Time-dependent thermal X-ray afterglows from GRBS

Time-dependent thermal X-ray spectra are calculated from physically plausible conditions around GRB. It is shown that account for time-dependent ionization processes strongly affects the observed spectra of hot rarefied plasma. These calculations may provide an alternative explanation to the observed X-ray lines of early GRBs afterglows (such as GRB 011211). Our technique will allow one to obtain independent constraints on the GRB collimation angle and on the clumpiness of circumstellar matter.     

Regularization of the Limit Problem of Satellite Librations in a Keplerian Orbit Taking Light Pressure into Account

The planar librations of a satellite whose center of mass moves along an elliptic orbit are considered. It is assumed that not only the gravitational moment but also the forces of light pressure act upon the satellite. Account is taken of the fact that the right-hand sides of the differential equations are nonanalytic functions of the phase variables. When e 1, e being the orbit's eccentricity, the deformations of solutions are considered for the case of a satellite moving along a highly elongated orbit. Such transformation of the initial system of differential equations is carried out so that the new system becomes regular up to the value e = 1. A limit problem corresponding to the case e = 1 is considered. When the azimuth angle of the light source coincides with the direction to the pericenter, the dynamical system is reversible. In this case, the known families of the periodic solutions to the problem can be continued up to the limit case.     

Stability and bifurcations of relative equilibria of a pendulum suspended on the equator

The problem of equilibria of a pendulum suspended at an equatorial point relative to the rotating Earth is considered. An altitude is determined at which the degree of instability of the inverted pendulum changes from two to unity. Relative equilibria are investigated that bifurcate from the radial one when its degree of instability changes. Their stability properties are studied.     

On plane oscillations of a pendulum with variable length suspended on the surface of a planet’s satellite

The problem of planar oscillations of a pendulum with variable length suspended on the Moon’s surface is considered. It is assumed that the Earth and Moon (or, in the general case, a planet and its satellite, or an asteroid and a spacecraft) revolve around the common center of mass in unperturbed elliptical Keplerian orbits. We discuss how the change in length of a pendulum can be used to compensate its oscillations. We wrote equations of motion, indicated a rule for the change in length of a pendulum, at which it has equilibrium positions relative to the coordinate system rotating together with the Moon and Earth. We study the necessary conditions for the stability of these motions. Chaotic dynamics of the pendulum is studied numerically and analytically.