scholarly journals Global searches of frozen orbits around an oblate Earth-like planet

Astrodynamics ◽  
2021 ◽  
Author(s):  
Yuechen Ma ◽  
Yanchao He ◽  
Ming Xu ◽  
Yaru Zheng
Keyword(s):  
Gravitational Field ◽  
Order Term ◽  
Equilibrium Points ◽  
Hamiltonian Function ◽  
Zonal Harmonic ◽  
Frozen Orbits ◽  
Frozen Orbit ◽  
Earth Like Planets ◽  
Orbital Periods ◽  
Delaunay Variables

AbstractA frozen orbit is beneficial for observation owing to its stationary apsidal line. The traditional gravitational field model of frozen orbits only considers the main zonal harmonic terms J2 and limited high-order terms, which cannot meet the stringent demands of all missions. In this study, the gravitational field is expanded to J15 terms and the Hamiltonian canonical form described by the Delaunay variables is used. The zonal harmonic coefficients of the Earth are chosen as the sample. Short-periodic terms are eliminated based on the Hori-Lie transformation. An algorithm is developed to solve all equilibrium points of the Hamiltonian function. A stable frozen orbit with an argument of perigee that equals neither 90° nor 270° is first reported in this paper. The local stability and topology of the equilibrium points are obtained from their eigenvalues. The bifurcations of the equilibrium points are presented by drawing their global long-term evolution of frozen orbits and their orbital periods. The relationship between the terms of the gravitational field and number of frozen points is addressed to explain why only limited frozen orbits are found in the low-order term case. The analytical results can be applied to other Earth-like planets and asteroids.

scholarly journals Aanalytical Solution for Motion Around Radiated Varying Mass Body

2019 ◽  
Vol 16 ◽  
pp. 8407-8419
Author(s):  
Marwa Abdullah Bin Humaidan ◽  
M. I. El-Saftawy ◽  
H. M. Asiri
Keyword(s):  
Time Series ◽  
Radiation Pressure ◽  
Pressure Effect ◽  
Hamiltonian Function ◽  
Perturbation Techniques ◽  
First Order ◽  
D Operator ◽  
Order Solution ◽  
Delaunay Variables ◽  
Varying Mass

In this work we will add the radiation pressure effect of varying mass body to the model of varying mass Hamiltonian function, including Periastron effect. The problem was formulated in terms of Delaunay variables. The solution of the problem was constructed based on Delava – Hansilmair perturbation techniques. Finally we find the first order solution for the problem as time series by calculating the desired order for the D operator and variables.

On the dynamics of buoyant and heavy particles in a periodic Stuart vortex flow

1993 ◽  
Vol 254 ◽  
pp. 671-699 ◽  
Author(s):  
Kek-Kiong Tio ◽  
Amable Liñán ◽  
Juan C. Lasheras ◽  
Alfonso M. Gañán-Calvo
Keyword(s):  
Dynamical System ◽  
Computational Study ◽  
Mass Density ◽  
Order Term ◽  
Equilibrium Points ◽  
Density Ratio ◽  
Viscous Drag ◽  
Perturbation Scheme ◽  
Heavy Particles ◽  
Drag Forces

In this paper, we study the dynamics of small, spherical, rigid particles in a spatially periodic array of Stuart vortices given by a steady-state solution to the two-dimensional incompressible Euler equation. In the limiting case of dominant viscous drag forces, the motion of the particles is studied analytically by using a perturbation scheme. This approach consists of the analysis of the leading-order term in the expansion of the ‘particle path function’ Φ, which is equal to the stream function evaluated at the instantaneous particle position. It is shown that heavy particles which remain suspended against gravity all move in a periodic asymptotic trajectory located above the vortices, while buoyant particles may be trapped by the stable equilibrium points located within the vortices. In addition, a linear map for Φ is derived to describe the short-term evolution of particles moving near the boundary of a vortex. Next, the assumption of dominant viscous drag forces is relaxed, and linear stability analyses are carried out to investigate the equilibrium points of the five-parameter dynamical system governing the motion of the particles. The five parameters are the free-stream Reynolds number, the Stokes number, the fluid-to-particle mass density ratio, the distribution of vorticity in the flow, and a gravitational parameter. For heavy particles, the equilibrium points, when they exist, are found to be unstable. In the case of buoyant particles, a pair of stable and unstable equilibrium points exist simultaneously, and undergo a saddle-node bifurcation when a certain parameter of the dynamical system is varied. Finally, a computational study is also carried out by integrating the dynamical system numerically. It is found that the analytical and computational results are in agreement, provided the viscous drag forces are large. The computational study covers a more general regime in which the viscous drag forces are not necessarily dominant, and the effects of the various parametric inputs on the dynamics of buoyant particles are investigated.

