The Astrophysical Journal, 573:829–844, 2002 July 10 # 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.
CALCULATING THE TIDAL, SPIN, AND DYNAMICAL EVOLUTION OF EXTRASOLAR PLANETARY SYSTEMS Rosemary A. Mardling1,2 and D. N. C. Lin2 Received 2001 July 20; accepted 2002 March 19
ABSTRACT Based on formulations by Heggie and by Eggleton, we present an efficient method for calculating self-consistently the tidal, spin, and dynamical evolution of a many-body system, here with particular emphasis on planetary systems. The star and innermost planet (or in general the closest pair of bodies in the system) are endowed with structure while the other bodies are treated as point masses. The evolution of the spin rates and obliquities of the extended bodies are calculated (for arbitrary initial obliquities), as is the tidal evolution of the innermost orbit. In addition, the radius of the innermost planet is evolved according to its ability to efficiently dissipate tidal energy. Relativistic effects are included to post-Newtonian order. For resonant systems such as GJ 876, the evolution equations must be integrated directly to allow for variation of the semimajor axes (other than from tidal damping) and for the possibility of instability. For systems such as Upsilon Andromedae in which the period ratio of the two inner planets is small, the innermost orbit may be averaged producing (in this case) a 50-fold reduction in the calculation time. In order to illustrate the versatility of the formulation, we consider three hypothetical primitive Earth-Moon-Sun-Jupiter systems. The parameters and initial conditions are identical in the first two models except for the Love number of the Earth, which results in dramatically different evolutionary paths. The third system is one studied by Touma & Wisdom and serves as a test of the numerical formulations presented here by reproducing two secular mean motion resonances (the evection and eviction resonances). The methods may be used for any system of bodies. Subject headings: planetary systems — planetary systems: formation — solar system: formation involves calculating the evolution of the (specific) orbital angular momentum vector of the inner orbit h, as well as its Runge-Lenz vector e. Since for an unperturbed orbit these vectors are constant, under external perturbations their components vary slowly compared to the orbital period (at least for stable orbits). The evolution of the orbital elements of the innermost planet due to tides and spin oblateness of the star and this planet, as well as their spin vectors, is calculated using a formulation devised by Eggleton, Kiseleva, & Hut (1998). This is based on the equilibrium tide model (see, e.g., Hut 1981), although a constant tidal lag time is not assumed a priori. Nonetheless, the resulting averaged expressions are the same for the spin-aligned case (see x 3.3). Touma & Wisdom (1994) have found that the evolution of the Earth-Moon system is not particularly sensitive to the model used for tidal friction, although there they consider partially rigid bodies, while many of our studies involve gaseous planets. Together, the Heggie and Eggleton formulations provide a powerful method for calculating the complex evolution of many-body planetary systems such as Upsilon Andromedae (Butler et al. 1999) and GJ 876 (Marcy et al. 2001; Laughlin & Chambers 2001). Other configurations may be studied such as the highly eccentric system HD 80606 with e ’ 0:927 and periastron separation 0.03 AU (Naef et al. 2001), for which a binary companion may be responsible for the high eccentricity via the Kozai effect, or which may have been left in this state after a close encounter with another planet (R. A. Mardling & D. N. C. Lin 2002b, in preparation). In x 2 we present the equations of motion for a four-body system in which the two closest bodies are tidally and rotationally evolving and one of the bodies has a relativistic potential. We refer to the numerical code associated with this as the ‘‘ direct code.’’ In x 3 we present evolution equa-
1. INTRODUCTION
The smallest ratio of semimajor axes of any two adjacent planets in the solar system is 0.29 for the orbits of Mars and Jupiter, with corresponding period ratio 0.16. However, apart from Mercury and Pluto, the eccentricities are all less than 0.1. Thus, perturbation analyses for planetary orbits in the solar system use eccentricity as the small parameter, leading to rather complex expansion expressions (Murray & Dermott 1999). The ratio of semimajor axes of the two innermost planetary orbits of the Upsilon Andromedae system (Butler et al. 1999; Mardling & Lin 2002) is 0.07, with corresponding period ratio 0.02. These are small enough to allow the use of the ratio of semimajor axes as the expansion parameter, considerably simplifying the analysis. Moreover, the small orbital period ratio justifies the use of orbit averaging so that for a three-body system the outer body responds to a ring of material. The Upsilon Andromedae system consists of three Jupiter-like planets and a Sun-like star. In addition to perturbations from the outer two planets, the innermost orbit is coupled to the star tidally and via the stellar spin and is significantly affected by the star’s post-Newtonian potential. Traditionally, people have studied orbit perturbation problems via a disturbing function and Lagrange’s equations (Murray & Dermott 1999) or Delaunay variables (Ford, Kozinsky, & Rasio 2000). Since there are many different perturbing effects to include in our analysis, we find it much simpler to use an approach due to D. C. Heggie (1995, private communication; see also Pollard 1966, p. 31). It
1 School of Mathematical Sciences, Monash University, Melbourne 3800, Australia;
[email protected]. 2 UCO/Lick Observatory, University of California at Santa Cruz, Santa Cruz, CA 95064;
[email protected].
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MARDLING & LIN
tions derived by averaging over the innermost orbit of the general system described in x 2. We refer to the numerical code associated with this as the ‘‘ averaged code.’’ In x 4 we present direct and averaged expressions for the self-consistent evolution of the radius of the tidally active planet. In x 5 we present equations for the secular evolution of the orbital elements of a coplanar three-body (point-mass) system. These are derived by averaging over both the inner and outer orbits of such a system. While we do not use these for numerical work, they give insight into the secular evolution of point-mass systems. In x 6 we present a comparison of the direct and averaged codes, as well as the tidal, spin, and dynamical evolution of three primitive Earth-Moon-SunJupiter systems, one serving the purpose of a test of the codes. Finally, in x 7 we present a summary. 2. EQUATIONS OF MOTION
Hierarchical (Jacobi) coordinates are used to calculate the orbits (see Fig. 1). These are such that the orbit of the innermost planet is referred to the star, the orbit of the second planet is referred to the center of mass of the star and the innermost planet, the orbit of the third planet is referred to the center of mass of the inner three bodies, etc. This system has the advantage that the relative orbits are simply perturbed Keplerian orbits so that the osculating orbital elements are easy to calculate (see Murray & Dermott 1999). However, note that astronomers often fit Kepler orbits to their data one planetary orbit at a time, assuming that each planet is isolated (D. A. Fischer 1999, private communication). For all calculations we use a Bulirsch-Stoer integrator that has been optimized for this problem. To illustrate its efficiency, only four integration steps per circular orbit are required to achieve relative errors of 1013 per step for the total angular momentum. However, as is often the case with direct orbit integrators, the problem of systematic error in the semimajor axes exists (but see Quinn & Tremaine 1990; He´non & Petit 1998). This amounts to a relative increase of 4 1013 per innermost orbit that is integrated directly.
