Annotation of the results obtained in 2021
The study of the dynamic structure of the near-Earth and near-Moon orbital spaces is a very urgent task, since knowledge of the dynamic features of various regions of space is necessary both in the development of new satellite systems and in determining the areas of disposal of spent objects. Within the framework of this project, we studied the dynamic structure of the near-Earth space (NES) starting from 8,000 km along the semi-major axis and up to the sphere of action of the Moon, as well as the near-Moon space (NMS) from 1911.8 km to 26,070 km along the semi-major axis. In both cases, the inclinations varied from 0 to 180 degrees. Modeling of the orbital evolution of objects in these areas was carried out using the software improved and developed during the previous stages of the project: numerical models of motion of the AES and AMS systems, as well as using methods of machine analysis, including a trained artificial neural network. Distribution maps of secular apsidal-nodal resonances associated with the Moon and the Sun in the NES and with the Earth and the Sun in the NMS were constructed. An analysis of the orbital evolution of the entire set of considered objects (more than 100 thousand) and resonance maps made it possible to draw a number of general conclusions about the nature of motion of objects in the studied areas. A significant part of the objects have a short lifetime, which is associated with an increase in the eccentricities of the orbits. All circumpolar objects, whose orbits have an inclination to the equator close to 90 degrees, possess a large and rapid increase in eccentricity and a short lifetime. This is explained by the action of secular resonances and is true for both NES and NMS. All stable apsidal-nodal resonances containing the rate of change in the longitude of the periapsis are grouped in the inclination range of 60–120 degrees. In NES, stable nodal resonances containing the rate of change of the longitude of the ascending node are grouped in the vicinity of three inclinations 0, 90, and 180 degrees, with most of them in the vicinity of 90 degrees. In NMS, this type of resonances is mainly concentrated in the range of inclinations from 70 to 110 degrees. The superposition of a large number of apsidal-nodal resonances leads to a rapid increase in the eccentricities of the objects' orbits, as a result of which the objects cease to exist either when they collide with the Earth/Moon or leave their spheres of gravity. Resonances associated with the mean motion of the perturbing bodies of the Moon and the Sun appear only in the lower part of the NES region under consideration. Secular resonances with the mean motion of the Sun are perturbed in the range of semi-major axes from 8,000 km to 21,000 km on both sides of an inclination of 90 degrees. Secular resonances with the mean motion of the Moon appear only in orbits whose semi-major axes are less than 16000 km and only in the dynamics of objects with reverse motion. No resonances associated with the mean motion of the perturbing bodies of the Earth and the Sun (2-5 orders) were found in the NMS. In addition to the influence of resonances on the AES and AMS motion, there are other factors that lead to a decrease in the lifetime. For example, in NES, starting from semi-major axes equal to 235,000 km, a significant part of objects have a short lifetime for most inclinations, and for semi-major axes exceeding 275,000 km, all objects have rapidly growing eccentricities and a short lifetime, regardless of inclination. This is due to the direct action of the Moon. Moreover, the closer the object's orbit to the equator, the greater the influence of the Moon. Studies of the dynamics of low-flying AMSs in the NMS have shown that the main reason of the eccentricity increase in low orbits is the direct action of the complex gravitational field of the Moon. Using for AES the ideas proposed by T. Galardo (2006) for asteroids, a comparative estimate of the forces of all orbital resonances acting in this space was obtained. This assessment showed that the lower the object, the greater the resonance force, and vice versa, the higher the object, the smaller the force. An analysis of the totality of the data obtained allows us to conclude that the action of almost all stable components of the orbital resonance is superimposed by the action of either the unstable components