A fast method to identify mean motion resonances
MMNRAS , 1–8 (2017) Preprint 30 August 2018 Compiled using MNRAS L A TEX style file v3.0
A fast method to identify mean motion resonances
E. Forg´acs-Dajka , , (cid:63) Zs. S´andor , , and B. ´Erdi Department of Astronomy, E¨otv¨os Lor´and University, P´azm´any P´eter s´et´any 1/A, H-1117 Budapest, Hungary Wigner RCP of the Hungarian Academy of Sciences, 29-33 Konkoly-Thege Mikl´os Str, H-1121 Budapest, Hungary Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences,H-1121 Budapest, Konkoly Thege Mikl´os ´ut 15-17., Hungary
Accepted 2018 March 7. Received 2018 March 7; in original form 2018 January 3
ABSTRACT
The identification of mean motion resonances in exoplanetary systems or in the SolarSystem might be cumbersome when several planets and large number of smaller bodiesare to be considered. Based on the geometrical meaning of the resonance variable, anefficient method is introduced and described here, by which mean motion resonancescan be easily find without any a priori knowledge on them. The efficiency of thismethod is clearly demonstrated by using known exoplanets engaged in mean motionresonances, and also some members of different families of asteroids and Kuiper-beltobjects being in mean motion resonances with Jupiter and Neptune respectively.
Key words: celestial mechanics – methods: numerical – (stars:) planetary systems
As a result of the ongoing discoveries, the number of exo-planets is continuously growing; up to now more than 3500planets are known in more than 2600 planetary systems.These numbers indicate that a significant amount of plan-ets form multi-planet systems, in which at least two planetsare revolving around a central star. Planets form in gas richprotoplanetary disks, therefore beside gravity, forces aris-ing from the ambient gaseous material play important roleas altering typically their semi-major axes during a processcalled orbital migration. If the orbital distance between twoplanets decreases in time due to their migration, the planetscan be orbitally locked at certain ratio of their semi-majoraxes, where the ratio of the mean angular velocities (meanmotions) can be expressed as a ratio of small positive in-teger numbers. If certain dynamical conditions are fulfilled,a resonant capture may happen meaning that the ratio ofthe semi-major axes and that of the mean motions does notchange during the further migration of the planets.Before the Kepler-era, mean motion resonances(MMRs) outside the Solar System have been detected be-tween giant planets. One of the most remarkable resonantsystems discovered at that time is around the M dwarfstar GJ 876 in which two planets are in 2:1 MMR (Marcyet al. 2001). Later on three planets in Laplace-type reso-nance have been identified (Rivera et al. 2010), being thefirst known resonant chain of exoplanets resembling to theJupiter’s Galilean satellites Io-Europa-Ganymede. Other ex-amples for the 2:1 MMR are the system HD 128311, in which (cid:63)
E-mail: [email protected] two massive planets are in resonance but with only one criti-cal argument librating (Vogt et al. 2005), and HD 73526, alsowith one librating critical argument (Tinney et al. 2006). Inthe system HD 60532 two giant planets are in 3:1 MMR(Desort et al. 2008). A further example for a planetary sys-tem that might be in the Laplace resonance is the systemHR 8799 in which three massive planets are orbiting aroundan A-type star (Go´zdziewski & Migaszewski 2014). Morerecently, more planets have been discovered to be dynami-cally locked in chains of MMRs among the Kepler systems.In Kepler-60 three planets with masses around ∼ M ⊕ are ina 5:4:3 Laplace-type MMR (Go´zdziewski et al. 2016). Thefour Neptune-mass planets in Kepler-223 have periods in ra-tios close to 3:4:6:8 (Mills et al. 2016) being another clearindication of planetary migration.Beside the exoplanetary systems, mean motion reso-nances also play an important role in shaping the dynamicsof the Solar System bodies. Mean motion resonances in theSolar System usually occur between a planet and small bod-ies: e.g. the members of the Hilda group of asteroids are ina 3:2, while the Trojan asteroids are in a 1:1 MMR withJupiter. MMRs between terrestrial planets and particularmembers