Velocity-density twin transforms in thin disk model

TL;DR

This paper explores the mathematical relationship between ring mass density and circular velocity in thin disk models, simplifying their integral transforms for practical computation and analysis.

Contribution

It presents a reduction of the integral transforms to one-fold integrals with identical weight functions, simplifying calculations compared to traditional formulas.

Findings

Transform simplification facilitates easier computation of disk properties

The new integral form avoids velocity derivative, improving numerical stability

Potential applications in galactic dynamics modeling

Abstract

Ring mass density and the corresponding circular velocity in thin disk model are known to be integral transforms of one another. But it may be less familiar that the transforms can be reduced to one-fold integrals with identical weight functions. It may be of practical value that the integral for the surface density does not involve the velocity derivative, unlike the equivalent and widely known Toomre's formula.