Clark and Eardley originally guessed that stable mass transfer could occur for a binary of mass ratio larger than a critical value (but smaller than a value determined by the presence of the ISCO; cf. Section 1.1), based on their analytic PN analysis . In their analysis, the decrease of the orbital separation associated with gravitational-wave emission, which is one of the important processes in compact binaries of close orbits (see Equation (2)), was not taken into account. Cameron and Iben showed the importance of the gravitational radiation reaction for the onset of the stable mass transfer even in the close binary of white dwarfs [38, 22]. In addition, Benz et al.  show that if the stripped mass of a mass-shedding star forms a disk around the companion primary star (or contributes to the spin-up of the companion), the condition for the onset of the stable mass transfer is further restricted (see below).
The condition for the onset of stable mass transfer is roughly derived applying the analysis method of [38, 22] to BH-NS binaries. Here we will derive the approximate condition, briefly showing the essence of their analysis method. In the following, the analysis is performed in the Newtonian-gravity framework (except for the incorporation of the gravitational radiation reaction), and we assume that the mass element is not lost from the system nor forms a common envelope for simplicity; the material stripped from a mass-shedding NS is assumed to fall into the companion BH or to form the disk surrounding the BH, because this is indeed the case as found in general relativistic simulations. We also assume that the NS radius does not change, because it indeed depends only weakly on the NS mass as long as the mass is larger than (e.g., [116, 186] and Figure 10). For a very low-mass NS, the radius steeply increases with decreasing NS mass. We do not consider such a low-mass NS here.
In the Newtonian framework, the total angular momentum of the system is written as
Using Equation (14) with , is written as
Mass shedding sets in when Condition (7) is satisfied. Thus, Equation (18) may be rewritten in the form
Next, we consider the case of . Then, the condition for stable mass transfer becomes
Many Newtonian simulations, even when including the gravitational radiation reaction by the quadrupole formula, have found that stable mass transfer occurs (e.g., the works by Janka et al. and Rosswog et al. [95, 171]). Their results agree in a sense with that in early predictions such as in . Janka et al.  show that the presence of the gravitational radiation reaction slightly prevents stable mass transfer, but this is not a very strong effect. However, their subsequent work [169, 174] shows that the discovery of stable mass transfer seems to be due to the lack of correct general relativistic physics. They performed an improved simulation in which general relativistic corrections to the gravity of the BH were phenomenologically taken into account via a pseudo-Newtonian prescription and showed that stable mass transfer was unlikely to occur, at least for the parameter space they considered. This shows that the gravitational radiation reaction alone does not prevent stable mass transfer, but this plus the strong two-body gravitational force in general relativity does. Remember that in the presence of general relativistic two-body effects, the onset of mass shedding outside the ISCO is possible only for a small value of , although for the stable mass transfer, a high value of is required. This conclusion agrees with the results in fully general relativistic simulations (see Section 3). General relativistic simulations, in which the mass accretion process to a BH is accurately followed, also show that the accretion of the stripped mass is used to spin up the BH. This is also an important property for preventing the onset of stable mass transfer (see Section 3).
It is worth noting that numerical simulations for BH-NS binaries with both a high BH spin with and a high mass ratio with have not been performed yet. For such a case, a strong repulsive force associated with spin-orbit coupling is likely to decrease the orbital radius of the ISCO and also to increase the inspiral time scale in a close orbit. This effect may help the onset of stable mass transfer, and thus, further studies are still required on this topic.
Living Rev. Relativity 14, (2011), 6
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