<figure><img src="image_14.png" /><figcaption>**Figure 11.** Maps of the relative differences between the projected gas mass distribution of the model that include only the gas and both the gas and the fundamental plane with the true distribution. The earlier and the latter are presented in the left- and right-panel panels, respectively.</figcaption></figure>

where the gas distribution is locally modifying the main halo ellipticity. However, the gap is reasonable as it is below 4 per cent, which indicates that models from previous studies are still providing a fairly accurate reproduction of the total mass. However, this could have an impact on analyses relying on an accurate estimation of the magnification, as a few per cent offset in mass can lead to significant differences in this quantity. It is also expected that in the case of a more perturbed cluster where the gas is not following the total mass, this underestimation would be greater thus biasing the resulting analysis.

## 7.1 Gas distribution reproduction

Fig. 11 shows the relative differences between the fitted gas distribution and the input one. As one can see, our method recovers it with excellent precision (≤2 per cent) as the discrepancy is mostly ≤1 per cent for the two models. Regarding the X-ray intrinsic error, our models are able to recover the input values (0.538 count) with good agreement (i.e. included in the "1σ"CI). Indeed, the model without the fundamental plane and the one with are presenting X-ray errors of $0.528_{-0.023}^{+0.022}$ and $0.529_{-0.024}^{+0.024}$ counts, respectively.

Regarding our goodness-of-fit procedure detailed in Section 6.1, the models with and without the fundamental plane show likelihoods for the best-fitting model that are inside the "1σ"CI and the "3σ"CI, respectively, of the expected likelihood distribution according to these count models. To assess with more confidence the previous results, we optimize 10 gas-only models with new realizations of X-ray constraints based on the same initial count model. The best-fitting models are showing 5 overfitting models, that are inside the "1σ"CI, and 5 underfitting models split between the limits of the "1σ"CI and the "3σ"CI. Hence, our two mock results made with a

joint fit are consistent with these gas-only models. As we are here in an idealized case, assuming a larger CI of "5σ" to define if a model reaches a good fit is a fair hypothesis according to the results on a mock cluster. We can also add that our goodness-of-fit score is biased by the non-consideration of the whole model posterior distribution for computation cost mitigation. We only use the best-fitting gas model to sample the expected likelihood distribution as it needs more than $10^5$ sample to provide reliable results on one model only, making percentile analysis amongst the whole distribution costly in terms of memory. Thus, we can expect that using the whole posterior distribution could provide different goodness-of-fit score than currently.

## 7.2 Recovering of the cluster member masses

To assess the efficiency of each model to estimate the mass of cluster members, we look at the recovered $r_{cut}$, $\sigma_0$ and their total mass $M$. For each model in the posterior distribution, we build a distribution from the ratio between the considered parameter and its true value amongst all cluster members. We compute the density associated with the previous distribution with the kernel density estimator implemented in Kumar et al. (2019) with a fixed bandwidth of 0.05. From the results on each model, we obtain the posterior distribution of this density. The plain lines and shaded areas in plots in Fig. 12 show the median and the "1σ" CI of the density distribution for each parameter. As one can see, when using fundamental plane, we recover the $r_{cut}$ and $\sigma_0$ more accurately than for the two other models, that rely on the Faber & Jackson solely. However, there are still offsets of ~20 per cent for the $r_{cut}$ when it is down to only a few per cent for the $\sigma_0$. In particular, this gap in the two parameters ($r_{cut}$ and $\sigma_0$) can be explained by the goodness of fit of the scaling factor, $v$, between the light and the mass distribution. Indeed, the model with gas distribution and the one without have $v$ values of $1.87_{-0.34}^{+0.35}$ and $2.45_{-0.36}^{+0.38}$ for an input one of 2.14, respectively. Hence, the biases on the posterior distribution are really similar to the ones on $r_{cut}$ which is expected due to the fact that $r_{cut} \propto v$ (see Appendix C) but also shows that the effect of $r_{core}^*/r_{cut}^*$ have a minor influence on the final estimation.

Regarding the total mass, $M$, shown on the right plot in Fig. 12, the differences when using of the fundamental plane with respect to the other models are less striking. Indeed, the results from the two models using the plane present similar biases than the models relying on the simpler scaling law. The main differences are in the precision of the estimation, which is lower for the two models using the plane, encompassing with a slightly higher probability density the areas around 1. As the $r_{core}$ estimation can not explain these biases (i.e.

<figure><img src="image_15.png" /><figcaption>**Figure 12.** Density distribution of the ratios between estimated dPIE parameters in the different models and their true values. Densities have been estimated for each model in the considered posterior distribution with the kernel density estimator implemented in Kumar et al. (2019) and a fixed bandwidth of 0.05. The shaded areas and the plain lines represent the "1σ"CI, and the median of the distribution of density. From left to right, the parameters considered are the cut radius, $r_{cut}$, the central velocity dispersion, $\sigma_0$, and the total mass, $M$.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Figure 13.** Density distribution of the ratios between estimated dPIE parameters of each cluster member sub-groups and their true values. The density has been estimated on the concatenated distribution of ratio amongst all posterior models. The kernel density estimator, as well as the order of the parameters, are the same as in Fig. 12.</figcaption></figure>

models with the fundamental plane are overestimating it), the only explanation is that good estimations of $r_{cut}$ or $\sigma_0$ are not obtained for the same model in the posterior frequently. Hence, estimating each parameter independently is more reliable than combining them, according to our results.

The method presented in this work used three different modelling schemes for the cluster members. We assess their efficiency in recovering the associated profile parameters in Fig. 13 with the model using the full method. Again, we build the distribution of the ratios presented previously for the three groups, but we have to adapt the presentation scheme as the distribution for one model of the $r_{cut}$ or $\sigma_0$ ratio leads to a Dirac distribution for the fundamental plane schemes. It is due to the proportional relations between the true values and the fitted ones for these parameters. Hence, we concatenate the distribution of each ratio from all models in the posterior, and we then estimate their density through the same method. Similarly to before, the model that use the fundamental plane provides a better estimation of $r_{cut}$ and $\sigma_0$. One can see that we have the same bias for both schemes on $r_{cut}$ estimation dominated by the $v$ estimation. Regarding $\sigma_0$ estimation, the two schemes show two opposite biases, with an underestimation when using $R_e$ as input and an overestimation when it is $\sigma_0$. In fact, the overestimation is easily explained from the behaviour of the projection factor $c_p$ (see Appendix C) that scales up or down with $v$ at fixed $r_{core}^*/r_{cut}^*$. Hence, as $\sigma_0 = \sigma_e/c_p$, if $v$ is lower than the true value, we will overestimate $\sigma_0$. It is more difficult to judge the pattern of the other scheme as also $v$ influences $\sigma_{e,FP}$. It scales exponentially with it according to the fundamental plane equation. Thus, we can deduce that the decrease in $\sigma_{e,FP}$ with $v$ is faster than the $c_p$ ones, leading to the global underestimation observed.

Regarding the total mass, $M$, estimation, we can see that the fundamental plane scheme with $\sigma_e$ as input is the best, as expected from our pre-modelling analysis in Section 3.2.3. It is followed by the Faber & Jackson scaling relation and, finally, the second fundamental plane. Interestingly, the latter scheme is the only one not to overestimate $\sigma_0$. Thus, it cannot compensate for $r_{cut}$ underestimation as the two others. Both fundamental plane schemes are not equally stable according to a random realization of constraints, as in their case, we have not modified the input observational parameters besides the multiple images. As shown by the previous analysis, the critical parameter in the wrong estimation pattern is $v$. Hence, physically motivated priors could solve this problem as it is linked to the mass-to-light ratio in our assumptions.

In comparison with the reference model that does not have a calibration of the cluster member scaling relation, we can see here that the Faber & Jackson scheme has a density centred on 1 for $M$ estimation. Hence, as seen in Fig. 8, the calibration helps to have a homogeneity between the different relations.

We assume that representing the real mass distribution of cluster members with the fundamental plane of elliptical galaxies is a fair hypothesis. Hence, the previous results show that incorporating knowledge of members kinematics as well as using a more complex relation than the Faber&Jackson scaling one allow us to recover more accurately and robustly the parameters of the cluster members mass profiles. This method effectively reduces the biases in the parameter estimation, as well as it improves the consistency of the mass model with existing observational information.

# 8 DISCUSSION

Our new scheme for the modelling of cluster member masses outlined in Section 4.3 makes several assumptions that we discuss in Section 8.1 in regard to the obtained mass models. Thanks to the addition of X-ray constraints, we obtain some important insights into the cluster gas thermodynamics. Finally, thanks to this new method, we discuss the hypothesis that AS1063 has undergone a merging event in Section 8.2, and its gas fraction mapping in Section 8.3.

## 8.1 Cluster member mass hypotheses

As detailed in Appendix C, our 3D model of the cluster member mass and light distribution assumes a constant mass-to-light ratio that has the same value as the scale factor, $v$, between the light and mass profiles. In particular, these assumptions allow us to link $R_e$ to $r_{cut}$, and to compute $c_p$. Still, there is no input information in the modelling neither on the total luminosity, $L$, nor the cut radius of the light distribution, $r_{cut, light}$. As the previous quantities are available, we address the consistency of our hypotheses in Fig. 14 presenting the dPIE total mass as a function of the cluster member luminosity in the left-hand panel, and the scaling between $r_{cut}$ and $r_{cut, light}$ in the right one. The assumed relations between these quantities and their estimations from our joint X-rays and lensing optimization are presented in orange, where the plain line and the shaded areas represent the median and the "1σ" CI of the posterior distribution, respectively. As the assumed relations are simplifications of the known reality (e.g. a constant mass-to-light ratio), we sample the more realistic ones, $M = \beta L^{1+\delta}$, and, $r_{cut} = \zeta r_{cut, light} + \eta$, on the data plotted. The obtained posterior distribution is represented in grey with the same scheme as previously. The parameter statistics for both relations are given in Table 3.

