Some orbital characteristics of lunar artificial satellites

In this paper we present an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites. We consider the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third-body in elliptical orbit (Earth is considered). Legendre polynomials are expanded in powers of the eccentricity up to the degree four and are used for the disturbing potential due to the third-body. We show a new approximated equation to compute the critical semi-major axis for the orbit of the satellite. Lie-Hori perturbation method up to the second-order is applied to eliminate the terms of short-period of the disturbing potential. Coupling terms are analyzed. Emphasis is given to the case of frozen orbits and critical inclination. Numerical simulations for hypothetical lunar artificial satellites are performed, considering that the perturbations are acting together or one at a time.     

Averaged model to study long-term dynamics of a probe about Mercury

This paper provides a method for finding initial conditions of frozen orbits for a probe around Mercury. Frozen orbits are those whose orbital elements remain constant on average. Thus, at the same point in each orbit, the satellite always passes at the same altitude. This is very interesting for scientific missions that require close inspection of any celestial body. The orbital dynamics of an artificial satellite about Mercury is governed by the potential attraction of the main body. Besides the Keplerian attraction, we consider the inhomogeneities of the potential of the central body. We include secondary terms of Mercury gravity field from \(J_2\) up to \(J_6\), and the tesseral harmonics \(\overline{C}_{22}\) that is of the same magnitude than zonal \(J_2\). In the case of science missions about Mercury, it is also important to consider third-body perturbation (Sun). Circular restricted three body problem can not be applied to Mercury–Sun system due to its non-negligible orbital eccentricity. Besides the harmonics coefficients of Mercury’s gravitational potential, and the Sun gravitational perturbation, our average model also includes Solar acceleration pressure. This simplified model captures the majority of the dynamics of low and high orbits about Mercury. In order to capture the dominant characteristics of the dynamics, short-period terms of the system are removed applying a double-averaging technique. This algorithm is a two-fold process which firstly averages over the period of the satellite, and secondly averages with respect to the period of the third body. This simplified Hamiltonian model is introduced in the Lagrange Planetary equations. Thus, frozen orbits are characterized by a surface depending on three variables: the orbital semimajor axis, eccentricity and inclination. We find frozen orbits for an average altitude of 400 and 1000 km, which are the predicted values for the BepiColombo mission. Finally, the paper delves into the orbital stability of frozen orbits and the temporal evolution of the eccentricity of these orbits.     

Artificial Martian frozen orbits and Sun-Synchronous orbits using continuous low-thrust control

This paper analyses three types of artificial orbits around Mars pushed by continuous low-thrust control: artificial frozen orbits, artificial Sun-Synchronous orbits and artificial Sun-Synchronous frozen orbits. These artificial orbits have similar characteristics to natural frozen orbits and Sun-Synchronous orbits, and their orbital parameters can be selected arbitrarily by using continuous low-thrust control. One control strategy to achieve the artificial frozen orbit is using both the transverse and radial continuous low-thrust control, and another to achieve the artificial Sun-Synchronous orbit is using the normal continuous low-thrust control. These continuous low-thrust control strategies consider J ₂, J ₃, and J ₄ perturbations of Mars. It is proved that both control strategies can minimize characteristic velocity. Relevant formulas are derived, and numerical results are presented. Given the same initial orbital parameters, the control acceleration and characteristic velocity taking into account J ₂, J ₃, and J ₄ perturbations are similar to those taking into account J ₂ perturbations for both Mars and the Earth. The control thrust of the orbit around Mars is smaller than that around the Earth. The magnitude of the control acceleration of ASFOM-4 (named as Artificial Sun-Synchronous Frozen Orbit Method 4) is the lowest among these strategies and the characteristic velocity within one orbital period is only 0.5219 m/s for the artificial Sun-Synchronous frozen orbit around Mars. It is evident that the relationship among the control thrusts and the primary orbital parameters of Martian artificial orbits is always similar to that of the Earth. Simulation shows that the control scheme extends the orbital parameters’ selection range of three types of orbits around Mars, compared with the natural frozen orbit and Sun-Synchronous orbit.     

