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Paper: Rotation Curves of Galaxies
Volume: 165, The Third Stromlo Symposium: The Galactic Halo
Page: 325
Authors: Kalnajs, Agris J.
Abstract: One can obtain a fairly good understanding of the relation between axially symmetric mass distributions and the rotation curves they produce without resorting to calculations. However it does require a break with tradition. The first step consists of replacing quantities such as surface density, volume density, and circular velocity with the mass in a ring, mass in a spherical shell, and the square of the circular velocity, or more precisely with 2 pi G r mu(r), 4 pi G r^2 rho(r), and Vc^2 (r). These three quantities all have the same dimensions, and are related to each other by scale-free linear operators. The second step consists of introducing ln(r) as the coordinate. On the log scale the scale-free operators becomes the more familiar convolution operations. Convolutions are easily handled by Fourier techniques and a surface density can be converted into a rotation curve or volume density in a small fraction of a second. A simple plot of 2 pi G r mu(r) as a function of ln(r) reveals the relative contributions of different radii to Vc^2(r). Such a plot also constitutes a sanity test for the fitting of various laws to photometric data. There are numerous examples in the literature of excellent fits to the tails that lack data or are poor fits around the maximum of 2 pi G r mu(r). I will discuss some exact relations between the above three quantities as well as some empirical observations such as the near equality of the maxima of 2 pi G r mu(r) and Vc^2 (r) curves for flat mass distributions.
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