Regional positioning using a low Earth orbit satellite constellation

Global and regional satellite navigation systems are constellations orbiting the Earth and transmitting radio signals for determining position and velocity of users around the globe. The state-of-the-art navigation satellite systems are located in medium Earth orbits and geosynchronous Earth orbits and are characterized by high launching, building and maintenance costs. For applications that require only regional coverage, the continuous and global coverage that existing systems provide may be unnecessary. Thus, a nano-satellites-based regional navigation satellite system in Low Earth Orbit (LEO), with significantly reduced launching, building and maintenance costs, can be considered. Thus, this paper is aimed at developing a LEO constellation optimization and design method, using genetic algorithms and gradient-based optimization. The preliminary results of this study include 268 LEO constellations, aimed at regional navigation in an approximately 1000 km \(\times \) 1000 km area centered at the geographic coordinates [30, 30] degrees. The constellations performance is examined using simulations, and the figures of merit include total coverage time, revisit time, and geometric dilution of precision (GDOP) percentiles. The GDOP is a quantity that determines the positioning solution accuracy and solely depends on the spatial geometry of the satellites. Whereas the optimization method takes into account only the Earth’s second zonal harmonic coefficient, the simulations include the Earth’s gravitational field with zonal and tesseral harmonics up to degree 10 and order 10, Solar radiation pressure, drag, and the lunisolar gravitational perturbation.     

Relativistic positioning: four-dimensional numerical approach in Minkowski space-time

We simulate the satellite constellations of two Global Navigation Satellite Systems: Galileo (EU) and GPS (USA). Satellite motions are described in the Schwarzschild space-time produced by an idealized spherically symmetric non rotating Earth. The trajectories are then circumferences centered at the same point as Earth. Photon motions are described in Minkowski space-time, where there is a well known relation, (Coll et al. in Class. Quantum Gravit. 27:065013, 2010a), between the emission and inertial coordinates of any event. Here, this relation is implemented in a numerical code, which is tested and applied. The first application is a detailed numerical four-dimensional analysis of the so-called emission coordinate region and co-region. In a second application, a GPS (Galileo) satellite is considered as the receiver and its emission coordinates are given by four Galileo (GPS) satellites. The bifurcation problem (double localization) in the positioning of the receiver satellite is then pointed out and discussed in detail.     

Computer-developed construction of analytic expressions for the coordinates and partial derivatives of Jupiter's Galilean satellites

In his effort to develop series expressions for the coordinates of the Galilean satellites accurate to one are second (Jovicentric), R. A. Sampson was forceda priori to adopt certain numerical values for several constants imbedded in his theory. His final numerical values for the series expressions are not amenable to adjustment of the constants of integration nor of physical constants which affect the motion of the satellites. A method which utilizes computer-based algebraic manipulation software has been developed to reconstruct Sampson's theory, to remove existing errors, to introduce neglected effects and to provide analytical expressions for the coordinates as well as for the partial derivatives with respect to orbital parameters, Jupiter and satellite masses, Jupiter's oblateness (J ₂,J ₄) and Jupiter's pole and period of rotation. The computer-based manipulations enable one to perform, for example, the approximately 10⁸ multiplications required in calculating some perturbations (and their partial derivatives) of Satellite II by Satellite III with ease, and provide algebraic expressions which can readily be adjusted to generate theories corresponding to revised constants of integration and physical parameters.     

Resonance in a geo-centric synchronous satellite under the gravitational forces of the Sun,the Moon and the Earth including it’s equatorial ellipticity

Resonances in a geo-centric synchronous satellite under the gravitational forces of the Sun, the Moon and the Earth including it’s equatorial ellipticity have been investigated. The resonance at two points resulting from the commensurability between the mean motion of the satellite and Γ (angle measured from the minor axis of the Earth’s equatorial ellipse to the projection of the satellite on the plane of the equator) is analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure of Brown and Shook. We have observed that the amplitude and the time period of the oscillation decrease as Γ increases in the first quadrant. The radial deviation (Δr) and the tangential deviation (r _c Δθ) have been determined. Here r _c represents the synchronous altitude. The effects of the arithmetic sum of amplitudes λ _i involved in the perturbation equations on orbital inclination 0^°≤α ₀≤90^° are shown. It is observed that $\sum_{i = 1}^{46} \lambda_{i}$ increases as α ₀ increases. We have also determined the displacement ΔD (called drift) due to the oscillatory terms under the summation sign involved in the equations of motion of the satellite. We have observed that the value of ΔD is less than 0.5^°.     