of the same resonance or the secular resonance, which leads to chaos. Moreover, the zones of increased chaoticity coincide with the superposition of secular resonances. Within the framework of this stage, a study of the dynamic structures of NES and NMS was also carried out, taking into account the effect of light pressure (LP) for various area-to-mass ratio (the ratio of the midsection area to the mass). The parameters of the numerical simulation and the method of distribution of objects are similar to those in the experiment carried out without taking into account the LP. Comparison of resonance maps constructed with and without taking into account the LP shows that in most cases, the resonance zones are preserved, although for a number of resonances, with an increase in the area-to-mass ratio, the zones can either become wider or completely disappear. In addition, an increase in the area-to-mass ratio value leads to an increase in the growth rate of orbital eccentricities and, as a result, to a decrease in the lifetime. The study of resonant motions of near-Earth asteroids is one of the important tasks on the way to solving the asteroid hazard problem. A stable mean motion resonance can serve as a protective mechanism against close approaches to the planets. An unstable resonance increases the risk of an asteroid approaching a planet, which in turn can lead to significant changes in the orbit elements of objects and possible approaches to the Earth. It is especially important to study the mean motion resonances for asteroids with small perihelion distances, since the orbits of these objects have an elongated shape and they are potentially capable of interacting with most planets in the Solar System. During the study, all asteroids with small perihelion distances moving in the vicinity of stable and unstable mean motion resonances with major planets were identified. Since the Yarkovsky effect (YE) can have a significant impact on the motion of these asteroids, when they pass near the Sun, the influence of the effect on the behavior of the resonant characteristics with time is considered. LP can also change the value of the semi-major axis, so its effect on the resonant characteristics was studied separately. The influence of the YE on the resonant characteristics – the resonant band and the critical argument – was estimated. The results of the study showed that the influence of the YE on the stable resonant relations is insignificant: it can insignificantly change the libration amplitude, but does not lead to the destruction of the resonance. In case of an unstable resonance, the number of passages through the exact commensurability changes, and for some objects, the resonance becomes even more stable. An estimation of the effect of LP on the behavior of characteristics of the mean motion resonance of asteroids showed that LP has less influence on the motion of asteroids under study then the YE, especially at diameters greater than 1 km. For a stable resonance, the influence of LP is similar to the influence of YE. For an unstable resonance, taking into account the LP leads to a change in the number of passages through the exact commensurability, in some cases partially destroying areas of stable resonant motion or, conversely, increasing the intervals of the resonant motion. A detailed study of the orbital evolution of asteroids 137924 2000 BD19 and 394130 2006 HY51 has been carried out both without taking into account the YE and LP and separately with taking into account each of them. The results of the study showed that taking into account the LP and YE has practically no effect on the evolution of the orbital elements of both asteroids, only slightly changing the behavior of the semi-major axis. Such a result of influence is reflected in the mutual position of asteroids and planets in the process of evolution and leads to a change in the number of approaches to major planets. Since the perturbations under study do not affect the longitude of the ascending node and the argument of the periapsis of the asteroid orbits, on the basis of which the characteristics of secular resonances are calculated, no changes in the behavior of secular resonances were revealed. The evolution of the OMEGNO parameter taking into account the YE and LP showed that these perturbations do not affect the motion predictability interval of asteroids 137924 2000 BD19 and 394130 2006 HY51.