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 The Relativistic Harmonic Oscillator in a Uniform Gravitational Field

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 294
Author(s):  
Michael M. Tung
Keyword(s):  
Gravitational Field ◽  
Harmonic Oscillator ◽  
Order Term ◽  
Classical Model ◽  
Relativistic Equation ◽  
Correct Treatment ◽  
Relativistic Limit ◽  
Starting Point ◽  
Relativistic Equation Of Motion ◽  
Lagrange Formalism

We present the relativistic generalization of the classical harmonic oscillator suspended within a uniform gravitational field measured by an observer in a laboratory in which the suspension point of the spring is fixed. The starting point of this analysis is a variational approach based on the Euler–Lagrange formalism. Due to the conceptual differences of mass in the framework of special relativity compared with the classical model, the correct treatment of the relativistic gravitational potential requires special attention. It is proved that the corresponding relativistic equation of motion has unique periodic solutions. Some approximate analytical results including the next-to-leading-order term in the non-relativistic limit are also examined. The discussion is rounded up with a numerical simulation of the full relativistic results in the case of a strong gravity field. Finally, the dynamics of the model is further explored by investigating phase space and its quantitative relativistic features.

scholarly journals The effect of zonal harmonics on dynamical structures in the circular restricted three-body problem near the secondary body

2020 ◽  
Vol 132 (9) ◽  
Author(s):  
Luke Bury ◽  
Jay McMahon
Keyword(s):  
Periodic Orbits ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Dynamic Environment ◽  
Smooth Transition ◽  
Mission Design ◽  
Jacobi Constant ◽  
Zonal Harmonic ◽  
Outer Planets ◽  
Three Body

Abstract The circular restricted three-body model is widely used for astrodynamical studies in systems where two major bodies are present. However, this model relies on many simplifications, such as point-mass gravity and planar, circular orbits of the bodies, and limiting its accuracy. In an effort to achieve higher-fidelity results while maintaining the autonomous simplicity of the classic model, we employ zonal harmonic perturbations since they are symmetric about the z-axis, thus bearing no time-dependent terms. In this study, we focus on how these perturbations affect the dynamic environment near the secondary body in real systems. Concise, easily implementable equations for gravitational potential, particle motion, and modified Jacobi constant in the perturbed model are presented. These perturbations cause a change in the normalized mean motion, and two different formulations are addressed for assigning this new value. The shifting of collinear equilibrium points in many real systems due to $$J_2$$ J 2 of each body is reported, and we study how families of common periodic orbits—Lyapunov, vertical, and southern halo—shift and distort when $$J_2$$ J 2 , $$J_4$$ J 4 , and $$J_6$$ J 6 of the primary and $$J_2$$ J 2 of the secondary body are accounted for in the Jupiter–Europa and Saturn–Enceladus systems. It is found that these families of periodic orbits change shape, position, and energy, which can lead to dramatically different dynamical behavior in some cases. The primary focus is on moons of the outer planets, many of which have very small odd zonal harmonic terms, or no measured value at all, so while the developed equations are meant for any and all zonal harmonic terms, only even terms are considered in the simulations. Early utilization of this refined CR3BP model in mission design will result in a more smooth transition to full ephemeris model.

scholarly journals Effects of Geopotential and Atmospheric Drag Effects on Frozen Orbits Using Nonsingular Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Paula Cristiane Pinto Mesquita Pardal ◽  
Rodolpho Vilhena de Moraes ◽  
Helio Koiti Kuga
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Orbital Evolution ◽  
Atmospheric Drag ◽  
Orbit Design ◽  
Frozen Orbits ◽  
Long Period ◽  
Frozen Orbit ◽  
And Control ◽  
Analytical Expressions

The concept of frozen orbit has been applied in space missions mainly for orbital tracking and control purposes. This type of orbit is important for orbit design because it is characterized by keeping the argument of perigee and eccentricity constant on average, so that, for a given latitude, the satellite always passes at the same altitude, benefiting the users through this regularity. Here, the system of nonlinear differential equations describing the motion is studied, and the effects of geopotential and atmospheric drag perturbations on frozen orbits are taken into account. Explicit analytical expressions for secular and long period perturbations terms are obtained for the eccentricity and the argument of perigee. The classical equations of Brouwer and Brouwer and Hori theories are used. Nonsingular variables approach is used, which allows obtaining more precise previsions for CBERS (China Brazil Earth Resources Satellite) satellites family and similar satellites (SPOT, Landsat, ERS, and IRS) orbital evolution.

scholarly journals Conservative evaluation of the uncertainty in the LAGEOS-LAGEOS II Lense-Thirring test

