1,721,026 research outputs found
An Efficient Method to Compute the Perturbation Spectrum in Linear Satellite Theory
No full textThe methods for the computation of the perturbation spectrum of the classical orbit elements presented in this work are based on the spectral representation of the celebrated Kaula's solution. Different algorithms are given for applications to the cases of orbits with circulating and frozen perigee. The main advantage to be found from the application fo these methods is the elimination of the searching over the set of frequencies which is characteristic of the usual computational approach
The Gravitational Perturbation Spectrum in Linear Satellite Theory
No full textA new method for calculating the perturbation spectrum in the framework of Kaula's linear satellite theory (LST) is introduced. The novelty of this approach consists in using recent results on the spectral decomposition of the perturbation frequencies in LST to provide a closed formulation for the amplitude and the phase of each line in the perturbation spectrum. The theory presented here can be applied to perturbations in the elements or in the radial and transverse directions due to the geopotential or to the tides. Separate algorithms are developed for application to orbits with circulating or frozen perigee
Translation and Rotation of the Spherical Harmonic Coefficients in the Expansion of the External Potential
No full textOrbit Injection Errors for the Proposed Lageos III Mission
No full textAn analysis is presented of the orbital injection errors for the Lageos III satellite mission. Several methods are introduced for the solution of the inverse problem in the theory of errors. The novelty of the present approach consists in the use of the full geopotential covariance matrix in the error propagation equations. The GEM-T1 covariance matrix is used. It is found that, by properly accounting for the correlation among the even zonal harmonic coefficients, the acceptable error bounds increase by an order of magnitude with respect to the case when only the variances are used. The most stringent constraint, even when using the full covariance, is on inclination, whose nominal value must be realized within approximately 0.1 deg for the recovery of the Lense-Thirring precession to be successful at the 3 percent level (accounting only for injection errors). The associated tolerance in the semimajor axis is about 30 km, while that in eccentricity is approximately 0.2. However, if the errors in semimajor axis and eccentricity can be kept to the routinely achievable levels of 10 km and 0.004, respectively, the tolerance in inclination can be relaxed to 0.2 deg
Position and Velocity Perturbations in the Orbital Frame in Terms of Classical Elements Perturbations
No full textThe transformation of classical orbit element perturbations to perturbations in position and velocity in the radial, transverse and normal directions of the orbital frame is developed. The formulation is given for the case of mean anomaly perturbations as well as for eccentric and true anomaly perturbations. Approximate formulas are also developed for the case of nearly circular orbits and compared with those found in the literature
Spectral Decomposition of Geopotential, Earth and Ocean Tidal Perturbations in Linear Satellite Theory
No full textThe equivalence-class structure of the frequency mapping is investigated for the nonresonant cases of both geopotential and tidal perturbations of orbits with circulating or frozen perigee. Kaula's (1961) linear satellite theory is the foundation for a technique for deriving the spectra of the perturbations as complex terms which depend on four indices. A set of algorithms are developed which describe the frequencies and terms with equivalent frequencies in the expansion of the perturbations in linear theory. The composition rule of frequency allows the generation of all the combinations of indices up to a maximum degree L and maximum value Q. An application of the algorithm is given for the spectral decomposition of geopotential perturbations. The algorithm permits the direct application of Kaula's theory without searching the set of frequencies
The Mapping of Kaula's Solution into the Orbital Reference Frame
No full textKaula's celebrated solution to the problem of satellite motion in the gravitational field of a rigid body is transformed to give the perturbation spectra in both position and velocity in the radial, transverse and normal directions of the orbital reference frame. This work is an extension and a refinement of the theory of orbital perturbations due to the geopotential previously published by Rosborough and Tapley (1987)
The Equations of Relative Motion in the Orbital Reference Frame
No full textThe analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill-Clohessy-Wiltshire (HCW) equations. Circular motion is not, however, a solution when the Earth's flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the J2 effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the J2 perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a J2-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the HCW equations for circular reference motion, or the de Vries/Tschauner-Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the J2 perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Clohessy-Wiltshire, or the HCW equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a J2-perturbed reference orbit is also introduced
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