Our model provides "2σ" CI for $v$ which overlap with the same CI for $\beta$ and $\zeta$. We note that our result lies between the estimate of $\beta$ and $\zeta$, indicating that our simplified model will need to be more complex to allow for such discrepancy. Indeed, sampling on both $\beta$ and $\zeta$ shows two disjoints "5σ" CI. As expected, $\delta > 0$
<figure><img src="image_16.png" /><figcaption>**Figure 13.** Density distribution of the ratios between estimated dPIE parameters of each cluster member sub-groups and their true values. The density has been estimated on the concatenated distribution of ratio amongst all posterior models. The kernel density estimator, as well as the order of the parameters, are the same as in Fig. 12.</figcaption></figure>

models with the fundamental plane are overestimating it), the only explanation is that good estimations of $r_{cut}$ or $\sigma_0$ are not obtained for the same model in the posterior frequently. Hence, estimating each parameter independently is more reliable than combining them, according to our results.

The method presented in this work used three different modelling schemes for the cluster members. We assess their efficiency in recovering the associated profile parameters in Fig. 13 with the model using the full method. Again, we build the distribution of the ratios presented previously for the three groups, but we have to adapt the presentation scheme as the distribution for one model of the $r_{cut}$ or $\sigma_0$ ratio leads to a Dirac distribution for the fundamental plane schemes. It is due to the proportional relations between the true values and the fitted ones for these parameters. Hence, we concatenate the distribution of each ratio from all models in the posterior, and we then estimate their density through the same method. Similarly to before, the model that use the fundamental plane provides a better estimation of $r_{cut}$ and $\sigma_0$. One can see that we have the same bias for both schemes on $r_{cut}$ estimation dominated by the $v$ estimation. Regarding $\sigma_0$ estimation, the two schemes show two opposite biases, with an underestimation when using $R_e$ as input and an overestimation when it is $\sigma_0$. In fact, the overestimation is easily explained from the behaviour of the projection factor $c_p$ (see Appendix C) that scales up or down with $v$ at fixed $r_{core}^*/r_{cut}^*$. Hence, as $\sigma_0 = \sigma_e/c_p$, if $v$ is lower than the true value, we will overestimate $\sigma_0$. It is more difficult to judge the pattern of the other scheme as also $v$ influences $\sigma_{e,FP}$. It scales exponentially with it according to the fundamental plane equation. Thus, we can deduce that the decrease in $\sigma_{e,FP}$ with $v$ is faster than the $c_p$ ones, leading to the global underestimation observed.

Regarding the total mass, $M$, estimation, we can see that the fundamental plane scheme with $\sigma_e$ as input is the best, as expected from our pre-modelling analysis in Section 3.2.3. It is followed by the Faber & Jackson scaling relation and, finally, the second fundamental plane. Interestingly, the latter scheme is the only one not to overestimate $\sigma_0$. Thus, it cannot compensate for $r_{cut}$ underestimation as the two others. Both fundamental plane schemes are not equally stable according to a random realization of constraints, as in their case, we have not modified the input observational parameters besides the multiple images. As shown by the previous analysis, the critical parameter in the wrong estimation pattern is $v$. Hence, physically motivated priors could solve this problem as it is linked to the mass-to-light ratio in our assumptions.

In comparison with the reference model that does not have a calibration of the cluster member scaling relation, we can see here that the Faber & Jackson scheme has a density centred on 1 for $M$ estimation. Hence, as seen in Fig. 8, the calibration helps to have a homogeneity between the different relations.

We assume that representing the real mass distribution of cluster members with the fundamental plane of elliptical galaxies is a fair hypothesis. Hence, the previous results show that incorporating knowledge of members kinematics as well as using a more complex relation than the Faber&Jackson scaling one allow us to recover more accurately and robustly the parameters of the cluster members mass profiles. This method effectively reduces the biases in the parameter estimation, as well as it improves the consistency of the mass model with existing observational information.

# 8 DISCUSSION

Our new scheme for the modelling of cluster member masses outlined in Section 4.3 makes several assumptions that we discuss in Section 8.1 in regard to the obtained mass models. Thanks to the addition of X-ray constraints, we obtain some important insights into the cluster gas thermodynamics. Finally, thanks to this new method, we discuss the hypothesis that AS1063 has undergone a merging event in Section 8.2, and its gas fraction mapping in Section 8.3.

## 8.1 Cluster member mass hypotheses

As detailed in Appendix C, our 3D model of the cluster member mass and light distribution assumes a constant mass-to-light ratio that has the same value as the scale factor, $v$, between the light and mass profiles. In particular, these assumptions allow us to link $R_e$ to $r_{cut}$, and to compute $c_p$. Still, there is no input information in the modelling neither on the total luminosity, $L$, nor the cut radius of the light distribution, $r_{cut, light}$. As the previous quantities are available, we address the consistency of our hypotheses in Fig. 14 presenting the dPIE total mass as a function of the cluster member luminosity in the left-hand panel, and the scaling between $r_{cut}$ and $r_{cut, light}$ in the right one. The assumed relations between these quantities and their estimations from our joint X-rays and lensing optimization are presented in orange, where the plain line and the shaded areas represent the median and the "1σ" CI of the posterior distribution, respectively. As the assumed relations are simplifications of the known reality (e.g. a constant mass-to-light ratio), we sample the more realistic ones, $M = \beta L^{1+\delta}$, and, $r_{cut} = \zeta r_{cut, light} + \eta$, on the data plotted. The obtained posterior distribution is represented in grey with the same scheme as previously. The parameter statistics for both relations are given in Table 3.

Our model provides "2σ" CI for $v$ which overlap with the same CI for $\beta$ and $\zeta$. We note that our result lies between the estimate of $\beta$ and $\zeta$, indicating that our simplified model will need to be more complex to allow for such discrepancy. Indeed, sampling on both $\beta$ and $\zeta$ shows two disjoints "5σ" CI. As expected, $\delta > 0$
<figure><img src="image_17.png" /><figcaption>**Figure 14.** Left-hand panel: Plot of the cluster member dPIE total mass as a function of their luminosity. Error bars represent the standard deviation of each quantity. Right-panel panel: Plot of the cluster member dPIE $r_{cut}$ as a function of the $r_{cut, light}$ estimated on their Sersic light profiles. For both panels, the grey area represents the actual relations fitted on the presented data, and the orange one represents the relations assumed in our modelling hypothesis, and its estimation through the joined X-rays and lensing fit.</figcaption></figure>

<table><caption>**Table 3.** Parameters of the more realistic relations linking *M* and *L* and also $r_{cut}$ and $r_{cut, light}$.</caption><thead><tr><th>Parameters</th><th>Median<sup>84</sup> per cent limit<br>16 per cent limit</th></tr></thead><tbody><tr><td>β</td><td>2.43<sup>+0.25</sup><sub>-0.21</sub></td></tr><tr><td>δ</td><td>0.074<sup>+0.073</sup><sub>-0.069</sub></td></tr><tr><td>ζ</td><td>1.24<sup>+0.11</sup><sub>-0.12</sub></td></tr><tr><td>η</td><td>3.85<sup>+0.26</sup><sub>-0.25</sub></td></tr></tbody></table>

is favoured, but our results prefer way lower values than the 0.35 estimated by Bolton et al. (2008) and Koopmans et al. (2009) from galaxy-galaxy lensing analysis; this value is actually the upper bound of the "5σ" CI for $\delta$. It is possible that $\delta$ measurements are biased by the galaxy mass estimation, which has been performed assuming the homology resulting in the observed differences with previous works. As seen on the right-panel panel in Fig. 14, the relation between $r_{cut}$ and $r_{cut, light}$ seems to present a non-zero offset with strong confidence indicating a turn for small $r_{cut, light}$. It is difficult to assess the meaning of this relation as the use of a dPIE for the light distribution is already a proxy of the real light distribution that simplifies the $c_p$ computation. Indeed, a dPIE can approximate a Sérsic profile (Suyu et al. 2014) partially, but it cannot reproduce its central and asymptotic behaviours.

Nevertheless, $\nu$ is neither providing a perfect representation nor a robust estimation of the previous relations. It is, however, a fair proxy to both of them as the measured relations are mainly taken into account in the wide $\nu$ posterior, and the modelling hypotheses are thus consistent with the actual results. Considering that cluster members are only accounting for a small percentage of the total mass and any increase in the model complexity will worsen the already high computational cost, this implementation of the fundamental plane provides more benefits than drawbacks. We also do not intend to propose a final description as it is only the second iteration of strong lensing mass modelling that includes the fundamental planes, with G22 being the first one. Hence, new schemes that would model the previous relations more closely to reality will allow us to improve the robustness of cluster members mass estimation.

