Drawing the line between explosion and collapse in electron-capture supernovae -- I. Impact of conductive flame speeds and ignition conditions on the explosion mechanism

Through 56 3D hydrodynamic simulations of ONe white dwarfs, this study identifies a critical transition density range and demonstrates that both ignition conditions and laminar flame speeds are decisive factors in determining whether electron-capture supernovae result in a thermonuclear explosion or a gravitational collapse.

The Gist

Imagine a star as a giant, cosmic pressure cooker. For most of its life, it's stable, balancing the crushing weight of gravity pushing in with the heat of nuclear fusion pushing out. But for a specific type of star—roughly 8 to 12 times the mass of our Sun—things get tricky at the very end.

This paper is about figuring out exactly when these stars decide to explode like a firecracker or collapse into a black hole (or a neutron star). The authors call this the "tipping point."

Here is the story of their discovery, broken down into simple concepts and analogies.

1. The Setup: The Ticking Time Bomb

These stars have a core made of Oxygen and Neon. As they age, they get squeezed tighter and tighter by their own gravity. Eventually, the pressure gets so high that electrons (tiny particles) start getting "squeezed" into protons.

Think of this like a crowded elevator. As more people (electrons) get squeezed in, they start disappearing into the walls (turning into neutrons). This is bad news for the elevator because the people were holding up the ceiling. Once they disappear, the ceiling starts to fall. This is called collapse.

However, there's a twist. When these electrons disappear, they release a burst of energy that can ignite a thermonuclear fire (like lighting a match in a gas tank). This fire wants to blow the star apart.

The Big Question: Does the fire win and blow the star up (a Thermonuclear Explosion), or does the gravity win and crush the star down (a Collapse)?

2. The Experiment: 56 Cosmic Simulations

The scientists didn't have a real star to play with, so they built 56 different "virtual stars" on supercomputers. They changed two main things in their simulations:

1. How dense the star was when the fire started (The "squeeze").
2. Where the fire started (Right in the center, or slightly off to the side).

They also tested two different "rules" for how fast the fire spreads (called flame speeds). One rule was the old standard, and the other was a newer, more complex rule based on modern physics.

3. The Four Outcomes: The Star's Fate

The researchers found that the star's fate isn't just a simple "yes or no." It's more like a spectrum with four distinct endings:

• The Prompt Explosion (The Firecracker): The fire starts, rises quickly, and blows the star apart before gravity can crush it. The star survives as a smaller, stable remnant.
• The Marginal Explosion (The Narrow Escape): The fire barely manages to stop the collapse. It's a tightrope walk. The fire burns the core, but it manages to blow the outer layers off just in time.
• The Marginal Collapse (The Crushed Fire): The fire starts, but gravity wins. The star collapses. However, the fire is still burning on the outside, creating a weird, messy explosion while the core is being crushed.
• The Prompt Collapse (The Instant Squeeze): Gravity is so strong that it crushes the star almost immediately. The fire doesn't even get a chance to grow; it gets smothered instantly.

4. The Secret Ingredient: The "Sinking Ash"

Here is the most surprising discovery, which the authors call the "Sinking Ash" effect.

Imagine you are baking a cake. Usually, when you mix in something heavy, it sinks to the bottom. In these stars, when the fire burns the fuel, it creates "ashes" (heavier elements).

• In normal stars: The ashes are hot and light, so they float up (buoyancy).
• In these dense stars: The ashes get so heavy (because of the electron capture process) that they sink back down to the center of the star.

Why does this matter?
These sinking ashes act like a poison pill. They carry "neutron-rich" material to the core, making the core even more unstable and prone to collapsing. It's like pouring water on a fire that is trying to save the building; the ashes actually help the building collapse faster.

5. The Twist: Faster Fire = More Collapse?

This is the counter-intuitive part that confused the scientists at first.

• They thought: "If the fire spreads faster, it should blow the star up better, right?"
• Reality: They found that if the fire spreads too fast (using the newer physics rules), it actually causes the star to collapse.

The Analogy:
Imagine trying to push a heavy boulder up a hill.