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Let the masses of the four bodies (star plus three planets) be m1, m2, m3, and m4, with m1 being the star, and let the relative position vectors of the three orbits be r12 r, r123 R, and r1234 , respectively. Furthermore, let the star and the innermost planet have structure that is specified by their radii S1 and S2,3 their moments of inertia I1 and I2, their quadrupole apsidal motion constants (or half the appropriate Love numbers for planets with some rigidity; see Murray & Dermott 1999) k1 and k2, and their Q-values Q1 and Q2.4 Bodies 3 and 4 are assumed to be structureless. Given N sources of external perturbation f 1 ; f 2 ; . . . ; f N , the equation governing the relative motion of the innermost pair is €r ¼
N X Gðm1 þ m2 Þ rþ fi : 3 r i¼1
ð1Þ
For the Upsilon Andromedae system, the accelerations f i are due to the relativistic potential of the star, the presence of the two other planets, the tidal and spin distortions of the star and innermost planet, and the tidal damping of the innermost orbit. Note, however, that the formulation presented here does not depend on any ratio of masses being small and hence may be used to study other configurations such as stellar hierarchies and star-planet-satellite systems (see D. N. C. Lin & R. A. Mardling 2002a, in preparation, for a study of the dynamical and tidal evolution of the Pluto-Charon system and M. Evonuk, D. N. C. Lin, & R. A. Mardling 2002, in preparation, for a study of extrasolar planetary satellite systems). Explicitly, the accelerations are as follows: 1. Those due to the third and fourth bodies (the second and third planets): f 3 ¼ Gm3 ðb 23 b 13 Þ ;
f 4 ¼ Gm4 ðb 24 b 14 Þ ;
ð2Þ
where b ij ¼ rij =r3ij . 2. The acceleration due to the quadrupole moment of body 1, composed of its spin distortion as well as the tidal distortion produced by the presence of body 2: S15 ð1 þ m2 =m1 Þk1 12Gm2 2 1 2 ^r 5ðX1 x ^rÞ 1 f QD ¼ r4 r3 ð3Þ 2ðX1 x ^rÞX1 ; where k1 is the apsidal motion constant (or half the tidal Love number for planets with some rigidity) of body 1, X1 is its spin rate, and ^r denotes a unit vector in the direction of r. 3. A similar expression for body 2 (with tidal distortion due to body 1). 4. The acceleration produced by the tidal damping of body 1: 5 8 6nk1 m2 S1 a f 1TF ¼ r Q1 m1 a x ½3ð^r r_ Þ^r þ ð^r µ r_ rX1 Þ µ ^r ; ð4Þ where a and n are, respectively, the semimajor axis and 3
The radius of the innermost planet can evolve. Defined to be the e-folding number of tidal oscillations needed to damp the tidal energy and related to the tidal lag angle by tan 2 ¼ 1=Q (see Goldreich & Soter 1966). 4
Fig. 1.—Hierarchical (Jacobi) coordinates. The symbol c123 denotes the center of mass of m1, m2, and m3, with similar definitions for c12 and c1234.
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mean motion of the mutual orbit of bodies 1 and 2. Note that for gaseous bodies an apsidal motion constant is normally utilized, while for solid bodies such as the Earth, the Love number is given in terms of the rigidity l1 and a modified Q-value, Q01 ¼ Q1 ð1 þ 19l1 =2g1 1 S1 Þ, where g1 is the gravitational acceleration at the surface of the planet and 1 is its mean density. The Love number is defined then as kL ¼ 2k1 ¼ 3Q1 =2Q01 (see Goldreich & Soter 1966; Peale 1999).5 5. A similar expression for body 2 with structure constant k2. 6. The orbital acceleration due to the post-Newtonian potential of the binary (see, e.g., Kidder 1995): Gm12 Gm12 3 2 _r ^r ð1 þ 3Þ_r x r_ 2ð2 þ Þ f rel ¼ 2 2 2 r c r 2ð2 Þ_rr_ ; ð5Þ where ¼ m1 m2 =m212 and c is the speed of light.
m123 ¼ m1 þ m2 þ m3 , and where m12 ¼ m1 þ m2 , m1234 ¼ m1 þ m2 þ m3 þ m4 . The equations of motion for the third and fourth bodies are € ¼ Gm123 ðm1 b m2 b þ m3 b Þ R 13 23 34 m12 Gm4 ðm1 b 14 þ m2 b 24 þ m3 b 34 Þ þ m12
ð7Þ
and €r1234 ¼
Gm1234 ðm1 b 14 þ m2 b 24 þ m3 b 34 Þ ; m123
ð8Þ
respectively. The total angular momentum is given by J tot ¼ l12 ðr µ r_ Þ þ l123 ðR µ R_ Þ þ l1234 ðr1234 µ r_ 1234 Þ þ I1 X 1 þ I2 X 2 ;
where E123 ¼
ð9Þ
where X1 and X2 are the spin vectors of bodies 1 and 2, respectively, l12 ¼ m1 m2 =m12 , l123 ¼ m12 m3 =m123 , and l1234 ¼ m123 m4 =m1234 . The total energy is given by 1 Gm1 m2 1 Gm12 m3 _ _ x x _ _ l R R E ¼ l12 r r þ 2 2 123 r R 1 Gm123 m4 l r_ 1234 x r_ 1234 þ 2 1234 r1234 1 1 þ E123 þ E1234 þ l12 1 þ l12 2 þ I1 21 þ I2 22 ð10Þ 2 2 5 Note that here we have used the formulation of Goldreich & Soter (1966), which assumes that the tidal lag angle is the same for all components of the tide.
Gm12 m3 Gm1 m3 Gm2 m3 R r13 r23
ð11Þ
and E1234 ¼
Gm123 m4 Gm1 m4 Gm2 m4 Gm3 m4 r1234 r14 r24 r34
ð12Þ
are interaction energies that are small for ‘‘ wide ’’ systems and S15 ð1 þ m2 =m1 Þk1 1 2 2Gm2 2 x 1 ¼ ðX1 ^rÞ 1 3 3 r3 r
ð13Þ
and 2 ¼
The accelerations given by equations (3) and (4) are due to Eggleton et al. (1998). The vectors rij rj ri are defined in terms of the Jacobi coordinates such that m2 m3 m4 r R r1234 ; r1 ¼ m12 m123 m1234 m1 m3 m4 r R r1234 ; r2 ¼ m12 m123 m1234 m12 m4 R r1234 r3 ¼ m123 m1234 m123 r4 ¼ r1234 ; ð6Þ m1234
831
S15 ð1 þ m1 =m2 Þk2 1 2 2Gm1 2 x^ ðX r Þ 2 3 2 r3 r3
ð14Þ
are potentials due to the quadrupole moments of the spin and tidal distortions of bodies 1 and 2, respectively. The total energy is conserved for nondissipative systems. Assuming that the spins of bodies 1 and 2 evolve solely as a result of tidal and spin torques, the evolution of the spin vector of body 1 is given by _ ¼ l r µ f 1 þ f 1 ð15Þ I1 X 12 QD TF ; 1 with a similar expression for body 2. The orbital elements of each orbit may be calculated from the position and velocity vectors following the procedure outlined in Murray & Dermott (1999, p. 53). A numerical scheme has been developed that takes all the effects described above into account, and we refer to this as the ‘‘ direct code.’’ For most systems of interest, orbit averaging can be used (see x 3), which generally reduces the computation time considerably. However, for dynamically unstable systems as well as stable resonant systems, both of which can involve significant energy exchange between orbits, the direct code must be employed. We have done this for the extrasolar planetary system GJ 876 in which the two (detected) planets have periods 30 and 61 days (D. N. C. Lin & R. A. Mardling 2002b, in preparation). The spin-orbit coupling and tidal accelerations are small for this system, as is the effect of relativistic apsidal advance, so that a pointmass code could have been used. However, we might expect to find shorter period resonant systems for which these effects are significant. In the case of the nonresonant system Upsilon Andromedae, the relativistic potential of the star dominates the rate of apsidal motion of the innermost planetary orbit, thereby ‘‘ detuning ’’ the large-amplitude secular variation of the eccentricity and reducing it by an order of magnitude (Mardling & Lin 2002). Consequently, the rate of tidal damping of the eccentricity (which is proportional to the eccentricity; see eqs. [28], [43], and [52]) is reduced, and hence so is the rate of planetary inflation and orbital decay. In contrast to this, however, for resonant systems the resonant contribution to the apsidal advance always dominates any quadrupole effect, including relativity. Comparing equations (53) and (83), the ratio of rates of apsidal advance due to the relativistic potential and the secular con-
MARDLING & LIN
