## Appendix A: mass transfer models with extreme mass ratiosThe results of Sects. 2.1and 2.2are examined here in the limiting
cases of In considering extreme values of the mass ratio, one makes the
reduced mass approximation, regarding all the mass as residing in one
star of fixed position and all the angular momentum in the other,
orbiting star. Errors are only of order Retention of the factors
It is now straightforward to compare the two models of mass transfer and loss in the extreme limits where they should be equivalent. Comparisons will be made first in the limit where the donor is a test-mass, where the unified and ring models correspond exactly. Second will be a treatment of the opposite limit, where the two models give different but reconcilable results. First, the wind and ring models are equivalent in the
limit, upon identification of
with and More interesting is that, in the strict limit, the parameters and are found only in the combinations and . The independence from other combinations of parameters can be understood by examining the ratio which tells the relative importance of angular momentum and mass
losses in the orbit's evolution. For small donor mass,
is small and mass loss without angular
momentum loss is unimportant. The strict mass loss term will be
important only when the coefficient of the
term, or , is of order
Even neglecting the questions of stability important to tidally
induced mass transfer, the situation is different when the mass-losing
star has all the mass and almost no angular momentum. In this case,
both mass and angular momentum loss are important. Since the more
massive body is the mass donor, a non-negligible fraction of the total
mass of the system may be ejected. Furthermore, the angular momentum
per reduced mass changes via isotropic re-emission and ring formation.
Therefore, both mass and angular momentum loss play a rôle in
the dynamics. A measure of relative importance is
, which in the limit of large The evolution of the angular momentum per reduced mass shows a significant difference the winds and ring models in their limit. In the wind case, one may write . The first part is the fractional rate of increase of , with loss of mass from the donor star. The second is the rate at which angular momentum is lost from the system, by isotropic re-emission. Since , the mass losing star is nearly stationary and mass lost in a direct wind removes no angular momentum. The ring + accretion case is slightly different, but may be
interpreted similarly. The term replaces
as the fraction of mass accreted onto the
first star; i.e.: . The term
is replaced by , as the
rate of loss of angular momentum. Although the donor star is
stationary in this ring case, angular momentum
is still lost. This is due to the particular construction of the ring
model, in which the angular momentum removed is proportional to the
system's , not to the specific angular momentum
of either body. Thus, in the ring model, even as
tends towards zero with increasing One might also notice for both models, that if angular momentum
loss is not overly efficient ( ## 1. Jeans's modesOften, one talks of the Jeans's mode of mass loss from a binary, in which there is a fast, sperically symmetric loss of mass. The Jeans' mode has two limits. One is a catastrophic and instantaneous loss of mass, as in a supernova event (see van den Heuvel (1994 ), for a discussion). In this case, if the orbit is initially circular and more than half the total mass is lost in the explosion, the system unbinds. This can be explained by the energetics. Initially, the system is virialised with . The loss of mass does not change the orbital velocities, so the kinetic energy per unit mass remains the same. The potential energy per unit mass is proportional to the total mass, so if more than half the mass is lost, and the orbit is unbound. A more detailed analysis may be done, and will give ratios of initial to final orbital periods and semi-major axes, for given initial to final mass ratios (Blaauw (1961 ); Flannery & van den Heuvel (1975 )). The equations in this paper will not give the 'standard' Jeans
solution and unbound orbits. Unbinding the orbit from elliptical to
hyperbolic requires , where
has been assumed from the outset. The argument
used in the preceding paragraph seemingly necessitates the unbinding
of the orbit with sufficient mass loss, but it is not applicable here,
as it also assumes a conservation of mechanical energy. Mechanical
energy is not conserved in the above calculations (Sect. (2.1), for
example), as the presence of dissipative forces to damp The other limit of Jeans's mode is mass loss by a fast wind, on a timescale slow compared to the orbital period. In this case, there is no preferred orientation for the Runge-Lenz vector (direction of semimajor axis in an eccentric orbit), and the orbit remains circular throughout the mass loss, with . This is the limit of the Jeans mode which our calculations reproduce. Jeans's mode is an example of a degenerate case (mentioned above; here, ), where . In this case, one may simply apply Eq. (3), and see that . ## Appendix B: extensions to models considered aboveIn the interest of completeness, a five-parameter model of mass transfer, combining winds from both stars, ring formation, and accretion is presented. The treatment in Sects. 2and 3.3is followed. Modifications necessary for inclusion of other forms of angular momentum loss, such as , due to gravitational radiation reaction, are also discussed. Construction of this model is straightforward, and results from the
inclusion of the various sinks of mass and angular momentum, due to
various processes. The nonconservative part of each model makes its
own contribution to the logarithmic derivatives of
and where each of the variables retains its old meaning. Replacing the old definition of in Eq. (9) with the new definition in Eq. (B1) makes Eqs. (10) - (13) applicable to this model, as well. Contributions from other evolutionary processes, such as gravitational wave radiation reaction; realistically prescribed stellar winds; tidal evolution, et c. can also be added. Each will make its own contribution to the angular momentum and total mass loss. For example, orbital decay by gravitational radiation reaction (Landau & Lifshitz, 1951 ) can be included as an other sink of angular momentum: In most cases, this type of physics can be modelled as an intrinsic
(the second term in Eq. (69)). Like the
intrinsic stellar expansion term of Eq. (68), this kind of evolution
occurs even in the absence of mass transfer. Therefore, the convenient
change of variables from We temporarily neglect these complications and consider mass transfer via isotropic wind, isotropic re-emission, and formation of a ring, with mass fractions , , and respectively. The remainder of the mass transfer (the fraction ) goes into accretion. The ratio is , where . Note that , , , and are all used as before; should still be regarded as the accreted fraction. Taking , = 0, gives formulae for a ring of strength : Where the relevant exponents are functions of the parameters and : It is also instructive to examine the model in the degenerate cases of and , where the functional forms change. When there is no accretion (), the standard becomes singular, while the term approaches 1. Defining the singular part of :
the equations governing binary evolution can be rewritten, for the case when no material is accreted: It might be worth noting that the above considerations are irrelevant for the pure models, where , , and vanish. There is no profound reason for this. In the case where all material is accreted (), there are seeming singularities in the coeficients and . Proper solution of the equations of evolution in this case, or setting before taking limiting values of the coeficients and , shows that there is no problem at all. In this case, the equations reduce to Eqs. (32), (33), and: © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |