It’s been a busy end to the year and this week I had two papers out on the arXiv. I briefly review them below.

## The location of the last stable orbit in Kerr spacetime

The first was with Leo Stein. One of the key differences between Newtonian and Einstein gravity is the presence of unstable and plunging orbits around compact objects. The surface in parameter space which separates stable and unstable orbits is called the ‘separatrix’. The location of the separatrix is non-trivial and in this work we present a new approach to finding it based upon finding roots of a high-order polynomial. This allows for analytic study of the separatrix. This allows us to easily derive many limits in previously calculated in the literature (such as the ISCO, ISSO, IBSO, etc). The polynomial is of order 12 in ‘p’ (the semi-latus rectum) so there is are closed form solutions in general but we provide a robust numerical scheme for finding the roots.

The above figure shows the an extended view in (p,e,x) space of the solutions of the separatrix polynomial. The physical separatrix is one of the sheets (restricted to 0 <= e <=1).

## Dissipation in extreme-mass ratio binaries with a spinning secondary

In this work we examine the gravitational-wave flux from a compact binary where the secondary is spinning. We derive the balance law in this case, which relates the flux to the local change in the secondary’s orbital constants of motion (energy, angular momentum). We then perform an explicit calculation for the case of a spin-aligned binary where the primary is a non-rotating (Schwarzschild) black hole. We make this calculation using the Teukolsky formalism reconstructing in the radiation gauge for the local calculation (we make a post-Newtonian and numerical calculation using this method), and a numerical Lorenz-gauge calculation. We find perfect agreement between the approaches and show explicitly that our balance law holds.

The above two plots show the comparison of the flux to infinity between the PN and numerical results, and the Lorenz-gauge metric perturbation for the (2,2)-mode for a spinning body. All the PN and numerical results can be found in the Black Hole Perturbation Toolkit.