# Probability Distribution Functions of Solar and Stellar Flares

## Abstract

**:**

^{−4}W m

^{−2}) in terms of the GOES (Geostationary Operational Environmental Satellites) flare class. The paper also studies superflares (more energetic than solar flares) from solar-type stars and found that their power exponent in the fitting of the gamma function distribution is around 1.05, which is much flatter than solar flares. The distribution function of stellar flare energy extrapolated downward does not connect to the distribution function of solar flare energy.

## 1. Introduction

^{33}erg [4]. Now it seems established that the energy release in solar flares is due to magnetic reconnection [5].

^{−α}; i.e., smaller flares are more numerous. This property led Parker [8] to propose that the solar corona might be heated [9] by energies supplied from numerous small flares, later called nanoflares (in contrast to another class of theory based on waves [10,11]). Here, the important criterion is whether the power law index α is larger or smaller than 2 [12]. The total energy brought by all the flares with energies between E

_{1}and E

_{2}is

_{1}→0) determine the total energy involved in the flare phenomenon. However, for the observed flares of moderate or large sizes it is generally believed that α < 2 (1.8 or about). Therefore, more contributions to W come from larger flares. On the large energy end, in terms of space weather it has been discussed how frequent very large (extreme) events would happen [4]. Recent discovery of energetic flares (superflares) from solar-type (old and slowly rotating) stars [13,14] has stimulated interest in solar extreme events.

^{4}flare events of peak flux above 10

^{−7}W m

^{−2}(1975–2020) and 3.4 × 10

^{4}events of fluence above 5 × 10

^{−5}J m

^{−2}(1997–2020) are plotted.

## 2. Data and Methods

^{−8}W m

^{−2}), B (10

^{−7}W m

^{−2}), C (10

^{−6}W m

^{−2}), M (10

^{−5}W m

^{−2}), and X (10

^{−4}W m

^{−2}) classes. Namely, an M5.5 flare means a flare of peak flux 5.5 × 10

^{−5}W m

^{−2}and an X12 flare has a peak flux of 12 × 10

^{−4}W m

^{−2}. The data are available from 1975, but the data before 1980 had one-digit accuracy (C2, M4, etc.); therefore, the data after 1980 which have two-digit accuracy (C2.1, M3.9, etc.) are used. The data after 1997 also include the fluence values (time integration of X-ray flux). In the following analysis flares with peak flux above M3 or 3 × 10

^{−5}W m

^{−2}(1980–2020, 1720 events) and also flares with fluence above 1 × 10

^{−2}J m

^{−2}(1997–2020, 1945 events) are picked up (the data are available as described in the Supplementary Materials). The analysis uses the old operational scale instead of the new science scale adopted by the National Oceanic and Atmospheric Administration (NOAA) [15] (operational values = 0.7 × science values). The peak flux data of 2020 and the background data of 2011–2020 which were in the science database had been converted to the operational scale.

_{obs}is the total duration of data (24 years). The dimension of f(F) is 1/(J m

^{−2}day). Figure 2 shows CCDF(F) (Figure 2a) and f(F) (Figure 2b).

_{p}, with N

_{p}= 1720 and τ

_{obs,p}= 41 years. The dimension of f(F

_{p}) is 1/(W m

^{−2}day).

_{0}(=1 × 10

^{−2}J m

^{−2}; F

_{0p}= 3 × 10

^{−5}W m

^{−2}for the peak flux data) is the lower boundary of the fitting. The other three are two-parameter models. The tapered power law distribution is defined by [20]

## 3. Results

_{0}× KS is large and the corresponding KS p-value is small, one can conclude that the model is rejected. However, it is found here that the KS measures are generally small, and the p-values are not so small even for the power-law models. This happens because the K-S test is not sensitive to misfitting at the tail of the distributions. If F

_{0}value is reduced, the fitting degrades and eventually all the models tend to be rejected by the K-S test. The ambiguities in setting the lower bound F

_{0}of the probability distribution function are discussed in Section 4.

#### 3.1. Tapered Power Law and Gamma Function Distributions

#### 3.2. Weibull Distribution

_{0}, the Weibull PDF approaches a power law with exponent 1 − k. Therefore, the fluence PDF behaves like F

^{−(1−k)}= F

^{−0.935}, which is significantly flatter than the tapered power law and gamma function distributions. This is another reason why the Weibull distribution is not favored compared to the tapered power law and gamma function distributions.

#### 3.3. Comparison with Published Results

#### 3.4. Prediction of Extreme Events

_{X}, is given in terms of the X-ray flare fluence, F, as

_{X}= 2π × (1 au)

^{2}× F × 10

^{7}.

