Luni-solar Gravity Research Articles

Duality-Based Near-Fuel-Optimal Impulsive Trajectory Computation for Spacecraft Rendezvous Under Perturbations

In this study, a novel impulsive trajectory optimization algorithm for near-fuel-optimal rendezvous under perturbations is presented. The algorithm, called algorithm for impulsive rendezvous trajectory optimization under perturbations (AIRTOP), is designed such that the accuracy of rendezvous constraints is improved as much as possible while retaining the first-order optimality, under high-fidelity dynamics models that encompass various orbital perturbations (e.g., nonspherical gravity, air drag, luni-solar gravity, and solar radiation pressure). To eliminate rendezvous constraint errors, AIRTOP solves the linearized impulsive rendezvous problem recursively using the nonsingular dual-primal optimization algorithm. To improve the computational efficiency, AIRTOP applies the approximate analytical gradient of the constraint error norm during its error elimination process. We present three numerical simulations near circular and elliptical orbits. The simulation results show that i) AIRTOP can successfully solve impulsive rendezvous problems under realistic orbital perturbations with minimal user intervention, and ii) it can generate more fuel-efficient solutions in similar or shorter computation time than two other methods, namely, the differential corrector of the General Mission Analysis Tool and the global optimization approach, which can also take complex perturbations into account.

Effects of Orbital Perturbations on Deployment Dynamics of Tethered Satellite System Using Variable-Length Element

Orbital perturbations caused by the space environment will induce deviations compared with the prescribed motion trajectory or dynamic performance of the tethered satellite system (TSS). To comprehensively investigate the effects of the orbital perturbations on the TSS, a precise variable-length element that can describe the large deformation and flexibility of the tether is adopted in this paper to predict the deployment dynamics considering orbital perturbations, including atmospheric drag, solar pressure, lunisolar gravitation, and J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> perturbation. To this end, the variable-length element using arbitrary LagrangianEulerian (ALE) description in the frame work of absolute nodal coordinate formulation (ANCF) is developed firstly. Subsequently, the dynamic governing equations for the TSS are established in the context of ANCF with the employment of ANCF reference node (ANCF-RN). Effects of orbital perturbations including atmospheric drag, solar pressure, lunisolar gravity and J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> perturbation on the dynamics of an optimal deployment are analyzed numerically. It is shown that the orbital perturbations except the J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> perturbation will induce the out-of-plane motion of the TSS. Additionally, the tether tension is less affected by the orbital perturbations. By contrast, the lunisolar gravitation will induce a deviation on the motion path of satellite, which should be considered in the design phase of the TSS.

Comparison of disposal options for Tundra orbits in terms of delta-V cost and long-term collision risk

Tundra orbits are inclined, moderately eccentric orbits with a 24-h period. These orbits undergo large excursions in eccentricity due to luni-solar gravity perturbations. For inclinations above 50°, eccentricity can grow to a value that causes perigee to reach the Earth's atmosphere, resulting in vehicle re-entry. In the current study, a range of potential disposal options for Tundra orbits were compared. Disposal options considered include moving to a disposal orbit near the Tundra mission orbit, lowering apogee just below GEO, and moving to near-circular orbits below and above GEO. Comparison metrics include delta-V cost (a measure of maneuver effort) and long-term collision risk. For the eccentric disposal orbit options, initial inclination and right ascension of ascending node (RAAN) were parametrically varied. For the near-circular disposal orbit options, initial argument of perigee and RAAN were parametrically varied. Study results indicate that a near-Tundra disposal orbit option appears to be the favorable selection since it has much lower delta-V cost than the other options and still has the second lowest overall collision risk, well below the threshold of 0.001 that is cited in U.S. standards on debris mitigation and space safety. In terms of orbital sustainability, for all RAAN values there are inclination values that enable re-entry within 200 years.

