Characteristic initial data for a star orbiting a black hole

We take further steps in the development of the characteristic approach to enable handling the physical problem of a compact self-gravitating object, such as a neutron star, in close orbit around a black hole. We examine different options for setting the initial data for this problem and, in order to shed light on their physical relevance, we carry out short time evolution of this data. To this end we express the matter part of the characteristic gravity code so that the hydrodynamics are in conservation form. The resulting gravity plus matter relativity code provides a starting point for more refined future efforts at longer term evolution. In the present work we find that, independently of the details of the initial gravitational data, the system quickly flushes out spurious gravitational radiation and relaxes to a quasiequilibrium state with an approximate helical symmetry corresponding to the circular orbit of the star.

Figure 1

For the case $m={10}^{-5}$, $M=0$, $R_{*}=3$, $a=9$ and grid sizes given by ${45}^2\times 63$, ${65}^2\times 93$, and ${85}^2\times 123$, the different self-convergence factors are obtained. The behavior observed is consistent with at least first order convergence.

Figure 2

Behavior of $||J||$ for the case with no black hole. Three scenarios are plotted: Full evolution (hydrodynamics $+$ gravity) using initial data $J=0$ (dashed line with “x” symbols); full evolution using quasi-Newtonian initial data (solid line with circular symbols); and gravitational evolution with frozen internal hydrodynamics using quasi-Newtonian data (dotted lines with triangular symbols). After some transient behavior, lasting until about $u\sim 2$, all curves roughly approach the same value.

Figure 3

Behavior of $||J||$ for the case of an $M=1$ black hole. Again three scenarios are plotted: Full evolution using initial data $J=0$ (dashed line with “x” symbols); full evolution using quasi-Newtonian initial data (solid line with circular symbols); and gravitational evolution with frozen hydrodynamics using quasi-Newtonian data (dotted lines with triangular symbols). After some transient behavior lasting until $u\sim 2$, all curves roughly approach the same value.

Figure 4

$L_2$ norm of the difference between the computed values of $J$ for the quasi-Newtonian and spherically symmetric initial data.

Figure 5

Absolute value of the difference between the computed values of $J$ at ${\mathcal{I}}^{+}$ evolved from quasi-Newtonian and from shear-free initial data. The panels [(a), (b), (c), (d)]correspond to the time sequence $u=0$, $u=0.75M$, $u=1.5M$, and $u=2.5M$. Panel (a) has been truncated for comparison purposes, as its height is $\approx 9\times {10}^{-5}$.

Figure 6

Comparison of $F_{\kappa }$ using quasi-Newtonian initial data.

Figure 7

Comparison of $F_{\kappa }$ using shear-free initial data.

Figure 8

For the case $m={10}^{-5}$, $R_{*}=3M$, $a=9M$ we show density profiles in the plane with stereographic coordinate $p=0$. The motion takes place in the $(r,q)$ plane. In all figures, the vertical axis is $x$ (the compactified radial coordinate) and the horizontal axis is $q$. The top panels correspond to the fixed coordinates (a) and corotating coordinates (b) at the initial time $u=0$. Figures (c) and (d) show the corresponding density profiles at $u=2.8$. The corotating coordinates, panel (d), perform well in keeping the coordinate location of the star fixed. For comparison, panel (c) shows the actual displacement of the star.

Figure 9

Time behavior of the minimum expansion $e^{-2\beta }$ at future null infinity vs initial location of the star. As the star is placed farther from the black hole its focusing effect on the central null rays increases. However, the expansion stays close to its initial value for a given separation $a$.

Figure 10

Behavior of the minimum of the expansion $e^{-2\beta }$ at future null infinity vs location of the star. As the star’s initial location is placed farther from the black hole, its focusing effect on the central null rays becomes considerably enhanced.