  One of the most mysterious astrophysical states is the common envelope (CE)
phase of binary evolution, in which two stars are enshrouded by the envelope
shed by one of them. Interactions between the stars and the envelope shrinks
the orbit. The CE can lead to mergers or to a subsequent phase of interactions.
Mergers may involve any combination of two compact objects and/or stars. Some
involving white dwarfs, may produce Type Ia supernovae, while merging neutron
stars may yield gamma-ray bursts, and merging compact objects of all kinds
produce gravitational radiation. Since CEs can arise from a variety of
different initial conditions, and due to the complexity of the processes
involved, it is difficult to predict their end states. When many systems are
being considered, as in population synthesis calculations, conservation
principles are generally employed. Here we use angular momentum in a new way to
derive a simple expression for the final orbital separation. This method
provides advantages for the study of binaries and is particularly well-suited
to higher order multiples, now considered to be important in the genesis of
potential mergers. Here we focus on CEs in binaries, and the follow-up paper
extends our formalism to multiple star systems within which a CE occurs.