of the asteroid family Hungaria can also be found.Moreover, as the existence of the Kirkwood gaps clearly in-dicates, the dynamical structure of the main belt is shapedby the MMRs between the asteroids and Jupiter (thoughtheir depletion is not only due to MMRs). Regarding largerobjects, between Neptune and the dwarf planet Pluto thereis a 3:2 MMR, which is a protective resonance keeping themon non-near-approaching orbital positions. This is of highimportance, since due to the high eccentricity of Pluto’s or-bit ( e = . ) the projection of its perihelion to Neptune’s © a r X i v : . [ a s t r o - ph . E P ] M a r E. Forg´acs-Dajka et al. orbital plane lies inside the orbit of Neptune thus the twoorbits are crossing each other (in projection). Interestingly,there are many other Kuiper-belt objects that are locked inthe 3:2 MMR with Neptune, these are the plutinos sharingtheir orbits with Pluto.When studying the dynamics of exoplanetary systemswith planets engaged in chains of mean motion resonances,or the motion of small celestial bodies, asteroids or trans-Neptunian objects (TNOs), one often finds the problem howto identify the possible mean motion resonances among thebodies involved. For example, during the formation of thedynamical structure of the trans-Neptunian region, reso-nances swept through the primordial disk due to the mi-gration of the planets, and planetesimals were captured intomean motion resonances. Modelling this process, a monitor-ing is necessary to see how the captures take place, whichplanetesimals are captured to which resonances (Hahn &Malhotra 2005; Levison et al. 2008). Similarly, studying thedynamical evolution of an asteroid family, one has to checkpossible resonances with the planets (Galiazzo et al. 2013).This needs the computation of the resonance variables, tocheck their possible libration, for all reasonable (often veryhigh order) resonant ratios of the mean motions of all smallbodies with all considered planets, and this has to be re-peated several times during the investigated time-span re-sulting in a considerable computational workload.In order to make easier the identification of the differ-ent mean motion resonances, we present a method we name
FAIR (as FA st I dentification of mean motion R esonances)that is easy to use and by which the identification of theMMRs is possible without any a priori knowledge on them.Our paper is organized as the following: in the next sectionwe introduce FAIR for inner and outer type MMRs, then intwo further sections particular examples are presented forSolar System objects, and for known exoplanetary systems,also including the study of resonant chains of planets.
FAIR
Let us denote by a , a (cid:48) the semi-major axes of two celestialbodies locked in a MMR, and by n , n (cid:48) the correspondingmean motions. For the sake of simplicity let us deal withthe situation of an asteroid and a planet, in this case theprimed quantities refer to the planet. This notation enablesus to distinguish between inner and outer resonances fromthe point of view of the asteroid. In the case of two giantplanets neither of the bodies are in such distinguished po-sitions, the non-primed quantities refer to the planet beingthe body to which respect the MMR is studied.Considering an asteroid and a planet, they can be eitherin (i) inner mean motion resonance, if a < a (cid:48) and the ratioof the mean motions is approximately nn (cid:48) = p + qp , (1)or in(ii) outer mean motion resonance, if a > a (cid:48) and approxi-mately n (cid:48) n = p + qp , (2) where p and q are relative prime integers. Here q is theorder of the resonance. The resonant perturbations are pro-portional to the q th power of the orbital eccentricities, thusfor small eccentricities low order resonances are the mostimportant. However, for large eccentricities high order reso-nances can also be significant. (These refer to the so-calledeccentricity-type resonance, when the longitudes of the per-ihelion are also involved, as in the cases studied below.)Resonances can be studied by using resonance vari-ables. For a given mean motion resonance there can be sev-eral types of resonance variables (Murray & Dermott 1999).From among them we consider those for which the method FAIR is applicable.
