definition, the same results as GR and no differential results could be detected. It would be interesting that data for other systems like EK Cep, BW Aqr, AG Per, QX Car, etc are be used.

To summarize, the NST by Moffat (1984) is not able to fit the observations of apsidal motion since it predicts too slow shifts in the periastron position. On the other hand, NST89 formulation depends on too much free parameters; some of them very hard to determine. We have also detected some inconsistency in the calculations of the ratio. Moreover, NST89 is not able to predict the apsidal motion a priori. The predictions of the General Relativity compare very well with the present data (also with those for non-relativistic systems following Claret & Giménez 1993a, b) and it is independent of previous calibrations. Concerning DI Her - GR is not able to predict the correct - some possibilities to explain its strange behaviour are investigated in a separate paper (Claret 1997b)

© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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