3D full-GR simulations of magnetorotational core-collapse supernovae on GPUs: A systematic study of rotation rates and magnetic fields

Using GPU-accelerated 3D general-relativistic magnetohydrodynamics simulations on the Frontier supercomputer, this systematic study of a $25\, M_\odot$ progenitor reveals that while magnetic fields of $10^{11}\, \mathrm{G}$ fail to drive explosions, fields of $10^{12}\, \mathrm{G}$ combined with varying rotation rates produce diverse outcomes ranging from spherical, neutrino-like explosions to high-velocity jets characteristic of broad-lined type Ic supernovae.

The Gist

The Big Picture: A Cosmic Firework Show

Imagine a massive star (about 25 times heavier than our Sun) reaching the end of its life. Its core collapses, creating a tiny, super-dense ball called a neutron star. Usually, this collapse results in a standard supernova explosion, powered by a "neutrino wind" (ghostly particles) that pushes the outer layers of the star away.

But some stars explode much more violently. These are called Hypernovae or Type Ic-bl supernovae. They shoot out material at incredible speeds (up to 30,000 km/s) and have broad, blurry lines in their light spectrum. Scientists believe these aren't just pushed by a gentle wind; they are launched by a magnetic jet, like a cosmic water hose turned on full blast.

This paper asks: What conditions are needed to turn a standard star collapse into a super-powered, jet-driven explosion?

The Experiment: A Cosmic "What-If" Machine

The researchers used a supercomputer (specifically, the "Frontier" supercomputer at Oak Ridge National Lab) to run 12 different simulations. Think of this as running a video game 12 times, but changing two specific "settings" every time to see what happens:

1. The Spin (Rotation Rate): How fast is the star spinning before it collapses? (From a slow spin to a very fast spin).
2. The Magnetism (Magnetic Field): How strong is the star's magnetic field? (They tested a "weak" field and a "strong" field).

They used a new, super-fast code called GRaM-X that runs on graphics cards (GPUs), which allowed them to run these complex 3D simulations much faster than ever before.

The Results: The "Goldilocks" Zone of Explosions

Here is what happened when they tweaked the settings:

1. The Weak Magnet (The "Dud" Fireworks)

• The Setup: They used a magnetic field strength of $10^{11}$ Gauss (strong, but not "super-magnetar" strong).
• The Outcome: No matter how fast the star was spinning, nothing happened. The shockwave stalled, and the star failed to explode.
• The Analogy: Imagine trying to start a campfire with a weak spark and a damp log. You can blow on it (spin it) all you want, but without a strong enough spark (magnetic field), the fire just fizzles out.

2. The Strong Magnet + Slow Spin (The "Stalled" Engine)

• The Setup: They used a super-strong magnetic field ($10^{12}$ Gauss) but kept the spin slow.
• The Outcome: Still no explosion. The magnetic field was strong enough to create some turbulence, but without the spin to "wind it up," it couldn't launch a jet.
• The Analogy: You have a powerful rocket engine (the magnetic field), but you forgot to turn the key in the ignition (the spin). The engine sits there, humming, but the rocket doesn't move.

3. The Strong Magnet + Medium Spin (The "Wobbly" Jet)

• The Setup: Strong magnetic field + a moderate spin.
• The Outcome: The star did explode, but the jet was messy. Instead of shooting straight up like a laser beam, the jet bent sideways, twisted, and fell back down.
• The Analogy: Imagine a garden hose with the nozzle turned on high, but the water pressure is fighting against a strong wind. The stream of water shoots out, but it wobbles, bends, and sprays in a wide, messy circle rather than a straight line.
• The Twist: Because the jet bent so much, the explosion looked spherical (round) from the outside. If an alien astronomer saw this, they might think it was a normal, neutrino-driven explosion, even though it was actually powered by a magnetic jet!

4. The Strong Magnet + Fast Spin (The "Rocket" Launch)

• The Setup: Strong magnetic field + very fast spin.
• The Outcome: BAM! A clean, powerful jet shot straight out the poles at speeds over 15,000 km/s.
• The Analogy: This is like a perfectly tuned rocket. The magnetic field acts as the fuel, and the rapid spin acts as the nozzle that focuses the energy into a tight, high-speed beam. This is exactly what we think creates the "Hypernovae" we see in the universe.

