Reconstructing the projected gravitational potential of galaxy clusters
  from galaxy kinematics
  We develop a method for reconstructing the two-dimensional, projected
gravitational potential of galaxy clusters from observed line-of-sight velocity
dispersions of cluster galaxies. It is the third of an intended series of
papers aiming at a unique reconstruction method for cluster potentials
combining lensing, X-ray, Sunyaev-Zel'dovich and kinematic data. The observed
galaxy velocity dispersions are deprojected using the Richardson-Lucy
algorithm. The obtained radial velocity dispersions are then related to the
gravitational potential by using the tested assumption of a polytropic relation
between the effective galaxy pressure and the density. Once the gravitational
potential is obtained in three dimensions, projection along the line-of-sight
yields the two-dimensional potential. For simplicity we adopt spherical
symmetry and a known profile for the anisotropy parameter of the galaxy
velocity dispersions. We test the method with a numerically simulated galaxy
cluster and galaxies identified therein. We extract a projected
velocity-dispersion profile from the simulated cluster and pass it through our
algorithm, showing that the deviation between the true and the reconstructed
gravitational potential is less then 10% within approximately 1.2 Mpc/h from
the cluster centre.
