THRESHOLDED POWER LAW SIZE DISTRIBUTIONS OF INSTABILITIES IN ASTROPHYSICS
THRESHOLDED POWER LAW SIZE DISTRIBUTIONS OF INSTABILITIES IN ASTROPHYSICS
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DOI：
10.1088/0004-637x/814/1/19
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发表时间：
2015-10
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影响因子：
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通讯作者：
M. Aschwanden
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作者：
M. Aschwanden
类似幂律的大小分布在天体物理不稳定性中无处不在。至少有四种自然效应会导致偏离理想的幂律大小分布，我们在这里以一种广义的方式对其进行建模：(1)不稳定性的物理阈值；(2)小于阈值x0的最小事件的不完全采样；(3)与事件无关的背景xb污染；(4)由于系统大小有限，在最大事件时的截断效应。这些效应可以用“阈值幂律”分布函数（也称为广义Pareto [type II]或Lomax分布），N (x) dx∝(x + x 0) - a dx, ？其中x0 > 0对于阈值效应是正的，而x0 < 0对于背景污染是负的。我们从一个指数增长演化模型中解析导出了这个阈值幂律分布函数的函数形状，该模型只有在扰动超过临界阈值x0时才会产生雪崩。我们将阈值幂律分布函数应用于地面、太阳（HXRBS、BATSE、RHESSI）和恒星耀斑（Kepler）数据集。我们发现阈值幂律模型对大多数观测数据提供了足够的拟合。该模型的主要优点是自动选择幂律拟合范围、诊断背景污染、物理不稳定性阈值、仪器检测阈值和有限的系统尺寸限制。当测试预测理想幂律的自组织临界模型时，我们建议包括这些自然截断效应。
Power-law-like size distributions are ubiquitous in astrophysical instabilities. There are at least four natural effects that cause deviations from ideal power law size distributions, which we model here in a generalized way: (1) a physical threshold of an instability; (2) incomplete sampling of the smallest events below a threshold x0; (3) contamination by an event-unrelated background xb; and (4) truncation effects at the largest events due to a finite system size. These effects can be modeled in the simplest terms with a “thresholded power law” distribution function (also called generalized Pareto [type II] or Lomax distribution), N ( x ) dx ∝ ( x + x 0 ) − a dx , ?> where x0 > 0 is positive for a threshold effect, while x0 < 0 is negative for background contamination. We analytically derive the functional shape of this thresholded power law distribution function from an exponential growth evolution model, which produces avalanches only when a disturbance exceeds a critical threshold x0. We apply the thresholded power law distribution function to terrestrial, solar (HXRBS, BATSE, RHESSI), and stellar flare (Kepler) data sets. We find that the thresholded power law model provides an adequate fit to most of the observed data. Major advantages of this model are the automated choice of the power law fitting range, diagnostics of background contamination, physical instability thresholds, instrumental detection thresholds, and finite system size limits. When testing self-organized criticality models that predict ideal power laws, we suggest including these natural truncation effects.