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

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 allowing an explicit formulation of the age as a function of element concentrations. The values of the binary proportionality function for the 238U-235U-Pb system and the ternary proportionality function for the 232Th-238U-235U-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 expressions 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 programming 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 propagation rules, finally allowing the calculation of the standard error of the result. The formulas and parameters derived can be used in a calculation spreadsheet.