Annually resolved measurements of the radiocarbon content in tree-rings have revealed rare sharp rises in carbon-14 production. These ‘Miyake events’ are likely produced by rare increases in cosmic radiation from the Sun or other energetic astrophysical sources. The radiocarbon produced is not only circulated through the Earth’s atmosphere and oceans, but also absorbed by the biosphere and locked in the annual growth rings of trees. To interpret high-resolution tree-ring radiocarbon measurements therefore necessitates modelling the entire global carbon cycle. Here, we introduce ‘ ticktack ’ ( https://github.com/SharmaLlama/ticktack/ ), the first open-source Python package that connects box models of the carbon cycle with modern Bayesian inference tools. We use this to analyse all public annual   14 C tree data, and infer posterior parameters for all six known Miyake events. They do not show a consistent relationship to the solar cycle, and several display extended durations that challenge either astrophysical or geophysical models.

.The sensitivity limits of space telescopes are imposed by uncalibrated errors in the point spread function, photon-noise, background light, and detector sensitivity. These are typically calibrated with specialized wavefront sensor hardware and with flat fields obtained on the ground or with calibration sources, but these leave vulnerabilities to residual time-varying or non-common path aberrations and variations in the detector conditions. It is, therefore, desirable to infer these from science data alone, facing the prohibitively high dimensional problems of phase retrieval and pixel-level calibration. We introduce a new Python package for physical optics simulation,   ∂  Lux, which uses the machine learning framework Jax to achieve graphics processing unit acceleration and automatic differentiation (autodiff), and apply this to simulating astronomical imaging. In this first of a series of papers, we show that gradient descent enabled by autodiff can be used to simultaneously perform phase retrieval and calibration of detector sensitivity, scaling efficiently to inferring millions of parameters. This new framework enables high dimensional optimization and inference in data analysis and hardware design in astronomy and beyond, which we explore in subsequent papers in this series.

Radiocarbon measurements from tree rings allow us to recover measurements of cosmic radiation from the distant past, and exquisitely calibrate carbon dating of archaeological sites. But in order to infer cosmic production rates from raw ΔC 14 data, we need to model the entire global carbon cycle, from the production of radiocarbon in the stratosphere and troposphere to its uptake by the oceans and biosphere. Many such competing models exist, in which the Earth system is partitioned into 'boxes' with reservoirs of C 12 , C 14 , and coefficients of flow between them.