Astron. Astrophys. 330, 533-540 (1998)

3. Alternative explanations

It is known that several phenomena can change the rate of the apsidal motion. For example, if a third companion is present, the times of minima will vary periodically and this will affect the observed apsidal motion. In this section we discuss some of these physical mechanisms in order to try to explain the case of DI Her. Some of these processes have been discussed already by other authors. However, we shall comment them briefly for completeness. New alternatives are also investigated.

3.1. Fast circularization of the orbit (Guinan & Maloney 1985)

The time of minima can change with time if a rapid decreasing of the eccentricity is present. If the discrepancy in the apsidal motion of DI Her is attributed to this mechanism a high derivative is needed ( -1.4 10^-4 /year). We have studied recently the circularization in close binary systems (Claret et al. 1995, Claret & Cunha 1997) and for the specific case of DI Her we have found that the derivative, given by -e/ where is the time scale for circularization, is around -10^-14 /year. This result was obtained using the radiative damping mechanism. Taking into account both times we can deduce that the efficiency in circularizing the orbit is several order of magnitude smaller than that required to fit the observations. This conclusion holds for the case of the hydrodynamical or radiative damping processes, though both mechanisms predict different critical times for circularization.

3.2. The third body (Guinan & Maloney 1985)

An extensive analysis of this possibility is given in the paper quoted above. The typical contribution to the apsidal motion is around 1 10^-3 degrees/cycle, i.e., of the order of those due to other causes. From the dynamical point of view the hypothetical third body would change the orbital parameters and affect directly the depth of the eclipses. Such a behaviour has not been detected yet. However, it is difficult to discard or to confirm it definitively. Meanwhile the question is still open.

3.3. Revision of the gravitation theory (NST84 and NST89 (Moffat 1984, 1989; this paper)

A different apsidal motion rate was proposed by Moffat 1984 using a new theory of gravitation based on a non-symmetrical tensor. The corresponding rate of variation of is given by

with

where P is given in days and the masses are in solar units.

Note that Eq. 6 reduces to the GR prediction when tends to 1. The apsidal motion rate can even be reversed depending on the value of l which is inferred from a pre-calibration. A linear fitting to the calibration performed by Moffat gives 10⁹ cm with the masses given in solar units. Using the observed values for DI Her and considering the above value for l one finds = + = 2.16 10^-4 degrees per cycle. This value seems to be in good agreement with the observational data within 2 . However, when one applies Eq. 6 to other systems the corresponding shift in the periastron positions are systematically too slow when compared with observed ones (Claret 1997).

Later Moffat (1989) modified his equations. The main difference concerning apsidal motion with respect to the 1984 formulation is that for equal masses and chemical compositions they predict the same results as GR. However before to adopting it as definitive some remarks should be done:

1. The theory does not predict the apsidal motion a priori
2. DI Her (and other problematic systems) were used in the calibration
3. The contribution to the apsidal motion is strongly dependent on the number of cosmions which it is not well constrained
4. The theory present too many free parameters
5. There is a strong dependence of the calibration on the systems used (DI Her is always used)
6. The primaries and secondaries of systems used in the 1989 calibration follow different mass-number of cosmions relationship. In fact the masses of the secondary of DI Her and the primary of AG Per differ only 0.2%. However, the respective number of cosmions differ in 65%.

The results based on the NST84(89) should be taken with care due to the problems indicated above. In the particular case of DI Her they do not formally explain this enigmatic system since it was used in the calibration. A more detailed discussion on this subject can be found in Claret (1997).

3.4. The effect of the circumstellar material (Guinan & Maloney 1985)

As already pointed out for non-relativistic systems (Claret & Giménez 1993a) the discrepancies between observed and theoretical apsidal motion rate can not be fully attributed to the change of the gravitational field of the stars due the presence of a circumstellar matter. The formula given by Hadjidemetriou (1967) gives

where is the density and U" is the associated period of apsidal motion. If the observed discrepancy is due to this effect the mean density of the circumstellar cloud would be as large as 10^-10 gcm^-3. This density is high for typical interstellar medium between components of close binaries. Moreover, as reported by Guinan & Maloney (1985), there is no observational evidence of such a high density cloud around the components of DI Her.

3.5. Inclination of the axes of rotation (Sakura 1984, Guinan & Maloney 1985, Company et al. 1989)

Depending upon the orientation of the rotation axis of the stars with respect to the angular orbital momentum there will be a correction to the rotational term of . This correction may even be negative, i.e., it is possible a regression of the position of the periastron mainly if the system is a young one. This is the case of DI Her in spite of the uncertainties in the determination of its age. Thus, this possibility seems to be very attractive for explaining its anomalous apsidal motion.

