Behaviour of a Two-planetary System on a Cosmogonic Time-scale

The orbital evolution of planetary systems similar to our Solar one represents one of the most important problems of Celestial Mechanics. In the present work we use Jacobian coordinates, introduce two systems of osculating elements, construct the Hamiltonian expansions in Poisson series for all the elements for the planetary three-body problem (including the problem Sun-Jupiter-Saturn). Further we construct the averaged Hamiltonian by the Hori-Deprit method with accuracy up to second order with respect to the small parameter, the generating function, the change of variables formulae, and the right-hand sides of the averaged equations. The averaged equations for the Sun-Jupiter-Saturn system are integrated numerically over a time span of 10 Gyr. The Liapunov Time turns out to be 14 Myr (Jupiter) and 10 Myr (Saturn). © 2005 International Astronomical Union.This work was partly supported by the RFBR, Grant 02-02-17516, and the Leading Scientific School, Grant NSh-1078.2003.02

Family of Metrics in the Space of Keplerian Orbits

Five-dimensional space of non-rectilinear Keplerian orbits is considered, as well as four its quotient spaces. In the last ones orbits are identified irrespective of values of longitudes of nodes, values of arguments of pericentres, values of both longitudes of nodes and arguments of pericentres, values of longitudes of nodes and arguments of pericentres under fixed values of longitudes of pericentres. All these spaces (except the last one) becomes metric spaces by introducing a suitable metric. Usable formulae for calculation of distances between orbits via their Keplerian elements are given. As to the last quotient space, the constructed for it function of a pair of orbits satisfies first two axioms of metric spaces. The validity of the third axiom (triangle axiom) is not demonstrated or disproved yet. The introduced orbital spaces, together with metrics, serve as a good tool for problems of searching close orbits, and identification of parent bodies in comet-asteroid-meteoroid complexes.Рассматриваются пятимерное пространство непрямолинейных кеплеровых орбит и четыре его фактор-пространства. В последних отождествляются орбиты вне зависимости от значений долгот узлов, значений аргументов перицентров, значений долгот узлов и аргументов перицентров, значений долгот узлов и аргументов перицентров при фиксированных долготах перицентров. Все указанные пространства (за исключением последнего) превращаются в метрические введением подходящих метрик. Приводятся рабочие формулы для вычисления расстояний между орбитами по их кеплеровым элементам. Что касается последнего факторпространства, то построенная для него функция пары орбит удовлетворяет первым двум аксиомам метрического пространства. Справедливость третьей аксиомы (аксиомы треугольника) пока не доказана и не опровергнута. Введенные пространства орбит вместе с метриками являются хорошим инструментом для задач поиска близких орбит и отождествления родительских тел в кометно-астероидно-метеороидных комплексах.Работа выполнена при финансовой поддержке РНФ (грант 18-12-00050)

The preventive destruction of a hazardous asteroid

One means of countering a hazardous asteroid is discussed: destruction of the object using a nuclear charge. Explosion of such an asteroid shortly before its predicted collision would have catastrophic consequences, with numerous highly radioactive fragments falling onto the Earth. The possibility of exploding the asteroid several years before its impact is also considered. Such an approach is made feasible because the vast majority of hazardous objects pass by the Earth several times before colliding with it. Computations show that, in the 10 years following the explosion, only a negligible number of fragments fall onto the Earth, whose radioactivity has substantially reduced during this time. In most cases, none of these fragments collides with the Earth. Thus, this proposed method for eliminating a threat from space is reasonable in at least two cases: when it is not possible to undergo a soft removal of the object from the collisional path, and to destroy objects that are continually returning to near-Earth space and require multiple removals from hazardous orbits

Laplace series for ellipsoidal structure’s bodies and level ellipsoid

Теория фигур равновесия активно развивалась в XIX столетии, когда выяснились причины, по которым наблюдаемые массивные небесные тела (Солнце, планеты, спутники) обладают близкой к эллипсоидальной формой. Было установлено, что существуют и в точности эллипсоидальные фигуры. Гравитационный потенциал таких фигур представляется рядом Лапласа. Его коэффициенты (гармонические коэффициенты, или постоянные Стокса In) определяются одним из двух способов. Во-первых, с помощью некоторого интегрального оператора, если известно распределение плотности внутри тела. Во-вторых, с помощью преобразования внешнего гравитационного потенциала, если известен последний. В представленной работе первым способом найдена асимптотика In для эллипсоида, эквиденситы (поверхности равной плотности) которого являются эллипсоидами вращения с возрастающим от центра к периферии сжатием. Оказалось, что асимптотика зависит только от средней плотности, плотности на поверхности внешнего эллипсоида и его сжатия. Вторым способом найдены In и их асимптотика для уровенного эллипсоида. Эти асимптотики совпадают только для эллипсоидов Маклорена. Следовательно, если уровенный эллипсоид не является маклореновским, его эквиденситы не могут быть эллипсоидами.Theory of the figures of equilibrium was developed actively during XIX century when causes were discovered making the form of observable massive celestial bodies (the Sun, planets, moons) almost ellipsoidal. The existence of exactly ellipsoidal figures was established. The gravitational potential of such figures can be presented by the Laplace series. Its coefficients (harmonic coefficients, or Stokes constants In) are determined via one of two ways, first, by a definite integral operator if density distribution inside the body is known, second, by a certain transformation of the outer gravitational potential if it is known. In the present paper asymptotics of In is found using the first approach for an ellipsoid if its equidensites (surfaces of equal density) are ellipsoids of revolution. It is supposed that equidensites’ oblateness increases from the centre to the periphery. It turned up that asymptotics depend on the mean density, density on the surface of the boundary ellipsoid, and its oblateness only. Coefficients In and their asymptotics are found using the second approach for a level ellipsoid. Both asymptotics coincide for Maclaurin ellipsoids only. Hence, if the level ellipsoid is not a Maclaurin one then its equidensites cannot be ellipsoids.Работа выполнена при поддержке гранта РФФИ 18–02–00552

