The face-on projection of the Miyamoto & Nagai disks

TL;DR

This paper derives an analytical formula for the face-on projected density of Miyamoto & Nagai disks with arbitrary flattening, using elliptic integrals, and verifies it against numerical results, correcting some errors in classical tables.

Contribution

It provides the first explicit analytical expression for the face-on projection of Miyamoto & Nagai disks, including corrections to known elliptic integral identities.

Findings

Derived an analytical formula using elliptic integrals

Validated the formula against numerical integration

Corrected errors in classical elliptic integral tables

Abstract

The face-on projected density profile of the Miyamoto & Nagai disks of arbitrary flattening is obtained analytically in terms of incomplete elliptic integrals of first and second type, by using two complementary approaches, and then checked against the results of numerical integration. As computer algebra systems do not seem able to obtain the resulting formula in any straightforward way, the relevant mathematical steps are provided. During this study, three wrong identities in the Byrd & Friedman tables of elliptic integrals have been identified, and their correct expression is given.