Particularly impressive is that, if you read the fine print, Jerry Ostriker was an NSF graduate fellow (meaning he was a grad student!) when he wrote this paper, which solves differential equations analytically at a level of technical virtuosity undoubtedly beyond anyone reared in the age of Mathematica. Ostriker obtains solutions for polytropic cylinders, of which an isothermal cylinder is the case where . One has

,

which leads to the fundamental equation

.

Using the transformation

,

,

one has a version of the Lane-Emden equation

For n=0, 1, and infinity there is a closed form solution, for other n a power series solution. In the particular case of an isothermal cylinder, the EOS is that for an ideal gas, and one has

and

.

Using the fundamental equation noted above and manipulating yields

.

Letting and , we find

.

Letting the equation may be integrated to give

.

Using these results in expressions Ostriker provides in the paper that give general forms for density and mass, one has

and .

The expression for density should look familiar from the Pineda paper: it gets integrated to yield his eqn. (1) for the surface density of the cylinder.

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