4. Determination of the rotational velocity
In order to perform an accurate abundance analysis via synthetic spectrum fitting, it is of great importance to use a well-determined rotational velocity (). This work is based on a rotation alvelocity determined by fitting individual spectral lines with those calculated by the synthetic spectrum program SYNTHE (Kurucz 1993). Ideally, the spectral lines chosen for such an analysis should not be broadened by hyperfine structure (hfs), isotopic shift (IS), Zeeman effect and line blending unless these effects can be modelled.
The effect of hfs originates from the interaction between nuclear spin and the angular momentum of the electrons. It is absent for those atoms with an even mass number, as these isotopes do not have a net nuclear spin. If the star is assumed to have the same isotopic mixture as our solar system, then iron is highly concentrated to one isotope (92% ). Therefore, iron lines do not show significant hyperfine structure or isotopic shift and are particularly useful when investigating rotational velocities. However, in the presence of a magnetic field the spectral lines may be broadened by the Zeeman effect. How much the spectral lines are affected by the magnetic field depends on the magnitude of the field, the transition wavelength and the Landé factors for the corresponding energy states. If we have transitions where both the upper and lower energy level have Landé factors equal to zero, the spectral line will be unaffected by the magnetic field. Ideally these lines should be used when investigating the rotational velocity for magnetic stars. We prefer to use iron lines to avoid IS and hfs and are then constrained to transitions between and states. These energy levels have Landé factors equal to zero, which is confirmed by experimental measurements (g 0.05) (Moore 1971). Unfortunately, there are few unblended transitions corresponding to these levels, and we have not been able to use any magnetically unaffected spectral lines since they are weak and blended. There are a few useful spectral lines which correspond to energy levels with relatively small Landé factors (g 0.5), and since they are located in the blue wavelength region the Zeeman effect is reduced relative to longer wavelengths. The FeII 4508 and 4491 lines are both unblended for stars with solar-like elemental abundance distributions and little affected by the magnetic field.
The rotational velocity for HR 1094 was determined to have an upper limit value of 17 km s-1, using the FeII line at 4508 Å. As shown in Figs. 1 and 2 the iron line appears to be only slightly blended in the blue wing, and therefore a good choice for this purpose. The error margin is estimated to be less than 1 km s-1, based on the visual appearance of the synthetic spectrum fits to that part of the observed profile not apparently affected by the noted blend. HR 1094 is observed to have a reasonably high rotational velocity and a magnetic field with a small magnitude, consequently the rotation is the dominating broadening mechanism for its spectrum. For stars with a more dominating magnetic field and a lower rotational velocity, for example HR 5049, special care should be taken when choosing lines for investigating the rotational velocity, since the field is capable of changing the line profile dramatically. The Zeeman components have been included in our synthetic spectrum analysis, regardless of whether the spectral lines FeII 4491 and 4508 are significantly broadened by the magnetic field.
HR 5049 presents a sharp-lined spectrum with a mean magnetic field of 4676 G (Mathys et al. 1997), and as shown in Fig. 2 the FeII 4508 line is also useful for the analysis of this object. The rotational velocity for HR 5049 was determined to be 4 km s-1. Dworetsky et al. (1980) suggested to be less than 6 km s-1 which is in agreement with our result. The magnetic field is determined to influence the value of the rotational velocity by less than 0.5 km s-1. The FeII 4491 line has a CoII line in its short wavelength wing, which jeopardises the investigation of . For slow rotational velocity stars ( 10) observed at high spectral resolution, the CoII line and the FeII line are resolved. The influence of the magnetic field on the line profiles has been investigated by including the Zeeman components in our synthetic spectrum and as shown in Fig. 3 only a small amount of the missing opacity can be explained by the Zeeman effect. We suggest that full knowledge of the hfs pattern for this line in collaboration with the magnetic broadening would fully explain this line profile. For stars with no enhancement of CoII , the FeII line at 4491 Å is suitable for accurately determining the rotational velocity, as exemplified by the spectrum of HR 3383 in Fig. 4.
In the analysis of HR 1094 by Sadakane the rotational velocity, , was taken to be less than 10 km s-1. This value was obtained by using the observed FWHM for the MgII 4481 line in a reference frame compiled by Sletteback et al. (1975). Outside of using synthetic spectra a determination of is often dependent on strong unblended lines such as MgII 4481 that are calibrated by a reference frame of standard stars. The risk in using such frames is the necessity to be sure that only stars similar to the investigated object are used for the reference frame.
For completeness, we have also analyzed the MgII 4481 feature in the spectrum of HR 1094 (Fig. 5). The result obtained by using this line is in disagreement with that obtained from unblended FeII lines. From the FeII lines used was determined to have an upper limit of 17 km s-1. The corresponding value from the MgII line is 20 km s-1. The 3 km s-1 difference is too large to be within the error margins of the determination. The modelled MgII feature is actually three lines from the same multiple (3d 2D - 4f 2F). For HR 5049 and HR 3383 the analysis using this feature yields a smaller value of than obtained from unblended FeII lines. It is difficult to simultaneously fit the MgII feature in both depth and width, which may be a consequence of the failure of the local thermodynamic equilibrium assumption of the calculations. Other explanations might be isotopic shift, hyperfine structure or Zeeman effect. Magnesium has three stabile isotopes and their relative abundances are, 24Mg : 25Mg : 26Mg = 79 : 10 : 11, where only one of the isotopes possesses hfs. The strength of the Landé factors for the upper (=2.5, 3.5) and lower (=1.5, 2.5) (Moore 1971) levels for the transitions implies a noticeable Zeeman structure, but the wavelength of the spectral feature indicates this effect to be negligible. Since it is difficult to account for all above mentioned effects, the use of the MgII 4481 Å is not recommended for the most accurate determination of .
© European Southern Observatory (ESO) 2000
Online publication: March 28, 2000