Common-Envelope Evolution in Binary Stars

• CEE is a dynamic phase in stellar binary evolution characterized by rapid, unstable mass transfer that leads to envelope engulfment and swift orbital contraction.
• Modeling CEE employs energy (α–λ) and angular momentum (SCATTER) formalisms to determine envelope ejection efficiency and predict binary survival outcomes.
• Hydrodynamic simulations and observational surveys confirm CEE mechanisms through signatures like luminous red novae and bipolar planetary nebulae.

Common-Envelope Evolution (CEE) is a critical, brief, and dynamically complex phase in the evolution of close binaries and higher-order multiples, where drag forces, mass transfer, and rapid orbital contraction reshape the architecture of formed systems and underpin the production of compact binaries, gravitational-wave progenitors, and a wide variety of luminous transients and exotica.

1. Physical Foundations and Stages of Common-Envelope Evolution

CEE is triggered when an evolved star (typically a red giant or asymptotic giant branch star) overflows its Roche lobe, leading to rapid, unstable mass transfer that engulfs both stellar components in a common envelope. The ensuing evolution is conventionally divided into several stages (Ivanova et al., 2012):

1. Loss of Corotation: Initiated by dynamical instability, e.g., runaway Roche-lobe overflow, Darwin instability, or expansion on nuclear/thermal timescales.
2. Dynamical Plunge-In: The companion spirals rapidly inward due to intense drag, generating spiral shocks and expelling some envelope mass on the dynamical timescale.
3. Self-Regulated Spiral-In: If the envelope is not ejected, the inspiral slows; ongoing friction drives further mass loss over thermal timescales.
4. Termination/Shell Detachment: Final envelope removal via delayed ejection or additional mass loss, potentially halting the inspiral and leaving a close binary or merger.
5. Post-CEE Evolution: The remnant binary contracts or merges; residual circumbinary or fallback material can affect eccentricity, spin, and subsequent observables.

CEE is responsible for forming close white dwarf binaries, neutron star/black hole binaries, post-AGB systems, hot subdwarfs, and is implicated in the progenitors of type Ia supernovae and double compact mergers (Ge et al., 2023).

2. Energy and Angular Momentum Formalisms

The outcomes of CEE are classically modeled using parametric prescriptions grounded in conservation principles, with the α–λ "energy formalism" being dominant, and more recently, angular momentum-based approaches like the SCATTER prescription gaining traction (Stefano et al., 2022).

2.1 α–λ Energy Formalism

The envelope is ejected if the orbital energy released as the orbit decays is sufficient, with efficiency factor α_CE:

$$\alpha_{\rm CE}\left[\frac{G\,M_{\rm core}\,M_2}{2\,a_f} - \frac{G\,M_1\,M_2}{2\,a_i}\right] = -E_{\rm bind}$$

where $E_{\rm bind}$ is typically modelled with a structure parameter λ:

$$E_{\rm bind} \approx -\frac{G M_1 M_{\rm env}}{\lambda R_1}$$

Here, α_CE reflects the conversion of released orbital energy to envelope ejection. λ encapsulates structural and microphysical details (internal energy, recombination, core definitions) (Hall et al., 2014). The physical definition of the core radius and post-CE stellar structure can significantly alter merger and survival outcomes, with detailed treatments required for precise modeling (Hall et al., 2014).

2.2 SCATTER Formalism and Angular-Momentum Approaches

SCATTER ("Single Components’ Angular momenTum TransfER") formulation expresses post-CEE separations solely in terms of global angular momentum budget:

$$a(f) = a(0)\left[\frac{(1+q)(1+q_{ec})^2}{1 + q(1+q_{ec})}\right] \exp\left\{ -2\eta \left[ \frac{q q_{ec}}{1+q} \right] F(q, \delta) \right\}$$

Here, η encodes the inefficiency of angular momentum transfer, Q(q, δ) partitions envelope mass draining from each star, F(q, δ) summarizes the mass-weighted deposition, and δ reflects the envelope mass partition "dimensionality." SCATTER naturally generalizes to higher-order multiples and directly ties to observed post-CE populations via fitting to mass and separation distributions (Stefano et al., 2022).