tribution from another body, respectively, is (for star-planet systems) 4Z 2 M ao 3 sec ¼ ; ð16Þ 1 e2i Mo ai where ai and ao are the semimajor axes of the inner and outer orbit, respectively, and Z ¼ ai ni =c is the ratio of the orbital speed of the inner orbit to the speed of light. For resonant systems, the ratio of apsidal advance due to the relativistic potential and the resonant contribution from another body, respectively, is res ¼ ei sec , while for nonresonant systems res ’ 0 (G. Bryden et al. 2002, in preparation). For systems with M ’ M and ai < 1 AU, Z ’ ð1 5Þ 104 . When M =Mo ’ 1000 and ei is small to moderate, we have 3 ao ; ð17Þ sec ’ 104 ai with res a factor ei smaller for resonant systems. For Upsilon Andromedae, ao =ai ’ 14, so that sec ’ 1:5. In contrast to this, 2 : 1 resonant systems have ao =ai ’ 1:6, so that res ’ 3 104 ei , never allowing the general relativistic potential to dominate the apsidal advance for realistic inner eccentricities. Nonetheless, the tidal dissipation in the inner planet of such a resonant system may be substantial and will be amplified via the resonant pumping of the eccentricity. The direct code should be used for such a system. The inclusion of quadrupole effects will also reduce the amplitudes of Kozai cycles, which can be induced when the relative inclination of two orbits is favorable (see, e.g., Holman, Touma, & Tremaine 1997; R. A. Mardling & D. N. C. Lin 2002b, in preparation). 3. ORBIT AVERAGING
Since the integration time step is governed by the shortest period orbit, for many applications it is often more efficient to time-average over this orbit so that the time step is governed by the second shortest period orbit. For example, in the case of the Upsilon Andromedae system this produces a 50-fold reduction in calculation time. In the following we present such an averaging procedure. We refer to the associated numerical scheme as the ‘‘ averaging code.’’ Following D. C. Heggie (1995, private communication), the secular evolution of the elements is calculated via the orbital angular momentum vector (per unit mass) of the inner orbit h ¼ r µ r_ and the Runge-Lenz vector e, a vector in the direction of periastron with magnitude equal to the eccentricity and defined by e¼
r_ µ h ^r : Gðm1 þ m2 Þ
ð18Þ
The rate of change of these vectors is then given by dh X ¼ r µ fi dt i and de ¼ dt
P
i ½2ðf i
x r_ Þr ðr x r_ Þf i
Gm12
ðf i x rÞ_r
:
ð19Þ
ð20Þ
Equations (19) and (20) are time-averaged over the inner orbit, and the resulting differential equations are integrated
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numerically to obtain the secular evolution of the orbital elements (eccentricity e, semimajor axis a, inclination i, argument of periastron !, and longitude of the ascending node ). These may be obtained from e and h via e ¼ j ej ; a¼
ð21Þ h2
Gðm1 þ m2 Þð1 e2 Þ cos i ¼ k x h^ ; cos ! ¼ ^e x ^n ; sin ! ¼ ^e x ðh^ µ ^nÞ ;
;
cos ¼ i x ^n ; sin ¼ j x n^ ;
ð22Þ ð23Þ ð24Þ ð25Þ ð26Þ ð27Þ
where i, j, and k form a basis for the reference frame with k being perpendicular to the reference plane and n ¼ k µ h. The rates of change of e, i, and are given by (Eggleton et al. 1998) de ¼ e_ x ^e ; dt di ðsin ! ^e þ cos ! ^qÞ x h_ ¼ ; dt h d ðcos ! ^e sin ! ^qÞ x h_ ¼ ; dt h sin i
ð28Þ ð29Þ ð30Þ
where ^q ¼ ^h µ ^e. The rate of apsidal advance is given by e_ _ x^ !_ þ cos i ¼ q; ð31Þ e while the average rate of change of the semimajor axis can be obtained from da a _ ¼ Eorb ; dt Eorb where the orbital energy is Eorb ¼ 12 Gm1 m2 =a and X E_ orb ¼ l12 r_ x f i
ð32Þ
ð33Þ
i
(note that we do not use these equations in the present formulation). The rate of transfer of angular momentum to or from the orbit is given by h_ x ^h. 3.1. Octopole Expansions For most of this section we restrict the analysis to that for three bodies only and assume that only body 1 has structure. The inclusion of a fourth body will be discussed separately. In order to perform the time-averaged integrations analytically, equation (2) is expanded in terms of Legendre polynomials. For example, the acceleration of the inner orbit produced by the third body (second planet) may be written in terms of Jacobi coordinates as follows: R 1 r R þ 2 r f 3 ¼ Gm3 jR 1 rj3 jR þ 2 rj3 " # l l1 1 Gm3 X ml1 r 2 ðm1 Þ ^ Þ ; ð34Þ Pl ð^r x R ¼ r R R l¼2 ðm1 þ m2 Þl1 D
832
where i ¼ mi =m12 , and for which the quadrupole and octo-
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EXTRASOLAR PLANETARY SYSTEMS
pole contributions are, respectively, f quad ¼
Gm3 ^ rÞ ð3xR R3
ð35Þ
and f oct
Gm3 m2 m1 3 2 ^ 3xr ; 5x r2 R ¼ 4 2 R m12
ð36Þ
^ . Note that when m1 ¼ m2 , the octopole where x ¼ r x R terms do not contribute to the perturbation. To octopole order, the equation of motion of the third body (eq. [7] without the fourth body) becomes ^ þ bquad þ boct ; € ¼ Gm123 R R R2 bquad
Gl12 m123 3 2 2 ^ 5x r R 3xr ; ¼ 4 R m12 2
and de ¼ hgquad i þ hgoct i þ hgQD i þ hgTF i þ hgGR i ; dt
ð42Þ
ð43Þ
where 3 h Gm12 m3
a 3 ^ 2R ^ 3^e d quad ¼ 1 e2 R R 2 a m12 i
^ 1R ^ 3 ^q þ 5e2 R ^ 1R ^ 2 ^h ; 1 þ 4e2 R
ð44Þ
3 pffiffiffiffiffiffiffiffiffiffiffiffiffi h m3 a 3 ^ ^ e 1 e 2 5R e; hgquad i ¼ n 1 R2^ R 2 m12 i
^ 3 ^h ; ^ 2 ^q þ R ^ 2R ^2 þ R þ 1 4R
ð45Þ
4 Gm12 m3 m1 m2 a 15e hd oct i ¼ R 16 a m12 m12 n h
^ 1R ^3 ^ 2R ^ 3^e þ 4 þ 3e2 R 10 1 e2 R i
2
2 ^2 ^ ^ R ^ 5 3 þ 4e2 R q 1 3 5 1 e R2 R3 ^ h
2 ^ 2 þ 5 1 þ 6e2 R ^ R ^ 4 þ 3e2 R 1 2 i o
3 ^ ^h ; þ 5 1 e2 R 2
ð46Þ
ð38Þ
1
Gl12 m123 m1 m2 R5 m12 m12
5 ^ 3 5x2 r2 r : 7x3 3xr2 R 2 2
The resulting averaged equations are dh ¼ d quad þ hd oct i þ d QD þ hd TF i dt
ð37Þ
and
833
boct ¼
ð39Þ
3.2. Single Averaging Perturbations of the inner orbit that are due to the presence of an outer body depend on the ratio r/R, while perturbations due to tides and spin depend on the ratio S1/r, where S1 is the radius of body 1. Orbit averaging is performed by assuming that the (inner) orbit is Keplerian and its orbital elements do not change in one orbital period. For perturbations that involve powers of r/R, the eccentric anomaly E is usually used as the integration variable. For example, * + Z 2 2 r 1 T r dt ¼ R T 0 R 2 Z 2 1 a ¼ ð1 e cos EÞ3 dE 2 R 0 2 3 2 a ¼ 1þ e ; ð40Þ 2 R
2
and 4 pffiffiffiffiffiffiffiffiffiffiffiffiffi m3 m1 m2 a 15 1 e2 hgoct i ¼ n R 16 m12 m12 nh
^ 2 þ 5 1 þ 6e2 R ^ 2R ^ 4 þ 3e2 R 1 2 i
3 ^ ^e þ 5 1 e2 R 2 h
2 ^ ^ 1R ^2 þ 4 þ 9e R1 5 1 3e2 R 2 o i
^ 3 ^q þ 10e2 R ^ 1R ^ 2R ^ 3 ^h ; 5 1 þ 2e2 R 1
ð47Þ
where we have used r ¼ að1 e cos EÞ and E_ ¼ n= ð1 e cos EÞ, with T being the orbital period, n ¼ 2=T the mean motion, and a the semimajor axis. For perturbations that involve powers of S1/r, the true anomaly ’ is usually used as the integration variable. For example, * + Z S1 2 1 T S1 2 dt ¼ T 0 r r pffiffiffiffiffiffiffiffiffiffiffiffiffiS1 2 ¼ 1 e2 ; ð41Þ a
where e and a are the eccentricity and semimajor axis of the inner orbit. The coordinate frame is defined by the basis vec^ x ^e, R ^ x^ ^2 ¼ R ^1 ¼ R q, tors ð^e; ^q; ^hÞ, where ^q ¼ ^h µ ^e, so that R ^ ^ ^ and R3 ¼ R x h. The quadrupole terms produce significant secular evolution of the eccentricity only for systems in which the inner and outer orbits are inclined, while the octopole terms can produce significant secular evolution only when m1 4m2 or m1 5 m2 . In addition, the semimajor axis is constant to all orders, and therefore the averaged equations of motion cannot produce unstable orbits in the sense that the ultimate configuration is unbound. The system is nonetheless nonlinear and can produce chaotic motion when the effects of tides and spin are included.
