A simple determination of the halo size $L$ from $^{10}$Be/$^9$Be data

The AMS-02 and HELIX experiments should soon provide $\mathrm{^{10}Be/^9Be}$ cosmic-ray data of unprecedented precision. We propose an analytical formula to quickly and accurately determine $L$ from these data. Our formula is validated against the full calculation performed with the propagation code \usine{}. We compare the constraints on $L$ set by Be/B and $\mathrm{^{10}Be/^9Be}$, relying on updated sets of production cross-sections. The best-fit $L$ from AMS-02 Be/B data is shifted from 5 kpc to 3.8 kpc when using the updated cross-sections. We obtained consistent results from the Be/B analysis with USINE, $L=3.8^{+2.8}_{-1.6}$ kpc (data and cross-section uncertainties), and from the analysis of $\mathrm{^{10}Be/^9Be}$ data with the simplified formula, $L=4.7\pm0.6$ (data uncertainties) $\pm2$ (cross-section uncertainties) kpc. The analytical formula indicates that improvements on $L$ thanks to future data will be limited by production cross-section uncertainties, unless either $\mathrm{^{10}Be/^9Be}$ measurements are extended up to several tens of GeV/n, or nuclear data for the production of $\mathrm{^{10}Be}$ and $\mathrm{^9Be}$ are improved; new data for the production cross-section of $\mathrm{^{16}O}$ into Be isotopes above a few GeV/n are especially desired.