The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities

We study the long term evolution of the distance between two Keplerian confocal trajectories in the framework of the averaged restricted 3-body problem. The bodies may represent the Sun, a solar system planet and an asteroid. The secular evolution of the orbital elements of the asteroid is computed by averaging the equations of motion over the mean anomalies of the asteroid and the planet. When an orbit crossing with the planet occurs the averaged equations become singular. However, it is possible to define piecewise differentiable solutions by extending the averaged vector field beyond the singularity from both sides of the orbit crossing set. In this paper we improve the previous results, concerning in particular the singularity extraction technique, and show that the extended vector fields are Lipschitz-continuous. Moreover, we consider the distance between the Keplerian trajectories of the small body and of the planet. Apart from exceptional cases, we can select a sign for this distance so that it becomes an analytic map of the orbital elements near to crossing configurations. We prove that the evolution of the 'signed' distance along the averaged vector field is more regular than that of the elements in a neighborhood of crossing times. A comparison between averaged and non-averaged evolutions and an application of these results are shown using orbits of near-Earth asteroids.Comment: 29 pages, 8 figure

Effect of Sun and Planet-Bound Dark Matter on Planet and Satellite Dynamics in the Solar System

We apply our recent results on orbital dynamics around a mass-varying central body to the phenomenon of accretion of Dark Matter-assumed not self-annihilating-on the Sun and the major bodies of the solar system due to its motion throughout the Milky Way halo. We inspect its consequences on the orbits of the planets and their satellites over timescales of the order of the age of the solar system. It turns out that a solar Dark Matter accretion rate of \approx 10^-12 yr^-1, inferred from the upper limit \Delta M/M= 0.02-0.05 on the Sun's Dark Matter content, assumed somehow accumulated during last 4.5 Gyr, would have displaced the planets faraway by about 10^-2-10^1 au 4.5 Gyr ago. Another consequence is that the semimajor axis of the Earth's orbit, approximately equal to the Astronomical Unit, would undergo a secular increase of 0.02-0.05 m yr^-1, in agreement with the latest observational determinations of the Astronomical Unit secular increase of 0.07 +/- 0.02 m yr^-1 and 0.05 m yr^-1. By assuming that the Sun will continue to accrete Dark Matter in the next billions year at the same rate as in the past, the orbits of its planets will shrink by about 10^-1-10^1 au (\approx 0.2-0.5 au for the Earth), with consequences for their fate, especially of the inner planets. On the other hand, lunar and planetary ephemerides set upper bounds on the secular variation of the Sun's gravitational parameter GM which are one one order of magnitude smaller than 10^-12 yr^-1. Dark Matter accretion on planets has, instead, less relevant consequences for their satellites. Indeed, 4.5 Gyr ago their orbits would have been just 10^-2-10^1 km wider than now. (Abridged)Comment: LaTex2e, 17 pages, no figures, 7 tables, 61 references. Small problem with a reference fixed. To appear in Journal of Cosmology and Astroparticle Physics (JCAP

On the representation of the gravitational potential of several model bodies

A Laplace series of spherical harmonics Yn(θ, λ) is the most common representation of the gravitational potential for a compact body T in outer space in spherical coordinates r, θ, λ. The Chebyshev norm estimate (the maximum modulus of the function on the sphere) is known for bodies of an irregular structure:〈Yn〉 ≤ Cn–5/2, C = const, n ≥ 1. In this paper, an explicit expression of Yn(θ, λ) for several model bodies is obtained. In all cases (except for one), the estimate 〈Yn〉 holds under the exact exponent 5/2. In one case, where the body T touches the sphere that envelops it,〈Yn〉 decreases much faster, viz.,〈Yn〉 ≤ Cn–5/2pn, C = const, n ≥ 1. The quantity p < 1 equals the distance from the origin of coordinates to the edge of the surface T expressed in enveloping sphere radii. In the general case, the exactness of the exponent 5/2 is confirmed by examples of bodies that more or less resemble real celestial bodies [16, Fig. 6]. © 2016, Allerton Press, Inc