Laplace-Lagrange Secular Theory Research Articles

Free Inclinations for Trans-Neptunian Objects in the Main Kuiper Belt

Abstract There is a complex inclination structure present in the trans-Neptunian object (TNO) orbital distribution in the main classical-belt region (between orbital semimajor axes of 39 and 48 au). The long-term gravitational effects of the giant planets make TNO orbits precess, but nonresonant objects maintain a nearly constant “free” inclination (I free) with respect to a local forced precession pole. Because of the likely cosmogonic importance of the distribution of this quantity, we tabulate free inclinations for all main-belt TNOs, each individually computed using barycentric orbital elements with respect to each object’s local forcing pole. We show that the simplest method, based on the Laplace–Lagrange secular theory, is unable to give correct forcing poles for objects near the ν 18 secular resonance, resulting in poorly conserved I free values in much of the main belt. We thus instead implemented an averaged Hamiltonian to obtain the expected nodal precession for each TNO, yielding significantly more accurate free inclinations for nonresonant objects. For the vast majority (96%) of classical-belt TNOs, these I free values are conserved to < 1° over 4 Gyr numerical simulations, demonstrating the advantage of using this well-conserved quantity in studies of the TNO population and its primordial inclination profile; our computed distributions only reinforce the idea of a very coplanar surviving “cold” primordial population, overlain by a large I-width implanted “hot” population.

Transit Duration Variations in Multiplanet Systems

Abstract A planet’s orbital orientation relative to an observer’s line of sight determines the chord length for a transiting planet, i.e., the projected distance a transiting planet travels across the stellar disk. For a given circular orbit, the chord length determines the transit duration. Changes in the orbital inclination, the longitude of ascending node, or both, can alter this chord length and thus result in transit duration variations (TDVs). Variation of the full orbital inclination vector can even lead to de-transiting or newly transiting planets for a system. We use Laplace-Lagrange secular theory to estimate the fastest nodal eigenfrequencies for over 100 short-period planetary systems. The highest eigenfrequency is an indicator of which systems should show the strongest TDVs. We further explore five cases (TRAPPIST-1, Kepler-11, K2-138, Kepler-445, and Kepler-334) using direct N-body simulations to characterize possible TDVs and to explore whether de-transiting planets might be possible for these systems. A range of initial conditions are explored, in which each realization is consistent with the observed transits. We find that tens of percent of multiplanet systems have fast enough eigenfrequencies to expect large TDVs on decade timescales. For the directly integrated cases, we find that de-transiting planets might occur on decade timescales, and TDVs of 10 minutes per decade are expected to be common.

Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance

Extrasolar systems with planets on eccentric orbits close to or in mean-motion resonances are common. The classical low-order resonant Hamiltonian expansion is unfit to describe the long-term evolution of these systems. We extend the Laplace-Lagrange secular approximation for coplanar systems with two planets by including (near-)resonant harmonics, and realize an expansion at high order in the eccentricities of the resonant Hamiltonian both at orders one and two in the masses. We show that the expansion at first order in the masses gives a qualitative good approximation of the dynamics of resonant extrasolar systems with moderate eccentricities, while the second order is needed to reproduce more accurately their orbital evolutions. The resonant approach is also required to correct the secular frequencies of the motion given by the Laplace-Lagrange secular theory in the vicinity of a mean-motion resonance. The dynamical evolutions of four (near-)resonant extrasolar systems are discussed, namely GJ 876 (2:1 resonance), HD 60532 (3:1), HD 108874 and GJ 3293 (close to 4:1).

Comparison between Laplace-Lagrange secular theory and numerical simulation: the case of $$\upsilon $$ υ Andromedae planetary system

The large increase in exoplanet discoveries in the last two decades showed a variety of systems whose stability is not clear. In this work we chose the $\upsilon$ Andromedae system as the basis of our studies in dynamical stability. This system has a range of possible masses, as a result of detection by radial velocity method, so we adopted a range of masses for the planets $c$ and $d$ and applied the secular theory. We also performed a numerical integration of the 3-body problem for the system over a time span of 30 thousand years. The results exposed similarities between the secular perturbation theory and the numerical integration, as well as the limits where the secular theory did not present good results. The analysis of the results provided hints for the maximum values of masses and eccentricities for stable planetary systems similar to $\upsilon$ Andromedae.