where we have used r ¼ að1 e2 Þ=ð1 þ e cos ’Þ and 1=2 r2 ’_ ¼ a2 nð1 e2 Þ . In addition to these expressions, we 1=2 have r_ ¼ aen sin E=ð1 e cos EÞ ¼ an ½e=ð1 e2 Þ sin ’. Mathematica was used to calculate the many averaging integrals; however, see Eggleton et al. (1998) for more details.
3.3. Tides, Spin, and Relativity Referring to equations (3), (19), and (20), the averaged contributions to h_ and he_ i produced by the quadrupole
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MARDLING & LIN
distortion of one body (here body 1) are, respectively, 3 m2 S1 1 2 d QD ¼ S1 k1 1 þ m1 a ð1 e2 Þ3=2 q h^e þ e h ^ q þ 0^ h
Referring to equation (15), the evolution of the spin vector is given simply by D E
_ ð54Þ I1 X 1 ¼ l12 d QD þ hd TF i ; ð48Þ
and D E k e ð1 þ m =m Þ S 5
1 1 2 1 1 0^ e þ 2e þ 2q gQD ¼ 2 n ð1 e2 Þ 2 a Gm2 q þ q h ^ þ 2h þ 3 f0 ðeÞ ^ h ; ð49Þ a so that the quadrupole distortion due to the tides contribq, h ¼ X1 x ^h, utes only to he_ i. Here e ¼ X1 x ^e, q ¼ X1 x ^ and f0 ðeÞ ¼
15½1 þ
ð3=2Þe2 ð1
þ
ð1=8Þe4
3 e2 Þ
5 m2 S1 11 h ^e e 9 f1 ðeÞ f2 ðeÞ 18 m1 a n e ^ h ; ð52Þ þ 0^ q þ 12 f2 ðeÞ n
6k1 n hgTF i ¼ Q1
where
g1 ðeÞ ¼ g2 ðeÞ ¼ g3 ðeÞ ¼ g4 ðeÞ ¼
1 þ ð15=4Þe2 þ ð15=8Þe4 þ ð5=64Þe6 13=2
ð 1 e2 Þ 1 þ ð3=2Þe2 þ ð1=8Þe4
; 5 ð 1 e2 Þ ð1=2Þ½1 þ ð3=2Þe2 þ ð1=8Þe4 9=2
ð 1 e2 Þ ð1=2Þ½1 þ ð9=2Þe2 þ ð5=8Þe4
11=2
;
h1 ðeÞ ¼
1 þ ð31=2Þe2 þ ð255=8Þe4 þ ð185=16Þe6 þ ð25=64Þe8 15=2
ð 1 e2 Þ
;
ð56Þ h2 ðeÞ ¼
4
1 þ ð15=2Þe þ ð45=8Þe þ ð5=16Þe ð1 e2 Þ
6
6
:
ð57Þ
3.4. The Back Effect on the Outer Orbit Averaging over the perturbation to the outer orbit gives € ¼ Gm123 R ^ þ bquad þ hboct i ; R 2 R
ð58Þ
where 4 nh Gm12 m123
m1 m2 a 3 ^1 bquad ¼ 2 4 þ 11e2 R R 4 a m12 m212 i
3
^ 5 1 e2 R ^ 1R ^ 2 ^e 5 1 þ 4e2 R 1 2 h
2 ^ 2 ^2 ^ þ 4 þ e R2 5 1 þ 4e R1 R2 h
3i
^ ^q þ 2 þ 3e2 R ^3 5 1 e2 R 2 i o
2
2 ^2 ^ ^ ^ R ^ 5 1 þ 4e2 R ð59Þ 1 3 5 1 e R2 R3 h
hboct i ¼
;
6
ð 1 e2 Þ
where
and
; 9=2 ð 1 e2 Þ 1 þ ð15=2Þe2 þ ð45=8Þe4 þ ð5=16Þe6 ð 1 e2 Þ 1 þ 2e2 ð21=8Þe4 ð3=8Þe6
ð55Þ
2
f2 ðeÞ ¼
where I1 is the moment of inertia of body 1. The rate of change of the orbital energy is given by E_ orb ¼ hl12 f TF x r_ i 5 S1 3k1 h 2 3 m2 ¼ l12 a n h1 ðeÞ h2 ðeÞ ; m1 a Q1 n
ð50Þ
:
The acceleration due to tidal friction in one body is given by equation (4). Substituting this expression into equations (19) and (20) and averaging produces the contributions to h_ and he_ i from tidal friction: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6k1 nm2 S1 9=2 e ^e g1 ðeÞ hd TF i ¼ Gm12 S1 Q1 m1 a n q h ^ ^ q þ g3 ðeÞ þ g4 ðeÞ h ; ð51Þ þ g2 ðeÞ n n
f1 ðeÞ ¼
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;
:
Note that these expressions approach zero as e ! 0, e ! 0, q ! 0, and h ! n. Referring to equation (5), the contribution to he_ i due to the relativistic potential of the star is 3a2 n3 e ^ q; ð53Þ hgGR i ¼ 2 c 1 e2 while the contribution to h_ is zero. Thus, the post-Newtonian potential produces only apsidal advance of the orbit.
