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

Abstract: Context: Superbursts are very energetic explosions in the crust of neutron stars in Low-Mass X-ray Binaries (LMXBs). These are triggered by unstable carbon burning at $T\leq 10<sup>{9}$ K. In recent years, there has been a re-examination of the carbon fusion rate, finding that at these temperatures it might be either smaller or higher with respect to the classic rate from Caughler & Fowler (1988) by a factor $10<sup>{3}$. Aims: We explore the consequences changing the carbon fusion rate has over the physics of superbursts. Methods: For simulating superbursts, we employ the public code MESA v.24.08.1, as well as four versions of the carbon fusion reaction rate. Results: An enhancement of the reaction rate by a factor $10<sup>{3}$ at $T\leq 10<sup>9$~K reduces the recurrence and decay times of the superburst, as well as the column depth at ignition. The opposite behavior is observed when the carbon fusion rate is reduced by the same factor. The maximum temperature reached during the explosion is also sensitive to these changes, leading to either an enhancement or a reduction in the synthesis of $α$-nuclides. These changes are comparable to the effect of reducing the amount of base heating at the bottom of the envelope.

• The paper demonstrates that variations in carbon fusion rates significantly alter superburst recurrence intervals and ignition column depths.
• MESA simulations reveal that enhanced rates (e.g., LUNA) lead to shorter recurrence times, lower peak temperatures, and modified nucleosynthesis.
• Degeneracies between fusion rate and base envelope heating complicate the interpretation of superburst lightcurves for constraining nuclear physics.

Impact of Carbon Fusion Rate Variations on Superburst Properties in Neutron Stars

Overview and Motivation

Superbursts are rare, high-energy thermonuclear explosions occurring in the crusts of neutron stars within low-mass X-ray binaries (LMXBs), triggered by unstable carbon burning at temperatures below $10^9$ K. For decades, theoretical modeling of these events has relied on the classical carbon fusion rate determined by Caughlan & Fowler (1988), but recent experimental work suggests the actual rate at relevant astrophysical temperatures may diverge by up to three orders of magnitude, with both suppression and enhancement scenarios possible. This study utilizes the MESA code to systematically evaluate the consequences of these uncertainties in the carbon fusion rate for superburst dynamics, recurrence, energetics, and nucleosynthetic yields.

Simulation Framework and Reaction Rate Choices

The modeling was performed using MESA (v24.08.1), a multi-zone stellar evolution code validated against KEPLER for thermonuclear bursts. Two mass accretion rates were adopted: $5.88 \times 10^{-9}$ $M_\odot$ yr$^{-1}$ (moderate, $\sim0.3\,M_\mathrm{Edd}$) and $2 \times 10^{-8}$ $M_\odot$ yr$^{-1}$ (high, $\sim M_\mathrm{Edd}$), with envelope base luminosities calibrated to ensure unstable burning. Four carbon fusion rates were evaluated:

• CF88: Classical rate (default in most past models).
• LUNA: Rate measured with the Trojan Horse method [Tumino et al., 2018], up to $10^3$ times higher than CF88 at low $5.88 \times 10^{-9}$0.
• HIN/HIN-RES: Empirical hindrance-based rates from [Monpribat et al., 2022], up to $5.88 \times 10^{-9}$1 times lower than CF88 at low $5.88 \times 10^{-9}$2; HIN-RES includes a speculative low-energy resonance.

All rates incorporate updated branching ratios for $5.88 \times 10^{-9}$3 channels. The reaction network was primarily limited to $5.88 \times 10^{-9}$4, with $5.88 \times 10^{-9}$5 cases tested for completeness at high accretion rates.

Numerical Results: Superburst Recurrence, Ignition, and Thermal Properties

Recurrence and Decay Times

Simulations demonstrate that the carbon fusion rate is a primary factor controlling superburst recurrence intervals, decay profiles, and ignition column depths. The data reveal:

• Elevated fusion rate (LUNA): Recurrence and decay times are shortened, and ignition occurs at shallower column depths. For instance, at moderate accretion (CF88: 241.5 days, LUNA: 100.2 days), and at high accretion (CF88: 16.4 days, LUNA: 10.8 days).
• Suppressed fusion rate (HIN): Recurrence and decay times lengthen, and ignition occurs at deeper column depths (e.g., at moderate accretion, recurrences rise to 364.3 days).
• CF88 vs HIN-RES: These yield nearly indistinguishable results at $5.88 \times 10^{-9}$6 K.

