SECULAR EVOLUTION OF HIERARCHICAL PLANETARY SYSTEMS Man Hoi Lee and S. J. Peale

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1 The Astrophysical Journal, 59:0 6, 003 August # 003. The American Astronomical Society. All rights reserved. Printed in U.S.A. SECULAR EVOLUTION OF HIERARCHICAL PLANETARY SYSTEMS Man Hoi Lee and S. J. Peale Department of Physics, University of California, Santa Barbara, CA 9306; mhlee@europa.physics.ucsb.edu, peale@io.physics.ucsb.edu Received 00 December ; accepted 003 April 4 ABSTRACT We investigate the dynamical evolution of coplanar, hierarchical, two-planet systems where the ratio of the orbital semimajor axes ¼ a =a is small. Hierarchical two-planet systems are likely to be ubiquitous among extrasolar planetary systems. We show that the orbital parameters obtained from a multiple-kepler fit to the radial velocity variations of a host star are best interpreted as Jacobi coordinates and that Jacobi coordinates should be used in any analyses of hierarchical planetary systems. An approximate theory that can be applied to coplanar, hierarchical, two-planet systems with a wide range of masses and orbital eccentricities is the octopole-level secular perturbation theory, which is based on an expansion to order 3 and orbit averaging. It reduces the coplanar problem to degree of freedom, with e (or e ) and $ $ as the relevant phase-space variables (where e, are the orbital eccentricities of the inner and outer orbits, respectively, and $ ; are the longitudes of periapse). The octopole equations show that if the ratio of the maximum orbital angular momenta, ¼ L =L ðm =m Þ =, for given semimajor axes is approximately equal to a critical value crit, then libration of $ $ about either 0 or 80 is almost certain, with possibly large amplitude variations of both eccentricities. From a study of the HD and HD 66 systems and their variants using both the octopole theory and direct numerical orbit integrations, we establish that the octopole theory is highly accurate for systems with d0: and reasonably accurate even for systems with as large as 3, provided that is not too close to a significant mean-motion commensurability or above the stability boundary. The HD system is not in a secular resonance, and its $ $ circulates. The HD 66 system is the first extrasolar planetary system found to have $ $ librating about 80. The secular resonance means that the lines of apsides of the two orbits are on average antialigned, although the amplitude of libration of $ $ is large. The libration of $ $ and the large amplitude variations of both eccentricities in the HD 66 system are consistent with the analytic results on systems with crit. The evolution of the HD 66 system with the best-fit orbital parameters and sin i ¼ (iis the inclination of the orbital plane from the plane of the sky) is affected by the close proximity to the : mean-motion commensurability, but small changes in the orbital period of the outer planet within the uncertainty can result in configurations that are not affected by mean-motion commensurabilities. The stability of the HD 66 system requires sin i > 0:3. Subject headings: celestial mechanics planetary systems planets and satellites: general. INTRODUCTION Extrasolar planet searches using high-precision radial velocity observations to measure the reflex motion of the host stars along the line of sight have now yielded over 00 extrasolar planets. The discoveries include 0 multipleplanet systems with either two or, in the cases of And and 55 Cnc, three detected planets. There are also indications that about half of the stars with one known planet are likely to have additional detectable distant companions (Fischer et al. 00). Many of the known multiple-planet systems exhibit interesting dynamics. The two planets about GJ 876 are deep in three orbital resonances at the : mean-motion commensurability (Laughlin & Chambers 00; Lee & Peale 00), and there are possibly two other systems with meanmotion resonances: : for the two planets about HD 8943 (Goździewski & Maciejewski 00) and 3 : for the inner two planets about 55 Cnc (Marcy et al. 00; Lee & Peale 003). The outer two planets of the And system are apparently locked in a secular resonance with the lines of apsides of the two orbits being aligned on average (Rivera & See, e.g., for a continuously updated catalog. See 0 Lissauer 000; Lissauer & Rivera 00; Chiang, Tabachnik, & Tremaine 00), and as we demonstrate in this paper, the two planets of the HD 66 system are also locked in a secular resonance, but with the lines of apsides being antialigned on average. The resonances are often vital in ensuring the long-term stability of the multiple-planet systems. Although the origin, evolution, and stability of the orbital configurations of multiple-planet systems can usually be analyzed with direct numerical integrations of the full equations of motion, a theoretical understanding of the dynamics often requires the development and application of theories with analytic approximations. The approximate theories also allow one to explore the parameter space much more rapidly than direct numerical integrations in the regions where the approximations are valid. Because the orbital eccentricities and inclinations of the planets in the solar system are generally small, while the ratios of the orbital semimajor axes of adjacent planets are generally large (a j =a jþ e0:5, except for the Mars-Jupiter pair), the classical perturbation theory developed for the solar system is based on an expansion of the disturbing functions that describe the mutual gravitational interactions of the planets in powers of the eccentricities and inclina-

2 0 LEE & PEALE Vol. 59 tions. In particular, the Laplace-Lagrange secular solution for the evolution of a two-planet system that is not affected by mean-motion commensurabilities is based on retaining just the secular terms (i.e., those not involving the mean longitudes) in the disturbing functions up to second order in the eccentricities and inclinations (see, e.g., Murray & Dermott 999). Since high orbital eccentricities are common among extrasolar planets and the distribution of the orbital eccentricities of the extrasolar planets in the known multiple-planet systems is similar to that of the single planets (Fischer et al. 003), the classical Laplace- Lagrange secular perturbation theory is not, in general, adequate for describing the secular evolution of the extrasolar planetary systems that are not affected by mean-motion commensurabilities. On the other hand, many of the known multiple-planet systems are hierarchical in the sense that the ratio(s) of the semimajor axes (a /a for the two-planet systems and a /a and/or a /a 3 for the three-planet systems) is small. There are pairs of