Planetary Desert around Compact Binaries: Dynamical Instability Triggered by Resonance-Induced Eccentricity Excitation

Abstract

Compact binaries with orbital periods shorter than about 7 days show an absence of transiting planets, a feature known as the ``circumbinary planet desert". The physical mechanism behind this desert remains unclear. We investigate its origin by simulating the long-term dynamics of multi-planet circumbinary systems with evolving inner binaries. Our simulations are based on the single-averaged secular equations that average only over the binary orbital period and fully incorporate planet-planet interactions. When an eccentric binary decays via tides, an outer planet can be captured into resonance advection in eccentricity, a state in which its apsidal precession locks with that of the binary, driving extreme eccentricity growth. While such growth can occur in a binary-single planet system, the parameter space is limited and may not necessarily induce instability. In a multi-planet system, however, the excited orbit inevitably crosses those of its neighbors, which triggers violent planet-planet scatterings and produces collisions or ejections. Crucially, these mutual gravitational interactions amplify the ``localized" instability of a single planet into a system-wide chain reaction, drastically reshaping the orbital architecture and potentially clearing out the inner regions of planetary systems. Our results suggest that the resonance-induced instability provides a natural explanation for the observed circumbinary planet desert.

Figures (6)

• Figure 1: Evolution of a stellar binary (black) with four (non-interacting) circumbinary planets in a coplanar configuration. The masses of the inner stellar binary are $m_1=0.56M_\odot$ and $m_2=0.46M_\odot$. The initial semimajor axis and eccentricity of the inner binary are $a_{b,0}=0.09\mathrm{AU}$ and $e_{b,0}=0.8$. The outer orbits, ordered by increasing distance, have initial semimajor axes $a_p=0.72\mathrm{AU}$, $1.24\mathrm{AU}$, $2.13\mathrm{AU}$ and $2.81\mathrm{AU}$. All four begin with eccentricity $e_{p,0}=0.001$. The longitudes of pericenter for both the binary and the planetary orbits are initialized to $\varpi_{b,0}=\varpi_{p,0}=0$. Each binary–planet system is evolved independently (i.e., without planet–planet interactions) by integrating the single-averaged (SA) secular equations (Equations \ref{['eq:Full Kozai 1']}-\ref{['eq:third body']}), where the tidal dissipation is included in the binary evolution (top and middle panels). The bottom panel shows the corresponding apsidal precession rates computed from Equations (\ref{['eq: precession rate inner']}) and (\ref{['eq: precession rate outer']}).
• Figure 2: Parameter space in the $a_p-e_{b,0}$ plane showing the regime for resonance capture and advection during the orbital decay of the stellar binary. The orbital period $T_p$ for planets at different distances is labeled at the top. We choose the same stellar binary as in Figure \ref{['fig:SA evolution']}, varying only the initial binary eccentricity $e_{b,0}$. The orange region marks systems that are dynamically unstable according to Equation (\ref{['eq: aoutc']}) by considering initially circular planetary orbits. The initial orbits of the planets are spaced according to the mutual Hill instability criterion with $K_{\mathrm{c}} = 10$ in Equation (\ref{['eq: Hill radius']}). The green region corresponds to systems in which the initial apsidal precession rate of the inner binary exceeds the precession rate of the planetary orbit. The pink region represents systems for which the binary semimajor axis never shrinks to the value required to trigger apsidal precession resonance before tidal decay ceases. Systems in the green and pink regions would not experience resonance during the binary orbital decay. For each planetary orbit, we integrate the single-averaged (SA) secular equations until tidal evolution ends and record the final eccentricity $e_{p,\mathrm{f}}$, whose value is represented by the size of the circles. The red crosses indicate orbits that become unbound in the numerical integration—either the semimajor axis turns negative or the eccentricity exceeds unity.
• Figure 3: Similar to Figure \ref{['fig:SA evolution']}, but including mutual gravitational interaction between two CBPs. The initial longitude of pericenter of each planet is randomly chosen between $0$ and $2\pi$. The evolution of the system can lead to either collision or ejection. Left panel: Two planets with initial semimajor axes $a_p=2.13\mathrm{AU}$ and $1.24\mathrm{AU}$ collide into a single body. Right panel: One planet initially located at $a_p=2.13\mathrm{AU}$ is ejected from the system.
• Figure 4: Same system as in Figure \ref{['fig:SA evolution']}, but with mutual gravitational interactions among all four planets. The initial longitude of pericenter of each CBP's orbit is randomly chosen in the range [$0-2\pi$]. The eccentricity of the planetary orbit grows significantly, even for the planet that does not undergo resonant eccentricity-excitation in isolation. The resulting orbital crossings trigger strong interactions between planets, which reshape the orbital architecture and lead to either an ejection event (left panel) or a collision event (right panel).
• Figure 5: The evolution of $9-$planet system during tidal decay of the host binary, with mutual planet-planet gravitational interactions included. The system parameters are taken from Figure \ref{['fig:parameter space']}, with $e_{b,0}=0.8$ as a fiducial value: The planets are ordered by increasing distance with semimajor axes $a_p=0.55\mathrm{AU}$, $0.72\mathrm{AU}$, $0.94\mathrm{AU}$, $1.24\mathrm{AU}$, $1.63\mathrm{AU}$, $2.13\mathrm{AU}$, $2.81\mathrm{AU}$, $3.69\mathrm{AU}$, $4.85\mathrm{AU}$. Mutual interactions lead to frequent planetary collisions, with some planets experiencing repeated collisions, e.g., O2 and O4 merge into O4', O1 and O5 merge into O5', and then O4' and O5' collide to form O5".