Application of Orbital Stability and Tidal Migration Constraints for Exomoon Candidates
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Application of Orbital Stability and Tidal Migration Constraints for Exomoon Candidates
Billy Quarles , Gongjie Li , and Marialis Rosario-Franco
2, 3 Center for Relativistic Astrophysics, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332 USA National Radio Astronomy Observatory, Socorro NM 87801, USA University of Texas at Arlington, Department of Physics, Arlington TX 76019, USA (Received October 1, 2020; Revised October 1, 2020; Accepted October 1, 2020)
Submitted to ApJLABSTRACTSatellites of extrasolar planets, or exomoons, are on the frontier of detectability using current tech-nologies and theoretical constraints should be considered in their search. In this Letter, we applytheoretical constraints of orbital stability and tidal migration to the six candidate KOI systems pro-posed by Fox & Wiegert (2020) to identify whether these systems can potentially host exomoons. Thehost planets orbit close to their respective stars and the orbital stability extent of exomoons is limitedto only ∼
40% of the host planets Hill radius ( ∼
20 R p ). Using plausible tidal parameters from thesolar system, we find that four out of six systems would either tidally disrupt their exomoons or losethem to outward migration within the system lifetimes. The remaining two systems (KOI 268.01 andKOI 1888.01) could host exomoons that are within 25 R p and less than ∼
3% of the host planets mass.However, a recent independent transit timing analysis by Kipping (2020) found that these systems failrigorous statistical tests to validate them as candidates. Overall, we find the presence of exomoons inthese systems that are large enough for TTV signatures to be unlikely given the combined constraintsof observational modeling, tidal migration, and orbital stability. Software to reproduce our results isavailable in the GitHub repository: Multiversario/satcand. INTRODUCTIONThe
Kepler data has discovered a myriad of exoplanets, however a substantial number of viable planet satellite(exomoon) candidates have not been uncovered. The best exomoon candidate (Kepler 1625b-I, Teachey & Kipping(2018)) is hosted by a Jupiter-sized exoplanet on a fairly wide orbit ( ∼
287 days). Fox & Wiegert (2020) recentlyidentified six KOIs (Kepler Objects of Interest) that exhibit transit timing variations (TTVs, Kipping 2009a,b) whichcould possibly be explained by the reflex motion of an exomoon. If validated, such a discovery would represent a giantleap forward in the detection of exomoons (Kipping et al. 2012, 2013b,a, 2014, 2015; Teachey et al. 2018). A majordifference between these KOIs and Kepler-1625b is the proximity to their host star, where gravitational tides and/orgeneral relativity effects can be important. We provide an analysis focusing on the orbital stability limits for exomoons(Rosario-Franco et al. 2020) and the possible outcomes of tidal migration considering the tidal influence between theplanet-star and planet-satellite (Sasaki et al. 2012).The search for exomoons using photometric data (Sartoretti & Schneider 1999; Cabrera & Schneider 2007) nowhas a long history due to the
Kepler mission, where additional constraints beyond TTVs are usually required (e.g.,transit duration variations, or TDVs, Kipping (2009a)), or techniques that make use of sampling effects Heller (2014);Hippke (2015); Heller et al. (2016). Kipping (2020) performed an independent analysis of the KOIs proposed by Fox& Wiegert (2020) and found no compelling for evidence among the six candidates using rigorous statistical hypothesistesting. Kepler-1625b passes 2 out of 3 such tests and remains the best exomoon candidate despite its own history(Heller 2018; Heller et al. 2019; Kreidberg et al. 2019). Kipping & Teachey (2020) have introduced constraints fromtidal interactions (Barnes & O’Brien 2002) that place limits on allowable ranges from TTVs or TDVs, however tidal
