Long-period Perturbations Research Articles

Ocean tidal perturbations on the orbit of the satellite “STARLETTE”

A continuous 300-day, world-wide laser-ranging data of the satellite STARLETTE between October 1976 and July 1977 was used to analyse the perturbation of the orbit by ocean tide. The long-period (longer than 20d) perturbations in the orbital inclination by the four main tide components K 1, P 1, K 2, S 2 were clearly identifiable in the maximum entropy power spectrum and periodogram. By comparing orbits including and excluding the tidal terms, the effect of the tide on the position of the satellite over 5 days was found to be of the order of one meter.

On the stability of hierarchical four-body systems

Generalized Jacobian coordinates can be used to decompose anN-body dynamical system intoN-1 2-body systems coupled by perturbations. Hierarchical stability is defined as the property of preserving the hierarchical arrangement of these 2-body subsystems in such a way that orbit crossing is avoided. ForN=3 hierarchical stability can be ensured for an arbitrary span of time depending on the integralz=c 2 h (angular momentum squared times energy): if it is smaller than a critical value, defined by theL 2 collinear equilibrium configuration, then the three possible hierarchical arrangements correspond to three disconnected subsets of the invariant manifold in the phase space (and in the configuration space as well; see Milani and Nobili, 1983a). The same definitions can be extended, with the Jacobian formalism, to an arbitrary hierarchical arrangement ofN≥4 bodies, and the main confinement condition, the Easton inequality, can also be extended but it no longer provides separate regions of trapped motion, whatever is the value ofz for the wholeN-body system,N≥4. However, thez criterion of hierarchical stability applies to every 3-body subsystem, whosez ‘integral’ will of course vary in time because of the perturbations from the other bodies. In theN=4 case we decompose the system into two 3-body subsystems whosec 2 h ‘integrals’,z 23 andz 34, att=0 are assumed to be smaller than the corresponding critical values $$\tilde z_{23} $$ and $$\tilde z_{34} $$ , so that both the subsystems are initially hierarchically stable. Then the hierarchical arrangement of the 4 bodies cannot be broken until eitherz 23 orz 34 is changed by an amount $$\tilde z_{ij} - z_{ij} \left( 0 \right)$$ ; that is the whole system is hierarchically stable for a time spain not shorter than the minimum between $$\Delta t_{23} = {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} {\dot z_{23} }}} \right. \kern-\nulldelimiterspace} {\dot z_{23} }}$$ and $$\Delta t_{34} = {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} {\dot z_{34} }}} \right. \kern-\nulldelimiterspace} {\dot z_{34} }}$$ . To estimate how long is this stability time, two main steps are required. First the perturbing potentials have to be developed in series; the relevant small parameters are some combinations of mass ratios and length ratios, theɛ ij of Roy and Walker. When an appropriate perturbation theory is based on theɛ ij , the asymptotic expansions are much more rapidly decreasing than the usual expansions in powers of the mass ratios (as in the classical Lagrange perturbation theory) and can be extended also to cases such as lunar theory or double binaries. The second step is the computation of the time derivatives $$\dot z_{ij} $$ (we limit ourselves to the planar case). To assess the long term behaviour of the system, we can neglect the short-periodic perturbations and discuss only the long-periodic and the secular perturbations. By using a Poisson bracket formalism, a generalization of Lagrange theorem for semimajor axes and a generalization of the classical first order theories for eccentricities and pericenters, we prove that thez ij do not undergo any secular perturbation, because of the interaction with the other subsystem, at the first order in theɛ ik . After the long-periodic perturbations have been accounted for, and apart from the small divisors problems that could arise both from ordinary and secular resonances, only the second order terms have to be considered in the computation of Δt 23, Δt 34. A full second order perturbative theory is beyond the scope of this paper; however an order-of-magnitude lower estimate of the Δt ij can be obtained with the very pessimistic assumption that essentially all the second order terms affect in a secular way thez ij . The same method could be applied also toN≥5 body systems. Since almost everyN-body system existing in nature is strongly hierarchical, the product of twoɛ ij is very small for almost all the real astronomical problems. As an example, the hierarchical stability of the 4-body system Sun, Mercury, Venus, and Jupiter is investigated; this system turns out to be stable for at least 110 million years. Although this hierarchical stability time is ∼10 times less than the real age of the Solar System, taking into account that many pessimistic assumptions have been done we can conclude that the stability of the Solar System is no more a forbidden problem for Celestial Mechanics.

Time-optimized north-south stationkeeping

Introduction T long-period perturbations of the solar-lunar gravitational fields cause a steady precession of the orbital plane and exhibit a large number of feriodic terms having different frequencies appearing in the orbital elements. For a nearly equatorial satellite orbit this increase will average 0.86 deg annually. In order to effectively control the performance of a geosynchronous satellite, plane change maneuvers (north-south Stationkeeping) must be performed to minimize peak latitude excursions. It is not atypical for this need to account for 95% of maneuver propulsion. Typically these maneuvers are performed at the ascending node of the orbit and as a consequence of the periodic regression of the lunar orbital nodes the next maneuver is likely to occur earlier than operationally desired. A control strategy is developed here that will produce the maximum time between maneuvers and the minimum annual velocity change.

