Orbital Dynamics of Exoplanetary Systems Kepler-62, HD 200964 and Kepler-11 3

arXiv:1601.02110v2 [astro-ph.EP] 12 Feb 2016 (BSK). ⋆ contain systems multiplanetary of handful exoplanetary Saturn– 5:2, a the Jupiter–Saturn, only system In in systems, systems. Solar occur Uranus–Neptune own MMRs and our near Uranus 2:1 In of and librating. 3:1 combination al. is et argument(certain (Petrovich angles) resonant integers orbital small the two and of pe- ratio orbital 2013), two a when near occur are MMRs riods 2011). sys- Wyatt feature exoplanetary common & are (Mustill similar resonances(MMRs) and motion system mean tems, al. Solar et our Schneider In by (2011)). developed systems exoplanetary configura- with for online base an orbital system http://exoplanet.eu/catalog/, resonant some e.g., near (see and have to- star tions planet planets each one a confirmed than where two is more systems least there multiplanetary at known exoplanets, has 504 known e.g., of these (see tal De- Among exoplanets 2015 exoplanetary 1000 cember. by meth- discovered of than been timing discovery have pul- pulse More http:/www.exoplanet.eu) the of decades, the increases. last help orbiting systems the the objects Over with ods. low-mass B1257+12 very PSR two sar (1992) Frail found ex- & Wolszczan multiple and and by first system discovered The was Solar 1989. system since our oplanet detected outside been have planets which are exoplanets The INTRODUCTION 1 o.Nt .Ato.Soc. Astron. R. Not. Mon. ai Mia Rajib HD Kepler-11 Kepler-62, and Systems 200964 Exoplanetary of Dynamics Orbital eateto ple ahmtc,Ida colo Mines, of School Indian Mathematics, Applied of Department -al [email protected] R) [email protected] (RM); [email protected] E-mail: ⋆ n aa ig Kushvah Singh Badam and 457 0910 21)Pitd1 eray21 M L (MN 2016 February 15 Printed (2016) 1089–1100 , w lnt fec ytmKpe-2 D206 n elr1 ar Kepler-11 words: and Key 200964 respectively. HD r this, resonances, Kepler-62, of the motion view system mean of In each 5:4 evolution value. of constant the around planets plotted librate two have they We that to Kepler-11. found sy method and and exoplanetary numerical three 200964 consider by HD we obtained work, 62, this of In solution that result. The with analytical motion. trun compared of the is equations three- the analytically from of solve function we perturbations disturbing approach, the the analytical of expansion obtain the and Using function. function disturbing of lmnso h lnt.Mil,w td h nuneo planetary ( of evolution identify influence We time the eccentricity. study and by we axis Mainly, systems semimajor planets. th exoplanetary the consider of of to elements us behaviour motivates exoplanetary dynamical which mechanics in the celestial (MMRs) of resonances field mean-motion exciting of presence The ABSTRACT srmty-clsilmcais-paeaysystems. planetary - mechanics celestial - astrometry data hna 204 hrhn,India Jharkhand, 826004, Dhanbad ⋆ rmErh ti h rteolntr ytmconsisting system exoplanetary first the is f Kepler-11 years It 2011). light The 2000 al. Earth. approximately et AU. from located (Johnson 0.35 star, MMR Sun-like only MMR giant a 4:3 is two by a 2:1 have in separated the are 200964 are in planets HD they star be and The to exoplanets 2013). supposed al. Kepler- are et them (Borucki of Kepler-62f two and planets, five per- has HZ 62e these 60 system star’s discovered Among roughly recently host planets. is This five It its Earth. 0.83. of than index larger part similarity likely cent Earth inner most has the is it in and Kepler-62e planet method. terrestrial was transit exoplanet a the This Kepler- Kepler-62. using dis 2013) star HZ. found al. around the et orbit (Borucki in a in exoplanet covered confirmed like only super-Earth been them a is have among 62e planets are exoplanets, there of thousand Although the dozen planet. one in the permanently than of exist surface more can the water at state which liquid in star a around or- the of eccentricities the the on and primaries depending bits. the 2015), of 2012 Kushvah ratio or & mass (Szebehely Pal circular problem 2005; using Papadakis three-body points Lagrangian restricted the elliptical of 3:2 instability in etc. and found 2015) al. others et Also (Santos MMR. MMR 2:1 4:3 in 1993b), 2:1 (Malhotra are in 82943 are 128311 HD which 876, HD of GL and example, majority For The 2006). MMRs. al. (Beaugé et in MMR planet of pair a natooy h aial oe(Z saregion a is (HZ) zone habitable the astronomy, In stability the regarding performed has work of lot A r ): 1) + A T E tl l v2.2) file style X r M em nteexpression the in terms MMR tm aeyKepler- namely stems oypolmadan and problem body u pno sthat is opinion our hc sobtained is which ytm sanew a is systems etrain on perturbations n21 : and 4:3 2:1, in e disturbing cated swr ostudy to work is snn angles esonant fteorbital the of aiaeour validate ar - ; 2 Rajib Mia and Badam Singh Kushvah six transiting planets. Although, none of these planets be in for orbital stability low-ratio orbital resonances, Kepler-11b and Kepler-11c are 3 M1 + M2 ∆ in 5:4 MMR (Lissauer et al. 2011). µ = 0.5 < µcrit = 0.175 3 , µ 6 1, (1) M 2 Malhotra (1993b) presented a detail theoretical analy- ⋆ (2 − ∆) sis of three-body effects in the putative planetary system of where M1 and M2 are masses of planetary companions and PSR 1257+12. She provided explicit elements for the time M⋆ is the mass of the star. The parameter ∆ is the ratio dependence of the osculating elements that are needed for an of the separation distance between the companions of their improved timing model for the system. She has also shown mean distance from the star. Specifically they defined that the 3:2 MMR of the orbital periods affect periodic vari- R − 1 a ations of the Keplerian orbital parameters. She has obtained ∆ = 2 , R = 2 , (2) R + 1 a the expansion of disturbing function and identified only two 1 first-order 3:2 resonance term. But we identify (r + 1) : r where a1 and a2 are the semimajor axes of the inner and MMR terms in the expression of disturbing function and outer orbits, respectively. The system will be unstable for obtained the perturbations from the truncated disturbing the case µ > µcrit within a few tens of planetary orbits. function. In this manuscript, we have taken three exoplan- In recent years, several exoplanetary systems have etary systems namely Kepler-62, HD 200964 and Kepler-11 been discovered and it is very interesting to know and as per our knowledge they contain planets in 2:1, 4:3 their dynamical stability. There are several authors and 5:4 near MMRs, respectively. We apply theory discussed (Fabrycky & Murray-Clay 2010; Couetdic et al. 2010; in Sections 3 and 4 individually in our three exoplanetary Davies et al. 2014; Adams & Bloch 2015; Petrovich 2015) systems that we have chosen. who have studied about the stability of exoplanetary This paper is organized as follows. In Section 2, we in- systems. Davies et al. (2014) have reviewed the long-term troduce the general three-body problem and the stability dynamical evolution of the planetary systems. They have of exoplanetary system. The expansions of the disturbing discussed the planet–planet interactions that take place function and perturbation equations of orbital elements are within our own Solar system and in more tightly–packed introduced in Section 3. In Section 4, we present our secular planetary systems. Some system becomes dynamically resonance dynamics of exoplanetary system. Applications of unstable because of planet–planet interactions build up the model for the case of 2:1 resonance are shown in Sec- and this lead to strong encounters and ultimately either tion 5 for Kepler-62 system. The case of the 4:3 resonance ejections or collisions of planets. It was shown that the Solar is discussed in Section 6 for HD 200964 system. In Section system is chaotic, but the four giant-planet sub-system of 7, the case of 5:4 is discussed for Kepler-11 system. Finally, the Solar system is stable although the terrestrial-planet Section 8 is devoted to conclusions. sub-system is marginally unstable with a small change of planet–planet encounters during the lifetime of the Sun. Likewise here we discuss the dynamical stability of two- planet sub-system of Kepler-62 system, HD 200964 system and two-planet sub-system of Kepler-11 system. Petrovich 2 GENERAL THREE BODY-PROBLEM AND (2015) provides an independent review on the stability of DYNAMICAL STABILITY two-planet systems. They studied the dynamical stability Suppose the barycentric position vectors of star and two and fates of hierarchical (in semimajor axis) two-planets systems with arbitrary eccentricities and mutual inclinations. planets be R~⋆, R~ 1 and R~ 2 respectively. Let two planets of They proposed the following new criteria for dynamical sta- masses M1 and M2 orbiting a star of mass M⋆. Let the Ja- bility cobi coordinates ~r1 and ~r2 be the position of M1 relative to M⋆ and position of M2 relative to the centre of mass aout(1 − eout) rap = > Y, (3) M⋆ and M1. We notice that this system is different from ain(1 + ein) the restricted problems in which one of the bodies must where, have negligible mass. Also we ignore the effects of oblate- 1 1 3 aout 2 ness of star and the planets. We assume that the planets in Y = 2.4[max(µin, µout)] ( ) + 1.15, all system are coplanar. This approximation of coplanarity ain makes sense because for exoplanets, their mutual inclination ain and aout are the semimajor axes of the inner and outer is not known with any precision, and is taken equal to zero planet and µin and µout are the planet-to-star mass ratios (Beaugéet al. 2003). Let ai,ei,λi and ωi be the semimajor of the inner and outer planets, respectively. The hierarchical axis, eccentricity, mean longitude and the longitude of the two planets systems are stable if they satisfy the condition in pericentre of the ith planet for i = 1, 2. equation (3) and systems that do not satisfy equation (3) are In the three-body problem, one of the interesting topic expected to be unstable. The fate of the unstable systems is the stability of the Solar system or exoplanetary system. classified according to planetary masses as when µin > µout, Although, in celestial mechanics it is the unsolved prob- system lead to ejections and for µin < µout, there is a slightly lem till now, many scientist studied this because of its im- favouring of collisions with the star. portance in many areas like space science, astronomy and Now the goal is to apply this latest stability criteria of astrophysics. In the context of planets and exoplanets for- Petrovich (2015) to Kepler-62, HD 200964, and Kepler-11 mulation, Graziani & Black (1981) studied conditions under system. For the case of Kepler-62 system considering only which model planetary systems which consisting of a star two planets Kepler-62e and Kepler-62f, using the masses and two planets with coplanar and initially circular orbits. and initial orbital elements, we obtain rap = 1.34767, Y = Based on their results, they obtained a necessary condition 1.31762 with µin > µout. Hence, stability criteria in equation Orbital Dynamics of Exoplanetary Systems Kepler-62, HD 200964 and Kepler-11 3

