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Large planets may not form fractionally large moons

Abstract

One of the unique aspects of Earth is that it has a fractionally large Moon, which is thought to have formed from a Moon-forming disk generated by a giant impact. The Moon stabilizes the Earth’s spin axis at least by several degrees and contributes to Earth’s stable climate. Given that impacts are common during planet formation, exomoons, which are moons around planets in extrasolar systems, should be common as well, but no exomoon has been confirmed. Here we propose that an initially vapor-rich moon-forming disk is not capable of forming a moon that is large with respect to the size of the planet because growing moonlets, which are building blocks of a moon, experience strong gas drag and quickly fall toward the planet. Our impact simulations show that terrestrial and icy planets that are larger than ~1.3−1.6R produce entirely vapor disks, which fail to form a fractionally large moon. This indicates that (1) our model supports the Moon-formation models that produce vapor-poor disks and (2) rocky and icy exoplanets whose radii are smaller than ~1.6R are ideal candidates for hosting fractionally large exomoons.

Introduction

Earth is the only known life-hosting planet and has a number of features unique to our solar system, including active plate tectonics, a strong magnetic field, and a large moon with respect to the size of Earth. The presence of the Moon controls the length of the day and ocean tides, which affects the terrestrial biological cycles. The Moon also stabilizes the Earth’s spin axis by at least several degrees1,2,3. Thus, at least for Earth, the Moon also contributes to Earth’s stable climate and potentially offers an ideal environment for life to develop and evolve. It should be noted, however, that the stability depends on several factors, such as the initial obliquity of the planet4,5.

The origin of the Moon and conditions to host a massive satellite have been under active debate. According to the canonical hypothesis of the lunar origin, the Moon formed from a partially vaporized disk generated by a collision between Earth and a Mars-sized impactor approximately 4.5 billion years ago6,7,8. This giant impact hypothesis was originally thought to be able to explain the geochemical observation that the Earth and Moon have nearly identical isotopic ratios9, assuming that a large impact can homogenize the two bodies. However, numerical simulations find that the disk is primarily made of the impactor materials and it is challenging to mix the two reservoirs10. This requires the impactor (“Theia”) to have isotopic ratios nearly identical to Earth by chance11,12 or homogenization during the disk stage13.

Alternatively, the Moon could have formed by a more energetic impact that would have homogenized the system more easily. Proposed energetic impacts include a collision between two half-Earth-sized objects14, and other high-energy and high-angular momentum cases15,16. One of the potential problems of the energetic models is that the final angular momentum of the system is two to three times higher than the current value, which may not be easily removed from the system17. Moreover, these impacts would have homogenized the Earth’s mantle, which may not be consistent with geochemical observations18. Another option is that the Moon could have formed by multiple impacts19. Small- and high-speed impactors can strip and launch Earth’s mantle materials into orbit, but if some of the impactors are large or slow, the disk and therefore the Moon would still hold isotopic signatures originating from the impactors.

These impacts generate the Moon-forming disks with various vapor mass fractions (VMF). In general, energetic impacts lead to higher VMF of the disk. The VMF for the canonical model is 0.1–0.320, and those for half-Earths as well as high velocity and high-angular momentum cases are 0.7–1.020, and those for the multiple moon model are 0.1–0.519. As the disk cools, VMF decreases. It is conventionally thought that moonlets (or lunatesimals) accrete from the liquid portion of the disk and moonlets accrete into the Moon over time. We revisit this process later in this work.

Given that planetary collisions are common in the young solar system and may have already been detected in extrasolar systems (e.g., refs. 21,22,23,24), it is expected that impact-induced moons (exomoons) are common (e.g., ref. 25). Unfortunately, we currently have a very limited understanding of the origin of exomoons due to a lack of confirmed exomoons. While several detection methods have been proposed for exomoons, including direct imaging26 and microlensing27, the Hunt for Exomoons with Kepler (HEK) (e.g., ref. 28) is leading the effort for exomoon detection, analyzing the Kepler data for an indication of any transit timing and transit duration variations (TTV and TDV) of exoplanets due to exomoons. Despite the extensive search (57 exoplanets were searched during HEK I-V28) and a follow-up study with the Hubble Space Telescope29, no exomoon has been confirmed to date30,31,32. Detecting an exomoon is very challenging, and the lifetime of moons around close-in planets, such as most of the Kepler planets, could be short due to the strong tide from the stars33. Moreover, close-in planets have small Hill radii, which is the distance where the gravity of the planet is dominant, and some moons that migrate outward can be lost due to orbital instability33. In the canonical “graze and merge” moon-forming scenario8, the return of the projectile occurs only if the projectile loops inside the 1/3 of the Hill radius34. Hence closer-in planets (with smaller Hill radii) would be less likely to experience a similar graze-and-merge impact. Nonetheless, planetary impacts that would lead to moon formation should be common in the late stages of planetary accretion, and the lack of a confirmed exomoon detection has not been fully explained.

