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Astron. Astrophys. 327, 620-635 (1997)

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Appendix A: mass transfer models with extreme mass ratios

The results of Sects. 2.1and 2.2are examined here in the limiting cases of q tending toward zero and infinity, to explore the relations between the different modes, and the connections to well-known toy models. This is done to develop an intuition which may be used in comparing the stability of mass transfer by the various modes.

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 q (or [FORMULA], if this is small). One might keep the Solar System in mind as a concrete example. The Sun has all the mass, and the total angular momentum is well approximated by Jupiter's orbital angular momentum. The total mass [FORMULA] and the reduced mass [FORMULA], with fractional errors [FORMULA].

Retention of the factors A (Eqs. (19), et seq. of Sect. 2.1) and [FORMULA] (Eq. (38), of Sect. 2.2) allow for broader comparisons between the two models. Formulae appropriate to the two extreme limits are in Table 4. Columns 2 and 3 pertain to the wind models; 4 and 5 to ring models.


Table 4. Formulae describing orbital evolution in the limits of extreme mass ratios, [FORMULA]

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 [FORMULA] limit, upon identification of [FORMULA] with [FORMULA] and A with [FORMULA] 6. This symmetry is not difficult to understand. In the [FORMULA] limit, only the direct isotropic wind and ring remove angular momentum. In this case, isotropic re-emission is an isotropic wind from a stationary source, and accretion is always conservative of both mass and angular momentum. In each torquing case (direct wind and ring), the ejected mass removes specific angular momentum at an enhanced rate - A in the winds model, [FORMULA] in the ring model. Thus, it makes sense that wherever A and [FORMULA] appear in the equations, they are in the products [FORMULA] and [FORMULA], the rate of angular momentum loss per unit mass lost from [FORMULA].

More interesting is that, in the strict [FORMULA] limit, the parameters [FORMULA] and [FORMULA] are found only in the combinations [FORMULA] and [FORMULA]. 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, [FORMULA] 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 [FORMULA] term, [FORMULA] or [FORMULA], is of order q or less. This happens, for example, in the Jeans' mode of mass loss ([FORMULA], [FORMULA]), discussed below.

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 [FORMULA], which in the limit of large q, goes as [FORMULA]. Again, in the case of extreme mass ratios, changes in [FORMULA] dominate the course of evolution. As before, there are times when the coefficient of the [FORMULA] term, [FORMULA] or [FORMULA], is of order q, or less. In this case, the above arguments fail and the strict mass loss term [FORMULA] must be included.

The evolution of the angular momentum per reduced mass shows a significant difference the winds and ring models in their [FORMULA] limit. In the wind case, one may write [FORMULA]. The first part is the fractional rate of increase of [FORMULA], 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 [FORMULA], 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 [FORMULA] replaces [FORMULA] as the fraction of mass accreted onto the first star; i.e.: [FORMULA]. The term [FORMULA] is replaced by [FORMULA], as the rate of loss of angular momentum. Although the donor star is stationary in this [FORMULA] 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 [FORMULA], not to the specific angular momentum of either body. Thus, in the ring model, even as [FORMULA] tends towards zero with increasing q, ejected mass will carry away angular momentum. This is the final caveat: the ring model is just a mechanism for the rapid removal of angular momentum from a binary system, and one should keep this in mind, particularly when [FORMULA].

One might also notice for both models, that if angular momentum loss is not overly efficient (A and [FORMULA] not too much greater than 1), then mass loss from the test mass widens the orbit and mass loss from the more massive star shrinks the orbit, just like in conservative mass transfer. Also, [FORMULA] is not everywhere small. In particular, when q is of order unity, [FORMULA] is also and the [FORMULA] term becomes as important as [FORMULA] in the equations of motion.

1. Jeans's modes

Often, 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 [FORMULA]. 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, [FORMULA] 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 [FORMULA], where [FORMULA] 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 e to 0 have been assumed.

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 [FORMULA]. 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, [FORMULA]), where [FORMULA]. In this case, one may simply apply Eq. (3), and see that [FORMULA].

Appendix B: extensions to models considered above

In 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 [FORMULA], 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 [FORMULA] and L:


where each of the variables retains its old meaning. Replacing the old definition of [FORMULA] 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 [FORMULA] (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 t to q used in Sect. (2) introduces singularities when evolution takes place in the absence of mass transfer. The equations of Sect. (2) still hold, but only for those phases of the binary's evolution durring which tidally-driven mass transfer takes place.

We temporarily neglect these complications and consider mass transfer via isotropic wind, isotropic re-emission, and formation of a ring, with mass fractions [FORMULA], [FORMULA], and [FORMULA] respectively. The remainder of the mass transfer (the fraction [FORMULA]) goes into accretion. The ratio [FORMULA] is [FORMULA], where [FORMULA]. Note that [FORMULA], [FORMULA], [FORMULA], and [FORMULA] are all used as before; [FORMULA] should still be regarded as the accreted fraction.


Taking [FORMULA], [FORMULA] = 0, gives formulae for a ring of strength [FORMULA]:


Where the relevant exponents are functions of the parameters [FORMULA] and [FORMULA]:


It is also instructive to examine the model in the degenerate cases of [FORMULA] and [FORMULA], where the functional forms change. When there is no accretion ([FORMULA]), the standard [FORMULA] becomes singular, while the term [FORMULA] approaches 1. Defining the singular part of [FORMULA]:



Table 5. This reference table is divided into three parts. First are the model parameter definitions. An index of equations for the coeficients [FORMULA], [FORMULA], and [FORMULA], relevant to each particular model, follows. Last are the various formulae of orbital evolution derived in this paper

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 [FORMULA] models, where [FORMULA], [FORMULA], and [FORMULA] vanish. There is no profound reason for this.

In the case where all material is accreted ([FORMULA]), there are seeming singularities in the coeficients [FORMULA] and [FORMULA]. Proper solution of the equations of evolution in this case, or setting [FORMULA] before taking limiting values of the coeficients [FORMULA] and [FORMULA], shows that there is no problem at all. In this case, the equations reduce to Eqs. (32), (33), and:


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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998