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 $\eta$, the fractional weight of the donor velocity in the effective velocity of the transferred mass: $v_{\rm trans} = \eta \, v_1 + (1-\eta) \, v_2$. We derive a closed-form expression for the angular momentum change per transfer event, $\Delta L/L = \delta m \, [(1-\eta)/M_2 - \eta/M_1]$. The two endpoint prescriptions ($\eta = 1$ and $\eta = 0$) produce angular momentum changes of opposite sign, yielding qualitatively different orbital evolution at every mass ratio. Conservative mass transfer ($\Delta L = 0$) corresponds uniquely to $\eta = M_1/M_{\rm tot}$, i.e. $v_{\rm trans} = v_{\rm COM}$. For constant $\eta$, we derive the general closed-form solution $a_f/a_0 = [(1-f)^{2(1-\eta)}(1+f q_0)^{2\eta}]^{-1}$.