Secular Orbital Evolution of Compact Planet Systems
aa r X i v : . [ a s t r o - ph . E P ] O c t Draft version October 30, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
SECULAR ORBITAL EVOLUTION OF COMPACT PLANET SYSTEMS
Ke Zhang , Douglas P. Hamilton , and Soko Matsumura Draft version October 30, 2018
ABSTRACTRecent observations have shown that at least some close-in exoplanets maintain eccentric orbitsdespite tidal circularization timescales that are typically much shorter than stellar ages. We exploregravitational interactions with a more distant planetary companion as a possible cause of these unex-pected non-zero eccentricities. For simplicity, we focus on the evolution of a planar two-planet systemsubject to slow eccentricity damping and provide an intuitive interpretation of the resulting long-termorbital evolution. We show that dissipation shifts the two normal eigenmode frequencies and eccen-tricity ratios of the standard secular theory slightly, and we confirm that each mode decays at its ownrate. Tidal damping of the eccentricities drives orbits to transition relatively quickly between periodsof pericenter circulation and libration, and the planetary system settles into a locked state in whichthe pericenters are nearly aligned or nearly anti-aligned.Once in the locked state, the eccentricities of the two orbits decrease very slowly because of tidesrather than at the much more rapid single-planet rate, and thus eccentric orbits, even for close-inplanets, can often survive much longer than the age of the system. Assuming that an observed close-in planet on an elliptical orbit is apsidally-locked to a more distant, and perhaps unseen companion,we provide a constraint on the mass, semi-major axis, and eccentricity of the companion. We findthat the observed two-planet system HAT-P-13 might be in just such an apsidally locked state, withparameters that obey our constraint reasonably well. We also survey close-in single planets, somewith and some without an indication of an outer companion. None of the dozen systems that weinvestigate provides compelling evidence for unseen companions. Instead, we suspect that (1) orbitsare in fact circular, (2) tidal damping rates are much slower than we have assumed, or (3) a recentevent has excited these eccentricities. Our method should prove useful for interpreting the results ofboth current and future planet searches.
Subject headings: planetary systems - planets and satellites: dynamical evolution and stability - planetsand satellites: general INTRODUCTION
In the past decade, many mechanisms have been pro-posed to explain a wide range of eccentricities seen amongexoplanets including planet-planet scattering, planet-disk interaction, mean-motion resonance passage, andKozai resonances (see, e.g., a review by Namouni (2007)and references there-in). All of these mechanisms canexcite orbital eccentricities effectively. While the orbitsof long-period planets could stay eccentric for billions ofyears, those of most close-in planets are likely to be cir-cularized within stellar ages because of tidal interactionsbetween the stars and the planets, except in some inter-esting special cases considered by Correia et al. (2012).In Figure 1, we plot the eccentricities of all 222 known(as of 2012 February) close-in planets with orbital pe-riods of less than 20 days. The average eccentricityfor these planets ( ∼ . ) is significantly smaller thanthat of all of the confirmed planets ( ∼ . ), whichindicates that the tidal interaction is an effective ec-centricity damping mechanism and that circularizationtimescales are typically shorter than stellar ages (about1-10 Gyr). Nevertheless, nearly half of these close-inplanets have non-zero orbital eccentricities. Recent stud- Corresponding Authors: [email protected], [email protected] Department of Astronomy, University of Maryland, CollegePark, MD 20742, USA School of Engineering, Physics, and Mathematics, Universityof Dundee, Scotland DD1 4HN Period [days]0.00.10.20.30.40.50.60.7 E cc e n t r i c i t y Fig. 1.—
Eccentricities of close-in planets with orbital periodsless than 20 days. The blue circles and orange triangles correspondto single- and multiple-planet systems, respectively. Error bars areplotted for both period and eccentricity. However, errors in periodare not apparent because they are very small. Orbital data, cour-tesy of the Exoplanet Orbit Database ( http://exoplanets.org/ ). ies have shown, however, that the orbital fits tend tooverestimate eccentricities (e.g., Shen & Turner 2008;Zakamska et al. 2011; Pont et al. 2011), so some mea-sured eccentric orbits for close-in planets might in factbe circular. However, it is unlikely that all close-in plan-ets have perfectly circular orbits and thus at least someof the non-zero eccentricities require a physical explana-tion. The main possibilities include (1) systems have onlyrecently attained their orbital configurations and eccen-tricities are damping quickly, (2) planetary tidal qualityfactors are much larger than those of the giant planets inour solar system and so eccentricities damp slowly, and(3) eccentricity excitation caused by an exterior planetslows the orbital circularization. In this paper, we ex-plore the third option and investigate the gravitationalinteractions between a close-in planet and a more distantcompanion.In systems with more than one planet, the most sig-nificant orbit-orbit interactions are often mean-motionresonances, which have been studied in detail for thesatellite systems of the giant planets (see reviews byGreenberg 1977; Peale 1986). Resonances in extra-solar planetary systems have also received increasedattention (e.g., Chiang 2003; Beaugé et al. 2003; Lee2004; Ketchum et al. 2013; Batygin & Morbidelli 2013).Mean-motion resonance passages during planetary mi-gration can be effective in exciting orbital eccentricities.These resonance passages can be divergent, in which caseimpulsive changes to the orbital elements result, or con-vergent, in which case trapping into resonance usuallyoccurs (Hamilton 1994; Zhang & Hamilton 2007, 2008).Although resonance capture typically leads to excited ec-centricities, we do not consider this process further here.Instead, we focus on the more prosaic secular interactionswhich are capable of maintaining orbital eccentricities fora greater variety of orbital configurations.Secular perturbations have been studied for cen-turies in the context of the Solar System (see, e.g.,Brouwer et al. 1950). With the discovery of extrasolarmulti-planet systems (Butler et al. 1999), applica-tions for secular theory have expanded rapidly (e.g.,Wu & Goldreich 2002; Barnes & Greenberg 2006b;Adams & Laughlin 2006a; Mardling 2007; Batygin et al.2009; Greenberg & Van Laerhoven 2011; Laskar et al.2012). Wu & Goldreich (2002) were the first to usesecular interactions to explain the non-zero eccentricityof a hot Jupiter. They showed that a close-in planetwith a companion could sustain a substantial orbitaleccentricity even though tidal damping was efficient.Zhang & Hamilton (2003) and Zhang (2007) confirmedthe results of Wu & Goldreich (2002) and obtainedmost of the results contained in Section 2 of the presentarticle. Mardling (2007) developed a detailed octopole-order secular theory to study orbital evolution of close-inplanets with companions and also confirmed the resultsof Wu & Goldreich (2002). Laskar et al. (2012) pointedout the importance of precession due to tidal effects. Ifthese well-studied secular interactions play an importantrole in delaying eccentricity damping, we might expectdifferences in the eccentricity distributions betweensingle- and multiple-planet systems. There is no obviousdifference between these two groups of planets, however,as can be seen in Figure 1. Perhaps future observationsmight reveal such a difference, but it is also plausiblethat many single close-in planets are actually accom-panied by unobserved companions that help maintaintheir eccentricities against tidal dissipation.In this paper, we revisit the problem of secular inter-actions with a distant companion in maintaining the ec-centricities of close-in planets. Our goals are to developan intuitive interpretation of the secular theory of a two-planet system and to test the model against observed planetary systems. In the next section, we present a lin-ear Laplace-Lagrange model for secular orbital evolutionduring tidal dissipation, starting with a review of secu-lar orbital interactions in a stable non-dissipative systemconsisting of a star and two planets. We then add tidaldissipation of eccentricities, and solve the coupled sys-tem to investigate how eccentricity damping affects theapsidal state of the two orbits. We add the precessiondue to planetary and stellar tidal and rotational bulgesas well as general relativity (GR) terms which can be sig-nificant for close-in planets. In Section 3, we apply thismodel to extrasolar planetary systems, illustrating howit might help to guide planet searches. Last, we discussand summarize our work in Section 4. MODEL
Stable Non-dissipative Systems with Two Planets
