2. Formation of absorption lines in the spectra of self-irradiated discs
Let us first describe the method used to calculate the total radiation spectrum of self-irradiated, optically thick, geometrically thin, accretion discs (Suleimanov 1992, 1996). We denote as F the final spectrum that results from the addition of the individual local ring spectra :
where i is the inclination angle of the disc with respect to the line of sight, R is the radius of the local ring and and are the inner and outer boundary disc radii, respectively.
For , we will adopt the spectrum of a stellar atmosphere with the same effective temperature and log g, provided that the ring effective temperature is less than 50000 K and the relation of incident X-ray flux to intrinsic ring flux is less than a certain critical parameter A. If these conditions do not hold, the ring is assumed to radiate as a black body with a temperature equal to .
The intrinsic ring flux is determined by the following equation (Shakura & Sunyaev 1973):
where M is the mass of the central compact object and is the accretion rate.
The incident flux is described following Ko & Kellman (1991) as:
where is the energy conversion factor, c is the speed of light and f is the fraction of the total incident flux on the disc at a given radius R.
Using formulae (2) and (3), the boundary disc radius can be obtained:
where it has been assumed that / equals A. Here . We note that the ratio to is greater than A if R is larger than .
The value of the parameter A depends on the structure of the irradiated atmosphere of the disc and it is different for different lines. Since no precise calculations are available, A should be regarded as a free parameter for each line. Sakhibullin & Shimansky (1996) found from exact LTE calculations of irradiated stellar model atmospheres, that lines of ions are more affected by external X-ray radiation than neutral element lines. The effect of X-ray irradiation on the lines of neutral elements is even lower if departures from LTE are taken into account (Sakhibullin & Shimansky 1995). The results obtained by these authors support our approach here.
The second parameter that characterizes a spectral line is the line temperature, , determined as the T of the stellar model giving the highest equivalent width of the line. As a consequence of the adopted self-irradiation model, it is required that is higher than the temperatures of the outer disc and boundary disc radii. These conditions can be re-written by using the usual parameters of accretion discs, , M, and relative disc luminosity :
The previous conditions allow us to establish which absorption lines might be observed in the spectra of X-ray novae if the accretion disc parameters M, L, and were known. Conversely, the values of these parameters may be estimated from the observation of absorption lines in these spectra.
For example, the relation (6) can be rewritten as
where the Balmer lines temperature was taken as 10000 K. Since the current estimates of the masses of the compact primaries in X-ray novae lie in the range 1-10 M and relative luminosities of X-ray novae can be estimated as 0.1-0.01 during outburst decay, the value of the parameter cannot be greater than 10-4 -10-5 if the Balmer lines have absorption wings as seen in the spectra of GRO J0422+32 and XN Vel 1993 (see references above). We note that direct calculations of give similar values (Suleimanov 1996). This implies that the parameter A has a value close to 1 or smaller if scattering radiation in a wind or corona were important in the irradiation of the accretion disc.
© European Southern Observatory (ESO) 1998
Online publication: January 27, 1998