Corresponding author: Billy [email protected] a r X i v : . [ a s t r o - ph . E P ] S e p Quarles, Li, & Rosario-Franco interactions that change the planetary rotation also need to be included because of the non-negligible effect on themoon lifetimes (Sasaki et al. 2012, see their Figure 13).Gravitational tidal models depend on parameters (e.g., tidal Love number k , tidal time lag ∆ t , moment of inertia α , or tidal quality factor Q ) that are unconstrained for most (if not all) exoplanets and even not well constrained forplanets in our own solar system (Goldreich & Soter 1966; Lainey 2016). Models based upon equilibrium tides witha constant time lag (Hut 1981; Eggleton et al. 1998; Fabrycky & Tremaine 2007) or with a constant Q (Goldreich &Soter 1966; Ward & Reid 1973) are qualitatively similar in their predictions of moon lifetimes (Tokadjian & Piro 2020),where discrepancies may arise long after the main sequence lifetime of the host stars. Although these parameters arenot well known for exoplanets, the tidal migration largely depends on the ratio k / ( αQ ) and reasonable extremes canbe estimated from the solar system planets.In this Letter, we determine the plausibility of exomoons orbiting the six candidates from Fox & Wiegert (2020)using orbital stability (Rosario-Franco et al. 2020), a constant Q tide model (Sasaki et al. 2012), and results from arecent TTV analysis (Kipping 2009b). In Section 2, we demonstrate how orbital stability limits can be used to placeupper limits on physical parameters of exomoons. We evaluate a constant Q tide model and estimate the lifetime ofexomoons in Section 3. We combine our analysis of exomoon orbital stability and tidal migrations with the upperlimits from Kipping (2020) in Section 4. Our results are summarized in Section 5, where we also identify how Kepler1625b-I fits within our analysis. ORBITAL STABILITYAn exomoon gravitationally interacts with both its host planet and the planet’s host star, where the combinationof these forces limits the orbital separation between the exomoon and its host planet. The limiting planet-satelliteseparation, or stability limit, is a fraction f crit of the the Hill radius R H (= a p [(M p +M sat )/(3M (cid:63) )] / ), which dependson the planetary semimajor axis a p , planetary mass M p , satellite mass M sat , and the stellar mass M (cid:63) . Our recentwork (Rosario-Franco et al. 2020) identified f crit ≈ . a sat , planet eccentricity e p and satellite mean anomaly M A sat . We define thestability limit as: a crit = f crit R H (1 − . e p ) in terms of the Hill radius, where the additional factor is necessary toaccount for changes in the Hill radius for eccentric orbits of the planet.Although the planetary semimajor axis is well-determined, there is a significant uncertainty in the stellar mass for thesix exomoon candidate systems proposed by Fox & Wiegert (2020). Moreover, the planetary mass is undetermined andwe must rely on probabilistic determinations (Chen & Kipping 2017) based upon statistical relationships uncoveredfrom the confirmed Kepler planets with radial velocity mass measurements. We summarize the current values anduncertainties obtained from the
Kepler
Exoplanet Archive (DR25) for the stellar mass M (cid:63) , planetary radius R p ,planetary semimajor axis a p , and system age τ in Table 1. Updated values are used based upon studies that implementasteroseismology (Silva Aguirre et al. 2015) or better isochrone fitting (Morton et al. 2016) for the stellar age. Bergeret al. (2018) identifies better constraints on the planet radius R p due to precise astrometric measurements from Gaia,where we update appropriately. The planetary mass is estimated using Forecaster from Chen & Kipping (2017)based upon our best knowledge of the planet radius and the satellite mass is small compared to the planetary mass.Using our formalism for the stability limit and the best known system parameters (Table 1), we identify the locationof a crit in units of the planetary radius R p and as a function of the planetary eccentricity in Figure 1. The red curvemarks the determination of the stability limit using the mean system values and the gray curves illustrate the variancein the stability limit due to the uncertainties in the system values. The black region denotes the combinations of satellitesemimajor axis a sat and planet eccentricity e p that permit long-term stability. We use a lower boundary on a sat = 2 R p ,but the lower boundary should be defined by the Roche limit. The Roche limit depends on unknown properties (massor density) of the exomoon candidates and their host planets. Using the mean values of the probabilistic planetarymasses, we can estimate some sensible values for the Roche limit. The Roche limit for KOI 1925.01 is ∼ .
75 R p ,while the Roche limit for all the other KOIs is less than 2 R p . Despite the unknowns, we can estimate the stabilitylimit a crit within a factor of ∼