The secular solution of a stable non-dissipative 2-planet system is known as Laplace-Lagrange theory andis discussed in great detail in Murray & Dermott (1999).In this section, we will develop a graphical interpretationof the solution that will help us understand the morecomplicated systems studied later in this paper. Afteraveraging the disturbing functions of each planet on theother, R j , over both planets’ orbital periods, Lagrange’splanetary equations can be linearized for small eccentric-ities and inclinations to (Murray & Dermott 1999, Sec-tion 7.1) ˙ a j = 0;˙ e j = − n j a j e j ∂ R j ∂̟ j , ˙ ̟ j = + 1 n j a j e j ∂ R j ∂e j ; (1) ˙ I j = − n j a j I j ∂ R j ∂ Ω j , ˙Ω j = + 1 n j a j I j ∂ R j ∂I j . Here, n j , a j , e j , I j , Ω j , and ̟ j are the mean motion,semi-major axis, eccentricity, inclination, longitude of as-cending node, and argument of the pericenter of the j th planetary orbit, respectively.Planetary semi-major axes remain constant and henceno long-term energy transfer between the orbits occursbecause we have averaged a conservative perturbationforce over each planet’s orbital period. Note that be-cause of our assumption of small eccentricities and in-clinations, the evolution equations of ( e j , ̟ j ) and ( I j , Ω j ) are completely decoupled and can be analyzed sep-arately. In this paper, we neglect the inclination andnode pair since they are less easily observable for extra-solar planets. Because the two sets of equations take thesame form, however, our eccentricity results below canbe easily applied to secular coupling of vertical motions.The disturbing function R is defined for a planetas the non-Keplerian potential at its location(Murray & Dermott 1999, Section 6). For two-planet systems without any external perturbations,it is simply the gravitational potential caused by theother planet. Here, we consider a simple planetarysystem consisting of a central star and two planetsin co-planar orbits. In terms of osculating elements(see, e.g., Murray & Dermott 1999, Section 2.9) and tosecond order in small eccentricities, Murray & Dermott(1999) show that the orbit-averaged disturbing functionsare given by: R = n a σq (cid:20) e − β e e cos( ̟ − ̟ ) (cid:21) , (2) R = n a σ √ α (cid:20) e − β e e cos( ̟ − ̟ ) (cid:21) , (3)where the subscript “1” refers to the inner planet and “2”to the outer one. We define the mass ratio between thetwo planets q = m /m , and the semi-major ratio of thetwo orbits α = a /a . The remaining parameters aredefined as follows: β = b (2)3 / ( α ) b (1)3 / ( α ) and σ = 14 n m m ∗ α b (1)3 / ( α ) , where m ∗ is the stellar mass and b (1)3 / ( α ) and b (2)3 / ( α ) are two of the Laplace coefficients (Murray & Dermott1999). The parameter σ has units of frequency, which wewill soon see characterizes the secular precession rates.Both β and σ decrease with increasing planetary separa-tion and, for small α , the Laplace coefficients reduce to b (1)3 / ( α ) ≈ α and b (2)3 / ( α ) ≈ α / .Following Brouwer et al. (1950), we transform a ( e j , ̟ j ) pair into a complex Poincaré canonical variable h j with the mapping: h j = e j exp( i̟ j ) , (4)where i = √− . Substituting Equations (2) and (3)into Equation (1) and rewriting in terms of h j yields aset of linear homogeneous ordinary differential equationssimilar to those for a double pendulum system: ˙ h j = i X k =1 A jk h k , (5)where the coefficient matrix A = σ (cid:26) q − qβ −√ αβ √ α (cid:27) . Two orthogonal special solutions of Equation. (5), orthe secular eigenmodes of the system, are given by (cid:18) ˆ h ± ˆ h ± (cid:19) = (cid:18) η s ± (cid:19) exp( ig s ± t ) , where the eigen-frequencies g s ± and eigenvector parame-ters η s ± can be obtained from the matrix A : g s ± = σ (cid:20) q + √ α ∓ q ( q − √ α ) + 4 q √ αβ (cid:21) , (6) η s ± = q − √ α ± p ( q − √ α ) + 4 q √ αβ qβ . (7)We use the superscript “ s ” to indicate that those param-eters are for a “static”, or non-dissipative, system. Theeigenmode components ˆ h ± and ˆ h ± depend only on thefixed constants α and q through Equations. (6) and (7),and not on the initial eccentricities and pericenter angles.The physical meaning of the two modes can be elu-cidated by transforming the solution of h j back to the Fig. 2.—
Aligned and anti-aligned secular modes. The planets(solid dots) follow nearly elliptical orbits about the central star.Arrows point from the star to orbital pericenters. . . . . . √ a / q − − h s − h s + b = 0.1 0.2 0.5 0.9 b = 0.90.50.20.1 Fig. 3.—
Eccentricity ratios vs. √ α/q in the pure secular modesas given by Equation (7). Different curves represent different β values. Along the dashed line q = √ α (or m a = m a ) andorbits have e = e in either mode. orbital elements ( e , ̟ ) with Equation (4). If the systemis fully in either the “+” or the “–” mode, we find thefollowing: ˙ ̟ ± = ˙ ̟ ± = g s ± , (8) ( e /e ) ± = | η s ± | , (9) cos(∆ ̟ ± ) = η s ± / | η s ± | , (10)where ∆ ̟ = ̟ − ̟ is the difference between the twopericenter angles. In either mode, the two orbits precesstogether at the rate g ± (Equation (8)), and their eccen-tricities keep a fixed ratio (Equation (9)). Furthermore,Equation (7) shows that η s + > while η s − < . Thus,Equation (10) states that the pericenters of the two or-bits are always aligned in the “+” mode ( cos(∆ ̟ ) = 1 ),and anti-aligned in the “–” mode ( cos(∆ ̟ ) = − ). Inan eigenmode, the system behaves as a rigid body withthe shapes and relative orientation of the elliptical orbitsremaining fixed (Figure 2). The frequency of the anti-aligned mode is faster ( g − > g + ) because of closer ap-proaches between the planetary orbits and thus strongerperturbations (Figure 2).It is instructive to consider the small q limit whichcorresponds to a tiny outer mass. In this case, Equa-tions (6) and (7) simplify to ( g s + = σq (1 − β ) , η s + = β )and ( g s − = σ √ α , η s − = −√ α/qβ ). For a true outer testparticle q → and only the first mode is possible. Thisaligned mode is stationary ( g + = 0 ) and, since η s + < ,the planet’s eccentricity exceeds that of the test particle.Similarly, with an inner test particle only the alignedmode survives, it is stationary, and the massive planetagain has the higher eccentricity.Equation (9) gives the ratio between the eccentricitiesin a perfect secular eigenmode. In Figure 3, we plot η s ± as a function of √ α/q for different β values. Althoughrare in real systems, q = √ α (or m a = m a ) makesan interesting special case. When this condition is met,the planets have similar angular momenta and the innerand outer planet orbital precession rates (the diagonalterms of the matrix A) are equal. In addition, η s ± = ± (Equation (7)), and therefore the inner and outer orbitshave the same eccentricity (Equation (9)). When q > √ α ( m a < m a ), the inner planet precesses fastest, hasa lower eccentricity in the aligned mode and a highereccentricity in the anti-aligned mode. The opposite istrue for q < √ α .In general, a system is in a mixed state composed of alinear combination of the two modes: h j = e + exp( iϕ + ) ˆ h j + + e − exp( iϕ − ) ˆ h j − . (11)Here the mode amplitudes e ± and phases ϕ ± are deter-mined by the initial orbits. Plotting the solution Equa-tion (11) on the complex plane yields a phase plot of e cos ̟ versus e sin ̟ as shown in Figure 4a. The e and e vectors in the plot represent ( e , ̟ ) pairs for thetwo orbits at a given time. The length of a vector isthe instantaneous eccentricity, and its polar angle is theinstantaneous longitude of pericenter. Each eccentric-ity vector is a vector sum of an aligned component ( e + and η s + e + are parallel) and an anti-aligned component( e − and η s − e − are antiparallel). The lengths of all ofthese vectors are determined by initial conditions. Ina pure aligned eigenmode, e − = | e − | = 0 and the e and e vectors are parallel. As time progresses, these ec-centricity vectors rotate together at rate g s + while main-taining their lengths. This solution corresponds to thealigned orbits in Figure 2. For the anti-aligned eigen-mode, e + = | e + | = 0 so that e and e are anti-paralleland rotate together at rate g s − . This state is depicted bythe pair of anti-aligned orbits in Figure 2. In the mostgeneral system, both motions occur simultaneously: theparallel eccentricity vectors ( e + and η s + e + ) rotate at rate g s + , while the anti-parallel vectors ( e − and η s − e − ) rotateat rate g s − . The resulting lengths of the eccentricities e and e , thus vary periodically as illustrated in Figure 4b.The maximum value of e occurs when e + is parallel to e − . At the same time, however, η s + e + and η s − e − areanti-parallel to each other, leading to a minimum valuefor e . The simultaneous maximum for e and minimumfor e seen in Figure 4b is a general result guaranteed byangular momentum conservation. The two eccentricities,in mathematical form are as follows: e = q e + e − + 2 e + e − cos( g s − − g s + ) t,e = q ( e + η s + ) + ( e − η s − ) + 2 e + e − η s + η s − cos( g s − − g s + ) t (Murray & Dermott 1999). Both eccentricities oscillateat the same frequency ( g s − − g s + ) as shown in Figure 4b. Secular Modes with Eccentricity Damping
