Computation of bound orbits in the plane of a galaxy with a flat rotation curve

Abstract

A standard topic in an advanced undergraduate classical mechanics course is the determination of the orbits in a gravitational field. In the present paper we report on the calculation of bound orbits in the gravitational field of a spiral galaxy. Calculations such as these could serve to focus attention on an area of cutting edge astrophysics and could serve as an instructive exercise for advanced undergraduates. In the computations given in this paper, use is made of real data on the flat rotation curve of NGC 3198 obtained by Begeman et al (van Albada et al 1985 Astrophys. J. 295 305–13; Begeman 1989 Astron. Astrophys. 223 47–60; Begeman 1987 PhD Thesis University of Groningen http://irs.ub.rug.nl/ppn/291578543), and a fitting of that data to a theoretical model outlined in a previous paper (Bacon and Sharrar 2010 Am. J. Phys. at press). The galaxy is modelled as a thin exponential disc of baryonic matter combined with a spherically symmetric dark matter halo. The bound orbits in the plane of the galaxy are investigated. The computations are carried out using an icon-driven systems-modelling program that avoids the need for extensive programming expertise. The range of orbits investigated includes bound circular orbits and bound closed and open orbits that precess. The bound closed and open orbits are bounded by circles generated by the loci of the apsides of the orbit.

A New Approach to the Yang-Mills Gauge Theory of Gravity and its Applications

“…We will see immediately that this agrees exactly with the empirical formula given by Salucci et al [30] who analyzed a large number of spiral galaxies and drew up a formula to describe their rotation curves. We have also extracted the values of G ⋆ (which is the product of the primed gravitational constant and the primed matter density in the halo) and m ′ /m (which is the ratio of the primed mass density to the regular mass density in the halo) by fitting some well-known galaxies [31,32,33,34,35], and it is amazing to find that a universal value of G ⋆ ≈ 10 −2 kpc −2 and a universal ratio of m ′ m ≈ 2 × 10 −9 fit very well with the observed results when r ranges from 3 kpc to 30 kpc. Note also that the value of GΣ 0 and h are more or less the same as those observed.…”

A Yang-Mills Type Gauge Theory of Gravity and the Dark Matter and Dark Energy Problems

“…We will see immediately that this agrees exactly with the empirical formula given by Salucci et al [13] who analyzed a large number of spiral galaxies and drew up a formula to describe their rotation curves. We have also extracted the values of G ⋆ (which is the product of the primed gravitational constant and the primed matter density in the halo) and m ′ /m (which is the ratio of the primed mass density to the regular mass density in the halo) by fitting some well-known galaxies [14][15][16][17][18], and it is amazing to find that a universal value of G ⋆ ≈ 10 −2 kpc −2 and a universal ratio of m ′ m ≈ 2 × 10 −9 fit very well with the observed results when r ranges from 3 kpc to 30 kpc. Note also that the value of GΣ 0 and h are more or less the same as those observed.…”

Quadratic Gravitational Lagrangian with Torsion Can Give Possible Explanations of the Form of Galactic Rotation Curves, of the Amount of Intergalactic Lensings, and of the Accelerating Expansion of the Universe

“…[10], we find that the theoretic predictions agree with that observed. Figure 1 and Table I give the fitting curves and fitting parameters [11][12][13][14][15].…”