Oblate Planet Research Articles

Integration of the primer vector in a central force field

This paper examines the primer vector which governs optimal solutions for orbital transfer when the central force field has a more general form than the usual inverse-square-force law. Along a null-thrust are that connects two successive impulses, the two sets of state and adjoint equations are decoupled. This allows the reduction of the problem to the integration of a linear first-order differential equation, and hence the solution of the optimal coasting are in the most general central force field can be obtained by simple quadratures. Immediate applications of the results can be seen in solving problems of escape in the equatorial plane of an oblate planet, satellite swing by, or station keeping around Lagrangian points in the three-body problem.

A Second-Order Artificial Satellite Theory Based on an Intermediate Orbit

An analytical second-order theory is developed for the motion of a satellite of an oblate planet whose gravitational potential includes the second, third, and fourth zonal harmonics. it is assumed that J2 is a small quantity of the first order, and that J2 and J4 are of the second order. The secular and the periodic perturbations are obtained to the third and to the second order, respectively. The former are contained in the Delaunay variables ", g", h", which are linear functions of the time, while the latter are given as additions to the Hill variables r", r", G", u", h" in the form of trigonometric series with constant coefficients. The theory is distinguished by a relative simplicity and compactness of the final algorithm achieved by the use of the following special devices and techniques: (i) an intermediate orbit, (ii) Hori's perturbation method, and (iii) the Hill variables. A comparison with the results of numerical integration of the equations of motion indicates that the theory is capable of predicting the position of a close Earth satellite to better than one meter over one-hundred revolutions.

Motion of a Satellite of a Very Oblate Planet

view Abstract Citations (20) References (1) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Motion of a Satellite of a Very Oblate Planet Danby, J. M. A. Abstract An axisymmetric potential with only the second harmonic included is investigated. The question of the integrability of the resulting dynamical system is open. But by considering large oblateness and using numerical methods, all the symptoms of nonintegrability appear. However the manner of their appearance indicates that for oblateness of astronomical interest, possible nonintegrability would be unimportant. Publication: The Astronomical Journal Pub Date: December 1968 DOI: 10.1086/110765 Bibcode: 1968AJ.....73.1031D full text sources ADS |

Rigorous and formal stability of orbits about an oblate planet

Idealized point mass motion in axisymmetric gravitational field, discussing orbital stability about oblate planet

Satellite motion for all inclinations around an oblate planet

A uniformly valid solution for the motion of a satellite around an oblate planet is presented. The Two-Variable Expansion Procedure was applied to obtain a solution valid for all inclinations including the critical. This solution is correct to order ε, where ε is a small parameter proportional to the oblateness parameterJ2. The reciprocal of the radius vector, eccentricity, perigee, inclination, and node of the satellite orbit are given as functions of the central angleφbetween node and satellite. The results are based on a potential which includes the second and fourth zonal harmonics. The solution for the case of critical inclination is first obtained separately and then matched with the solution of the noncritical case to establish a solution uniformly valid for all inclinations.

Invariant properties of the spheroidal potential of an oblate planet

Gravitational potential of oblate planet, expressed in terms of oblate spheroidal coordinates, generalized by means of transformation of associated Cartesian system

Satellite Motion about an Oblate Planet for Low and Moderate Eccentricities.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Satellite Motion about an Oblate Planet for Low and Moderate Eccentricities. Douglas, B. C. ; Ingram, D. S. Abstract Periodic variations in the elements a, i, ~, e sin co, e cos ~o, (M+co) up to order e J2 have been obtained. The terms of order J2 are similar to those given by Kozai (Astron. J. 66, 8, 1961). The addition of terms of order e J2 only slightly increases the complexity of the solution. For eccentricity =0.01, comparison with an exact numerical integration shows errors in geocentric distance of few tens of feet and errors in velocity of a few hundredths of a ft/sec. Publication: The Astronomical Journal Pub Date: 1965 DOI: 10.1086/109667 Bibcode: 1965AJ.....70T.673D full text sources ADS |