Since a planet close to its host star experiences a dragforce caused by planetary tides raised by the star, we seeka way to include tides into the mathematical formalismof the previous section. For small eccentricities for which the secular solution Equation (11) is valid, tidal changesin a are usually negligible compared with the dampingin e (Goldreich 1963). Since the damping timescales forclose-in planets (in the order of yr) are much longerthan the secular timescales (typically ∼ yr), we treatthe damping effect as a small perturbation to the secularsolution.Stars and planets raise tides on each other, which, inturn, perturb the planet’s orbit. For very close-in plan-ets, the combination of stellar tides raised by the planetand planetary tides raised by the star typically act to de-crease the planet’s orbital period and eccentricity. Tidaldissipation in the star usually leads to orbital decay andcircularization on a timescale much longer than the ageof most planetary systems, while that tides in the planetcan damp its orbital eccentricity rather quickly (e.g.,Goldreich 1963; Burns 1977; Rasio et al. 1996). Thus,we assume that the orbital circularization is largely dom-inated by planetary tides, and approximate the eccen-tricity damping rate as follows (e.g., Murray & Dermott1999): λ = − ˙ ee = 634 1 Q ′ p m ∗ m p (cid:18) R p a (cid:19) n , (12)where Q ′ p ≡ . Q p /k is the modified tidal quality factor, Q p is the tidal quality factor, and k is the Love numberof degree 2.The addition of the constant tidal damping (Equa-tion (12)) adds an extra term to the eccentricity equationin Equation (1), which now reads as follows: ˙ e j = − n j a j e j ∂ R j ∂̟ j − λ j e j , with λ j being the eccentricity damping rate. The coeffi-cient matrix of Equation (5) is now the following: A = σ (cid:26) q + i ξ − qβ −√ αβ √ α + i ξ (cid:27) , (13)where the dimensionless ξ j = λ j /σ ≪ parameterizesthe damping strength.Eccentricity damping causes the eigen-frequencies ofmatrix A to have both real and imaginary parts. Asin other dynamical systems, the real parts ( g ± ) of theeigen-frequencies still represent the precession rates ofthe secular modes, while the imaginary parts ( γ ± ) indi-cate that the amplitudes of the modes change over time.This becomes clearer if we rewrite the two orthogonalspecial solutions of the system as follows: (cid:18) ˆ h ± ˆ h ± (cid:19) = (cid:18) η ± (cid:19) exp( − γ ± t ) exp( ig ± t ) , (14)where η ± is the new eccentricity ratio for each mode.For ξ j ≪ , we solve the new matrix for the complexfrequencies and find the following: g ± = g s ± ± q √ αβ [( q − √ α ) + 4 q √ αβ ] / σ ( ξ − ξ ) , (15) γ ± = 12 " λ + λ ± √ α − q p ( q − √ α ) + 4 q √ αβ ( λ − λ ) . (16)Eccentricity damping increases the precession rate of v cos e s i n e v h + e + e − e + e e h − e − g − t+ j − g − t+ j − g + t+ j + g + t+ j + . . . . e . . . . . e (a) (b) Fig. 4.—
General solution for a two-planet system. (a) The solution (Equation (11)) on a phase plot. The arrows represent theeccentricity vector e exp( iω ) . The total eccentricity of each orbit ( e or e ) is the magnitude of the vector sum of aligned ( + ) and anti-aligned ( − ) components. The two aligned components, e + and η s + e + , rotate (precess) at rate g s + , while the two anti-aligned components( e − and η s − e − ) rotate at rate g s − . (b) Secular evolution of orbital eccentricities from an N-body simulation. Plot shows the eccentricitiesof two planetary orbits in a computer-simulated system consisting of a 1 Solar-mass star, a 1 Jupiter-mass hot-Jupiter at 0.05 AU, and a0.8 Jupiter-mass companion at 0.2 AU. The simulation shows an oscillation period of ∼ yr, in good agreement with the prediction ofthe secular model: π/ ( g − − g + ) = 1670 yr. the aligned mode and decreases that of the anti-alignedmode, but only by an extremely small amount (of or-der ξ j ). These tiny frequency changes were neglectedby Laskar et al. (2012), but otherwise our damping ratesare in perfect agreement. The tiny frequency changesare due to the slightly different orbital configurations inthe secular modes as we shall describe below. The neweigen-vectors of the matrix A are as follows: η ± = η s ± " ± i ξ − ξ p ( q − √ α ) + 4 q √ αβ , (17)where we have neglected terms of second- and higher-order power of ξ j .In general, the η ± ’s are complex with small imaginarycomponents. If we ignore the imaginary parts for themoment, then η ± are real and Equation (14) shows thatthe pericenter angles of the two orbits are the same (pos-itive η + , aligned mode) or ◦ apart (negative η − , anti-aligned mode). For complex η ± , however, the two orbitsare not exactly aligned or anti-aligned any longer. In-stead, ∆ ̟ ± shifts from ◦ and ◦ by a small angle ǫ = tan − ξ − ξ p ( q − √ α ) + 4 q √ αβ ! ≈ ξ − ξ p ( q − √ α ) + 4 q √ αβ . In the “aligned” mode, the new pericenter difference is ∆ ̟ + = ǫ so that the inner exoplanet’s pericenter slightlylags that of the outer exoplanet. The lag is maximizedfor q = √ α , the case with equal eccentricities and equalprecession rates for the two planets. Because of this mis-alignment, the minimum distance between the two orbitsis slightly less than that in the undamped case (see Fig-ure 2). This causes the average interaction between thetwo orbits to be stronger, leading to an increase of theprecession frequency as indicated by Equation (15). Sim-ilarly, ∆ ̟ − = 180 ◦ − ǫ in the “anti-aligned” mode; theslight rotation results in a weaker average interaction anda slightly slower mode-precession rate. The deviation an-gle ǫ is tiny, and the eccentricity ratios in the two modes, | η ± | , are nearly the same as | η s ± | . Thus, we continue touse “aligned” and “anti-aligned” to refer to the two modes.Nevertheless, the small deviation angle is physically im-portant because it is what enables monotonic dampingof the outer planet’s eccentricity.In addition to the slight mis-alignment, each mode am-plitude also damps at the rate given by Equation (16).If only planetary tides contribute to eccentricity damp-ing, Equation (12) shows that the damping rate decreasesrapidly with the planet’s semi-major axis ( λ ∝ a − . ). Inthe absence of secular interactions between the planets,the outer orbit is hardly affected. With this interaction,however, the damping applied to the eccentricity of theinner orbit is partially transmitted to the outer planet,causing a decrease of its eccentricity as well. The damp-ing rates of the two modes are different, unless q = √ α .An interesting result from Equation (16) is that the sumof the two mode-damping rates is equal to the sum of thetwo individual eccentricity damping rates: γ + + γ − = λ + λ . The physical interpretation of this expression is that sec-ular interactions between the planets simply act to redis-tribute where the damping occurs.In Figure 5, we compare our analytical results with nu-merical integration of both the secular equations and thedirect N-body equations, with an artificial eccentricitydamping added only to the inner planet in all cases. Theeccentricity evolution curves of the inner planet are plot-ted for two different cases, and e-folding rates are mea-sured and labeled for all curves. Figure 5a illustrates asystem with q > √ α . The top panel shows results fromsecular equations, and the bottom panel shows the cor-responding N-body simulations. Each panel plots threecurves: (1) the single planet case, in which the eccen-tricity of the inner planet damps at rate λ , (2) a two-planet aligned mode with damping rate γ + , and (3) atwo-planet anti-aligned mode (damping rate γ − ). For q > √ α , Figure 5a, eccentricities damp much faster inthe anti-aligned mode than in the aligned mode, as pre- (a) (b) Fig. 5.—
Eccentricity damping of systems in different secular modes found by integration of the secular equations (top panels) and directN-body simulations (bottom panels). The plots show the eccentricity evolution of a hot-Jupiter (1 Jupiter-mass planet at 0.05 AU froma 1 solar-mass star) with a companion; γ + and γ − represent the mode damping rates, which are e-folding times measured directly fromthe curves. Also plotted is a single hot-Jupiter subject to an artificial eccentricity damping with a rate λ = 7 . × − yr − . (a) A 0.8Jupiter-mass companion is located at 0.2 AU ( q > √ α ), with predicted mode damping rates γ + = 1 . × − yr − , γ − = 6 . × − yr − (Equation (16)). (b) A 0.3 Jupiter-mass companion is located at 0.2 AU ( q < √ α ), with predicted γ + = 6 . × − yr − and γ − = 1 . × − yr − . dicted by Equation (16). A comparison between the topand bottom panel shows close agreement (within 2%) be-tween full-scale N-body simulation and integration of theapproximate secular equations. Damping rates predictedby Equation (16) match the observed secular decay ratesalmost perfectly. Figure 5b shows a system with q < √ α ,for which the aligned mode damps faster than the anti-aligned mode. The faster damping rate of the alignedmode in the N-body simulation is within 0.5% of the pre-diction, but that of the slow anti-aligned mode, however,is ∼ off. This discrepancy might be the result ofunmodeled tidal perturbations to the inner body’s semi-major axis. The smaller m of Figure 5b weakens thesecular interaction, thereby emphasizing these tidal ef-fects.The different damping rates for the two modes are par-ticularly interesting, especially for well-separated nearly-decoupled orbits for which α and β are small. In thiscase, the q √ αβ term under the square root of Equa-tion (16) is much smaller than the other term. As aresult, if the eccentricity damping on one orbit is muchfaster than on the other ( λ ≫ λ ), as in the case of tides,one mode damps rapidly at nearly the single-planet tidalrate λ . The system evolves quickly into a single modewhich decays substantially more slowly.Because of the different damping rates for the twomodes, a system will evolve into a single mode even ifit starts in a mixed state. Figure 6 shows the eccentric-ity and apsidal angle evolution of the systems depicted inFigure 5, but with initial conditions that lead to mixedstates. Figure 6a shows the case of q > √ α . Before4 Myr, the system is in a mixed state, so both eccentric-ities, as well as their ratio, oscillate (cf. Figure 4). Asthe short-lived anti-aligned mode damps away, the orbitsbegin to librate around ∆ ̟ ≈ ◦ , the two eccentricitiesoscillate less and less, and in the end, the eccentricityratio settles to the aligned mode ratio | η + | predicted by Equation (17). Figure 6b shows the corresponding plotsfor q < √ α . The orbital elements undergo similar evo-lution, except that the aligned mode damps quickly andthe system ends up in the anti-aligned mode. Apsidal Circulation and Libration