Some of the comments addressed here could also be relevant in the case of G22-like modelling as our scheme using $\sigma_{0,FP}$ is quite similar to theirs. Indeed, we also have $R_e$ proportional to $r_{cut}$ through admittedly a different parametrization, but a translation is

possible. The other differences are the association of the light and mass distributions, and the computation of $c_p$ if taken into account. In any case, if a value of $\nu$ or $c_p$ is assumed (e.g. $\nu = 1$ for B19), a parameter transformation between approaches can be put in place.

## 8.2 Merging events

As previously mentioned in Section 6.2, it is suspected that AS1063 is undergoing a merging event. It was first proposed by Gómez et al. (2012) in the form of a major merger (1:4 mass ratio) happening perpendicular to the LOS with arguments based on the kinematics of cluster members, a high intra-cluster gas temperature and the asymmetry of the X-ray emission. Since then, further studies have brought up new arguments in favour of this scenario. de Oliveira, Jiménez-Teja & Dupke (2022) analysed the intra-cluster light distribution which shows evidence for a cluster merger. Xie et al. (2020) and Rahaman et al. (2021) reported a giant radio halo and a gradient in the temperature distribution with a bow shape, respectively. Both are indicators of a merging event. The spectral analysis of the cluster members made by Mercurio et al. (2021) revealed a bimodal redshift distribution amongst the 1200 objects measured. Their results on the kinematics properties of the galaxies near the NE halo are also in favour of a different origin than the rest of the cluster members. In that context, we want to show in this section the ability of our method products and by-products to combine pieces of evidence from both lensing and X-ray analyses to help confirm such hypotheses.

Our modelling of the X-ray surface brightness is only able to fit details at the cluster scale; thus, signs of gas sloshing from an infalling object can be visible in the map of the residual counts. Indeed, the results presented in Section 6.2 show a shallow pattern of overestimation and underestimation of the observed counts. Fig. 15 presents the residual map smoothed with a Gaussian kernel of 2 pixel size, as well as the count map smoothed with an adaptive Gaussian gradient method<fnref n="10" /> (Sanders et al. 2022). The earlier shows more clearly the pattern previously mentioned, which could be the result of the initial object containing the NE halo that has fallen into AS1063 following the SW to NE direction. The axis of this trajectory is the

<footnote n="10">¹⁰https://github.com/jeremysanders/ggm</footnote>
<figure><img src="image_17.png" /><figcaption>**Figure 14.** Left-hand panel: Plot of the cluster member dPIE total mass as a function of their luminosity. Error bars represent the standard deviation of each quantity. Right-panel panel: Plot of the cluster member dPIE $r_{cut}$ as a function of the $r_{cut, light}$ estimated on their Sersic light profiles. For both panels, the grey area represents the actual relations fitted on the presented data, and the orange one represents the relations assumed in our modelling hypothesis, and its estimation through the joined X-rays and lensing fit.</figcaption></figure>

<table><caption>**Table 3.** Parameters of the more realistic relations linking *M* and *L* and also $r_{cut}$ and $r_{cut, light}$.</caption><thead><tr><th>Parameters</th><th>Median<sup>84</sup> per cent limit<br>16 per cent limit</th></tr></thead><tbody><tr><td>β</td><td>2.43<sup>+0.25</sup><sub>-0.21</sub></td></tr><tr><td>δ</td><td>0.074<sup>+0.073</sup><sub>-0.069</sub></td></tr><tr><td>ζ</td><td>1.24<sup>+0.11</sup><sub>-0.12</sub></td></tr><tr><td>η</td><td>3.85<sup>+0.26</sup><sub>-0.25</sub></td></tr></tbody></table>

is favoured, but our results prefer way lower values than the 0.35 estimated by Bolton et al. (2008) and Koopmans et al. (2009) from galaxy-galaxy lensing analysis; this value is actually the upper bound of the "5σ" CI for $\delta$. It is possible that $\delta$ measurements are biased by the galaxy mass estimation, which has been performed assuming the homology resulting in the observed differences with previous works. As seen on the right-panel panel in Fig. 14, the relation between $r_{cut}$ and $r_{cut, light}$ seems to present a non-zero offset with strong confidence indicating a turn for small $r_{cut, light}$. It is difficult to assess the meaning of this relation as the use of a dPIE for the light distribution is already a proxy of the real light distribution that simplifies the $c_p$ computation. Indeed, a dPIE can approximate a Sérsic profile (Suyu et al. 2014) partially, but it cannot reproduce its central and asymptotic behaviours.

Nevertheless, $\nu$ is neither providing a perfect representation nor a robust estimation of the previous relations. It is, however, a fair proxy to both of them as the measured relations are mainly taken into account in the wide $\nu$ posterior, and the modelling hypotheses are thus consistent with the actual results. Considering that cluster members are only accounting for a small percentage of the total mass and any increase in the model complexity will worsen the already high computational cost, this implementation of the fundamental plane provides more benefits than drawbacks. We also do not intend to propose a final description as it is only the second iteration of strong lensing mass modelling that includes the fundamental planes, with G22 being the first one. Hence, new schemes that would model the previous relations more closely to reality will allow us to improve the robustness of cluster members mass estimation.

Some of the comments addressed here could also be relevant in the case of G22-like modelling as our scheme using $\sigma_{0,FP}$ is quite similar to theirs. Indeed, we also have $R_e$ proportional to $r_{cut}$ through admittedly a different parametrization, but a translation is

possible. The other differences are the association of the light and mass distributions, and the computation of $c_p$ if taken into account. In any case, if a value of $\nu$ or $c_p$ is assumed (e.g. $\nu = 1$ for B19), a parameter transformation between approaches can be put in place.

## 8.2 Merging events

As previously mentioned in Section 6.2, it is suspected that AS1063 is undergoing a merging event. It was first proposed by Gómez et al. (2012) in the form of a major merger (1:4 mass ratio) happening perpendicular to the LOS with arguments based on the kinematics of cluster members, a high intra-cluster gas temperature and the asymmetry of the X-ray emission. Since then, further studies have brought up new arguments in favour of this scenario. de Oliveira, Jiménez-Teja & Dupke (2022) analysed the intra-cluster light distribution which shows evidence for a cluster merger. Xie et al. (2020) and Rahaman et al. (2021) reported a giant radio halo and a gradient in the temperature distribution with a bow shape, respectively. Both are indicators of a merging event. The spectral analysis of the cluster members made by Mercurio et al. (2021) revealed a bimodal redshift distribution amongst the 1200 objects measured. Their results on the kinematics properties of the galaxies near the NE halo are also in favour of a different origin than the rest of the cluster members. In that context, we want to show in this section the ability of our method products and by-products to combine pieces of evidence from both lensing and X-ray analyses to help confirm such hypotheses.

Our modelling of the X-ray surface brightness is only able to fit details at the cluster scale; thus, signs of gas sloshing from an infalling object can be visible in the map of the residual counts. Indeed, the results presented in Section 6.2 show a shallow pattern of overestimation and underestimation of the observed counts. Fig. 15 presents the residual map smoothed with a Gaussian kernel of 2 pixel size, as well as the count map smoothed with an adaptive Gaussian gradient method<fnref n="10" /> (Sanders et al. 2022). The earlier shows more clearly the pattern previously mentioned, which could be the result of the initial object containing the NE halo that has fallen into AS1063 following the SW to NE direction. The axis of this trajectory is the

<footnote n="10">¹⁰https://github.com/jeremysanders/ggm</footnote>
<figure><img src="image_18.png" /><figcaption>**Figure 15.** Left-hand panel: Map of the smoothed residual (Gaussian kernel of 2 pixel size) between the observation and the best-fitting model. Blue and red contours highlight the region of negative and positive excess, respectively. These regions are used to probe a possible gas sloshing quantitatively with ICM properties. Right-panel panel: Adaptively smoothed count map between 0.5 and 7.0 keV with a Gaussian gradient magnitude method (Sanders et al. 2022).</figcaption></figure>

same as indicated in previous works (Gómez et al. 2012; Rahaman et al. 2021). The latter map and its ‘U’ shaped pattern are an indication of ram pressure stripping, in particular, that an object which seems to match with the BCG position is moving in the Intra-Cluster Medium (ICM). If we invert the reference frame, it can be interpreted as the ICM moving around the BCG as a consequence of an infalling object passing in the cluster.

In addition to the information related to the X-ray surface brightness and its fit, we have in hand maps of the ICM properties that are direct products of our method or which can easily be deduced from them. Indeed, to define our plasma emission model, we need a map of the cluster temperature that is produced along with the Norm of the APEC model. From these two quantities, we deduce a pseudo-entropy, a pseudo-pressure and a deprojected gas density. The three latter quantities, as well as the temperature for a SN threshold of 6, are shown in Fig. 16, with the ‘spiral’ pattern highlighted with white contours. Globally, these maps show a unimodal distribution with an asymmetry towards the NE, but they are clearly in disagreement of two distinguishable structures as seen in the case of a major merger with a close mass ratio such as the Bullet cluster (Million & Allen 2009). Indeed, there only seems to be one low entropy area associated with the AS1063 centre. Thus there is no trace of a remnant low entropy gas of a second cluster core, and the pseudo-pressure is also not showing a complex structure, as expected in the case of a major merger event. However, AS1063 does not look totally relaxed, as the temperature map presents some gradients in agreement with a perturbed state. From this analysis and the mass ratio of 300 between the two dPIEs haloes representing the main mass component and the NE one, the remaining plausible scenario is the fall of a much less massive object, such as a galaxy group, into AS1063. Such events will produce some sloshing of the gas, as previously proposed, which can be constrained quantitatively from the ICM properties in the under/over estimation area of the X-ray surface brightness (Ueda et al. 2021). In particular, it is expected that the entropy and temperature are higher in the negative excess region than in the positive one, with a difference higher for the first quantity. It is the opposite for the density, and the pressure should be unchanged. Due to maps binning being on a scale similar to the considered area, they are not the best tools to measure the differences between these regions.

