POSSIBLE RESONANCES IN THE ¹²C + ¹²C FUSION RATE AND SUPERBURST IGNITION

Superbursts are long, energetic, and rare thermonuclear flashes on accreting neutron stars in low-mass X-ray binaries. Their durations (∼hours), fluences (∼10⁴² erg), and recurrence times (∼years) distinguish superbursts from their typical hydrogen- and helium-triggered counterparts (for reviews, see Kuulkers 2004; Cumming 2005; Strohmayer & Bildsten 2006). As of this writing, astronomers have detected 15 superbursts from 10 sources (Kuulkers 2004; in't Zand et al. 2004; Remillard et al. 2005; Kuulkers 2005; Keek et al. 2008, and references therein).

The proposal (Cumming & Bildsten 2001; Strohmayer & Brown 2002) that thermally unstable ¹²C fusion (Woosley & Taam 1976; Taam & Picklum 1978; Brown & Bildsten 1998) triggers superbursts offers a reasonable explanation of their origin. Cooling model fits to superburst light curves (Cumming & Macbeth 2004; Cumming et al. 2006) as well as observed fluences and recurrence times (e.g., Keek et al. 2006) suggest ignition column depths Σ_ign ≈ 10¹² g cm⁻², where Σ ≡ ∫ρ dz is the radially integrated density. Previous superburst ignition models (Cumming & Bildsten 2001; Strohmayer & Brown 2002; Cumming 2003; Brown 2004; Cooper & Narayan 2005; Cumming et al. 2006; Gupta et al. 2007) demonstrated that ¹²C ignites at Σ ≈ 10¹² g cm⁻² only if ¹²C is abundant and the ocean temperature T ≈ 6 × 10⁸ K at that column depth; within existing models of nuclear heating in the neutron star crust, such a large temperature requires an inefficient neutrino emission mechanism in the neutron star core and a low thermal conductivity in the neutron star crust, so that the crust is much hotter than the core.

Recent observations, simulations, and experiments have exposed three fundamental problems with this scenario. First and foremost is the inference that the ocean is in fact too cold for ¹²C ignition at the inferred column depth Σ_ign ≈ 10¹² g cm⁻². This comes from fits (Shternin et al. 2007; Brown & Cumming 2009) to the quiescent cooling of the quasi-persistent transient KS 1731 − 260 (Wijnands et al. 2002; Rutledge et al. 2002; Cackett et al. 2006), a system that also exhibited a superburst (Kuulkers et al. 2002b). The timescale for the quiescent luminosity to decrease suggests that the crust's thermal conductivity is high; as a result, the inner crust temperature remains close to that of the core even during the accretion outburst. Cackett et al. (2008) reach the same conclusion for MXB 1659 − 29 (see also Brown & Cumming 2009). In fact, molecular dynamics simulation results (Horowitz et al. 2007, 2009; Horowitz & Berry 2009) suggest that the neutron star crust is arranged in a regular lattice and therefore has a high thermal conductivity. Neither shear-induced viscous heating (Piro & Bildsten 2007; Keek et al. 2009) nor deep crustal heating due to electron captures, neutron emissions, and pycnonuclear reactions (e.g., Haensel & Zdunik 2008; Horowitz et al. 2008; Gupta et al. 2008) can account for the heat necessary to raise the ocean temperature to the required level (although see Page & Cumming 2005; Blaschke et al. 2008, who consider heating in strange stars and hybrid stars, respectively).

Second, evidence of heavy-ion fusion hindrance at extreme sub-Coulomb-barrier energies (Jiang et al. 2002, 2007, 2008) implies that the cross section and thereby the ¹²C + ¹²C reaction rate may be orders of magnitude smaller than that assumed in the aforementioned superburst ignition models. When included in superburst ignition models, heavy-ion fusion hindrance increases Σ_ign by at least a factor of 2 (Gasques et al. 2007).

Third, the means by which nuclear burning on the stellar surface produces sufficient quantities of ¹²C to trigger superbursts is poorly understood. Superburst models require large ¹²C mass fractions for ignition (Cumming & Bildsten 2001; Cumming 2003; Cooper & Narayan 2005; Cooper et al. 2006; Cumming et al. 2006). All systems that exhibit superbursts show helium-triggered type I X-ray bursts as well (e.g., Galloway et al. 2008), but theoretical models of such bursts yield ¹²C mass fractions far smaller than those required for ignition (Joss 1978; Schatz et al. 2001, 2003b; Koike et al. 2004; Woosley et al. 2004a; Fisker et al. 2005, 2008; Peng et al. 2007; Parikh et al. 2008). Most systems that exhibit superbursts apparently undergo long periods of stable nuclear burning between successive helium-triggered bursts (Kuulkers et al. 2002a; in't Zand et al. 2003; Keek et al. 2008); stable burning generates much more ¹²C than unstable burning, but the calculated yield is insufficient to trigger superbursts in all systems, particularly those accreting at a high rate (Taam & Picklum 1978; Schatz et al. 1999, 2003b; Cooper et al. 2006; Fisker et al. 2006).

Detection of a superburst from the classical transient 4U 1608 − 522 (Remillard et al. 2005; Kuulkers 2005) with Σ_ign ≈ 10¹² g cm⁻² (Keek et al. 2008) exacerbates all three problems: (1) the transient's inferred ocean temperature is lower than those of other systems exhibiting superbursts; (2) heavy-ion fusion hindrance is greater at lower temperatures; and (3) most of the matter accreted onto the neutron star prior to the observed superburst likely burned during helium-triggered type I X-ray bursts, which current theoretical calculations suggest would generate far less ¹²C than that required for ignition.

Reconciling superburst observations with the current theoretical model is impossible. This motivates both a critical assessment of the current ignition model and a search for alternative ignition mechanisms.

Kuulkers et al. (2002b) proposed such an alternative mechanism. Electron captures onto protons and the subsequent captures of the resulting neutrons onto heavy nuclei liberate ≈ 7 MeV/m_u (Bildsten & Cumming 1998). Pre-threshold captures of super-Fermi electrons are very temperature sensitive and therefore could trigger an energetic thermonuclear flash. An attractive feature of this mechanism is that ignition occurs always at the same electron chemical potential; thus Σ_ign would be similar for all superbursts, in accord with observations. Unfortunately, the calculated Σ_ign ≈ 2 × 10¹⁰ g cm⁻² is much smaller than the inferred superburst Σ_ign, making it an unlikely trigger mechanism. This motivates our investigation of exothermic electron captures onto heavy nuclei, which occur throughout the ocean and crust of an accreting neutron star, including the superburst ignition region (Sato 1979; Blaes et al. 1990; Haensel & Zdunik 1990, 2003, 2008; Gupta et al. 2007). In Section 2, we determine the thermal stability of electron captures onto heavy nuclei in accreting neutron stars. We show that instability requires unrealistically large reaction $\mathcal {Q}$-values, where $\mathcal {Q}$ is the energy released per capture; thus we conclude that electron captures in accreting neutron star oceans are thermally stable. We then consider the relevance of α captures onto light elements such as ¹²C in Section 3. We find that none of these reactions is a feasible mechanism and thereby confirm the proposal that ¹²C + ¹²C triggers superbursts.

In Section 4, we assess whether the ¹²C + ¹²C reaction rate could be much larger than the fiducial rate. We investigate (Section 4.1) the screening enhancement factor, including a careful evaluation of corrections to the liner mixing rule, and show that uncertainties in the plasma screening enhancement are unlikely to change the ¹²C + ¹²C reaction rate enough. We then consider the nuclear cross section. We find that a strong resonance at an energy near 1.5 MeV in the ¹²C + ¹²C system, which theoretical nuclear physics models predict, could increase the reaction rate in the astrophysically relevant temperature range by over 2 orders of magnitude. In Section 5, we show that the existence of such a resonance could decrease the predicted Σ_ign by a factor ≈2–4 and thereby alleviate the discrepancy between superburst models and observations. We conclude in Section 6 by discussing the implications of our findings.

Consider the accretion-driven compression of a matter element containing a nucleus of mass M(A, Z), where A is the mass number and Z is the proton number. The degenerate electrons' chemical potential μ_e rises as the nucleus advects to higher pressures. Eventually M(A, Z) + μ_e/c² exceeds M(A, Z − 1) and electron capture becomes energetically favorable. Such captures often occur in equilibrium and release a negligible amount of energy; however, some captures can heat the ocean in two mutually inclusive ways (e.g., Gupta et al. 2007). (1) An electron captures into an exited state of the daughter nucleus if, for example, the daughter nucleus's ground state is forbidden. The daughter nucleus then radiatively de-excites and thereby heats the ocean. (2) If the parent nucleus is even–even, then M(A, Z − 1)>M(A, Z − 2) due to the nuclear pairing energy, and a second electron capture immediately ensues. The latter, post-threshold electron capture occurs out of equilibrium and thus releases heat.

