Satellite motion in a non-singular gravitational potential

We study the effects of a non-singular gravitational potential on satellite orbits by deriving the corresponding time rates of change of its orbital elements. This is achieved by expanding the non-singular potential into power series up to second order. This series contains three terms, the first been the Newtonian potential and the other two, here R ₁ (first order term) and R ₂ (second order term), express deviations of the singular potential from the Newtonian. These deviations from the Newtonian potential are taken as disturbing potential terms in the Lagrange planetary equations that provide the time rates of change of the orbital elements of a satellite in a non-singular gravitational field. We split these effects into secular, low and high frequency components and we evaluate them numerically using the low Earth orbiting mission Gravity Recovery and Climate Experiment (GRACE). We show that the secular effect of the second-order disturbing term R ₂ on the perigee and the mean anomaly are 4″.307×10⁻⁹/a, and −2″.533×10⁻¹⁵/a, respectively. These effects are far too small and most likely cannot easily be observed with today’s technology. Numerical evaluation of the low and high frequency effects of the disturbing term R ₂ on low Earth orbiters like GRACE are very small and undetectable by current observational means.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.     

Long-term evolution of orbits about a precessing oblate planet. 2. The case of variable precession

We continue the study undertaken in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary’s equator in the cause of its spin-axis variations. As demonstrated by Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)], this opinion had stemmed from an inexact interpretation of a correct result by Goldreich [Astron. J. 70, 5–9 (1965)]. Although Goldreich [Astron. J. 70, 5–9 (1965)] mentioned that his result (preservation of the initial inclination, up to small oscillations about the moving equatorial plane) was obtained for non-osculating inclination, his admonition had been persistently ignored for forty years. It was explained in Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)] that the equator precession influences the osculating inclination of a satellite orbit already in the first order over the perturbation caused by a transition from an inertial to an equatorial coordinate system. It was later shown in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] that the secular part of the inclination is affected only in the second order. This fact, anticipated by Goldreich [Astron. J. 70, 5–9 (1965)], remains valid for a constant rate of the precession. It turns out that non-uniform variations of the planetary spin state generate changes in the osculating elements, that are linear in , where is the planetary equator’s total precession rate that includes the equinoctial precession, nutation, the Chandler wobble, and the polar wander. We work out a formalism which will help us to determine if these factors cause a drift of a satellite orbit away from the evolving planetary equator.By “precession,” in its most general sense, we mean any change of the direction of the spin axis of the planet—from its long-term variations down to nutations down to the Chandler wobble and polar wander.     

The construction of a third-order secular analytical J-S-U-N theory by the Hori-Lie technique

We expand both parts, the principal and indirect, of the Hamiltonian function up to the third order in the masses for the four major planets Jupiter-Saturn-Uranus-Neptune. Accordingly we write down the secular terms ofF ₁,F ₂,F ₃ and the critical terms ofF ₁,F ₂ in terms of the canonical variables of H. Poincaré neglecting powers higher than the second inH, K, P, Q.     

Analytical Approach to the Motion of a Lunar Artificial Satellite

The canonical equations of motion of an artificial lunar satellite are formulated including the effects of the asphericity of the Moon comprising the harmonics J ₂, J ₂₂, J ₃, J ₃₁, J ₄ andJ ₅, the oblateness of the Earth up to the second zonal harmonic, as well as the disturbing function due to the attractions of the Earth and of the Sun (terms are retained up to order 10^-6 for the higher orbits and 10^-8 for the lower orbits). This revised version was published online in July 2006 with corrections to the Cover Date.     

Perturbation of a satellite orbiter by the oblateness of the primary planet

The perturbation of an orbiter around a large satellite of a giant planet (Jupiter, Saturn, Uranus or Neptune) produced by the oblateness of the planet is investigated. The perturbing force of theJ ₂-term (general case) and theJ ₄-term (special case of small eccentricity and inclination) is expanded in an appropriate form and the main term and the parallactic term are given explicitly. The variations of the orbital elements are derived using the stroboscopic method. An example shows that the perturbation of the orbit cannot be neglected.     