To obtain a quantitative answer on the presence of gas sloshing, we extract two spectra based on the blue (negative excess) and red (positive excess) contours shown in Fig. 15 and fit them with the same method as described in Section 3.1. We present in Fig. 17 the temperature, entropy, pressure, and density measurements with a cosmetic normalization to plot them next to each other. As one can see, it shows the expected pattern for gas sloshing. If we look more precisely in detail at the posterior distribution of all measurements, the "5σ" CI of the entropy and density agree with that phenomenon, and it is "3σ" CI for the temperature. Hence, we can assess that we do detect the existence of gas sloshing in AS1063. Such a process is in agreement with a minor merger event between the cluster and a much smaller infalling object like a galaxy group. Our method and the tools we developed are able to give us important insights into the AS1063 dynamical state without requiring a disjoint analysis, thus showing an example of the possible study that will benefit from this kind of modelling.

## 8.3 Hydrostatic equilibrium and gas fraction

As we model with separate haloes, the intra-cluster gas and the rest of the matter, we do not assume the hydrostatic equilibrium. Thus, our method is well suited to analyse the hydrostatic bias by comparing its results with another method on the same observables. To perform this comparison with the associated hydrostatic mass of AS1063, we fit a Navarro–Frenk–White profile (Navarro, Frenk & White 1996) on the X-ray observations with the HYDROMASS<fnref n="11" /> package based on Ettori et al. (2010) method. We use the same count, background, and exposure maps as well as the plasma emission model, but we use a wider FoV than the one used for the LENSTOOL optimization. Indeed, we fit the X-ray surface brightness up to almost 1 Mpc as we are less dominated by the background in the circular binning. To obtain a temperature profile, we use the same fitting method presented in Section 3.1, but instead of using a Voronoï tessellation, we make circular binning with 3000 counts each in the 4–7 keV band.

Fig. 18 presents the integrated projected mass profiles and the deprojected gas fraction of the hydrostatic reconstruction, B19 and

<footnote n="11">¹¹https://hydromass.readthedocs.io/en/latest/intro.html</footnote>
<figure><img src="image_18.png" /><figcaption>**Figure 15.** Left-hand panel: Map of the smoothed residual (Gaussian kernel of 2 pixel size) between the observation and the best-fitting model. Blue and red contours highlight the region of negative and positive excess, respectively. These regions are used to probe a possible gas sloshing quantitatively with ICM properties. Right-panel panel: Adaptively smoothed count map between 0.5 and 7.0 keV with a Gaussian gradient magnitude method (Sanders et al. 2022).</figcaption></figure>

same as indicated in previous works (Gómez et al. 2012; Rahaman et al. 2021). The latter map and its ‘U’ shaped pattern are an indication of ram pressure stripping, in particular, that an object which seems to match with the BCG position is moving in the Intra-Cluster Medium (ICM). If we invert the reference frame, it can be interpreted as the ICM moving around the BCG as a consequence of an infalling object passing in the cluster.

In addition to the information related to the X-ray surface brightness and its fit, we have in hand maps of the ICM properties that are direct products of our method or which can easily be deduced from them. Indeed, to define our plasma emission model, we need a map of the cluster temperature that is produced along with the Norm of the APEC model. From these two quantities, we deduce a pseudo-entropy, a pseudo-pressure and a deprojected gas density. The three latter quantities, as well as the temperature for a SN threshold of 6, are shown in Fig. 16, with the ‘spiral’ pattern highlighted with white contours. Globally, these maps show a unimodal distribution with an asymmetry towards the NE, but they are clearly in disagreement of two distinguishable structures as seen in the case of a major merger with a close mass ratio such as the Bullet cluster (Million & Allen 2009). Indeed, there only seems to be one low entropy area associated with the AS1063 centre. Thus there is no trace of a remnant low entropy gas of a second cluster core, and the pseudo-pressure is also not showing a complex structure, as expected in the case of a major merger event. However, AS1063 does not look totally relaxed, as the temperature map presents some gradients in agreement with a perturbed state. From this analysis and the mass ratio of 300 between the two dPIEs haloes representing the main mass component and the NE one, the remaining plausible scenario is the fall of a much less massive object, such as a galaxy group, into AS1063. Such events will produce some sloshing of the gas, as previously proposed, which can be constrained quantitatively from the ICM properties in the under/over estimation area of the X-ray surface brightness (Ueda et al. 2021). In particular, it is expected that the entropy and temperature are higher in the negative excess region than in the positive one, with a difference higher for the first quantity. It is the opposite for the density, and the pressure should be unchanged. Due to maps binning being on a scale similar to the considered area, they are not the best tools to measure the differences between these regions.

To obtain a quantitative answer on the presence of gas sloshing, we extract two spectra based on the blue (negative excess) and red (positive excess) contours shown in Fig. 15 and fit them with the same method as described in Section 3.1. We present in Fig. 17 the temperature, entropy, pressure, and density measurements with a cosmetic normalization to plot them next to each other. As one can see, it shows the expected pattern for gas sloshing. If we look more precisely in detail at the posterior distribution of all measurements, the "5σ" CI of the entropy and density agree with that phenomenon, and it is "3σ" CI for the temperature. Hence, we can assess that we do detect the existence of gas sloshing in AS1063. Such a process is in agreement with a minor merger event between the cluster and a much smaller infalling object like a galaxy group. Our method and the tools we developed are able to give us important insights into the AS1063 dynamical state without requiring a disjoint analysis, thus showing an example of the possible study that will benefit from this kind of modelling.

## 8.3 Hydrostatic equilibrium and gas fraction

As we model with separate haloes, the intra-cluster gas and the rest of the matter, we do not assume the hydrostatic equilibrium. Thus, our method is well suited to analyse the hydrostatic bias by comparing its results with another method on the same observables. To perform this comparison with the associated hydrostatic mass of AS1063, we fit a Navarro–Frenk–White profile (Navarro, Frenk & White 1996) on the X-ray observations with the HYDROMASS<fnref n="11" /> package based on Ettori et al. (2010) method. We use the same count, background, and exposure maps as well as the plasma emission model, but we use a wider FoV than the one used for the LENSTOOL optimization. Indeed, we fit the X-ray surface brightness up to almost 1 Mpc as we are less dominated by the background in the circular binning. To obtain a temperature profile, we use the same fitting method presented in Section 3.1, but instead of using a Voronoï tessellation, we make circular binning with 3000 counts each in the 4–7 keV band.

Fig. 18 presents the integrated projected mass profiles and the deprojected gas fraction of the hydrostatic reconstruction, B19 and

<footnote n="11">¹¹https://hydromass.readthedocs.io/en/latest/intro.html</footnote>
<figure><img src="image_19.png" /><figcaption>**Figure 16.** Maps of the ICM properties obtained from the X-ray emission, with the temperature (top left), pseudo-entropy (top right), pseudo-pressure (bottom left), and pseudo-density (bottom right). The white or black contour highlights the 'spiral' pattern seen in the X-ray fit residual.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 17.** Comparison of the ICM properties between the positive and negative excess regions, which are presented in Fig. 15 with the red and blue contour, respectively. The error bars represent the "1σ" CI of the posterior distribution.</figcaption></figure>

our models. Plain lines show the median profiles amongst 4000 models from the posterior distribution when the shaded areas highlight "3σ" CI of the distribution. As one can see, the hydrostatic model is in quite good agreement with both lensing models, in particular beyond ~60 kpc, where they are both included in the "3σ" CI of the hydrostatic mass. Below this radius, lensing mass profiles are almost the same and, on average less than 15 per cent away from each other. The closest multiple image from the profile centre (i.e. BCG) is at a radius of 75 kpc, thus, the lensing reconstructions are more determined by modelling assumptions than the actual constraints in the central area. It is then hard to assess if this discrepancy is meaningful. If we only consider the profiles beyond this radius, the hydrostatic mass bias is then inferior to 1 per cent, which is expected for a slightly perturbed cluster (Ansarifard et al. 2020). This shows that AS1063 is close to being in hydrostatic equilibrium, which would not be the case if it were undergoing a major merger event. It is consistent with the scenario of a minor merger presented previously, and already discussed by Sartoris et al. (2020). Regarding the gas fraction, our model provides results similar to the hydrostatic equilibrium beyond ~220 kpc as both "3σ" CI overlapped. Overall, the average of the absolute differences between both is of 10 per cent
<figure><img src="image_19.png" /><figcaption>**Figure 16.** Maps of the ICM properties obtained from the X-ray emission, with the temperature (top left), pseudo-entropy (top right), pseudo-pressure (bottom left), and pseudo-density (bottom right). The white or black contour highlights the 'spiral' pattern seen in the X-ray fit residual.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 17.** Comparison of the ICM properties between the positive and negative excess regions, which are presented in Fig. 15 with the red and blue contour, respectively. The error bars represent the "1σ" CI of the posterior distribution.</figcaption></figure>