Here, we explore the possibility of not all the planets can form impact-induced moons, which can be fractionally large with respect to their host planets. We constrain the condition to form impact-induced moons in order to (1) constrain the lunar origin, and (2) explain the lack of confirmed exomoons to date. We conduct impact simulations with the smoothed particle hydrodynamics (SPH) method, which describes fluid as a collection of particles20,35, to characterize moon-forming disks and moon accretion process. We consider two endmembers of planetary compositions: a rocky planet consisting of 70 wt% of forsterite mantle and 30 wt% iron core and an icy planet consisting of 70 wt% water ice and 30 wt% forsterite core. Gas giants and mini-Neptunes are not the primary focus of this study, but we discuss implications for these planets later. Our simulations show that a moon-forming disk with a high vapor mass fraction (VMF) is not capable of forming a moon that is large with respect to the size of the planet because growing moonlets of 100 m–100 km in size in the disk experience strong gas drag from vapor, lose their angular momentum and fall onto the planet on a short timescale, failing to grow further. This does not occur in a vapor-poor disk (i.e., small VMF) because the gas drag effect is much weaker. An initially vapor-rich disk can start forming stable moonlets once it cools and the VMF is small enough, but by the time a significant portion of the disk mass is lost, failing to form a fractionally large moon. Consequently, our study supports the moon-formation models that produce initially small VMF, such as the canonical model. Moreover, we find that rocky planets larger than 6 M (~1.6 R, where R is the Earth radii), where M is the Earth mass, and icy planets larger than 1 M (~1.3 R) produce completely vapor disks, and therefore these planets are not capable of forming fractionally large impact-induced moons. For this reason, we propose that future exomoon observations should focus on exoplanets smaller than ~1.6 R to detect impact-induced exomoons.

Results

Effect of vapor on growing moonlets

If a moon-forming disk started from a completely vapor state (i.e., VMF = 1.0), liquid droplets would start forming as the disk cools. These liquid droplets fall toward the midplane on a short timescale36. They would experience strong gas drag from the vapor because droplets orbit around the planet with the Keplerian velocity vK, whereas vapor moves slower due to the radial pressure support. This radial drift velocity is described as \({v}_{r}=-2\eta {v}_{{{{{{{{\rm{K}}}}}}}}}{\tau }_{{{{{{{{\rm{f}}}}}}}}}/(1+{\tau }_{{{{{{{{\rm{f}}}}}}}}}^{2})\)37, where \(\eta =-\frac{1}{2}{(\frac{{c}_{{{{{{{{\rm{s}}}}}}}}}}{{v}_{{{{{{{{\rm{K}}}}}}}}}})}^{2}\frac{\partial {{{{{{\mathrm{ln}}}}}}}\,P}{\partial {{{{{{\mathrm{ln}}}}}}}\,r}\) is the pressure gradient parameter, which describes the strength of the radial pressure gradient of the vapor, P is the pressure, cs is the sound velocity, vK is the Keplerian velocity, τf is the dimensionless stopping time expressed as \({\tau }_{{{{{{{{\rm{f}}}}}}}}}=\frac{8}{3{C}_{{{{{{{{\rm{D}}}}}}}}}\eta }\frac{{\rho }_{{{{{{{{\rm{p}}}}}}}}}}{{\rho }_{{{{{{{{\rm{g}}}}}}}}}}\frac{{R}_{{{{{{{{\rm{p}}}}}}}}}}{r}\), where CD is the drag coefficient, ρp is the particle density, ρg is the gas density, Rp is the particle radius, and r is the distance from the planet. The radial fall velocity is largest when τf = 1 and potentially the particle can fall toward the planet very quickly, depending on the disk condition. This gas drag effect has been considered for small droplets, but not for growing moonlets16, which are building blocks of a moon.

In fact, this was a major challenge for planet formation in the protoplanetary disk. The very same mechanism would have removed 1 m-sized planetesimals, which are building blocks of planets, at 1 AU on the timescale of 80 years37,38, which is a much shorter timescale than the planet formation timescale (a few to tens of millions of years). This is the so-called “meter-barrier problem” and was an outstanding problem in planetary science for decades.

While this is the first time to investigate the effect of gas drag in a moon-forming disk in detail, previous work also points out another issue of a vapor-rich disk. The previous work39 suggests that an initially vapor-rich moon-forming disk may not be dynamically stable due to shocks caused by density gradients in the disk and loses a significant portion of its mass in a short time, concluding that a vapor-rich moon-forming disk is not capable of forming a large moon.

SPH simulations

We conduct SPH simulations to quantify VMF as well as the pressure graduate parameter η to characterize the strength of the gas drag. In the case of rocky planets, including Earth, a high VMF of a moon-forming disk means that the disk is primarily made of silicate vapor and a small amount of liquid (magma). For an icy planet, a high VMF can mean a steam disk potentially with embedded silicate droplets. The parameters for SPH simulations are the total (target and impactor) mass MT (M ≤ MT ≤ 6 M for rocky planets and 0.1 M ≤ MT ≤ 1 M for icy planets, where M is the Earth mass), the impactor-to-total-mass ratio γ (0.13 ≤ γ ≤ 0.45), impact velocity vimp, and impact angle θ. In this work, we use the fixed impact angle (θ = 48.6) and the impact velocity (vimp = vesc), where vesc is the mutual escape velocity. The impact angle and velocity are similar to those for the canonical moon-forming impact models8,10. The reason why we explore different mass ranges for the rocky and icy planets is that the required mass for complete vaporization is different between them, as we discuss in detail below.

