1.6 Stable mass transfer occurs or not

One interesting question for the final phase of BH-NS binaries is whether the stable mass transfer occurs in a close orbit; specifically, the question is whether the mass accretion from a NS to its companion BH proceeds for a time scale longer than the orbital period while escaping tidal disruption or merger. This could occur if the orbital separation does not decrease due to the presence of mass accretion. As discussed in several works and also indicated in the following, the answer is not trivial at all.

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 [43Jump To The Next Citation Point]. 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 (2View Equation)), 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 [38Jump To The Next Citation Point, 22Jump To The Next Citation Point]. In addition, Benz et al. [22Jump To The Next Citation Point] 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 ∼ 0.4M ⊙ (e.g., [116Jump To The Next Citation Point, 186] and Figure 10View Image). 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

∘ ---- Gr Jorb = ---MBHMNS, (14 ) m0
where the mass of the disk around the BH is included in MBH. The time evolution of the angular momentum obeys the equation
˙ ˙ ˙ Jorb = −JGW − S, (15 )
where J˙GW is the loss rate of the angular momentum by the gravitational-wave emission, written assuming that the system is composed of point masses, as 6.4m0 (μ ∕m0 )2(Gm0 ∕c2r)7∕2, and S˙(≥ 0) denotes the total increase rate of the angular momentum of the BH and a disk of the BH. ˙ S is totally unknown; to determine it, a numerical simulation is necessary. Because it should be approximately proportional to M˙NS (the mass-shedding rate of the NS) and MBH, we write ˙S = αsGc −1M˙BHMBH, where we use M˙BH = − M˙NS. αs is a constant of order unity if all the stripped mass elements contribute to spin-up of the BH or to forming a disk; on the other hand, α = 0 s if the stripped mass element does not contribute to the spin-up and the formation of a disk. The assumption αs = 0 corresponds to the case in which we simply assume that the orbital momentum of the accreting material is simply added to the companion BH (the orbital angular momentum of the BH is increased).

Using Equation (14View Equation) with M˙ = − M˙ > 0 BH NS, J˙ orb is written as

˙r Q − 1M ˙ J˙orb = ---Jorb + ---------NSJorb, (16 ) 2r Q MNS
and then, we obtain
˙ r˙Jorb = Q-−--1|MNS-|Jorb − J˙GW − ˙S. (17 ) 2r Q MNS
For stable mass transfer to occur, ˙r has to satisfy ˙r ≥ 0. After a short calculation, the condition is written as
16π Q2 6Gm M |MN˙S | ≥ --√------------------( ----0)5∕2(--NS-) 45 6(Q − 1)(Q + 1)2 c2r Porb ----αsQ---- 6Gm0-- 1∕2 +√ -- ( c2r ) |M ˙NS |. (18 ) 6 (Q − 1)
Here, the first and second terms on the right-hand side denote the effects of the gravitational radiation reaction, and of the spin-up or disk formation of the BH, respectively.

Mass shedding sets in when Condition (7View Equation) is satisfied. Thus, Equation (18View Equation) may be rewritten in the form

16 π Q7 ∕6(Q + 1)1∕2 5∕3 𝒞 5∕2 MNS |M ˙NS | ≥ --√-----------------CΩ (----) (-----) 45 6 (Q − 1) 1∕6 Porb αsC1Ω∕3Q5 ∕6(Q + 1)1∕2 𝒞 1∕2 + --√-------------------(---) |M ˙NS |. (19 ) 6 (Q − 1) 1∕6
First, we consider the case of αs = 0. Then the condition can be written in the form
|M˙NS-|Porb -16π--Q7∕6(Q-+-1-)1∕2- 5∕3 -𝒞--5∕2 M ≥ √ -- (Q − 1) C Ω (1∕6) . (20 ) NS 45 6
The left-hand side denotes the fraction of mass shedding in one orbital period. For stable mass transfer to occur, it should be much smaller than unity; at least smaller than unity. Here, 3.6 ≲ Q7 ∕6(Q + 1)1∕2∕(Q − 1) ≲ 5.4 for 2 ≤ Q ≤ 10, and C Ω is likely to be ≲ 0.3. Thus, for a realistic value of 𝒞 = 0.13 –0.21, the right-hand side of Equation (20View Equation) is ≲ 0.1. This suggests that stable mass transfer may occur in principle with a short time scale (in ∼ 10 orbits). However, we note that for a too large value of Q, mass shedding is unlikely to occur outside the ISCO, as illustrated in Section 1.1 (but for a high-spin BH, this may be avoided).

Next, we consider the case of α = O (1) s. Then, the condition for stable mass transfer becomes

˙ 7∕6 1∕2 1∕3 5∕6 1∕2 |MNS-|Porb ≥ -16√π-Q---(Q--+-1)---C5∕3(-𝒞--)5∕2[1 − αsCΩ-√Q---(Q--+-1)---(-𝒞-)1∕2]− 1. (21 ) MNS 45 6 (Q − 1) Ω 1∕6 6(Q − 1) 1∕6
Here, 2.3 ≲ Q5 ∕6(Q + 1)1∕2∕(Q − 1) ≲ 3.1 for 2 ≤ Q ≤ 10, and C1 ∕3≲ 0.7 Ω. Thus, the term [⋅⋅⋅]−1 is likely to be much larger than unity for α = O (1 ) s, and because of this presence of the term associated with αs = O(1), stable mass transfer is more strongly restricted. This indicates that the possibility for the occurrence of stable mass transfer depends strongly on the accretion process of the mass-shedding fluid elements, e.g., if the accreting material contributes mainly to the spin-up of the companion BH with αs = 1, the last term (the term [⋅⋅⋅]−1) is ≳ 3, and thus, the mass accretion rate has to be quite high for stable mass transfer to occur. However, with such a high mass-shedding rate, stable mass transfer would only for a short time scale, if at all. One interesting point is that to realize the onset of the stable mass transfer, the formation of the disk surrounding the BH or the spin-up of the BH has to be significantly suppressed.

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. [95Jump To The Next Citation Point, 171]). Their results agree in a sense with that in early predictions such as in [43]. Janka et al. [95Jump To The Next Citation Point] 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 Q, although for the stable mass transfer, a high value of Q 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 a > 0.9 and a high mass ratio with Q ≥ 10 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.


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