On the conditions under which instantaneous mass-transfer mappings reproduce conservative limits

This paper analyzes a specific instantaneous mass-transfer mapping in which a discrete mass element is relocated from the donor to the accretor and assigned the accretor’s velocity. We show that this mapping changes orbital angular momentum at each transfer event by an amount proportional to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>−</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mi>/</mml:mi> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> even when total linear momentum is exactly conserved. We derive the cumulative drift in closed form: after transferring a fraction f of the donor mass, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mi>/</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> = (1 - f) (1 + f <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is the initial mass ratio. A consequence of this result is that the unclosed mapping leaves the orbital separation unchanged ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mi>/</mml:mi> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> = 1), because the angular momentum drift and reduced-mass change cancel exactly—the orbit is frozen while mass is redistributed. In illustrative experiments transferring 15% of the donor mass, the resulting orbital deviation from the conservative prediction reaches 1–22% depending on mass ratio, and scales ~ linearly with the total transferred mass, in quantitative agreement with the analytical formula. The drift is a mathematical property of the mapping definition, not a discretization artifact, and, for this mapping, does not vanish as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>→</mml:mo> </mml:math> 0. We derive and implement a closure condition—a uniform rescaling of tangential velocities after each transfer event—that restores angular momentum conservation and recovers the continuous conservative limit. With this closure active, the isolated two-body simulations achieve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:math> % agreement with analytical predictions for all mass ratios tested. The closure condition may serve as a diagnostic or correction in numerical simulations that represent mass transfer through discrete position–velocity updates, such as interacting binaries embedded in dynamical N -body simulations. As a proof of concept, we also test the closure in hierarchical three-body systems and find that it suppresses spurious transfer-induced angular-momentum jumps in the inner binary while preserving the expected qualitative orbital response under both weak and stronger external perturbations.