Since beside the selected asteroid and planet there are alsoother planets in a planetary system, we should also take intoconsideration the secular rates (cid:219) (cid:36) , (cid:219) (cid:36) (cid:48) of the perihelion lon-gitudes, caused by the mutual gravitational perturbations.In this case one can obtain the following resonance vari-ables: θ ≡ ( p + q ) λ (cid:48) − p λ − q (cid:36), (3) θ ≡ ( p + q ) λ (cid:48) − p λ − q (cid:36) (cid:48) , (4)where λ = M + (cid:36) , (cid:36) = ω + Ω , λ is the mean orbital longi-tude, M the mean anomaly, ω the argument of perihelion, Ω the longitude of the ascending node, (cid:36) the longitude of theperihelion of the asteroid. The primed variables refer to theplanet with similar meaning.Eqs (3) and (4) refer to the case of exact resonance. Neara resonance, Eqs (3) and (4) are satisfied approximately,and the resonance variables oscillate (librate) around a meanvalue, which is usually ◦ or ◦ , with amplitudes ∆ θ < ◦ .Actually, in the neighbourhood of a resonance, the be-haviour of a resonance variable can be quite complex. De-pending on the actual system and initial conditions, it canexhibit libration, circulation, alternation between librationand circulation, or chaotic fluctuations. When searching forresonance, our aim is to establish that the resonant ratioof the mean motions (or their combinations with the sec-ular frequencies) is maintained and the corresponding res-onance variable librates. These are necessary and sufficientconditions that a resonance exist. We note that the aboveintroduced resonance variables appear as critical argumentsin the series expansion of the perturbed two-body potentialin terms that contain the q th powers of e and e (cid:48) . There aremixed types of eccentricity-type resonances too, correspond-ing to critical arguments (and resonance variables) that con-tains coefficients of ee (cid:48) type. In this work, however, we donot consider the latter cases of mixed eccentricity-type res-onances.Now let us consider Eq. (3) in two cases. First, when λ = λ (cid:48) , that is at the conjunction of the asteroid and theplanet, it follows from (3) that θ c = q ( λ − (cid:36) ) = qM , (5)where θ c is the value of θ at conjunction. Writing ¯ θ = θ MNRAS000
Since beside the selected asteroid and planet there are alsoother planets in a planetary system, we should also take intoconsideration the secular rates (cid:219) (cid:36) , (cid:219) (cid:36) (cid:48) of the perihelion lon-gitudes, caused by the mutual gravitational perturbations.In this case one can obtain the following resonance vari-ables: θ ≡ ( p + q ) λ (cid:48) − p λ − q (cid:36), (3) θ ≡ ( p + q ) λ (cid:48) − p λ − q (cid:36) (cid:48) , (4)where λ = M + (cid:36) , (cid:36) = ω + Ω , λ is the mean orbital longi-tude, M the mean anomaly, ω the argument of perihelion, Ω the longitude of the ascending node, (cid:36) the longitude of theperihelion of the asteroid. The primed variables refer to theplanet with similar meaning.Eqs (3) and (4) refer to the case of exact resonance. Neara resonance, Eqs (3) and (4) are satisfied approximately,and the resonance variables oscillate (librate) around a meanvalue, which is usually ◦ or ◦ , with amplitudes ∆ θ < ◦ .Actually, in the neighbourhood of a resonance, the be-haviour of a resonance variable can be quite complex. De-pending on the actual system and initial conditions, it canexhibit libration, circulation, alternation between librationand circulation, or chaotic fluctuations. When searching forresonance, our aim is to establish that the resonant ratioof the mean motions (or their combinations with the sec-ular frequencies) is maintained and the corresponding res-onance variable librates. These are necessary and sufficientconditions that a resonance exist. We note that the aboveintroduced resonance variables appear as critical argumentsin the series expansion of the perturbed two-body potentialin terms that contain the q th powers of e and e (cid:48) . There aremixed types of eccentricity-type resonances too, correspond-ing to critical arguments (and resonance variables) that con-tains coefficients of ee (cid:48) type. In this work, however, we donot consider the latter cases of mixed eccentricity-type res-onances.Now let us consider Eq. (3) in two cases. First, when λ = λ (cid:48) , that is at the conjunction of the asteroid and theplanet, it follows from (3) that θ c = q ( λ − (cid:36) ) = qM , (5)where θ c is the value of θ at conjunction. Writing ¯ θ = θ MNRAS000 , 1–8 (2017) fast method to identify mean motion resonances and denoting the amplitude of θ by ∆ θ , since ¯ θ − ∆ θ ≤ θ c ≤ ¯ θ + ∆ θ, it follows that ¯ θ − ∆ θ ≤ qM ≤ ¯ θ + ∆ θ. Here M is the mean anomaly of the asteroid at the momentof conjunctions, and