Common Envelope Evolution in Binary Stars

• Common envelope evolution is a short-lived, critical phase in binary systems marked by unstable mass transfer and rapid orbital shrinkage.
• High-resolution simulations demonstrate that hydrodynamic drag, recombination energy, and angular momentum exchange are key to efficient envelope ejection.
• Advanced models employing αλ, γ, and SCATTER formalisms, along with jet feedback considerations, refine our understanding of post-CE binary outcomes.

A common envelope (CE) event is a short-lived but dynamically crucial phase in binary and higher-multiple stellar evolution, wherein two (or more) stellar cores orbit inside a shared, extended envelope originating from an evolved primary star. CE evolution plays a foundational role in the progenitors of close compact binaries—white dwarf (WD) binaries, neutron star (NS) binaries, double compact-object mergers, type Ia supernovae, X-ray binaries, and luminous red novae. The scenario involves unstable mass transfer, rapid orbital shrinkage, dramatic mass ejection, and complex energetics and angular-momentum exchange. Despite its centrality in compact binary formation, many aspects of CE evolution remain uncertain, with population-synthesis studies still relying on parametrized “formalisms” calibrated by hydrodynamic simulations and observations.

1. Fundamental Principles and Physical Ingredients

The CE event is triggered when the donor star, typically a red giant or asymptotic giant branch star, overfills its Roche lobe. Dynamically unstable mass transfer leads to the engulfment of the companion (stellar, sub-stellar, or compact object) within the envelope. The classical scenario is governed by hydrodynamical drag, dynamical friction, and tidal torques, which extract orbital energy and angular momentum from the embedded binary and communicate this energy to the envelope.

The critical physical quantities are:

• Envelope binding energy:

$$E_\mathrm{bind} = \int_{M_\mathrm{core}}^{M_\mathrm{tot}}\left[-G\frac{m(r)}{r} + u(r)\right]\,dm$$

or, in the αλ-formalism,

$$E_\mathrm{bind} = \frac{G\,M_\mathrm{d}\,M_\mathrm{env}}{\lambda\,R_\mathrm{d}}$$

where $M_\mathrm{d}$ is the donor's initial mass, $M_\mathrm{env}$ the envelope mass, $R_\mathrm{d}$ its radius, and $\lambda$ a structural parameter accounting for density profile and internal energy inputs (Ivanova, 2011, Ricker et al., 2011).

• Orbital energy change:

$$\Delta E_{\rm orb} = -G\frac{M_\mathrm{core} M_\mathrm{comp}}{2a_\mathrm{f}} + G\frac{M_\mathrm{d} M_\mathrm{comp}}{2a_\mathrm{i}}$$

with initial ($a_\mathrm{i}$) and final ($a_\mathrm{f}$) binary separations.

• Efficiency parameter:

$$\alpha_\mathrm{CE} = \frac{E_{\rm bind}}{\Delta E_{\rm orb}}$$

represents the fraction of released orbital energy used to unbind the envelope (Ivanova, 2011, Valsan et al., 2023).

During inspiral, the principal sink of orbital energy is gravitational (dynamical) drag, with tidal torques dominating the extraction of angular momentum from the orbit. Hydrodynamic (ram) drag is typically much less significant. Mass transfer onto the companion is usually highly sub-Eddington—Bondi-Hoyle-Lyttleton prescriptions are found to overestimate the accretion rate by orders of magnitude in realistic 3D flows (Ricker et al., 2011).

2. Hydrodynamics, Energetics, and Mass Ejection

High-resolution 3D hydrodynamical studies reveal that the CE event is inherently asymmetric and non-linear. Key results include:

• Partial envelope ejection and the fate of the binary:

In low-mass binaries (e.g., $1.05\, M_\odot$ red giant + $0.6\, M_\odot$ companion), only $\sim$25–26% of the envelope is typically unbound during the rapid inspiral, and the orbital separation shrinks by a factor $\sim7$ over $\sim5$ orbits (Ricker et al., 2011). Typical instantaneous mass-loss rates at late times are $\dot M_{\rm out} \sim 2\,M_\odot\,\mathrm{yr}^{-1}$. In these simulations, only $\sim25\%$ of the released orbital energy goes into ejection of the envelope; the remainder is distributed as heating (thermal plus kinetic) in the remaining bound envelope (α$_\mathrm{CE}\simeq0.2$–0.3 for similar mass ratios).

• Flow morphology:

Gas flow near the cores in the orbital plane is subsonic, nearly corotating with discrete spiral arms forming in the outer envelope. Outflows are highly equatorially concentrated—90% of ejected mass within $\pm30^\circ$ of the orbital plane (Ricker et al., 2011, Merlov et al., 2021).

