4. The method of calculation and results
Having the fluxes of the components of the binary AG Dra we can compose a system of two equations with two unknown quantities - its distance and the radius of its primary. The effective temperature of this component was estimated by Smith et al. (1996) and amounts to be 4300 K. The cool giant's continuum can be fitted with a function giving the energy distribution of Boo, since, according to Griffin & Lynas-Grey (1999) the effective temperature of this star is 429030 K the same as that of AG Dra. This function consists of two parts, the first part is related to the radiation of a blackbody with a temperature of 4300 K and the second one to the energy distribution of Boo. At the wavelength of the U band the second part has a value of . Then the first equation is:
where R is the cool giant's radius and d - the distance.
Let us now consider the second equation. We suppose that the emitting circumbinary nebula is formed by a wind with spherical symmetry and a constant velocity and has an inner boundary the radius of the giant, as only a small portion occulted by it is not an ionized region. To give an expression to the flux of the nebula the state of ionization of helium is need to be known. We calculated the ratio of the emission measures of the neutral and ionized helium, allowing that the lines of the are pure recombination lines. That is really the case when the electron temperature is 15 000 K (Mikolajewska et al. 1995). We used visual line fluxes of and the flux of the He II 4686 line from the paper of Gonzalez-Riestra et al. (1999) at a phase, close to the maximum light. Since these data are related to quiescence of the system, we used the sum of the fluxes of the narrow and broad emission components of the line He II 4686. In this way we obtained a ratio HeHe+ of about 0.5. This result shows that the singly ionized helium is dominant in the nebula and we assume that the nebular emission mostly is continuum emission of hydrogen and neutral helium. For this flux we have
where V is the volume of the nebula. The quantities are related to the emission coefficients of hydrogen and neutral helium and are determined by recombinations and free-free transitions. The particle density in the giant's wind is a function of the distance to the center and can be expressed via the continuity equation , where is the mass-loss rate; - the wind velocity, km s-1 (Mikolajewska et al. 1995) and - the mean molecular weight, (Nussbaumer & Vogel 1987). The inner boundary of the region of integration is the radius of the star and the outer one - infinity. It is also necessary to have the quantities . The position of the U photometric system is close to the Balmer limit, and the spectral observations in this region of Tomova & Tomov (1999) show another characteristic feature of the emission of AG Dra: the blending of the Balmer lines with high numbers produces an apparent continuum longward of the Balmer limit near 3650 - 3660 Å which has the same flux as the Balmer continuum excess shortward of 3650 Å. For this reason we used the values of the emission coefficients on the short wavelength side (Osterbrock 1974; Pottasch 1984). We adopt an electron temperature of 15 000 K as proposed by Mikolajewska et al. (1995) and helium abundance of 0.1 (Vogel & Nussbaumer 1994). Solving the system of equations, with the adopted values of the mass-loss rate the distance is obtained to be in the range 15601810 pc, and the stellar radius - in the range 2832 . This result shows that the size of the giant star is small compared with its Roche lobe.
Our estimates can be compared with the estimates, based on other methods. For example Smith et al. (1996) studying the IR absorption spectrum and performing an abundance analysis of the giant, concluded that its bolometric magnitude and radius are in the intervals and 50 . Mikolajewska et al. (1995) came to the conclusion that the distance is 2.5 kpc. On the other hand the observational data of HIPPARCOS satellite provided a lower limit to the distance of about 1 kpc (Viotti et al. 1997).
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001