Survival of Exomoons Around Exoplanets

Regular moons are predicted to form generally in planetary systems, as a direct outcome of planet formation. During the core-accretion, gravitational perturbations between planet embryos imply a series of constructive impacts up to the formation of a fully grown planet (Nagasawa et al. 2007). This phase may happen between a few Myr to a few 100 Myr after the star formation (see e.g., Chambers & Wetherill 1998), and in these processes, satellites may form around the growing planets. Such giant impacts are advocated for the formation of the Earth's Moon (Canup 2004) and for the formation of Uranus and Neptune's satellites (Morbidelli et al. 2012). Satellites may also be formed in rings, which could be natural outcomes of giant impacts or tidal disruptions, either in the planet formation phase or later in a relaxed planetary system (see e.g., Charnoz et al. 2009; Canup 2012; Crida & Charnoz 2012).

An alternative scenario invokes the accretion of moons in the gaseous circumplanetary envelopes that surround the most massive giant planets during their growth in the gaseous protoplanetary disks (see e.g., Canup & Ward 2006; Sasaki et al. 2010; Miguel & Ida 2016). Inside the circumplanetary disk, the solid material is replenished by the surrounding protoplanetary disk. This solid material is thought to accrete in the form of large satellites in a way similar to planets. Then the young satellites migrate inward and sometimes sink into the planet. When the circumplanetary disk disappears, the satellite system remains and then undergoes long-term tidal evolution.

After the planets have been formed, the orbits of moons evolve due to the planet's tides. Tidal forces lead to the dissipation of energy. On the one hand, this usually results in the gradual expansion of the moon's orbits, if the evolution starts outside the planet's synchronous radius. (A synchronous orbit is where the moon's orbital period is equal to the planet's rotational period.) On the other hand, moons inside the synchronous orbit will migrate inwards and can reach the planet's Roche limit where the moons are disrupted by tidal forces.

The timescale of orbital evolution depends mainly on the intensity of dissipation in the planet's interior, which relies on the material composition and internal structure of the planet. The dissipation coefficient also varies with the changing planet radius (for gas-rich planets Recent papers attempting to compute the quality factor, Q, (also known as tidal dissipation) show that Q could even be frequency dependent, and thus may strongly depend on the orbital distance (see e.g., Ogilvie & Lin 2004). However, in the current stage of our knowledge, the large uncertainties in the origin of the dissipation process and in the internal structure of detected exoplanets preclude any theoretical calculation of Q for a specific planet. For this reason, in the present paper, a classical approach is adopted where Q will be considered as a constant, and many values will be explored. The dissipation scales as Q⁻¹ and we can assume it to be in a steady-state when exploring those scenarios which lead to moons on the long-term stability. For rocky planets, Q is low and the dissipation is high (Q ∼ 10–100), conversely for giant planets Q is high (∼10⁴–10⁵) and dissipation is low (Goldreich & Soter 1966; Alvarado-Montes et al. 2017).

The quest for exomoon discovery is a challenging task in itself, and upon finding an exomoon we could empirically constrain the Q value of the host planet by considering the planetary distance and the age of the star. This will provide invaluable information about the exoplanet interior, too, since high/low dissipation rates may depend on the planet core and interface with mantle (Lainey et al. 2012; Remus et al. 2012).

Recently, Alvarado-Montes et al. (2017) investigated the orbital changes of exomoons around close-in giant planets due to tidal effects. In addition to the tidal evolution, they implemented the changes of the giant planet's physical parameters into the model. The radius of the planet decreases over time, and this change is most prominent in the first billion years after the formation of the planet. The value of k₂/Q slowly increases, which also has an impact on the orbital evolution of the moon.

Probably the most intriguing question concerning the moons is the lack of a firmly confirmed moon in exoplanetary systems. Before novel observation projects are designed for exomoon detection, first we need to find the answer to the question of where the exomoons would be. Did we use suboptimal signal detection strategies, or did we simply look for moons in systems where their existence is not too likely? This latter possibility is the central question of this paper.

Barnes & O'Brien (2002) showed that exomoons are more stable at longer planetary orbital periods, and also that the orbit of exomoons can tidally evolve first outward and then inward (toward the planet) as the planetary rotation slows due to tidal interaction with the star. Heller (2012) studied the orbital stability of habitable moons using the method of Kipping (2009) and it was found that for stars of masses below ≈0.2M_⊙, moons are not likely to stay on stable orbits around planets which are in the habitable zone. Later Zollinger et al. (2017) showed that for planets in the habitable zones of ≲0.5M_⊙ stars, the orbit of moons is not stable as they migrate toward the planet.

