DDI Mass Transfer in Binary Stars

• Delayed Dynamical Instability (DDI) mass transfer is a process in binary systems where stable, thermally regulated mass exchange shifts to a rapid, dynamically unstable runaway phase.
• Critical changes in the donor's envelope—such as core mass increase and recombination energy release—alter mass ratios and Roche lobe dynamics, precipitating instability.
• DDI plays a key role in forming compact objects, luminous red novae, and ultra-compact binaries, offering insights into transient astrophysical phenomena.

Delayed dynamical instability (DDI) mass transfer is a fundamental process in binary and multiple star systems, governing the pathway to rapid orbital contraction, merger, and transient phenomena. DDI mass transfer arises when a binary, initially transferring mass on a secular (thermal or nuclear) timescale, evolves such that changes in stellar structure or orbital parameters precipitate a transition to dynamical instability—often leading to a runaway phase, common envelope evolution, and merger. The delayed nature, as opposed to prompt dynamical instability, is a consequence of the donor’s envelope initially stabilizing mass transfer until structural changes, such as core mass increase or recombination energy release, drive a rapid expansion or contraction that cannot be contained by the Roche geometry. DDI mass transfer is critical for modeling the formation of double compact objects (white dwarfs, neutron stars, black holes), understanding the progenitors of luminous red novae, and explaining the orbital and mass distributions of evolved binaries and post-merger systems.

1. Physical Principles and Definitions

DDI mass transfer occurs in binaries which, after a period of secular and often stable mass exchange, transition to instability. Let $q \equiv M_{\rm donor}/M_{\rm accretor}$ be the mass ratio, $R$ the donor radius, and $R_L$ the Roche lobe radius. DDI is characterized by:

• An initial phase where mass transfer proceeds at a rate self-regulated by the donor’s ability to adjust its structure (thermal timescale).
• Secular evolution may be governed by $\zeta = (\partial \ln R / \partial \ln M)$, the mass–radius exponent of the donor, with stability provided when $\zeta > \zeta_L$, where $\zeta_L$ describes the Roche lobe response.
• Over time, mass loss alters the donor envelope (often peeling radiative layers, growing a convective zone, or causing recombination energy release) and $\zeta$ drops below the critical value. The system crosses a threshold (delayed) for dynamical instability, initiating a rapid, dynamically-driven mass loss phase: the DDI.

For convective main sequence/subgiant donors, dynamical simulations indicate a DDI threshold with $m_c \lesssim 0.15$ (fractional core mass), marking the transition where mass transfer becomes dynamically unstable after a shallow contact phase (Jr. et al., 2010).

Key formulas encapsulating stability include:

$$\begin{align*} r_{\rm fc} &\simeq 2.66 + 0.08(1 - m_c)^4 \quad \text{(first contact)} \ r_{\rm sec} &\simeq 2.69 - 3m_c \quad \text{(secular instability threshold)} \ r_{\rm Roche} &\simeq 2.11 + 0.25(1 - m_c)^4 \quad \text{(Roche limit)} \end{align*}$$

where $r$ is the binary separation scaled to the unperturbed stellar radius $R$. These criteria permit mapping phase transitions from stable contact, through secular/dynamical instability, to merger.

2. Stellar Structure Evolution and Onset of Instability

The delayed nature of DDI is tightly coupled to donor envelope physics. In intermediate/high-mass stars, thermal time scale mass transfer typically precedes dynamical runaway. This pre-instability phase can significantly alter the envelope’s entropy profile and binding energy, and recombination plays a critical role in the donor’s response (Pavlovskii et al., 2014, Temmink et al., 11 Sep 2025).

