Constraints on the Cosmic-Ray Ionization Rate in the $z\sim2.3$ Lensed
  Galaxies SMM J2135$-$0102 and SDP 17b from Observations of OH$^+$ and
  H$_2$O$^+$
  Cosmic rays are predominantly accelerated in shocks associated with star
formation such as supernova remnants and stellar wind bubbles, so the
cosmic-ray flux and thus cosmic-ray ionization rate, $\zeta_{\rm H}$, should
correlate with the star-formation rate in a galaxy. Submillimeter bright
galaxies (SMGs) are some of the most prolific star forming galaxies in the
Universe, and gravitationally lensed SMGs provide bright continuum sources
suitable for absorption line studies. Abundances of OH$^+$ and H$_2$O$^+$ are
useful for inferring $\zeta_{\rm H}$ when combined with chemical models, and
have been used for this purpose within the Milky Way. At redshifts $z\gtrsim2$
transitions out of the ground rotational states of OH$^+$ and H$_2$O$^+$ are
observable with ALMA, and we present observations of both molecules in
absorption toward the lensed SMGs SMM J2135$-$0102 and SDP 17b. These
detections enable an exploration of $\zeta_{\rm H}$ in galaxies with extreme
star formation and high supernova rates, both of which should significantly
enhance cosmic-ray production. The observed OH$^+$ and H$_2$O$^+$ absorption is
thought to arise in massive, extended halos of cool, diffuse gas that surround
these galaxies. Using a chemical model designed to focus on the reaction
network important to both species, we infer cosmic-ray ionization rates of
$\zeta_{\rm H}\sim10^{-16}$-$10^{-14}$ s$^{-1}$ in these extended gaseous
halos. As our estimates come from gas that is far away from the sites of
cosmic-ray acceleration, they imply that cosmic-ray ionization rates in the
compact regions where star formation occurs in these galaxies are orders of
magnitude higher.