adjacent planets in the 0 known multipleplanet systems (two of which have three planets). Among the nine pairs that are not known or suspected to be in mean-motion resonances, five have a j =a jþ d0:, and all but one (47 UMa) have a j =a jþ d 3. Thus, a secular perturbation theory for a two-planet system that is based on an expansion in a /a and valid for high eccentricities should provide an accurate description of the secular evolution of many extrasolar planetary systems. It is useful to consider hierarchical two-planet systems in the context of the general hierarchical triple systems in which a third body orbits an inner binary on a much wider orbit. A hierarchical triple system can be treated as two binaries on slowly perturbed Kepler orbits by using Jacobi coordinates, where the position of the secondary of mass m of the inner binary is relative to the primary of mass m 0 and the position of the third body of mass m is relative to the center of mass of m 0 and m. A hierarchical two-planet system is simply a hierarchical triple system with m and m much smaller than m 0.Inx we derive the orbital parameters in Jacobi coordinates obtained by the observers from a two- (or more generally multiple-) Kepler fit to the radial velocity variations of a host star and show that Jacobi coordinates should be used in any analyses of hierarchical (and possibly other types of) planetary systems. Star-centered, or astrocentric, coordinates can introduce significant high-frequency variations in orbital elements that should be nearly constant on orbital timescales. The high-frequency variations can then lead to erroneous sensitivity of the evolution of the orbital elements to the starting epoch. Secular perturbation theories based on an expansion in ¼ a =a have been developed for hierarchical triple systems. The expansion to order, called the quadrupole approximation, was developed by Kozai (96) and Harrington (968). This quadrupole-level secular perturbation theory is successful in explaining the Kozai mechanism, whereby the perturbations between the inner binary and the third body can lead to large variations in the eccentricity of the inner binary and the mutual inclination angle between the inner and outer binaries if the initial mutual inclination is sufficiently high. However, the quadrupole approximation is not adequate for studying hierarchical two-planet systems. Although there is no direct information on the mutual inclination angles in the known hierarchical two-planet systems, the formation of planets from a common disk of materials surrounding the host star makes nearly coplanar orbits the most probable configuration. When the orbits are coplanar, the conservation of total angular momentum of the system and the secularly constant semimajor axes mean that the eccentricities of the two orbits are coupled and oscillate out of phase. The quadrupole term does not contribute to these eccentricity oscillations for coplanar orbits. Marchal (990), Krymolowski & Mazeh (999), and Ford, Kozinsky, & Rasio (000) have extended the approximation to octopole (i.e., 3 ) order. Blaes, Lee, & Socrates (00) have recently applied this octopole-level secular perturbation theory, with a sign error corrected (see footnote 5) and with modifications to include the effects of general relativistic precession and gravitational radiation on the inner binary, to study the dynamical evolution of hierarchical triples of supermassive black holes. Unlike the quadrupole term, the octopole term does produce eccentricity oscillations for coplanar orbits. In this paper we use both the octopole-level secular perturbation theory and direct numerical orbit integrations to investigate the dynamical evolution of coplanar, hierarchical, two-planet systems. The applicability of the octopole theory is limited by its use of an expansion in and an averaging over the inner and outer orbital motions. In particular, the orbit averaging eliminates the effects of meanmotion commensurabilities and the possible development of instabilities. We establish the validity and limits of the octopole theory by comparison with direct numerical orbit integrations. In x 3 we summarize the derivation of the octopole theory, compare the octopole theory to the classical Laplace-Lagrange secular solution, and deduce some useful results from the octopole equations analytically. In x 4 we study the dynamical evolution of the HD and HD 66 systems and their variants. The HD system is not in a secular resonance, and its secular resonance variable $ $ circulates, where $, are the longitudes of periapse of the inner and outer orbits, respectively. For the HD system and a wide variety of systems with 0: (including some for which the octopole theory predicts rather unusual dynamical behaviors), we show that the octopole results are in excellent agreement with the direct integration results. Direct integrations of two-planet systems similar to HD 68443, but with different initial a (and hence different initial ), are used to show that systems with initial above a critical value are generally unstable. As anticipated by the analytic results derived in x 3, the HD 66 system is in a secular resonance with $ $ librating about 80, and it shows large amplitude variations of both eccentricities. We show that the evolution of the HD 66 system with the best-fit orbital parameters and sin i ¼ (i is the inclination of the orbital plane from the plane of the sky) is affected by the proximity to the : mean-motion commensurability, but that small changes in the orbital period of the outer planet within the uncertainty can result in configurations that are not affected by meanmotion commensurabilities. For the latter type of configurations, we show that the octopole results are in reasonably good agreement with the direct integration results, even though (0.3) is quite large. We also consider the effects of varying the inclination i and show that the HD 66 system is unstable if sin id0:3. Our conclusions are summarized in x 5.