When the donor’s outer layers become sufficiently depleted or the convective envelope grows, the mass–radius exponent $\zeta_{\rm ad}$ computed from adiabatic models ($\partial \ln R / \partial \ln M$ at conserved entropy) is a key determinant of DDI onset. The transition often occurs after stripping the radiative envelope, revealing a quasi-isentropic core and sharply dropping the stability threshold $q_{\rm ad}$ (Ge et al., 2015). For evolved donors, the quasi-adiabatic critical mass ratio $q_{\rm qad}$ systematically decreases as the star ages and the envelope transitions from radiative to convective (Temmink et al., 2022). In MESA simulations, the critical value for Hertzsprung gap donors is $q_{\rm qad} \sim 0.25$, and for convective giants it can decrease to $\sim 0.1$ (Temmink et al., 2022).

Notably, a new criterion is gaining traction: DDI occurs not strictly when $\zeta < \zeta_L$, but when the donor overfills its outer Lagrangian lobes (L2/L3), marking irreversible mass loss and instability (Pavlovskii et al., 2014).

3. Hydrodynamic and Population Synthesis Modeling

Numerical simulations and analytic population synthesis models (e.g., using MESA, SPH, or refined polytropic/adabatic sequences) are integral for capturing DDI evolution. Three-dimensional hydrodynamic modeling is essential when extreme mass transfer rates are achieved, especially in compact object binaries (DWD, WD–NS) (Marcello et al., 2014):

• Super-Eddington mass transfer models include detailed treatments of fluid motion, Newtonian self-gravity, and radiation-transport (via FLD or implicit methods).
• Conservation of angular momentum and energy is critical—in particular, mechanical disc winds are found to extract angular momentum efficiently, reducing the stability threshold and promoting DDI in WD–NS binaries (Bobrick et al., 2017).
• Simulation grids yield critical mass ratios, evolutionary tracks, and merger outcomes, informing population synthesis models. Stability criteria largely determine which channels (stable MT vs. CE plunge) dominate binary evolution outcomes, affecting compact object merger rates and observable population properties (Li et al., 2022, Ge et al., 26 Nov 2024).

Table: Example DDI regimes by evolutionary state and critical mass ratios.

Donor Evolutionary State	Typical $q_{\rm crit}$	DDI Manifestation
Main Seq., radiative	$>$ 3	Delayed runaway as envelope lost
Hertzsprung gap	$\sim$ 0.25 (post-MS)	Dimming then brightening
Giant, convective	$1.5$–$2.2$ (Pavlovskii et al., 2014)	Delayed, but rapid expansion
White dwarf donors	$0.2$	Tidal disruption post-DDI (Bobrick et al., 2017)

4. Observational Signatures, Transients, and Astrophysical Outcomes

DDI mass transfer underpins numerous observable phenomena:

• Luminous Red Novae (LRNe): Pre-instability evolution produces slow dimming followed by rapid brightening powered by recombination; DDI models for M31-2015 and V838 Mon match observed outburst behaviors and photometry, with the latter case requiring a more massive donor and new binary interpretation due to tertiary dominance in pre-outburst photometry (Temmink et al., 11 Sep 2025).
• Double Neutron Stars and White Dwarfs: The inspiral and coalescence of twin binaries after DDI mass transfer can lead to the formation of double neutron stars and planetary nebulae with double degenerate cores (Jr. et al., 2010).
• Ultra-compact X-ray Binaries (UCXB): The stability threshold set by DDI determines which white dwarf–neutron star binaries persist as UCXBs and which are destroyed by merger (Bobrick et al., 2017).
• Asteroseismic fingerprints: Mass transfer via DDI leaves distinctive chemical abundance profiles and Brunt–Väisälä frequency features in mass gainers, allowing identification via g-mode period spacing diagnostics (Wagg et al., 8 Mar 2024).
• Cataclysmic Variables and Dwarf Novae: Enhanced mass transfer events (with delayed disc response) manifest as “stunted” outbursts, with simulations reproducing light curves (e.g., EX Draconis, UU Aquarii) where the accretion disc’s viscous timescale mediates the delayed response (Baptista et al., 2011, Schlindwein et al., 28 Aug 2024).