Open Physics ◽  
2010 ◽  
Vol 8 (1) ◽  
Author(s):  
Lorenzo Iorio
Keyword(s):  
Gravitational Field ◽  
Systematic Error ◽  
Gravitational Potential ◽  
Global Solutions ◽  
The Other ◽  
Zonal Harmonic ◽  
General Relativistic ◽  
The Difference ◽  
Diurnal Rotation ◽  
Multipolar Expansion

AbstractWe deal with the test of the general relativistic gravitomagnetic Lense-Thirring effect currently being conducted in the Earth’s gravitational field with the combined nodes Ω of the laser-ranged geodetic satellites LAGEOS and LAGEOS II. One of the most important sources of systematic uncertainty on the orbits of the LAGEOS satellites, with respect to the Lense-Thirring signature, is the bias due to the even zonal harmonic coefficients J ℓ of the multipolar expansion of the Earth’s geopotential which account for the departures from sphericity of the terrestrial gravitational potential induced by the centrifugal effects of its diurnal rotation. The issue addressed here is: are the so far published evaluations of such a systematic error reliable and realistic? The answer is negative. Indeed, if the difference ΔJ ℓ among the even zonals estimated in different global solutions (EIGEN-GRACE02S, EIGEN-CG03C, GGM02S, GGM03S, ITG-Grace02, ITG-Grace03s, JEM01-RL03B, EGM2008, AIUB-GRACE01S) is assumed for the uncertainties δJ ℓ instead of using their more-or-less calibrated covariances $$ \sigma _{J_\ell } $$, it turns out that the systematic error δμ in the Lense-Thirring measurement is about 3 to 4 times larger than in the evaluations so far published based on the use of the covariances of one model at a time separately, amounting up to 37% for the pair EIGEN-GRACE02S/ITG-Grace03s. The comparison among the other recent GRACE-based models yields bias as large as about 25–30%. The major discrepancies still occur for J 4; J 6 and J 8, which are just to which the zonals the combined LAGEOS/LAGOES II nodes are most sensitive.

scholarly journals Electric Vehicle Sales Catastrophe Averted (?)

2020 ◽  
Vol 14 (3) ◽  
pp. 1
Author(s):  
Timothy Sands
Keyword(s):  
Financial Analysis ◽  
Order Term ◽  
Equilibrium Points ◽  
Gasoline Prices ◽  
Vehicle To Grid ◽  
Charging Station ◽  
Forcing Function ◽  
Real Behavior ◽  
Time Propagation ◽  
Data Extrapolation

The literature indicates catastrophic potential for electric vehicles around the year 2019. Amidst the predicted time, this research revisits the prequel analysis with state-of-the-art deterministic artificial intelligence methodologies to posit the potential for catastrophe in the upcoming year. The methodology has proven effective with motion mechanics, electrodynamics, and even financial analysis of sales date in the prequels, since the model commences with simple regression for mathematical model formulation asserting the certainty equivalence principle, followed by derivative modeling and eventually catastrophe analysis of the derivative models. The prequel analysis paradigms are retained in this sequel utilizing both monthly and cumulative sales data in simple least squares algorithms for predictive curve fitting to establish context and help correctly model the mathematical degree of the data. Extrapolation by forward time-propagation established predictions for models of various mathematical degrees (again merely for context). Next, catastrophe analysis (of the derivative form) revealed stable and unstable equilibrium points and then parametric variation was induced to evaluate the resulting behavior of the derivative models, highlighting the importance of the coefficient of the second order term (the acceleration or change of rate of sales as a forcing function). While the forcing function typically embodies both gasoline prices and vehicle charging proliferation, the relative stability of gas prices together with factors such as vehicle-to-grid elevate charging-station proliferation as the primary forcing function of slow-dynamics in catastrophe analysis. This brief manuscript revisits the prequel research to test the validity of those conclusions and with the benefit of the passage of time, reveal how well the mathematical modeling predicted real behavior. The main finding is the predicted potential catastrophe is less likely to occur and recommendations are made to insure catastrophe is averted.

scholarly journals Analytical Effects of Gravitational Waves on the Motion of an Artificial Satellite

1997 ◽  
Vol 165 ◽  
pp. 431-436
Author(s):  
M.H. Youssef ◽  
M.K. Ahmed
Keyword(s):  
Gravitational Field ◽  
Artificial Satellite ◽  
Perturbation Technique ◽  
Canonical Transformations ◽  
Orbital Elements ◽  
Long Period ◽  
Short Period ◽  
Lie Transformations ◽  
Delaunay Variables ◽  
Analytical Expressions

AbstractThe motion of an artificial satellite in the Earth’s gravitational field is discussed in the post-Newtonian framework including the effect of weak gravitation waves using the perturbation technique of the canonical Lie-transformations. Two successive canonical transformations are used to derive analytical expressions for the short-period, long-period and secular perturbations of orbital elements. The solution is expressed in terms of the Delaunay variables.

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