A second-order satellite orbit theory, with compact results in cylindrical coordinates

First-order perturbations of a low-eccentricity satellite orbit due to the general harmonic coefficientJ1mof the Earth’s gravitational field are derived, and compactly expressed (see equations (92)-(94)) in cylindrical polar coordinates. For the dominant harmonicJ2, the perturbations are taken to second order, and it is shown how formulae for the second-order variation of the orbital elements depend on the definition of the mean elements used for reference. With a particular choice of mean elements, the formulae for perturbations in cylindrical coordinates are again very compact (see equations (297), (315) and (321)). The general approach also yields first-order perturbations due to lunisolar gravity and eclipse-free solar radiation. The paper finishes with a set of untruncated expressions (valid for any eccentricity) forJ2/2 secular and long-periodic perturbations.

On long-periodic perturbations of Trojan asteroids

view Abstract Citations (2) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On long-periodic perturbations of Trojan asteroids Erdi, B. ; Presler, W. H. Abstract Previous analytical results (Erdi 1979) concerning the long-periodic perturbations of the eccentricity and of the longitude of the perihelion of Trojan asteroids are compared with numerical integration of the equations of motion of the corresponding plane elliptic restricted three-body problem. Thirty asteroids around the Lagrangian points L4 and L5 are tested. The periodicity of the eccentricity is approximately 3600 yr. The perihelion circulates (20 asteroids) or librates (ten asteroids) with the same period. There is good agreement between the analytical and numerical results. Publication: The Astronomical Journal Pub Date: December 1980 DOI: 10.1086/112846 Bibcode: 1980AJ.....85.1670E Keywords: Asteroids; Eccentricity; Numerical Integration; Orbit Perturbation; Orbital Elements; Three Body Problem; Trojan Orbits; Equations Of Motion; Jupiter (Planet); Lagrangian Equilibrium Points; Perihelions; Astronomy full text sources ADS |

A New Approach for the Construction of Long-Periodic Perturbations

AbstractThe aim of this work is to study perturbations of planets of a period of some thousands of years. The use of an analytical method allows us to separate all different influences, e.g. near resonances and is combined with the very precise method of the numerical integration. The truncation to low orders can be avoided which is made by analytical methods in using developments with respect to the small parameters inclinations and eccentricities. For this purpose a special form of the Lagrange Equations is used where the terms containing the inverse distancefrom the planet to the perturbing one are separated as it is the most difficult to compute. To develop this a specific formulation has been found where the short periodic terms can precisely be determined. Although the development seems to be of a certain complexity the small numbers of quantities used can be tabulated once and for all in a specific problem. It should be possible to integrate the new form of the Lagrange Equations within a reasonable computer-time to determine the long periodic perturbations.

Short-period and long-period perturbations of a spherical satellite due to direct solar radiation

On the basis of expressions derived by Kozai, and new ones developed here, a detailed, semianalytic algorithm is presented for calculating radiation-pressure perturbations in the Keplerian elements. Through some simple modifications, the algorithm is also made to hold whene=0 and/ori=0. The perturbations are obtained by summing over the sunlit segment of the satellite's orbit during each revolution or part thereof. The end points of this segment are evaluated numerically once per revolution. The effect of the inherent uncertainties in the boundaries of the Earth's shadow is discussed. The algorithm is tested by means of numerical integration of the equations of motion and through comparisons with observations of the balloon satellite 1963 30D during a 200-day interval.

The resonant orbit of the lost city meteorite

An integration of the long-period perturbations of Jupiter and Saturn on the orbit of the Lost City meteorite shows that the average nodal rate of the meteorite was nearly the same as the nodal rates of Jupiter and Saturn. This near equality of rates led to a large resonant variation in the inclination of the meteorite's orbit with a period on the order of 1 million years. The resonance makes the orbital evolution too sensitive to the deficiencies of the perturbation calculations and to the values of the elements at impact to allow definitive calculations. It is shown that an orbit similar to the meteorite's orbit can spend approximately 88% of its time with an eccentricity too small to allow for intersections with the earth's orbit. The ordinary cosmic ray exposure age of the Lost City meteorite does not suggest that its collisional lifetime was unusual, but the role of resonances in determining the lifetime of typical meteorites is unknown.

Stability of the solar system: Evidence from the asteroids

An analysis of the distribution of the orbital periods of the asteroids has shown that there is a preference for these periods to be near-commensurate with that of Mars. We suggest that this preference is associated with a formation process and implies that the orbital period of Mars has not changed greatly since the time of asteroid formation. We deduce from this that the solar system is highly stable and long-period gravitational perturbations have probably had little influence on the gross evolution of the solar system.

Long-range variations of orbits with arbitrary inclination and eccentricity

Secular and long-periodic perturbations for satellites and asteroids with arbitrary inclination and eccentricity are discussed by analyzing the energy integral.

