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Magnetic helicity budget of solar active regions prolific of eruptive and confined flares

We compare the coronal magnetic energy and helicity of two solar active regions (ARs), prolific in major eruptive (AR~11158) and confined (AR~12192) flaring, and analyze the potential of deduced proxies to forecast upcoming flares. Based on nonlinear force-free (NLFF) coronal magnetic field models with a high degree of solenoidality, and applying three different computational methods to investigate the coronal magnetic helicity, we are able to draw conclusions with a high level of confidence. Based on real observations of two solar ARs we checked trends regarding the potential eruptivity of the active-region corona, as suggested earlier in works that were based on numerical simulations, or solar observations. Our results support that the ratio of current-carrying to total helicity, $|H_\mathrm{J}|/|H_\mathrm{V}|$, shows a strong ability to indicate the eruptive potential of a solar AR. However, $|H_\mathrm{J}|/|H_\mathrm{V}|$ seems not to be indicative for the magnitude or type of an upcoming flare (confined or eruptive). Interpreted in context with earlier observational studies, our findings furthermore support that the total relative helicity normalized to the magnetic flux at the NLFF model's lower boundary, $H_\mathrm{V}/ϕ^2$, represents no indicator for the eruptivity.