Role of low-lying resonances for the ${}^{10}\mathrm{Be}(p,\alpha ){}^7\mathrm{Li}$ reaction rate and implications for the formation of the Solar System

Evidence for the presence of short-lived radioactive isotopes when the Solar System formed is preserved in meteorites, providing insights into the conditions at the birth of our Sun. A low-mass core-collapse supernova had been postulated as a candidate for the origin of ${}^{10}\mathrm{Be}$, reinforcing the idea that a supernova triggered the formation of the Solar System. We present a detailed study of the production of ${}^{10}\mathrm{Be}$ by the $\nu$ process in supernovae, which is very sensitive to the reaction rate of the major destruction channel, ${}^{10}\mathrm{Be}{(p,\alpha )}^7\text{Li}$. With data from recent nuclear experiments that show the presence of a resonant state in ${}^{11}\mathrm{B}$ at $\approx 193$ keV, we derive new values for the ${}^{10}\mathrm{Be}{(p,\alpha )}^7\text{Li}$ reaction rate, which are significantly higher than previous estimates. We show that, with the new ${}^{10}\mathrm{Be}{(p,\alpha )}^7\text{Li}$ reaction rate, a low-mass CCSN is unlikely to produce enough ${}^{10}\mathrm{Be}$ to explain the observed ${}^{10}\mathrm{Be}/{}^9\mathrm{Be}$ ratio in meteorites, even for a wide range of neutrino spectra considered in our models.

Figure 1

Peak temperature profiles for the supernova models used here. The C shell, where the production of ${}^{10}\mathrm{Be}$ occurs, is highlighted in gray. The peak temperature in the C shell ranges from 0.6–1.5 GK.

Figure 2

Mass fraction profile of the 15 $M_{\odot }$ model. The top panel shows the presupernova mass fractions of several important isotopes to indicate the star's compositional layers. The middle panel shows the profiles of the ${}^{10}\mathrm{Be}$ mass fraction at 200 s (Final) and at the time of shock arrival, i.e., when the peak temperature is reached. Note that this time is different for each location. The bottom panel shows the absolute value of the integrated net reaction fluxes, $\mathcal{F}$, as defined in Eq. (4) due the two most important destructive reactions.

Figure 3

Impact of the variation of the ${}^{10}\mathrm{Be}(p,\alpha )$ reaction rate on the ${}^{10}\mathrm{Be}$ yields and ratio relative to the result with the JINA-REACLIB reaction rate. The top two panels are for high $E_{\nu }$ and the bottom two panels for low $E_{\nu }$. The results with the neutrino spectra from the 3D simulation of the 11.8 $M_{\odot }$ model is also included in the top panel (11.8 $M_{\odot }$*).

Figure 4

Time evolution of the mass fractions of free protons and ${}^{10}\mathrm{Be}$ for the 15 $M_{\odot }$ model in the outer C shell at mass coordinate of 2.32 $M_{\odot }$ and with the ${}^{10}\mathrm{Be}{(p,\alpha )}^7\text{Li}$ rate enhanced by a factor 50 for low and high $E_{\nu }$. With high $E_{\nu }$ the proton mass fraction is enhanced by almost factor of 10, leading to a lower final ${}^{10}\mathrm{Be}$ mass fraction than in the low $E_{\nu }$ case.

Figure 5

Mass fraction profile of ${}^{10}\mathrm{Be}$ for the 15 $M_{\odot }$ model with high $E_{\nu }$ for the range of values of the ${}^{10}\mathrm{Be}{(p,\alpha )}^7\text{Li}$ reaction rate multiplier $\mathcal{R}$. The contribution of the bottom of the C shell decreases with increasing $\mathcal{R}$.

Figure 6

$R$-matrix calculations to assess the impact of the 193 keV $1/2^{+}$ resonance on the ${}^{10}\mathrm{Be}(p,\alpha )$ reaction cross section. The top panel shows the $S$ factor and the bottom panel the reaction rate as a function of temperature. Results without the 193 keV $1/2^{+}$ resonance are shown (red line) as well as several results assuming different resonance widths and assumptions about the interference with the $3/2^{+}$ resonance (see text). For comparison, the rate from Ref. [22] is also shown (green dashed line). The result with ${\mathrm{\Gamma }}_p$ = 6 keV, ${\mathrm{\Gamma }}_{\alpha }$ = 6 keV, ($+$−) is the new recommended rate.

Figure 7

${}^{10}\mathrm{Be}$ yields for the models studied with high and low $E_{\nu }$. The red and blue shaded bands indicate the range of results between the lower and recommended rate from Sec. 4. For comparison, the dashed lines indicate the results with the JINA-REACLIB rate from Ref. [22] from models with the corresponding neutrino spectra. Note that we assumed and explosion energy of $0.2\times {10}^{51}$ erg for the 11.8 $M_{\odot }$ and $1.2\times {10}^{51}$ erg for all other models.

Figure 8

${}^{10}\mathrm{Be}$ yield for the different models of the neutrino spectra listed in Table 3 for the recommended and lower ${}^{10}\mathrm{Be}{(p,\alpha )}^7\text{Li}$ reaction rate. The green band indicates the yield that is required to be in agreement with the observed ESS ${}^{10}\mathrm{Be}/{}^9\mathrm{Be}$ ratio when assuming the scenario of Ref. [17].

Figure 9

Reaction rate fit with the REACLIB parametrization, using a resonant and nonresonant contribution. The parametrization and reaction rate calculation agree within 5% for all temperatures above 0.3 GK, which are the most relevant for CCSN explosions.