5 Gm12 m123 m1 m2 m1 m2 a 15e R 16 a2 m12 m12 m212 h
^ 2 5 1 e2 R ^2 4 þ 3e2 35 1 þ e2 R 1 2
2 2i 140 2 ^ 4 ^ R ^ e e R1 þ 35 1 e2 R þ 35 þ 1 2 ^ 3 h
^ 3R ^ 1R ^ 2 þ 35 þ 140 e2 R ^ þ 5 6 þ e2 R 1 2 3 i h
^ 1R ^ 1R ^ 3 ^q þ 5 4 þ 3e2 R ^3 þ 35 1 e2 R 2 i
140 2 ^ 3 ^ 2 ^ ^2 ^ ^ e R1 R3 þ 35 1 e R1 R2 R3 h : þ 35 þ 3 ð60Þ
The total angular momentum J ¼ l12 h þ I1 X1 þ l123 R µ R_ is then conserved.
ð61Þ
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EXTRASOLAR PLANETARY SYSTEMS 3.5. Including a Fourth Body
The expression for f 4 , the acceleration of the inner orbit produced by the fourth body (third planet), may be written in terms of Jacobi coordinates as follows: r124 1 r r124 þ 2 r f 4 ¼ Gm4 jr124 1 rj3 jr124 þ 2 rj3 " # l1 1 ml1 Gm4 X r l 2 ðm1 Þ Pl ð^r x ^r124 Þ ; ¼ r r124 r124 l¼2 ðm1 þ m2 Þl1
2 the moments of inertia of the two with I1;2 ¼ 1;2 m1;2 S1;2 bodies. Taking the star and planet to be polytropes of indices 3 and 1, respectively, 1 ¼ 0:076 and 2 ¼ 0:261. Assuming that the binding energy and spin frequency of the star do not change, we have
D
€r1234 ¼ where
Gm1234 m12 r124 Gm1234 m3 r34 þ cquad þ coct þ ; ð63Þ 3 m123 m123 r334 r124
Gl12 m1234 3 2 2 ^ 5y r 3yr ; ð64Þ r 124 r4124 m123 2
Gl m1234 m1 m2 5 3 7y 3yr2 ^r124 ¼ 5 12 2 m12 r124 m123
3 5y2 r2 r ; ð65Þ 2
cquad ¼ coct
and y ¼ r x ^r124 . This procedure may be generalized to any number of bodies.
4. PLANETARY INFLATION
For a multiplanet system such as Upsilon Andromedae in which the innermost planet is extremely close to the host star, it is possible for the radius of this planet to grow as it attempts to damp the eccentricity that is continually excited by the other planets. Let the radius of this planet, S2(t), be a function of time. The prescription for its rate of change is then as follows. Assuming a one-planet system, the rate of change of the total energy of the system is given by E_ tot ¼ E_ bind þ E_ spin ¼ L ;
Ebind ¼
1 Gm21 1 Gm22 1 Gm1 m2 ; 2 S1 2 S2 ðtÞ 2 aðtÞ
ð67Þ
and Espin includes the spin energies of the star and the planet, Espin ¼ 12 I1 21 þ 12 I2 22 ;
ð68Þ
ð69Þ
E_ spin ¼ I2 2 _ 2 þ 2 m2 S2 S_ 2 22 :
ð70Þ
Tidal dissipation within the planet leads to changes in a and 2. Defining the quantity (see Eggleton et al. 1998, eq. [39a]) 1 Gm1 m2 a_ þ I2 2 _ 2 ¼ l12 f TF x ð_r X2 µ rÞ E_ tide ¼ 2 aðtÞ a
ð71Þ
and assuming that expansion does not directly modify the value of 2, the rate of change of the radius of the planet is given by S_ 2 ¼
L þ E_ tide : ð1=2ÞGm22 =S22 þ 2 m2 S2 22
ð72Þ
The equilibrium radius (for a nonrotating planet) is given by equation (1) in Bodenheimer, Lin, & Mardling (2001) as a function of the rate at which tidal energy is deposited into the planet. In this case (in the absence of spin) S_ 2 ¼ 0, so that by equation (72) L ¼ E_ tide . The luminosity of the planet L is therefore given by the inverse of equation (1) in Bodenheimer et al. (2001) [in fact, we refit the data for the inverse and calculate a fifth-order polynomial for log ðL=L Þ vs. log ðS2 =R Þ]. While the expression for E_ tide (eq. [71]) is used as it stands in equation (72) in the direct code, it is averaged over the inner orbit for use in the averaged code. This results in the following expression: 5 m1 S2 6k2 E_ tot ¼ l12 a2 n m2 a Q2 1 2 e h1 ðeÞ þ 2q h2 ðeÞ þ 2h h3 ðeÞ 2 ð73Þ 2nh h4 ðeÞ þ n2 h5 ðeÞ ; where h1 ðeÞ ¼ h2 ðeÞ ¼
ð66Þ
where L is the rate of loss of energy from the planet’s surface. Ebind is the binding energy in the system, which includes the orbital binding energy as well as the binding energies of the star and planet,
1 Gm22 S_ 2 1 Gm1 m2 a_ þ E_ bind ¼ 2 S2 ðtÞ S2 2 aðtÞ a and
ð62Þ where r124 ¼ ðm3 =m123 ÞR þ r1234 (see Fig. 1). The quadrupole and octopole contributions are then the same as for the third body (eqs. [35] and [36]) but with m3 replaced by m4 and R replaced by r124 . Similarly, an expression to octopole order for the second group of terms in equation (7) for the evolution of the middle orbit may be obtained by replacing m123 with m4 and R with r124 . Finally, the equation of motion for the outer body may be expanded to octopole order as follows:
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h3 ðeÞ ¼ h4 ðeÞ ¼
1 þ ð3=2Þe2 þ ð1=8Þe4 9=2
ð 1 e2 Þ 1 þ ð9=2Þe2 þ ð5=8Þe4 9=2
ð 1 e2 Þ 1 þ 3e2 þ ð3=8Þe4
;
ð74Þ
;
ð75Þ
; 9=2 ð1 e 2 Þ 1 þ ð15=2Þe2 þ ð45=8Þe4 þ ð5=16Þe6
ð76Þ
; ð77Þ 6 ð 1 e2 Þ 1 þ ð31=2Þe2 þ ð255=8Þe4 þ ð185=16Þe6 þ ð25=64Þe8 h5 ðeÞ ¼ ; 15=2 ð1 e 2 Þ ð78Þ
where now e ¼ X2 x ^e, q ¼ X2 x ^q, and h ¼ X2 x ^ h. Note that this expression approaches zero as e ! 0, e ! 0,
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q
hhd quad ii ¼ 0 ;
e
R
Q
where
E
m3 ~ ωo- ~ ωi ψ ~ω i
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i
Fig. 2.—Frames of reference used for coplanar double averaging
q ! 0, and h ! n. Also note that the prescription does not conserve angular momentum identically.
