Planet-Planet Scattering Explains the Mass-Eccentricity Relation of Warm Jupiters

Abstract

Warm giant planets with orbital periods of tens of days exhibit a positive correlation between mass and eccentricity. We interpret this trend as the outcome of planet-planet scattering, representing a transition from collision-dominated interactions among low-mass planets to ejection-dominated interactions among high-mass planets. This framework has important implications for warm Jupiter origins. It suggests that warm Jupiters originate from compact, multi-planet configurations. The dynamical interactions that shape their present-day architectures likely occur near their current semimajor axes, regardless of whether warm Jupiters formed through convergent disk-driven migration or in-situ formation. We argue that several observed properties of warm Jupiter systems, including the eccentricity bimodality, the mass-eccentricity relation, and generally low stellar obliquities, can be explained by this picture. We further predict that not only circular warm Jupiters, but also eccentric warm Jupiters, should frequently have additional planetary companions that are detectable through radial velocity observations. Finally, scattering can produce eccentricities high enough to trigger high-eccentricity tidal migration, potentially explaining the emerging population of proto-hot Jupiters on tidal migration tracks.

Figures (3)

• Figure 1: Panel (a): Warm Jupiter eccentricity as a function of planet--star mass ratio. Panel (b): Warm Jupiter eccentricity as a function of semimajor axis. Blue circles denote warm Jupiters with confirmed planetary companions, while orange circles indicate those in single or uncertain system architectures. Additional symbols mark the type of companions: circles for Jovian planets ($M_c \geqslant 0.3\,M_{\rm Jup}$) and squares for sub-Jovian planets ($M_c < 0.3\,M_{\rm Jup}$). The color of these symbols indicates the companion separation, with black corresponding to $< 10$ mutual Hill radii and grey to $\geqslant 10$ mutual Hill radii. Mutual Hill radii are computed using companion masses inferred from transit-timing variations (TTVs) or RVs. Companion masses from RVs retain the usual $M_c\sin{i}$ degeneracy. Error bars show $1\sigma$ measurement uncertainties. The dashed and dash-dotted curves represent the maximum eccentricity expected from planet--planet scattering, assuming scattering of equal-mass planets with the additional properties marked on the curve. The shaded grey region marks the parameter space consistent with high-eccentricity tidal migration, bounded by tidal dissipation efficiency and the Roche limit.
• Figure 2: Similar to Figure \ref{['fig:obs_e']}, but now showing $e / e_{\rm sc}$ on the vertical axis. With this normalization, the previously observed paucity of high-eccentricity warm Jupiters at low mass ratios or small semimajor axes is no longer apparent. The hatched regions mark the forbidden parameter space where $e > 1$, computed for a Jupiter-radius planet at $0.2\,\mathrm{au}$ in panel (a) and for a Jupiter-mass, Jupiter-radius planet in panel (b).
• Figure 3: $e/e_{\rm sc}$ as a function of $e_{\rm sc}$ for different companion mass assumptions: $M_c/M_p = 1$, $M_c/M_p \sim \log U(0.5,2)$, and $M_c/M_p \sim \log U(0.1,10)$. Symbols are the same as those used in Figure \ref{['fig:obs_e']}. The hatched region indicates the nonphysical regime with $e > 1$. In all cases, the distributions are nearly uniform in $e/e_{\rm sc}$, showing no clear dependence on $e_{\rm sc}$.