Simulating the magnetorotational collapse of supermassive stars: Incorporating gas pressure perturbations and different rotation profiles

Collapsing supermassive stars (SMSs) with masses $M\gtrsim {10}^{4–6}{M}_{\odot }$ have long been speculated to be the seeds that can grow and become supermassive black holes (SMBHs). We previously performed general relativistic magnetohydrodynamic (GRMHD) simulations of marginally stable $\mathrm{\Gamma }=4/3$ polytropes uniformly rotating at the mass-shedding limit and endowed initially with a dynamically unimportant dipole magnetic field to model the direct collapse of SMSs. These configurations are supported entirely by thermal radiation pressure and reliably model SMSs with $M\gtrsim {10}^6{M}_{\odot }$. We found that around 90% of the initial stellar mass forms a spinning black hole (BH) remnant surrounded by a massive, hot, magnetized torus, which eventually launches a magnetically-driven jet. SMSs could be therefore sources of ultra-long gamma-ray bursts (ULGRBs). Here we perform GRMHD simulations of $\mathrm{\Gamma }\gtrsim 4/3$, polytropes to account for the perturbative role of gas pressure in SMSs with $M\lesssim {10}^6{M}_{\odot }$. We also consider different initial stellar rotation profiles. The stars are initially seeded with a dynamically weak dipole magnetic field that is either confined to the stellar interior or extended from its interior into the stellar exterior. We calculate the gravitational wave burst signal for the different cases. We find that the mass of the black hole remnant is 90%–99% of the initial stellar mass, depending sharply on $\mathrm{\Gamma }-4/3$ as well as on the initial stellar rotation profile. After $t\sim 250–550M\approx 1-2\times {10}^3(M/{10}^6{M}_{\odot })\mathrm{s}$ following the appearance of the BH horizon, an incipient jet is launched and it lasts for $\sim {10}^4–{10}^5(M/{10}^6{M}_{\odot })\mathrm{s}$, consistent with the duration of long gamma-ray bursts. Our numerical results suggest that the Blandford-Znajek mechanism powers the incipient jet. They are also in rough agreement with our recently proposed universal model that estimates accretion rates and electromagnetic (Poynting) luminosities that characterize magnetized BH-disk remnant systems that launch a jet. This model helps explain why the outgoing electromagnetic luminosities computed for vastly different BH-disk formation scenarios all reside within a narrow range ($\sim {10}^{52\pm 1}\mathrm{erg}{\mathrm{s}}^{-1}$), roughly independent of $M$.

Figure 1

Gas-to-radiation pressure ratio $\beta$ (upper panel) and SMS mass (lower panel) as a function of polytropic index $n$ using Eqs. (1) and (4). For $n$ within the range where $0<\beta <0.1$, gas pressure is a small perturbation, yet the mass varies by orders of magnitude.

Figure 2

3D volume rendering of the rest-mass density normalized to its initial maximum value ${\rho }_{0,\max }=1.66(M/{10}^6{M}_{\odot }{)}^{-2}\mathrm{g}{\mathrm{cm}}^{-3}$ at select times for the n29-EXTINT-DIFF case (see Table 1). Solid lines indicate the magnetic field lines and arrows show plasma velocities with length proportional to their magnitude. The bottom left panel displays the collimated, helical magnetic field and outgoing plasma, whose zoomed-in view near the horizon is shown in the bottom right panel. Here $M=4.9(M/{10}^6{M}_{\odot })\mathrm{s}=1.47\times {10}^6(M/{10}^6{M}_{\odot })\mathrm{km}$.

Figure 3

3D volume rendering of the rest mass density normalized to the corresponding initial maximum value ${\rho }_{0,\max }$ in log scale (for details see Table 1) for cases n3-EXTINT, n295-EXTINT, n29-EXTINT, and n29-EXTINT-0.75SPIN, shown from top to the bottom, respectively. The initial and final configurations for these cases are shown in left and right panels, respectively. See Table 3 for a summary of global parameters describing the final outcome of these cases. Solid lines indicate the magnetic field lines while arrows display plasma velocities with length proportional to their magnitude. Here $M=4.9(M/{10}^6{M}_{\odot })\mathrm{s}=1.47\times {10}^6(M/{10}^6{M}_{\odot })\mathrm{km}$.

Figure 4

Dependence of the black hole mass (top panel), black hole dimensionless spin parameter (middle panel), and the accretion mass disk (bottom panel) on different EOSs, magnetic field configurations, and rotation profiles for models in Table 1. The mass of the black hole remnant, and hence the mass of the disk, is sharply sensitive to changes in the EOS as well as the initial rotation profile of the SMS.

Figure 5

Rest mass accretion rate $\dot{M}$ vs time for cases listed in Table 1. Notice that the time has shifted to the black hole formation time and it is normalized to $(M/{10}^6{M}_{\odot })$.

Figure 6

Evolution of the electromagnetic Poynting luminosity $L_{\mathrm{EM}}$ crossing a sphere at coordinate radius $R_{\mathrm{ext}}=100M-175M\sim 1.4-2.4\times {10}^8(M/{10}^6{M}_{\odot })\mathrm{km}$ for all SMS models seeded with an external-interior magnetic field configuration (see Table 1). Horizontal dashed lines indicate the expected BZ values computed via Eq. (13). Here $t_{\mathrm{jet}}$ is the time at which the jet front has reached $\sim 100M$ above the black hole pole.

Figure 7

Real part of the $(l,m)=(2,0)$ mode of ${\mathrm{\Psi }}_4$ as a function of $t-t_{\mathrm{BH}}$ for the cases with interior and exterior B-field at an extraction radius $R_{\mathrm{ext}}\sim 100M$. The cyan curve represents the n3-EXTINT case displayed in [27]. Cases with other B-field configurations share similarity with their counterparts in the figure.