Regularized luni-solar gravity dynamics on resident space objects

Resident space object population in highly elliptical high perigee altitude (> 600 km) orbits is significantly affected by luni-solar gravity. Using regularization, an analytical orbit theory with luni-solar gravity effects as third-body perturbations in terms of Kustaanheimo-Stiefel regular elements is developed. Numerical tests with different cases resulted in good accuracy for both short- and long-term orbit propagations. It is observed that the luni-solar perturbations affect the accuracy of the analytical solution seasonally. The analytical theory is tested with the observed orbital parameters of the few objects in highly elliptical orbits. The analytical evolution of osculating perigee altitude is found to be concurrent with observed data. Solar perturbation, when compared with lunar perturbation, is established to be dominant over such orbits.

Orbital lifetime and collision risk reduction for inclined geosynchronous disposal orbits

A growing number of satellites are operating on inclined geosynchronous orbits (IGSOs) with inclination typically much higher than that of traditional geosynchronous orbits (GEOs). Several recent studies have considered the long-term evolution of IGSOs. Unlike traditional GEO disposal orbits, IGSO disposal orbits can undergo large excursions in eccentricity due to the effect of luni-solar gravity perturbations. For specific ranges of initial orbital elements, perigee can reach the Earth's atmosphere, resulting in vehicle reentry. A previously published study by the authors on Tundra orbits, a specific class of IGSO with critical inclination and moderate eccentricity, demonstrated that orbital lifetime can be reduced below 200 years (in some cases below 25 years) and that the corresponding collision probability with background objects can be significantly reduced below that for traditional GEO disposal orbits. This paper presents a study of a broad range of IGSO disposal orbits. Three cases of IGSO disposal orbit were considered: (1) near circular orbits (motivated by the BeiDou and IRNSS constellations); (2) orbits with intermediate eccentricity (motivated by the QZS constellation); and (3) orbits with larger eccentricity (motivated by the Sirius Tundra constellation). For each case, a large number of long-term propagations using the high-precision code TRACE were performed. Disposal orbit initial inclination and right ascension of ascending node (RAAN) were parametrically varied. The Aerospace Debris Environment Projection Tool (ADEPT) suite was used to determine collision probability with inactive background objects and with operational satellites in GEO, medium Earth orbit (MEO), and low Earth orbit (LEO). Study results show that orbital lifetime of IGSOs can be reduced to less than 200 years for all three IGSO cases considered if initial inclination is high enough. The minimum required inclination depends on initial RAAN. An orbital lifetime less than 200 years offers effective reduction of probability of collision with inactive background objects (by 0.7–1.8 orders of magnitude) and can be achieved in a wider disposal orbit design space than would be needed to reduce orbital lifetime to less than 25 years. For near circular IGSOs, collision probability with operational satellites will be higher at high inclinations than at low inclinations, but there are options for mitigation. When inclination is not high enough to achieve reentry, a high eccentricity storage disposal orbit may be a consideration if collision probability is lower than for other options.

Dynamical cartography of Earth satellite orbits

We have carried out a numerical investigation of the coupled gravitational and non-gravitational perturbations acting on Earth satellite orbits in an extensive grid, covering the whole circumterrestrial space, using an appropriately modified version of the SWIFT symplectic integrator, which is suitable for long-term (∼120 years) integrations of the non-averaged equations of motion. Hence, we characterize the long-term dynamics and the phase-space structure of the Earth-orbiter environment, starting from low altitudes (∼400 km) and going up to the GEO region and beyond. This investigation was done in the framework of the EC-funded “ReDSHIFT” project, with the purpose of enabling the definition of passive debris removal strategies, based on the use of physical mechanisms inherent in the complex dynamics of the problem (i.e., resonances). Accordingly, the complicated interactions among resonances, generated by different perturbing forces (i.e., lunisolar gravity, solar radiation pressure, tesseral harmonics in the geopotential) are accurately depicted in our results, where we can identify the regions of phase space where the motion is regular and long-term stable and regions for which eccentricity growth and even instability due to chaotic behavior can emerge. The results are presented in an “atlas” of dynamical stability maps for different orbital zones, with a particular focus on the (drag-free) range of semimajor axes, where the perturbing effects of the Earth’s oblateness and lunisolar gravity are of comparable order. In some regions, the overlapping of the predominant lunisolar secular and semi-secular resonances furnish a number of interesting disposal hatches at moderate to low eccentricity orbits. All computations were repeated for an increased area-to-mass ratio, simulating the case of a satellite equipped with an on-board, area-augmenting device. We find that this would generally promote the deorbiting process, particularly at the transition region between LEO and MEO. Although direct reentry from very low eccentricities is very unlikely in most cases of interest, we find that a modest “delta-v” (ΔV) budget would be enough for satellites to be steered into a relatively short-lived resonance and achieve reentry into the Earth’s atmosphere within reasonable timescales (∼50 years).