Figure 1 shows snapshots of giant impacts with SPH. The top two rows represent an impact between rocky planets and the bottom two rows represent an impact between icy planets. The red-yellow and blue-sky-blue colors represent mantle entropy. Gray (iron) and orange (silicate) represent the core materials. In both simulations, the same input parameters are used (MT = 1 M, γ = 0.13, θ = 48.5, vimp = vesc). The icy planets are larger than rocky planets because ice is much less dense than silicate. The overall dynamics during an impact are remarkably similar between the two systems (Fig. 1). In contrast, thermodynamics during the impacts differ significantly. The overall disk vapor mass fraction is approximately 0.3 (i.e., 70 wt% of the mass is in the liquid phase) for the rocky planet case, whereas it is 1.0 for the icy planet case. This is primarily because the latent heat of water (~2.3 × 106 J kg−1) is much smaller than the latent heat of silicate (~1.2 × 107 J  kg−140).

Fig. 1: Snapshots of giant impacts (Runs ID2 and ID20, Table 1).
figure 1

The top two rows represent an impact between two rocky planets. The red-orange colors represent the entropy of the mantle material (forsterite). The iron core is shown in gray. The bottom two rows represent an impact between two icy planets. The blue-sky-blue colors represent the entropy of water ice, and the orange color represents forsterite. The scale represents 107 m.

The SPH output is summarized in Table 1. Figure 2 shows the disk mass MD and disk angular momentum LD as a function of the total mass MT for rocky and icy planets. The colors represent γ, and the dashed lines represent analytical estimates of the disk mass25. Generally, the disk mass agrees with the analytical model, especially at small γ. There is a slight upward trend as a function of the total mass in the disk mass and angular momentum at γ ≥ 0.30. This may be because the larger extent of tidal deformation of the colliding bodies at the time of the impact with larger MT, which may make the impact angle slightly shallower and generate a larger disk mass and a larger angular momentum. However, the deviation from the analytical solution is relatively small and more simulations with larger MT are needed to confirm this trend.

Table 1 Summary of the results.
Fig. 2: Disk masses and angular momenta based on our SPH simulations.
figure 2

a, b The disk mass MD normalized by the target mass Mt as a function of the total mass MT normalized by the Earth mass for rocky and icy planets, respectively. The colors represent the values of γ. The dashed lines represent the disk mass estimate from previous work assuming a rocky composition25. c, d The disk angular momentum LD normalized by the total angular momentum for rocky and icy planets LT, respectively.

Figure 3 shows the VMF of a moon-forming disk as a function of γ and MT. The panels (a) and (b) represent rocky and icy planetary impacts. The vapor mass fraction is over 0.96 at the planetary mass ≥6 M for rocky bodies and ≥1 M for icy planets. The larger MT is, the higher VMF is in general. This is because the kinetic energy involved in such an impact generally increases as the planetary mass increases (see “Methods”). Moreover, a fractionally larger impactor (i.e., larger γ) contributes to higher impact-induced heating and higher VMF as well. An icy planet produces a disk with a higher VMF than a rocky planet with the same mass, as discussed above.

Fig. 3: The vapor mass fraction of the disk as a function of the impactor-to-total mass ratio and the total mass normalized by the Earth mass.
figure 3

The panels (a) and (b) represent rocky and icy planet cases, respectively. Brown colors represent higher vapor mass fractions, whereas blue colors represent lower values.

The disk structures at the midplane formed by the rocky and icy planet collisions are shown in Fig. 4 (Run ID6, MT = 5 M, γ = 0.13) and Fig. 5 (Run ID20, MT = 1 M, γ = 0.13), respectively. The dotted line in panel (a) represents the surface density directly obtained from the SPH simulation. We find that this initial disk structure is unstable because it does not meet the Rayleigh criterion where the disk angular momentum Lz needs to monotonically increase in the radial direction (dLz/dr > 0, where r is the radial distance from the planet)20,41. Thus, the disk would dynamically adjust itself within a few days. Two potential disk surface density structures shown in the solid lines satisfy this condition while conserving the disk angular momentum and mass. The skye-blue line represents an exponential model and the green line represents a model that is a polynomial model (see “Methods”). The disk temperature (panel b), vapor mass fraction (panel c) and the pressure gradient parameter η (panel d) are shown as a function of r/R, where R is the Earth radius. The temperature, vapor mass fraction, and η are calculated assuming that the disk is in the hydrostatic equilibrium20. The pressure parameter η in the disk is typically ~0.02–0.06 for rock and icy moon-forming disks, which is ~10 times higher than the typical values in the protoplanetary disk42.

Fig. 4: Disk structure formed by an impact between rocky planets (MT = 5M, γ = 0.13, θ = 48. 5, vimp = vesc, and Run ID6).
figure 4

All of these models have the same angular momenta and disk masses. The panels show (a) the surface density, (b) temperature, (c) vapor mass fraction, and (d) pressure gradient parameter η as a function of the radial distance from the planet r normalized by the Earth radius R. The dotted line represents the direct outcome from the SPH simulation. The sky-blue line represents an exponential model and the green line represents a polynomial disk model. The planetary radius is 1.74 R.

Fig. 5: Disk structure at the midplane formed by an impact between icy planets (MT = 1M, γ = 0.13, θ = 48. 5, vimp = vesc, Run ID20).
figure 5

The line descriptions are the same as those in Fig. 4. The panels show (a) the surface density, (b) temperature, (c) vapor mass fraction, and (d) pressure gradient parameter η as a function of r/R. The planetary radius is 1.42 R.

