From Common Envelope Evolution to Luminous Red Novae I: A One-dimensional Radiation Hydrodynamic Model

1. Problem Statement

Common Envelope Evolution (CEE) is a critical but poorly understood phase in binary star evolution where a companion star spirals into the envelope of a giant star. This process determines the final separation of binary cores, influencing the formation of gravitational wave sources, X-ray binaries, and Type Ia supernovae progenitors.

The specific physical challenge addressed in this work is the envelope unbinding mechanism during the "plunge-in" phase. While 3D hydrodynamic simulations exist, they often rely on adiabatic assumptions that neglect radiative energy transport. Conversely, 1D models often lack self-consistent radiation pressure. The authors aim to bridge this gap by investigating how radiation pressure, gas pressure, and recombination energy interact to drive mass ejection and produce observable signatures like Luminous Red Novae (LRNe).

2. Methodology

The authors developed and utilized a one-dimensional Radiation Hydrodynamic (RHD) code named "Guangqi" to simulate the dynamics of the expanding envelope.

• Governing Equations: The model solves the RHD equations in the laboratory frame using the Flux-Limited Diffusion (FLD) approximation. This approach bridges the gap between optically thick (diffusion) and optically thin (free-streaming) regimes.
  • Equations cover mass, momentum, total energy, and radiation energy conservation.
  • A General Equation of State (EoS) includes species for $H_2$, $H$, $H^+$, $He$, $He^+$, $He^{2+}$, and electrons, accounting for recombination energies.
• Numerical Scheme:
  • Hydrodynamics: Solved using an HLLC Riemann solver with a MUSCL scheme and operator unsplit method.
  • Radiation-Gas Coupling: Thermodynamic coupling is handled implicitly using a GMRES Krylov iterative solver (Petsc) to manage the stiff timescales between hydrodynamics and radiation.
  • Sub-cycling: To ensure accuracy, the thermodynamic system is integrated with sub-steps ($n_{sub}=6$) relative to the hydrodynamic timestep.
• Simulation Setup:
  • Domain: $[450 R_\odot, 1.5 \times 10^4 R_\odot]$ with a central fixed mass of $15 M_\odot$.
  • Boundary Conditions: The inner boundary injects mass and energy for 50 days (expanding phase), then switches to an inflow-only condition to allow fallback. The outer boundary assumes free-streaming (optically thin).
  • Parameter Survey: The authors performed a grid survey varying three key parameters:
    1. Radiation-to-gas internal energy ratio ($E/e_g$): $[0.2, 3.2]$.
    2. Injection velocity ratio ($\bar{v}_{ej}$): $[0.3, 0.6]$ (relative to local escape velocity).
    3. Mass injection rate ($\dot{M}$): $[2.5, 5, 10] M_\odot/\text{yr}$.
  • Duration: Simulations ran for 500 days to capture the full dynamical evolution and light curve decay.

3. Key Contributions

1. Self-Consistent 1D RHD Model: The paper presents a robust 1D model that self-consistently couples radiation pressure with gas dynamics, overcoming the limitations of purely adiabatic 3D models or simplified 1D models.
2. Quantification of Unbinding Efficiency: The study quantifies the unbound mass fraction ($\eta$) as a function of physical parameters, specifically identifying the non-linear relationship between total envelope energy ($\zeta$) and mass ejection.
3. Identification of Driving Mechanisms: The work isolates the specific roles of radiation pressure versus gas pressure gradients in different opacity regimes.
4. Light Curve Physics: The model successfully reproduces complex light curve features, including a secondary plateau driven by molecular hydrogen recombination ($H \to H_2$).

4. Key Results

• Radiation Pressure Dominance: In high-opacity, high-luminosity regions immediately below the recombination front, radiation pressure becomes the dominant driver of mass ejection. The ratio of radiative acceleration to gravity ($a_{rad}/g$) exceeds unity in these layers, accelerating material even in deep layers.
• Non-linear $\eta - \zeta$ Relation: The unbound mass fraction ($\eta$) exhibits a non-linear dependence on the ratio of total envelope energy to gravitational binding energy ($\zeta$).
  • Unbinding is inefficient when $\zeta < 0.7$.
  • In the range $\zeta \in [0.7, 1.05]$, acceleration becomes effective.
  • Mass Injection Rate Effect: Higher $\dot{M}$ leads to higher $\eta$ because the dense envelope traps radiation (high optical depth), allowing the gas to acquire more momentum. In tenuous envelopes, radiation decouples and escapes, reducing the driving force.
• Light Curve Morphology:
  • Peak Luminosity: Higher $E/e_g$ ratios lead to higher peak luminosities but faster fading due to rapid expansion and density drops.
  • Secondary Plateau: A distinct secondary plateau appears in the light curve (e.g., days 140–280) caused by the recombination of atomic hydrogen to molecular hydrogen ($H \to H_2$). This releases latent heat, sustaining the luminosity. This mechanism may explain "SPRITEs" (infrared transients) observed by Spitzer.
• Convergence and Accuracy: Error analysis shows the model is accurate ($|\delta E_{err}| \le 3\%$) for $E/e_g < 1$ or supersonic flows. In radiation-dominated or subsonic regimes, energy may be underestimated by up to $\sim 8\%$, but the core dynamical conclusions remain robust.

5. Significance

• Bridging Theory and Observation: The model provides a physical framework to interpret Luminous Red Novae (LRNe) as the observational signature of the CEE plunge-in phase.
• Refining Binary Evolution Models: The findings offer quantitative insights to refine the standard $\alpha_{CE}$ formalism (Webbink 1984), which currently relies on simplified energy conservation assumptions. The non-linear $\eta-\zeta$ relationship suggests that a simple linear energy budget is insufficient.
• Future Directions: The authors highlight that while 1D models capture the thermodynamics well, future work must incorporate multi-dimensional (2D/3D) effects to account for angular momentum transport, geometric asymmetries, and the coupling of the inner boundary to the actual binary inspiral dynamics.

In summary, this paper establishes that radiation pressure is a critical, often dominant, mechanism for envelope ejection in CEE, and that molecular recombination plays a vital role in the late-time observational signatures of these events.