Kopal (1978) has studied the problem. The correction depends on the quantity -1.5sin² ( +i)-0.5sin2( +i)tan(0.5i) where is the angle of inclination of the rotational axis of the component i in the invariable plane of the system. Depending on the angles involved we can have a recession of the line of the periastron. This moved Sakura (1984) to invoke it as a possible explanation for the disagreement with observations. Some years later Company et al. 1989 returned to the problem of no alignment of the spin and angular orbital momentum. Their Eq. 1 summarized their results

where the indices tidal j and rot j refer to the tidal and rotational contribution of the component j. The variable x and y are given by n.n₁ and n.n₂ respectively where n is the unitary vector in the direction of the angular orbital momentum. The vectors n₁ and n₂ are the unitary vectors in the direction of the spin of the components. Thus cos () = n₁ .ep / and cos () = n₂ .ep / where ep is the unitary vector in the direction of the periastron. A numerical analysis of the Eq. 8 carried out by the authors indicated that the discrepancy in the apsidal motion of DI Her could be attributed to the inclination of the rotation axis of the components if the corresponding angles are of the order of 70 degrees.

For DI Her the ratio of the orbital to rotational angular momentum is about 10³. This means that the time scales for synchronization and for the decay of the angle , defined by the orbital and equatorial planes, are of the same order of magnitude. We have integrated the corresponding differential equations and have found that t_s t 10⁸ years. The corresponding critical time for circularization is about 25% larger than the mentioned ones. When compared with the derived age for DI Her t is around 20 times larger, that is, following the tidal evolution theory, it is possible that the rotation axes of DI Her are still inclined with respect to the plane of the orbit (of course, only if the initial angles were different from zero). We have re-analyzed this possibility through Eq. 8 using new models and apsidal motion rate and the result is invariant, that is, the required angles are around 70 degrees.

Observations of eclipsing binary stars during the eclipses can help us to elucidate the position of the rotational axes in a binary system. As in these phases the different surfaces velocities are eclipsed, there appears a net effect in form of a Doppler shift in the center of the line (Rossiter effect). Maloney & Guinan 1989 have observed the primary eclipse of DI Her. They reported that the preliminary results indicated that the orbital and equatorial planes are coplanar. These observations constrain this hypothesis severely but as in the third body hypothesis, only more systematic observations may elucidate the situation.

3.6. The theoretical k₂ derived from stellar models (this paper)

The discrepancies between observations and theoretical predictions for the apsidal motion for non-relativistic systems were reduced drastically due to the application of new stellar models by considering evolutive effects on k₂ and new input physics. In fact, as the results by Claret & Giménez 1993ab indicate, modern stellar interior models are able to reproduce the observations of the apsidal motion rates, at least for those systems with well determined astrophysical properties. Therefore, the reasons for the disagreement present in DI Her should be attributed to other effects: even in the hypothetical case of no distortional contribution, the ratio between the observed and the theoretical rate (provided in this case only by the General Relativity) would be around 2.3.

3.7. Effects of the dynamic tides (this paper)

The classical apsidal motion rate is deduced in the framework of the equilibrium tides. Smeyers et al. 1991 and Ruymaekers 1993 extended it to the case of dynamic tides. Their main results are, that for short periods and sufficiently large eccentricities, apsidal motion rates derived in the framework of the dynamic tides are smaller than those predicted in the framework of equilibrium tides.

Recently Marshall et al. 1995 have examined the observations of DI Her outside the minimum realized during the years 1993-94. They reported "possible low amplitude light variations". However, these results and the associated periods are still uncertain due to a possible variability of the comparison star used.

However, concerning DI Her, one should keep in mind that even in the hypothetical case that such corrections leaded to k₂ =0 the discrepancy would still remain (see previous Subsection).

3.8. Effects of stellar rotation (this paper)

As the ratio of rotational to tidal distortion contribution is important in DI Her (around 0.56 and 0.58 for the primary and secondary respectively) it is convenient to investigate the changes in the predicted apsidal motion due to rotation. We have introduced rotation into our code of stellar evolution and we investigated its influence on the radius, luminosity and k₂ (Claret & Giménez 1993a). Within the quasi spherical approximation the expected correction due to stellar rotation on the theoretical k₂ depends on the parameter = 2v² R/3GM, and for nearly homogeneous models we have found . The theoretical correction is too small to explain the observed discrepancy in DI Her although for some systems it can help to remove the eventual disagreements.

