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.     

Geometric Derivation of the Delaunay Variables and Geometric Phases

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T³ on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S ¹ action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J ₂-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J ₂-Hamiltonian is a collective Hamiltonian of the T³ momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J ₂ effect as geometric phases.     

The Earth-figure perturbations in the Lunar theory

We compute the perturbations on the motion of the Moon due to the shape of the Earth. The zonal terms inJ ₂,J ₃, andJ ₄ are considered. The accuracy is estimated at 3×10^–5 and the results compared with previous theories.     

Relativistic positioning: errors due to uncertainties in the satellite world lines

Global navigation satellite systems use appropriate satellite constellations to get the coordinates of an user—close to Earth—in an almost inertial reference system. We have simulated both GPS and GALILEO constellations. Uncertainties in the satellite world lines lead to dominant positioning errors. In this paper, a detailed analysis of these errors is developed inside a great region surrounding Earth. This analysis is performed in the framework of the so-called relativistic positioning systems. Our study is based on the Jacobian (J) of the transformation giving the emission coordinates in terms of the inertial ones. Around points of vanishing J, positioning errors are too large. We show that, for any 4-tuple of satellites, the points with J=0 are located at distances, D, from the Earth centre greater than about 2R/3, where R is the radius of the satellite orbits which are assumed to be circumferences. Our results strongly suggest that, for D-distances greater than 2R/3 and smaller than 10⁵ km, a rather good positioning may be achieved by using appropriate satellite 4-tuples without J=0 points located in the user vicinity. The way to find these 4-tuples is discussed for arbitrary users with D<10⁵ km and, then, preliminary considerations about satellite navigation at D<10⁵ km are presented. Future work on the subject of space navigation—based on appropriate simulations—is in progress.     

Construction d'orbites periodiques de satellites dans un systeme d'axes tournants,I

Résumé On développe une méthode de construction d'orbites périoldiques dans un système d'axes tournants, pour un satellite gravitant autour d'un sphéroide. Les orbites sont quasi circulaires,i est l'inclinaison sur le plan équatorial de la planète. Pour les petites inclinaisons, la solution est donnée jusqu'aux termes enJ ₂ ² etJ ₄.Ce modèle peut être appliqué aux satellites de Saturne. Des valeurs observées des longitudes des noeuds ascendants de Mimas et Téthys, on donne une estimation des valeurs deJ ₂ etJ ₄ du potentiel de Saturne. La valeur deJ ₂ est très sensible aux valeurs adoptées pour le rayon équatorial de la planète.

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Construction of periodic orbits of satellites in a moving system of axes, I

We give an algorithm for the construction of periodic orbits in a rotating frame for the cases of satellites moving around an oblate planet.The orbits are near to the circular case; the asymptotic developments of the periodic solutions are completely calculated for the termsJ ₂ andJ ₄ of the potential. The solutions for small inclinations are given up toJ ₂ ² .The families of solutions depend on three parameters: the semi-major axis, the inclination of the generating orbit and the initial position on this orbit.These solutions can be applied to the motion of the Saturnian satellites. From the observed longitudes of the ascending nodes of Mimas and Tethys, we estimate the valuesJ ₂ andJ ₄ of the Saturnian potential, the value ofJ ₂ very strongly depends on the adopted value of the planet's equatorial diameter.

     

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.     

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.     

Impact of the Quadrupole Moment of the Sun on the Dynamics of the Earth-Moon System

