## 2. Mass transfer and the evolution of orbital parametersIn this section, expressions for the variation of orbital parameters with loss of mass from one of the stars are derived. In what follows, the two stars will be referred to as and , with the latter the mass losing star. A binary composed of two stars with radii of gyration much less
than the semimajor axis where the period Here, the reduced mass , the mass ratio
, and Eqs. (1) and (2) combine to give expressions for the semimajor axis and orbital period, in terms of the masses and angular momentum: There are various modes (or, borrowing from the terminology of nuclear physics, channels) of mass transfer associated with the different paths taken by, and destinations of mass lost from . The details of mass transfer must be considered, in order to calculate the orbital evolution appropriate to each mode. Several of these modes are described here, roughly following Sect. 2.3 of the review article by van den Heuvel (1994 ). The mode which is most often considered is accretion, in which
matter from is deposited onto
. The accreted component (slow mode) conserves
total mass and orbital angular momentum
The second mode considered is Jeans's mode, which van den Heuvel calls, after Huang (1963 ), the fast mode; the third is isotropic re-emission. Jeans's mode is a spherically symmetric outflow from the donor star in the form of a fast wind. The best known example of Jeans's mode: the orbital evolution associated with Type II supernovae in binaries (Blaauw 1961 ) differs markedly from what is examined here. In that case, mass loss is instantaneous. A dynamically adiabatic outflow () is considered here. An interesting variant of Jeans's mode is isotropic re-emission. This is a flow in which matter is transported from the donor star to the vicinity of the accretor, where it is then ejected as a fast, isotropic wind. The distinction between Jeans's mode and isotropic re-emission is important in considerations of angular momentum loss from the binary. Mass lost by the Jeans mode carries the specific angular momentum of the mass loser; isotropically re-emitted matter has the specific angular momentum of the accretor. Wind hydrodynamics are ignored. The fourth case considered is the intermediate mode, or mass loss to a ring. No mechanism for mass loss is hypothesised for this mode. The idea is simply that the ejecta has the high specific angular momentum characteristic of a circumbinary ring. The differences among each of these modes is in the variation of
the quantities In the following sections, formulae for the orbital evolution of a binary under different modes of mass loss are presented and integrated to give expressions for the change in orbital parameters with mass transfer. The different modes are summarised in Table (1). Due to the formality of the following section and the cumbersome nature of the formulae derived, one should try to develop some kind of intuitition, and understanding of the results. The two sets of results (i.e.: one for a combination of winds, the other for ring formation) are examined in the limits of extreme mass ratios, compared to one another, and their differences reconciled in the first Appendix.
## 2.1. Unified wind modelConsider first a model of mass loss which includes isotropic wind, isotropic re-emission, and accretion, in varying strengths. Conservative mass transfer, as well as pure forms of the two isotropic winds described above, can then be recovered as limiting cases by adjusting two parameters of this model: Here, is an infinitesimal mass of wind from star 2 (isotropic wind), and an infinitesimal mass of wind from star 1 (isotropic re-emission). Both amounts are expressed in terms of the differential mass , lost from star 2. A sign convention is chosen such that is positive if the star's mass increases, and is positive if removing matter (from ). Given the above, one can write formulae for the variation of all the masses in the problem: Neglecting accretion onto the stars from the ISM and mass currents originating directly from the accretor, all transferred mass comes from the mass donating star () and , , and the accreted fraction all lie between 0 and 1, with the condition of mass conservation , imposed by Eq. (9). For the remainder of this section, the subscript on will be eschewed. If mass is lost isotropically from a nonrotating star, it carries
no angular momentum in that star's rest frame. In the center of mass
frame, orbital angular momentum If a wind-emitting star is rotating, then it loses spin angular
momentum The above consideration becomes important in the case of a tidally locked star, such as a Roche lobe filling red giant. In this case, if magnetic braking removes spin angular momentum from the star at some enhanced rate, that star will begin to spin asynchronously to the orbit and the companion will establish tides to enforce corotation. These tides are of the same form as those between the oceans and the Moon, which are forcing the length of the day to tend towards that of the month. Strong spin-orbit coupling changes the evolution of orbital angular
momentum in two ways. First, for a given orbital frequency, there is
an extra store of angular momentum, due to the inertia of the star.
Thus, for a given torque, This enhancement will be treated formally, by taking the angular
momentum loss due to the fast mode to occur at a rate Keeping the above discussion in mind, the angular momentum lost from the system, due to winds is as follows: which can be simplified by substituting in expressions for the : So far, the equations have been completely general;
, , and Furthermore, if the enhancement factor where the exponents
Finally, substitution of Eqs. (22), (23), and (24) into Eqs. (3)
and (4) gives expressions for the evolution of the semimajor axis and
orbital period in terms of the changing mass ratio The derivatives of these functions may be evaluated either by logarithmic differentiation of the above expressions (Eqs. (28) and (29)), or by substitution of Eqs. (11), (12), (13), and (19) into Eqs. (5) and (6). Either way, the results are: The reader should be convinced that the above equations are correct. First, and by fiat, they combine to give Kepler's law. Second, they reduce to the familiar conservative results: in the limit. Finally, and again by construction, the formulae are composable. Ratios such as are all of the functional form , so if one forms e.g.: , the result is immediately independent of the arbitrarily chosen intermediate point . Note that the results for isotropic re-emission obtained by Bhattacharya & van den Heuvel (1991 , Eq. (A.6)), and again by van den Heuvel (1994 , Eq. (40)) are not composable, in the sense described above. The correct expression for in the case of isotropic re-emission may be obtained by setting : Also note that Eq. (A.7) of Bhattacharya & van den Heuvel (1991 ) should have its exponential in the numerator, as opposed to the denominator. In van den Heuvel (1994 ), Eq. (37) should read . In Eq. (38), an equals sign should replace the plus. (These corrections were also found by Tauris (1996 )). Thus, if there is mass loss from one star, with constant fractions
of the mass going into isotropic winds from the donor and its
companion, one can express the variation in the binary parameters,
, ## 2.2. Formation of a coplanar ringNow consider a model in which mass is transferred by accretion and ring formation, as described above. For concreteness, follow a standard prescription (cf: van den Heuvel (1994 )) and take the ring's radius, to be a constant multiple of the binary semimajor axis. This effectively sets the angular momentum of the ring material, since for a light ring, Formulae describing orbital evolution can be obtained acording to the prescription of the previous section. If a fraction of the mass lost from is used in the formation of a ring, then Eqs. (9) and (19) should be replaced as follows: Following the procedure used in Sect. (2.1), and with (): Unless the ring is sufficiently wide (), it
will orbit in a rather uneven potential, with time-dependant tidal
forces which are comparable to the central force. In such a potential,
it would likely fragment, and could fall back upon the binary.
Consequently, stability probably requires the ring to be at a radius
of at least a few times © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |