Self-Limited Accretion onto Embedded Binaries in a Uniform Medium

Abstract

We study accretion from a uniform gas at rest onto equal-mass binaries -- the binary Bondi problem -- as a function of adiabatic index~$γ$ and compactness $ξ\equiv R_B/a$, where $R_B$ is the Bondi radius of the binary and $a$ is the component separation. We present three-dimensional hydrodynamic simulations spanning $ξ= \{0.1, 1, 10\}$ at $γ= \{1, 4/3, 5/3\}$. Isothermal gas ($γ= 1$) accretes cooperatively at high compactness, with efficiency $η\equiv \dot{M}_{\rm binary}/\dot{M}_{\rm Bondi} \to 1$ for $ξ\gg 1$ and a stable sonic surface that screens the orbital modulation. Adiabatic gas ($γ> 1$) is self-limiting: the orbit drives shocks that generate entropy, producing convective turbulence that suppresses accretion to $η\approx 0.3$ ($γ= 4/3$) and $η\approx 0.1$ ($γ= 5/3$), burying the orbital signature in broadband noise. We derive a stability criterion from first principles: the sonic surface is the separatrix of the Bondi saddle point, and the binary annihilates it in $N \propto (γ-1)^{-1}(\sqrt{ξ/ξ_m} - 1)$ orbits, where $ξ_m = 4/(5{-}3γ)$ is the container threshold at which the sonic surface first encloses the binary, and the $(γ-1)^{-1}$ divergence follows from the lack of entropy generation at isothermal shocks. For $γ= 5/3$, no saddle point exists at any~$ξ$ and the neutrally stratified Bondi profile is convectively unstable by a distinct mechanism. The single comparison $t_{\rm cool}$ versus $NT$ -- where $T$ is the orbital period -- determines whether an embedded binary accretes cooperatively or throttles its own fuel supply; simulations confirm the analytic thresholds and scaling.

Figures (7)

• Figure 1: Density gradient schlieren $|\nabla \ln \rho|$ with velocity streamlines overlaid, showing flow morphology across compactness $\xi$ for isothermal ($\gamma=1$, top) and relativistic ($\gamma=4/3$, bottom) gas. Isothermal flows remain stationary and organized at all $\xi$: independent accretion at $\xi=0.1$, coherent spiral structure at $\xi=1$, and radial monopole-like inflow at $\xi=10$. Polytropic flows exhibit similar organization at low $\xi$ but become turbulent at high $\xi$: $\xi=10$ shows complete loss of coherent structure with chaotic, multi-scale density fluctuations. The transition from organized to turbulent occurs between $\xi=1$ and $\xi=10$ for $\gamma=4/3$, consistent with the predicted threshold Eq. \ref{['eq:xi_merge']} where shock-generated entropy overwhelms the sonic buffer. All above plots are shown at $t \geq 3t_B$ for each respective $\xi$.
• Figure 2: Density gradient schlieren $|\nabla\ln\rho|$ in the orbital plane at $t = 3\,t_B$ for $\xi = 10$, zoomed to $r \leq 3a$. Top: for $\gamma = 1$, the flow is laminar; the only structure is the high-density bridge connecting the two accretors. Bottom: for $\gamma = 4/3$, shocks co-rotating wih the binary at $r \sim 2a$ marking the disrupted sonic surface. Inside it, the flow is fully turbulent -- filaments, vortices, and secondary shocks fill the interior. This is the entropy instability in action: orbital stirring drives shocks that generate entropy, inflating the post-shock gas and destroying the transonic structure.
• Figure 3: Density gradient schlieren $|\nabla \ln\rho|$ for $\gamma = 5/3$ in the orbital plane at $t \geq 3\,t_B$. The flow is disordered at all $\xi$: no sonic surface exists to shield the outer flow. Turbulent structure extends to $\sim 30\,a$ at $\xi = 10$.
• Figure 4: Time- and angle-averaged radial profiles at $\xi = 10$: density (top left), squared sound speed (top right), absolute radial velocity (bottom left), and turbulent velocity dispersion via Reynolds decomposition (bottom right). Gray dotted lines show the analytic Bondi profiles; reference power laws are labeled. Lighter traces show the temporal variance of the angle-averaged profiles.. For $\gamma = 1$ (solid, pink), all profiles track the Bondi solution and $\delta v$ is negligible: the flow is laminar. For $\gamma = 4/3$ (dash-dotted yellow), the density profile obeys the Bondi character reasonably well, but the $\gamma = 5/3$ flattens towards $\rho \propto r^{-1}$ inside $r \sim 5a$ while the temperature profiles $c^2 \propto r^{-3(\gamma-1)/2}$ for both are preserved. The radial velocity is suppressed by factors of a few below freefall; the turbulent dispersion exceeds it by a factor of a few to $\sim10$, confirming that the interior is a turbulence-dominated reservoir with accretion as a residual.
• Figure 5: Angular power spectrum of the entropy proxy $\kappa = P/\rho^\gamma$ at $\xi = 10$. For each of seven logarithmically spaced shells in $r/a \in [1, 10]$, the fluctuation field $\delta\kappa = \kappa - \langle\kappa\rangle_{\rm shell}$ is interpolated onto an equirectangular $(\theta,\phi)$ grid and decomposed via 2D FFT with $\sin\theta$ weighting; the resulting $C_\ell$ are averaged across shells and mapped to spatial wavenumber $k = \ell/r$. The isothermal case ($\gamma = 1$) returns $C_\ell \sim 10^{-38}$ at all scales, confirming that the adiabatic spectra reflect genuine entropy production rather than numerical or geometric artifacts. Both $\gamma = 5/3$ and $\gamma = 4/3$ follow $k^{-7/2}$ over the resolved range $k \lesssim k_a$, steeper than Kolmogorov ($k^{-5/3}$) or Burgers ($k^{-2}$) scaling, consistent with a spectrum set by isolated shock fronts and convective boundaries rather than a volume-filling cascade. The $\gamma = 5/3$ spectrum carries more power at all scales, reflecting larger per-shock entropy jumps in the stiffer equation of state. The upturn above $k_a$ is a pixel window artifact of the equirectangular interpolation and should not be interpreted as physical.