Natural orbit approximations in single power-law potentials

Abstract

In a previous paper, I demonstrated the accuracy of simple, precessing, power ellipse (p-ellipse) approximations to orbits of low-to-moderate eccentricity in power-law potentials. Here I explore several extensions of these approximations to improve accuracy, especially for nearly radial orbits. 1) It is found that moderately improved orbital fits can be achieved with higher order perturbation expansions (in eccentricity), with the addition of `harmonic' terms to the solution. 2) Alternately, a matching of the extreme radial excursions of an orbit can be imposed, and a more accurate estimate of the eccentricity parameter is obtained. However, the error in the precession frequency is usually increased. 3) A correction function of small magnitude corrects the frequency problem. With this correction, even first order approximations yield excellent fits at quite high eccentricity over a range of potential indices that includes flat and falling rotation curve cases. 4) Adding a first harmonic term to fit the breadth of the orbital loops, and determining the fundamental and harmonic coefficients by matching to three orbital positions further improves the fit. With a couple of additional small corrections one obtains excellent fits to orbits with radial ranges of more than a thousand for some potentials. These simple corrections to the basic p-ellipse are basically in the form of several successive approximations, and can provide high accuracy. They suggest new results including that the apsidal precession rate scales approximately as $log(1-e)$ at very high eccentricities $e$. New insights are also provided on the occurrence of periodic orbits in various potentials, especially at high eccentricity.

arXiv comment12 pages, 8 figures, MNRAS accepted, typos and minor grammatical errors corrected in resubmission

References in corpus (11)

• Action-space clustering of tidal streams to infer the Galactic potential
• A map for eccentric orbits in triaxial potentials
• The symmetries and scaling of tidal tails in galaxies
• On the Orbit Structure of the Logarithmic Potential
• Simple, Accurate, Approximate Orbits in the Logarithmic and a Range of Power-Law Galactic Potentials
• Hamiltonians of Spherically Symmetric, Scale-Free Galaxies in Action-Angle Coordinates
• Analytic Central Orbits and their Transformation Group
• A Study of the Orbits of the Logarithmic Potential for Galaxies
• Periodic orbits in the logarithmic potential
• Closed, spirograph-like orbits in power law central potentials
• Torus-fitting method for obtaining action variables in two-dimensional Galactic potentials

Cited by in corpus (8)

• Timelike geodesics in Naked Singularity and Black Hole Spacetimes
• Towards an observational test of black hole versus naked singularity at the galactic center
• Particle motion and tidal force in a non-vacuum-charged naked singularity
• Lindblad Zones: resonant eccentric orbits to aid bar and spiral formation in galaxy discs
• Retrograde Precession of Relativistic Orbits and the Quest for Charged Black Holes
• A Warp in Progress: H I and Radio Continuum Observations of the Spiral NGC 3145
• Orbits in elementary, power-law galaxy bars: 1. Occurence and role of single loops
• Post-Newtonian properties of EMRI with Power Law Potential