Radial fall of moonlets

Assuming that η ~ 0.04, the radial fall timescale for this disk is ~1 day for a 2-km-sized body for a silicate vapor disk and ~1 day for a 1.3 km for a water vapor disk (see Methods). This is comparable to the vertical fall timescale of particles (~a few days36) and much shorter than the accretion timescale of solid and liquid materials (~1−10s years43). The typical lifetime for the moon-forming disk is estimated to be ~100 years44, and therefore the radial fall timescale is extremely short for a few km-sized moonlets. This process can prevent moonlets from growing larger than km in size, which we call here the “km-barrier problem” for moon-forming disks. One of the potential differences between the meter-barrier and km-barrier problems is that mutual gravity can facilitate the growth of the moonlets in the latter scenario, but this effect is likely minor and would not solve the radial fall issue.

An initially 100 wt% vapor disk cannot form a moon that is large with respect to the size of the planet due to the rapid infall timescale, according to our arguments above. We, therefore, propose that the moon-forming disk needs to be initially vapor-poor, supporting the canonical moon-formation hypothesis whose vapor mass fraction is small (0.1–0.3). Moreover, we can constrain the large moon-forming region as shown in Fig. 6; rocky planets larger than 6 M and icy planets larger than 1 M may not form a large satellite because their VMFs are 1.0. Here, a “large” moon is defined as a satellite whose mass is approximately a few to 10 wt% of the planetary mass. This is consistent with the Earth–Moon (VMF ~0.1–0.320), and Pluto–Charon (icy planets, VMF ~ 0–0.0845), given that these moon-forming impacts produce small VMFs. The radii of a 6 M rocky planet and a 1 M icy planet are ~1.3–1.6 R46,47. We also predict that gaseous mini-Neptunes may also have trouble producing large moons because of their large masses as well as potentially more vapor-rich disks if the H/He gas becomes part of the disk48.

Fig. 6: Schematic view of the mass range in which a fractionally large exomoon can form by an impact.
figure 6

The horizontal axis represents the mantle composition and the vertical axis represents the planetary mass normalized by the Earth mass M. Rocky planets smaller than 6 M and icy planets smaller than 1 M are capable of forming fractionally large moons as indicated by the orange shading. Our prediction is consistent with planet–moon systems in the solar system.

Discussion

It should be noted that our model does not exclude forming small moons around large planets. As the disk cools and the VMF decreases, the gas drag effect weakens and moonlets start to form. However, during the cooling process, the vapor-rich disk would viscously expand and a significant portion of the disk would be lost to the planet (see “Methods” for detailed discussion). This can be an explanation for fractionally small moons of Uranus (1.02 × 10−4Mp)49, which could have formed by a large impact48,49,50.

A large impact is not the only way to form a moon. A satellite can also form within a circumplanetary disk. The Galilean satellites and Titan are thought to have formed from dust in such circumplanetary disks. These disks typically have smaller dust-to-gas ratios (~10−2–10−4 (see ref. 51)). The typical satellite total mass formed in a circumplanetary disk is estimated to be 10−4Mp51 due to the two competing processes: supply of dust materials to the disk and satellite loss due to their radial fall caused by the gas. This total mass estimate is also consistent with the newly observed exomoon disk mass (~10−4Mp)52. A moon can form by gravitational capture. It is unclear if there is a maximum satellite size in the case of gravitational capture, but at least in the solar system, gravitationally captured moons have fractionally small masses (e.g., Triton, 2.46 × 10−4Mp53). A significant fraction of the orbital kinetic energy of a heliocentric body needs to be dissipated during capture, which requires certain conditions53. In contrast, impacts, especially at relatively low velocity, can naturally produce fractionally large moons (the Pluto–Charon system has 0.118 Mp). Under specific conditions, a nearly intact impactor can also be captured as a moon45,54, but this requires very grazing incidence and very low relative velocity and is thus likely less frequent than moons formed in impact-induced disks, given that formation of an impact-induced disk is common55.

There are several aspects that need to be considered further. The presence of the Roche radius near the moon-forming region makes the accretion process complex. Moonlets that enter inside the Roche radius become small fragments due to the planetary tide and they become well coupled with vapor. The evolution of the fragments may be similar to the evolution of the vapor disk (see “Methods”), but more detailed disk modeling is necessary to assess the effect of the Roche radius. In addition, if moonlets can skip this dangerous km size problem by forming a large moonlet very quickly, even a vapor-rich disk can potentially form a large moon. A promising mechanism is so-called streaming instability, which is a clump-formation mechanism due to the spontaneous concentration of particles followed by gravitational collapse. This mechanism was originally proposed to solve the planet formation problem in the protoplanetary disk56. Whether the same mechanism works for a moon-forming disk will be considered in our future study57.

We only explore limited parameter space (e.g., impactor size and its velocity), but of course other parameters are possible. Nevertheless, the parameters chosen here tend to produce a disk that is massive enough to form a large satellite and that is less shock-heated than other scenarios (especially when γ = 0.13). Moreover, it is likely that high VMF cases (e.g., ≥0.8) are enough to inhibit the accretion of large moons, but the threshold VMF is difficult to determine. For this reason, we assume that VMF = 1 is the upper limit to form an impact-induced large moon.

