  We present a novel approach for a combined analysis of X-ray and
gravitational lensing data and apply this technique to the merging galaxy
cluster MACS J0416.1$-$2403. The method exploits the information on the
intracluster gas distribution that comes from a fit of the X-ray surface
brightness, and then includes the hot gas as a fixed mass component in the
strong lensing analysis. With our new technique, we can separate the
collisional from the collision-less diffuse mass components, thus obtaining a
more accurate reconstruction of the dark matter distribution in the core of a
cluster. We introduce an analytical description of the X-ray emission coming
from a set of dual Pseudo-Isothermal Elliptical (dPIE) mass distributions,
which can be directly used in most lensing softwares. By combining
\emph{Chandra} observations with Hubble Frontier Fields imaging and MUSE
spectroscopy in MACS J0416.1$-$2403, we measure a projected gas over total mass
fraction of approximately $10\%$ at $350$ kpc from the cluster center. Compared
to the results of a more traditional cluster mass model (diffuse halos plus
member galaxies), we find a significant difference in the cumulative projected
mass profile of the dark matter component and that the dark matter to total
mass fraction is almost constant, out to more than $350$ kpc. In the coming era
of large surveys, these results show the need of multi-probe analyses for
detailed dark matter studies in galaxy clusters.