3. Hydrodynamic Simulations and Microphysics of Envelope Ejection

Advances in 3D hydrodynamics, both grid-based and SPH, have elucidated the structure, mass loss, and physical processes governing CEE (Frank et al., 2018, Bronner et al., 2023, Prust et al., 2019).

• Energy Deposition and Timescales: Initial plunge-in rapidly deposits orbital energy and angular momentum, but simulations show only partial ejection during the dynamical phase; remaining envelope often persists as a bound, inflated structure (Glanz et al., 2018, Ivanova et al., 2012).
• Role of Recombination Energy: Incorporation of recombination energy of H/He is essential; 3D moving-mesh and 1D/3D hybrid simulations demonstrate that recombination energy supplemented by orbital energy can account for ≳60% envelope unbinding, with α_CE~1 in favorable regimes (Prust et al., 2019).
• Accretion, Disk Formation, and Jet Launching: High rates of super-Eddington accretion onto the companion are observed in simulations only if a pressure-release mechanism exists, most plausibly a bipolar jet (Chamandy et al., 2018, Chamandy et al., 2018). Absence of such a valve leads to accretion stagnation and limited envelope loss.

Hybrid approaches leveraging parametrized drag and heating capture 3D-inspired energetics and ejection efficiency at lower computational cost for moderate mass ratios (q≲0.5), but break down for higher mass-ratio systems due to 3D effects (Bronner et al., 2023).

4. Ejection Mechanisms: Jets and Dust-Driven Winds

Multiple mechanisms beyond orbital energy participate in envelope unbinding:

4.1 Jets Powered by Accretion Disks

Observations of post-CEE planetary nebulae (PNe) show that jets are the most robust observable marker: 72% of well-studied post-CEE PNe exhibit jets, while only 50% display equatorial tori/disks. Correcting for observational bias, the intrinsic frequency of jets launched by companions is at least three times higher than that of dense equatorial outflows (Soker, 2024).

Physical modeling demonstrates:

• Mass Accretion/Jets: Main-sequence companions accrete envelope gas via Bondi–Hoyle–Lyttleton flow. A significant fraction (η≈0.1–0.3) of accreted mass is launched as high-velocity jets (v_jet~several 10^2–10^3 km/s). The energy and momentum imparted by jets can unbind a nontrivial fraction of the envelope, and angular momentum deposited is comparable to or even exceeds that extracted by the orbital decay itself (Hillel et al., 2023, Schreier et al., 2022).
• Morphology/Turbulence: Jets inflate low-density, bipolar lobes, seed large-scale turbulence and vorticity, and deposit angular momentum aligned (or misaligned) with the orbital plane. In the case of inclined jets (due to tertiary-induced spin-orbit misalignment), up to ≈23° envelope spin-orbit misalignment can be produced, relevant for double compact mergers (Schreier et al., 2022). Jet-driven turbulence enhances energy transport and mixing (Hillel et al., 2023).

4.2 Dust-Driven Winds

If the rapid (dynamical) ejection fails, the expanded, cool, and dusty envelope—analogous to AGB stars—can drive slow, radiatively accelerated winds over ≈10^4–10^5 years. Grains condense as the envelope cools to ≲1500 K, and radiative pressure on dust grains initiates mass loss. These winds are capable of unbinding the residual envelope with mass-loss rates Ṁ ≈ 10^–5 M_⊙/yr and velocities ~10 km/s (Glanz et al., 2018). Observational constraints from wide triples indicate that such slow, dust-driven CEE ejection is required to maintain outer tertiaries, with best-fit ejection timescales 10^3–10^5 yr (Michaely et al., 2018).