hhgquad ii ¼ n
qffiffiffiffiffiffiffiffiffiffiffiffiffi 3 e 1 e2 i i m3 ai 3
^q ; ð83Þ m12 ao 4 ð1 e2o Þ3=2 4 Gm12 m3 m1 m2 ai 15 hhd oct ii ¼ a m12 m12 ao 64
ei eo 4 þ 3e2i sinð$i $o Þ^h ; ð84Þ 5=2 ð1 e2o Þ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffi m3 m1 m2 ai 15 eo 1 e2i hhgoct ii ¼ n m12 m12 ao 64 " 2 4 þ 3ei sinð$i $o Þ^e 5=2 ð1 e2o Þ # 4 21e2i cosð$i $o Þ^q ; ð85Þ þ 5=2 ð1 e2o Þ
where $i is the longitude of periastron of inner orbit and n is its mean motion. Referring to equations (37), (38), and (39), the rates of change of the outer orbital angular momentum H and the outer Runge-Lenz vector E are given by dH ¼ Rµ F dt
ð86Þ
dE 2 F x R_ R R x R_ F ðF x RÞR_ ¼ ; Gm123 dt
ð87Þ
5. DOUBLE AVERAGING
In order to gain some insight into how the various terms contribute to the secular evolution of a three-body system, we average over the outer orbit of a coplanar (point-mass) triple. The evolution of the orbital elements for the reduced coplanar problem with small eccentricities is presented in Murray & Dermott (1999), while that for the general noncoplanar problem is studied in Ford et al. (2000). In the latter, it is difficult to separate the various contributions because of the complicating effect of nonzero inclination. In the following, the double-averaged rates of change of e and h are derived, as well as those for the outer orbital angular momentum H and Runge-Lenz vector E. The reference frames used for this procedure are shown in Figure 2. Note that here we assume that the origins of the fe; qg and fE; Qg frames coincide, which amounts to ignoring indirect terms. However, in the application of this analysis (Mardling & Lin 2002), we are interested in quantities such as the difference in the longitudes of periastra for which the indirect terms do not appear. Averaging over the outer orbit polar angle (such that $o is the true anomaly of the outer orbit, with $o the longitude of periastron of the outer orbit) and putting ei ¼ e and ai ¼ a for the inner orbit and R¼
ao ð1 e2o Þ ; 1 þ eo cosð $o Þ
ð79Þ
where eo and ao are the outer eccentricity and semimajor axis, respectively, equations (42) and (43) become
dh ð80Þ ¼ hhd quad ii þ hhd oct ii dt and
de dt
¼ hhgquad ii þ hhgoct ii ;
ð81Þ
ð82Þ
and
where F ¼ bquad þ boct . The double-averaged rates of change of these quantities are then given by
dH ð88Þ ¼ hhDquad ii þ hhDoct ii dt and
dE dt
¼ hhG quad ii þ hhG oct ii ;
ð89Þ
where hhDquad ii ¼ 0 ; ð90Þ
3=2 7=2 2 m123 m1 m2 ai 3 eo 2 þ 3ei ^ hhG quad ii ¼ n Q; 8 ð1 e2o Þ2 m12 ao m212 ð91Þ 2 4 Gm12 m123 m1 m2 m1 m2 ai hhDoct ii ¼ 2 ai m12 m a m12 12 o pffiffiffiffiffiffiffiffiffiffiffiffiffi
15 ^ ; ei eo 1 e2o 4 þ 3e2i sinð$i $o ÞH 64 ð92Þ 3=2 9=2 m123 m1 m2 m1 m2 ai hhG oct ii ¼ n m12 m12 ao m212
h
15 ^ ei 4 þ 3e2i 1 e2o sinð$i $o ÞE 64 i
^ : ð93Þ þ 1 þ 4e2o cosð$i $o ÞQ
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^ is a unit vector in the direction of periastron of the where E ^ ¼H ^µE ^ , where H ^ is a unit vector perouter orbit and Q pendicular to the outer orbit. Note that according to equation (32), dai ¼0; dt dao ¼0: dt
ð94Þ ð95Þ
Only the octopole terms contribute to a secular exchange of angular momentum between the inner and outer orbits. Since the semimajor axes are constant, this corresponds to variations in the inner and outer eccentricities. Similarly, only the octopole terms produce a secular variation of the inner eccentricity, in contrast to inclined systems for which the quadrupole terms can produce significant variation. Coplanar systems may be described entirely via the equations for e_ and E_ . Also note that for the case m1 ¼ m2 , the octopole terms are also zero for coplanar motion. Thus, for example, a planet orbiting an equal-mass binary in the same plane as the binary would not experience any secular variation of eccentricity. Numerical experiments indicate that this is true to all orders. In order to compare our formulation with that of Murray & Dermott (1999), consider a system consisting of a star of mass m1 ¼ M and inner and outer planets of masses m2 ¼ Mi and m3 ¼ Mo , with Mi ; Mo 5 M . Assume that the inner and outer orbital eccentricities ei and eo are small. To first order in ei and eo, equations (83), (85), (91), and (93) reduce to 3 3n Mo ai ei ^ hhgquad ii ¼ q; ð96Þ 4 M ao 4 15n Mo ai hhgoct ii ¼ eo sinð$i $o Þ^e 16 M ao q ; ð97Þ þ cosð$i $o Þ^ 7=2 3n Mi ai ^ ; hhG quad ii ¼ eo Q ð98Þ 4 M ao 9=2 15n Mi ai ^ ei sinð$i $o ÞE hhG oct ii ¼ 16 M ao ^ : ð99Þ þ cosð$i $o ÞQ According to equations (28) and (31), the rates of change of the eccentricities and the rates of apsidal advance are given by 4 dei 15n Mo ai ¼ eo sinð$i $o Þ ; ð100Þ 16 M dt ao 9=2 deo 15n Mi ai ¼ ei sinð$i $o Þ ; ð101Þ 16 M dt ao 3 d$i 3n Mo ai 5 eo ai ¼ 1 cosð$i $o Þ ; 4 M 4 ei dt ao ao ð102Þ 7=2 d$o 3n Mi ai 5 ei ai ¼ 1 cosð$i $o Þ : 4 M 4 eo dt ao ao ð103Þ Equations (100) and (102) can now be compared with equa-
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tions (6.170) and (6.171) of Murray & Dermott (1999). Their analysis uses eccentricity as the expansion parameter, a natural choice for configurations such as the solar system, resulting in complex expressions involving Laplace coefficients, which are functions of the ratio of semimajor axes. Writing equations (100) and (102) as dei ai Mo C3 eo sinð$i $o Þ ; ¼n ð104Þ dt ao M d$i ai Mo eo ¼n cosð$i $o Þ ; C3 2C1 þ C3 dt ao M ei ð105Þ we have that 3 ai 2 ; C1 ¼ 8 ao
15 ai 3 C3 ¼ : 16 ao
ð106Þ
These are simply the leading-order terms of the power series expansions of the Murray & Dermott (1999) expressions for C1 and C2 and agree well when ei, eo, and ai/ao are small. 6. APPLICATIONS
The two schemes presented in this paper may be used to study a wide range of problems. Some examples for which the direct code is useful include (1) the 2 : 1 resonant system GJ 876 (Marcy et al. 2001; Laughlin & Chambers 2001; G. Bryden et al. 2002, in preparation); (2) the 2 : 1 resonant system HD 82943 in which the star appears to have an abnormally high 6Li content, possibly as a result of the tidal disruption of a planet (Israelian et al. 2001; Sandquist et al. 2002); and (3) star-planet-moon systems for which dynamical and tidal stability are of interest (M. Evonuk et al. 2002, in preparation). Note that since dynamical stability can be influenced by damping processes as well as higher moments of matter distributions, it is often vital to include such effects when studying stability. The averaged code is particularly useful when the period of the ‘‘ tidal ’’ orbit is much smaller than other orbital periods in the system. Examples include (1) the early and present Earth-Moon-Sun-Jupiter system (12 : 1 [present]; 20 Myr); (2) the Pluto-Charon-Sun system (with Neptune if necessary; 13,619 : 1; 4.5 Gyr; D. N. C. Lin & R. A. Mardling 2002a, in preparation); (3) Upsilon Andromedae and its system of three planets (50 : 1; 12 Myr), which includes the evolution of the radius of the innermost planet (R. A. Mardling & D. N. C. Lin 2002); and (4) the highly eccentric system HD 80606 (e ¼ 0:927, periastron separation 0.03 AU; Naef et al. 2001) together with a binary companion star capable of pumping up the eccentricity via the Kozai mechanism (R. A. Mardling & D. N. C. Lin 2002b, in preparation). Here the numbers in parentheses refer to the period ratios of the inner two orbits and the time reached in 9 hr of CPU time on a Compaq ES40 (833 MHz), respectively, the latter reflecting the period of the orbit that governs the size of the integration time step (the second orbit). In addition, the codes may be used to study purely stellar systems. We now present some tests and examples of applications of the two codes. 6.1. A Comparison of the Direct and Averaged Codes: Point Masses Figure 3 shows the evolution of the eccentricities of a
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Fig. 3.—Comparison of the ([a] and [c]) direct and ([b] and [d]) averaged codes for a point-mass hierarchy of four equal-mass bodies whose initial eccentricities are zero and whose initial semimajor axes are 1, 6, and 30. The innermost orbit (solid lines) is initially inclined by 49 to the outermost orbit, while the middle orbit (dot-dashed lines) is coplanar with the outermost orbit (dotted lines). Panels (c) and (d ) compare the relative inclination of the inner two orbits. See text for discussion.
