Spin and Obliquity Distributions of Low-mass Planets Shaped by Dynamical Instability

TL;DR

This paper investigates how dynamical instability and planet-planet collisions shape the spin magnitudes and obliquities of low-mass exoplanets (super-Earths and mini-Neptunes). Using large ensembles of N-body simulations with sticky-sphere merger prescriptions for two- and three-planet systems, the authors find that collision products typically exhibit obliquities with cos(θ_SL) distributed nearly uniformly, while the spin magnitude distribution is approximately linear in |S|/S_max. Parameter studies show that larger planetary radii or masses and smaller initial inclinations tend to polarize obliquities toward ±1 and flatten the high-spin tail, with trends explained via an analytic framework relating collision geometry to angular momentum transfer. The results generalize across two- and three-planet configurations and offer insights into the rotational states of post-collision planets, informing later tidal evolution and observational interpretation of exoplanet spin properties, despite simplifications such as neglecting mass loss and multi-collision histories.

Abstract

Exoplanetary systems hosting multiple low-mass planets are thought to have experienced dynamical instability, during which planet-planet collisions and mergers occur; these collisions can impart substantial amount of angular momentum to the merger remnants, changing the obliquities of the resulting planets significantly. In this work, we carry out a series of $N$-body experiments to investigate the spin magnitude $(|\vec{S}|)$ and obliquity $(θ_{\rm SL})$ distributions of low-mass exoplanets that have gone through planetary collisions. In our fiducial super-Earth (with $m=3M_{\oplus}$, $R=1.3R_{\oplus}$) and mini-Neptune systems (with $m=9M_{\oplus}$, $R=2.5R_{\oplus}$), the collision products follow a nearly uniform distribution in $\cos{θ_{\rm SL}}$ and the spin-magnitude distribution is approximately linear in $|\vec{S}|$. Parameter studies and theoretical analysis show that increasing planetary radii or masses, or decreasing the initial planet-planet mutual inclinations, tend to polarize the obliquity distribution toward alignment or anti-alignment (i.e., excess probability near $\cos{θ_{\rm SL}}=\pm1$). Experiments with initially two-planet and three-planet systems produce qualitatively similar outcomes, suggesting that the trends in this study may generalize to systems with higher planetary multiplicities.

Figures (6)

• Figure 1: Spin and obliquity distributions due to planet-planet collisions in our fiducial simulations. The top row is from the super-Earth simulations (i.e., with $m_1 = m_2 = 3M_{\oplus}$ and $R_1=R_2=1.3R_{\oplus}$), while the bottom row is the mini-Neptune simulations (i.e., with $m_1 = m_2 = 9M_{\oplus}$ and $R_1=R_2=2.4R_{\oplus}$). The first column shows the spin magnitudes and obliquities of the merger products found in our runs. The second and third Columns show the obliquity and spin magnitude distributions of the merger products. All simulations adopt $a_1=0.1$au and initial inclination $i_{1,2} \in (0^\circ, 2^\circ)$. The black lines are the analytical distributions from LJR.2020.ApJ, see also Equations \ref{['eq:f_cos']} and \ref{['eq:f_S']} from Section \ref{['subsec:reason']}.
• Figure 2: Parameter study for the spin and obliquity distributions of the collision products. Top: for planets with $m_1 = m_2 = 4.0M_{\earth}$, $i_{\rm max}=0.5^{\circ}$, and different physical radii. Middle:: for planets with $R_1 = R_2 = 2.0R_{\earth}$, $i_{\rm max}=0.5^{\circ}$, and different masses. Bottom: for planets with $R_1 = R_2 = 2.0R_{\earth}$, $m_1 = m_2 = 4.0M_{\earth}$, and different $i_{\rm max}$.
• Figure 3: The obliquity and spin distribution of merger products with different fixed $i_{\rm init}$ when $R_1 = R_2 = 1.3R_{\earth}$ and $m_1 = m_2 =3M_{\earth}$.
• Figure 4: Obliquity and spin magnitude distribution of merger products from the fiducial three-planet simulations (see the upper panel of Table \ref{['tab:three-planet-sim-list']}). The top row shows the results for the super-Earth systems, and the bottom row is for the mini-Neptune systems. All simulations adopt $a_1=0.1$au and initial inclination $i_{1,2} \in (0^\circ, 2^\circ)$. The distributions shown separately for three different collision channels: mergers between the inner pair ($m_1$ and $m_2$, green curves), the innermost and outermost planets ($m_1$ and $m_3$, purple histograms), and the outer pair ($m_2$ and $m_3$, orange curves).
• Figure 5: Final orbital elements in our fiducial three-planet simulations after collisions. The results for the super-Earth and mini-Neptune cases are shown as filled and stepped histograms, respectively. Top: Semi-major axis distributions of the planets. The initial values of $a_{1,2,3}$ are marked as vertical lines. Middle: Mutual orbital inclination distributions of the planets. Bottom: Eccentricity distributions of the planets, where we plot the results for the merger products and the non-collided planets separately.