On the long-term secular increase in sunspot number

Correlated with the maximum amplitude (R _max) of the sunspot cycle are the sum (R _sum) and the mean (R _mean) of sunspot number over the duration of the cycle, having a correlation coefficient r equal to 0.925 and 0.960, respectively. Runs tests of R _max, R _sum, and R _mean for cycles 0–21 have probabilities of randomness P equal to 6.3, 1.2, and 9.2%, respectively, indicating a tendency for these solar-cycle related parameters to be nonrandomly distributed. The past record of these parameters can be described using a simple two-parameter secular fit, one parameter being an 8-cycle modulation (the so-called Gleissberg cycle or long period) and the other being a long-term general (linear) increase lasting tens of cycles. For each of the solar-cycle related parameters, the secular fit has an r equal to about 0.7–0.8, implying that about 50–60% of the variation in R _max, R _sum, and R _mean can be accounted for by the variation in the secular fit.Extrapolation of the two-parameter secular fit of R _max to cycle 22 suggests that the present cycle will have an R _max = 74.5 ± 49.0, where the error bar equals ± 2 standard errors; hence, the maximum amplitude for cycle 22 should be lower than about 125 when sunspot number is expressed as an annual average or it should be lower than about 130 when sunspot number is expressed as a smoothed (13-month running mean) average. The long-term general increase in sunspot number appears to have begun about the time of the Maunder minimum, implying that the 314-yr periodicity found in ancient varve data may not be a dominant feature of present sunspot cycles.     

1990 photometry of a small amplitude W UMa system LS Delphini

DifferentialUBV observations, carried out in 1990 observing season, of a small amplitude (0 _. ^m 15 inV andB) W UMa system LS Del = HD 199497 are presented. Wavelength-dependent light variations from cycle to cycle indicate that the system is in a very active phase, probably due to magnetic flare activity or mass transfer in the system. An analysis of the minima times indicate a probable secular increase of the photometric period which requires a mass transfer from less massive to more massive component. If this is true then the reverse-algol model by Liuet al. (1988) for this system would not be valid.     

Secular variations of major planets' orbital elements

The secular variations of the orbital elements of principal planets are calculated by means of classical Lagrange's method. The terms of the second order with respect to mass, introduced by Hill (1897) and Brouwer and van Woerkom (1950), have been taken into account as well. The best contemporary values of planetary masses and mean elements (Bretagnon, 1982) served as the starting data set for this calculation. Considerable differences with respect to previous solutions of the same type (Brouwer and van Woerkom, 1950: Sharaf and Boudnikova, 1967) were found in the coefficientsA ₅₅,A ₅₆, andA ₆₆ of the system of equations of variation of elements and in the roots (frequencies)r ₅ andr ₆. Results are compared with some higher order/higher degree solutions and their accuracy discussed. It is confirmed that the solutions like that of Brouwer and van Woerkom, although not being completely inferior to all higher order/higher degree ones, can be considered as the first approximation only. Hence, they should be replaced by more accurate ones (Duriez, 1979: Bretagnon, 1984: Laskar, 1984) in the future applications.     

Long period variations of the motion of a satellite due to non-resonant tesseral harmonics of a gravity potential

The main effects of tesseral harmonics of a gravity potential expansion on the motion of a satellite, are short period variations as well as long period variations due to resonances. However, other smaller long period and secular variations can arise from interactions between tesseral terms of the same order. The analytical integration of these effects is developed, using numerical evaluation of Kaula eccentricity and inclination functions. Examples for some Earth's geodetic satellites show that secular effects can reach a few decameters per year. The secular variations can even reach several hundred of meters per year for the Mars natural satellite Phobos.     

Expansion of the perturbing function of a satellite restricted four-body problem

A satellite four-body problem is the problem of motion of an artificial satellite of a planet in a region of the space where perturbations due to the gravitational field of the planet are of the same order as perturbations due to influences of two perturbing bodies. In this paper an expansion of the perturbing function into a Fourier series in terms of angular Keplerian elements ( _j , _j ,M _j :j=0,1,2) (designations are standard) is obtained taking into account a sharp commensurability of the typen/ ₀=(p+q)/p (n is the mean motion of the artificial satellite and ₀ is the angular velocity of rotation of the planet,p andq are integers).The coefficients of the Fourier series are the functions of the positional Keplerian elements (a _j ,e _j ,i _j ;j=0, 1, 2) (designations are standard) and, in particular, are series in terms ofe _j that, generally speaking, can be written out to an accuracy ofe _j ¹⁹ .The expansion obtained can be used for the construction of a semianalytical theory of motion of resonant satellites on the basis of conditionally periodic solutions of the restricted four-body problem.     

A new light on the L 1, L 2 Lagrangian points

The relation between the locations of L ₁, L ₂ Lagrangian points and the boundary to their respective satellite system is brought forth, in that, the Lagrangian points L ₁, L ₂ are seen to lie just on the boundary to their respective satellite system.     

The Impact of the Static Part of the Earth's Gravity Field on Some Tests of General Relativity with Satellite Laser Ranging

In this paper we calculate explicitly the classical secular precessions of the node and the perigee of an Earth artificial satellite induced by the even zonal harmonics of the static part of the geopotential up to degree l = 20. Subsequently, their systematic errors induced by the mismodelling in the even zonal spherical harmonics coefficients J _l are compared to the general relativistic secular gravitomagnetic and gravitoelectric precessions of the node and the perigee of the existing laser-ranged geodetic satellites and of the proposed LARES. The impact of the future terrestrial gravity models from CHAMP and GRACE missions is discussed as well. Preliminary estimates with the recently released EIGEN-1S gravity model including the first CHAMP data are presented.     

