Chemical U-Th-Pb geochronology: A precise explicit approximation of the age equation and associated errors

Abstract

In a single decaying system, the age determined from the exponential decay law is directly related to its linear Maclaurin approximation. This relationship can be additively extended to several decaying systems resulting in the same daughter element, by using proportionality functions, thus al-lowing an explicit formulation of the age as a function of element concentrations. The values of the binary proportionality function for the 238 U- 235 U-Pb system and the ternary proportionality function for the 232 Th- 238 U- 235 U-Pb system were determined by iterations of the exponential decay formula up to 4 Ga, with a step of 10 Ma, for a set of 24 different U/Th ratios. From the iteration data, the expres-sions of the two functions and the associated coefficients were determined by polynomial regression and mathematical programing on conveniently separated time and compositional intervals. Additional time- and composition-dependent age corrections optimized by mathematical program-ming of the residuals lead to an accuracy of 0.005 Ma of the resulting age. The error propagation can be traced through all the operations defined by explicit formulas according to simple error propaga-tion rules, finally allowing the calculation of the standard error of the result. The formulas and param-eters derived can be used in a calculation spreadsheet. © 2012 Silesian University of Technology, Gliwice, Poland.