Malfunction of satellite navigation systems GPS and GLONASS caused by powerful radio emission of the Sun during solar flares on December 6 and 13, 2006, and October 28, 2003

Poor quality of functioning of GPS during solar flares on December 6 and 13, 2006 is analyzed in this paper. These flares were accompanied by extremely high (unexampled) level of the solar radio emission flux. A comparison is made of these events with the solar flare on October 28, 2003. Statistically reliable experimental evidence is obtained that GPS positioning was partially paralyzed on the sunlit side of the Earth during the strongest bursts of solar radio emission. The obtained results give a serious ground to revise the role played by space weather factors in operation of modern satellite systems and to take these factors into account more carefully, when such systems are designed and exploited.     

The motion of a particle in perturbed field of an attracting center

The integrable case of the perturbed two-body problem is considered. The perturbation is determined by the potential of a special form. The L-matrix is chosen in such a way that partial separation of variables should take place in regular coordinates. Integration of the equations of motion of the problem under consideration is made. The solutions are expressed through elliptic functions. The orbits for various cases are constructed. The results of numerical calculations are given.     

Variational equations in parametric variables and transformation of their solutions

An analytical relationship is derived which connects the matrix of partial derivatives of the current state vector with respect to the initial conditions with the matrix of solutions to variational equations in parametric variables. These equations correspond to the equations of motion obtained using the generalized coordinate-time transformation admitting an increase of the number of unknown parameters. Particular cases of transformations leading to the variational equations in regularizing variables of Sperling-Burdet and Kustaancheimo-Stiefel are considered. All formulas necessary in these cases for determination of the matrix of isochronous derivatives are obtained.     

The family of potentials admitting integration of the perturbed two-body problem in regular coordinates

An infinite system of potentials is presented that admits separation of regular variables in the perturbed two-body problem. The regular coordinates are constructed using a specially selected L-matrix. An explicit solution to the problem in the elliptical case is constructed. In the general case the solution is reduced to inversion of hyper-elliptic integrals. The cases of motion with and without constraints are considered. The results of numerical experiments are presented.     

Stochastic motion of geosynchronous satellites

Long-period evolution of the orbital elements of geosynchronous objects moving in the neighborhood of separatrices of the region of libration resonance is studied. At the phase plane “the longitude of the subsatellite point-semi-major axis” the regions of migration of separatrices where the separatrices can move due to the influence of perturbations are isolated. The dimensions of these regions and the distances between them have the same order of magnitude, several kilometers along the semi-major axis. The regions of migration do not overlap at the initial value of the orbit inclination 0°. When inclination increases, gradual overlapping of these regions takes place. On average, the width of the zones of stochasticity increases when the area to mass ratio increases and decreases when inclination of the orbit increases. The zones of stochasticity corresponding to different separatrices are separated from each other. A catalog of the passive geosynchronous objects moving in the neighborhood of separatrices is compiled. The evolution of the objects of the catalog is studied.     

Specific features of motion of geosynchronous satellites in the neighborhood of unstable stationary points

A study of peculiarities of the motion of equatorial circular geosynchronous satellites is performed in the neighborhoods of unstable stationary points with longitudes of 165° and 345°. The maps of initial conditions are constructed on the phase plane “longitude of subsatellite point-semi-major axis of orbit” corresponding to various types of regular motions and to quasi-random solutions. The dimensions of zones of the quasi-random solutions are within the limits from decimal fractions of a degree to a few degrees (in longitude) and from hundreds of meters up to several kilometers (along the semi-major axis).