The Model of Radio Two-way Time Comparison between Satellite and Station and Experimental Analysis

Time synchronization between satellite and station is the key technique of satellite navigation system and the foundation of realization of satellite navigation and positioning. Aiming at solving the problems of time synchronization, we have discussed a new method of radio two-way time comparison between satellite and station, deduced in detail the reduction model of up- and down-link pseudo ranges between satellite and station, and provided a practical calculation model of clock error between satellite and station. By calculating the differences between up- and down-link pseudo ranges, this method has eliminated the influences of common errors, such as the tropospheric delay, satellite ephemeris errors, ground station coordinates errors and so on. The ionospheric delay relevant to signal frequency is also weakened largely, thus this improves the accuracy of time comparison greatly. Finally, experimental analysis is conducted by using observational data, and the results show that the accuracy of radio two-way time comparison between satellite and station can attain about 0.34 ns, which validates the correctness of theoretical method and model.     

Longitudinal variation of equatorial electrojet parameters derived from POGO satellite observations

A method of analysis has been developed making it possible to obtain electrojet parameters from vertical profiles of the POGO electrojet data. This has provided an unprecedented quantity of electrojet parameters—about 500 each of its peak eastward current intensity J₀, its total eastward current I₊, and its half width w—spanning through 360° longitude round the Earth, 5 h of daytime and September equinoctial months of the years 1967, 1968 and 1969. The daytime all-longitude averages of the parameters for the 3 years are 235 ± 14 km for w, 232 ± 63A km^?1 for J₀ and (54 ± 11) × 10³A for I₊. This first coverage round the globe shows that J₀ and I₊ vary considerably with longitude unlike w. While confirming the expected maximum of J₀ and I₊ at about 280°E longitude, we find that each of them also has an unexpected major maximum at about 100°E and a subsidiary maximum at 190°E longitude. The major maxima are found to be prominent only at hours near local noon. It is argued that the surprising longitudinal variations of J₀ and I₊ cannot be explained satisfactorily by variations in the upper mantle conductivity along the dip equator but possibly by longitudinal variations of wind systems and gradients at electrojet altitudes.     

Impact cratering and break up of the small bodies of the Solar System

We consider the largest impact craters observed on small satellites and asteroids and the impact disruption of such bodies. Observational data are considered from 21 impact-like structures on 13 satellites and 8 asteroids (target body radii in the range 0.7-265 km). If the radius of the target body is R and the diameter of the largest crater observed on this body D, the ratio D/R is then the main observational parameter of interest. This is found on the observed bodies and compared to data obtained in the laboratory. Taking the largest observed value for D/R as a proxy for the ratio Dc/R (where Dc is the diameter of the largest crater that can be formed on a body without shattering it) it was found that for the observed icy satellites Dc,icy≈1.2R and for the asteroids and the rocky satellites Dc,rocky≈1.6R. In laboratory experiments with ice targets at impactor speeds of 1 to 3 km s⁻¹ we obtained Dc,icy≈1.64R.     

CEMP星HE 1005-1439中子俘获元素天体物理来源的研究

HE1005-1439是一颗金属丰度极低([Fe/H]～-3.0)的碳增丰贫金属星(Carbon Enhanced Metal-Poor,CEMP),该星的s-过程元素显著超丰([Ba/Fe]=1.16±0.31,[Pb/Fe]=1.98±0.19),而r-过程元素温和超丰([Eu/Fe]=0.46±0.22),使用单一的s-过程模型和i-过程模型均不能拟合该星中子俘获丰度分布.采用丰度分解的方法探究该星化学元素的天体物理来源可有助于理解CEMP星的形成和化学演化.利用s-过程和r-过程的混合模型对其中子俘获元素的丰度分布进行拟合,发现该星的中子俘获元素主要来源于低质量低金属丰度AGB伴星的s-过程核合成,而r-过程核合成也有贡献.     

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.     

Satellite orbital precessions caused by the first odd zonal J 3 multipole of a non-spherical body arbitrarily oriented in space

An astronomical body of mass M and radius R which is non-spherically symmetric generates a free space potential U which can be expanded in multipoles. As such, the trajectory of a test particle orbiting it is not a Keplerian ellipse fixed in the inertial space. The zonal harmonic coefficients J ₂,J ₃,… of the multipolar expansion of the potential cause cumulative orbital perturbations which can be either harmonic or secular over time scales larger than the unperturbed Keplerian orbital period T. Here, I calculate the averaged rates of change of the osculating Keplerian orbital elements due to the odd zonal harmonic J ₃ by assuming an arbitrary orientation of the body’s spin axis \(\hat{\boldsymbol{k}}\) . I use the Lagrange planetary equations, and I make a first-order calculation in J ₃. I do not make a-priori assumptions concerning the eccentricity e and the inclination i of the satellite’s orbit.     

