arXiv:2105.12040v1 [astro-ph.EP] 25 May 2021 Keywords uvvlo xmosaon exoplanets around exomoons of Survival ehd:nmrcl–Paesadstlie:dnmclev dynamical satellites: and Planets – numerical Methods: : ls-npaeswihaemsl aordb urn detec current by escape favoured mostly to are moons which causes planets close-in instability Tidal planets. close-in ftepae.Frtemjrt ftecoei lnt ( planets close-in the of majority the known For not of survival planet. if term the or long of star, the the determining of factor age survive important most the moon is the (which when calcul time cases We integration investigated the exoplanets. of known fraction around the orbits moon of stability 2 Netherlands The Groningen, tbeobtfrmos ewe 0ad30dy efidatransi a find we days 300 and 70%. 10 to zero Between about moons. for orbit stable 10 oeadbcmsdsutdb togtdlforces. tidal strong in by migrates disrupted moon becomes the and which lobe in scenario”, “disruption (3) and 1 Szab .Dobos V. 3 Hungary Budapest, 1121, E¨otv¨os (ELKH), Network Sciences, Lor´and Research 5 cnro:()“okn” nwihtemo a tbeorbi stable a c fo has specific orbits moon few the moon a which study of in We evolution “locking”, the moon. (1) test where scenarios: one time and integration planet years one star, a of 4 hc r h otlkl ohs moons. host to likely most the strategies, are search which optimize to order detect In previously exomoons. characterize discover to aim CHEOPS like missions Abstract. 2020 January E-mail: Hungary 112, u. h. Imre okl hg ilo srnmclIsiue eerhCe Research Institute, Mikl´os Astronomical Thege Konkoly aty srnmclIsiue nvriyo Groningen, of University Institute, Astronomical Kapteyn T-LEEolntRsac ru,90,SetIr .u 11 u. h. Imre Szent 9700, Group, Research Exoplanet MTA-ELTE LEEovo o´n nvriy ohr srpyia O E¨otv¨os Astrophysical ELTE Gothard Lor´and University, nvri´ ePrs ntttd hsqed lb eParis de Globe du Physique de Institut Paris, Universit´e de 9 u eut ieapsil xlnto otelc fsucces of lack the to explanation possible a give results Our pligtemdlt elcssfo neolntcatalogue exoplanet an from cases real to model the Applying eivsiaetetdleouino yohtclmo orb moon hypothetical of evolution tidal the investigate We er) 2 ecp cnro hr h onlae h plan the leaves moon the where scenario” “escape (2) years); o ´ 3 [email protected] , 5 1 , ept ueosatmt,n xmo a rl enconfirm been firmly has exomoon no attempts, numerous Despite 2 , 3 .Charnoz S. , 4 .P A. , al ´ 2 .Roque-Bernard A. , < 0dy ria eid)teei no is there periods) orbital days 10 ene odtrietoeplanets those determine to need we intechniques. tion rbigtdlydsutdaround disrupted tidally being or NS -50 ai,France Paris, F-75005 CNRS, , 77A,Lnlvn12, Landleven AD, 9747 sraoy zmahl,Szent Szombathely, bservatory, okl hg ilo ´ut Mikl´os 15-17, Thege Konkoly neooni h ria period orbital the is exomoon an deolnt,adptnilyto potentially and exoplanets, ed ad ni trahsteRoche the reaches it until wards tefrAtooyadEarth and Astronomy for ntre rudtepae o h full the for planet the around d hnteaeo h Sun).The the of age the then , t h uvvlrt hc is which rate survival the ate fleoondsoeisfor discoveries exomoon sful lto n stability and olution naln iesae( scale time long a on t lw n fteethree these of one llows tsgaiainldomain; gravitational et’s t nssesconsisting systems in its ini uvvlrt from rate survival in tion 4 ,Sobtey Hungary Szombathely, 2, esuytelong-term the study we , n y M. Gy. and sswt e billion ten with ases dt ae New date. to ed & Survival of exomoons around exoplanets 2
1. Introduction
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 Myrs to a few 100 Myrs after the star formation (see e.g. Chambers and 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 and 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 and Ward, 2006; Sasaki et al., 2010; Miguel and 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 undergo 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 dependant, and thus may strongly depend on the orbital distance (see e.g. Ogilvie and 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 1 constant, and many values will be explored. The dissipation scales as Q− and we can assume it to be in 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 and Soter, 1966; ∼ Alvarado-Montes et al., 2017). Survival of exomoons around exoplanets 3
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 year
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 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 and 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 (towards 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 are not ⊙ stable as they migrate towards 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 details and with high accuracy, which would be meaning less 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 and O’Brien (2002) which are formulated in Eqs. 1–3 (Section 2). We apply this method to identify different evolution pathways of moons (Section 3) and then we calculate survival rates of hypothetical moons around known exoplanets with the aim of pointing out to specific planets as observation targets which might have satellites (Section 4 and Appendix A). 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. Guimar˜aes and Valio (2018) had a similar goal with the focus on the photometric detectability of hypothetical moons around Kepler planets. They selected 54 planets which could possibly host observable moons. Although we also list planets with Survival of exomoons around exoplanets 4
survival rates for moons, we focus our investigations on the survival statistics.
