Understanding the Drag Torque in Common Envelope Evolution

TL;DR

This work advances our understanding of drag torque during common-envelope evolution by testing binarity-aware torque models against a new 3D CE simulation in which the companion mass matches the AGB core, enabling near-symmetric, interpretable dynamics. It demonstrates that a local, corotating gas region within roughly 1.5 inter-particle separations dominates the torque and that a uniform-density prolate spheroid model or a Kim+08-like double-perturber model can reproduce the measured torque with physically motivated parameter choices. The ellipsoid approach yields torque predictions closely matching the simulation when using mean interior density and fitted geometric parameters, while the double-perturber approach matches well with a density scaling factor $\xi \approx 0.44$, illustrating complementary routes to simple, time-dependent CE torque prescriptions. Together, these results support developing 1D drag-force prescriptions for CE evolution and inform binary population synthesis, with potential relevance to luminous red novae and the broader dynamics of binary-gas interactions.

Abstract

Common envelope (CE) evolution is largely governed by the drag torque applied on the in-spiralling stellar components by the envelope. Previous work has shown that idealized models of the torque based on a single body moving in rectilinear motion through an unperturbed atmosphere can be highly inaccurate. Progress requires new models for the torque that account for binarity. Toward this end we perform a new 3D global hydrodynamic CE simulation with the mass of the companion point particle set equal to the mass of the asymptotic giant branch star core particle to maximize symmetry and facilitate interpretation. First, we find that a region around the particles of a scale comparable to their separation contributes essentially all of the torque. Second, the density pattern of the torque-dominating gas and, to an extent, this gas itself, is roughly in corotation with the binary. Third, approximating the spatial distribution of the torquing gas as a uniform-density prolate spheroid whose major axis resides in the orbital plane and lags the line joining the binary components by a constant phase angle reproduces the torque evolution remarkably well, analogous to studies of binary supermassive black holes. Fourth, we compare the torque measured in the simulation with the predictions of a model that assumes two weak point-mass perturbers undergoing circular motion in a uniform background without gas self-gravity, and find remarkable agreement with our results if the background density is taken to be equal to a fixed fraction (~0.44) of the density at the spheroid surface. Overall, this work makes progress toward developing simple time-dependent models of the CE phase, for example by informing the development of drag force prescriptions for 1D spherically symmetric CE simulations, which could be used to explore the parameter space of luminous red novae or in binary population synthesis studies.

Figures (12)

• Figure 1: Separation between the AGB core and companion particles as a function of time. The dashed line shows twice the softening radius, $2r_\mathrm{soft}$, and the inset shows the orbit of the primary core (black) and companion (red).
• Figure 2: The $z$-component of torque on the binary system about the particle CM (left axis). Top: Shown is the torque (i) measured directly from the simulation (black), (ii) including only contributions out to the contour $\rho=\rho_\mathrm{c}=0.006\rho_\mathrm{max}(t)$ (magenta), (iii) reconstituted using equation \ref{['eq:torque_z']} with $\overline{\rho}$, $\Delta\phi$ and $\widetilde{B}/\widetilde{A}$ measured from the simulation (cyan), (iv) reconstituted using equation \ref{['eq:torque_z_mean']} for a co-rotating spheroid with $\langle\Delta\phi\rangle = 14.9^\circ$ and $\langle\widetilde{B}/\widetilde{A}\rangle=0.654$ (orange). The orbital separation of the particles, $a$, is shown on the right axis (dashed red). The inset shows a zoom-in of the torque at late times. Bottom: The $z$-component of the torque (i) measured directly from the simulation (same as in the top panel, black), (ii) calculated from equation \ref{['torque_Kim08_general_explicit']} with $c_\mathrm{s,0}$ taken as the mean sound speed $\overline{c}_\mathrm{s}$ inside the surface $\rho=\rho_\mathrm{c}$ and $\rho_0$ taken as the density on this surface (dashed magenta), and (iii) the same but now $\rho_0$ is taken to be $0.44$ times the value on the surface (solid magenta). The particle Mach number $\mathcal{M}_\mathrm{p}$, obtained by dividing the particle tangential speed in the particle CM frame by $\overline{c}_\mathrm{s}$, is shown on the right axis in solid green.
• Figure 3: Top left: Density contours at $\rho=0.01\rho_\mathrm{max}(t)$, $0.006\rho_\mathrm{max}(t)$, and $0.005\rho_\mathrm{max}(t)$ in the orbital plane at the time $t=188.7\,{\rm d}$, with ellipse fitted to the contour $\rho=\rho_\mathrm{c}=0.006\rho_\mathrm{max}(t)$, which was found to enclose effectively all of the gas producing significant torque (see also Figure \ref{['fig:torque']}). The ellipse is phase-shifted by an angle $\Delta\phi$ with respect to the axis that passes through the particles. Top right: Similar to the top-left panel but now showing the plane perpendicular to the orbital plane and rotated clockwise by the angle $\Delta\phi$ about the orbital axis, shown by the dashed line in the top left panel. The length of the ellipse major axis is set equal to that in the orbital plane, but the length of the minor axis is allowed to differ. Bottom left: Adapted from figure 1 of Kim+08, showing the steady state for $\mathcal{M}_\mathrm{p}=0.6$ sliced through the orbital plane in their idealized double-perturber model. The black circle shows the orbit of the point masses that perturb the background density. Colour shows the density contrast $\log\mathcal{D}$, where $\mathcal{D}=(c_\mathrm{s,0}^2a/Gm)\lambda$ with $m$ the binary mass and $\lambda=(\rho-\rho_0)/\rho_0$. Overplotted for comparison is the time-averaged best fit ellipse in the orbital plane from our simulation, with parameter values noted on the plot (see also Figure \ref{['fig:params']}). Bottom right: Similar to the bottom left panel but now showing $\log\mathcal{D}$ for our simulation, at the same time as the top row, when $\rho_0=0.44\rho_\mathrm{c}=3.16\times10^{-6}\,{\rm g\,cm^{-3}}$, and $c_\mathrm{s,0}=\overline{c}_\mathrm{s}=93.0\,{\rm km\,s^{-1}}$. The region outside $0.44\rho_\mathrm{c}$ has negative values of $\mathcal{D}$ and is coloured grey. The contour $\rho=\rho_\mathrm{c}$ is plotted in yellow. The white contours near the softening radius (black circles, AGB core on the left and companion on the right) show contours of $\log\mathcal{D}=1.2\,\mathrm{and}\,0.8$, while the contours outside $\rho=\rho_\mathrm{c}$ show $\log\mathcal{D}=-0.4, -0.8\,\mathrm{and}\,-1.2$.
• Figure 4: Relative root mean squared error (RMSE) of ellipse fitting in both planes for the ellipsoid model discussed in Section \ref{['sec:spheroid']}. (the highest value is set to unity and other values are calculated with respect to it). We observe a higher RMSE in the perpendicular plane as the major axis in this plane is forced to have the same value as that in the orbital plane. We start our analysis at $t=125\,{\rm d}$.
• Figure 5: Time evolution of key fit parameters for the lagging spheroid model: (i) ratio of semi-major axis $A$ to separation $a$, (ii) ratios of semi-minor axis ($B$ in the orbital plane and $C$ in the perpendicular plane) to semi-major axis $A$, and (iii) lag angle $\Delta \phi$ between the binary axis and the major axis of the fitted ellipsoid (right axis). The simulation output (thin lines) oscillates rapidly with time. Thick lines show $10\,{\rm d}$-moving averages and the dotted lines show mean values over the time domain of the analysis (125 days onward).