our models. Plain lines show the median profiles amongst 4000 models from the posterior distribution when the shaded areas highlight "3σ" CI of the distribution. As one can see, the hydrostatic model is in quite good agreement with both lensing models, in particular beyond ~60 kpc, where they are both included in the "3σ" CI of the hydrostatic mass. Below this radius, lensing mass profiles are almost the same and, on average less than 15 per cent away from each other. The closest multiple image from the profile centre (i.e. BCG) is at a radius of 75 kpc, thus, the lensing reconstructions are more determined by modelling assumptions than the actual constraints in the central area. It is then hard to assess if this discrepancy is meaningful. If we only consider the profiles beyond this radius, the hydrostatic mass bias is then inferior to 1 per cent, which is expected for a slightly perturbed cluster (Ansarifard et al. 2020). This shows that AS1063 is close to being in hydrostatic equilibrium, which would not be the case if it were undergoing a major merger event. It is consistent with the scenario of a minor merger presented previously, and already discussed by Sartoris et al. (2020). Regarding the gas fraction, our model provides results similar to the hydrostatic equilibrium beyond ~220 kpc as both "3σ" CI overlapped. Overall, the average of the absolute differences between both is of 10 per cent
<figure><img src="image_21.png" /><figcaption>**Figure 18.** Integrated profiles of the 2D total mass density (left-hand panel), and gas fraction (right-panel panel) integrated in spheres of increasing radii. For both panels, plain lines and shaded regions represent the median and the "3σ" CI of the profiles distribution amongst the 4000 models of the original posterior distributions.</figcaption></figure>

below 600 kpc, mainly dominated by the area of disagreement close to the BCG, which shows a gas fraction 17 per cent lower than ours. We note that this X-ray fit provides a different result on the gas mass than the multiscale deprojection presented in Section 6.3. In that case, our model presents a slightly lower deprojected gas mass than the NFW fit. Hence, both the total mass discrepancy and the previous one lead to this disagreement on the gas fraction, even if both of them are in quite close agreement. There is a significant difference with the B19 model, which is a direct consequence of their gas mass estimation which is higher than the two others, as seen in Fig. 9.

We present the 2D distribution of the projected gas fraction in Fig. 19 for B19 and our best-fitting model. If, globally, the observed gas fraction is similar with a median of the relative difference of 11 per cent between both maps, there are some differences in the morphology. Indeed, the ellipticity discrepancies between the DM and the gas can be clearly seen in our model when it is less evident in B19. In particular, the lobes of a lower or higher gas fraction have different shapes between both models on areas that are mainly outside the constrained one. It can be explained by the differences in the modelling and the assumptions on the parameters, such as dPIEs' cut radius, $r_{cut}$. In addition, more complexity can be observed in our 2D distribution that has been added by the B-spline surfaces in the area mentioned in Section 6.3. We note that the 2D analysis of the gas fraction is possible with our modelling method without the need for any extra-tools, thus allowing simple access to the position and shape of the different cluster components.

In recent work, Cerini, Cappelluti & Natarajan (2023), derived and demonstrated the utility of power-spectrum analysis of fluctuations in the lensing maps and X-ray surface brightness maps to assess the dynamical state of the gas in clusters. They were able to demonstrate that departure from hydrostatic equilibrium and the existence of small-scale turbulent processes, like shocks, gas sloshing, etc., could be extracted from combined analysis using these new metrics. Similarly, our work also amply demonstrates the gain from the combination of multiwavelength data to characterize the dynamical state of clusters. What is powerful about such methods is the leveraging a set of observables that are independent of the dynamical state (lensing data) along with other observables (X-ray data) to derive new insights into the dynamics and assembly of galaxy clusters, specifically the role of ongoing mergers and their signatures.

## 9 SUMMARY AND CONCLUSION

In this paper, we present a method to reconstruct the mass of galaxy clusters from the combination of strong lensing, X-ray emission, and cluster member morphology and kinematics data. We incorporate all these observables into a single process by performing a joint optimization of multiple image positions and X-ray photon counts. Measurements on galaxies are added in the form of physically motivated priors on the three galaxy scaling relations. This new method has been implemented in the software LENSTOOL and test it on the well-observed cluster AS1063. Both software and mass models are released publicly to the community.

We find that our method is able to recover the strong lensing constraints like other available state-of-the-art methods. Furthermore, our model of the gas captures the asymmetric shape of AS1063 X-ray emission efficiently. Thus, our model is able to increase the complexity of the reconstruction by deviating from the forced elliptical symmetry of the main mass haloes with the addition of the gas distribution. The presented treatment of cluster members takes advantage of the fundamental plane of elliptical galaxies, which allows the addition of a scatter in the $\sigma_0$-mass plane in comparison to previous treatments. Thanks to our combined calibration process, we propose a coherent modelling of the cluster member mass through the fundamental plane and the Faber & Jackson scaling relation.

We use a mock cluster based on AS1063 to assess the robustness of our method to the assumed uncertainties on the constraints as well as the benefit of the enhancement presented in this work. We find that we recover the gas mapping up to less than 3 per cent of error, as well as the intrinsic error added to the gas emission models which is inside the "1σ" CI. In the case of AS1063, the non-inclusion of the gas-only leads to a maximum of a few per cent error in the total mass. In particular, it is not possible to capture the deviation from the main elliptical symmetry with a solely parametric treatment. We expect that this might be more significant for perturbed clusters where the gas distribution is less well correlated with the DM content. We compare the ability of our current cluster member model to previous work and report that using the combination of the fundamental plane and Faber & Jackson scaling relations with a calibration from actual measurements allows us to reduce the bias on the estimation of dPIE's $\sigma_0$ and $r_{cut}$. However, the improvement on the posterior distribution of the cluster member masses propagated from these two quantities is less important.
<figure><img src="image_21.png" /><figcaption>**Figure 18.** Integrated profiles of the 2D total mass density (left-hand panel), and gas fraction (right-panel panel) integrated in spheres of increasing radii. For both panels, plain lines and shaded regions represent the median and the "3σ" CI of the profiles distribution amongst the 4000 models of the original posterior distributions.</figcaption></figure>

below 600 kpc, mainly dominated by the area of disagreement close to the BCG, which shows a gas fraction 17 per cent lower than ours. We note that this X-ray fit provides a different result on the gas mass than the multiscale deprojection presented in Section 6.3. In that case, our model presents a slightly lower deprojected gas mass than the NFW fit. Hence, both the total mass discrepancy and the previous one lead to this disagreement on the gas fraction, even if both of them are in quite close agreement. There is a significant difference with the B19 model, which is a direct consequence of their gas mass estimation which is higher than the two others, as seen in Fig. 9.

We present the 2D distribution of the projected gas fraction in Fig. 19 for B19 and our best-fitting model. If, globally, the observed gas fraction is similar with a median of the relative difference of 11 per cent between both maps, there are some differences in the morphology. Indeed, the ellipticity discrepancies between the DM and the gas can be clearly seen in our model when it is less evident in B19. In particular, the lobes of a lower or higher gas fraction have different shapes between both models on areas that are mainly outside the constrained one. It can be explained by the differences in the modelling and the assumptions on the parameters, such as dPIEs' cut radius, $r_{cut}$. In addition, more complexity can be observed in our 2D distribution that has been added by the B-spline surfaces in the area mentioned in Section 6.3. We note that the 2D analysis of the gas fraction is possible with our modelling method without the need for any extra-tools, thus allowing simple access to the position and shape of the different cluster components.

In recent work, Cerini, Cappelluti & Natarajan (2023), derived and demonstrated the utility of power-spectrum analysis of fluctuations in the lensing maps and X-ray surface brightness maps to assess the dynamical state of the gas in clusters. They were able to demonstrate that departure from hydrostatic equilibrium and the existence of small-scale turbulent processes, like shocks, gas sloshing, etc., could be extracted from combined analysis using these new metrics. Similarly, our work also amply demonstrates the gain from the combination of multiwavelength data to characterize the dynamical state of clusters. What is powerful about such methods is the leveraging a set of observables that are independent of the dynamical state (lensing data) along with other observables (X-ray data) to derive new insights into the dynamics and assembly of galaxy clusters, specifically the role of ongoing mergers and their signatures.

## 9 SUMMARY AND CONCLUSION

In this paper, we present a method to reconstruct the mass of galaxy clusters from the combination of strong lensing, X-ray emission, and cluster member morphology and kinematics data. We incorporate all these observables into a single process by performing a joint optimization of multiple image positions and X-ray photon counts. Measurements on galaxies are added in the form of physically motivated priors on the three galaxy scaling relations. This new method has been implemented in the software LENSTOOL and test it on the well-observed cluster AS1063. Both software and mass models are released publicly to the community.

We find that our method is able to recover the strong lensing constraints like other available state-of-the-art methods. Furthermore, our model of the gas captures the asymmetric shape of AS1063 X-ray emission efficiently. Thus, our model is able to increase the complexity of the reconstruction by deviating from the forced elliptical symmetry of the main mass haloes with the addition of the gas distribution. The presented treatment of cluster members takes advantage of the fundamental plane of elliptical galaxies, which allows the addition of a scatter in the $\sigma_0$-mass plane in comparison to previous treatments. Thanks to our combined calibration process, we propose a coherent modelling of the cluster member mass through the fundamental plane and the Faber & Jackson scaling relation.