3 No., 003 EVOLUTION OF HIERARCHICAL PLANETARY SYSTEMS 03. JACOBI ORBITAL PARAMETERS FROM MULTIPLE-KEPLER FITS TO RADIAL VELOCITY OBSERVATIONS Except for systems such as GJ 876, where the perturbations between the two planets are significant on orbital timescales and a dynamical fit to the stellar radial velocity variations is essential (Laughlin & Chambers 00; Rivera & Lissauer 00; Nauenberg 00), it is often adequate to fit the radial velocity variations of a star with two (or more) planets over the time span of the available observations by assuming that the planets are on unperturbed Kepler orbits. Many authors have assumed that the orbital parameters obtained from the multiple-kepler fits are in astrocentric coordinates, but Lissauer & Rivera (00; see also the note added in proof of Rivera & Lissauer 000) have pointed out that Jacobi coordinates better emulate the assumption of unperturbed Kepler orbits. In this section we derive explicitly the orbital parameters in Jacobi coordinates obtained from a two- Kepler fit. It is straightforward to generalize the derivation to an N-Kepler fit. We show that especially for hierarchical systems such as HD 68443, the use of astrocentric coordinates can introduce erroneous features in the evolution of the orbital elements. Let us consider a system consisting of a central star of mass m 0, an inner planet of mass m, and an outer planet of mass m and use Jacobi coordinates, with r being the position of m relative to m 0 and r being the position of m relative to the center of mass of m 0 and m. We shall refer to the orbit of m relative to m 0 as the inner orbit and the orbit of m relative to the center of mass of m 0 and m as the outer orbit. If the inner orbit is an unperturbed Kepler orbit, the lineof-sight (LOS) component of the velocity of m relative to m 0 is h i V ;r ¼ _r sinð! þ f Þþr f _ cosð! þ f Þ sin i ¼ a qffiffiffiffiffiffiffiffiffiffiffiffi ½cosð! þ f Þþe cos! Šsin i ; ðþ P e where a dot over a symbol denotes d/dt and a, e, i,!, f, and a 3= P ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gðm 0 þ m Þ are, respectively, the semimajor axis, eccentricity, inclination, argument of periapse, true anomaly, and period of the inner orbit. In equation () G is the gravitational constant. Note that the reference plane is the plane of the sky, and the planet approaches the observer at the ascending node, but the radial velocity of an approaching planet is negative. Then the LOS component of the velocity of m 0 relative to the center of mass of m 0 and m is V0;r 0 ¼ m V ;r ¼ K ½cosð! þ f Þþe cos! Š ; m 0 þ m ð3þ where the amplitude K ¼ G =3 m sin i q ffiffiffiffiffiffiffiffiffiffiffiffi P ðm 0 þ m Þ =3 e : ð4þ ðþ Similarly, if the outer orbit is an unperturbed Kepler orbit, the LOS component of the velocity of m relative to the center of mass of m 0 and m is V ;r ¼ a qffiffiffiffiffiffiffiffiffiffiffiffi ½cosð! þ f Þþe cos! Šsin i ; ð5þ P e where a, e, i,!, f, and a 3= P ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gðm 0 þ m þ m Þ are, respectively, the semimajor axis, eccentricity, inclination, argument of periapse, true anomaly, and period of the outer orbit. Then the LOS component of the velocity of the center of mass of m 0 and m relative to the center of mass of the whole system is V0;r 0 ¼ m V ;r ¼ K ½cosð! þ f Þþe cos! Š ; m 0 þ m þ m where the amplitude K ¼ G =3 m sin i q ffiffiffiffiffiffiffiffiffiffiffiffi : ð8þ P ðm 0 þ m þ m Þ =3 e Thus, if the orbits of the planets in Jacobi coordinates are unperturbed Kepler orbits, the radial velocity of the star m 0 is V r ¼ V0;r 0 þ V 0;r 0 þ V ¼ K ½cosð! þ f Þþe cos! Š þ K ½cosð! þ f Þþe cos! ŠþV ; ð9þ where V is the LOS velocity of the center of mass of the whole system relative to the observer. Equation (9) is exactly the formula used by observers in two-kepler fits, but with the amplitudes defined in equations (4) and (8) and the orbital periods defined in equations () and (6). 3 Since the true anomaly f j depends on P j, e j, and the time of periapse passage T peri,j, a two-kepler fit directly yields five parameters, P j, K j, e j,! j, and T peri,j, for each orbit. Table shows these parameters for the HD planets from Marcy et al. (00) and the HD 66 planets from D. A. Fischer (00, private communication). If the stellar mass m 0 is known, equations (), (4), (6), and (8) can be used to derive m j and a j for assumed sin i j. (A common but less accurate practice is to derive m j sin i j and a j by assuming that m and m are negligible compared to m 0.) Table also shows m j (in units of Jupiter mass, M J ) and a j for sin i j ¼. Throughout this paper, we refer to the substellar companions of HD as planets, but it should be noted that their sin i j ¼ (i.e., minimum) masses are about 7.7M J and 7M J, which are near or above the deuterium-burning limit. The alternative assumption that the orbits of the planets in astrocentric coordinates are unperturbed Kepler orbits would also yield equation (9) for the radial velocity of the 3 Note that the planetary masses in the equations for the amplitudes and theporbital periods are the physical masses m j and not the Jacobi masses m j j k¼0 m k= P j k¼0 m k, as stated by Lissauer & Rivera (00). ð6þ ð7þ

4 04 LEE & PEALE Vol. 59 TABLE Orbital Parameters of the HD and HD 66 Planets HD HD 66 Parameter Inner Outer Inner Outer P (days) K (m s ) e ! (deg) T peri (JD)...,450,047.58,450,50.6,449,943.7,449,673.9 m (M J ) a (AU) Note. The parameters P, K, e,!, and T peri are from two-kepler fits by Marcy et al. 00 and D. A. Fischer 00, private communication. The parameters m and a are derived for sin i ¼ and adopted stellar masses of.0 and.07 M for HD and HD 66, respectively. host star, but with K j ¼½Gðm 0 þ m j Þ=P j Š =3 m j sin i j ðm 0 þ m þ m Þ ð e j Þ = and P j ¼ a 3= j ½Gðm 0 þ m j ÞŠ =. However, as we show next, while the Jacobi orbits of the planets of a hierarchical system are nearly Keplerian on orbital timescales, the astrocentric orbit of the outer planet can deviate significantly from a Kepler orbit on orbital timescales. Figure a shows the variations in the semimajor axes and eccentricities of the HD planets if we assume that the orbital parameters obtained from the two-kepler fit are in astrocentric coordinates and plot the astrocentric a j and e j. The planets are assumed to be on coplanar orbits with sin i ¼ (note that coplanar orbits have the same inclinations, i ¼ i ¼ i, and the same longitudes of ascending node, ¼, referenced to the plane of the sky). The solid and dotted lines are from direct numerical orbit integrations starting at T peri, and T peri, of Table, respectively. The direct integrations were performed using the Wisdom- Holman (99) integrator contained in the SWIFT 4 software package, with input and output in astrocentric 4 See Fig. a Fig. b Fig.. (a) Variations in the orbital semimajor axes, a and a, and eccentricities, e and e, of the HD planets with sin i ¼, if we assume that the best-fit orbital parameters obtained from the two-kepler fit are in astrocentric coordinates and plot the astrocentric a j and e j. The solid and dotted lines are from direct numerical orbit integrations starting at T peri, and T peri, of Table, respectively. (b) Same as (a), but we interpret the best-fit orbital parameters obtained from the two-kepler fit as orbital parameters in Jacobi coordinates and plot the Jacobi a j and e j. The solid and dotted lines are almost indistinguishable. The dashed lines in the lower two panels of (b) are from the octopole-level secular perturbation theory.

