ON THE MIGRATION OF JUPITER AND SATURN: CONSTRAINTS FROM LINEAR MODELS OF SECULAR RESONANT COUPLING WITH THE TERRESTRIAL PLANETS

The apsidal orbital precession of a system of planets can be described using a set of linear first-order differential equations, whose solution is a system of eigenmodes (see, e.g., Brouwer & Clemence 1961; Murray & Dermott 1999). The equations of motion for the evolution of the planets' eccentricities (e) and longitude of perihelia (ϖ) can be constructed from the disturbing function by assuming that terms involving the mean longitudes are small and retaining lowest-order terms in the eccentricity and inclinations. For planet i, Lagrange's equations for the eccentricity vector (h_i, k_i) = e_i(sin ϖ_i, cos ϖ_i) are

$$\begin{equation} \frac{dh_{i}}{dt} = \sum _{l} A_{il}k_{l} \end{equation} \tag{ 1 }$$

$$\begin{equation} \frac{dk_{i}}{dt} = - \sum _{l} A_{il}h_{l}, \end{equation} \tag{ 2 }$$

where

$$\begin{equation} A_{ii}= \frac{n_{i}a_{i}}{4} \sum _{l\ne i} \left(\frac{m_l}{M_{\odot }+m_i} \right) \frac{\alpha }{ a_{>} } b_{3/2}^{1} + \frac{3GM_{\odot }}{c^2a_{i}} n_i \end{equation} \tag{ 3 }$$

$$\begin{equation} A_{il} = - \frac{n_i a_i}{4} \left(\frac{m_l}{M_{\odot } + m_i} \right) \frac{\alpha }{ a_{>} } b_{3/2}^{2}, \end{equation} \tag{ 4 }$$

and m_i, a_i, and n_i are a planet's mass, semimajor axis, and mean motion, respectively. The standard terms where α = a_</a_>, a_> = max(a_i, a_l), a_< = min(a_i, a_l), and Laplace coefficients (b^s_j) have been used. Post-Newtonian corrections for relativistic precession are included in the diagonal matrix elements where c is the speed of light (see, e.g., Adams & Laughlin 2006).

Solving these equations for a planetary system is an eigenvalue problem whose eigenfrequencies (g_j) are real and represent the precession frequencies of the modes. The solutions are of the form

$$\begin{equation} h_{i} = \sum _{j} e_{ij}\sin (g_{j}t + \beta _{j}) \end{equation} \tag{ 5 }$$

$$\begin{equation} k_{i} = \sum _{j} e_{ij}\cos (g_{j}t + \beta _{j}), \end{equation} \tag{ 6 }$$

where the values of modal eccentricity amplitudes e_ij and the initial phases of the modes β_j are constants determined by projecting the system onto some initial state at time (t = 0).

In our use of this secular model there are two convenient and complementary formulations of the eigenvalue solution, each with its own advantages. The modal eccentricity amplitude (e_ij) solution (above) clearly identifies the orbital variation of each planet i resulting from the coupling with the other planets via mode j. As the orbital elements of a system are its principal dynamical coordinates, this formulation is easily interpreted and can be compared with the results of N-body simulations (e.g., using Fourier analysis to identify fundamental frequencies and amplitudes). However, this representation of the system's modes tends to mask the structural signature of the eigenvectors and the physical interpretation of a mode's amplitude.

When considering the secular evolution of an isolated system the orbital energies and semimajor axes of the planets are constant and the system angular momentum is conserved. These two constraints may be combined to identify the so-called angular momentum deficit (AMD) as an important integral of a secularly evolving system. The AMD is defined as

$$\begin{equation} \mbox{AMD} = \sum _i m_i n_i a_i^2\big(1-\sqrt{1-e_i^2}\cos I_i\big), \end{equation} \tag{ 7 }$$

where i runs over the planets and I_i is a planet's orbital inclination (see, e.g., Laskar 1996). Physically, the AMD represents the angular momentum that must be added to the system to make all the planets' orbits circular and coplanar. Because secular interactions do not alter the semimajor axes of the planets, a system's AMD quantifies the amount of angular momentum that is available for exchange between planets via gravitational interactions and limits the amplitude of eccentricity and inclination oscillations of the planets.

The linear secular system can also be formulated with the following transformation of variables and matrix elements:

$$\begin{equation} H_i = \sqrt{m_i n_i a_i^2} e_i\sin \varpi _i = \psi _i h_i \end{equation} \tag{ 8 }$$

$$\begin{equation} K_i = \sqrt{m_i n_i a_i^2} e_i\cos \varpi _i = \psi _i k_i, \end{equation} \tag{ 9 }$$

and A_il → (ψ_i/ψ_l)A_il, where the factor ψ_i is the square root of a planet's circular orbital angular momentum (see, e.g., Brouwer & Clemence 1961).

The system's solution is again expressed as a superposition of the modes

$$\begin{equation} H_i \equiv \sum _j H_{ij} = \sum _j C_j v_{ij} \sin (g_jt + \beta _j) \end{equation} \tag{ 10 }$$

$$\begin{equation} K_i \equiv \sum _j K_{ij} = \sum _j C_j v_{ij} \cos (g_jt + \beta _j), \end{equation} \tag{ 11 }$$

where $\mathbf {v}_j\equiv \lbrace v_{ij}\rbrace$ is the jth orthonormal eigenvector and C_j its modal amplitude computed from a set of orbital elements and masses. The system quantity $\frac{1}{2}\sum _im_in_ia_i^2e_i^2 = \frac{1}{2}(H_i^2 + K_i^2)^2=\frac{1}{2}\sum _jC_j^2$ corresponds to the horizontal component of the system's AMD in our secular model. The secular modal amplitudes {C_j} are integrals of the system and reveal the fraction of the system AMD partitioned into a single mode. Conversion back and forth between the two formulations is easily accomplished using

$$\begin{equation} e_{ij} = \frac{C_j v_{ij}}{\psi _i}. \end{equation} \tag{ 12 }$$

As the mass distribution of a planetary system evolves (e.g., due to planetary migration, stellar spin down, dispersal of the gaseous nebula, etc.), the system's eigenfrequencies, eigenvectors, and modal amplitudes become time-dependent. However, if the dynamical tuning of the system is slow with respect to the eigenfrequencies {g_j} and the system is not near a secular resonance (i.e., g_j − g_k is not small), the variation in mode amplitudes tends to oscillate rapidly and average out (see Ward 1981, for a complete development). For slow enough changes in the system, the modal amplitudes {C_j} can be considered as effectively constant. We exploit this convenient aspect of this formulation when exploring the consequences of secular resonant sweeping during planetary migration.

We have calculated the normal mode solution for the eccentricity evolution of a six-planet system consisting of Jupiter and Saturn and the terrestrial planets. Uranus and Neptune have been omitted from this treatment for simplicity as their influence on Jupiter and Saturn is weak and not efficiently communicated to the inner solar system.

The system's orthonormal eigenvectors $(\mathbf {v}_j)$ are shown in Figure 1 with their associated eigenfrequencies (g_j). As is the convention for the solar system, the eigenmode index (j) identifies the planet that has the largest component in the eigenvector (i.e., Mercury has the largest component in the j = 1 mode and its evolution is strongly governed by this mode, Venus has the largest component in the j = 2 mode, etc.). Large components for multiple planets in a single eigenvector indicate strong coupling between those planets. The sign of an eigenvector component (v_ij) indicates the relative apsidal orientation within the mode, with like signs indicating alignment and opposite signs anti-alignment.

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The eigenvectors of this six-planet system can be divided into two groups, those where the terrestrial planets have the largest components (j = 1–4) and those where Jupiter and Saturn have the dominant components (j = 5–6). This trend in the structure of the eigenvectors is indicative of the weak coupling between the terrestrial and giant planets in their current configuration. The eigenvector signatures of the terrestrial modes (j = 1–4) each have large components for at least two planets indicating significant coupling in their evolution.

Using the VSOP82 ephemeris of Bretagnon (1982), we have computed the initial amplitudes and phases of this system's normal modes. The fraction of the system's AMD present in each mode (C²_j/2) is shown in Figure 2. The modal eccentricity amplitudes are listed in Table 1. The j = 5, 6 modes of Jupiter and Saturn contain the majority of the system's AMD.⁴ Among the modes that control the evolution of the terrestrial planets the j = 1 and j = 4 mode amplitudes contain ∼85% of the terrestrial subsystem's AMD (i.e., the sum Σ⁴_{j = 1}C_j²/2). The j = 1 and j = 4 modes strongly affect the evolution of Mars and Mercury and are responsible for the large eccentricities of these planets. Similarly, the j = 2, 3 modes strongly affect the evolution of Venus and Earth. The relatively small amplitudes of these modes are manifested as smaller eccentricities of Venus and Earth.

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This dynamical configuration and partitioning of the AMD between the terrestrial planets is peculiar. The terrestrial eigenfrequencies with the largest amplitudes are both in close proximity to an eigenfrequency with a much smaller amplitude. For example, the j = 1 and j = 2 frequencies are close with g₁ = 586 yr⁻¹ and g₂ = 742 yr⁻¹, and the amplitude ratio is C₁/C₂ ≈ 5. Similarly, the j = 3 and j = 4 modes that strongly affect Earth and Mars are close in value with g₃ = 1739 yr⁻¹ and g₄ = 1804 yr⁻¹ and have an amplitude ratio of C₄/C₃ ≈ 2.

The frequencies of the terrestrial modes (i.e., j = 1–4) all take values between those that dominate the present-day evolution of Jupiter and Saturn (i.e., g₅ < g_1–4 < g₆). As a result the giant planet precession frequencies need not be grossly modified to resonate with terrestrial modes (e.g., g₅ = g₁). Meteorites and asteroids placed in or near a secular resonance with the giant planets may be strongly perturbed to eccentric orbits (Farinella et al. 1993), and may even be driven into the Sun (e.g., Gladman et al. 1997; Chambers & Wetherill 1998).

