Large planets may not form fractionally large moons

Earth possesses several unique features within our solar system, including active plate tectonics, a strong magnetic field, and a disproportionately large moon relative to its size. The Moon significantly influences Earth's rotational period and ocean tides, affecting terrestrial biological cycles and contributing to the planet's climate stability. The Moon's stabilizing effect on Earth's axial tilt is crucial, minimizing variations that could otherwise lead to extreme climate fluctuations. However, the extent of this stabilization is influenced by other factors such as the planet's initial obliquity. The Moon's origin has been a subject of extensive research. The leading hypothesis posits that the Moon formed from a partially vaporized disk generated by a collision between the early Earth and a Mars-sized object. This 'giant impact' hypothesis initially appeared to explain the Earth and Moon's nearly identical isotopic ratios. However, numerical simulations have challenged this, showing the disk predominantly consists of the impactor's material, making isotopic homogenization difficult. This challenge requires either the impactor ('Theia') having a coincidental isotopic match with Earth or homogenization occurring during the disk stage. Alternative, more energetic impact scenarios have been proposed to explain the isotopic similarities more readily, involving collisions of half-Earth-sized objects. Yet these models often result in excessive final angular momentum for the Earth-Moon system. In contrast, the multiple-impact model suggests that the Moon could have formed from multiple impacts over time which helps in resolving certain issues, but it may introduce complications with isotopic signatures. The impact energy and type (size, speed, composition of impactor) significantly impact the composition of the resulting moon-forming disk, specifically its vapor mass fraction (VMF). Energetic impacts tend to produce disks with a higher VMF. The prevalence of planetary collisions during the early stages of solar system formation suggests that exomoons (moons around extrasolar planets) should be abundant. However, despite extensive searches, such as the Hunt for Exomoons with Kepler (HEK), no exomoons have been definitively confirmed. The lack of confirmation raises questions about the conditions required for the formation of large moons and the potential challenges in their detection. The limited success in finding exomoons may also be attributed to the short lifespans of moons close to their host planets due to stellar tidal forces and the inherent challenges in detecting small exomoons.

The existing literature offers diverse models for moon formation. The canonical giant impact model suggests a collision between the early Earth and a Mars-sized body formed a partially vaporized disk that later accreted into the Moon. While this model addresses the isotopic similarities, it struggles with reproducing the observed angular momentum. Other models propose more energetic impacts, involving half-Earth-sized bodies, that may solve the angular momentum issue but lead to more complete homogenization of the Earth's mantle than is geochemically observed. Multiple-impact scenarios offer another alternative, proposing the Moon aggregated from debris ejected by multiple collisions. The vapor mass fraction (VMF) of the resultant disk, ranging from vapor-poor (0.1–0.3) in the canonical model to vapor-rich (0.7–1.0) in energetic impact models, is a key parameter influencing moon formation. The prevailing view assumes that moonlets accrete from the liquid portion of the disk, eventually merging to form the Moon. Studies have explored the challenges of exomoon detection, highlighting factors like the short orbital lifetimes of moons around close-in planets and the small Hill radii of such planets that lead to orbital instability and loss of moons.

This study uses smoothed particle hydrodynamics (SPH) simulations to investigate the effect of vapor mass fraction (VMF) on moon formation. Two end-member planetary compositions are considered: a rocky planet (70% forsterite mantle, 30% iron core) and an icy planet (70% water ice, 30% forsterite core). Gas giants and mini-Neptunes are not the primary focus, though implications are discussed. The simulations vary total mass (M_T), impactor-to-total-mass ratio (γ), and impact velocity (V_imp) while keeping impact angle fixed. The key parameter investigated is the effect of gas drag on growing moonlets (100 m–100 km in size) within the moon-forming disk. The SPH simulations model the giant impact events, tracking the formation and evolution of the resulting disks. Parameters such as disk mass (M_D), angular momentum (L), and VMF are extracted from the simulations. The simulations utilize the M-ANEOS semi-analytical equation of state for silicate and iron, and a five-phase EOS for water for the ice planets. The impact simulations model the dynamics and thermodynamics of giant collisions, focusing on the vapor mass fraction (VMF) of the resulting disks. The authors track disk parameters (mass, angular momentum, surface density) derived from the SPH simulations and compare them to theoretical estimates. They analyze the gas drag experienced by growing moonlets within the disk, considering the radial drift velocity caused by pressure gradients in the vapor component. The simulations examine different total masses of the colliding planets, impactor-to-total mass ratios, and impact velocities to determine the conditions under which a significant amount of vapor is generated in the impact and disk. A key factor in this analysis is calculating the pressure gradient parameter (η) in the disk. The researchers further analyze the evolution of the disk, simulating the radial infall of moonlets under the influence of gas drag. They calculate radial infall timescales for moonlets of varying sizes and compare these timescales to the accretion timescale. They create simple models (exponential and polynomial) of the surface density profiles to analyze the stability and evolution of the disk. Using these models, they track the viscous spreading of the initially vapor-rich disks and determine how the mass and distribution of the disk change over time, and how much mass is lost to the planet.