5 No., 003 EVOLUTION OF HIERARCHICAL PLANETARY SYSTEMS 05 coordinates. Figure b is similar to Figure a, but it shows the variations in the Jacobi a j and e j, with the initial orbital parameters also in Jacobi coordinates. The direct integrations were performed using the modified Wisdom-Holman integrator described in x 4. In astrocentric coordinates (Fig. a), the evolution of the orbital elements is sensitive to the starting epoch, and while a is nearly constant and e oscillates only on the secular timescale, both a and e show significant fluctuations on the orbital timescales (note that we have reduced overcrowding in Fig. a by using a sampling interval of 50 yr, which is long compared to the orbital timescales). Plots of the variations in the orbital elements of the HD planets in Marcy et al. (00), Nagasawa, Lin, & Ida (003), and Udry, Mayor, & Queloz (003, from calculations by W. Benz) show similar behaviors and are probably due to the use of astrocentric coordinates. In contrast, in Jacobi coordinates (Fig. b), the evolution is not sensitive to the starting epoch (the solid and dotted lines in Fig. b are almost indistinguishable); both a and a are nearly constant, while both e and e oscillate only on the secular timescale, with the maximum in e coinciding with the minimum in e and vice versa. As we shall see, these behaviors can be understood with the octopole-level secular perturbation theory for coplanar, hierarchical, two-planet systems, which is also based on Jacobi coordinates. From the facts that the Jacobi a j are nearly constant and that the Jacobi e j oscillate only on the secular timescale for hierarchical systems, it is easy to estimate the fractional fluctuation in the astrocentric a, which is primarily due to the velocity of the star m 0 relative to the center of mass of m 0 and m. It is approximately 4ðm =m 0 Þ½ða =a Þð þ e ;max Þ=ð e ;max ÞŠ = when the Jacobi e is at its maximum e,max (and e is at its minimum, which is assumed to be small and is neglected). The fluctuations in the astrocentric a in Figure a are in good agreement with this estimate. Note that the fractional fluctuation is larger for smaller a /a, i.e., for systems that are more hierarchical. In their study of the three-planet And system, Rivera & Lissauer (000) have also reported that the use of Jacobi coordinates eliminates the high-frequency variations in the semimajor axes and eccentricities of the outer two planets and reduces the sensitivity of the evolution to the initial epoch. It is clear that Jacobi coordinates should be used in the analysis of hierarchical systems such as HD (where a =a 0:0) and And (where a =a 0:07 and a =a 3 0:33). It is likely that Jacobi coordinates should also be used in the analysis of other types of planetary systems, especially those for which multiple-kepler fits are adequate. 3. OCTOPOLE-LEVEL SECULAR PERTURBATION THEORY As we mentioned in x, an approximate theory that describes the secular evolution of coplanar, hierarchical, two-planet systems in Jacobi coordinates, such as that shown in Figure b for the HD system, is the octopole-level secular perturbation theory. In this section we summarize the derivation of this octopole theory, compare it to the classical Laplace-Lagrange secular perturbation theory, and deduce some results from the octopole equations analytically. 3.. Equations As in x, we consider a system consisting of a central star of mass m 0, an inner planet of mass m, and an outer planet of mass m and use Jacobi coordinates, with r being the position of m relative to m 0 and r being the position of m relative to the center of mass of m 0 and m. The Hamiltonian of this system is H ¼ Gm 0m Gðm 0 þ m Þm a a Gm 0 m Gm m ; ð0þ r 0 r r r where a j is the osculating semimajor axis of the jth orbit (with j ¼ and for the inner and outer orbits, respectively) and r k is the distance between m k and m. With r < r, both /r 0 and /r can be expanded in powers of r /r, leading to the expanded Hamiltonian H ¼ Gm 0m Gðm 0 þ m Þm a a G X k r k a kþ M k P k ðcos Þ ; ðþ a a r k¼ where ¼ a =a, M k ¼ m 0 m m m k 0 ð m Þ k ðm 0 þ m Þ k ; ðþ P k is the Legendre polynomial of degree k, and is the angle between r and r. The first two terms in equations (0) and () represent the independent Kepler motions of the inner and outer orbits, while the remaining terms represent the perturbations to the Kepler motions. For hierarchical systems with r 5 r, the Jacobi decomposition leads to two slowly perturbed Kepler orbits, even if m and m are not much smaller than m 0. In the general case in which mutually inclined orbits are allowed, it is convenient to use the Delaunay variables with the invariable plane as the reference plane. The coordinates are the mean anomalies l j, the arguments of periapse g j ¼! j, and the longitudes of ascending node h j ¼ j,and the conjugate momenta are L ¼ m 0m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gðm 0 þ m Þa ; ð3þ m 0 þ m L ¼ ðm 0 þ m Þm m 0 þ m þ m G j ¼ L j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gðm 0 þ m þ m Þa ; ð4þ qffiffiffiffiffiffiffiffiffiffiffiffi e j ; ð5þ H j ¼ G j cos i j ; ð6þ where e j and i j are the eccentricities and inclinations of the orbits. The momenta L j, G j, and H j are, respectively, the magnitude of the maximum possible angular momentum (if the orbit were circular), the magnitude of the angular momentum, and the z-component of the angular momentum of the jth orbit. By expressing cos ¼ r x r =ðr r Þ in terms of the Delaunay variables, it is easy to show that the Hamiltonian contains h and h only in the combination

6 06 LEE & PEALE Vol. 59 h h,andh h ¼ 80 if the invariable plane is used as the reference plane. Constant h h and constant total angular momentum H þ H mean that h j and H j can be eliminated from the problem usually referred to as the elimination of nodes and the system is reduced from 6 to 4 degrees of freedom (see, e.g., Marchal 990). Kozai (96) studied the secular evolution of hierarchical triple systems with m 5 m 5 m 0 and e ¼ 0 by retaining just the lowest order ( ) quadrupole term in the series in equation () and averaging the Hamiltonian over the inner and outer orbital motions. Harrington (968) generalized the quadrupole analysis to general hierarchical triple systems. Marchal (990), Krymolowski & Mazeh (999), and Ford et al. (000) extended the approximation by retaining also the octopole term of the order of 3 in the series in equation () and derived an octopole-level, orbit-averaged Hamiltonian. The approach used in these studies to average the Hamiltonian is the von Zeipel method, which involves the determination of a canonical transformation such that the