Effect Of Oblateness Research Articles

On the Motion of Short-Period Comets in the Neighbourhood of Jupiter

An intermediate orbit is developed for studying the hyperbolic jovicentric motion of a comet in the vicinity of Jupiter, where the effect of Jupiter's oblateness may become significant. A comparison is made with some of the results obtained by other investigators on the motions of specific periodic comets.

SOLUTION OF THE LOW‐ALTITUDE SATELLITE EQUATIONS

ABSTRACTThe system of nonlinear differential equations of motion of satellites under the influence of oblateness and drag are transformed into an appropriate system of equations containing two small parameters. The solution determined by the perturbation method of poincaré is obtained in power series in two small parameters exhibiting the coupling effect of oblateness and drag on the motion. The first‐order Z perturbation is compared and shown to agree with the known result. The higher‐order perturbations are presented in integral forms that can be readily evaluated. This method, by far the simplest, offers a technique of determining trajectories with accuracy suitable for practical applications.

Effect of oblateness and drag on equatorial orbits with small eccentricities.

Effect of oblateness and drag on equatorial orbits with small eccentricities.

Discussion of paper by S. J. Peale, ‘Dust belt of the Earth’

In a recent paper, Peale [1966] has suggested that part, or indeed all, of the zodiacal light could be due to dust particles in geocentric orbits rather than in heliocentric orbits as is generally believed. Such particles are subject to gravitational perturbations due to the earth's oblateness and the presence of the sun and moon, as well as to the effects of radiation pressure. Peale [1966] argues that radiation pressure will produce a diffuse concentration of dust particles near the ecliptic plane and that the asymmetry with respect to the ecliptic will be almost completely undisturbed by the gravitational perturbations beyond a geocentric distance of 7.7 earth radii, where the effects of oblateness and lunisolar forces are equal. On the other hand, the zodiacal light is known to be nearly symmetrical about the ecliptic; thus Blackwell and Ingham [1961] find that the plane of symmetry is inclined at 1°.5 to the ecliptic with its node at about 120° longitude, which very nearly coincides with the invariable plane of the solar system.

Contribution of earth oblateness to gravity torque on a satellite.

Earth oblateness effect on torque experienced in nonuniform geogravitational field by satellite with unequal principal moments of inertia

Modified Encke special perturbation method

view Abstract Citations (16) References (1) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Modified Encke special perturbation method Kyner, W. F. ; Bennett, M. M. Abstract The classical Encke method for integrating the equations of motion of a near-earth satellite is modified so that the difference between the nominal and true orbits does not contain the first-order effects of the earth's oblateness. The nominal trajectory is well defined for all inclinations and all bounded orbits. Nu- merical tests of the new method indicated that drag-free orbits can be integrated for at least 100 revolutions before the length of the difference vector exceeds 65 km. Publication: The Astronomical Journal Pub Date: September 1966 DOI: 10.1086/109965 Bibcode: 1966AJ.....71..579K full text sources ADS |

Effects of air drag on near-circular satellite orbits

solved, showing the oscillatory nature of the solutions. The solution is given in its more general form, which includes nonzero initial conditions and which shows that the oscillatory terms depend very strongly on the initial velocity, i.e., the initial departure from an exactly circular orbit. Next, the case of large descent into exponentially increasing atmosphere is solved. The solution includes the oscillatory terms that tend to die out with the descent. In the last section, the effects of diurnal variations and atmospheric oblateness are treated. It is shown that effects of diurnal variations can be pronounced, but that the effects of atmospheric oblateness are relatively small. Numerical examples are included.

Effect of atmospheric oblateness on the acceleration of a satellite

An appropriate theory has been developed to derive the effect of atmospheric oblateness on the acceleration n˙, defined as the rate of change of mean motion n. This effect is a periodic one, as would be expected, having half the period of the argument of perigee. The theoretical expression so derived was used to find the effect of atmospheric oblateness on one hypothetical satellite and three actual satellites: 1958γ, 1958ζ, and 1960γ2. The effect can be significant for low satellites and should be taken into account for a precise computation of densities, which can be in error by ±18 per cent at an altitude of 230 km if atmospheric oblateness is ignored. For higher satellites, however, this effect is much smaller. To apply this procedure to derive atmospheric oblateness, we need a few low satellites with small eccentricity in polar orbit having a lifetime in which the perigee has made several revolutions of the earth. The preliminary work done so far indicates that the atmospheric oblateness increases with altitude. An exact determination of the atmospheric oblateness must, however, await more extensive data.

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.

