Introduction

Radiogenic helium-geochronology is based on a summed set of differential equations:

$$\frac{d[He]}{dt} = -\sum_{i=1}^{n} \frac{d[P_i]}{dt} \frac{d[P_i]}{dt} = -\lambda_i [P_i]$$ (1)

where t = time, [He] = helium abundance, [P$_i$] = abundance of the i$^{th}$ parent nuclide and $\lambda_i$ = decay constant of this nuclide (for 1 $\leq$ i $\leq$ n). Despite the simplicity of Equation 1, there are several ways to solve it, three of which will be discussed in Section 2. A linear approximation is accurate to better than 1% for ages up to 100Ma, which can be considered satisfactory in comparison with the external reproducibility of (U-Th)/He dating (20-30%; e.g., Stock et al., 2006). Nevertheless, most researchers rightly decide to calculate an exact age by numerical iteration. This paper raises the point that the accuracy gained by doing so is easily lost by two common practices: (1) performing the $\alpha$-ejection after, rather than before age calculation and (2) using the arithmetic mean age to summarize a dataset of several single-grain measurements. After Section 3 presents two similarly biased alternatives to the arithmetic mean age that are appropriate for specific applications, Section 4 introduces the central age as the most accurate way to compute average (U-Th)/He ages. The accuracy gained by using the central age instead of the arithmetic mean age is comparable to that gained by iteratively solving the (U-Th)/He age equation instead of using the linear approximation. The only cost of the new procedure is computational complexity. To facilitate the calculations, they are implemented in an online calculator (http://pvermees.andropov.org/central) and illustrated on a published dataset of inclusion-bearing apatites. Finally, Section 5 presents a generalized method to calculate central ages for datasets that also include a fourth radioactive parent, $^{147}$Sm.