## Abstract

The large size and wide orbit of the recently announced exomoon candidate Kepler-1625b-i are hard to explain within traditional theories of satellite formation. We show that these properties can be reproduced if the satellite began as a circumstellar co-orbital body with the original core of the giant planet Kepler-1625b. This body was then drawn down into a circumplanetary orbit during the rapid accretion of the giant planet gaseous envelope, a process termed “pull-down capture.” Our numerical integrations demonstrate the stability of the original configuration and the capture process. In this model, the exomoon Kepler-1625b-i is the protocore of a giant planet that never accreted a substantial gas envelope. Different initial conditions can give rise to capture into other co-orbital configurations, motivating the search for Trojan-like companions to this and other giant planets.

## INTRODUCTION

The planetary satellites of the Solar system planets provide a wealth of information regarding the conditions under which the planets formed, reflecting the different pathways available in the inner and outer Solar system (*1*). Our attempts to understand the great variety of planetary types and architectures discovered around other stars would benefit greatly from similar information. Therefore, the recent announcement of a tentative detection of a bound companion to the transitting planet Kepler-1625b (*2*, *3*) is of potentially great importance. The precise properties of this observed system depend on the details of the analysis, but a representative value for the mass of the planet is ∼2*M _{J}*, the mass of the satellite ∼10

*M*

_{⊕}, and the present distance of the exomoon from the planet ∼45

*R*. As shown in Fig. 1, this implies that the proposed satellite (hereafter called Kepler-1625b-i) has a mass and angular momentum far in excess of anything seen in the satellites of the Solar system planets. The parameters of Kepler-1625b-i are comparable to those of planets recently discovered orbiting close to low-mass stars. It is therefore not obvious that Kepler-1625b-i formed in a similar way to the Solar system moons.

_{J}The standard model for the formation of moons around gaseous planets is to form them from a disk of solid material in orbit. The origin of this disk has been variously hypothesized to be a solid-enhanced disk left behind by the initial collapse (*4*) or one that is fed by continuous accretion from the protoplanetary nebula (*5*). In either variant, the nominal mass of the resulting moons is well below the observed values for Kepler-1625b-i [e.g., (*6*)]. Population synthesis studies of these models can just reach the observed values (*7*), but in such cases, the amount of solid material added to the planet itself is such that the planet is enriched in solids-to-mass fractions ∼50%. Such a large amount of solid enrichment would imply a planetary radius ∼0.8*R _{J}* (

*8*), which is smaller than the observed radius (

*2*). It appears as though that the Kepler-1625b-i system cannot be realized within the standard formalism developed for the Solar system giant planets.

Another suggested scenario is that the Kepler-1625b-i could be another planet that was captured by Kepler-1625b during a close encounter, in which tidal interactions dissipated enough energy to capture the planet. Such a process would capture the satellite into a highly eccentric orbit, which would eventually circularize to a much more compact configuration (*9*). There is, in principle, enough angular momentum in the spin of a newly formed giant planet (see Fig. 1) to drive the satellite outward (*10*), much as is believed to be the case in our own terrestrial system [e.g., (*11*)]. The problem here is one of time scale. Tidal interactions get rapidly weaker with distance, and the outward spiral of the orbit will be limited by the lifetime of the system. We can estimate the time scale to spiral out to the present orbit because of transfer of angular momentum from the planet to the satellite (*12*) as*M _{s}* is the mass of Kepler-1625b-i and

*a*is the semimajor axis of its planetocentric orbit. We have assumed a bulk tidal dissipation in the planet Kepler-1625b-i, σ

*, equivalent to that of Jupiter, σ*

_{p}*, as calculated on the basis of the orbital evolution of the Jovian moons (*

_{J}*13*). This is orders of magnitude too long, given the age of the system. To quantify the discrepancy, we can also reverse the above argument to estimate the value necessary to achieve the desired level of outspiral. If we set

*T*= 10

_{p}^{10}years, then we require σ

*∼ 1100 σ*

_{p}*. If we cast this in terms of the commonly quoted “tidal*

_{J}*Q*,” then it would imply

*Q*∼ 25 for the planet, a value more characteristic of a terrestrial planet than a gas giant. The conclusion that we draw from the above discussion is that all scenarios that assemble or capture Kepler-1625b-i after the host planet formed suffer from the problem that they produce moons that are either too small or too close.

