  Context. Galaxy clusters can be used as cosmological probes, but to this end,
they need to be thoroughly understood. Combining all cluster observables in a
consistent way will help us to understand their global properties and their
internal structure. Aims. We provide proof of the concept that the projected
gravitational potential of galaxy clusters can directly be reconstructed from
X-ray observations. We also show that this joint analysis can be used to
locally test the validity of the equilibrium assumptions in galaxy clusters.
Methods. We used a newly developed reconstruction method, based on
Richardson-Lucy deprojection, that allows reconstructing projected
gravitational potentials of galaxy clusters directly from X-ray observations.
We applied this algorithm to the well-studied cluster Abell 1689 and compared
the gravitational potential reconstructed from X-ray observables to the
potential obtained from gravitational lensing measurements. [...] Results.
Assuming spherical symmetry and hydrostatic equilibrium, the potentials
recovered from gravitational lensing and from X-ray emission agree very well
beyond 500 kpc. Owing to the fact that the Richardson-Lucy deprojection
algorithm allows deprojecting each line of sight independently, this result may
indicate that non-gravitational effects and/or asphericity are strong in the
central regions of the clusters. Conclusions. We demonstrate the robustness of
the potential reconstruction method based on the Richardson-Lucy deprojection
algorithm and show that gravitational lensing and X-ray emission lead to
consistent gravitational potentials. Our results illustrate the power of
combining galaxy-cluster observables in a single, non-parametric, joint
reconstruction of consistent cluster potentials that can be used to locally
constrain the physical state of the gas.
