Analytical estimates of secular frequencies for binary star systems
Ákos Bazsó, Elke Pilat-Lohinger
备注：
17 pages; Credit: Authors, Copyright: Proceedings of the First Greek-Austrian Workshop on Extrasolar Planetary Systems, 2017, pages 45-62, reproduced with permission
发表信息：
Proceedings of the First Greek-Austrian Workshop on Extrasolar Planetary Systems, 2017. Editors: T. I. Maindl, H. Varvoglis, R. Dvorak, ISBN/EAN13: 1544255632 / 9781544255637
Binary and multiple star systems are extreme environments for the formation and long-term presence of extrasolar planets. Circumstellar planets are subject to gravitational perturbations from the distant companion star, and this interaction leads to a long-period precession of their orbits. We investigate analytical models that allow to quantify these perturbations and calculate the secular precession frequency in the dynamical model of the restricted three-body problem. These models are applied to test cases and we discuss some of their shortcomings. In addition, we introduce a modified Laplace-Lagrange model which allows to obtain better frequency estimates than the traditional model for large eccentricities of the perturber. We then generalize this model to any number of perturbers, and present an application to the four-body problem.
We present an analytical formalism to study the secular dynamics of a system consisting of N-2 planets orbiting a binary star in outer orbits. We introduce a canonical coordinate system and expand the disturbing function in terms of canonical elliptic elements, combining both Legendre polynomials and Laplace coefficients, to obtain a general formalism for the secular description of this type of configuration. With a quadractic approximation of the development, we present a simplified analytical solution for the planetary orbits for both the single planet and the two-planet cases. From the two-…
We present a semi-analytical correction to the seminal solution for the secular motion of a planet's orbit under gravitational influence of an external perturber derived by Heppenheimer (1978). A comparison between analytical predictions and numerical simulations allows us to determine corrective factors for the secular frequency and forced eccentricity in the co-planar restricted three-body problem. The correction is given in the form of a polynomial function of the system's parameters that can be applied to first-order forced eccentricity and secular frequency estimates. The resulting secula…
We explore the effects of an undetected outer giant planet on the dynamics, observability, and stability of Systems with Tightly-packed Inner Planets (STIPs). We use direct numerical simulations along with secular theory and synthetic secular frequency spectra to analyze how analogues of Kepler-11 and Kepler-90 behave in the presence of a nearly co-planar, Jupiter-like outer perturber with semi-major axes between 1 and 5.2 au. Most locations of the outer perturber do not affect the evolution of the inner planetary systems, apart from altering precession frequencies. However, there are location…
We study the secular evolution of a particle in deep mean motion resonance (MMR) with a planet in the planar elliptic restricted three body problem. We do not consider any restriction neither in the planet's eccentricity $e_p$ nor in the particle's eccentricity $e$. The methodology used is based on a semi-analytical model that consists on calculating the averaged resonant disturbing function numerically, assuming for this that in the resonant scale of time all the orbital elements of the particle are constant. In order to obtain the secular evolution inside the MMR, we make use of the adiabati…
The long-term variations in the orbit of the Earth govern the insolation on its surface and hence its climate. The use of the astronomical signal, whose imprint has been recovered in the geological records, has revolutionized the determination of the geological timescales. However, the orbital variations beyond 60 Myr cannot be reliably predicted because of the chaotic dynamics of the planetary orbits in the Solar System. Taking this dynamical uncertainty into account is necessary for a complete astronomical calibration of geological records. Our work addresses this problem with a statistical …
Context. From numerical simulations, it is known that some secular resonances may affect the motion of near-Earth objects (NEOs). However, the specific location of the secular resonance inside the NEO region is not fully known, because the methods previously used to predict their location can not be used for highly eccentric orbits and when the NEOs cross the orbits of the planets. Aims. In this paper, we aim to map the secular resonances with the planets from Venus to Saturn in the NEO region, even for high values of the eccentricity. Methods. We used an averaged semi-analytical model that ca…