Qualitative properties of orbits about an oblate planet

Qualitative properties of orbits about an oblate planet

SATELLITE LIFETIMES UNDER THE INFLUENCE OF CONTINUOUS THRUST, ATMOSPHERIC DRAG, AND PLANET OBLATENESS

Approximate analytic solutions are obtained for the lifetimes of satellites in either circular or elliptical orbits under the influence of continuous tangential thrust, atmospheric drag, and planet oblateness. The cases treated are those in which the satellite mission requires the perigee of the orbit to remain within fixed bounds in altitude and position. For circular orbits it is found that, to increase greatly the useful life time over the nonthrusting case, an almost perfect matching of thrust to drag is required. Two solutions are obtained for elliptical orbits. The first requires that the thrust level over each revolution be the value needed to hold the perigee fixed. Extremely low thrust levels are found over most of the lifetime while increases in lifetime on the order of 30% over the nonthrusting case result. The second case considers thrusting at a constant level over a symmetrical arc about perigee; the length of the arc for each cycle being determined by the condition that the perigee remain fixed. Greatly increased lifetimes over the nonthrusting case and the preceding case are obtained with low mass consumption and small thrust-to-drag ratios. Expressions for the forces required to provide a secular change of the line of nodes, the line of apsides, and the inclination including the effect of planet oblateness are given and several examples are calculated.

Free modes of layered oblate planets

The equations governing the motion of an elastic gravitating nonhomogeneous isotropic field are considered, and a general solution for the Laplace or Fourier transform of the displacement vector is given in spherical coordinates. The perturbation method is used to set the boundary conditions for an oblate layered spheroid and it is seen that, except when there is rotational symmetry in the boundary conditions, in the geometry of the field, and in the displacements, the two different sets of usually uncoupled modes do not separate, and generally there is coupling in the displacements associated with these two sets of modes. If the displacements have axial symmetry these modes degenerate into the usual torsional and normal modes. The theoretical study refers to a model consisting of an indefinite number of homogeneous layers having different density and different elastic parameters, separated by spheroids of different flattening; but the computations of the first-order perturbation of the torsional periods caused by the oblateness have been made for two planet models, one consisting of a homogeneous shell limited inside by a sphere and outside by a spheroid and one consisting of a solid homogeneous spheroid. Since the influence of the flattening on the lower periods of oscillation in these two cases is of the order of only a few parts in a thousand no further computations have been made for the higher-order periods or harmonics or for more sophisticated models. Finally, it is shown that the broadening of the spectral lines caused by the flattening of the earth could not seriously interfere with the measurement of the dissipation function from the width of the same lines.

Perturbations of a Hyperbolic Orbit by an Oblate Planet

Perturbations of a Hyperbolic Orbit by an Oblate Planet

Addendum to "Some remarks on the motion of a satellite of an oblate planet,"

Addendum to "Some remarks on the motion of a satellite of an oblate planet,"

Formulae for an accurate intermediary orbit of an artificial satellite

view Abstract Citations (7) References (1) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Formulae for an accurate intermediary orbit of an artificial satellite Vinti, John P. Abstract Formulae are given for computing the drag-free orbit of an artificial satellite of an oblate planet moving in the field of a certain gravitational potential. This potential, expressed in oblate spheroidal coor- dinates and leading to separability of the Hamilton-Jacobi equation, fits the zeroth and second zonal harmonic exactly and, in the case of the earth, yields more than half the fourth zonal harmonic. The solution gives periodic terms through the second order in the oblateness parameter and secular terms exactly, for the given potential. It thus furnishes an intermediary orbit which should always remain very accurate. Publication: The Astronomical Journal Pub Date: November 1961 DOI: 10.1086/108452 Bibcode: 1961AJ.....66..514V full text sources ADS |

Satellite Motions About an Oblate Planet

The drag-free path of a vehicle in the vicinity of an oblate planet is discussed. Although the solution found is approximate in the sense that it is a truncated power series in the oblateness parameter / , there is no restriction on either the eccentricity or the inclination angle. Approximate analytic expressions are derived for the radial position, speed, angular momentum, deviation from the initial plane of motion, and rate of apsidal advance. In addition, the injection speed required for prescribed apses is derived. The differences between first-order results and inverse-square results are found from the analysis, and numerical results are given. A comparison of the analytical results with the results of integrating the equations of motion directly on a digital computer indicates excellent agreement over a wide range of inclination angles and eccentricities.

