On error bounds and initialization in satellite orbit theories

The order of magnitude of the error is investigated for a first-order von Zeipel theory of satellite orbits in an axisymmetric force field, i.e., first-order long period and short-period effects are included along with second order secular rates. The treatment is valid for zero eccentricity and/or inclination. In the case where initial position and velocity vectors are known, the in-track position error over time intervals of order 1/J ₂ is kept at 0(J ₂ ²), like the other position errors and velocity errors, by calibration of the mean motion with the aid of the energy integral. The results are specifically applicable to accuracy comparisons of the Brouwer orbit prediction method with numerical integration. A modified calibration is presented for the general asymmetric force field which includes tesseral harmonics.     

Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.     

Third-order secular Solution of the variational equations of motion of a satellite in orbit around a non-spherical planet

We constructed an analytical theory of satellite motion up to the third order relative to the oblateness parameter of the Earth (J ₂). Equations of secular variations was developed for the first three orbital elements (a, e, i) of an artificial satellite. The secular variations are solved in a closed form.     

Ice Mass Balance and Ice Dynamics from Satellite Gravity Missions

An overview of advances in ice research which can be expected from future satellite gravity missions is given. We compare present and expected future accuracies of the ice mass balance of Antarctica which might be constrained to 0.1–0.3 mm/year of sea level equivalent by satellite gravity data. A key issue for the understanding of ice mass balance is the separation of secular and interannual variations. For this aim, one would strongly benefit from longer uninterrupted time series of gravity field variations (10 years or more). An accuracy of 0.01 mm/year for geoid time variability with a spatial resolution of 100 km would improve the separability of ice mass balance from mass change due to glacial isostatic adjustment and enable the determination of regional variations in ice mass balance within the ice sheets. Thereby the determination of ice compaction is critical for the exploitation of such high accuracy data. A further benefit of improved gravity field models from future satellite missions would be the improvement of the height reference in the polar areas, which is important for the study of coastal ice processes. Sea ice thickness determination and modelling of ice bottom topography could be improved as well.     

Tesseral resonance effects on satellite orbits

Resonance effects on satellite orbits due to tesseral harmonics in the potential field have been studied by many authors. Most of these studies have been restricted to nearly circular 24-hour orbits and to the deep resonance regime, where there is exact commensurability between earth rotation and orbit period. Resonance effects have also been noted, however, on eccentric synchronous and subsynchronous orbits and on orbits with far from commensurate periods. These have received much less attention; the object of this paper is to study the whole spectrum of orbits with respect to resonance effects.     

A numerical-analytical method for studying the orbital evolution of distant planetary satellites

We describe an approximate numerical-analytical method for calculating the perturbations of the elements of distant satellite orbits. The model for the motion of a distant satellite includes the solar attraction and the eccentricity and ecliptic inclination of the orbit of the central planet. In addition, we take into account the variations in planetary orbital elements with time due to secular perturbations. Our work is based on Zeipel’s method for constructing the canonical transformations that relate osculating satellite orbital elements to the mean ones. The corresponding transformation of the Hamiltonian is used to construct an evolution system of equations for mean elements. The numerical solution of this system free from rapidly oscillating functions and the inverse transformation from the mean to osculating elements allows the evolution of distant satellite orbits to be studied on long time scales on the order of several hundred or thousand satellite orbital periods.     

Tesseral harmonic perturbations in radial transverse and binormal components

The formulae for the perturbations in radial, transverse and binormal components of the Earth artificial satellite motion have been derived. Perturbations due to the tesseral part of the geopotential are considered. The geopotential expressed in terms of the orbital elements has the form proposed by Wnuk (1988). The formulae for the perturbations have been obtained using the Hori (1966) method. They can be effectively applied in calculation of the perturbations in the components including the coefficients of the high order and degree tesseral harmonics. The derived formulae reveal no singularities at zero eccentricity.     

An analytical theory for tesseral gravitational harmonics

An analytical method has been developed for the treatment of tesseral harmonic perturbations. The procedure is an iterative Lie transformation technique which avoids the typical eccentricity expansions as well as the numerical singularities normally associated with resonance conditions. At each iteration, terms of the perturbing potential become multiplied by the ratio of the satellite's orbital period to the earth's rotational period. Following a suitable number of iterations, the potential is deemed to be sufficiently small that it may be ignored, with the tesseral effects captured in the transformation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Temporal Variations in the Second-Degree Stokes Tesseral Geopotential Coefficients from Topex/Poseidon Altimetry

The TOPEX/POSEIDON (T/P) altimetry data set covering the periodof January 1, 1993 to January 3, 2001 was used to derive monthlyseries of the second-degree tesseral geopotential coefficients.To account for the sea water temperature variations, rathersimple models have been devised and discussed, describinglocalized as well as areal variations of sea water temperatureand heights. The second-degree tesseral coefficients have alsobeen shown to be proportional to the pressureportions of the oceanic equatorial effective excitation functions,used in Ocean Angular Momentum (OAM) data. OAM datatogether with Atmospheric Angular Momentum (AAM) data canbe used to study observed polar motion (PM) series.The excess PM rates, derived from the T/P effective excitationfunctions, were compared to the corresponding observed PM rates,derived from the International Earth Rotation Service (IERS)Bulletin A and corrected with AAM also obtainedfrom IERS. The noise of the T/P derived PM rate series was foundto be significantly larger than the corresponding Bulletin A/AAMPM rate residuals as well as the PM rates derived from anindependent OAM series that was also available for the1993–2000 period.     

