3. Calculation of observed and expected apsidal motion
3.1. Theoretical formalism
Consider a not necessarily synchronously rotating binary system. If P denotes the orbital period and U the period of revolution of the apsidal line, the theoretical ratio for both rotational and second-harmonic tidal distortion of both stars can be expressed as (Kopal 1978, p. 243)
where the coefficients (i=1,2) are given by
A is the semimajor axis of the relative binary orbit, and , i=1,2, denote the actual angular velocity of rotation of the individual components; for the special case of synchronous rotation one obviously has = . The constants denote the so-called second internal structure constants of the two stars, which are related to the internal stellar density distribution (Kopal 1978). The extreme cases apply for a centrally condensed mass, i.e., a point mass, with , and for a star with uniform density distribution with .
where both the masses and as well as the semimajor axis A of the relative binary orbit are expressed in solar units.
In order to relate the expected rate of apsidal motion to the observable change between period between primary minima and secondary minima , I use the formula derived by Rudkjobing (1959):
3.2. Application to CrB
I now apply Eqs. 2- 6to the specific case of CrB using the numbers quoted in Table 1 and 4, and find - expressing in seconds - = 5.67 sec, with a relativistic contribution from Eq. 5of 0.95 sec or 17%. This theoretical estimate obviously agrees very well with the observed value of 4.8 2.1 seconds. Nevertheless one has to ask, whether this agreement is fortuitous or whether it indeed indicates a correct theoretical understanding of apsidal motion in CrB. Putting the question differently, one has to ask what are the errors in the theoretical estimate of .
Looking at Eq. 6, one realizes that there is a variety of possible sources of error for the theoretically expected value of , viz., errors in the orbital elements and e, errors in the derived radii, masses and periods, and errors in the internal structure constants. As to errors in and e, they lead to overall errors of less than 1 percent, and will therefore be ignored in the following. As to the second source of error, I note that the rotation period of the secondary is unknown. Using the nominal system parameters for the primary and secondary, I calculate a 1 percent contribution for the secondary assuming a rotation period of 5 days, and a 6 percent contribution for the secondary assuming a rotation period of 1 day. Given the age of CrB (it is a member of the Ursa Major stream) and its X-ray luminosity, a rotation period of 1 day is extremely unlikely, and I therefore conclude that the overall contributions to the total apsidal motion and its error budget is essentially determined by the A-type star alone.
How do the parameters , and enter into the expression for ? In the expression 3one notices that the terms proportional to dominate by far; hence the errors in the masses of the stars, basically determined from the photometry and Kepler's law, do not figure significantly. The parameters and are in turn determined from the light curve modeling and the measurement by Slettebak et al. (1975; unfortunately without error), and the coefficient turns out be proportional to through ; therefore any errors in propagate and are amplified. Specifically changing the radius of the primary component within the limits quoted by Tomkin & Popper (1986) and keeping all other parameters fixed, changes the expected values in the range 3.43 - 6.31 seconds. Obviously, the radius of primary is the single most important uncertain physical parameter in the CrB system.
The error in the relativistic contribution to the periastron advance depends only on the total mass, which is very accurately known, and the semimajor axis of the orbit, which again can be considered to be quite well known, since the new HIPPARCOS parallax for CrB confirms the parallax used by Tomkin & Popper (1986). Therefore the error on the calculated relativistic periastron advance of 0.95 sec should be quite small.
Finally, I turn to errors in the structure constants . Except for systematic "errors" in the stellar models, which one of course would like to determine from the observations of apsidal motion, the errors in depend on the observational uncertainties in mass, age, and chemical composition. I use the grid of models by Hejlesen (1987), who published the internal structure constants for stellar models as a function of mass and age for a few values of chemical composition. Obviously only the primary matters. First of all, varying the mass of the primary within the permissible range (10 %), I find the same uncertainty in the derived internal structure constants . Tomkin & Popper (1986) quote an age of yrs for CrB; over this time span, the primary has evolved about 0.4 magnitudes away from the zero age main sequence (ZAMS), while the secondary should be on the ZAMS for practical purposes. They further show that a chemical composition of X=0.7 and Z=0.02 yields acceptable fits to effective temperature and surface gravity of both stars.
Obviously, the internal structure constants change rapidly for evolution off the main sequence. Using specifically Hejlesen's (1987) model tracks, one finds for a ZAMS model of mass and chemical composition and , , while the same model at an age of yrs yields , i.e., a reduction of 80 %. Similar changes are found when varying the chemical composition, however, it appears that low metallicity models, for which one obtains the largest internal structure constants are excluded by the photometry and age of the system. Using the nominal system parameters listed in Table 1 and 4, I calculate corresponding periastron advances of 5.3 sec assuming a ZAMS model, and 2.95 sec for a year old model; adding a relativistic contribution of 0.95 sec, one obtains a total periastron advance of 6.25 and 3.91 sec respectively for the two cases. This range fits perfectly to the measured value of periastron advance of 4.8 2.1 sec. Thus, at the moment, one can only state that the observed periastron advance is certainly consistent with theoretical expectations.
© European Southern Observatory (ESO) 1998
Online publication: April 15, 1998