Precession-tracking coordinates for simulations of compact-object binaries

Binary black hole simulations with black hole excision using spectral methods require a coordinate transformation into a corotating coordinate system where the black holes are essentially at rest. This paper presents and discusses two coordinate transformations that are applicable to precessing binary systems, one based on Euler angles, the other on quaternions. Both approaches are found to work well for binaries with moderate precession, i.e., for cases where the orientation of the orbital plane changes by $\ll 90°$. For strong precession, performance of the Euler-angle parametrization deteriorates, eventually failing for a 90° change in orientation because of singularities in the parametrization (“gimbal lock”). In contrast, the quaternion representation is invariant under an overall rotation and handles any orientation of the orbital plane as well as the Euler-angle technique handles nonprecessing binaries.

Figure 1

Typical behavior of the Euler angles and their derivatives for a nearly polar orbit inclined at 85° with respect to the $x\mathrm{\text{−}}y$ plane.

Figure 2

Newtonian simulations with inclination of the orbital plane of angle $\beta =0°$, 10°, 70° from the $x\mathrm{\text{−}}y$ plane performed with both control systems. Time is measured in units of orbital period.

Figure 3

Post-Newtonian simulations with inclination of the orbital plane angle $\beta =0°$, 10°, 70° from the $x\mathrm{\text{−}}y$ plane performed with both control systems. Time is measured in units of initial orbital period. The binary is equal mass and nonspinning, with the initial coordinate separation of 20.

Figure 4

The inclination angle $\beta$ for the three systems under study. The curves stop just after a common horizon is detected.

Figure 5

The trajectories of the centers of the apparent horizons of the black holes in inertial coordinates for the three simulations. Top to bottom: d11.68q2.5, d12q2.5, d14.5q1.5. The left panels show the projection onto the $x\mathrm{\text{−}}y$ plane and the right, the $x\mathrm{\text{−}}z$ plane.

Figure 6

Three full NR simulations performed with the quaternion control systems with initial conditions listed in Table .

Figure 7

Translation control system output for case d14.5q1.5 done with the quaternion and the Euler-angle control systems.

Figure 8

Top: the real part of $r{h}_{22}$ extracted at $R=304$ for the simulation d14.5q1.5 done with both control systems. Bottom: the phase differences. The data have been time shifted by the extraction radius.

Figure 9

The fractional difference in orbital frequency estimated from quaternions and from trajectories (solid) and the orbital phase difference in radians (dashed). The data are from the run d14.5q1.5.