Here, we consider collisions between planets that have the same compositions, but a collision between an icy and rock planets is possible. This may make it possible for a large icy planet (≥1 M) to have a large rocky satellite if the disk is primarily made of the rocky impactor. Nevertheless, such impacts are likely less common unless there are extensive radial mixing events that lead to a large fraction of planetary objects crossing the ice line (e.g., Nice model and grand tack scenario58,59).

Our model predicts that the moon-forming disk needs to be initially liquid or solid rich, supporting the canonical moon-forming impact model8. Moreover, this work will help narrow down planetary candidates that may host exomoons; we predict that planets whose radii are smaller than ~1.6R would be good candidates to host fractionally large exomoons (≤6 M for rocky planets and ≤1 M for icy planets). These relatively small exoplanets are understudied (only four out of 57 exoplanets surveyed by the HEK project are under this radius limit), which can potentially explain the lack of exomoon detection to date. Super-Earths are likely better candidates than mini-Neptunes to host exomoons due to their generally lower masses and potentially lower H/He gas contribution to the disk. This narrower parameter space may help constrain exomoon search in data from various telescopes, including Kepler, the Hubble space telescope, CHaraterising ExOPlanet Satellite (CHEOPS)60, and the James Webb Space Telescope (JWST).

Methods

Estimate on radial fall timescale

In the moon-forming disk, the Newton drag force is written as \(F={C}_{{{{{{{{\rm{D}}}}}}}}}\pi {R}_{{{{{{{{\rm{p}}}}}}}}}^{2}{\rho }_{{{{{{{{\rm{g}}}}}}}}}{v}_{{{{{{{{\rm{rel}}}}}}}}}^{2}/2\), where CD is the drag coefficient (~0.44 for turbulent flows), Rp is the particle radius, ρg is the gas density, vrel( = ηvK) is the relative velocity between the particles and gas38, and vK is the Keplerian velocity. The friction time is expressed as \({t}_{{{{{{{{\rm{f}}}}}}}}}=\frac{8}{3{C}_{{{{{{{{\rm{D}}}}}}}}}}\frac{{\rho }_{{{{{{{{\rm{p}}}}}}}}}{R}_{{{{{{{{\rm{p}}}}}}}}}}{{\rho }_{{{{{{{{\rm{g}}}}}}}}}{v}_{{{{{{{{\rm{rel}}}}}}}}}}\) where ρp is the particle density. The dimensionless stopping time τf = ΩKtf, where ΩK is the Keplerian angular velocity (here \({{{\Omega }}}_{{{{{{{{\rm{K}}}}}}}}}=\sqrt{G{M}_{\oplus }/{(3{R}_{\oplus })}^{3}}=2.38\times 1{0}^{-4}\) s−1 at r = 3R). The dimensionless stopping time is scaled as

$${\tau }_{{{{{{{{\rm{f}}}}}}}}} ={{{\Omega }}}_{{{{{{{{\rm{K}}}}}}}}}{t}_{{{{{{{{\rm{f}}}}}}}}}=\frac{8}{3{C}_{{{{{{{{\rm{D}}}}}}}}}\eta }\frac{{\rho }_{{{{{{{{\rm{p}}}}}}}}}}{{\rho }_{{{{{{{{\rm{g}}}}}}}}}}\frac{{R}_{{{{{{{{\rm{p}}}}}}}}}}{r}\\ =1.18{\left(\frac{\eta }{0.04}\right)}^{-1}\left(\frac{{\rho }_{{{{{{{{\rm{p}}}}}}}}}}{3000\;{{{{{{{\rm{kg}}}}}}}}\ {{{{{{{{\rm{m}}}}}}}}}^{-3}}\right){\left(\frac{{\rho }_{{{{{{{{\rm{g}}}}}}}}}}{40\;{{{{{{{\rm{kg}}}}}}}}{{{{{{{{\rm{m}}}}}}}}}^{-3}}\right)}^{-1}\left(\frac{{R}_{{{{{{{{\rm{p}}}}}}}}}}{2\;{{{{{{{\rm{km}}}}}}}}}\right){\left(\frac{r}{3{R}_{\oplus }}\right)}^{-1}.$$
(1)

The radial drift rate is described as \({v}_{r}=-2\eta {v}_{{{{{{{{\rm{K}}}}}}}}}{\tau }_{{{{{{{{\rm{f}}}}}}}}}/(1+{\tau }_{{{{{{{{\rm{f}}}}}}}}}^{2})\)37. The timescale for radial fall is

