Initial Longitude Research Articles

Deepening of radiation belt particles in South Atlantic Anomaly Region: A scenario over past 120 years

• Weakened magnetic field is pushing radiation belt particles towards the Earth in SAA. • The average deepening rate of radiation belt particles is 4.3 km/year. • The deepening is increased by 434–590 km in last 120 years. • Deepening in the SAA region is depended on its initial longitude. • Deepening in the SAA region is independent of its equatorial pitch angle. The geomagnetic field has an unusual weak spot over South America and the South Atlantic Ocean, called South Atlantic Anomaly (SAA). We know that the magnetospheric particles trapped in the geomagnetic field move to the deeper altitudes over the SAA, resulting in the lower inner boundary of the radiation belt. However, over the past 400 years, the magnetic field in the SAA region has decreased consistently. In some recent studies, a rapid decrease of the magnetic field in the SAA region and the possibility of splitting of SAA have been reported. This weakened geomagnetic field has a bearing on the lower altitudinal reach (deepening) of the radiation belt particles. In this context, we performed simulations to examine and quantify the lowest altitudes encountered by the radiation belt particles over the SAA region. We found that particles of energies 100 keV-50 MeV, initially entering from an altitudes of 1–2 Earth radii are moving closer to the Earth at the rate of 3.8 - 4.7 km/year, which results in approximately 434 - 590 kms decrease in their altitude over the SAA region during 1900 to 2020. Furthermore, we noticed that the lowest altitude encountered by the particle in the SAA region is dependent on the initial longitude and independent of its initial local pitch angle corresponding to its entry in the Earth’s magnetosphere.

A Prediction Model of Online Car-Hailing Demand Based on K-means and SVR

The paper proposed a prediction model of online car-hailing demand based on K-means and support vector regression (SVR) methods. In the past few years, online car-hailing market demand has grown rapidly, and prediction of rapid demand growth has become a hot topic. This study takes the initial longitude and latitude of online car-hailing orders as the eigenvalues for K-means clustering. The clustering results are taken as the result of area division. The number and size of potential demand areas could be determined automatically. This method of area division solves the shortcomings of traditional artificial meshing division and existing administrative division methods. The model takes a small-sample data set as the application object and uses the SVR method for data regression. Finally, we conduct an empirical study on Didi’s real data set in the core area of Chengdu City, China. The final experimental results suggest that the area division method based on K-means is reasonable and that the demand prediction model based on K-means and SVR is effective.

Quantitative and Qualitative Results from Orbital Plane Precession of Medium-high Near-circular Orbits

Abstract The theoretical analysis of Allan & Cook on the mechanisms of orbital plane precession for medium-high circular orbits is briefly reviewed, and the fact that there are two types of possible precession mechanisms (not only the classical Laplace precession) for the long-term evolution of the orbital plane of medium-high near-circular orbits is especially emphasized. Based on two types of possible precession mechanisms, the analytical formulae of the quantitative variation ranges of orbital inclination derived by Zhao et al. are revised, and the complete analytical formulae for medium-high near-circular orbits are obtained. The qualitative characteristic formula for the variation of orbital inclination-longitude of the ascending node of geostationary orbits in Zhao et al. is generalized to medium-high near-circular orbits, and the more general analytical formulae are obtained. Taking the near-circular orbits with the altitudes as the geosynchronous and medium orbits of the BeiDou navigation satellites as examples, the theoretically quantitative and qualitative results from orbital plane precession are verified by the first-order semianalytical calculations with simplified models and numerical calculations with precise models to a certain extent. Moreover, the specific quantitative and qualitative results for medium-high near-circular orbits with the given initial inclinations and different initial longitudes of the ascending node are particularly presented, from which some conclusions are summarized. Due to the high similarity of orbital plane precession for natural satellites to that of artificial satellites, the results and methodology in this paper can be directly extended to the long-term evolution of the orbital plane for natural satellites and help to understand and explain some orbit dynamic phenomena of natural satellites.

