The paper discusses the development of scientific areas of mechanics in the research by the honorary professor of St. Petersburg State University, Honored Worker of Science and Technology of the Russian Federation, and Doctor of Physical and Mathematical Sciences Viktor Sergeevich Novoselov, founder of the scientific school of analytical mechanics, space dynamics, biomechanics, and applied mathematics. He extended the basic theorems of analytical dynamics to mechanical systems of variable composition. Using variational methods, he obtained a number of remarkable results on the dynamics of controllable systems. In particular, he proposed a general scheme for constructing analytical approximations, which has been used to solve a variety of equations.

The distance functions on the set of Keplerian orbits play an important role in solving the problems of searching for theparent bodies of meteoroid streams. A special kind of function is the distances in the quotient spaces of orbits. Three metricsof this type were developed earlier. These metrics allow disregarding the longitude of the ascending node or the argumentof pericenter, or both. Recently, we introduced one more quotient space, where two orbits are considered to be identical ifthey differ only in their longitudes of nodes and arguments of pericenters, but have the same sum of these elements (thelongitude of pericenter).

Several metrics transforming diverse spaces of Keplerian orbits into metric ones have been proposed in the last 15 years. They are used to estimate the proximity of orbits of celestial bodies (comets, asteroids, and meteoroid complexes). Quotient spaces, which allow us to leave out of account those orbital elements that change secularly under various perturbations, are of great importance. Three quotient spaces were examined previously. Nodes are ignored in one of these; arguments of pericenters are ignored in the second one; both nodes and arguments of pericenters are ignored in the third one. We introduce the fourth quotient space, where orbits with arbitrary longitudes of nodes and arguments of pericenters are identified under the condition that their sum (longitude of pericenter) remains fixed. Function $${{\varrho }_{6}}$$, which represents the distance between the indicated classes of orbits and satisfies the first two axioms of metric spaces, is determined. An algorithm for its calculation is proposed. In the general case, the most challenging part of the algorithm is finding the solution of a trigonometric equation of the third degree. The issue of validity of the triangle axiom for $${{\varrho }_{6}}$$ (at least in its relaxed version) will be examined later.

The NASA spacecraft New Horizons allowed us to refine many physical, chemical, topographic, and geological characteristics of main bodies of the Pluto – Charon system significantly. Their sizes have been clarified. Their shapes were approximated by triaxial ellipsoids. But measurement errors of the axes differences a−c, a−b have exceeded their values. So the asphericity of the figures was not revealed. The same concerns the asphericity of the gravitational fields of Pluto and Charon. So measurements during future space missions must be more accurate. To evaluate the necessary precision we need bilateral theoretical boundaries of the possible values of polar and equatorial oblatenesses α=(a−c)/a, β=(a−b)/a, as well as the second zonal and sectorial harmonics I2, I22 for both bodies. In the present paper we have obtained them under the assumption of the hydrostatic equilibrium of the Pluto and Charon figures. Pressure, centrifugal forces, and gravity (tidal interaction included) were taken into account. It turns out that the influence of rotation and tides are comparable; tides change the Charon shape more than the Pluto one; dimensionless quantities α, β, I2, I22 are of the order of 10-5÷10-4 for Pluto, and 10-4÷10-3 for Charon.

The results of the search for Apollo, Amor, and Aten asteroids with the orbits close to those of meteoroids of the δ-Cancrids meteor complex (code DCA), which consists of the northern (code NCC) and southern (code SCC) branches, are presented. The search for small bodies in close orbits was performed on the basis of a multifactorial method of combining several criteria: Drummond orbital similarity criterion, Kholshevnikov metric, and parameters of the dynamic evolution of the orbits using two catalogs of meteor orbits (Japan Meteor Society, SonatoCo, and CAMS Meteoroid Orbit Database v2.0, CAMS) obtained from television observations. Asteroids in close orbits with the meteoroid orbits of the northern NCC and southern SCC branches of the δ-Cancrids are identified only in the Apollo group. The following asteroids are common for the NCC and SCC branches: 2015 PU228, 2014 YQ34, 2017 YO4 (according to the CAMS orbit catalog); Hephaistos 1978 SB, 2003 RW11, 2006 BF56, 2011 SR12, 2014 RS17, 2001 YB5 (SonatoCo catalog). The asteroid 85182 (1991 AQ) is identified only with the northern NCC branch but in two catalogs of meteor orbits.