2. Kipping (2020) identified a large eccentricity ( e p ∼ .
6) for KOI 1925.01 through hisphotodynamical fits, which substantially truncates the stability limit for exomoons in the system so that the largestplanet-satellite separation is a sat (cid:46) −
12 R p . TIDAL MIGRATIONTidal migration timescales and/or distances can be used to constrain the possibility of an exoplanet to host exomoons(Barnes & O’Brien 2002; Sucerquia et al. 2019). The migration depends on several parameters that are unknown (tidal xomoon Constraints k p and tidal Quality factor Q p ), but we can identify plausible parameters using values from the solarsystem. Using the observed planetary radius R p , we assign either 0.299 (R p < ⊕ ; Lainey (2016)) or 0.12 (R p ≥ ⊕ ;Gavrilov & Zharkov (1977)) for the tidal Love numbers. A lower limit for Q p can be estimated using the system age τ and the critical mean motion n crit (= (cid:112) G ( M p + M sat ) /a ) determined from the stability limit a crit . We parameterizethe planet-satellite mass ratio as f m = M sat /M p and evaluate tidal models over a wide range (10 − ≤ f m ≤ − ).We implement a constant Q tidal model (Sasaki et al. 2012) that is directly applicable to planet-satellite mass ratios M sat /M p < .
1, which is akin to the Pluto-Charon system (Cheng et al. 2014). Through our tidal model, we areinterested in two regimes: 1) the satellite tidally migrates outward past the stability limit (see §
2) before the satellite’smean motion synchronizes with the planetary spin frequency (Ω p = n sat ) or 2) the satellite tidally migrates inwardtowards the Roche limit following angular momentum conservation after synchronization. Sasaki et al. (2012) providesan analytical decision tree algorithm that is based on the following differential equations:˙ n sat = − k p R Q p M sat M p n / [ G ( M p + M sat )] / sgn[Ω p − n sat ] , (1)˙ n p = − k p R Q p n / G ( M p + M sat )[ G ( M (cid:63) + M p + M sat )] / sgn[Ω p − n p ] , (2)˙Ω p = − k p R αQ p (cid:34) GM [ GM p ] n sgn[Ω p − n sat ] + n GM p sgn[Ω p − n p ] (cid:35) , (3)which depends on the exomoon’s mass M sat , planetary mean motion n p , and the moment of inertia constant α .Equations 1–3 are valid assuming that the exomoon’s orbit is not yet synchronized with the planetary rotation (Ω p >n sat ), the exomoon spin Ω sat synchronous with its mean motion (Ω sat = n sat ), and the planetary spin is large comparedto its mean motion (Ω p > n p ). Moreover, these equations are applicable for circular and coplanar orbits. Eccentricplanetary orbits are beyond our scope because only one of the candidates has an estimate for the planetary eccentricity,but these equations can be modified by including a polynomial function N ( e ) (e.g., Cheng et al. 2014).After synchronization between the satellite mean motion and planetary rotation (Ω p = n sat ), the planet-satellitesystem evolves through angular momentum conservation. The total angular momentum L consists of the sum of threeterms: 1) the planetary rotational angular momentum, 2) the planetary orbital angular momentum, and 3) the satelliteorbital angular momentum, which is represented by: L = αM p R p Ω p + M p [ G ( M (cid:63) + M p + M sat )] / n / p + µ [ G ( M p + M sat )] / n / sat , (4)which includes the reduced mass µ = ( M p M sat ) / ( M p + M sat ). Substituting Ω p = n sat and taking the first derivative˙ L , we obtain the differential equations that evolve due to angular momentum conservation as:˙ n sat = − M p [ G ( M (cid:63) + M p + M sat )] / n − / ˙ n p µ [ G ( M p + M sat )] / n − / − αR M p , (5)the argument for the sgn function in Equation 2 is replaced with [ n sat − n p ], and the planetary rotation follows thesatellite mean motion evolution ( ˙Ω p = ˙ n sat ), which spins up the planet as the satellite spirals inward. Equation 5 ismodified from Equation 14b in Sasaki et al. (2012) to include all of the masses, including a reduced-mass factor µ onthe exomoon’s orbital angular momentum (Cheng et al. 2014).Conditions for regime (1) can be determined by first integrating Equation 1 analytically and setting the result equalto the critical mean motion n crit . The tidal quality factor Q p is proportional to the total tidal migration timescale T ,where Q p has to be sufficiently large so that the exomoon can begin at a given a sat and remain bound for at least thesystem age τ . A similar approach is used by Barnes & O’Brien to prescribe limits for the satellite mass (Barnes &O’Brien 2002, see their Equation 8), where we solve for Q p instead. As a result, we obtain a lower limit for Q p as: Q crit ≥ k p R τ M sat (cid:112) G ( M p + M sat ) M p (cid:16) a / − a / (cid:17) , (6) Quarles, Li, & Rosario-Franco where a tidal quality factor below the critical value ( Q p < Q crit ) will migrate outward past the stability limit on atimescale less than the system lifetime τ . Figure 2 shows this lower limit Q crit (color-coded; log scale) for each of thesix exomoon candidate systems as a function of the planet-satellite mass ratio M sat /M p and initial separation a sat on alogarithmic scale. Tidally unstable conditions are colored white and unrealistic conditions Q crit > are colored gray.The lower limit Q crit is evaluated using the mean values from Table 1, where the observational uncertainties in theplanetary radius, planetary mass, and the system age shift these values slightly. Equation 6 shows that uncertaintiesin the planetary radius drive the largest changes and it is one of the better constrained observational quantities.We can also infer a plausible value for Q p from the planetary radius as long as the host planet is not in the anambiguous region (Rogers 2015; Chen & Kipping 2017). KOI 1925.01 is nearly Earth-sized, where we can estimatethat its Q p (cid:46)
200 and regions with Q crit (cid:38)
200 could be excluded (light blue to red). This is justified because all of theterrestrial planets in the solar system have Q p (cid:46)
100 and specifically for the Earth Q p ≈
12 (Lainey 2016). A similarapproach can be applied to the other KOIs using a very uncertain estimate for Neptune’s Q p ∼ Q crit (cid:38) p >> n sat )for the above conditions to hold, which is the case considering an initial Ω p near break-up. For slower planetary rotationrates, we must consider the planet-satellite system evolution using angular momentum conservation (Equation 5) andevaluate whether the infall timescale is less than the system age τ . Figure 3 illustrates a numerical solution of KOI1925.01 using Equations 1–3 (Ω p > n sat ) or Equation 5 with a modified Equation 2 (Ω p = n sat ) using a Runge-Kutta-Fehlberg integration scheme ( scipy ; Virtanen et al. 2020) with an absolute and relative tolerance of 10 − .The time evolution of Ω p and n sat are evaluated assuming that the host planet is Earth-like in its tidal Love number( k p = 0 . p , and we use the meanvalues for the stellar mass, planetary radius, and system age. We evaluate two values in Q p (10 and 100), as well as twomass ratios (0.0123 and 0.3) that are color-coded in the legend. For the Earth-Moon mass ratio ( M sat /M p = 0 . p = n sat and follows Equation 5 once synchronized. The planetary spin angular momentum isinsufficient to drive the satellite past the stability limit for a circular orbit (horizontal dash-dot line), but the infallphase ultimately destroys the satellite. The timescale for this evolution increases linearly with the assumed Q p and a Q p that is much larger than terrestrial values is necessary to prolong the satellite lifetime enough to be observed by Kepler . Moreover, if we use the truncated stability limit assume e p = 0 . ∼ years.As the planet-satellite mass ratio increases, the satellite mean motion synchronizes with the host planet spin rapidlyand nearly all of the evolution follows angular momentum conservation (Equation 5). Cheng et al. (2014) showed asimilar evolution with the Pluto-Charon system, where Pluto’s tidal Love number ( k p = 0 . M sat /M p = 0 . years, but eventually enters an inspiral phase, where a larger Q p delays the demise proportionally ( ˙ n sat ∝ ( n sat /n p ) / ˙ n p ). To prolong the satellite lifetime to equal the systemlifetime, a large dissipation factor is needed ( Q p ∼ COMBINING LIMITS FROM OBSERVATIONAL MODELING, ORBITAL STABILITY, AND TIDALMIGRATIONAnalysis of the
Kepler data can uncover the planetary radius, planetary orbital period, and even estimates for thestellar mass and age using asteroseismology (Silva Aguirre et al. 2015). Fox & Wiegert (2020) used transit timingvariations (TTVs) to suggest an unseen perturber within six KOI systems, which could be caused by gravitationalinteractions with an exomoon. Additionally, Fox & Wiegert (2020) prescribe a 1 R ⊕ transit depth threshold for theproposed satellite because it otherwise would have been detected in the Kepler data. This puts an upper limit on themass ratio to ∼ M sat /M p (cid:46) . ⊕ because such distortions are not apparent inthe light curves presented in Kipping (2020) and assume a Mars-like density to derive the respective satellite mass. A repository is available on GitHub (and archived on Zenodo) containing python scripts that reproduce our results and figures. xomoon Constraints σ constraints shown in Kipping (2020).From Section 2, we apply an orbital stability constraint (Rosario-Franco et al. 2020) assuming a circular planetaryorbit. In Section 3, we introduce constraints based upon tidal migration (Sasaki et al. 2012; Cheng et al. 2014), wherebound exomoons are possible for Q crit (cid:46) Q crit (cid:46)
200 (Earth-like) host planets.Figure 4 shows the combination of constraints as a function of the planet-satellite mass ratio M sat /M p and separation a sat on a logarithmic scale. The black regions indicate parameters that allow for possibly extant satellites, which remainbelow the stability limit for at least the system lifetime. The red and blue regions are excluded based upon orbitalstability and tidal migration constraints, respectively. The tidal migration constraints apply our constraint that Q crit <
