Implementation of gravitational shocks in two-dimensional Fokker-Planck models

¹ Department of Astronomy & Space ScienceKyung Hee University, Yongin, 446-701 Kyungki, Korea
e-mail: sungsoo.kim@khu.ac.kr
² School of Space Research (WCU program), Kyung Hee University, Yongin, 446-701 Kyungki, Korea
³ Department of Informational Society Studies, Saitama Institute of Technology, Fukaya, 369-0293 Saitama, Japan

Received: 21 December 2011
Accepted: 28 February 2012

Abstract

We derive analytical formulae for drift and dispersion terms of energy and angular momentum (⟨ΔE⟩,  ⟨(ΔE)²⟩,  ⟨ΔJ⟩, and  ⟨(ΔJ)²⟩) as well as their cross term (⟨ΔEΔJ⟩) for stellar systems under an impulsive perturbation. These terms are expressed as functions of E, J, and orbit averages of powers of radius (r) and the product of radius and velocity (rv), and we confirm our formulae with a numerical simulation. Then with another numerical simulation for a time-varying (Gaussian) perturbation, we find that the adiabatic corrections suggeted by Gnedin and Ostriker can be applied not only to the energy changes (drift and dispersion) but also to the angular momentum changes, if the changes are expressed as functions of energy only. The corrections do not describe the changes accurately when the changes are considered as functions of both energy and angular momentum. The deviations between the numerical simulation and analytical expectations are consierable only in the cluster core though, where the effect of perturbation is relatively weaker. These results are to be implemented in two-dimensional (E − J) Fokker-Planck models of the evolution of globular clusters.