We analyse the secular dynamics of planets on S-type coplanar orbits in tight binary systems, based on first- and second-order analytical models, and compare their predictions with full N-body simulations. The perturbation parameter adopted for the development of these models depends on the masses of the stars and on the semimajor axis ratio between the planet and the binary. We show that each model has both advantages and limitations. While the first-order analytical model is algebraically simple and easy to implement, it is only applicable in regions of the parameter space where the perturba…
We revisit the problem of secular resonance sweeping during the dissipation of a protoplanetary disk and its possible role in exciting the orbits of primordial asteroids, in the light of recent models of solar system evolution. We develop an integrator that incorporates the gravitational effect of a uniformly (or not) depleting, axisymmetric disk with arbitrary surface density profile; its performance is verified by analytical calculations. The secular response of fictitious asteroids, under perturbations from Jupiter, Saturn and a decaying disk, is thoroughly studied. Note that the existence …
Massive eccentric disks (gaseous or particulate) orbiting a dominant central mass appear in many astrophysical systems, including planetary rings, protoplanetary and accretion disks in binaries, and nuclear stellar disks around supermassive black holes in galactic centers. We present an analytical framework for treating the nearly Keplerian secular dynamics of test particles driven by the gravity of an eccentric, apsidally aligned, zero-thickness disk with arbitrary surface density and eccentricity profiles. We derive a disturbing function describing the secular evolution of coplanar objects, …
The aim of this work is to provide new insights on the dynamics associated to the resonances which arise as a consequence of the coupling of the effect due to the oblateness of the Earth and the Solar Radiation Pressure (SRP) effect for an uncontrolled object with moderate to high area-to-mass ratio. Analytical estimates for the location of the resulting resonant equilibrium points are provided, together with formulas to compute the maximum amplitude of the corresponding variation in the eccentricity, as a function of the initial conditions of the object and of its area-to-mass ratio. The peri…
Pairs of planets in a system may end up close to their host star on eccentric orbits as a consequence of planet planet scattering, Kozai or secular migration. In this scenario, general relativity and secular perturbations have comparable timescales and may interfere with each other with relevant effects on the eccentricity and pericenter evolution of the two planets. We explore, both analytically and via numerical integration, how the secular evolution is changed by general relativity for a wide range of different initial conditions. We find that when the faster secular frequency approaches th…
After the detection of extrasolar planets in binary and multiple star systems questions arose about their dynamics and habitability. In this study we investigate a five-dimensional parameter space for wide binary stars with a massive planet beyond the habitable zone (HZ). Our aim is to reveal those orbital and physical parameter combinations that expose bodies in the primary star's HZ to secular perturbations. Building upon an established semi-analytical model, we combine two separate analytical models into a new one to treat the restricted four-body problem. We apply this new Combined Analyti…
Gravitational perturbations in multi-planet systems caused by an accompanying star are the subject of this investigation. Our dynamical model is based on the binary star HD41004 AB where a giant planet orbits HD41004 A. We modify the orbital parameters of this system and analyze the motion of a hypothetical test-planet surrounding HD41004 A on an interior orbit to the detected giant planet. Our numerical computations indicate perturbations due to mean motion and secular resonances. The locations of these resonances are usually connected to high eccentricity and highly inclined motion depending…
In this work, we construct and test an analytical and a semianalytical secular models for two planets locked in a coorbital non-coplanar motion, comparing some results with the case of restricted three body problem. The analytical average model replicates the numerical N-body integrations, even for moderate eccentricities ($\lesssim$ 0.3) and inclinations ($\lesssim10^\circ$), except for the regions corresponding to quasi-satellite and Lidov-Kozai configurations. Furthermore, this model is also useful in the restricted three body problem, assuming very low mass ratio between the planets. We al…
The Lidov-Kozai mechanism allows a body to periodically exchange its eccentricity with inclination. It was first discussed in the framework of the quadrupolar secular restricted three-body problem, where the massless particle is the inner body, and later extended to the quadrupolar secular nonrestricted three body problem. In this paper, we propose a different point of view on the problem by looking first at the restricted problem where the massless particle is the outer body. In this situation, equilibria at high mutual inclination appear, which correspond to the population of stable particle…
