# On the Time Distribution of Supernova Antineutrino Flux

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## Abstract

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## 1. Introduction

What do we know for sure about the various phases of gravitational collapse?How to learn more using neutrino observations?

## 2. Open Issues after Supernova 1987A

- that the supernova precursor did not fit fully expectations, and perhaps was non-standard, with unexpected implications;
- that the neutron star had not been observed, even casting a shadow over the significance of the observed events;
- that the observed neutrino events were more directional than expected—directed from the supernova forward-raising similar concerns as in the previous point;
- that also, their average energies were lower than those calculated, perhaps indicating an instrumental problem;
- that the comparability of the energy spectra of Kamiokande-II, IMB and Baksan was not entirely clear;

- There is a growing consensus towards the idea that the precursor could have been a two-star system that had recently merged, although it is not clear whether this impacts expectations about core collapse and neutrinos significantly.
- The direction of the neutrino events, studied e.g., in Ref. [58], seems less problematic than occasionally claimed.
- It has been widely recognized that the theoretical uncertainties in the mean energies are much larger than those estimated in the past, and therefore, it is not currently claimed that there is any serious incompatibility with the theory.
- A complete and systematic study of the energy spectra has verified the compatibility of the energy spectra and confirmed the stability and substantial validity [44] of a standard interpretation, as the one initially summarized by Bahcall. There is a well-defined region of the parameter space that allows the interpretation of the events as due to gravity collapse, as being due to a non-atypical gravitational collapse; the average energy of the antineutrinos is ${\overline{E}}_{\nu}=12\mathrm{M}\mathrm{e}\phantom{\rule{-0.21251pt}{0ex}}\mathrm{V}$ and the total radiated energy is of the order of $5\times {10}^{52}\mathrm{erg}$, with errors of 10% and 30%, respectively.
- The only problem that remains unsolved is the meaning of the 5 low-energy events seen by the Mont Blanc/LSD detector [51], which precede those seen by the other three detectors, and which do not seem easy to attribute to the supernova.

## 3. Parameterized Spectrum of Electronic Antineutrinos

#### 3.1. Generalities

#### 3.2. Model with Two Emission Phases

we will assume that at any given time, the flux can be described by a sum of the accretion and cooling components,

each of which is quantified by a temperature and an intensity of the emission (in the way discussed in the next section) each of which is a function of time.

#### 3.2.1. Emission from Thermal Cooling

#### 3.2.2. Emission from Processes around the Accretion Zone

#### 3.3. Expectations

- The luminosity, the number of irradiated neutrinos and signal rates in the detectors are much higher during the accretion phase than during the cooling phase. This result is consistent with what has been discussed in the previous literature [42,43,79], and it is due to the volumetric character of the accretion emission highlighted above.
- If the cooling luminosity is about the same for all six neutrino types of neutrinos and antineutrinos, then in about 10 s $3\times {10}^{53}$ $\mathrm{erg}$ will be extracted from the core of the star.
- The number of electron antineutrino events from the accretion phase, which we expect to last a fraction of a second, will be a bit smaller but comparable with that of the cooling phase.

#### 3.4. Remark on Neutrino Flavor Transformation

## 4. A Model for the Time Evolution

#### 4.1. Description of Luminosity

- the position of the maximum of the curve ${t}_{0}$;
- the two timescales that drive the decrease in luminosity, ${\mathsf{\tau}}_{a}$ and ${\mathsf{\tau}}_{c}$, for accretion and cooling emission, respectively.

#### 4.2. Description of the Flux

## 5. Tests and Applications

#### 5.1. Illustration of the Expected Flux

#### 5.2. Comparison with SN1987A

#### 5.3. Predictions

## 6. Variants and Possible Improvements

#### 6.1. Variants Concerning the Cooling Component

#### 6.2. Variants Concerning the Accretion Component

#### 6.3. Variants Concerning the Other Neutrino Flavors

## 7. Discussion and Outlook

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Detector Response

#### Appendix A.1. Kinematics

- The threshold of the reaction, which is obtained when ${E}_{\mathrm{e}}^{CM}={m}_{\mathrm{e}}$, is$${E}_{\mathrm{thr}}={\mathsf{\delta}}_{+}\phantom{\rule{2.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}{\mathsf{\delta}}_{\pm}=\frac{{({m}_{\mathrm{n}}\pm {m}_{\mathrm{e}})}^{2}-{m}_{\mathrm{p}}^{2}}{2{m}_{\mathrm{p}}}.$$
- For ${E}_{\nu}>{E}_{\mathrm{thr}}$ we have$${E}_{\mathrm{e}}^{CM}=\frac{{E}_{\nu}-\mathsf{\delta}}{\sqrt{1+2{E}_{\nu}/{m}_{\mathrm{p}}}}\phantom{\rule{2.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}\mathsf{\delta}=\frac{{m}_{\mathrm{n}}^{2}-{m}_{\mathrm{e}}^{2}-{m}_{\mathrm{p}}^{2}}{2{m}_{\mathrm{p}}}.$$
- The corresponding momentum ${p}_{\mathrm{e}}^{CM}$ can be written as$${p}_{\mathrm{e}}^{CM}=\sqrt{\frac{({E}_{\nu}-{\mathsf{\delta}}_{+})({E}_{\nu}-{\mathsf{\delta}}_{-})}{1+2{E}_{\nu}/{m}_{\mathrm{p}}}},$$

#### Appendix A.2. Description of Some Supernova Neutrino Telescopes

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**Figure 1.**(

**a**) Sketch of one typical shape of the antineutrino luminosity, as indicated by a simulation by the Garching Group, as reported in Ref. [16]. After a fast growth phase (red), there is a very intense emission lasting a fraction of a second (highlighted in orange) followed by a less intense, slowly decreasing and long-lasting emission (from yellow to blue). (

**b**) Conceptual diagram for the emission. Around the nascent neutron star there are many emitting centres, due to the reaction between positrons and neutrons. From Ref. [79].

