Free-floating Planets Produced by Planet-Planet Scatterings: Ejection Velocity and Survival Rate of Their Moons

TL;DR

This study addresses whether moons can survive the ejection of planets produced by planet–planet scatterings and what the ejection velocities of the resulting free-floating planets are. It combines an analytic CR3BP approach based on Jacobi energy to bound $v_\infty$ with extensive $N$-body simulations for two- and three-planet systems, and compares to circumbinary ejections. Key contributions include a closed-form boundary for ejection velocity from $E_J$, a minimum planetary mass needed for ejection, and robust characterizations of $v_\infty$ distributions and moon-survival probabilities across system architectures; moon-survival proxies rely on the minimum planet–planet separation prior to ejection. The results show that a substantial fraction of moons can endure ejection, and the ejection velocity traces the parent system's masses and separations, offering a practical diagnostic for the origin of FFPs and guiding future exomoon searches around these nomad planets.

Abstract

The discovery of numerous free-floating planets (FFPs) has intensified interest in their origins and dynamical histories. A leading formation mechanism is planet-planet scatterings in unstable multi-planetary systems, which can naturally lead to planetary ejections. If these planets originally host moons, it remains an open question whether such satellites can remain gravitationally bound to FFPs after ejection. In this work, we investigate both the ejection velocity of FFPs produced by planet-planet scatterings and the survival rate of their potential moons; we estimate the latter by determining the statistics of the minimum planet-planet distance prior to planet ejection, and comparing it to the initial orbital radius of the moon relative to its host planet. Using the circular restricted three-body framework, we derive an analytical boundary for the ejection velocity based on Jacobi energy conservation, which agrees with the results of integrations. We also identify a minimum planetary mass required for successful ejection. For two-planet systems with finite planetary masses, we use simulations and analytical arguments to determine how the ejection velocity scales with the planetary mass and initial semi-major axis. We contrast our results to the ejection of planets around binaries in unstable orbits. Extending our analysis to three-planet systems yields similar results, reinforcing the robustness of our conclusions. These findings offer insights into the property of FFPs and inform future efforts to search for exomoons around them.

Figures (16)

• Figure 1: Velocity at infinity vs. parameter $\alpha$ for ejected test particles, where $v_{\infty}$ is in units of $\sqrt{G(M+m_1)/a_1}$, and $\alpha$ is related to the final pericenter distance of the particle. From left to right, the columns correspond to systems with $m_1=10M_{\rm J}$, $5M_{\rm J}$, $1M_{\rm J}$, respectively. In the upper panels, ejection events are classified into two groups: the black dots denote the cases where the scattered particle remains in the orbital plane of the inner planet, while the green dots represent non-coplanar configurations. In the bottom panels, the boundaries of $v_{\infty}$ for the co-planar group are fitted analytically by Equations (\ref{['eq_vana']}) and (\ref{['eq_qf']}), where the red and blue lines represent $E_J=E_{J,\max}$ and $E_{J,\min}$ respectively.
• Figure 2: The cumulative distribution (CDF) and complementary cumulative distribution (CCDF) of minimum separation between the planets, $r_{12,\min}$ (scaled by the mutual Hill radius $R_{\rm H}$) in systems of different planetary masses. The black, red and blue lines represent systems with $m_1=10M_{\rm J}$, $5M_{\rm J}$, $1M_{\rm J}$, respectively, with the stellar mass $M=1M_{\odot}$. Other parameters are the same for each system, with mass ratio $m_1/m_2=10$, initial semi-major axes $a_1=1$au, $a_2=a_1+2R_{\rm H}$, eccentricity $e_1=e_2=10^{-5}$, inclination $i_1=i_2=R_{\rm H}/a_1$. Initial argument of the pericenter, longitude of the ascending node, and mean anomaly are uniformly and randomly chosen in the range $[0, 2\pi]$.
• Figure 3: Same as Figure \ref{['figrm1']}, but in systems of different planetary mass ratios and with $m_1$ fixed at 10$M_{\rm J}$. The black, red, blue, and magenta lines represent systems of $m_2=1M_{\rm J}$, $2M_{\rm J}$, $5M_{\rm J}$, $10M_{\rm J}$, respectively, corresponding to mass ratio of 10, 5, 2, 1.
• Figure 4: Same as Figure \ref{['figrm1']}, but in systems of different initial inclinations and with $m_1$ fixed at $10M_{\rm J}$, $m_1/m_2$ fixed at 10. Initial inclinations of $i=i_1=i_2=R_{\rm H}/a_1$, $0.5R_{\rm H}/a_1$, $0.1R_{\rm H}/a_1$, $0.01R_{\rm H}/a_1$ are denoted by black, red, blue, and magenta lines, respectively.
• Figure 5: Same as Figure \ref{['figrm1']}, but in systems of different initial planetary separations and with $m_1$ fixed at $10M_{\rm J}$, $m_1/m_2$ fixed at 10. Initial planetary separation $a_2-a_1 =kR_{\rm H}$, with $k=2.5$, 2, 1.5, 1 are denoted by blue, black, red, and magenta lines, respectively.