The PAIRS project: a global formation model for planets in binaries. II. Gravitational perturbation effects from secondary stars

TL;DR

PAIRS extends the Bern planet formation framework to $S$-type binaries by embedding the secondary star's gravitational perturbations into a wide-binary N-body integrator and by incorporating disk truncation effects. Through three simulation sets (A: parameter grid, B: one-embryo in binary vs single-star, C: multi-embryo in binary), the study quantifies how disk truncation and stellar perturbations reshape formation pathways and final architectures. Key results show that planets forming beyond about $0.4R_{\rm trunc}$ are frequently lost, and higher binary eccentricities amplify planetary eccentricities and destabilize systems, with additional suppression of growth when embryos are further from the primary. Multi-embryo runs reveal that binary perturbations plus planet-planet interactions can dramatically alter system outcomes even inside nominally stable regions, underscoring the need for a fully coupled, population-level treatment of planet formation in binaries.

Abstract

Roughly half of Sun-like stars have at least one stellar companion, whereas it is widely assumed that most known exoplanets orbit single stars, largely due to observational biases. However, astrometric surveys, direct imaging, and speckle interferometry are steadily increasing the number of confirmed exoplanets in binaries. A stellar companion introduces additional effects, such as circumstellar disk truncation and gravitational perturbations, which can strongly impact planet formation. While global planet formation models, for example the Bern model, have been broadly applied to single stars, modeling S-type binaries requires key modifications to capture these effects. This study extends the Bern model by incorporating the gravitational influence of a stellar companion into its N-body integrator, allowing us to quantify how this perturbation affects planetary formation and final system architecture across a range of binary configurations. By comparing binary and single-star systems under identical initial conditions, we can assess the specific impact of binary-induced dynamics. We ran three sets of simulations: (i) a grid of in situ single-embryo cases to quantify gravitational effects; (ii) formation simulations with and without migration to compare outcomes with single-star analogs; and (iii) multi-embryo runs to evaluate impacts on multi-planetary systems. Planets forming beyond half the host star's Hill radius are much more likely to become unbound especially in systems with high binary eccentricity. Even within stable zones, growth is suppressed by both reduced material availability and increased eccentricity from stellar perturbations. Both disk truncation and stellar perturbations must be included to model planet formation in S-type binaries accurately. Neglecting either one will end up misrepresenting planetary growth and survival.

Figures (19)

• Figure 1: Simulation set A: Initial planet location relative to the disk truncation radius (top: $a_{\text{p,0}}/R_\text{trunc}$), and to the primary star’s Hill radius (bottom: $a_{\text{p,0}}/R_\text{Hill,1}$). Top panel: Color bar shows the final planet mass and the gray dashed line marks $0.4\cdot R_\text{trunc}$. Bottom panel: Color bar shows the binary mass ratio and the gray and black dashed lines mark $1/3$ and $1/2\cdot R_\text{Hill,1}$. Circles denote surviving planets, while crosses indicate planets lost by ejection or instability from close encounters with the central star.
• Figure 2: Simulation set A: Initial planet location relative to the binary periastron distance ($a_{\text{p,0}}/r_\text{periastron}$) versus mean planet eccentricity. The color bar shows the binary eccentricity and purple dashed lines indicate the equilibrium eccentricity, $e_\text{p}^\text{(eq)}$, Mardling2007 for three binary eccentricities. Circles denote surviving planets, while crosses indicate planets lost by ejection or instability from close encounters with the central star.
• Figure 3: Simulation set A: Initial planetary position relative to the disk truncation radius, $a_{\text{p,0}}/R_\text{trunc}$, versus the final planet mass. The color bar shows the binary periastron distance. Circle markers represent stable planets from our simulations (effect of disk truncation + gravitational perturbation inside the N-body integrator), while green crosses represent stable planets from Paper I simulations (effect of disk truncation only).
• Figure 4: Simulation set A: Initial planetary position relative to the binary separation, $a_{\text{p,0}}/a_\text{bin}$, versus the binary mass ratio (top panel) and initial planetary position relative to the disk truncation radius, $a_{\text{p,0}}/R_\text{trunc}$, versus the binary eccentricity (bottom panel). We display planets that became at least more massive than Mars and their color reflects their final planet mass. The shaded contours highlight regions of highest planet density, obtained through a two-dimensional kernel density estimation (KDE) considering planets with masses above 10 M$_\mathrm{\oplus}$.
• Figure 5: Simulation set B (in situ): time evolution of planet mass (top row) and eccentricity (bottom row), color-coded by binary separation. The embryo is placed at 5 au (left column) and 20 au (right column). In the top row, the dashed line represents the core mass, while the solid line shows the total planet mass. Note: the x-axis and y-axis scales are different in all panels in the top row.