A revision of the lifetime of submoons: tidal dynamics with the Euler-Lagrange equation

TL;DR

The study tackles the lifetime and stability of submoons in a nested star–planet–moon–submoon system by deriving the Euler–Lagrange equations under a Constant Geometric Lag tidal model and solving the resulting ODEs numerically. This full tidal-network approach extends prior two-body analyses, enabling exploration of how planet- and moon-tide migration interact to modulate submoon survivability over gigayear timescales. Key findings show that an Earth-like system could harbor an asteroid-sized submoon only if the moon’s orbit is modestly outward of its current position, while a Kepler-1625-like system could sustain submoons as massive as $\sim 1.8\,M_\oplus$ provided the moon’s orbit is wide (>$100\,R_p$); a valley of reduced stability appears at intermediate submoon masses. The results imply habitable submoons may be rare and underscore the need for more physically consistent tidal models; the framework, though exploratory, lays the groundwork for future extensions to more complex orbital geometries and dissipation laws with potential astrobiological implications.

Abstract

Submoons, moons orbiting other moons, may be exotic environments capable of hosting extraterrestrial life. We extend previous studies to revise the maximum lifetime of these objects due to planetary, lunar and sublunar tidal migration. Using the Euler-Lagrange equation with a tidal dissipation process as specified by the Constant Geometric Lag model, we derive and solve the governing equations numerically to map the semi-major axis parameter space for star-planet-moon-submoon systems in which the submoon could be massive enough to host life. We find that Earth could have hosted asteroid-sized submoons ($\sim10^{15}\mathrm{kg}$), whereas a submoon near the previously proposed upper limit ($\sim4.6\cdot10^{17}\mathrm{kg}$) would have driven the Moon $\sim30\%$ farther from Earth than its current orbit. A Warm Jupiter system like Kepler1625 has greater potential of hosting a massive submoon. We found that a submoon of around $10\%M_{\text{Luna}}$ could survive if Kepler1625b's hypothesized moon were $68\%$ farther away then what the best-fit model suggests ($67R_{\mathrm{p}}$ instead of $40R_{\mathrm{p}}$). Giant submoons of mass $1.8M_{\oplus}$ are stable in a Kepler1625-like system. In these cases, the moon orbit is wide ($> 100R_{\mathrm{p}}$). Decreasing the submoon mass to a habitability prerequisite of $0.5M_{\oplus}$, likely needed for a stable atmosphere and plate tectonics, leads to a smaller total number of stable iterations relative to the $m_{sm}=1.8M_{\oplus}$ case. In fact, we identified a minimum number of stable iterations on intermediate submoon mass-scales of around $0.1M_{\oplus}$. This is likely due to an interplay between small tidal forces at small submoon masses and small Roche-Limits at very high submoon masses. If submoon formation pathways in Warm Jupiter systems prefer such intermediate mass-scales, habitable submoons could be a rare phenomenon.

Figures (6)

• Figure 1: A satellite tidally disturbing its host. The tidal bulge forms at a location that is shifted by $\delta_j$ with respect to the position of the tide-raising satellite on the sky. In this case, the host's spin rate exceeds the satellite's orbital frequency, leading to the mass concentration being carried ahead of the satellite. It is the mass concentration at this angle that has the most important contribution to the non-central exterior potential and thus it enters in the tidal energy, Eq. \ref{['eq: i-j-th Tidal energy']}. The little mass packets making up the tidally disturbed body are characterized by the surface angle $\psi_j$ with the change in this angle being the host's spin frequency $\Omega_j$.
• Figure 2: Network of inter dependencies of the variables of interest. The variables are not only coupled through the differential equations, but also through the Hill-Radius, which is computed in order to calculate the upper escape bound, the critical semi-major axis, $a_{\text{crit}}$. Since the semi-major axes change in time, a numerical solver needs to calculate the upper bound $a_{\text{crit}}$ dynamically in each time step in order to not trigger premature termination.
• Figure 3: (a) Effect of changing the $k$ parameter in the $\mathrm{tanh}(k(\cdot))$ model, (b) Particularly unsmooth evolution exhibited by the $\mathrm{sgn}(\cdot)$ model and (c) small scale oscillations in the $\mathrm{sgn}(\cdot)$ model.
• Figure 4: Stability regions of an Earth-like submoon system. Shown are point clouds in the semi-major axes parameter space which are color-coded according to how long a system described by each of these points is stable before tides act to remove or destroy the moon, submoon or planet. The two rows describe systems of different submoon masses ($10^{15}\mathrm{kg}$ for upper and $4.6\cdot 10^{17}\mathrm{kg}$ for lower). The left columns show initial states, the right columns the tidally evolved initial states. In each plot, the green pillar marks the approximate configuration of our current Earth-system in this parameter space (with an unconstrained $z$-axis since Earth harbors no natural submoon). The units on the $y-$ and $z-$axes are Earth-radii and Luna-radii respectively.
• Figure 5: Stability regions of a Warm-Jupiter-like submoon system. Shown are point clouds in the semi-major axes parameter space which are color-coded according to how long a system described by each of these points is stable before tides act to remove or destroy the moon, submoon or planet. The two rows describe systems of different submoon masses ($m_{\mathrm{sm}}=10^{-1}M_{\text{Luna}}$ for upper and $m_{\mathrm{sm}}=1.8M_{\oplus}$ for lower). The left columns show initial states, the right columns the tidally evolved initial states. In each plot, the green pillar approximately marks the current configuration Kepler1625b in this parameter space (with an unconstrained $z$-axis since there is no evidence to the effect of an existing submoon). The units on the $y-$ and $z-$ axes are eleven Earth-radii and four Earth-radii respectively.