Numerical evolutions of a black hole-neutron star system in full general relativity: Head-on collision

We present the first simulations in full general relativity of the head-on collision between a neutron star and a black hole of comparable mass. These simulations are performed through the solution of the Einstein equations combined with an accurate solution of the relativistic hydrodynamics equations via high-resolution shock-capturing techniques. The initial data is obtained by following the York-Lichnerowicz conformal decomposition with the assumption of time symmetry. Unlike other relativistic studies of such systems, no limitation is set for the mass ratio between the black hole and the neutron star, nor on the position of the black hole, whose apparent horizon is entirely contained within the computational domain. The latter extends over $\sim 400M$ and is covered with six levels of fixed mesh refinement. Concentrating on a prototypical binary system with mass ratio $\sim 6$, we find that although a tidal deformation is evident the neutron star is accreted promptly and entirely into the black hole. While the collision is completed before $\sim 300M$, the evolution is carried over up to $\sim 1700M$, thus providing time for the extraction of the gravitational-wave signal produced and allowing for a first estimate of the radiative efficiency of processes of this type.

Figure 1

Left panel: Coordinate lines of $A$ and $B$ on the Cartesian grid. The arrow denotes the location of the singularity ($b=30$) and the sector is a sketch of the neutron star. Right panel: coordinate lines of $A$ and $B$ on their grid. The arrow denotes the location of the singularity and the sector is a sketch of the neutron star.

Figure 2

Fractional error in the evaluation of the ADM mass resulting from a finite-size computational domain. The two curves show the error in the different cases in which the lapse function in Eq. (21) is set to be one or not.

Figure 3

Computational setup of the mixed binary problem discussed here. Shown are the borders of the six mesh-refinement levels (The actual resolution is higher than sketched here). Because of the use of reflection symmetry at the $x\mathrm{\text{−}}y$ and the $x\mathrm{\text{−}}z$ plane, we only have to use grid points at positive $y$ and positive $z$ coordinates.

Figure 4

2-Norm of the Hamiltonian constraint along the $x$-axis for different resolutions of the spectral and the Cartesian grid. The dashed line corresponds to a fourth-order convergence rate and is offered as a comparison.

Figure 5

Shown is the maximal rest-mass density $\rho$ over time (solid line) and as comparison a trigonometric function with the frequency of the fundamental mode of the unperturbed star of which is $\approx 1.28\mathrm{kHz}$.

Figure 6

Isocontours of the rest-mass density and of the apparent horizon surface (dashed line) in the $(x,y)$ plane; the area shown as filled refers to the excised region of the computational domain. The different panels refer to times from $0M$ to $350M$ in steps of $50M\simeq 0.25\mathrm{ms}$.

Figure 7

Time evolution of a representative metric function (i.e., $g_{xx}$), in its cross-section onto the $(x,y)$ plane, with the circle across the $y=0$ axis showing the position of the apparent horizon. The different panels refer to times from $0M$ to $350M$ in steps of $50M\simeq 0.25\mathrm{ms}$ and we have reported only the solution evaluated on the coarsest grid.

Figure 8

Isocontours of the rest-mass density and of the apparent horizon surface (dashed line) in the $(x,y)$ plane; the area shown as filled refers to the excised region of the computational domain. The different panels refer to times from $265M$ to $335M$ in steps of $10M$. Note that the scale in the first four frames is different (larger) than the one in the last four frames.

Figure 9

Time evolution of the mass of the apparent horizon shown for a high-resolution simulation (solid line) and a low-resolution one (dashed line). Indicated with a horizontal dotted line is the expected ADM mass and the error-bar indicates the error introduced by a finite-size domain for the smallest resolution used. The inset shows the evolution around the time of the accretion of the neutron star onto the black hole.

Figure 10

Time evolution of the ratio between the polar and equatorial proper circumferences of the apparent horizon soon after the collision (solid line). Shown for a comparison with a dashed line is the QNM oscillation of a black hole with mass $\simeq 5.86M$, fundamental $\ell =2$ frequency of $f\approx 2.06\mathrm{kHz}$ and damping time of $0.32\mathrm{ms}$.

Figure 11

1D slice of the Hamiltonian constraint violation along the $x$-axis at a representative time $t=100M$. Each panel reports two curves indicating, respectively, the violations computed with a low-resolution grid (solid line) and with a high-resolution one (dashed line) scaled by a factor of 4 for the expected second-order accuracy. While the left panel shows the portion of the grid containing the two compact objects, the right one focuses on the violation around the black hole.

Figure 12

$+$ component of the gravitational-wave emission from the merging binary system, extracted at a radius of $70M$ from the final black hole. The $\times$ component is identically zero.