On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems

We study the secular evolution of several exoplanetary systems by extending the Laplace-Lagrange theory to order two in the masses. Using an expansion of the Hamiltonian in the Poincar\'e canonical variables, we determine the fundamental frequencies of the motion and compute analytically the long-term evolution of the keplerian elements. Our study clearly shows that, for systems close to a mean-motion resonance, the second order approximation describes their secular evolution more accurately than the usually adopted first order one. Moreover, this approach takes into account the influence of the mean anomalies on the secular dynamics. Finally, we set up a simple criterion that is useful to discriminate between three different categories of planetary systems: (i) {\it secular} systems (HD 11964, HD 74156, HD 134987, HD 163607, HD 12661 and HD 147018); (ii) systems {\it near a mean-motion resonance} (HD 11506, HD 177830, HD 9446, HD 169830 and $\upsilon$ Andromedae); (iii) systems {\it really close to} or {\it in a mean-motion resonance} (HD 108874, HD 128311 and HD 183263).

Extrasolar Planetary Dynamics with a Generalized Planar Laplace‐Lagrange Secular Theory

The dynamical evolution of nearly half of the known extrasolar planets in multiple-planet systems may be dominated by secular perturbations. The commonly high eccentricities of the planetary orbits calls into question the utility of the traditional Laplace-Lagrange (LL) secular theory in analyses of the motion. We analytically generalize this theory to fourth-order in the eccentricities, compare the result with the second-order theory and octupole-level theory, and apply these theories to the likely secularly-dominated HD 12661, HD 168443, HD 38529 and Ups And multi-planet systems. The fourth-order scheme yields a multiply-branched criterion for maintaining apsidal libration, and implies that the apsidal rate of a small body is a function of its initial eccentricity, dependencies which are absent from the traditional theory. Numerical results indicate that the primary difference the second and fourth-order theories reveal is an alteration in secular periodicities, and to a smaller extent amplitudes of the planetary eccentricity variation. Comparison with numerical integrations indicates that the improvement afforded by the fourth-order theory over the second-order theory sometimes dwarfs the improvement needed to reproduce the actual dynamical evolution. We conclude that LL secular theory, to any order, generally represents a poor barometer for predicting secular dynamics in extrasolar planetary systems, but does embody a useful tool for extracting an accurate long-term dynamical description of systems with small bodies and/or near-circular orbits.

The Secular Evolution and Dynamical Architecture of the Neptunian Triplet Planetary System HD 69830

We perform numerical simulations to study the secular orbital evolution and dynamical structure in the HD 69830 planetary system, using the best-fit orbital solutions by Lovis and coworkers. In the simulations, we show that a triplet Neptunian system is stable for at least 2 Gyr and that the stability would not be greatly influenced even if we varied the planetary masses. In addition, we employ Laplace-Lagrange secular theory to investigate the long-term behavior of the system, and the outcomes demonstrate that this theory can well describe and predict the secular orbital evolution for three Neptune-mass planets, where the secular periods and amplitudes of the eccentricities are in good agreement with those from direct numerical integrations. We first reveal that the secular periods of the eccentricity, e1 and e2, are identical. Moreover, we extensively explore the planetary configuration of three Neptune-mass companions with one massive terrestrial planet in 0.07 AU ≤ a ≤ 1.20 AU, to examine the potential asteroid architecture. We underline that there are stable zones lasting for at least 105 yr for low-mass terrestrial planets located between 0.3 and 0.5 AU and between 0.8 and 1.2 AU. We also find that the secular resonances can excite the eccentricities of the terrestrial bodies and that the accumulation or depletion of the asteroid belt is also shaped by orbital resonances of the outer planets; for example, the asteroidal gaps at the 2 : 1 and 3 : 2 mean motion resonances with planet c. In a dynamical sense, the proper candidate regions for the existence of potential terrestrial planets or habitable zones are 0.35 AU < a < 0.50 AU and 0.80 AU < a < 1.00 AU for relatively low eccentricities, implying the possible asteroidal structure in the system.

The dynamics of two massive planets on inclined orbits

The significant orbital eccentricities of most giant extrasolar planets may have their origin in the gravitational dynamics of initially unstable multiple planet systems. In this work, we explore the dynamics of two close planets on inclined orbits through both analytical techniques and extensive numerical scattering experiments. We derive a criterion for two equal mass planets on circular inclined orbits to achieve Hill stability, and conclude that significant radial migration and eccentricity pumping of both planets occurs predominantly by 2:1 and 5:3 mean motion resonant interactions. Using Laplace–Lagrange secular theory, we obtain analytical secular solutions for the orbital inclinations and longitudes of ascending nodes, and use those solutions to distinguish between the secular and resonant dynamics which arise in numerical simulations. We also illustrate how encounter maps, typically used to trace the motion of massless particles, may be modified to reproduce the gross instability seen by the numerical integrations. Such a correlation suggests promising future use of such maps to model the dynamics of more coplanar massive planet systems.