We use a mock cluster based on AS1063 to assess the robustness of our method to the assumed uncertainties on the constraints as well as the benefit of the enhancement presented in this work. We find that we recover the gas mapping up to less than 3 per cent of error, as well as the intrinsic error added to the gas emission models which is inside the "1σ" CI. In the case of AS1063, the non-inclusion of the gas-only leads to a maximum of a few per cent error in the total mass. In particular, it is not possible to capture the deviation from the main elliptical symmetry with a solely parametric treatment. We expect that this might be more significant for perturbed clusters where the gas distribution is less well correlated with the DM content. We compare the ability of our current cluster member model to previous work and report that using the combination of the fundamental plane and Faber & Jackson scaling relations with a calibration from actual measurements allows us to reduce the bias on the estimation of dPIE's $\sigma_0$ and $r_{cut}$. However, the improvement on the posterior distribution of the cluster member masses propagated from these two quantities is less important.
<figure><img src="image_22.png" /><figcaption>**Figure 19.** Maps of the gas fraction from our best-fitting model and B19 one in the left-hand and right-panel panels, respectively.</figcaption></figure>

We assess the consistency of our cluster member modelling hypotheses with available measurements to probe the connection between mass and light as well as the assumed scalings of their light and mass profiles. We find that with our simplification, we provide a fair but not robust, estimation of both as they present similar behaviour but with different scaling slopes. Future modelling improvements could aim at finding new schemes that would avoid our assumptions while still leveraging the empirically found correlations in the fundamental plane.

Thanks to our combination of X-ray and lensing, we obtain a much more exhaustive view of cluster lenses in comparison to a disjoint analysis. In particular, from the residual of the X-ray emission fit to the gas thermodynamics combined with an accurate representation of the mass distribution, we provide new evidence for a recent merging event in AS1063, rejecting the major merger possibility. Indeed, we find proof for gas sloshing thanks to the tools we put in place to map the intra-cluster gas temperature. In addition, as the cluster components are disentangled, we can observe the differences between the baryonic and non-baryonic components by mapping in 2D the gas fraction compared to earlier work Bonamigo et al. (2018), as our total mass reconstruction is more flexible thanks to the B-spline surface. We find that AS1063 is nearly at the hydrostatic equilibrium, in agreement with the analysis by Sartoris et al. (2020).

The new method presented here includes baryonic components explicitly and self-consistently and demonstrates its capabilities. The next step would be to extend this work to a model of a sample of cluster lenses, such as the complete set of BUFFALO clusters or other newly observed cluster lenses with the recently launched *James Webb Space Telescope (JWST)*. These will be well suited as some of the most extreme cluster lenses in these samples consist of highly perturbed clusters that have undergone or are actively undergoing major merger events. Thus, the inclusion of the modelling of the gas will improve the total mass distribution and help assess the dynamical state of the cluster. Performing the analysis, we present here on a larger sample of clusters, we would be able to measure offsets between the different cluster elements and provide constraints on a possible self-interaction of the DM. Thanks to the more exhaustive picture of the dynamical state of clusters, the comparison with results from simulated merging events would be easier and more complete.

Such a sample will offer a homogeneous window into the high-redshift universe that will strongly benefit from the enhanced consistency achieved using the multiwavelength observations of clusters. Hence, reducing biases on lensed galaxy measurements where a local bad estimation of the total mass can have here more impact than on analyses focused on cluster physics. A natural improvement to our method would be the inclusion of the weak lensing constraints to reduce the induced bias on the core and extend the scale on which the mass distribution is reconstructed, for example, a merging with *Hybrid-LENSTOOL* (Niemiec et al. 2020) method. The codes to use weak lensing measurements are already available, but the structures at the outskirts have to be identified beforehand with a free-form method. Modelling of the gas would benefit from taking inspiration from these methods with a multiscale reconstruction, losing some properties valuable to analyse the cluster state but providing a highly accurate mapping of the associated mass in exchange.

As demonstrated by Mahler et al. (2023), in the modelling of the *JWST* data of the first lensing cluster to be observed, SMACS J0723.3-7327, the flexibility to include an additional mass component that corresponds to the baryons locked into the ICL is likely to prove invaluable. Such addition could be done with a similar methodological extension as we presented here for the X-ray data in a self-consistent way. The prospects for comprehensive modelling of the multiple mass components of a cluster, robustly, to include all the baryonic contributors: stellar component of cluster members, X-ray gas and the ICL, in addition to the dominant DM component, is becoming feasible.

Finally, we aim to provide one of the first steps of a new cluster mass modelling standard, which is based on a self-consistent and simultaneous inclusion of multiwavelength data and multiple observables. This new methodology, we believe, provides the appropriate enhancement in the sophistication of cluster mass modelling aligned with the improvement in the availability of observational data as we transit from the *HST* to the *JWST* era of cluster and high-redshift studies.

## ACKNOWLEDGEMENTS

The authors are grateful to the referee for carefully reading the manuscript and for valuable suggestions and comments, which
<figure><img src="image_22.png" /><figcaption>Figure 19. Maps of the gas fraction from our best-fitting model and B19 one in the left-hand and right-panel panels, respectively.</figcaption></figure>

We assess the consistency of our cluster member modelling hypotheses with available measurements to probe the connection between mass and light as well as the assumed scalings of their light and mass profiles. We find that with our simplification, we provide a fair but not robust, estimation of both as they present similar behaviour but with different scaling slopes. Future modelling improvements could aim at finding new schemes that would avoid our assumptions while still leveraging the empirically found correlations in the fundamental plane.

Thanks to our combination of X-ray and lensing, we obtain a much more exhaustive view of cluster lenses in comparison to a disjoint analysis. In particular, from the residual of the X-ray emission fit to the gas thermodynamics combined with an accurate representation of the mass distribution, we provide new evidence for a recent merging event in AS1063, rejecting the major merger possibility. Indeed, we find proof for gas sloshing thanks to the tools we put in place to map the intra-cluster gas temperature. In addition, as the cluster components are disentangled, we can observe the differences between the baryonic and non-baryonic components by mapping in 2D the gas fraction compared to earlier work Bonamigo et al. (2018), as our total mass reconstruction is more flexible thanks to the B-spline surface. We find that AS1063 is nearly at the hydrostatic equilibrium, in agreement with the analysis by Sartoris et al. (2020).

The new method presented here includes baryonic components explicitly and self-consistently and demonstrates its capabilities. The next step would be to extend this work to a model of a sample of cluster lenses, such as the complete set of BUFFALO clusters or other newly observed cluster lenses with the recently launched *James Webb Space Telescope (JWST)*. These will be well suited as some of the most extreme cluster lenses in these samples consist of highly perturbed clusters that have undergone or are actively undergoing major merger events. Thus, the inclusion of the modelling of the gas will improve the total mass distribution and help assess the dynamical state of the cluster. Performing the analysis, we present here on a larger sample of clusters, we would be able to measure offsets between the different cluster elements and provide constraints on a possible self-interaction of the DM. Thanks to the more exhaustive picture of the dynamical state of clusters, the comparison with results from simulated merging events would be easier and more complete.

Such a sample will offer a homogeneous window into the high-redshift universe that will strongly benefit from the enhanced consistency achieved using the multiwavelength observations of clusters. Hence, reducing biases on lensed galaxy measurements where a local bad estimation of the total mass can have here more impact than on analyses focused on cluster physics. A natural improvement to our method would be the inclusion of the weak lensing constraints to reduce the induced bias on the core and extend the scale on which the mass distribution is reconstructed, for example, a merging with *Hybrid-LENSTOOL* (Niemiec et al. 2020) method. The codes to use weak lensing measurements are already available, but the structures at the outskirts have to be identified beforehand with a free-form method. Modelling of the gas would benefit from taking inspiration from these methods with a multiscale reconstruction, losing some properties valuable to analyse the cluster state but providing a highly accurate mapping of the associated mass in exchange.

As demonstrated by Mahler et al. (2023), in the modelling of the *JWST* data of the first lensing cluster to be observed, SMACS J0723.3-7327, the flexibility to include an additional mass component that corresponds to the baryons locked into the ICL is likely to prove invaluable. Such addition could be done with a similar methodological extension as we presented here for the X-ray data in a self-consistent way. The prospects for comprehensive modelling of the multiple mass components of a cluster, robustly, to include all the baryonic contributors: stellar component of cluster members, X-ray gas and the ICL, in addition to the dominant DM component, is becoming feasible.

Finally, we aim to provide one of the first steps of a new cluster mass modelling standard, which is based on a self-consistent and simultaneous inclusion of multiwavelength data and multiple observables. This new methodology, we believe, provides the appropriate enhancement in the sophistication of cluster mass modelling aligned with the improvement in the availability of observational data as we transit from the *HST* to the *JWST* era of cluster and high-redshift studies.

## ACKNOWLEDGEMENTS

The authors are grateful to the referee for carefully reading the manuscript and for valuable suggestions and comments, which
helped improve and clarify it. This work used data and catalogue products from HFF-DeepSpace, funded by the National Science Foundation and Space Telescope Science Institute (operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555). This research has been made possible by observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute. These observations are associated with program GO-15117. We also made use of observations collected at the European Southern Observatory under ESO programme(s) 60.A-9345(A), 095.A-0653(A) and 186.A-0798. ML acknowledges the Centre National de la Recherche Scientifique (CNRS) and the Centre National des Etudes Spatiale (CNES) for their support. MM acknowledges the project PCI2021-122072-2B, financed by MICIN/AEI/10.13039/501100011033, and the European Union 'NextGenerationEU'/RTRP. DJL is supported by STFC grant ST/T000244/1. DJL and MJ are supported by the United Kingdom Research and Innovation (UKRI) Future Leaders Fellowship 'Using Cosmic Beasts to uncover the Nature of Dark Matter' (grant number MR/S017216/1).