East-west stationkeeping requirements of nearly synchronous satellites due to Earth's triaxiality and luni-solar effects

A closed form solution, for longitude and semimajor axis deviations in the neighborhood of a prespecified station, is obtained for nearly synchronous satellites. The model use includes the important terms in Earth's zonal and tesseral harmonics as well as the luni-solar perturbations. The initial semimajor axis for two-maneuver east-west stationkeeping is then deduced. Due to the luni-solar effects, it is found that the initial semimajor axis deviation from synchronous orbit value is highly dependent on the initial position of the satellite relative to the Moon and the Sun. Verifications of the results by means of numerical integrations are also included.     

Secular variations of the photometric parameters of Mira Ceti variables and semiregular variables

The photometric properties of long-period variable stars and problems related to their traditional classification are analyzed. A general review of secular variations of such light-curve parameters studied in our previous papers as the period, amplitude, and mean brightness is given. Several types of Mira Ceti stars that differ in the secular period variation pattern are identified. A connection between the secular amplitude variations of certain variables with the effect of multiperiodicity is revealed. It is shown that the pattern of these variations and their numerical properties may be used to classify these variable stars.     

Secular effects in the orbits of the Galilean satellites

This paper presents the results of an investigation into the secular behavior of the orbits of the Galilean satellites of Jupiter. Kamel's perturbation method is used to remove all the explicitly periodic variables from the differential equations that describe the long period behavior of the orbits to third order in the masses, and the resulting differential equations for the secular behavior are then solved. Several numerical examples are given to illustrate the sensitivity of the solution to variations in the masses of the satellites.     

First period analyses of five neglected Algol-type eclipsing binaries: TT And, V342 Aql, RW Cap, BZ Cas and TW Lac

We present the first study of the orbital period variations of five neglected Algol-type eclipsing binaries TT And, V342 Aql, RW Cap, BZ Cas and TW Lac, using their O–C diagrams gathered from all available times of eclipse minima. These O–C diagrams indicate that short term periodic variations superimposed on secular period increases as expected in mass transferring Algols. However, due to short time coverage of the data, the secular period increase is not clear in the case of BZ Cas and V342 Aql. The secular period increase is interpreted in terms of the combined effect of mass transfer between the components of the system and the mass loss by a stellar wind from the system. The mass transfer rates from the less massive secondary components to the more massive primaries for non-conservative cases would be about 10⁻⁷M/yr and 10⁻⁸M/yr for RW Cap and V342 Aql, respectively, and 10⁻⁹M/yr for TT And and TW Lac. Therefore, the Algol systems RW Cap and V342 Aql have the largest mass transfer rate, which could be in Case AB type, while those of the Algol systems TT And and TW Lac display the slow mass transfer rate and they could be in Case B type. The sinusoidal forms of the orbital period variations of all five Algol systems can be due to either by the light-time effects due to unseen components in these systems, or by the cyclic magnetic activity effects of the cool secondary components. The possible third bodies in all five Algol binaries would have masses larger than one solar mass. If these hypothetical large massive third bodies were normal stars, they should be detectable. Therefore, new photometric and spectroscopic observations of these systems and careful analyses of those data are required. Otherwise, the cyclic magnetic activity effects of the secondary components could be the basis of a working hypothesis in explaining the cyclic period variations of these systems.     

A period investigation of two chromospherically active binary stars: RT Coronae Borealis and PW Herculis

Orbital period variations of two chromospherically active binary systems, RT CrB and PW Her, are presented. It is shown that the orbital period of RT CrB undergoes a cyclic oscillation with a period of 53.9 years. For PW Her, an alternate change, with a period of 42.7 years, is found to superimpose on a rapid secular increase (dP/dt=+3.53×10⁻⁶ days/year). If the period oscillations of those two systems are caused by the light-time effect of a third body, the analysis for RT CrB indicates that the third body would be a low-mass main-sequence star, while, for PW Her, the mass of the third body should be no less than 7.8 M_⊙. Since no spectral lines of the third body were seen in PW Her from the spectroscopic study by Popper [AJ 100 (1990) 247], if there is a third body in the system, it can only be a black hole. However, as both components in the two binary stars were showing strong chromospheric activity, the alternate period variations are more plausibly explained as the result of magnetic activity cycles. No secular period changes of RT CrB are found, which is in agreement with the detached evolved configuration of the system. The long-term period increase of PW Her may indicate that it is on an active phase of mass transfer (dm/dt=2.17×10⁻⁶ M_⊙/year).     

