  In a recent work [1] the authors studied the dynamics of the interface
separating a vacuum from an inviscid incompressible fluid, subject to the
self-gravitational force and neglecting surface tension, in two space
dimensions. The fluid is additionally assumed to be irrotational, and we proved
that for data which are size $\epsilon$ perturbations of an equilibrium state,
the lifespan $T$ of solutions satisfies $T \gtrsim \epsilon^{-2}$. The key to
the proof is to find a nonlinear transformation of the unknown function and a
coordinate change, such that the equation for the new unknown in the new
coordinate system has no quadratic nonlinear terms. For the related
irrotational gravity water wave equation with constant gravity the analogous
transformation was carried out by the last author in [3]. While our approach is
inspired by the last author's work [3], the self-gravity in the present problem
is a new nonlinearity which needs separate investigation. Upon completing [1]
we learned of the work of Ifrim and Tataru [2] where the gravity water wave
equation with constant gravity and constant vorticity is studied and a similar
estimate on the lifespan of the solution is obtained. In this short note we
demonstrate that our transformations in [1] can be easily modified to allow for
nonzero constant vorticity, and a similar energy method as in [1] gives an
estimate $T\gtrsim\epsilon^{-2}$ for the lifespan $T$ of solutions with data
which are size $\epsilon$ perturbations of the equilibrium. In particular, the
effect of the constant vorticity is an extra linear term with constant
coefficient in the transformed equation, which can be further transformed away
by a bounded linear transformation. This note serves as an appendix to the
aforementioned work of the authors.