## DATA AVAILABILITY

The LENSTOOL output files describing the mass model presented in this article are available in GitHub at https://github.com/njzifjoie/z/AS1063-model-Beauchesne-2023. These files can be interpreted without the need for the latest release of LENSTOOL. In particular, there is no need for the extensions presented in this paper, such as the fundamental plane relation or the X-ray modelling. We also released publicly the redshift catalogue produced with the new analysis of the MUSE data cube at the following link: https://cral-perso.univ-lyon.1.fr/abo/perso/johan.richard/MUSE_data_release/.

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* Faber S. M., Jackson R. E., 1976, ApJ, 204, 668

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* Fruscione A. et al., 2006, in Silva D. R., Doxsey R. E., eds, Proc. SPIE Conf. Ser. Vol. 6270, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series. SPIE, Bellingham, 62701V,

* Garmire G. P., Bautz M. W., Ford P. G., Nousek J. A., Ricker George R. J., 2003, in Truemper J. E., Tananbaum H. D., eds, Proc. SPIE Conf. Ser. Vol. 4851, X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy. SPIE, Bellingham, 28,

* Gehrels N., 1986, ApJ, 303, 336

* Gilman D., Birrer S., Treu T., 2020, A&A, 642, A194

* Gómez P. L. et al., 2012, AJ, 144, 79

* Granata G. et al., 2022, A&A, 659, A24 (G22)

* HI4PI Collaboration, 2016, A&A, 594, A116

* Harvey D., Massey R., Kitching T., Taylor A., Tittley E., 2015, Science, 347, 1462

* Häubler B. et al., 2013, MNRAS, 430, 330

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* Kaastra J. S., 2017, A&A, 605, A51

* Karman W. et al., 2015, A&A, 574, A11

* Karman W. et al., 2017, A&A, 599, A28

* Kneib J.-P., Natarajan P., 2011, A&AR, 19, 47

* Koopmans L. V. E. et al., 2009, ApJ, 703, L51

* Kravsov A. V., Borgani S., 2012, ARA&A, 50, 353

* Kumar R., Carroll C., Hartikainen A., Martin O., 2019, J. Open Source Softw., 4, 1143

* Lagattuta D. J. et al., 2017, MNRAS, 469, 3946

* Lagattuta D. J. et al., 2019, MNRAS, 485, 3738

* Lam T. Y., Nishimichi T., Schmidt F., Takada M., 2012, Phys. Rev. Lett., 109, 051301

* Limousin M., Kneib J.-P., Natarajan P., 2005, MNRAS, 356, 309

* Limousin M., Beauchesne B., Jullo E., 2022, A&A, 664, A90

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* Mamon GST. A., Biviano A., Boué G., 2013, MNRAS, 429, 3079

* Markevitch M., Gonzalez A. H., Clowe D., Vikhlinin A., Forman W., Jones C., Murray S., Tucker W., 2004, ApJ, 606, 819
helped improve and clarify it. This work used data and catalogue products from HFF-DeepSpace, funded by the National Science Foundation and Space Telescope Science Institute (operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555). This research has been made possible by observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute. These observations are associated with program GO-15117. We also made use of observations collected at the European Southern Observatory under ESO programme(s) 60.A-9345(A), 095.A-0653(A) and 186.A-0798. ML acknowledges the Centre National de la Recherche Scientifique (CNRS) and the Centre National des Etudes Spatiale (CNES) for their support. MM acknowledges the project PCI2021-122072-2B, financed by MICIN/AEI/10.13039/501100011033, and the European Union 'NextGenerationEU'/RTRP. DJL is supported by STFC grant ST/T000244/1. DJL and MJ are supported by the United Kingdom Research and Innovation (UKRI) Future Leaders Fellowship 'Using Cosmic Beasts to uncover the Nature of Dark Matter' (grant number MR/S017216/1).

## DATA AVAILABILITY

The LENSTOOL output files describing the mass model presented in this article are available in GitHub at https://github.com/njzifjoie/z/AS1063-model-Beauchesne-2023. These files can be interpreted without the need for the latest release of LENSTOOL. In particular, there is no need for the extensions presented in this paper, such as the fundamental plane relation or the X-ray modelling. We also released publicly the redshift catalogue produced with the new analysis of the MUSE data cube at the following link: https://cral-perso.univ-lyon.1.fr/abo/perso/johan.richard/MUSE_data_release/.

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## APPENDIX A: B-SPLINE MODELLING OF THE X-RAY BACKGROUND

The approach outlined here is empirical and aimed at modelling the number of counts directly (i.e. it implies a unitary instrument response) with B-spline functions of degree 3. We used it on Chandra observation, but it is a general method that can be applied to other X-ray observatories as long as they provide the same kind of data. The whole PYTHON code that we developed is publicly available in the LENSTOOL repository and takes benefit of the SHERPA fitting environment as well as the B-spline routines splev and

<figure><img src="image_23.png" /><figcaption>**Figure A1.** Spectra of blank-sky observations for each of the four CCD in the [0, 13] keV energy band associated with observations 18 611. Black histograms represent the photon counts, and plain red lines highlight the median of the background model. Shaded red regions represent the uncertainty on the parameter estimations.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Figure A2.** Left-hand panel: Black histogram presents the spectrum of the blank-sky background associated with the observation 18 611 on the core of AS1063. The red line and shaded region represent the background model and the uncertainties in the parameter estimation. The extraction region is a square of 8.3 arcmin around the cluster centre and is overlapped on different CCDs. Right-panel panel: Spectrum of AS1063 in the previously defined extraction regions combining the three observations considered for that work. The colour scheme is the same as before except for the model where red plain lines and shaded region represent the median and the "1σ" CI of the posterior distribution of the spectrum model.</figcaption></figure>
Meneghetti M. et al., 2017, MNRAS, 472, 3177

Meneghetti M. et al., 2020, Science, 369, 1347

Mercurio A. et al., 2021, A&A, 656, A147

Million E. T., Allen S. W., 2009, MNRAS, 399, 1307

Montes M., 2022, Nature Astron., 6, 308

Natarajan P., Kneib J.-P., 1997, MNRAS, 287, 833

Natarajan P. et al., 2017, MNRAS, 468, 1962

Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563

Nedkova K. V. et al., 2021, MNRAS, 506, 928

Newman A. B., Treu T., Ellis R. S., Sand D. J., 2013, ApJ, 765, 25

Niemiec A., Jauzac M., Jullo E., Limousin M., Sharon K., Kneib J.-P., Natarajan P., Richard J., 2020, MNRAS, 493, 3331

Oguri M., 2010, PASJ, 62, 1017

Pagul A. et al., 2023, preprint (arXiv:2307.04635)

Paterno-Mahler R., Blanton E. L., Randall S. W., Clarke T. E., 2013, ApJ, 773, 114

Peng C. Y., Ho L. C., Impey C. D., Rix H.-W., 2010, AJ, 139, 2097

Postman M. et al., 2012, ApJS, 199, 25

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Richard J. et al., 2010, MNRAS, 404, 325

Richard J. et al., 2021, A&A, 646, A83

Robertson A., Harvey D., Massey R., Eke V., McCarthy I. G., Jauzac M., Li B., Schaye J., 2019, MNRAS, 488, 3646

Rossetti M., Ghizzardi S., Molendi S., Finoguenov A., 2007, A&A, 463, 839

Sanders J. S. et al., 2022, A&A, 661, A36

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Xie C. et al., 2020, A&A, 636, A3

Zitrin A. et al., 2009, MNRAS, 396, 1985

de Oliveira N. O. L., Jiménez-Teja Y., Dupke R., 2022, MNRAS, 512, 1916

## APPENDIX A: B-SPLINE MODELLING OF THE X-RAY BACKGROUND

The approach outlined here is empirical and aimed at modelling the number of counts directly (i.e. it implies a unitary instrument response) with B-spline functions of degree 3. We used it on Chandra observation, but it is a general method that can be applied to other X-ray observatories as long as they provide the same kind of data. The whole PYTHON code that we developed is publicly available in the LENSTOOL repository and takes benefit of the SHERPA fitting environment as well as the B-spline routines splev and

<figure><img src="image_23.png" /><figcaption>**Figure A1.** Spectra of blank-sky observations for each of the four CCD in the [0, 13] keV energy band associated with observations 18 611. Black histograms represent the photon counts, and plain red lines highlight the median of the background model. Shaded red regions represent the uncertainty on the parameter estimations.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Figure A2.** Left-hand panel: Black histogram presents the spectrum of the blank-sky background associated with the observation 18 611 on the core of AS1063. The red line and shaded region represent the background model and the uncertainties in the parameter estimation. The extraction region is a square of 8.3 arcmin around the cluster centre and is overlapped on different CCDs. Right-panel panel: Spectrum of AS1063 in the previously defined extraction regions combining the three observations considered for that work. The colour scheme is the same as before except for the model where red plain lines and shaded region represent the median and the "1σ" CI of the posterior distribution of the spectrum model.</figcaption></figure>
splrep (Dierckx 1993) through their SCIPY wrappers (Virtanen et al. 2020).