200 for KOI 1925.01 and Q crit < σ boundaryin TTVs (Kipping 2020) and parameters above the curve (white region) are excluded because the TTV amplitudewould be too large. The gray region represents where the satellite tides could be significant as to prolong the lifetimeof the satellite, but in most cases those regions can be excluded because the satellite could produce detectable transitsor distortions (hatched white region). KOI 1925.01 is an exception, but we show in Figure 3 (cyan and magentacurves) that the combination of stellar tides with the planetary tides causes the satellite to spiral inwards onto itshost planet on a timescale less than the system age. Exomoons in KOI 1925.01 are completely excluded within ourparameter space, especially if the planet does indeed have a high eccentricity (Fig. 3). The other KOIs are significantlyconstrained to less than half of the unconstrained area alone (i.e., below the black curves).We use the current mean values from the respective parameters in Table 1, where the planetary mass and systemage are the most uncertain. The system age affects our calculation of Q crit (Equation 6) linearly and thus the heightof the black region in Figure 4 could change by a factor of ∼ ∼ a crit for orbital stability and Q crit for tidal migration. Doubling the planetary mass in each case increases the viabilityof exomoons, our assumptions on other planetary properties, such as the tidal Love number, should also be updateddue to the increased planetary density. Our results represent a snapshot of the current knowledge without preciseplanetary masses or eccentricities, where additional observations are needed to produce more accurate results. CONCLUSIONSKipping (2020) performed an independent analysis of the transit timing variations (TTVs) for the six KOI candidatesthat Fox & Wiegert (2020) proposed that such TTVs could result from unseen exomoons. Our study complementsthe work by Kipping & Teachey (2020) by exploring the theoretical constraints for exomoons in these systems basedon our previous study for the orbital stability of exomoons (Rosario-Franco et al. 2020) and other works that evaluatetidal migration scenarios (Sasaki et al. 2012; Chen & Kipping 2017). We find that ∼
50% of the parameter space canbe excluded due to instabilities that occur from orbital stability constraints ( a sat (cid:38)
20 R p ). Interior to the stabilitylimit, exomoons face additional hurdles due to the tidal migration within the system lifetime. Four of the KOIcandidate systems (KOI 303.01, 1925,01, 2728.01, and 3220.01) are significantly constrained due to tidal migrationtimescales, where the remaining two systems (KOI 268.01 and 1888.01) could allow for low-mass ( M sat /M p (cid:46) . a sat (cid:46)
20 R p ) exomoons within the current estimates of the system ages. Observational uncertaintycan affect our estimates, where the biggest differences arise through our estimate of the planetary mass M p using aprobabilistic framework with Forecaster (Chen & Kipping 2017). However, observational constraints due to theTTV amplitude and non-detection of exomoon transits limit the increases to the tidally allowed region due to thisuncertainty such that our results remain accurate within a factor of a few. Our models assume a circular planetaryorbit, where relaxing this condition typically halves the extent of exomoon separations due to a much smaller Hillradius at planetary periastron. Overall, it appears unlikely that the six KOI systems proposed by Fox & Wiegert(2020) can host large enough exomoons to explain the observed TTVs due to a tidal migration constraint on theplanet-satellite mass ratio.Although these six KOIs may not host exomoons, Kepler 1625b-I (Teachey et al. 2018) remains the best exomooncandidate system. Rosario-Franco et al. (2020) highlighted this assessment in that the host planet orbits much fartherfrom its host star, which diminishes the influence of stellar tides and significantly increases the Hill radius. UsingEquation 6, we find the lower limit for tidal dissipation Q crit ≥ Quarles, Li, & Rosario-Franco scenarios proposed. Exomoons, in general, are an evolving prospect where significant care needs to be used while theyremain on the bleeding edge of our detection capabilities.ACKNOWLEDGMENTSM.R.F acknowledges support from the NRAO Gr¨ote Reber Fellowship and the Louis Stokes Alliance for MinorityParticipation Bridge Program at the University of Texas at Arlington. This research has made use of the NASAExoplanet Archive, which is operated by the California Institute of Technology, under contract with the NationalAeronautics and Space Administration under the Exoplanet Exploration Program.
