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We present ALMA Cycle 4 observations of CO(1-0), CO(3-2), and $^{13}$CO(3-2)line emission in the brightest cluster galaxy of RXJ0821+0752. This is one ofthe first detections of $^{13}$CO line emission in a galaxy cluster. Half ofthe CO(3-2) line emission originates from two clumps of molecular gas that arespatially offset from the galactic center. These clumps are surrounded bydiffuse emission that extends $8~{\rm kpc}$ in length. The detected $^{13}$COemission is confined entirely to the two bright clumps, with any emissionoutside of this region lying below our detection threshold. Two distinctvelocity components with similar integrated fluxes are detected in the$^{12}$CO spectra. The narrower component ($60~{\rm km}~{\rm s}^{-1}$ FWHM) isconsistent in both velocity centroid and linewidth with $^{13}$CO(3-2)emission, while the broader ($130-160~{\rm km}~{\rm s}^{-1}$), slightlyblueshifted wing has no associated $^{13}$CO(3-2) emission. A simple localthermodynamic model indicates that the $^{13}$CO emission traces $2.1\times10^{9}~{\rm M}_\odot$ of molecular gas. Isolating the $^{12}$CO velocitycomponent that accompanies the $^{13}$CO emission yields a CO-to-H$_2$conversion factor of $\alpha_{\rm CO}=2.3~{\rm M}_{\odot}~({\rmK~km~s^{-1}})^{-1}$, which is a factor of two lower than the Galactic value.Adopting the Galactic CO-to-H$_2$ conversion factor in brightest clustergalaxies may therefore overestimate their molecular gas masses by a factor oftwo. This is within the object-to-object scatter from extragalactic sources, socalibrations in a larger sample of clusters are necessary in order to confirm asub-Galactic conversion factor.

A $^{13}$CO Detection in a Brightest Cluster Galaxy