3 Orbital solution

In order to determine a radial velocity curve for the two components in $\psi ^2$ Ori we used two spectra, in which the stars have a large separation, as templates for cross correlation. In the first template spectrum the primary has a radial velocity of 159 kms^-1, and -116 kms^-1 in the second template. For each template we have done a full analysis to obtain an orbital solution; comparison of the two sets of results allows us to estimate the errors of our orbital solution.

For the radial velocity curve of the primary we used 5 echelle orders in the cross-correlation process, covering the following wavelength regions: 4245-4295, 4545-4585, 4562-4615, 4634-4692, and 5010-5055 Å. We used the best 82 spectra for the orbit determination of the primary.

For the secondary we also used 5 orders: 4137-4150, 4382-4395, 4915-4931, 5010-5055, and 5865-5885 Å. As the weaker lines of the secondary are disturbed by the lines of the primary near conjuctions, we used only 47 spectra for the orbit determination of the secondary.

For each spectrum we used the median radial velocity of the 5 orders. The standard deviation of the 5 values is typically 4 kms^-1 for the primary, and 12 kms^-1 for the secondary. We used the error of the mean of the 5 values as error estimates in the orbit fits.

As we did not obtain a useful spectrum of a radial velocity standard, we used the HeI lines 4009.27, 4143.76, 4387.93, 4921.93, 5015.68, 5047.74, and 6678.15 Å, to calibrate the velocity shifts of the templates. This resulted in errors in mean for the 7 lines of 4 kms^-1and 7 kms^-1 for primary and secondary respectively. Because of this inaccuracy, our derivation of the system velocity has a similar error.

The radial velocity curve of the two components, and the result of the orbit fits, are plotted in Fig. 2. The results of the fits of the two templates are listed in Table 1. One can see that the differences between the two solutions are larger than the errors of the individual fits allow. Therefore we list an additional final set of orbital parameters, based on the mean values and the error in the mean values of the two solutions. Orbit fits with the period fixed to the value of 2.526 day (Lu 1985) did not give solutions that are significantly different than the ones in Table 1.

3.1 Geometry of the binary system

In order to estimate the radii of the stars in $\psi ^2$ Ori we need an accurate value of their rotational velocities. Lu (1985) has presented estimates of the projected rotational velocity of the two stars of the binary based on the FWHM of He lines: $v_1\sin i=95 \pm 5$ kms^-1 and $v_2\sin i=75 \pm 5$ kms^-1. We found that the estimate for the primary is somewhat too small to explain the line widths in our spectra. Therefore we fitted a model of a rotating star with pulsations with degree $\ell=6$ (see Sect. 5) to the mean profile of the spectra of the primary after shifting to the velocity frame relative to the primary (Fig. 1). We fitted the model (Schrijvers et al. 1997) to the SiIII 4574 and OII 4590 lines. The derived value of $v_1\sin i=105 \pm 6$ kms^-1 is robust against changes in pulsational parameters, intrinsic line width, and limb darkening. We have checked the previously determined rotation velocity of the secondary, but due to the fact that the lines of heavy elements are very shallow and that the helium lines suffer from normalization problems, we were not able to derive an improved value of $v_2\sin i$.

Assuming that the rotation rates of the stars are synchronised at periastron ( $\Omega^2_{\rm rot}=\frac{(1+e)}{(1-e)^3}\Omega^2_{\rm orb}$, Kopal 1978, i.e. $\Omega_{\rm rot}=1.11\Omega_{\rm orb}$), and that the stars have aligned their rotation axes perpendicular to the orbital plane, and using the measurements of $v_1\sin i=105 \pm 6$ kms^-1 and $v_2\sin i=75 \pm 5$ kms^-1, we estimate the equatorial radii of the two components of $\psi ^2$ Ori as $R_1\sin i=4.7 \pm 0.3~R_\odot$ and $R_2\sin i=3.3 \pm 0.3~R_\odot$.

 

 

Table 2: Orbital solutions from the literature.

	JD	orbital period	v0	K1	K2	e	$\omega$	apsidal period
		[day]	[kms-1]	[kms-1]	[kms-1]		[ $\hbox{$^\circ$ }$]	[year]
Plaskett (1908)	2 417 916	2.52588	12(1)	144		0.065(0.011)	185(11)
Beardsley (1969)	2 419 408		17(1)	142		0.05(0.01)	240(15)
Pearce (1953)	2 429 189	2.52596	16	143	235	0.07	93
Chopinet (1953)	2 434 024		21(2)	136		0.04(0.02)	221(22)
Lu (1985)	2 437 685	2.52596	26(1)	139	219	0.044(0.008)	285(14)
Abt & Levy (1978)	2 442 418		19(1)	142		0.08(0.01)	356( 9)	44.8
current	2 450 774	2.529(5)	19(5)	145	237	0.053(0.001)	172( 5)	47.5(0.7)

If the assumed rotation rates are correct, and the stars comply to the mass-radius relation for ZAMS stars (Landolt-Börnstein), we can estimate the inclination from the dynamically derived masses in Table 1 and the above radius estimates. The implied inclinations are $i_1 \sim 62\hbox{$^\circ$ }$ for the primary (with 1$\sigma$confidence interval 56 ) and $i_2 \sim 65\hbox{$^\circ$ }$ for the secondary (58 ), which are consistent with the orbital inclination $58\pm 8\hbox{$^\circ$ }$ as derived from light-curve modelling (Hutchings & Hill 1971). Note that the inclination cannot be too large, as no eclipses are observed. (Waelkens & Rufener (1983) present observations hinting at a small eclips, but this observation has never been confirmed.) Taking the estimates $R_1\sin i = 4.7~R_\odot$ and $R_2\sin i = 3.3~R_\odot$ as polar radii, eclipses are expected for .

With $R_1\sin i=4.7 \pm 0.3~R_\odot$ and $R_2\sin i=3.3 \pm 0.3~R_\odot$, $\psi ^2$ Ori is a detached system with relative radii R₁/A₁ = 0.66 and R₂/A₂ = 0.28 with respect to the semi-major axes of the orbits, and R₁/d₁ = 0.45 and R₂/d₂ = 0.39 with respect to the distances d between stellar center and the inner Lagrangian point L₁.

Figure 3: Apsidal motion: least-squares fit (dashed) and $\chi ^2$ fit (solid). See Table 2 for references.