2.1. Governing Equations

We construct a simple model of the accreted layer to determine the stability of exothermic electron captures to thermal perturbations. We assume spherical accretion onto a neutron star of mass M = 1.4 M_☉ and radius R = 10 km at an accretion rate per unit area $\dot{\Sigma }$. The accreted layer's scale height is much less than R, so we set the gravitational acceleration g = GM/R²(1 − 2GM/Rc²)^−1/2 = 2.43 × 10¹⁴ cm s⁻² throughout the layer. The layer is always in hydrostatic equilibrium, so the column depth Σ is a good Eulerian coordinate. To facilitate comparisons between microphysical and observationally inferred quantities, we express microphysical quantities in terms of the macroscopic coordinate Σ using the following approximate relation between mass density ρ and Σ for relativistic, degenerate electrons,³

$$\begin{equation} \rho \approx 5.9\times 10^{8}\;\mathrm{g}\;{\mathrm{c}\mathrm{m}}^{-3}\left(\frac{\langle A/Z \rangle }{2} \right) \Sigma _{12}^{3/4}, \end{equation} \tag{ 1 }$$

where 〈A/Z〉 is the mean molecular weight per electron and Σ = Σ₁₂ × 10¹² g cm⁻². We denote the Eulerian time and spatial derivatives as ∂/∂t and ∂/∂Σ, respectively, and the Lagrangian derivative following a matter element as D/Dt, where $D/Dt = \partial /\partial t \,\,{+}\,\, \dot{\Sigma }\partial /\partial \Sigma$. The governing transport, entropy, and continuity equations are

$$\begin{equation} F = \rho K \frac{\partial T}{\partial \Sigma }, \end{equation} \tag{ 2 }$$

$$\begin{equation} T\frac{Ds}{Dt} = \mathcal {E}X r_{\mathrm{ec}}+ \frac{\partial F}{\partial \Sigma }, \end{equation}\hspace{-4pc} \tag{ 3 }$$

$$\begin{equation} \frac{DX}{Dt} = -X r_{\mathrm{ec}}, \end{equation} \tag{ 4 }$$

where F is the flux, K is the thermal conductivity, s is the entropy, $\mathcal {E}= \mathcal {Q}/(Am_\mathrm{u})$ is the energy per gram released via electron captures, X is the parent nucleus mass fraction,

$$\begin{equation} r_{\mathrm{ec}}= \left(\frac{\ln 2}{{\langle f \! t \rangle }} \right) \frac{1}{(m_\mathrm{e}c^{2})^{5}} \int ^{\infty }_{Q} E^{2} (E-Q)^{2} f(E,\mu _{\mathrm{e}},T)dE, \end{equation} \tag{ 5 }$$

is the electron capture rate (Fuller et al. 1985), 〈ft〉 is the effective ft value (Fuller et al. 1980, 1985; Langanke & Martínez-Pinedo 2001), m_e is the electron mass, Q is the threshold energy, and

$$\begin{equation} f(E,\mu _{\mathrm{e}},T) = \frac{1}{1 + \exp [(E-\mu _{\mathrm{e}})/k_\mathrm{B}T]} \end{equation} \tag{ 6 }$$

is the Fermi–Dirac distribution function.

Consider pre-threshold electron captures, where μ_e < Q. For T = 0, all electrons have energies E ⩽ μ_e by Equation (6); electron capture is blocked. For T > 0, some electrons have E > Q and thus can capture. The number of electrons with E > Q increases with T, which makes pre-threshold electron capture temperature sensitive. In the pre-threshold limit (μ_e − Q)/k_BT ≪ 0,

$$\begin{equation} r_{\mathrm{ec}}(\mu _{\mathrm{e}},T) = \left(\frac{\ln 2}{{\langle f \! t \rangle }} \right) \frac{2Q^{2}(k_\mathrm{B}T)^{3}}{(m_\mathrm{e}c^{2})^{5}} \exp \left[ {-}\! \left(\frac{Q-\mu _{\mathrm{e}}}{k_\mathrm{B}T} \right) \right], \end{equation} \tag{ 7 }$$

(Fuller et al. 1985; Bildsten & Cumming 1998). Conversely, for μ_e > Q a majority of electrons has E > Q and hence can capture for any T, making r_ec relatively temperature insensitive and thus thermally stable. We therefore consider pre-threshold electron captures exclusively hereafter.

2.2. Pre-threshold Electron Capture

Pre-threshold electron captures occur within a thin layer in the deep ocean. To illustrate this, consider the height-integrated capture rate. Relativistic, degenerate electrons supply the pressure P = gΣ, so

$$\begin{equation} \Sigma \approx \frac{\mu _{\mathrm{e}}^{4}}{12\pi ^{2}g(\hbar c)^{3}} = 10^{12}\;\mathrm{g}\;{\mathrm{c}\mathrm{m}}^{-2}\left(\frac{\mu _{\mathrm{e}}}{3.43\;\mathrm{M}\mathrm{eV}} \right)^{4}. \end{equation} \tag{ 8 }$$

Integrating Equation (7) over Σ and using Equation (8),

$$\begin{equation} \int _{0}^{\mu _{\mathrm{e}}} r_{\mathrm{ec}}(\mu _{\mathrm{e}}^{\prime },T)\,d\Sigma ^{\prime } \approx \left(\frac{4 k_\mathrm{B}T}{\mu _{\mathrm{e}}} \right) r_{\mathrm{ec}}(\mu _{\mathrm{e}},T) \Sigma \end{equation} \tag{ 9 }$$

for μ_e ≫ k_BT. Equation (9) shows that pre-threshold electron captures occur in a narrow column depth range

$$\begin{equation} \frac{\Delta \Sigma }{\Sigma } \approx \frac{4 k_\mathrm{B}T}{\mu _{\mathrm{e}}} = 0.050\left(\frac{T_{8}}{5} \right) \Sigma _{12}^{-1/4}, \end{equation} \tag{ 10 }$$

where T = T₈ × 10⁸ K.

Now consider electron captures in steady state, such that electrons capture onto nuclei at the same rate as accretion advects the nuclei (see also the discussion in Bildsten & Cumming 1998). Equation (4) becomes

$$\begin{equation} \dot{\Sigma }\frac{\partial \ln X}{\partial \Sigma } = - r_{\mathrm{ec}}. \end{equation} \tag{ 11 }$$

Integrating Equation (11) from 0 to Q and using Equations (8) and (9), we find that most electron captures occur pre-threshold when

$$\begin{equation} \Sigma _{12}> 0.027 \left(\frac{T_{8}}{5} \right)^{-16/5} \left(\frac{\dot{\Sigma }}{0.3 \dot{\Sigma }_{\mathrm{Edd}}}\right)^{4/5} \left(\frac{{\langle f \! t \rangle }}{10^{3} \;\mathrm{s}} \right)^{4/5}, \end{equation} \tag{ 12 }$$

where $\dot{\Sigma }_{\mathrm{Edd}}\approx 10^{5} \ \mathrm{g}\ {\mathrm{c}\mathrm{m}}^{-2}\ {\mathrm{s}}^{-1}$ is the local accretion rate at which the accretion flux equals the Eddington flux.⁴ Superbursts ignite at column depths Σ₁₂ ∼ 1 and accretion rates $\dot{\Sigma }\approx 0.1$–$1 \;\dot{\Sigma }_{\mathrm{Edd}}$. Equation (12) shows that superallowed electron captures (for which 〈ft〉 ∼ 10³–10⁴ s) occur pre-threshold at superburst ignition depths.