$${t}_{{{{{{{{\rm{fall}}}}}}}}} =\frac{r}{{v}_{r}}=\frac{1}{2\eta }{\left(\frac{{r}^{3}}{GM}\right)}^{\frac{1}{2}}\frac{1+{\tau }_{{{{{{{{\rm{f}}}}}}}}}^{2}}{{\tau }_{{{{{{{{\rm{f}}}}}}}}}}\\ =\left\{\begin{array}{ll}12.3\,{{{{{{{\rm{h}}}}}}}}\left(\frac{{\rho }_{{{{{{{{\rm{g}}}}}}}}}}{40\;{{{{{{{\rm{kg}}}}}}}}\ {{{{{{{{\rm{m}}}}}}}}}^{-3}}\right){\left(\frac{{\rho }_{{{{{{{{\rm{p}}}}}}}}}}{3000\,{{{{{{{\rm{kg}}}}}}}}{{{{{{{{\rm{m}}}}}}}}}^{-3}}\right)}^{-1}{\left(\frac{M}{{M}_{\oplus }}\right)}^{-\frac{1}{2}}{\left(\frac{r}{3{R}_{\oplus }}\right)}^{\frac{5}{2}}{\left(\frac{{R}_{{{{{{{{\rm{p}}}}}}}}}}{2\,{{{{{{{\rm{km}}}}}}}}}\right)}^{-1}\hfill&\,{{\mbox{at}}}\,\,{\tau }_{f}\ll 1,\\ 29.1\,{{{{{{{\rm{h}}}}}}}}{\left(\frac{\eta }{0.04}\right)}^{-1}{\left(\frac{M}{{M}_{\oplus }}\right)}^{-\frac{1}{2}}{\left(\frac{r}{3{R}_{\oplus }}\right)}^{\frac{3}{2}}\hfill&\,{{\mbox{at}}}\,\,{\tau }_{f} \sim 1,\\ 17.3\,{{{{{{{\rm{h}}}}}}}}{\left(\frac{\eta }{0.04}\right)}^{-2}{\left(\frac{{\rho }_{{{{{{{{\rm{g}}}}}}}}}}{40\,{{{{{{{\rm{kg}}}}}}}}{{{{{{{{\rm{m}}}}}}}}}^{-3}}\right)}^{-1}\left(\frac{{\rho }_{{{{{{{{\rm{p}}}}}}}}}}{3000\,{{{{{{{\rm{kg}}}}}}}}\ {{{{{{{{\rm{m}}}}}}}}}^{-3}}\right){\left(\frac{M}{{M}_{\oplus }}\right)}^{-\frac{1}{2}}{\left(\frac{r}{3{R}_{\oplus }}\right)}^{-\frac{1}{2}}\left(\frac{{R}_{{{{{{{{\rm{p}}}}}}}}}}{2\;{{{{{{{\rm{km}}}}}}}}}\right)&\,{{\mbox{at}}}\,\,{\tau }_{{{{{{{{\rm{f}}}}}}}}}\gg 1.\\ \end{array}\right.$$
(2)

G is the gravitational constant, and M is the mass of the planet. This leads to tfall = 1.21 days at Rp = 2 km (τf = 1.18), tfall = 72 days at Rp = 200 km (τf = 118), tfall ~ 1.68 years at Rp = 1700 km (τf  ~ 1010) (the radius of the Moon is 1.731 × 103 km), and tfall ~ 280 years at Rp ~ 1 cm (τf = 5.9 × 10−6 and r = 2R is assumed). This indicates that any moonlets that reach ~km in size will fall inwards on a very short timescale (less than 1 day). For an icy moon-forming disk, τf = 1 at Rp = 1.3 km assuming ρp = 1000 kg m−3 and ρg = 10 kg m−3. We assume that liquid droplets can accrete and grow quickly and that turbulence at the midplane would not prevent growth36.

During the radial fall, a moonlet may evaporate due to heating by the gas drag. The total heating during the fall is approximately written as Fvveltfall. The evaporative mass fraction of the moonlet is written as Fvveltfall/MmoonletL, where Mmoonlet is the moonlet mass (\(=\frac{3}{4}{\rho }_{{{{{{{{\rm{p}}}}}}}}}\pi {R}_{{{{{{{{\rm{p}}}}}}}}}^{3}\)) and L is the latent heat. Evaporation is not a significant factor for a 2 km rocky moonlet in the moon-forming disk; only 20 wt% of the moonlet evaporate, whereas the icy moonlet completely evaporates (ρp = 1000 kg m−3, ρg = 10 kg m−3, Rp = 1.3 km). In this case, moonlets that evaporate increase the vapor fraction of the disk, slowing down the cooling process. It is possible that even more mass is lost to the planet during the cooling process. In addition, small droplets (mm and cm in size) falling from the outer (colder) part of the disk evaporate quickly due to the temperature difference between the droplets and vapor16, but this evaporation process is slow and therefore not significant for km-sized moonlets.

Disk evolution

Here, we describe a simple model that tracks the time evolution of a completely vapor disk. The disk would viscously spread while conserving the angular momentum. The analytical solution of the surface density of the disk is described as49

$${{\Sigma }}={{{\Sigma }}}_{0}{t}_{* 0}^{-21/22}{\left(\frac{r}{R}\right)}^{-3/4}\exp \left[-{\left(\frac{r}{{r}_{{{{{{{{\rm{d0}}}}}}}}}}\right)}^{5/4}{t}_{* 0}^{-15/22}\right],$$
(3)

where

$${t}_{* 0}=1+\frac{t}{{t}_{{{{{{{{\rm{diff0}}}}}}}}}}=1+\frac{t}{\frac{16}{75}{\left(\frac{{r}^{2}}{\nu }\right)}_{{r}_{{{{{{{{\rm{d0}}}}}}}}},t = 0}}.$$
(4)