I consider the dynamics of mean motion resonances between pairs of co-planar planets and derive a new integrable Hamiltonian model for planets' resonant motion. The new model generalizes previously-derived integrable Hamiltonians for first-order resonances to treat higher-order resonances by exploiting a surprising near-symmetry of the full, non-integrable Hamiltonians of higher-order resonances. Whereas past works have frequently relied on truncated disturbing function expansions to derive integrable approximations to resonant motion, I show that no such expansion is necessary, thus enabling …
Context. The dwarf planet Ceres and the asteroid Vesta have been studied by the Dawn space mission. They are the two heaviest bodies of the main asteroid belt and have different characteristics. Notably, Vesta appears to be dry and inactive with two large basins at its south pole. Ceres is an ice-rich body with signs of cryovolcanic activity. Aims. The aim of this paper is to determine the obliquity variations of Ceres and Vesta and to study their rotational stability. Methods. The orbital and rotational motions have been integrated by symplectic integration. The rotational stability has been …
Statistical analysis of the orbits of distant Kuiper Belt Objects (KBOs) have led to suggest that an additional planet should reside in the Solar System. According to recent models, the secular action of this body should cause orbital alignment of the KBOs. It was recently claimed that the KBOs concerned by this dynamics are presumably trapped in mean motion resonances with the suspected planet. I reinvestigate here the secular model underlying this idea. The original analysis was done expanding and truncating the secular Hamiltonian. I show that this is inappropriate here, as the series expan…
Using numerical integrations, we find that the orbital eccentricity of Saturn's moon Iapetus undergoes prominent multi-Myr oscillations. We identify the responsible resonant argument to be $\varpi-\varpi_{g5}+Ω-Ω_{eq}$, with the terms being the longitudes of pericenter of Iapetus and planetary secular mode $g_5$, Iapetus's longitude of the node and Saturn's equinox. We find that this argument currently (on a $10^7$~yr timescale) appears to librate with a very large amplitude. On longer timescales, the behavior of this resonant angle is strongly dependent on the resonant interaction between Sat…
We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula's expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane's expansion…
Two semi-analytical one-degree-of-freedom secular models are presented for the motion of small bodies beyond Neptune. A special attention is given to trajectories entirely exterior to the planetary orbits. The first one is the well-known non-resonant model of Kozai (1962) adapted to the transneptunian region. Contrary to previous papers, the dynamics is fully characterized with respect to the fixed parameters. A maximum perihelion excursion possible of 16.4 AU is determined. The second model handles the occurrence of a mean-motion resonance with one of the planets. In that case, the one-degree…
Context: Seasonal variations and climate stability of a planet are very sensitive to the planet obliquity and its evolution. This is of particular interest for the emergence and sustainability of land-based life, but orbital and rotational parameters of exoplanets are still poorly constrained. Numerical explorations usually realised in this situation are thus in heavy contrast with the uncertain nature of the available data. Aims: We aim to provide an analytical formulation of the long-term spin-axis dynamics of exoplanets, linking it directly to physical and dynamical parameters, but still gi…
The long-term dynamics of the geostationary Earth orbits (GEO) is revisited through the application of canonical perturbation theory. We consider a Hamiltonian model accounting for all major perturbations: geopotential at order and degree two, lunisolar perturbations with a realistic model for the Sun and Moon orbits, and solar radiation pressure. The long-term dynamics of the GEO region has been studied both numerically and analytically, in view of the relevance of such studies to the issue of space debris or to the disposal of GEO satellites. Past studies focused on the orbital evolution of …
Spatially resolved images of debris disks frequently reveal complex morphologies such as gaps, spirals, and warps. Most existing models for explaining such morphologies focus on the role of massive perturbers (i.e. planets, stellar companions), ignoring the gravitational effects of the disk itself. Here we investigate the secular interaction between an eccentric planet and a massive, external debris disk using a simple analytical model. Our framework accounts for both the gravitational coupling between the disk and the planet, as well as the disk self-gravity -- with the limitation that it ign…
