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