ERRATUM: "MODULES FOR EXPERIMENTS IN STELLAR ASTROPHYSICS (MESA): BINARIES, PULSATIONS, AND EXPLOSIONS" (2015, ApJS, 220, 15)

Errors that were initially present in Schwab et al. (2015)—for which a separate erratum has been submitted—were propagated into Section 8 of this paper. We correct these errors here.

In Equation (72), the argument in the exponential pre-factor has the wrong sign. This term should be $\mathrm{exp}(\pi \alpha Z)$. The error is repeated in Equations (73), (75), (76), and (80). Correcting this error—which was also present in the computer implementation of these equations—increases the weak rates calculated using the on-the-fly capabilities by a factor of $\mathrm{exp}(2\pi \alpha Z)$. For Z = 10, this corresponds to an increase of approximately 60%.

There is also an error in the use of $({ft})$ values. The same $({ft})$ value was used for both the electron-capture and the beta-decay transition between two states. However, these $({ft})$ values should differ by the ratio of the spin degeneracy of the states: that is, the $({ft})$ values for electron-capture and beta-decay have the relationship

$$\begin{eqnarray}&&\displaystyle \frac{{({ft})}_{\beta }}{2{J}_{\beta }+1}=\displaystyle \frac{{({ft})}_{\mathrm{ec}}}{2{J}_{\mathrm{ec}}+1},\end{eqnarray} \tag{ 1 }$$

where J is the spin of the parent (initial) nuclear state. This error can cause the rate of an individual transition to be underestimated or overestimated by up to a factor of three (depending on the spins). This error manifests itself in Equation (80), which does not differentiate between the $({ft})$ values. A fully corrected version of Equation (80) reads

$$\begin{eqnarray}&&{C}_{\mathrm{max}}=\displaystyle \frac{7{\pi }^{4}\mathrm{ln}2}{60}\left(\displaystyle \frac{{m}_{{\rm{e}}}{c}^{2}}{{({ft})}_{\beta }+{({ft})}_{\mathrm{ec}}}\right){\left(\displaystyle \frac{{k}_{{\rm{B}}}T}{{m}_{{\rm{e}}}{c}^{2}}\right)}^{4}{\left(\displaystyle \frac{Q}{{m}_{{\rm{e}}}{c}^{2}}\right)}^{2}\mathrm{exp}(\pi \alpha Z).\end{eqnarray} \tag{ 2 }$$

This error in the use of $({ft})$ values extended to the input nuclear data. For the calculations shown in Figures 35, 36, and 37, the $({ft})$ values for beta decay were used for both electron capture and beta decay, so the electron-capture rates were incorrect; for the calculations shown in Figures 34 and 38, the $({ft})$ values for electron capture were used for both electron capture and beta decay, so the beta-decay rates were incorrect.

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The primary purpose of Section 8 is to demonstrate that the on-the-fly rates improve upon coarse table interpolation in several applications. This conclusion is unaffected by these errors. Correcting these errors has quantitative, but not qualitative, effects on Figures 34–38. No arguments or conclusions were based on specific values shown in the figures. We describe the effects of these errors on the figures and provide corrected versions.

Figures 34 and 35: The electron-capture and beta-decay rates change by the previously described factors. Due to the large range and logarithmic scale of the y-axis, these shifts are visually small.

Figure 36: Equation (80) and the numerical rate implementation contained the same errors, thus both the on-the-fly points (circles) and the analytic estimate (dashed line) shift, but they continue to display the same level of agreement.

Figure 37: The increased beta-decay rates lead to additional Urca-process cooling. The qualitative evolution is similar, but the central temperature immediately after the A = 23 cooling is reduced by approximately 0.1 dex to $\mathrm{log}(T/{\rm{K}})\approx 8.6$.

Figure 38: The increased electron-capture rates cause the increases in central temperature associated with the electron captures to occur at slightly lower central densities. The shift is comparable in magnitude to the linewidth in the figure.

Since the publication of Paxton et al. (2015), the rates used by Toki et al. (2013) have been made publicly available as part of Suzuki et al. (2016). After correcting our errors, we compared the rates from MESA with these tabulated rates by other authors. Within the range of validity of MESA's treatment, we found good agreement (typically within 10%).

We have corrected these errors in the current version of MESA (Release 8118). We thank Gabriel Martínez-Pinedo and Heiko Möller for helpful communications regarding these errors.