Incorporation of Heliospheric Imagery Into the CME Analysis Tool for Improvement of CME Forecasting

AbstractCoronal mass ejections (CMEs) cause the largest geomagnetic disturbances at Earth, which impact satellites, wired communication systems, and power grids. The CME Analysis Tool (CAT) is used to determine a CME's initial longitude, latitude, angular width, and radial speed from coronagraph images. These are the initial conditions for the Wang‐Sheeley‐Arge Enlil solar wind model, along with the ambient solar wind properties derived from magnetograms. However, the coronagraph imagery is limited by field of view. We have incorporated heliospheric imagery (HI) from the Solar Terrestrial Relations Observatory into CAT to create the CME Analysis Tool with Heliospheric Imagery (CAT‐HI). These HI images have a larger field of view, allowing tracking of CMEs to greater distances from the Sun. We have compared the performances of CAT and CAT‐HI by examining the expected arrival times of CMEs at the L1 Lagrange point and found them to be consistent. However, CAT‐HI is advantageous because it could be used to prune ensemble forecasts and issue routine updates for CME arrival time forecasts. Finally, we discuss CAT‐HI in the context of an operational mission at the L4 or L5 Lagrange points.

An estimation of Envisat’s rotational state accounting for the precession of its rotational axis caused by gravity-gradient torque

The rotational state of Envisat is re-estimated using the specular glint times in optical observation data obtained from 2013 to 2015. The model is simplified to a uniaxial symmetric model with the first order variation of its angular momentum subject to a gravity-gradient torque causing precession around the normal of the orbital plane. The sense of Envisat’s rotation can be derived from observational data, and is found to be opposite to the sense of its orbital motion. The rotational period is estimated to be (120.674±0.068)·exp(4.5095±0.0096)×10-4·ts, where t is measured in days from the beginning of 2013. The standard deviation is 0.760 s, making this the best fit obtained for Envisat in the literature to date. The results demonstrate that the angle between the angular momentum vector and the negative normal of the orbital plane librates around a mean value of 8.53°±0.42° with an amplitude from about 0.7° (in 2013) to 0.5° (in 2015), with the libration period equal to the precession period of the angular momentum, from about 4.8 days (in 2013) to 3.4 days (in 2015). The ratio of the minimum to maximum principal moments of inertia is estimated to be 0.0818±0.0011, and the initial longitude of the angular momentum in the orbital coordinate system is 40.5°±9.3°. The direction of the rotation axis derived from our results at September 23, 2013, UTC 20:57 is similar to the results obtained from satellite laser ranging data but about 20° closer to the negative normal of the orbital plane.

Dynamics in the vicinity of (101955) Bennu: solar radiation pressure effects in equatorial orbits

Here we study the dynamical effects of the solar radiation pressure (SRP) on a spacecraft that will survey the near-Earth rotating asteroid (101955) Bennu when the projected shadow is accounted for. The spacecraft's motion near (101955) Bennu is modelled in the rotating frame fixed at the centre of the asteroid, neglecting the sun gravity effects. We calculate the solar radiation pressure at the perihelion, semi-major axis and aphelion distances of the asteroid from the Sun. The goals of this work are to analyse the stability for both homogeneous and inhomogeneous mass distribution and study the effects of the solar radiation pressure in equatorial orbits close to the asteroid (101955) Bennu. As results, we find that the mascon model divided into ten equal layers seems to be the most suitable for this problem. We can highlight that the centre point $E$8, which was linearly stable in the case of the homogeneous mass distribution, becomes unstable in this new model changing its topological structure. For a Sun initial longitude $\psi_0 = -180^o$, starting with the spacecraft longitude $\lambda = 0$, the orbits suffer fewer impacts and some (between 0.4 and 0.5 km), remaining unwavering even if the maximum solar radiation is considered. When we change the initial longitude of the Sun to $\psi_0 = -135^o$, the orbits with initial longitude $\lambda = 90^0 $ appear to be more stable. Finally, when the passage of the spacecraft in the shadow is accounted for, the effects of solar radiation pressure are softened, and we find more stable orbits.

Long-term dynamical evolution of Tundra-type orbits

Tundra-type orbits are elliptical geosynchronous orbits located at the critical inclination. The long-term dynamical evolution of this type of special orbit is investigated in this paper. First, the effect of Earth’s gravitational potential is examined. A simplified Hamiltonian of Tundra-type orbits subjected to Earth’s gravitational potential is presented through a strict magnitude comparison of the involved terms. Based on this simplified Hamiltonian with two degrees of freedom, the equilibrium points of the orbits subjected to Earth’s gravitational potential and their stabilities are discussed. This simplified Hamiltonian is then reduced to a one-degree-of-freedom system dominating the intermediate-period motion of the orbits approximately. In particular, the main characteristic parameters of the intermediate-period motion for nominal Tundra-type orbits and the corresponding specific results for three Sirius satellites in such orbits are presented. Second, the effect of lunisolar perturbations is examined. A magnitude comparison elementarily illustrates that the effect of tesseral harmonics of the Earth’s gravitational potential on the long timespan evolution of Tundra-type orbits is negligible compared to that of lunisolar perturbations. A simplified dynamical model including lunisolar perturbations is then presented. Based on this simplified dynamical model, the influences of lunar precession, the initial longitude of the ascending node, the initial argument of perigee, and the initial epoch on the long-term dynamical evolution of the orbits are comparatively analyzed. Finally, numerical calculations with exact perturbation models are conducted to verify the theoretical analysis and to provide more information about the dynamical evolution of Tundra-type orbits.