On the stability of dust orbits in mean-motion resonances perturbed by from an interstellar wind

Circumstellar dust particles can be captured in a mean-motion resonance (MMR) with a planet and simultaneously be affected by non-gravitational effects. It is possible to describe the secular variations of a particle orbit in the MMR analytically using averaged resonant equations. We derive the averaged resonant equations from the equations of motion in near-canonical form. The secular variations of the particle orbit depending on the orientation of the orbit in space are taken into account. The averaged resonant equations can be derived/confirmed also from Lagrange’s planetary equations. We apply the derived theory to the case when the non-gravitational effects are the Poynting–Robertson effect, the radial stellar wind, and an interstellar wind. The analytical and numerical results obtained are in excellent agreement. We found that the types of orbits correspond to libration centers of the conservative problem. The averaged resonant equations can lead to a system of equations which holds for stationary points in a subset of resonant variables. Using this system we show analytically that for the considered non-gravitational effects, all stationary points should correspond to orbits which are stationary in interplanetary space after an averaging over a synodic period. In an exact resonance, the stationary orbits are stable. The stability is achieved by a periodic repetition of the evolution during the synodic period. Numerical solutions of this system show that there are no stationary orbits for either the exact or non-exact resonances.     

Precision Analytical Calculation of Geodynamical Effects on Satellite Motion

A new analytical method for calculating satellite orbital perturbations due to different disturbing forces is developed. It is based on the Poincaré method of small parameter but takes advantages of modern high-performance computers and of the tools of computer algebra. All perturbations proportional up to and including the 5th-order of small parameters are obtained. The method can precisely calculate the effects of all geodynamical forces on satellite motion given by the most up-to-date IAU and IERS models, such as non-central Earth gravity potential, precession and nutation of the geoequator, polar motion and irregularities in the Earth's rotation, effect of ocean and solid Earth tides, pole tide, and secular variations of gravity coefficients.Numerical tests prove the method's accuracy to be equivalent to 1–2 cm when calculating positions of high altitude geodetic satellites (like ETALON), and/or of GLONASS navigational spacecraft. The accuracy is stable over 1 year at least and comparable to that of the best tracking measurements of satellites.Positions of low altitude geodynamical satellites (like STARLETTE) by the analytical method are calculated to an accuracy of about 70cm over a month's interval. The method is developed for future use in GLONASS/GPS on-board ephemeris computation where it can improve the current scheme of their flight control.This revised version was published online in October 2005 with corrections to the Cover Date.     

A semi-analytic theory for the motion of a lunar satellite

A semi-analytical solution to the problem of the motion of a satellite of the moon is presented. Perturbative effects which are considered include those due to the attraction of the moon, earth, and sun, the non-sphericity of the moon's gravitational field, coupling of lower-order terms, solar radiation pressure, and physical libration. Short-period terms and intermediate-period terms, terms with the period of the moon's longitude, are produced by means of von Zeipel's method; it is proposed to obtain the secular perturbations, and those depending only on the argument of perilune, by numerical integration of the equations of motions. The short-period terms and intermediate-period terms are developed up to second order, where first order is 10^–2. The secular perturbations and perturbations dependent on the argument of perilune are obtained to third order.     

Satellite resonance with longitude-dependent gravity—II Effects involving the eccentricity

For a satellite in a nominally circular orbit at arbitrary inclination whose mean motion is commensurable with the Earth's rotation, the dependence of gravity on longitude leads to a resonant variation in eccentricity as well as the long-period oscillation in longitude. Provided forces capable of processing perigee are present, it is shown that the change in eccentricity for a satellite captured in librational resonance is not secular but periodic.

There are corresponding resonance effects for a satellite in a nominally equatorial but eccentric orbit. Here the commensurability condition is that the longitudes of the apses shall be nearly repetitive relative to the rotating Earth. There will be a long-period oscillation in longitude which can take the form of either a libration (trapped) or a circulation (free), and there will also be an oscillation of the orbital plane having the same period as the precession of perigee relative to inertial space.     

Monitoring Secular Orbital Period Variations of Some Eclipsing Binaries at the Ankara University Observatory

This study concerns the long-term monitoring of the secular variation character in the orbital period of some short-period eclipsing binaries observed at the Ankara University Observatory. Among the systems of our observing list are CK Boo, V502 Oph and V836 Cyg that show long-term secular variations in their orbital periods. We use classical O-C diagram analysis technique as a tool to reveal the character of the period variations of these binary systems.