scholarly journals A set of orbital elements to fully represent the zonal harmonics around an oblate celestial body

2021 ◽  
Author(s):  
David Arnas ◽  
Richard Linares
Keyword(s):  
Linear System ◽  
Celestial Body ◽  
Gravitational Force ◽  
Polynomial System ◽  
Orbital Elements ◽  
First Order ◽  
Zonal Harmonics ◽  
J2 Perturbation ◽  
Unperturbed Problem ◽  
Complete Linear System

Abstract This work introduces a new set of orbital elements to fully represent the zonal harmonics problem around an oblate celestial body. This new set of orbital elements allows to obtain a complete linear system for the unperturbed problem and, in addition, a complete polynomial system when considering the perturbation produced by the zonal harmonics from the gravitational force of an oblate celestial body. These orbital elements present no singularities and are able to represent any kind of orbit, including elliptic, parabolic and hyperbolic orbits. In addition, an application to this formulation of the Poincaré-Lindstedt perturbation method is included to obtain an approximate first order solution of the problem for the case of the J2 perturbation.

scholarly journals Curves with Two Pencils and Associated Maps to Projective Spaces

2006 ◽  
Vol 13 (3) ◽  
pp. 411-417
Author(s):  
Edoardo Ballico
Keyword(s):  
Linear System ◽  
Projective Spaces ◽  
Free Resolution ◽  
Homogeneous Ideal ◽  
Projective Curve ◽  
Minimal Free Resolution ◽  
Complete Linear System

Abstract Let 𝑋 be a smooth and connected projective curve. Assume the existence of spanned 𝐿 ∈ Pic𝑎(𝑋), 𝑅 ∈ Pic𝑏(𝑋) such that ℎ0(𝑋, 𝐿) = ℎ0(𝑋, 𝑅) = 2 and the induced map ϕ 𝐿,𝑅 : 𝑋 → 𝐏1 × 𝐏1 is birational onto its image. Here we study the following question. What can be said about the morphisms β : 𝑋 → 𝐏𝑅 induced by a complete linear system |𝐿⊗𝑢⊗𝑅⊗𝑣| for some positive 𝑢, 𝑣? We study the homogeneous ideal and the minimal free resolution of the curve β(𝑋).

scholarly journals On new solutions of linear system of first -order fuzzy differential equations with fuzzy coefficient

2016 ◽  
Vol 2016 ◽  
pp. 110-117 ◽  
Author(s):  
A. Karimi Dizicheh ◽  
S. Salahshour ◽  
F. Ismail ◽  
A. Ahmadian Hosseini
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
First Order

Motion of a satellite in the earth’s gravitational field

1960 ◽  
Vol 254 (1276) ◽  
pp. 48-65 ◽  
Keyword(s):  
Gravitational Field ◽  
Spherical Harmonics ◽  
Main Part ◽  
Equations Of Motion ◽  
Orbital Elements ◽  
Partial Derivatives ◽  
Rates Of Change ◽  
Zonal Harmonics ◽  
The Earth ◽  
Secular Terms

The equations of motion of a satellite are given in a general form, account being taken of the precession and nutation of the earth. The main part of the paper deals with the motion arising from the gravitational field of the earth, expressed as a general expansion in spherical harmonics. By evaluating the partial derivatives in Lagrange’s planetary equations, • expressions are obtained for the rates of change of the orbital elements. Particular consideration is given to the form of the expressions for the secular terms arising from the first four zonal harmonics.

ON THE DEGREE AND THE BIRATIONALITY OF THE SECOND ADJUNCTION MAPPING

1999 ◽  
Vol 10 (06) ◽  
pp. 707-719 ◽  
Author(s):  
MAURO C. BELTRAMETTI ◽  
ANDREW J. SOMMESE
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Mild Condition ◽  
Three Dimensional ◽  
Ample Line Bundle ◽  
Kodaira Dimension ◽  
Projective Manifold ◽  
Very Ample Line Bundle ◽  
System 2 ◽  
Complete Linear System

Let ℒ be a very ample line bundle on ℳ, a projective manifold of dimension n ≥3. Under the assumption that Kℳ + (n-2) ℒ has Kodaira dimension n, we study the degree of the map ϕ associated to the complete linear system |2(KM + (n-2) L)|, where (M, L) is the first reduction of (ℳ, ℒ). In particular we show that under a number of conditions, e.g. n ≥ 5 or Kℳ + (n-3)ℒ having nonnegative Kodaira dimension, the degree of ϕ is one, i.e. ϕ is birational. We also show that under a mild condition on the linear system |KM + (n-2) L| satisfied for all known examples, ϕ is birational unless (ℳ, ℒ) is a three dimensional variety with very restricted invariants. Moreover there is an example with these invariants such that deg ϕ= 2.

scholarly journals Analytical and Semi-Analytical Treatment of the Satellite Motion in a Resisting Medium

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
S. E. Abd El-Bar ◽  
F. A. Abd El-Salam
Keyword(s):  
Fourth Order ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Orbital Dynamics ◽  
Atmospheric Drag ◽  
Satellite Motion ◽  
Orbital Elements ◽  
Analytical Treatment ◽  
First Order ◽  
Resisting Medium