point-mass hierarchy of four equal-mass bodies whose initial eccentricities are zero and whose initial semimajor axes are 1, 6, and 30. The innermost orbit is inclined by 49 to the outermost orbit, while the middle orbit is coplanar with the outermost orbit. The innermost eccentricity is substantially excited in the direct case (Fig. 3a) but is hardly excited when the innermost orbit is averaged. The eccentricity of the middle orbit is similar in both cases, although in the direct case, its modulation reflects that of the innermost orbit (the modulation period is the inner orbital period). The outermost orbit is essentially the same for both cases. Figures 3c and 3d show the evolution of the relative inclination of the orbits. These are similar in both cases although it is modulated on a timescale of half an inner orbital period in the direct case. Figure 4 shows the evolution of the eccentricities for the same configuration as in Figure 3, but for a planetary system consisting of a solar mass star and three Jupiter mass planets. Here the amplitude of the innermost eccentricity is similar in both cases, although in the direct case it is modulated on the timescale of the inner orbital period. 6.2. Spin-Orbit Coupling and Tidal Damping in a Four-Body System: The Primitive Earth-Moon-Sun-Jupiter System In order to illustrate the effects of spin-orbit coupling and tidal damping in a four-body system, we consider two hypothetical primitive Earth-Moon-Sun-Jupiter systems, which
are calculated for 22 Myr using the averaging code. Since this only involves an averaging over the lunar orbit, it does not preclude the effects of secular mean motion resonances. Such resonances have been shown to be important (Touma & Wisdom 1998), and since capture into resonance is a hysteretic process, models that are integrated backward in time are not reliable (see, e.g., Touma & Wisdom 1994). Finally, in order to verify the validity of the code, we use the initial conditions of Touma & Wisdom (1998) to reproduce their results regarding capture into the evection and eviction resonances. The former occurs when the period of apsidal advance of the Moon’s orbit is around 1 yr. The initial total angular momentum of the Earth-Moon system is taken to be its present-day value, as is the orbit of its barycenter around the Sun. The orbital and intrinsic parameters we have chosen for the initial configurations are listed in Table 1, the only difference between the two models being the Love number of the Earth. Figures 5, 6, and 7 show the evolution of various orbital elements and angles for model 1. Figure 5a shows the evolution of the spin periods of the Earth and the Moon (TE and TM, respectively), together with the lunar orbital period Torb. The orbital period increases rapidly as the Earth spins down via the tidal torque of the Moon, releasing its angular momentum to the orbit. The Moon rapidly becomes quasi-synchronous at perigee with a rotation period of around 70 hr, while the Earth’s rotation period increases from 7.45 to only 11 hr in the time shown. Figure 5b compares the component of
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Fig. 4.—Same configuration as in Fig. 2, but for a planetary system consisting of a solar mass star and three Jupiter mass planets
angular momentum perpendicular to the orbit for the Earth’s spin and the Earth-Moon orbit, as a fraction of the magnitude of the total angular momentum of the EarthMoon system. Initially the Earth’s spin dominates, but they swap over at around 1:2 105 yr. These quantities start to oscillate at around 15 Myr, corresponding to the largeamplitude variation in the Earth’s obliquity (Fig. 6c). Figure 5c shows the growth of the semimajor axis as the orbit gains angular momentum and energy from the spin of the Earth (and to a minor extent, the Moon). Figure 5d shows the evolution of the eccentricity of the Moon’s orbit around the Earth eM (discussed in the next paragraph), as well as the Earth’s orbit around the Sun eE. The latter is modulated by the secular perturbation from Jupiter and has a modulation period of around 186,000 yr. Figure 6a plots the obliquity of the Earth relative to the ecliptic that oscillates around a mean value equal to the initial value (6 ). The libration period and amplitude increase until around 15 Myr, at which time the Earth ‘‘ falls over ’’ (relative to the ecliptic) to oscillate around a mean value of around 34 , the amplitude and period from then on decreasing. Figure 6b shows the inclination of the Moon’s orbit relative to the ecliptic, the period of nutation due to the Sun’s torque (here not fully resolved) being of the order of 200 yr. Figure 6c shows the obliquities of the Earth and Moon relative to the lunar orbit, the short-period oscillations reflecting the nutation of the Moon’s orbit. The Moon’s obliquity reduces swiftly from 70 to an average quasi-equilibrium of
around 3 . Figure 6d plots the Moon’s eccentricity as a function of h/n, where h is the component of the Earth’s spin vector in the direction of the lunar orbit normal and n is the lunar mean motion. The Moon’s eccentricity is modulated by the Sun’s perturbation, the amplitude of which increases as the orbit widens. The mean eccentricity will continue to increase until the sign of e_ M ¼ ^e x e_ changes (see eq. [43]). For this model, the contribution from tidal friction, 5 54kn m2 S1 ^e x gTF e_ TF ¼ Q m1 a 11 h e f1 ðeÞ þ f2 ðeÞ ; ð107Þ 18 n dominates the evolution of eM, this term depending on the current eccentricity as well as h/n. Thus, in the same panel we also plot the critical value of the eccentricity ecrit at which e_ TF can change sign as a function of h/n (see eq. [52]). That is, given a value of h/n, ecrit satisfies 11 h f1 ðecrit Þ þ f2 ðecrit Þ ¼0: ð108Þ 18 n Corresponding to the severe change in average obliquity of the Earth at 15 Myr, h/n begins to oscillate, as can be seen in Figure 7a. Although the calculation ends at 22 Myr, we can guess at the subsequent evolution based on the behavior of models for which the evolution is much faster. Eventually
TABLE 1 Data for Three Primitive Earth-Moon-Sun Systems Parameter
Model 1
Eccentricity............................................... Perigee separation (R) ............................. Lunar orbital perioda ................................ Spin period of Earth (hr) ........................... Spin period of Moon (hr) .......................... Lunar obliquity of Earth (deg) .................. Lunar obliquity of Moon (deg) ................. Relative inclination of orbits (deg) ............ Q-value of Earth ....................................... Q-value of Moon....................................... Love number of Earth............................... Love number of Moon ..............................
0.1 10 2.16 7.45 6 0 70 6 10 100 0.04 0.004
a
Model 2
0.0004
Test Model 0.01 3.5 9.2 5.2 Synchronous 0 0 10 11000 105 0.947 0.095
Value for model 1 is in units of days, whereas value for test model is in hours.