Conjunction challenges of low-thrust geosynchronous debris removal maneuvers

The conjunction challenges of low-thrust engines for continuous thrust re-orbiting of geosynchronous (GEO) objects to super-synchronous disposal orbits are investigated, with applications to end-of-life mitigation and active debris removal (ADR) technologies. In particular, the low maneuverability of low-thrust systems renders collision avoidance a challenging task. This study investigates the number of conjunction events a low-thrust system could encounter with the current GEO debris population during a typical re-orbit to 300km above the GEO ring. Sensitivities to thrust level and initial longitude and inclination are evaluated, and the impact of delaying the start time for a re-orbiting maneuver is assessed. Results demonstrate that the mean number of conjunctions increases hyperbolically as thrust level decreases, but timing the start of the maneuver appropriately can reduce the average conjunction rate when lower thrust levels are applied.

Orbital stability of systems of closely-spaced planets, II: configurations with coorbital planets

We numerically investigate the stability of systems of 1 $${{\rm M}_{\oplus}}$$ planets orbiting a solar-mass star. The systems studied have either 2 or 42 planets per occupied semimajor axis, for a total of 6, 10, 126, or 210 planets, and the planets were started on coplanar, circular orbits with the semimajor axes of the innermost planets at 1 AU. For systems with two planets per occupied orbit, the longitudinal initial locations of planets on a given orbit were separated by either 60° (Trojan planets) or 180°. With 42 planets per semimajor axis, initial longitudes were uniformly spaced. The ratio of the semimajor axes of consecutive coorbital groups in each system was approximately uniform. The instability time for a system was taken to be the first time at which the orbits of two planets with different initial orbital distances crossed. Simulations spanned virtual times of up to 1 × 108, 5 × 105, and 2 × 105 years for the 6- and 10-planet, 126-planet, and 210-planet systems, respectively. Our results show that, for a given class of system (e.g., five pairs of Trojan planets orbiting in the same direction), the relationship between orbit crossing times and planetary spacing is well fit by the functional form log(t c /t 0) = b β + c, where t c is the crossing time, t 0 = 1 year, β is the separation in initial orbital semimajor axis (in terms of the mutual Hill radii of the planets), and b and c are fitting constants. The same functional form was observed in the previous studies of single planets on nested orbits (Smith and Lissauer 2009). Pairs of Trojan planets are more stable than pairs initially separated by 180°. Systems with retrograde planets (i.e., some planets orbiting in the opposite sense from others) can be packed substantially more closely than can systems with all planets orbiting in the same sense. To have the same characteristic lifetime, systems with 2 or 42 planets per orbit typically need to have about 1.5 or 2 times the orbital separation as orbits occupied by single planets, respectively.

10.1007/s11208-008-2001-0

Meteorite impacts onto a small satellite lead to the ejection of a regolith mass, which is much greater than the impactor mass, into cosmic space. Assume that an isotropic ejection with velocities smaller than the maximum possible velocity b took place at the time moment t 0. Since the orbital periods are unequal, the particle trajectories will densely fill a certain domain D. The same domain will be filled after an explosion of an artificial satellite moving in a high orbit. One to three months later, the node and pericenter longitudes will be distributed over the entire circle and the domain D will become a body of revolution, a topological solid torus. We examine the domain of possible particle motion and its boundary S immediately after the impact event (an unperturbed case) and the same domain under the assumption that the initial longitudes of nodes and pericenters were already a result of considerable changes (a perturbed case). In both cases, we managed to construct the domain D and its boundary S analytically: parametric equations containing only relatively simple functions were obtained for S. The basic topologic and differential-geometric properties of S were studied completely.

