Approximating the inspiral of test bodies into Kerr black holes

We present a new approximate method for constructing gravitational radiation driven inspirals of test bodies orbiting Kerr black holes. Such orbits can be fully described by a semilatus rectum p, an eccentricity e, and an inclination angle $\iota ,$ or, by an energy E, an angular momentum component $L_z,$ and a third constant Q. Our scheme uses expressions that are exact (within an adiabatic approximation) for the rates of change $(\mathrm{ṗ},ė,\dot{\iota })$ as linear combinations of the fluxes $(Ė,L_z,\mathrm{Q̇}),$ but uses quadrupole-order formulas for these fluxes. This scheme thus encodes the exact orbital dynamics, augmenting it with an approximate radiation reaction. Comparing inspiral trajectories, we find that this approximation agrees well with numerical results for the special cases of eccentric equatorial and circular inclined orbits, far more accurate than corresponding weak-field formulas for $(\mathrm{ṗ},ė,\dot{\iota }).\mathrm{}$ We use this technique to study the inspiral of a test body in inclined, eccentric Kerr orbits. Our results should be useful tools for constructing approximate waveforms that can be used to study data analysis problems for the future Laser Interferometer Space Antenna gravitational-wave observatory, in lieu of waveforms from more rigorous techniques that are currently under development.