We propose instead that Kepler-1625b-i became a satellite by the process of pull-down capture during the rapid accretion of the giant planet gaseous envelope (*14*). In the standard theory of giant planet formation by core accretion (*15*), the rocky cores of giant planets grow slowly by the accretion of planetesimals while maintaining a hydrostatically supported gaseous envelope. Eventually, the core grows large enough that gas pressure in the envelope can no longer support it against the planetary gravity and hydrostatic equilibrium in the envelope breakdown. At this point, the planet accretes gas on a dynamical time scale, until such time as it has opened a gap in the gaseous disk, which limits further supply. The final mass of the planet may grow further as gas is supplied to the planet by viscous transport through the disk (*16*), but a substantial fraction of the mass is accreted rapidly during this episode. The rapid growth in the planetary mass affects the dynamics of nearby bodies that share similar orbits to the growing planet. We will show that some fraction of these bodies can be drawn down into stable configurations that correspond to traditional satellite orbits.

## MATERIALS AND METHODS

The original pull-down scenario (*14*) was used to describe the capture of low-mass irregular satellites to the giant planets of the solar system. Our first step was therefore to demonstrate that this process also operates when the initial co-orbital pair was of comparable mass. The numerical integrations used in this paper were performed by integrating the equations of motion for the direct gravitational interactions of three bodies using a Bulirsch-Stoer integrator. Although there are several programs in wide use for the integration of planetary system dynamics, they often use mass-dependent coordinate system transformations (such as Jacobi coordinates) to increase the efficiency and accuracy of the integration schemes. The use of these transformations can introduce subtle errors when the planetary masses are a function of time. Hence, we opted to numerically integrate a general three-body problem in a fixed coordinate system, only transferring to heliocentric coordinates at the end. In the Supplementary Materials, we demonstrate that the starting conditions for the pull-down scenario remain valid in this limit and that pull-down capture proceeds in a qualitatively similar fashion.

Figure 2 shows an example of such a pulldown. We started at one planet, the future giant planet, with a mass *m*_{2} = 30*M*_{⊕}, as expected from a giant planet core as it transitions from a hydrostatically supported envelope to runaway gas accretion (*15*). The other was assumed to have a mass *m*_{3} = 10*M*_{⊕}, as a representative value for Kepler-1625b-i. This can be viewed as a second protocore that lagged the former in growth rate and so has not yet reached the point of envelope instability. The pair was assumed to be in orbit about a 1-*M*_{⊙} (solar mass) star with a semimajor axis of 1 astronomical unit. The initial positions and velocities were chosen such that the pair exhibits a quasi-periodic, retrograde orbit when viewed in the frame co-rotating with the larger of the pair. This planet was then assumed to grow according to*T* = 500. This describes a mass accretion rate that decays exponentially and was intended to describe the initially rapid accretion of mass that tails off as a gap is opened in the protoplanetary disk. The smaller body is dragged down by the changing gravitational potential to an orbit of final semimajor axis of 31*R _{J}*, which lies within the estimated range for Kepler-1625b-i. Farther or closer orbits are easily obtained by starting with an initial binary of different separation.

To understand the dynamics, let us briefly return to the restricted problem (*m*_{3} = 0) so that we may make use of a conserved integral, the Jacobi constant *C*. As the mass of *m*_{2} grows, the orbit of *m*_{2} about *m*_{1} shrinks to conserve angular momentum. At any given moment in the evolution, the system can still be described in the context of the restricted three-body problem, but the value of *C* changes. The critical values of *C* that regulate the orbital dynamics also evolve with time but do not scale similarly with mass, as do the orbits. This means that the mass growth of *m*_{2} can lead to a qualitative change in the orbit of a test particle if the value of *C* crosses a threshold value that separates one family of orbits from another.

To demonstrate this, we fixed our definition of the Jacobi constant to the original coordinate system so that the absolute value will evolve as the mass does, namely

This has the same form as the traditional definition, but the numerical value will evolve as *m*_{2}. The quantities *x*, *y* (the components of the position of *m*_{3} in the co-rotating orbital plane), and *V* (the velocity of *m*_{3} in the co-rotating frame) are all defined relative to the center of mass and so will also evolve as the mass ratio changes. The semimajor axis *a* also evolves as the orbit shrinks. This was calculated directly from the equation for the orbital energy. The critical values of *C*, as a function of *m*_{2}, were calculated from the extrema of the instantaneous pseudopotential. Figure 3 shows the derivative ∂*C*/∂*m*_{2} for the case *m*_{2} = 9 × 10^{−5} and *m*_{3} = 0. We see that the derivative peaks near the planet (*17*), so that particles that spend a long time in the vicinity of the planet will be most affected by the mass growth. Figure 4 shows the evolution of three different initial orbits, indicated as filled circles in Fig. 3. The orbit shown in Fig. 4A experiences a substantial growth in *C* and soon crosses the critical values that define the L_{2} and L_{1} Lagrangian points. This is the formal criterion for pull-down capture, meaning that the planetary orbit is ultimately confined to the Hill sphere of *m*_{2}. The orbit shown in Fig. 4B demonstrates that the Lagrangian L_{4} point is stable and that particles exhibiting this kind of tadpole orbit only become more tightly bound upon mass growth of *m*_{2}. In Fig. 4C, we see that horseshoe orbits that pass around the L_{3} point can be also pulled down into tadpole orbits, in this case about the L_{5} point.

