3.2 Mass transfer modes and mass loss in binary systems

GW emission is the sole factor responsible for the change of orbital parameters of a pair of compact (degenerate) stars. However, at the early stages of binary evolution, it is the mass transfer between the components and the loss of matter and its orbital momentum that play a dominant dynamical role. Strictly speaking, these processes should be treated hydro-dynamically and they require complicated numerical calculations. However, binary evolution can also be described semiqualitatively, using a simplified description in terms of point-like bodies. The change of their integrated physical quantities, such as masses, orbital angular momentum, etc. governs the evolution of the orbit. This description turns out to be successful in reproducing the results of more rigorous numerical calculations (see, e.g., [295] for more detail and references). In this approach, the key role is allocated to the total orbital angular momentum Jorb of the binary.

Let star 2 lose matter at a rate ˙ M2 < 0 and let β (0 ≤ β ≤ 1) be the fraction of the ejected matter which leaves the system (the rest falls on the first star), i.e. M˙1 = − (1 − β )M˙2 ≥ 0. Consider circular orbits with orbital angular momentum given by Equation (14View Equation). Differentiate both parts of Equation (14View Equation) by time t and exclude dΩ ∕dt with the help of Kepler’s third law (7View Equation). This gives us the rate of change of the orbital separation:

˙a ( M β M ) M˙ J˙ --= − 2 1 + (β − 1) --2− ----2- --2-+ 2 -orb-. (30 ) a M1 2 M M2 Jorb
In the previous equation ˙a and M˙ are not independent variables if the donor fills its Roche lobe. One defines the mass transfer as conservative if both β = 0 and ˙ Jorb = 0. 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:

  1. conservative mass transfer,
  2. non-conservative Jeans mode of mass loss (or fast wind mode),
  3. non-conservative isotropic re-emission,
  4. sudden mass loss from one of the component during supernova explosion, and
  5. 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.

3.2.1 Conservative accretion

In the case of conservative accretion, matter from M2 is fully deposited onto M1. The transfer process preserves the total mass (β = 0) and the orbital angular momentum of the system. It follows from Equation (30View Equation) that

√ -- M1M2 a = const,

so that the initial and final binary separations are related as

af ( M1i M2i)2 -- = -------- . (31 ) ai M1f M2f
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.

3.2.2 The Jeans (fast wind) mode

In this mode the ejected matter completely escapes from the system, that is, β = 1. 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 J2 = (M1 ∕M )Jorb (we neglect a possible proper rotation of the star, see [425]). For the loss of orbital momentum J˙ orb it is reasonable to take

M2˙ ˙Jorb = ----J2. (32 ) M2
In the case β = 1, Equation (30View Equation) can be written as
(Ω˙a2) ˙Jorb M1 M˙2 ---2--= ----− ------. (33 ) Ωa Jorb M M2
Then Equation (33View Equation) in conjunction with Equation (32View Equation) gives Ωa2 = const, that is, √ ------ GaM = const. Thus, as a result of such a mass loss, the change in orbital separation is
af Mi --= ---. (34 ) ai Mf
Since the total mass decreases, the orbit always widens.

3.2.3 Isotropic re-emission

The matter lost by star 2 can first accrete to star 1, and then, a fraction β 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 [236119] 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 M1, rather than by the orbital momentum of the mass-losing star M2. The orbital momentum loss can be written as

˙ ˙J = β M2-J , (35 ) orb M1 1
where J = (M ∕M )J 1 2 orb is the orbital momentum of the star M 1. In the limiting case when all the mass attracted by M1 is expelled from the system, β = 1, Equation (35View Equation) simplifies to
J˙orb M˙2M2 ---- = -------. (36 ) Jorb M1M
After substitution of this formula into Equation (30View Equation) and integration over time, one arrives at
( )2 ( ) af= Mi- M2i- exp − 2 M2i-−-M2f . (37 ) ai Mf M2f M1
The exponential term makes this mode of the mass transfer very sensitive to the components mass ratio. If M1 ∕M2 ≪ 1, the separation a between the stars may decrease so much that the approximation of point masses becomes invalid. The tidal orbital instability (Darwin instability) may set in, and the compact star may start spiraling toward the companion star center (the common envelope stage; see Section 3.5 below).
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