2. Methods
2.1. Orbital evolution of satellites
We calculate the orbital evolution of the planet and the moon from the following equations (Barnes and O’Brien, 2002; Sasaki et al., 2012).
5 16/3 dn 9 k2pR GMmnm m = p Ω n , 8/3 sgn( p m) (1) dt −2 Qp(GMp) −
5 dn 9 k2pR p = p 16/3 Ω 2/3 np sgn( p np) , (2) dt −2 QpGMp (GM⋆) −
3 2 3 dΩ 3 k2pR (GM ) 3 k2pR 1 p = p m 4 Ω p 4 Ω 3 nmsgn( p nm) npsgn( p np) ,(3) dt −2 Qp α(GMp) − − 2 Qp αGMp − where nm and np are the mean motions of the moon and the planet, k2p is the second order
Love number of the planet which relates to the dilatation due to tidal response, Qp is the quality factor of the planet which describes the dissipation of energy in the body, G is the gravitational constant, Rp is the radius of the planet, Mm, Mp and M⋆ are the masses of the moon, the planet and the host star, respectively, Ωp is the rotation frequency of the planet, and 2 α is the prefactor of the moment of inertia (αMpR ). Circular orbits are assumed for both the planets and the satellites. p Since Equations (1) and (2) have the form ofy ˙ = C y (where y is either nm or np, 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 nm and np are known, this set of equations can be solved analytically on the domains until nm, np or Ωp do not surpass each other; in this case, the presence of the 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 nm, np or Ωp while in the distinct pieces, the exact soluion 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 to evolve until the current age of the star, or if it is not know 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 1/3 Rcrit = 0.5 RHill = 0.5 ap Mp/(3M⋆) . If at the end of the calculation the moon is still orbiting the planet, then it is counted as a ”survival” case. Survival of exomoons around exoplanets 5
2.2. Parameters used for the integrations
Stellar mass (M⋆), planet mass (Mp), semi-major axis (ap) and radius of the planet (Rp) were taken from the The Extrasolar Planets Encyclopaedia (http://exoplanet.eu, data obtained on March 25, 2021). Only those planets are considered where the M or the M sin i or the R p p · p value is known (at least one of them). Planets with higher masses than 13 Jupiter mass are ignored as well as those with M⋆ < 0.08M . In addition, the selected planets also had to have ⊙ either the orbital period (Pp) or the semi-major axis data. After this selection 4064 planets remained. For both the radius and mass (or M sin i if M is not given) of the planet, the uncertainties p· p were taken into account wherever possible. If there was no uncertainty in the data, then simply the given mass or radius was used. If there was a symmetric uncertainty given, then the mass or radius values were generated by choosing a random value with a Gauss distribution. The symmetry was determined with the same method as described in Chen and Kipping (2017, Section 2.2) where the difference between the upper and lower errors must be below 10%. For these symmetric cases the mean of these two values was taken as the standard deviation of the data. If the difference was higher than 10%, then the errors were ignored and simply the mean value was used. We applied exactly the same method for those planets where only the M sin i value was known, not the real mass. In these cases the error of the M sin i p · p · was taken into account wherever possible. This way we could generate realistic data for all planets, taken into account the uncertainties in the measurements. There are 1007 cases (out of 4064) where both the radius and the mass (M or M sin i) of p p · the planet are known. For all the other planets, the Forecaster model was used for generating realistic values for the missing parameter (Chen and Kipping, 2017). This tool predicts the mass (or radius) of the planet based on the radius (or mass). As an input, the generated (above- mentioned Gauss distributed) mass (or radius) values were used as posteriors. This way, for each run the planet had a different mass or radius, or both (if the original data had symmetric errors). See the following description of the defined ranges and distribution of the chosen random values for the rest of the (initial) parameters. Mass of the moon: The mass of the hypothetical moons is selected randomly with a • uniform distribution between 1% and 10% of the host planet’s mass, for each planet. Semi-major axis of the moon: The initial semi-major axis of the moon is chosen • randomly with a uniform distribution on a logarithmic scale between 2 Rp and Rcrit. Radius of the moon: Estimating the radius of exomoons is very challenging, because only • the Solar System serves as an example. We choose to define it through their densities (ρm) using their mass which is already chosen. Based on the big moons (radius >200 km) in the Solar system, we define three categories depending on the locations with respect to
the snowline (asnow). This is because beyond the snowline there are more icy, hence less dense bodies in the Solar system. The snowline is calculated as a = T 2 R /T 2 where snow eff · ⋆ 0 T ff is the effective temperature of the star and T 230 K is the equilibrium temperature e 0 ≈ at the planet’s sub-stellar point (Cowan and Agol, 2011). A fourth category is also given Survival of exomoons around exoplanets 6
