Molecules with ALMA at Planet-forming Scales (MAPS) XIII: HCO$^+$ and
  disk ionization structure
  We observed HCO$^+$ $J=1-0$ and H$^{13}$CO$^+$ $J=1-0$ emission towards the
five protoplanetary disks around IM Lup, GM Aur, AS 209, HD 163296, and MWC 480
as part of the MAPS project. HCO$^+$ is detected and mapped at
0.3\arcsec\,resolution in all five disks, while H$^{13}$CO$^+$ is detected
(SNR$>6 \sigma$) towards GM Aur and HD 163296 and tentatively detected (SNR$>3
\sigma$) towards the other disks by a matched filter analysis. Inside a radius
of $R\sim 100$ au, the HCO$^+$ column density is flat or shows a central dip.
At outer radii ($\gtrsim 100$ au), the HCO$^+$ column density decreases
outwards, while the column density ratio of HCO$^+$/CO is mostly in the range
of $\sim 10^{-5}-10^{-4}$. We derived the HCO$^+$ abundance in the warm CO-rich
layer, where HCO$^+$ is expected to be the dominant molecular ion. At $R\gtrsim
100$ au, the HCO$^+$ abundance is $\sim 3 \times 10^{-11} - 3\times 10^{-10}$,
which is consistent with a template disk model with X-ray ionization. At the
smaller radii, the abundance decreases inwards, which indicates that the
ionization degree is lower in denser gas, especially inside the CO snow line,
where the CO-rich layer is in the midplane. Comparison of template disk models
with the column densities of HCO$^+$, N$_2$H$^+$, and N$_2$D$^+$ indicates that
the midplane ionization rate is $\gtrsim 10^{-18}$ s$^{-1}$ for the disks
around IM Lup, AS 209, and HD 163296. We also find hints of an increased
HCO$^+$ abundance around the location of dust continuum gaps in AS 209, HD
163296, and MWC 480. This paper is part of the MAPS special issue of the
Astrophysical Journal Supplement.