In Figure 6, the apsidal motion of the two orbitschanges from libration of ∆ ̟ about ◦ or ◦ to circu-lation of ∆ ̟ through a full ◦ , and to libration againduring the eccentricity damping. In order to understandwhat determines the apsidal state of the orbits, we plotthe aligned and anti-aligned components of the eccen-tricities for Figure 6a in the top two panels of Figure 7.Recall that the total eccentricities of the orbits at anytime can be obtained from the components as illustratedin Figure 4. The lower panels in the figure show the phasediagrams of both orbits ( e cos(∆ ̟ ) versus e sin(∆ ̟ ) ) atfive different time indicated in the top two panels. Thetwo orbits move along the phase curves, which them-selves change slowly over time. Two critical instants,labeled s − and s + , are circulation-libration separatrices,which represent the transitions of the apsidal state fromanti-aligned libration to circulation ( s − ), and from circu-lation to aligned libration ( s + ). These two points dividethe evolution curves into three regions.In region I ( t < yr), the anti-aligned componentsare stronger than the aligned ones for both orbits ( e − >e + and | η s − | e − > η s + e + ). In the phase plots, both the e and the e curves are closed and stay on the left sideof the vertical e sin(∆ ̟ ) axis, indicating the libration of ∆ ̟ about ◦ . With the decrease in the amplitudes ofall components, especially the faster damping of the anti-aligned ones, the curves move closer toward the origin,resulting in an increased libration width of ∆ ̟ .The anti-aligned separatrix s − crossing occurs at t =10 yr, when the two components for the outer orbitare equal ( | η s − | e − = η s + e + ) and e might drop to zero, (a) (b) Fig. 6.—
Secular evolution of the same systems shown in Figure 5, but with different initial conditions so that the systems begin inmixed states. (a) For q > √ α , the anti-aligned mode damps quickly at rate γ − from Figure 5a, and the system evolves to the aligned mode( ∆ ̟ ≈ ◦ ). (b) For q < √ α , the aligned mode damps more rapidly and the system evolves to the anti-aligned mode ( ∆ ̟ ≈ ◦ ). resulting in a phase curve for the orbit that is tangent tothe vertical axis at the origin ( s − curve on the inner phaseplot of Figure 7). The phase curve for e at s − is a half-oval whose straight edge includes the origin. Accordingly,when e drops to zero, ∆ ̟ jumps from ◦ to − ◦ forthe largest possible full libration amplitude of ◦ .As the system moves past s − , the phase curves for bothorbits enclose the origin and circulation results. The cir-culation region (II) is located between the two separa-trices (i.e., when yr < t < . × yr), wherethe anti-aligned component of the inner orbit is strongerthan the aligned one ( e − > e + ), while it is weaker for theouter orbit ( | η s − | e − < η s + e + ). With the continuous fastdamping of the anti-aligned mode, the system crosses thealigned separatrix s + at t = 3 . × yr, when the twocomponents for the inner orbit are equal ( e − = e + ). Thetwo separatrices occur at those times when each phasecurve in Figure 7 touches the origin.After s + , both phase curves are to the right of thevertical axis, indicating libration about the aligned mode(region III). The two anti-aligned components are bothsignificantly damped and the system now has both e −
Evolution of the apsidal states during eccentricity damping. The top two panels show the time evolution of the eccentricitycomponents for the system in Figure 6(a) for which m a < m a and the anti-aligned mode damps fastest. The inner orbit has an alignedcomponent e + and an anti-aligned one e − , while η s + e + and | η s − | e − are the components for the outer orbit. Two circulation-librationseparatrices ( s − and s + ) divide the evolution curves into three parts: anti-aligned libration (region I), circulation (region II), and alignedlibration (region III). The bottom panels show phase diagrams of the inner and outer orbits on the complex e exp( i ∆ ̟ ) plane at thecorresponding points indicated in the top panels. The shape of the phase curves depends on the relative strength of the two componentsfor each orbit. Banana shapes result from the large difference between the two components: e − ≪ e + at (a) and | η s − | e − ≪ η s + e + at (c). In conclusion, the apsidal state of a two-planet secularsystem depends on the sign of the simple product: P = ( e + − e − )( η s + e + + η s − e − ) . Libration occurs when the same mode components arestronger for both orbits (
P > ), and circulation occurswhen one mode is stronger for the inner orbit, but weakerfor the outer one ( P < ). This result is in full agree-ment with a slightly more complicated formula given byBarnes & Greenberg (2006b).Eccentricity damping is effective in changing the apsi-dal state of the orbits because the two modes damp atdifferent rates. Eccentricity excitation is equally capableof moving the two orbits across libration-circulation sep-aratrices. This can be easily visualized by running theplots in Figures 7 and 9 backward in time. For systemswith m a < m a (see Figure 7), eccentricity excitationwould eventually bring the orbits into anti-aligned libra- tion (region I), while eccentricity damping brings theminto aligned libration (region III). The opposite is truefor systems with m a > m a (Figure 9). All mecha-nisms that change eccentricities slowly cause planetarysystems to move toward apsidal libration. Additional Apsidal Precessions
In Sections 2.2 and 2.3, we have considered tides asan orbital circularization mechanism. Tides, however,also cause apsidal precession (see, e.g., Ragozzine & Wolf2009): ˙ ̟ T,p = 152 k p (cid:18) Ra (cid:19) m ∗ m n, ˙ ̟ T, ∗ = 152 k ∗ (cid:18) R ∗ a (cid:19) mm ∗ n. v cos e e e s i n e v v cos e s i n e v e e v cos e s i n e v e e DvDv (II)(I) (III) Dv Fig. 8.—
Eccentricity component diagrams for different regionsin Figure 7. These diagrams are similar to Figure 4, but nowshown in a frame rotating at the same rate as the anti-alignedmode so that the horizontal vectors are e − and η − e − . Here, theanti-aligned mode damps faster than the aligned mode ( γ − > γ + )so that the circles move horizontally toward the origin faster thantheir radii shrink. The system starts with ∆ ̟ ≈ ◦ in region I andevolves to ∆ ̟ ≈ ◦ in region III. Here, ˙ ̟ T,p and ˙ ̟ T, ∗ are the tidal precession rates dueto planetary and stellar tides, respectively; k p is theLove number of the planet; and k ∗ is that of the star.These expressions assume that tidal bulges are directlyunderneath the distant body (non-dissipative tides) butcould adjusted to account for slight angular offsets in thetidal bulges (dissipative tides).For close-in exoplanets, orbital precession caused byGR effects is also important. To lowest order in eccen-tricity, the precession rate (e.g., Danby 1988) is the fol-lowing: ˙ ̟ GR = 3 a n c , where c is the speed of light.In addition, the rotational bulges raised on the planetand its star also lead to orbital precession, with respec-tive rates (Ragozzine & Wolf 2009): ˙ ̟ R,p = k p (cid:18) Ra (cid:19) m ∗ m Ω n n, ˙ ̟ R, ∗ = k ∗ (cid:18) R ∗ a (cid:19) Ω ∗ n n. Here, Ω and Ω ∗ are the spin rates of the planet and thestar, respectively.To account for these additional orbital precessions, wedefine a dimensionless quantity: κ = ˙ ̟ T,p + ˙ ̟ GR + ˙ ̟ R,p + ˙ ̟ T, ∗ + ˙ ̟ R, ∗ σ , which is the ratio of the sum of all additional precessionsto the characteristic secular precession. These additionalprecessions add extra terms to Equation (1), which nowreads as follows: ˙ ̟ j = + 1 n j a j e j ∂ R j ∂e j + κσ. Since κ is independent of e for small eccentricities, the ex-tra terms do not change the form of Equation (5), and soall discussion of the general secular modes still holds. In particular, the system still has aligned and anti-alignedmodes and the two modes damp separately. The modefrequencies, damping rates, and eccentricity ratios, how-ever, need to be revised. Now the diagonal terms of thecoefficient matrix A jk should be adjusted to the follow-ing: A = σ (cid:26) q + κ + iξ − qβ −√ αβ √ α (1 + α κ ) + iξ (cid:27) , which gives the new mode frequencies and eccentricityratios: g s ± = 12 σ (cid:26) ( q + κ ) + √ α (1 + α κ ) ∓ q [ q + κ − √ α (1 + α κ )] + 4 q √ αβ (cid:27) , (18) η s ± = q + κ − √ α (1 + α κ ) ± p [ q + κ − √ α (1 + α κ )] + 4 q √ αβ qβ , (19) g ± = g s ± ± q √ αβ { [ q + κ − √ α (1 + α κ )] + 4 q √ αβ } / σ ( ξ − ξ ) , (20) γ ± = 12 " λ + λ ± √ α (1 + α κ ) − ( q + κ ) p [ q + κ − √ α (1 + α κ )] + 4 q √ αβ ( λ − λ ) , (21) η ± = η s ± ( ± i ξ − ξ p [ q + κ − √ α (1 + α κ )] + 4 q √ αβ ) . (22) Note that these equations can be obtained from Equa-tions (6), (7), (15), (16), and (17) with the simple sub-stitution ( q ± √ α ) → [ q + κ ± √ α (1 + α κ )] .These additional apsidal precessions increase both sec-ular rates g s ± (Equation 18) since they cause the orbitsto precess in the same direction as the secular interactiondoes. The mode eccentricity ratio | η s + | (Equation 19) in-creases significantly with any source of precession thatfavors the inner orbit, indicating that it is more difficultto force the eccentricity of a rapidly precessing inner orbitin the aligned mode. The ratio | η s − | , however, decreasesslightly; increasing the precession of the inner orbit inthe anti-aligned mode actually strengthens secular cou-pling. As for the mode damping rates (Equation 21),additional precession decreases the aligned mode damp-ing rate, but increases that of the anti-aligned mode. Italso decreases ǫ , the deviation angle of the mode apsidallines from perfect alignment (Equation 22). APPLICATIONS TO THE OBSERVED SYSTEMS
We now apply the aforementioned theory to close-inexoplanets, beginning with known two-planet systems.We then proceed to systems in which there is linear trendin the star’s radial velocity (RV) that might signal thepresence of an outer companion and finally we considersystems with no hint of a companion.
