In this paper we exploit the fact that the collapsing core of a massive star, at the endpoint of its thermonuclear life, rapidly becomes hydrostatic after bouncing at nuclear densities. The energy transferred by pdV work from the inner unshocked core as it becomes hydrostatic to the bounce shock in the outer core is, to a good approximation, the initial energy of the supernova. The magnitude of this energy, which goes into the entropization, dissociation, and kinetic energy of the shocked matter, determines in large measure whether the purely hydrodynamic mechanism of Type II supernovae works. It can be shown semi-analytically that this energy is equal, with minor qualifications, to the binding energy of the hydrostatic remnant. We present analytical formulae for the structure and binding energy of a hydrostatic residue modeled as a composite of two nested polytropes. It is shown that the binding energy of such a configuration, when taken as a function of mass, is independent of the details of the equation of state of the stiff inner polytrope and hence insensitive to the poorly known nuclear equation of state. The nuclear physics enters directly mainly through the ratio of pressure to density at the transition densitymore » where matter becomes stiff. Since all the relevant masses of the remnant can be determined by self-similar or hydrostatic analysis, th initial shock energy can be derived without recourse to hydrodynamic calculation. This perspective allows us to extract most of the essentials of bounce energetics.« less