Let star 2 lose matter at a rate and let be the fraction of the ejected matter which leaves the system (the rest falls on the first star), i.e. . Consider circular orbits with orbital angular momentum given by Equation (14). Differentiate both parts of Equation (14) by time and exclude with the help of Kepler’s third law (7). This gives us the rate of change of the orbital separation:
It is important to distinguish some specific cases (modes) of mass transfer:
As specific cases of angular momentum loss we consider GW emission (see Section 3.1.3 and 3.1.4) and a magnetically coupled stellar wind (see Section 3.1.5) which drive the orbital evolution for short-period binaries. For non-conservative modes, one can also introduce some sub-cases, such as, for example, a ring-like mode in which a circumbinary ring of expelled matter is being formed (see, e.g., ). Here, we will not go into the details of such sub-cases.
In the case of conservative accretion, matter from is fully deposited onto . The transfer process preserves the total mass () and the orbital angular momentum of the system. It follows from Equation (30) that
so that the initial and final binary separations are related as
In this mode the ejected matter completely escapes from the system, that is, . The escape of matter can take place either in a spherically symmetric way or in the form of bipolar jets moving from the system at high velocity. In both cases, matter carries away some amount of the total orbital momentum proportional to the orbital angular momentum of the mass losing star (we neglect a possible proper rotation of the star, see ). For the loss of orbital momentum it is reasonable to take
The matter lost by star 2 can first accrete to star 1, and then, a fraction of the accreted matter, can be expelled from the system. This happens, for instance, when a massive star transfers matter to a compact star on the thermal timescale ( 106 years). The accretion luminosity may exceed the Eddington luminosity limit, and the radiation pressure pushes the infalling matter away from the system, in a manner similar to the spectacular example of the SS 433 binary system. Another examples may be systems with helium stars transferring mass onto relativistic objects [236, 119] or hot white dwarf components of cataclysmic variables losing mass by optically thick winds . The same algorithm may be applied to the time-averaged mass loss from novae . In this mode of mass-transfer, the binary orbital momentum carried away by the expelled matter is determined by the orbital momentum of the accreting star , rather than by the orbital momentum of the mass-losing star . The orbital momentum loss can be written as
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