^{−2}, E

_{X}is in erg, and 1 astronomical unit (au) = 1.5 × 10

^{11}m. From this, E

_{rad}, the total radiated energy all over the electromagnetic spectrum (contributed mostly from ultra-violet (UV)) in erg, is estimated as [34]

_{rad}= 1.03 × 10

^{9}× E

_{X}

^{0.766}.

_{p}(J m

^{−2}), approximately by

_{p}≈ 7.93 × 10

^{−4}× F

^{0.945},

^{9}years), the largest flare we experience would be X224, or X374 for one-sigma errors.

## 4. Discussion

_{0}value of the GOES flare fluence, F

_{0}= 1 × 10

^{−2}J m

^{−2}, is translated to the radiated energy of 3.8 × 10

^{30}erg. It can be well seen that the PDF of solar flares does not connect to the stellar flare PDF but decays at an energy of approximately 10

^{33.5}erg. Likewise, the PDF of stellar flares has a flatter distribution and does not connect to the PDF of solar flares. Although solar and stellar flares are both believed to be powered by magnetic energy (most likely by magnetic reconnection) of spotted regions, the distributions of the size and magnetic field strength in starspots would not be the same as in sunspots and may have a wider variety, because of different internal structures and rotation periods. Stellar flares may have a variety of power-exponent values, and the present results of α = 1.95 for solar flares and α = 1.05 for stellar flares are just two examples and there might be a continuous distribution of α between 1 and 2.

## 5. Conclusions

## Supplementary Materials

^{−5}J m

^{−2}(1997–2020) and peak flux above 1 × 10

^{−7}W m

^{−2}(1980–2020), respectively.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Solar soft X-ray flares detected by the Geostationary Operational Environmental Satellites (GOES): peak flux (

**a**) and fluence of flares (

**b**) as a function of year.

**Figure 2.**Complementary cumulative distribution function (

**a**) and occurrence rates (

**b**) of flare fluence, F. The histogram in plot (

**b**) is built with a bin-size of Δlog

_{10}F = 0.2. Short vertical bars indicate statistical errors of √n on the bins with counts n.

**Figure 3.**(

**a**) The blue line shows the power-law fit to the observed CCDF of GOES flare fluence data. (

**b**) The derived power-law distribution overplotted on the observed flare occurrence rate histogram. The vertical dashed line indicates the lower boundary (1 × 10

^{−2}J m

^{−2}) of data used for fitting.

**Figure 4.**The CCDFs of flare fluence data fitted by (

**a**) a tapered power law, (

**b**) a gamma function distribution, and (

**c**) a Weibull distribution. The vertical dashed line indicates the lower boundary (1 × 10

^{−2}J m

^{−2}) of data used for fitting.

**Figure 5.**This analysis results compared to earlier findings [17,33] of the flare occurrence rates (PDFs) for fluence (

**a**) and peak flux (

**b**). The thick black histograms are the observed PDFs. Red, olive, and teal curves indicate, respectively, tapered power law, gamma function, and Weibull distributions fitted to the data. Purple and green curves are power law [17] and Weibull [33] fits, respectively. The vertical dashed lines indicate the lower boundary in the data used for fitting, namely 1 × 10

^{−2}J m

^{−2}for fluence and M3 (3 × 10

^{−5}W m

^{−2}) for peak flux, respectively. Veronig et al.’s [17] parameters of power-law fits are: N = 8400, τ

_{obs}= 4 years, α = 2.03, and F

_{0}= 2 × 10

^{−3}J m

^{−2}(for flare fluence), and N

_{p}= 49409, τ

_{obs,p}= 25 years, F

_{0p}= 2 × 10

^{−6}W m

^{−2}, and α = 2.11 (for flare peak flux). Gopalswamy’s [33] parameters of Weibull distribution fit are: N

_{p}= 55285, τ

_{obs,p}= 48 years, F

_{0p}= 9.1 × 10

^{−7}W m

^{−2}, k = 0.167, and β = 3.77. See text for details.

**Figure 6.**Statistical distributions of flares on solar-type stars. (

**a**) CCDF with a power-law (blue) and gamma function (olive) distribution fits. (

**b**) The histogram of flare occurrence rates with the derived power law and gamma function distributions. The data are from [39]. The vertical dashed line indicates the lower boundary (3 × 10

^{34}erg) of the data used for fitting.

**Figure 7.**Comparison of solar and stellar flare energy distributions. Black and purple histograms show the occurrence rates of solar X-ray flares and stellar flares [39], respectively. The olive curves are fit to these data by the gamma function distributions. The vertical dashed lines indicate the lower boundary for fitting adopted for the solar flare data (3.8 × 10

^{30}erg of radiated energy) and for the stellar flare data (3 × 10

^{34}erg).

**Table 1.**Derived parameters for X-ray fluence distributions (F

_{0}= 1 × 10

^{−2}J m

^{−2}, N = 1945). See text for details.

