The Coupled Tidal Evolution of the Moons and Spins of Warm Exoplanets

TL;DR

This work investigates the coupled dynamics of warm exoplanets and their moons, addressing how star–planet tides and moon–planet tides shape spin states and obliquities. Using analytical criteria and self-consistent secular simulations, it demonstrates that many warm planets undergo Laplace-plane instability causing moon loss, while surviving moons migrate inward and are disrupted near the Roche limit, often ejecting the planet from tidal spin equilibria. Moons can drive long-lived high obliquities, especially when moon-driven spin changes dominate, and even lost moons leave lasting imprints on spin evolution, complicating obliquity predictions. The results imply that exomoon demographics are a crucial, currently uncertain factor in anticipating present-day obliquities and potential detectability of moons around close-in exoplanets.

Abstract

Context: The Solar System giant planets harbour a wide variety of moons. Moons around exoplanets are plausibly similarly abundant, even though most of them are likely too small to be easily detectable with modern instruments. Moons are known to affect the long-term dynamics of the spin of their host planets; however, their influence on warm exoplanets (i.e.\ with moderately short periods of about $10$ to $200$~days), which undergo significant star-planet tidal dissipation, is still unclear. Aims: Here, we study the coupled dynamical evolution of exomoons and the spin dynamics of their host planets, focusing on warm exoplanets. Methods: Analytical criteria give the relevant dynamical regimes at play as a function of the system's parameters. Possible evolution tracks mostly depend on the hierarchy of timescales between the star-planet and the moon-planet tidal dissipations. We illustrate the variety of possible trajectories using self-consistent numerical simulations. Results: We find two principal results: i) Due to star-planet tidal dissipation, a substantial fraction of warm exoplanets naturally evolve through a phase of instability for the moon's orbit (the `Laplace plane' instability). Many warm exoplanets may have lost their moon(s) through this process. ii) Surviving moons slowly migrate inwards due to the moon-planet tidal dissipation until they are disrupted below the Roche limit. During their last migration stage, moons -- even small ones -- eject planets from their tidal spin equilibrium. Conclusions: The loss of moons through the Laplace plane instability may contribute to disfavour the detection of moons around close-in exoplanets. Moreover, moons (even those that have been lost) play a critical role in the final obliquities of warm exoplanets. Hence, the existence of exomoons poses a serious challenge in predicting the present-day obliquities of observed exoplanets.

Figures (12)