qM should be taken mod π . Thus ¯ θ + k π − ∆ θ q ≤ M ≤ ¯ θ + k π + ∆ θ q , where k is an integer. This gives q centres ¯ θ q + k π q , k = , , . . . , q − along the orbit of the asteroid around which the conjunctionsof the asteroid with the planet can take place within regionsof half-size ∆ θ / q . For example, for q = and ¯ θ = ◦ thecentres are at ◦ , ◦ , and ◦ , while for ¯ θ = ◦ theseare at ◦ , ◦ , and ◦ .In the second case, when M = ◦ , that is at the perihe-lion of the asteroid, (cid:36) = λ , and it follows from (3) that θ p = ( p + q )( λ (cid:48) − λ ) , where θ p is the value of θ at the perihelion. Since ¯ θ − ∆ θ ≤ θ p ≤ ¯ θ + ∆ θ, it follows that ¯ θ − ∆ θ ≤ ( p + q )( λ (cid:48) − λ ) ≤ ¯ θ + ∆ θ. As before, it also follows that ¯ θ + k π − ∆ θ p + q ≤ λ (cid:48) − λ ≤ ¯ θ + k π + ∆ θ p + q giving p + q centres ¯ θ p + q + k π p + q , k = , , . . . , p + q − around which in regions with half-size ∆ θ /( p + q ) the asteroidcan be found with respect to the planet when the former isat perihelion.The previous considerations can be summed up in stat-ing that plotting λ (cid:48) − λ against M in a rectangular coordinatesystem with M as horizontal and λ (cid:48) − λ as vertical axis, therewill be q centres on the horizontal, and p + q centres on thevertical axis.The same reasoning can be repeated with Eq. (4), con-cluding to that plotting λ (cid:48) − λ against M (cid:48) there will be q centres on the horizontal, and p centres on the vertical axis. In the case of an outer resonance, the resonance variablesfor the eccentricity-type resonances are θ = ( p + q ) λ − p λ (cid:48) − q (cid:36), (6)and θ = ( p + q ) λ − p λ (cid:48) − q (cid:36) (cid:48) . (7)By similar considerations as before, it follows from Eq. (6)that plotting λ − λ (cid:48) against M , there will be q centres on thehorizontal, and p centres on the vertical axis. In the case of Table 1.
Resonance variables. The third column shows the vari-ables to be plotted versus each other, the fourth and fifth thenumber of centres on the horizontal and vertical axes.type resonance variable plot hor vertinner ( p + q ) λ (cid:48) − p λ − q (cid:36) λ (cid:48) − λ vers M q p + q inner ( p + q ) λ (cid:48) − p λ − q (cid:36) (cid:48) λ (cid:48) − λ vers M (cid:48) q p outer ( p + q ) λ − p λ (cid:48) − q (cid:36) λ − λ (cid:48) vers M q p outer ( p + q ) λ − p λ (cid:48) − q (cid:36) (cid:48) λ − λ (cid:48) vers M (cid:48) q p + q Eq. (7), plotting λ − λ (cid:48) against M (cid:48) , there will be q centres onthe horizontal, and p + q centres on the vertical axis. Table 1gives a summary of the different cases in the inner and outerresonances. We have investigated the eccentricity-type resonances so far,however, it can be easily shown that with minor modifica-tions, the method
FAIR also works for inclination-type res-onanances. Let us consider the non-mixed inclination-typeresonances of order q , where the critical arguments appearin terms containing the q th power of either I or I (cid:48) (being theinclinations of the body and of the perturber, respectively)as coefficients in the series expansion of the perturbing po-tential: θ I , = ( p + q ) λ (cid:48) − p λ − q Ω , (8) θ I , = ( p + q ) λ (cid:48) − p λ − q Ω (cid:48) , (9)where Ω and Ω (cid:48) are the longitude of nodes of the body andthe perturber, respectively. Equating the mean longitudes, λ = λ (cid:48) , similarly to Equation (5), conjunctions happen when θ I , c = q ( λ − Ω ) = q ( ω + M ) . (10)This suggests that in the inclination-type resonances oneshould replace M with M + ω , and by a similar argumen-tation to the eccentricity-type resonances, there will be q centres on the horizontal, and ( p + q ) centres on the verticalaxis when plotting λ (cid:48) − λ versus M + ω in a rectangular coor-dinate system. The above considerations can also be appliedto the outer inclination-type resonances, when λ (cid:48) − λ shouldbe plotted either versus M + ω or M (cid:48) + ω (cid:48) . Thus by replacingin Table 1 M and M (cid:48) with M + ω and M (cid:48) + ω (cid:48) , moreover (cid:36) and (cid:36) (cid:48) with Ω and Ω (cid:48) respectively, one can easily obtainthe resonance variable of the corresponding inclination-typeresonance. The properties of the resonance variables, described in Sec-tion 2, can be used to find easily whether a resonance existsbetween a planet and an asteroid. Deciding from the semi-major axes that the looked for resonance is inner or outer,then searching for inner resonances, one plots λ (cid:48) − λ versus M and M (cid:48) , (or in the cases of inclination-type MMRs versus M + ω and M (cid:48) + ω (cid:48) ) computed by numerical integration of MNRAS , 1–8 (2017)
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Figure 1.
