Issue 
A&A
Volume 570, October 2014



Article Number  A25  
Number of page(s)  16  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201423730  
Published online  09 October 2014 
Online material
Appendix A: The perturbing forces and
Consider a primary star of mass M_{1}. At time t = t_{0}, it has an orbital velocity v_{1} = v_{0}, and an orbital angular velocity ω. At a later time t = t′, a particle is ejected from the primary at the innerLagrangian point, located at a distance r_{ℒ1} from the primary’s mass centre (see Fig. A.1), and so the primary’s mass becomes , where . The particle’s absolute velocity is . As a result, the centre of mass is shifted by (A.1)with respect to its unperturbed location, and its new orbital velocity is . During a time interval δt, the change in the primary’s orbital velocity is (see Hadjidemetriou 1969b; Sepinsky et al. 2007b, for further details) (A.2)where Q_{1} is the primary’s momentum, primed quantities indicate values at time t′, and V_{ej} = W_{ej} − v_{1} is the relative velocity of the ejected material with respect to the primary’s mass centre.
At selfaccretion, the primary accretes the particle of mass . Just before selfaccretion occurs at time t = t′′, the momentum of the primary and ejected particle, , is (A.3)where W_{acc} is the absolute velocity of the selfaccreted particle. The orbital velocity, , is the sum of the nonperturbed orbital velocity at time t′′, , and the perturbation to the velocity because of the original ejection episode, so (A.4)
Fig. A.1
Illustration of the selfaccretion process. At time t = t_{0}, the donor star moves along its orbital path (solid black curve), with an orbital velocity v_{0} (cyan arrow). At t′, a particle of mass is ejected from the innerLagrangian point located at r_{ℒ1} with respect to the donor’s mass centre. The ejection shifts the centre of mass by δr_{1}, and the donor follows a new orbit (long dashed curve) with velocity v′. At t′′ just before the particle is reaccreted (at ), the orbital velocity is v′′. Subsequently, selfaccretion shifts the donor’s mass centre by δr_{self}, and it follows the orbit indicated by the dotdashed curve, with a velocity v^{′′′}. The dashed circles represent the locations of the donor if no mass ejection had taken place, while the dotted circle indicates the donor’s location had no selfaccretion occurred. 

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where we have used Eq. (A.1) and ignored terms larger than firstorder. Inserting Eq. (A.4) into Eq. (A.3) gives (A.5)The particle is selfaccreted at time t^{′′′}, at a location with respect to the primary’s mass centre. This shifts the mass centre by an amount (A.6)and the primary’s momentum is now (A.7)The new orbital velocity, , is the sum of the unperturbed orbital velocity, , and the perturbations to the velocity arising from the ejection and selfaccretion processes, i.e. (A.8)where we have used Eq. (A.6), and once again ignored terms higher than firstorder. Inserting Eq. (A.8) into Eq. (A.7) yields (A.9)Taking the difference between Eqs. (A.9) and (A.5), dividing the result by δt and using the fact that yields (A.10)Following Sepinsky et al. (2007b), the absolute acceleration of the primary’s mass centre is the sum of Eq. (A.10), the relative acceleration of the primary’s mass centre, , and the Coriolis acceleration, to give (A.11)where is the relative velocity of the selfaccreted particle with respect to the primary’s mass centre.
Summing Eqs. (A.11) and (A.2) gives the acceleration of the primary’s mass centre from both the ejection and selfaccretion process. In the limit δt → 0, and remembering that , then the acceleration of the primary’s mass centre is (A.12)where R_{1} is the position vector of the primary with respect to an inertial reference frame, and F_{1} is the sum of all external forces acting on the primary, which writes as (A.13)where r = R_{2} − R_{1}, R_{2} is the position vector of the secondary, and f_{1} is the force acting on the primary via the matter stream. Since the secondary is not accreting, its equation of motion is (A.14)where f_{2} is the force acting on the secondary by the accretion stream. Subtracting Eq. (A.12) from Eq. (A.14) gives the equation of motion of the secondary with respect to the primary, which is (A.15)which takes the form (A.16)Here, ê_{r} is a unit vector pointing along r, and ê_{t} is a unit vector perpendicular to ê_{r} in the direction of the orbital motion. Taking the dot product of Eq. (A.16) with ê_{r} and ê_{t} respectively, yields (A.17)and (A.18)which are the same as Eqs. (11) and (12), noting that ω′ = ω′′ = ω for circular orbits. The quantity ψ^{∗} is the angle between ê_{r} and the impact site, and the subscripts “r” and “t” indicate components along ê_{r} and ê_{t} respectively.
Appendix B: Torque arising from mass transfer,
Consider a primary star of mass M_{1}, and a secondary of mass M_{2}, separated by a distance r. They respectively orbit the common centre of mass with an orbital velocity v_{1} and v_{2}. The velocity of the secondary with respect to the primary is v = v_{2} − v_{1}, and the orbital angular momentum is given by (B.1)where e is the eccentricity, m = M_{1}M_{2}/M is the reduced mass, M = M_{1} + M_{2}, and v_{t} is the orbital velocity along ê_{t}, given by (B.2)and ν is the true anomaly. Taking the time derivative of the last equality in Eq. (B.1) and noting that ṙ = 0 for an osculating orbit (see, e.g. Bonačić Marinović et al. 2008), yields (B.3)Similarly to Eq. (A.16), the equation of motion of a binary acted on by perturbing forces and reads (B.4)Taking the dot product of Eq. (B.4) with ê_{t} gives (B.5)Inserting Eqs. (B.5) and (B.2) into Eq. (B.3), and using the first equality in Eq. (B.1), gives the torque applied onto the orbit from mass transfer, (B.6)The net change of the primary’s mass is the sum of mass transferred to the companion via RLOF, , and mass ejected by the wind, Ṁ_{1,loss}< 0, i.e. (B.7)Similarly, for the secondary (B.8)where the first term on the right hand side gives the accretion rate and Ṁ_{2,loss} includes the mass ejected from the system during nonconservative mass transfer. Substituting Eqs. (B.7) and (B.8) into Eq. (B.6) gives (B.9)where q = M_{1}/M_{2}, is the torque acting on the orbit as a consequence of RLOF, while is the torque applied by the material leaving the system. The corresponding torque applied onto the transferred mass is just (B.10)Using Eqs. (B.9) and (B.10) with e = 0 gives Eq. (15).
Next, we demonstrate the consistency of Eq. (B.9) by showing that in the classical formalism for conservative mass transfer . If all material is transferred to the secondary (α_{self} = 0, β = 1), if the stars are treated as point masses (r_{ℒ1} = 0, r_{acc} = 0), and if we neglect the gravitational attraction by the accretion stream (f_{1} = 0, f_{2} = 0), Eq. (10) reduces to (B.11)where for conservative mass transfer. Sepinsky et al. (2007b) and Luk’yanov (2008) demonstrated that, if the
orbital angular momentum is conserved, in a circular orbit V_{1,t} and V_{2,t} are related by (B.12)Substituting Eq. (B.12) into Eq. (B.11), and that result into Eq. (B.6) gives for a circular orbit (B.13)where we have used M_{1}/M = q/ (1 + q). Using Eq. (B.1), Eq. (B.13) reduces to zero, as required.
© ESO, 2014
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