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Solar flares strongly affect the Sun's atmosphere as well as the Earth'senvironment. Quantifying the maximum possible energy of solar flares of thepresent-day Sun, if any, is thus a key question in heliophysics. The largestsolar flares observed over the past few decades have reached energies of a fewtimes 10^{32} ergs, possibly up to 10^{33} ergs. Flares in active Sun-likestars reach up to about 10^{36} ergs. In the absence of direct observations ofsolar flares within this range, complementary methods of investigation areneeded. Using historical reports for solar active region, we scaled to observedsolar values a realistic dimensionless 3D MHD simulation for eruptive flares,which originate from a highly sheared bipole. This enabled us to calculate themagnetic fluxes and flare energies in the model in a wide paramater space.Firstly, commonly observed solar conditions lead to modeled magnetic fluxes andflare energies that are comparable to those estimated from observations.Secondly, we evaluate from observations that 30% of the area of sunspot groupsare typically involved in flares. This is related to the strong fragmentationof these groups, which naturally results from sub-photospheric convection. Whenthe model is scaled to 30% of the area of the largest sunspot group everreported, with its peak magnetic field being set to the strongest value evermeasured in a sunspot, it produces a flare with a maximum energy of ~ 6x10^{33}ergs. The results of the model suggest that the Sun is able to produce flaresup to about six times as energetic in total solar irradiance fluence as thestrongest directly-observed flare from Nov 4, 2003. Sunspot groups larger thanhistorically reported would yield superflares for spot pairs that would exceedtens of degrees in extent. We thus conjecture that superflare-productiveSun-like stars should have a much stronger dynamo than in the Sun.

The standard flare model in three dimensions. II. Upper limit on solar flare energy