Testing the Universality of Self-organized Criticality in Galactic, Extragalactic, and Black Hole Systems

Abstract In this study, we test whether the power law slopes ( α F ) for fluxes ( F ), and ( α E ) for energies ( E ) are universal in their size distributions, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>N</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∝</mml:mo> <mml:msup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>N</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∝</mml:mo> <mml:msup> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msup> </mml:math> , in astrophysical observations of galactic, extragalactic, and black hole systems. This is a test of fundamental importance for self-organized criticality (SOC) systems. The test decides whether (i) power laws are a natural consequence of the scale-freeness and inherent universality of SOC systems, or (ii) if they depend on more complex physical scaling laws. The former criterion allows quantitative predictions of the power-law-like size distributions, while the latter criterion requires individual physical modeling for each SOC variable and data set. Our statistical test, carried out with 61 published data sets, is consistent with the former option, which implies that observed power laws can simply be derived from the scale-freeness and do not require specific physical models to understand their statistical distributions. The observations show a mean and standard deviation of α F = 1.78 ± 0.29 for SOC fluxes and α E = 1.66 ± 0.22 for SOC fluences, and thus are consistent with the prediction of the fractal-diffusive SOC model, with α F = 1.80 and α E = 1.67.