(3) implies that system is stable. In the planetary system HD where

200964, the inner and outer planets HD 200964b and HD α (r+1) K α r D b 1 α , 200964c are in eccentric orbit with ein ≃ 0.040, eout ≃ 0.181 1( ) = − ( +1)+ ( ) (9) n 2 o 2 and ain ≃ 1.601 AU, aout ≃ 1.950 AU. Using the initial 1 α (r) d orbital elements and masses we found rap = 0.959166 and K2(α) = (r + )+ D b 1 (α), D = . (10) 2 2 2 dα Y = 1.43346 with µin > µout. So, stability criteria in equation (3) implies that HD 200964 system is unstable against Now using the truncated disturbing function of Eq.(8), we ejections. Also for the exoplanetary system Kepler-11 con- obtain the perturbation equations for the time variation of sidering only two planets, Kepler-11b and Kepler-11c, we the orbital elements as: the time variation of the semi-major found rap = 1.059964 > Y = 1.24015 with µin < µout. axes are Hence, this system is unstable and there is a slightly favour- a˙1 M2 ing of collisions with star than ejections. = 2 n1α [∂ψQ(ψ, α)+ α sin ψ a1 M⋆ α (r+1) −r r +1+ D b 1 (α)e1 2 2 × sin{(r + 1)λ2 − rλ1 − ω1} 3 EXPANSION OF THE DISTURBING 1 α (r) FUNCTION AND PERTURBATION +r r + + D b 1 (α)e2 2 2 2 EQUATIONS OF ORBITAL ELEMENTS × sin{(r + 1)λ2 − rλ1 − ω2}] , (11) The expansion of the disturbing function in the orbital eccentricities to ﬁrst order for the periodic terms and to second a˙2 M1 order for the secular terms is given as (Malhotra 1993a) = −2 n2 [∂ψQ(ψ, α)+ α sin ψ a2 M⋆ α (r+1) ∂ −(r + 1) r +1+ D b 1 (α)e1 a2V = Q(ψ, α) − α cos ψ − e1 cos(λ1 − ω1) α Q(ψ, α) ∂α 2 2 × sin{(r + 1)λ − rλ − ω } ∂ 2 1 1 +α cos ψ]+ e1 sin(λ1 − ω1) 2 Q(ψ, α) + 2α sin ψ 1 α (r) ∂ψ +(r + 1) r + + D b 1 (α)e2 2 2 2 ∂ +e2 cos(λ2 − ω2) (1 + α )Q(ψ, α) − 2α cos ψ (4) × sin{(r + 1)λ − rλ − ω }] . (12) ∂α 2 1 2 ∂ The time variation of the eccentricities: −e2 sin(λ2 − ω2) 2 Q(ψ, α) + 2α sin ψ ∂ψ M2 1 (2) 1 (1) (2) e˙1 = n1α αb 3 (α)e2 sin(ω1 − ω2) 2 2 2 + α b 3 (α)(e1 + e2) − 2b 3 (α)e1e2 cos(ω1 − ω2) , M⋆ 4 8 2 2 h i α (r+1) − r +1+ D b 1 (α) where 2 2 × sin{(r + 1)λ2 − rλ1 − ω1}] , (13) λj = Mj + ωj ,

ψ = λ1 − λ2, (5) 1 M1 1 (2) 2 − 2 e˙2 = − n2 αb 3 (α)e1 sin(ω1 − ω2) Q(ψ, α) = (1 − 2α cos ψ + α ) . M⋆ 4 2

(j) 1 α (r) Also the Laplace coeﬃcients bs (α) and Fourier series ex- − r + + D b 1 (α) pansion of Q(ψ, α) are deﬁned as 2 2 2