Range of values of the Sun's mass quadrupole moment of coefficient J₂ arising both from experimental and theoretical determinations enlarge across literature on two orders of magnitude, from around 10^-7 until to 10^-5. The accurate knowledge of the Moon's physical librations, for which the Lunar Laser Ranging data reach an outstanding precision level, prove to be appropriate to reduce the interval of J₂ values by giving an upper bound of J₂. A solar quadrupole moment as high as 1.1 10^-5 given either from the upper bounds of the error bars of the observations, or from the Roche's theory, is not compatible with the knowledge of the lunar librations accurately modeled and observed with the LLR experiment. The suitable values of J₂ have to be smaller than 3.0 10^-6. As a consequence, this upper bound of 3.0 10^-6 is accepted to study the impact of the Sun's quadrupole moment of mass on the dynamics of the Earth-Moon system. Such as effect (with J₂ = 5.5±1.3 × 10^-6) has been already tested in 1983 by Campbell & Moffat using analytical approximate equations, and thus for the orbits of Mercury, Venus, the Earth and Icarus. The approximate equations are no longer sufficient compared with present observational data and exact equations are required. As if to compute the effect on the lunar librations, we have used our BJV relativistic model of solar system integration including the spin-orbit coupled motion of the Moon. The model is solved by numerical integration. The BJV model stems from general relativity by using the DSX formalism for purposes of celestial mechanics when it is about to deal with a system of n extended, weakly self-gravitating, rotating and deformable bodies in mutual interactions. The resulting effects on the orbital elements of the Earth have been computed and plotted over 160 and 1600 years. The impact of the quadrupole moment of the Sun on the Earth's orbital motion is mainly characterized by variations of , , and . As a consequence, the Sun's quadrupole moment of mass could play a sensible role over long time periods of integration of solar system models. This revised version was published online in July 2006 with corrections to the Cover Date.     

Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations

Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J ₂ is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J ₂-perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J _2n , n ≥ 1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential.     

On the existence of J 2 invariant relative orbits from the dynamical system point of view

The present paper addresses the existence of J ₂ invariant relative orbits with arbitrary relative magnitude over the infinite time using the Routh reduction and Poincaré techniques in the J ₂ Hamiltonian problem. The current research also proposes a novel numerical searching approach for J ₂ invariant relative orbits from the dynamical system point of view. A new type of Poincaré mapping is defined from different central manifolds of the pseudo-circular orbits (parameterized by the Jacobi energy E, the polar component of momentum H _z and the measure of distance Δr between the fixed point and its central manifolds) to the nodal periods T _d and the drifts of longitude of the ascending node during one period (ΔΩ), which differs from Koon et al.’s (AIAA 2001) definition on central manifolds parameterized by the same fixed point. The Poincaré mapping is surjective because it compresses the three-dimensional variables into two-dimensional images, and the mapping degenerates into a bijective mapping in consideration of the fixed points. An iteration algorithm to the degenerated bijective mapping is proposed from the continuation procedure to perform the ergodic representation of E- and H _z -contour maps on the space of T _d –ΔΩ. For the surjective mapping with Δr ≠ 0, different pseudo-circular or elliptical orbits may share the same images. Hence, the inverse surjective mapping may achieve non-unique variables from a single image, which makes the generation of J ₂ invariant relative orbits possible. The pseudo-circular or elliptical orbits generated from the surjective mapping will be defined in different meridian planes. Hence, the critical contribution of the present paper is the assignment of J ₂ invariant relative orbits to different invariant parameters E and H _z depending on the E- and H _z -contour map, which will hold J ₂ invariant relative orbits for extended durations. To investigate the high-order nonlinearity neglected by previous studies, a formation configuration with a large magnitude of 500 km is successfully generated from the theory developed in the present work, which is beyond the scope of the linear conditions of J ₂ invariant relative orbits. Therefore, the existence of J ₂ invariant relative orbit with an arbitrary relative magnitude over the infinite time is achieved from the dynamical system point of view.     

Prebiotic organic synthesis in early Earth and Mars atmospheres: Laboratory experiments with quantitative determination of products formed in a cold plasma flow reactor