It is well known that asteroids and comets fall into the Sun. Metal pollution of white dwarfs and transient spectroscopic signatures of young stars like $β$-Pic provide growing evidence that extra solar planetesimals can attain extreme orbital eccentricities and fall onto their parent stars. We aim to develop a general, practically implementable, semi-analytical theory of secular eccentricity excitation of small bodies in mean motion resonances with an eccentric planet valid for arbitrary values of the eccentricities and including the short-range force due to General Relativity. Our semi-analy…
We extend the formalism of a previous paper to include the effects of flybys and instantaneous perturbations such as supernovae on the long-term secular evolution of hierarchical multiple systems with an arbitrary number of bodies and hierarchy, provided that the system is composed of nested binary orbits. To model secular encounters, we expand the Hamiltonian in terms of the ratio of the separation of the perturber with respect to the barycentre of the multiple system, to the separation of the widest orbit. Subsequently, we integrate over the perturber orbit numerically or analytically. We ve…
The evolution of protoplanetary discs and the related process of planet formation is regulated by angular momentum transport and mass-loss processes. Over the past decade, the paradigm of viscosity has been challenged and MHD disc winds appear as a compelling scenario to account for disc accretion. In this work, we aim to construct the equivalent of the widely used analytical description of viscous evolution for the MHD wind case. The transport of angular momentum and mass induced by the wind is parameterized by an $α$-like parameter and by the magnetic lever arm parameter $λ$. Extensions of t…
In a multiplanet system, tides acting on the inner planet can significantly affect the orbital evolution of the entire system. While tides usually damp eccentricities, a novel mechanism identified by Correia et al. (2012) tends to raise eccentricities as a result of the tides' effect on the inner planet's rotation. Our analytical description of this spin-driven tidal (SDT) effect shows that, while the inner planet's eccentricity undergoes pumping, the process is more completely described by an exchange of strength between the two eigenmodes of the dynamical system. Our analysis allows derivati…
Recent observations have shown that at least some close-in exoplanets maintain eccentric orbits despite tidal circularization timescales that are typically shorter than stellar ages. We explore gravitational interactions with a distant planetary companion as a possible cause of these non-zero eccentricities. For simplicity, we focus on the evolution of a planar two-planet system subject to slow eccentricity damping and provide an intuitive interpretation of the resulting long-term orbital evolution. We show that dissipation shifts the two normal eigenmode frequencies and eccentricity ratios of…
Planet-planet perturbations can cause planets' orbital elements to change on secular timescales. Previous work has evaluated the nodal precession rate for planets in the limit of low $α$ (semi-major axis ratio, 0$<$$α$$\leq$1). Our simulations show that systems at high $α$ (or low period ratio), similar to multiplanet systems found in the Kepler survey, have a nodal precession rate that is more strongly dependent on eccentricity and inclination. We present a complete expansion of the nodal precession rate to fourth order in the disturbing function and show that this analytical solution much be…
An ever-growing observational aggregate of extrasolar planets has revealed that systems of planets that reside in or near mean-motion resonances are relatively common. While the origin of such systems is attributed to protoplanetary disk-driven migration, a qualitative description of the dynamical evolution of resonant planets remains largely elusive. Aided by the pioneering works of the last century, we formulate an approximate, integrable theory for first-order resonant motion. We utilize the developed theory to construct an intuitive, geometrical representation of resonances within the cont…
We consider the solid-solid interactions in the two body problem. The relative equilibria have been previously studied analytically and general motions were numerically analyzed using some expansion of the gravitational potential up to the second order, but only when there are no direct interactions between the orientation of the bodies. Here we expand the potential up to the fourth order and we show that the secular problem obtained after averaging over fast angles, as for the precession model of Boue and Laskar [Boue, G., Laskar, J., 2006. Icarus 185, 312-330], is integrable, but not trivial…
We present a survey on exoplanetary systems of binary stars with stellar separations less than 100 au. For a sample of 11 binaries that harbour detected circumstellar giant planets we investigate the frequency of systems with secular resonances (SR) affecting the habitable zone (HZ). Secular resonances are connected to dynamically unstable or chaotic regions by enforcing highly eccentric motion. We apply a semi-analytical method to determine the locations of linear SR, which is based on finding the apsidal precession frequencies of the massive bodies. For configurations where the giant planet …