Linear secular theory has a number of properties that make it a useful tool for exploring the implications of giant planet migration. As described above, the secular frequencies {g_j} and eigenvectors $\lbrace \mathbf {v}_j\rbrace$ are functions of the planet masses {m_i} and semimajor axes {a_i}. This allows us to compute the evolution of the system's eigenfrequencies, and their sweeping through the solar system, for various orbital migration histories of the planets. When a system's mass distribution does not change, its mode amplitudes {C_j} are integrals of the system's time evolution. For the solar system, the planetary precession periods are ∼10⁴–10⁵ yr. As long as the mass distribution of the system evolves on longer timescales and the system is not near resonance, the modal amplitudes {C_j} may be considered effectively constant (Ward 1981).

In the standard Laplace–Lagrange model, the direct terms of the disturbing function containing the mean longitudes and all the indirect terms in the disturbing function may be neglected when averaging over long timescales. However, near commensurabilities between the mean motions of planets, resonant terms in the disturbing function can make significant contributions to orbit precession. A well-known example of this in the solar system is the so-called Great Inequality which results from the close proximity of Jupiter and Saturn to their mutual 5:2 MMR. Jupiter and Saturn's nearness to this resonance modifies the g₅ and g₆ frequencies by ≈ + 05 yr⁻¹ and +6'' yr⁻¹, respectively (see, e.g., Brouwer & Clemence 1961; Murray & Dermott 1999). The lowest-order mean motion resonances encountered during the divergent migration of Jupiter and Saturn are the 2:1 and 3:2. Both are first-order resonances and may strongly affect the secular evolution of the planets.

Malhotra et al. (1989) developed an approach for including the influence of first-order mean motion resonances in the secular model and used it to account for the evolution of the Uranian satellites. Their treatment of the problem includes the averaged influence of the direct terms from the disturbing function related to first-order (ℓ: ℓ − 1) mean motion resonances. The 2:1 is unusual among the first-order resonances as it also gives rise to indirect terms in the disturbing function of both planets (see Murray & Dermott 1999).

Figure 3 shows the apsidal eigenfrequencies of Jupiter and Saturn as a function of Saturn's semimajor axis determined using several different methods. In each case, Jupiter's semimajor axis was held constant at 5.2 AU; the semimajor axis of Saturn was increased from just outside the mutual 3:2 mean motion resonance at about 6.8 AU, across the 2:1 at 8.25 AU, to 9.7 AU.

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To determine the eigenfrequencies of the Jupiter–Saturn system as a function of their semimajor axes we performed a set of individual N-body simulations, each with a different initial semimajor axis for Saturn. In each simulation the planets were assigned nearly circular, nearly coplanar orbits (i.e., e ≃ 0.005, I = e/2), random phase elements and were evolved for about 11 Myr. The SyMBA integrator (Duncan et al. 1998) was used in these simulations and the precessional effects of general relativity were included (e.g., Quinn et al. 1991).

We determined the eccentricity amplitudes (e_ij) and eigenfrequencies of the system by analyzing the evolution of a planet's eccentricity vector (h_i, k_i) using a Frequency Modified Fourier Transform code (hereafter FMFT; Sidlichovsky & Nesvorny 1997).⁵ For each simulation the two frequencies with the largest amplitudes in the evolution of Jupiter's eccentricity vector (h₅, k₅) are shown in Figure 3 with gray triangles. The close proximity of the measured frequencies to the eigenfrequencies of the Jupiter–Saturn system identifies them as the g₅ and g₆. Since the N-body simulations include all gravitational interactions between the planets, resonant, and otherwise, these FMFT results serve as "measurements" of the apsidal eigenfrequencies of the actual system and can be used to assess the fidelity of more approximate, but analytically tractable, treatments.

As Saturn's orbit is expanded, the values of the g₅ and g₆ frequencies show two important characteristics. First, near both the 3:2 and 2:1 MMRs the value of the g₆ frequency diverges. The behavior of the g₅ is somewhat different, showing divergence behavior only near the 2:1 mean motion resonance. For orbital separations outside the 2:1 resonance (i.e., the semimajor axis of Saturn greater than 8.25 AU in this case), the two eigenfrequencies decrease as Jupiter and Saturn continue to diverge.

We have used secular theory to compute the modal frequencies in three different ways. The standard Laplace–Lagrange treatment is shown as a gray solid line in both figures. This treatment contains no information regarding mean longitudes of the planets and fails to account for divergence in the g_{5, 6} frequencies near the 3:2 and 2:1 mean motion resonances.

We also computed the apsidal eigenfrequencies of Jupiter and Saturn, correcting the Laplace–Lagrange treatment with the direct terms of Malhotra et al. (1989). The value of g₅ and g₆ are shown with a dash-dotted line. The corrections due to the direct terms account for divergence of the g₆ frequency when approaching the 3:2 or 2:1 resonances, but fail to account for the changes in the g₅ frequency due to the 2:1 resonance.

Finally, we compute the apsidal eigenfrequencies using the correction of both direct and indirect terms from the first-order mean motion resonances. The contributions of the indirect terms associated with the 2:1 MMR may be included in the model of Malhotra et al. (1989) by adding −2αδ_{j, 1} to their Equation (26b), where δ_j1 is the Kronecker delta function (R. Malhotra 2011, private communication).⁶ As the 2:1 is the only first-order resonance with indirect terms it is also the only resonance affected by this amended theory.

The inclusion of the indirect terms dramatically improves the agreement between the secular frequencies computed with FMFT-measured values and the corrected secular theory and accounts for the behavior of the g₅ frequency around the 2:1 MMR. This model assumes that the resonant argument is rapidly circulating and consequently becomes less accurate near the resonant separatrix where the circulation rate slows. The continuous behavior of the corrected g₅ frequency as the 2:1 MMR is crossed, its inflection and local maxima are model artifacts of these assumptions. For Jupiter and Saturn more than ∼0.1 AU outside the 2:1 MMR resonance, the results of the N-body simulations and the modified secular theory are in close agreement.

The influence of additional mean motion resonances between Jupiter and Saturn are also visible in FMFT measurements of g₅ and g₆. These appear as deviation from the smooth general trends indicated by secular theory. The effects of the 5:3 and 7:4 MMRs when Saturn is near 7.3 and 7.55 AU and the 5:2 MMR as Saturn approaches its current orbit at 9.54 AU are evident. These second- and third-order resonances are relatively weak compared to the 3:2 and 2:1 and their influence on the eigenfrequencies is both smaller and localized to configurations close to exact resonance.

These results clearly demonstrate that the divergent migration of Jupiter and Saturn across their 2:1 MMR, as suggested by Tsiganis et al. (2005) and Gomes et al. (2005), results in secular resonant sweeping of the g₅ across the values of all four of the current terrestrial eigenfrequencies. Thus, this migration scenario may subject the terrestrial planets to potentially destabilizing secular resonant perturbations.

Consider a test particle in a system where planetary migration causes its orbital precession frequency (g) to pass through a resonance with one of the system's eigenfrequencies (g_j). In a reference frame that rotates with frequency g_j, the equations of motion of h = esin (ϖ − g_jt − β_j) and k = ecos (ϖ − g_jt − β_j) for the planetesimal interacting with the resonant mode are

$$\begin{equation} \frac{dh}{dt}-(g-g_{j})k = \mu _{j} \end{equation} \tag{ 13 }$$

$$\begin{equation} \frac{dk}{dt}+(g-g_{j})h = 0, \end{equation} \tag{ 14 }$$

where

$$\begin{equation} \mu _{j} = \frac{n a}{4} \sum _i e_{ij} \left(\frac{m_i}{M_{\odot }}\right) \frac{\alpha }{a_{>}} b_{3/2}^2. \end{equation} \tag{ 15 }$$

An analytic solution for resonance passage can be obtained by introducing a new independent variable ϕ ≡ ∫Δgdt, where Δg = g − g_j and transforming Equations (13) and (14) into

$$\begin{equation} \frac{dh}{d\phi } - k = \frac{\mu _{j}}{\Delta g} \end{equation} \tag{ 16 }$$

$$\begin{equation} h + \frac{dk}{d \phi }=0. \end{equation} \tag{ 17 }$$

The particular solution of this expression is

$$\begin{equation} h = (C_1 + c)\cos \phi + (S_1 + s) \sin \phi \end{equation} \tag{ 18 }$$

$$\begin{equation} k = -(C_1 + c)\sin \phi + (S_1 + s) \cos \phi \end{equation} \tag{ 19 }$$

with

$$\begin{equation} C_1 = \int \left(\frac{\mu _j}{\Delta g} \right) \cos \phi \, d\phi \end{equation} \tag{ 20 }$$

$$\begin{equation} S_1 = \int \left(\frac{\mu _j}{\Delta g} \right) \sin \phi \, d\phi. \end{equation} \tag{ 21 }$$

Defining t = 0 as the time of exact resonance (g = g_j) and setting $\Delta \dot{g} =\dot{g} -\dot{g}_j$ where $\dot{g}_j=|dg_j/dt|$, the variable $\phi =-t^{\prime 2}=-\frac{1}{2}\Delta \dot{g}t^2=-\frac{1}{2}(\Delta g)^2/\Delta \dot{g}$, the solutions for h, k following resonance passage are

$$\begin{eqnarray} h_{{\rm res}} &=& {-}\mu _{j} \left(\frac{\pi }{\Delta \dot{g}} \right)^{1/2}\nonumber\\ &&\times \left[ \left(C_1(t^{\prime }) + \frac{1}{2} \right) \cos t^{\prime 2} + \left(S_1(t^{\prime }) + \frac{1}{2} \right) \sin t^{\prime 2} \right]\qquad \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} k_{{\rm res}} &=& \mu _{j} \left(\frac{\pi }{\Delta \dot{g}} \right)^{1/2} \nonumber\\ &&\times\left[ \left(S_1(t^{\prime }) + \frac{1}{2} \right) \cos t^{\prime 2} - \left(C_1(t^{\prime }) + \frac{1}{2} \right) \sin t^{\prime 2} \right], \qquad \end{eqnarray} \tag{ 23 }$$

where S₁ and C₁ are the Fresnel sine and cosine integrals and we have required that h, k → 0 as t → −∞.