2.3. Thermal Stability Analysis

We now derive a one-zone model (e.g., Fujimoto et al. 1981; Paczyński 1983; Bildsten 1998b) from the governing Equations (2)–(4) to determine the stability of pre-threshold electron captures to thermal perturbations and thereby ascertain whether electron captures trigger superbursts. We consider only temperature perturbations and ignore the accretion-induced entropy advection through the bottom of the zone. Therefore, we set ∂/∂t = 0 in Equation (4) and approximate Ds/Dt = ∂s/∂t in Equation (3). Perturbations occur at constant pressure since the scale height ∼Σ/ρ ≪ R; therefore, we write Tds = C_PdT, where C_P is the specific heat at constant pressure. Equations (2)–(4) become

$$\begin{eqnarray} F &=& \rho K \frac{\partial T}{\partial \Sigma }, \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} C_P \frac{\partial T}{\partial t} &=& \mathcal {E}X r_{\mathrm{ec}}+ \frac{\partial F}{\partial \Sigma }, \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} \dot{\Sigma }\frac{\partial X}{\partial \Sigma } &=& -\!X r_{\mathrm{ec}}. \end{eqnarray} \tag{ 15 }$$

We simplify Equations (13)–(15) as follows. We set ρ, F, and X to be constant throughout the layer; specifically, we adopt step-like profiles for F and X,

$$\begin{equation} F(\Sigma,t) = F_0(t) \Theta (\Sigma _{\mathrm{ec}}-\Sigma),\quad X(\Sigma) = X_0 \Theta (\Sigma _{\mathrm{ec}}-\Sigma), \end{equation} \tag{ 16 }$$

where Θ is the Heaviside step function, Σ_ec denotes the column depth at the bottom of the layer, and F₀ and X₀ denote the values at the top of the layer, where Σ ≪ Σ_ec. We assume the ocean consists of a single ion species and set X₀ = 1. Electron-ion scattering sets the ocean's thermal conductivity (Yakovlev & Urpin 1980; Itoh et al. 1983; Potekhin et al. 1999)

$$\begin{equation} K \approx \frac{9.8\times 10^{18}}{ Z^{2/3} A^{1/3}} \!\left(\frac{T_{8}}{5} \!\right)\!\left(\frac{\rho }{6\times 10^{8}\;\mathrm{g}\;{\mathrm{c}\mathrm{m}}^{-3}} \right)^{1/3} \mathrm{erg\ cm^{-1} s^{-1} K^{-1}}, \end{equation} \tag{ 17 }$$

where we set the Coulomb logarithm Λ_ei = 1, a value appropriate for a plasma at T₈ ≈ 5 and ρ ≈ 6 × 10⁸ g cm⁻³. Since K ∝ T, we rewrite Equation (13) as

$$\begin{equation} F = \frac{\rho K}{2 T} \frac{\partial T^{2}}{\partial \Sigma }. \end{equation} \tag{ 18 }$$

Equations (16) and (18) then imply

$$\begin{equation} T^{2}(\Sigma,t) = T_0^{2} + \left[T_{\mathrm{ec}}(t)^{2}-T_0^{2} \right]\frac{\Sigma }{\Sigma _{\mathrm{ec}}}. \end{equation} \tag{ 19 }$$

Integrating Equations (14)–(15) over Σ and using Equations (9), (10), (16), (18), and (19), we find

$$\begin{eqnarray} C_P \frac{\partial T_{\mathrm{ec}}}{\partial t} &=& \frac{\Delta \Sigma }{\Sigma _{\mathrm{ec}}}\mathcal {E}r_{\mathrm{ec}}- \frac{\rho K}{2 T}\left(\frac{T_{\mathrm{ec}}^{2}-T_0^{2}}{\Sigma _{\mathrm{ec}}^{2}} \right), \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} r_{\mathrm{ec}}&=& \frac{\dot{\Sigma }}{\Delta \Sigma }. \end{eqnarray} \tag{ 21 }$$

Equation (21) shows that electron capture occurs when the lifetime of a parent nucleus 1/r_ec equals the time t_accΔΣ/Σ, an advecting element spends within the capture region, where

$$\begin{equation} t_{\mathrm{acc}}\equiv \Sigma /\dot{\Sigma } \end{equation} \tag{ 22 }$$

is the accretion timescale. Note that this differs from the usual assumption (Blaes et al. 1990; Bildsten 1998a; Bildsten & Cumming 1998; Ushomirsky et al. 2000) that electron capture occurs when 1/r_ec = t_acc.

Finally, conducting a linear stability analysis on Equation (20) and using Equation (21), the thermal instability criterion is

$$\begin{equation} \nu \frac{\mathcal {E}}{t_{\mathrm{acc}}} > \frac{\rho K T_{\mathrm{ec}}}{\Sigma _{\mathrm{ec}}^{2}}, \end{equation} \tag{ 23 }$$

where the temperature sensitivity of the height-integrated electron capture rate

$$\begin{equation} \nu \equiv \frac{\partial \ln (r_{\mathrm{ec}}\Delta \Sigma)}{\partial \ln T} = 4 + \ln \left[8\ln 2\frac{t_{\mathrm{acc}}}{{\langle f \! t \rangle }}\frac{Q^{2}(k_\mathrm{B}T)^{4}}{(m_\mathrm{e}c^{2})^{5}\mu _{\mathrm{e}}} \right] \end{equation} \tag{ 24 }$$

from Equations (7) and (9). Noting that electrons capture when μ_e/Q ≈ 1, we write Equation (24) as

$$\begin{equation} \nu = 8.14+\ln \left[ \left(\frac{T_{8}}{5} \right)^{4} \Sigma _{12}^{5/4} \left(\frac{\dot{\Sigma }}{0.3 \dot{\Sigma }_{\mathrm{Edd}}}\right)^{-1} \left(\frac{{\langle f \! t \rangle }}{10^{3} \;\mathrm{s}} \right)^{-1} \right]. \end{equation} \tag{ 25 }$$

Using Equations (22) and (17), the thermal instability criterion (23) becomes

$$\begin{equation} \left(\frac{\nu }{8.14}\right) \mathcal {Q}> 20 \left(\frac{T_{8}}{5} \right)^{2}\left(\frac{\dot{\Sigma }}{0.3 \dot{\Sigma }_{\mathrm{Edd}}}\right)^{-1} \left(\frac{A}{2Z} \right)^{2} \;\mathrm{M}\mathrm{eV}, \end{equation} \tag{ 26 }$$

where $\mathcal {Q}$ is the energy released per electron capture.

We tested the accuracy of Equation (23) using the suitably modified global linear stability analysis of Cooper & Narayan (2005). The minimum $\mathcal {Q}$ for instability derived from the global stability analysis differed from that of Equation (26) by less than 30% for each of the 12 test cases.

Typically $\mathcal {Q}< Q$ because the daughter nucleus is generally more massive than the parent nucleus, although exceptions exist. Gupta et al. (2007) find $\mathcal {Q}< 6.2\;\mathrm{M}\mathrm{eV}$ for all electron captures that occur for μ_e < 6 MeV, or equivalently, Σ₁₂ < 10 (Equation (8)). From Equation (26), it follows that electron captures are thermally stable for the accretion rates and column depths at which superbursts occur. Therefore, we conclude that electron captures do not trigger superbursts.

In this section, we examine whether light-element fusion reactions trigger superbursts. Hydrogen has an electron capture threshold energy Q = 1.2933 MeV and thus depletes at Σ₁₂ ≲ 2 × 10⁻² (Equation (8)). The helium abundance at Σ_ign is less certain, and we discuss it below. The paucity of stable isotopes of Z = 3–5 nuclei leaves ¹²C as the next reasonable alternative, which we address in Section 4. Finally, nuclei with Z > 6 are unlikely candidates because the extra Coulomb repulsion causes the fusion rates to be significantly lower than that of ¹²C.

Thus, other than ¹²C + ¹²C, α capture reactions such as ¹²C(α, γ)¹⁶O are the only plausible fusion reactions that might trigger superbursts. The conditions for these reactions to produce superbursts are similar to those for electron captures, namely, that (1) the reaction rate r_nuc is sufficiently temperature dependent to produce unstable burning, and (2) α particles must survive to the inferred Σ_ign. Below, we show that the latter condition is not met; thus α particles deplete too quickly to trigger superbursts.

The condition for α particles to survive at a column depth Σ is $Y / \sum _i \left(r_{\mathrm{nuc}}\right)_i > \Sigma /\dot{\Sigma }$ (see Section 2.3), where Y is the helium mass fraction and the sum is over all reactions that consume α particles. Using the triple-α reaction rate of Fushiki & Lamb (1987) and setting ρ = 6 × 10⁸ g cm⁻³, T₈ = 5, and Y = 1, we find r_nuc = 2.2 × 10⁶s⁻¹, which implies a lifetime of 4.6 × 10⁻⁷s. The reaction rate r_nuc ∝ Y³, so the lifetime is much larger for smaller helium abundances. The accretion timescale $t_{\mathrm{acc}}= 10^{7} [\Sigma _{12}/(\dot{\Sigma }/\dot{\Sigma }_{\mathrm{Edd}}) ] \;\mathrm{s}$ (Equation (22)), indicating that Y < 10⁻⁷ for helium to survive. At this low Y, the rise in temperature from consuming the helium via, e.g., ¹²C(α, γ), is ≲10⁶ K ≪ T and hence insufficient to trigger a thermal instability.