The key finding is that high-VMF disks, prevalent in impacts involving planets larger than ~1.3–1.6 Earth radii (for icy and rocky planets, respectively), inhibit the formation of fractionally large moons. The strong gas drag from the vapor causes moonlets to lose angular momentum and rapidly spiral into the planet, preventing further growth. This phenomenon, termed the "km-barrier problem," contrasts with the known "meter-barrier problem" in protoplanetary disk formation. While an initially vapor-rich disk might begin forming stable moonlets once it cools and the VMF reduces, significant mass loss occurs during this cooling period, preventing the formation of a sizeable moon. The study thus supports models producing initially vapor-poor disks, such as the canonical giant-impact model for the Moon's formation. The SPH simulations demonstrate a clear correlation between planetary mass and VMF. The VMF increases as the total mass of the colliding planets increases, because greater impact kinetic energy leads to more complete vaporization. Similarly, larger impactors (higher γ) lead to higher VMF. Icy planets generate disks with higher VMF than comparable-mass rocky planets due to water's lower latent heat of vaporization. The simulations confirm that the surface density profiles of the initially produced disks are unstable and evolve rapidly (within days) towards more stable configurations. The radial infall timescale for km-sized moonlets in these vapor-rich disks is drastically shorter (on the order of a day) than the accretion timescale (years), effectively preventing growth beyond km sizes. This prevents the formation of a fractionally large moon (defined as a few to 10% of the planet's mass). The study identifies a critical size threshold: rocky planets larger than ~6 Earth masses (~1.6 Earth radii) and icy planets larger than ~1 Earth mass (~1.3 Earth radii) produce completely vaporized disks (VMF≈1.0), incapable of forming large moons. The results are consistent with the Earth-Moon system (vapor-poor disk) and the Pluto-Charon system (icy, relatively vapor-poor). The analysis also suggests gas giants and mini-Neptunes may face similar challenges in forming large moons due to their large masses and potentially high H/He gas contribution to the disk. The model does not necessarily preclude smaller moon formation around larger planets. As disks cool and VMF decreases, moonlets can form, but significant mass loss is likely to occur. This model could explain the fractionally small moons of Uranus.

The findings suggest that the initial conditions of the moon-forming disk are crucial in determining the size of the resulting moon. A vapor-poor disk, as predicted by the canonical model, is essential for forming a fractionally large moon. The "km-barrier problem", caused by gas drag in vapor-rich disks, highlights a significant obstacle to moon formation in the simulations. The results provide valuable constraints on the lunar origin, favoring scenarios that produce initially vapor-poor disks. The study also offers an explanation for the lack of confirmed exomoons: larger planets may be less likely to form fractionally large moons due to the vapor-rich nature of their impact-generated disks. This suggests that focusing exomoon searches on smaller exoplanets (radii <~1.6 Earth radii) could be a more fruitful approach. Super-Earths appear as better candidates than mini-Neptunes due to their lower masses and potential for less H/He gas in their disks. This work has implications for exomoon detection strategies. The study advocates for focusing on smaller exoplanets (≤6 Earth masses for rocky planets and ≤1 Earth mass for icy planets), a region currently understudied. This narrowed parameter space will improve the efficiency of future exomoon searches with various telescopes, including Kepler, Hubble, CHEOPS, and JWST. The study’s limitations include the exploration of only a limited parameter space (impactor size and velocity) and the simplification of certain physical processes. However, the chosen parameters are typical for moon-forming impacts and less shock-heated than some other scenarios. Also, while this study focuses on collisions between planets of the same composition, future work should investigate the scenarios for collisions between planets of different compositions.

This research demonstrates that the vapor mass fraction of a moon-forming disk critically impacts the final moon size, particularly emphasizing the challenges in forming fractionally large moons around larger planets. The simulations support the canonical giant impact model for Moon formation and suggest that smaller exoplanets are more likely candidates for hosting fractionally large exomoons. Future research should explore a wider range of parameters, incorporating factors like the Roche limit and streaming instability, and investigate impacts between planets with different compositions. The findings refine exomoon search strategies by highlighting smaller exoplanets as prime targets.

The study explored a limited parameter space of impact conditions, focusing on a specific impact angle and a range of planet masses and impactor sizes. The model simplifies several aspects of disk evolution, such as the detailed treatment of the Roche limit and the influence of streaming instability. The simulations also assumed uniform disk entropy and hydrostatic equilibrium, which might not accurately capture the complexities of real-world disk evolution. Future work should address these limitations by incorporating more complex disk models and exploring a wider range of impact scenarios.