Long after resonance passage, as t → ∞, the test particle's new free eccentricity is

$$\begin{equation} e_{{\rm res}} = \mu _{j} \left(\frac{2 \pi }{|\Delta \dot{g}|} \right)^{1/2}. \end{equation} \tag{ 24 }$$

In essence, resonance passage excites a new contribution to the free eccentricity of the body. The amplitude of this response is determined by the rate of passage through resonance and is a function of the migration rate. It is convenient to use an exponentially decaying migration rate of the form

$$\begin{equation} \frac{da_i}{dt}=\frac{\Delta a_i}{\tau }, \end{equation} \tag{ 25 }$$

where Δa_i = a_{i, f} − a_i is the distance of the planet from its final semimajor axis (a_{i, f}) and τ is the characteristic migration timescale. We treat τ as a global migration constant and use this to parameterize migration speed. We note that Equation (25) can also be used to convert between analytic constraints, that are easily expressed as an exponential timescale (τ), and migration rates (da_i/dt) which might be more easily measured from N-body simulations of planetary migration. Using the exponential migration model,

$$\begin{equation} \Delta \dot{g} = \sum _i \left(\frac{dg}{da_i} - \frac{dg_j}{da_i} \right) \frac{\Delta a_i}{\tau }, \end{equation} \tag{ 26 }$$

where all terms except the timescale are functions of the planetary masses and semimajor axes at the time of resonance. This can also be used to identify a critical migration timescale (τ*) required to excite the eccentricity to some prescribed value e_res

$$\begin{equation} \tau ^{*} = \frac{1}{2\pi } \left|\sum _i \left(\frac{dg}{da_i} - \frac{dg_j}{da_i} \right) \Delta a_i \right|_r \left(\frac{e_{{\rm res}}}{\mu _j} \right)^2, \end{equation} \tag{ 27 }$$

where the mass and orbital state of the perturbing planets are encapsulated in μ_j (see Equation (15)). If the test particle's free eccentricity prior to resonance passage was zero, then Equation (24) is the new eccentricity after passage of the secular resonance. However, when the free eccentricity is not initially zero, the initial secular coordinates h_o, k_o add vectorally with those obtained from resonance passage h_res, k_res. The two contributions may interfere constructively or destructively depending on their relative phase γ (Ward et al. 1976; Minton & Malhotra 2008) and the resulting eccentricity can be determined from geometric considerations

$$\begin{equation} e_f = \big(e_o^2 + e_{{\rm res}}^2 + 2e_o e_{{\rm res}} \cos \gamma\big)^{1/2} \end{equation} \tag{ 28 }$$

(Heppenheimer 1980). Recognizing the phase between contributions as effectively random and isotropic allows Equation (28) to be used to as a distribution function of resulting eccentricities. For example, given amplitudes for the initial and resonant eccentricity vectors, γ = 0, π/2, and π yield the maximum, median, and minimum eccentricities expected.

Due to their much smaller mass and modal amplitudes, the terrestrial planets have a negligible influence on the evolution of the giant planets. The subsystem approximation of Ward (1981) treats the terrestrial planets and giant planets as separate and independently evolving secular systems and then examines their interaction. A brief summary of this model is described below (see Ward 1981 for the complete development).

In this treatment the secular evolution matrix $\mathbf {A}$, whose elements are defined in Equations (3) and (4), is divided into submatrices that describe coupling within each terrestrial and giant planet subgroup and between the subgroups

$$\begin{equation} \mathbf {A} = \left(\begin{array}{@{}cc@{}}\mathbf {A}_T & \mathbf {P} \\ \ms \tilde{\mathbf {P}} & \mathbf {A}_G \end{array} \right). \end{equation} \tag{ 29 }$$

Here $\mathbf {A}_{T}$ is the terrestrial 4 × 4 submatrix and $\mathbf {A}_{G}$ is the 2 × 2 submatrix describing Jupiter and Saturn. The direct influence of the giant planets on the terrestrial planets' precession frequencies is included through the diagonal elements of $\mathbf {A}_{T}$ (see Equation (3)). The coupling between the terrestrial and giant planet modes are included through the 4 × 2 and 2 × 4 off-diagonal submatrices of $\mathbf {A}$ and are labeled as $\tilde{\mathbf {P}}$ and $\mathbf {P}$.

The solutions for each subgroup are assumed to be independent. As before we label the solution to the 4 × 4 terrestrial subgroup ($\mathbf {A}_{T}$)

$$\begin{equation} \left\lbrace \begin{array}{@{}c@{}}H\\ \ms K \end{array} \right\rbrace = C_{j} \mathbf {v}_{j} \left\lbrace \begin{array}{@{}c@{}}\sin (g_jt+\beta _j)\\ \ms \cos (g_jt+\beta _j) \end{array} \right\rbrace \end{equation} \tag{ 30 }$$

with j = 1–4. Similarly, the solution for the Jupiter–Saturn subgroup ($\mathbf {A}_{G}$) is

$$\begin{equation} \left\lbrace \begin{array}{@{}c@{}}H\\ \ms K \end{array} \right\rbrace = D_n \mathbf {u}_{n} \left\lbrace \begin{array}{@{}c@{}}\sin (\nu _nt+\delta _n)\\ \ms \cos (\nu _nt+\delta _n) \end{array} \right\rbrace, \end{equation} \tag{ 31 }$$

where ν_n and $\mathbf {u}_n$ are the two-planet eigenfrequencies and eigenvectors, D_n and δ_n are the amplitude and phase of the two-planet modes, and we retain the index labels n = 5, 6 conventionally used for these modes in the solar system. Because the terrestrial planets only weakly affect the precession of the giant planets, the eigenmodes of the two- and four-planet subgroups are nearly identical to those of the six-planet system (i.e., ν_n = g_n, for n = 5, 6) shown in Figures 1 and 2. Hereafter we refer to the giant planet eigenfrequencies with the symbol ν_n.

Figure 4 shows the four eigenfrequencies of the terrestrial subgroup along with the ν₅ frequency as a function of Saturn's semimajor axis. As before, Jupiter is held fixed near its current orbit at 5.2 AU. The terrestrial frequencies are predominantly determined by their mutual interactions and Jupiter's proximity and are only weakly influenced by Saturn's orbital position. This is illustrated in Figure 4 as the terrestrial frequencies (g_1–4 shown in gray) are nearly constant as Saturn's orbital radius expands.

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The ν₅ frequency decreases from about 12'' yr⁻¹ to about 85 yr⁻¹ as Saturn's orbit expands from just outside the 3:2 mean motion resonance with Jupiter to about 7.8 AU. The small gaps in the FMFT measurement for the ν₅ frequency when Saturn is at 7.3 AU and 7.55 AU result from the dispersion of forcing frequencies introduced by the nearby 5:3 and 7:4 MMRs between Jupiter and Saturn. As Saturn's orbit expands beyond 7.8 AU toward the 2:1 with Jupiter near 8.25 AU, the influence of the 2:1 MMR dominates over the linear secular terms and the ν₅ frequency increases from 85 yr⁻¹ → 20'' yr⁻¹. It resonates with the g₃ and g₄ terrestrial frequencies when Saturn is near 8.17 AU and 8.20 AU, respectively.

As Saturn's orbital radius is increased outside the 2:1, the effect of the mean motion resonance weakens with distance and the ν₅ frequency again decreases from about 20'' yr⁻¹ to 4'' yr⁻¹ as Saturn's orbit expands from 8.3 AU to 9.54 AU. The ν₅ resonates a second time with both the g₄ and g₃ just exterior to the 2:1 MMR when Saturn is near 8.30 AU and 8.33 AU. As Saturn's orbit is expanded further, the ν₅ resonates with the g₂ and g₁ when Saturn is near 8.63 AU and 8.79 AU. In total we have identified six individual secular resonances between the ν₅ frequency and a terrestrial eigenmode that occurs as a result of the divergent migration, of Jupiter and Saturn, from just outside their mutual 3:2 mean motion resonance to their present orbits. The semimajor axes of Saturn for each of these resonances is listed in Table 2. In their numerical exploration of this problem, Brasser et al. (2009) parameterized the giant planet configuration at the time of resonance with the ratio of the orbital periods of Saturn to Jupiter (P_S/P_J). Because the value of the period ratio (P_S/P_J) at g_j − ν₅ resonances is weakly dependent on Jupiter's semimajor axis this ratio can be used as a convenient independent variable to approximately identify resonant configurations. Its value for each secular resonance is listed in Table 2.

Table 2. Secular Resonances During the Divergent Migration of Jupiter and Saturn (See, e.g., Figure 4)

Mode		Saturn	Period Ratio	Frequency	Resonance Width	Critical Migration Parameter
j	n	aS	PS/PJ	gj	Fjn	$\hat{\tau }^*_{jn}$
		(AU)		('' yr−1)	('' yr−1)	(Myr)
3	5	8.17	1.968	17.59	0.040	0.598
4	5	8.20	1.979	18.34	−0.028	3.609
4	5	8.30	2.016	18.30	−0.025	1.840
3	5	8.33	2.025	17.57	0.036	0.329
2	5	8.63	2.138	7.51	−0.052	0.050
1	5	8.79	2.196	5.87	0.013	0.683

Note. In this model Jupiter's semimajor axis is constant at 5.2 AU.