transformed Hamiltonian is independent of the transformed mean anomalies l and l and such that the original and transformed variables differ only by highfrequency terms that are multiplied by powers of the small parameter. Since the transformed Hamiltonian is independent of the transformed l j, the transformed L j and the corresponding semimajor axes are constant. In this paper we focus on hierarchical two-planet systems with coplanar orbits, since the formation of planets from a common disk of materials surrounding the host star makes nearly coplanar orbits the most probable configuration. In the limit of coplanar orbits (i ¼ i ¼ 0 ), the Delaunay variables described above are not well defined, since there are no ascending nodes with respect to the invariable plane, but the longitudes of periapse $ j are well defined. Thus, for coplanar orbits, it is convenient to use the canonical variables l j, $ j, L j,andg j. In the limit of coplanar orbits, by noting that the expression for the angle between the directions of periapse reduces to cos ¼ cosðg g Þ¼ cosðg g þ 80 Þ¼cosð$ $ Þ (since h h ¼ 80 ), the doubly averaged octopole-level Hamiltonian derived by Marchal (990), Krymolowski & Mazeh (999), and Ford et al. (000) can be written as 5 H oct ¼ G m 3 0 m3 ðm 0 þ m ÞL G ðm 0 þ m Þ 3 m 3 ðm 0 þ m þ m ÞL C ð þ 3e ÞþC 3 e e ð4 þ 3e Þ cosð$ $ Þ ; ð7þ where C ¼ G ðm 0 þ m Þ 7 m 7 L 4 6 ðm 0 þ m þ m Þ 3 ðm 0 m Þ 3 L 3 G3 ; ð8þ 5 There is a sign error common to both Krymolowski & Mazeh (999) and Ford et al. (000). We follow Ford et al. and denote the coefficients of the quadrupole and octopole terms by C and C 3, respectively. Krymolowski & Mazeh denote these coefficients by C and C, respectively. In the averaged Hamiltonian and the subsequent equations of motion of Ford et al., the coefficient C 3 should be replaced by C 3. Similarly, in the averaged Hamiltonian and the subsequent equations of motion of Krymolowski & Mazeh, the coefficient C should be replaced by C. C 3 ¼ 5 G ðm 0 þ m Þ 9 m 9 ðm 0 m Þ L 6 64 ðm 0 þ m þ m Þ 4 ðm 0 m Þ 5 L 3 ; ð9þ G5 and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e j ¼ G j : ð0þ L j As in Ford et al. (000), we do not include in equation (7) terms of the order of 7/ induced by the canonical transformation of the von Zeipel method, which were included partially by Marchal (990) and fully by Krymolowski & Mazeh (999). As we mentioned above, the orbit averaging eliminates l j from H oct, and hence L j (and a j ) is constant. Furthermore, H oct contains $ and $ only in the combination $ $, and choosing this difference as a new variable in a canonical transformation simply reveals that the total angular momentum G þ G is an integral of motion. Thus, in the octopole approximation, the coplanar problem is reduced to degree of freedom, with the last two terms in equation (7) as an integral of motion. We can also see from equation (7) that the quadrupole approximation is not adequate for studying coplanar systems. If we drop the octopole term with coefficient C 3 in equation (7), the Hamiltonian is independent of $ and $, meaning that both e and e are constant, which is not generally true. (For mutually inclined orbits, only e is constant in the quadrupole approximation; Harrington 968.) From the equations of motion d$ j H oct dg j and j dt H oct ; j we obtain the following equations for the variation of e j and $ j : de dt ¼ A ð e e Þ= þ 3e =4 sinð$ $ Þ ; ðþ ð e Þ5= de dt ¼A þ 3e e =4 sinð$ $ Þ ; ð3þ ð e Þ d$ ð e ¼A Þ= dt ð e A e ð e Þ= þ 9e =4 Þ3= e ð e Þ5= d$ dt cosð$ $ Þ ; þ 3e ¼A = e ð þ 4e A Þ þ 3e =4 ð e Þ e ð e Þ3 cosð$ $ Þ ; where the constant coefficients A ¼ 4C 3 G 5 ¼ 5 L L 6 n m m 0 m 4 ; m 0 þ m m 0 þ m A ¼ 4C 3 G 5 ¼ 5 L L 6 n m 0 m m 0 m ðm 0 þ m Þ 3 ; m 0 þ m A ¼ C G 3 ¼ 3 L L 4 n m 3 ; m 0 þ m A ¼ C G 3 ¼ 3 L L 4 n m 0 m ðm 0 þ m Þ ; ð4þ ð5þ ð6aþ ð6bþ ð6cþ ð6dþ

7 No., 003 EVOLUTION OF HIERARCHICAL PLANETARY SYSTEMS 07 with ¼ a =a and the mean motions n ¼ Gðm 0 þ m Þ=a 3 = ; n ¼ Gðm 0 þ m þ m Þ=a 3 = : ð7þ Consistent with the fact that the system can be reduced to degree of freedom, either equation () or (3) is redundant, since e and e are related by the conservation of total angular momentum, and equations (4) and (5) can be combined as dð$ $ Þ dt " # ð e ¼A Þ= ð e L þ 3=e Þ3= L ð e " Þ e ð e A Þ = þ 9=4e e ð e Þ5= L e ð þ 4e Þ þ # 3=4e L e ð e Þ3 cosð$ $ Þ : ð8þ If we relax the assumption of coplanar orbits and allow a small mutual inclination angle i mu between the inner and outer orbits, the number of degrees of freedom is increased from to, but an expansion of the orbitaveraged octopole-level Hamiltonian of Marchal (990), Krymolowski & Mazeh (999), and Ford et al. (000) in powers of i mu shows that equations () (5) are only modified by terms of the order of i mu and higher. One of the additional terms for de /dt (which has only an octopole term in the coplanar limit; see eq. []) is a quadrupole term with coefficient C i mu. Thus, we expect a small mutual inclination angle i mu 5 = to have little effect on the secular evolution of e j and $ j of a hierarchical system. This is confirmed by direct numerical orbit integrations in x 4. It is important to note that the octopole-level secular perturbation theory is derived under the assumptions that there are no mean-motion commensurabilities and that (or more precisely r /r )5. As we shall see in x 4, close proximity to even a rather high order commensurability (e.g., : ) can affect the evolution of a system. We shall also see in x 4 that although there is a limit to how large can be for a system to be stable, the octopole theory can provide a reasonable description of the secular evolution for as large as 3 if the planets are not too massive. 