Effective closure temperature in leaky and/or saturating thermochronometers

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The classical equation of closure temperature (TC) in thermochronometry (Dodson, 1973), assumed (i) no storage limitation for the accumulating radiogenic product, (ii) a negligible product concentration at the initial temperature of cooling T0, and (iii) a negligible product loss at the final (present-day) temperature TP. A subsequent extension (Ganguly and Tirone, 1999) provided a simple correction for systems cooling from an arbitrary T0, at which presence of an initial concentration profile may affect final concentrations. Here, we use a combination of analytical and numerical solutions to derive a general expression for the effective closure temperature in (i) systems which cool between arbitrary initial and final temperatures, potentially still suffering from thermal product loss at TP (termed ‘leaky’), and (ii) systems which may contain a physical limit on the maximum amount of product that can be stored (termed ‘saturating’). While all conservative results can be easily reproduced, an extended use of our formulation provides meaningful effective closure temperatures even when the standard calculation schemes fail. For a first-order loss radiometric system governed by K(T)=sexp(−E/RT), where E [J mol−1] and s [s−1] are the Arrhenius parameters and R is the gas constant, we find that the effective closure temperature TC(T0,TP) is given by:TC(T0,TP)={1TP−R/Eτλ−τKPln[1+τλ−τKP(τKP)τλe−τKP(Γ(τλ,τKP)−Γ(τλ,τK0))]}−1 where K0 and KP [s−1] are shorthand for K(T0) and K(TP), respectively, λ [s−1] the production rate, τ [s] a time constant, and Γ(a,z) the upper incomplete gamma function. Under conventional conditions, our solution reduces to Dodsonʼs formula. Although the solution strictly applies only to systems where 1/T increases linearly with time, it is nevertheless a useful approximation for a broad range of cooling functions in systems where closure occurs close to the systemʼs initial/final thermal boundary conditions. We clarify the use and the meaning of TC(T0,TP) by drawing a comparison between (i) a hypothetical application of apatite U–Pb dating (TC≈450°C) on Venus (mean surface temperature of 450 °C, leaky behaviour), and (ii) the recently introduced thermochronometric application of optically stimulated luminescence (OSL) dating on Earth (both leaky and saturating behaviour).