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Using the subgroup approximation, Ward (1981) showed that tuning the system through a linear secular resonance between the terrestrial and giant planets (i.e., ν_n = g_j) excites the terrestrial mode amplitude by an amount

$$\begin{equation} C_{j,{\rm res}} = D_n F_{jn} \left(\frac{2\pi }{|\Delta \dot{g}_{jn}|} \right)^{1/2}, \end{equation} \tag{ 32 }$$

where $\Delta \dot{g}_{jn} = \dot{g}_j - \dot{\nu }_n$. Here F_jn is an effective width of the resonance given by

$$\begin{equation} F_{jn}=\tilde{\mathbf {v}}_j \,{\bm \cdot}\, \mathbf {P} \,{\bm \cdot}\, \mathbf {u}_n, \end{equation} \tag{ 33 }$$

where $\tilde{\mathbf {v}}_j$ is the transpose of $\mathbf {v}_j$. Equation (32) is the generalization of the test particle response of resonance passage (Equation (24)) to a four-planet terrestrial subsystem. Here the mode amplitudes C_j, D_n take the place of modal eccentricities, and the functional dependence on the migration rate through $\Delta \dot{g}_{jn}$ is identical to the test particle case.

The AMD exchanged with the terrestrial planets through the resonance is

$$\begin{equation} C_{j,{\rm res}}^2 = D_n^2 \tau \left[2\pi F_{jn} \left|\sum _i \left(\frac{dg_j}{da_i} - \frac{d\nu _n}{da_i} \right) \Delta a_i \right|^{-1}_r \right]. \end{equation} \tag{ 34 }$$

The excitation of the terrestrial mode depends on three principal factors: the configuration of the planetary system at resonance (determined by the value of the terms in square brackets), the amplitude of the resonant giant planet mode (D₅), and the migration timescale (τ).

It is convenient to combine the forcing amplitude and migration timescale into a single migration parameter

$$\begin{equation} \hat{\tau } = \left(\frac{D_5}{D_{5,{\rm obs}}}\right)^2 \tau = \left(\frac{e_{55}}{0.0433} \right)^2 \tau. \end{equation} \tag{ 35 }$$

The value of the migration parameter is proportional to the AMD exchange and can be used to identify the family of giant planet eccentricity states (e.g., e₅₅) and migration timescales (τ) that produce a certain level of excitation. In this definition, we have scaled the migration parameter so that $\hat{\tau }=\tau$ when Jupiter and Saturn migrate with eccentricities equal to their present-day values.

Using the above relations we identify the critical migration parameter for each resonance ($\hat{\tau }_{jn}^*$) capable of exciting a terrestrial mode to its observed amplitude

$$\begin{equation} \hat{\tau }_{jn}^* = \frac{1}{2 \pi F_{jn}^2} \left(\frac{C_{j,{\rm obs}}}{D_{n,{\rm obs}}}\right)^2 \left|\sum _i \left(\frac{dg_j}{da_i} - \frac{d\nu _n}{da_i} \right) \Delta a_i \right|_r. \end{equation} \tag{ 36 }$$

In a sense, passage through the secular resonance may be used as a dynamical "speed trap" to constrain the eccentricity state and maximum migration timescale (or minimum migration rates) of Jupiter and Saturn at the time of resonance. For each secular resonance we have computed the giant planet orbits, resonance widths (F_jn), and critical migrations parameters ($\hat{\tau }_{jn}^*$) for exciting the resonant terrestrial mode to its observed amplitude. The characteristics describing these resonances are listed in Table 2. Equation (32) indicates that resonant excitation is dependent on the slope dν₅/da₆. The g₃ − ν₅ and g₄ − ν₅ secular resonances occur when Jupiter and Saturn are very near the 2:1 MMR and the modified secular model becomes less accurate. For these, we use values for the ν₅ frequency gradient interpolated from the FMFT results in Figure 4.

The effective width of a resonance (F_jn) is determined by the projection of the resonant terrestrial eigenvector ($\mathbf {v}_j$) onto the off-diagonal matrix $\mathbf {P}$ that accounts for coupling between the terrestrial and giant planet subgroups. As these terms scale with planet mass (see, e.g., Equation (4)), the effective width tends to be larger for resonances with modes where more massive planets have larger eigenvector components (v_kj). The resonance widths are shown in Table 2. Since the j = 2, 3 modes have the largest eigenvector components for Earth and Venus (see Figure 1) resonances with these modes have greater effective widths than resonances with the j = 1, 4 modes.

Also, the greater effective widths of the resonances with the j = 2, 3 modes and Equation (32) indicate that for a given frequency sweeping rate at resonance ($\Delta \dot{g}_{jn}$), the j = 2, 3 modes experience a greater excitation of their amplitudes than the j = 1, 4 modes. This susceptibility to excitation appears in contrast to the mode amplitude partitioning of the terrestrial planets, where C₂ and C₃ are smaller than C₁ and C₄ (see Figure 2).

This general model can be used to predict the excitation of modal eccentricity amplitudes due to passage through a secular resonance (e_{ij, res})

$$\begin{equation} e_{ij,{\rm res}} = e_{ij,{\rm obs}}\left(\frac{\hat{\tau }}{\hat{\tau }_{j5}^*}\right)^{1/2}, \end{equation} \tag{ 37 }$$

where e_{ij, obs} are the eccentricity amplitudes of the present-day system listed in Table 1. Note that in both the test particle and subgroup approximation the resonant forcing of eccentricity scales as $\hat{\tau }^{1/2}$ (or equivalently e_res∝e₅₅τ^1/2).

These relations and the observed state of the terrestrial planets can be used to constrain the giant planet migration parameter at the time of passage through a secular resonance. For example, if we know Jupiter's eccentricity at the time of resonance (e.g., the current value) then we can constrain the migration timescale of Jupiter and Saturn (and vice versa).

The critical migration parameters $(\hat{\tau }^*_{jn})$ required to excite the terrestrial modes to their observed amplitudes are listed in Table 2. All of these migration parameters are shorter than the migration timescales observed in self-consistent models of planetesimal-driven giant planet migration (e.g., τ ∼ 5–20 Myr; Hahn & Malhotra 1999; Gomes et al. 2004) by a factor ranging from a few to 200. These secular resonances constrain migration to occur with a smaller e₅₅ Jovian eccentricity amplitude and/or short migration timescales to avoid driving the terrestrial eccentricities to large values. Further, we note that resonances between the ν₅ frequency and the g₂ and g₃ terrestrial modes, predominantly responsible for the small eccentricities of Earth and Venus, place the strongest constraints on the migration of Jupiter and Saturn. We compare these model predictions with the results of orbital integrations below.

To clearly illustrate the role of these events using orbital integrations it is useful to start with terrestrial mode amplitudes (C_1–4) that are initially very small. The initial conditions for these simulations were prepared in the following way. Jupiter was placed on its current orbit and Saturn at 7.2 AU. Both giant planets were given initial eccentricities of 0.01 and comparable inclinations. The terrestrial planets were given their current semimajor axes, small eccentricities, and inclinations (e = 0.01) and randomly chosen angular phases. This configuration was processed further by integrating it forward in time while applying an additional force to damp the eccentricities and inclinations of the terrestrial planets with a decay timescale of three million years. The timescale was chosen to be slow compared to the secular timescale so that it effectively damped the terrestrial mode amplitudes rather than the elements of a single planet (Agnor & Ward 2002). For these initial conditions the FMFT-measured eccentricity amplitudes are e_ij ≲ 0.005 for i, j = 1–4 (for the terrestrial planets) and e₅₅ ≃ 0.015 for Jupiter, about one-third its observed value. This value of e₅₅ is comparable to that suggested in Tsiganis et al. (2005). For these initial conditions, the critical migration timescales needed to excite the terrestrial modes to their observed values are about a factor of nine longer than the migration parameter listed in Table 2 (see Equation (35)).

We have performed a series of N-body orbital integrations using this initial state. In these models Saturn is forced to migrate outward across the 2:1, halting its migration at 8.5 AU. The migration force applied produces exponential migration with an e-folding timescale (τ) similar to that of Malhotra (1993). The starting and ending points of Saturn's migration leave the ν₅ frequency between the g₂ and g₃ frequencies and avoids depositing the system near a secular resonance (see, e.g., Figure 4). This allows us to isolate the effects of individual events during migration and use FMFT analysis of the final states to measure the ultimate eccentricity amplitudes that result.

In this set of simulations we varied the migration timescale from τ = 10⁴ → 4 × 10⁷ years. The low end of this range is near the secular timescale and illustrates the response of the system to rapid, essentially non-adiabatic, evolution. The upper end encompasses migration timescales suggested by N-body simulations of planetesimal-driven giant planet migration (e.g., Hahn & Malhotra 1999; Gomes et al. 2004).

Figure 5 shows an example of the coupling between the gas giant and terrestrial planets as Jupiter and Saturn are forced to migrate across their mutual 2:1 mean motion resonance with an e-folding timescale of τ = 10 Myr. The top frame shows the evolution of the semimajor axes, pericenters, and apocenters of Jupiter and Saturn. Their divergent crossing of the 2:1 MMR occurs near t/τ = 0.425 and the excitation of the giant planet eccentricities is readily apparent as the increased separation of the pericenter and apocenter curves from the semimajor axes. The lower panel shows these same orbital parameters for the terrestrial planets. Mars' eccentricity is excited to a value similar to the observed one. As Earth and Venus have significant components in both the j = 3 and j = 4 modes (see, e.g., Figure 1), their excitation increases eccentricities of these planets as well. This excitation is coincident with Jupiter and Saturn migrating across the 2:1 MMR.

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Figure 6 shows the eccentricity excitation due to the first passage through the g₃ − ν₅ and g₄ − ν₅ secular resonances as a function of the migration timescale τ. The theoretical predictions for the j = 3 and j = 4 eccentricity amplitudes of Earth and Mars computed with Equation (37) are shown with dotted and dashed lines. The maximum eccentricity for planet i results when the contributions from these two modes are in phase, yielding a value that is simply their sum e_{i, max} = e_i3 + e_i4. This is shown with a solid line for both planets. We compare these predictions with the results of N-body integrations by identifying the maximum osculating eccentricity of Mars and Earth observed in each simulation prior to the time when Jupiter and Saturn crossed their 2:1 MMR. These values are shown with filled circles.