Tobias Mayer's observations of the Sun: Evidence against a secular decrease of the solar diameter

Evidence for a significant secular decrease of the solar diameter has recently been presented by Eddy and Boornazian (1979). With regard to the enhanced interest in periodic or non-periodic variations of figure and size of the Sun, a very reliable series of 129 transit observations made by Tobias Mayer in 1756–1760 has been analyzed. The necessity for applying adequate corrections to measurements of this kind is stressed again. Mayer's observations yield R = (960.16±0.13). This is in excellent agreement with more recent photoelectric transit observations and lends no support whatsoever to the assumption of a secular decrease.     

An analytic satellite theory using gravity and a dynamic atmosphere

An analytical solution is given for the motion of an artifical Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational effects of the zonal harmonicsJ ₂,J ₃, andJ ₄ are included, and the drag effects of any arbitrary dynamic atmosphere are included. By a dynamic atmosphere, we mean any of the modern empirical models which use various observed solar and geophysical parameters as inputs to produce a dynamically varying atmosphere model. The subtleties of using such an atmosphere model with an analytic theory are explored, and real world data is used to determine the optimum implementation. Performance is measured by predictions against real world satellites. As a point of reference, predictions against a special perturbations model are also given.     

Long-Term Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession

It was believed until very recently that a near-equatorial satellite would always keep up with the planet’s equator (with oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich [Astronomy & Astrophysics (2004) Vol. 415, pp. 1187–1199], this misconception originated from a wrong interpretation of a (mathematically correct) result obtained in terms of non-osculating orbital elements. A similar analysis carried out in the language of osculating elements will endow the planetary equations with some extra terms caused by the planet’s obliquity change. Some of these terms will be non-trivial, in that they will not be amendments to the disturbing function. Due to the extra terms, the variations of a planet’s obliquity may cause a secular drift of its satellite orbit inclination. In this article we set out the analytical formalism for our study of this drift. We demonstrate that, in the case of uniform precession, the drift will be extremely slow, because the first-order terms responsible for the drift will be short-period and, thus, will have vanishing orbital averages (as anticipated 40 years ago by Peter Goldreich), while the secular terms will be of the second order only. However, it turns out that variations of the planetary precession make the first-order terms secular. For example, the planetary nutations will resonate with the satellite’s orbital frequency and, thereby, may instigate a secular drift. A detailed study of this process will be offered in a subsequent publication, while here we work out the required mathematical formalism and point out the key aspects of the dynamics. In this article, as well as in (Efroimsky 2004), we use the word ‘‘precession’’ in its most general sense which embraces the entire spectrum of changes of the spin-axis orientation -- from the long-term variations down to the Chandler Wobble down to nutations and to the polar wonder.     

On the stabilization of the collinear libration points of the restricted circular three-body problem

The possibility of stabilizing the collinear libration points of the circular restricted three-body problem by using an additional jet acceleration (constant in magnitude) is investigated. Three stabilization laws are considered when the jet acceleration is either directed continuously to one of the primariesm ₁,m ₂ or is parallel to the line joining them. The solution of the problem formulated is based on the method of the driving forces structure analysis created by W. Thomson and P. Tait. It is shown that none of the stabilization laws mentioned ensures the existence of the isolated minimum of changed potential energy, and therefore the secular stability of the collinear libration points is impossible. In the 3rd and 4th paragraphs the possibility of a gyroscopic stabilization of these points is considered. It is shown that the gyroscopic stabilization of the external libration points is possible only when jet acceleration is either directed to the distant mass or is parallel to the line joining the primaries. The necessary and sufficient conditions of the gyroscopic stabilization are given. It is also shown that the internal libration points cannot be stabilized by any of the laws considered. For the Earth-Moon system the numerical data of time-existence of the satellite in the vicinity of the libration point situated near the Moon are given.     

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.     

A second order analytical atmospheric drag theory based on the TD88 thermospheric density model

A second order atmospheric drag theory based on the usage of TD88 model is constructed. It is developed to the second order in terms of TD88 small parameters K _n,j . The short periodic perturbations, of all orbital elements, are evaluated. The secular perturbations of the semi-major axis and of the eccentricity are obtained. The theory is applied to determine the lifetime of the satellites ROHINI (1980 62A), and to predict the lifetime of the microsatellite MIMOSA. The secular perturbations of the nodal longitude and of the argument of perigee due to the Earth’s gravity are taken into account up to the second order in Earth’s oblateness.