Two-planet Systems
There are more than 30 multi-planet systems thathost one or more close-in planets with the orbital pe-riod P orb . days. Out of these, many systems in-cluding Gliese 876, 55 Cnc, and υ And have three ormore planets, which makes apsidal analysis more com-plicated (e.g., Barnes & Greenberg 2006a). Moreover,most Kepler-detected planets have unknown eccentric-ities, and are not the best candidates for our analysis.After removing multiple planet systems and those withpoorly determined eccentricities, we are left with eighttwo-planet systems, with which we can test the lineartidal model.0
Fig. 9.—
Evolution of the apsidal state during eccentricity damping. Similar to Figure 7, but using data from Figure 6(b) for which m a > m a . The aligned mode damps fastest and the system moves from aligned libration to circulation, and finally to anti-alignedlibration. First, we compute eccentricity damping timescalesfor the inner planets ( τ e ), and compare τ e with thestellar age ( τ Age ) to determine whether an orbit hashad sufficient time to circularize. For all systems, weadopt conventional planetary and stellar tidal qualityfactors Q p = 10 and Q ∗ = 10 , respectively (see, e.g.,Jackson et al. 2008; Matsumura et al. 2010, and refer-ences therein). We convert these to modified tidal qual-ity factors as Q ′ = 1 . Q/k . For giant planets in oursolar system (Jupiter, Saturn, Uranus, and Neptune),the measured gravitational moments agree well with an n ∼ polytrope (Hubbard 1974; Bobrov et al. 1978)which corresponds to the Love number k = 0 . (Motz1952). For most stars, on the other hand, the n = 3 polytrope is a good approximation (e.g., Horedt 2004),which yields k = 0 . (Motz 1952). We finally ob-tain Q ′ p = 2 . × and Q ′∗ = 5 . × . We esti-mate the eccentricity damping timescale τ e by integrat-ing a set of tidal equations based on the equilibrium tidemodel from the measured planetary eccentricity down to e = 10 − , and further assume that the tidal quality fac- tors evolve proportional to the inverse of the mean mo-tion Q ∝ /n (see, e.g., Matsumura et al. 2010). Someof our results appear in the final column of Table 1. Notethat these simulation results compare favorably to thesimpler approximation of Equation (12) when one prop-erly accounts for the multiple e-folding times needed todamp eccentricities to e = 10 − . Only 2 of the 8 two-planet systems (HD 187123 and HAT-P-13) have shorttidal circularization times compared with the stellar ages;physical and orbital parameters of these two systems canbe found at the top of Table 1. Since τ e << τ Age , thesystems are likely to have been significantly modified bytides.Next, we check the strength of secular interactions forthe two systems. In the HD 187123 system, the innerplanet is at ∼ . AU while the outer one is at ∼ AU.With this configuration, the inner planet is stronglybound to the star and its interaction with the outerplanet is weak. The inner and outer planets of HAT-P-13, however, are more closely spaced at ∼ . and ∼ . AU, respectively. Furthermore, since m >> m TABLE 1Properties of Planets and Stars Discussed in This Paper
Planet m p ( m J ) R p ( R J ) a (AU) e ω ( ◦ ) m ∗ ( m ⊙ ) Age (Gyr) τ e (Gyr)HAT-P-13 b . ± . . ± .
079 0 . ± . . ± . +27 − . +0 . − . +2 . − . ∼ . HAT-P-13 c . ± .
691 1 . ± . . ± . . ± .
35 1 . +0 . − . +2 . − . HD 187123 b . ± . ∗ . ± . . ± . . . +0 . − . . ∼ . HD 187123 c . ± .
152 4 . ± .
367 0 . ± . ± . . +0 . − . . GJ 436 b . ± . . +0 . − . . ± . . ± .
019 351 ± . . +0 . − . +4 − ∼ BD -10 3166 b . ± . ∗ . ± . . +0 . −
334 0 . +0 . − . . ∼ . HAT-P-26 b . ± . . ± .
052 0 . ± . . ± .
06 100 ±
165 0 . ± .
033 9 +3 − . ∼ . WASP-34 b . ± . . +0 . − . . ± . . ± . ± . . ± .
07 6 . +6 . − . ∼ . HD 149143 b . ± . ∗ . ± . . ± .
01 0 ∼ . ± . . ± . ∼ . HAT-P-21 b . ± .
173 1 . ± .
092 0 . ± . . ± .
016 309 ± . ± .
042 10 . ± . ∼ HAT-P-23 b . ± .
122 1 . ± .
09 0 . ± . . ± .
044 120 ±
25 1 . ± .
035 4 ± ∼ . HAT-P-32 b . ± .
169 2 . ± .
099 0 . ± . . ± .
061 50 ±
29 1 . +0 . − . . +1 . − . ∼ . HAT-P-33 b . ± .
117 1 . ± .
29 0 . ± . . ± .
081 100 ±
119 1 . ± .
096 2 . ± . ∼ . HD 88133 b . ± . ∗ . ± . . ± .
072 349 1 . ± . ∼ . References . — Data for HAT-P-13 system are from Winn et al. (2010); HD 187123 from Wright et al. (2009); HAT-P-21 and HAT-P-23from Bakos et al. (2011); HAT-P-26 Hartman et al. (2011a); HAT-P-32 and HAT-P-33 from Hartman et al. (2011b); HD 88133 and BD -10 3166 from Butler et al. (2006); GJ 436 from Maness et al. (2007); WASP-34 from Smalley et al. (2011); HD 149143 from Fischer et al.(2006). The planetary mass m p assumes the lower limit m p sin i for listed planets without measured planetary radius R p . The circularizationtimescale τ e is estimated by integrating a set of tidal equations (e.g., Matsumura et al. 2010) from the tabulated eccentricity to e = 10 − .An asterisk signifies an assumed value. Fig. 10.—
Current apsidal state of the HAT-P-13 system. Weshow the orbits of both planets on the e exp( i ∆ ̟ ) plane, similar tothe lower plots in Figures 7 and 9. The orange curves represent thesolution of secular equations, and the blue curves are obtained froman N-body simulation. The orbits librate about ∆ ̟ = 0 ◦ , with alibration amplitude ∼ ◦ predicted by secular theory and ∼ ◦ measured from the N-body simulation. The narrowness of the outerplanet’s arc in both cases is due to the fact that m >> m . and e = 0 . is large (Table 1), the secular forcing ofthe inner planet is substantial. Thus, out of the eightsystems, only HAT-P-13 has both strong secular interac-tions and a short tidal damping time; it is, accordingly,the best test case for our theory.We now utilize the linear secular theory without tidesdeveloped in Section 2.1, and show that the model pre-dicts the current apsidal state of HAT-P-13 reasonablywell. In Figure 10, we compare the apsidal state of HAT-P-13 estimated by secular theory with that obtained froma direct N-body simulation done by the HNBody code (Rauch & Hamilton 2002). Since HAT-P-13c’s large ec-centricity of ∼ . violates the assumption of the linearsecular theory, it is understandable that the discrepancybetween the two integrations is ∼ (Figure 10). Itappears that the system librates with a large amplitudeabout ∆ ̟ = 0 ◦ so that the system is not far from thealigned separatrix s + (Figure 7). Here, we do not take ac-count of any additional apsidal precessions. By taking ac-count of the GR effect, we find that the results stay sim-ilar — the system librates with a large amplitude. How-ever, by considering all of the additional apsidal preces-sions, the system appears to circulate. Thus, HAT-P-13is likely to be yet another example of a multi-planet sys-tem being near a secular separatrix (Barnes & Greenberg2006b). The near-separatrix state of this system is some-what surprising given that the tidal decay time is a tinyfraction of the stellar age (Table 1) and so the systemshould have long ago damped to the apsidally-lockedstate. Perhaps, given the uncertainty in Q ′ , our esti-mation of τ e is off by a large factor. Alternatively, theremight be a third as-yet-undiscovered planet affecting thissecular system.We next compare the current state of the system withthe expectation from tidal-damping theory as developedin Section 2.2. We assume that the parameters of theinner planet’s orbit are known, assume further that onemode has fully damped away (despite the contrary evi-dence of Figure 10), and proceed to predict parametersof the outer planet. We begin by asking which modeis favored, which depends on the damping rates givenby Equation (16), or Equation (21) when other apsidalprecession effects are important.When a system is locked into one of the secular eigen-modes, the outer and inner orbits have a predictable ec-centricity ratio | η ± | = ( e /e ) ± (see Equations (17) and(22)). In Figure 11, we show contour plots of the pre-dicted eccentricity of the outer planet e for the moreslowly damped eigen-mode in the parameter space ( q , α ). The dashed curve divides the space into a region inwhich the slow anti-aligned mode persists (top left, red2 /m a = a / a . . . . . . . . . . . . . . . . . . HAT-P-13 A n ti - a li g n e d L i b r a ti o n Aligned Libration /m a = a / a . . . . . . . . . . . . . . . . . HAT-P-13 A n ti - a li g n e d L i b r a ti o n Aligned Libration (a) without additional precessions (b) with General Relativity /m a = a / a . . . . . . . . . . . . . . . . . HAT-P-13 A n ti - a li g n e d L i b r a ti o n Aligned Libration /m a = a / a . . . . . . . . . . . . . . . . . HAT-P-13 A n ti - a li g n e d L i b r a ti o n Aligned Libration (c) with GR and planetary tides (d) with all additional precessions
Fig. 11.—
Orbital states of possible companions for HAT-P-13b (a) with no additional apsidal precessions, (b) with GR precession,(c) with GR and planetary tidal precessions, and (d) with different assumptions about the orbital precession terms from Section 2.4. Eachpoint in these q - α plots represents an outer companion with corresponding mass and semi-major axis. The dashed curve divides the planeinto regions in which the aligned mode damps faster (top left) and the anti-aligned mode damps faster (bottom right). Furthermore, inthe shaded regions, either the aligned mode (blue shading) or the anti-aligned mode (red shading) can survive tidal dissipation and lastlonger than the age of the system ( Gyr). Last, the solid contour lines represent the eccentricity of the outer planet assuming that it is inthe long-lived mode. The location of HAT-P-13c is marked by a white symbol with error bars. area) and a region in which the slow aligned mode sur-vives (bottom right, blue area). In these shaded areas,the lifetime of the slower mode is longer than the age ofthe system. Conversely, in the white central area, bothmodes should have already damped away and both or-bits would be circular by now. Given the inner planet’snon-zero e = 0 . (Table 1) and that τ e << τ Age , weexpect the outer planet to be in one of the shaded re-gions. Furthermore, since e > correspond to unboundorbits, these parts of the shaded areas in Figure 11 arealso off limits.The boundaries of these colored areas are determinedby equating the age of the system (5 Gyr for HAT-P-13)to the circularization time τ e in Table 1. An older systemage τ Age and/or faster damping timescale τ e would ex-pand the white area outward away from the dashed line.If tides have not been active over the full stellar age, as is possible for a recent resonance crossing or a planet-planetscattering event, the white area would shrink inward to-ward the dashed line.The effect of the additional apsidal precessions is sub-stantial and can be quantified by comparing panels (a)-(d) in Figure 11. The panels (a)-(d) show the estimatesfrom the linear secular theory (a) without additional pre-cessions, (b) with GR precession, (c) with GR and plan-etary tidal precessions, and (d) with all apsidal preces-sions, respectively. By comparing panel (a) with panels(b) and (c), we find that GR and planetary tidal preces-sions significantly change the q − α plot. Conversely, fromthe comparison of panels (c) and (d), we can tell that theother precessions have negligible effects on HAT-P-13.In Figure 12, we compare the GR precession rate witheach of other precession rates for all of the systems listedin Table 1. For these close-in systems, we find that3 Fig. 12.—
