We analyze the effect of gravitational radiation reaction on generic orbitsaround a body with an axisymmetric mass quadrupole moment Q to linear order inQ, to the leading post-Newtonian order, and to linear order in the mass ratio.This system admits three constants of the motion in absence of radiationreaction: energy, angular momentum, and a third constant analogous to theCarter constant. We compute instantaneous and time-averaged rates of change ofthese three constants. For a point particle orbiting a black hole, Ryan hascomputed the leading order evolution of the orbit's Carter constant, which islinear in the spin. Our result, when combined with an interaction quadratic inthe spin (the coupling of the black hole's spin to its own radiation reactionfield), gives the next to leading order evolution. The effect of thequadrupole, like that of the linear spin term, is to circularize eccentricorbits and to drive the orbital plane towards antialignment with the symmetryaxis. In addition we consider a system of two point masses where one body has asingle mass multipole or current multipole. To linear order in the mass ratio,to linear order in the multipole, and to the leading post-Newtonian order, weshow that there does not exist an analog of the Carter constant for such asystem (except for the cases of spin and mass quadrupole). With mild additionalassumptions, this result falsifies the conjecture that all vacuum, axisymmetricspacetimes posess a third constant of geodesic motion.
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