From the results of this section and Section 2, we conclude that ¹²C fusion triggers superbursts.

A possible solution to the superburst ignition problem is that the true ¹²C + ¹²C fusion rate is larger than assumed. ¹²C ignition at the inferred Σ_ign requires a ∼10⁴ reaction rate enhancement for an ocean temperature of 4 × 10⁸ K (Cumming et al. 2006) or a ∼10² enhancement for 5 × 10⁸ K. The two sources of uncertainty in the fusion rate are (1) plasma screening effects and (2) the nuclear cross section σ(E). In the following subsections, we investigate whether either source could account for such a large increase in the fusion rate.

4.1. Plasma Screening

Superbursts ignite in a strongly coupled Coulomb plasma. Two dimensionless parameters determine the plasma's state. The first is the Coulomb coupling parameter

$$\begin{equation} \Gamma \equiv \frac{Z^{2}e^{2}}{ak_\mathrm{B}T} = 6.0 \left(\frac{T_{8}}{5} \right)^{-1} \Sigma _{12}^{1/4} \left(\frac{Z}{6} \right)^{5/3}, \end{equation} \tag{ 27 }$$

where a = (3Z/4πn_e)^1/3 is the ion-sphere radius, n_e is the electron number density, and we used Equation (1), which assumes the gravitational acceleration g = 2.43 × 10¹⁴ cm s⁻². For Γ ≪ 1, Coulomb coupling is weak and the ions constitute a Maxwell–Boltzmann gas. As Γ increases, the ions gradually become a Coulomb liquid. When Γ>175, the ions crystallize (Potekhin & Chabrier 2000). Equation (27) implies that superbursts ignite in a Coulomb liquid. The second dimensionless parameter,

$$\begin{equation} \zeta \equiv \frac{3\Gamma }{\tau } \approx 0.17 \left(\frac{T_{8}}{5} \right)^{-2/3} \Sigma _{12}^{1/4} \left(\frac{2Z}{A} \right)^{1/3}, \end{equation} \tag{ 28 }$$

is the ratio of the classical turning point to the ion separation, where

$$\begin{equation} \tau = \left(\frac{27 \pi ^2 \mu Z^{4} e^{4}}{2 k_\mathrm{B}T \hbar ^{2}}\right)^{1/3} = \left(\frac{27 \pi ^2 Am_\mathrm{u}Z^{4} e^{4}}{4 k_\mathrm{B}T \hbar ^{2}}\right)^{1/3}, \end{equation} \tag{ 29 }$$

and μ is the reduced mass of the reacting nuclei. Specifically, ζ = r_TP/a, where r_TP is the radius at which the Coulomb energy Z²e²/r_TP equals the classical Gamow peak energy (Clayton 1983)

$$\begin{equation} E^{\mathrm{pk}}= \frac{\tau k_\mathrm{B}T}{3}. \end{equation} \tag{ 30 }$$

Many-body interactions in a strongly coupled Coulomb plasma modify the Coulomb potential between two reacting nuclei (for a review, see Ichimaru 1993). From these many-body interactions, one derives an effective two-body potential

$$\begin{equation} V(r) = \frac{Z^{2}e^{2}}{r} - H(r), \end{equation} \tag{ 31 }$$

where r is the distance between the reacting nuclei. From H(r), one derives the plasma screening enhancement to the reaction rate exp(〈H(r)〉/k_BT), where 〈H(r)〉 is a path-integral average of H(r) (e.g., Ichimaru 1993). One can expand the static mean-field potential H(r) as a power series in (r/a)² (Widom 1963). Neglecting quantum effects in H(r), the leading order term H(0) is a thermodynamic quantity; H(0) equals the difference between the Coulomb (or excess) Helmholtz free energy before and after the reaction (DeWitt et al. 1973).

Monte Carlo simulations and hypernetted chain calculations of binary ionic mixtures (Hansen & Vieillefosse 1976; Hansen et al. 1977; Chabrier & Ashcroft 1990; Ogata et al. 1993; Rosenfeld 1995, 1996; DeWitt et al. 1996; DeWitt & Slattery 2003) suggest that the excess free energy obeys the linear mixing rule to high accuracy in the regime Γ>1 (Potekhin et al. 2009). Therefore, authors usually invoke the linear mixing rule when deriving the plasma screening enhancement to the reaction rate. In this case, the total free energy of an ionic mixture

$$\begin{equation} \frac{F^{\mathrm{ex}}}{k_\mathrm{B}T} = \displaystyle \sum _{i} N_i f^{\mathrm{ex}}(\Gamma _i), \end{equation} \tag{ 32 }$$

where N_i is the number of ions with charge Z_i, f^ex ≡ F^ex,OCP/Nk_BT is the well-determined reduced excess free energy per ion of a one-component plasma (e.g., Chabrier & Potekhin 1998; Potekhin & Chabrier 2000), and Γ_i ∝ Z^5/3_i is the Coulomb coupling parameter for species i (Equation (27)).

From Equation (32), H(0)/k_BT = 2f^ex(Γ) − f^ex(2^5/3Γ) (Jancovici 1977, see also the Appendix), where Γ is that of the reacting ions; using the ion-sphere model result f^ex ≈ −0.9Γ (Salpeter 1954), H(0)/k_BT = 0.9(2^5/3 − 2)Γ ≈ 1.0573Γ, so the lowest-order screening enhancement to the ¹²C + ¹²C reaction rate

$$\begin{equation} \exp \left(\frac{H(0)}{k_\mathrm{B}T} \right) = 5.5\times 10^{2} \exp \left[ \left(\frac{T_{8}}{5} \right)^{-1} \Sigma _{12}^{1/4} - 1 \right]. \end{equation} \tag{ 33 }$$

Despite its simplicity, Equation (33) is adequate for most applications (Gasques et al. 2005; Yakovlev et al. 2006; Chugunov et al. 2007). We discuss corrections to the screening enhancement below.

4.1.1. Deviation from Linear Mixing Rule

The excess free energy F^ex of a multicomponent plasma exhibits small deviations from the linear mixing rule (Equation (32)). In general,

$$\begin{equation} \frac{F^{\mathrm{ex}}}{k_\mathrm{B}T} = \displaystyle \sum _{i} N_i f^{\mathrm{ex}}(\Gamma _i) + N \Delta f^{\mathrm{ex}}, \end{equation} \tag{ 34 }$$

where Δf^ex ⩾ 0 is a function of both the charges Z_i and concentrations x_i ≡ N_i/N of the ionic species (e.g., DeWitt et al. 1996). Using the hypernetted chain calculations of DeWitt et al. (1996) and the ansatz Δf^ex ∝ x₁x₂(Z₂/Z₁)^3/2 (DeWitt & Slattery 2003), we find

$$\begin{equation} \Delta f^{\mathrm{ex}}= (0.0091 \ln \Gamma _1 + 0.018) x_1 x_2 \left(\frac{Z_2}{Z_1}\right)^{3/2} \end{equation} \tag{ 35 }$$

for a binary ionic mixture (see also Potekhin et al. 2009).

To incorporate linear mixing rule deviations into H(0) calculations, previous authors assumed a one-component plasma (consisting of ¹²C ions in this case). Fusion of two ¹²C ions generates a compound nucleus ²⁴Mg and thereby forms a binary ionic mixture. One determines H(0) by finding the difference in F^ex before and after the reaction in the limit that the compound nucleus concentration x₂ → 0 (Ichimaru 1993). Using this assumption and Equations (34) and (35), we find the correction to H(0),

$$\begin{equation} \frac{\Delta H(0)}{k_\mathrm{B}T} = -0.026\ln \Gamma - 0.051, \end{equation} \tag{ 36 }$$

in good agreement with DeWitt et al. (1996); for a one-component plasma, linear mixing rule deviations reduce the plasma screening enhancement factor by ∼10%.

