Improved strong lensing modelling of galaxy clusters using the Fundamental Plane: Detailed mapping of the baryonic and dark matter mass distribution of Abell S1063

Aims. From accurate photometric and spectroscopic information, we build the Fundamental Plane (FP) relation for the early-type galaxies of the cluster Abell S1063. We use this relation to develop an improved strong lensing model of the cluster, and we decompose the cluster's cumulative projected total mass profile into its stellar, hot gas, and dark matter mass components. We compare our results with the predictions of cosmological simulations. Methods. We calibrate the FP using Hubble Frontier Fields photometry and data from the Multi Unit Spectroscopic Explorer on the Very Large Telescope. The FP allows us to determine the velocity dispersions of all 222 cluster members included in the model from their measured structural parameters. As for their truncation radii, we test a proportionality relation with the observed half-light radii. Fixing the mass contribution of the hot gas component from X-ray data, the mass density distributions of the diffuse dark matter haloes are optimised by comparing the observed and model-predicted positions of 55 multiple images of 20 background sources distributed over the redshift range 0.73 - 6.11. We determine the uncertainties on the model parameters with Monte Carlo Markov chains. Results. We find that the most accurate predictions of the positions of the multiple images are obtained when the truncation radii of the member galaxies are approximately 2.3 times their effective radii. Compared to earlier work on the same cluster, our model allows for the inclusion of some scatter on the relation between the total mass and the velocity dispersion of the cluster members. We notice a lower statistical uncertainty on the value of some model parameters. For instance, the main dark matter halo of the cluster has a core radius of 86 +/- 2 kpc: the uncertainty on this value decreases by more than 30% with respect to previous work. Taking advantage of a new estimate of the stellar mass of all cluster members from the HST multi-band data, we measure the cumulative two-dimensional mass profiles out to a radius of 350 kpc for all baryonic and dark matter components of the cluster. At the outermost radius of 350 kpc, we obtain a baryon fraction of 0.147 +/- 0.002. We study the stellar-to-total mass fraction of the high-mass cluster members in our model, finding good agreement with the observations of wide galaxy surveys and some disagreement with the predictions of halo occupation distribution studies based on N-body simulations. Finally, we compare the features of the sub-haloes as described by our model with those predicted by high-resolution hydrodynamical simulations. We obtain compatible results in terms of the stellar over total mass fraction. On the other hand, we report some discrepancies both in terms of the maximum circular velocity, which is an indication of the halo compactness, and the sub-halo total mass function in the central cluster regions.