ABOUT A POSSIBLE CONNECTION BETWEEN EARTHQUAKES AND LUNAR-SOLAR GRAVITY VARIATIONS

A possible correlation between the destructive earthquakes of magnitude M = 7 and above and luni-solar gravity variations between 1975 and 2015 has been analyzed. The lunar-solar variations are characterized by three extreme points: the maximum and minimum values of gravity, and the maximum rate of change of variations. At this time, there is an extreme impact of lunar-solar attraction on the earth’s crust and the Earth as a whole. Variations can be a source of irreversible deformation in the earth’s crust. If in this case, there is an additional external impact of space factors, the probability of an earthquake is increased. In a time, the earthquakes are grouped near extremes of lunar-solar variations: half of the events are associated with the maximum gradient of variations change, and the second half is equally confined to the maximum and minimum value of gravity variations. Lunar-solar variations of gravity in conjunction with other cosmic influences can cause earthquakes.

Peter Barlow's insights and contributions to the study of tidal gravity variations and ultra-weak light emissions in plants.

A brief review is given of Peter W. Barlows' contributions to research on gravity tide-related phenomena in plant biology, or 'selenonastic' effects as he called them, including his early research on root growth. Also, new results are presented here from long-term recordings of spontaneous ultra-weak light emission during germination, reinforcing the relationship between local lunisolar tidal acceleration and seedling growth. The main ideas and broad relevance of the work by Barlow and his collaborators about the effects of gravity on plants are reviewed, highlighting the necessity of new models to explain the apparent synchronism between root growth and microscale gravity changes 107 times lower than that exerted by the Earth's gravity. The new results, showing for the first time the germination of coffee beans in sequential tests over 2 months, confirm the co-variation between the patterns in ultra-weak light emission and the lunisolar tidal gravity curves for the initial growth phase. For young sprouts (<1 month old), the rhythm of growth as well as variation in light emission exhibit the once a day and twice a day periodic variations, frequency components that are the hallmark of local lunisolar gravimetric tides. Although present, this pattern is less pronounced in coffee beans older than 1 month. The apparent co-variation between ultra-weak light emission and growth pattern in coffee seedlings and the lunisolar gravity cycles corroborate those previously found in seedlings from other species. It is proposed here that such patterns may attenuate with time for older sprouts with slow development. These data suggest that new models considering both intra- and intercellular interactions are needed to explain the putative sensing and reaction of seedlings to the variations in the gravimetric tide. Here, a possible model is presented based on supracellular matrix interconnections.

FROM ORDER TO CHAOS IN EARTH SATELLITE ORBITS

ABSTRACT We consider Earth satellite orbits in the range of semimajor axes where the perturbing effects of Earth’s oblateness and lunisolar gravity are of comparable order. This range covers the medium-Earth orbits (MEO) of the Global Navigation Satellite Systems and the geosynchronous orbits (GEO) of the communication satellites. We recall a secular and quadrupolar model, based on the Milankovitch vector formulation of perturbation theory, which governs the long-term orbital evolution subject to the predominant gravitational interactions. We study the global dynamics of this two-and-a-half degrees-of-freedom Hamiltonian system by means of the fast Lyapunov indicator (FLI), used in a statistical sense. Specifically, we characterize the degree of chaoticity of the action space using angle-averaged normalized FLI maps, thereby overcoming the angle dependencies of the conventional stability maps. Emphasis is placed upon the phase-space structures near secular resonances, which are of primary importance to the space debris community. We confirm and quantify the transition from order to chaos in MEO, stemming from the critical inclinations and find that highly inclined GEO orbits are particularly unstable. Despite their reputed normality, Earth satellite orbits can possess an extraordinarily rich spectrum of dynamical behaviors and, from a mathematical perspective, have all the complications that make them very interesting candidates for testing the modern tools of chaos theory.