However, the plasma at the superburst ignition depth is likely a mixture of ¹²C and heavier ions with Z ≲ 46 (e.g., Koike et al. 1999, 2004; Schatz et al. 2001, 2003b; Woosley et al. 2004a). Generalizing Equation (35) for a multicomponent plasma with Z_i < Z_j for i < j, we find (Ogata et al. 1993)

$$\begin{eqnarray} \begin{array}{@{}ll@{}} &\Delta f^{\mathrm{ex}}\,{=}\, \displaystyle \sum _{i<j}x_i x_j \Delta f^{\mathrm{ex}}_{ij};\\ &\Delta f^{\mathrm{ex}}_{ij} \ {=}\ (0.0091 \ln \Gamma _i\ {+}\ 0.018) \!\left(\frac{Z_j}{Z_i}\right)^{3/2}\!. \end{array} \end{eqnarray} \tag{ 37 }$$

To illustrate the effect spectator ions have on the screening enhancement, consider for simplicity a ternary ionic mixture of ¹²C, ²⁴Mg, and a representative spectator ion ⁵⁶Fe. Using Equations (34) and (37) and again taking the ²⁴Mg concentration x₂ → 0, so that x₁ + x₃ = 1, we find (see Appendix)

$$\begin{equation} \frac{\Delta H(0)}{k_\mathrm{B}T} = - \Delta f^{\mathrm{ex}}_{12}+x_3 \left[ \Delta f^{\mathrm{ex}}_{12} + (1+x_3)\Delta f^{\mathrm{ex}}_{13} -\Delta f^{\mathrm{ex}}_{23} \right]. \end{equation} \tag{ 38 }$$

Note that Equation (38) reduces to (36) in the limit x₃ → 0, as it should. Equation (38) shows that, since the bracketed term is positive, heavy spectator ions increase the screening enhancement factor (i.e., ∂ΔH(0)/∂x₃ > 0). For the fiducial ¹²C mass fraction 0.2 and ⁵⁶Fe mass fraction 0.8 (such that x₁ = 7/13 and x₃ = 6/13; Cumming et al. 2006), linear mixing rule deviations increase the plasma screening enhancement by ≈10%.

Potekhin et al. (2009) developed an analytic formula for Δf^ex that is more accurate than Equation (37). We derive the corresponding formula for ΔH(0)/k_BT in the Appendix. In Figure 1, we plot ΔH(0)/k_BT as a function of the spectator ion number fraction x₃ using both Equation (38) and the expression derived from Potekhin et al. (2009). Figure 1 confirms that heavy spectator ions increase the plasma screening enhancement to the reaction rate, although the two expressions for ΔH(0)/k_BT differ quantitatively.

Zoom In Zoom Out Reset image size

4.1.2. Corrections to 〈H(r)〉

The next term in the expansion of H(r) goes as (r/a)² ∝ ζ²; its contribution is small because ζ² ≪ 1 (Equation (28)). From Jancovici (1977), we find

$$\begin{equation} \frac{\langle H(r)\rangle -H(0)}{k_\mathrm{B}T} \approx -\frac{5}{32}\Gamma \zeta ^{2} = -0.027 \left(\frac{T_{8}}{5} \right)^{-7/3} \Sigma _{12}^{3/4}. \end{equation} \tag{ 39 }$$

This result agrees very well with more accurate calculations (Alastuey & Jancovici 1978; Ogata et al. 1991; Ogata 1997). Higher order terms are even smaller; therefore, we conclude that corrections to 〈H(r)〉 are unimportant in calculating the plasma screening enhancement for ¹²C ignition.

4.1.3. Electron Screening Corrections

In the above analysis, we tacitly assumed a uniform electron density. Although highly degenerate, electrons nonetheless slightly concentrate around positively charged ions. Electron polarization mitigates the Coulomb repulsion between ions relative to an unpolarized configuration. This has two counteracting effects: it (1) lowers the Coulomb repulsion between the two reacting ions, which increases the reaction rate, and (2) attenuates the many-body Coulomb interactions and thereby H(r), which decreases the reaction rate. The Yukawa potential Z²e²/rexp(−r/r_TF) describes the two-body potential, where r_TF is the Thomas–Fermi screening length. For relativistic, degenerate electrons, r_TF/a = 3.0(Z/6)^−1/3 (e.g., Haensel et al. 2007), so electron screening is weak; from the results of Sahrling & Chabrier (1998), electron screening changes the reaction rate by ≲1%.

Corrections to the lowest-order plasma screening enhancement (Equation (33)) change the ¹²C + ¹²C reaction rate by a factor < 2. Therefore, we conclude that uncertainties in the plasma screening enhancement are too small to explain the discrepancy between superburst observations and theoretical model results.

4.2. The Nuclear Cross Section

Although the plasma screening enhancement to the ¹²C + ¹²C reaction rate is well determined for superburst conditions, the nuclear cross section σ(E) is not. Many groups have measured σ(E) at various center-of-mass energies E down to ≈2.1 MeV (Patterson et al. 1969; Mazarakis & Stephens 1972, 1973; Spinka & Winkler 1974; High & Čujec 1977; Kettner et al. 1977, 1980; Erb et al. 1980; Treu et al. 1980; Becker et al. 1981; Dasmahapatra et al. 1982; Satkowiak et al. 1982; Rosales et al. 2003; Barrón-Palos et al. 2004, 2006; Aguilera et al. 2006; Spillane et al. 2007). However, the energy range of interest is centered at the classical Gamow peak energy (cf. Equation (29)–(30)),

$$\begin{equation} E^{\mathrm{pk}}= 1.5\left(\frac{T_{8}}{5} \right)^{2/3 }\;\mathrm{M}\mathrm{eV}, \end{equation} \tag{ 40 }$$

and has a full width

$$\begin{equation} \Delta E^{\mathrm{pk}}= 4 \left(\frac{E^{\mathrm{pk}}k_\mathrm{B}T}{3} \right)^{1/2} = 0.59 \left(\frac{T_{8}}{5} \right)^{5/6}\;\mathrm{M}\mathrm{eV}. \end{equation} \tag{ 41 }$$

Thus, σ(E) in the astrophysically relevant energy range is experimentally unknown.

This situation is common in nuclear astrophysics: to determine the astrophysical reaction rate, one either extrapolates the experimental data to lower energies or calculates the rate theoretically (e.g., Caughlan & Fowler 1988; Gasques et al. 2005). In doing so, one tacitly assumes that the astrophysical rate has the nonresonant form, i.e., no prominent resonances exist in the compound nucleus within the relevant energy range.⁵ However, several groups have detected strong resonances in the ¹²C + ¹²C system at energies below the Coulomb barrier. Resonances exist throughout the entire energy range probed so far, and the spacing between adjacent resonances⁶ is ≈0.3 MeV. Therefore, a resonance probably exists near E^pk (Bromley et al. 1960; Almqvist et al. 1960; Galster et al. 1977; Korotky et al. 1979; Erb et al. 1980; Treu et al. 1980; Spillane et al. 2007). Indeed, Michaud & Vogt (1972) and Perez-Torres et al. (2006) predict that a resonance exists in the ¹²C + ¹²C system with energy E_R ≈ 1.5 MeV. If the resonance is strong, the thermally averaged reaction rate 〈σv〉 would be much larger than assumed.

To illustrate the effect a strong resonance within the Gamow window would have on the ¹²C + ¹²C reaction rate, we follow the prediction of Perez-Torres et al. (2006) and assume the existence of a single, narrow resonance with E_R = 1.5 MeV. Then

$$\begin{eqnarray} \langle \sigma v \rangle &=& \langle \sigma v \rangle _{\mathrm{NR}} + \langle \sigma v \rangle _{\mathrm{R}}, \end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray} &&\nonumber \\ \langle \sigma v \rangle _{\mathrm{R}} &=& \left(\frac{\pi }{3 m_\mathrm{u}k_\mathrm{B}T}\right)^{3/2} \hbar ^{2} (\omega \gamma)_{\mathrm{R}}\exp \left(-\frac{E_{\mathrm{R}}}{k_\mathrm{B}T} \right), \end{eqnarray} \tag{ 43 }$$

where 〈σv〉_NR is the nonresonant contribution to the total reaction rate as given in, e.g., Caughlan & Fowler (1988), 〈σv〉_R is the resonant contribution,

$$\begin{equation} (\omega \gamma)_{\mathrm{R}}= 2(2J + 1) \frac{\Gamma _{\mathrm{C}}(\Gamma _{\mathrm{R}}-\Gamma _{\mathrm{C}})}{\Gamma _{\mathrm{R}}} \approx 2(2J+1)\Gamma _{\mathrm{C}} \end{equation} \tag{ 44 }$$

is the resonance strength, J is the total angular momentum of the resonance, Γ_C is the entrance channel width, and Γ_R ≫ Γ_C is the resonance width.

