4. Results and discussion
The local electron and ion density of the elements given in Table 1 have been calculated for a wide range of the parameter values. In this paper, the data for sodium and calcium producing the strongest visual lines (see Table 2) are examined, other results will be presented elsewhere.
4.1. Local density and ionization degree
The obtained electron density and the relative abundances of ions n (C I)/n (C), n (Na I)/n (Na), and n (Ca II)/n (Ca) are presented in Fig. 2. They were calculated for two models with a different degree of the matter concentration in the clumps (see the upper part of the figure). Variations of the relative abundances outside the zone shown in Fig. 2 are rather small. In the shadows produced by the clumps the fractions of C I, Na I and Ca II are slightly higher than ahead of them.
From Fig. 2 one can see that carbon, sodium and calcium in the interclump medium are mainly in the state of the C II, Na II, and Ca III ions. Note that the calcium is twice ionized and the size of the C II region is rather large because the ionizing radiation of the chromosphere was included in our model.
The ionization degree of the gas in the shell outside the clumps is and practically does not depend on the distance from the star up to AU. At larger distances, the ionization of hydrogen by cosmic rays starts to increase the ionization degree.
In the clumps, the ionization degree is mainly determined by the gas density and the distance of the clump from the star (see Fig. 3). The values of shown in Fig. 3a were obtained by variations of the gas to dust ratio in Eq. (4). For the model with the values of parameters from Table 3, cm-3. A similar (but not the same) dependence occurs if we increase or decrease . The knees seen in Fig. 3b are related to the narrow layer where the diffuse dust appears in the shell ().
4.2. Lines forming in the shell
If there are no clumps in the line of sight, the absorption lines should be blueshifted with the typical wind velocity. Our calculations made with the standard values of the parameters demonstrate that the column density of Na I and Ca II are small and the lines should be extremely weak (see the last row of Table 4).
Table 4. Dependence of column density, line optical thickness and equivalent width on the terminal wind velocity
In our model, the shell lines are mainly formed at the distances 1 - 5 AU where only the negligible part of sodium and calcium is in the state of Na I and Ca II. In order to enlarge the amount of neutral sodium and the Ca II ion one can decrease the ionizing flux or/and increase the gas density. The latter may be made if we invoke an anisotropy or a strong deceleration of the wind, or possibly a shock where the stellar wind encounters the surrounding envelope.
As our calculations show, if the final wind velocity is as low as = 5 km s-1 the absorption lines can become rather strong (see Table 4). However, in this case it is difficult to explain the lines highly shifted from their rest wavelengths and the short-time line variability.
4.3. Lines forming in clumps
Some results of our calculations for the models with a clump in the line of sight are presented in Table 5. The values of the parameters which are not mentioned there have been chosen to be equal to the standard values given in Table 3. The changes of the density distribution in the clump (the parameter ) influence the results only slightly and are not illustrated in Table 5.
Table 5. Dependence of column density, line optical thickness and equivalent width on model parameters
Some model parameters listed in Table 3 affect the results strongly. This group includes the gas to dust ratio, the element depletion and the velocity of large-scale gas motions. At present, the values of these parameters are unknown, and they could be estimated from a comparison of the calculated and observed equivalent width of the Na I and Ca II lines. The results presented in Table 5 demonstrate that the saturated lines of Na I and Ca II can be formed in the clumps, and that the lines could be observed with the current telescopes.
It should be noted that the calculated column density of Na I and Ca II are of the same order. This result would change drastically provided we neglect the radiation of the chromosphere and the H II region. In this case, calcium will be mainly in the state of the Ca II ion, and the lines of the ion will be considerably enhanced in comparison with the sodium lines. Such a model including only the photospheric radiation has been considered by Sorelli et al. (1996).
4.3.1. Large and small clumps
If the projected size of the clump is comparable with the stellar diameter (), the correlation between the stellar brightness variations and the behaviour of the absorption lines should be expected. Then, the data from Table 5 may be used to estimate some characteristics of large clumps.
However, in many cases large variations of the absorption components of the sodium and calcium lines are not accompanied by significant brightness variations (see, e.g., Fig. 5 in the paper of Grinin et al. 1994). In order to damp down the brightness variations, the projected size of the clump must be much smaller than the stellar diameter. If the observed variation of stellar brightness is , then the upper limit to the clump size is
The numerical coefficient in Eq. (11) is obtained if we accept the standard value of (Table 5) provided the observed star obscuration is . If the size of the clump is equal to its length, i.e. AU, the gas density at the clump center is cm-3. The lines forming in such a clump would be rather strong (see Table 5). Their variability may be connected with the changes of the physical conditions in the moving clump (turbulent velocity, element depletion).