The Librating Companions in HD 37124, HD 12661, HD 82943, 47 Ursa Majoris, and GJ 876: Alignment or Antialignment?

We investigated the apsidal motion for the multi-planet systems. In the simulations, we found that the two planets of HD 37124, HD 12661, 47 Uma and HD 82943 separately undergo apsidal alignment or antialignment. But the companions of GJ 876 and $\upsilon$ And are only in apsidal lock about $0^{\circ}$. Moreover, we obtained the criteria with Laplace-Lagrange secular theory to discern whether a pair of planets for a certain system are in libration or circulation.

Could the 55 Cancri Planetary System Really Be in the 3[rcolon]1 Mean Motion Resonance?

We integrate the orbital solutions of the planets orbiting 55 Cnc. In the simulations, we find that not only three resonant arguments $\theta_{1}=\lambda_{1}-3\lambda_{2}+2\tilde\omega_{1}$, $\theta_{2}=\lambda_{1}-3\lambda_{2}+2\tilde\omega_{2}$ and $\theta_{3}=\lambda_{1}-3\lambda_{2}+(\tilde\omega_{1}+\tilde\omega_{2})$ librate respectively, but the relative apsidal longitudes $\Delta\omega$ also librates about $250^{\circ}$ for millions of years. The results imply the existence of the 3:1 resonance and the apsidal resonance for the studied system. We emphasize that the mean motion resonance and apsidal locking can act as two important mechanisms of stabilizing the system. In addition, we further investigate the secular dynamics of this system by comparing the numerical results with those given by Laplace-Lagrange secular theory.

Stability of the HD 12661 Planetary System

In this paper we study the stability of the HD 12661 planetary system in the framework of the N- body problem. Using the initial conditions found and announced by the California & Carnegie Planet Search Team, (http://exoplanets.org/almanacframe.html), we estimate the dynamical limits on orbital parameters that provide sta- ble (quasi-periodic) motions of the system. We investigate the orbital stability by combining the MEGNO indicator analysis with short-term integrations of the orbital dynamics. The MEGNO technique, invented by Cincotta & Simo (2000), makes it possible to distinguish efficiently between chaotic and regular dynamics of a conservative dynamical system. The orbital evo- lution, derived simultaneously with MEGNO, helps to identify sources of instability. The nominal initial condition leads to a chaotic solution, with a Lyapunov time � 1300 yr. In spite of this, the system motion seems to be bounded. This was examined directly, by long-term, 1 Gyr integrations. During this time, the eccentricities vary in the range (0.1, 0.4). The system is locked in apsidal resonance with the critical argument librating about 180 ◦ , with a full amplitude varying typically between 40 ◦ and 180 ◦ . Using MEGNO, we found that the HD 12661 system evolves on a border of the 11:2 mean motion resonance. This reso- nance is stable and results in a quasi-periodic evolution of the system. From the viewpoint of global dynamics, the crucial factor for system stability is the presence of the apsidal resonance. We detected this resonance in a wide neighborhood of the initial condition in the space of orbital parameters of the system, and in wide ranges of relative inclination and masses of the planets. The center of libration can be 180 ◦ (as in the nominal system) or 0 ◦ . The regime depends on the initial values of the apsidal longitudes. Statistically, the system prefers almost exclusively one of these two resonance regimes. The HD 12661 system gives a very evident example of the dynamical role of secular resonances, and their influence on the stability of exosystems containing Jupiter-like planets. Data derived by numerical experiments are compared with the results of Laplace-Lagrange secular theory. The analytical theory gives a crude approximation of the secular dynamics, because the eccentricities and masses of the planets are large, and the nominal system is near the 11:2 mean motion resonance.

Unstable modes of Keplerian discs

We derive the softened analogue of the Laplace-Lagrange secular theory of planetary motion, and use it to show that a small fraction of counter-rotating stars is all it takes for a hot Keplerian disc to grow unstable lopsided modes.

Some instabilities in the 3/1 and 2/1 resonances

In a very simple way, it is possible to show the existence of small regions of instability, inside the observed 3/1 and 2/1 Kirkwood Gap, by using the classical Laplace-Lagrange secular theory.