Lunar frozen orbits revisited

Lunar frozen orbits, characterized by constant orbital elements on average, have been previously found using various dynamical models, incorporating the gravitational field of the Moon and the third-body perturbation exerted by the Earth. The resulting mean orbital elements must be converted to osculating elements to initialize the orbiter position and velocity in the lunar frame. Thus far, however, there has not been an explicit transformation from mean to osculating elements, which includes the zonal harmonic \(J_2\), the sectorial harmonic \(C_{22}\), and the Earth third-body effect. In the current paper, we derive the dynamics of a lunar orbiter under the mentioned perturbations, which are shown to be dominant for the evolution of circumlunar orbits, and use von Zeipel’s method to obtain a transformation between mean and osculating elements. Whereas the dynamics of the mean elements do not include \(C_{22}\), and hence does not affect the equilibria leading to frozen orbits, \(C_{22}\) is present in the mean-to-osculating transformation, hence affecting the initialization of the physical circumlunar orbit. Simulations show that by using the newly-derived transformation, frozen orbits exhibit better behavior in terms of long-term stability about the mean values of eccentricity and argument of periapsis, especially for high orbits.     

High-precision repeat-groundtrack orbit design and maintenance for Earth observation missions

The focus of this paper is the design and station keeping of repeat-groundtrack orbits for Sun-synchronous satellites. A method to compute the semimajor axis of the orbit is presented together with a station-keeping strategy to compensate for the perturbation due to the atmospheric drag. The results show that the nodal period converges gradually with the increase of the order used in the zonal perturbations up to \(J_{15}\). A differential correction algorithm is performed to obtain the nominal semimajor axis of the reference orbit from the inputs of the desired nodal period, eccentricity, inclination and argument of perigee. To keep the satellite in the proximity of the repeat-groundtrack condition, a practical orbit maintenance strategy is proposed in the presence of errors in the orbital measurements and control, as well as in the estimation of the semimajor axis decay rate. The performance of the maintenance strategy is assessed via the Monte Carlo simulation and the validation in a high fidelity model. Numerical simulations substantiate the validity of proposed mean-elements-based orbit maintenance strategy for repeat-groundtrack orbits.     

Sun-synchronous orbits near critical inclination

The long period dynamics of Sun-synchronous orbits near the critical inclination 116.6° are investigated. It is known that, at the critical inclination, the average perigee location is unchanged by Earth oblateness. For certain values of semimajor axis and eccentricity, orbit plane precession caused by Earth oblateness is synchronous with the mean orbital motion of the apparent Sun (a Sun-synchronism). Sun-synchronous orbits have been used extensively in meteorological and remote sensing satellite missions. Gravitational perturbations arising from an aspherical Earth, the Moon, and the Sun cause long period fluctuations in the mean argument of perigee, eccentricity, inclination, and ascending node. Double resonance occurs because slow oscillations in the perigee and Sun-referenced ascending node are coupled through the solar gravity gradient. It is shown that the total number and infinitesimal stability of equilibrium solutions can change abruptly over the Sun-synchronous range of semimajor axis values (1.54 to 1.70 Earth radii). The effect of direct solar radiation pressure upon certain stable equilibria is investigated.     

Balanced earth satellite orbits

An analytic model for third-body perturbations and for the second zonal harmonic of the central body's gravitational field is presented. A simplified version of this model applied to the Earth-Moon-Sun system indicates the existence of high-altitude and highly-inclined orbits with their apsides in the equator plane, for which the apsidal as well as the nodal motion ceases. For special positions of the node, secular changes of eccentricity and inclination disappear too (balanced orbits). For an ascending node at vernal equinox, the inclination of balanced orbits is 94.56°, for a node at autumnal equinox 85.44°, independent of the eccentricity of the orbit. For a node perpendicular to the equinox, there exist circular balanced orbits at 90° inclination. By slightly adjusting the initial inclination as suggested by the simplified model, orbits can be found — calculated by the full model or by different methods — that show only minor variations in eccentricity, inclination, argument of perigee, and longitude of the ascending node for 10⁵ revolutions and more. Orbits near the unstable equilibria at 94.56° and 85.44° inclination show very long periodic librations and oscillations between retrogade and prograde motion.Retired from IBM Vienna Software Development Laboratory.     

