Combining magneto-hydrostatic constraints with Stokes profile inversions. IV. Imposing $\nabla\cdot{\bf B}=0$ condition

Inferences of the magnetic field in the solar atmosphere by means of spectropolarimetric inversions (i.e., Stokes inversion codes) yield magnetic fields that are non-solenoidal($\nabla\cdot{\bf B} \ne 0$). Because of this, results obtained by such methods are sometimes put into question. We aim to develop and implement a new technique that can retrieve magnetic fields that are simultaneously consistent with observed polarization signals and with the null divergence condition. The method used in this work strictly imposes $\nabla\cdot{\bf B}=0$ by determining the vertical component of the magnetic field ($B_{\rm z}$) from the horizontal ones ($B_{\rm x},B_{\rm y}$). We implement this solenoidal inversion into the FIRTEZ Stokes inversion code and apply it to spectropolarimetric observations of a sunspot observed with the Hinode/SP instrument. We show that the solenoidal inversion retrieves a vertical component of the magnetic field that is consistent with the vertical component of the magnetic field inferred from the non-solenoidal one. We demonstrate that the solenoidal inversion is capable of a better overall fitting to the observed Stokes vector than the non-solenoidal inversion. In fact, the solenoidal magnetic field fits Stokes $V$ worse, but this is compensated by a better fit to Stokes $I$. We find a direct correlation between the worsening in the fit to the circular polarization profiles by the solenoidal inversion and the deviations in the inferred $B_{\rm z}$ with respect to the non-solenoidal inversion. These results support the idea that common Stokes inversion techniques fail to reproduce $\nabla\cdot{\bf B}=0$ mainly as a consequence of the uncertainties in the determination of the individual components of the magnetic field.