scholarly journals Orbital stability of Earth Trojans

2019 ◽  
Vol 622 ◽  
pp. A97 ◽  
Author(s):  
Lei Zhou ◽  
Yang-Bo Xu ◽  
Li-Yong Zhou ◽  
Rudolf Dvorak ◽  
Jian Li
Keyword(s):  
Orbital Stability ◽  
Dynamical Behaviour ◽  
Secular Resonance ◽  
Lagrange Points ◽  
Secular Resonances ◽  
Frequency Space ◽  
Yarkovsky Effect ◽  
Horseshoe Orbits ◽  
The Stability ◽  
Primordial Earth

The only discovery of Earth Trojan 2010 TK7 and the subsequent launch of OSIRIS-REx have motived us to investigate the stability around the triangular Lagrange points of the Earth, L4 and L5. In this paper we present detailed dynamical maps on the (a0, i0) plane with the spectral number (SN) indicating the stability. Two main stability regions, separated by a chaotic region arising from the ν3 and ν4 secular resonances, are found at low (i0 ≤ 15°) and moderate (24 ° ≤i0 ≤ 37°) inclinations, respectively. The most stable orbits reside below i0 = 10° and they can survive the age of the solar system. The nodal secular resonance ν13 could vary the inclinations from 0° to ∼10° according to their initial values, while ν14 could pump up the inclinations to ∼20° and upwards. The fine structures in the dynamical maps are related to higher degree secular resonances, of which different types dominate different areas. The dynamical behaviour of the tadpole and horseshoe orbits, reflected in their secular precession, show great differences in the frequency space. The secular resonances involving the tadpole orbits are more sensitive to the frequency drift of the inner planets, thus the instabilities could sweep across the phase space, leading to the clearance of tadpole orbits. We are more likely to find terrestrial companions on horseshoe orbits. The Yarkovsky effect could destabilize Earth Trojans in varying degrees. We numerically obtain the formula describing the stabilities affected by the Yarkovsky effect and find the asymmetry between the prograde and retrograde rotating Earth Trojans. The existence of small primordial Earth Trojans that avoid being detected but survive the Yarkovsky effect for 4.5 Gyr is substantially ruled out.

scholarly journals Existence of Asteroids in the Inner Solar System

2004 ◽  
Vol 202 ◽  
pp. 238-240
Author(s):  
S. A. Tabachnik ◽  
N. W. Evans
Keyword(s):  
Solar System ◽  
Terrestrial Planets ◽  
Secular Resonance ◽  
Lagrange Points ◽  
The Earth ◽  
Numerical Integrations ◽  
Test Particles ◽  
Horseshoe Orbits

Ensembles of in-plane and inclined orbits in the vicinity of the Lagrange points of the terrestrial planets are integrated for up to 100 million years. Mercurian Trojans probably do not exist, although there is evidence for long-lived, corotating horseshoe orbits with small inclinations. Both Venus and the Earth are much more promising, as they possess rich families of stable tadpole and horseshoe orbits. Our survey of in-plane test particles near the Martian Lagrange points shows no survivors after 60 million years. Low inclination test particles do not persist, as their inclinations are quickly increased until the effects of a secular resonance with Jupiter cause de-stabilisation. Numerical integrations of inclined test particles for timespans of 25 million years show stable zones for inclinations between 14° and 40°. Both Martian Trojans 5261 Eureka and 1998 VF31 lie deep within the stable zones, which suggests they may be of primordial origin.

scholarly journals Systematic survey of the dynamics of Uranus Trojans

2020 ◽  
Vol 633 ◽  
pp. A153 ◽  
Author(s):  
Lei Zhou ◽  
Li-Yong Zhou ◽  
Rudolf Dvorak ◽  
Jian Li
Keyword(s):  
Solar System ◽  
Stability Region ◽  
Large Fraction ◽  
Instability Strip ◽  
Stability Regions ◽  
Lagrange Points ◽  
Secular Resonances ◽  
Secondary Resonances ◽  
The Stability ◽  
Planetary Migration

Context. The discovered Uranus Trojan (UT) 2011 QF99 and several candidate UTs have been reported to be in unstable orbits. This implies that the stability region around the triangular Lagrange points L4 and L5 of Uranus should be very limited. Aims. In this paper, we aim to locate the stability region for UTs and find out the dynamical mechanisms responsible for the structures in the phase space. The null detection of primordial UTs also needs to be explained. Methods. Using the spectral number as the stability indicator, we constructed the dynamical maps on the (a0, i0) plane. The proper frequencies of UTs were determined precisely with a frequency analysis method that allows us to depict the resonance web via a semi-analytical method. We simulated radial migration by introducing an artificial force acting on planets to mimic the capture of UTs. Results. We find two main stability regions: a low-inclination (0° −14°) and a high-inclination regime (32° −59°). There is also an instability strip in each of these regions at 9° and 51°, respectively. These strips are supposed to be related with g − 2g5 + g7 = 0 and ν8 secular resonances. All stability regions are in the tadpole regime and no stable horseshoe orbits exist for UTs. The lack of moderate-inclined UTs is caused by the ν5 and ν7 secular resonances, which could excite the eccentricity of orbits. The fine structures in the dynamical maps are shaped by high-degree secular resonances and secondary resonances. Surprisingly, the libration centre of UTs changes with the initial inclination, and we prove it is related to the quasi 1:2 mean motion resonance (MMR) between Uranus and Neptune. However, this quasi-resonance has an ignorable influence on the long-term stability of UTs in the current planetary configuration. About 36.3% and 0.4% of the pre-formed orbits survive fast and slow migrations with migrating timescales of 1 and 10 Myr, respectively, most of which are in high inclination. Since low-inclined UTs are more likely to survive the age of the solar system, they make up 77% of all such long-life orbits by the end of the migration, making a total fraction up to 4.06 × 10−3 and 9.07 × 10−5 of the original population for fast and slow migrations, respectively. The chaotic capture, just like depletion, results from secondary resonances when Uranus and Neptune cross their mutual MMRs. However, the captured orbits are too hot to survive until today. Conclusions. About 3.81% UTs are able to survive the age of the solar system, among which 95.5% are on low-inclined orbits with i0 <  7.5°. However, the depletion of planetary migration seems to prevent a large fraction of such orbits, especially for the slow migration model. Based on the widely adopted migration models, a swarm of UTs at the beginning of the smooth outward migration is expected and a fast migration is favoured if any primordial UTs are detected.