The goal of this study was to explore prebiotic chemistry in a range of plausible early Earth and Mars atmospheres. To achieve this laboratory continuous flow plasma irradiation experiments were performed on N₂/H₂/CO/CO₂ gas mixtures chosen to represent mildly reducing early Earth and Mars atmospheres derived from a secondary volcanic outgassing of volatiles in chemical equilibrium with magmas near present day oxidation state. Under mildly reducing conditions (91.79% N₂, 5.89% H₂, 2.21% CO, and 0.11% CO₂), simple nitriles are produced in the gas phase with yield (G in molecules per 100 eV), for the key prebiotic marker molecule HCN at G∼1×¹⁰⁻³ (0.1 nmol J⁻¹). In this atmosphere localized HCN concentrations possibly could approach the 10⁻² M needed for HCN oligomerization. Yields under mildly oxidizing conditions (45.5% N₂, 0.1% H₂, 27.2% CO, 27.2% CO₂) are significantly less as expected, with HCN at G∼3×¹⁰⁻⁵ (). Yields in a Triton atmosphere which can be plausibly extrapolated to represent what might be produced in trace CH₄ conditions (99.9% N₂, 0.1% CH₄) are significant with HCN at G∼1×¹⁰⁻² (1 nmol J⁻¹) and tholins produced. Recently higher methane abundance atmospheres have been examined for their greenhouse warming potential, and higher abundance hydrogen atmospheres have been proposed based on a low early Earth exosphere temperature. A reducing (64.04% N₂, 28.8% H₂, 3.60% CO₂, and 3.56% CH₄), representing a high CH₄ and H₂ abundance early Earth atmosphere had HCN yields of G∼5×¹⁰⁻³ (0.5 nmol J⁻¹). Tholins generated in high methane hydrogen gas mixtures is much less than in a similar mixture without hydrogen. The same mixture with the oxidizing component CO₂ removed (66.43% N₂, 29.88% H₂, 0% CO₂, and 3.69% CH₄) had HCN yields of G∼1×¹⁰⁻³ (0.1 nmol J⁻¹) but more significant tholin yields.     

Effect of 3rd-degree gravity harmonics and Earth perturbations on lunar artificial satellite orbits

In a previous work we studied the effects of (I) the J ₂ and C ₂₂ terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J ₂ + J ₃’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.     

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.     

Estimation of low degree geopotential coefficients using SLR data

Geodetic satellites have been providing the low frequency part of the geopotential models used for precise orbit determination purposes (e.g. JGM3, EGM96, …). Nevertheless they can be used to estimate the temporal variation of selected coefficients, helping to clarify the complex interrelations in the earth-ocean-atmosphere system. In this paper we present the two years long analysis of SLR data from the seven available geodetic satellites (Lageos I–II, Stella, Starlette, Ajisai, Etalon I–II) to recover monthly estimates of low degree geopotential coefficients; the results are obtained analysing the satellites separately and in proper combination. An accurate modelling of the satellite orbits is required in order to separate the geopotential coefficients: we assume as a priori geopotential the JGM3 model together with its associated tides and we take care of non-gravitational effects on the satellites by means of proper empirical estimated accelerations. The time series of the estimated coefficients (J₂, J₃, J₄, J₅) are inspected to detect the sub-annual perturbations related to seasonal variation of mass distribution. Huge residual seasonal signals in the orbit of Stella indicate a strong model deficiency related to the Sun's influence on the environment. The remaining six satellites are homogeneously modelled and build up a three cycles per year oscillation on J₂ and a seasonal oscillation (1 year and six month periods) revealed on the J₄. The origin and possible causes of these signals are further discussed in the text. We also present a preliminary estimate, using twelve years of Lageos-I and Lageos-II observations, that is compared with previous obtained values.     

A New Pumping Mechanism for A Series of Class II Methanol Masers in the J 0 - J -1 E Transitions

A new pumping mechanism – methanol masers without population inversion is presented in this paper. It can be used to explain the formation of a series of J ₀ → J _-1 E methanol masers, while the 2₁ → 3₀ A ⁺ methanol masers are regarded as a driving coherent micrwave field. In the new mechanism, the intensities of J ₀ - J _-1 E methanol masers are increased with the decreasing transition frequencies (or with rotational number J, approximately). These results agree with Slysh et al. (1995) and Slysh et al. (1999) J ≤ 5 observations for G3345.01+1.79 and W48, in which both J ₀ → J _-1 E and 2₁ → 3₀ A ⁺ methanol masers are detected coincidentally. Other astronomical conditions, such as magnetic field, 2₁ → 3₀ A ⁺ coherent radition, incoherent pumping rate by thermal radition and so on are also discussed. The new mechanism can operate as a complement to other ordinary maser pumping mechanisms for some class II methonal maser sources. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

