scholarly journals Polar planets around highly eccentric binaries are the most stable

2020 ◽  
Vol 494 (4) ◽  
pp. 4645-4655
Author(s):  
Cheng Chen ◽  
Stephen H Lubow ◽  
Rebecca G Martin
Keyword(s):  
Semimajor Axis ◽  
Orbital Stability ◽  
Equal Mass ◽  
Escape Time ◽  
Mass Ratio ◽  
Eccentric Orbit ◽  
Polar Orbit ◽  
Formation And Evolution ◽  
Binary Orbit ◽  
The Stability

ABSTRACT We study the orbital stability of a non-zero mass, close-in circular orbit planet around an eccentric orbit binary for various initial values of the binary eccentricity, binary mass fraction, planet mass, planet semimajor axis, and planet inclination by means of numerical simulations that cover 5 × 104 binary orbits. For small binary eccentricity, the stable orbits that extend closest to the binary (most stable orbits) are nearly retrograde and circulating. For high binary eccentricity, the most stable orbits are highly inclined and librate near the so-called generalized polar orbit which is a stationary orbit that is fixed in the frame of the binary orbit. For more extreme mass ratio binaries, there is a greater variation in the size of the stability region (defined by initial orbital radius and inclination) with planet mass and initial inclination, especially for low binary eccentricity. For low binary eccentricity, inclined planet orbits may be unstable even at large orbital radii (separation ${\gt}5 \, a_{\rm b}$). The escape time for an unstable planet is generally shorter around an equal mass binary compared with an unequal mass binary. Our results have implications for circumbinary planet formation and evolution and will be helpful for understanding future circumbinary planet observations.

scholarly journals Analysis of the orbital stability close to the binary asteroid (90) Antiope

2020 ◽  
Vol 496 (2) ◽  
pp. 1645-1654
Author(s):  
S Aljbaae ◽  
A F B A Prado ◽  
D M Sanchez ◽  
H Hussmann
Keyword(s):  
Gravitational Field ◽  
Equilibrium Points ◽  
Orbital Stability ◽  
Solar Radiation Pressure ◽  
Equal Mass ◽  
Orbital Dynamics ◽  
Full Potential ◽  
Mass Ratio ◽  
Binary Asteroid ◽  
Polyhedral Shape

ABSTRACT We provide a generalized discussion on the dynamics of a spacecraft around the equal-mass binary asteroid (90) Antiope, under the influence of solar radiation pressure at the perihelion and aphelion distances of the asteroid from the Sun. The polyhedral shape of the components of this asteroid is used to accurately model the gravitational field. Five unstable equilibrium points are determined and classified into two cases that allow classifying of the motion associated with the target as always unstable. The dynamical effects of the mass ratio of our binary system are investigated. We tested massless particles initially located at the periapsis distance on the equatorial plane of the primary of our binary asteroid. Bounded orbits around our system are not found for the longitudes λ ∈ {60, 90, 120, 240, 270, 300}. We also discuss the orbital dynamics in the full potential field of (90) Antiope. The tested motions are mainly dominated by the binary’s gravitational field; no significant effects of the SRP are detected. For λ = 180°, less perturbed orbits are identified between 420 and 700 km from the centre of the system, that corresponds to orbits with Δa < 30 km and Δe < 0.15. All the orbits with initial periapsis distance smaller than 350 km either collide with components of our asteroid or escape from the system.

scholarly journals An integrable model for first-order three-planet mean motion resonances

2021 ◽  
Vol 133 (8) ◽  
Author(s):  
Antoine C. Petit
Keyword(s):  
Integrable Model ◽  
Degree Of Freedom ◽  
Mass Ratio ◽  
Zeroth Order ◽  
Mean Motion Resonances ◽  
First Order ◽  
Mean Motion ◽  
Formation And Evolution ◽  
The Stability ◽  
Three Body

AbstractRecent works on three-planet mean motion resonances (MMRs) have highlighted their importance for understanding the details of the dynamics of planet formation and evolution. While the dynamics of two-planet MMRs are well understood and approximately described by a one-degree-of-freedom Hamiltonian, little is known of the exact dynamics of three-body resonances besides the cases of zeroth-order MMRs or when one of the bodies is a test particle. In this work, I propose the first general integrable model for first-order three-planet mean motion resonances. I show that one can generalize the strategy proposed in the two-planet case to obtain a one-degree-of-freedom Hamiltonian. The dynamics of these resonances are governed by the second fundamental model of resonance. The model is valid for any mass ratio between the planets and for every first-order resonance. I show the agreement of the analytical model with numerical simulations. As examples of application, I show how this model could improve our understanding of the capture into MMRs as well as their role in the stability of planetary systems.