π × sin{(r + 1)λ2 − rλ1 − ω2}] . (14) 1 1 2 cos jp dp b(j)(α)= , (6) s π 2 s 2 2 Z0 (1 − 2α cos p + α ) The time variation of the periastrons: ∞ 1 Q(ψ, α)= b(j)(α) cos jψ. (7) M2 1 (1) 1 (2) e2 2 s ω˙1 = n1α αb 3 (α) − αb 3 (α) cos(ω1 − ω2) j=X−∞ M⋆ 4 2 4 2 e1 1 α (r+1) From equations (4) and (7), we can determine two terms as- − r +1+ D b 1 (α) e 2 2 sociated with the two ﬁrst-order r +1 : r arguments, namely 1 × cos{(r + 1)λ2 − λ1 − ω1}] , (15) θ1 = (r + 1)λ2 − rλ1 − ω1 and θ2 = (r + 1)λ2 − rλ1 − ω2, where λ1 and λ2 are the mean longitudes of planet 1 and planet 2, respectively. Therefore, the truncated disturbing M1 1 (1) 1 (2) e1 ω˙2 = n2 αb 3 (α) − αb 3 (α) cos(ω1 − ω2) function is M⋆ 4 2 4 2 e2

1 1 α (r) a2V = Q(ψ, α) − α cos ψ + r + + D b 1 (α) e2 2 2 2 +K1(α)e1 cos{(r + 1)λ2 − rλ1 − ω1} × cos{(r + 1)λ2 − rλ1 − ω2}] . (16) +K2(α)e2 cos{(r + 1)λ2 − rλ1 − ω2}

1 (1) 2 2 (2) If we put r = 2 in the above results, disturbing function + α b 3 (α)(e1 + e2) − 2b 3 (α)e1e2 8 n 2 2 and the time variations of orbital elements agree with that × cos(ω1 − ω2)} , (8) of Malhotra (1993b). 4 Rajib Mia and Badam Singh Kushvah

4 SECULAR RESONANCE DYNAMICS OF Table 1. Orbital parameters of the Kepler-62 system. EXOPLANETARY SYSTEM The data are taken from Borucki et al. (2013). In this section we discuss the long-term variations of the eccentricities of the exoplanets by secular theory with MMR. Parameter Kepler-62e Kepler-62f It is convenient to deﬁne the vertical and horizontal compo- P (days) 122.3874 ± 0.0008 267.291 ± 0.005 nents of eccentricity by a(AU) 0.427 ± 0.004 0.718 ± 0.007 i(deg) 89.98 ± 0.02 89.90 ± 0.03 pj = ej sin ωj , qj = ej cos ωj . (17) T0(BJD–2454900) 83.404 ± 0.003 522.710 ± 0.006 ± − ± These new variables have the advantage that they can re- e cos ω 0.05 0.17 0.05 0.14 e sin ω −0.12 ± 0.02 −0.08 ± 0.10 move the singularities at zero eccentricity in Eqs.(13)-(16). After some calculation, we obtain the equations for the variation of pj and qj (j = 1, 2) as

2 p˙j = A q + Ej cos{(r + 1)λ − rλ }, (18) jk k 2 1 which are similar to the classical Laplace–Lagrange secular Xk=1 solutions for the eccentricities (Murray & Dermott 1999). 2 We present variations of the semimajor axes and eccentrici- q˙j = − Ajkpk − Ej sin{(r + 1)λ2 − rλ1}. (19) ties of Kepler-62, HD 200964 and Kepler-11 systems in fol- Xk=1 lowing sections. These are the first-order differential equations and hence the problem of secular perturbations reduces to the eigenvalue problem, where the coefficient matrix A is given by

A11 A12 A21 A22 and 5 KEPLER-62 SYSTEM M2 2 (1) M2 2 (2) A11 = n1α b 3 (α),A12 = − n1α b 3 (α), 4M⋆ 2 4M⋆ 2 5.1 The 2:1 MMR M1 (2) M1 (1) A21 = − n2αb 3 (α),A22 = n2αb 3 (α).(20) In Section 3, we have discussed r + 1 : r MMR case. We 4M⋆ 2 4M⋆ 2 are now concentrating on the dynamics of 2:1 resonance of Also planets Kepler-62e and Kepler-62f orbiting Kepler 62. From

M2 α (r+1) Table 1, it is clear that periods of Kepler-62e and Kepler- E1 = − n1α (r +1)+ D b 1 (α), 62f are 122.3874 and 267.291 d, respectively. So, there exists M⋆ 2 2 nearly 2:1 resonance between these two planets. In this case M1 1 α (r) E2 = n2 (r + )+ D b 1 (α). (21) the two terms associated with the two first-order 2:1 argu- M⋆ 2 2 2 ments are θ1 = 2λ2 − λ1 − ω1 and θ2 = 2λ2 − λ1 − ω2, where The solutions are given by the superposition of free oscilla- λ1, λ2 are the mean longitudes and ω1,ω2 are periastron tions and forced oscillations as longitudes of Kepler-62e and Kepler-62f, respectively. Also 2 the apsidal lock between the orbiting companions is another pj (t) = eji sin(git + βi)+ Fj sin((r + 1)λ2 − rλ1), important feature for the resonant systems. The relative ap- Xi=1 sidal longitudes is defined as ∆ω = ω1 − ω2. If atleast one 2 of the resonant angles among θ1 and θ2 librates around a qj (t) = eji cos(git + βi)+ Fj cos((r + 1)λ2 − rλ1(22)), constant value, then it is said to be in 2:1 MMR. Moreover, Xi=1 the system is said to be in apsidal co-rotation if ∆ω also where the frequencies gi (i = 1, 2) are the eigenvalues of the librates. In Fig. 6, we depict the evolution of the resonant coefficient matrix A and eji are the component of the two angles θ1,θ2 and ∆ω. From this figure, one can observe the corresponding eigenvectors. The normalization of eigenvec- behaviour of the resonant angles against time. We see that tors and phases βi can be determined by the initial condi- the two resonant angles θ1 and θ2 are librating about 0 rad, tions. The amplitude of forcing is given as also ∆ω librates around 0 rad. This results imply that the two planets Kepler-62e and Kepler-62f of Kepler-62 system F E 1 = −B−1. 1 , (23) are nearly 2:1 MMR and besides in apsidal co-rotation. It is F2 E2 also noticed that the peak-to-valley amplitude of libration of θ1,θ2 and ∆ω are around 12, 6 and 6.1 rad, respectively. The where B = [A − {(r + 1)n2 − rn1}I] and I denotes a 2 × 2 identity matrix. Now if there is no MMR, then equation (22) resonance angle θ1 goes to large amplitude oscillations. It reduces to may be due to the planet’s large angular displacement from the periastron of its orbits (Ketchum et al. 2013). Now, if 2 we put r = 1 in Section 3, we can obtain the perturba- pj (t)= eji sin(git + βi), tion equations for the time variation of the orbital elements. Xi=1 2 First we solve the equations for the semimajor axes. For this, we integrate equations (11) and (12) and then varia- qj (t)= eji cos(git + βi). Xi=1 tions of the semimajor axes are given by a1(t)= a1,0+δa1(t), Orbital Dynamics of Exoplanetary Systems Kepler-62, HD 200964 and Kepler-11 5

0.002 0.002 HIL HIIL 0.001 0.001 1 1 a a

1 0.000 1 0.000 a a ∆ ∆ -0.001 -0.001 -0.002 -0.002 0 2 4 6 8 10 0 2000 4000 6000 8000 10 000 tHyearsL tHyearsL

Figure 1. Perturbative solution for the time variation of the semimajor axes: (I) for t ∈ [0, 10] , (II) for long time t ∈ [0, 10000] of inner planet Kepler-62e.