Facility:
Exoplanet Archive
Software:
Forecaster Chen & Kipping (2017); scipy Virtanen et al. (2020); matplotlib Hunter (2007)REFERENCES
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Quarles, Li, & Rosario-Franco
Table 1.
Parameters for the 6 exomoon candidate KOIs.KOI M (cid:63) R p M p † a p τ References(M (cid:12) ) ( R ⊕ ) (M ⊕ ) (AU) (Gyr)268.01 1.175 +0 . − . +0 . − . +11 . − . +0 . − . a,b303.01 0.871 +0 . − . +0 . − . +6 . − . +3 . − . a,c1888.01 1.406 +0 . − . +0 . − . +16 . − . +0 . − . a,b1925.01 0.890 +0 . − . +0 . − . +0 . − . +0 . − . a,b2728.01 1.450 +0 . − . +0 . − . +9 . − . +0 . − . a,b,d3220.01 1.340 +0 . − . +0 . − . +24 . − . +0 . − . a,b,d a Kepler Exoplanet Archive DR25 b Silva Aguirre et al. (2015) c Morton et al. (2016) d Berger et al. (2018) † Planet masses M p are estimated probabilistically using the planet radius R p (Chen& Kipping 2017). xomoon Constraints Figure 1.
The range in exomoon semimajor axis a sat for each of the six Kepler
KOIs proposed by Fox & Wiegert (2020) isconstrained using our updated outer stability limit formula (Rosario-Franco et al. 2020) as a function of the planetary radius R p , where the black region marks the stable exomoon regime as a function of assumed planetary eccentricity and the whiteregion denotes parameters that are quickly lost due to gravitational perturbations. The red curve shows the outer stability limitusing the mean parameters for each system (see Table 1) and the gray curves indicate how the outer limit changes in responseto observational or modeling uncertainties. The estimated Roche limit for most of the KOI candidates is below 2 R p , except forKOI 1925.01, where its Roche limit is marked with a horizontal dashed white line. Quarles, Li, & Rosario-Franco
Figure 2.
The minimum planetary tidal quality factor Q crit (color-coded) that allows for an exomoon to survive beyond thecurrent system lifetime τ for each of the six candidate KOIs. The mean values are used for the stellar mass, planetary radius,and planetary mass from Table 1, where k p = 0 .
299 for Earth-like planets (Lainey 2016) for KOI 1925.01 and k p = 0 . Q crit > , which is unrealisticgiven our knowledge of the solar system giant planets. xomoon Constraints Figure 3.
Evolution using the mean parameters from KOI 1925.01 for a putative satellite’s mean motion n m (solid) and theplanet’s spin frequency Ω p (dashed) using a constant Q tidal model (Sasaki et al. 2012), where the initial satellite separation is5 R p and the planetary rotation period begins at 10 hours. The mean values are used for the stellar mass, planetary radius, andplanetary mass from Table 1, where a vertical solid (black) line marks the mean system lifetime τ and a horizontal (dash-dot)line denotes the critical mean motion n crit corresponding to the outer stability limit (Rosario-Franco et al. 2020). The satellite’smean motion and planetary rotation synchronize (Ω p = n m ) causing the solid and dashed curves to overlap (solid with whitedots). For the high mass ratio case ( M sat /M p = 0 . Quarles, Li, & Rosario-Franco
Figure 4.
Limits on the planet-satellite mass ratio M sat /M p and satellite separation a sat , where regions of parameter spacecan be excluded based upon orbital stability (red), tidal migration (blue and gray), and observational modeling (white). Theblack curve marks the 3 σ upper limits adapted from Kipping (2020). The cyan dashed line delineates the orbital stabilityboundary. The hatched (white) regions mark regions that we exclude because the satellite radius R m is large enough to producea detectable transit within the Kepler data ( R m (cid:38) . R ⊕ ) assuming a Mars-like satellite bulk density ( ρ sat = 3 .
93 g/cm3