3.. Comparison with Classical Laplace-Lagrange Secular Perturbation Theory As we mentioned in x, the classical secular perturbation theory developed for the solar system is based on an expansion in the eccentricities and inclinations. While it is valid to all orders in the ratio of the semimajor axes, it can be applied only to systems with small planetary masses on nearly circular and nearly coplanar orbits. In contrast, the octopole-level secular perturbation theory is based on an expansion in, and it can be applied to hierarchical systems with a wide range of masses, eccentricities, and, in its most general form, inclinations. We now show that the octopole equations () (5) agree with the equations for the classical Laplace-Lagrange secular solution for coplanar systems where the planetary masses, eccentricities, and are all small. To the lowest order in the eccentricities, equations () (5) reduce to where de dt ¼ A e sinð$ $ Þ ; de dt ¼A e sinð$ $ Þ ; d$ e ¼A A cosð$ $ Þ ; dt e d$ e ¼A A cosð$ $ Þ ; dt e A ¼ð5=6Þðm =m 0 Þn 4 ; A ¼ð5=6Þðm =m 0 Þn 3 ; A ¼ 3 4 ðm =m 0 Þn 3 ; and A ¼ 3 4 ðm =m 0 Þn ð9þ ð30þ ð3þ ð3þ in the limit m ; m 5 m 0. The classical Laplace-Lagrange secular solution is based on retaining just the secular terms in the disturbing functions up to second order in the eccentricities and inclinations. For a coplanar planetary system, the equations for the variation of e j and $ j (see, e.g., Murray & Dermott 999) are of the same form as equations (9) (3), but with the A jk replaced by m A 0 ¼ 4 n b ðþ m 0 þ m 3=ðÞ ; ð33aþ A 0 ¼ 4 n m b ðþ m 0 þ m 3=ðÞ ; ð33bþ A 0 ¼ 4 n m b ðþ m 0 þ m 3=ðÞ ; ð33cþ A 0 ¼ 4 n m b ðþ m 0 þ m 3=ðÞ ; ð33dþ where b ðjþ 3=ðÞ is the Laplace coefficient. (It should be noted that the classical theory uses astrocentric coordinates instead of Jacobi coordinates, but that the distinction between the two sets of coordinates vanishes in the limit m =m 0 5 = =4; see x.) Since b ðþ 3= ðþ ¼ð5=4Þ and b ðþ 3=ðÞ ¼3 to the lowest order in, the classical and octopole secular perturbation equations are identical in the limit of small planetary masses, eccentricities, and Some Analytic Results Although it does not appear that the octopole-level secular perturbation equations can be solved analytically, some useful results can be deduced analytically. We begin by rewriting equations (), (3), and (8) as de d ¼ e ð e Þ= þ 3=4e sinð$ $ Þ ; ð34þ ð e Þ5=

8 08 LEE & PEALE Vol. 59 de d ¼ e þ 3=4e sinð$ $ Þ ; ð35þ ð e " Þ dð$ $ Þ ¼ ð e Þ= d ð e þ # 3=e Þ3= ð e " Þ e ð e Þ = þ 9=4e e ð e Þ5= e ð þ 4e Þ þ # 3=4e e ð e Þ3 cosð$ $ Þ ; ð36þ where ¼ A ¼ A ¼ 5 m 0 m ; A A 4 m 0 þ m ð37þ ¼ A ¼ A ¼ L ; A A L ð38þ ¼ t=t e ; ð39þ and t e ¼ ¼ 4 m 0 þ m A 3 3 : m n ð40þ Recall that either e or e can be eliminated from the above equations by the conservation of total angular momentum, which can be written in dimensionless form as ¼ðG þ G Þ=ðL þ L Þ ¼½ð e Þ= þð e Þ= Š=ð þ Þ ¼constant : We can see from equations (34) (36) that coplanar, hierarchical systems with the same and (which are constant) and the same initial e, e, and $ $ have the same trajectory in the phase-space diagram of e (or e ) versus $ $ and can differ only in the period of eccentricity oscillations. In particular, in the limit m ; m 5 m 0, since ð5=4þ, ðm =m Þ =, and t e 4ðm 0 =m Þ=ð3 3 n Þ, systems with the same, m /m, and initial e, e, and $ $, but different m 0, m, and/or a, differ only in the period of eccentricity oscillations, which is proportional to ðm 0 =m Þ=n. As we discussed in x, a two-kepler fit yields e j and! j, and in the limit m ; m 5 m 0, m j sin i j, and a j, but not the inclinations i j of the orbital planes to the plane of the sky. (Hereafter, all inclinations are relative to the plane of the sky and not to the invariable plane.) If we assume that the orbits are coplanar, the above results mean that the trajectory in e (or e ) versus $ $ and the amplitudes of eccentricity oscillations should be independent of the unknown inclination i of both orbits and that the period of eccentricity oscillations should be proportional to sin i. The phase-space structure of the secular evolution of coplanar, hierarchical, two-planet systems can be understood by plotting the trajectories of systems with the same,, and in a diagram of e (or e ) versus $ $ (see, e.g., Figs. and 6 below). The fixed points in the phasespace diagram can be found by solving de =d ¼ de =d ¼ dð$ $ Þ=d ¼ 0. It is clear from equations (34) and (35) that de =d ¼ de =d ¼ 0 requires sinð$ $ Þ¼0, or $ $ ¼ 0 or 80. As we shall see, there are usually two libration islands, one about a fixed point at $ $ ¼ 0 and another about a fixed point at $ $ ¼ 80 ; additional fixed points are possible if the total angular momentum is low. At $ $ 90 and 70, since cosð$ $ Þ0, equation (36) reduces to dð$ $ Þ=d ð e Þ = =ð e Þ 3= ð þ 3e =Þ=ð e Þ : Therefore, to the lowest order in the eccentricities, if, dð$ $ Þ=d 0 and $ $ should be nearly constant, while both e and e change. According to equations (34) and (35), e should be decreasing (from near its maximum possible value for the given total angular momentum to near zero) and e should be increasing (from near zero to near its maximum possible value for the given total angular momentum) at $ $ 90, where sinð$ $ Þ, and vice versa at $ $ 70, where sinð$ $ Þ. These behaviors are most consistent with a phase space that is dominated by two large libration islands, one about a fixed point at $ $ ¼ 0 and another about a fixed point at $ $ ¼ 80. Note also that for a given total angular momentum the maximum possible values of e and e are comparable if. Finite eccentricities change the condition for dð$ $ Þ=d 0at$ $ 90 and 70 to ð e Þ= ð e Þ= = ð þ 3e =Þ, which is slightly less than for moderate eccentricities. To estimate how much smaller than the critical value of is for a given dimensionless total angular momentum, we substitute into the above condition the value of the eccentricities when they are equal [e ¼ e ¼ð Þ = ] and obtain crit ¼ =ð5 3 Þ. Note that crit ¼ is recovered for ¼. Therefore, large libration islands and large amplitude variations of both e j are likely if crit ¼ =ð5 3 Þ. 4. NUMERICAL RESULTS In this section we study the dynamical evolution of the HD and HD 66 systems and their variants and demonstrate the validity and limits of the octopole-level secular perturbation theory by comparison with direct numerical orbit integrations. Except for the direct integrations discussed in x 4.3, the planets are assumed to be on coplanar orbits. Since the octopole-level secular perturbation equations do not involve high-frequency terms, numerical integrations of these equations are rapid. We integrated equations (35) and (36), with e found from conservation of total angular momentum, using a Bulirsch-Stoer integrator. The direct numerical orbit integrations were performed using a modified version of the Wisdom-Holman (99) integrator contained in the SWIFT software package. In addition to changing the input and output to Jacobi orbital elements, we divide the Hamiltonian into a part that describes the Kepler motions of the inner and outer orbits and a part that describes the perturbations to the Kepler motions using equation (0) instead of the division used by Wisdom & Holman (99), which moves the term Gm m / r from the perturbation Hamiltonian to the Kepler Hamiltonian of the outer orbit. This modified integrator can handle hierarchical systems where m and m are not small. 4.. HD The best-fit orbital parameters of the HD planets from Marcy et al. (00) and the inferred planetary masses and semimajor axes for sin i ¼ are listed in Table.