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.

PNEUMATIC VIBROISOLATION SYSTEM OF THE BASE DESK WITH NATURAL FREQUENCY REGULATION

2021 ◽  
Vol 113 ◽  
pp. 91-100
Author(s):  
Radka JÍROVÁ ◽  
Lubomír PEŠÍK
Keyword(s):  
Natural Frequency ◽  
Automotive Industry ◽  
Natural Frequencies ◽  
Dynamical Behaviour ◽  
Operating Conditions ◽  
Frequency Regulation ◽  
The Stability ◽  
Short Time

Vibroisolation systems of base desks for machine and testing facilities usually cannot effect efficient changing of their own frequencies according to operating conditions. Especially in the case of the automotive industry, the possibility of changing natural frequencies is very desirable. During varying operating conditions, the vibroisolation system needs to be regulated easily and quickly regarding the minimisation of dynamical forces transmitted to the ground and to ensure the stability of the testing process. This paper describes one of the options of tuning the base desk at a relatively short time and by sufficient change of own frequencies, which decides the dynamical behaviour of the whole system.

scholarly journals Stability Mechanism of Planetary System of ν Andromedae

2004 ◽  
Vol 202 ◽  
pp. 202-204 ◽  
Author(s):  
Hiroshi Kinoshita ◽  
Hiroshi Nakai
Keyword(s):  
Perturbation Theory ◽  
Planetary System ◽  
Secular Resonance ◽  
Secular Perturbation ◽  
Outer Planets ◽  
The Stability

Three planets are detected around ν Andromedae. The stability of Upsilon Andromedae Planetary system is maintained by the co rotation of the pericenters of the two outer planets. If the pericenters of the two outer planets move independently, the planetary system becomes unstable. The corotation of the pericenters is explained by the secular perturbation theory. This corotation is not a secular resonance.

Extended local Rytov Fourier migration method

Geophysics ◽  
1999 ◽  
Vol 64 (5) ◽  
pp. 1535-1545 ◽  
Author(s):  
Lian‐Jie Huang ◽  
Michael C. Fehler ◽  
Peter M. Roberts ◽  
Charles C. Burch
Keyword(s):  
Phase Changes ◽  
Fast Fourier Transform Algorithm ◽  
Scalar Wave ◽  
Depth Migration ◽  
Frequency Space ◽  
Rytov Approximation ◽  
Migration Method ◽  
The Stability ◽  
Wavefield Extrapolation ◽  
Lateral Variations

We develop a novel depth‐migration method termed the extended local Rytov Fourier (ELRF) migration method. It is based on the scalar wave equation and a local application of the Rytov approximation within each extrapolation interval. Wavefields are Fourier transformed back and forth between the frequency‐space and frequency‐wavenumber domains during wavefield extrapolation. The lateral slowness variations are taken into account in the frequency‐space domain. The method is efficient due to the use of a fast Fourier transform algorithm. Under the small angle approximation, the ELRF method leads to the split‐step Fourier (SSF) method that is unconditionally stable. The ELRF method and the extended local Born Fourier (ELBF) method that we previously developed can handle wider propagation angles than the SSF method and account for the phase and amplitude changes due to the lateral variations of slowness, whereas the SSF method only accounts for the phase changes. The stability of the ELRF method is controlled more easily than that of the ELBF method.

Orbital stability of standing waves for nonlinear fractional Schrödinger equation with unbounded potential

2019 ◽  
Vol 12 (03) ◽  
pp. 1950043
Author(s):  
Xiaohua Liu
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Orbital Stability ◽  
Fractional Schrödinger Equation ◽  
Unbounded Potential ◽  
Nonlinear Fractional Schrödinger Equation ◽  
The Stability ◽  
First Time ◽  
Fractional Schrodinger Equation

In this paper, the orbital stability of standing waves for nonlinear fractional Schrödinger equation is considered. By constructing the constrained functional extreme-value problem, the existence of standing waves is studied. With the help of the orbital stability theories presented by Grillakis, Shatah and Strauss, the orbital stability of standing waves is determined by the sign of a discriminant. To our knowledge, it is the first time that the abstract orbital stability theories presented by Grillakis, Shatah and Strauss are applied to study the stability of solutions for fractional evolution equation.

scholarly journals The Stability of the Planetary Triangular Lagrange Points

1990 ◽  
Vol 123 ◽  
pp. 533-536
Author(s):  
Seppo Mikkola ◽  
K.A. Innanen
Keyword(s):  
Solar System ◽  
Progress Report ◽  
Scientific Applications ◽  
Medium Term ◽  
Lagrange Points ◽  
The Earth ◽  
Self Consistent ◽  
The Stability

AbstractNumerical, self-consistent, n-body integrations of the solar system show significant indications of medium-term (i.e. several million-year) stability for the various planet-Sun L4,L5 configurations. A progress report of our computations, emphasizing the inner solar system, will be given. There exist interesting possibilities for these locations (including the Earth) as the sites for longer term scientific applications, both pure and applied.

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