The averaging method is widely used in celestial mechanics, in which a mean orbit is introduced and slightly deviates from an osculating one, as long as disturbing forces are small. The difference $$\delta {\mathbf{r}}$$ in the celestial body positions in the mean and osculating orbits is a quasi-periodic function of time. Estimating the norm $$\left\| {\delta {\mathbf{r}}} \right\|$$ for deviation is interesting to note. Earlier, the exact expression of the mean-square norm for one problem of celestial mechanics was obtained: a zero-mass point moves under the gravitation of a central body and a small perturbing acceleration $${\mathbf{F}}$$. The vector $${\mathbf{F}}$$ is taken to be constant in a co-moving coordinate system with axes directed along the radius vector, the transversal, and the angular momentum vector. Here, we solved a similar problem, assuming the vector $${\mathbf{F}}$$ to be constant in the reference frame with axes directed along the tangent, the principal normal, and the angular momentum vector. It turned out that $${{\left\| {\delta {\mathbf{r}}} \right\|}^{2}}$$ is proportional to $${{a}^{6}}$$, where $$a$$ is the semi-major axis. The value $${{\left\| {\delta {\mathbf{r}}} \right\|}^{2}}{{a}^{{ - 6}}}$$ is the weighted sum of the component squares of $${\mathbf{F}}$$. The quadratic form coefficients depend only on the eccentricity and are represented by the Maclaurin series in even powers of $$e$$ that converge, at least for $$e < 1$$. The series coefficients are calculated up to $${{e}^{4}}$$ inclusive, so that the correction terms are of order $${{e}^{6}}$$.

In this article, we prove the relaxed triangle inequality for Southworth and Hawkins, Drummond and Jopek orbital similarity criteria on the set of non-rectilinear Keplerian orbits with the eccentricity bounded above. We give estimates of the minimal coefficients in the inequality for each criterion and show that one of the calculated coefficients is exactly minimal. The obtained inequalities can be used for the acceleration of algorithms involving pairwise distances calculations between orbits. We present an algorithm for calculation of all distances not exceeding a fixed number in a quasi-metric space and demonstrate that the algorithm is faster than the complete calculation on the set of meteors orbits. Finally, we estimate the correlation dimensions of the set of main belt asteroids orbits and meteors orbits with respect to various orbital metrics and quasi-metrics.

The diversion of a hazardous asteroid on a collisional trajectory with the Earth using a transversal thruster is considered. The thruster could be either mounted on the asteroid or used as a “gravitational tractor”. The aim of the study is to establish the fundamental possibility (or impossibility) of diverting an asteroid to a safe distance over a time of order months or years. This is acceptable, since the impact of an asteroid with a diameter of order 100 m on the Earth just after its discovery is very improbable. A model formulation of the problem in which the thruster provides a constant transversal acceleration of the asteroid is used. The corresponding Euler-type equations are transformed using the method of previous averaging. These equations are solved using the method of “slow time” power series, and the adequacy of the solutions over time scales of decades is demonstrated. An asteroid up to 55 m in diameter can be deflected over a year using a 1 N thruster. Asteroids with diameters up to 50 m can be diverted over a month, and with diameters up to 150 m over a year, using a 20 N thruster. Moving larger asteroids requires more time or more powerful engines.

The theory of figures of celestial bodies, which are in the state of hydrostatic equilibrium under the action of pressure, gravitational, and centrifugal forces, took the form of a rigorous mathematical theory in the second part of the 20th century. Fundamental physical laws served as its basis. The Huygens–Roche figure (its total mass is concentrated in the center, while the rotating atmosphere takes the equilibrium form) plays an important role in the theory. Properties of the figure are carefully examined. In particular, it is known that each isobar (surface of equal pressure) itself is one of the three-parameter family of Huygens—Roche surfaces. However, as far as we know, convexity (or its absence) has not been discussed in the literature. Meanwhile, there are non-convex figures between equilibrium ones. In the present paper, we find the curvature of the meridional section of an arbitrary Huygens—Roche figure both in closed form and in the form of a series in powers of the Clairaut parameter, which is basic in the theory of equilibrium figures. We succeeded in proving that the curvature is positive and is bounded away from zero. Hence, every surface of the family of Huygens—Roche figures is convex and has no points of flattening. Moreover, none of the curves on its surface has points of straightening.

