Improved gravitational radiation time-scales II: spin-orbit
  contributions and environmental perturbations
  Peters' formula is an analytical estimate of the time-scale of gravitational
wave (GW)-induced coalescence of binary systems. It is used in countless
applications, where the convenience of a simple formula outweighs the need for
precision. However, many promising sources of the Laser Interferometer Space
Antenna (LISA), such as supermassive black hole binaries and extreme mass-ratio
inspirals (EMRIs), are expected to enter the LISA band with highly eccentric
($e \gtrsim$ 0.9) and highly relativistic orbits. These are exactly the two
limits in which Peters' estimate performs the worst. In this work, we expand
upon previous results and give simple analytical fits to quantify how the
inspiral time-scale is affected by the relative 1.5 post-Newtonian (PN)
hereditary fluxes and spin-orbit couplings. We discuss several cases that
demand a more accurate GW time-scale. We show how this can have a major
influence on quantities that are relevant for LISA event-rate estimates, such
as the EMRI critical semi-major axis. We further discuss two types of
environmental perturbations that can play a role in the inspiral phase: the
gravitational interaction with a third massive body and the energy loss due to
dynamical friction and torques from a surrounding gas medium ubiquitous in
galactic nuclei. With the aid of PN corrections to the time-scale in vacuum, we
find simple analytical expressions for the regions of phase space in which
environmental perturbations are of comparable strength to the effects of any
particular PN order, being able to qualitatively reproduce the results of much
more sophisticated analyses.