Why This Matters

Before this study, scientists mostly ran these simulations in 2D (like looking at a flat slice of a cake) or could only afford to run a few models. This paper is a breakthrough because:

1. It's 3D: Real life is 3D. Jets can twist, bend, and wobble in ways a flat simulation can't show.
2. It's Systematic: They didn't just guess; they tested a whole grid of possibilities to find the exact recipe for an explosion.
3. It's Fast: By using modern graphics cards (GPUs), they proved we can now run these massive, complex simulations to study the universe in detail.

The Bottom Line

To get a "Hypernova" (a super-fast, jet-driven explosion), you need two things working together:

1. A super-strong magnetic field (the fuel).
2. A very fast spin (the nozzle).

If you have the fuel but no spin, nothing happens. If you have the spin but weak fuel, nothing happens. But if you have both, you get a cosmic firework that shoots material across the galaxy at relativistic speeds.

The researchers also noted that their "fastest" models were still a bit slower than the fastest explosions seen in real life. They suspect this is because the star they simulated was a bit too "heavy" and dense. If they simulate a lighter, less dense star, they expect the jets to go even faster, potentially reaching the 30,000 km/s speeds seen in the most extreme supernovae.

Technical Summary

1. Problem Statement

Core-collapse supernovae (CCSNe) are generally understood to be driven by neutrino heating. However, a small fraction of observed supernovae, specifically broad-lined Type Ic supernovae (SNe Ic-bl) and hypernovae, exhibit kinetic energies ($\sim 10^{52}$ erg) and ejecta velocities ($15,000–30,000$ km/s) that cannot be explained by the neutrino mechanism alone. These events are hypothesized to be powered by the magnetorotational mechanism, where rapid core rotation and strong magnetic fields launch relativistic jets.

Key Challenges:

• Initial Conditions: The pre-collapse rotation profiles and magnetic field strengths of massive stars are poorly constrained.
• Computational Cost: Capturing the Magnetorotational Instability (MRI) and jet formation requires full 3D General Relativistic Magnetohydrodynamics (GRMHD) simulations with high resolution. Previous studies were limited to a few models (typically 2D or very sparse 3D sets) due to the immense computational cost.
• Resolution vs. Physics: Resolving the MRI wavelength requires $\sim 50–100$ m resolution, while typical CCSN simulations resolve $\sim 300$ m. Furthermore, maintaining high resolution as the shock wave expands to large radii is computationally prohibitive in standard setups.

2. Methodology

The authors performed a systematic parameter study using a newly developed, GPU-accelerated code.

• Progenitor Model: A single $25\,M_\odot$ Zero-Age Main Sequence (ZAMS) star from Aguilera-Dena et al. (2022). This progenitor has a high compactness parameter ($\xi_{2.5} = 0.47$), making it difficult to explode via neutrino mechanisms alone.
• Code: GRaM-X, a GPU-accelerated dynamical-spacetime ideal-GRMHD code within the Einstein Toolkit framework. It utilizes the Z4c formulation for Einstein's equations and the Valencia formulation for GRMHD.
• Numerical Setup:
  • Grid: Adaptive Mesh Refinement (AMR) using CarpetX. The shock is dynamically tracked and always contained within AMR level 6, ensuring a minimum resolution of 1.48 km throughout the shocked region (a significant improvement over static grids where resolution degrades as the shock expands).
  • Physics: Finite-temperature LS220 Equation of State (EoS). Neutrino transport is approximated using the computationally efficient M0 scheme (since neutrinos are energetically sub-dominant in magnetorotational explosions).
  • Initial Conditions: 12 models varying two parameters:
    • Initial Magnetic Field ($B_0$): $10^{11}$ G and $10^{12}$ G.
    • Initial Rotation Rate ($\Omega_0$): $0.14, 0.5, 1.0, 1.5, 2.0, 2.5$ rad/s.
  • Simulation Duration: 190–260 ms post-bounce.

