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Combining Strong Lensing and Dynamics in Galaxy Clusters: integrating MAMPOSSt within LENSTOOL I. Application on SL2S J02140-0535

We present a new framework were we simultaneously fit strong lensing (SL) and dynamical data. The SL analysis is based on LENSTOOL, and the dynamical analysis uses MAMPOSSt code, which we have integrated into LENSTOOL. After describing the implementation of this new tool, we apply it on the galaxy group SL2S\,J02140-0535 ($z_{\rm spec}=0.44$), which we have already studied in the past. We use new VLT/FORS2 spectroscopy of multiple images and group members, as well as shallow X-ray data from \xmm. We confirm that the observed lensing features in SL2S\,J02140-0535 belong to different background sources. One of this sources is located at $z_{\rm spec}$ = 1.017 $\pm$ 0.001, whereas the other source is located at $z_{\rm spec}$ = 1.628 $\pm$ 0.001. With the analysis of our new and our previously reported spectroscopic data, we find 24 secure members for SL2S\,J02140-0535. Both data sets are well reproduced by a single NFW mass profile: the dark matter halo coincides with the peak of the light distribution, with scale radius, concentration, and mass equal to $r_s$ =$82^{+44}_{-17}$ kpc , $c_{200}$ = $10.0^{+1.7}_{-2.5}$, and $M_{200}$ = $1.0^{+0.5}_{-0.2}$ $\times$ 10$^{14}$M$_{\odot}$ respectively. These parameters are better constrained when we fit simultaneously SL and dynamical information. The mass contours of our best model agrees with the direction defined by the luminosity contours and the X-ray emission of SL2S\,J02140-0535. The simultaneous fit lowers the error in the mass estimate by 0.34 dex, when compared to the SL model, and in 0.15 dex when compared to the dynamical method.The combination of SL and dynamics tools yields a more accurate probe of the mass profile of SL2S\,J02140-0535 up to $r_{200}$. However, there is tension between the best elliptical SL model and the best spherical dynamical model.