# A New Mass Model for Nuclear Astrophysics: Crossing 200 keV Accuracy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Gaussian Process Regression

## 3. Nuclear Masses

#### 3.1. Augmenting the DZ10 Model with a GP

#### 3.2. Extrapolation Using the DZ10-GP Model

#### 3.3. Comparison with AME2020

## 4. Outer Crust

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Gaussian Process Regression Equations

## References

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**Figure 1.**Colours online. Examples of the structure of prior functions for various choices of the ℓ parameter. The shaded area represents the $1\sigma $ confidence interval.

**Figure 2.**Demonstration of Gaussian process regression. The true function is $y=sin\left(x\right)$, and the data points are at $x=\{0,0.5,2,3.5,6\}$. The solid line represents the GP mean, and the shaded areas give the $2\sigma $ confidence intervals. The optimised kernel parameters are ${\eta}^{2}=0.602,\phantom{\rule{3.33333pt}{0ex}}\ell =1.063$. See text for details.

**Figure 3.**Left panel: residuals as a function of nucleon number A for the DZ10 model, for measured masses. In the right panel are the same residuals shown as a histogram, with a Gaussian fit overlaid (for which the mean is fixed to 0, and the standard deviation to that of the residuals). See text for details.

**Figure 4.**Posterior distributions of GP parameters obtained through MCMC sampling. The horizontal and vertical solid lines indicate the optimal parameter values obtained by maximising the likelihood. The vertical dotted lines on each 1D histogram indicate the mean and $1\sigma $ confidence intervals obtained through MCMC sampling. See text for details.

**Figure 5.**Distributions of the residuals for the DZ10 and DZ10-GP models for measured masses. Gaussian fits to the residuals are also shown, with the mean fixed to 0 and the standard deviation fixed to that of the residuals. See text for details.

**Figure 6.**The same as Figure 3 but for the DZ10-GP model. See text for details.

**Figure 7.**Same as Figure 5 but for extrapolated masses. See text for details.

**Figure 8.**Residuals for the DZ10 and DZ10-GP models, for the $Z=28$ and $Z=29$ isotopic chains. The vertical dashed lines represent the transition from nuclei used for training to nuclei for which predictions are made. See text for details.

**Figure 9.**GP correction for $Z=28$ and $Z=29$. The vertical dashed lines represent the transition from nuclei used for training to nuclei for which predictions are made. The shaded ares represent the GP $1\sigma $ error bars. See text for details.

**Figure 10.**Distributions of the residuals for the DZ10 and DZ10-GP models, for new masses presented in AME2020 [45]. Gaussian fits to the residuals are also shown, with the mean fixed to 0 and the standard deviation fixed to that of the residuals. See text for details.

**Figure 11.**Colours online. Existence probability of a given nucleus within the outer crust as a function of the pressure, obtained via a Monte Carlo sampling using the DZ10-GP mass table. See text for details.

**Figure 12.**Variations of Z and N with pressure in the outer crust for the BSk20 and BPS models. The shaded area represents the regions covered by the Monte Carlo procedure detailed in the text and obtained using the DZ10-GP model. See text for details.

**Figure 13.**Equation of state, including statistical uncertainties, of the outer crust of a NS, calculated using the DZ10-GP mass model. See text for details.

**Table 1.**Percentage of nuclei included in the total error bars for the DZ10-GP model for three different sectors of the nuclear chart.

$1\mathit{\sigma}$ | $2\mathit{\sigma}$ | $3\mathit{\sigma}$ | |
---|---|---|---|

Full chart | 61% | 88.8% | 96.2% |

$50\le A\le 150$ | 59.2% | 89.1% | 97.3% |

$20\le Z\le 50$ | 54.4% | 84.1% | 95.5% |

**Table 2.**Composition of the outer crust of a NS using the DZ10 and DZ10-GP mass models. In the first and fourth columns, we report the maximum value of pressure at which the nucleus is found using the minimisation procedure. The horizontal line separates the measured and extrapolated masses reported in AME2016 [32].

DZ10 | DZ10-GP | ||||
---|---|---|---|---|---|

${\mathbf{P}}_{\mathrm{max}}$[MeVfm${}^{-\mathbf{3}}$] | N | Z | ${\mathbf{P}}_{\mathrm{max}}$ [MeVfm${}^{-\mathbf{3}}$] | N | Z |

3.30 $\times {10}^{-10}$ | 30 | 26 | 3.30 $\times {10}^{-10}$ | 30 | 26 |

4.36$\times {10}^{-8}$ | 34 | 28 | 4.36 $\times {10}^{-8}$ | 34 | 28 |

3.56 $\times {10}^{-7}$ | 36 | 28 | 3.56 $\times {10}^{-7}$ | 36 | 28 |

4.02 $\times {10}^{-7}$ | 38 | 28 | 4.02 $\times {10}^{-7}$ | 38 | 28 |

1.03 $\times {10}^{-6}$ | 50 | 36 | 1.03 $\times {10}^{-6}$ | 50 | 36 |

5.59 $\times {10}^{-6}$ | 50 | 34 | 5.59 $\times {10}^{-6}$ | 50 | 34 |

1.76 $\times {10}^{-5}$ | 50 | 32 | 5.59 $\times {10}^{-6}$ | 50 | 32 |

1.77 $\times {10}^{-5}$ | 50 | 30 | |||

1.58 $\times {10}^{-4}$ | 50 | 28 | 3.22 $\times {10}^{-5}$ | 50 | 28 |

1.82 $\times {10}^{-4}$ | 82 | 42 | 1.21 $\times {10}^{-4}$ | 82 | 42 |

3.31 $\times {10}^{-4}$ | 82 | 40 | 1.81 $\times {10}^{-4}$ | 82 | 40 |

4.83 $\times {10}^{-4}$ | 82 | 38 | 3.31 $\times {10}^{-4}$ | 82 | 38 |

4.86 $\times {10}^{-4}$ | 82 | 36 | 4.84 $\times {10}^{-4}$ | 82 | 36 |

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Shelley, M.; Pastore, A.
A New Mass Model for Nuclear Astrophysics: Crossing 200 keV Accuracy. *Universe* **2021**, *7*, 131.
https://doi.org/10.3390/universe7050131

**AMA Style**

Shelley M, Pastore A.
A New Mass Model for Nuclear Astrophysics: Crossing 200 keV Accuracy. *Universe*. 2021; 7(5):131.
https://doi.org/10.3390/universe7050131

**Chicago/Turabian Style**

Shelley, Matthew, and Alessandro Pastore.
2021. "A New Mass Model for Nuclear Astrophysics: Crossing 200 keV Accuracy" *Universe* 7, no. 5: 131.
https://doi.org/10.3390/universe7050131