• Figure 1: Orbital inclinations of Laplace state $\mathrm{P}_1$ and its twin $\mathrm{P}_{-1}$ (colorbar, given by Eq. \ref{['eq:i_p1']}, \ref{['eq:def_im1prime']}) as functions of $a_{\rm m} / r_{\rm M}$ and the planet's obliquity $\theta$. In both panels, the location of $a_{\rm m} = r_{\rm M}$ is denoted with a vertical black dashed line, and the $\mathrm{E}_1$ region (where $\mathrm{P}_1$ and $\mathrm{P}_{-1}$ are unstable) is shown by a white hatched zone. Across the two panels, a representative evolutionary history is shown via the three cyan points and arrows (S1, S2, S3). While the inclinations of $\mathrm{P}_1$ and $\mathrm{P}_{-1}$ only depend on $\theta$ and $a_{\rm m} / r_{\rm M}$, we also illustrate the evolution using the following concrete parameters for interpretability: $a_{\rm p} = 0.4\;\mathrm{au}$, $e_{\rm p}=0$, $R_{\rm p} = 2R_\oplus$, and $m_{\rm p} = 10M_\oplus$; We then denote the corresponding values of the planet's spin rate along the top-left axis (relevant to planetary tidal evolution) and the lunar semi-major axis along the top-right axis (relevant to lunar migration). In the left panel, the planet is initially rapidly rotating with a period of $7.5$ hours, and the moon is located at $a_{\rm m} = 5R_{\rm p}$ (S1). Then, as the planet despins and the obliquity damps, $r_{\rm M}$ decreases while the moon is still located at $a_{\rm m} = 5R_{\rm p}$, and the system reaches S2 (we have neglected lunar tidal migration for illustrative purposes). Going from S1 to S2, the inclination of the lunar orbit smoothly flips from prograde to retrograde (assuming that the moon occupies $\mathrm{P}_{-1}$). In the right panel, as the moon inspirals, starting from S2, $a_{\rm m}$ decreases until the moon is disrupted by the planet when $a_{\rm m} \simeq R_{\rm p} \gtrsim r_{\rm M}$ (S3).
• Figure 2: Example of obliquity evolution of the planet HIP 41378 f and a hypothetical moon. The mass of the moon is $m/M = 7\times10^{-4}$, and it is initialised at a distance $a_\mathrm{m}=5$$R_\mathrm{p}$. For the red, blue, and black curves, the lunar migration rate is analogous to that measured for Io and Jupiter Lainey-etal_2009. Dark colors show the trajectory when the moon is still present; light colors show the trajectory after the loss of the moon, assumed to be instantly disrupted once its pericenter goes below the Roche limit. For the red curve, the star-planet tidal dissipation is taken into account with tidal time-lag $\Delta t = 3\times 10^{-9}$ years. For the blue curve, the star-planet tidal dissipation is artificially increased using $\Delta t = 2\times 10^{-4}$ years. For the black curve, the star-planet tidal dissipation is artificially increased using $\Delta t = 2\times 10^{-2}$ years. For the green curve, the moon's migration is turned off, and the star-planet tidal dissipation is taken into account using $\Delta t = 2\times 10^{-4}$ years.
• Figure 3: Locations and widths of secular spin-orbit resonances for the exoplanet HIP 41378 f for different parameters of a hypothetical moon. The moon is placed on its Laplace equilibrium plane at a fixed distance from the planet (see labels). The three resonances reachable appear in blue, red, and green. They correspond to the resonances labelled $s_3$, $s_6$, and $s_1$ by Saillenfest-etal_2023. The hatched blue zone is the $\mathrm{E}_1$ region where the classic Laplace plane of the moon is unstable. The magenta circle highlights the fully relaxed state of the planet ($\omega=n_\mathrm{p}$ and $\theta\approx 0$). Panel c additionnaly shows the black trajectory from Fig. \ref{['fig:HIP41378f']}: the system evolves from right to left; the moon is lost in the $\mathrm{E}_1$ region, then the planet fully relaxes due to tidal dissipation.
• Figure 4: Plot of the $\mathrm{E}_1$ region for two different values of $a_{\rm m} / R_{\rm p}$. For a planet born rapidly rotating, no moon will be found in the shaded regions of parameter space, which will pass through the $\mathrm{E}_1$ region during the planet's tidal despinning and alignment. The $\mathrm{E}_1$ region can also be encountered due to lunar migration if the lunar migration timescale is faster than the planet's despinning timescale. In this case, the moon is lost if the planet's obliquity lies within about $69^\circ$ and $111^\circ$ (i.e. between the black dashed lines) independent of $a_{\rm m} / R_{\rm p}$.
• Figure 5: The different mechanisms of moon loss as a function of the system's parameters. The top panel shows the instability limit ($a_{\rm m} = 0.36\,r_{\rm H}$) for a Super Earth-like and Jupiter-like planet in solid lines, and the midpoint radius $r_{\rm M}$ (evaluated with the planet spinning at $2/3$ its maximal rate) in dashed lines. The second panel shows the regions of parameter space for the lunar mass ratio ($m_{\rm m} / m_{\rm p}$) and planet semi-major axis ($a_{\rm p}$) for a typical super Earth (SE). The grey region above the black dashed line is where moons are lost anyway through runaway migration barnes2002_moonstab. The vertical green line denotes the boundary between where planets do and do not despin within $\sim 1$ Gyr (Eq. \ref{['eq:taup']}). The blue solid curve denotes the boundary where lunar tidal migration (Eq. \ref{['eq:taum']}) and tidal planet despinning result in comparable changes to $a_{\rm m} / r_{\rm M}$ (Eq. \ref{['eq:E1_ratio']}): above this curve, $\mathrm{E}_1$ is crossed due to lunar migration; below this curve, $\mathrm{E}_1$ is crossed due to planetary despinning. The blue dashed line denotes the boundary where lunar tidal migration results in significant changes to $a_{\rm m} / r_{\rm M}$ in $\sim$ Gyr timescales. Accordingly, the region where moons are lost due to lunar migration (with or without capture in secular spin-orbit resonance) is shaded blue, and the region where moons are lost due to planetary despinning is shaded green. The vertical purple line denotes the point at which $r_{\rm M} = 5R_{\rm p}$ (Eq. \ref{['eq:rM_Rp']}). The middle panel is the same for the parameters of Kepler-79d, and the red cross denotes the fiducial parameters adopted for a hypothetical moon (see Fig. \ref{['fig:Kepler79d']}). The bottom panel is the same for a Jupiter-mass planet.