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Significant discoveries in X-ray astronomy made over the last two decades have stimulated astronomers to seek particular evolutionary explanations for each type of observational appearance of WD, NS and BH. Taken as a whole, these explanations constitute a general evolutionary scheme which is known as the ``evolutionary scenario''. We follow the basic ideas on stellar evolution to describe the evolution of binaries both with normal and compact companions based upon the original ideas that appeared in the papers by Paczynski (1971), Tutukov and Yungelson (1973) and van den Heuvel and Heise (1972) (see recent review by van den Heuvel, 1994).
To avoid extensive numerical calculations in the statistical simulations, we treat the continuous evolution of each binary component as a sequence of a finite number of basic evolutionary states (for example, main sequence, red (super)giant, helium (Wolf-Rayet) star, hot WD), in which stellar parameters significantly differ from each other. The evolutionary state of the entire binary can be thus determined as a combination of the states of each component, and alters once the faster evolving component goes into the next state.
Figure 1: Modes of the first mass transfer as defined by Webbink (1979). 
At each such stage we assume that the star does not change its physical parameters (mass, radius, luminosity, rate and velocity of stellar wind, etc.) which effect the evolution of its companion (especially in the case of compact magnetized stars). Every time the faster evolving component goes over into the next stage we recalculate its parameters. Depending on the evolutionary stage, the state of the slower evolving star is changed accordingly or may remain the same. With some exceptions (such as the common envelope stage and supernova explosion), no simultaneous change of state for both components can occur. Whenever possible we use analytical approximations for all parameters found in the literature.
Prior to describing the main evolutionary states of the normal component, we note that unlike for a single star, the evolution of a binary component is not solely determined by the initial mass and chemical composition solely. This is due to the well-known fact that the primary can fill its Roche lobe either when still on the main sequence, or having a well-developed (degenerate) core comprising helium, carbon or heavier elements, etc. This in turn determines the rate at which it overflows the Roche lobe during the first mass transfer to the secondary, and the type of remnant left behind. We follow Webbink (1979) in treating the first mass exchange modes for normal binary components, which account for the physical state of the star, in more detail than the simple types of mass exchanged (A, B, C introduced by Kippenhahn and Weigert, 1967). We use Kippenhahn's notations A, B, C and D (for very wide systems with independently evolving companions) for evolutionary types of binary as a whole, and Webbink's designations for mass exchange modes for each component [Ia], [Ib], [IIa], [IIb], [IIIa], etc. (Figure 1).
We consider binaries with and constant chemical (solar) composition. The process of mass transfer between components is treated as conservative when appropriate, that is the total angular momentum of the binary system is treated as constant. If the accretion rate from one star to another is sufficiently high (say, the mass transfer occurs on a time-scale a few times shorter than the thermal Kelvin-Helmholz time for a normal companion) or a compact object is engulfed by the giant atmosphere of the companion, the common envelope (CE) stage of binary evolution can begin (see Paczynsky, 1976; van den Heuvel, 1983).
Other cases of non-conservative evolution (for example, stages with strong stellar wind or those where a loss of binary angular momentum occurs due to gravitational radiation or magnetic stellar wind) are treated using well-known prescriptions (see, e.g. Verbunt and Zwaan, 1981; Rappaport et al., 1982; Lipunov and Postnov, 1987b,c, 1988[112, 113, 116]).
Figure 2: Schematic representation of the typical evolutionary track of a massive binary system with normal and compact star stages indicated.