Dynamically Relevant Local Coordinates for Halo Orbits

Gustafson, Eric; Scheeres, Daniel J.

doi:10.2514/6.2008-6432

Cited by 5 publications

(8 citation statements)

References 2 publications

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“…thus leading to the relationship u(t) = Φ(t, 0)u(0) [58]. Thus, after every orbit period the eigenvectors repeat their direction, and expand or contract according to whether |λ| is greater or less than one.…”

Section: Stability Of Periodic Orbitsmentioning

confidence: 99%

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Orbital Motion in Strongly Perturbed Environments

Scheeres

2012

190

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Section: Stability Of Periodic Orbitsmentioning

confidence: 99%

“…To evaluate this the concept of a generalized eigenvector must be introduced, discussed in some detail in [58] for astrodynamical systems. The presence of these unity eigenvalues for periodic orbits in time-invariant systems complicate the solution process for finding periodic orbits.…”

Section: Unity Eigenvalues For Time Invariant Systemsmentioning

confidence: 99%

Orbital Motion in Strongly Perturbed Environments

Scheeres

2012

190

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“…This is always the case with ∇H , since ∇H (ȳ 0 ) = 0 would imply thatȳ 0 is an equilibrium 5. This is the case of the problems considered for the numerical tests in Section 5 (see, e.g.,[17]). More in general, this is true for problems admitting only one first integral, and possessing the so called scaling property (see[27] for details).…”

mentioning

confidence: 94%

Energy-conserving methods for Hamiltonian boundary value problems and applications in astrodynamics

2014

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We introduce new methods for the numerical solution of general Hamiltonian boundary value problems. The main feature of the new formulae is to produce numerical solutions along which the energy is precisely conserved, as is the case with the analytical solution. We apply the methods to locate periodic orbits in the circular restricted three body problem by using their energy value rather than their period as input data. We also use the methods for solving optimal transfer problems in astrodynamics

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“…The center magnitude ρ and center phase γ are computed using the center eigenvector v c in Eqs. (18). The left eigenvectors are normalized to have magnitude one, whereas the orbiter state difference retains its magnitude.…”

Section: Manifold Coordinatesmentioning

confidence: 99%

“…To test whether orbits closer to the stable manifold exhibit longer lifetimes, the eigenvectors of the state transition matrix associated with the stable and unstable manifolds are computed and used to create a set of local dynamic coordinates associated with these manifolds [18]. A right eigenvector u is a column vector that satisfies the following property:…”

Section: B Eigenvectorsmentioning

confidence: 99%

Long-Life Europa Geodesy Orbits Accounting for Navigation Uncertainties

Boone

Scheeres

2014

Journal of Guidance, Control, and Dynamics

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This work investigates the properties of phase space near a periodic orbit and their applications to orbit design and orbit determination. A low-altitude, near-polar periodic orbit is computed and an error covariance is generated by processing range-rate and altimetry measurements over seven days. The resulting covariance is used to disperse the orbit initial conditions in a Monte Carlo simulation. The distribution of the Monte Carlo run is biased toward longer orbit lifetimes, due to the stable and unstable manifolds associated with the periodic orbit. The orbit determination covariance is mapped into manifold coordinates and long lifetime orbits are shown to be aligned with the stable manifold. Periodic orbit continuation is used to generate a family of orbits with similar phase-space characteristics to understand the variation of manifold structure with orbit elements and Jacobi energy. The effect of Europa eccentricity on the phase-space location of long lifetime orbits is discussed and conclusions are given regarding the connection between orbit lifetime, design, and determination. Nomenclature a, e, i, Ω, ω, M = Keplerian orbital elements, km, [ ], deg, deg, deg, deg, respectively a s = stable manifold coordinate a u = unstable manifold coordinate C = Jacobi energy, km 2 ∕s 2 C = consider parameter state vector C nm ∕S nm = Stokes coefficients h 2 , k 2 = second degree tidal Love number n E = Europa mean motion, rad∕s R = inertial frame satellite position vector, km R e = Europa radius, km r = rotating frame satellite position vector, km t = time, s U = gravitational potential, km 2 ∕s 2 u = right eigenvector v = left eigenvector X = satellite state vector γ = center manifold phase, deg θ = manifold intersection angle, deg Λ = information matrix λ = eigenvalue μ = standard gravitational parameter, km 2 ∕s 2 ρ = center manifold magnitude Φ = state transition matrix

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Dynamically Relevant Local Coordinates for Halo Orbits

Cited by 5 publications

References 2 publications

Orbital Motion in Strongly Perturbed Environments

Orbital Motion in Strongly Perturbed Environments

Energy-conserving methods for Hamiltonian boundary value problems and applications in astrodynamics

Long-Life Europa Geodesy Orbits Accounting for Navigation Uncertainties

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