3.9. Viscosity of the stellar interior (this paper)

The usual formalism for the apsidal motion rate was derived under the assumption that stellar viscosity is low (see Kopal 1978). Hosokawa (1985) using two extreme cases for the stellar viscosity - inviscid and rigid body - derived an approximation for . The main difference with respect to Eq. 4 is that a correction is introduced in the tidal term. The correction can be written as (1- ) where is the "viscosity". Its extreme values are zero for a perfect fluid and 1.4 for a rigid body.

In the hypothetical and improbable case that stars behave like rigid bodies, the correction would lead to = 7.4 10^-4 degrees per cycle which is still a large value when compared with the observed rate.

3.10. Observational aspects of the apsidal motion rates (this paper)

The range of periods of apsidal motion is very large: some decades up to almost 10⁵ years. The time spent in observing eccentric systems which exhibit apsidal motion is comparable in some case with U itself but in other ones the ratio is very small. Let us examine some observational conditions under which the apsidal motion rates are obtained and that may affect its interpretation.

1) A common problem of the systems showing apsidal motion is that, during the recollection of time of minima to obtain , different techniques (mainly data reduction) have been used with different instruments. This means that different detectors with different levels of confidence were used. Indeed as quoted by Guinan & Milone 1985 the differences in for DI Her obtained using photographic, visual and photo-electric measurements and only photometric data can reach 270% (1.75 to 6.5 10^-3 degrees per year).

2) It should be mentioned that these observations were carried out during about 27 years only, that is, they only cover 0.1% of the apsidal motion period (in case of photometric observations). This is another important point that, in our opinion, can limit definitive conclusions on DI Her and on other systems with slow apsidal motion.

3) The new observations of obtained by Guinan et al. in 1994 reduce the difference observed-predicted rates. They found that the ratio / - which was 6.7 - is reduced to 4.2. This is essentially due to the new observations and data analysis of since the theoretical values for k₂ and observed values of radii, eccentricity and masses are essentially the same.

4) In addition, there is some controversy about the best method to analyze the data of times of minimum in order to obtain the apsidal motion rate (Maloney & Guinan 1991). These authors found a observational apsidal motion rate for AS Cam different than that by Krzesinski et al. 1990. The later authors found that the discrepancy for AS Cam was reduced if a different value of eccentricity was used. Maloney & Guinan interpreted this result as an erroneous use of the least square method. This system also presents discrepancies with respect to the GR prediction.

5) Another alternative to understand the behavior of DI Her could be based on the ratio between the time scales involved: the apsidal motion period and the time interval since apsidal motion have been measured . As mentioned before, for DI Her this ratio is very small. This means that the time spent in observing the system is very short when compared with the apsidal motion period. Of course this could lead to poor results. In order to illustrate this situation in the light of the present discussion we plot in Fig. 3 the discrepancies between observed and theoretical apsidal motion rates as a function of log U. All systems with good absolute dimensions determination - relativistic and non-relativistic ones - are represented. In spite of the uncertainties in their absolute parameters V541 Cyg and AS Cam are also presented since these systems present high deviations.
This figure shows very interesting trends. Up to U 10000 years there is no evidence of any systematic dependence of the discrepancies on the period U. However, for DI Her, V541 Cyg and AS Cam there is a dependence of the deviations with the period of apsidal motion. These systems, which present the largest deviations, also present the largest U's.
Perhaps a way to see such effects more clearly is to examine Fig. 4, where we plot the relative discrepancies as a function of log (U/ ) where is the age of the system. For DI Her, for example, the orbital time scale begins to be comparable with the stellar evolution time scale.

Fig. 3. Relative differences between the observed and the predicted internal structure constant as a function of the apsidal motion period U for relativistic and non-relativistic systems. The positions of DI Her, V541 Cyg and AS Cam are indicated.

Fig. 4. Relatives differences between the observed and predicted apsidal motion rates as a function of the relative orbital and evolutionary time scales.

6) An additional observational difficulty may comes from the position of the longitude of periastron. The observations of apsidal motion are expected to be unfavorable for near 0, and 2 . In Fig. 5 we represent the quantity D, given by

Fig. 5. D (Eq. 9) as a function of the longitude of periastron for DI Her, AS Cam and V541 Cyg.

as a function of . Recall that is dependent on the time derivative of D. In that figure we show the position of V541 Cyg, AS Cam and DI Her in the plane D. Although AS Cam does not lie in a unfavorable position the corresponding values of D are too small. On the other hand V541 Cyg - which present the smallest deviation - is in a more favorable position and the values of D are larger than in the previous case. DI Her is not in a very favorable position to obtain apsidal motion information and the values of D are of the same order of that for V541 Cyg.

© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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