The resonant contribution 〈σv〉_R ∝ (ωγ)_R. Using the Breit–Wigner single resonance formula,

$$\begin{equation} \sigma (E) = \frac{\pi \hbar ^{2}}{12 E m_\mathrm{u}} \frac{(\omega \gamma)_{\mathrm{R}}\Gamma _{\mathrm{R}}}{(E-E_{\mathrm{R}})^{2} + (\Gamma _{\mathrm{R}}/2)^{2}} \end{equation} \tag{ 45 }$$

(e.g., Clayton 1983). Evaluating Equation (45) at E = E_R,

$$\begin{eqnarray} (\omega \gamma)_{\mathrm{R}}&=& 3.4\times 10^{-8} \left(\frac{\Gamma _{\mathrm{R}}}{100\;\mathrm{k}\mathrm{eV}} \right)\nonumber\\ &&\times\left(\frac{\sigma (E_{\mathrm{R}})}{10^{-13}\;\mathrm{barn}} \right) \left(\frac{E_{\mathrm{R}}}{1.5\;\mathrm{M}\mathrm{eV}} \right) \;\mathrm{eV}, \end{eqnarray} \tag{ 46 }$$

where we normalize the resonance width Γ_R to that typical of known resonances (see, e.g., Table IV of Aguilera et al. 2006) and the cross section at resonance σ(E_R) to the approximate value Perez-Torres et al. (2006) predict. For this work, we adopt (ωγ)_R = 3.4 × 10⁻⁸ eV as the fiducial resonance strength.

To determine an upper limit for (ωγ)_R, we demand that the resonance's contribution to the astrophysical S-factor at a given energy E, S_R(E), be less than the experimentally measured value S_exp(E) for all E ≳ 2.1 MeV, the lowest energy probed at the time of this writing. The S-factor for ¹²C + ¹²C is

$$\begin{equation} S(E) = \sigma (E) E \exp \left[87.21 \left(\frac{E}{\mathrm{M}\mathrm{eV}} \right)^{-1/2} + 0.46 \left(\frac{E}{\mathrm{M}\mathrm{eV}} \right)\right] \end{equation} \tag{ 47 }$$

(Patterson et al. 1969; Clayton 1983). From Equations (45) and (47), we write

$$\begin{equation} S_{\mathrm{R}} (E)= S(E_{\mathrm{R}}) \frac{(\Gamma _{\mathrm{R}}/2)^{2}}{(E-E_{\mathrm{R}})^{2} + (\Gamma _{\mathrm{R}}/2)^{2}}. \end{equation} \tag{ 48 }$$

Using Equations (45), (47), and (48), demanding that S_R(E) < S_exp(E), and noting that (E − E_R)² ≫ (Γ_R/2)², we find

$$\begin{eqnarray} (\omega \gamma)_{\mathrm{R}}&<& 5.5\times 10^{-8} \left(\frac{\Gamma _{\mathrm{R}}}{100\;\mathrm{k}\mathrm{eV}} \right)^{-1}\nonumber \\ &&\times \left[ \left(\frac{E-E_{\mathrm{R}}}{\mathrm{M}\mathrm{eV}} \right)^{2} \left(\frac{S_{\mathrm{exp}}(E)}{10^{16} \;\mathrm{M}\mathrm{eV}\;\mathrm{barn}} \right) \right] \;\mathrm{eV} \end{eqnarray} \tag{ 49 }$$

for a resonance at E_R = 1.5 MeV. Equation (49) must be satisfied for all E. According to the experimental data, the minimum value of the bracketed term is ≈1 (see, e.g., Figure 4 of Spillane et al. 2007), so

$$\begin{equation} (\omega \gamma)_{\mathrm{R}}< 5.5\times 10^{-8} \left(\frac{\Gamma _{\mathrm{R}}}{100\;\mathrm{k}\mathrm{eV}} \right)^{-1} \;\mathrm{eV}. \end{equation} \tag{ 50 }$$

If Γ_R ≈ 100 keV, then our fiducial strength (ωγ)_R = 3.4 × 10⁻⁸ eV is comparable to the maximum possible strength. Γ_R may be much smaller, however; the resonance at 2.14 MeV, the lowest-energy resonance known as of this writing, has a width Γ_R < 12 keV (Spillane et al. 2007). Therefore, we set (ωγ)_R = 3.4 × 10⁻⁷ eV, which is 10 times larger than our fiducial rate, as a reasonable upper limit.

Figure 2 shows the effect a 1.5 MeV resonance has on the reaction rate 〈σv〉. For the fiducial (ωγ)_R value, the resonance increases 〈σv〉 by a factor ≳25 at temperatures relevant to superbursts; for the (ωγ)_R upper limit, the resonance increases 〈σv〉 by a factor ≳250. These increases are of the order required to reconcile the observationally inferred Σ_ign with that calculated from theoretical models for a specific range of assumed crust thermal conductivities and core neutrino emissivities. In the following section, we compute the superburst Σ_ign with the effect of this resonance.

Zoom In Zoom Out Reset image size

We use the global linear stability analysis of Cooper & Narayan (2005) to determine the effect a strong resonance in the ¹²C + ¹²C system would have on the superburst ignition depth Σ_ign. We assume steady spherical accretion onto a neutron star of mass M = 1.4 M_☉ and radius R = 10 km. The accreted matter composition is that of the Sun: the hydrogen mass fraction X = 0.7, helium mass fraction Y = 0.28, and heavy-element mass fraction Z = 0.02. Furthermore, we follow Cumming et al. (2006) and assume the ¹²C mass fraction X_C = 0.2 at the base of the accreted layer.

We make the following two modifications to the model of Cooper & Narayan (2005). (1) Cooper & Narayan (2005) followed Brown (2000) and assumed the energy generated by electron captures, neutron emissions, and pycnonuclear reactions in the crust was distributed uniformly between Σ₁₂ = 6 × 10³ and 2 × 10⁵. We now follow Haensel & Zdunik (2008) and distribute the energy according to their Table A.3. (2) Plasma screening reduces the entrance channel width Γ_C. Therefore, the plasma screening enhancement for the resonant contribution to the reaction rate includes a correction factor that reduces the overall enhancement (Salpeter & Van Horn 1969; Mitler 1977), although the reduction is only a few percent for the conditions relevant for superbursts (see, e.g., Figure 1 of Cussons et al. 2002). We now use the formalism of Itoh et al. (2003) for the plasma enhancement factors of both the resonant and nonresonant contributions.

The ¹²C + ¹²C reaction rate, accretion rate $\dot{\Sigma }$, and ocean temperature profile together determine Σ_ign. The temperature profile is a strong function of the crust's thermal conductivity and core's neutrino emissivity, both of which are poorly constrained. We parametrize these uncertainties by implementing two conductivity and three core neutrino emissivity prescriptions that likely bracket their true values in accreting neutron stars. The thermal conductivity is a decreasing function of the impurity parameter Q_imp = 〈Z²〉 − 〈Z〉² (Itoh & Kohyama 1993, see also Daligault & Gupta 2009). Schatz et al. (1999) found Q_imp ∼ 100 from steady state nucleosynthesis calculations, although subsequent calculations suggest that Q_imp should be smaller (Schatz et al. 2003a; Woosley et al. 2004a; Koike et al. 2004; Horowitz et al. 2007, 2009). In addition, fits to the quiescent light curves of KS 1731 − 260 (Shternin et al. 2007; Brown & Cumming 2009) and MXB 1659 − 29 (Brown & Cumming 2009) require that Q_imp ∼ 1. Since both observations and molecular dynamics simulations imply that the crust forms an ordered lattice, we adopt Q_imp = 3 and 100 as the two bracketing values. The core neutrino emissivity, and thereby the core cooling rate, depends on the unknown ultradense matter equation of state (for reviews, see Yakovlev & Pethick 2004; Page et al. 2006). We consider one "fast" cooling model for which the pion condensate process dominates and two "slow" cooling models for which either the modified Urca or nucleon–nucleon bremsstrahlung process dominates (see, e.g., Table 1 of Page et al. 2006); these roughly correspond to cases "A," "B," and "D" of Cumming et al. (2006, see their Table 2). The respective core temperatures for these models are approximately 3 × 10⁷ K, 3 × 10⁸ K, and 6 × 10⁸ K.