0.0010 0.0010 HIL HIIL 0.0005 0.0005

2 0.0000 2 0.0000 a a - - 2 0.0005 2 0.0005 a a ∆ -0.0010 ∆ -0.0010 -0.0015 -0.0015 -0.0020 -0.0020 0 2 4 6 8 10 0 2000 4000 6000 8000 10 000 tHyearsL tHyearsL

Figure 2. Perturbative solution for the time variation of the semimajor axes: (I) for t ∈ [0, 10] , (II) for long time t ∈ [0, 10000] of outer planet Kepler-62f.

a2(t)= a2,0 + δa2(t), where where

θj (t)=(2n2 − n1)t + 2(σ2 + ω2) − (σ1 + ω1) − ωj , δa1(t) 2M2α n1 = [Q(ψ(t), α) − Q(ψ0, α) ψ(t)=(n1 − n2)t +(σ1 + ω1) − (σ2 + ω2). (26) a1,0 M⋆ n1 − n2 n1 α Using the data given in Table 1, we can determine α ψ t ψ D 2π − (cos ( ) − cos 0)] − − 2+ nj = , j = 1, 2, where j = 1 means planet Kepler-62 2n2 − n1 h 2 Pj j (2) 3 α and = 2 for Kepler-62f. Now from equations (24) and (25), ×b 1 (α)e1(cos θ1(t) − cos θ1,0)+ + D we observe that there are two components for the variations 2 2 2 in the semimajor axes. In the ﬁrst component, period is (1) 2π ×b 1 (α)e2(cos θ2(t) − cos θ2,0) , (24) equal to the period of the planets, ≈225.757 d and 2 |n1−n2| i M n corresponding fractional amplitude j 1 ≈ 2 × 10−4 M⋆|n1−n2| 2π and in the second component period is |2n2−n1| ≈ 1452.87 Mj n1 −3 days with fractional amplitude M n n ≈ 1 × 10 . In δa2(t) 2M1 n2 ⋆|2 2− 1| = − [Q(ψ(t), α) − Q(ψ0, α) Fig.1, curve (I) represents the time variation of semimajor a2,0 M⋆ n1 − n2 axis of planet Kepler-62e for the time interval t ∈ (0, 10) 2n α −α(cos ψ(t) − cos ψ )] − 2 − 2+ D and curve (II) for long time t ∈ (0, 10000). For the same 0 n n 2 2 − 1 h 2 time interval, we have shown the time variation of the semi- (2) 3 α (1) major axis of planet Kepler-62f (in Fig.2). Also, a compari- ×b 1 (α)e1(cos θ1(t) − cos θ1,0)+ + D b 1 (α) 2 2 2 2 son between numerical and analytical solution for the time ×e2(cos θ2(t) − cos θ2,0)]) , (25)variation of the semimajor axes is shown in Fig.3 to vali- 6 Rajib Mia and Badam Singh Kushvah

0.4276 0.4276 (II) (III) 0.4274 0.4274 0.4274 (I)

1 0.4272 1 0.4272 1 0.4272 a a a 0.4270 0.4270 0.4270 0.4268 0.4268 0.4268 0 2000 4000 6000 8000 10 000 0 2000 4000 6000 8000 10 000 9990 9992 9994 9996 9998 10 000 tHyearsL tHyearsL tHyearsL

0.7185 0.7184 0.7185 (III) (I) 0.7182 0.7180 (II) 0.7180 0.7180

2 0.7175 2 2 0.7175 0.7178 a a a 0.7176 0.7170 0.7170 0.7174 0.7165 0.7165 0.7172 0 2000 4000 6000 8000 10 000 0 2000 4000 6000 8000 10 000 9990 9992 9994 9996 9998 10 000 tHyearsL tHyearsL tHyearsL

Figure 4. Comparison between numerical and analytical solution of the semimajor axes of planets of Kepler-62 system for long time t ∈ [0, 10000]. The upper panel corresponds to Kepler-62e and the lower panel corresponds to Kepler-62f. In each panel, (I) represents the result by analytical theory, (II) represents the numerical solution while (III) represents the zoom part of (I) and (II) for the time interval t ∈ [9990, 10000]. date the result. The thick line represents result by analyt- The solutions for the eccentricities can be written as ical theory and the thin line represents numerical solution. 2

Also for long time t ∈ (0, 10000), a comparison between nu- pj (t)= eji sin(git + βi)+ Fj sin(2λ2 − λ1), merical and analytical solution for the time variation of the Xi=1 orbital semimajor axes of planets of Kepler-62 system are 2 shown in Fig. 4. In this ﬁgure, the upper panel corresponds qj (t)= eji cos(git + βi)+ Fj cos(2λ2 − λ1), (28) to Kepler-62e and while lower panel corresponds to Kepler- Xi=1 62f. In each panel, frame (I) and (II) represent the result by where B = [A − (2n2 − n1)I]. Using the theory as dis- analytical theory and numerical solution while frame (III) cussed in Section 4, we obtain two eigenfrequencies, g1 = shows the same behaviour of changes in semimajor axes for −3 −1 −4 −1 1.57246 × 10 rad yr and g2 = 2.65518 × 10 rad yr t ∈ (9990, 10000) (as in Fig. 3). We see that for long time, together with β1 = −1.27048 rad, β2 = 1.14194 rad, and analytical solution of semimajor axis of Kepler-62e lies be- −3 −3 F1 = 1.09302 × 10 , F2 = −1.29804 × 10 , where eji are tween (0.4267 and 0.4276) and numerical solution lies be- given in Table 3. The evolution of the eccentricities of the tween (0.4268 and 0.4275) and in case of Kepler-62f analyti- planets Kepler-62e and Kepler-62f are depicted in Fig. 5. It is cal solution lies between (0.7172 and 0.7185) and numerical clear that when one eccentricity reaches its maximum value, solution lies between (0.7164 and 0.7185). the other remains at its minimum value and conversely when one eccentricity is minimum, then other reaches exactly its minimum value. Also the eccentricity of Kepler-62e oscillates between 0.03230 and 0.05391 and eccentricity of Kepler-62f oscillates between 0.04154 and 0.04787.

