5 Comprehensive characteristics of the systems

Since the physical parameters of the binaries listed in Table 1 are known the present analysis of the orbital period changes allows for a search for possible general relations between the system parameters and the period changes.

The so called r-q diagram, although used just for assessment of the properties of the accretion disks (e.g. Peters 1989), plays an even more important role since it contains information not only about the fate of the transferred matter, but also about the evolutionary status of the binary, tidal effects and the influence of the mass stream on the gainer. This diagram displays $r=R_{\rm g}/a$, which is the radius of the gainer $R_{\rm g}$expressed in terms of fraction of the orbital separation a, versus the mass ratio q. Position of a system in this diagram allows to determine the fate of the stream of the transferred matter in the ballistic approximation. As Lubow & Shu (1975) calculated two curves can be plotted. The curve $\omega$$_{\rm min}$ denotes the minimum distance of the infalling stream from the center of the gainer. A permanent disk can develop in systems lying below this curve because the stream completely misses the gainer. The curve $\omega_{\rm d}$ gives the radius of the disk which is formed from the infalling matter. Systems lying between $\omega$$_{\rm min}$ and $\omega_{\rm d}$can have only transient disks which are mixture of the matter of the stream and the photosphere splashed by impact of the infalling stream. On the other hand, the impact of the stream onto the gainer is almost tangential to its photosphere in systems with large r. Generally, the higher above the curve $\omega$$_{\rm min}$ the system lies, the more difficult conditions for development of the disk. Also the interaction of the gainer with the stream becomes stronger in systems with larger r.

All systems analysed here are plotted in the r-q diagram in Fig. 17a. Different symbols were used to resolve the systems with variable periods from those which periods can be regarded as constant during the whole interval of observations. It can be seen that all systems lie above the curve $\omega$$_{\rm min}$, i.e. the stream collides at least partly with the gainer. Moreover, most of these binaries (thirteen of sixteen) are situated inside the area of direct impactors above $\omega_{\rm d}$ and some even have gainer almost filling its lobe. These facts suggest that the tidal interaction is appreciable in most binaries of this set and also the gainers are strongly affected by the impact of the transferred matter.

  

Figure 17: The positions of the analysed binaries in the r-q diagram a) and in the P-q diagram b). The dot-dashed line marks the equivalent radius of the Roche lobe as a fraction of the orbital separation in a) and the meaning of the curves $\omega$$_{\rm min}$ and $\omega_{\rm d}$ is explained in the text. Systems with variable periods are marked by filled circles while empty circles denote binaries where no period change was observed during the whole interval covered by the data. Unsure cases are marked by crosses. See the text for details

Similar diagram can be seen in Fig. 17b where the mass ratio is plotted versus the orbital period. The same symbols as in Fig. 17a were used. Most systems with variable periods occupy the upper-left corner of the P-q diagram. This fact may be to some extent caused by the selection effect since the binaries with longer periods have larger orbital separation and the probability of observing deep eclipses is generally lower. Notice that there is a well visible tendency for the large mass ratios to occur in systems with the shortest periods.

The absolute value of the rate of the period change $\Delta$P/P plotted versus the mass ratio q can be seen in Fig. 18. Only those systems where the stability of the period or its change could be determined with a good degree of accuracy were plotted. Some trends can be distinguished here with the help of the r-q and P-q diagrams: (a) most systems (six of eight) with P<7 days ($R_{\rm g}/a\gt\omega_{\rm d}$)and with q>0.4 display variable period within $\vert\Delta$$P/P\vert~= 1.2\ 10^{-10}$ days$^{\rm -1}$ to $3.22\ 10^{-9}$ days$^{\rm -1}$;(b) periods of binaries with P<7 days ($R_{\rm g}/a\gt\omega_{\rm d}$) but with q<0.4 can be considered constant on the time scale of several decades; (c) two systems ($\beta$ Lyr and RY Sct) with $\omega$$_{\rm min} <R_{\rm g}/a <\omega_{\rm d}$ (P>11 days) and with q<0.4 display extremely large period changes; although the parameters of AQ Cas are similar to these two systems the absence of period change moves it to group (b). We admit that the number of binaries used is not high but as we noted above the limitation is given by the requirement of both a good model and sufficient coverage by the timings for each system.

  

Figure 18: Dependence of the rate of the period change $\Delta$P/P (in days$^{\rm -1}$) on the mass ratio q. Absolute value of the period change was used and the systems which have $\Delta$P/P<0 are marked by circled crosses. See the text for details

The loss of matter from the loser leads to considerable changes of its parameters through the epoch of the mass transfer. The status of the loser and the degree of progress of the mass exchange can be illustrated by the position of this star in the mass-luminosity diagram (Fig. 19). The mass of the loser decreases in the course of the epoch of the outflow of matter. The star therefore moves to the left (towards smaller masses) in this diagram. The luminosity initially decreases in the phase of the rapid mass transfer but begins to grow again as the system enters the more advanced evolutionary stadium in which the mass transfer rate slows down. Luminosity of each loser was calculated from its radius and $T_{\rm eff}$ given in Table 1. In the case of IZ Per the parameters of the loser could be determined only from a combination of the statistical mass of the primary for its spectral type (taken from Harmanec 1988) and the light curve solution by Wolf & West (1993). The mass and luminosity of the loser in IZ Per must be therefore taken with caution but even an error of 50% would not significantly alter the position of this star in Fig. 19.

  

Figure 19: Positions of the losers in the mass-luminosity diagram. The systems with constant periods are plotted as open circles lying in the xy plane. Filled circles denote the binaries with variable periods and are shifted from the xy plane by the distance which corresponds to the logarithm of absolute value of $\Delta$P/P (in days$^{\rm -1}$), plotted on the vertical (z) axis. Notice the prevailing location of the binaries with constant periods in the lower-left corner. The curve labeled as EMT marks the end of mass transfer for case B given by De Greve (1993)