Figure 3 shows the superburst ignition column depth Σ_ign as a function of $\dot{\Sigma }/\dot{\Sigma }_{\mathrm{Edd}}$ for various neutron star models.⁷ A 1.5 MeV resonance in the ¹²C + ¹²C system lowers Σ_ign by a factor ≈2 and ≈4 for the fiducial and maximum (ωγ)_R values, respectively; the lowered Σ_ign values are in accord with the observationally inferred values for a range of realistic neutron star model parameters. Therefore, we conclude that (1) a strong resonance may exist at an energy ≈1.5 MeV above the ¹²C + ¹²C ground state, and (2) if such a resonance exists, it will mitigate the discrepancy between observationally inferred superburst ignition depths and those calculated from theoretical models.

Zoom In Zoom Out Reset image size

For the low-mass X-ray transient 4U 1608 − 522, which exhibited a superburst, the thermal quiescent luminosity constrains the core temperature to be ≈2.5 × 10⁸ K. Fits to the superburst light curve find an ignition column Σ_ign = (1.5–4.1) × 10¹² g cm⁻². A resonance at 1.5 MeV could make the ignition temperature over this range as low as (4.1–4.8) × 10⁸ K, which is marginally consistent with the calculated crust temperature at the time of the superburst (Keek et al. 2008).

For the transient KS 1731 − 260, the timescale for the effective temperature to decrease implies Q_imp ≲ 1, and the lowest observed effective temperature implies that the core temperature is ≲10⁸ K (Shternin et al. 2007; Brown & Cumming 2009). Under these conditions, the temperature at Σ ≈ 10¹² g cm⁻² is unlikely to be >3 × 10⁸ K and therefore too cold to match the inferred ignition depth, even if the proposed resonance exists. Recent theoretical calculations of nuclear reactions in the neutron star crust suggest that the pycnonuclear fusion of neutron-rich, low-Z ions such as ²⁴O (Horowitz et al. 2008) or reactions triggered by β-delayed neutron emissions (Gupta et al. 2008) may provide a strong source of heating for the neutron star outer crust. Indeed, fits to the quiescent light curves of KS 1731 − 260 and MXB 1659 − 29 suggest that the heating in the outer crust is larger than can be accounted for from electron captures (Brown & Cumming 2009). Although a survey of neutron star models with this additional heating is outside the scope of this paper, we note that a strong resonance does alleviate the discrepancy in Σ_ign even if it does not entirely resolve it.

In this work, we reexamined the superburst trigger mechanism to address the discrepancy between observationally inferred superburst ignition column depths Σ_ign and those calculated in theoretical models. Motivated by the suggestion of Kuulkers et al. (2002b) and the similarity between inferred ignition column depths from different sources, we first explored the viability of thermally unstable electron captures as the trigger mechanism in Section 2. We found that electron captures are always thermally stable in accreting neutron star oceans; thus electron captures do not trigger superbursts. We then investigated the viability of nuclear fusion reactions other than ¹²C + ¹²C. Accretion-induced nuclear reactions deplete ions with Z < 6 at column depths Σ ≪ Σ_ign, whereas ions with Z > 6 fuse at Σ ≫ Σ_ign. We therefore confirmed the proposal (Cumming & Bildsten 2001; Strohmayer & Brown 2002) that ¹²C + ¹²C triggers superbursts.

We then examined the ¹²C + ¹²C fusion rate in Section 4, noting that superburst model results would be in accord with observations if the true fusion rate were greater than the standard rate by a factor ≳10². Two factors determine the fusion rate: plasma screening effects and the nuclear cross section σ(E). Uncertainties in, and corrections to, the plasma screening enhancement to the reaction rate alter the usual enhancement by a factor <2 and thus cannot resolve the discrepancy between superburst observations and theoretical models. However, uncertainties in σ(E) are much larger; indeed, σ(E) is experimentally unknown at astrophysically relevant energies. We find that a strong resonance in the ¹²C + ¹²C system at an energy near 1.5 MeV could increase the fusion rate by a few orders of magnitude at the temperatures relevant to superbursts. Both theoretical optical potential models and extrapolations of existing experimental data suggest that a resonance exists at an energy near 1.5 MeV. If this is true and the resonance strength (ωγ)_R is sufficiently large, it could eliminate the discrepancy between observationally inferred superburst ignition column depths and theoretical model results (see Figure 3).

In Section 1, we outlined three fundamental problems that exist with superburst ignition. We address these problems below in the context of our results.

• 1.  
  The results of all previous superburst models imply that ocean temperatures are too low for ¹²C ignition at the inferred Σ_ign ≈ 10¹² g cm⁻². A strong resonance near 1.5 MeV in the ¹²C + ¹²C system would decrease the temperature required for ignition at Σ_ign ≈ 10¹² g cm⁻² from ≈6 × 10⁸ K to ≈5 × 10⁸ K.
• 2.  
  Heavy-ion fusion hindrance would imply that the standard S-factor overestimates the true ¹²C + ¹²C fusion rate. This existence of such hindrance is currently speculative for both ¹²C + ¹²C in particular (Jiang et al. 2007) and exothermic fusion reactions in general (Jiang et al. 2008; Stefanini et al. 2008). Furthermore, the effect heavy-ion fusion hindrance has on resonant reactions is unknown. Therefore, it is unclear whether heavy-ion fusion hindrance poses a problem for superburst ignition.
• 3.  
  The ¹²C yield from nucleosynthesis models is often lower than that required for a thermal instability. A strong resonance would reduce the minimum ¹²C abundance required for a superburst, but by only a small amount. Thus, this problem would be attenuated but not resolved.

Our result has implications for ¹²C + ¹²C reactions in other contexts, namely Type Ia supernovae and massive stellar evolution. In addition, we have also presented a general prescription for understanding the screening enhancement factor in a multicomponent plasma. We briefly describe each of these topics before concluding with an outlook on future measurements.

6.1. Implications for Type Ia Supernovae and Massive Stellar Evolution

The fusion of ¹²C is an important stage in the post-main-sequence evolution of a massive star, and it is the reaction that ignites a white dwarf and triggers a thermonuclear (Type Ia) supernova. In both systems, the competition between heating from the ¹²C + ¹²C reaction and cooling from neutrino emissions determines ignition. To explore the implications of a resonance in the reaction cross section on these phenomena, we construct ignition curves (Figure 4), defined as _nuc(ρ, T) = _ν(ρ, T), for a ¹²C-¹⁶O plasma with X_C = 0.5. We compute the neutrino emissivity for the pair, photo, plasma, and bremsstrahlung processes using analytical fitting formulae (Itoh et al. 1996). For _nuc, we use the effective reaction $\mathcal {Q}$-value of 9.0 MeV (Chamulak et al. 2008), which includes heating from both the p- and α- branches and subsequent reactions; the ignition curve is insensitive to the choice of $\mathcal {Q}$. Three curves are plotted in Figure 4 for different choices of the ¹²C + ¹²C rate: the standard nonresonant rate (Caughlan & Fowler 1988, dotted line), a resonance at E_R = 1.5 MeV with our fiducial strength (ωγ)_R = 3.4 × 10⁻⁸ eV (solid line), and a resonance at E_R = 1.5 MeV with our maximum strength (ωγ)_R = 3.4 × 10⁻⁷ eV (dashed line).

Zoom In Zoom Out Reset image size

As is evident from Figure 4, the effect of a resonance at E_R = 1.5 MeV is minimal for the ignition of Type Ia supernovae, which are thought to ignite at central densities >10⁹ g cm⁻³ (see, e.g., Woosley et al. 2004b). It is interesting to speculate that a resonance at a lower energy might shift the ignition curve to lower densities. This would reduce the in situ neutronization of the nickel-peak material synthesized in the explosive burning and the neutronization during the pre-explosive convective burning (Piro & Bildsten 2008; Chamulak et al. 2008); moreover, numerical simulations (Röpke et al. 2006) find that a lower central density reduces the growth of the turbulent flame velocity (because the lower gravitational acceleration decreases the growth rate of the Rayleigh–Taylor instability), which leads to a less vigorous explosion and a decreased production of iron-peak elements. Although lower densities are necessary to avoid overproduction of neutron-rich isotopes such as ⁵⁴Fe and ⁵⁸Ni (Woosley 1997; Iwamoto et al. 1999), this may be ameliorated by improved electron capture rates onto pf-shell nuclei (Martínez-Pinedo et al. 2000; Brachwitz et al. 2000). Moreover, there is observational evidence that the majority of supernovae do undergo electron captures in the innermost ≈0.2 M_☉ of ejecta (Mazzali et al. 2007). Given the uncertainties in modeling the progenitor evolution, flame ignition, and explosion, and in the dependence of the ignition density on the accretion history of the white dwarf (see Lesaffre et al. 2006, for a recent discussion), we do not think it possible to constrain the existence of such a resonance from observations at this time, but future modeling efforts should clearly allow for this possibility.