5. Observational Counterparts and Post-CEE Outcomes

CEE has direct, empirically testable observables:

• Luminous Red Novae (LRNe): Direct counterparts of CEE transients, identifiable via recombination-controlled, plateau-shaped red light curves spanning days to months, characteristic expansion velocities (~200–1000 km/s), and plateau luminosities/energies matching model predictions (Ivanova et al., 2013).
• Post-CEE Binaries: sdB/sdO, WD+WD/MS, and planetary nebulae with binary central stars exhibit orbital period and eccentricity distributions diagnostic of the ejection mechanism. Notably, nonzero post-CEE eccentricities (up to e≈0.18) persist for initial eccentric orbits, contrary to previous assumption of complete circularization (Glanz et al., 2021).
• Bipolar Planetary Nebulae: Simulations confirm that aspherical, toroidal ejecta from CEE produce the necessary pre-existing density distribution for highly collimated, shock-focused, bipolar nebular outflows—a hallmark of many PNe morphologies (Frank et al., 2018). Jet and wind dynamics induce symmetry breaking in nebular lobes, with detailed dependency on the momentum injection rate and 3D structure.

6. Parametric Constraints, Efficiency, and Population Synthesis

The parametric CE-ejection efficiency α_CE is not a universal constant. Observational constraints from post-CEE sdB+WD systems yield sample averages around ⟨α_CE⟩ ≃ 0.32, with trends indicating α_CE increases strongly with the initial mass ratio q_i. A least-squares fit across well-determined sdB+WD binaries gives:

$$\log_{10}\alpha_{\rm CE} = -2.2927 + 2.6169 \log_{10} q_i$$

Empirically, systems with more extreme mass ratios require significantly higher α_CE to achieve envelope ejection (Ge et al., 2023). Incorporation of rejuvenation effects from prior Roche-lobe overflow (RLOF) events reduces the envelope binding energy by ≈42–96%, further increasing post-CE survival fractions and yielding final separations up to ≈50% wider than for single-star models (Renzo et al., 2022).

Improvement and validation of core-radius definitions, as well as adoption of self-consistent energy and angular momentum prescriptions, are required to produce accurate population synthesis predictions for double-compact merger rates, type Ia supernovae, and LISA/LIGO source populations (Hall et al., 2014, Stefano et al., 2022).

7. Outstanding Challenges and Future Directions

Despite decades of dedicated work, full numerical convergence and comprehensive modeling of CEE remain out of reach. Essential outstanding issues include:

• Multi-Physics 3D Simulations: Integrating jet formation, magnetohydrodynamics, recombination, dust formation, and radiative transfer across multiple phases of CEE, covering dynamical plunge-in through protracted wind-driven ejection (Soker, 2024, Hillel et al., 2023).
• Time-Dependent Envelope Ejection: Observations of bound outer tertiaries dictate that CE mass loss can be protracted (10^3–10^5 yr), incompatible with purely dynamical models. Hybrid wind-driven/jet-driven models need quantitative calibration and validation (Michaely et al., 2018).
• 1D/3D Model Interface: Robust 1D CE models with dynamically calibrated drag and energy deposition matching 3D results, especially for q≳0.5, are required for efficient synthesis calculations (Bronner et al., 2023).
• Observational Surveys: Deeper, time-resolved, multi-wavelength imaging and spectroscopy of post-CEE nebulae, LRNe, and wide triple populations to refine dynamical and structural constraints.
• Population Synthesis with Physically-Tied Parameters: Adoption of variable α_CE(q), self-consistent binding energies, and incorporation of jet/dust energy sources, informed by the latest hydrodynamic and empirical data.

CEE remains a unifying challenge linking multi-phase stellar evolution, binary dynamics, transient astronomy, and gravitational-wave astrophysics. Advances hinge on detailed integration of microphysical ejection mechanisms, observational systematics, and multidimensional simulations guided by empirical constraints (Ivanova et al., 2012, Soker, 2024).