Here we investigate the evolution of hypothetical satellites orbiting the currently known exoplanets with numerical integration, under the influence of tidal interaction. We map a wide range of parameters, including planet periods, moon to planet mass ratios, and k₂/Q. Instead of studying a few cases in large detail and with high accuracy, which would be meaningless due to our lack of knowledge on both the physical origin of tidal dissipation and on the planet's internal structure, we are more interested in the statistical survival rate of moons. In order to investigate a large range of configurations over billions of years of evolution, we used the average description of Barnes & O'Brien (2002) which are formulated in Equations (1)–(3) (Section 2). We apply this method to identify different evolutionary pathways of moons (Section 3) and then we calculate survival rates of hypothetical moons around known exoplanets with the aim of pointing out specific planets as observation targets that might have satellites (Section 4 and the supplementary data).

Independently of our work, Sucerquia et al. (2019) used similar methods to investigate the orbit stability of exomoons around close-in planets, and estimated their detectability through TDV and TTV effects. Guimaraes & Valio (2018) had a similar goal with a focus on the photometric detectability of hypothetical moons around Kepler planets. They selected 54 planets that could possibly host observable moons. Although we also list planets with survival rates for moons, we focus our investigations on survival statistics.

2.1. Orbital Evolution of Satellites

We calculate the orbital evolution of the planet and the moon from the following equations (Barnes & O'Brien 2002; Sasaki et al. 2012).

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}{n}}_{{\rm{m}}}}{{\rm{d}}{t}}=-\displaystyle \frac{9}{2}\displaystyle \frac{{k}_{2{\rm{p}}}{R}_{{\rm{p}}}^{5}{{GM}}_{{\rm{m}}}{n}_{{\rm{m}}}^{16/3}}{{Q}_{{\rm{p}}}{\left({{GM}}_{{\rm{p}}}\right)}^{8/3}}\mathrm{sgn}({{\rm{\Omega }}}_{{\rm{p}}}-{n}_{{\rm{m}}}),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}{n}}_{{\rm{p}}}}{{\rm{d}}{t}}=-\displaystyle \frac{9}{2}\displaystyle \frac{{k}_{2{\rm{p}}}{R}_{{\rm{p}}}^{5}}{{Q}_{{\rm{p}}}{{GM}}_{{\rm{p}}}{\left({{GM}}_{\star }\right)}^{2/3}}{n}_{{\rm{p}}}^{16/3}\mathrm{sgn}({{\rm{\Omega }}}_{{\rm{p}}}-{n}_{{\rm{p}}}),\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}{{\rm{\Omega }}}_{{\rm{p}}}}{{\rm{d}}{t}} & = & -\displaystyle \frac{3}{2}\displaystyle \frac{{k}_{2{\rm{p}}}{R}_{{\rm{p}}}^{3}}{{Q}_{{\rm{p}}}}\displaystyle \frac{{\left({{GM}}_{{\rm{m}}}\right)}^{2}}{\alpha {\left({{GM}}_{{\rm{p}}}\right)}^{3}}{n}_{{\rm{m}}}^{4}\mathrm{sgn}({{\rm{\Omega }}}_{{\rm{p}}}-{n}_{{\rm{m}}})\\ & & -\displaystyle \frac{3}{2}\displaystyle \frac{{k}_{2{\rm{p}}}{R}_{{\rm{p}}}^{3}}{{Q}_{{\rm{p}}}}\displaystyle \frac{1}{\alpha {{GM}}_{{\rm{p}}}}{n}_{{\rm{p}}}^{4}\mathrm{sgn}({{\rm{\Omega }}}_{{\rm{p}}}-{n}_{{\rm{p}}}),\end{array}\end{eqnarray} \tag{ 3 }$$

where n_m and n_p are the mean motions of the moon and the planet, k_2p is the second-order Love number of the planet which relates to the dilatation due to tidal response, Q_p is the quality factor of the planet which describes the dissipation of energy in the body, G is the gravitational constant, R_p is the radius of the planet, M_m, M_p and M_⋆ are the masses of the moon, the planet, and the host star, respectively, Ω_p is the rotation frequency of the planet, and α is the prefactor of the moment of inertia (α M_p R²). Circular orbits are assumed for both the planets and the satellites.