Asteroid 153 Hilda in a 3:2 MMR with Jupiter. the equations of motion for sufficiently long time. If there isa resonance, some stripes will appear on the plots (see theexamples below), and counting the number of their intersec-tions with the horizontal and vertical axis will provide the p and q values of the resonance. Searching for outer reso-nances, the procedure is the same with plotting λ − λ (cid:48) versus M and M (cid:48) and counting the numbers of intersections on thehorizontal and vertical axis. Since in the case of a resonancethe appearance of the proper number of intersections is anecessary condition, once p and q are found, the ratio of themean motions and the libration of the resonance variableshould be checked with them.When processing the above plots, there can be raisedtwo technical questions: (i) how long the numerical integra-tions should be carried out, and (ii) how frequently shouldthe points be plotted in order to obtain good quality plotssuitable for determination of a MMR. Without giving a def-inite answer, we have empirically found that regarding theplotting frequency, the one hundredth of the period of theinner body certainly will give good result when using an in-tegration time that corresponds to one hundred periods ofthe outer body. In this section we provide a few examples for the applica-tion of the method
FAIR to Solar System bodies involvedin various mean motion resonances. The orbits have beennumerically calculated by using an own-developed adap-tive step size Runge-Kutta-Nystrom 6/7 N-body integrator(Dormand & Prince 1978). In our integrations all planetsexcept Mercury have been taken into account, and as initialconditions we used heliocentric coordinates and velocitiestaken from JPL Horizons at the epoch JD 2457754.50.
The main dynamical characteristic of the Hilda group ofasteroids (named after 153 Hilda) is that its members arelocked in a 3:2 MMR with Jupiter (Broˇz & Vokrouhlick´y2008). As a first example on how the method
FAIR worksis given by plotting λ (cid:48) − λ versus M (see Figure 1, leftpanel), where λ (cid:48) is the mean longitude of the perturbingbody (Jupiter, in this case), and λ , M are the mean lon-gitude, and mean anomaly of the asteriod 153 Hilda, re-spectively. Since this is a MMR of inner type, and λ (cid:48) − λ is plotted against M , according to the first row of Table 1the counting of the number of intersecting stripes with the Figure 2.
The 3:2 MMR between Neptune and Pluto
Figure 3.
The 2005TN trans-Neptunian object in a 5:3 MMRwith Neptune. horizontal and vertical axis gives q = and p + q = , seethe left panel of Figure 1. Thus, the ratio of the mean mo-tions is ( p + q )/ p = / , and the corresponding resonancevariable (or resonant angle) is θ = λ (cid:48) − λ − (cid:36) , librating inthis case around ◦ as shown in the right panel of Figure 1.We note that in this case, the ratio of the mean motions andthe resonance variable are determined solely by counting thecrossings of the stripes with the horizontal and vertical axes. As a further well known example, we consider the case ofNeptune and Pluto, being in a 3:2 MMR (Cohen & Hub-bard 1965). In this case, the more massive Neptune (theperturber) orbits closer to the Sun, thus the MMR understudy is of outer type. In the left panel of Figure 2 we dis-played λ − λ (cid:48) versus M . Since this is a MMR of outer type,according to the third row of Table 1 the counting of thecrossings of the stripes with the horizontal and vertical axisgives q = and p = . Thus the ratio of the mean motionsis ( p + q )/ p = / , and the corresponding resonance variable, θ = λ − λ (cid:48) − (cid:36) librates around ◦ .We note that although the above two well known meanmotion resonances can be identified easily without the ap-plication of the method FAIR , we were able to identify themwithout knowing their character. In what follows, we showsome more sophisticated cases, in which the identificationof the resonances and the corresponding resonance variablesmight be more complicated. is a trans-Neptunian object (TNO) discovered bySheppard et al. (2005). In order to demonstrate the effec- MNRAS000
The 2005TN trans-Neptunian object in a 5:3 MMRwith Neptune. horizontal and vertical axis gives q = and p + q = , seethe left panel of Figure 1. Thus, the ratio of the mean mo-tions is ( p + q )/ p = / , and the corresponding resonancevariable (or resonant angle) is θ = λ (cid:48) − λ − (cid:36) , librating inthis case around ◦ as shown in the right panel of Figure 1.We note that in this case, the ratio of the mean motions andthe resonance variable are determined solely by counting thecrossings of the stripes with the horizontal and vertical axes. As a further well known example, we consider the case ofNeptune and Pluto, being in a 3:2 MMR (Cohen & Hub-bard 1965). In this case, the more massive Neptune (theperturber) orbits closer to the Sun, thus the MMR understudy is of outer type. In the left panel of Figure 2 we dis-played λ − λ (cid:48) versus M . Since this is a MMR of outer type,according to the third row of Table 1 the counting of thecrossings of the stripes with the horizontal and vertical axisgives q = and p = . Thus the ratio of the mean motionsis ( p + q )/ p = / , and the corresponding resonance variable, θ = λ − λ (cid:48) − (cid:36) librates around ◦ .We note that although the above two well known meanmotion resonances can be identified easily without the ap-plication of the method FAIR , we were able to identify themwithout knowing their character. In what follows, we showsome more sophisticated cases, in which the identificationof the resonances and the corresponding resonance variablesmight be more complicated. is a trans-Neptunian object (TNO) discovered bySheppard et al. (2005). In order to demonstrate the effec- MNRAS000 , 1–8 (2017) fast method to identify mean motion resonances Figure 4.
