The Effective Velocity of Transferred Mass: How Momentum Prescriptions Determine Binary Orbital Evolution

Abstract

In binary stellar evolution, the orbital response to mass transfer depends on how angular momentum is redistributed. We introduce a one-parameter family of prescriptions characterized by $η$, the fractional weight of the donor velocity in the effective velocity of the transferred mass: $v_{\rm trans} = η\, v_1 + (1-η) \, v_2$. We derive a closed-form expression for the angular momentum change per transfer event, $ΔL/L = δm \, [(1-η)/M_2 - η/M_1]$. The two endpoint prescriptions ($η= 1$ and $η= 0$) produce angular momentum changes of opposite sign, yielding qualitatively different orbital evolution at every mass ratio. Conservative mass transfer ($ΔL = 0$) corresponds uniquely to $η= M_1/M_{\rm tot}$, i.e. $v_{\rm trans} = v_{\rm COM}$. For constant $η$, we derive the general closed-form solution $a_f/a_0 = [(1-f)^{2(1-η)}(1+f q_0)^{2η}]^{-1}$.