Expansion of the principal functions of Keplerian motion using complex variables

A special system of canonical variables is considered. An algorithm for expanding the principal functions of Keplerian motion in new elements is presented. The advantage of the proposed system is a relatively small number of terms in the classical expansions of the unperturbed two-body problem. A method for expanding the time derivatives of the rectangular coordinates is proposed. Some estimates of the number of terms in the presented expansions have been obtained through numerical experiments.     

Analysis of the Stability of a Planetary System on Cosmogonic Time Scales

Astronomy Letters - We consider the dynamical evolution of planetary systems whose structure is nearly circular and coplanar. The analysis is performed by the Hori–Deprit averaging method...     

Electric State of the Near-Surface Atmosphere according to the Results of Tethered Balloon Observations

The electric state of the near-surface atmosphere up to a height of 400 m is investigated using a tethered balloon with a measuring platform and a ground-based information-measuring complex of the Borok middle-latitude geophysical observatory. For the first time, measurements were taken simultaneously for vertical profiles of the atmospheric electric field, polar electrical conductivities, size distribution of aerosol particles, and the volume activity of radon, which have allowed estimating the average values and variability of the space charge density and conduction current in the atmosphere. The height dependence of the electric potential with respect to the Earth’s surface and electrical resistance of the near-surface atmospheric column under different conditions of the temperature stratification is studied.

     

On the Definition of the Astrocentric Coordinates in the Planetary Problem

Astronomy Letters - Two ways of defining the Poincaré astrocentric coordinates in the planetary problem are considered and compared. Both ways are shown to lead to the same system of equations...     

Expansion of the Hamiltonian for the planetary problem into a Poisson series in the heliocentric reference frame

An expansion of the Hamiltonian for the N-planet problem into a Poisson series using a system of modified (complex) Poincare´ canonical elements in the heliocentric coordinate system is constructed. The Lagrangian and Hamiltonian formalisms are used. The first terms in the expansions of the principal and complementary parts of the disturbing function are presented. Estimates of the number of terms in the presented expansions have been obtained through numerical experiments. A comparison with the results of other authors is made.     

Averaging the Equations of a Planetary Problem in an Astrocentric Reference Frame

A system of averaged equations of planetary motion around a central star is constructed. An astrocentric coordinate system is used. The two-planet problem is considered, but all constructions are easily generalized to an arbitrary number N of planets. The motion is investigated in modified (complex) Poincarécanonical elements. The averaging is performed by the Hori–Deprit method over the fast mean longitudes to the second order relative to the planetary masses. An expansion of the disturbing function is constructed using the Laplace coefficients. Some terms of the expansion of the disturbing function and the first terms of the expansion of the averaged Hamiltonian are given. The results of this paper can be used to investigate the evolution of orbits with moderate eccentricities and inclinations in various planetary systems.