Since Equations (1) and (2) have the form of $\dot{y}=C\,{y}^{p}$ (where y is either n_m or n_p, p is the exponent of 16/3 while C contains all of the astrophysical constants) and Equation (3) can directly be integrated once the time evolution of n_m and n_p are known, this set of equations can be solved analytically on the domains until n_m, n_p or Ω_p do not surpass each other; in this case, the presence of the $\mathrm{sgn}()$ function would not allow the fully analytical solution. We, therefore, implemented the solution in a piecewise-analytical manner, monitoring the sign changes in the differences between n_m, n_p or Ω_p while in the distinct pieces, the exact solution is evaluated. Thus, besides the vicinity of the points where the signs commute, the solution does not even depend on this stepsize, allowing us to highly speed up the computations.

For calculating the survival rate of moons around known exoplanets, we let the orbits of the planet and the moon evolve until the current age of the star, or if it is not known then until the age of the Sun (4.57 Gyr). We solve the equations analytically with fixed time steps to monitor whether the moon reaches the Roche lobe of the planet or the critical distance (half of the Hill sphere). In these cases, the integration ends because the moon is considered to be lost either because of tidal disruption or escape from the planet's gravitational bond (for the latter case see Domingos et al. 2006). The critical distance is calculated by ${R}_{{\rm{crit}}}=0.5\,{R}_{{\rm{Hill}}}=0.5\,{a}_{{\rm{p}}}{\left({M}_{{\rm{p}}}/(3{M}_{\star })\right)}^{1/3}$.

If at the end of the calculation the moon is still orbiting the planet, then it is counted as a "survival" case.

2.2. Parameters used for the Integrations

In the example cases (see Figures 1 and 2), the system configuration consists of a Solar-mass star with an Earth-like planet and a moon that has a different mass in the different cases. The quality factor of the planet is Q_p = 10 in all cases. The initial parameters used for the different runs are summarized in Table 1. Note that the time shown in the horizontal axis is significantly shorter in Figure 1 (where it ends at 5 × 10⁶ yr) than in Figure 2 (where it ends at 10¹⁰ yr).

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3.1. Evolution Outcomes

3.2. Timescales

We calculated the evolution of possible model moons in the known planetary systems with the aim of pre-selecting those exoplanets which can physically have a satellite. These planets might worth a detailed follow-up observation for a moon survey. Planets and their model moons were simulated according to the recipe described in Section 2. The results are listed in Table A.1 in the supplementary data, available online at stacks.iop.org/PASP/133/094401/mmedia, ordered by decreasing survival rate. (In the table only those cases which have non-zero survival rates are shown.) The table also contains the mean of the integration times for each planet. If the survival rate is a hundred percent, then the mean of the integration times will be equal to the age of the star (or of the Sun if the star's age is not known, as described in Section 2). If in some cases the integration stops earlier, then the mean of the integration times will be shorter. This parameter together with the survival rate can be considered as a measure for the stability of moon orbits.

Figure 3 shows the survival rate for test exomoons as a function of the orbital period of the planet. Because the critical escaping distance is small below 10 days orbital period, only a few exomoons stay in a stable orbit around the planet. Above ∼300 days, the survival rate is about 70%–90%, as a consequence of the large critical distance. Between about 5 and 300 days we find a "transition zone" where the increasing size of the critical escaping distance results in higher and higher survival rates.

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The saturation at ∼85% survival rate for large orbital periods is caused by the chosen initial semimajor axes of the moons which in some cases is below the synchronous orbit. In these cases, the moons spiral inwards to the planet until they reach the Roche sphere.

Only a fraction of known exoplanets are promising exomoon hosts according to this work. This is primarily because the orbital period behaves as a moon "censor": not many satellites are expected (survival rate mostly below 40%) for planets in shorter period orbits than 10 days. This is not surprising, since planets, this close to the star have small Hill radii, hence stable orbits for a moon are limited to a very narrow space. As the orbit of the moon evolves, sooner or later it will reach the instability limit and escape from the planet's gravitational bond.

Interestingly, the planets with known mass and radius data (see the black dots in Figure 3) are showing smaller deviation above ∼100 days orbital period, compared to those for which the radii or masses are estimated from their other parameter (teal and yellow colored dots, respectively). For the planets with known mass and radius value, the survival rate reaches 50% at around 100 day orbital period of the planet, while for the other planets it is much more diffused in between approximately a hundred and a thousand days. Choosing a mass or radius value for the planet is an extra free parameter that naturally causes higher uncertainty in the results. This might be the reason for the shift of the transition zone from 5–100 days to 30–300 days orbital period for the two kinds of planets. For those where we have to add random mass or radius values, we will find more unstable cases, hence the survival rate starts increasing only at higher orbital periods where the critical escape distance from the planet is larger. Another explanation for the shift could be an observation bias, since the planets for which both mass and radius data are measured are those for which both transit and radial velocity observations were made. This means that they are orbiting close to the star and in the meantime, they are also massive. This might also explain why the transition zone is closer to the star for these planets.