• Slow Fire (Old Rules): The fire spreads slowly. This allows turbulence (chaos, swirling winds) to form. These swirls act like a turbocharger, making the fire spread super fast later on. This turbo-charged fire can reach the cooler, lighter parts of the star and blow it apart before gravity crushes it.
• Fast Fire (New Rules): The fire spreads so smoothly and quickly that it doesn't create any turbulence. It stays "laminar" (smooth). Because it's smooth, it stays trapped in the dense, heavy center of the star. It burns the core, the ashes sink, the core gets poisoned, and gravity wins.

So, paradoxically, a "slower" fire that creates chaos (turbulence) is better at saving the star than a "faster," smoother fire.

6. The Location Matters

The scientists also found that where the fire starts is crucial.

• If the fire starts dead center, it's very hard to save the star. The ashes sink immediately, and the star collapses.
• If the fire starts off-center (like a few miles away from the middle), the star has a much better chance of exploding. The fire has time to grow and reach the lighter outer layers before the core gets too heavy.

The Bottom Line

This paper tells us that the fate of these stars is a delicate dance between gravity, fire, and fluid dynamics.

• To explode: You need a fire that starts off-center and creates enough chaos (turbulence) to blow the star apart before the core gets poisoned by sinking ashes.
• To collapse: If the fire is too smooth, starts in the exact center, or the star is too dense, the ashes sink, the core gets poisoned, and the star implodes.

Why should we care?
If these stars do explode (which this paper says is possible under the right conditions), they create unique chemical elements (like Calcium and Titanium) that we see in our solar system today. If they collapse, they leave behind neutron stars. Understanding this "tipping point" helps us understand where the elements in our bodies came from and what happens when stars die.

Technical Summary

Here is a detailed technical summary of the paper "Drawing the line between explosion and collapse in electron-capture supernovae: I. Impact of conductive flame speeds and ignition conditions on the explosion mechanism" by Holas et al.

1. Problem Statement

Electron-capture supernovae (ECSNe) occur in intermediate-mass stars ($8-12 M_\odot$) when an Oxygen-Neon (ONe) core reaches the Chandrasekhar mass limit. Traditionally, these events were thought to result in a gravitational collapse to a neutron star (cECSN) triggered by electron captures on$^{20}\text{Ne}$and$^{24}\text{Mg}$. However, recent theoretical work suggests that under specific conditions, a thermonuclear explosion (tECSN) could occur instead, potentially explaining certain isotopic anomalies (e.g.,$^{48}\text{Ca}$,$^{50}\text{Ti}$).

The central problem addressed in this study is the lack of a mapped transition regime between these two outcomes. It remains unclear how the central density at ignition ($\rho_c$) and the ignition location influence whether the star explodes or collapses. Furthermore, the role of laminar flame speeds and the resulting turbulence in determining this outcome had not been systematically investigated in 3D hydrodynamic simulations.

2. Methodology

The authors conducted a comprehensive parameter study using 56 full 3D hydrodynamic simulations of ONe white dwarfs (WDs) using the Leafs code.

• Initial Conditions:
  • Models consisted of homogeneous ONe WDs ($65%$$^{16}\text{O}$,$35%$$^{20}\text{Ne}$) with a fixed electron fraction$Y_e = 0.493$.
  • Central Density ($\rho_c$): Varied between $\log \rho_c \approx 9.95$ and $10.4$g cm$^{-3}$.
  • Ignition Geometry: Five different ignition radii were tested ($2.5, 10, 20, 35, 50, 73$ km from the center) to simulate different convective scenarios. The ignition spot was modeled as a composite of nine overlapping spheres to introduce realistic perturbations.
• Physics & Numerics:
  • Flame Model: A level-set technique was used to track the flame front.
  • Flame Speed Parameterizations: Two different prescriptions for laminar flame speed ($v_{lam}$) were compared:
    1. TW92 (Timmes & Woosley 1992): Assumes fixed $Y_e = 0.5$.
    2. S20 (Schwab et al. 2020): Includes modern microphysics and accounts for varying $Y_e$ (crucial for neutron-rich ashes).
  • Turbulence: A subgrid-scale model (Schmidt et al. 2006) was used to calculate turbulent flame speeds ($v_{turb}$).
  • Gravity & EOS: Full Poisson equation solved for gravity; Equation of State included relativistic electrons/positrons, ions, radiation, and Coulomb corrections.
  • Neutrinos: Thermal and weak neutrino losses were included. Electron captures were tracked via tabulated $\dot{Y}_e$ values, limited to $Y_e \ge 0.25$.