3. Key Contributions

• Largest 3D GRMHD Parameter Study: This work presents the largest set of 3D full-GR GRMHD simulations for magnetorotational CCSNe to date (12 models), enabled by GPU acceleration on the Frontier supercomputer.
• Systematic Resolution Strategy: Unlike previous studies where shock resolution degraded as the shock expanded, this study maintained a constant high resolution (1.48 km) over the entire shock region via dynamic AMR boundary adjustment.
• Computational Feasibility: Demonstrated that systematic 3D parameter studies are now feasible, with a total cost of $\sim 35,000$ GPU node-hours (less than 10% of a typical Tier-0 allocation).

4. Key Results

A. Explosion Outcomes based on Magnetic Field ($B_0$)

• $B_0 = 10^{11}$ G (Models B11): No explosions. Regardless of rotation rate, these models failed to launch a successful jet. The shock stalled at $\sim 150–250$ km, remaining nearly spherical. The magnetic fields did not reach the magnetar strength required to drive an explosion within the simulated timeframe.
• $B_0 = 10^{12}$ G (Models B12): Explosions occurred for sufficiently high rotation rates, with outcomes heavily dependent on $\Omega_0$.

B. Explosion Outcomes based on Rotation Rate ($\Omega_0$) for $B_0 = 10^{12}$ G

1. Slow Rotation ($\Omega_0 \le 0.5$ rad/s):
   • Models R01B12 and R05B12 failed to launch a clean jet.
   • R05B12 showed signs of jet formation (low plasma-$\beta$, asymmetric shock) but did not fully explode by the end of the simulation.
2. Intermediate Rotation ($\Omega_0 = 1.0, 1.5$ rad/s):
   • Models R10B12 and R15B12 successfully launched bipolar jets.
   • Crucial Finding: The jets were distorted by MHD instabilities (specifically the kink instability), causing them to bend sideways and tilt toward the equatorial plane.
   • Observational Implication: These explosions appear spherically symmetric from an observational standpoint, potentially mimicking neutrino-driven explosions despite being magnetorotationally powered.
3. Fast Rotation ($\Omega_0 \ge 2.0$ rad/s):
   • Models R20B12 and R25B12 launched strong, steady jets along the rotation axis.
   • Shock Velocity: Reached peak polar velocities of $15,000–22,500$ km/s, consistent with observed SNe Ic-bl.
   • Asphericity: These models exhibited high asphericity ($Z \approx 0.3$).

C. Energetics and Remnants

• Explosion Energy: At $\sim 200$ ms, explosion energies ranged from $0.3 \times 10^{50}$ to $0.9 \times 10^{50}$ erg. While lower than the canonical $10^{52}$ erg for hypernovae, the authors note that energy continues to grow and has not saturated.
• PNS Evolution: The Proto-Neutron Star (PNS) mass and radius evolution were remarkably similar across all models, reaching $\sim 1.75–1.8\,M_\odot$ and shrinking to $\sim 45–55$ km, respectively. Accretion rates dropped below $1\,M_\odot$/s by 200 ms.

5. Significance and Conclusions

• Mechanism Validation: The study confirms that the magnetorotational mechanism can drive explosions in high-compactness progenitors, provided the initial magnetic field is $\ge 10^{12}$ G and rotation is rapid ($\Omega_0 \gtrsim 1.0$ rad/s).
• Observational Diversity: The results explain the diversity of observed supernovae.
  • SNe Ic-bl: Likely correspond to the high-rotation, high-field models (R20B12, R25B12) with high-velocity, asymmetric jets.
  • "Neutrino-like" Magnetorotational Events: Intermediate rotation models (R10B12, R15B12) produce explosions that look spherical and have lower velocities, potentially confusing observers about the underlying engine.
• Resolution Importance: The study highlights that maintaining high resolution in the shocked region is critical. Previous studies using static grids saw shock saturation due to resolution loss; this study shows continuous shock expansion when resolution is preserved.
• Future Work: The authors plan to extend simulations to longer timescales ($\gtrsim 1-2$ s) to determine final explosion energies and remnant fates (neutron star vs. black hole) and to employ more sophisticated M1 neutrino transport for nucleosynthesis studies.

In summary, this paper establishes a new benchmark for 3D GRMHD supernova simulations, demonstrating that GPU acceleration enables comprehensive parameter studies that reveal complex dependencies between rotation, magnetic fields, and explosion morphology.