Trajectories in Logarithmic Potentials

Trajectories in logarithmic potentials are investigated by taking as example the motion of an electron within a cylindrical capacitor. The solution of the equation of motion in plane polar coordinates, (r,{\phi}) is attained by forming a series expansion of r and of 1/r as a function of {\phi}. The terms of the series contain polynomials, the recurrence relation of which is given, together with some further characteristics. By the comparison-theorem of infinite series, the convergence of the solution is demonstraded. The simplest trajectories in logarithmic potentials are represented by rosette type orbits with a period of 4{\pi}/3, and by circular paths.Comment: 17 pages, 5 figure

Analytical model for non-thermal pressure in galaxy clusters - III. Removing the hydrostatic mass bias

Non-thermal pressure in galaxy clusters leads to underestimation of the mass of galaxy clusters based on hydrostatic equilibrium with thermal gas pressure. This occurs even for dynamically relaxed clusters that are used for calibrating the mass-observable scaling relations. We show that the analytical model for non-thermal pressure developed in Shi & Komatsu 2014 can correct for this so-called 'hydrostatic mass bias', if most of the non-thermal pressure comes from bulk and turbulent motions of gas in the intracluster medium. Our correction works for the sample average irrespective of the mass estimation method, or the dynamical state of the clusters. This makes it possible to correct for the bias in the hydrostatic mass estimates from X-ray surface brightness and the Sunyaev-Zel'dovich observations that will be available for clusters in a wide range of redshifts and dynamical states.Comment: 9 pages, 8 figures, published in MNRA