3. Key Contributions

• First 3D Parameter Study: This is the first study to systematically map the transition between explosion and collapse in ECSNe using 3D hydrodynamics, moving beyond previous 1D and 2D approximations.
• Flame Speed Sensitivity: The study rigorously demonstrates that the choice of laminar flame speed parameterization (specifically the treatment of $Y_e$) is a critical determinant of the outcome, often overriding the initial density conditions.
• Mechanism of Ash Sinking: The authors provide a detailed 3D visualization and analysis of the "sinking ashes" phenomenon, where neutron-rich burning products sink into the core due to density inversion, accelerating collapse.
• Transition Density Mapping: They establish a quantitative boundary for the transition from tECSN to cECSN based on ignition location and central density.

4. Key Results

A. Four Distinct Modes

The simulations revealed four distinct evolutionary modes:

1. Prompt Explosion: Low density; flame rises buoyantly, halts collapse, and unbinds the star.
2. Marginal Explosion: Intermediate density; the flame barely halts collapse. Crucially, ashes sink into the core, neutronizing it and reducing the effective Chandrasekhar mass ($M_{Ch,eff}$), but the explosion still succeeds.
3. Marginal Collapse: Slightly higher density; gravity wins, but the rising flame still incinerates the outer shell.
4. Prompt Collapse: High density; gravity overpowers burning almost immediately.

B. The Counter-Intuitive Role of Flame Speed

A major finding is that faster laminar flame speeds (S20) favor collapse, while slower speeds (TW92) favor explosion in the transition regime.

• Mechanism: Faster laminar flames (S20) burn a larger volume quickly, leading to rapid neutronization of the ashes. These ashes sink into the core, causing a density inversion that accelerates gravitational contraction.
• Turbulence Suppression: The faster laminar speed suppresses the formation of Rayleigh-Taylor (RT) instabilities. Without these instabilities, the flame cannot transition to the much faster turbulent flame speed.
• Consequence: The flame remains confined to high-density regions where nuclear burning is inefficient (due to photodisintegration and neutrino losses). In contrast, the slower TW92 flame allows instabilities to grow, enabling the flame to transition to turbulent propagation, reach lower-density regions, and generate enough energy to halt collapse.

C. Transition Density and Ignition Location

• Transition Range: The transition from explosion to collapse occurs roughly between $\log \rho_c = 10.0$ and $10.15$g cm$^{-3}$.
• Ignition Offset: A more off-center ignition allows for higher central densities to still result in an explosion.
  • Reasoning: Off-center ignitions delay the sinking of ashes into the core, giving the flame more time to grow and potentially transition to turbulent burning before the core becomes unstable.
• Flame Speed Impact: The transition density is lower for the S20 (faster) flame speed compared to TW92. For S20, explosions are only possible at lower densities because the suppression of turbulence prevents the flame from escaping the high-density core.

5. Significance and Implications

• Observational Predictions: The study suggests that tECSNe are possible but require very specific conditions (low central density at ignition or highly off-center ignition). If they occur, they produce bound remnants with neutron-rich iron-group cores and potentially high kick velocities due to asymmetry.
• Stellar Evolution Constraints: To predict whether a specific progenitor results in an explosion or collapse, stellar evolution models must accurately determine the ignition location and the central density at that moment. Small changes in convection or electron capture rates could shift the outcome.
• Remnant Fate: The "sinking ashes" mechanism implies that even successful thermonuclear explosions in high-density ONe WDs leave behind a bound remnant with a highly neutronized core. The long-term stability of these remnants is an open question.
• Future Work: The authors emphasize the need for multidimensional stellar evolution simulations to constrain ignition conditions and further study the fate of the bound remnants and the composition of the ejecta.

In summary, the paper establishes that the fate of an ECSN is a delicate balance between gravitational collapse and thermonuclear burning, heavily mediated by the interaction between laminar flame speed, turbulence generation, and the sinking of neutron-rich ashes.