Bound central orbits

Bound near-equatorial orbits around neutron stars

Recent discovery of kilohertz quasi-periodic brightness oscillations of low mass X-ray binaries (LMXBs) has attracted attention to highly relativistic periodic motion near accreting neutron stars. Most models proposed so far involve (almost) inertial motion in the vicinity of the stars’ innermost stable circular orbits. In the present paper we study general-relativistic circular and eccentric o...

Orbits in Central Force Fields III

Analytic Central Orbits and their Transformation Group

A useful crude approximation for Abelian functions is developed and applied to orbits. The bound orbits in the power-law potentials Ar take the simple form (l/r) = 1 + e cos(mφ), where k = 2 − α > 0 and l and e are generalisations of the semilatus-rectum and the eccentricity. m is given as a function of ‘eccentricity’. For nearly circular orbits m is √ k, while the above orbit becomes exact at ...

Hydrodynamic capture of microswimmers into sphere-bound orbits.

Self-propelled particles can exhibit surprising non-equilibrium behaviors, and how they interact with obstacles or boundaries remains an important open problem. Here we show that chemically propelled micro-rods can be captured, with little change in their speed, into close orbits around solid spheres resting on or near a horizontal plane. We show that this interaction between sphere and particl...

Central Symmetries of Periodic Billiard Orbits in Right Triangles

We show that the midpoint of each periodic perpendicular beam hits the right-angle vertex of the triangle. The beam returns to itself after half its period with the opposite orientation, i.e. a Möbeius band.