 

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Annotation of the results obtained in 2019
Near-Earth space (NES) is increasingly used in the interests of humanity. More and more satellite systems are being created and deployed in space. The spent objects of these systems, as a rule, stay in near-Earth orbits and turn into space debris. Currently, according to the estimations of the European Space Agency in the NES there are more than 30 thousand objects of space debris larger than 10 cm, about 900 thousand objects are in the range from 1 to 10 cm, and about 128 million have size about 1 mm, and this number is constantly increasing. The orbital dynamics of these objects is influenced by many different factors that can significantly change both the spatial orientation and configuration of orbit under investigation. Such factors include secular and orbital resonances. Knowledge of the resonant structure of the orbital space allows one to predict possible catastrophic events with great certainty, to optimize the choice of the placement of new satellite systems and the parking of spent objects. The study of the dynamic structure of the NES requires extensive numerical experiments, accompanied by the analysis of large amounts of data. In this regard, the important tasks of the project are the implementation of automation of a number of research stages that require big data processing, including using machine learning methods and artificial neural networks (ANNs), as well as the development and improvement of methods and software for studying the dynamics of near-Earth objects of artificial origin. It was carried out primarily during the implementation of the first stage of the project in accordance with the schedule. During the implementation of the first stage of the project, studies were carried out on various methods of using ANNs to identify the presence of resonances in the NES and determine their type (stable / unstable). As a result, a trained ANN was obtained, which showed an accuracy of 99.3% in determining the presence of resonance and its classification. The algorithm HDBSCAN for automatic clustering of data was applied, which made it possible to quickly localize objects that demonstrate atypical and interesting dynamics. We developed the algorithms and software for detecting the Lidov–Kozai effect and automated the selection of objects with chaotic dynamics by the MEGNO parameter. We modified the software package “Numerical Model of Motion of Artificial Earth Satellite Systems” implemented on the SKIF Cyberia cluster of Tomsk State University. The productivity of the software package was improved due to the replacement of the integrator with a new and more efficient integrator developed at the Scientific Research Institute of Applied Mathematics and Mechanics TSU and a change in the parallelization algorithm. This modification of the numerical model made it possible to increase the computational speed by almost 3 times at preserving the accuracy of numerical orbit simulation. The important results of the first stage of the project implementation include a modification of the technique for identifying and studying secular resonances in the dynamics of near-Earth objects. The peculiarity of the technique is that, in contrast to the traditionally used approximate analytical techniques, all the frequencies in satellite motion caused by the action of perturbing factors are determined numerically by exact formulas. This approach gives more reliable results than a purely analytical technique when studying the resonant structure of orbital motion with a large or rapidly growing eccentricity. Using the software developed and modified during the implementation of the project, including the use of ANNs, we constructed maps of the dynamic structure for the LEO region (where the effect of the atmosphere is not so strong and it makes sense to consider the influence of secular resonances) and for the part of the MEO region with the semi-major axis up to 25500 km (to the zone of operation of the GLONASS system). The regions are identified where only secular resonances affect the motion of objects. It was shown that the features of the orbital evolution of objects in the zone that does not contain orbital resonances are determined, first of all, by the action of secular first-order apsidal-nodal resonances: the Lidov – Kozai apsidal resonance and nodal resonance which relates the change rate of longitudes of the ascending node of satellite and perturbing bodies. It was revealed that the first-order nodal resonance passes through the entire region under consideration and exhibits stability at 0, 90, and 180 degrees. The results of the study showed that in the considered region of the orbital space the effect of the Lidov–Kozai mechanism becomes noticeable for the first time. Features of its influence on the dynamics of objects depend on the interaction of this mechanism with the effect of the Earth oblateness. It was revealed that the Lidov–Kozai mechanism begins to appear in orbits with semimajor axis of about 10 000 km and then appears in all areas of the MEO region. In addition, it was shown that the combined influence of several secular resonances is not common in the dynamics of objects in the zone under consideration. The overlap of several stable secular resonances of different types does not lead to chaotic character in the motion of objects, and an increase of the chaotic character of motion is observed under the combined action of stable and unstable secular resonances. The features of the evolution of objects of the