The Kepler equation for inspiralling compact binaries

Compact binaries consisting of neutron stars / black holes on eccentric orbit undergo a perturbed Keplerian motion. The perturbations are either of relativistic origin or are related to the spin, mass quadrupole and magnetic dipole moments of the binary components. The post-Newtonian motion of such systems decouples into radial and angular parts. We present here for the first time the radial motion of such a binary encoded in a generalized Kepler equation, with the inclusion of all above-mentioned contributions, up to linear order in the perturbations. Together with suitably introduced parametrizations, the radial motion is solved completely

Evidence for nonstatistical dynamics in the Wolff rearrangement of a carbene

Two 13C-labeled isomers of the formal Diels−Alder adduct of acetylmethyloxirene to tetramethyl 1,2,4,5-benzenetetracarboxylate have been synthesized. Flash vacuum thermolysis of these adducts leads to various isotopic isomers of acetylmethylketene, the ratios of which have been determined by NMR. The surprising finding that the principal product comes from methylpyruvoyl carbene rather than its more stable isomer diacetylcarbene is explained by MPWB1K density functional calculations, which show that the reactant probably undergoes a unimolecular rearrangement to a norcaradiene derivative prior to its fragmentation. Coupled-cluster calculations on the methylpyruvoyl carbene show that it is capable of undergoing three unimolecular isomerizations. The fastest is 1,2-acetyl migration to give acetylmethylketene directly. The next is rearrangement via acetylmethyloxirene to diacetylcarbene and thence by Wolff rearrangement to acetylmethylketene. The least-favorable reaction is degenerate rearrangement via 1,3-dimethyl-2-oxabicyclo[1.1.0]butan-4-one (the epoxide of dimethylcyclopropenone). The combined experimental and computational results indicate that Wolff rearrangement of the diacetylcarbene occurs with a 2.5:1 ratio of the methyl groups despite the fact that they are related by a twofold axis of symmetry in the carbene. Preliminary molecular dynamics simulations are consistent with this conclusion. Taken together, the results suggest that the Wolff rearrangement is subject to the same kind of nonstatistical dynamical effects detected for other kinds of thermally generated reactive intermediates