  In this paper, we propose a high order semi-implicit well-balanced finite
difference scheme for all Mach Euler equations with a gravitational source
term. To obtain the asymptotic preserving property, we start from the
conservative form of full compressible Euler equations and add the evolution
equation of the perturbation of potential temperature. The resulting system is
then split into a (non-stiff) nonlinear low dynamic material wave to be treated
explicitly, and (stiff) fast acoustic and gravity waves to be treated
implicitly. With the aid of explicit time evolution for the perturbation of
potential temperature, we design a novel well-balanced finite difference WENO
scheme for the conservative variables, which can be proven to be both
asymptotic preserving and asymptotically accurate in the incompressible limit.
Extensive numerical experiments were provided to validate these properties.