scholarly journals Planet formation and stability in polar circumbinary discs

2019 ◽  
Vol 628 ◽  
pp. A119 ◽  
Author(s):  
Nicolás Cuello ◽  
Cristian A. Giuppone
Keyword(s):  
Three Dimensional ◽  
Orbital Plane ◽  
Equal Mass ◽  
Wide Range ◽  
Rotation Vectors ◽  
Hydrodynamical Simulations ◽  
Binary Orbit ◽  
The Stability ◽  
P Type

Context. Dynamical studies suggest that most circumbinary discs (CBDs) should be coplanar (i.e. the rotation vectors of the binary and the disc should be aligned). However, some theoretical works show that under certain conditions a CBD can become polar, which means that its rotation vector is orthogonal with respect to the binary orbital plane. Interestingly, very recent observations show that polar CBDs exist in nature (e.g. HD 98800). Aims. We test the predictions of CBD alignment around eccentric binaries based on linear theory. In particular, we compare prograde and retrograde CBD configurations. Then, assuming planets form in these systems, we thoroughly characterise the orbital behaviour and stability of misaligned (P-type) particles. This is done for massless and massive particles. Methods. The evolution of the CBD alignment for various configurations was modelled through three-dimensional hydrodynamical simulations. For the orbital characterisation and the analysis stability, we relied on long-term N-body integrations and structure and chaos indicators, such as Δe and MEGNO. Results. We confirm previous analytical predictions on CBD alignment, but find an unexpected symmetry breaking between prograde and retrograde configurations. More specifically, we observe polar alignment for a retrograde misaligned CBD that was expected to become coplanar with respect to the binary disc plane. Therefore, the likelihood of becoming polar for a highly misaligned CBD is higher than previously thought. Regarding the stability of circumbinary P-type planets (also know as Tatooines), polar orbits are stable over a wide range of binary parameters. In particular, for binary eccentricities below 0.4 the orbits are stable for any value of the binary mass ratio. In the absence of gas, planets with masses below 10−5 M⊙ have negligible effects on the binary orbit. Finally, we suggest that mildly eccentric equal-mass binaries should be searched for polar Tatooines.

scholarly journals Orbital dynamics of circumbinary planets

2019 ◽  
Vol 490 (4) ◽  
pp. 5634-5646 ◽  
Author(s):  
Cheng Chen ◽  
Alessia Franchini ◽  
Stephen H Lubow ◽  
Rebecca G Martin
Keyword(s):  
Numerical Simulations ◽  
Semimajor Axis ◽  
Quantitative Agreement ◽  
Eccentric Orbit ◽  
Secular Dynamics ◽  
Evolutionary Changes ◽  
Planetary Orbits ◽  
Monotonically Decreasing Function ◽  
Formation And Evolution ◽  
Analytic Models

ABSTRACT We investigate the dynamics of a non-zero mass, circular orbit planet around an eccentric orbit binary for various values of the binary eccentricity, binary mass fraction, planet mass, and planet semimajor axis by means of numerical simulations. Previous studies investigated the secular dynamics mainly by approximate analytic methods. In the stationary inclination state, the planet and binary precess together with no change in relative tilt. For both prograde and retrograde planetary orbits, we explore the conditions for planetary orbital libration versus circulation and the conditions for stationary inclination. As was predicted by analytic models, for sufficiently high initial inclination, a prograde planet’s orbit librates about the stationary tilted state. For a fixed binary eccentricity, the stationary angle is a monotonically decreasing function of the ratio of the planet-to-binary angular momentum j. The larger j, the stronger the evolutionary changes in the binary eccentricity and inclination. We also calculate the critical tilt angle that separates the circulating from the librating orbits for both prograde and retrograde planet orbits. The properties of the librating orbits and stationary angles are quite different for prograde versus retrograde orbits. The results of the numerical simulations are in very good quantitative agreement with the analytic models. Our results have implications for circumbinary planet formation and evolution.

scholarly journals Semi-Analytical Estimates for the Orbital Stability of Earth’s Satellites

2021 ◽  
Vol 31 (6) ◽  
Author(s):  
Irene De Blasi ◽  
Alessandra Celletti ◽  
Christos Efthymiopoulos
Keyword(s):  
Normal Form ◽  
Semimajor Axis ◽  
Orbital Stability ◽  
Stability Estimates ◽  
Term Stability ◽  
Long Term Stability ◽  
Lunisolar Perturbations ◽  
The Stability ◽  
Laplace Plane