5.2 Secular resonance dynamics of Kepler-62 system 6 HD 200964 SYSTEM Now we draw attention to the secular theory of Kepler-62 6.1 The 4:3 MMR system by considering two planets, Kepler-62e and Kepler- 62f, where the two planets are in 2 : 1 MMR. We avoid For HD 200964 system data are taken from (Johnson et al. the contributions from planets Kepler-62b, Kepler-62c and 2011). As in the previous Section 5, we now focus on the Kepler-62d because their contributions are much less than dynamics of 4:3 resonance of planets HD 200964b and HD that of the mutual eﬀect of two outer planets. Hence, we dis- 200964c orbiting the star HD 200964. In this case, also we cuss secular resonance dynamics of planets Kepler-62e and may determine two terms associated with the two ﬁrst- order Kepler-62f by ignoring the other planets. In this case 4:3 resonance with the arguments namely θ3 = 4λ2 −3λ1−ω1 and θ4 = 4λ2 − 3λ1 − ω2, and the relative apsidal longitude M2 α (2) is ∆ω = ω1 − ω2, where λ1 and λ2 are the mean longitudes E1 = − n1α 2+ D b 1 (α), M⋆ 2 2 of HD 200964b and HD 200964c, respectively. In Fig. 12, M1 3 α (1) curve (a) and curve (b) show the behaviour of the resonant E2 = n2 + D b 1 (α). (27) M⋆ 2 2 2 angles θ1 and θ2 and curve (c) represents the plot of the Orbital Dynamics of Exoplanetary Systems Kepler-62, HD 200964 and Kepler-11 7

6 3 3 L L 4 (a) L 2 (b) 2 (c) 2 1 1 0 0 0 radians radians radians -2 - H -

H 1 1 H 1 2

Θ - 4 Θ -2 -2 DΩ -6 -3 -3 0 20 000 40 000 60 000 80 000 100 000 0 20 000 40 000 60 000 80 000 100 000 0 20 000 40 000 60 000 80 000 100 000 tHyearsL tHyearsL tHyearsL

Figure 6. The evolution of the resonant angles θ1 (a), θ2 (b) and apsidal angle ∆ω (c) of Kepler-62 system. Note that θ1, θ2 and ∆ω librate around 0 and the peak-to-valley amplitude of these librations are around 12, 6 and 6.1 rad, respectively.

0.4276 0.13

0.4274 0.12

0.4272 0.11 1 a 0.4270 0.10 eccentricity 0.4268 0.09

0.08 0.4266 0 2 4 6 8 10 0 5000 10 000 15 000 20 000 25 000 30 000 H L t years tHyearsL

Figure 5. Planet’s eccentricity: curve (I) is eccentricity of Kepler- 0.7185 62e , curve (II) is eccentricity of Kepler-62f for long time t ∈ [0, 30000].

0.7180

2 a1(t)= a1,0 + δa1(t), a2(t)= a2,0 + δa2(t), where a

0.7175 δa1(t) 2M2α n1 = [Q(ψ(t), α) − Q(ψ0, α) a1,0 M⋆ n1 − n2

3n1 α −α(cos ψ(t) − cos ψ0)] − − 4+ D 0.7170 4n2 − 3n1 h 2 0 2 4 6 8 10 (4) 7 α (3) ×b 1 (α)e1(cos θ3(t) − cos θ3,0)+ + D b 1 (α) tHyearsL 2 2 2 2 ×e2(cos θ4(t) − cos θ4,0)]) , (29) Figure 3. Comparison between numerical and analytical solution δa2(t) 2M1 n2 for the time variation of the semimajor axes of Kepler-62 system: = − [Q(ψ(t), α) − Q(ψ , α) a M n n 0 the thick line represents the result by analytical theory and the 2,0 ⋆ 1 − 2 thin line represents the numerical solution. 4n2 α −α(cos ψ(t) − cos ψ0)] − − 4+ D 4n2 − 3n1 h 2 (4) 7 α (3) apsidal angle ∆ω against time. From this figure, we see that ×b 1 (α)e1(cos θ3(t) − cos θ3,0)+ + D b 1 (α) 2 2 2 2 θ1 librates around 0 rad with a larger amplitude of ±11.34 ×e2(cos θ4(t) − cos θ4,0)]) , (30) rad and θ2 librates around 0 rad with an amplitude of ±8 rad. This larger amplitude variations may be due to the where planet’s large angular displacements from the periastrons of its orbits (Ketchum et al. 2013). The libration of these two θj (t)=(4n2 − 3n1)t + 4(σ2 + ω2) resonant angles confirm that there exists 4:3 MMR between − 3(σ1 + ω1) − ωj , (j = 3, 4) (31) the two planets of the system HD 200964. Also the apsidal ψ(t)=(n1 − n2)t +(σ1 + ω1) − (σ2 + ω2). (32) angle ∆ω librates about 0 rad with an amplitude of ±2.8 rad, which indicates that there exists an apsidal libration 2π We can determine nj = , j = 1, 2 using the data between two planets HD 200964b and HD 200964c. Pj Now, the perturbation equations for the time variation given in the Table (2) where j = 1 represents planet HD of the semimajor axes can be obtained by substituting r = 3 200964b and j = 2 for HD 200964c. For this case we observe (in Section 3). We solve the equations for the semimajor that there are two components for the variations in the semi- axes then the variations of the semimajor axes are given by major axes (see equation (29)). In the first component, pe- 8 Rajib Mia and Badam Singh Kushvah

0.10 0.10 HIL HIIL 0.05 0.05

1 0.00 1 0.00 a a - - 1 0.05 1 0.05 a a ∆ -0.10 ∆ -0.10 -0.15 -0.15 -0.20 -0.20 0 20 40 60 80 100 120 140 0 2000 4000 6000 8000 10 000 tHyearsL tHyearsL

Figure 7. Perturbative solution for the time variation of the semimajor axes: (I) for t ∈ [0, 150] , (II) for long time t ∈ [0, 10000] of inner planet HD 200964b.

0.4 0.4 HIIL 0.3 HIL 0.3 0.2 0.2 2 2 a a

2 0.1 2 0.1 a a ∆ 0.0 ∆ 0.0 -0.1 -0.1 -0.2 -0.2 0 20 40 60 80 100 120 140 0 2000 4000 6000 8000 10 000 tHyearsL tHyearsL

Figure 8. Perturbative solution for the time variation of the semimajor axes: (I) for t ∈ [0, 150] , (II) for long time t ∈ [0, 10000] of outer planet HD 200964c.

Table 2. Physical and orbital parameters of the HD 200964 system ical and analytical solution for the time interval t ∈ (0, 100) corresponding to the best ﬁt of (Johnson et al. 2011). of the semimajor axes are shown in Fig.9 to validate the result. The thick line represents results obtained by analytical Parameter HD 2600964 HD 200964b HD 200964c theory and the thin line represents the numerical solution. Also for long time t ∈ (0, 10000), a comparison between nu- Mp sin i 1.44M⊙ 1.85MJ 0.895MJ merical and analytical solution for the time variation of the . . Period(d) 613 8 825 0 orbital semimajor axes of planets of HD 200964 system are a(AU) 1.601 1.950 shown in Fig. 10. In this ﬁgure, the upper panels are for HD e 0.040 0.181 ω(deg) 288.0 182.6 200964b and the lower panels are for HD 200964c. In each Tp(JD) 2454900 2455000 panel, frame (I) and (II) represent the result by analytical theory and numerical solution, respectively, while frame (III) shows the same behaviour of changes in semimajor axes for t ∈ (9990, 10000) as seen in Fig. 9. We see that for long 2π riod is equal to the period of planets, |n1−n2| ≈2397.66 d and time, analytical solutions of semimajor axis of HD 200964b M n corresponding fractional amplitude j 1 ≈ 2×10−3. In lies between (1.37 and 1.66) and numerical solutions lies be- M⋆|n1−n2| 2π tween (1.42 and 1.66) and while in the case of HD 200964c the second component, period is |4n2−3n1| ≈ 25575 d with M n analytical solution lies between (1.8 and 2.5) and numerical fractional amplitude j 1 ≈ 7 × 10−2. In Fig.7, curve M⋆|2n2−n1| lies between (1.84 and 2.4). (I) represents the time variation of semimajor axis of planet HD 200964b for the time interval t ∈ (0, 150) and curve (II) is for long time t ∈ (0, 10000). For the same time interval, we have shown the time variation of the semimajor axis of planet HD 200964c in Fig.8. A comparison between numer- Orbital Dynamics of Exoplanetary Systems Kepler-62, HD 200964 and Kepler-11 9