Analysis on the long term orbital evolution of Molniya satellites

Long term evolution of the Molniya satellites are investigated by means of historical data analysis, theoretical analysis and numerical integration. Both the mean motion resonance problem and the critical inclination problem are studied. The period and the amplitude of the semi-major axis for each satellite are obtained analytically and compared with the observational data. In addition, the reason of the observed sudden changes in the center and the amplitude of the oscillating semi-major axes is determined as the effect of the atmosphere drag. For the long period perigee motion, the dominant perturbations come from the luni-solar gravity. A two-degree-of freedom system is established by adding the two periodic terms of the neighbor resonances to the Hamiltonian of the classical single resonance model. In theory, the resulting resonance overlap model is responsible for the chaotic layer between the libration region and the circulation region. In practice, it is applied to explain the quick decay of the earliest Molniya satellites and to study the satellites that still orbiting the Earth at present.

Fourth order expansions of the luni-solar gravity perturbations along rotating axes for trajectory optimization

A fourth order extension of the analytic form of the accelerations due to the luni-solar gravity perturbations along rotating axes is presented. These derivations are carried out in order to increase the accuracy of the dynamic modeling of perturbed optimal low-thrust transfers between general elliptic orbits, and enhance the fidelity of trajectory optimization software used in simulations and mission analyses. A set of rotating axes attached to the thrusting spacecraft is used such that both the thrust and perturbation accelerations due to Earth's geopotential and the luni-solar gravity are mathematically resolved along these axes prior to numerical integration of the actual trajectory. This Gaussian form of the state as well as the adjoint differential equations form a set of equations that are readily integrated and an iterative process is used to achieve convergence to a desired transfer. This analysis further reveals that further extensions to higher orders, say to the fifth order and beyond, are not needed to extract even more accuracy in the solutions because the minimum-time transfer solutions become fully stabilized in the sense that they do not exhibit any differences beyond a fraction of a second, or at most a few seconds even in the more extreme cases of very large orbits with apogee heights around 100,000 km with strong lunar influence.

On prediction of re-entry time of an upper stage from GTO

The evolution of objects in geostationary transfer orbit (GTO) is determined by a complex interplay of atmospheric drag and luni-solar gravity. These orbits are highly eccentric (eccentricity >0.7) and have large variations in velocity and perturbations during a revolution. The periodic changes in the perigee altitudes of these orbits are mainly due to the gravitational perturbations of the Sun and the Moon. The re-entry time of the objects in such orbits is sensitive to the initial conditions. The aim of this paper is to study the re-entry time of the cryogenic stage of the Indian geo-synchronous launch vehicle, GSLV-F04/CS, which has been decaying since 2 September 2007 from initial orbit with eccentricity equal to 0.706. Two parameters, initial eccentricity and ballistic coefficient, are chosen for optimal estimation. It is known that the errors are more in eccentricity for the observations based on two line elements (TLEs). These two parameters are computed with response surface method using a genetic algorithm for the selected eight different zones, based on rough linear variation of the mean apogee altitude during 200 days orbit evolution. The study shows that the GSLV-F04/CS will re-enter between 5 December 2010 and 7 January 2011. The methodology is also applied to study the re-entry of six decayed objects (cryogenic stages of GSLV and Molniya satellites). Good agreement is noticed between the actual and the predicted re-entry times. The absolute percentage error in re-entry prediction time for all the six objects is found to be less than 7%. The present methodology is being adopted at Vikram Sarabhai Space Centre (VSSC) to predict the re-entry time of GSLV-F04/CS.