Comparisons of apsidal precession rates for planetarysystems listed in Table 1. Blue and orange symbols represent theratios of rotational and GR precessions ( ̟ R /̟ GR ) and tidal andGR precessions ( ̟ T /̟ GR ), respectively. Circles and triangles cor-respond to precessions due to stellar and planetary deformations,respectively. For all of the close-in systems listed here, either GRor planetary tidal precession dominates the additional apsidal pre-cession. either GR or planetary tidal precession dominates theadditional apsidal precession, while the effects of stellartidal and rotational precessions tend to be much smaller.For HAT-P-13, the precession rate due to tidal deforma-tion of the inner planet is a factor of a few larger thanthe GR precession rate, while the other precessions aremuch smaller than GR.As discussed in Section 2.4, adding extra preces-sions diminishes the anti-aligned area (red shading) dueto faster damping rates but significantly expands thealigned area (blue shading) in accordance with Equa-tion (21). Notice that, as expected, the lower left quad-rant of the plot (i.e., small q and small α ) experiences thegreatest changes from panel (a) to panels (b)-(d). Con-versely, the changes to the shading of the other threequadrants are relatively minor. The dashed lines inFigures 11(a) and (b)-(d) are given by q = √ α and q + κ = √ α (1 + α κ ) , respectively, where κ includesonly the corresponding terms in (b)-(d). These expres-sions simply compare the real diagonal elements of therespective A matrices. The anti-aligned mode dampsmost quickly if the inner planet precesses faster while thealigned mode damps first if the outer planet precessesfaster. Along this dashed line, the difference betweenthe mode precession rates ( g s + − g s − ) is minimized andthe mode damping rates γ ± = ( λ + λ ) / are identical.Last, note that including additional apsidal precessionssignificantly changes the contours for the outer planet’seccentricity. In the region of aligned libration, the ec-centricity contours are moved upward by the inclusion ofthese precessions, indicating that a more eccentric outerplanet is required to maintain the mode, for a given α and q . In the anti-aligned region, the contours move tothe left, indicating that a lower eccentricity on the outerplanet is needed to preserve the mode. The reasons forthese changes were discussed in Section 2.4.The actual location of the outer planet HAT-P-13cis marked in all of the panels with white error bars.The planet resides well within the more slowly dampedaligned mode region, as expected from linear secular the-ory (see also Figure 10). Figures 11(b)-(d) suggest that the eccentricity of the outer planet exceeds one, whilethe observed value is actually . ± . (Table 1).It is clear that e ≥ is an unphysical result, whichmight be attributable to: (1) using linear secular theorydespite large eccentricities, and (2) assuming that onemode dominates despite the evidence from Figure 10.Although the quantitative agreement is not very good,the figure does predict apsidal alignment and a large ec-centricity for the outer planet. We accordingly concludethat the secular perturbations between the two planetsin the HAT-P-13 system might be responsible for thenon-zero eccentricity of the inner planet.This section shows the power and pitfalls of ourmethod. If estimates for the stellar ages and tidal damp-ing timescales are accurate, we can determine whethera given system should currently be near a single eigen-mode. The HAT-P-13 system with a stellar age 40 timeslonger than the estimated damping time should have hadample time to reach such a state, and yet Figure 10 showsthat it has not. Perhaps the time estimates are inaccu-rate or perhaps there was a recent disruptive event inthe system. In either case, this suggests that a certainamount of caution is warranted when proceeding to inves-tigate single-planet systems. With this in mind, in thefollowing two sections, we study single-planet systemswith and without observed linear trends in the stellarradical velocity that might be indicative of a companion.We investigate whether the observed non-zero eccentrici-ties could be explained by unseen potential companions. Single-planet Systems with a Hint of a Companion
As shown in Sections 2.2 and 2.3, a two-planet systemshould have evolved into either an aligned or an anti-aligned apsidally locked state when the tidal dissipationis strong enough. Equations (19) and (21) thus provide asingle constraint on the three parameters of the unknownouter companion: the mass ratio q , the semi-major axisratio α , and the eccentricity ratio of the two planets.Therefore, we can predict a range of possible companionsthat might force a non-zero eccentricity on an observedclose-in planet. We illustrate our method with severalexamples here and in the next section.There are 16 single, close-in planet systems with P orb ≤ days and non-zero eccentricities that have an observedlinear trend in the stellar RV, which indicates the pos-sible existence of a companion on a more distant orbit.For these systems, we can place a unique constraint onthe potential companion. For simplicity, we exclude thetwo systems that have a large projected stellar obliquity(HAT-P-11 and WASP-8), and compare the estimated τ e with the stellar age τ Age for each of the remainingsystems, as described earlier. We find that 5 of 14 sys-tems (GJ 436, BD -10 3166, HAT-P-26, WASP-34, andHD 149143) have τ e < τ Age . We list parameters for thesesystems in the middle section of Table 1. Because theproperties of the putative companions are unknown, wecannot test for a strong secular interaction as in the pre-vious section and so we investigate all five systems.Figure 13(a) is similar to Figure 11(d), but for theplanetary system around GJ 436. The thick, dashedcurve is an upper limit to the outer planet’s massestimated from the observed linear trend. Here, wesimply assume that the minimum mass of a poten-tial outer planet is expressed as m = a a lt /G , where4 /m a = a / a . . . . . . . . . . . GJ436 R V L i m i t: m / s Linear Trend /m a = a / a . . . . . . . . . . . . . . . . . WASP-34 R V L i m i t: m / s Linear Trend (a) (b)
Fig. 13.—
Eccentricities of possible companions for hot Jupiters (a) GJ 436, and (b) WASP-34 including all apsidal precession effectsas in Figure 11(d). In the white area, both modes should damp away within the age of the system of 6 and 6.7 Gyr, respectively. Thecentral stars of these systems each have an observed linear trend in their radial velocities, 1.36 and 55 m s − yr − , respectively, that mightbe indicative of second planets. The thick, dashed curves represent these constraints. For WASP-34, the region to the right of the solidportion of the curve represents where a potential candidate is expected to exist (Smalley et al. 2011). The dotted curves represent the RVobservation limit of − ; only planets to the right of the nearly vertical solid part of this curve are detectable with current technologyin a 1 yr observational period. a lt = 1 . ± . − yr − is an observed linear trend(Maness et al. 2007). The dotted curve indicates anobservation limit for the RV method. The solid por-tion of this curve is plotted as a reference, and it showsthe limit estimated for a 1 yr observation period. Tothe right of this solid curve, a full orbit of a hypothet-ical outer planet is observable within a year. To plotthis, we express the mass of a potential outer planet as m sin i = m ∗ v ∗ sin i/ p Gm ∗ /a , where i is the viewingangle, and assume the RV limit of v ∗ sin i = 1 m s − .The observed planet GJ 436 b is about a Neptune-mass object ( m sin i = 0 . m J ∼ . m N ) whichis . AU away from the central star (orbital period ∼ . days), and has an orbital eccentricity of . ± . (Table 1). If this eccentricity is due to another planet andthe system has damped to an eigen-mode, then eccen-tricity contours in Figure 13(a) show that broad alignedand anti-aligned regions are allowed for a potential com-panion, except for small ( m . . m ∼ . m N )and/or distant ( a & . a ∼ . AU) planets. Fur-thermore, the plotted RV limit indicates that nearly allhypothetical outer planets which could be responsiblefor the high eccentricity of GJ 436 b should be observ-able within a year. The curve representing the maximumlinear trend, however, is far below the e = 1 contour,implying that this potential planet cannot be responsi-ble for the current eccentricity of the observed planet;given its great distance, the secular interactions are sim-ply too weak. This result is consistent with the compari-son of the secular and tidal circularization timescales byMatsumura et al. (2008). Since the system has a tidaldissipation timescale ( τ e ∼ ) comparable to the stel- This long-term trend has not been confirmed by HARPS(Bonfils et al. 2013). lar age ( τ Age ∼ +4 − Gyr ), a non-zero eccentricity of thisplanet might also be explained within the uncertaintiesof the stellar age.Another example is shown in Figure 13(b) for WASP-34, which has an observed planet of m sin i = 0 . m J , a = 0 . AU, and e = 0 . ± . (see Table 1).Again, most of the parameter space of the q − α planeis available for a possible secular companion, except forsmall ( m . . m E ) and/or distant ( a & . AU)planets. The solid portion of the RV limit again indi-cates that such a companion should be observable withina year. The system has an observed linear trend of ± − yr − (Smalley et al. 2011). Since the long-term trend has not reached its maxima or minima, theorbital period of the outer body has to be greater thantwice the RV data baseline. The solid portion of thelinear trend corresponds to this limit of a & . AUand m & . m J (Smalley et al. 2011). A yet-to-be-observed companion should lie in the triangular arearight of the solid part of the linear trend and below theobservable region of the RV Limit. It is clear that such aregion does not have any overlap with the critical e = 1 contour. However, they lie relatively close to each otherso that the uncertainties in both observations and thesecular model could bring them closer to have an over-lap. Our model predicts that the companion planet re-sponsible for both this trend and the eccentricity of theWASP-34 b would have a very high eccentricity.If no companion is present, how do we explain WASP-34? For this system, although the estimated eccentricitydamping time ( τ e ∼
700 Myr ) is short compared withthe stellar age ( τ Age ∼ . +6 . − . Gyr ), the eccentric orbitmodel gives only a slightly better fit than the circular one(Smalley et al. 2011). Thus, the inner orbit might well becircular. Alternatively, if the orbit is truly eccentric, we5would need to assume about an order of magnitude lessefficient tidal dissipation to explain this system. Last,the system could also have undergone some dynamicalevent lately which changed the original eccentricities.The other systems with a linear trend (BD -10 3166,HD 149143, and HAT-P-26) show a similar result toGJ 436 (Figure 13(a)), and thus a potential planet is toofar to force the eccentricity of the inner planet to itscurrent value. It is interesting that the observed eccen-tricities are low and consistent with zero for BD -10 3166and HD 149143 (Butler et al. 2000; Fischer et al. 2006),and poorly constrained for HAT-P-26 (Hartman et al.2011a). BD -10 3166 has a short tidal dissipationtimescale ( τ e ∼