1.65 1.65 1.65 (III) 1.60 (I) 1.60 (II) 1.60 1.55

1 1.55 1 1 1.55 a 1.50 a a 1.50 1.50 1.45 1.45 1.40 1.45 1.40

0 2000 4000 6000 8000 10 000 0 2000 4000 6000 8000 10 000 9900 9920 9940 9960 9980 10 000 tHyearsL tHyearsL tHyearsL

2.4 (I) 2.3 (III) (II) 2.4 2.2 2.2 2 2 2 2.2 2.1 a a a 2.0 2.0 2.0 1.9 1.8 1.8 0 2000 4000 6000 8000 10 000 0 2000 4000 6000 8000 10 000 9900 9920 9940 9960 9980 10 000 tHyearsL tHyearsL tHyearsL

Figure 10. Comparison between numerical and analytical solution of the semimajor axes of planets of HD 200964 system for long time t ∈ [0, 10000]. The upper panel corresponds to HD 200964b and the lower panel corresponds to HD 200964c. In each panel, (I) represents the analytical solution, (II) represents the numerical solution, respectively, and (III) represents the zoom part of (I) and (II) for the time interval t ∈ [9990, 10000].

6.2 Secular resonance dynamics of HD 200964 Table 3. Values of eji for the Kepler-62, HD200964 and system Kepler-11 systems. As in Section 5.2, we discuss the long-term variations of the e i i eccentricities of the planets by secular theory with MMR. System ji = 1 = 2

Now we discuss the secular theory of HD 200964 system Kepler-62 e1i 0.026594 0.103945 by considering all two planets. After some calculation, the e2i −0.0200403 0.109099 solutions for the eccentricities can be written as HD 200964 e . − . 2 1i 0 10331 0 0655869 e2i −0.198811 −0.0638386 ej sin ωj = eji sin(git + βi)+ Fj sin(4λ2 − 3λ1), Kepler-11 Xi=1 e1i 0.00810542 0.043162 2 e2i −0.00229914 0.044661 ej cos ωj = eji cos(git + βi)+ Fj cos(4λ2 − 3λ1)(33), Xi=1 where the frequencies gi (i = 1, 2) are the eigenvalues Table 4. Physical and orbital parameters of the Kepler-11 system of the coefficient matrix A and eji are the components corresponding to the best fit of (Lissauer et al. 2011) of the two corresponding eigenvectors. The normalization of the eigenvectors and the phases βi can be determined Parameter Kepler-11b Kepler-11c by the initial conditions. The amplitude of forcing will be same as in Eq.(23) but B will be changed which is B = Mass 4.3M⊕ 13.5M⊕ M2 α (4) Period(d) 10.30375 13.02502 [A−(4n2 −3n1)I] and E1 = − M n1α 4+ 2 D b 1 ,E2 = ⋆ 2 . a(AU) 0.091 ± 0.003 0.106 ± 0.004 M1 7 α (3) n2 + D b 1 . For this system, we obtain the eigen- e cos ω 0.0534 ± 0.0383 0.0416 ± 0.0332 M⋆ 2 2 2 −2 −1 e sin ω −0.0039 ± 0.0072 −0.0007 ± 0.0060 frequencies, g1 = 2.16821×10 rad yr and g2 = 8.94711× −4 −1 Epoch(BJD) 2454971.5052 2454971.1748 10 rad yr together with β1 = 0.247833, β2 = 0.502108 −2 −2 and F1 = 4.3939 × 10 , F2 = −9.84327 × 10 . The evolution of the eccentricities of the two planets HD 200964b and HD 200964c are depicted in Fig. 11. orbiting the same star and yield a somewhat weaker perturbations. The inner pair of observed planets, Kepler-11b and Kepler-11c, lie near a 5:4 orbital period resonance and strongly interact with one another. So, we avoid the contri- 7 KEPLER-11 SYSTEM butions from planets Kepler-11d, Kepler-11e, Kepler-11f and 7.1 The 5:4 MMR Kepler-11g because their contribution are much less than that of the mutual effect of two inner planets. We are now Lissauer et al. (2011) observed perturbations of planets concentrating on the dynamics of 5:4 resonance of planets Kepler-11d and Kepler-11f by planet Kepler-11e and con- Kepler-11b and Kepler-11c orbiting Kepler-11. In this case, firm that all three sets of transits are produced by planets the two terms associated with the two first-order 5:4 argu- 10 Rajib Mia and Badam Singh Kushvah

3 10 (a) (b) (c) L L L 5 2 5 1 0 0 0 radians radians radians

H - H -5 H 1 1 2 -5 Θ Θ -2 -10 DΩ -3 0 2000 4000 6000 8000 10 000 0 2000 4000 6000 8000 10 000 0 2000 4000 6000 8000 10 000 tHyearsL tHyearsL tHyearsL

Figure 12. The evolution of the resonant angles θ1 (a), θ2 (b) and apsidal angle ∆ω (c) of HD 200964 system. Note that θ1, θ2 and ∆ω librate around 0 rad with an amplitude of ±11.34, ±8 and ±2.8 rad, respectively.

0.005 0.005 HIL HIIL 0.004 0.004 0.003 1 1 0.003 a a

1 0.002 1 a a

∆ ∆ 0.002 0.001 0.000 0.001 -0.001 0.000 0.0 0.2 0.4 0.6 0.8 1.0 0 2000 4000 6000 8000 10 000 tHyearsL tHyearsL

Figure 13. Perturbative solution for the time variation of the semimajor axes: (I) for t ∈ [0, 1] , (II) for long time t ∈ [0, 10000] of inner planet Kepler-11b.