For known exoplanets with long orbital periods, the host stars are quite faint, and the S/N ratio required for a successful moon detection is not achievable by even as precise instruments like the Kepler telescope. This finding is the resolution of the seeming paradox of missing exomoons in Kepler data. Moons are more likely to have a stable orbit around long orbital period planets, but these cases are difficult to observe. This result is also an alert for later transit finding surveys to focus on planets around bright stars with ≳10–100 days orbital period.

The successful detection of a moon around an exoplanet relies basically on two independent factors: the moons must exist (occurrence), and those moons must be observable. The occurrence is a product of two factors: the formation rate and the stability timescale. Formation studies of moons (and planets or stars) are beyond the scope of this paper, however, with the described formalism, we estimate the survival rate of moons around currently known planets.

The survival rate of model moons in known exoplanetary systems shows that satellites can only stay in orbit around the host planet if the orbital period of the planet is larger than five days. In our investigations, we found a transition zone between 5 and 300 days orbital periods where the survival rate is growing with larger orbital periods. Above 300 days orbital period the moon survival rate reaches 70%–90% for the investigated planets. These specific numbers however depend on the range of chosen initial parameters. Also, increasing the integration time could result in decreasing the survival rates in some cases. This would be only a quantitative change as it would not alter the relation between the survival rate of moons and the orbital period of the planets. Our investigation serves statistical purposes, and it shows a clear trend in survival rates that increase with the planetary orbital period.

This result is in good agreement with the findings of Alvarado-Montes et al. (2017), who developed a more detailed model including the changes of the planet's physical parameters over time as an additional factor influencing the orbital evolution of satellites. Their results showed that moons of planets with lower than ∼60 day orbital periods collide with the planet, but for planets with larger stellar distances, the satellites slowly migrate outwards and stay inside the Hill-sphere of the planet during the whole 4 Gyr integration time ("realistic" case, Figure 9 in their article). Including the change in the planet radius and k₂/Q tidal parameter over time in their model increased the chance of moon survival. This implies that neglecting these factors in our study might underestimate the moon survival rates.

Our results are in line with those of Sucerquia et al. (2019), too, who concluded that long-period planets are more favorable moon hosts than short-period ones. They used a very similar calculation method with the aim of estimating the TTV and TDV signals for detection. They conclude that in their investigated phase space no system produces TDV signals strong enough to be observed, but there are very few cases where the TTV signal would be observable with TESS and Kepler. Larger moon-to-planet ratios and stellar distances could help observations.

With a similar aim, but using a different method, Guimara˙es & Valio (2018) also investigated known planets which could host moons. Besides checking the possible semimajor-axes of moons, they also took into account their detectability, providing a list of 54 Kepler planets and candidates which could have detectable moons.

Recently, Tokadjian & Piro (2020) made a similar work to ours in which they investigated the orbit evolution of moons around known exoplanets using simplified tidal lag models. They identified 36 habitable zone planets around which a moon might be stable for long timescales (at least for 1 Gyr). We found that from these planets 29 have at least a 50% survival rate for exomoons (typically between 50 and 65%.). There are also four planets with survival rates higher than 65%: these are Kepler-459 b (69%), Kepler-456 b (72%), Kepler-1654 b (70%) and Kepler-1647 b (69%). They also found that for 21 planets the timescale before losing the moon is larger than the Hubble time in all investigated cases. Our results show moon survival rates typically between 55% and 70% for these planets with mean integration times (t_mean) between 2 and 3 Gyr (see the supplementary data). There is one outstanding case (Kepler-1628 b) that resulted in only a 9% survival rate in our study. The explanation of our lower stability levels is that a much bigger phase space was explored for the initial parameters compared to the study by Tokadjian & Piro (2020). Also, since the age of these planets is typically lower than 5 Gyrs (and in some cases, it is not known), our calculation stopped much earlier.

Martínez-Rodríguez et al. (2019) investigated the orbital stability and habitability of exomoons around known planets of M dwarfs. They found four planets that were in the habitable zone and for which the orbit of the moon is stable for a long timescale (the migration time of the moon is larger than the Hubble time). These four planets are HIP 12961 b, HIP 57050 b, GJ 876 b and c (in their paper under the names of CD-23 1056b, Ross 1003b, IL Aqr b, and c, respectively). Interestingly, for these planets our model gave 0%, 0%, 6% and 0% survival rates, respectively. One of these four planets, however, HIP 57050 b was thoroughly investigated also by Trifonov et al. (2020) (planet name: GJ 1148b in their paper) where they were using secular theory and direct N-body integrations to study the possibility of a moon. Similar to our results, they have found that it is unlikely that a moon would stay in a stable orbit around the planet.

