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While it is important to design stellarators with high magneto-hydrodynamic(MHD) stability $\beta$-limit, it is also crucial to ensure that good magneticsurfaces exist in a large range of $\beta$ values. As $\beta$ increases,pressure-driven currents perturb the vacuum magnetic field and often lead tothe emergence of magnetic field line chaos, which can worsen the confinementand is the cause of another kind of $\beta$-limit, the so-called equilibrium$\beta$-limit. In this paper, we explore numerically the dependence of theequilibrium $\beta$-limit on the bootstrap current strength using the SteppedPressure Equilibrium Code (SPEC). We develop a diagnostic to determine whetheror not magnetic islands are expected to participate significantly to radialtransport, and we build an analytical model to predict the expected equilibrium$\beta$-limit, which recovers the main features of the numerical results. Thisresearch opens the possibility to include additional targets in stellaratoroptimization functions, provides additional understanding on the existence ofmagnetic surfaces at large $\beta$, and is a step forward in the understandingof the equilibrium $\beta$-limit.

Equilibrium $\beta$-limits dependence on bootstrap current in classical stellarators