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Astron. Astrophys. 330, 999-1004 (1998)

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3. The Li I [FORMULA] 6708 line in spectra of self-irradiated accretion discs

According to our assumptions in Sect. 2, the equivalent width, [FORMULA], of a spectral line in the spectrum of a self-irradiated accretion disc can be determined by the expression

[EQUATION]

where [FORMULA] (R) is the equivalent width of the line in the spectrum of the disc ring of radius R and [FORMULA] is the local continuum flux at the line wavelength. In our case [FORMULA] is adopted as the [FORMULA] in the spectrum of a star with the same [FORMULA] and log g, as the surface of the accretion disc at radius R. Because the equivalent width of the lithium depends weakly on gravity, an analytical fit of the dependence [FORMULA] from [FORMULA] with log g = 4 has been obtained:

[EQUATION]

Here [FORMULA], [FORMULA] and [FORMULA] = 4530 K are the fit parameters, which depend on the Li abundance. This analytical approximation has been found from our LTE calculations of the equivalent widths of the resonance doublet which were performed using Kurucz (1993) stellar model atmospheres. A comparison of our results with those of Gerbaldy et al. (1995) is shown in Fig. 1. The accuracy of the fit is better than 20 [FORMULA] in the [FORMULA] range 4500 K to 6000 K, and it is of the same order for higher temperatures (up to 9000 K) if [FORMULA] mÅ.

[FIGURE] Fig. 1. Comparison of the equivalent width of the Li I [FORMULA] 6708 Å  line computed by us and Gerbaldi et al. (1995) as a function of [FORMULA].

A rough estimate of the equivalent width of the Li I line in the spectrum of the accretion disc can be obtained by adopting a black body approximation for the flux in the continuum. We have from Eq. (8)

[EQUATION]

where [FORMULA] and the integral in the denominator has been calculated assuming an infinitely large disc (Lynden-Bell 1969). A first-order approximate integration yields:

[EQUATION]

[EQUATION]

which is accurate to better than 12 [FORMULA] as compared with the exact results from Eq. (10). This last expression can be used as a crude estimate of the Li I [FORMULA] 6708 line equivalent width in spectra of X-ray novae. In order to illustrate this we show in Fig. 2 the dependence of the smallest outer disc radii which allows the equivalent width of the Li I line to be greater than 10 mÅ versus the relative disc luminosity for three values of the mass of the compact central object. We assume log N (Li) =2, [FORMULA] and a temperature of 5000 K for the Li I line.

[FIGURE] Fig. 2. Dependencies of the smallest outer disc radii, which allows for [FORMULA] of the Li I [FORMULA] 6708 Å  line to be greater than 10 mÅ, on log of relative disc luminosity together with the smallest [FORMULA] obtained from condition (5) for three central object masses.

In order to explore the behaviour of the Li I [FORMULA] 6708 line equivalent widths we have applied Eqs. (10) and (11) in a wide range of accretion parameters and Li abundances. Masses were ranged from 1 to 10 [FORMULA], [FORMULA] from 0.001 to 0.3, [FORMULA] from 10-5 to 10-3 and log N (Li) from 2 to 4. We found that if the disc luminosity is less than 0.03 [FORMULA] and [FORMULA] is less than 10-4, then the [FORMULA] of the Li I line is higher than 10 [FORMULA] for log N (Li) greater than 2, and the equivalent width can be as large as 50-100 [FORMULA] for log N (Li)=3. Such absorption lines could be detectable in the spectra of X-ray novae during outburst decay.

It was then decided to conduct a more precise calculation of the equivalent widths of the Li I resonance doublet and [FORMULA] in the spectra of the self-irradiated geometrically thin optically thick accretion [FORMULA] -discs around Schwarzschild black holes. The spectra were calculated following the prescriptions given in Sect. 2 and in previous work by Suleimanov (1996) using the computer code STARDISK based on the code ATLAS5 (1970). The value of [FORMULA] was calculated assuming A = 1, but only directly impacting X-ray radiation was taken into account. The disc half-thickness [FORMULA] was obtained with real opacity and disc structure along the z -coordinate taken into consideration (Suleimanov 1992). In this approach the radiation field was calculated exactly, from the radiative transfer equation, in those rings where [FORMULA] and [FORMULA] K.

The computations were done for a set of reasonable disc model parameters: M =10 [FORMULA], [FORMULA] =1, [FORMULA] = 9 [FORMULA] 105 cm, [FORMULA] = [FORMULA] = 4500 K) but no more than 8 [FORMULA] 1010 cm, log N (Li)=3, [FORMULA] = cos i = 1 and relative disc luminosities in the range 0.1-0.001 Eddington luminosity. One of the parameters of this typical set or the Li abundance was varied whereas the other parameters were kept constant. In Figs. 3-7 the dependence of the [FORMULA] of Li I and [FORMULA] lines on the relative disc luminosity are shown for different masses of the primary (Fig. 3), different outer disc radii (Fig.4), Li abundances (Fig. 5), [FORMULA] parameters (Fig. 6) and disc inclinations to the line of sight (Fig. 7).


[FIGURE] Fig. 3. Equivalent width of the Li I [FORMULA] 6708 Å  and [FORMULA] lines vs. log of relative disc luminosity for typical set of disc models (see text), but for different masses of central object.

[FIGURE] Fig. 4. Equivalent width of the Li I [FORMULA] 6708 Å  and [FORMULA] lines vs. log of relative disc luminosity for a typical set of disc models, (see text) and different outer disc radii.

[FIGURE] Fig. 5. Equivalent width of the Li I [FORMULA] 6708 Å  and [FORMULA] lines vs. log of relative disc luminosity for typical set of disc models (see text), and different Li abundances

[FIGURE] Fig. 6. Equivalent width of the Li I [FORMULA] 6708 Å  and [FORMULA] lines vs. log of relative disc luminosity for a typical set of disc models (see text), but for different [FORMULA] parameters

[FIGURE] Fig. 7. Equivalent width of the Li I [FORMULA] 6708 Å  and [FORMULA] lines vs. log of relative disc luminosity for a typical set of disc models (see text), but for different disc inclination angles

As it is evident from Figs. 3-7, the Li I resonance line may be observed provided the disc luminosity is less than 0.01 [FORMULA]. The equivalent width depends slightly on [FORMULA] and the central object mass, but the larger [FORMULA] and the Li abundance the greater the [FORMULA] of the Li I line. And, vice versa, the smaller the disc inclination the greater the [FORMULA] of the Li I line. According to our calculations if the equivalent width of [FORMULA] is greater than 10 [FORMULA] the Li I line has [FORMULA] [FORMULA] 20 m [FORMULA] and can be detected.

The exact results seem to be identical to estimate calculations and have close correlation with conditions (5) and (6). This means that conditions (5) and (6) may be used for other lines too.

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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