• Role of recombination energy:

In moving-mesh simulations that include detailed equations of state, including ionization/recombination energy, ejected-envelope fractions can be as high as 63–66%. Recombinational energy reduces the envelope's effective binding by $10^{47}$–$10^{48}$ erg, comparable to the orbital energy available, substantially increasing the unbinding efficiency (Prust et al., 2019).

• Timeline of ejection and expansion:

In 3D moving-mesh studies, a $2\,M_\odot$ red giant with a $1\,M_\odot$ companion ejects 80% of its envelope within $\sim400$ days and completely unbinds the envelope within $\sim1500$ days. The envelope enters a phase of homologous expansion shortly thereafter, simplifying subsequent radiative-transfer calculations for observed transients (Valsan et al., 2023).

3. Computational and Analytical Formalisms

Theoretical modeling relies on a hierarchy of approaches:

• αλ-formalism:

This parametrizes the efficiency of energy deposition (α) and the structure of the envelope (λ). The simplicity makes it suitable for binary population synthesis, but both α and λ are dependent on initial conditions, mass ratios, and envelope structure (Ivanova, 2011, Ricker et al., 2018).

• Alternative prescriptions:

The $\gamma$-formalism uses angular-momentum conservation, while the recent SCATTER formalism refines angular-momentum tracking by partitioning mass and angular momentum exchange between core, envelope, and companions, and provides analytic expressions for final separations (e.g.,

$$a_f = a_0 \cdot \frac{(1+q)(1+q_\mathrm{ec})^2}{1+q(1+q_\mathrm{ec})}\exp\left[-2\eta\frac{q q_\mathrm{ec}}{1+q}F(q,\delta)\right]$$

with $q=M_c/M_2$, $q_\mathrm{ec}=M_e/M_c$, and a dimensionless partitioning parameter $\delta$) (Stefano et al., 2022).

• Adiabatic mass-loss models:

These track the self-consistent binding energy as the donor loses mass adiabatically. The difference between adiabatic and classical binding-energy calculations can reach a factor of 2 for long-period binaries, affecting predictions for the post-CE separation and the inferred efficiency parameter (e.g., $\beta_\mathrm{CE}\lesssim\alpha_\mathrm{CE}$ for wide orbits) (Ge et al., 2022).

• Simulation codes:

3D hydrodynamic CE evolution is modeled with adaptive mesh refinement (AMR) techniques, moving-mesh codes (e.g., MANGA), and smoothed-particle hydrodynamics (SPH), with state-of-the-art studies including detailed microphysics, recombination energy, and, more recently, jet feedback and dust-driven winds (Ricker et al., 2011, Prust et al., 2019, Valsan et al., 2023, Hillel et al., 2023, Glanz et al., 2018).

4. Variants: Multiplicity, Eccentricity, and Special Channels

• Triple and multiple-star CEE:

Hierarchical triples can undergo highly non-linear CE evolution. The “parasite” scenario involves an outer giant engulfing a close WD–MS binary, and outer-CEE channeling the energy needed to unbind the inner binary's envelope. The standard WD–MS inner binary lacks sufficient orbital energy to unbind its envelope, but the triple system can deliver the necessary energy via triple-dynamical interactions. Rare channels such as the parasite CEE occur in about $10^{-3}$ of all evolved triples (Soker et al., 2021). Hydrodynamic simulations find that triple CEE typically yields higher envelope ejection and much more aspherical ejecta than binary CE (Glanz et al., 2020).

• Evolution with eccentric binaries:

Simulations demonstrate that commonly assumed full circularization during CEE is inaccurate. Initial eccentricities ($e_i$) up to 0.95 lead to post-CE eccentricities $e_f$ as high as 0.18 for “grazing” pericenters, and up to 0.4 for deep plunges. Enhanced dynamical mass-loss occurs in the most eccentric cases, and the residual eccentricity can have observable consequences, affecting delay-time distributions for supernovae and binary mergers by up to tens of percent (Glanz et al., 2021).

• Sub-stellar and planetary companions:

CE involving planetary-mass objects (e.g., WD 1856 b) is possible due to lowered envelope binding during red giant branch (RGB) He-flash. Additional internal energy from wave deposition can inflate the envelope and lower $E_\mathrm{bind}$, allowing even planetary-mass companions to unbind the envelope and survive as close-in orbits around WDs (Merlov et al., 2021, Chamandy et al., 2020).

• Massive-star CEE and implications for gravitational wave sources:

For high-mass, low-metallicity binaries (e.g., $\sim80\,M_\odot$ He core + $30\,M_\odot$ BH), only 40–50% of the envelope is unbound and final orbital separations are generally too wide ($a_f/a_i\sim0.3$–$0.7$) to merge as GW sources within a Hubble time unless augmented by extra energy deposition (e.g., accretion feedback, jets) (Ricker et al., 2018).

