  Compressible Euler-Poisson equations are the standard self-gravitating models
for stellar dynamics in classical astrophysics. In this article, we construct
periodic solutions to the isothermal ($\gamma=1$) Euler-Poisson equations in
$R^{2}$ with possible applications to the formation of plate, spiral galaxies
and the evolution of gas-rich, disk-like galaxies. The results complement
Yuen's solutions without rotation (M.W. Yuen, Analytical Blowup Solutions to
the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars, J. Math.
Anal. Appl. 341(2008), 445--456.). Here, the periodic rotation prevents the
blowup phenomena that occur in solutions without rotation. Based on our
results, the corresponding $3$D rotational results for Goldreich and Weber's
solutions are conjectured.