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For short migration timescales (τ ≲ 10⁵ yr), the initial eccentricities (e_ij ≃ 0.005) dominate over contributions from the secular resonances. For longer migration timescales, the excitation of the resonances is more apparent. The N-body results always fall below the theoretical maximum and above the predicted amplitudes due to single modes. The time required to reach the maximum eccentricity of a two-mode system is the about half the beat period of two eigenfrequencies. For the j = 3 and j = 4 modes this is ∼1 Myr. For τ ≳ 1 Myr there is sufficient time in the simulation to realize the maximum eccentricity prior to the Jupiter–Saturn 2:1 crossing and the values measured from N-body simulations cluster tightly near the theoretical maximum. For τ ≳ 10⁵ yr the maximum eccentricities of both planets follow the τ^1/2 scaling of Equation (37) and are within 0.005 (the initial amplitude) of the theoretical prediction. Clearly, the theory described above provides an accurate description of excitation during first passage through the g₃ − ν₅ and g₄ − ν₅ secular resonances.

We note that the predicted excitation of both modal eccentricity amplitudes of Mars (e₄₃ and e₄₄) are comparable (see Figure 6(a)). For a given migration timescale τ, the rate of frequency sweeping through the g₃ − ν₅ resonance is less (see |dν₅/da₆| in Figure 4), and the effective resonant width (F₃₅) greater, than for sweeping through the g₄ − ν₅ resonance. Both factors contribute to the greater excitation of C₃ relative to C₄. Finally, the eigenvector components of Mars in the j = 3 and j = 4 modes are comparable in magnitude (i.e., |v_{4, 3}| ≃ |v_{4, 4}|, see Figure 1). The net effect is that for any migration timescale (τ), the eccentricity excitation of the j = 3 component in Mars' eccentricity is actually greater than that of the j = 4 mode (i.e., e₄₃ ≃ 1.1e₄₄, see Figure 6(a)). This appears in contrast to the observed eccentricity partitioning for Mars, where e₄₃ = 0.020 and e₄₄ = 0.067.

The increase in the eccentricities of Jupiter and Saturn as they diverge across their mutual 2:1 resonance might be communicated to the terrestrial planets. This change in the eccentricities and longitude of perihelia occurs over the circulation period of the 2:1 resonant argument ($\mathcal {O}(10^3)$ yr). As this change occurs over a timescale much shorter than the secular periods, this eccentricity excitation might be viewed as a sudden change in system's mode amplitudes C_j and phases β_j. Examination of the 2:1 MMR crossing finds that this impulsive event fairly purely excites the C₆ mode amplitude and that the excitation of the C₅ mode, is weak ($e_{55}\sim \mathcal {O}(10^{-3})$) and insufficient to explain its observed value (Morbidelli et al. 2009).⁸ In general, this resetting of the giant planets' eccentricities should alter the entire system's mode amplitudes. However, in the observed solar system, the j = 6 mode weakly contributes to the eccentricity evolution of the terrestrial planets (see the small values of e_i6 in Table 1). So, the terrestrial eccentricity amplitudes (e_ij, where i, j = 1–4) are modified by small amounts to accommodate an increase in the j = 6 amplitude. On this basis, we suggest that the impulsive excitation of the C₆ mode amplitude is only weakly communicated to the terrestrial planets and its effect on the terrestrial j = 1–4 modes is small relative to contributions from the g_j − ν₅ secular resonances.

The excitation obtained from the second crossing of the g₄ − ν₅ and g₃ − ν₅ resonances can also be predicted using Equation (37). For a given migration timescale the excitation of the j = 3, 4 resonances encountered inside of the Jupiter–Saturn 2:1 MMR are comparable to the excitation due to these secular resonances outside the 2:1.

Depending on the relative phasing between excitation from the first and second resonance crossings, these two contributions may interfere constructively or destructively and result in eccentricities larger or smaller than predicted for passage through a single resonance. Figure 7 shows examples from N-body simulations that illustrate both constructive and destructive interference due to successive passages through the g₃ − ν₅ and g₄ − ν₅ resonances as Jupiter and Saturn cross their 2:1 MMR.

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In both examples the migration timescale τ is sufficiently long that the first secular resonance crossing near t/τ = 0.4 excites both j = 3 and j = 4 modes to amplitudes larger than their observed values. For the case shown in (a) the second passage through resonance results in a further increase in eccentricity amplitudes. In the second example (b), the contributions from the two resonances interfere destructively resulting in a final eccentricity state consistent with the observed terrestrial planets. This occurs despite a migration parameter greater than the critical value.

The latter result illustrates that Jupiter and Saturn's crossing of the 2:1 MMR can temporarily excite the eccentricity of Mars to values of 0.2 or larger. If Mars' orbit was excited in this way it may have facilitated the depletion of otherwise dynamically stable and unpopulated regions exterior to Mars' a = 1.7–2.0 AU (Evans & Tabachnik 2002; Bottke et al. 2010).

For each simulation in this set of models we conducted an additional simulation to measure the eccentricity amplitudes (e_ij) of the terrestrial planets via Fourier analysis. The FMFT-measured values for Earth (e₃₃) and Mars (e₄₄) are shown in Figure 8 with filled circles as a function of the migration timescale τ. Using linear secular theory we have computed the eccentricity amplitude for each planet due to secular resonance crossing both interior and exterior to the 2:1 MMR. As expected the net effect of the contributions from these two resonances is stochastic and depends on the relative phase between them. Using Equation (28) as a distribution function we show the median eccentricity expected with a black solid line. The eccentricity amplitudes of the 25th and 75th percentiles (P₂₅ and P₇₅) are shown with dotted lines. Each line follows the τ^1/2 scaling.

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Due to the stochastic influence of multiple resonance passages there is significant scatter in the eccentricities produced. However, about half of the FMFT-measured eccentricity amplitudes lie between the 25th and 75th percentile curves, indicating that the results are in good accord with the distribution suggested by Equation (28) and demonstrate that the linear model effectively predicts characteristic levels of excitation and interference.

If the ν₅ frequency swept through resonances with both the g₃ and g₄ frequencies then the large amplitude of the j = 4 mode relative to the j = 3 mode requires explanation. Perhaps the relative amplitudes of the j = 3, 4 modes resulted from the interference between multiple contributions. Linear secular theory and our results above suggest that the final modal eccentricity amplitudes e_ij can be calculated as the vector sum of multiple contributions (as in Equation (28)). The g₃ − ν₅ and g₄ − ν₅ resonances are close together and are each encountered twice in rapid succession as Jupiter and Saturn diverge across their 2:1 MMR. These resonant contributions combine with any primordial amplitude that resulted from the planet formation epoch.

Studies of the gas-free, late-stage accretion of terrestrial planets have had difficulty simultaneously accounting for the small eccentricities of Earth and Venus and the smallish masses of Mercury and Mars. Further, terrestrial planet formation simulations often produce planets with masses 0.7–1.0 M_⊕ with orbital eccentricities larger than Earth or Venus (see, e.g., Chambers & Wetherill 1998; Agnor et al. 1999; Chambers 2001; O'Brien et al. 2006; Raymond et al. 2009) and suggest that a broad range of eccentricities might plausibly arise from accretion processes.

Here we examine the relative probability of eccentricity amplitudes combining to yield a dynamical state comparable to the observed one for the terrestrial planets. We consider the contributions to each mode individually below. We compute the final value of the e₃₃ eccentricity amplitude from the vector sum of an assumed initial amplitude (e.g., one resulting from formation) and two contributions from passage through the g₃ − ν₅ resonances on both sides of the 2:1 Jupiter–Saturn MMR. For a given value of the migration parameter ($\hat{\tau }$), the amplitudes of resonant contributions are determined using Equation (37). We then choose their relative orientations at random, combine them and compute the resultant eccentricity vector. For each value of the migration parameter this procedure is repeated 10⁴ times and the cases where the final eccentricity amplitude is similar to the observed state of terrestrial planets are counted.

When examining the results for the j = 3, 4 modes, we consider final eccentricity states with e₃₃ < 0.02 and 0.05 < e₄₄ < 0.09 to be "close" to the observed dynamical state of the terrestrial planets. While this definition is somewhat arbitrary, the first constraint is within a factor of two of the small e₃₃ amplitude, and the second is within ≃ 30% of the observed e₄₄ amplitude (see Table 1 for observed values).

4.5.1. Constraints on the Origin of e₃₃

In Figure 9(a), we show the percent of cases that result in e₃₃ < 0.02 as a function of the migration parameter for small (0.01), modest (0.05), and large (0.10) initial modal eccentricity amplitudes. Note that the critical migration parameter for the g₃ − ν₅ resonances interior and exterior to the 2:1 are $\hat{\tau }^*_{35}\simeq 0.60 \mbox{ and } 0.33$ Myr, respectively. For migration parameters larger than 0.7 Myr, crossing each g₃ − ν₅ resonance contributes a component larger than the observed amplitude and significant cancellation is required to achieve the observed small amplitude. For small initial eccentricity amplitudes (0.01), the percent of outcomes with e₃₃ < 0.02 decreases from 100%, to about 20% as the migration parameter increases from 0.1 to 2.0 Myr.

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When either the initial amplitude (e₃₃ = 0.05, 0.10) or resonant contributions are substantially larger than the observed e₃₃ value (e.g., for $\hat{\tau }>\hat{\tau }_{35}^*$), significant cancellation between all three contributions is required to produce a small final eccentricity. Figure 9(a) shows that strong cancellation and e₃₃ < 0.02 can happen, but is an infrequent occurrence for large initial eccentricity amplitudes e₃₃ = 0.05, 0.10 or migration parameters greater than 2–5 Myr. On the other hand, the origin of the small e₃₃ amplitude is broadly consistent with a small primordial amplitude (i.e., e₃₃ ≃ 0.01) and migration parameters ≲ 1–2 Myr.⁹ It might also be explained by the cancellation of large resonant and initial contributions, but this becomes less likely when the contributions are substantially larger than the observed amplitude.