420 Myr ) compared with the stellarage ( τ Age ∼ .
18 Gyr ), so the circular orbit assump-tion makes sense. HD 149143 and HAT-P-26 have rel-atively long dissipation timescales ( τ e ∼ .
75 Gyr and ∼ . , respectively) compared with the stellar ages( τ Age ∼ . ± . and ∼ +3 − . Gyr , respectively).Thus, uncertainties in the age estimates and/or in thetidal dissipation rates could allow close-in planets tomaintain their eccentricities without assistance.In summary, we have found no single-planet systems,where the none-zero eccentricities could be explained byperturbations from hypothetical planets correspondingto the observed linear trends. However, their orbitscould be circular (WASP-34, BD -10 3166, HD 149143,and HAT-P-26), or the eccentric orbit could be ex-plained within the uncertainties in the estimated stellarage (GJ 436).
Single-planet Systems with no Hint of a Companion
Given that there are many single, close-in planet sys-tems without linear trends, we focus on planets whosenon-zero eccentricities are hardest to explain — theclosest-in exoplanets. There are eight single-planet sys-tems with an orbital period P orb ≤ days, eccentricity ≥ . , and a small or unknown stellar obliquity. Compar-ing the tidal timescale to the stellar age, we find that fiveof eight such systems have τ e < τ Age (HAT-P-21, HAT-P-23, HAT-P-32, HAT-P-33, and HD 88133, see Table 1).All of these are hot Jupiter systems. For the remainingthree systems, the stellar ages of KOI-254 and Kepler-15are unknown, while GJ 674 has a very long τ e > Gyrcompared with the stellar age ( ∼ . Gyr).Figure 14(a) is similar to Figure 11(d), but for the plan-etary system around HD 88133. The planet is . AUaway from the central star with an orbital period of . days and an eccentricity of . ± . (see Ta-ble 1). Since the circularization time is estimated tobe very short ( τ e ∼
380 Myr ) compared with the stel-lar age ( τ Age ∼ .
56 Gyr ), a moderately high eccentric-ity of this planet is surprising. Figure 14(a) excludesthe entire anti-aligned region for a potential compan-ion. The figure also excludes most companions withsmall mass ( m . . m J ) and/or long orbital period( a & . AU). This is understandable because theseplanets would have weak secular interactions with theinner planet. Thus, a potential companion is expectedto be in the aligned region, massive, and close to thestar. As the RV observation limit shows, such massiveplanets in the aligned libration region would be observ-able within a year, although no such planet has been found. HD 88133 has a low stellar jitter ( ∼ . − ),and the eccentric orbit assumption works only slightlybetter than the circular one (Fischer et al. 2005). Thus,unless the tidal quality factor for this system is very dif-ferent from what we have assumed here, we argue thatthe true eccentricity of HD 88133 b is actually near zero.Follow-up observations would better constrain the eccen-tricity and the existence or absence of a potential com-panion for this system.The case for HAT-P-21 is shown in Figure 14(b). Theplanet is . AU away from the central star (orbitalperiod ∼ . days), and has an orbital eccentricity of . ± . (Table 1). Figure 14(b) allows a very broadparameter space for a possible secular companion; thewhite zone indicating efficient eccentricity damping isnearly absent. However, this undercuts our assumptionthat the system has had time to damp into a pure eigen-mode and, accordingly, the eccentricity contours are notreliable. If we proceed with the dubious assumption of asingle mode, the figure does not allow most companionswith m . . m J and/or a & . AU. Furthermore,the − RV limit observationally precludes almost anyplanet that can be significantly coupled to HAT-P-21 b.Accordingly, we seek another explanation for the eccen-tricity of this system; it can be naturally explained iftidal dissipation were just slightly less efficient than wehave assumed, since the estimated tidal circularizationtime is relatively long ( τ e ∼ . Gyr) compared with itsstellar age ( τ Age ∼ +3 − . Gyr).The other systems (HAT-P-23, HAT-P-32, and HAT-P-33) have a similar trend to HD 88133 (Figure 14(a)),with the entire anti-aligned region being excluded for apotential companion. Since all of their circularizationtimes are more than 1-2 orders of magnitude shorterthan the estimated stellar ages, the moderately-high ec-centricities ( e > . ) of these planets need to be ex-plained. From figures similar to Figure 14, we find thatpotential companions for these systems tend to be moremassive than the observed planets and thus are likelyto be observable by the RV method. However, no com-panions have been found. It is interesting that all ofthese systems have high stellar jitters (Bakos et al. 2011;Hartman et al. 2011b). Although this could mean thatpotential companion planets are difficult to observe, highjitters also lead to poorly-constrained orbital eccentric-ities. All of these planets can also be fit well with thecircular orbit model. Our model suggests that the circu-lar orbits are probably the most likely solution. Futureobservations that yield a more accurate solution for theeccentricity are needed.For our analysis of exoplanetary systems in this sec-tion, we did not explicitly take account of the effects ofuncertainties in orbital or stellar parameters. In par-ticular, errors in stellar ages and eccentricities are oftenlarge and might change our results significantly. We havetested such effects for all of the systems we discussedin this section, and found that our conclusions will notchange within the currently estimated uncertainties inparameters. Also, the assumption of an apsidal lock thatwe made in Sections 3.2 and 3.3 might be too strong; in-stead, it is possible that not enough time has elapsed forcomplete damping of one secular mode. In this case, fora close-in planet with known mass, semi-major axis, and6 /m a = a / a . . . . . . . . . . . HD88133 /m a = a / a . . . . . . . . . . . . HAT-P-21 (a) (b)
Fig. 14.—
Eccentricities of possible companions for hot Jupiters (a) HD 88133, and (b) HAT-P-21 including all apsidal precession effectsas in Figure 11(d). In the white area, both modes should damp away within the age of the system of 9.56 and . ± . Gyr, respectively.Because the inner HAT-P-21 planet has a long damping time of about 6 Gyr, all eccentricities can be sustained and almost no white areais visible. The dotted curves represent the RV observation limit of − , and the solid portion indicates a 1 yr observing period. Allpotential HD 88133 outer planets reside in the aligned zone are detectable. Aligned and anti-aligned solutions exist for HAT-P-21, andnearly all outer planets are detectable. eccentricity, the constraint on m , a , and e is approxi-mate rather than exact. DISCUSSIONS AND CONCLUSIONS
The eccentric orbits of single, close-in planets are gen-erally circularized on timescales shorter than the stellarages. Close-in planets in multiple-planet systems havemuch longer tidal circularization timescales and thusare able to maintain eccentric orbits for the stellar agesor longer. Given this difference in tidal circularizationtimes, we might expect a difference in the eccentricitydistributions of close-in planets with and without knowncompanions. The eccentricities forced by secular pertur-bations from an outer planet are typically small, however,and so it is perhaps not surprising that no such differencehas yet been observed.In this paper, we have explored the possibility that thenon-zero eccentricities of close-in planets are due to ob-served or hypothetical planetary companions. We haveprovided an intuitive interpretation of a simple secularevolution model of a coplanar two-planet system that in-cludes both the effect of the orbital circularization (Sec-tion 2.2) and apsidal precessions due to GR corrections aswell as tidal and rotational deformations (Section 2.4).We have tested our model by comparing the evolutionof apsidal states and orbital eccentricities with N-bodysimulations, and found that the agreement between themodel and the simulations is very good. We have also ap-plied our model to all of the relevant two-planet systems(Section 3.1), as well as single-planet systems with andwithout a long-term trend in the RV to