Peak luminosities ($5.88 \times 10^{-9}$7 erg s$5.88 \times 10^{-9}$8) are relatively insensitive to fusion rate, but decay profiles and ignition depths shift in accordance with the rate, paralleling changes induced by base envelope heating.

Thermal and Nucleosynthetic Evolution

Temperature profiles during superbursts show marked sensitivity to the adopted fusion rate:

• LUNA-enhanced rate: Maximum temperatures reduced ($5.88 \times 10^{-9}$9 K), faster cooling and reduced $M_\odot$0-nuclide synthesis.
• HIN rate: Maximum temperature increased ($M_\odot$1 K), slower cooling, enhanced nucleosynthetic flow toward heavier nuclei.

Notably, multi-zone models reveal two-stage post-burst cooling: a brief phase of sustained high temperatures (~2 hrs), followed by rapid decline, with burning confined to the region near ignition depth.

Abundance Distribution and Nucleosynthetic Yields

Final ash distributions consistently show a double-peaked structure centered at $M_\odot$2 (e.g., Si) and $M_\odot$3 (Fe), largely invariant across fusion rate and accretion scenarios, with secondary maxima at intermediate $M_\odot$4-nuclides. Accretion rates modulate the $M_\odot$5 region, where LUNA yields significantly larger amounts of $M_\odot$6-nuclides compared to HIN or CF88. At high accretion ($M_\odot$7), enhanced synthesis of $M_\odot$8O and $M_\odot$9 nuclei is observed, with negligible production beyond $^{-1}$0, even with extended networks.

Sensitivity Analyses and Robustness

Interplay of Base Heating and Fusion Rate

Systematic variation of base envelope luminosity $^{-1}$1 demonstrates degeneracy with the fusion rate: enhanced fusion or increased $^{-1}$2 yield similar reductions in recurrence and decay times, and vice versa. However, fusion rate modifications alone cannot mimic effects of varying accretion rate $^{-1}$3. Thus, observed superburst lightcurves cannot uniquely constrain the fusion rate if $^{-1}$4 is uncertain.

Composition and Network Effects

Replacing $^{-1}$5Fe with an inert high-$^{-1}$6 nuclide (e.g., $^{-1}$7Ge) in the accreted envelope, or removing photodisintegration reactions, highlights that $^{-1}$8 nuclide synthesis is predominantly governed by carbon burning. The iron-peak distribution is only partially sensitive to accreted composition, with nucleosynthetic endpoint shifting to $^{-1}$9 in absence of $\sim0.3\,M_\mathrm{Edd}$0Fe. Examination of extended reaction networks ($\sim0.3\,M_\mathrm{Edd}$1) confirms negligible flow to higher $\sim0.3\,M_\mathrm{Edd}$2 at $\sim0.3\,M_\mathrm{Edd}$3, reinforcing the appropriateness of $\sim0.3\,M_\mathrm{Edd}$4 truncation for this regime.

The n-channel of carbon fusion, although comprising up to $\sim0.3\,M_\mathrm{Edd}$5 of the rate at high $\sim0.3\,M_\mathrm{Edd}$6, produces only minor abundance shifts and does not substantially affect gross burst properties.

Implications and Future Directions

This study substantially refines predictions for superburst phenomenology by quantifying the effects of carbon fusion rate uncertainties. The results indicate that base envelope heating and fusion rate are equally influential in setting superburst timescales when accretion rate is well-constrained, but their effects are degenerate from an observational perspective. The double-peaked ash distribution is robust, stemming from the combination of thermonuclear conditions and accreted composition, while nucleosynthetic flow beyond $\sim0.3\,M_\mathrm{Edd}$7 is rare in typical superburst environments.

The findings caution against using superburst lightcurves to uniquely determine low-temperature carbon fusion rates, particularly given the non-observability of envelope base heating. Future developments may require more precise constraints on neutron star parameters, accretion rates, and direct spectroscopic measurements of burst ashes or pre-burst composition to disambiguate nuclear physics dependencies. Additionally, scenarios with higher accretion (hyperbursts) or electron captures (neon seeds) merit further exploration, especially with extended reaction networks and inclusion of neutron diffusion and pycnonuclear regimes.

Conclusion

The impact of carbon fusion rate uncertainties on superburst properties is quantitatively comparable to changes in envelope base heating, governing recurrence, decay, and ignition depth. Nucleosynthetic yields, particularly in the iron and silicon peaks, are robust across plausible rates and accretion regimes. Degeneracies between fusion rate and heating limit the interpretive power of superburst lightcurves for constraining nuclear physics, underscoring the need for integrated modeling and observational strategies in future studies.