for those cases when Teff or R⋆ is not known, hence asnow cannot be calculated. The ρm values are selected randomly with a Gauss distribution with the following parameters. 3 3 – If ap < asnow: the mean value of ρm is3g/cm , σ = 1/3g/cm . 3 3 – If asnow ap < 2asnow: the mean value of ρm is 2.5 g/cm , σ = 1/3g/cm . ≤ 3 3 – If 2asnow ap: the mean value of ρm is 2.5 g/cm , σ = 1/6g/cm . ≤ 3 3 – If asnow is not known: the mean value of ρm is 2.5 g/cm , σ = 1/2g/cm . Rotation period of the planet: In the Solar system the planet that has the slowest spin • around its axis is Jupiter with 9.9 hours. Scholz et al. (2018) defines an empirical spin– mass relation for planets and brown dwarfs. According to their work, the spin velocity scales as v √M, and from this relation it can be concluded that the spin period does not ∼ exceed 5 days even for brown dwarfs. (Spin–orbit resonances are not taken into account.) Based on these information and considering possible deviations, we set the spin period to a random value with uniform distribution between 10 hours and 5 days. Quality factor of the planet: For the quality factor (Q ) three categories are definied. • p – Rocky planets if Rp < 2R : 10 < Qp < 500 (Goldreich and Soter, 1966). Since ⊕ lower values are more probable, the distribution of the randomly chosen Qp value is uniform on a logarithmic scale. 3 6 – Ice/gas giants if Rp 2R and Pp > 10 days: 10 < Qp < 10 (Goldreich and Soter, ≥ ⊕ 1966; OgilvieandLin, 2004; Laineyetal., 2012, 2017). For these planets
intermediate values are the most probable in this range, hence Qp values are chosen 4.5 0.5 with a log-normal distribution with a mean of Qp = 10 and σ = 10 . 6 – Hot jupiters if Rp 2R and Pp 10 days: Qp 5 10 (Ogilvie and Lin, 2004). ≥ ⊕ ≤ ∼ · We do not have much information on the possible Qp value of these planets, hence we use a Gauss distribution with Q = 5 106 as the mean value and σ = 2 106. p · ·
Qp is a critical parameter, as it controls the tidal evolution of the planet, however its value is very poorly constrained, and, in the current state of our knowledge it could vary by several order of magnitudes even for similar planets (Remus et al., 2012). For the giant planets in the Solar System, Q is mostly constrained by the secular acceleration of a big moon orbiting relatively close to the planet and also by theoretical considerations. For these planets Q is estimated to be in the range of 104–105 (see e.g. Goldreich and Soter, 1966; Ogilvie and Lin, 2004; Lainey et al., 2009). However, if Q is not treated as a constant but as a function of the tidal frequency, then we obtain much lower values for Saturn with Q about 2500 (Lainey et al., 2012, 2017). Q is mostly unknown for ice giants like Uranus and Neptune, but theoretical considerations put them in the range > 104 (Goldreich and Soter, 1966). However this estimate is based on some unconstrained assumption on the moon formation scenarios and ages. There is also a more recent estimation for Neptune which puts the Q value between 9000 and 36000 (Zhang and Hamilton, 2008). Conversely, the Q of terrestrial planets is very low, in comparison to gas giants, and is typically between ten and a few hundreds (see e.g. MacDonald, 1964; Goldreich and Soter, 1966; Smith and Born, 1976; Ray et al., 2001; Rainey and Aharonson, 2006; Lainey et al., 2007). The Q for these planets is better constrained because of the possibility of laboratory Survival of exomoons around exoplanets 7
tests of rocks (where possible). While Q might go up to 500 for rocky planets, it is typically below 200 in the Solar System. It is highly challenging to draw conclusions about the possible Q value of exoplanets. It seems reasonable to assume that rocky planets have similar values as in the Solar System. We can also speculate that maybe the Q of giant planets is in the same order of magnitude as their Solar System counterparts, however, theoretically, with low-density ice-rich cores, Q in the range a few hundreds would also be possible for a Saturn-like planet (Lainey et al., 2017). In hot-jupiters, because of the assumed synchronous spin and circularized orbit, Q might be around 5 106 (Ogilvie and Lin, 2004). · The Love number k2 varies modestly from one planet to another in the Solar System but is between 0.1 and 0.6 (Gavrilov and Zharkov, 1977). We choose k2p = 0.5. However k2 never appears alone, but only in k2/Q. So we fix the value of k2 and vary the value of Q to save a free degree of liberty that would lengthen the computation time. For the same reason, the parameter used for calculating the gyration radius is also fixed to α = 0.3. It is expected that planets with different compositions and structures have different gyration radii, but by keeping α fixed, a free parameter is removed. For each planet ten thousand runs are made to get a wide sample of the different initial parameters. What we call the ”survival rate” is simply the ratio of surviving moons (those that stayed in orbit around the planet until the end of the integration) and the total number of tested configurations. For a few example cases, the parameters of the planet and the moon are set to specific values which are described in details in Section 3.