Analytic expansions of luni-solar gravity perturbations along rotating axes for trajectory optimization—Part 2: The multipliers system and simulations

An efficient self-contained trajectory optimization software is generated by making use of de Pontécoulant's analytic lunar theory removing the need for an outside third body ephemeris program to compute the lunar and solar position vectors at each integration step. The accelerations being further resolved along the rotating Euler–Hill frame after expansion to third order in the spacecraft radial distance, the adjoint differential equations are derived in a direct manner complementing the generation of the dynamic system of equations for full compatibility. Because the variation of parameters equations are cast in terms of the nonsingular equinoctial elements with the perturbation accelerations resolved in their analytic form along the rotating axes, the adjoint equations are also derived in the same manner providing a highly efficient and accurate system of equations for rapid computations in conjunction with Aerospace Corporation's NLP2 nonlinear programming codes to search for the initial values of the multipliers that steer the spacecraft towards its target orbit in minimum time. Numerical simulations show that the solutions obtained by the analysis developed in this paper are essentially identical to the more indirect approach based on the use of inertial accelerations obtained from a separate ephemeris generator and subsequent conversions to the thrust frame and equinoctial system.

Analytic expansions of luni-solar gravity perturbations along rotating axes for trajectory optimization: Part 1: The dynamic system

An analytic form of the accelerations due to the luni-solar perturbations resolved along the rotating Euler-Hill frame is devised by using the expansion method. The addition of higher order terms to the main gravity gradient term linear in the spacecraft radial distance, carried out to the third order, provides a very high level of accuracy in accounting for the gravity perturbations experienced by a vehicle in orbit due to the sun and the moon. The nodal precession as well as the perigee advance of the lunar orbit is taken into account analytically by using the analytic lunar theory of de Pontécoulant. The analytic description of the apparent solar orbit and the motion of the moon remove the need to call an epherneris generator at each integration step during the numerical integration of the spacecraft trajectory, leading to the self-contained software for rapid and efficient optimal trajectory generation through iterations. Equinoctial elements are used to describe the spacecraft state and the luni-solar accelerations are given in terms of the apparent solar and lunar longitudes as well as Eulerian angles of the spacecraft orbit with respect to the inertial ecliptic system. The analysis is useful in optimal low-thrust orbit transfers complementing previous analyses carried out by this author, in which thrust and Earth zonal perturbations such as J 2, J 3 and J 4 in terms of the nonsingular equinoctial elements are included.

Inclination Adjustment Maneuver and Frozen Orbit Keeping of The Advanced Land Observing Satellite (ALOS)

The Advanced Land Observing Satellite Daichi (ALOS) was successfully injected into its mission orbit on March 26, 2006. ALOS' missions require the orbit to be precisely recurrent regarding both the ground track and altitude. Therefore, the ALOS mission orbit is a sun-synchronous sub-recurrent frozen orbit. A sun-synchronous orbit has to be maintained by compensating inclination change caused by lunisolar gravity perturbation. In the ALOS case, inclination adjustment is planned every two and a half years. The inclination adjustment should be done by out-of-plane maneuvers. During the inclination adjustment, it is also required that the ground track and altitude error should be minimized. However, it is difficult because there are some restrictions on the out-of-plane maneuvers; such as necessity of attitude change maneuver, extra in-plane accelerations by the attitude change maneuver. JAXA planned and performed the inclination adjustment which kept the ground track and altitude error as small as possible by estimating extra in-plane accelerations and exploiting rotation of eccentricity vector caused by gravity perturbations. In this paper, we review the ALOS requirements for orbit keeping, and discuss the challenges of ALOS inclination adjustment maneuvers due to the restrictions. Our strategies and the theoretical background are also presented with subsequent evaluation of flight results.

Detection of Long-Period Core Modes by Using the Data from Global Superconducting Gravimeters

