3D modelling of HCO$^+$ and its isotopologues in the low-mass proto-star
  IRAS16293$-$2422
  Ions and electrons play an important role in various stages of the star
formation process. By following the magnetic field of their environment and
interacting with neutral species, they slow down the gravitational collapse of
the proto-star envelope. This process (known as ambipolar diffusion) depends on
the ionisation degree, which can be derived from the \hco abundance. We present
a study of \hco and its isotopologues (H$^{13}$CO$^+$, HC$^{18}$O$^+$, DCO$^+$,
and H$^{13}$CO$^+$) in the low-mass proto-star IRAS16293$-$2422. The structure
of this object is complex, and the HCO$^+$ emission arises from the
contribution of a young NW-SE outflow, the proto-stellar envelope and the
foreground cloud. We aim at constraining the physical parameters of these
structures using all the observed transitions. For the young NW-SE outflow, we
derive $T_{\rm kin}=180-220$ K and $n({\rm H_2})=(4-7)\times10^6$ cm$^{-3}$
with an HCO$^+$ abundance of $(3-5)\times10^{-9}$. Following previous studies,
we demonstrate that the presence of a cold ($T_{\rm kin}$$\leqslant$30 K) and
low density ($n({\rm H_2})\leqslant1\times10^4$ cm$^{-3}$) foreground cloud is
also necessary to reproduce the observed line profiles. We have used the
gas-grain chemical code \textsc{nautilus} to derive the HCO$^+$ abundance
profile across the envelope and the external regions where
X(HCO$^+$)$\gtrsim1\times10^{-9}$ dominate the envelope emission. From this, we
derive an ionisation degree of
$10^{-8.9}\,\lesssim\,x(e)\,\lesssim\,10^{-7.9}$. The ambipolar diffusion
timescale is $\sim$5 times the free-fall timescale, indicating that the
magnetic field starts to support the source against gravitational collapse and
the magnetic field strength is estimated to be $6-46 \mu$G.