Orbital stability of systems of closely-spaced planets

An investigation of the stability of systems of 1 M ⊕ (Earth-mass) bodies orbiting a Sun-like star has been conducted for virtual times reaching 10 billion years. For the majority of the tests, a symplectic integrator with a fixed timestep of between 1 and 10 days was employed; however, smaller timesteps and a Bulirsch–Stoer integrator were also selectively utilized to increase confidence in the results. In most cases, the planets were started on initially coplanar, circular orbits, and the longitudinal initial positions of neighboring planets were widely separated. The ratio of the semimajor axes of consecutive planets in each system was approximately uniform (so the spacing between consecutive planets increased slowly in terms of distance from the star). The stability time for a system was taken to be the time at which the orbits of two or more planets crossed. Our results show that, for a given class of system (e.g., three 1 M ⊕ planets), orbit crossing times vary with planetary spacing approximately as a power law over a wide range of separation in semimajor axis. Chaos tests indicate that deviations from this power law persist for changed initial longitudes and also for small but non-trivial changes in orbital spacing. We find that the stability time increases more rapidly at large initial orbital separations than the power-law dependence predicted from moderate initial orbital separations. Systems of five planets are less stable than systems of three planets for a specified semimajor axis spacing. Furthermore, systems of less massive planets can be packed more closely, being about as stable as 1 M ⊕ planets when the radial separation between planets is scaled using the mutual Hill radius. Finally, systems with retrograde planets can be packed substantially more closely than prograde systems with equal numbers of planets.

The orbital dust torus as an enveloping surface of a family of trajectories of isotropically ejected particles

Meteorite impacts onto a small satellite lead to the ejection of a regolith mass, which is much greater than the impactor mass, into cosmic space. Assume that an isotropic ejection with velocities smaller than the maximum possible velocity b took place at the time moment t0. Since the orbital periods are unequal, the particle trajectories will densely fill a certain domain D. The same domain will be filled after an explosion of an artificial satellite moving in a high orbit. One to three months later, the node and pericenter longitudes will be distributed over the entire circle and the domain D will become a body of revolution, a topological solid torus. We examine the domain of possible particle motion and its boundary S immediately after the impact event (an unperturbed case) and the same domain under the assumption that the initial longitudes of nodes and pericenters were already a result of considerable changes (a perturbed case). In both cases, we managed to construct the domain D and its boundary S analytically: parametric equations containing only relatively simple functions were obtained for S. The basic topologic and differential-geometric properties of S were studied completely.

Eulerian velocity reconstruction in ideal atmospheric dynamics using potential vorticity and potential temperature

An approach for the reconstruction of the velocity field in ideal atmospheric dynamics for given density and the two tracers potential vorticity and potential temperature (entropy) is presented. The method is based on the fundamental equations without approximation. The key step is to satisfy the continuity equation by the inclusion of a third Lagrangian tracer χ. This field is determined by closure conditions for density, potential vorticity and by boundary conditions. The reconstruction is, using the exterior calculus, with the four-dimensional 1-form based on the velocity components (1, u, v, w), density , potential vorticity Q and potential temperature θ. In the mean atmospheric flow χ represents the initial longitude of a fluid particle. For stationary flows χ is related to the Bernoulli function. Examples with analytical solutions are presented for a Rossby wave and zonal and rotational shear flow.

Long term evolution of the Galileo constellation due to gravitational forces

Abstract The Galileo system will be based on a constellation of 30 satellites planned to fly on circular orbits at an altitude of 23,616 km and an inclination of 56° to the equator. At this altitude, satellites orbit in 14 h and 21 min: the Galileo satellites will orbit in a (5,3)-commensurability with the Earth’s rotation which causes resonance effects with the Earth’s gravity field on the orbital motion. Mixed with the effect of third bodies (Moon and Sun), this commensurability is on the basis of a strong instability of the Galileo constellation. This paper is devoted to a modelling of that 4-body system (satellite, Earth, Moon, Sun). Some interesting characteristics of the whole constellation can be deduced. We show that resonance effects considerably modify the initial configuration of the constellation, strongly depending on initial longitudes of satellites if the motion is free-of-manoeuvres. As a result, these resonance effects must be taken into account to perform the necessary Galileo constellation analysis, and to study its long-term evolution.

Spatial Variations in Atmospheric Predictability

Abstract Calculations are made with the NCAR six-layer general circulation model to determine the time evolution of errors initially confined to a region 4000 km in diameter superimposed upon real global data. Three experiments are made to distinguish between the effects of an error located initially on the northern or southern sides of the jet stream or in the tropical area. Results show that the largest error centers generally evolve in the jet stream; however, the propagation rate is much less than advection effects would suggest. Coverage of the Northern Hemisphere is accomplished as much by propagation across the north pole and via the tropical belt as it is via the jet stream. It is not complete even after seven days. As a whole the tropics are more sensitive than the middle latitudes to initial errors. Cross-equatorial effects are most pronounced at and just east of the initial longitude of the error.