Dwarf planet 136108 Haumea in a 12:7 MMR withNeptune
Figure 5.
Neptune and 2001QR in 1:1 MMR tiveness of the method
FAIR , let us forget for now that thisbody is in a 5:3 MMR with Neptune. The only informationwe suppose to have is that this TNO orbits outside Neptune,therefore when seeking for a resonant behaviour we shouldconsider a MMR of outer type. To do so we display the plot λ − λ (cid:48) versus M , where λ (cid:48) is the mean longitude of Neptune,while λ and M are the mean longitude and mean anomalyof 2005 TN , respectively. Studying the left panel of Figure3, we can count the numbers of crossings of the stripes withthe axes, yielding q = and p = (see the third row of Table1). Thus the ratio of the mean motions is ( p + q )/ p = / ,and the resonance variable θ = λ − λ (cid:48) − (cid:36) librates around ◦ , see the right panel of Figure 3. λ − λ (cid:48) against M , see Figure 4. As usual, λ (cid:48) refersto Neptune’s mean orbital longitude, while the non-primedquantities to Haumea’s orbital elements. Investigating Fig-ure 4, one can identify q = crossings with the horizontal,and p = crossings with the vertical axis. This yields toa mean motion ratio of ( p + q )/ p = / , and a resonancevariable θ = λ − λ (cid:48) − (cid:36) librating around approximately ◦ . Figure 6.
The HD 60532 system in 3:1 inner MMR
The method
FAIR can also be applied to Trojan-type mo-tions. The main characteristic of the Trojan-type, or in otherwords co-orbital motion is the 1:1 MMR meaning that thebodies involved are sharing similar orbits. The best knownexamples for Trojan-type motion are the Trojan asteroidspopulating the neighbourhood of the stable triangular La-grangian points L and L of the Sun-Jupiter system. Theresonance variable in these cases is λ − λ (cid:48) which librates ei-ther around ◦ , or ◦ depending on whether the asteroidis in the vicinity of the L , or L point. In the λ (cid:48) − λ versus M plots of the Neptune’s Trojan 2001 QR (see Figure 5, leftpanel) we find one strip parallel to the horizontal axis of thecoordinate system yielding q = (see Table 1, first row), e.g.a zeroth order resonance. The only crossing with the verti-cal axis means p + q = , that is p = . Thus the resonanceis ( p + q )/ p = / , and the resonance variable is θ = λ (cid:48) − λ ,librating around ◦ (Figure 5, right panel). Finally, with two additional examples we demonstrate theapplicability of the method
FAIR to identify resonances be-tween pairs of exoplanets, or chains of resonances in whichmany of them are involved. The numerical integration of or-bits have been performed by using the already mentionedRKN 6/7 integrator.
Until the discovery of the system around HD 60532, an F-type star with mass M ∗ = . M (cid:12) by Desort et al. (2008),there were not known any giant planets involved in the 3:1MMR. The final confirmation that giant planets can be inthe 3:1 MMR was given by Laskar & Correia (2009), whilea formation study favouring a migration based scenario wasperformed by S´andor & Kley (2010). In the latter studythe authors demonstrated that the convergent migration, MNRAS , 1–8 (2017)
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Figure 7.
