Some conservative nonlinear integral operators

Zadanie pt. „Digitalizacja i udostępnienie w Cyfrowym Repozytorium Uniwersytetu Łódzkiego kolekcji czasopism naukowych wydawanych przez Uniwersytet Łódzki” nr 885/P-DUN/2014 zostało dofinansowane ze środków MNiSW w ramach działalności upowszechniającej naukę

Dynamics and Heating of the Magnetic Network on the Sun: Efficiency of mode transformation

We aim to identify the physical processes which occur in the magnetic network of the chromosphere and which contribute to its dynamics and heating. Specifically, we study the propagation of transverse (kink) MHD waves which are impulsively excited in flux tubes through footpoint motions. When these waves travel upwards, they get partially converted to longitudinal waves through nonlinear effects (mode coupling). By solving the nonlinear, time-dependent MHD equations we find that significant longitudinal wave generation occurs in the photosphere typically for Mach numbers as low as 0.2 and that the onset of shock formation occurs at heights of about 600 km above the photospheric base. We also investigate the compressional heating due to longitudinal waves and the efficiency of mode coupling for various values of the plasma $\beta$, that parameterises the magnetic field strength in the network. We find that this efficiency is maximum for field strengths corresponding to $\beta\approx 0.2$, when the kink and tube wave speeds are almost identical. This can have interesting observational implications. Furthermore, we find that even when the two speeds are different, once shock formation occurs, the longitudinal and transverse shocks exhibit strong mode coupling.Comment: 8 pages, 3 figure

The three-body problem

The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more than 300 years. In this paper, we present a review of the three-body problem in the context of both historical and modern developments. We describe the general and restricted (circular and elliptic) three-body problems, different analytical and numerical methods of finding solutions, methods for performing stability analysis, search for periodic orbits and resonances, and application of the results to some interesting astronomical and space dynamical settings. We also provide a brief presentation of the general and restricted relativistic three-body problems, and discuss their astronomical applications.Comment: 49 pages, 10 figures, Published in Reports on Progress in Physic

Orbital Stability of Planets in Binary Systems: A New Look at Old Results

About half of all known stellar systems with Sun-like stars consist of two or more stars, significantly affecting the orbital stability of any planet in these systems. This observational evidence has prompted a large array of theoretical research, including the derivation of mathematically stringent criteria for the orbital stability of planets in stellar binary systems, valid for the "coplanar circular restricted three-body problem". In the following, we use these criteria to explore the validity of results from previous theoretical studies.Comment: 3 pages, 1 figure; submitted to: Exoplanets: Detection, Formation and Dynamics, IAU Symposium 249, eds. Y.-S. Sun, S. Ferraz-Mello, and J.-L. Zhou (Cambridge: Cambridge University Press