The motion of a zero-mass point under the action of a gravitational force toward a central body $${\cal S}$$ and a perturbing acceleration P′ whose magnitude is inversely proportional to the square of the distance to $${\cal S}$$ is considered. The direction of P′ is constant in one of the three coordinate systems most widely used in astronomy: the main inertial system $${\cal O}$$ and two orbiting systems $${{\cal O}_s}$$ with their x axes along the radius vector for s = 1 and along the velocity vector for s = 2.The ratio of |P′|to the main acceleration due to the gravitation of the central body is taken to be small. An averaging transformation in a first approximation in a small parameter of the problem is applied to the equations of motion in the osculating elements. Closed expressions are obtained for the right-hand sides of the equations of motion in the mean elements. These are expressed in terms of elementary functions in the systems $${\cal O}$$ and $${{\cal O}_1}$$ ; complete elliptical integrals arise in the system $${{\cal O}_2}$$ . Closed expressions are obtained for the change-of-variable functions. All the functions encountered in the systems $${\cal O}$$ and $${{\cal O}_1}$$ are elementary functions, apart from those determining the variations of the mean anomaly. The latter is given by an integral of an elementary function, as well as a series in powers of the eccentricity that converges absolutely and uniformly when 0 ⩽ e ⩽ 1. All functions in the system $${{\cal O}_2}$$ apart from those determining the variations of the mean anomaly can be expressed in terms of incomplete elliptical integrals. The variations of the mean anomaly are calculated using a Fourier series in the mean anomaly. Integration of the averaged equations of motion will be considered in future papers. Possible applications of this model problem include the motion of an asteroid taking into account the Yarkovsky-Radziewski effect, and the motion of a spacecraft with a solar sail, when the perturbing action is inversely proportional to the square of the distance from the Sun. It stands to reason that determining the components of the vector P′ requires knowledge of the thermal-physical characteristics of the body in question and the parameters of its rotational motion in the former case, and of the orientation of the solar sail in the latter case.

One possible means of counteraction against a hazardous asteroid is discussed: destruction of the object with a nuclear device during an earlier encounter with the Earth. This is feasible, since virtually all hazardous asteroids appear in near-Earth space several times before they hit the Earth. Computations show that this method is effective and essentially harmless if certain conditions are satisfied. Two possibilities are acceptable. In the first, a spacecraft overtakes the asteroid in a heliocentric orbit. In the second, the asteroid overtakes the spacecraft, which demands a substantially lower characteristic geocentric velocity for the spacecraft. This method for eliminating this cosmic threat is reasonable in two cases: when it is not possible to achieve a soft departure of the object from the collision orbit, and when the object continuously returns to the Earth. A soft departure from a collision orbit can be required multiple times, whereas the destruction of such an object must occur only once.

Abstract Ural Federal University is one of the main and most effective centers for educating young astronomers in Russia. The traditional student scientific conferences “Physics of Space” have been successfully held annually in Kourovka Astronomical Observatory of the Ural Federal University for 50 years and have gained recognition not only in Russia. The conference initiated many educational initiatives both in the Ural region and, in general, in Russia. The astronomy education system implemented by UrFU and partners includes the following activities: 1) education and career guidance of schoolchildren in the Lyceum of UrFU, 2) activities to attract applicants, 3) training at the speciality, undergraduate, and graduate level, 4) participation in the student conferences “Physics of Space”, 5) postgraduate studies, 6) cooperation in the field of education. This activity ensures the attraction of promising youth to scientific research.

The theory of figures of equilibrium was extensively studied in the nineteenth century, when the reasons for which observed massive celestial bodies (such as the Sun, planets, and satellites) are almost ellipsoidal were discovered. The existence of exactly ellipsoidal figures was established. The gravitational potential of such figures can be represented by a Laplace series whose coefficients (the Stokes constants In) are determined by a certain integral operator. In the case of an ellipsoid of revolution with homothetic equidensites (surfaces of constant density), the general term of this series was found, and for some of the other mass distributions, the first few terms of the series were determined. In this paper, the general term of the series is found in the case where the equidensites are ellipsoids of revolution with oblateness increasing from the center to the surface. Simple estimates and asymptotics of the coefficients In are also found. It turns out that the asymptotics depends only on the mean density, the density on the surface of the outer ellipsoid, and the oblateness of the outer ellipsoid.