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

2. The system DI Her and the observed apsidal motion rate

The components of DI Her are two stars with masses m₁ =5.185 0.108 and m₂ =4.534 0.06 M . The radii are 2.680 0.046 and 2.477 0.045 R respectively while the period is 10.55 days (Popper 1982). The observed apsidal motion rate is 1.88 10^-4 5.20 10^-5 degrees per cycle (Guinan & Maloney 1985). Recent determination of times of minima gives = 3.00 10^-4 4.3 10^-5 degrees per cycle (Guinan et al. 1994). The eccentricity is 0.489 0.002.

Concerning evolution, DI Her is a young system, as we can see in Fig. 1 where we represent the variation of the radius as a function of the time for both components. A common age of 5 10⁶ years is found (log =6.7). The present models were computed with X = 0.7045 and Z=0.015 and the mixing-length parameter was 1.52 while for core overshooting we have adopted = 0.20. For a description of the stellar evolutionary model see Claret 1995. The age determination for very young systems is not straightforward since in these cases it is difficult to decide what is the "good" isochrone. The error bars concerning DI Her in Fig. 1 show this fact, since we can find a range of acceptable ages within them. The uncertainty in age determination leads to an error around 10% in the theoretical k₂ . As we do not know what is the observed chemical composition of the system the above values are used only as an approximation. We have performed tests running models with other chemical compositions and we have found that the values of k₂ are not too much dependent on composition (Claret 1995, Fig. 7). Given the magnitude of the discrepancy between theory and observation and the comments made above such uncertainties in the models do not affect seriously the present analysis. For the same models the theoretical values of k₂ can be inferred. We obtain k₂₁ = 8.68 10^-3 and k₂₂ = 8.08 10^-3 for the primary and secondary respectively (Fig. 2).

Fig. 1. Models for DI Her. The radii are shown as a function of time. An acceptable common age of 5 106 years is found. The adopted chemical composition was X=0.7045 and Z=0.015 with =1.52 and =0.20. The continuous line represents the primary, and the dotted one denotes the secondary.

Fig. 2. The theoretical apsidal motion constants k2 as a function of the time for the same models of Fig. 1. Note that inferred values of k2 are for the same age found in Fig. 1.

With these data we can compute the expected value for corrected by the relativistic contribution as

were the symbol dist denotes tidal and rotational contribution and GR indicates the relativistic contribution to the periastron advance. The classical part due to distortions of the components can be written using the following equations:

where P is the orbital period and U is the apsidal motion period. The c are given by

The auxiliary functions f(e) and g(e) can be written as

and and denote the angular velocity of the component i and the keplerian one, A is the semi major axis of the orbit, R_i and m_i are the radius and mass of the component i respectively.

Following (Levi-Civita 1937, Kopal 1978) the relativistic contribution to the advance of the periastron is given by

where m_i and A are given in solar units. Introducing numerical values in Eqs. 2, 3, 4 and 5 and using Eq. 1 we get = 1.25 10^-3 degrees per cycle. We have obtained this value using for the values 3.50 and 3.78 respectively for the primary and secondary. The resulting theoretical apsidal motion rate is about 4 times larger than the observed one corrected by the relativistic correction.

© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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