A Correlation Between the Final Separation and Mass Ratio from Common Envelope Simulations

TL;DR

This study challenges the universality of the energy-formalism parameter $\alpha$ in common envelope evolution (CEE) by showing an empirical linear correlation between the post-plunge-in separation $a_{\rm f}$ and the mass ratio $q$ across RGB and higher-mass giants using 13 three-dimensional simulations with corotation variations and Riemann-solver tests. The results indicate that while $a_{\rm f}$ scales roughly linearly with $q$ (normalized by the giant radius $R_1$) for $q>0.15$, this trend cannot reliably predict the final post-CEE separation because significant evolution occurs after the plunge-in, during the slow spiral-in, and because numerical choices strongly affect outcomes. The paper demonstrates that energy-formalism-based predictions may instead reflect a dynamical threshold where enough orbital energy is liberated to unbind the envelope, suggesting a multi-phase, phase-driven description of CEE rather than a single, monolithic event. By comparing with previous simulations and observations, the authors argue for a modular, phase-resolved modeling approach that can better capture the diversity of CEE outcomes and guide population-synthesis applications.

Abstract

Analytical models for common envelope evolution (CEE), particularly the energy formalism, are used in binary population synthesis to predict post-CEE configurations. This formalism is based on an efficiency parameter alpha, which relates the orbital energy released during CEE to that required to unbind the envelope of the giant. However, one of the main challenges is that CEE is a multiscale, multiphysics process. As a result, there may not be a universal value for alpha, or even a general expression. Using 13 3D simulations of CEE with RGBs (1 and 2 M$_\odot$ primary; four mass ratios; with and without corotation), we present an empirical linear correlation between the post-plunge-in separation and the mass ratio, normalized by the giant radius. This trend for the plunge-in phase of CEE persists across RGB, AGB, and supergiant simulations in the literature, even for partially bound envelopes. Therefore, alpha from simulations should not be used to predict the final separation, but rather as a diagnostic of whether sufficient orbital energy has been liberated to completely eject the envelope immediately after the radial plunge. If this condition is not met, further in-spiral is expected in later stages of CEE, which may explain why the final separation of post-CEE observations is generally smaller than those predicted by the linear fit. Our results reinforce the idea that a better description could emerge if CEE is treated as a sequence of distinct phases, rather than treating it as a single event governed by alpha.

Figures (7)

• Figure 1: Left: Change of orbital separation between the core of the giant and the companion over time. The dashed lines represent the instantaneous orbital separation $r$, while the solid lines show the semi-major axis $a$, as defined by equation \ref{['eq:semi-major']}. Since the orbit is initially open, $a$ is not shown for the first $\sim 50$ days. The stars show the end of the plunge-in, as defined in the text. Because snapshots from simulations are saved only every 10 days, which is longer than the orbital period, the apparent periodicity in the separation is an sampling artifact rather than the true orbital period. Right: Unbound fraction over time. We included the explanation about the sampling.
• Figure 2: Best linear fits for the simulation sets listed in Tab. \ref{['tab:our_simulations']}. For the $2~\text{M}_\odot$ case with corotation, two curves are shown: the separation at its minimum (purple), and the separation when the plunge ends, as defined in the text (turquoise).
• Figure 3: Density in the $y$–$x$and $y$–$z$ planes at $30.2$, $201.3$, $402.7$, $604$ and $704.7$ days for the $2~\text{M}_\odot$ RGB star, $q=1$ and corotation. The orbit around the companion and the core is mostly cleared by $\sim 700$ days, coinciding with the orbital separation stop increasing. The cyan 'x' is the companion and the '+' is the core.
• Figure 4: Left: AGB, RGB, and Supergiants - SGs (Tabs. \ref{['tab:A3']} and \ref{['tab:new_sims']}) simulations from the literature, along with the best collective fit (black). Right: Comparison of the linear fit for different sets of simulations from the literature that studied different q keeping other parameters fixed 2012ApJ...744...52P2018MNRAS.477.2349I2020AA...644A..60S2022AA...660L...8O. The shadowed colored markers represent all simulated systems.
• Figure 5: Simulations are grouped into 22 equally spaced q-bins. Each point represents the mean absolute deviation normalized by the fitted value, with error bars showing the full range (min to max) within each bin, not the standard deviation. The shadowed colored markers represent all simulated systems.