4.5.2. Constraints on the Origin of e₄₄

The small observed e₃₃ amplitude provides a stronger constraint on giant planet migration than the e₄₄ amplitude. Specifically, the primordial, pre-migration C₃ mode amplitude must be less than or comparable to its current value (i.e., e₃₃ ≲ 0.01) and the migration parameter $\hat{\tau }\lesssim 2$ Myr to account for j = 3 amplitude without requiring the low probability cancellation between large contributions.

If we consider the evolution of the j = 3 and j = 4 modes as independent, we can combine the analysis of the individual modes to ascertain the probability of these resonance producing a terrestrial system with both e₃₃ and e₄₄ close to their observed values as the product of the probability of two independent events. For initial amplitudes of e₃₃ ≲ 0.01 and e₄₄ ≲ 0.05 there is a reasonable range of migration parameters (0.6–1 Myr) where the constraints on both e₃₃ and e₄₄ can individually be satisfied in ≳ 50% of cases. While our analysis is approximate, it suggests that crossing the 2:1 MMR might produce a state similar to the observed eccentricity partitioning of the j = 3 and j = 4 terrestrial modes in ≃ 25% of cases with these initial amplitudes and migration parameters.

We again isolate the influence of secular resonances by constructing initial conditions where the terrestrial mode amplitudes are very small, the giant planets' modal amplitudes are near their present values, Jupiter's semimajor axis is fixed at 5.2 AU, and Saturn's initial semimajor axis of 8.5 AU places the ν₅ frequency between the g₃ and g₂ frequencies. We prepared this initial condition using a series of orbital integrations. First, we performed an integration starting with only Jupiter and Saturn with their observed orbits and slowly forced Saturn to migrate inward to a semimajor axis of 8.5 AU. Because the forced migration is slow relative to the secular timescale, the D_{5, 6} modal amplitudes are effectively constant during migration to the new orbits. The terrestrial planets were then added to the system and we conducted a second integration applying slow eccentricity damping to decrease the terrestrial mode amplitudes. The resulting state of the Jupiter–Saturn-terrestrial planet system was then used as the initial condition in the numerical experiments that follow. FMFT analysis of this initial condition finds the terrestrial eccentricity amplitudes with e_ij ≲ 0.005 and the e₅₅ amplitude of Jupiter very near its current value (i.e., e₅₅ ≃ 0.0435). Thus, for this set of simulations $\hat{\tau }=\tau$.

Figure 10 shows the eccentricity evolution of the terrestrial planets as Saturn migrates from 8.5 AU → 9.5 AU with an exponential timescale of τ = 2 Myr. The g₂ − ν₅ and g₁ − ν₅ resonances are crossed at about t/τ =0.3 and 0.6 when Saturn is near 8.7 and 8.9 AU, respectively. Excitation of both terrestrial eigenmodes is evident in the relative timing of the maximum and minimum eccentricity observed in the orbits of Mercury, Venus, and Earth. Mercury's osculating eccentricity reaches a maximum when contributions from both the j = 1 and j = 2 modes are in phase. For this same orientation between modes, the eigenvector signatures for Earth and Venus are anti-aligned (see Figure 1) resulting in cancellation between modal contributions and the osculating eccentricities of Earth and Venus simultaneously reach a minima. For this cancellation to yield eccentricities near zero, the eccentricity amplitudes from each mode must be comparable. In agreement with secular theory, the eccentricity amplitudes of Mercury, Venus, and Earth are each driven to values in excess of the present-day values. Mars is only weakly coupled to the other terrestrial planets through the j = 1, 2 modes (see Figure 1) and is minimally excited via these resonances.

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Following each migration simulation we conducted a separate orbital integration and performed FMFT analysis of the resulting dynamical state. The modal eccentricity amplitudes (e_ij) for Mercury, Venus, and Earth are shown in Figure 11 as a function of the migration timescale (τ). The predicted excitation of the j = 1, 2 modes are shown as solid and dashed lines, respectively. Also the FMFT-measured eccentricity amplitudes for the j = 1, 2 modes are shown with diamonds and filled circles, respectively.

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For migration timescales τ = 0.1 → 0.4 Myr, the eccentricity amplitudes closely match the predictions of secular theory and follow the τ^1/2 scaling. For Mercury the excitation of e₁₁ ≃ 0.20 requires τ = 0.6 Myr. Note the j = 2 component of Mercury's eccentricity is excited to half that of the j = 1 mode (e₁₂ ≈ 0.5e₁₁) for any timescale τ. For slower migration, Mercury may be excited to very large eccentricity due to the combined effect of both resonances. In these cases general relativity and higher order effects conspire to increase Mercury's precession rate and the value of the g₁ eigenfrequency (Laskar 2008). For τ = 0.3 → 1.0 Myr, the FMFT measurement of the g₁ eigenfrequency increases from 584 yr⁻¹ → 63 yr⁻¹. This tuning of the g₁ toward the g₂ = 743 yr⁻¹ alters the structure of both eigenvectors. As two eigenfrequencies converge, their eigenvector signatures become increasingly similar (see detailed discussions of this issue in Ward 1981). As the value of g₁ increases toward the g₂, the magnitude of the components of Earth and Venus in the j = 1 eigenvector increase and those of Mercury slightly decrease. The j = 2 eigenvector evolves in a complementary fashion. The component of Mercury in the j = 2 mode (v₁₂) increases while the components of Venus and Earth (v₂₂ and v₃₂) decrease. In this case, the modest discrepancies in FMFT-measured eccentricity amplitude and secular theory in Figure 11 result from changes in the eigenfrequencies and eigenvector signatures. We note that these effects become important at eccentricities well above the observed values in the present solar system.

For migration timescales τ < 0.1 Myr, the entire epoch of migration may occur in less than a precession cycle. Such migration is not slow with respect to the ν₅ frequency. Since the eccentricity amplitudes of the terrestrial planets were all e_ij ≲ 0.005 prior to migration, Figure 11 shows that rapid non-adiabatic giant planet migration can also contribute to the eccentricity amplitudes and AMD in the terrestrial modes. For the j = 2 mode, the eccentricity amplitudes of e₁₂ = 0.052, e₂₂ = 0.033, and e₃₂ = 0.025 for Mercury, Venus, and Earth are systematically achieved and appear independent of the migration timescale. These values correlate with the planet's distance from the ν₅ resonance in the present-day solar system and are likely related to the forced eccentricity in the region. These values are also greater than the observed values for the j = 2 eccentricity amplitudes of the terrestrial planets.

For the fastest migration timescale explored above, this change in eccentricity amplitudes results from a single rapid jump from one orbital state of Jupiter and Saturn to another. However, giant planet migration due to scattering between the planets would involve several to many stochastic jumps in semimajor axes and modal amplitudes (Morbidelli et al. 2009). Repeated scattering events among the giant planets may induce stochastic and diffusive exchange of AMD between the terrestrial and giant planet mode amplitudes. If the eccentricities shown in Figure 11 are indicative of the size of the steps, then this type of evolution may also perturb the terrestrial system to eccentricities larger than the observed values. As in the case of slower planetesimal-driven migration, selective cancellation between multiple contributions may be required for the terrestrial planets to emerge from this type of giant planet evolution with an eccentricity state resembling the observed one, even if the initial j = 2, 3 mode amplitudes are small. Brasser et al. (2009) show an N-body simulation result that illustrates this type of evolution. A systematic exploration of the diffusion of AMD between the giant and terrestrial planets during an epoch of giant planet scattering is needed to more generally evaluate the influence this style of giant planet migration has on the terrestrial planets.

There is about an order of magnitude difference between the critical migration parameters for the j = 2 and 1 modes ($\hat{\tau }^*_{25}= 0.05$ Myr and $\hat{\tau }^*_{15}= 0.7$ Myr, see Table 2) with both values shorter than migration timescales suggested for planetesimal-driven migration by at least an order of magnitude. Could cancellation between the resonant contributions yield a dynamical state consistent with the terrestrial planets for migration timescales of 2–10 Myr? To evaluate this hypothesis we model the addition of multiple contributions to the j = 2, 1 modes using the Monte Carlo technique described in Section 4.5. Because the g₂ − ν₅ and g₁ − ν₅ resonances are crossed once during the last ∼1 AU of divergence between Jupiter and Saturn, the final mode amplitude results from just two contributions, the secular resonant one and the primordial amplitude. When comparing results with the terrestrial planets, we consider "close" to the observed dynamical state to mean e₂₂ < 0.03 and 0.12 < e₁₁ < 0.23 for the j = 2 and j = 1 modes, respectively. As above, these ranges are somewhat arbitrary, but are comparable to the observed e₂₂ = 0.02 amplitude and within 30% of the observed e₁₁ = 0.18 amplitude.

Figure 12 shows the percentage of outcomes consistent with the observed e₂₂ and e₁₁ as a function of the migration parameter for several different assumed primordial eccentricity amplitudes. Cancellation can produce systems with e₂₂ < 0.03 for migration parameters larger than the critical one (see, e.g., Figure 12(a)), but the requisite destructive interference requires primordial and resonant contributions of comparable size. Also, the probability of the required cancellation decreases as the size of contributions (and migration parameter) increases. For example, migration parameters of 1 Myr contribute a vector of about 0.10 in magnitude to e₂₂ (see Equation (37) and Table 2). Combining this with an initial eccentricity amplitude of 0.10 will produce a sufficiently small final value of e₂₂ in ≲ 10% of cases.

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Figure 12(b) shows results for the e₁₁ amplitude. Cancellation with an initial e₁₁ amplitude generally broadens the range of migration parameters that are able to produce systems close to the observed state of the terrestrial planets. However, even large initial eccentricities of 0.10 do not permit migration parameters much larger than 1 Myr, and are not consistent with migration timescales suggested for planetesimal migration, unless the amplitude of the e₅₅ amplitude is lower than its observed value as these resonances are encountered.