ments are θ1 = 5λ2 − 4λ1 − ω1 and θ2 = 5λ2 − 4λ1 − ω2, δa2(t) 2M1 n2 = − [Q(ψ(t), α) − Q(ψ0, α) where λ1 and λ2 are the mean longitudes of Kepler-11b and a2,0 M⋆ n1 − n2 Kepler-11c, respectively. For this system, the plots of the 5n α −α(cos ψ(t) − cos ψ )] − 2 − 5+ D θ θ 0 resonant angles 1, 2 and the relative apsidal angle against 5n2 − 4n1 h 2 time are shown in curves (a), (b) and (c) (in Fig. 16). Sim- (5) 9 α ilarly, we see that for this system also θ librates around 0 ×b 1 (α)e1(cos θ1(t) − cos θ1,0)+ + D 1 2 2 2 rad with an amplitude of ±1.6 rad and θ2 librates around 0 (4) rad with an amplitude of ±1.3 rad. Obviously, these results ×b 1 (α)e2(cos θ2(t) − cos θ2,0) , (35) 2 conﬁrm that the two inner pair of observed planets, Kepler- i 11b and Kepler-11c of Kepler-1l system lie in 5:4 MMR. The where libration of ∆ω around 0 rad with an amplitude of ±0.32 rad also means that the system is in apsidal co-rotation. θj (t)=(5n2 − 4n1)t + 5(σ2 + ω2) − 4(σ1 + ω1) − ωj , ψ(t)=(n − n )t +(σ + ω ) − (σ + ω ). (36) Similarly, if we put r = 4 in Section 3, we can obtain 1 2 1 1 2 2 the perturbation equations for the time variation of the or- Using the numerical values of parameters given in Table 4, bital elements. In this case, we also integrate Eqs.(11) and 2π we can calculate mean motion nj = , j = 1, 2 where (12) and then variations of the semimajor axes are given by Pj j = 1 means planet Kepler-11b and j = 2 for Kepler-11c. a (t)= a , + δa (t), a (t)= a , + δa (t), where 1 1 0 1 2 2 0 2 Similar to the previous two systems in this case, we have also seen that there are two components for the variations in the semimajor axes. The period in the ﬁrst component is 2π equal to the period of the planets, |n1−n2| ≈49.3176 d and M n corresponding fractional amplitude j 1 ≈ 2 × 10−4 M⋆|n1−n2| δa1(t) 2M2α n1 π = [Q(ψ(t), α) − Q(ψ0, α) and in the second component period is 2 ≈ 230.861 a1,0 M⋆ n1 − n2 |5n2−4n1 | M n d with fractional amplitude j 1 ≈ 3 × 10−3. 4n1 α M⋆|2n2−n1| −α(cos ψ(t) − cos ψ0)] − − 5+ D 5n2 − 4n1 h 2 In Fig.13, curve (I) represents the time variation of (5) 9 α semimajor axis of planet Kepler-11b for the time interval ×b 1 (α)e1(cos θ1(t) − cos θ1,0)+ + D 2 2 2 t ∈ (0, 1) and curve (II) is for long time t ∈ (0, 10000). (4) For the same time interval, we have shown the time varia- ×b 1 (α)e2(cos θ2(t) − cos θ2,0) , (34) tion of the semimajor axis of planet Kepler-11c (in Fig.14). 2 i Orbital Dynamics of Exoplanetary Systems Kepler-62, HD 200964 and Kepler-11 11

0.0000 0.0000 -0.0002 HIL -0.0002 HIIL -0.0004 -0.0004 2 2 a a - - 2 0.0006 2 0.0006 a a ∆ ∆ -0.0008 -0.0008 -0.0010 -0.0010 -0.0012 -0.0012 0.0 0.2 0.4 0.6 0.8 1.0 0 2000 4000 6000 8000 10 000 tHyearsL tHyearsL

Figure 14. Perturbative solution for the time variation of the semimajor axes: (I) for t ∈ [0, 1] , (II) for long time t ∈ [0, 10000] of outer planet Kepler-11c.

0.09105 0.09135 (I) (III) 0.09100 (II) 0.0914 0.09130 0.09095 0.0913

1 0.09125 1 1 0.09090 a a 0.09120 a 0.0912 0.09085 0.09115 0.09080 0.0911 0.09110 0.09075 0.0910 0 200 400 600 800 1000 0 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 tHyearsL tHyearsL tHyearsL

0.10598 0.1062 0.1060 (I) (II) 0.1059 0.10596 0.1060 0.1058 2 2 2 0.10594 0.1058 0.1057 a a a 0.1056 0.10592 0.1056 0.1055 (III) 0.10590 0.1054 0.1054 0 200 400 600 800 1000 0 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 tHyearsL tHyearsL tHyearsL

Figure 15. Comparison between numerical and analytical solution of the semimajor axes of planets of Kepler-11 system for long time t ∈ [0, 10000]. The upper panel corresponds to Kepler-11b and the lower panel corresponds to Kepler-11c. In each panel, (I) represents the result by analytical theory, (II) represents the numerical solution and (III) shows comparison between numerical and analytical solution for the time interval t ∈ [0, 1] of the semimajor axes of Kepler-11 system: the thick line represents the result by analytical theory and the thin line represents the numerical solution.

Comparison between numerical and analytical solution of terms and to second order for the secular terms neglecting the semimajor axes of planets of Kepler-11 system for long the higher order terms, the discrepancies between the ana- time t ∈ (0, 10000) are shown in Fig. 15. The upper panel is lytical and numerical results are expected (Fig.15). given for Kepler-11b and the lower panel is for Kepler-11c. In each panel, (I) represents the result by analytical theory, (II) represents the numerical solution and (III) shows comparison between numerical and analytical solution for the time interval t ∈ [0, 1] of the semimajor axes of Kepler-11 system. The thick line represents the result by analytical 7.2 Secular solution of Kepler-11 system theory and the thin line represents the numerical solution. We see that for long time, analytical solution of semimajor Now we discuss the secular theory of Kepler-11 system con- axis of Kepler-11b lies between (0.09110 and 0.09135) and sidering two planets Kepler-11b and Kepler-11c, where the numerical solution lies between (0.09079 and 0.09105) while two planets are in 5 : 4 MMR. In this case in the case of Kepler-11c analytical solutions lies between (0.10590 and 0.10598) and numerical solution lies between M2 α (5) E1 = − n1α 5+ D b 1 (α), (0.1054 and 0.1062). Since we have used truncated disturb- M⋆ 2 2 ing function to ﬁrst order in the eccentricities for the periodic M1 9 α (4) E2 = n2 + D b 1 (α). (37) M⋆ 2 2 2 12 Rajib Mia and Badam Singh Kushvah

0.060 1.65 0.055 HIL 1.60 0.050 1.55 HIIL 1 0.045 a 1.50 0.040 1.45 eccentricity 0.035 1.40 0.030 0 20 40 60 80 100 tHyearsL 0 200 400 600 800 1000 tHyearsL

Figure 17. Planet’s eccentricity: curve (I) is eccentricity of 2.4 Kepler-11b , curve (II) is eccentricity of Kepler-11c for long time t ∈ [0, 500].

2.2 2

a in Section 4, for this system, we obtain g1 = 7.31037 × −2 −1 −3 −1 2.0 10 rad yr and g2 = 1.53994 × 10 rad yr together with β1 = −0.397394 rad, β2 = −0.0312697 rad, and F = 2.84641 × 10−3, F = −9.39297 × 10−4. The evolu- 1.8 1 2 tion of the eccentricities of the two planets Kepler-11b and 0 20 40 60 80 100 Kepler-11c are depicted in Fig. (17) over a time span of 1000 yr which is derived from the above solution for eccentricities. tHyearsL It is clear from the ﬁgure that the periodicity occurs in the variation of eccentricities. It is also shows that the minimum Figure 9. Comparison between numerical and analytical solu- in eccentricity of Kepler-11b coincides with the maximum tions for the time variation of the semimajor axes of HD 200964 of Kepler-11c and reverse is also true. Moreover, the eccen- system: the thick line represents the result by analytical theory tricity of Kepler-11b oscillates between 0.07633 and 0.1313 and the thin line represents the numerical solution. and eccentricity of Kepler-11c oscillates between 0.08769 and 0.1302.