## RESULTS

An important potential discriminant between the pull-down capture model and traditional satellite formation models is that this model generically produces final satellite orbits that have substantial eccentricity and orbital inclination (defined here as the angle between the angular momentum vectors of the *m*_{3} to *m*_{2} orbit relative to the *m*_{2} to *m*_{1} orbit). Although there is no observational constraint on the orbital eccentricity of Kepler-1625b-i, there is weak evidence for substantial orbital inclination (*2*). During pull-down capture, the absolute value of any vertical oscillation about the planetary orbital plane is largely preserved, but the planetocentric radius of the satellite orbit shrinks markedly. This means that the inclination of the satellite orbit about the planet is amplified during the pulldown. Figure 5 shows the evolution of a pair that is initially misaligned by only 0.6° in the circumstellar orbital plane. These initial cores both orbit the star in the prograde direction, but the mutual orbit is retrograde in the frame of the growing planet, so the initial inclination in the rotating frame is ∼180°. The mass growth leads to rapid changes in orbital inclination as the orbit shrinks, eventually settling into a retrograde orbit but with substantial final inclination relative to the planetary orbital plane. This planet has the same mass growth as in Fig. 2.

Captured orbits also show a wide range of orbital eccentricities. The most eccentric of these get close enough to the planet that tidal dissipation in the satellite will likely circularize the orbit during the age of the star. However, we estimate, in the Supplementary Materials, that this is likely to occur only for those satellites whose orbits get within ∼16*R _{J}* of the planet. Given the estimated parameters of Kepler-1625b-i (semimajor axis in the range of 23 to 55

*R*), this suggests that the orbit of the satellite will still retain its original eccentricity. This is important, as it can affect the predictions for future transit events. It also implies that there may be a population of similarly massive satellites on more compact, circular orbits, awaiting discovery around giant planets.

_{J}The time scale of the planetary growth has an important effect on the outcome. Figure 6 shows the final semimajor axis and inclination for three different integrations of the pulldown of a *m*_{3} = 3 × 10^{−5} satellite. In each case, the initial value of the planet mass is *m*_{2} = 9 × 10^{−5}, and the final planet mass is 2.2 × 10^{−3}. The top shows a characteristic growth time scale *T*_{0} = 10, comparable to the dynamical time of the gas at this location. The middle shows the case for *T*_{0} = 100 (this is the case shown in Fig. 3), and the bottom shows the case for *T*_{0} = 1000. In the first two cases, the integration is for *T* = 500, and in the third case, the integration is for *T* = 5000. As *T*_{0} increases, the final states are more tightly bound and of lower inclination. The efficiency of capture also drops significantly with increasing *T*_{0}. The initial separations of the simulations in Fig. 6 are drawn uniformly from 2 to 10 initial Hill radii, and the velocities are drawn uniformly from the range that allows stable initial orbits. Inclinations are drawn uniformly from 0° to 2^{∘}. Integrating forward, in each case, 10^{4} samples from these conditions yield a capture efficiency of 8.5% for *T*_{0} = 10, which drops to 2.1% for *T*_{0} = 100 and 0.3% for *T*_{0} = 1000. The latter two values produce very few systems in the observed range of separation and inclination. If systems like Kepler-1625b-i turn out to be common, then they would argue in favor of a model in which the bulk of the mass of gas giant planets is accreted within a few local dynamical times.

## DISCUSSION

The inferred radius and mass of the satellite Kepler-1625b-i (*2*) are consistent with the expected properties of a giant planet core before the dynamical instability stage of the core accretion scenario [e.g., (*15*)]. We suggest that multiple potential giant planet cores were growing on similar orbits in this system. Such a crowded location is a consequence of the proposal that planet growth is enhanced at particular locations in a protoplanetary disk, so-called “planet traps” (*18*), where hydrodynamical and/or chemical conditions conspire to enhance the growth rates of planetary cores and to slow their radial migration. Similar conditions may arise from the assembly of planetary cores in the presence of eccentricity dissipation (*19*). When the first planet to enter the dynamically unstable phase starts to grow, the changing gravitational interactions will perturb the remaining cores and can lead to a variety of outcomes. As discussed in Results, the dynamical evolution of co-orbital bodies varies depending on their position relative to the accreting planet. The objects that show the most marked evolution are those that spend the most time in close proximity to the growing planet. It was shown that a stable class of these orbits exists in the context of the Hill problem (*20*) and in the more general context of bodies with finite masses (*21*). In the Supplementary Materials, we describe the properties of this orbital family in the mass range of interest (that of giant planet cores of mass ∼10 to 30*M*_{⊕}). During the mass growth, these nearby objects are dragged down closer to the planet, resulting in a final orbit well within the Hill sphere, i.e., a permanently captured satellite. Objects that exhibit tadpole or horseshoe orbits before the accretion remain stable and end up more tightly bound in these configurations (*22*). Other orbits can become destabilized in the process. It has been proposed that Uranus and Neptune may also be proto-giant planet cores that were originally formed in the Jupiter-Saturn region and scattered outward (*23*) as Jupiter and Saturn grew, and they may represent the outcome of this pathway.

