Choice of high-apogee AES orbits on the basis of the qualitative methods of the theory of perturbations and situational analysis. Part I. Situational studies based on orbital tori

The paper discusses the problems of the choice of high-apogee orbits of artificial Earth satellites (AES), proceeding from the tasks of space experiments aimed at studying near-earth space and taking into account the features of the orbital evolution and ballistic lifetime. The suggested methods of the choice of orbits consist of two components. The first is based on the use of mathematical models of studied regions of near-earth space and various techniques of situation analysis, among which the annual and daily orbital tori developed by the author about 35 years ago are key. The second component is based on qualitative methods of the theory of perturbations of high-apogee AES orbits developed by M.L. Lidov more than 50 years ago.     

On the features of long-term evolution of a high-apogee orbit of the SPECTR-R spacecraft

The practical tasks related to qualitative investigation of long-term evolution of high-apogee orbits of artificial Earth satellites (AES), for which the main perturbing factors are gravitational perturbations from the Moon and the Sun, are considered. Attention is given to the problem of the ballistic lifetime of similar orbits, and the issues associated with possibilities of the correction of orbits for ensuring the required duration of their ballistic lifetime are considered. The orbit of the SPECTR-R spacecraft launched in July of 2011 is considered as an example.     

Orbit design for the Spektr-R spacecraft of the ground-space interferometer

In order to carry out tasks of the RadioAstron mission, a high-apogee orbit was designed. On average, the period of its satellite’s orbit around the Earth is 8.5 days with evolution due to gravitational perturbations produced by the Moon and the Sun. The perigee and apogee of this orbit vary within the limits 7500–70000 km and 270000–333000 km, respectively. The basic evolution of the orbit represents a rotation of its plane around the line of apsides. Over 3 years, the plane normal to the orbit draws on the celestial sphere an oval with a semi-major axis of about 150° and semi-minor axis of about 45°.     

Fast measurements of parameters of the Solar Wind using the BMSW instrument

Design of the plasma spectrometer BMSW (Fast Monitor of the Solar Wind, possessing high temporal resolution) is described in the paper, as well as its characteristics and modes of operation. Some examples of measurements of various properties of the solar wind, made with this instrument installed onboard the high-apogee satellite Spektr-R, are presented.     

Investigation of Periods of Evolution for Elliptical Orbits in the Double-Averaged Hill Problem

The geometric analysis applied to the solutions obtained by M.L. Lidov [1] of the satellite version of a restricted circular double-averaged three-body problem allowed us fresh view at the comparative analysis of evolution periods for various elliptical orbits. Based on the similarity theory, the parameters of similitude are invoked for orbits and perturbations. An expression for the period of evolution of three orbit elements through the appropriate similitude parameters is derived. On this basis, a regularity is formulated, the consequences of which are analyzed.     

Application of periodic solutions of three-dimensional three-body problem to designing the orbit of a space telescope

A method of constructing three-dimensional orbits with a necessary evolution in the system the Sun — (Earth + Moon) is described. The orbit (promising from the viewpoint of solving formulated research problems) of the Millimetron spacecraft is suggested. Feasibility of such an orbit is demonstrated, as well as a possibility to observe with its help the majority of objects on the celestial sphere and to transmit the data to the Earth.     

Gravitation Maneuver Using the Families of Super-Unstable Orbits around the Libration Points

The families of periodic solutions to an autonomous Hamiltonian system in that part where the solutions are unstable have their specific field of influence. Under strong instability, the orbits that have fallen in such a field of influence are drawn into the family as in a whirlpool and then are thrown away from it. In the particular case of the restricted three-body problem, the orbits around the libration points L ₁ and L ₂ are super-unstable and the Keplerian elements in motion over these orbits change drastically. When the orbits fall into such a domain, for some time the motion is close to the motion around the libration point, and after being thrown out of this domain, the Keplerian orbital elements also change essentially.     

Observations of auroral kilometric radio emission on the <Emphasis Type="Italic">Interball-1</Emphasis> satellite in the active Sun period 1999–2000

Results of almost two years (January 1999–October 2000) of continuous observations of auroral kilometric radio emission with the instrument AKR-X onboard the high-apogee satellite of the Earth Interball-1 are presented. The observations were conducted at the growth stage (in 1999) and in the maximum (2000) of solar activity within the 100–1500 kHz frequency band. The results of AKR detection in the vicinity of the maximum of its spectrum at a frequency of 252 kHz are presented. Both similarity (for example, the character of global directivity) and important differences from the AKR emission observed during the solar activity minimum [5] are found. Together with very high sporadicity, strong seasonal changes in the intensity are typical for the emission. It is completely absent in the spring-summer period in the Northern Hemisphere and is strongly suppressed in this period in the Southern Hemisphere. Probable nature of these features of AKR is discussed.     

