Publikace ÚTF

The dynamical system studied in previous papers of this series, namely a bound time-like
geodesic motion in the exact static and axially symmetric space–time of an (originally)
Schwarzschild black hole surrounded by a thin disc or ring, is considered to test whether
the often employed ‘pseudo-Newtonian’ approach (resorting to Newtonian dynamics in gravitational
potentials modified to mimic the black hole field) can reproduce phase-space properties
observed in the relativistic treatment. By plotting Poincar´e surfaces of section and using two
recurrence methods for similar situations as in the relativistic case, we find similar tendencies
in the evolution of the phase portrait with parameters (mainly with mass of the disc/ring and
with energy of the orbiters), namely those characteristic to weakly non-integrable systems.
More specifically, this is true for the Paczy´nski–Wiita and a newly suggested logarithmic potential,
whereas the Nowak–Wagoner potential leads to a different picture. The potentials and
the exact relativistic system clearly differ in delimitation of the phase-space domain accessible
to a given set of particles, though this mainly affects the chaotic sea whereas not so much
the occurrence and succession of discrete dynamical features (resonances). In the pseudo-
Newtonian systems, the particular dynamical features generally occur for slightly smaller
values of the perturbation parameters than in the relativistic system, so one may say that the
pseudo-Newtonian systems are slightly more prone to instability. We also add remarks on
numerics (a different code is used than in previous papers), on the resemblance of dependence
of the dynamics on perturbing mass and on orbital energy, on the difference between the
Newtonian and relativistic Bach–Weyl rings, and on the relation between Poincar´e sections
and orbital shapes within the meridional plane.