Zollinger et al. (2017) investigated the tidal evolution and habitability of exomoons assuming possible configurations and found that moons in the habitable zone of M dwarfs experience a tidal heating rate comparable to the tidal heat flux of Jupiter's moon, Io (∼2 W m⁻²). For stars with masses ≲0.2M_⊙, because of the strong gravitational effect of the nearby star, the tidal heating rate of a Mars-like moon around a Jupiter-like planet is much stronger than that of Io's, implying that these moons might not be habitable. The maximum limit of tidal heating for habitability, however, is not known, and atmospheric effects can influence the habitability of bodies, as well (Barnes et al. 2013).

Recent hydrodynamical simulations showed that icy moons can form from circumplanetary disks of ice giants such as Uranus and Neptune, indicating that there might be a large population of moons around exoplanets beyond the snowline (Szulágyi et al. 2018). According to our findings, these moons, if formed, can stay in stable orbits for a long period of time.

Older stars host more evolved systems, in which there is a higher possibility that most satellites have disappeared due to either disruption (at the Roche radius of the planet) or escape. The presence of a massive exomoon may prevent synchronization of the planet with the star and promotes synchronization with the moon. In this case, satellites also evolve into a 1:1 spin–orbit resonance, getting to a mutually synchronised planet-moon system. Less dissipative planets (gas giants and ice giants) are more prone to keep satellites because of slower orbital migration (see e.g., Sasaki et al. 2012). If an exomoon is observed around a planet, it can provide a strong constraint on the value of the quality factor, Q, which describes the dissipation in the planet and gives hint on its interior.

Since younger stars host less evolved systems, there is a higher chance for planets to still have satellites (which will escape or be destroyed later). But satellites on less distant orbits are more difficult to observe (Szabó et al. 2006; Simon et al. 2012). If more than one moon is found, then their orbital architecture may give information on their formation, and provide a strong constrain on Q. It may also be possible to infer the presence of rings.

There is no confirmed exomoon detection to date, but the quest for finding moons in Kepler data is still in progress. Teachey et al. (2018) reported an exomoon candidate, Kepler-1625 b I (see also Teachey & Kipping 2018), but this discovery is under debate (Kreidberg et al. 2019; Teachey & Kipping 2019). Regardless of the existence of this moon, we found stable orbits for satellites around this planet. Recently, the presence of an exomoon candidate was suggested based on observations of sodium and potassium lines around planet WASP-49 b (Oza et al. 2019). Further observations are needed for confirmation.

As a result of our calculations, we conclude that planets with long orbital periods support high survival rates for moons, and close-in planets, which are easier to observe, are less likely to host moons. This can serve as an explanation for not having a confirmed exomoon discovery so far.

Thanks to high accuracy photometry and long-term follow-up missions, CHEOPS is measuring the radius of numerous known exoplanets through transits with unprecedented accuracy. This paves the way for characterizing the close environment of exoplanets and can unveil the presence of exomoons and exorings. Moons and rings are natural companions of planets. Neptune-sized planets are especially promising targets owing to their (putative) lower dissipation and large mass compared to Earth-like planets. Such targets will be in the reach of the CHEOPS mission. Following the CHEOPS mission, ESA's PLATO 2.0 mission, capable of discovering and following exoplanet systems and host stars, and the ARIEL mission will continue the task of collecting precise exoplanet light curves. Whereas exomoons have never been firmly detected up to date, they should exist and their successful detections will provide invaluable information on the planet's interior and formation process (Crida & Charnoz 2012). Detecting the first exomoons will be a major discovery as moons could provide habitable environments, in particular, if they orbit volatile-rich planets (like giant planets) and are located inside, or close to, the habitable zone (Forgan & Dobos 2016; Dobos et al. 2017).

We thank the referee, Alessandro Trani for the useful comments which improved the manuscript.

The COFUND project oLife has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 847675. This project has been supported by the Hungarian National Research, Development and Innovation Office (NKFIH) grant GINOP-2.3.2-15-2016-00003, the City of Szombathely under Agreement No. 67.177-21/2016 and the grant KEP-7/2018 of the Hungarian Academy of Sciences. V.D. has been supported by the Hungarian National Research, Development, and Innovation Office (NKFIH) grants PD-131737 and K-131508. S.C. acknowledges support for the CHEOPS mission from the CNES agency, and EXOATMOS program from LabEx UnivEarthS (ANR-10-LABX-0023 and ANR-18-IDEX-0001).