9 No., 003 EVOLUTION OF HIERARCHICAL PLANETARY SYSTEMS 09 Fig.. Trajectories in the phase-space diagrams of e vs. $ $ (top panels) and e vs. $ $ (bottom panels) for two-planet systems with the same masses, initial semimajor axes, and total angular momentum as the sin i ¼ HD system. The left- and right-hand panels show the results from direct numerical orbit integrations and the octopole-level secular perturbation theory, respectively. The trajectories through the squares are those of the HD system, with the squares showing the current parameters of the system. The initial conditions for the other trajectories are $ $ ¼ 0 and e ¼ 0:06, 0.08, 0.0, 0.5, 0.5, 0.40, 0.65, or 0.70, or $ $ ¼ 80 and e ¼ 0:706 or 0.707, with e being determined from the total angular momentum. All the direct integrations (except that for the HD system) start with the outer planet at apoapsis and the inner planet at opposition. This system has ¼ a =a ¼ 0:0, ¼ 5ðm 0 m Þ= ½4ðm 0 þ m ÞŠ ¼ 0:6, ¼ L =L ¼ 0:43, and ¼ ðg þ G Þ=ðL þ L Þ¼0:963. Figure b shows the variations in the Jacobi a j and e j, with the solid and dotted lines (almost indistinguishable) from direct numerical orbit integrations starting at T peri, and T peri, of Table, respectively, and the dashed lines in the bottom two panels from the octopole-level secular perturbation theory. The trajectories in Figure marked by the squares (which denote the current parameters of the HD system) are the trajectories of the sin i ¼ HD system in the phase-space diagrams of e versus $ $ (top panels) and e versus $ $ (bottom panels), with the left- and right-hand panels showing the results from direct integrations and the octopole theory, respectively (see below for a discussion of the other trajectories in Fig. ). The direct integration results agree with the octopole theory in that the evolution is not sensitive to the starting epoch and that a and a are nearly constant. In addition, the trajectories in the diagram of e (or e ) versus $ $ from the direct integrations are in excellent agreement with that from the octopole theory. The only noticeable difference is that the period of eccentricity oscillations predicted by the octopole theory is about 3% longer than that found in the direct integrations (:8 0 4 yr). Krymolowski & Mazeh (999) have reported that the oscillation period predicted by the octopole theory can be improved by the inclusion of the terms of the order of 7/ induced by the canonical transformation of the von Zeipel method and neglected by us. To study the effects of sin i, we have performed two direct integrations of the coplanar HD system with sin i ¼ 0:4, one starting at T peri, and the other T peri,. (Although the two-kepler fit to the radial velocity observations does not yield sin i j, the lack of evidence for stellar wobble in the Hipparcos astrometric data limits sin i e0:4 for the outer planet of HD 68443; Marcy et al. 00.) In both cases, the amplitudes of eccentricity oscillations and the trajectory in e (or e ) versus $ $ are almost identical to those shown in Figures b and for sin i ¼, and the factor by which the eccentricity oscillation period shortens agrees with sin i ¼ 0:4 to better than These results are in good agreement with the analytic results derived in x 3.3 in the limit m ; m 5 m 0, even though m is almost 44M J (and m =m 0 0:04) when sin i ¼ 0:4. In addition to the trajectories of the sin i ¼ HD system, Figure also shows the trajectories of systems with the same masses, initial semimajor axes, and total angular momentum as the sin i ¼ HD system. (Recall that the octopole results are also valid for other systems with the same,, and.) There is excellent agreement between the direct-integration and octopole results in all cases. The HD system is not in a secular resonance, and its $ $

10 0 LEE & PEALE Vol. 59 circulates. Indeed, it is far from the relatively small libration islands about the fixed points at ð$ $ ; e Þ¼ ð0 ; 0:046Þ and (80, 0.70) in the phase-space diagram of e versus $ $. The small libration islands and modest eccentricity variations in Figure are consistent with the fact that these systems have ¼ 0:43 far from crit ¼ 0:836 for the dimensionless total angular momentum ¼ 0:963 of these systems. The above e -values for the fixed points were obtained from the octopole theory as the roots of dð$ $ Þd ¼ 0 at $ $ ¼ 0 and 80, with dð$ $ Þ=d from equation (36) and e from the conservation of total angular momentum. As we discussed in x 3.3, the octopole theory predicts that the fixed points must be at $ $ ¼ 0 or 80. To demonstrate how the octopole theory can be used to explore the parameter space rapidly, we have used the same root-finding procedure to determine the number and positions of fixed points as a function of the dimensionless total angular momentum ¼ðG þ G Þ=ðL þ L Þ for systems with the same and as the sin i ¼ HD system. The e -values for the fixed points at $ $ ¼ 0 and 80 as a function of are shown in the left- and righthand panels of Figure 3, respectively. For all values of shown in Figure 3, there is an elliptic fixed point at $ $ ¼ 0 with relatively small e (bottom left-hand panel) and an elliptic fixed point at $ $ ¼ 80 with e close to the maximum possible value indicated by the dashed line (right-hand panel). These two fixed points are the ones seen above for systems with the same (=0.963; dotted lines in Fig. 3) as the sin i ¼ HD system. For 0:87dd0:888, there are two additional fixed points at $ $ ¼ 0 with e very close to (top left-hand panel of Fig. 3). These additional fixed points emerge at 0:888 with the same e -value (square in the top lefthand panel of Fig. 3) and the one with an e -value that increases (decreases) with decreasing is an elliptic (hyperbolic) fixed point. Direct numerical orbit integrations confirm that there are indeed stable librations about an elliptic fixed point close to the additional one predicted by the octopole theory. For example, for ¼ 0:88, the octopole theory predicts that the additional elliptic fixed point is at e ¼ 0:9948 (and e ¼ 0:30). A direct integration of a system with these values of e, e, and $ $ as initial conditions (and with the masses and semimajor axes of the sin i ¼ HD system and mean anomalies l ¼ 0 and l ¼ 80 ) shows that this system has $ $ librating about 0 with an amplitude of about 6 and e varying between and It should be noted that the treatment of this system with e 0:9946 and a 0:3 AUas point masses interacting via Newtonian gravity is inadequate since the periapse distance of the inner planet is in fact less than the stellar radius. However, based on the octopole theory, we expect variants of this system with much larger a (so that the periapse distance of the inner planet is well outside the stellar radius) but the same to show similar libration behaviors. On the other hand, the inner planet could still come sufficiently close to the star for general relativistic precession and tidal effects to be important. For d0:87, the additional elliptic fixed point vanishes, leaving only the hyperbolic fixed point (top left-hand panel of Fig. 3). Both octopole and direct-integration calculations show that systems with d0:87, initial $ $ ¼ 0, and initial e above the hyperbolic fixed point have e increasing to unity in a finite time. Again, we expect variants of these Fig. 3. Values of e for the fixed points at $ $ ¼ 0 (left-hand panels) and 80 (right-hand panel) in the phase-space diagram of e vs. $ $ (like Fig. ) as a function of the dimensionless total angular momentum ¼ðG þ G Þ=ðL þ L Þ, as determined by the octopole theory for systems with the same (=0.6) and (=0.43) as the sin i ¼ HD system. The dotted lines indicate the dimensionless total angular momentum ¼ 0:963 of the sin i ¼ HD system, and the dashed lines indicate the maximum possible e as a function of. See text for the meaning of the square in the top left-hand panel.

