  We consider the motion of the interface separating a vacuum from an inviscid,
incompressible, and irrotational fluid, subject to the self-gravitational force
and neglecting surface tension, in two space dimensions. The fluid motion is
described by the Euler-Poission system in moving bounded simply connected
domains. A family of equilibrium solutions of the system are the perfect balls
moving at constant velocity. We show that for smooth data which are small
perturbations of size $\epsilon$ of these static states, measured in
appropriate Sobolev spaces, the solution exists and remains of size $\epsilon$
on a time interval of length at least $c\epsilon^{-2},$ where $c$ is a constant
independent of $\epsilon.$ This should be compared with the lifespan
$O(\epsilon^{-1})$ provided by local well-posdness. The key ingredient of our
proof is finding a nonlinear transformation which removes quadratic terms from
the nonlinearity. An important difference with the related gravity water waves
problem is that unlike the constant gravity for water waves, the self-gravity
in the Euler-Poisson system is nonlinear. As a first step in our analysis we
also show that the Taylor sign condition always holds and establish local
well-posedness for this system.