On the possible resonant orbits of exoplanets

Within the framework of the restricted three-body problem, the possible orbits of a small-mass exoplanet in the system with a massive exoplanet on an elliptic orbit are investigated. Possible quasi-circular orbits are sought. The dependence of the Kozdai-Lidov effect (the Kozdai resonance) on the eccentricity of the orbit of a massive planet is discussed. The effect of the commensurabilities of the mean motions on the value of the eccentricity perturbations is considered.     

Long-Period Variations of the Eccentricity Vector Valid also for Near Circular Orbits around a Non-Spherical Body

This paper studies the long period variations of the eccentricity vector of the orbit of an artificial satellite, under the influence of the gravity field of a central body. We use modified orbital elements which are non-singular at zero eccentricity. We expand the long periodic part of the corresponding Lagrange equations as power series of the eccentricity. The coefficients characterizing the differential system depend on the zonal coefficients of the geopotential, and on initial semi-major axis, inclination, and eccentricity. The differential equations for the components of the eccentricity vector are then integrated analytically, with a definition of the period of the perigee based on the notion of “free eccentricity”, and which is also valid for circular orbits. The analytical solution is compared to a numerical integration. This study is a generalization of (Cook, Planet. Space Sci., 14, 1966): first, the coefficients involved in the differential equations depend on all zonal coefficients (and not only on the very first ones); second, our method applies to nearly circular orbits as well as to not too eccentric orbits. Except for the critical inclination, our solution is valid for all kinds of long period motions of the perigee, i.e., circulations or librations around an equilibrium point.     

Analysis of the orbit of 1968-90A (Cosmos 248)

The satellite 1968-90A (Cosmos 248), was launched in October 1968 into an orbit inclined at 62.25° to the equator, with an initial perigee height of 475 km, apogee height 543 km, and orbital period 94.8 min. The orbit has been determined at 57 epochs over nearly one and a quarter cycles of the argument of perigee from January 1972 until December 1975 with the aid of the RAE orbit refinement program PROP, using nearly 3000 observations. For most of these orbits the standard deviations in inclination are less than 0.0009° (corresponding to about 100m in cross-track distance). The values of eccentricity give perigee heights accurate to between 30 and 120m.The main purpose of the orbit determination was to provide accurate values of the eccentricity for use in determining the odd zonal harmonics in the Earth's gravitational potential. These values have been analysed to determine the amplitude of the oscillation in eccentricity, which is found to be 0.00433 ± 0.00001.     

Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields: a first approach

This article provides a method for finding initial conditions for perturbed frozen orbits around inhomogeneous fast rotating asteroids. These orbits can be used as reference trajectories in missions that require close inspection of any rigid body. The generalized perturbative procedure followed exploits the analytical methods of relegation of the argument of node and Delaunay normalisation to arbitrary order. These analytical methods are extremely powerful but highly computational. The gravitational potential of the heterogeneous body is firstly stated, in polar-nodal coordinates, which takes into account the coefficients of the spherical harmonics up to an arbitrary order. Through the relegation of the argument of node and the Delaunay normalization, a series of canonical transformations of coordinates is found, which reduces the Hamiltonian describing the system to a integrable, two degrees of freedom Hamiltonian plus a truncated reminder of higher order. Setting eccentricity, argument of pericenter and inclination of the orbit of the truncated system to be constant, initial conditions are found, which evolve into frozen orbits for the truncated system. Using the same initial conditions yields perturbed frozen orbits for the full system, whose perturbation decreases with the consideration of arbitrary homologic equations in the relegation and normalization procedures. Such procedure can be automated for the first homologic equation up to the consideration of any arbitrary number of spherical harmonics coefficients. The project has been developed in collaboration with the European Space Agency (ESA).     