Radiative and Photochemical Processes in Mesospheric Dynamics: Part II, Vertical Propagation of Long Period Disturbances at the Equator

This paper considers the vertical propagation of a long-period, small-amplitude perturbation in a medium in which radiative transfer and photochemistry play important roles. The perturbation and the basic field are assumed to be axially symmetric and symmetric about the equator; the basic wind field is geostrophic and the basic temperature field is in radiative equilibrium. It is found that long-period perturbations can only propagate by virtue of the physical effects of radiative transfer and photochemistry. The computed wave propagates downwards and, for a period of 2.2 years, the phase speed is close to the observed speed of 1.5 km month−1 for the “26-month” equatorial oscillation. The observed relative phases of velocity and temperature fields, and the sharp attenuation of the oscillation below 20 to 25 km are also found in the model wave. There are discrepancies between the model and the observed “26-month” oscillation, which are to be expected in view of the nonlinearity of the observed phenomenon. However, it appears that, for complex reasons, the observed wave may satisfy equations similar to those occurring in the linear theory.

Perturbations applied to the Izsak-Borchers relations

view Abstract Citations (1) References (7) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Perturbations applied to the Izsak-Borchers relations Allen, William A. ; Knolle, Werner E. Abstract This paper illustrates the improvement gained for a typical analytical near-earth satellite orbit when the J22 and J3 long-periodic perturbations are adjoined to the second-order Izsak-Borchers relations. The standards of comparison were two numerically integrated ephemerides. One standard consisted of the Vinti potential function with only J22 term superposed; the other standard was also the Vinti potential, but with only the J3 term superposed. With reference to the J22 standard, the respective rms fits in position for a 24 h ephemeris were 3.07X 10- earth radii units (eru) for the Izsak equations and only 0.29X 10- eru when the J22 perturbations were included. The corresponding J22 perturbations applied to the velocities resulted in improvement from 3.20X 10-8 eru/sec to 0.53X 10-8 eruIsec. With reference to the J3 standard, the respective rms fits in position were 3.63X 10-8 eru for the Izsak equations and 0.15X l0- eru when the J3 perturbations were applied. The corresponding J3 perturbations applied to the velocities resulted in im- provement from 2.31 X 10-8 eru/sec to 0.25X 10-8 eru/sec. Publication: The Astronomical Journal Pub Date: March 1965 DOI: 10.1086/109705 Bibcode: 1965AJ.....70..152A full text sources ADS |

On the long-period effects in the motion of an artificial satellite caused by the ellipticity of the equator of the Earth

If a satellite has a mean daily motion nearly commensurable with the angular velocity of rotation of the earth, long-period perturbations will influence its motion. The eccentricity of such a commensurable orbit is not necessarily small, as the orbit of Explorer VI shows. This circumstance caused the author to develop a semianalytical method of treating these perturbations, which avoids the development of the disturbing function into powers of the eccentricity and consequently includes the cases of elongated orbits and orbits with moderate eccentricities.

Tesseral harmonics of the gravitational potential of the earth as derived from satellite motions

view Abstract Citations (19) References (5) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Tesseral harmonics of the gravitational potential of the earth as derived from satellite motions Kozai, Yoshihide Abstract Some constants associated with tesseral harmonics of the gravitational potential of the earth are derived by use of photoreduced Baker-Nunn observations of satellites t958 2, 1959 , and 1959 . Since the periods of the long-periodic perturbations due to the tesseral harmonics are known, amplitudes and phases of the periodic terms are obtained by the differential orbit improvement program for each element of each satellite. On the assumption that tesseral harmonics of higher than fourth degree cannot cause any appre- ciable effect on the satellite motion, constants of the tesseral harmonics are derived from the observed amplitudes and phases, and are given in Table V. Publication: The Astronomical Journal Pub Date: September 1961 DOI: 10.1086/108434 Bibcode: 1961AJ.....66..355K full text sources ADS |

Densities of the Upper Atmosphere Derived From Discoverer Satellites

motion as determined by the variations in co). The mean radius r0, the mean eccentricity e and the mean rate of rotation of perigee du/dfi, each experiences long-period perturbations. The traditional osculating values of these quantities (5) would, in addition, reflect the short-period perturbations which here have been relegated to the additive term Jv in 1/r. This form, in which the longand short-period perturbations are separately disphyed, serves to clarify the total description of the satellite motion. When e is small (of order J), the total orbit bears little resemblance to an ellipse. In fact, Jv is then very nearly given by the single term

On the long-period lunisolar effect in the motion of the artificial satellite

This article contains two systems of formulas for the determination of the long-period perturbations caused by the sun and the moon in the motion of an artificial satellite. The first system can be used for the determination of the lunar effect for all satellites. The second method is more convenient for finding the lunar effect for close satellites and the solar effect for all satellites. The knowledge of these effects is essential for the determination of the stability of the orbit of the satellite. The basic equations of both systems are arranged in a form that permits the use of numerical integration. The two theories are more accurate and more adaptable to the use of electronic machines than are the analytical developments obtained previously.