We study the dynamics of the 3-D three-body problem of a small body moving under the attractions of a star and a giant planet which orbits the star on a much wider and elliptic orbit. In particular, we focus on the influence of an eccentric orbit of the outer perturber on the dynamics of a small highly inclined inner body. Our analytical study of the secular perturbations relies on the classical octupole hamiltonian expansion (third-order theory in the ratio of the semi-major axes), as third-order terms are needed to consider the secular variations of the outer perturber and potential secular …
Although the discovery of the chaotic motion of the inner planets in the solar system dates back to more than thirty years ago, the secular chaos of their orbits still dares more analytical analyses. Apart from the high-dimensional structure of the motion, this is probably related to the lack of an adequately simple dynamical model. Here, we consider a new secular dynamics for the inner planets, with the aim of retaining a fundamental set of interactions responsible for their chaotic behaviour, while being consistent with the predictions of the most precise orbital solutions currently availabl…
We calculate the eccentricity excitation of asteroids produced by the sweeping $ν_6$ secular resonance during the epoch of planetesimal-driven giant planet migration in the early history of the solar system. We derive analytical expressions for the magnitude of the eccentricity change and its dependence on the sweep rate and on planetary parameters; the $ν_6$ sweeping leads to either an increase or a decrease of eccentricity depending on an asteroid's initial orbit. Based on the slowest rate of $ν_6$ sweeping that allows a remnant asteroid belt to survive, we derive a lower limit on Saturn's m…
Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. i) We demonstrate the long term stability of the semimajor axis within the framework of the $J_2$ problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $H_{J_2}$ . ii) We demonstrate the stability of the eccentricity an…
The SecularMultiple code, presented in two previous papers of this series, integrates the long-term dynamical evolution of multiple systems with any number of bodies and hierarchical structure, provided that the system is composed of nested binaries. In the formalism underlying SecularMultiple, we previously averaged over all orbits in the system. This approximation significantly speeds up numerical integration of the equations of motion, making large population synthesis studies possible. However, the orbit averaging approximation can break down when the secular evolution timescale of the sys…
Since the original work of Hansen and Tisserand in the XIXth century, there have been many variations in the analytical expansion of the three-body disturbing function in series of the semi-major axis ratio. With the increasing number of planetary systems of large eccentricity, these expansions are even more interesting as they allow us to obtain for the secular systems finite expressions that are valid for all eccentricities and inclinations. We revisited the derivation of the disturbing function in Legendre polynomial, with a special focus on the secular system. We provide here expressions o…
This work focuses on providing closed form analytical expressions to define frozen orbits under the effects of the zonal harmonics of an Earth-like planet. Particularly, the perturbation effects from the terms J2, J3, J4, J5, J6, and J7 are considered in this work. This is done using a power series expansion in the small parameter that allow not only to provide an approximate solution to the system, but also to determine the analytical expressions that define the initial osculating conditions that generate frozen orbits. As a result of that, the proposed methodology allows to study the bifurca…
Öpik-type collision frequency for Kozai-driven projectiles: Target bodies on inclined circular orbits
School of Aeronautics and Astronautics, Shenzhen Campus of Sun Yat-sen University（中山大学深圳校区航空航天学院）；Shenzhen Key Laboratory of Intelligent Microsatellite Constellation（深圳市智能微卫星星座重点实验室）
The herein presented analytical framework fully describes the motion of coplanar systems consisting of a stellar binary and a planet orbiting both stars on orbital as well as secular timescales. Perturbations of the Runge-Lenz vector are used to derive short period evolution of the system, while octupole secular theory is applied to describe its long term behaviour. A post Newtonian correction on the stellar orbit is included. The planetary orbit is initially circular and the theory developed here assumes that the planetary eccentricity remains relatively small (e_2<0.2). Our model is tested a…
Circumbinary planets in coplanar triple-star systems: I. Minimum eccentricity variation region
Facultad de Matemática, Astronomía, Física y Computación (FAMAF), Universidad Nacional de Córdoba (UNC)（科尔多瓦国立大学数学、天文学、物理与计算学院）；Instituto de Astronomía Teórica y Experimental (IATE), CONICET-UNC（理论与实验天文研究所）；Observatorio Astronómico de Córdoba (OAC)（科尔多瓦天文台）；Univ. Grenoble Alpes, CNRS, IPAG（格勒诺布尔阿尔卑斯大学法国国家科学研究中心IPAG实验室）