PERTURBATIONAL VARIATIONS IN A BALLISTIC MISSILE OR SATELLITE ORBIT ABOUT AN OBLATE EARTH

This paper treats the problem of the oblateness effect on Earth satellites and ballistic missiles iii terms of the expected deviations in position prediction relative to an ideal Keplerian orbit. The resulting closed form solutions in the variations in radius, in-plane angle, and lateral displacement have the following significant features: 1) easily instrumented in realtime tracking and prediction systems; 2) no restrictions on eccentricity, including orbits with escape velocities; and 3) offers a physical insight into the oblateness effect on freeflight trajectories. In addition, the second-order solutions are applicable for all angles of inclination with the possible exception of the region of critical inclination. Nomenclature V = oblate potential function GM = gravitational product of Earth re = mean value of the equatorial radius of Earth r = instantaneous geocentric radius of 'the satellite <P = colatitude angle of the satellite measured from Earth's rotating axis / = second harmonic oblateness coefficient dr = variation in the satellite radius caused by oblateness 0 = orbital in-plane angle of the satellite measured relative to the nodal line /8 = nondimensional coefficient of angular momentum perturbation i = inclination angle of the orbital plane c = eccentricity of the orbit I = a(l — e2), length of the semilatus rectum At = angle of perigee of the orbit relative to the nodal line ^ = lateral deflection angle of the satellite relative to the reference orbit plane L — Lagrangian function E = energy

Effect of Earth's Oblateness on the Rotational Motion of an Artificial Earth Satellite

Effect of Earth's Oblateness on the Rotational Motion of an Artificial Earth Satellite

Propulsion Requirement to Counteract the Effect of Earth's Oblateness

Earth satell ites, particularly those at the lower alt i tudes, are subject to external forces that , in general, undergo cont inuous changes because of the asphericity of Earth's shape. These variations induce secular and periodic changes in the parameters of the satellite's orbit which, under certain c ircumstances , may be undesirable. Therefore, the performance requirements necessary to counteract the effect of Earth's oblateness should be est imated in order to properly design t h e satell ite system. The following analysis presents requirements based on the present best e s t imate of Earth's potential . These requirements are given in terms of integral equations that , because of their nature, can only be evaluated by digital techniques . A simplified closed form solution is o b tained assuming constant mass (per revolution) and orbit circularity. Performance es t imates based on this type of solution can be used as rules of t h u m b for preliminary design purposes.

A Determination of the Atmospheric Oblateness from the Motion of Two Low Satellites.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Determination of the Atmospheric Oblateness from the Motion of Two Low Satellites. Nigam, R. C. Abstract An appropriate theory has been developed to derive the effect of the atmospheric oblateness on the acceleration n, defined as the rate of change of mean motion n. By mean motion is meant the number of revolutions made by the satellite in one day from one perigee to another. This effect is a periodic one, as one would expect, with half the period of the argument of perigee. For small values of eccentricity (e <0.2), it is expressed by [n2a2(1;;{{;) sin2i1 3 15H 67H 1 2e -e2- - O(e3) cos2 , 2 8ae 8a where k (r) = 2' (CDA /m) p (r), = ac/H, H is the scale height, f the atmospheric oblateness, q the geocentric perigee distance, etc. In order to get reliable values for the atmospheric oblateness, one needs a few low satellites in polar orbit having a lifetime in which the perigee has made several revolutions of the earth. Further, if the satellites are not spherical, reliable information about the mode of tumbling should also be available from some independent source. As none of the above-mentioned characteristics could be met in the satellites available for this investigation, we chose satellites 1958 and 1958 , both of which had their perigee altitude less than 200 km, for a preliminary check of the derived theoretical expression for the effect of atmospheric oblateness on the accelerations, and to derive the value for the atmospheric oblateness. The atmospheric oblateness at the altitudes of 176 and 186 km, which are the mean altitudes of the perigee points of the satellites 1958 and 1958 , respectively, are obtained as 1/284 and 1/238, respectively. The value of the atmospheric oblateness computed theoretically assuming the solid-body rotation of the atmosphere for an altitude of 176 km is 1/291. The quantitative results on the atmospheric oblateness are, therefore, far from being exact. These, on taking into consideration the uncertainty inherent in the two determinations, are, however, in conformity with the theoretical atmospheric models near 200 km, as given by T. E. Sterne (Astron. J. 63, 81,1958) and F. S. Johnson (J. Geophys. Research 65, 2227,1960). The results therefore appear to suggest that the atmospheric oblateness increases with altitude. An exact determination of the atmospheric oblateness must however await more extensive data. Publication: The Astronomical Journal Pub Date: September 1961 DOI: 10.1086/108423 Bibcode: 1961AJ.....66..292N full text sources ADS |

On Mr King-Hele's Theory of the Effect of the Earth's Oblateness on the Orbit of a Close Satellite

The relations are obtained between α and Ω, the parameters defining King-Hele's plane μ, and the osculating inclination and longitude of node. The secular motion of the node is thereby obtained in terms of mean osculating elements in a form comparable to that arising in Brouwer's treatment of the canonical equations. A modification is made to King-Hele's theory which removes the singularities in the Je terms of dΩ/dΨ, and it is also found that there are no terms of order Je2 in α or Ω.

