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

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.