Probability Distribution Functions of Solar and Stellar Flares
Abstract
:1. Introduction
2. Data and Methods
3. Results
3.1. Tapered Power Law and Gamma Function Distributions
3.2. Weibull Distribution
3.3. Comparison with Published Results
3.4. Prediction of Extreme Events
4. Discussion
5. Conclusions
Supplementary Materials
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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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 |
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 |
X-ray Fluence (J m−2) | Approx. GOES Flux | Total Energy (erg) | Interval (Years) |
---|---|---|---|
1.0 × 10−2 | M1.0 | 1.5 × 1031 | 1.3 × 10−2/1.3 × 10−2 |
1.0 × 10−1 | M9.0 | 8.8 × 1031 | 1.4 × 10−1/1.4 × 10−1 |
1.0 × 100 | X7 | 5.1 × 1032 | 3.0 × 100/2.3 × 100 |
2.0 × 100 | X15 | 8.7 × 1032 | 1.2 × 101/7.3 × 100 |
5.0 × 100 | X36 | 1.8 × 1033 | 1.8 × 102/5.6 × 101 |
6.3 × 100 | X45 † | 2.1 × 1033 | 4.6 × 102/1.1 × 102 |
X-ray Fluence (J m−2) | Approx. GOES Flux | Total Energy (erg) | Interval (Years) |
---|---|---|---|
6.5 × 100/9.9 × 100 | X46/X69 | 2.2 × 1033/3.0 × 1033 | 1.0 × 103 |
1.0 × 101/1.6 × 101 | X70/X108 | 3.0 × 1033/4.3 × 1033 | 1.0 × 104 |
1.4 × 101/2.3 × 101 | X95/X152 | 3.8 × 1033/5.6 × 1033 | 1.0 × 105 |
1.8 × 101/3.0 × 101 | X122/X197 | 4.7 × 1033/6.9 × 1033 | 1.0 × 106 |
2.7 × 101/4.6 × 101 | X177/X293 | 6.3 × 1033/9.5 × 1033 | 1.0 × 108 |
3.4 × 101/5.9 × 101 | X224/X374 | 7.7 × 1033/1.2 × 1034 | 4.6 × 109 |
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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
Sakurai T. Probability Distribution Functions of Solar and Stellar Flares. Physics. 2023; 5(1):11-23. https://doi.org/10.3390/physics5010002
Chicago/Turabian StyleSakurai, Takashi. 2023. "Probability Distribution Functions of Solar and Stellar Flares" Physics 5, no. 1: 11-23. https://doi.org/10.3390/physics5010002