  Binary evolution is influenced by dynamical scattering with other stars in
dense environments. Heggie's law states that, due to their environments, hard
binaries (whose orbital energy surpasses the typical energy of single stars)
tend to harden (increase their orbital energy), while soft binaries typically
soften. Here, we show that Heggie's law sometimes needs to be revised, by
accounting for an external potential, for example, for binaries in nuclear
stellar discs or AGN discs, that are affected by the central massive black
hole, or binary planetesimals in proto-planetary discs, affected by the host
star. We find that in such environments, where the Hill radius is finite,
binary-single scattering can have different outcomes. In particular, a
three-body encounter could be cut short due to stars being ejected beyond the
Hill radius, thereby ceasing to participate in further close interactions. This
leads to a systematic difference in the energy changes brought about by the
encounter, and in particular slows binary hardening, and even causes some hard
binaries to soften, on average, rather than harden. We use our previously
derived analytical, statistical solution to the bound chaotic three-body
problem to quantitatively characterise the revision of the hardening-softening
phase transition and evolution of binaries. We also provide an analytical
calculation of the mean hardening rate of binaries in any environment (also
reproducing the results of detailed N-body simulations). We show that the
latter exhibits a non-trivial dependence on the Hill radius induced by the
environment.
