Self-Limited Accretion onto Embedded Binaries in a Uniform Medium

We study accretion from a uniform gas at rest onto equal-mass binaries -- the binary Bondi problem -- as a function of adiabatic index~$\gamma$ and compactness $\xi \equiv R_B/a$, where $R_B$ is the Bondi radius of the binary and $a$ is the component separation. We present three-dimensional hydrodynamic simulations spanning $\xi = \{0.1, 1, 10\}$ at $\gamma = \{1, 4/3, 5/3\}$. Isothermal gas ($\gamma = 1$) accretes cooperatively at high compactness, with efficiency $\eta \equiv \dot{M}_{\rm binary}/\dot{M}_{\rm Bondi} \to 1$ for $\xi \gg 1$ and a stable sonic surface that screens the orbital modulation. Adiabatic gas ($\gamma > 1$) is self-limiting: the orbit drives shocks that generate entropy, producing convective turbulence that suppresses accretion to $\eta \approx 0.3$ ($\gamma = 4/3$) and $\eta \approx 0.1$ ($\gamma = 5/3$), 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 $N \propto (\gamma-1)^{-1}(\sqrt{\xi/\xi_m} - 1)$ orbits, where $\xi_m = 4/(5{-}3\gamma)$ is the container threshold at which the sonic surface first encloses the binary, and the $(\gamma-1)^{-1}$ divergence follows from the lack of entropy generation at isothermal shocks. For $\gamma = 5/3$, no saddle point exists at any~$\xi$ 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.