An analytic mean-field $\alpha^2$-dynamo with a force-free corona

Stellar dynamos are affected by boundary conditions imposed by stellar coronae. Under some approximations it is possible to find analytical solutions. Interior dynamo models often consider a current-free coronae without taking into account the constraints imposed by the presence of currents in the corona. We aim to analytically evaluate the effect of coronal currents and of an outer boundary condition on the efficiency of an $alpha^2$ dynamo. We intend to estimate the change in geometry and dinamo excitation numbers with respect to the current-free case. We analytically solve the turbulent dynamo induction equation for a homogeneous, non-mirror symmetric turbulence, in a spherical domain surrounded by a linear force-free corona with magnetic field B satisfying curl B = beta B. The main result is that the dynamo number is a decreasing function of beta. Moreover, if the current is parallel to the field (beta u003e 0) the dynamo number is smaller than the force-free case. On the contrary (beta u003c 0) the dynamo number is greater than the force-free case. We conclude that the presence of currents in the corona needs to be taken into account because it affects the condition for excitation of a dynamo.