Detection and Prediction of Absorbed Radiation Doses from Solar Proton Fluxes onboard Orbital Stations

We consider cases of simultaneous detection of the absorbed doses produced by proton fluxes of powerful solar events onboard the Mir and ISS orbital stations and the Ekspress A3 geosynchronous satellite. Experimental data are analyzed using a software package that takes into account the energy spectra of protons at the Earth's orbit depending on the time of event evolution, as well as their penetration to near-earth orbits and through the protective shields of spacecraft. Based on a comparison of the experimental data of dosimeters with the calculation of absorbed doses under the action of solar proton events, we developed a method of estimating the effective thickness of the shielding of dosimeters and made some estimates. A possibility is considered for predicting the radiation hazard onboard orbital stations upon the appearance of solar proton events using dosimeter data from a geosynchronous orbit.     

Periodic solutions in three-dimensional restricted three-body problem: 2. Loss of symmetry at the 1:1 resonance

Within the framework of the circular restricted three-body problem a family of inverse periodic orbits around the two attracting bodies (the Egorov’s family) and families generated by it at the 1:1, 2:1, and 3:1 resonances for three-dimensional orbits in the Sun-Earth and Earth-Moon systems are considered. Their relationship with families generated by orbits around the libration points L ₁, L ₂ and L ₃ is investigated. One of the families contains periodic solutions that seem promising as possible orbits for the space radio telescope of the Millimetron project.     

Stable spatial orbits around collinear libration points

A technique of generation of spatial periodic solutions to the restricted circular three-body problem from periodic orbits of the planar problem has been used for the families of orbits around collinear libration points L ₁ and L ₂. Developing the families obtained at the 1: 1 resonance, we have obtained stable solutions both in the Earth-Moon system and in the Sun-Earth system. Of course, the term “around the libration point” is rather conventional; the obtained orbits become more similar to the orbits around the smaller attracting body. The further development of the family of orbits “around” the libration point L ₂ in the Sun-Earth system made it possible to find the orbits satisfying the new, much more rigorous constraints on cooling the spacecraft of the Millimetron project.     

同步双小行星系统共振轨道设计

研究了同步双小行星系统中共振轨道的设计方法及演化规律。首先，基于双椭球模型建立探测器运动方程，并给出共振轨道初值选取方法。然后，利用改进并行打靶法，提出一种双小行星系统平面共振轨道两步修正方法。同时结合稳定性理论及分岔理论，给出双小行星系统三维共振轨道生成和延拓方法；最后，以双小行星系统1999KW4为例，设计了共振比为1∶1,1∶2,1∶3,1∶4,2∶3的平面和空间共振轨道族，并分析了共振轨道的特性及轨道周期和轨道能量的变化规律。给出的双小行星系统中共振轨道的设计方法具有普适性，对未来双小行星系统探测任务中的轨道设计具有一定的参考意义与借鉴价值。     

Optimization of the trajectory of the spacecraft insertion into the system of heliocentric orbits

The possibility of the spacecraft insertion into the system of operational heliocentric orbits has been analyzed. It has been proposed to use a system of several operational heliocentric orbits. On each orbit, the spacecraft makes one or more revolutions around the Sun. These orbits are characterized by a relatively small perihelion radius and relatively high inclination, which allows one to investigate the polar regions of the Sun. The transition of the spacecraft from one orbit to another has been performed using an unpowered gravity assist maneuver near Venus and does not require the cruise propulsion operation. Each maneuver transfers the spacecraft into the sequence of operational heliocentric orbits. We have analyzed several systems of operational heliocentric orbits into which the spacecraft can be inserted by means of the considered transportation system with electric propulsion (EP). The mass of the spacecraft delivered to these systems of operational orbits has been estimated.     