Σ0 is the initial surface density at r = R, where r is the orbital distance and R is the planetary radius. rd0 is the typical disk size, which we define 3R. t is time, and tdiff0 is the diffusion time at r = rd0 and t = 0 and ν is the viscosity. Traditionally, the viscosity of the moon-forming disk is assumed to be controlled either by the instability-driven viscosity61,62 or thermally driven viscosity44. The instability-driven viscosity is caused by self-gravitating clumps forming from a gravitationally unstable liquid layer inside the Roche radius. Inside the Roche radius, any self-gravitating bodies will be destroyed by the planetary tide. The thermally driven viscosity is caused by balancing heat generation by gravitational instability and radiative cooling. Here, we consider a completely vapor disk, which is gravitationally stable, and therefore no clumping is possible. Thus, the viscosities described above are not representative in our scenario. Here, we assume the turbulent kinematic viscosity described as \(\nu =\alpha {c}_{{{{{{{{\rm{s}}}}}}}}}^{2}{{{\Omega }}}_{{{{{{{{\rm{K}}}}}}}}}^{-1}\) (see refs. 63,64), where α is a parameter that describes the turbulent strength, cs is the sound speed, and ΩK is the Keplerian angular velocity. α is unknown, but 10−3 has been used for a protoplanetary disk and 10−6–10−2 has been used for the magneto-rotational instability (MRI)-driven lunar forming disk65,66,67. Here, we use α = 5 × 10−3 as a conservative value. \({c}_{{{{{{{{\rm{s}}}}}}}}}=\sqrt{\tilde{\gamma }{R}_{{{{{{{{\rm{gas}}}}}}}}}T/m}\), where γ is the heat capacity ratio (\(\tilde{\gamma }=1.4\)), Rgas is the gas constant, T(= 4000 K for rocky planets and 1000 K for icy planets) is the temperature, m is the mean molecular weight (30 g/mol for silicate44 and 18 g/mol for water).

The timescale for the disk lifetime depends on the model. For the Moon-forming disk, it is often considered to be ~100 years43,44 but the timescale estimates range from 10s of years to 1000s of years16,44,64,65,66. Here, we assume that the disk lifetime is comparable to the disk condensation timescale. Roughly speaking, the timescale τ is described as

$$\tau =\frac{fML}{16\pi {R}^{2}{\sigma }_{SB}{T}_{{{{{{{{\rm{ph}}}}}}}}}^{4}}=100\left(\frac{f}{0.02}\right)\left(\frac{M}{{M}_{\oplus }}\right){\left(\frac{R}{{R}_{\oplus }}\right)}^{-2}\left(\frac{L}{1.2\times 1{0}^{7}\;{{{{{{{\rm{J}}}}}}}}\ {{{{{{{{\rm{kg}}}}}}}}}^{-1}}\right){\left(\frac{{T}_{{{{{{{{\rm{ph}}}}}}}}}}{1410\,{{{{{{{\rm{K}}}}}}}}}\right)}^{-4}{{{{{{{\rm{years}}}}}}}},$$
(5)

where σSB is the Stefan Boltzmann constant, f is the disk mass with respect to the planetary mass M, and R is the planetary radius, where it is assumed that the typical disk radius is 3R. We also assume MR3.7 for rocky planets28 and MR3 for icy planets47. L is the latent heat, where L = 2.3 × 106 J kg−1 for icy planets and L = 1.2 × 107 J kg−1 for rocky planets. Tph is the photospheric temperature. Traditionally, the photospheric temperature of the Moon-forming disk has been assumed to be ~2000 K44, but the temperature can be lower if heat transport to the photosphere is not efficient65. In this calculation, we assume the photospheric temperature for silicate is 1410 K, which gives τ ~ 100 years for rocky systems. Tph is 270 K for icy planets68.

Figure 7a shows the surface densities of the Moon-forming disk at 0 year (purple solid line), 10 years (light purple dash-dot line), and 100 years (yellow dashed line), assuming f = 0.02, M = M, R = R, L = 1.2 × 107 J kg−1, Tph = 1410 K. The x−axis represents the distance from the planet normalized by the Earth radius R. As time passes, the disk extends radially and the surface density decreases. After 100 years, the disk mass becomes 0.155 of the original disk mass. Figure 7b shows the final disk mass MD,final normalized by the initial disk mass MD,initial = fM as a function of various planetary masses (f = 0.02). The purple solid line represents icy planets, and the yellow dashed-dotted line represents rocky planets. The disk evolution calculations are stopped when the disk lifetime is reached. MD,final/MD,initial ~ 0.2 for rocky planets in M = 0.1–6 M, whereas MD,final/MD,initial < 0.06 for icy planets in the same mass range. This is because the lifetime of an icy moon-forming disk is longer than that of a rocky moon-forming disk due to its much smaller Tph. Thus, the disk loses its significant portion of the mass by viscous evolution. Needless to say, this is a very simple model that ignores processes such as condensation, vapor-droplet interaction, and the Roche radius. More detailed investigations are needed to track the evolution of the disk.

Fig. 7: Evolution of completely vapor disks.
figure 7

a Evolution of the surface density of the Moon-forming disk as a function of r/R. The purple solid line, light purple dashed-dotted line, and the yellow dashed line represent the surface density at 0 year, 10 years, and 100 years, respectively. b The final disk mass normalized by the initial disk mass for icy planets (solid purple line) and rocky planets (dashed yellow line) as a function of the planetary mass M normalized by M. For the details, see the main text.

SPH simulations setting

We use an SPH code that we developed from the ground up. This code has been extensively tested18,20,35. We use the semi-analytical equation of state called M-ANEOS to represent silicate rock (forsterite) and iron. The forsterite EOS we use here is “SPH-N” in previous work69, where VMF can depend on the choice of an M-ANEOS input file. For ice, we used the five-phase EOS for water70 for the range between 0 and 20,000 K and density smaller than 5000 kg m−3. Outside of this range, the water EOS is interpolated using the table and SESAME 7154 EOS.

