On the impact of the carbon fusion rate over the properties of superbursts -- Numerical simulations of superbursts with MESA

Technical Summary: On the impact of the carbon fusion rate over the properties of superbursts

Problem Statement
Superbursts are energetic thermonuclear explosions occurring in the crusts of neutron stars within Low-Mass X-ray Binaries (LMXBs), triggered by unstable carbon burning at temperatures $T \le 10^9$ K. While previous multi-zone simulations (e.g., Keek & Heger 2011, 2012) have successfully modeled superburst recurrence times and light curves, they relied on the classic carbon fusion rate from Caughlan & Fowler (1988, CF88). Recent nuclear physics investigations have challenged this rate, suggesting that at relevant temperatures ($T \le 10^9$ K), the actual rate could be significantly different. Specifically, experimental data from the LUNA collaboration (Tumino et al. 2018) suggests the rate could be enhanced by a factor of $\sim 10^3$, while hindrance models (Monpribat et al. 2022) suggest it could be reduced by a similar factor. The impact of these substantial variations on superburst observables—such as recurrence times, decay times, ignition depths, and nucleosynthesis yields—remains unexplored in the context of modern stellar evolution codes.

Methodology
The authors employed the public stellar evolution code MESA v.24.08.1 to simulate the evolution of neutron star envelopes undergoing mass accretion. The simulations utilized a reduced nuclear reaction network, Approx_A56 (covering nuclides up to $A=56$), with select runs using Approx_A64 to verify truncation effects.

Key parameters and configurations included:

• Accretion Rates: Two fiducial rates were tested: a "moderate" rate ($\dot{M}_{moderate} \approx 5.88 \times 10^{-9} M_\odot \text{yr}^{-1}$) and a "high" rate ($\dot{M}_{high} \approx 2 \times 10^{-8} M_\odot \text{yr}^{-1}$).
• Composition: The accreted material was modeled as 20% $^{12}\text{C}$ and 80% $^{56}\text{Fe}$, consistent with literature requirements for triggering superbursts. The initial envelope was pure $^{56}\text{Fe}$.
• Base Heating: A fixed base luminosity ($L_b$) was applied at the bottom of the envelope to ensure unstable carbon ignition, calibrated to match previous KEPLER simulations.
• Carbon Fusion Rates: Four distinct reaction rates were compared:
  1. CF88: The classical rate (fiducial model).
  2. LUNA: Based on LUNA measurements, predicting a rate $\sim 10^3$ times higher than CF88 at $T \sim 2\text{--}3 \times 10^8$ K.
  3. HIN: Based on the hindrance model, predicting a rate $\sim 10^3$ times lower than CF88.
  4. HIN-RES: The HIN model including a low-lying resonance, yielding results intermediate between HIN and CF88.
• Physical Treatments: The simulations included radiative opacities (Wolf prescription), Mixing Length Theory for convection, and fixed surface pressure boundary conditions. General relativistic corrections were omitted.

Key Results
The study quantifies how variations in the carbon fusion rate alter superburst properties:

1. Timescales and Ignition Depth:

   • Enhanced Rate (LUNA): Leads to shorter recurrence times and decay times, and triggers ignition at a shallower column depth ($y_{ignition}$). For $\dot{M}_{moderate}$, the recurrence time drops from $\sim 241$ days (CF88) to $\sim 100$ days.
   • Reduced Rate (HIN): Results in longer recurrence and decay times, with ignition occurring at deeper column depths. For $\dot{M}_{moderate}$, the recurrence time increases to $\sim 364$ days.
   • Intermediate Rate (HIN-RES): Produces results nearly indistinguishable from the CF88 fiducial model.

2. Temperature and Light Curves:

   • The maximum temperature reached during the explosion is sensitive to the rate. The HIN rate yields the highest peak temperatures ($\sim 5 \times 10^9$ K), while LUNA yields lower peaks ($\sim 3 \times 10^9$ K).
   • The luminosity decay profiles shift accordingly: LUNA simulations show faster decay, while HIN simulations show slower decay.
   • At $\dot{M}_{high}$, peak luminosity is largely insensitive to the rate choice, but decay times and recurrence intervals still vary significantly.

3. Nucleosynthesis and Ash Composition:

   • The distribution of ashes exhibits a characteristic double-peaked structure (around $^{28}\text{Si}$ and $^{56}\text{Fe}$) across all models.
   • Rate Sensitivity: For $A \le 44$, the abundance of $\alpha$-nuclides is highly sensitive to the rate. The LUNA rate (higher temperature/efficiency) produces significantly more $\alpha$-nuclides compared to CF88/HIN-RES, while the HIN rate produces fewer.
   • High Accretion Rate Effects: At $\dot{M}_{high}$, there is a notable enhancement in the synthesis of nuclei with $28 \le A \le 50$ and a specific accumulation of $\sim 0.1\%$ $^{16}\text{O}$, a feature less prominent at moderate rates.
   • Photodisintegration: Simulations replacing the inert $^{56}\text{Fe}$ with $^{72}\text{Ge}$ confirmed that the iron-peak ash distribution is largely driven by $^{56}\text{Fe}$ photodisintegration rather than carbon fusion alone. Removing photodisintegration channels halted nucleosynthesis beyond $^{40}\text{Ca}$.

4. Degeneracy with Base Heating:

   • The study demonstrates that changing the carbon fusion rate has an effect on superburst timescales and ignition depth comparable to changing the base luminosity ($L_b$) of the envelope. For instance, reducing the rate (HIN) and increasing $L_b$ produces similar recurrence times. This suggests a degeneracy where light curve fitting alone cannot uniquely constrain both the fusion rate and the base heating.

Significance and Conclusions
The paper concludes that the carbon fusion rate at $T \le 10^9$ K is a critical parameter governing superburst physics. The authors assert that:

• Uncertainties in the carbon fusion rate can explain variations in observed recurrence and decay times without necessarily invoking changes in the mass accretion rate or base heating.
• The "canonical" ash composition derived from previous KEPLER simulations (using CF88) is robust regarding the double-peaked structure but sensitive to the specific reaction rate regarding $\alpha$-nuclide abundances.
• The degeneracy between the carbon fusion rate and base heating implies that fitting a single superburst light curve cannot unambiguously determine the underlying nuclear physics or the thermal state of the neutron star crust.
• The results validate the use of MESA for superburst modeling, showing consistency with previous KEPLER results while offering a public code alternative for exploring nuclear physics uncertainties.

The authors modestly note that while their findings suggest the carbon fusion rate could alleviate the need for high base heating in models (if the LUNA rate is correct), the uncertainty in mass accretion rates for specific sources (like 4U 1636-53) prevents definitive conclusions about which rate is physically realized. Furthermore, they clarify that their treatment of carbon burning as purely thermonuclear is justified by the densities and temperatures reached, distinct from pycnonuclear regimes.