non-resonant zone (not containing orbital resonances) are compared with the evolution of objects moving in the areas of orbital resonance. It was revealed that the motion in the resonance zones is more chaotic. In a non-resonant zone the chaotic character of motion is rarely observed, but it is a characteristic property of resonance zones. Moreover, the determining factor in the occurrence of chaotic character in the motion of objects is the presence of unstable components of the orbital resonance in the dynamics. The contribution of secular resonances to the occurrence of chaotic character in resonance zones is secondary, but they affect the orbital evolution, which is manifested by an increase in the amplitudes of long-term oscillations of positional variables. First of all, this applies to secular resonances of the first order, which were mentioned above. Along with space debris, objects of natural origin, namely asteroids, may pose a danger to the Earth. Near-Earth asteroids have complex motion due to the presence of approaches to large planets and possible unstable orbital and secular resonances. The study of the dynamics of such objects is possible only by numerical methods, since it requires taking into account the influence of many forces, in particular non-gravitational effects, such as light pressure and the Yarkovsky effect. The study of probabilistic orbital evolution requires the use of multiprocessor computing systems and the processing of large amounts of information. During the implementation of the first stage of the project, a transition was made to the automated processing of calculation results. In particular, we automated graphing process using scripts and their analysis using machine learning methods. The IDA (Investigation of dynamics of asteroids) software package has been modified by including algorithms for determining the transverse acceleration coefficient A2, created by the Yarkovsky effect in the motion of asteroids. We realized and analyzed two methods for obtaining this parameter from the condition of the minimum mean square error. In the first method, different values of A2 are scanned within a given interval with a certain step. For each parameter value, the least-squares problem is solved and the mean square error is determined. For the desired value of the parameter transverse acceleration A2 one is taken that corresponds to the minimum value of the mean-square error. This technique has a number of disadvantages: it limits the researcher to the interval of variation the parameter A2 and the discrete step, and it requires solving the least squares problem at each step, which significantly increases the program run time. In the second method, the parameter is included in the number of estimated parameters during solving the least squares problem. In this technique, the problem is solved once, and, in addition, it allows one to obtain the uncertainties in determining the parameter A2. The developed algorithms were tested on several asteroids with small perihelion distances, since the Yarkovsky effect can have a significant impact on their dynamics due to the regular passage of objects near the Sun. In particular, we compared the methods for determining the transverse acceleration parameter A2 for asteroids having small perihelion distances and a different structure of observations. Due to the fact that some asteroids have radar observations, the ability to process them was included in the IDA software package. The example of asteroids 425755 2011 CP4 and 3200 Phaethon shows the ability to obtain the required parameters using the developed software, with a precision comparable to the precision of NASA. It should be noted that these two asteroids have radar observations. Thus, the study showed that both methods are in good agreement with each other, but due to the described disadvantages of the first technique, we can conclude that the second method is more effective. Further, the orbital evolution of the asteroids 504181 2006 TC and 2007 PR10 were studied in detail with and without the Yarkovsky effect. We revealed the approaches of 2006 TC and of 2007 PR10 with all the planets of the inner group (Mercury, Venus, the Earth, Mars), considered secular resonance dynamics (a number of apsidal-nodal resonances were revealed), and estimate the predictability time of the asteroid motion using the OMEGNO parameter. The study results showed that the Yarkovsky effect has a significant influence on the semi-major axis of the asteroids orbits, that leads to changes in the number and distance of approaches. As for the estimation of predictability time using the OMEGNO parameter, the motion of the asteroid 504181 2006 TC can be considered regular only in the time interval (1440, 2270) years, and 2007 PR10 is in the interval (1390, 4310) years. Outside of these intervals, features of chaotic character appear at the orbits, that is likely the result in multiple and close approaches of the asteroids with to major planets. In addition, the new version of IDA software package provides the ability to determine the light pressure parameter from observations. The technique was tested on the example of asteroid 3200 Phaethon. Our results are consistent with NASA data. In preparation for the second stage, we built a simplified model of the numerical motion of the Moon’s satellites, which includes the main potential (LP150Q) and a point model of the influence of external bodies (the Earth and the Sun). The complication of the model and bringing it to the high-precision level are planned at the second stage of the project.