Possibility of detection of the long-period core modes has been discussed in this paper by using the long-term gravity observations recorded with 13 superconducting gravimeters (SGs) in a worldwide network of the Global Geodynamics Projects. Identical techniques are used to remove the luni-solar gravity tides, the long-term trends of instrument drift, and the effects of atmospheric pressure from all 13 SG data sets. The power spectral density of each data set, as well as the product spectral density of all the data sets, is evaluated. The background noise levels in various non-tidal bands are estimated and analyzed. The results indicate that the background noise level of the global SG observations is 0.0649nm/s2 in inter-tidal band from 0.047 to 0.075cph, 0.0350nm/s2 in another inter-tidal band from 0.089 to 0.117cph, and 0.0138nm/s2 in the sub-tidal band from 0.172 to 0.333cph. The magnitude thresholds of any global harmonic signals, which may be detected by the global SGs, in these three non-tidal bands are 0.0416, 0.0231 and 0.0098nm/s2, respectively. It implies that the signals, related to the long-period core modes, may be identified in the global SG observations, if they exist. The spectral peaks of the global harmonic signals with periods of 16.55, 15.79, 11.00 and 10.09h are detected in the two inter-tidal bands of the global SG records, and they may come from the long-period oscillations in the fluid outer core of the Earth. In the sub-tidal band, there are no obvious peaks relating to the Slichter triplet claimed by Smylie in 1992. Instead, 8 spectral peaks of global harmonic signatures significantly emerge from the background noise. Comparing their periods with those of the Slichter triplet predicted theoretically, it may be concluded that the Slichter triplet is the possible sources of three peaks of them. Of course, a more certain conclusion cannot be drawn now until further knowledge of the deep interior of the Earth is obtained and more acceptable theoretical technique for the dynamic study of the fluid outer core is developed.

A study of gravity variations caused by polar motion using superconducting gravimeter data from the GGP network

Long-term continuous gravity observations, recorded at five superconducting gravimeter (SG) sta- tions in the Global Geodynamic Project (GGP) net- work, as well as data on orientation variations in the Earth's rotation axis (i.e. polar motion), have been used to investigate the characteristics of gravity variations on the Earth's surface caused by polar motion. All the SG gravity data sets were pre-processed using identical techniques to remove the luni-solar gravity tides, the long-term trends of the instrumental drift, and the effects of atmospheric pressure. The analysis indicates that the spectral peaks, related to the Chandler and annual wobbles, were identified in both the power and product spectral density estimates. The magnitude of gravity variations, as well as the gravimetric amplitude factor associated with the Chandler wobble, changed significantly at different SG stations and during different observation periods. However, when all the SG obser- vations at these five sites were combined, the gravimetric parameters of the Chandler wobble were retrieved accurately: 1.1613 ± 0.0737 for the amplitude factor and )1� .30 ± 1� .33 for the phase difference. The value of the estimated amplitude factor is in agreement with that predicted theoretically for the zonal tides of an elastic Earth model.

Consideration of lifetime limitation for spent stages in GTO

The evolution of objects in geostationary transfer orbit (GTO) is determined by a complex interplay of atmospheric drag and luni-solar gravity. These orbits are characterized by periodic changes in perigee altitudes caused by gravitational perturbations of the Sun and the Moon. The aim of this paper is to investigate the basic physics of the luni-solar perturbations in GTO and their interaction with drag perturbation. A detailed analysis is made to study the influence of launch time on the orbital lifetime. This study reveals that even minute changes in the launch time can result in very large changes in the lifetime. The analysis is applied to the orbital evolution of the orbiting upper stage of the first development flight of the Indian Geo Stationary Satellite Launch Vehicle GSLV-D1, which took place on 18 April 2001. By appropriate selection of the launch time, re-entry in around 600 days is expected.

Earth satellite orbits with resonant lunisolar perturbations - II. Some resonances dependent on the semi-major axis, eccentricity and inclination

Earth satellite orbits resonant with respect to lunisolar gravity and direct solar radiation pressure perturbations are discussed with particular reference to those resonances satisfying commensurability conditions of the following form: ψ 4 = α ώ + γ (ώ p + Ṁ p ) ≈ 0 and ψ 5 = β Ω . + γ (ώ p + Ṁ p ) ≈ 0, where ω is the argument of perigee of the satellite’s orbit, Ω is the longitude of its ascending node, ω p is the argument of perigee of the lunar or solar orbits, and M p is the mean anomaly of the lunar or solar orbits; α, β and γ are integers. Certain simple relations are derived connecting the satellite’s semi-major axis, eccentricity and inclination; they must be satisfied, if the satellite is to exist in the commensurabilities ψ 4 ≈ 0 and ψ 5 ≈ 0. Tables are also given which contain the predominant resonant terms in the lunisolar gravity and direct solar radiation pressure disturbing function expansions for every commensurability of the type ψ 4 ≈ 0 and ψ 5 ≈ 0. Finally some important examples of these resonances are discussed.

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.