The efficiency of the capture depends primarily on the speed with which the giant planet mass grows during the epoch at which the satellite crosses into the Hill sphere. Hence, stability depends only weakly on the final planet mass, although late-time mass growth will draw the satellite closer to the planet. A massive satellite may also restrict the inflow of gas to form more traditional satellites if these are formed from a disk that is supplied continuously (*5*) but may still allow formation of a traditional moon population in the case where the disk is a consequence of the initial collapse (*4*).

It is too early to tell whether such massive satellite configurations are rare or common, as the observed system is right at the edge of detectability and may represent the tip of an iceberg still to be recovered. However, there are hints that Neptune mass planets, comparable to Kepler-1625b-i, may be more numerous than expected from current models of giant planet formation. These claims have recently been made on the basis of microlensing surveys (*24*) and studies of transitting planets (*25*). Such an overabundance is consistent with our scenario in which many cores inhabit the giant planet formation region initially but only some are allowed to grow to giant planet masses. In such an event, we also anticipate that objects with mass similar to Kepler-1625b-i may also be captured into other stable configurations, such as Trojan orbits. This predicted population is also presently at the limit of detectability, but potential signatures of this type have been reported recently (*26*, *27*).

In this model, the parameters of Kepler-1625b-i also provide an important glimpse into the properties of a poorly understood phase of the core accretion model. Descriptions of the initial growth of giant planet envelopes assume hydrostatic equilibrium, an approximation that fails when the core reaches ∼100*M*_{⊕} (*15*). The final mass of the planet is determined by the properties of the gas disk and the rate at which mass can be supplied viscously to the planetary location (*16*). However, in between these two limits, the evolution of the planetary mass is a consequence of the dynamical instability of the disk gas in the vicinity of the growing core, a phase that is still poorly captured by current models although it is responsible for a substantial fraction of the mass growth of the planet. The efficiency of the pull-down process is a function of the speed of growth in this phase. If we repeat the calculation, from the same initial conditions but with a longer growth time, then the pull-down capture becomes increasingly inefficient for growth times >30 orbital times. As shown in the Supplementary Materials, this is a consequence of the fact that the phase space for stable orbits narrows when the planetocentric radius is of the order of the Hill sphere radius. If the mass growth is fast enough, then the satellite can be pulled down through the unstable region before orbital instabilities have a chance to grow. If the mass growth is too slow, then the orbit becomes unstable and the satellite undergoes a close encounter with the planet and is scattered out of the co-orbital region. The range of final inclinations is also larger for shorter growth times, as shown in Fig. 6. A larger sample of these orbits, with constraints on orbital eccentricity and inclination, could provide a probe of the growth time scale.

If Kepler-1625b-i were a rocky body, then it might provide an interesting alternative environment for studying planetary habitability, as the level of insolation that it receives from the host star is compatible with broad definitions of planetary habitable zones (*2*). If our interpretation is correct, then this body is more accurately described as a proto-giant planet core, isolated from the gas disk before it had a chance to exceed the threshold for runaway gas accretion. However, even in this stage, it should host a gaseous envelope of similar mass to its rocky inventory (*10*), which is consistent with the estimated radius and too thick to enable Earth-like conditions. On the other hand, studies of objects such as Kepler-1625b-i can provide an invaluable snapshot into a phase of giant planet evolution that is usually hidden beneath several hundred Earth masses of gas.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/10/eaaw8665/DC1

Supplementary Methods and Methods

Fig. S1. Retrograde stable orbits.

Fig. S2. Initial conditions for stable orbits.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**Some of the simulations described here were performed on the UCLA Hoffman2 shared computing cluster and using the resources provided by the Bhaumik Institute. This research has made use of NASA’s Astrophysics Data System.

**Author contributions:**B.M.S.H. conceived of the idea, performed the calculations, and wrote the manuscript.

**Competing interests:**The author declares that he has no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).