5. Feedback, Jets, and Non-orbital Energy Sources

• Jet feedback and turbulence:

Simulations explicitly injecting jets from compact-object companions (e.g., NSs or BHs) during in-spiral show that jet-deposited energy and angular momentum can match or exceed that lost from the orbital decay itself. Jet-driven turbulence (characteristic effective viscosity $\nu_t\sim10^{18}$–$10^{20}\,\mathrm{cm}^2\,\mathrm{s}^{-1}$) can transport angular momentum and unbind envelope material more efficiently than hydrodynamical drag alone. The creation of low-density bubbles, spiral vortex patterns, and enhanced non-spherical geometry is prominent (Hillel et al., 2023).

• Dust-driven winds post-CEE:

In systems where dynamical ejection does not remove the entire envelope, a subsequent dust-driven wind phase is possible. Post-inspiral, the extended, cool envelope resembles that of an AGB star; radiation pressure on newly formed dust can accelerate slow but massive winds that erode the remnant envelope over $10^4$–$10^5$ years. The efficiency of this process is strongly dependent on dust properties, condensation radius, envelope structure parameter $\lambda$, and progenitor mass ratio (Glanz et al., 2018).

• Accretion onto the companion:

Accretion rates estimated by standard Bondi-Hoyle-Lyttleton theory overpredict actual accretion by $\sim 100\times$. Measured rates from 3D simulations show subsonic flows, and sustained super-Eddington accretion requires a pressure-release valve, typically bipolar jets or, for NSs, neutrino cooling, to prevent stalling due to gas pile-up. The feedback from accretion-powered jets may be the dominant additional unbinding mechanism in many real systems (Chamandy et al., 2018).

6. Observational Consequences and Astrophysical Applications

• Close compact binaries and transients:

CE evolution naturally leads to close double WDs (type Ia SN progenitors), NS–WD binaries, recycled pulsar + He star systems, and merging double neutron stars. Observational features, such as the rapidly evolving transients ("luminous red novae") and highly aspherical planetary nebulae, are consistent with CE-driven eruption of envelope material (Garcia-Segura, 8 Dec 2025, Yang et al., 21 May 2025).

• Planetary nebulae dynamics:

Analytical solutions for post-CE nebular dynamics explain both the fastest and slowest planetary nebulae, in contrast to single AGB models which only cover intermediate expansion velocities. The critical parameter is the fraction $\varepsilon$ of bound mass left in the equatorial shell, which modulates the shell expansion speed and observed velocities in the range $2$–$70$ km/s (Garcia-Segura, 8 Dec 2025).

• SN Ia channel demographics:

Population-synthesis studies of CE involving WDs and AGB stars predict that core-degenerate (CD) and double-degenerate (DD) channels (distinguished by whether the merger occurs during or after CE) contribute comparably to normal and peculiar type Ia supernovae. The presence of massive circumstellar shells at the time of explosion is a natural prediction of CE scenarios, especially when SNe occur within $10^6$ years of envelope ejection (Canals et al., 2018).

• Constraints from hot subdwarf binaries:

Analysis of 142 hot subdwarf B (sdB) binaries using adiabatic-mass-loss models demonstrates that classical and self-consistent binding-energy prescriptions can differ by up to a factor of 2, with a clear declining efficiency of envelope ejection for wider orbits. Linear correlations between mass ratio and CE efficiency inform population-synthesis model inputs (Ge et al., 2022).

7. Theoretical Challenges and Future Directions

While the αλ energy formalism remains foundational for practical modeling in population-synthesis studies, the associated efficiency parameters must be treated as functions of mass ratio, evolutionary state, and mass-loss physics. No single value of α or λ fits all observed systems, and hydrodynamic simulations emphasize the necessity of including recombination energy, jet feedback, dust-driven wind phases, and non-conservative mass and angular-momentum loss.

Boundary definitions for the remnant “core,” the fate of envelope convection vs. radiative stratification, and the role of multiple components (beyond binaries) represent major outstanding issues. The increasingly quantitative connection to observed populations (e.g., sdB binaries, sub-luminous transients, gravitational-wave sources) is a focus of current research.

State-of-the-art simulations, analytical models, and formalisms like SCATTER are now being extended to triple and higher-multiplicity hierarchies, offering tractable tools for the next generation of population synthesis and observational interpretation (Stefano et al., 2022, Glanz et al., 2020). Refinement of these approaches remains a leading priority for resolving uncertainties in the endpoint architectures of post-CE stellar systems.