Continuum Model for Space Debris Evolution with Account of Collisions and Orbital Breakups

The paper discusses the mathematical modeling of long-term orbital debris evolution taking into account mutual collisions of space debris particles of different sizes. Investigations and long-term forecasts of orbital debris environment evolution in low Earth orbits are essential for future space mission hazard evaluation and for adopting rational space policies and mitigation measures. The paper introduces a new approach to space debris evolution mathematical modeling based on continuum mechanics incorporating partial differential equations. This is an alternative to the traditional approaches of celestial mechanics incorporating ordinary differential equations to model fragments evolution. The continuum approach to orbital debris evolution modeling has essential advantages for describing the evolution of a large number of particles, because it replaces the traditional tracking of space objects by modeling the evolution of their density of distribution.     

Solution of the flyby problem for large space debris at sun-synchronous orbits

the paper considers the flyby problem related to large space debris (LSD) objects at low earth orbits. The data on the overall dimensions of known last and upper stages of launch vehicles makes it possible to single out five compact groups of such objects from the NORAD catalog in the 500–2000 km altitude interval. The orbits of objects of each group have approximately the same inclinations. The features of the mutual distribution of the orbital planes of LSD objects in the group are shown in a portrait of the evolution of deviations of the right ascension of ascending nodes (RAAN). In the case of the first three groups (inclinations of 71°, 74°, and 81°), the straight lines of relative RAAN deviations of object orbits barely intersect each other. The fourth (83°) and fifth (97°–100°) LSD groups include a considerable number of objects whose orbits are described by straight lines (diagonals), which intersect other lines many times. The use of diagonals makes it possible to significantly reduce the temporal and total characteristic velocity expenditures required for object flybys, but it complicates determination of the flyby sequence. Diagonal solutions can be obtained using elements of graph theory. A solution to the flyby problem is presented for the case of group 5, formed of LSD objects at sun-synchronous orbits.     

Synergistic use of debris environment models for flexible, very-long term projections

Long-term debris environment projections are of great importance for assessing the necessity and effectiveness of debris mitigation measures. Two types of models have been developed to predict these environments. Environment evolution models like the EVOLVE code are using detailed mission model data to input spacecraft, upper stages, and operational debris into specific orbits at specific times; debris from fragmentations are placed in orbits defined by the state vector of the fragmenting object(s) and the breakup model. The second type, typified by the CHAIN program, uses a particle-in-box model that bins the environment in size and altitude rather than following the orbit evolution of individual debris fragments. A 3-Step approach using both the EVOLVE and CHAIN model in a synergistic way was used to increase the reliability of long term environment projections. EVOLVE historical projections 1957–1995 could be validated by comparison to measurements. The comparison of 100 year projection runs of EVOLVE and CHAIN for different traffic scenarios showed a good agreement. In this paper, for the first time, CHAIN projections up to 10,000 years, based on validated boundary conditions derived by EVOLVE are presented, indicating clearly the need of early implementation of effective mitigation measures to prevent exponential population growth by collisional cascading effects.     

Biaxial Mode of Rotation of a Satellite in the Orbit Plane

A mode of motion of a satellite with respect to its center of mass is studied, which is called the biaxial rotation in the orbit plane. In this mode of rotation, an elongated and nearly dynamically symmetric satellite rotates around the longitudinal axis, which, in turn, rotates around the normal to the plane of an orbit; the angular velocity of rotation around the longitudinal axis is several times larger than the orbital angular velocity, deviations of this axis from the orbit plane are small. Such a rotation is convenient in the case when it is required to secure a sufficiently uniform illumination of the satellite's surface by the Sun at a comparatively small angular velocity of the satellite. The investigation consists of the numerical integration of equations of the satellite's motion, which take into account gravitational and restoring aerodynamic moments, as well as the evolution of the orbit. At high orbits, the mode of the biaxial rotation is conserved for an appreciable length of time, and at low orbits it is destroyed due to the impact of the aerodynamic moment. The orbit altitudes and the method of constructing the initial conditions of motion that guarantee a sufficiently prolonged period of existence of this mode are specified.     

A model for the evolution of the on-orbit man-made debris environment

A model for the evolution of the low Earth orbit man-made debris population is presented and the results of several test cases discussed. Debris sources include normal operations in space, explosions occurring on spacecraft in orbit, and collisions between objects in orbit; the stochastic occurrence of these deposition events is modeled using Monte Carlo techniques. A technique for discriminating between objects populating long-life vs rapid-decay orbits is discussed and applied to the analysis of debris contributions from collisions of comparable sized objects. In varying degrees, each of the cases presented indicate there is cause for concern for spacecraft and space operations from the 1990s onward-man-made debris will play a role which may vary from presenting a considerable hazard to certain operations or certain spacecraft to effectively prohibiting the use of certain spaceccraft or space operations.