Recall that the N-body simulations with migration timescales τ = 0.01 Myr are fast with respect to the secular timescale. Also for very fast migration the resulting e₁₁ and e₂₂ amplitudes are comparable to those obtained by migration parameters $\hat{\tau }\simeq 0.1$ Myr. We use this level of excitation to estimate the minimum forcing expected during instability-driven migration (i.e., planet–planet scattering). For example, Figure 12 shows that for modest initial eccentricity amplitudes (i.e., e₁₁ ≃ e₂₂ ≃ 0.05) and fortuitous interference between primordial and resonant contributions, both the large observed value of e₁₁ and the small observed value of e₂₂ might be produced in ∼10% of cases for migration parameters of 0.10–0.4 Myr. This fast exponential migration modeled here might be thought of as the rapid divergence of Jupiter and Saturn by a single strong scattering event, or one large jump in semimajor axis to the observed orbit. When Jupiter and Saturn experience multiple scattering events the forcing of the terrestrial planets may be greater and the success rate in producing the observed state of the inner planets lower.

Because the e₁₁ modal amplitude is about an order of magnitude larger than e₂₂ and both are excited in comparable ways by the sweeping ν₅ resonance (see Equation (37), Table 2, and Figure 11), it is difficult to explain their very different amplitudes as a natural outcome of planetary migration, without invoking strong interference between multiple contributions or additional processes.

To clarify the dominant processes at work, we have made a number of simplifying assumptions. First, we have neglected the influence of Uranus and Neptune. The distant locations and relatively small mass of the ice giants render them weak perturbers of Jupiter and Saturn. Their omission from the secular model does not significantly modify the ν₅ and ν₆ frequencies of Jupiter and Saturn or those of the terrestrial planet subsystem (e.g., $\Delta g_1/g_1\sim \mathcal {O}(10^{-3})$).

Similarly, we have neglected any modification of the system's eigenfrequencies that results from interaction with the planetesimal disk fueling migration. The modification of the ν₅ and ν₆ frequencies by the planetesimal disk is a function of the disk mass and location. When considering the late migration of Jupiter and Saturn, the planetesimal disk must have survived for at least the 30–100 Myr required to form the terrestrial planets. Bodies with orbits near Jupiter and Saturn have a median dynamical lifetime of about 6–9 Myr (Tiscareno & Malhotra 2003; Bailey & Malhotra 2009). This suggests that the Jupiter–Saturn region was largely devoid of planetesimals by the time the terrestrial planets were completely assembled (see, e.g., Gomes et al. 2005). A planetesimal disk exterior to Jupiter and Saturn with mass comparable to that of Uranus and Neptune (i.e., ∼30 M_⊕) affects the ν₅ and ν₆ frequencies by an amount comparable to that of the ice giant planets and can be similarly neglected.

For rapid migration (τ < 1–2 Myr), a large fraction of the planetesimal disk must be on crossing orbits with Jupiter and Saturn. To roughly estimate the effect of this disk on the giant planet eigenfrequencies we treat it as a single 10 M_⊕ planet between the gas giants. This increases both the ν₅ and ν₆ frequencies by ≃ 20%. However, this contribution decays as migration proceeds and the planetesimal disk is depleted. We suggest that these deviations in the ν₅ frequency are not likely to prevent resonances with the terrestrial modes or grossly modify the factors that determine the excitation experienced by the inner planets.

In exploring the problem we held Jupiter's semimajor axis constant to reduce the number of model parameters. More generally, the configuration of the planetary system when secular resonances occur depends on the semimajor axes of all the planets. So, the examples shown and values listed in Table 2 are not unique. Self-consistent N-body simulations of planetesimal-driven migration of the solar system typically show Jupiter to migrate inward a few tenths of an AU and Saturn migrating outward a few AU on the same timescale (e.g., Hahn & Malhotra 1999; Gomes et al. 2004).

We have used the secular analytic model to examine how Jupiter's migration affects the planetary configuration where resonances occur and the critical migration parameter for each resonance. Considering Jupiter with an initial semimajor axis of 5.4 AU, Saturn starting just outside the 3:2 MMR with Jupiter, and both planets migrating to their observed orbits, all six of the secular resonances reported in Table 2 are encountered, albeit with slightly different values of the planetary semimajor axes, resonant frequencies, and critical migration timescales. While the ν_{5, 6} frequencies are strongly affected by the orbital divergence of Jupiter and Saturn, the effect of Jupiter's inward migration of ∼0.2 AU on the terrestrial eigenfrequencies (g_1–4) is small and only slightly larger than the effect of Saturn's outward migration (see, e.g., Figure 4). Consequently, the orbital period ratio between Saturn and Jupiter (P_S/P_J) at a particular g_j − ν₅ secular resonances is nearly independent of Jupiter's semimajor axis, or whether it is migrating. This allows the period ratio to be used as a single independent variable to identify secular resonances (see, e.g., Brasser et al. 2009) and for comparison with scenarios where both gas giants are migrating.

Further, the secular model shows that the frequency gradient at resonance due to Jupiter (|d(g_j − ν₅)/da₅|) is comparable to that of Saturn (|d(g_j − ν₅)/da₆|). In this situation, the planet with the greater migration rate (i.e., da/dt) most strongly determines the level of terrestrial excitation and the critical migration parameter (see Equation (36)). Due to its larger mass Jupiter tends to be less mobile and its planetesimal-driven migration rate generically slower than Saturn's. Consequently, Saturn's migration rate most strongly determines excitation of the inner solar system. Finally, for the cases where Jupiter migrates inward a few tenths of an AU, the critical migration parameters vary from those listed in Table 2 by 20%–30%.

In total, the various assumptions we have made to simplify the model and its presentation modestly affect the secular system. We consider the critical migration parameters listed in Table 2 as broadly representative of many plausible migration scenarios suggested for the solar system.

As Jupiter and Saturn diverge across their mutual 2:1 MMR they encounter the g₃ − ν₅ and g₄ − ν₅ secular resonances on both sides of the resonance (i.e., for P_S/P_J ≃ 1.97–2.03). Due to several factors, including the small amplitude of the j = 3 terrestrial mode, the g₃ − ν₅ resonances near the 2:1 MMR of Jupiter and Saturn constrain the giant planet migration parameter to satisfy $\hat{\tau } \lesssim \hat{\tau }_{35}^{*} = 0.60 \mbox{ \,Myr.}$ Following the 2:1 crossing, the g₂ − ν₅ and g₁ − ν₅ resonances are encountered as Jupiter and Saturn diverge outside their 2:1 MMR (when P_S/P_J ≃ 2.1–2.2, see Table 2). In a similar manner the migration history considered must satisfy $\hat{\tau } \lesssim \hat{\tau }_{25}^{*} = 0.05 \mbox{ \,Myr}$ when P_S/P_J ≃ 2.1 and $\hat{\tau } \lesssim \hat{\tau }_{15}^{*} = 0.68 \mbox{ \,Myr}$ when P_S/P_J ≃ 2.2. We note that greater values of the migration parameters may also produce acceptable states for secular resonances with the terrestrial modes (j = 1–4). However, achieving agreement with the observed terrestrial system relies on cancellation of multiple contributions (see, e.g., Figures 7 and 8). When the magnitudes of all contributions are known, we have shown that the probability of producing a particular eccentricity state can be estimated using analytical arguments (see Equation (28), Figures 9 and 12). We have also shown that the probability of yielding the required cancellation between multiple contributions decreases significantly when the critical migration parameter $\hat{\tau }_{j5}^*$ is exceeded by a factor of two to three. With these caveats we suggest that these values represent characteristic constraints on the migration of Jupiter and Saturn due to passage through these secular resonances. Further, because the solar system's giant planets migrate as a coupled system, constraints on the migration of Jupiter and Saturn should be considered as indirect constraints on the formation and migration of Uranus and Neptune and the evolution of the giant planet system as a whole.

For each g_j − ν₅ secular resonance identified, planetesimal-driven migration of the giant planets (i.e., τ ≃ 5–20 Myr) with the Jovian eccentricity comparable to the present value would excite the terrestrial system to eccentricities and modal amplitudes well beyond their observed values. The constraints on giant planet migration imposed by these secular resonances and the dynamical structure of the terrestrial planets may be satisfied in several ways.

First, if the giant planets migrated with eccentricity amplitudes comparable to their present-day values the strong forcing of the terrestrial planets via the secular resonances may have interfered destructively with an initial amplitude of the j = 1–4 modes (e.g., from the accretion epoch). This could deposit the terrestrial planets in a dynamical state similar to the observed one. The general scenario requires the cancellation of two large components and we have shown that this is accordingly of low probability (see also Brasser et al. 2009).

Using the resonant excitation model and geometric arguments we can identify additional constraints required on this type of evolution. For cancellation to be viable, the initial and resonant contributions must be of a similar size and the excitation from resonance crossing necessarily less than an amount that would render the terrestrial subsystem unstable. For the g₃ − ν₅ and g₄ − ν₅ secular resonances encountered near the 2:1 MMR, the terrestrial planets are driven to crossing orbits if the migration timescale of Jupiter and Saturn is τ ≳ 10 Myr and the e₅₅ has the present-day value. For the g₂ − ν₅ and g₁ − ν₅ resonances, $\hat{\tau } \gtrsim 2$ Myr leads to crossing orbits and instabilities (see, e.g., Figure 10). This latter constraint indicates that planetesimal-driven migration through the g_{2, 1} − ν₅ secular resonances is not viable unless the Jovian eccentricity amplitude e₅₅ is significantly smaller than the observed value.

Second, if the Jovian eccentricity (e₅₅) was smaller during migration, then longer timescales might be permissible. For τ ∼ 10 Myr, the g₃ − ν₅ secular resonance constrains the Jovian eccentricity amplitude e₅₅ ≲ 0.011, or about a quarter of the current value, when Jupiter and Saturn migrated across the 2:1 MMR. This eccentricity is comparable to those typical of models of planetesimal-driven migration. However, since it is less than the observed e₅₅ = 0.0433 value it also requires that the D₅ mode amplitude be excited later via some other process. Morbidelli et al. (2009) examined the origin of the D₅ mode amplitude in detail and found that repeated planet–planet scattering between Jupiter or Saturn and an ice giant planet may explain the observed D₅ amplitude and account for its excitation via an instability in the giant planets' orbits after the 2:1 MMR of Jupiter and Saturn is crossed. This analytic result is consistent with the numerical simulations of Brasser et al. (2009) and the conclusions drawn from them. Using the secular model we have also shown for initial eccentricity amplitudes e₃₃ ≲ 0.01 and e₄₄ ≲ 0.05, the eccentricity partitioning of the j = 3 and j = 4 terrestrial modes can be expected with a roughly 25% probability for migration parameters in the range $\hat{\tau }=0.6\hbox{--}1.0$ Myr. Again, these values are broadly consistent with those of planetesimal-driven migration.