Model | α | β | ΔAIC | √N × KS | K-S p-Value |
---|---|---|---|---|---|

Power law | 2.015 ± 0.023 | 11.6 | 0.69 | 0.72 | |

Tapered power law | 1.973 ± 0.021 | 0.00948 ± 0.0031 | 0.00 | 0.47 | 0.98 |

Gamma function | 1.949 ± 0.032 | 0.00448 ± 0.0020 | 1.10 | 0.57 | 0.90 |

Weibull | k = 0.0648 ± 0.0242 | 14.7 ± 5.9 | 7.20 | 0.69 | 0.73 |

**Table 2.**Derived parameters for X-ray peak flux distributions (F

_{0p}= 3 × 10

^{−5}W m

^{−2}, N = 1720). See text for details.

Model | α | β | ΔAIC | √N × KS | K-S p-Value |
---|---|---|---|---|---|

Power law | 2.162 ± 0.028 | 15.7 | 0.89 | 0.41 | |

Tapered power law | 2.077 ± 0.032 | 0.026 ± 0.007 | 0.00 | 0.40 | 0.99 |

Gamma function | 2.040 ± 0.045 | 0.014 ± 0.005 | 1.48 | 0.44 | 0.99 |

Weibull | k = 0.104 ± 0.030 | 10.2 ± 3.3 | 7.00 | 0.57 | 0.90 |

**Table 3.**Predicted flare intervals as a function of X-ray fluence values (gamma function distribution, α = 1.949 ± 0.032, β = 0.00448 ± 0.0020). The second numbers after a slash are derived by assuming one-sigma errors (α = 1.949 + 0.032, β = 0.00448 − 0.0020). See text for details.

X-ray Fluence (J m ^{−2}) | Approx. GOES Flux | Total Energy (erg) | Interval (Years) |
---|---|---|---|

1.0 × 10^{−2} | M1.0 | 1.5 × 10^{31} | 1.3 × 10^{−2}/1.3 × 10^{−2} |

1.0 × 10^{−1} | M9.0 | 8.8 × 10^{31} | 1.4 × 10^{−1}/1.4 × 10^{−1} |

1.0 × 10^{0} | X7 | 5.1 × 10^{32} | 3.0 × 10^{0}/2.3 × 10^{0} |

2.0 × 10^{0} | X15 | 8.7 × 10^{32} | 1.2 × 10^{1}/7.3 × 10^{0} |

5.0 × 10^{0} | X36 | 1.8 × 10^{33} | 1.8 × 10^{2}/5.6 × 10^{1} |

6.3 × 10^{0} | X45 ^{†} | 2.1 × 10^{33} | 4.6 × 10^{2}/1.1 × 10^{2} |

^{†}Carrington event (1859 Sept.1) [34].

**Table 4.**Predicted flare sizes as a function of flare intervals (gamma function distribution, α = 1.949 ± 0.032, β = 0.00448 ± 0.0020). The second numbers after a slash are derived by assuming one-sigma errors (α = 1.949 + 0.032, β = 0.00448 − 0.0020). See text for details.

X-ray Fluence (J m ^{−2}) | Approx. GOES Flux | Total Energy (erg) | Interval (Years) |
---|---|---|---|

6.5 × 10^{0}/9.9 × 10^{0} | X46/X69 | 2.2 × 10^{33}/3.0 × 10^{33} | 1.0 × 10^{3} |

1.0 × 10^{1}/1.6 × 10^{1} | X70/X108 | 3.0 × 10^{33}/4.3 × 10^{33} | 1.0 × 10^{4} |

1.4 × 10^{1}/2.3 × 10^{1} | X95/X152 | 3.8 × 10^{33}/5.6 × 10^{33} | 1.0 × 10^{5} |

1.8 × 10^{1}/3.0 × 10^{1} | X122/X197 | 4.7 × 10^{33}/6.9 × 10^{33} | 1.0 × 10^{6} |

2.7 × 10^{1}/4.6 × 10^{1} | X177/X293 | 6.3 × 10^{33}/9.5 × 10^{33} | 1.0 × 10^{8} |

3.4 × 10^{1}/5.9 × 10^{1} | X224/X374 | 7.7 × 10^{33}/1.2 × 10^{34} | 4.6 × 10^{9} |

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Sakurai, T.
Probability Distribution Functions of Solar and Stellar Flares. *Physics* **2023**, *5*, 11-23.
https://doi.org/10.3390/physics5010002

**AMA Style**

Sakurai T.
Probability Distribution Functions of Solar and Stellar Flares. *Physics*. 2023; 5(1):11-23.
https://doi.org/10.3390/physics5010002

**Chicago/Turabian Style**

Sakurai, Takashi.
2023. "Probability Distribution Functions of Solar and Stellar Flares" *Physics* 5, no. 1: 11-23.
https://doi.org/10.3390/physics5010002