11 No., 003 EVOLUTION OF HIERARCHICAL PLANETARY SYSTEMS systems with larger a (so that the periapse distance of the inner planet is well outside the stellar radius) but the same to show similar increase in e. However, before e reaches unity, either the effects of tides and general relativistic precession would eventually stop the increase in e or the inner planet would collide with the star. 4.. HD 66 The best-fit orbital parameters of the HD 66 planets from D. A. Fischer (00, private communication) and the inferred planetary masses and semimajor axes for sin i ¼ are listed in Table. Figure 4 shows the variations in the Jacobi a j and e j, with the solid and dotted lines from direct numerical orbit integrations starting at T peri, and T peri, of Table, respectively, and the dashed lines in the bottom two panels from the octopole-level secular perturbation theory. The direct integration results are not consistent with the secular theory in that the evolution of the orbital elements is sensitive to the starting epoch. It turns out that a system with exactly the best-fit orbital parameters is very close to the : mean-motion commensurability (P =P ¼ 5:486 ¼ 0:9975 =). The direct integration starting at T peri, (solid lines in Fig. 4) shows irregular fluctuations in a j and e j (note, e.g., the irregular jumps in the mean values of Fig. 4. Variations in the Jacobi semimajor axes, a and a, and eccentricities, e and e, of the sin i ¼ HD 66 system with the best-fit orbital parameters listed in Table. The solid and dotted lines are from direct integrations starting at T peri, and T peri, of Table, respectively, and the dashed lines in the bottom two panels are from the octopole theory. This system with initial P =P ¼ 0:9975 = is affected by the : mean-motion commensurability, and the direct integration results are sensitive to the starting epoch. The direct integration starting at T peri, shows irregular fluctuations in a j and e j (note, e.g., the irregular jumps in the mean values of a and a at successive minima of e ) and is most likely chaotic. The most noticeable effect of the : commensurability on the direct integration starting at T peri, is the reduction in the amplitudes of the eccentricity variations. a and a at successive minima of e ) and is most likely chaotic. The direct integration starting at T peri, (dotted lines in Fig. 4) does not show any obvious irregular jumps in a j or e j and may be either regular or very weakly chaotic. But its smaller amplitudes of eccentricity variations are due to the : commensurability. The chaos in one (and possibly both) of these calculations is due to the overlap of the resonances at the : commensurability (see Holman & Murray 996 and Murray & Holman 997 for a similar situation in the planar elliptic restricted three-body problem). For both of the direct integrations shown in Figure 4, we have examined the 0 eccentricity-type mean-motion resonance variables at the : commensurability and confirmed that some of them alternate between circulation and libration. We have extended the direct integrations shown in Figure 4 and found that both are stable for at least 0 6 yr, but we cannot rule out the possibility that the chaos would lead to instability on longer timescales. The orbital period P (and the other orbital parameters) of the outer planet of HD 66 are not currently known to high precision, because the time span of the available observations is comparable to P. To study the effects of varying P, we have performed two sets of direct integrations with different initial P, one starting at T peri, and the other at T peri,. The initial values of P were chosen such that ðp =P Þ=ð=Þ ¼0:98, 0.99,.0,.0, and.03. The initial values of the other orbital parameters that can be obtained from the two-kepler fit (P, K,, e,,!,,and T peri,, ) were fixed at the values listed in Table, and the planetary masses and initial semimajor axes were derived assuming that sin i ¼. We find that the cases with initial ðp =P Þ=ð=Þ ¼:0 are also affected by the : meanmotion commensurability, while those with initial ðp =P Þ=ð=Þ ¼0:98, 0.99, and.0 show regular secular evolution. Although the variations in a j for the cases with initial P =P ¼ :03 = ¼ 0:9997 7=3 indicate that these cases are probably affected by the 7 : 3 commensurability, the variations in e j and the trajectories in the diagram of e versus $ $ are qualitatively indistinguishable from those showing regular secular evolution (for at least 0 5 yr). In Figures 5 and 6 we show examples of the cases with regular secular evolution. Figure 5 shows the variations in a j and e j for the sin i ¼ HD 66 system with P =P ¼ 0:99 =. The solid and dotted lines are from the direct integrations starting at T peri, and T peri,, respectively, and the dashed lines are from the octopole-level secular perturbation theory. The trajectories of this system in the phase-space diagrams of e versus $ $ and e versus $ $ are the trajectories marked by the squares (which denote the current parameters of the HD 66 system) in Figure 6. The direct integration results are not sensitive to the starting epoch. The amplitudes of eccentricity oscillations and the trajectories in the diagram of e (or e ) versus $ $ predicted by the octopole theory are in reasonably good agreement with those from direct integrations, even though (=0.33) is quite large. The main error of the octopole theory is in the period of eccentricity oscillations, with the predicted period (: 0 4 yr) about 75% longer than that found in the direct integrations ( : 0 4 yr). Figures 5 and 6 show some interesting properties of the HD 66 system that are shared by all of the cases studied above with different initial P, including those affected by the close proximity to a mean-motion commensurability.