基于不同轨道高度iHCO通信卫星的CAPS星座优化研究

基于通信卫星的导航系统可以利用比地球静止轨道（Geostationary Earth Orbit,GEO）高约200 km的倾斜高圆轨道（inclined Highly Circular Orbit,iHCO）通信卫星组成导航星座.结合两种轨道高度的倾斜高圆轨道通信卫星,仿真分析了利用倾斜高圆轨道卫星组成的中国区域定位系统（Chinese Area Positioning System,CAPS）的导航性能,并讨论了利用倾斜高圆轨道卫星组成的中国区域定位系统实现中国区域覆盖的最佳星座布局.     

High resolution spectral analyses of the Jovian decametric radiation. I. Burst morphology and drift rates

Morphological analyses of high resolution spectral recordings of Jovian decametric radiation show a regime of phenomena not seen at lower resolutions. Observed emissions range from narrowband (50 kHz) simple quasiperiodic bursts to wideband emissions (extending over a 500 kHz passband) exhibiting complex structural detail. Assuming gyroemission from electrons in a dipole field for which the magnetic moment is 10 Gauss R_J³, drift rate measurements of the bursts indicate that the source size is of the order of 600 km and its location is near R = 1.3 R_J at a colatitude of 27.3°. The measurements suggest that the emitting electrons belong to a population having a very specific equatorial pitch angle near 3.5°. This study concludes that it may be possible to verify gyroemission as the mechanism responsible for the decametric radiation.     

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.     

Propagation Characteristics of CMEs Associated with Magnetic Clouds and Ejecta

We have investigated the characteristics of magnetic cloud (MC) and ejecta (EJ) associated coronal mass ejections (CMEs) based on the assumption that all CMEs have a flux rope structure. For this, we used 54 CMEs and their interplanetary counterparts (interplanetary CMEs: ICMEs) that constitute the list of events used by the NASA/LWS Coordinated Data Analysis Workshop (CDAW) on CME flux ropes. We considered the location, angular width, and speed as well as the direction parameter, D. The direction parameter quantifies the degree of asymmetry of the CME shape in coronagraph images, and shows how closely the CME propagation is directed to Earth. For the 54 CDAW events, we found the following properties of the CMEs: i) the average value of D for the 23 MCs (0.62) is larger than that for the 31 EJs (0.49), which indicates that the MC-associated CMEs propagate more directly toward the Earth than the EJ-associated CMEs; ii) comparison between the direction parameter and the source location shows that the majority of the MC-associated CMEs are ejected along the radial direction, while many of the EJ-associated CMEs are ejected non-radially; iii) the mean speed of MC-associated CMEs (946 km?s^?1) is faster than that of EJ-associated CMEs (771 km?s^?1). For seven very fast CMEs (≥?1500 km?s^?1), all CMEs with large D (≥?0.4) are associated with MCs and the CMEs with small D are associated with EJs. From the statistical analysis of CME parameters, we found the superiority of the direction parameter. Based on these results, we suggest that the CME trajectory essentially determines the observed ICME structure.     

The impact cratering record of Fennoscandia

The current database of craterform structures in Fennoscandia contains 22 structures of impact origin and about fifty other structures which lack sufficient evidence for impact. The discovery rate of new structures has been one or two per year during the past ten years. The proven impact structures are located in southern Fennoscandia and the majority have been found in Proterozoic target rocks. The age of the structures varies from prehistoric to ≤ 1000 Ma and their diameters (D) from 0.04 km to 55 km. Nine of the structures contain impact melt. A characteristic feature of the Fennoscandian impact record is a relatively large number of small (≤ 5 km) but old (> 200 Ma) structures: this is a result of success of geophysical methods to discover small but old impact structures in an eroded shield covered with relatively thin overburden. Some of the large circular structures in satellite images and/or in geophysical maps may represent deeply eroded scars of very old impacts, but due to the lack of shock metamorphic features, impact-generated rocks or identified ejecta layers, they cannot yet be classified as impact sites. Two huge structures are proposed here as possible impact sites on the basis of circular satellite images and distinct geophysical anomalies: the Lycksele structure in northern Sweden (D ~ 120 km, see also Witschard, 1984) and the Valga structure in Latvia/Estonia (D ~ 180 km). However, endogeneous explanations, like buried granites, basement domings, or fault-bounded blocks are also possible for these structures. Hints, such as distal ejecta layers or impact produced breccia dykes, of an Archaean or Early Proterozoic impact structure have not been found in Fennoscandia so far. New ways of searching for these structures are proposed with particular emphasis on high-resolution integrated geophysical methods. The impact cratering rate in Fennoscandia is ~ 2.0 · 10^?14 km^?2 a^?1 (for craters with D > 3 km) corresponding to about two events per every 100 Ma for the last 700 Ma. Due to erosion, this is a minimal estimate but is higher than the global rate probably due to strong research activity for finding impact structures in Fennoscandia.     