4.3.2. Clumps at different distances
The equivalent width of the sodium and calcium lines for the clumps located at different distances from the star are presented in Fig. 4. The solid lines show the results of calculations for the model with the standard values of parameters (see Table 3). The different values of W obtained are connected with the decrease of the ionizing radiation flux with distance (see Eq. (8)). The dashed lines were calculated for the model where the turbulent velocity is inversely proportional to the distance: with = 1 km s-1 at = 50 AU and = 10 km s-1 at = 2 AU. The latter value of can be estimated from the width of the calcium and sodium lines ( FWHM/1.66; Kaplan & Pikel'ner 1979) observed by Catala et al. (1986a). The turbulent velocity increasing with the decrease of leads to the growth of W when 10 - 15 AU. Closer to the star, a significant increase of the flux of ionizing radiation occurs. This results in a decline of the number of neutral sodium and Ca II ions. Note that the destruction of dust grains in the clumps may produce additional Na and Ca atoms that could change the element depletion and increase the gas to dust ratio.
4.3.3. Moving clumps
The behaviour of the absorption lines in spectra of HAeBe stars should depend on the clump orbits (radial velocities) and the physical conditions in the clumps (the width and strength of the lines). In the above discussion, the latter topic has been treated. Now, we consider some problems related to the clump orbital motion.
It is naturaly to adopt that the periastron of the clump orbit lies between the star and an observer, and the line of apsides does not coincide with the line of sight. Evidently, a very small part of the clump orbit only is projected on the stellar disk and can be observable.
When a clump moving from the distant circumstellar environments appears on the stellar disk, the absorption line should be blueshifted and narrow. Closer to the periastron, the radial velocity drops down to a minimum value. Because of the growth of the turbulent velocity, the line width should increase. Meantime, the growing number of ions may make the line deeper.
If a clump is observed after its passage of the periastron, the line is expected to be redshifted. The redshifted components must be broader than the blueshifted ones because the clump was disturbed near the star. Note that very wide lines may be a result of the presence of several clumps in the line of sight. The numerous mechanisms of clump destruction operating near HAeBe stars (see Friedemann et al. 1995 and Mitskevitch 1995 for discussion) can cause the fragmentation and destruction of large clumps in the immediate stellar environments. It is expected that after the periastron the fragments continue their way moving away from the star or falling onto them. In both cases, the absorption lines should be redshifted.
4.3.4. 90 km s-1 component in the spectrum of AB Aur
Some speculative estimates of the clump orbit can be made from the observations of AB Aur made in December 1991 by Catala et al. (1993). The authors observed the absorption components of the Na I D, Ca II K and Fe II lines blueshifted by 90 km s-1. The equivalent width of the sodium and iron lines was 0.03 - 0.10 Å (Catala 1996). It was found that the lifetime of the 90 km s-1 component was longer than 24 hours (from the observations of the Na I D lines, ), but less than 67 hours (from the observations of the Fe II lines, ). We attribute the first interval to the time of the sodium ionization. Then, the upper limit to the electron density inside the clump is
Here, is the rate of radiative recombination of neutral sodium given by Eq. (B1). The numerical value was calculated for = 500 K.
The photometric history of AB Aur gives evidence that its visual brightness was varying considerably in thirties (see discussion in Voshchinnikov et al. 1996). A far infrared variability has been found by Prusti & Mitskevitch (1994). Clumps may be responsible for both effects. However, during many last years the brightness of the star is stable. Then, the size of the clump producing the - 90 km s-1 component must be small (see Eq. (11)). As follows from Fig. 3, a clump with the central density in the limits cm-3 satisfies the condition of Eq. (12). The clump's location in the shell may be arbitrary between 2 AU and 50 AU in our model. However, the value of near the lower limit (2 AU) seems to be more probable because the observed equivalent width W (Na I D2) = 0.03 - 0.05 Å may be easily reproduced in this case (see Fig. 4).
The total clump velocity
allows to estimate the type of Keplerian orbit. A body moving near a star is known to have an elliptic orbit (see, e.g., Roy 1978) if
Here, we used the value of AU. So, we can conclude that the clump orbit seems to be hyperbolic.
4.3.5. Ten days absorption event in the spectrum of HR 5999
For HAeBe star HR 5999 Tjin A Djie et al. (1989) published the radial velocities and the equivalent width of sodium, calcium, iron and other lines together with the simultaneous photometry. Here, we interpret the behaviour of W (Na I D2) within a ten days period: from 21.05.78 till 01.06.78. In this time, there were made 9 observations and the stellar brightness decreased from V = 6:m9 to 7:m2. The variations of the radial velocity and the equivalent width with brightness are shown in Fig. 5 by open circles. The observations obtained on the sequent dates are connected by dashed line.
For our modelling, we choose the photospheric fluxes corresponding to the Kurucz (1979) model with the parameters = 7800 K and = 3.5 which are close to those adopted for HR 5999 (Thé et al. 1994). The changes of could be explained as an absorption in the clump approaching the periastron. However, the distance that the clump can pass during ten days is too small to expect large changes of the ionizing flux and the sodium abundance. Therefore, we treat the observed variations of W (Na I D2) as a result of the stellar obscuration by a clump with a variable turbulent velocity which is smaller in its denser parts and larger in its rarefied parts. The density profile may look like that shown in Fig. 2 for the clump with a moderate gas concentration. The observed slope of data can be reproduced if we take the value = 6 km s-1 in the outer parts of the clump and = 2 km s-1 in the inner ones. The observations could be fitted better if one chooses the gas to dust ratio = 5 cm-2 mag-1, that is two orders larger than the standard interstellar one.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998