0.4

HIIL 8 CONCLUSIONS 0.3 In this work we have used general three-body problem as a model for the study of dynamics of exoplanetary sys- 0.2 tems. We have applied the latest stability criteria given by Petrovich (2015) to examine the stability of exoplane-

eccentricity 0.1 tary systems Kepler-62, HD 200964, and Kepler-11. We have identiﬁed (r + 1) : r MMR terms in the expression of dis- HIL turbing function and obtained the perturbations from the 0.0 truncated disturbing function. It is found that the evolution 0 200 400 600 800 1000 of the resonant angles librates around constant value. Thus, tHyearsL according to this study, it is our opinion that 2:1, 4:3 and 5:4 near MMRs occur between Kepler-62e and Kepler-62f, HD Figure 11. Planet’s eccentricity: curve (I) is eccentricity of HD 200964b and HD 200964c and Kepler-11b and Kepler-11c, 200964b , curve (II) is eccentricity of HD 200964c for long time respectively. We have obtained the orbital solution of planets t ∈ [0, 1000]. of Kepler-62 system in the 2:1 MMR, planets of HD 200964 in 4:3 MMR and planets of Kepler-11 in the 5:4 MMR. We have found that the periodicity occurs in the variation of ec- The solutions for the eccentricities can be written as centricities. It is also observed that a minimum in eccentric- 2 ity of Kepler-62e coincides with a maximum of Kepler-62f pj (t)= eji sin(git + βi)+ Fj sin(5λ2 − 4λ1), and opposite is also true. Moreover maximum in Kepler- Xi=1 11b’s eccentricity coincides with a minimum in Kepler-11c’s 2 eccentricity and conversely. A comparison is presented be- qj (t)= eji cos(git + βi)+ Fj cos(5λ2 − 4λ1), (38) tween the analytical solution and numerical solution. Xi=1 Furthermore, the derived explicit analytical expressions also for this case B = [A − (5n2 − 4n1)I]. With the numer- would be used for the study of others newly discovered ical values (from Table 4) and using the theory discussed exoplanetary systems. Moreover, the work would be ex- Orbital Dynamics of Exoplanetary Systems Kepler-62, HD 200964 and Kepler-11 13

1.5 (b) 0.3 (a) 1.0 L (c) L L 1.0 0.2 0.5 0.5 0.1 0.0 0.0 0.0 radians radians radians - - 0.5 - H 0.1 H H 0.5 2 1 -1.0 -0.2 Θ Θ

-1.0 DΩ -1.5 -0.3 0 200 400 600 800 0 200 400 600 800 0 200 400 600 800 tHyearsL tHyearsL tHyearsL

Figure 16. The evolution of the resonant angles θ1 (a), θ2 (b) and apsidal angle ∆ω (c) of Kepler-11 system. Note that θ1, θ2 and ∆ω librate around 0 rad with an amplitude of ±1.6, ±1.3 and ±0.32 rad, respectively. tended by considering additional planets (such as Kepler- Malhotra R., 1993a, in Phillips J. A., Thorsett S. E., Kulka- 62c, Kepler-11e) which negligibly perturb the pair of mod- rni S. R., eds, Planets Around Pulsars Vol. 36 of Astro- elled planets in each system with the help of fully N-body nomical Society of the Pacific Conference Series, Orbital simulations. dynamics of PSR1257+12 and its two planetary companions. pp 89–106 Malhotra R., 1993b, ApJ, 407, 266 Murray C., Dermott S., 1999, Solar System Dynamics. ACKNOWLEDGEMENTS Cambridge Univ. Press, Cambridge Mustill A. J., Wyatt M. C., 2011, in Sozzetti A., Lattanzi We are thankful to Inter-University Centre for Astron- M. G., Boss A. P., eds, The Astrophysics of Planetary omy and Astrophysics (IUCAA), Pune, India for support- Systems: Formation, Structure, and Dynamical Evolution ing library visits and for the use of computing facilities. Vol. 276 of IAU Symposium, Hamiltonian model of cap- Badam Singh Kushvah(BSK) is also grateful to the In- ture into mean motion resonance. pp 300–303 dian Space Research Organization (ISRO), Department of Pal A. K., Kushvah B. S., 2015, MNRAS, 446, 959 Space, Government of India, for providing financial sup- Papadakis K. E., 2005, Ap&SS, 299, 67 port through the RESPOND Programme (Project No - Petrovich C., 2015, ApJ, 808, 120 ISRO/RES/2/383/2012-13). Petrovich C., Malhotra R., Tremaine S., 2013, ApJ, 770, 24 Santos M., Correa-Otto J., Michtchenko T., Ferraz-Mello S., 2015, A&A, 573, A94 REFERENCES Schneider J., Dedieu C., Le Sidaner P., Savalle R., Zolo- tukhin I., 2011, A&A, 532, A79 Adams F. C., Bloch A. M., 2015, Monthly Notices of the Szebehely V., 2012, Theory of Orbit: The Restricted Prob- Royal Astronomical Society, 446, 3676 lem of Three Bodies. Elsevier Science: New York and Lon- BeaugéC., Ferraz-Mello S., Michtchenko T., 2003, The As- don. trophysical Journal, 593, 1124 Wolszczan A., Frail D. A., 1992, Nature, 355, 145 BeaugéC., Michtchenko T., Ferraz-Mello S., 2006, Monthly Notices of the Royal Astronomical Society, 365, 1160 Borucki W. J., Agol E., Fressin F., Kaltenegger L., Rowe J., Isaacson H., Fischer D., Batalha N., Lissauer J. J., Marcy G. W., et al., 2013, Science, 340, 587 Couetdic J., Laskar J., Correia A. C. M., Mayor M., Udry S., 2010, A&A, 519, A10 Davies M. B., Adams F. C., Armitage P., Chambers J., Ford E., Morbidelli A., Raymond S. N., Veras D., 2014, in Beuther H., Klessen R. S., Dullemond C. P., Henning T., eds, Protostars and Planets VI. Univ. Arizona Press, Tucson, AZ, pp 787–808 Fabrycky D. C., Murray-Clay R. A., 2010, ApJ, 710, 1408 Graziani F., Black D. C., 1981, ApJ, 251, 337 Johnson J. A., Payne M., Howard A. W., Clubb K. I., Ford E. B., Bowler B. P., Henry G. W., Fischer D. A., Marcy G. W., Brewer J. M., Schwab C., Reffert S., Lowe T. B., 2011, AJ, 141, 16 Ketchum J. A., Adams F. C., Bloch A. M., 2013, The As- trophysical Journal, 762, 71 Lissauer J. J., Fabrycky D. C., Ford E. B., Borucki W. J., Fressin F., Marcy G. W., Orosz J. A., Rowe J. F., Torres G., Welsh W. F., et al., 2011, Nature, 470, 53