Intriguingly, the largest effect is at ρ ≲ 10⁵ g cm⁻³, which is the region encountered by post-main-sequence massive stars (see Woosley et al. 2002, and references therein). Stellar evolutionary calculations, which include the shock-induced explosive nucleosynthesis, find that a decrease in the ¹²C + ¹²C rate leads to enhancements in ²⁶Al and ⁶⁰Fe abundances (Gasques et al. 2007). Further calculations are needed to determine whether an enhanced ¹²C + ¹²C rate would produce interesting changes in nucleosynthesis.

6.2. Ignition in a Multicomponent Plasma

6.3. Outlook for Future Measurements

We thank Lars Bildsten, Philip Chang, Andrey Chugunov, Richard Cyburt, Daniel Kasen, Hendrik Schatz, Michael Wiescher, Dima Yakovlev, Remco Zegers, and the anonymous referee for their advice and feedback. The Joint Institute for Nuclear Astrophysics (JINA) supported this work under NSF-PFC grant PHY 02-16783. A.W.S. and E.F.B. are supported by NASA/ATFP grant NNX08AG76G, and R.L.C. is supported by the National Science Foundation under grant No. NSF PHY05-51164.

In this appendix, we derive ΔH(0), the correction to the plasma screening enhancement factor due to linear mixing rule deviations. Consider a strongly coupled Coulomb plasma consisting of N ≡ ∑_iN_i ions, where N_i is the number of ions with charge Z_i, x_i ≡ N_i/N is the number fraction of species i, and species 1 and 2 are the reactant and product of the reaction, respectively, so that Z₂ = 2Z₁. DeWitt et al. (1973) found that, neglecting quantum contributions, H(0) equals the difference in the Coulomb free energy before and after the reaction; since a fusion reaction destroys two reactant ions and creates one product ion,

$$\begin{eqnarray} H(0)& =& F^{\mathrm{ex, initial}} - F^{\mathrm{ex, final}} = F^{\mathrm{ex}}(N_1, N_2,..., N_n)\nonumber\\ &&-F^{\mathrm{ex}}(N_1-2, N_2+1,..., N_n). \end{eqnarray} \tag{ A1 }$$

Writing Equation (A1) in a more general form,

$${\fontsize{9}{10}\selectfont{\begin{equation} H(0) = - \frac{F^{\mathrm{ex}}(N_1-2\Delta N_2, N_2+\Delta N_2,..., N_n) - F^{\mathrm{ex}}(N_1, N_2,..., N_n) }{\Delta N_2}, \end{equation}}} \tag{ A2 }$$

where ΔN₂ is the number of products created. In the limit ΔN₂ ≪ N₁, N₂, Equation (A2) simplifies to

$$\begin{equation} H(0)= -\!\left(\frac{\partial }{\partial N_2} - 2\frac{\partial }{\partial N_1}\right) F^{\mathrm{ex}}. \end{equation} \tag{ A3 }$$

Using the general expression for F^ex (Equation (34)), we find

$$\begin{equation} \frac{H(0)}{k_\mathrm{B}T} = 2f^{\mathrm{ex}}(\Gamma _1) - f^{\mathrm{ex}}(\Gamma _2) + \lbrace \Delta f^{\mathrm{ex}}- \mathcal {D}\Delta f^{\mathrm{ex}}\rbrace, \end{equation} \tag{ A4 }$$

where the term 2f^ex(Γ₁) − f^ex(Γ₂) is the well-known result for a plasma obeying the linear mixing rule (e.g., Jancovici 1977), the bracketed term is ΔH(0)/k_BT, and we have defined the operator

$$\begin{equation} \mathcal {D}\equiv N \left(\frac{\partial }{\partial N_2} - 2\frac{\partial }{\partial N_1} \right) = \frac{\partial }{\partial x_2} -2 \frac{\partial }{\partial x_1} + \displaystyle \sum _{i=1}^{n} x_i \frac{\partial }{\partial x_i}. \end{equation} \tag{ A5 }$$

Potekhin et al. (2009) derived an accurate, analytic fitting formula for Δf^ex (see their Equation (16)). Using their formula for an unpolarized electron background, which is appropriate for a strongly coupled Coulomb liquid, Δf^ex = Δf^ex(Γ, 〈Z〉, 〈Z²〉, 〈Z^5/2〉), where Γ = ∑_ix_iΓ_i and 〈Z^k〉 = ∑_ix_iZ^k_i. From Equations (A4) and (A5) of this work and Equations (12), (14), and (16) of Potekhin et al. (2009), we find

$$\begin{eqnarray} \frac{\Delta H(0)}{k_\mathrm{B}T} &=& \Delta f^{\mathrm{ex}}\left\lbrace 1 - \left[\frac{\partial \ln \Delta f^{\mathrm{ex}}}{\partial \ln \Gamma } \mathcal {D}\ln \Gamma + \frac{\partial \ln \Delta f^{\mathrm{ex}}}{\partial \ln \langle Z \rangle } \mathcal {D}\ln \langle Z \rangle\right.\right.\nonumber\\ &&\left.\left. +\ \frac{\partial \ln \Delta f^{\mathrm{ex}}}{\partial \ln \langle Z^{2} \rangle } \mathcal {D}\ln \langle Z^{2} \rangle +\frac{\partial \ln \Delta f^{\mathrm{ex}}}{\partial \ln \langle Z^{5/2} \rangle } \mathcal {D}\ln \langle Z^{5/2} \rangle \right] \right\rbrace,\nonumber\\ \end{eqnarray} \tag{ A6 }$$

where

$$\begin{eqnarray} \frac{\partial \ln \Delta f^{\mathrm{ex}}}{\partial \ln \Gamma } &=& {-}\frac{abc \Gamma ^{b}}{1+a \Gamma ^{b}}, \quad \quad \mathcal {D}\ln \Gamma = 1 + (2^{5/3}-2)\frac{\Gamma _{1}}{\Gamma }, \nonumber\\ && \mathcal {D}\ln \langle Z^{k} \rangle = 1 + (2^{k}-2) \frac{Z_1^{k}}{\langle Z^{k} \rangle }, \nonumber \\&&[-2pt] \nonumber \\ \frac{\partial \ln \Delta f^{\mathrm{ex}}}{\partial \ln \langle Z \rangle } &=& {-}\frac{1}{2}\frac{\zeta ^{\mathrm{DH}}}{\zeta ^{\mathrm{DH}}-\zeta ^{\mathrm{LM}}} - \frac{ac \Gamma ^{b}}{1+a \Gamma ^{b}} \left[ \frac{1.1+34\delta ^{3}}{2.2 \delta + 17 \delta ^{4}} (1-\delta)\right.\nonumber\\[-2pt] &&\left. {+}\ 0.4b \left(\ln \Gamma + \frac{1}{1-b} \right) \right] + \frac{d}{3} \ln (1+a\Gamma ^{b}), \nonumber \\ &&\nonumber \\ \frac{\partial \ln \Delta f^{\mathrm{ex}}}{\partial \ln \langle Z^{2} \rangle } &=& \frac{3}{2}\frac{\zeta ^{\mathrm{DH}}}{\zeta ^{\mathrm{DH}}-\zeta ^{\mathrm{LM}}} + \frac{ac \Gamma ^{b}}{1+a \Gamma ^{b}} \left[ \frac{3.3+102\delta ^{3}}{2.2 \delta + 17 \delta ^{4}} (1-\delta)\right.\nonumber\\ &&\left. +\ 0.2b \left(\ln \Gamma + \frac{1}{1-b} \right) \right] - \frac{d}{6} \ln (1+a\Gamma ^{b}), \nonumber \\ &&\nonumber \\[-2pt] \frac{\partial \ln \Delta f^{\mathrm{ex}}}{\partial \ln \langle Z^{5/2} \rangle } &=& {-}\frac{\zeta ^{\mathrm{LM}}}{\zeta ^{\mathrm{DH}}-\zeta ^{\mathrm{LM}}} - \frac{ac \Gamma ^{b}}{1+a \Gamma ^{b}} \left[ \frac{2.2+68\delta ^{3}}{2.2 \delta + 17 \delta ^{4}} (1-\delta) \right].\nonumber\\ \end{eqnarray} \tag{ A7 }$$