The resonant pairs of the system Kepler 60, the threeplanets being in 5:4:3 Laplace-type MMR thus capture in the 3:1 MMR was only possible for largerplanetary masses corresponding to Fit II of Laskar & Correia(2009). In this work we checked both Fit I with low andFit II with high planetary masses to show the efficiency ofthe method
FAIR . Our results are displayed in Figure 6,where in the upper left panel the ( λ (cid:48) − λ ) versus M plotis displayed for Fit I, while in the bottom left panel forFit II. These resonances are of inner types (as the primedorbital elements are refering to the outer giant planet), andby calculating the number of the crossings of the strips withthe horizontal and vertical axis we have (see Table 1, firstrow) q = and p + q = . Thus p = , and the ratio of themean motions is ( p + q )/ p = / . This means that without anya priori knowledge, just by a careful analysis of the ( λ (cid:48) − λ ) versus M plot we could identify the mean motion resonance,and also write down the corresponding resonant variable,which in these cases is θ = λ (cid:48) − λ − (cid:36) , librating around ◦ (Figure 6, right panels). It is noteworthy that due to therelatively large libration amplitude, the strips in the ( λ (cid:48) − λ ) versus M plot are quite broad. As we have already mentioned, there are planetary systems,typically discovered by the Kepler mission, in which theplanets are captured in chains of mean motion resonances.A prominent example of these systems is Kepler 60, wherethree planets with masses around M ⊕ are in the 5:4:3Laplace-type MMR (Go´zdziewski et al. 2016). Here we ap-ply the method FAIR to the Fit II of the cited paper, in which the pairs of the planets in MMRs are investigated.First, without any a priori knowlegde on the MMRs, we canstudy planets b and c. We display λ − λ as the functionof M , see the upper left panel of Figure 7. We note thatthe index 1 stays for the inner, while index 2 for the outerplanet, planets b and c in this configuration. Consideringthat λ , (cid:36) , M correspond to λ , (cid:36) , M and λ (cid:48) , (cid:36) (cid:48) , M (cid:48) cor-respond to λ , (cid:36) , M , and that this is an outer type MMR,according to Table 1 (third row) the numbers of crossingswith the horizontal and vertical axis give q = and p = .The mean motions ratio in this case is thus ( p + q )/ p = / corresponding to a resonant variable θ , = λ − λ − (cid:36) ,librating around ◦ (see the upper right panel of Figure7). As a next step, we can also check, whether the planetsc and d are in a MMR too. To do so, we plot λ − λ asthe function of M , shown in the middle left panel of Figure7. Now the non-primed elements correspond to those withindex 2, and the primed ones to those with index 3. Accord-ing to Table 1 (third row), we have q = and p = , thatis there is a ( p + q )/ p = / MMR with a resonant variable θ , = λ − λ − (cid:36) , librating around ◦ (middle rightpanel of Figure 7).Finally, we also plot λ − λ as the function of M (bot-tom left panel of Figure 7). Counting the numbers of cross-ings of the stripes with the axes, we find q = and p = corresponding to a mean motion ratio ( p + q )/ p = / , andindeed, the resonant variable θ , = λ − λ − (cid:36) libratesaround ◦ (bottom right panel of Figure 7). Thus usingthe system Kepler 60, we gave an evidence that the method FAIR is also applicable for a quick identification of MMRsbetween pairs of exoplanets being in resonant chains. Theabove analysis can also be performed for inner type reso-nances.
FAIR toinclination-type MMR
Similarly to the eccentricity-type MMRs, the method
FAIR can also be applied to detect inclination-type resonances.The best known example for inclination-type MMR in theSolar System is between the Saturnian satellites Mimas andThetis corresponding to a 2:1 commensurability. Inclination-type MMRs can also be developed between migrating giantplanets if the eccentricity damping of the outer migratingplanet is modest, allowing the rapid growth of the innerplanet’s eccentricity. This can additionally lead to the fastgrowth of the inner planet’s inclination (see Libert & Tsiga-nis 2009). Here we consider the following example that leadsto a 3:1 MMR: the inner planet with mass m = M J (where M J is the mass of Jupiter) is started from 5 au from nearlycircular and planar, initially non-migrating orbit. An outerplanet with mass m = M J is started from 16.5 au from acircular orbit with negligible inclination with respect to theplane of reference of the coordinate system, and is forced tomigrate in timescale of τ a = × years, with the same ec-centricity damping timescale. According to Libert & Tsiga-nis (2009), the two planets enter first into a 3:1 eccentricity-type, and later on into an inclination-type MMR, as thelibration of the following eccentricity-type and inclination- MNRAS000
FAIR can also be applied to detect inclination-type resonances.The best known example for inclination-type MMR in theSolar System is between the Saturnian satellites Mimas andThetis corresponding to a 2:1 commensurability. Inclination-type MMRs can also be developed between migrating giantplanets if the eccentricity damping of the outer migratingplanet is modest, allowing the rapid growth of the innerplanet’s eccentricity. This can additionally lead to the fastgrowth of the inner planet’s inclination (see Libert & Tsiga-nis 2009). Here we consider the following example that leadsto a 3:1 MMR: the inner planet with mass m = M J (where M J is the mass of Jupiter) is started from 5 au from nearlycircular and planar, initially non-migrating orbit. An outerplanet with mass m = M J is started from 16.5 au from acircular orbit with negligible inclination with respect to theplane of reference of the coordinate system, and is forced tomigrate in timescale of τ a = × years, with the same ec-centricity damping timescale. According to Libert & Tsiga-nis (2009), the two planets enter first into a 3:1 eccentricity-type, and later on into an inclination-type MMR, as thelibration of the following eccentricity-type and inclination- MNRAS000 , 1–8 (2017) fast method to identify mean motion resonances Figure 8.
