We examine the reconstruction of galaxy cluster radial density profiles obtained from Chandra and XMM X-ray observations, using high quality data for a sample of twelve objects covering a range of morphologies and redshifts. By comparing the results obtained from the two observatories and by varying key aspects of the analysis procedure, we examine the impact of instrumental effects and of differences in the methodology used in the recovery of the density profiles. We find that the final density profile shape is particularly robust. We adapt the photon weighting vignetting correction method developed for XMM for use with Chandra data, and confirm that the resulting Chandra profiles are consistent with those corrected a posteriori for vignetting effects. Profiles obtained from direct deprojection and those derived using parametric models are consistent at the 1% level. At radii larger than $sim$6, the agreement between Chandra and XMM is better than 1%, confirming an excellent understanding of the XMM PSF. We find no significant energy dependence. The impact of the well-known offset between Chandra and XMM gas temperature determinations on the density profiles is found to be negligible. However, we find an overall normalisation offset in density profiles of the order of $sim$2.5%, which is linked to absolute flux cross-calibration issues. As a final result, the weighted ratios of Chandra to XMM gas masses computed at R2500 and R500 are r=1.03$pm$0.01 and r=1.03$pm$0.03, respectively. Our study confirms that the radial density profiles are robustly recovered, and that any differences between Chandra and XMM can be constrained to the $sim$ 2.5% level, regardless of the exact data analysis details. These encouraging results open the way for the true combination of X-ray observations of galaxy clusters, fully leveraging the high resolution of Chandra and the high throughput of XMM.
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