4 Apsidal motion

Table 2 lists the observed values of the periastron angle $\omega$. In Fig. 3 we present a least squares and a $\chi ^2$ fit to these data. For the measurement dated JD 2429189 we used the average of all other errors on $\omega$ as an error estimate. The least squares fit gives for the apsidal motion period U = 46 year, and the $\chi ^2$ fit gives $U = 47.5 \pm 0.7$ year. This value is in good agreement with the first estimate by Batten et al. (1978; $\sim$40 year), and with the value determined by Abt & Levy (1978; 44.8 year). It is in contrast, however, with the value of 149 year determined by Monet (1980).

The observed apsidal motion is due to a not purely Keplerian potential of the binary system. This can be caused by the presence of a third body orbiting $\psi ^2$ Ori, by effects of general relativity, or by tidal and rotational forces in the binary.

Figure 4: CLEANed periodogram of the wavelength region of the OII 4590.8 and 4596.2 Å lines, with the corresponding summed periodogram in the right panel. The bottom panel shows the mean of the 78 spectra used to compute this periodogram; the left panel shows their window function. The signal was prewhitened for the orbital frequency and its first four harmonics. The double peak at 10.6c/d, with its one-day aliases, is due to non-radial pulsations.

As the orbital velocities of the two stars in $\psi ^2$ Ori are mildly relativistic, we can approximate the expected apsidal motion with the expressions given by Giménez (1985) or Stairs et al. (1998). We find that for mass estimates of 13.9 $M_\odot$ and 8.5 $M_\odot$ ( $i = 60\hbox{$^\circ$ }$) the period of apsidal motion for this binary expected from the theory of general relativity is about 1000 years. This is in contrast with the observed period of 47.5 year, indicating that other perturbations of the Keplerian potential are more important.

The reported values for the system velocity of $\psi ^2$ Ori range from 12 kms^-1 to 26 kms^-1, which indicates that the value is variable although it is not clear if for all determinations the velocities were transformed to the heliocentric frame. The spread of the points as a function of time does not allow a proper period search in order to find the orbital period of a possible third body. Using Kopal (1959) and Wolf et al. (1999) we find that for a hypothetical $M_3=9~M_\odot$ third component the orbital period P₃ must be as short as about $P_3 \sim 5P = 12.5$ day, in order to give an apsidal motion period similar to that observed. For a less-massive third component, and for a longer orbital period P₃, the apsidal motion period becomes longer ( $U \propto P_3^2(M_1+M_2+M_3)/M_3$). It is clear that such a close third body is unlikely and in contrast with the observations of $\psi ^2$ Ori.

We conclude that the observed apsidal motion is due to tidal and rotational forces in this close binary, and that effects of general relativity and a possible third body can be neglected.

4.1 Internal structure constant

Assuming $R_1\sin i=4.7 \pm 0.3~R_\odot$ and $R_2\sin i=3.3 \pm 0.3~R_\odot$, i.e. assuming periastron synchronisation for both components, we computed the internal structure constant as averaged over the two stars, $\overline{k_2}$= P/(U(c₁+c₂)), with $c_i\propto(R_i/(A_1+A_2))^5$ as defined in e.g. Claret & Giménez (1993), giving c₁ = 0.011 and c₂ = 0.005. Neglecting the influences of general relativity and a possible third body we find $\log\overline{k_2} = -2.02 \pm 0.16$. Accounting for general relativity we find $\log\overline{k_2} = -2.00 \pm 0.16$.

We use Tables 17-20 in Claret & Giménez (1992) to compare the observed value of $\overline{k_2}$ with that expected from theory. Using the age of subgroup 1a of the Orion OB1 association, $11.4 \pm 1.9$ Myr, as the age of the stars in the binary, we find $\log\overline{k_{2,1}} = -2.4$ for the primary and $\log\overline{k_{2,2}} = -2.1$ for the secondary. Combining these numbers with those of c₁ and c₂ leads to a theoretical value of $\log\overline{k_2} = -2.25$. Note that with the adopted age of 11.4 Myr the primary is very near to the end of the main sequence, if it is as massive as 15 $M_\odot$. For this reason the tabulated value of $\log\overline{k_{2,1}}$ is high. The discrepancy between observed and theoretical values of $\log\overline{k_2}$ therefore indicates that the binary is somewhat younger than assumed, or that the primary is less massive than 13 $M_\odot$ which would imply .