On Mr King-Hele's Theory of the Effect of the Earth's Oblateness on the Orbit of a Close Satellite

Summary> The relations are obtained between α and Ω, the parameters defining King-Hele's plane π, and the osculating inclination and longitude of node. The secular motion of the node is thereby obtained in terms of mean osculating elements in a form comparable to that arising in Brouwer's treatment of the canonical equations. A modification is made to King-Hele's theory which removes the singularities in the Jeterms of dΩ/dψ, and it is also found that there are no terms of order Je2in α or Ω

Secular and Periodic Motions of the Node of an Artificial Earth-Satellite

IN a recent paper on the effect of the Earth's oblateness on the orbit of a near satellite, King-Hele1 mars an otherwise commendable presentation by abundant criticism of some earlier papers on this subject by myself and collaborators2,3. Although King-Hele's remarks were not so intended, his statements have had the effect of raising serious doubts about the validity of my early solutions and in particular about the usefulness of my approach to the problem of satellite orbit perturbations.

Theoretical Evaluation of Atmospheric Drag Effects in the Motion of an Artificial Satellite.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Theoretical Evaluation of Atmospheric Drag Effects in the Motion of an Artificial Satellite. Brower, Dirk ; Hori, Gen-Ichiro Abstract iciii o HoRI, Yale University Observatory. If the atmospheric density above perigee height is represented by p = A exp ( - ar), the well-known expressions for the drag accelera tions are X3= -CDA (%Q)V exp(-ar), 2= ~-a1l - 2w (xix2- x2x,) + (x,2x22), xj(j = 1, 2, 3) being the Cartesian coordinates; i= ~X2, 2= Xi, i=0. In the Delaunay variables L , 1k (for L, C, H, I, g, h), the equations become dLk F dlk F -Qk, dt l dt L 8x1 Th=E 3-, Qk= X,-. Similarly, the equations after the elimination of the periodic terms, both short and long, become dLk" dlk" F** dt dt L " Pk ~= P3_ lj Lj + Q3 Qk"= Pj____ Lj LQ3 By ignoring the rotation of the atmosphere (i.e., putting co=0), let Pk, q , p1c", q " be the functions obtained by replacing X5 by Xj. Then it is easily found that 2a 2 L2 p,=L1 -1 q1=2esinE-sinf r e L1 2 p2=L2 q2=-sinf e p3=L3 q =Q The solution of the problem wi h co = 0 is therefore reduced to that of the equations dLk" dt = -CDAVPk" exp(- ), dlk" OF** ___= - ,,$CDA Vq " exp(-o r). dt OLk The quantities Pk", q " have been obtained from the theory (Brouwer, 1959, Astron. J. 64, 378) to the first power of k2 (or J) in the periodic terms, to the second power in the secular terms. No infinite series appear in these expressions. In the development of V exp ( - ar) infinite series expressions in e are unavoidable. A development to the fourth power of e has been made. The final differential equations are solved by iteration. The investigation has not been completed, but it appears that the procedure adopted is a powerful method for dealing with the simultaneous analytic solution of the effects of oblateness and drag in the motion of an artificial satellite. Publication: The Astronomical Journal Pub Date: 1960 DOI: 10.1086/108070 Bibcode: 1960AJ.....65Q.342B full text sources ADS |

Oblateness correction to impact points of ballistic rockets

A method is developed for finding the effect of the Earth's oblateness on the impact points of ballistic rockets. Suitable descriptive variables are chosen so that the oblateness perturbations can be obtained by quadrature, and explicit results are obtained correct to first order in an oblateness parameter; specifically, changes in range angle, a lateral displacement angle, and time of flight. The short range limiting forms of the corrections are obtained and interpreted, and an illustrative numerical example is given for a long range trajectory.

Effect of Earth's Oblateness on the Period of a Satellite

The effect of the earth's oblateness on the period of the lat i tudinal mot ion of a satellite in a near-circular orbit is investigated, and it is shown that in addition to the usual dependence on alt i tude the period depends as well on the incl ination of the orbit to the equator. For orbits in the neighborhood of the equator, the period is less than for a n orbit at the same he ight around a spherical earth. The period increases wi th orbit incl inat ion, and for polar orbits the period is greater than for an orbit at the same he ight over a spherical earth. For a satell ite at a he ight of 300 miles the difference in periods between polar and nearequatorial orbits is 12 sec.