J2 Invariant Relative Orbits for Spacecraft Formations

An analytic method is presented to establish J ₂ invariant relative orbits. Working with mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the J₂ influence. Two first order conditions are established between the differences in momenta elements (semi-major axis, eccentricity and inclination angle) that guarantee that the drift rates of two neighboring orbits are equal on the average. Differences in the longitude of the ascending node, argument of perigee and initial mean anomaly can be set at will, as long as they are setup in mean element space. For near polar orbits, enforcing both momenta element constraints may result in impractically large relative orbits. It this case it is shown that dropping the equal ascending node rate requirement still avoids considerable relative orbit drift and provides substantial fuel savings.This revised version was published online in October 2005 with corrections to the Cover Date.     

Asymptotic solution for the two-body problem with constant tangential thrust acceleration

An analytical solution of the two body problem perturbed by a constant tangential acceleration is derived with the aid of perturbation theory. The solution, which is valid for circular and elliptic orbits with generic eccentricity, describes the instantaneous time variation of all orbital elements. A comparison with high-accuracy numerical results shows that the analytical method can be effectively applied to multiple-revolution low-thrust orbit transfer around planets and in interplanetary space with negligible error.     

Analytical direct planetary perturbations on the orbital motion of satellites

An analytical expansion of the disturbing function arising from direct planetary perturbations on the motion of satellites is derived. As a Fourier series, it allows the investigation of the secular effects of these direct perturbations, as well as of every argument present in the perturbation. In particular, we construct an analytical model describing the evection resonance between the longitude of pericenter of the satellite orbit and the longitude of a planet, and study briefly its dynamic. The expansion developed in this paper is valid in the case of planar and circular planetary orbits, but not limited in eccentricity or inclination of the satellite orbit.     

Orbits near critical inclination,including lunisolar perturbations

An improved theory is presented of long period perigee motion for orbits near the critical inclinations 63.4° and 116.6°. Inclusion of lunisolar perturbations andall measured zonal harmonic coefficients from a recent Earth model are significant improvements over existing theories. Phase portraits are used to depict the interaction between eccentricity magnitude and argument of perigee. The Hamiltonian constant can be chosen as the parameter to display a family of phase plane trajectories consisting of libration, circulation, and asymptotic motion along separatrices near equilibrium points. A two parameter family of phase portraits is defined by the other two integrals, the average semimajor axis and component of angular momentum resolved along the Earth's polar axis. There are regions of the parameter space where the stability and total number of equilibria can change, or two separatrices can coalesce. These phenomena signal large qualitative changes in phase portrait topology. Numerical studies show that lunisolar perturbations control stability of equilibria for orbits with semimajor axes exceeding 1.4 Earth radii. Moreover, a theory which includes lunisolar perturbations predicts larger maximum fluctuations in eccentricity and faster oscillations near stable equilibria compared to a theory which models only the zonal harmonics.     

The gravitational perturbation spectrum in linear satellite theory

A new method for calculating the perturbation spectrum in the framework of Kaula's linear satellite theory (LST) is introduced. The novelty of this approach consists in using recent results on the spectral decomposition of the perturbation frequencies in LST to provide a closed formulation for the amplitude and the phase of each line in the perturbation spectrum. The theory presented here can be applied to perturbations in the elements or in the radial and transverse directions due to the geopotential or to the tides. Separate algorithms are developed for application to orbits with circulating or frozen perigee.     