The 3-D lattice theory of Flower Constellations

Flower Constellations (FCs) have been extensively studied for use in optimal constellation design. The Harmonic FCs (HFCs) subset, representing the symmetric configurations, have recently been reformulated into 2-D Lattice Flower Constellations (2D-LFCs), encompassing the complete set of HFCs. Elliptic orbits are generally avoided due to the deleterious effects of Earth’s oblateness on the constellation, but here we present a novel concept for avoiding this problem and enabling more effective global coverage utilizing elliptic orbits. This new 3D Lattice Flower Constellations (3D-LFCs) framework generalizes the 2D-LFCs, Walker constellations, elliptical Walker constellations, and many of Draim’s global coverage constellations. Previous studies have shown FCs can provide improved performance in global navigation over existing Global Navigation Satellite Systems (GNSS). We found a 3D-LFC design that improved the average positioning accuracy by 3.5 % while reducing launch $\varDelta v$ Δ v requirements when compared to the existing Galileo GNSS constellation.     

Speckle interferometry of asteroids II. 532 Herculina

Speckle interferometry of 532 Herculina performed on January 17 and 18, 1982, yields triaxial ellipsoid dimensions of (263 ± 14) × (218 ± 12) × (215 ± 12) km, and a north pole for the asteroid within 7° of RA = 7^b47^m and DEC = ?39° (ecliptic coordinates γ = 132° β = ?59°). In addition, a “spot” some 75% brighter than the rest of the asteroid is inferred from both speckle observations and Herculina's lightcurve history. This bright complex, centered at asterocentric latitude ?35°, longitude 145–165°, extends over a diameter of 55° (115 km) of the asteroid's surface. No evidence for a satellite is found from the speckle observations, which leads to an upper limit of 50 km for the diameter of any satellite with an albedo the same as or higher than Herculina.     

The rotation of small asteroids

The addition of the unbiased sample of R. P. Binzel and J. D. Mulholland (Icarus56, 519–533) nearly triples the sample size of photoelectrically determined rotational parameters for main belt asteroids with estimated diameters (D) ≤30 km. Nonparametric stattistical tests which require no assumptions about the distributions or variances of the samples are used to examine rotational parameters for all D ≤ 30 km asteroids. A comparison of photoelectric and photographic results shows that the techniques have a highly significant difference in the range of detected frequencies. This difference does not allow photographic and photoelectric observations to be combined for meaningful statistical tests since a photographic bias toward smaller sample variances can induce statistical results that appear overly significant. Photographic observations also show a highly significant bias toward detecting asteroids with larger lightcurve amplitudes. The fit of a Maxwellian to the observed rotational frequency distribution can be rejected at a highly significant confidence level but the observed distribution can be acceptably fit by two Maxwellian distributions, which is consistent with the hypothesis that there are separate populations of slow and fast rotating asteroids. The frequency distributions of <15 km main belt asteroids and Earth and Mars crossers are not found to differ significantly. However, the larger mean lightcurve amplitude of the Earth and Mars crossing asteroids is found to be statistically significant. This latter result is interesting in view of the lack of any strong inverse amplitude versus diameter relation for small asteroids. No significant diameter dependence on rotational frequency is seen among only D ≤ 30 km asteroids. However, the inverse frequency versus diameter relation for D ≤ 120 km asteroids found by S. F. Dermott, A. W. Harris, and C. D. Murray (Icarus, in press) is found to be statistically significant using a linear least-squares analysis of photoelectric data only. No significant diameter dependence on rotational lightcurve amplitude is seen using linear least-squares analysis of photoelectric data for D≤30 and D≤90 km asteroids. However, a significant inverse amplitude versus diameter relation is found when this analysis is extended to D≤120 km asteroids. This finding may be consistent with the hypothesis of Dermott et al. that near 120 km there is a transition between primordial asteroids and their collisional fragments.     

