Explaining the physical origin of BÃ¥th's law

BÃ¥th's law is one of the three well-known scaling laws for earthquakes. It states that the difference in magnitudes of the mainshock and its largest aftershock is approximately constant, independent of the magnitude of the mainshock. Despite the progress in understanding the nature of BÃ¥th's law, the question of whether this law has a physical basis, or is simply a consequence of basic statistical features of aftershock sequences, has remained controversial. In this article we show that BÃ¥th's law can be derived within the Cosserat continuum theory from equations describing fault interaction. Our equations can describe both (1) the interacting mainshocks and aftershocks, and (2) the interacting foreshocks and mainshocks. We also derive (1) spatial extension of BÃ¥th's law to the normalized distance between the locations of the interacting mainshocks and aftershocks (or foreshocks and mainshocks), and (2) temporal extension of BÃ¥th's law to the difference between the time of the interacting mainshocks and aftershocks (or foreshocks and mainshocks).