On beryllium-10 production in gaseous protoplanetary disks and implications on the astrophysical setting of refractory inclusions

Calcium-Aluminum-rich Inclusions (CAIs), the oldest known solids of the solar system, show evidence for the past presence of short-lived radionuclide beryllium-10, which was likely produced by spallation during protosolar flares. While such $^{10}$Be production has hitherto been modeled at the inner edge of the protoplanetary disk, I calculate here that spallation at the disk surface may reproduce the measured $^{10}$Be/$^9$Be ratios at larger heliocentric distances. Beryllium-10 production in the gas prior to CAI formation would dominate that in the solid. Interestingly, provided the Sun's proton to X-ray output ratio does not decrease strongly, $^{10}$Be/$^9$Be at the CAI condensation front would increase with time, explaining the reduced values in a (presumably early) generation of CAIs with nucleosynthetic anomalies. CAIs thus need not have formed very close to the Sun and may have condensed at 0.1-1 AU where sufficiently high temperatures originally prevailed.