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Common-envelope evolution is a stage in binary system evolution in which agiant star engulfs a companion. The standard energy formalism is an analyticalframework to estimate the amount of energy transferred from the companion'sshrinking orbit into the envelope of the star that engulfed it. We showanalytically that this energy transfer is larger than predicted by the standardformalism. As the orbit of the companion shrinks, the mass it encloses becomessmaller, and the companion is less bound than if the enclosed mass had remainedconstant. Therefore, more energy must be transferred to the envelope for theorbit to shrink further. We derive a revised energy formalism that accounts forthis effect, and discuss its consequences in two contexts: the formation ofneutron star binaries, and the engulfment of planets and brown dwarfs by theirhost stars. The companion mass required to eject the stellar envelope issmaller by up to $50\%$, leading to differences in common-envelope evolutionoutcomes. The energy deposition in the outer envelope of the star, which isrelated to the transient luminosity and duration, is up to a factor of$\approx7$ higher. Common-envelope efficiency values above unity, as defined inthe literature, are thus not necessarily unphysical, and result at least partlyfrom an incomplete description of the energy deposition. The revised energyformalism presented here can improve our understanding of stellar merger andcommon-envelope observations and simulations.

A revised energy formalism for common-envelope evolution: repercussions for planetary engulfment and the formation of neutron star binaries