The numerous attempts to reproduce the lithium line profile suggest that two different ways can be followed. In the first way we adjust the local effective temperature of a near-polar lithium spot, assuming a given lithium abundance. The calculations show that it is necessary to strongly decrease the temperature (to 6000 K), even assuming a maximum lithium abundance (3.2 in the scale of N(H) = 12.0, i.e. the initial abundance, before lithium is depleted). But so large temperature difference between the photosphere and the spots () does not agree with the small photometric variability of the star.
The other way is the assumption of lithium stratification in the stellar atmosphere and its concentration in the surface layers, which can be provided by the mechanism of ambipolar diffusion (Babel, 1993).
If we assume that the brightness variations are synchronous with the rotation of the star, we can put definite limitations on possible differences between the physical conditions in the spots and in the photosphere. For HD 83368 the spot structure must be in accordance with the fact, that the B and V light curves are in opposite phases. We know three possible ways to explain a variation of the star brightness when a spot appears on the visible hemisphere:
Therefore we accepted this third hypothesis of identical temperature conditions for the spot regions and the photosphere assuming enhanced lithium abundance in the spots.
The star rotation results in broadening all spectral lines in the stellar spectrum because of the Doppler effect. The degree of this broadening depends on the projection of the equatorial velocity on the line of sight .
For calculating the lithium line profile we splitted the visible hemisphere of the star surface in 32 32 points with rectangular coordinates , each of which is the Gaussian (quadrature) division point. The line profile in this case can be calculated according to Tassoul (1978).
Let be the intensity of radiation emitted by the point on the disk of a non-rotating star in the spectral line at the distance from the central wavelength . For a rotating star it is necessary to use , where is the radial velocity in the point and c the velocity of light. Since (where is the equatorial velocity of the star, and i is the inclination angle of the rotational axis to the line of sight), the intensity of the radiation emitted by the point on the disk of a uniformly rotating star is given by , and the line profile is calculated accordingly
where is the intensity of the continuous radiation, emitted by the point , and is the line intensity, which is computed for each point of the visible surface taking into account the presence of spots with different chemical compositions. In this work we assume there are two symmetrical circular spots with a homogeneous abundance of the selected chemical element (in this case lithium) inside the spot. The coordinates and the radius of each spot are given in spherical coordinates relative to the observer (longitude l and latitude ).
6.2. The analysis of the Li line profiles
In the paper of North et al. (1998) it was shown the presence of two diametrically opposite spots. In this work we have calculated the coordinates and surface areas of spots supposed them as circular. The best method to reveal the spotted structure of a stellar surface is the following: spot parameter retrieval from the line profiles for a number of rotational phases. However this method demands the availability of high quality spectra, distributed over a larger number of rotational phases (more than ten), than we have. Therefore we have used the reverse method, i.e. direct spectrum modelling with choice of spot parameters in order to obtain the best fit to the observed spectra.
After clearing up questions, connected with the differences of physical conditions in the spots and its surrounding photosphere (see above), our free parameters are:
The choice of the parameters is made in order to provide the best agreement between the theoretical and observed spectra.
To derive the spots locations we set as a first approximation the longitudes of the two spots to the points where RV curve of the Li line crosses the value of the mean radial velocity of the star. Then, changing the value of i with the spots' latitudes equal to zero, we find the best agreement between the computed and observed profiles of the Li line for all phases. A further improving of the fit we reached by varying the latitudes and minute corrections of abundances. In general a good fit was reached for phases near subsolar position of a spot, while worse one for a spot location near a limb. The procedure of fitting observed and calculated spectra was carried out until the discrepancy of both spectra reached its minimum. The variations of the parameters in the whole range of possible values permit us to obtain a single solution for the atmospheric model with and photospheric Li abundance ,
Fig. 8a shows observed and calculated with these spot parameters the Li line profiles for all phases. In the calculations we included 10 components of the Li line fine structure for a magnetic field value, corresponding to each phase. But the influence of magnetic splitting for 2 kG on Li abundance is insignificant ( <0.1 dex). We used our abundance estimates for Fe and other electron-contributors elements, which differ from the solar one. The choice of these abundances affects essentially the Li estimate, because lithium is mainly ionized (Li II ) in a A-type stellar atmosphere.
We also have calculated the Li profile taking into account the nearest rare earth elements lines, namely 6706.051 Å Ce II , 6706.705 Å Pr III , 6707.473 Å Sm II and 6708.099 Å Ce II . The gf-value for the last line, = -2.210, was estimated from the spectrum of Przybylski's star HD 101065, using two nearby lines of Ce II 6704.524 Å and 6706.051 Å with known gf-values. The Kurucz' atmospheric model with and [M/H] = 0. was used for this star. We can note that Ce II lines are stronger than the Li line, in the spectrum of the Przybylski star, unlike HD 83368.
Fig. 8b shows the spectrum at phase 0.689, where we see both Li spots and the rare earth lines are rather strong. If Li spot locations have been computed for all phases together, the REE spot location were chosen for each phase independently. Our calculations for all the 8 phases confirm the hypothesis that rare earth elements may be concentrated in rings around the lithium spots. So, if lithium spots, as it was determined from the mapping procedure, belong on the equator at longitudes l = and , rare earths are seen on l = to , to , to and to .
Note, that we varied the Li spot latitudes, but the best results were found under zero latitudes. For REE (and Fe II spots, Sect. 4) we have adopted the same latitudes. It should be noted also that our analysis of the REE spots for all phases reveals the same features at adjacent phases. The data about the rare earth spots (or rings), visible at the phase 0.689 (see Fig. 8b), are given in the Table 3.
© European Southern Observatory (ESO) 2000
Online publication: June 5, 2000