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Fig. 5.—Model 1: (a) lunar orbital period Torb and spin periods of the Moon TM and Earth TE; (b) lunar orbital angular momentum Jorb and spin angular momentum of Earth JE, as fractions of the magnitude of the total angular momentum of Earth-Moon system Jtot; (c) expansion of the lunar semimajor axis, aM, in units of R, SE; (d ) orbital eccentricities of the Moon eM and the Earth eE.
the eccentricity will become greater than ecrit and its average rate of change will become negative. It will reduce in such a way that it stays above the curve for ecrit, involving also a reduction in h/n. The latter reduction occurs because the semimajor axis starts to shrink in response to a small (minimum) perigee separation (Fig. 7b). It is not clear, however, how the Earth’s obliquity will evolve, if indeed the system remains stable. The corresponding set of plots for model 2 shown in Figures 8 and 9 dramatically illustrates the effect of reducing the Love number (or, equivalently, increasing the Q-value) of the Earth, corresponding to a more rigid Earth and (or) a slower damping rate (see eq. [52]). In model 1 dissipation in the Earth dominates the evolution of the eccentricity, while in model 2 dissipation in the Moon dominates. Our chosen values for the quadrupole Love numbers should be compared to the accepted present-day values of kE ¼ 0:299 and kM ¼ 0:030 (Yoder 1995). The eccentricity of the lunar orbit (Fig. 8a) is now dominated by the Sun’s perturbation so that jh^e x goct ij > jh^e x gTF ij (see eq. [43]). The general trend is for eM to reduce since, on average, the quantity sinð$i $o Þ is positive (see eq. [97]). The Earth slowly surrenders its spin angular momentum to the orbit (Fig. 8b), with the latter assuming dominance at around 6 Myr. The spin of the Moon very quickly becomes quasi-synchronous with the orbit (Fig. 8c). While the rate at which the orbit widens is reduced by a factor of around 13 compared to model 1, one of the most dramatic differences between the two models is the obliquity of the Earth relative to the ecliptic (Fig. 8d).
The average value is around 18=5 (not unlike the presentday value of 23=5), and the modulation period and amplitude decrease exponentially with an e-folding time of around 13.6 Myr. This is in contrast to model 1 where initially the amplitude and period of the obliquity modulation increase until the system undergoes a kind of phase change so that the behavior becomes as in model 2 (but with an average obliquity of 34 ). Finally, in Figure 9a we plot the lunar obliquities of the Earth and Moon. The small oscillations superimposed on the Earth’s lunar obliquity correspond to the modulation of the lunar orbit produced by the Sun’s torque (Fig. 9b), while these are absent from the Moon’s lunar obliquity. This simply indicates that the inertial orientation of the Earth is dominated by the Sun’s torque rather than the Moon’s, while the Moon responds preferentially to that of the Earth. It should be stressed that the models above have been used for illustration only, and any study of the early EarthMoon system would require a much more thorough investigation. However, it is clear that the present numerical schemes are useful for (for example) parameter-space studies of the structural and dynamical history of planetary and satellite systems. Such a study is presently being undertaken by the authors for the Pluto-Charon-Sun system. 6.2.1. A Numerical Test
In order to demonstrate the validity of the codes, we reproduce some of the results of Touma & Wisdom (1998),
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Fig. 6.—Model 1: (a) ecliptic obliquity of the Earth; (b) inclination of the lunar orbit to the ecliptic; (c) lunar orbit obliquities of the Earth and Moon; (d ) Moon’s eccentricity eM as a function of the component of the Earth’s spin in the direction of the orbit normal h, in units of the mean motion n. Also plotted in panel (d ) is the critical eccentricity ecrit, above which the rate of change of the eccentricity due to tides is negative (see text).
in which the authors study resonances in the early EarthMoon system. The Moon is initially in the equatorial plane of the Earth with an eccentricity of 0.01 and a semimajor axis of 3.5 R. The initial ecliptic obliquity is 10 , and the initial rotation period of the Earth is 5.2 hr (see Touma &
Wisdom 1998). The Love number of the Earth is chosen to give the known value of J2 (Touma & Wisdom 1994, eq. [56]). The Love number of the Moon is taken to be an order of magnitude smaller. The Q-value of the Earth is taken to be such that the initial recession rate of the Earth is 1 km
Fig. 7.—Model 1: (a) ratio h/n as a function of time; (b) lunar perigee separation in units of R
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Fig. 8.—Model 2: (a) orbital eccentricities of the Moon eM and the Earth eE; (b) lunar orbital angular momentum Jorb and spin angular momentum of the Earth JE, as fractions of the magnitude of the total angular momentum of the Earth-Moon system Jtot; (c) lunar orbital period Torb and spin periods of the Moon TM and Earth TE; (d ) ecliptic obliquity of the Earth.
yr1. The parameters are listed in Table 1. Figures 10a and 10b plot the Moon’s eccentricity and inclination to the Earth’s equator as functions of the lunar semimajor axis (in units of R) and show the passage of the system through the evection and eviction resonances, respectively. The averaged code was used to produce these plots. Extreme evection occurs when the period of apsidal advance of the Moon’s orbit is around 1 yr and is thus a secular mean motion resonance. Eviction (excitation of the inclination of the Moon’s orbit to the spin axis of the Earth, i.e., the Earth’s lunar
obliquity) occurs following escape from the evection resonance and is associated with the large eccentricity at this point. Simple models of these processes may be found in Touma & Wisdom (1998). Figures 10c and 10d plot the same quantities except that the data were produced by the direct code. The results are essentially identical except for a few details corresponding to escape from resonance. The plots should be compared with Figures 1 and 2 of Touma & Wisdom (1998). There are slight differences that reflect the fact that precise values for Love numbers and Q-values are
Fig. 9.—Model 2: (a) lunar orbit obliquities of the Earth and Moon; (b) inclination of the lunar orbit to the ecliptic
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Fig. 10.—Test of the codes. (a) and (b) averaged code; (c) and (d ) direct code. (a) and (c) evection resonance; (b) and (d ) eviction resonance.
not supplied in that paper; however, it is clear that the present codes are reliable.
7. SUMMARY
We have presented two powerful methods for calculating the complex tidal, spin, and dynamical evolution of extrasolar planetary systems. In the first scheme the equations of motion for the orbits and spins are integrated directly, allowing for any dynamical instabilities and resonances to develop naturally. Nonresonant systems for which the ratio of the two innermost orbital periods is small are more efficiently integrated using the second scheme. This involves calculating the evolution of the single-averaged specific angular momentum and Runge-Lenz and spin vectors, as well as directly integrating any other orbits in the system. Both schemes treat the inner two bodies as extended objects, endowing them with damping efficiencies, structure via moments of inertia and apsidal motion constants or (in the case of bodies with some rigidity) Love numbers, and variable spins and obliquities. In addition, the general relativistic post-Newtonian potential of one body is included. As at the time of writing, these numerical schemes are best suited for short- to medium-term studies, except in cases in which an orbital period ratio is extreme, as it is for the Pluto-Charon-Sun system. However, as CPUs become faster (and at the same time as more planetary systems are discovered), these schemes will become more and more useful.
We are currently developing modified versions of these codes that allow for the evolutionary effects of a protoplanetary disk. Other possible variations include allowing the tidal damping efficiencies (the Q-values) to vary in time, as well as the apsidal motion constants or Love numbers. In particular, these structure constants could be made to depend on their associated Q-values. Permanent quadrupole moments can also be included. General relativistic (nonplanetary) systems involving the spin of one or more bodies for which energy and angular momentum loss via gravitational waves is important may be studied by including higher order relativistic terms (see, e.g., Kidder 1995). Additional bodies may be added using the Jacobi coordinate system. Additional structured bodies may easily be added in the direct formulation; however, special care would have to be taken in the case of the averaged formulation because different averaging periods would be involved. Finally, we would like to emphasize once again that the schemes presented here are entirely general and can be used for any system of bodies. We thank S. Aarseth, P. Eggleton, and D. Heggie for useful conversations and the referee for useful suggestions. This research has been supported in part by NSF through grants AST 96-18548 and AST 99-87417; NASA through grants NAG 5-7515, NAG 5-8196, and NAG 5-10727; and an astrophysics theory program that supports a joint Center for Star Formation Studies at NASA Ames Research Center, University of California at Berkeley, and University of California at Santa Cruz.
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