In most of the simulations, the initial entropy values of the mantles are 3165 J K−1 kg−1 and 3696 J K−1 kg−1 for rocky and icy planets, respectively. These values correspond to surface temperatures ~2000 K and 300 K for an Earth-sized planet, respectively. The exception is Run ID1 (Table 1) uses 1096 J K−1 kg−1. This corresponds to ~300 K at the surface.

The vapor mass fraction of a disk is determined assuming that the disk has a uniform entropy regardless of the radial distance and that the disk is in a hydrostatic equilibrium. The detailed method is described in our previous publication20. The disk’s mass, surface density, and angular momentum are directly calculated from the SPH simulation and are listed in Table 1.

Disk model

Figure 4 shows the disk structure for a rocky planetary impact. The skyeblue line represents an exponential surface density model (\(\sigma (r)=({c}_{1}+{c}_{2}r)\exp (-{c}_{3}r)\)) and here we pick values that satisfy dσ/dr = 0 at the inner edge, where r is the radial distance from the Earth’s spin axis. We conserve the disk mass and angular momentum within 5% (c1 = 5.857 × 107 kg m−2, c2 = −0.948 m−1, and c3 = −2.023 × 10−8 m−1). The green line represents a polynomial model (\(\sigma (r)={c}_{1}{(r/{r}_{0})}^{{c}_{2}}+{c}_{3}\)) and the error is less than 2% (c1 = 1.0 × 108 kg m−2, c2 = −0.809, c3 = −1.617 × 107 kg m−2). Similarly, the structures of an icy disk shown in Fig. 5. The parameters we use for the exponential disk are c1 = 2.6505 × 107 kg m−2, c2 = −0.5916 m−1, c3 = −2.79017 × 10−8 m−1 and those for the polynomial model are c1 = 5.5599 × 107 kg m−2, c2 = −0.5060, c3 = 1.920 × 107 kg m−2. The error is less than 5% and 2%, respectively.

M T dependence of VMF and local clumps

Roughly speaking, the vapor mass fraction produced by an impact is scaled as \(\frac{\gamma }{(1-\gamma )L}{v}_{{{{{{{{\rm{imp}}}}}}}}}^{2}\propto \frac{\gamma }{(1-\gamma )L}{M}_{{{{{{{{\rm{t}}}}}}}}}^{2/3}\), where Mt is the target mass, L is the latent heat, and vimp is the impact velocity, which is assumed to be proportional to the escape velocity of the target,\(\sqrt{2G{M}_{{{{{{{{\rm{t}}}}}}}}}/{R}_{{{{{{{{\rm{t}}}}}}}}}}\). For simplicity, \({R}_{{{{{{{{\rm{t}}}}}}}}}\propto {M}_{{{{{{{{\rm{t}}}}}}}}}^{1/3}\) is assumed and γ is assumed to be small. Therefore, larger planets and more ice-rich planets produce high VMF disks in general.

It is possible to have the non-negligible amount of solid or liquid in the disk if its VMF is smaller than 1 and MT is large. For example, in Run ID19, the disk liquid mass is 0.129 × 6M × (1 − γ) × (1 − VMF) ~ 0.001 M. If the entire mass is in one object, the radius is ~640 km. Its residence time would be ~230 days, which can be large enough to make a difference in the moon accretion process. Nevertheless, this is an unlikely scenario given that VMF tend to be uniform throughout the disk (Figs. 4 and 5 as well as previous work20), making it unlikely to have a local concentration of melt.

Data availability

Data produced in this study are summarized in Table 1. Raw data are not shown due to their large size but are available from the corresponding author upon reasonable request.

Code availability

The numerical codes that are used in this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank David J. Stevenson, John Tarduno, David Kipping, and Alycia Weinberger for the insightful discussion. M.N. was supported in part by the National Aeronautics and Space Administration (NASA) Headquarters under the NASA Earth and Space Science Fellowship Program Grant NNX14AP26H, the Carnegie DTM fellowship, NASA grant numbers 80NSSC19K0514 and 80NSSC21K1184 and Grant-in-Aid for JSPS Fellows JSPS KAKENHI Grant Number 10J09549. Partial funding for M.N. was provided by the Center for Matter at Atomic Pressures (CMAP), a National Science Foundation (NSF) Physics Frontier Center, under Award PHY-2020249. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation. M.N. was also supported in part by the Alfred P. Sloan Foundation under grant number G202114194. S.I. was supported by JSPS Kakenhi 21H04512. H.G. was supported by MEXT KAKENHI Grant No. JP17H06457. Numerical computations were partly carried out on GRAPE system at Center for Computational Astrophysics, the National Astronomical Observatory of Japan. Figure 6 is created by Michael Franchot.

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M.N. contributed to the idea of the project, performed the simulations, and wrote the manuscript. H.G. contributed to the project design and the SPH code development, E.A. contributed to the EOS implementation, and S.I. contributed to the disk dynamics and evolution discussion. All authors contributed to the discussion, interpretation of the results, and revising the manuscript.

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Correspondence to Miki Nakajima.

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Nakajima, M., Genda, H., Asphaug, E. et al. Large planets may not form fractionally large moons. Nat Commun 13, 568 (2022). https://doi.org/10.1038/s41467-022-28063-8

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