 

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Annotation of the results obtained in 2020
During the second stage of the project, a study of the near-Earth space (NES) area was carried out up to the sphere of influence of the Moon in relation to the Earth (from 35,000 to 315,000 km along the semi-major axis). The resonant structure of this near-Earth space has been studied in detail. It is shown that a significant part of the objects in the region under consideration have a short lifetime, which is fully explained by an increase in the eccentricities of the orbits. All circumpolar objects, whose orbits have an inclination to the equator close to 90 degrees, possess a large and rapid increase in the eccentricity and a short lifetime. Starting from the semi-major axes equal to 235,000 km, a significant part of the objects have a short lifetime for most inclinations, and for the semi-major axes exceeding 275,000 km, all objects have rapidly growing eccentricities and a short lifetime regardless of the inclination. This is due to the direct influence of the Moon. The closer the object's orbit to the equator, the greater the influence of the Moon. The nature of the evolution of objects not subjected to the direct influence of the Moon is explained by the action of numerous secular resonances, both stable and unstable. The distribution maps of secular apsidal-nodal resonances associated with the Moon and the Sun were constructed in the considered NES region. These maps, as well as the maps of superposition of various types of resonances for the Moon and the Sun were obtained using artificial neural networks (ANN). Analysis of all the constructed maps allowed us to draw the following conclusions: all stable apsidal-nodal resonances containing the rate of change in the periapsis longitude are grouped in the inclinations range of 60–120 degrees, and stable nodal resonances containing the rate of change in the longitude of the ascending node are grouped in the vicinity of three inclinations of 0, 90, and 180 degrees, with most of them grouping in the vicinity of 90 degrees. This explains why all objects in the circumpolar region have a rapidly growing eccentricity and a short lifetime. In order to identify in this NES area the distinctive features in the motion of near-Earth objects in the regions affected by orbital resonances and outside their action, several regions of orbital resonances are considered. Earlier, when studying the regions of action of low-order orbital resonances, we came to the conclusion that the regions of chaotic motion are determined, first of all, by the superposition of stable and unstable components of the orbital resonance. The superposition of secular resonances only enhances the chaos. However, when considering the regions of high-order resonances, for example, 2:1 resonance with the Earth's rotation speed, we saw a situation exactly opposite to that observed in the region of low-order orbital resonances. Comparing the data on all types of resonances acting in the region of 2:1 resonance with the Earth's rotation speed, we found that the orbital resonance is manifested itself only by a slight increase in the chaos of motion of objects falling into this resonance. The orbital evolution is mainly influenced by secular resonances. Thus, in this part of the NES, the dynamics of the region containing high-order orbital resonances differs little in its features from the regions that do not contain these resonances. A numerical model of motion of artificial satellites of the Moon, improved due to the refinement of the model of forces and the use of a more efficient integrator, has been developed. In addition, the numerical model was supplemented with the ability to calculate the averaged parameter MEGNO, which makes it possible to judge about the chaos of motion of space objects. The model is implemented in two versions: for a personal computer and in a parallel computing environment on the "Skif Cyberia" cluster of Tomsk State University. Comprehensive testing of the model has been carried out, which has confirmed its high accuracy. Testing of the automatic data processing system was carried out using an artificial neural network by selecting the marked data for the circumlunar objects. The testing has shown that in order to correctly recognize the presence or absence of resonance, in addition to the time series associated with the critical argument, it is necessary to consider as signals two more time series associated with the changes in the eccentricity and inclination of the orbit. Using the developed software, the analysis of the effect of the selenium potential structure on the motion of circumlunar objects was carried out, which showed that the cause of the increase in eccentricity in low orbits is the direct action of the selenium potential. The analysis of the resonance structure of the circumlunar orbital space is carried out. Based on the results of the study, the maps of the action of the secular apsidal-nodal resonances of 1-4 orders and semi-secular resonances of 2-5 orders with average motions of the disturbing bodies of the Earth and the Sun on the circumlunar objects were compiled. It is shown that the main reason for the increase in the eccentricity and a significant decrease in the orbital lifetime of objects at the medium and high altitudes should be considered the influence of the secular resonance of the Lidov – Kozai type and resonances close to it on the orbital dynamics of the AMS. Semi-secular resonances with the average