indicate a pos-sible second planet (Sections 3.2 and 3.3, respectively).The following is a summary of our main results.1. In the lowest-order secular theory, the evolution ofnon-dissipative two-planet systems is described bya linear combination of two modes characterized by pericenter alignment and anti-alignment. Ec-centricity damping slightly shifts the two normalmodes from perfect symmetry, which speeds up theprecession rate of the aligned mode and slows thatof the anti-aligned mode (see Section 2.2).2. Eccentricity damping affects the two modes at dif-ferent rates. Accordingly, the apsidal state ofa two-planet system transitions between librationand circulation, and eventually is locked to eitheran aligned or anti-aligned state (see Section 2.3).The eccentricity of both planets subsequently de-cays at a very slow rate.3. GR, tidal, and rotational effects increase the pre-cession rates of both aligned and anti-alignedmodes. As a result, they decrease the aligned-modedamping rate, and increase the anti-aligned modedamping rate (see Section 2.4).We confirm results of previous studies(e.g., Wu & Goldreich 2002; Mardling 2007;Greenberg & Van Laerhoven 2011) and show thatclose-in planets in multiple-planet systems can maintainnon-zero orbital eccentricities substantially longer thancan single ones. We find, however, that there arecurrently few two- or one-planet systems that showsigns of secular interactions that are strong enough tosignificantly slow tidal circularization. In Section 3.1,we find that only one out of eight systems (HAT-P-13)shows tentative evidence that secular interactions areslowing orbital circularization. In Section 3.2, we applyour model to 14 single-planet systems with a lineartrend and find that 5 of 14 have τ e < τ Age . Our secularmodel predicts that none of their eccentricities is likelyto be affected by hypothetical planets that could causethe long-term linear trends. We have further studied7eight very close-in ( P orb ≤ days), significantly eccentric( e ≥ . ) single-planet systems in Section 3.3. We findfive of eight systems that cannot be explained by asingle-planet orbital circularization with the conven-tional tidal quality factors. Potential companions forall of these systems are massive planets in the apsidallyaligned region and should be observable with currenttechnology. Since all of the host stars have high stellarjitters, it is possible that the planetary eccentricities aresystematically overestimated or that outer planets aremore difficult to observe than we have assumed here.Our model has some limitations and caveats. Wehave adopted the leading-order secular theory for two-planet systems, and in principle our model cannot beapplied to high-eccentricity or high-inclination systems.However, the model predicts the general trend of ap-sidal states fairly well even for a highly eccentric case(see Section 3.1). Moreover, recent observations indi-cate that multiple-planet systems tend to be well-aligned(e.g., Figueira et al. 2012; Fabrycky et al. 2012). Nev-ertheless, it is useful to extend this kind of a studyto higher eccentricities and inclinations and to systemswith more than two planets. Also, we have ignoredthe slow decay in semimajor axis due to tides, whichshould be weaker than eccentricity damping by a fac-tor of e ; this is consistent with the low e assumptionmade by linear secular theory. Nevertheless, previousstudies have shown that the eccentricity of the innerplanet does damp faster than that of the outer planet as a result of inward migration (e.g., Wu & Goldreich2002; Greenberg & Van Laerhoven 2011). Another con-sequence of different damping rates is that the eccen-tricity ratio does not remain constant as the apsidallylocked state evolves. Our expression for the mode-damping rates (Equation (21)) is consistent with thatof (Greenberg & Van Laerhoven 2011, see their Equa-tion 19), in the limit of no migration. The tidal androtational deformations of planets and stars also changeorbital precession rates, and these effects might becomemore important than the GR effect for very close-in plan-ets or rapidly spinning stars. These effects can be easilyadded to our model using the techniques of Section 2.4(e.g., Laskar et al. 2012).Overall, our study indicates that secular interactionsslow down the tidal circularization of the inner planetwhile speeding up that of the outer planet. Our survey oflikely systems available to us in 2012 indicate that secularinteractions might not be a dominant cause for the cur-rently observed hot, eccentric planets. The lack of close-in planets with strong secular interactions might be par-tially explained by inward orbital decay that is acceler-ated by non-zero eccentricities (e.g., Adams & Laughlin2006b). Further research should determine whether thescarcity of compact secular systems will persist. Withimproved statistics and precision measurements of close-in exoplanet eccentricities, it might become possible tofind diagnostic differences in the eccentricity distribu-tions for single- and multiple-planet systems. REFERENCESAdams, F. C., & Laughlin, G. 2006a, ApJ, 649, 992, 992—. 2006b, ApJ, 649, 1004, 1004Bakos, G. Á., Hartman, J., Torres, G., et al. 2011, ApJ, 742, 116,116Barnes, R., & Greenberg, R. 2006a, ApJ, 652, L53, L53—. 2006b, ApJ, 638, 478, 478Batygin, K., Laughlin, G., Meschiari, S., et al. 2009, ApJ, 699, 23,23Batygin, K., & Morbidelli, A. 2013, AJ, 145, 1, 1Beaugé, C., Ferraz-Mello, S., & Michtchenko, T. A. 2003, ApJ,593, 1124, 1124Bobrov, A. M., Vasil’Ev, P. P., Zharkov, V. N., & Trubitsyn,V. P. 1978, Soviet Ast., 22, 489, 489Bonfils, X., Delfosse, X., Udry, S., et al. 2013, A&A, 549, A109,A109Brouwer, D., van Woerkom, A., & Jasper, J. 1950Burns, J. A. 1977, in Planetary Satellites, ed. J. A. Burns(Tuscon, AZ, USA: Univ. of Arizona Press), 113–156Butler, R. P., Marcy, G. W., Fischer, D. A., et al. 1999, ApJ, 526,916, 916Butler, R. P., Vogt, S. S., Marcy, G. W., et al. 2000, ApJ, 545,504, 504Butler, R. P., Wright, J. T., Marcy, G. W., et al. 2006, ApJ, 646,505, 505Chiang, E. I. 2003, ApJ, 584, 465, 465Correia, A. C. M., Boué, G., & Laskar, J. 2012, ApJ, 744, L23,L23Danby, J. M. A. 1988Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2012, ArXive-prints, arXiv:1202.6328Figueira, P., Marmier, M., Boué, G., et al. 2012, A&A, 541, A139,A139Fischer, D. A., Laughlin, G., Butler, P., et al. 2005, ApJ, 620,481, 481Fischer, D. A., Laughlin, G., Marcy, G. W., et al. 2006, ApJ, 637,1094, 1094Goldreich, R. 1963, MNRAS, 126, 257, 257Greenberg, R. 1977, in Planetary Satellites, ed. J. A. Burns(Tuscon, AZ, USA: Univ. of Arizona Press), 157–168 Greenberg, R., & Van Laerhoven, C. 2011, ApJ, 733, 8, 8Hamilton, D. P. 1994, Icarus, 109, 221, 221Hartman, J. D., Bakos, G. Á., Kipping, D. M., et al. 2011a, ApJ,728, 138, 138Hartman, J. D., Bakos, G. Á., Torres, G., et al. 2011b, ApJ, 742,59, 59Horedt, G. P., ed. 2004, Astrophysics and Space Science Library,Vol. 306, Polytropes - Applications in Astrophysics and RelatedFieldsHubbard, W. B. 1974, Icarus, 23, 42, 42Jackson, B., Greenberg, R., & Barnes, R. 2008, ApJ, 678, 1396,1396Ketchum, J. A., Adams, F. C., & Bloch, A. M. 2013, ApJ, 762,71, 71Laskar, J., Boué, G., & Correia, A. C. M. 2012, A&A, 538, A105,A105Lee, M. H. 2004, ApJ, 611, 517, 517Maness, H. L., Marcy, G. W., Ford, E. B., et al. 2007, PASP, 119,90, 90Mardling, R. A. 2007, MNRAS, 382, 1768, 1768Matsumura, S., Peale, S. J., & Rasio, F. A. 2010, ApJ, 725, 1995,1995Matsumura, S., Takeda, G., & Rasio, F. A. 2008, ApJ, 686, L29,L29Motz, L. 1952, ApJ, 115, 562, 562Murray, C. D., & Dermott, S. F. 1999Namouni, F. 2007, in Lecture Notes in Physics, Berlin SpringerVerlag, Vol. 729, Lecture Notes in Physics, Berlin SpringerVerlag, ed. D. Benest, C. Froeschle, & E. Lega, 233Peale, S. J. 1986, in Satellites, ed. J. A. Burns & M. S. Matthews(Tuscon, AZ, USA: Univ. of Arizona Press), 159–223Pont, F., Husnoo, N., Mazeh, T., & Fabrycky, D. 2011, MNRAS,414, 1278, 1278Ragozzine, D., & Wolf, A. S. 2009, ApJ, 698, 1778, 1778Rasio, F. A., Tout, C. A., Lubow, S. H., & Livio, M. 1996, ApJ,470, 1187, 1187Rauch, K. P., & Hamilton, D. P. 2002, in Bulletin of the AmericanAstronomical Society, Vol. 34, Bull. Am. Astron. Soc., 938Shen, Y., & Turner, E. L. 2008, ApJ, 685, 553, 5538