AbstractNormal 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 $$J_2$$ J 2 problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $${\mathcal {H}}_{J_2}$$ H J 2 . (ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the ‘geolunisolar’ Hamiltonian $${\mathcal {H}}_\mathrm{gls}$$ H gls ), after a suitable reduction of the Hamiltonian to the Laplace plane. (iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $${\mathcal {H}}_{J_2}$$ H J 2 and $${\mathcal {H}}_\mathrm{gls}$$ H gls models, which reflect necessary conditions for the holding of Nekhoroshev’s theorem on the exponential stability of the orbits. We find that the $${\mathcal {H}}_{J_2}$$ H J 2 model is non-convex, but satisfies a ‘three-jet’ condition, while the $${\mathcal {H}}_\mathrm{gls}$$ H gls model restores quasi-convexity by adding lunisolar terms in the Hamiltonian’s integrable part.

scholarly journals Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1398
Author(s):  
Natalia Kolkovska ◽  
Milena Dimova ◽  
Nikolai Kutev
Keyword(s):  
Solitary Waves ◽  
Orbital Stability ◽  
Second Derivative ◽  
Cubic Nonlinearity ◽  
Abstract Theory ◽  
Power Type ◽  
Analytical Formulas ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Double Dispersion

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

scholarly journals Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems

1999 ◽  
Vol 172 ◽  
pp. 443-444
Author(s):  
Massimiliano Guzzo
Keyword(s):  
Degenerate System ◽  
Escape Time ◽  
Kam Theorem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Arbitrary Potential ◽  
Hamiltonian Perturbation Theory ◽  
Three Body ◽  
Nekhoroshev Stability ◽  
Degenerate Hamiltonian Systems

Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

scholarly journals Forming bulges during galaxy minor mergers

2007 ◽  
Vol 3 (S245) ◽  
pp. 63-66 ◽  
Author(s):  
T. J. Cox ◽  
J. Younger ◽  
L. Hernquist ◽  
P. F. Hopkins
Keyword(s):  
Black Hole ◽  
Large Mass ◽  
Elliptical Galaxies ◽  
Equal Mass ◽  
Mass Ratio ◽  
Black Hole Growth ◽  
Major Merger ◽  
Hole Growth ◽  
Minor Mergers ◽  
Large Mass Ratio

AbstractThe hierarchical formation of structure suggests that dark halos, and the galaxies they host, are shaped by their merging history. While the idea that mergers between galaxies of equal mass, i.e., major merger, produce elliptical galaxies has received considerable attention, he galaxies that result from minor merger, i.e., mergers between galaxies with a large mass ratio, is much less understood. We have performed a large number of numerical simulations of minor mergers, including cooling, star formation, and black hole growth in order to study this process in more detail. This talk will present some preliminary results of this study, and in particular, the morphology and kinematics of minor merger remnants.

On the Stability of the Linearly Related Modes of Certain Nonlinear Two-Degree-of-Freedom Systems

1961 ◽  
Vol 28 (1) ◽  
pp. 71-77 ◽  
Author(s):  
C. P. Atkinson
Keyword(s):  
Nonlinear Differential Equations ◽  
Mass Ratio ◽  
Positive Mass ◽  
Fourth Degree ◽  
Real Roots ◽  
General Stability ◽  
The Stability ◽  
The Masses ◽  
Spring Constants ◽  
Two Degree Of Freedom

This paper presents a method for analyzing a pair of coupled nonlinear differential equations of the Duffing type in order to determine whether linearly related modal oscillations of the system are possible. The system has two masses, a coupling spring and two anchor springs. For the systems studied, the anchor springs are symmetric but the masses are not. The method requires the solution of a polynomial of fourth degree which reduces to a quadratic because of the symmetric springs. The roots are a function of the spring constants. When a particular set of spring constants is chosen, roots can be found which are then used to set the necessary mass ratio for linear modal oscillations. Limits on the ranges of spring-constant ratios for real roots and positive-mass ratios are given. A general stability analysis is presented with expressions for the stability in terms of the spring constants and masses. Two specific examples are given.

A Study of the Orbital and Intrinsic Variability of the Double-Lined Spectroscopic Binary β Centauri

2002 ◽  
Vol 185 ◽  
pp. 86-87
Author(s):  
M. Ausseloos ◽  
C. Aerts ◽  
K. Uytterhoeven
Keyword(s):  
High Resolution ◽  
Orbital Motion ◽  
High Signal ◽  
Mass Ratio ◽  
Eccentric Orbit ◽  
Intrinsic Variability ◽  
Signal To Noise ◽  
Orbital Parameters ◽  
Spectroscopic Binary ◽  
First Time

AbstractWe introduce our observational study of the orbital motion of β Cen. Using 463 high signal-to-noise, high-resolution spectra obtained over a timespan of 12 years it is shown that the radial velocity of β Cen varies with an orbital period of 357.0 days. We derive for the first time the orbital parameters of β Cen and find a very eccentric orbit (e = 0.81) and similar component masses with a mass ratio M1/M2 = 1.02. Both the primary and the secondary exhibit periodic line-profile variations.

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