Two migrating giant planets captured into a 3:1eccentricity-type MMR. The method
FAIR is applied to the wholelength of numerical integration.
Figure 9.
Two migrating giant planets captured into a 3:1inclination-type MMR. The method
FAIR is applied to the wholelenght of numerical integration. type variables clearly indicates: θ = λ − λ − (cid:36) , (11) θ I , = λ − λ − Ω . (12)The resonance variable θ begins its libration roughly after t ∼ . × years being the time when the resonant capturehappens (see Figure 8). Interestingly, the center of librationis shifted from ◦ to lower values between ◦ − ◦ .The inclination of the inner planet gets excited around t ∼ . × years, that coincides to the libration of the resonancevariable θ I , (see Figure 9). After this epoch the system isboth in 3:1 eccentricity-type and inclination-type MMR, seethe right panels of Figures 8 and 9. Studying the left panelsof these figures that display the plots ( λ − λ ) versus M for the eccentricity-type, and ( λ − λ ) versus ( M + ω ) forinclination-type MMRs, one can see that the method FAIR is able to identify both eccentricity-type and inclination-typeMMRs providing the correct resonance variables. We notethat the plots have been made for the whole timespan ofthe numerical integration, also before the planets are gotcaptured in resonances, thus there are scattered points inthe figures being not yet settled in the stripes indicating theMMRs.
Finally, we also show that the method
FAIR is able to detecttemporary capture into a MMR. It can happen, for instance,that during their migration planets enter into a MMR, seeour results in the previous part. The left panels of Figures 8
Figure 10.
Temporary capture of 2007 RW into a 1:1 MMR,being a co-orbital companion of Neptune for a while. and 9 contain scattered points corresponding to that epochswhen the planets are not engaged into the 3:1 MMR. On theother hand it is important that the scattered points shouldnot repress those settled in the stripes.Beside the cases of migrating planets, there are exam-ples of temporary capture of bodies in co-orbital orbits, too.This happens to the asteroid 2007 RW , which was assumedas Neptune Trojan (de la Fuente Marcos & de la Fuente Mar-cos 2012). The right panel of Figure 10 shows the temporarylibration of the synodic longitude λ (cid:48) − λ , around ◦ . Theleft panel of this figure shows the ( λ (cid:48) − λ ) versus M plot,in which one parallel stripe to the horizontal axis becomesvisible around λ (cid:48) − λ = ◦ among the scattered points in-dicating the case of the 1:1 MMR, see additionally Figure5. In this paper we presented a novel method which is suit-able to easily decide whether two planets are involved insome mean motion resonance, being either of eccentricityor inclination type, without any a priori knowledge of itscharacter. Based on the geometrical meaning of the reso-nance variables, first we described in detail how our methodworks. Next we demonstrated through a few examples of realbodies of the Solar System and of exoplanetary systems, in-volved in various MMRs, the straightforward applicabilityand efficiency of the method
FAIR . We have found that thismethod is able to detect MMRs also in those cases whenthe involved bodies are only temporarily captured, such asmigrating pairs of planets or co-orbital companions to giantplanets. Our method can also serve as a practical help forfuture analysis of celestial bodies which may be envolved inmean motion resonances.
ACKNOWLEDGEMENTS
We thank the anonymous referee for her/his suggestions andcomments helping us to improve the work. This researchhas been supported by the Hungarian National Research,Development and Innovation Office, NKFIH grant K-119993and the HAS Wigner RCP – GPU-Lab. Zs. S´andor thanksthe support of the J´anos Bolyai Research Scholarship of theHungarian Academy of Sciences.
MNRAS , 1–8 (2017)
E. Forg´acs-Dajka et al.
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