K. V. Kholshevnikov

Impact-generated dust clouds around planetary satellites: spherically symmetric case

Abstract An analytic model of an impact-generated, steady-state, spherically symmetric dust cloud around an atmosphereless planetary satellite (or planet—Mercury, Pluto) is constructed. The projectiles are assumed to be interplanetary micrometeoroids. The model provides the expected mass, density, and velocity distributions of dust in the vicinities of parent bodies. Applications are made to Jupiters moon Ganymede and six outer satellites of Saturn. In the former case, the model is shown to be consistent with the measurements of the dust detector system onboard the Galileo spacecraft. In the latter case, estimates are given and recommendations are made for the planned experiment with the Cassini cosmic dust analyzer (CDA) during targeted flybys of the spacecraft with the moons. The best CDA pointing to maximize the number of detections is in the ram direction. With this pointing, measurements are possible within a few to about 20 min from the closest approach, with maximum minute impact rates ranging from about 1 for Phoebe and Hyperion to thousands for Enceladus. Detections of the ejecta clouds will still be likely if CDAs angular offset from the ram direction does not exceed 45°. The same model can be applied to dust measurements by other space missions, like New Horizons to Pluto or BepiColombo to Mercury.

Manifolds and Metrics in the Relative Spacecraft Motion Problem

This paper establishes a methodology for obtaining the general solution to the spacecraft relative motion problem by utilizing the Cartesian configuration space in conjunction with classical orbital elements. The geometry of the relative motion configuration space is analyzed, and the relative motion invariant manifold is determined. Most importantly, the geometric structure of the relative motion problem is used to derive useful metrics for quantification of the minimum, maximum, and mean distance between spacecraft for commensurable and noncommensurable mean motions. A number of analytic solutions as well as useful examples are provided, illustrating the calculated bounds. A few particular cases that yield simple solutions are given. Nomenclature a = semimajor axis E = eccentric anomaly E = follower orbit e = eccentricity F = follower perifocal frame f = true anomaly I = inertial frame i = inclination Jk = Bessel function L = leader-fixed frame M = mean anomaly n = mean motion n0 = fundamental frequency R = leader position vector R = relative motion invariant manifold r = follower position vector W = distance function α = normalized semimajor axis μ = gravitational constant ρ = relative position vector � = right ascension of the ascending node ω = argument of periapsis ω = angular velocity vector |·| = vector norm �·� = signal norm Superscripts � = leader ∗ = relative orbital element

On the Distance Function Between Two Keplerian Elliptic Orbits

The problem of finding critical points of the distance function between two Keplerian elliptic orbits is reduced to the determination of all real roots of a trigonometric polynomial of degree 8. The coefficients of the polynomial are rational functions of orbital parameters. Using computer algebra methods we show that a polynomial of a smaller degree with such properties does not exist. This fact shows that our result cannot be improved and it allows us to construct an optimal algorithm to find the minimal distance between two Keplerian orbits.

The Expansion of the Hamiltonian of the Two-Planetary Problem into a Poisson Series in All Elements: Estimation and Direct Calculation of Coefficients

This is the second paper in a series of articles devoted to one of the basic problems of celestial mechanics: the evolution of solar-type planetary systems. In the first paper (Kholshevnikov et al., 2001), we reviewed the history and the current state of the issue, outlined the scheme of the study, introduced Jacobi coordinates and related osculating elements, and indicated the form of the Hamiltonian expansion into a Poisson series in all elements. In this paper, the expansion coefficients are found according to a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. At the first stage, we restricted our analysis to the two-planetary problem (Sun–Jupiter–Saturn). The general case will be investigated in a forthcoming paper.

The Expansion of the Hamiltonian of the Planetary Problem into the Poisson Series in All Keplerian Elements (Theory)

The study of the evolution of planetary systems, primarily of the Solar System, is one of the basic problems of celestial mechanics. The stability of motion of giant planets on cosmogonic time scales was established by numerical and analytical methods, but the question about the evolution of orbits of terrestrial planets and arbitrary solar-type planetary systems remained open. This work initiates a series of papers allowing one to advance in solving the problem of the evolution of the solar-type planetary systems on cosmogonic time scales by using powerful analytical tools. In the first paper of this series, we choose the optimum reference system and obtain the Poisson series expansion of the Hamiltonian of the problem in all Keplerian elements. We propose to use the integral representation of the corresponding coefficients or the Poisson processor means instead of conventionally addressing any possible special functions. This approach extremely simplifies the algorithm. The next paper of this series deals with the calculation of the expansion coefficients.

A concept of a space hazard counteraction system: Astronomical aspects

The basic science of astronomy and, primarily, its branch responsible for studying the Solar System, face the most important practical task posed by nature and the development of human civilization—to study space hazards and to seek methods of counteracting them. In pursuance of the joint Resolution of the Federal Space Agency (Roscosmos) and the RAS (Russian Academy of Sciences) Space Council of June 23, 2010, the RAS Institute of Astronomy in collaboration with other scientific and industrial organizations prepared a draft concept of the federal-level program targeted at creating a system of space hazard detection and counteraction. The main ideas and astronomical content of the concept are considered in this article.

Dynamical evolution of weakly disturbed two-planetary system on cosmogonic time scales: The Sun-Jupiter-Saturn system

In the present paper, we used the Hori-Deprit method to construct the averaged Hamiltonian of the two-planetary problem accurate to the second order of a small parameter, the generating function of the transform, the change of variables formulas, and the right-hand sides of the equations in average elements. The evolution of the two-planet Sun-Jupiter-Saturn system was investigated by numerical integration over 10 billion years. The motion of the planets has an almost periodic character. The eccentricities and inclinations of Jupiter’s and Saturn’s orbits remain small but different from zero. The short-term disturbances remain small over the entire period considered in the study.

On linking coefficient of two Keplerian orbits

We define a function of the set of pairs of Keplerian ellipses so that the sign of the function will be a topological invariant of their configuration. The sign is negative if and only if the related ellipses are linked. Two modifications of the coefficient which are more reliable in the case of closed to coplanar orbits are proposed. Explicit formulae representing the linking coefficients as functions of orbital elements are deduced. Extension in the case of unbounded orbits is obtained. We suggest different ways to use these coefficients for determining intersections of pairs of osculating Keplerian orbits. If we study dynamical behaviour of geometric configuration of pairs of Keplerian orbits, we can fix the moments of their intersections. These moments correspond exactly to the vanishing of linking coefficients.

On properties of integrals of the Legendre polynomial

AbstractProperties of the integrals

Estimating the derivative of the Legendre polynomial

P_{n0} (x) = P_n (x),P_{nk} (x) = \int\limits_{ - 1}^x {P_{n,k - 1} (y)dy}