5. Factors Influencing Stability and DDI Thresholds

DDI and mass transfer instability thresholds are sensitive to several physics:

• Envelope Structure: Convective vs. radiative nature, superadiabatic layers, and recombination energy release—all modify $\zeta_{\rm ad}$ and the mass transfer threshold (Pavlovskii et al., 2014, Ge et al., 2015).
• Metallicity: Metal-poor radiative donors experience lower $q_{\rm ad}$ for instability than metal-rich ones; for convective donors, the trend reverses (Ge et al., 29 Aug 2024).
• Non-conservativeness: Inefficient accretion (with mass/AM loss from the system) decreases $q_{\rm ad}$ and advances the dynamical regime in massive binaries (Ge et al., 29 Aug 2024, Ge et al., 26 Nov 2024).
• Accretor Response: The tendency for accretors to expand when rapidly accreting (especially for MS/HG stage, as opposed to donor’s response alone) results in lower $q_{\rm crit}$ and increased CE risk (Zhao et al., 9 Apr 2024).
• Angular Momentum Transport: Feedback from disc winds and direct impact streams, especially in compact object accretors, enhances orbital shrinkage and DDI susceptibility (Bobrick et al., 2017, Jia et al., 2016).

6. Mathematical and Theoretical Frameworks

Quantitative assessment of DDI employs:

• Energy formalism to compute envelope binding energy,

$$E_{\rm bind} = \int_{M_{\rm He}}^{M_{\rm d}} (e_{\rm int}(m) - \frac{G m}{r(m)})\, dm$$

and the “α–prescription” for post-CE separation,

$$E_{\rm bind} = \alpha \Delta E_{\rm orb} = \alpha \left[\frac{G M_a (M_{d,i} - M_{\rm env})}{2 a_{\rm post-CE}} - \frac{G M_a M_{d,i}}{2a_i}\right]$$

(Temmink et al., 11 Sep 2025).

• Thresholds for instability:

$\zeta_{\rm ad} = \zeta_L(q_{\rm ad})$

where $q_{\rm ad}$ decreases as the donor envelope evolves, and non-conservative mass transfer per

$$q_{\rm crit} = \frac{M_{\rm donor}}{M_{\rm accretor}}$$

with the critical value set by donor/accretor response, metal abundance, and mass transfer efficiency (Ge et al., 26 Nov 2024).

• Disc viscosity and responses:

The characteristic timescale for viscous adjustment is parametrized via

$\nu = \alpha c_s H$

and disc expansion and contraction velocities constrained through observations and simulations (Baptista et al., 2011, Schlindwein et al., 28 Aug 2024).

7. Broader Implications and Ongoing Research

DDI mass transfer informs critical understanding of compact binary populations, gravitational wave source rates, and observed transients:

• Updated thresholds from adiabatic models indicate a broader stability domain, enlarging the phase space of systems which avoid premature common envelope evolution and altering predicted merger rates for white dwarf and black hole binaries (Li et al., 2022, Ge et al., 26 Nov 2024).
• Population synthesis algorithms must incorporate physically accurate DDI criteria (especially $q_{\rm crit}$ behavior as a function of donor evolutionary phase and chemical composition) to convert microphysical models to macroscopic observables and event rates.
• Observational campaigns (e.g., Gaia) can identify DDI progenitors pre-outburst via photometric and asteroseismic diagnostics, providing tests for theoretical models (Temmink et al., 11 Sep 2025, Wagg et al., 8 Mar 2024).
• Asteroseismic and photometric studies of mass gainers and binary transients are sharpening the ability to distinguish accretion-induced rejuvenation and instability-induced outbursts from stable evolutionary pathways.

In summary, delayed dynamical instability mass transfer is a pivotal process controlling the pathways to binary merger, compact object formation, and transient events. Its nuanced physics is governed by donor and accretor envelope evolution, angular momentum removal mechanisms, chemical composition, and the transfer efficiency, all of which must be modeled with hydrodynamic fidelity and incorporated into population synthesis simulations for accurate prediction and interpretation of the outcomes across astrophysical environments.