Planar Motions About an Oblate Planet

Two cases of plane motion (equatorial and polar) of a vehicle about an oblate planet are discussed. Further, the results of these drag free plane motion studies may be applied to the determination of radial position, speed, and angular momentum for a motion whose initial velocity vector is inclined at an arbitrary angle to the equatorial plane. Approximate analytic results are found by expanding the solution of the system of nonlinear differential equations of the path in powers of the oblateness parameter /. Although only the first power is retained in this solution, no restriction is placed on eccentricity. Thus, for example, the results for an eccentricity of 0.5 may be found from the analy­ sis. In fact, results for speeds in excess of escape speed (e.g., e = 1.1) may be found. Comparison with results obtained by numerically integrating the equations of motion indicates excellent agree­ ment with the analysis.

Some remarks on the motion of a satellite of an oblate planet

Some remarks on the motion of a satellite of an oblate planet

First-order perturbations in the motion of artificial satellites due to the oblateness of the earth: V. F. Proskurin and Iu. V. Batrakov: (pp. 32–38)

The problem of a spacecraft orbiting the Neptune–Triton system is presented. The new ingredients in this restricted three body problem are the Neptune oblateness and the high inclined and retrograde motion of Triton. First we present some interesting simulations showing the role played by the oblateness on a Neptune’s satellite, disturbed by Triton. We also give an extensive numerical exploration in the case when the spacecraft orbits Triton, considering Sun, Neptune and its planetary oblateness as disturbers. In the plane a × I (a = semi-major axis, I = inclination), we give a plot of the stable regions where the massless body can survive for thousand of years. Retrograde and direct orbits were considered and as usual, the region of stability is much more significant for the case of direct orbit of the spacecraft (Triton’s orbit is retrograde). Next we explore the dynamics in a vicinity of the Lagrangian points. The Birkhoff normalization is constructed around L2, followed by its reduction to the center manifold. In this reduced dynamics, a convenient Poincaré section shows the interplay of the Lyapunov and halo periodic orbits, Lissajous and quasi-halo tori as well as the stable and unstable manifolds of the planar Lyapunov orbit. To show the effect of the oblateness, the planar Lyapunov family emanating from the Lagrangian points and three-dimensional halo orbits are obtained by the numerical continuation method.

The orbit of a satellite of an oblate planet

view Abstract Citations (59) References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The orbit of a satellite of an oblate planet Garfinkel, Boris Abstract . The pape gives a solution of the problem of motion of a particle in the potential field V - I/r + 2kP2 (sin O)/r3 + k'P4 (sin O)/r1, where k and k' are small parameters. The first approximation is furnished by a V8 that incorporates a major portion of the second spherical harmonic and leads to a closed solution with no secular variations of order k. The perturbations of this non-Keplenan intermediary orbit are derived here by the von-Zeipel modification of the method of Delaunay. The secular variations are carried up to orders k2 and k', and the periodic variations to orders k and k'/k. A computational summary is included; the effect of the third spherical harmonic is appended. Publication: The Astronomical Journal Pub Date: November 1959 DOI: 10.1086/107956 Bibcode: 1959AJ.....64..353G full text sources ADS |

On the motion of a satellite of an oblate planet

On the motion of a satellite of an oblate planet

The gravitational orbit of a satellite of an oblate planet

view Abstract Citations (31) References (1) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The gravitational orbit of a satellite of an oblate planet Sterne, Theodore E. Abstract . At Berkeley, California, in August, 1956, the author described to the American Astronomical Society an exactly solvable Hamiltonian that approximates, more closely than can an ellipse, the gravitational motion of a close satellite of an oblate planet. At the New York meeting in December, 1956, the author described an improved Hamiltonian, and mentioned that just before the talk he had discovered still another that appeared to provide canonical constants with no secular first-order perturbations at all. This last solution is here presented; although the canonical constants have apparently no first-order secular perturbations, the mean anomalistic motion is changed because of them by a small and easily computed amount. The present unperturbed solution with the changed mean motion provides an ephemeris nearly as simply as the ellipse of the two-body problem but involves distinct and very accurate anomalistic, nodical, and sidereal periods. It may suffice for the finding ephemeris of an artificial earth satellite. With periodic first-order perturbations included, the results are expected to be considerably more accurate than a conventional ellipse with only first-order perturbations of its elements. Publication: The Astronomical Journal Pub Date: January 1958 DOI: 10.1086/107673 Bibcode: 1958AJ.....63...28S full text sources ADS |