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:

In the previous equation and are not independent variables if the donor fills its Roche lobe. One defines the mass transfer as conservative if both and . The mass transfer is called non-conservative if at least one of these conditions is violated.It is important to distinguish some specific cases (modes) of mass transfer:

- conservative mass transfer,
- non-conservative Jeans mode of mass loss (or fast wind mode),
- non-conservative isotropic re-emission,
- sudden mass loss from one of the component during supernova explosion, and
- common-envelope stage.

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., [381]). 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

The well-known “rule of thumb” for this case says that the orbit shrinks when the more massive component loses matter, and the orbit widens in the opposite situation. During such a mass exchange, the orbital separation passes through a minimum, if the masses become equal in course of mass transfer.

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 [425]). For the loss of orbital momentum it is reasonable to take

In the case , Equation (30) can be written as Then Equation (33) in conjunction with Equation (32) gives , that is, . Thus, as a result of such a mass loss, the change in orbital separation is Since the total mass decreases, the orbit always widens.

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 ( 10^{6} 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 [189]. The same algorithm may be applied to the time-averaged mass loss from
novae [466]. 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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