g Dependent particle concentration due to sedimentation

Sedimentation of particles in a fluid has long been used to characterize particle size distribution. Stokes’ law is used to determine an unknown distribution of spherical particle sizes by measuring the time required for the particles to settle a known distance in a fluid of known viscosity and density. In this paper, we study the effects of gravity on sedimentation by examining the resulting particle concentration distributed in an equilibrium profile of concentration C _m,n above the bottom of a container. This is for an experiment on the surface of the Earth and therefore the acceleration of gravity had been corrected for the oblateness of the Earth and its rotation. Next, at the orbital altitude of the spacecraft in orbit around Earth the acceleration due to the central field is corrected for the oblateness of the Earth. Our results show that for experiments taking place in circular or elliptical orbits of various inclinations around the Earth the concentration ratio C _m,n /C _m,ave , the inclination seems to be the most ineffective in affecting the concentration among all the orbital elements. For orbital experiment that use particles of diameter d _p =0.001 μm the concentration ratios for circular and slightly elliptical orbits in the range e=0–0.1 exhibit a 0.009 % difference. The concentration ratio increases with the increase of eccentricity, which increases more for particles of larger diameters. Finally, for particles of the same diameter concentration ratios between Earth and Mars surface experiments are related in the following way .     

Artificial halo orbits for low-thrust propulsion spacecraft

We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L ₁ and L ₂ in the circular restricted three body problem, where the third body is a low-thrust propulsion spacecraft in the Sun–Earth system. Although such halo orbits about artificial equilibrium points can be generated using a solar sail, there are points inside L ₁ and beyond L ₂ where a solar sail cannot be placed, so low-thrust, such as solar electric propulsion, is the only option to generate artificial halo orbits around points inaccessible to a solar sail. Analytical and numerical halo orbits for such low-thrust propulsion systems are obtained by using the Lindstedt Poincaré and differential corrector method respectively. Both the period and minimum amplitude of halo orbits about artificial equilibrium points inside L ₁ decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L ₂ in contrast show an increase in period with an increase in low-thrust acceleration. However, the minimum amplitude first increases and then decreases after the thrust acceleration exceeds 0.415 mm/s². Using a continuation method, we also find stable artificial halo orbits which can be sustained for long integration times and require a reasonably small low-thrust acceleration 0.0593 mm/s².     

The masers of E-type methanol in orion KL and SGR B2

Using a simplified model the statistical equilibrium and radiative transfer equations of E-type-CH₃OH are solved for Orion KL and SgrB2. According to our calculation results and the observation data taken by Matsakiset al. (1980) and Morimotoet al. (1985a, b), the physical conditions of both sources are estimated. In theJ ₂-J ₁ E methanol maser region of Orion KL, the density, kinetic temperature, dust temperature, and the fractional abundance are 0.8–2×10⁶ cm^–3, 150, 30–90 K, 0.8–8×10^–6. In the 4_–1-3₀ E and 5_–1-4₀ E methanol maser region of Sgr B2 the correspondance physical conditions above are 10⁴ cm³, 45, 23 K, and 7×10^–7, respectively.     

Meteorites from the asteroid 6 Hebe

We have numerically integrated the orbits of 18 fictitious fragments ejected from the asteroid 6 Hebe, an S-type object about 200km across which is located very close to theg=g ₆ (orv ₆) secular resonance at a semimajor axis of 2.425AU and a (proper) inclination of 15° .0. A realistic ejection velocity distribution, with most fragments escaping at relative speeds of a few hundredsm/s, has been assumed. In four cases we have found that the resonance pumps up the orbital eccentricity of the fragments to values >0.6, which result into Earth-crossing, within a time span of 1Myr; subsequent close encounters with the Earth cause strongly chaotic orbital evolution. The closest Earth and Mars encounters recorded in our integration occur at miss distances of a few thousandths ofAU, implying collision lifetimes <10⁹ yr. Some other fragments affected by the secular resonance become Mars-crossers but not Earth-crossers over the integration time span. Two bodies are injected into the 3 : 1 mean motion resonance with Jupiter, and also display macroscopically chaotic behaviour leading to Earth-crossing. 6 Hebe is the first asteroid for which a realistic collisional/dynamical evolutionroute to generate meteorites has been fully demonstrated. It may be the parent body of one of the ordinary chondrite classes.