When examining the g₂ − ν₅ and g₁ − ν₅ resonances, a Jovian modal eccentricity amplitude of e₅₅ ≲ 0.003 is required to allow a planetesimal-driven migration timescale of τ ≃ 10 Myr through these resonances without exciting the j = 2 terrestrial mode amplitude beyond its observed value. This value of e₅₅ is quite small, and again requires the late excitation of the Jovian eccentricity (e.g., via scattering with a ≳ 10 M_⊕ body to excite the D₅ amplitude during the last 0.5–0.8 AU of divergent migration between Jupiter and Saturn; Morbidelli et al. 2009).

Third, rapid, stochastic migration in large steps in the semimajor axis is also possible when the system of planets is unstable and planet–planet scattering ensues. Such global instabilities may allow secular resonances to be jumped over and the whole migration epoch to be much shorter than suggested by planetesimal-driven migration (Thommes et al. 1999; Tsiganis et al. 2005; Morbidelli et al. 2007; Batygin & Brown 2010). If such rapid migration was driven by scattering between planets, this process must cause the orbits of Jupiter and Saturn to diverge. For Saturn to move outward it must scatter an ice giant inward. Similarly for Jupiter to move inward, it must scatter an ice giant outward. This style of instability-driven migration might allow the ν₅ and ν₆ secular resonances to move quickly through the inner solar system. N-body simulations of this process by Brasser et al. (2009) suggest that it is possible that the terrestrial planets might emerge from this style of planet migration with orbits comparable to their observed ones.

On the other hand, our simulations with very fast migration (τ < 0.2 Myr) indicate that the j = 2 terrestrial mode can be excited to amplitudes in excess of the observed values. If the D₅ mode amplitude originated via repeated planet–planet scattering (Morbidelli et al. 2009) it appears plausible that instability-driven giant planet migration may perturb the terrestrial planets to an overly excited state. In this regard, additional work assessing the relative frequency of characteristic outcomes (e.g., those similar to the observed state) would be welcome.

Finally, if planetary migration was rapid as suggested by the constraints of these secular resonances (τ ≲ 5–10 Myr), then entire epoch of giant planet formation and migration, could have been largely complete before terrestrial planet accretion was finished. In this case the dynamical structure of the inner solar system was determined later (e.g., by processes related to accretion and evolution) and the formation timescale of the terrestrial planets (i.e., 30–100 Myr) constrains the timing by which giant planet formation and migration was largely complete. Recently, Walsh & Morbidelli (2011) have investigated the effect of the early planetesimal-driven migration of Jupiter and Saturn on terrestrial planet formation. Their N-body accretion simulations with a migration timescale of τ = 5 Myr for Jupiter and Saturn produce terrestrial planet systems with AMDs comparable to the observed terrestrial planets. While several open problems of terrestrial planet formation persist in these simulations (e.g., explaining the small mass of Mars and the clearing and dynamical structure of the asteroid belt), these results suggest that effects of the sweeping ν₅ resonance through the terrestrial region may be tempered by terrestrial accretion dynamics.

In the Nice model explanation of the LHB (Gomes et al. 2005), an instability among the giant planets is initiated by Jupiter and Saturn crossing their mutual 2:1 MMR. This instability may be delayed by the very slow divergent migration of Jupiter and Saturn of a few tenths of an AU toward the 2:1 MMR over 600–700 Myr. This slow migration roughly corresponds to exponential migration timescales of τ ≳ 3 × 10⁹ yr. The secular resonant model presented here predicts that crossing the g₃ − ν₅ and g₄ − ν₅ resonances this slowly requires the Jovian eccentricity amplitude $e_{55} \lesssim \mathcal {O}(10^{-4})$ to leave the terrestrial planets in a state comparable to the observed one. A modestly larger Jovian eccentricity (e₅₅ ≳ 0.001) may perturb the terrestrial planets to crossing orbits and drive a global instability of the terrestrial system.

Requiring such small values of the Jovian eccentricity is a very strong constraint on the evolution of the gas giants and we interpret this as an indication that the slow divergent migration of Jupiter and Saturn toward the 2:1 MMR is not generally consistent with the coexistence and dynamical structure of the terrestrial planets. As a result, the specific LHB initiating scenario described in Gomes et al. (2005) appears unlikely. Brasser et al. (2009) have drawn similar conclusions based on the results of N-body simulations.

If the LHB resulted from the large-scale migration of the giant planets, then an instability and giant planet scattering may be required to avoid strongly perturbing the terrestrial planets via secular resonances (Brasser et al. 2009). Alternative modes of triggering a late instability (e.g., Uranus and Neptune's divergent migration across a mean motion resonance) are possible and may offer viable alternative modes of initiating the LHB (Morbidelli et al. 2007).

However, the j = 5 mode of Jupiter and Saturn has components in terrestrial planets comparable to those of j = 2, 3 terrestrial modes (see, e.g., Table 1). If giant planet scattering is responsible for exciting the j = 5 mode amplitude to the observed value, then this excitation may also be communicated to the terrestrial planets (e.g., stochastic diffusion of AMD between j = 5 mode and the terrestrial j = 1–4 modes). Our simulations of very fast migration (i.e., τ ≲ 0.10 Myr) suggest that excitation of the terrestrial planets via this process is likely, but more work to examine the net exchange of AMD between the terrestrial and giant planet systems is needed to assess the dynamical implications of instability-driven giant planet migration for the terrestrial planets.

We have shown that if the terrestrial planets witnessed an epoch when the gross orbital structure of the giant planets changed, then the dynamical structure of the inner solar system was likely altered. In this scenario, the ultimate dynamical state of the terrestrial planets is a product of several processes (e.g., accretion dynamics, passage through secular resonances, AMD diffusion via giant planet scattering, and chaotic diffusion of the system's AMD over long timescales) that act in concert and whose individual influences are challenging to isolate and disentangle.

The small amplitudes of the j = 2, 3 terrestrial eigenmodes, that are closely associated with the small eccentricities of Earth and Venus, provide strong constraints on the migration of Jupiter and Saturn. While more work examining the characteristic forcing of the terrestrial planets by instability-driven migration is needed, explaining the observed dynamical state of the terrestrial planets as a natural and frequent outcome of the late migration of the giant planets remains a challenge.

An alternative and perhaps simpler possibility is that giant planet formation and migration was largely complete before the dynamical structure of the inner solar system was determined. Formation models and isotopic evidence suggest that terrestrial planet accretion was complete in 30–100 Myr. The bulk of giant planet migration could easily be accommodated in shorter time intervals via some combination of tidal interaction with a gas disk, planet–planet scattering, and planetesimal-driven migration. Any strong perturbation of the inner solar system resulting from giant planet migration might then be tempered through a variety of processes that depend on the state of the forming planets and the protoplanetary disk at the time of the disturbance. Additional mechanisms that may act at this earlier time include dynamical friction with a disk of planetesimals, orbital damping due to interactions with a remnant gas disk, or a combination of dynamical processes (Nagasawa et al. 2005; Thommes et al. 2008a).

In the sequential accretion scenario, the formation timescale for giant planet cores (∼10 M_⊕) in the outer regions of a minimum mass nebula appear to be considerably longer than the observationally inferred depletion timescale of protostellar disks. Ice giant formation at smaller orbital radii (and shorter orbital periods) is often invoked to overcome this problem, but requires the subsequent outward migration of the ice giant planets. However, it is also possible that disk gaps near Jupiter and Saturn's orbits may have provided migration barriers which eventually led to the accumulation of planet-building materials and the emergence of Uranus and Neptune within a few million years (see Section 1).

The resonant structure of the Kuiper Belt is often interpreted as a product of planetesimal-driven (Malhotra 1995; Ida et al. 2000; Hahn & Malhotra 2005) or instability-driven migration (Levison et al. 2008) of the giant planets. However, the giant planets' outward migration may also be induced by their tidal interaction with a viscously expanding gaseous nebula (Lin & Papaloizou 1986) or by a disk which is undergoing photoevaporation. In these cases, giant planet migration may be mostly complete in ∼10 Myr and could predate the final assembly of the terrestrial planets. Such an early migration scenario would avoid the excitation of the terrestrial planets' eccentricities.

This contingency requires that the delivery of LHB impactors to the Moon be achieved after the formation, large-scale migration, and/or global orbital restructuring of the giant planets. Within the context of the Nice model migration scenarios, only explaining the LHB requires that the bulk of giant migration occurred late. If giant planet migration was completed before the dynamical state of the inner solar system was determined, other aspects of the Nice model remain viable (e.g., the capture of Jupiter's Trojans and the orbital properties of the giant planets). Obviously, the delivery of the LHB impactors must then be accounted for via some other means (e.g., the Planet V hypothesis described in Chambers 2007).

Additional progress in unraveling the sequence of events and processes that produced the observed solar system might be made by examining the formation and early evolution of the gas and ice giant planets. We note that the nature and details of gas and ice giant formation in the solar system remain poorly understood. The basic assumptions and initial conditions of current migration models (e.g., that Jupiter and Saturn formed in a much more compact configuration) may change as the formation and early history of the giant planets are clarified.

Finally, we note that migration near and through mean motion resonances is germane to the orbital evolution of planetary systems. Consequently, the resonant sweeping processes examined here have applications among extrasolar planetary systems, satellite systems, and other dynamical characteristics of the solar system. Analyses of these applications will be presented elsewhere.