An Analytical Model for the Eccentricity Cascade: Hot Jupiter Formation via S-type Instability

Abstract

A widely explored pathway for hot Jupiter (HJ) formation is high-eccentricity migration driven by von Zeipel-Lidov-Kozai cycles induced by an exterior companion. However, for a distant or low-mass companion, this mechanism typically demands that the planet's initial orbit be very nearly perpendicular to that of the companion. In previous work (Yang et al. 2025), we demonstrated that such fine-tuning can be circumvented in the HAT-P-7 system due to the presence of an intermediate body that efficiently couples the orbits of the planet and the distant companion -- a mechanism we termed the eccentricity cascade (EC). In this work, we analytically characterize the dynamics governing the EC and delineate the parameter space within which it effectively operates. Our qualitative results are as follows: (i) The proto-HJ's eccentricity is most efficiently excited when the inner triple is on the verge of dynamical instability, (ii) the addition of a distant fourth body allows this instability to be approached gradually, and (iii) the instability mechanism is closely related to the stability of circumstellar (S-type) planets in binaries. By deriving an analytic criterion for S-type instability, we obtain closed-form expressions describing the onset of the EC. Our results show that efficient HJ formation via the EC occurs across a broad range of intermediate perturbers, highlighting its potential as a robust migration channel.

Figures (11)

• Figure 1: Schematic of HJ formation via the EC mechanism. The planet (blue; subscript "p") and inner companion (gray; subscript "1") begin on nearly coplanar orbits, while the outer companion (red) is inclined by $\gtrsim 40^\circ$. The outer companion induces ZLK oscillations of the inner companion, periodically exciting its eccentricity and inclination. During high-eccentricity phases, the inner companion repeatedly perturbs the planet through weak scatterings, driving chaotic diffusion in the planet's eccentricity. This process ultimately triggers the planet's migration.
• Figure 2: Instability timescale for the planet's eccentricity to reach an extreme value of $e_p=0.99$ as a function of the stochasticity parameter $K$ defined in Equation \ref{['eq:K']}. Systems that remain stable for the full integration ($10^6 P_1$) appear as a ceiling in the distribution.
• Figure 3: Comparison between our analytical stability criterion (Equation \ref{['eq:e_crit']}) and the empirical stability boundary from HW99. The figure shows the critical companion eccentricity $e_{1,\mathrm{crit}}$ as a function of the semimajor axis ratio $a_p/a_1$ for binary mass ratios $\mu = 0.1$ and $\mu = 0.5$.
• Figure 4: Example of HJ formation in the HAT-P-7 system via the EC mechanism, adapted from Yang2025. The three panels show the evolution of semimajor axes, eccentricities and mutual inclinations. The dashed line in the top panel denotes the present-day orbital location of HAT-P-7b.
• Figure 5: Same system configuration as in Figure \ref{['fig:ec_example']}, but with a lower-mass inner companion on a wider orbit, resulting in $\Omega_{12}/\omega_{p1}=0.3$ (compared to 0.001 previously). The planet and inner companion are no longer strongly coupled, allowing their mutual inclination to increase as the inner companion undergoes ZLK oscillations induced by the outer companion. Around 13 Myr, the mutual inclination exceeds the ZLK threshold, initiating high-e migration of the planet via the classical ZLK mechanism.