Formation of the galilean satellites in a gaseous nebula

A model for Galilean satellite formation was analyzed in which the satellites accrete in the presence of a dense, gaseous disk-shaped nebula and rapidly form optically thick, gravitationally bound primordial atmospheres. Upper-bound temperatures expected during accretion lead to partially differentiated structures for both Ganymede and Callisto, although with Ganymede much more differentiated than Callisto. When allowance is made for the aerodynamic breaking of infalling planetesimal fragments, lower surface temperatures result, and the amount of partial differentiation of Callisto is small, possibly approaching zero for a narrow size distribution of infalling planetesimals. The model is chosen to be consistent with the observed densities of the Galilean satellites and our current understanding of Jupiter formation. The retention of ices more volatile than H₂O is considered but not modeled in detail. A nominal nebula of ～0.1 Jupiter masses is constructed by consideration of likely surface density profiles and existing Jupiter collapse calculations. This nebula is optically thick (even if grain opacity is ignored) in both radial and vertical directions and has a temperature profile T ～ 3600 (R_J/R), where R_J is Jupiter's radius and R is the radial distance in the disk midplane. Satellites accrete very rapidly (dynamical time scales being 10²–10⁴ years) and their optically thick gaseous envelopes are unable to eliminate the heat of accretion by radiation. Water-saturated, convective, adiabatic envelopes form, through which planetesimals fall, break up, and partially disseminate their mass. The resulting satellite surface temperatures during accretion are calculated. Possible implications of these models for the subsequent evolution of Ganymede and Callisto are explored and it is suggested that the extensive differentiation undergone by Ganymede may provide the right environment for subsequent resurfacing, whereas the relative lack of extensive differentiation for Callisto may explain the inferred absence of endogenic tectonism.     

Formation of the regular satellites of giant planets in an extended gaseous nebula I: subnebula model and accretion of satellites

We model the subnebulae of Jupiter and Saturn wherein satellite accretion took place. We expect each giant planet subnebula to be composed of an optically thick (given gaseous opacity) inner region inside of the planet’s centrifugal radius (where the specific angular momentum of the collapsing giant planet gaseous envelope achieves centrifugal balance, located at rCJ ∼ 15R_J for Jupiter and rCS ∼ 22R_S for Saturn) and an optically thin, extended outer disk out to a fraction of the planet’s Roche-lobe (R_H), which we choose to be ∼R_H/5 (located at ∼150 R_J near the inner irregular satellites for Jupiter, and ∼200R_S near Phoebe for Saturn). This places Titan and Ganymede in the inner disk, Callisto and Iapetus in the outer disk, and Hyperion in the transition region. The inner disk is the leftover of the gas accreted by the protoplanet. The outer disk may result from the nebula gas flowing into the protoplanet during the time of giant planet gap-opening (or cessation of gas accretion). For the sake of specificity, we use a solar composition “minimum mass” model to constrain the gas densities of the inner and outer disks of Jupiter and Saturn (and also Uranus). Our model has Ganymede at a subnebula temperature of ∼250 K and Titan at ∼100 K. The outer disks of Jupiter and Saturn have constant temperatures of 130 and 90 K, respectively.Our model has Callisto forming in a time scale ∼10⁶ years, Iapetus in 10⁶-10⁷ years, Ganymede in 10³-10⁴ years, and Titan in 10⁴-10⁵ years. Callisto takes much longer to form than Ganymede because it draws materials from the extended, low density portion of the disk; its accretion time scale is set by the inward drift times of satellitesimals with sizes 300-500 km from distances ∼100R_J. This accretion history may be consistent with a partially differentiated Callisto with a ∼300-km clean ice outer shell overlying a mixed ice and rock-metal interior as suggested by Anderson et al. (2001), which may explain the Ganymede-Callisto dichotomy without resorting to fine-tuning poorly known model parameters. It is also possible that particulate matter coupled to the high specific angular momentum gas flowing through the gap after giant planet gap-opening, capture of heliocentric planetesimals by the extended gas disk, or ablation of planetesimals passing through the disk contributes to the solid content of the disk and lengthens the time scale for Callisto’s formation. Furthermore, this model has Hyperion forming just outside Saturn’s centrifugal radius, captured into resonance by proto-Titan in the presence of a strong gas density gradient as proposed by Lee and Peale (2000). While Titan may have taken significantly longer to form than Ganymede, it still formed fast enough that we would expect it to be fully differentiated. In this sense, it is more like Ganymede than like Callisto (Saturn’s analog of Callisto, we expect, is Iapetus). An alternative starved disk model whose satellite accretion time scale for all the regular satellites is set by the feeding of planetesimals or gas from the planet’s Roche-lobe after gap-opening is likely to imply a long accretion time scale for Titan with small quantities of NH₃ present, leading to a partially differentiated (Callisto-like) Titan. The Cassini mission may resolve this issue conclusively. We briefly discuss the retention of elements more volatile than H₂O as well as other issues that may help to test our model.