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

This paper introduces a unified one-parameter framework, defined by the fractional weight $\eta$ of the donor's velocity in the transferred mass's effective velocity, to derive closed-form expressions that determine how angular momentum redistribution governs the orbital evolution of binary stars during mass transfer.

Original authors: Jerry Li (Yuze)

Published 2026-08-20

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CC BY 4.0

Original authors: Jerry Li (Yuze)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Stars rarely exist in isolation. Many are born in pairs, locked in a gravitational embrace where they orbit a common center. Over billions of years, these binary systems can undergo a dramatic transformation: one star, having exhausted its fuel, swells up and begins to spill its outer layers onto its companion. This process, known as mass transfer, is a fundamental engine of stellar evolution. It determines whether the pair will drift apart, spiral inward to merge, or survive as a stable system. For decades, astronomers have relied on computer models to predict these outcomes, but a critical piece of the puzzle has remained hidden in the assumptions of those models. The question is not just how much mass moves, but how that moving mass carries its momentum. When a chunk of gas leaves one star and lands on another, does it arrive with the speed of the star it left, the speed of the star it is joining, or something in between? The answer to this seemingly technical detail dictates whether the orbit shrinks or expands, ultimately deciding the fate of the binary and the types of cosmic explosions or gravitational waves it might produce.

A new study by Jerry Li at Marc Garneau Collegiate Institute in Toronto brings this hidden variable into the light, offering a clear framework to understand how momentum is shared during these stellar exchanges. The research focuses on a specific parameter, a single number that describes the velocity of the transferred material. Imagine the two stars as dancers moving in a circle; as one passes a hand to the other, the speed at which that hand moves determines the balance of the dance. The study introduces a scale to measure this speed, ranging from a scenario where the transferred mass keeps the exact speed of the donor star to a scenario where it instantly adopts the speed of the accretor. By running precise computer simulations and deriving exact mathematical solutions, the author demonstrates that these two extreme choices lead to completely opposite results. If the mass keeps the donor's speed, the orbit shrinks, pulling the stars closer together. If the mass adopts the accretor's speed, the orbit expands, pushing the stars apart.

The most significant finding is that the long-standing standard model used by astronomers, which assumes the orbit's total spin remains perfectly unchanged, corresponds to a very specific, narrow point on this scale. It is not a neutral default but a precise condition where the transferred mass must move at the exact speed of the system's center of mass. The study proves that this condition is unique; any deviation from it, even a small one, creates a net torque that either drives the stars together or pushes them apart. The researchers mapped out the entire spectrum between the two extremes, showing that the final distance between the stars changes smoothly and predictably as the momentum prescription shifts. This means that the uncertainty in how we model this momentum transfer is not a minor detail; it can alter the predicted final separation of a binary system by nearly eighty percent, a margin of error comparable to other major uncertainties in stellar physics.

To ensure these results were not just theoretical artifacts, the study subjected them to rigorous testing. The author ran simulations with varying levels of detail, changing how frequently the mass transfer was calculated and how finely the time steps were divided. The results remained consistent to six significant figures, confirming that the dramatic difference between the shrinking and expanding orbits is a fundamental physical consequence of the momentum choice, not a glitch in the computer code. The study also looked at what might happen in a real, physical system. By estimating the speed of gas flowing through the point between the stars where it escapes the donor's gravity, the author found that real systems might naturally fall somewhere in the middle of the scale, but often closer to the side that causes the orbit to shrink. This suggests that many binary systems might be evolving toward a merger faster than current models predict.

This work does not propose a new mechanism for mass transfer but rather clarifies the rules governing how existing models handle it. It provides a new tool for astronomers to test the sensitivity of their predictions. By adjusting this single parameter, researchers can now bracket the range of possible outcomes for binary evolution, from the formation of compact objects to the rate of gravitational wave events. The study concludes that the assumption of a perfectly conservative transfer is a specific physical claim, not an inevitability. For the first time, the community has a clear, quantitative way to ask how much the momentum of the transferred gas matters, and the answer is that it matters profoundly. The fate of a binary star system is written in the speed of the gas that flows between them, and understanding that speed is now a matter of explicit calculation rather than implicit assumption.

Technical Summary: The Effective Velocity of Transferred Mass

Problem Statement
In the study of binary stellar evolution, the long-term orbital response to mass transfer is governed by the redistribution of angular momentum. While standard population synthesis codes (e.g., BSE, COSMIC, COMPAS) typically assume conservative mass transfer where total orbital angular momentum is conserved ( $\dot{L}_{orb} = 0$ ), this assumption implicitly encodes a specific physical condition regarding the velocity of the transferred material. Specifically, standard derivations assume the transferred mass carries the center-of-mass (COM) velocity, yet this velocity-based condition is rarely stated explicitly as a tunable parameter. Consequently, the degree of freedom regarding how momentum is partitioned for mass that remains within the system (as opposed to mass lost to the system, which is parameterized by Soberman et al., 1997) remains uncharacterized. This lack of explicit parameterization obscures the sensitivity of orbital evolution outcomes—such as orbital expansion versus contraction, common-envelope inspiral, or the formation of compact-object binaries—to the underlying momentum closure assumptions.

