Astron. Astrophys. 344, 607-613 (1999)

4. The wind structure of SY Mus

The column densities of neutral hydrogen determined as function of the orbital phase from the observed spectra, allow to investigate the cool wind structure of the giant component. This was first shown by V91 in a pioneer study of the symbiotic binary EG And. The analysis of the observed column densities led Vogel to determine an empirical velocity law for the cool wind in that system. The law turned out to be remarkable different from the "-laws" usually employed to describe stellar winds. In fact, it is characterized by low velocity values near the giant component followed by a very rapid increase at a distance of approximately 3 giant's stellar radii. The emission lines in the UV spectra of SY Mus are evidence that some portion of the giant's stellar wind is ionized by the radiation stemming from the hot component. The modeling of EG And by V91 indicated nevertheless that the assumption of wind neutrality is a fairly good first approximation for the study of the red giant's wind. Assuming further that the wind is spherically symmetric and that the eccentricity of the system is negligible (Schmutz et al., 1994) one can write for the column density

where b is the impact parameter, or projected separation between the components, and is the wind velocity at distance r from the giant's center. In this equation b and r are adimensional quantities, expressed in units of the giant's radius R. Similarly the velocity is expressed in units of the terminal velocity . The dimension of the parameter a, also called parameter of the system, given by

where is the mass-loss rate of the red giant and µm_H is the average molecular weight, determines therefore the physical dimensions of the hydrogen column density, (b).

The impact parameter, b, is related to the inclination angle i of the orbital plane and the phase by the formula

p being the separation (in units of R) between the components of the binary. For SY Mus we employ p=4.3 (Schmutz et al., 1994) and i=95.8 (Harries & Howarth, 1996a). In order to find the velocity from the observed column densities , Eq. (4) has to be inverted. This is an integral equation of Fredholm's type but it can be brought to Abel's type by means of suitable transformations. Such inversion problems are frequently ill-posed and demand special methods for their investigation. Specifically, the empirical determination of stellar wind velocity by means of Eq. (4) has been discussed by Kessler (1991) and KDV93 who have developed procedures for its inversion. In Kessler (1991) the sought-for function (the "source function") is approximated by a piecewise linear function and the velocity computed numerically for each value of the column density (the "data function"). In KDV93 the inversion is accomplished through explicit diagonalization of the Abel's operator. In this method the "data function" is represented by a Taylor expansion and the velocity field is then obtained from the inverted Abel's operator. We shall employ both approaches to determine the cool wind velocity profile of SY Mus and the resulting column densities to be compared with the observations.

For the velocity obtained from the numerical inversion of the observed column densities we have found that a convenient analytical description is provided by the expression

where and are the complete and incomplete gamma functions respectively and are given by

and

For a fixed value of the parameter of the system a, the parameters A and B in Eq. (7) can be chosen so as to obtain the best description of the velocity. The parameters A and B fix also the coordinates (r₀, v₀) of the inflection point:

In Fig. 4 we show the velocity profile obtained in this way with A=14.20 and B=3.383. Of course, it must be remembered that this behavior is an extrapolation due to the adopted form of the function and to its fitting to the observed points in the region .

Fig. 4. Wind velocity profile from Eq. (7) with A=14.2 and B=3.383 (solid line). Some profiles according to the -law (dashed lines) are also shown for comparison.

With the value a=210²⁰ cm^-2 of the parameter of the system, this law translates in column densities which best match the observations (Fig. 5).

Fig. 5. Observed column densities as a function of the impact parameter. The curve represents the computed column densities using the velocity profile given formula (7) (solid line), using Vogel's law (dotted line) and using formula (13) (dashed line).

We have compared the column densities furnished by the velocity profile (Eq. 7) with those resulting from Vogel's law (V91), first applied to investigate the wind structure of the symbiotic system EG And,

In Fig. 5, we plot the column densities implied by Vogel's function with parameters m=11, r₀=3.75, v₀=0.2 and a=410²⁰ cm^-2.

The agreement between the curves is quite good particularly for large values of impact parameter where they practically coincide.

Following KDV93 we consider now the finite Taylor expansion

and best-fit this expression to the observed data. The inversion method of KDV93 permits then to invert the Abel's operator and to determine the corresponding Taylor expansion for the "source function" from which the velocity field can be reconstructed. We remark however, that the column density is a decreasing function of the impact parameter b, asymptotically tending to the value which implies a non-decreasing velocity profile converging to a constant terminal value. The Knill method requires that the observational points cover all the region until this terminal value is reached, in order to avoid a random behavior of the polynomial outside this observational region. This is not our case, in view of the restricted range of the observations. In fact we did not find any best-fitting polynomial through all the points and having at the same time the required behavior of the wind velocity profile. Neverthelees, including in the fitting only those points giving such a behavior we can get an upper bound for the parameter a. The polynomial (N=4) that we obtained fits the four inner points plus the point at b=2.658 with the lowest possible value of that guarantees that the polymonial does not have any positive root. This polynomial is shown in Fig. 5 and the bound given by it is a=6.010²⁰ cm^-2.

The relation 5 shows that the parameter of the system a is related to important physical quantities such as the wind terminal velocity, and the mass-loss rate of the red giant. Assuming a wind terminal velocity =30 km s^-1, a giant radius of 86 (Schmutz et al., 1994) and making use of the values of the parameter a from our modeling, we obtain for the mass-loss rate: =0.8310^-9 /yr for the velocity profile given by expression (7); =1.710^-9 /yr for Vogel's law as applied to SY Mus. The upper limit of the parameter a results in the bound =2.510^-9 /yr for the mass-loss rate. The models by Harries & Howarth (1996) of Raman scattering in symbiotic stars suggest a terminal velocity of about 60 km s^-1 for SY Mus. This would give =1.610^-9 /yr if we use formula (7). We notice that for the assumed terminal velocities the estimated the mass-loss rate is rather low if compared with most of the S-type symbiotic systems with radio measurements (see Seaquist & Taylor 1990). This is especially the case for the first two models from which it was possible to obtain simultaneously an estimate of the parameter a and a good matching of the observed neutral hydrogen column densities.

© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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