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We analyze the Miyamoto--Nagai substitution, which was introduced over fortyyears ago to build models of thick disks and flattened elliptical galaxies.Through it, any spherical potential can be converted to an axisymmetricpotential via the replacement of spherical polar $r^2$ with $R^2 + ( a +\!\sqrt{z^2+b^2} )^2$, where ($R,z$) are cylindrical coordinates and $a$ and$b$ are constants. We show that if the spherical potential has everywherepositive density, and satisfies some straightforward constraints, then thetransformed model also corresponds to positive density everywhere. This is insharp contradistinction to substitutions like $r^2 \rightarrow R^2 + z^2/q^2$,which leads to simple potentials but can give negative densities. We use theMiyamoto--Nagai substitution to generate a number of new flattened models withanalytic potential--density pairs. These include (i) a flattened model with anasymptotically flat rotation curve, which (unlike Binney's logarithmic model)is always non-negative for a wide-range of axis ratios, (ii) flattenedgeneralizations of the hypervirial models which include Satoh's disk as alimiting case and (iii) a flattened analogue of the Navarro--Frenk--White halowhich has the cosmologically interesting density fall-off of (distance)$^{-3}$.Finally, we discuss properties of the prolate and triaxial generalizations ofthe Miyamoto-Nagai substitution.

Self-consistent potential-density pairs of thick disks and flattened galaxies