Analysis of the orbit of cosmos 387 (1970-111A) near 15th-order resonance

Cosmos 387 (1970-111A) was launched on 16 December 1970 into a near-circular orbit with an average height of 540 km and an inclination of 74.0°. On 5 November 1971 the orbit, in its slow contraction under the influence of air drag, passed through 15th-order resonance, when the ground track repeats after 15 revolutions. The orbit has been determined with the aid of the RAE orbit refinement program PROP at 19 epochs between May 1971 and June 1972, using 1500 optical and radar observations. The average accuracy is about 70 m in perigee height and 0.001° in inclination.The variation of orbital inclination while the satellite was experiencing 15th-order resonance, as given by these 19 orbits and 55 U.S. Navy orbits, has been analysed to obtain equations accurate to 4 per cent for the geopotential coefficients of order 15 and odd degree (15, 17, 19 …). These equations have subsequently been used (with others) in determining individual coefficients of order 15 and odd degree.The variation of eccentricity with argument of perigee showed unexpected complexity, including a tight loop near resonance (Fig. 4). Analysis of the variation in eccentricity has yielded, for the first time, accurate equations for the geopotential coefficients of order 15 and even degree (16, 18 …), thus opening the way to the evaluation of individual coefficients of this type. The variations in the argument of perigee and right ascension of the node have also been analysed.     

Lunar perturbations of artificial satellites of the earth

Lunisolar perturbations of an artificial satellite for general terms of the disturbing function were derived by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. As Kozai (1966, 1973) noted, both inclination and node of the Moon's orbit with respect to the equator of the Earth are not simple functions of time, while the same elements with respect to the ecliptic are well approximated by a constant and a linear function of time, respectively. In the present work, we obtain the disturbing function for the Lunar perturbations using ecliptic elements for the Moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The Moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the Earth. We can then apply the results to numerical integration by using coordinates of the Moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the Moon's motion represented by a precessing and rotating elliptical orbit.     

A dynamical investigation of the conjecture that Mercury is an escaped satellite of Venus

The possibility that Mercury might once have been satellite of a Venus, suggested by a number of anomalies, is investigated by a series of numerical computer experiments. Tidal interaction between Mercury and Venus would result in the escape of Mercury into a solar orbit. Only two escape orbits are possible, one exterior and one interior to the Venus orbit. For the interior orbit, subsequent encounters are sufficiently distant to avoid recapture or large perturbations. The perihelion distance of Mercury tends to decrease, while the orientation of perihelion librates for the first few thousand revolutions. If dynamical evolution or nonconservative forces were large enough in the early solar system, the present semimajor axes could have resulted. The theoretical minimum quadrupole moment of the inclined rotating Sun would rotate the orbital planes out of coplanarity. Secular perturbations by the other planets would evolve the eccentricity and inclination of Mercury's orbit through a range of possible configurations, including the present orbit. Thus the conjecture that Mercury is an escaped satellite of Venus remains viable, and is rendered more attractive by our failure to disprove it dynamically.     

A vectorial approach to determine frozen orbital conditions

Taking into consideration a probe moving in an elliptical orbit around a celestial body, the possibility of determining conditions which lead to constant values on average of all the orbit elements has been investigated here, considering the influence of the planetary oblateness and the long-term effects deriving from the attraction of several perturbing bodies. To this end, three equations describing the variation of orbit eccentricity, apsidal line and angular momentum unit vector have been first retrieved, starting from a vectorial expression of the Lagrange planetary equations and considering for the third-body perturbation the gravity-gradient approximation, and then exploited to demonstrate the feasibility of achieving the above-mentioned goal. The study has led to the determination of two families of solutions at constant mean orbit elements, both characterised by a co-planarity condition between the eccentricity vector, the angular momentum and a vector resulting from the combination of the orbital poles of the perturbing bodies. As a practical case, the problem of a probe orbiting the Moon has been faced, taking into account the temporal evolution of the perturbing poles of the Sun and Earth, and frozen solutions at argument of pericentre 0\(^{\circ }\) or 180\(^{\circ }\) have been found.