Please note special time and location of this talk.  The motion of the free boundary of an incompressible fluid body subject to its self gravitational force can be described by a free boundary problem of the Euler-Poisson system. This problem differs from the water wave problem in that the constant gravity in water waves is replaced by a nonlinear self-gravity. In this talk, we present some recent results on the well-posedness of this problem and give a lower bound on the lifespan of smooth solutions. In particular, we show that the Taylor sign condition always holds leading to local well-posedness, and for smooth data of size $\epsilon$ a unique smooth solution exists for time greater than or equal to $O(1/{\epsilon}^2)$. This is achieved by constructing an appropriate quantity and a coordinate transformation such that the new quantity in the new coordinate system satisfies an equation without quadratic nonlinearities.  This is joint work with L. Bieri, S. Shahshahani and S. Wu.