The orbital dynamics of an artificial satellite in the Earth's atmosphere is considered. An analytic first-order atmospheric drag theory is developed using Lagrange's planetary equations. The short periodic perturbations due to the geopotential of all orbital elements are evaluated. And to construct a second-order analytical theory, the equations of motion become very complicated to be integrated analytically; thus we are forced to integrate them numerically using the method of Runge-Kutta of fourth order. The validity of the theory is checked on the already decayed Indian satellite ROHINI where its data are available.

scholarly journals An Implicitly Balanced Hurricane Model with Physics-Based Preconditioning

2005 ◽  
Vol 133 (4) ◽  
pp. 1003-1022 ◽  
Author(s):  
J. M. Reisner ◽  
A. Mousseau ◽  
A. A. Wyszogrodzki ◽  
D. A. Knoll
Keyword(s):  
Linear System ◽  
Physical Model ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Solution Procedure ◽  
Sound Waves ◽  
Time Step ◽  
Efficient Manner ◽  
First Order ◽  
Implicit Approach

A numerical framework for simulating hurricanes based upon solving a nonlinear equation set with an implicitly balanced solution procedure is described in this paper. The physical model is the Navier–Stokes equations plus a highly simplified and differentiable microphysics parameterization package. Because the method is fully implicit, the approach is able to employ time steps that result in Courant–Friedrichs–Lewy (CFL) numbers greater than one for advection, gravity, and sound waves; however, the dynamical time scale of the problem must still be respected for accuracy. The physical model is solved via the Jacobian-free Newton–Krylov (JFNK) method. The JFNK approach typically requires the approximate solution of a large linear system several times per time step. To increase the efficiency of the linear system solves, a physics-based preconditioner has been employed. To quantify the accuracy and efficiency of the new approach against traditional approaches, the implicitly balanced solver was first compared against semi-implicit approaches for the simulation of a precipitating moist bubble. The moist-bubble simulations demonstrated the ability of the implicitly balanced approach to achieve a given level of accuracy in a more efficient manner than either a first-order semi-implicit approach or a traditional leapfrog semi-implicit approach. This behavior is further illustrated in first-of-a-kind three-dimensional implicitly balanced hurricane simulations that reveal the first-order-in-time semi-implicit algorithm needs to take a time step at least 60 times smaller than the implicitly balanced algorithm to produce a comparable accuracy.

Experimental Study on Tire Non-Steady State Slip Properties

2013 ◽  
Vol 753-755 ◽  
pp. 1736-1744
Author(s):  
Jie Liu ◽  
Xiao Ling Jia
Keyword(s):  
Linear System ◽  
Dynamic Performance ◽  
Lateral Force ◽  
Slip Angle ◽  
Test Results ◽  
Relaxation Length ◽  
Side Slip Angle ◽  
First Order ◽  
Depth Analysis ◽  
Yaw Angle

As for the two typical inputs of pure side slip angle and pure yaw angle, this paper presents the in-depth analysis of lateral force, aligning torque and relaxation length respectively within the domains of distance and spacial frequency, and also explains the test results by theoretical model. Within the small side slip angle, tire is a first-order linear system. Relaxation length is equivalent to the time constant of linear system, which decreases as slip angle increases. It indicates the dynamic performance of tire system.

scholarly journals The exponential eigenmodes of the carbon-climate system

2012 ◽  
Vol 3 (2) ◽  
pp. 1107-1158
Author(s):  
M. R. Raupach
Keyword(s):  
Linear System ◽  
Co2 Emissions ◽  
Co2 Emission ◽  
Climate Model ◽  
Climate System ◽  
State Variables ◽  
First Order ◽  
The Past ◽  
Emissions Scenarios ◽  
Co2 Sink

Abstract. Several basic ratios describing the carbon-climate system are observed to adopt relatively steady values. Examples include the CO2 airborne fraction (the fraction of the total anthropogenic CO2 emission flux that accumulates in the atmosphere) and the ratio T/QE of warming (T) to cumulative total CO2 emissions (QE). This paper explores the reason for such near-constancy in the past, and its likely limitations in future. The contemporary carbon-climate system is often approximated as a first-order linear system, for example in response-function descriptions. All such linear systems have exponential eigenfunctions in time (an eigenfunction being one that, if applied to the system as a forcing, produces a response of the same shape). This implies that, if the carbon-climate system is idealised as a linear system (Lin) forced by exponentially growing CO2 emissions (Exp), then all ratios among fluxes and perturbation state variables are constant. Important cases are the CO2 airborne fraction (AF), the cumulative airborne fraction (CAF), other CO2 partition fractions and cumulative partition fractions into land and ocean stores, the CO2 sink uptake rate (kS, the combined land and ocean CO2 sink flux per unit excess atmospheric CO2), and the ratio T/QE. Further, the AF and the CAF are equal. The Lin and Exp idealisations apply approximately (but not exactly) to the carbon-climate system in the period from the start of industrialisation (nominally 1750) to the present, consistent with the observed near-constancy of the AF, CAF and T/QE in this period. A nonlinear carbon-climate model is used to explore how the likely future breakdown of both the Lin and Exp idealisations will cause the AF, CAF and kS to depart significantly from constancy, in ways that depend on CO2 emissions scenarios. However, T/QE remains approximately constant in typical scenarios, because of compensating interactions between emissions trajectories, carbon-cycle dynamics and non-CO2 gases. This theory assists in establishing both the basis and limits of the widely-assumed proportionality between T and QE, at about 2 K per trillion tonnes of carbon.

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