Tracing the origin of presolar grains: algorithms to parametrize supernovae

Certain meteorites contain grains of stardust with isotopic compositions that suggest supernova origins. Furthermore, individual grains often contain signatures from different zones of the star, from the iron-rich core to the hydrogen- and helium-rich envelope. This implies large-scale mixing between layers, and a grain’s specific combination of isotopic ratios can be used to constrain these mixing mechanisms. To this end we have developed an algorithm which, given a grain’s set of measured isotopic ratios and a theoretical supernova model, assigns ‘mixing fractions’ to various zones of the supernova, indicating what fraction of the grain comes from each zone. The algorithm employs a cocktail of numerical methods to locate the ‘best’ set of mixing ratios, defined by a least squares relative error objective function measuring the disagreement between the constrained supernova model and the measured grain data. The algorithm, as is, includes the Gauss-Newton method, gradient descent and an exhaustive combinatorial search. A correction is added to maintain the condition C/O \u3e 1, a necessary condition for graphite and SiC condensation. When used on a supernova model divided into seven or fewer zones, these methods consistently approximate all of a grain’s isotopic ratios to within an order of magnitude, and the agreement improves when fewer ratios must be fit. Incorporating more layers increases the dimension of the unknown variable, and rapidly increases computation time and reduces accuracy for the algorithm designed for 8 layers. For the higher dimensional problem, a Markov Chain Monte Carlo method is in development. While previous attempts at the problem have succeeded in fitting all but one or two measured ratios, our method prioritizes each measurement equally, distributing the relative error equally among the various isotopic ratios