Bridging relativistic jets from black hole scales to long-term electromagnetic radiation distances: A moving-mesh general relativistic hydrodynamics code with the HLLC Riemann solver

Relativistic jets accompany the collapse of massive stars, the merger of compact objects, or the accretion of gas in active galactic nuclei. They carry information about the central engine and generate electromagnetic radiation. No self-consistent simulations have been able to follow these jets from their birth at the black hole scale to the Newtonian dissipation phase, making the inference of central engine property through astronomical observations undetermined. We present the general relativistic moving-mesh framework to achieve the continuity of jet simulations throughout space and time. We implement the general relativistic extension for the moving-mesh relativistic hydrodynamic code, jet, and develop a tetrad formulation to utilize the Harten-Lax-van Leer Contact Riemann solver in the general relativistic moving-mesh code. The new framework is able to trace the radial movement of relativistic jets from central regions where strong gravity holds all the way to distances of jet dissipation.

Figure 1

Top: radial rest-mass density profile of the Bondi flow for the medium resolution $\mathtt{Nt}=256$. The mass accretion rate is ${\dot{M}}_{\text{bondi}}={10}^{-4}M$. The blue dotted line indicates the initial analytical solution at $T=0M$, while the yellow dashed line denotes the numerical solution at $T=50M$. The green dash-dotted line shows their difference. The black dotted vertical line indicates the radial location of the event horizon of the black hole at $r=0.78M$ in the maximally sliced trumpet coordinate. Middle: the radial fluid velocity $-W{v}^r$ profile. $W$ is the Lorentz factor, and $v^r$ is the fluid velocity with respect to the Eulerian observer. Bottom: the L1-norm of error for the rest-mass density as a function of the azimuthal grid number $\mathtt{Nt}$. The dash line indicates second-order convergence.

Figure 2

Normalized variation for the central maximum density as a function of dynamical time for the TOV star at two resolutions ($\mathtt{Nt}=64$ and $\mathtt{Nt}=128$). Top: the time evolution for the stationary case. Bottom: the time evolution for the pressure-depleted star whose equilibrium pressure has been reduced by ten percent globally.

Figure 3

Contour plot of a stationary torus around a static black hole. The simulation adopts an axisymmetric spherical domain. The azimuthal angle extends from $\pi /3$ to $2\pi /3$. The radial grid extends from $r_{\min }=4$ to $r_{\max }=20$. The top panel shows the initial profile of the logarithmic density. The middle panel represents the density profile at the end of the simulation $t=2000$. The bottom panel represents the contour plot of the radial velocity $W{v}^r$.

Figure 4

Time snapshots for the density contour of a hot low-density bubble embedded inside a cold Bondi flow. The left panel shows the simulation results with the HLLE Riemann solver while the right panel presents the comparative results from the simulation with the HLLC Riemann solver. The simulation video for the case with HLLC Riemann solver is available from Youtube at [116].

Figure 5

Profiles for the spherical shock tube test at $t=0.3$ in fixed-mesh simulations with the HLLC (solid line) and HLLE (dotted line) Riemann solver.

Figure 6

(a) Shock tube test in spherical coordinate for the fixed-mesh (black-dot) and moving-mesh (MM) (red-square) simulations. The profiles are presented at $t=0.3$. Initial discontinuity locates at $x=0.25$. The inner (outer) state of the shock tube is $\rho =1$, $P=1$, $v=0$ ($\rho =0.1$, $P=0.1$, $v=0$). The EOS follows the ideal gas law with $\mathrm{\Gamma }=4/3$. (b) Shock tube test in spherical coordinate for the moving-mesh simulations with different control schemes for the cell’s aspect ratio. The black-dot line represents the simulation where the cell’s maximum aspect ratio has not been set. The red-square line shows the simulation where the maximum aspect ratio of the cell has been set to 1.5.

Figure 7

Profiles for the spherical shock tube test at $t=0.3$ in the moving-mesh simulations with different grid resolution. The purple diamond shows results from the simulation with resolution $\mathtt{Nt}=2048$ while the red square and blue circle represent results from the $\mathtt{Nt}=128$ simulation and the $\mathtt{Nt}=32$ simulation, respectively.

Figure 8

Jet launching process from a black hole-torus system, visualized at four-time snapshots. From left to right, the contour plots represent the logarithmic density, temperature-like variable $\mathrm{\Theta }$, normalized radial velocity $W{v}^r/c$, and the cell’s radial resolution (i.e., inverse aspect ratio). The inner contour plots zoom in on the central region of the domain. The inner boundary is stationary before $t=30[\mathrm{ms}]$. The simulation video is available from Youtube at [133] with high-definition video quality available.

Figure 9

The full-time-domain evolution of the relativistic jet emerged from a black hole-torus system. The inner boundary moves outward at a fraction of the local maximum velocity. The simulation video is available from Youtube at [133] with high-definition video quality available.