6 Spectral type and orbital period

For M giants, the spectral type is strongly correlated with mass loss, temperature, and radius (see e.g. Reimers 1975; Dyck et al. 1996; Dumm & Schild 1998). Stellar radius and mass loss are known to be key parameters for triggering interaction phenomena in binaries. As we have now a large sample of accurate spectral types for the cool giants in symbiotic systems at hand we can search for correlations with binary parameters and outburst properties.

We searched the literature for orbital parameters. We found orbital periods for about 30 systems whereas additional parameters are less frequently determined. The periods are listed in Table 6. About half of the periods are determined from radial velocity measurements, the other half is deduced from periodic brightness variations that are supposed to be due to the binary revolution (see also Mikoajewska 1997).

  

Table 6: Published orbital periods of symbiotic systems. The activity classes are labeled in the following way: ClSS = Z And type outbursts (classical symbiotic variables); SyNe = symbiotic novae; SyRNe = recurrent novae; stable = no strong outburst is recorded (to our knowledge)

  

Figure 7: Orbital periods as a function of the spectral type of the cool binary components. The symbols denote different activity classes: recurrent novae ($\times$), variables with Z And type outbursts ($\triangle$), symbiotic novae ($\diamondsuit$), and objects without recorded strong outbursts ($\square$). The full line markes the period limit discussed in the text, the dashed line connects the relatively widest systems

In Fig. 7 we plot the periods from Table 6 versus the respective spectral types from Table 5. The shortest period system, at the lower right corner of the plot, is the symbiotic recurrent nova T CrB. T CrB is an extra-ordinary symbiotic, and we will discuss it further below. Neglecting this object for the moment, we discover that the systems that contain an M giant cluster in Fig. 7 on a straight line. While there are some system above this line, there is none far below. Therefore, the line represents a minimum orbital period, $P_{\rm min}$, for symbiotic systems of a given spectral type. The line that is drawn in the figure corresponds to the formula

$$P_{\rm min}=1.31^S\cdot190~{\rm d}$$

where S stands for the M subtype (e.g. S=3 for spectral type M3).

This limit can be interpreted in terms of binary geometry, because the spectral type is related to the radius R of the giant and the period is related to the binary separation:

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Radii of cool M giants of different spectral subtypes are given by Dumm & Schild (1998). We adopted their respective (median) values.

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The orbital period is related to the binary separation through Kepler's third law. Instead of the binary separation we computed the distance, $\ell_1$, from the center of the cool star to the inner Lagrangian point L₁. For this we had to adopt a circular orbit and masses for both stars. We adopted $M_{\rm wd}=0.6~\mbox{\,$M_{\odot}$}$ and $M_{\rm rg}=1.4~\mbox{\,$M_{\odot}$}$ for the hot and the cool components respectively. These are typical values (see e.g. Schmutz et al. 1994; Schild et al. 1996). An increase in the total mass would have the effect of shifting $\ell_1$ to slightly larger values: doubling the total mass, results in an increase of $\ell_1$ by $\Delta\log(\ell_1)=0.1$. Reversing the mass ratio has approximately the same effect, but in the opposite direction.

In Fig. 8 we show $\ell_1$ vs. R for the systems of spectral type M. The diagram looks still similar to Fig. 7 (except for the confinement to spectral class M). The three dashed lines indicate the locii $\ell_1=R$, $\ell_1=2\cdot R$, and $\ell_1=4\cdot R$. The systems clustering at the period limit are now located close to $\ell_1=2\cdot R$, with only T CrB far below. Hence, the limit evidently corresponds to a configuration where the red giant's photosphere reaches about half way out to L₁:

$$R\ \raisebox{-0.4ex}{$\stackrel{<}{\scriptstyle \sim}$}\ \frac12\cdot\ell_1 .$$

We therefore believe that the period limit is in fact a restriction to the radius of the red giant. Under these conditions the system should be well detached. If the photosphere of the cool giant further approaches the Lagrangian critical surface the symbiotic phenomenon either disappears or becomes short living. We speculate that dynamical mass exchange would start, leading to an object that is no more regarded as a symbiotic star.

  

Figure 8: The distance of the inner Lagrangian point, $\ell_1$, versus the cool star's radius, R, as determined from the spectral type, for M-type systems. Symbols are as in Fig. 7. The dotted lines denote $\ell_1=R$, $\ell_1=2\cdot R$, and $\ell_1=4\cdot R$,respectively

On the other hand, the cool star in T CrB, according to Fig. 8, fills its Roche lobe. This interpretation fits perfectly with its periodic light variations that are attributed to the ellipsoidal shape of the red giant due to the tidal distortions caused by the companion (e.g. Belczynski & Mikoajewska 1998). Strong ellipsoidal light variations are rarely observed for symbiotic stars. T CrB is a peculiar symbiotic, in that it belongs at the same time also to the class of recurrent novae with fast, large amplitude outbursts. It differs also from other symbiotics by the fact that the hot companion is more massive than the cool giant. According to current mass transfer theories, this property prevents dynamically unstable Roche lobe overflow from the cool giant (see Belczynski & Mikoajewska 1998). A similar but much weaker ellipticity effect has also been found in the light curve of EG And (Wilson & Vaccaro 1997). This seems to suggest that $R\gt\ell_1/2$ for this object (e.g. $R\approx 0.8\ell_1$ when adopting a large inclination as expected for an eclipsing system) in contradiction with Fig. 8. However, as discussed in Wilson & Vaccaro (1997) such a large radius for the red giant is in contradiction with current estimates for the distance of the EG And system. One possible solution of this problem could be, that radiation pressure lowers the effective surface gravity of the red giant, so that the tidal distortions are significantly enhanced and could therefore already be visible for $R\approx \ell/2$ (see e.g. Drechsel et al. 1995).

There exist systems with $\ell_1\gg2\cdot R$,but the symbiotic phenomena are possibly less pronounced unless the wider separation is compensated by another property (e.g. a higher luminosity of the white dwarf). In terms of orbital period, the widest systems lie on the dashed line in Fig. 7. This line corresponds to a period $P_{\rm max}=1.20^S\cdot1660~{\rm d}$.

There is possibly a tendency for the classical symbiotic variables (Z And type) to cluster closer to the $P_{\rm min}$ limiting ratio than the symbiotic novae and those objects with less prominent outburst activity.