3. Example cases
In the example cases (see Figs. 1 and 2), the system configuration consists of a Solar-mass star with an Earth-like planet and a moon which has different mass in the different cases. The quality factor of the planet is Qp = 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 Fig. 1 (where it ends at 5 106 years) than in Fig. 2 (where it ends at 1010 years). ·
3.1. Evolution outcomes
In the integrations, the moon has been observed to follow three different families of evolutionary tracks, and depending on the actual parameter set, all of these three tracks can evolve rapidly, resulting in an unstable system, or the evolution can be slow enough to lead to a surviving moon. Both the morphology and the stability depends on the position of the Hill sphere, thus eventually, on the masses and orbital periods of both the planet and the moon, the
stellar mass, and Qp.
3.1.1. Disruption scenario When the planet is close enough to the star, the tidal torque on the planet from the star very effectively spins down the planet. The tidal bulge is shifted Survival of exomoons around exoplanets 8
Fig.1upper Fig.1lower Fig.2upper Fig.2middle Fig.2bottom panel panel panel panel panel
Mm/Mp 1/85 1/85 1/85 1/150 1/300
ap [AU] 0.3 0.3 1.0 1.0 0.8 6 1 nm [10− s− ] 38.0 6.1851 2.6640 2.6640 2.6640
Table 1. Initial parameters used for the example cases shown in the different panels of Figs. 1 and 2. The star mass is M⋆ = 1M , the planet mass is Mp = 1M and the quality factor of the ⊙ ⊕ planet is Qp = 10 in each case. slightly toward the trailing side of the moon’s orbit, the moon quickly looses energy, spirals inward and is disrupted at the Roche lobe. The time scale of this migration is very short (in the order of 1–200 million years, mostly depending on the stellar distance and on the moon– planet mass ratio) compared to the 10 Gyr integration time, therefore we consider that these moons are unstable. This scenario is described in details in Alvarado-Montes et al. (2017), and our results are completely compatible with their findings (Fig. 1, top panel).
3.1.2. Escape scenario If the planet’s orbit is close to the star, then the Hill radius is smaller. On any orbit inside the Hill sphere, the moon suffers strong tidal forces, especially if the planet is relatively massive, which makes the outward migration very fast. As a consequence, if the moon formed close to the critical distance from the planet (half of the Hill sphere), then it will quickly escape, before synchronization could happen (Fig. 1, lower panel).
3.1.3. Tidal locking In connection with tidal locking, we define three different scenarios.
Moon–planet locking: the Selenoid scenario. The evolution track is somewhat similar to the Earth–Moon case. First, the moon migrates outwards, until a spin-orbit synchronisation is reached between the orbital period of the moon and the rotation of the planet. In the later evolution, the moon slowly turns back, the typical time scale before colliding with the planet was in the order of 10 billion years in our integrations. (More specifically, the end state of the moon is reaching the Roche radius of the planet, where the moon will be disrupted). Therefore, we consider this scenario as stable (Fig. 2, top panel). We note that the similarity with the Earth–Moon case is limited to the synchronization between the two bodies, but there is a qualitative difference: here the planet spin is synchronizing with the orbit of the moon, while in the Earth–Moon case the spin of the Moon is synchronized with the orbit of the Earth. Star–planet locking. This scenario is similar to the Selenoid, but rather than synchronising with the moon, the planet is synchronised to the central star. We did find this possibility in several test runs, and seems to be typical for certain configurations (close- in systems with lightweight moon). In these cases, the moon’s migration is slow and has a similar time scale than in the Selenoid case. The path is qualitatively identical: first the moon migrates outwards and then later inward (Fig. 2, middle panel). This configuration is also stable on the long term, and the moon can be observed (with difficulties coming from its low Survival of exomoons around exoplanets 9