Methodology
The author introduces a one-parameter family of momentum prescriptions, the $\eta$ -family, to explicitly model the effective velocity of transferred mass. The velocity of a mass element $\delta m$ transferred from a donor ( $M_1$ ) to an accretor ( $M_2$ ) is defined as a linear combination of the donor's velocity ( $\mathbf{v}_1$ ) and the accretor's velocity ( $\mathbf{v}_2$ ):
$\mathbf{v}_{trans} = \eta \mathbf{v}_1 + (1 - \eta) \mathbf{v}_2$
where $\eta \in [0, 1]$ is a dimensionless parameter.

$\eta = 1$ : The donor-velocity prescription (transferred mass carries the donor's instantaneous velocity).
$\eta = 0$ : The accretor-velocity prescription (transferred mass carries the accretor's instantaneous velocity).

The study derives closed-form analytical expressions for the change in orbital angular momentum ( $\Delta L$ ) and the final orbital separation ( $a_f$ ) as functions of $\eta$ , the initial mass ratio ( $q_0 = M_{1,0}/M_{2,0}$ ), and the transferred mass fraction ( $f$ ). These analytical results are verified using direct two-body numerical simulations integrating the equations of motion with the velocity Verlet algorithm, employing impulsive mass transfer events.

Key Contributions

Explicit Velocity Parameterization: The paper identifies that conservative mass transfer ( $\Delta L = 0$ ) corresponds uniquely to the condition where the transferred mass carries the center-of-mass velocity, which occurs when $\eta = M_1/M_{tot}$ . This reformulates the standard "conservative" assumption from a global constraint into a specific, mass-ratio-dependent point in prescription space.
Closed-Form Solutions: The author derives general analytical expressions for the cumulative angular momentum change and the final orbital separation ratio for any constant $\eta$ :
$\frac{a_f}{a_0} = \left[ (1-f)^{2(1-\eta)} (1 + f q_0)^{2\eta} \right]^{-1}$
This solution interpolates continuously between the two endpoint prescriptions and recovers the standard conservative result only when $\eta$ tracks the instantaneous mass ratio.
Monotonicity Proof: The paper demonstrates that for fixed mass ratio and transfer fraction, the orbital response is strictly monotonic with respect to $\eta$ . This establishes that the two endpoint prescriptions ( $\eta=0$ and $\eta=1$ ) bracket all intermediate behaviors, and that the conservative case is not a neutral default but a specific physical claim.

Results

Opposite Orbital Responses: The two endpoint prescriptions produce angular momentum changes of opposite signs at every mass ratio. The donor-velocity prescription ( $\eta=1$ ) always leads to orbital contraction ( $\Delta L < 0$ ), while the accretor-velocity prescription ( $\eta=0$ ) always leads to orbital expansion ( $\Delta L > 0$ ).
Magnitude of Uncertainty: The spread in predicted final orbital separations between the endpoint prescriptions is significant. For a mass ratio $q_0 = 2.0$ and transfer fraction $f=0.15$ , the difference in final separation is approximately 80% ( $a_f/a_0$ ranges from 0.592 to 1.384).
Physical Estimates: By estimating the effective velocity of a mass-transfer stream at the $L_1$ point under the assumption of corotation, the author derives an approximate effective parameter $\eta_{eff} \approx 1 - R_L/a$ . For typical mass ratios ( $q \sim 0.3–3$ ), this yields $\eta_{eff} \approx 0.5–0.7$ . Crucially, this physically motivated value does not generally coincide with the conservative condition ( $\eta_{cons} = q/(1+q)$ ); depending on the mass ratio, the corotation limit can predict either orbital contraction or expansion.
Numerical Convergence: Numerical simulations confirm the analytical results to six significant figures, demonstrating that the sign reversal and magnitude of orbital changes are determined by the analytical prescription rather than discretization artifacts.

Significance
The paper argues that the choice of momentum prescription is a critical, previously unquantified degree of freedom in binary population synthesis. The assumption of "conservative transfer" in existing codes implicitly sets $\eta = M_1/M_{tot}$ , a specific physical condition regarding stream dynamics that is not guaranteed by hydrodynamics. By introducing the $\eta$ -family, the author provides a compact framework to:

Make implicit assumptions in unresolved mass-transfer calculations explicit and testable.
Quantify the sensitivity of population synthesis predictions (e.g., merger rates, gravitational-wave source populations) to deviations from the conservative assumption.
Serve as a natural interface for calibrating population synthesis models against resolved hydrodynamic simulations, where the effective $\eta$ can be measured directly from the stream momentum.

The work does not propose a new mass-transfer mechanism but rather a structural reinterpretation of the momentum closure, enabling a systematic survey of how different physical assumptions about stream velocity impact the evolutionary fate of binary systems.

Technical Summary: The Effective Velocity of Transferred Mass

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