| Abstract
| - In radiative layers of rotating stars the luminosity carried by circulation currents through a surface of constant entropy (circulation luminosity) is shown to be positive. The corresponding decrease in the temperature gradient is important in the secondary of contact binaries. This result removes the deadlock in the theory of contact binaries.
The resulting treatment of contact binaries is investigated, assuming thermal equilibrium. The sources of the circulation luminosity in the secondary can be written as the product of a circulation function (a normalized non-negative function of the fractional mass) and an amplitude. If the amplitude is adjusted to give a prescribed temperature difference $\Delta T_{\rm e}=T_{{\rm e}1}-T_{{\rm e}2}$, the choice of the circulation function is (in a broad range) unimportant. This invariance extends in a close approximation to all observable properties as well as to the internal structure. The temperature difference is bound to be positive. The fractional extent of radiative regions is larger in the secondary than in the primary. In the course of evolution the period increases and the mass ratio decreases. Comparing thermodynamic quantites on level surfaces, pressure and density are larger in the secondary than in the primary. The specific entropy is larger in the primary than in the secondary. The temperature difference is remarkably small and almost vanishing when averaged over the level surfaces occupied in both components.
The only free parameter (apart from $\Delta T_{\rm e}$) is the efficiency fE of the energy transfer from the primary to the secondary. Using standard values for the parameters, a survey of unevolved and evolved contact configurations is presented. Observational tests are passed. In stable systems the degree of contact is small. Stable systems in the period-colour diagram, unevolved and evolved, cover the strip (and only the strip) of observed systems in this diagram. Lower limits for period and effective temperature, compatible with the observed limits, are caused by the requirement of thermal stability. Stable systems with mass ratios very close to unity are possible, in accordance with recent observations. Since stability considerations are essential in these observational tests the results support the assumption of thermal equilibrium as well as the treatment of the stability problem.
Models for individual observed systems with reliable data are well-determined (apart from some freedom in $\Delta T_{\rm e}$) and can be used to calibrate the efficiency and to determine metallicity and age. All models obtained so far are stable. This again supports the assumption of thermal stability. The results show that evolutionary effects are important and that the efficiency is very small ( $f_{\rm E}=10^{-3}\ldots 10^{-5}$).
Arguments are presented that the velocity field in the common envelope has a reversing layer, with motions from the secondary to the primary in the layers just above the critical surface and from the primary to the secondary in the surface layers.
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