Upper limits on the water vapour content of the $\beta$ Pictoris debris disk

The debris disk surrounding $\beta$~Pictoris has been observed with ALMA to contain a belt of CO gas with a distinct peak at $\sim$85 au. This CO clump is thought to be the result of a region of enhanced density of solids that collide and release CO through vaporisation. The parent bodies are thought to be comparable to solar system comets, in which CO is trapped inside a water ice matrix. Since H$_2$O should be released along with CO, we aim to put an upper limit on the H$_2$O gas mass in the disk of $\beta$ Pictoris. We use archival data from the Heterodyne Instrument for the Far-Infrared (HIFI) aboard the Herschel Space Observatory to study the ortho-H$_2$O 1$_{10}$-1$_{01}$ emission line. The line is undetected. Using a python implementation of the radiative transfer code RADEX, we convert upper limits on the line flux to H$_2$O gas masses. The resulting lower limits on the CO/H$_2$O mass ratio are compared to the composition of solar system comets. Depending on the assumed gas spatial distribution, we find a 95% upper limit on the ortho-H$_2$O line flux of $7.5 \times 10^{-20}$ W m$^{-2}$ or $1.2 \times 10^{-19}$ W m$^{-2}$. These translate into an upper limit on the H$_2$O mass of $7.4 \times 10^{16}$-$1.1 \times 10^{18}$ kg depending on both the electron density and gas kinetic temperature. The range of derived gas-phase CO/H$_2$O ratios is marginally consistent with low-ratio solar system comets.