**Figure 2.**Expected neutrino flux (1), differential in time and neutrino energy, from Equations (9) and (11), given the parameters (33) and the model emission (28) and (29). The reference distance $D=10\mathrm{k}\mathrm{pc}$ is assumed. (

**a**) First second, flux in linear scale. (

**b**) First 100 s, time in logarithmic scale.

**Figure 4.**Top row: (

**a**) Time position of the events from SN1987A, as observed in Kamiokande-II (black vertical lines) compared with the differential counting rate predicted in the model (blue curve, signal + background) for the value of the parameters indicated in the text [42,43,44]. (

**b**) Same but with respect to the energy. Bottom row: Cumulative distribution functions (CDF) for the same flux, regarded as a model of SN1987A emission. We show in black the data, in blue the theoretical expectation. (

**c**) Cumulative time distributions (${t}_{\mathrm{off}}=50$ ms, p-value = 56%). (

**d**) Cumulative energy distribution (p-value 51%).

**Figure 5.**Expected differential counting rate of electron antineutrino events in Super–Kamiokande differential in neutrino energy, as per definitions (5) and (6), given the parameters (33) and the model emission (28) and (29). The reference distance $D=10\mathrm{k}\mathrm{pc}$ is assumed. (

**a**) The first second of emission; each contour marks steps of 25 s

^{‒1}MeV

^{‒1}. (

**b**) Global distribution. Note the similarity with the flux, as shown in Figure 2.

**Figure 6.**Counting rate in IceCube, as expected from model emission (28) and (29) given the parameters (33). The gray line marks the 30 $\mathrm{k}$$\mathrm{Hz}$ threshold given by background fluctuations in $\mathsf{\delta}t=1.6384\mathrm{m}\mathrm{s}$ time bins (see Appendix A.2). Note the similarity with the luminosity curve shown in Figure 3, driven by the similarities in the definitions (4) and (6).

**Table 1.**Reference values for luminosity ($\mathcal{L}$), number of neutrinos per second (${\dot{N}}_{\nu}$), average energy (${\overline{E}}_{\nu}$), rates in detectors (${\mathcal{R}}_{\mathrm{SK}}$, ${\mathcal{R}}_{\mathrm{Ice}3}$), for the two phases of cooling and of accretion defined in Section 3.2. All the quantities are referred to as the electron antineutrinos (${\nu}_{\mathrm{e}}$) species. For the given quantities, we specify benchmark values and power law indices, as defined in Equations (15) and (17) respectively.

${\mathcal{L}}^{\u2605}$ | ${\dot{\mathit{N}}}_{\mathit{\nu}}^{\u2605}$ | ${\overline{\mathit{E}}}_{\mathit{\nu}}^{\u2605}$ | ${\mathcal{R}}_{\mathbf{SK}}^{\u2605}$ | ${\mathcal{R}}_{\mathbf{Ice}3}^{\u2605}$ | |
---|---|---|---|---|---|

benchmark values [equation (15)] | |||||

[$\mathrm{erg}$/$\mathrm{s}$] | [${\overline{\nu}}_{\mathrm{e}}$/$\mathrm{s}$] | [$\mathrm{M}\mathrm{e}\phantom{\rule{-0.21251pt}{0ex}}\mathrm{V}$] | [$\mathrm{Hz}$] | [$\mathrm{Hz}$] | |

cooling | $5.2\times {10}^{51}$ | $2.3\times {10}^{56}$ | $14.2$ | $6.7\times {10}^{2}$ | $7.9\times {10}^{4}$ |

accretion | $5.0\times {10}^{52}$ | $2.4\times {10}^{57}$ | $13.0$ | $5.2\times {10}^{3}$ | $4.7\times {10}^{5}$ |

power law indices [equation (17)] | |||||

cooling | $4.0$ | $3.0$ | $1.0$ | $5.1$ | $6.0$ |

accretion | $5.5$ | $4.6$ | $0.9$ | $6.7$ | $7.5$ |

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Vissani, F.; Gallo Rosso, A.
On the Time Distribution of Supernova Antineutrino Flux. *Symmetry* **2021**, *13*, 1851.
https://doi.org/10.3390/sym13101851

**AMA Style**

Vissani F, Gallo Rosso A.
On the Time Distribution of Supernova Antineutrino Flux. *Symmetry*. 2021; 13(10):1851.
https://doi.org/10.3390/sym13101851

**Chicago/Turabian Style**

Vissani, Francesco, and Andrea Gallo Rosso.
2021. "On the Time Distribution of Supernova Antineutrino Flux" *Symmetry* 13, no. 10: 1851.
https://doi.org/10.3390/sym13101851