movements of the Earth and the Sun are localized in the middle and upper regions of the circumlunar orbit space, starting with the semi-major axis of 6000 km. The action of these resonances is unstable. In the change of the resonant arguments, a permanent transition from libration to circulation and vice versa can be traced. Therefore, they do not have a significant impact on the orbital evolution. The class of asteroids with small perihelion distances (less than 0.15 AU) is of great interest for scientists from the point of view of the possible danger to the Earth from these objects. The dynamics of asteroids, as they pass near the Sun, can be significantly influenced by the non-gravitational perturbing factors, such as the Yarkovsky effect and solar radiation pressure. Not taking into account these perturbations in the force model, in a number of cases can lead to an unreliable prediction of motion, including erroneous estimates of the collision probability. The main problem with taking into account these perturbations is that for most asteroids, the physical properties required to determine the parameters of the Yarkovsky effect and solar radiation pressure are poorly known to date. In this situation, one of the ways to determine the parameters is to obtain them from the observations of asteroids. In the course of the second stage of the project, the parameters of the Yarkovsky effect A2, solar radiation pressure ks, and their root-mean-square errors were obtained for all asteroids with small perihelion distances using the modified IDA software package. According to the data of the Minor Planet Center site, 50 such asteroids are known as of January 2021. In the process of fitting their orbits, the parameters A2 and ks were included in the number of estimated ones, along with the coordinates and velocity components. The observations were taken from the Minor Planet Center site. Estimation of the precision of determining the parameter of solar radiation pressure ks showed that for most asteroids, the root-mean-square error of ks coincides in order of magnitude with the value itself, and for 8 recently discovered asteroids, the order of the error exceeds the order of ks. Only for 3200 Phaethon, the error is one order of magnitude less than the parameter value. On the example of asteroids with known physical parameters (3200 Phaethon, 137924 2000 BD19, 276033 2002 AJ129, 386454 2008 XM, 394130 2006 HY51, and 394392 2007 EP88), it is shown that the use of the improved ks value insignificantly affects the value of the mean-square-error, which can be due to the weak influence of the solar radiation pressure on the motion of the studied asteroids. Evaluation of the influence of the solar radiation pressure on the motion of these objects made it possible to conclude that the influence of the solar radiation pressure is less than that the Sun oblateness and relativistic effects of the Sun. The obtained values of the Yarkovsky effect parameter and its root-mean-square error for asteroids with small perihelion distances allowed us to draw the following conclusions. Like the coefficient of solar radiation pressure ks, the values of the parameter A2 and its root-mean-square error for most objects are of the same order of magnitude. Analysis of the results of determining the parameter A2 for the studied asteroids showed that with an increase in the observation interval, the precision of the parameter being determined significantly improves. This conclusion was confirmed by the experiments with model and real observations. On the example of asteroids 3200 Phaethon and 137924 2000 BD19, an experiment was carried out to reduce the number of real observations from the present time to the past, and on the example of asteroids 2008 MG1 and 2020 BU13, an experiment to increase the arc length by using model observations was performed. However, for each asteroid, there is a limiting arc length, upon reaching which the precision of the determining the parameter ceases to improve. The analysis of the results also showed that for objects with the arc length less than 30 days, the determination of the Yarkovsky effect parameter value from the observations is not reliable enough and for its application, it is worth waiting for new observations of asteroids. The probabilistic orbital evolution of asteroids 3200 Phaethon and 2007 PR10 with and without taking into account the Yarkovsky effect has been constructed using the IDA software package. For both asteroids, multiple and close encounters with Mercury, Venus, Earth, and Mars were revealed, and the chaoticity of their orbits was estimated using the OMEGNO parameter (Orthogonal Mean Exponential Growth factor of Nearby Orbit). Taking into account the Yarkovsky effect in both cases led to significant changes in the evolution of the semi-major axis, which was reflected in the number of encounters of asteroids with planets and the distances to them. As a result of such changes, the confidence regions of asteroids behave differently without taking into account the effect and with it. The search for the resonances in the motion of asteroids showed that 3200 Phaethon is moving in the vicinity of the 3/7 mean motion resonance with Venus. Without taking into account the Yarkovsky effect, the asteroid is in resonance with the planet for a longer time, which indicates the need to take it into account for a more correct picture. No mean motion resonances were found in the dynamics of asteroid 2007 PR10.

 

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