We start our modelling by fitting each CCD spectra on the [0, 13] keV energy band on each observation. This range has been chosen to ensure that the background model will have the right value on the edge of the [0.5, 12] keV one. Each fit is made in two steps; the first is an optimization with a $\chi^2$ statistic with the Gehrels (1986) approximation implemented in SHERPA through a Levenberg-Marquardt method. During this optimization, only the B-spline coefficients are not fixed, while the knots positions and their numbers are obtained by the splrep routine. The coefficient values given by the preceding routine are used to define starting points for the $\chi^2$ minimization. Finally, the second step is a simplex optimization from the results of the last one with the XSPEC Cstat. Initially, we wanted only to include this last step as a Poisson likelihood is more adapted to the considered data sets, but the optimization was not converging to satisfying models in some cases. The resulting spectra models for the observation of 18 611 on each CCD are presented in Fig. A1.

To adapt the previous spectra model to the region where is extracted the source spectrum, we extract the one from the same area in the blank-sky background. We then fit it with a linear combination of the CCD model that contributes using a simplex method with the Cstat statistic. By allowing the coefficient of the linear combination to deviate from the ratio of the overlapping area of the CCD on the region of interest on the total region area, we aimed to take into account spatial differences on a unique CCD. Fig. A2 shows the model obtained for the whole cluster fit of Section 3.1 on the observation 18 611 in the left-hand panel. Before using this background model, we re-scale the coefficient of the linear combination with the ratio of counts on the [9.5, 12] keV range between the observation and the blank-sky per CCD. In this energy band, it mostly accounts for the particle background as the effective area of Chandra is almost null. We make this re-scaling to take into account the differences per CCD due to the discrepancy in exposure times. Finally, the spectrum model is used in the nested sampling method with a normalization per observation.

The right-panel panel of Fig. A2 presents the result of the fitting procedure on the whole cluster. It highlights one of the limitations of this approach, which is the fluctuation of the amplitude of the fluorescent emission lines amongst a CCD for high energies. Hence, to improve this modelling, we could allow them to vary, and a second enhancement would be the disentanglement of the instrumental and sky background. However, as we are mainly interested in the core of clusters where we have a high SN and the energy range of interest for the source is [0.5, 7] keV, both of these issues should not affect our results.

<table><caption>**Table B1.** Prior distributions on the LOSVD parameters used to generate a sample of simulated spectra from AS1063 cluster members.</caption><thead><tr><td>Parameters</td><td>Distribution</td></tr></thead><tbody><tr><td>V</td><td>U(-250, 250)</td></tr><tr><td>σ<sub>True</sub></td><td>U(10, 400)</td></tr><tr><td>h<sub>1</sub></td><td>U(-0.15, 0.15)</td></tr><tr><td>h<sub>2</sub></td><td>U(-0.15, 0.15)</td></tr></tbody></table>

## APPENDIX B: BIAS ON THE MEASUREMENT OF THE LOSVD

Following Bergamini et al. (2019), we created mock spectra to test and correct our estimation of the LOSVD measurements with PPXF and PYMULTINEST libraries. To obtain spectra that will represent the galaxy population of AS1063, we use the linear combination of template spectra with weights obtained from the best-fitting solution for each galaxy considered. We then use PPXF to convolve them with different LOSVD parameters. In particular, the velocity V, the velocity dispersion σ, and the two Hermites moments h₁ and h₂ are sampled randomly from the distribution shown in Table B1 and the Legendre polynomial coefficients are taken from the best-fitting solution. For each galaxy (i.e. 107 elements), we sampled 100 mock spectra that we transformed to MUSE-like measurements by degrading the spectral resolution and FWHM to match the real ones. We finally add noise based on the real spectra variance that we adjusted in amplitude to obtain an SN sampled with a uniform probability between 3 and 50. In that way, we have mock errors taking into account specific variations, such as the subtraction of skylines in the real extracted spectra.

The mock spectra have been fitted with the exact same methodology as outlined in Section 3.2 and $\sigma_{fit}$ which is the median of obtained posterior distribution on $\sigma$ is plotted against the true value $\sigma_{True}$ in Fig. B1. Our fit globally follows the input LOSVD parameters with a small bias that we estimate by fitting a polynomial of degree 2 on binned values of $\sigma_{fit}$. The binned data and the fitted polynomial are shown by the cyan scatter and lines in the previous graph and the polynomial expression is the following:

$$ \sigma_{\text{fit,Corr}} = -2.029 \times 10^{-4} \sigma_{\text{fit}}^2 + 1.035 \sigma_{\text{fit}} + 12.22 \quad (\text{B1}) $$

Thus, $\sigma_{fit}$ is suffering a small underestimation of around 3 per cent that is increased at low $\sigma_{True}$ value (i.e. $\sigma_{True} < 100$ km/s). It is well visible in the central plot in Fig. B1 that shows the ratio between $\Delta\sigma = \sigma_{fit} - \sigma_{True}$ and the statistical errors from the posterior distribution $\delta\sigma$ in function of $\sigma_{True}$. However, after applying equation (B1) on all the velocity dispersion posterior distribution, we obtain a $\sigma_{fit,Corr}$ and $\delta\sigma_{corr}$ that provides a better estimation of $\sigma_{True}$ as we can see on the plot on the right column of Fig. B1. Indeed, the ratios

<figure><img src="image_25.png" /><figcaption>**Figure B1.** Left-hand panel: Plot of the true velocity dispersion, $\sigma_{True}$, and the best-fitting one, $\sigma_{fit}$. Middle Panel: Plot of the ratio between the $\Delta\sigma$ (i.e. $\sigma_{True} - \sigma_{fit}$) and the statistical uncertainty measured on the posterior distribution in function of $\sigma_{True}$. Right-panel panel: Same as middle panel but $\sigma_{fit}$ has been corrected with relations given in equation (B1).</figcaption></figure>
splrep (Dierckx 1993) through their SCIPY wrappers (Virtanen et al. 2020).

We start our modelling by fitting each CCD spectra on the [0, 13] keV energy band on each observation. This range has been chosen to ensure that the background model will have the right value on the edge of the [0.5, 12] keV one. Each fit is made in two steps; the first is an optimization with a $\chi^2$ statistic with the Gehrels (1986) approximation implemented in SHERPA through a Levenberg-Marquardt method. During this optimization, only the B-spline coefficients are not fixed, while the knots positions and their numbers are obtained by the splrep routine. The coefficient values given by the preceding routine are used to define starting points for the $\chi^2$ minimization. Finally, the second step is a simplex optimization from the results of the last one with the XSPEC Cstat. Initially, we wanted only to include this last step as a Poisson likelihood is more adapted to the considered data sets, but the optimization was not converging to satisfying models in some cases. The resulting spectra models for the observation of 18 611 on each CCD are presented in Fig. A1.

To adapt the previous spectra model to the region where is extracted the source spectrum, we extract the one from the same area in the blank-sky background. We then fit it with a linear combination of the CCD model that contributes using a simplex method with the Cstat statistic. By allowing the coefficient of the linear combination to deviate from the ratio of the overlapping area of the CCD on the region of interest on the total region area, we aimed to take into account spatial differences on a unique CCD. Fig. A2 shows the model obtained for the whole cluster fit of Section 3.1 on the observation 18 611 in the left-hand panel. Before using this background model, we re-scale the coefficient of the linear combination with the ratio of counts on the [9.5, 12] keV range between the observation and the blank-sky per CCD. In this energy band, it mostly accounts for the particle background as the effective area of Chandra is almost null. We make this re-scaling to take into account the differences per CCD due to the discrepancy in exposure times. Finally, the spectrum model is used in the nested sampling method with a normalization per observation.

The right-panel panel of Fig. A2 presents the result of the fitting procedure on the whole cluster. It highlights one of the limitations of this approach, which is the fluctuation of the amplitude of the fluorescent emission lines amongst a CCD for high energies. Hence, to improve this modelling, we could allow them to vary, and a second enhancement would be the disentanglement of the instrumental and sky background. However, as we are mainly interested in the core of clusters where we have a high SN and the energy range of interest for the source is [0.5, 7] keV, both of these issues should not affect our results.

<table><caption>**Table B1.** Prior distributions on the LOSVD parameters used to generate a sample of simulated spectra from AS1063 cluster members.</caption><thead><tr><th>Parameters</th><th>Distribution</th></tr></thead><tbody><tr><td>V</td><td>U(-250, 250)</td></tr><tr><td>σ<sub>True</sub></td><td>U(10, 400)</td></tr><tr><td>h<sub>1</sub></td><td>U(-0.15, 0.15)</td></tr><tr><td>h<sub>2</sub></td><td>U(-0.15, 0.15)</td></tr></tbody></table>

## APPENDIX B: BIAS ON THE MEASUREMENT OF THE LOSVD

Following Bergamini et al. (2019), we created mock spectra to test and correct our estimation of the LOSVD measurements with PPXF and PYMULTINEST libraries. To obtain spectra that will represent the galaxy population of AS1063, we use the linear combination of template spectra with weights obtained from the best-fitting solution for each galaxy considered. We then use PPXF to convolve them with different LOSVD parameters. In particular, the velocity V, the velocity dispersion σ, and the two Hermites moments $h_1$ and $h_2$ are sampled randomly from the distribution shown in Table B1 and the Legendre polynomial coefficients are taken from the best-fitting solution. For each galaxy (i.e. 107 elements), we sampled 100 mock spectra that we transformed to MUSE-like measurements by degrading the spectral resolution and FWHM to match the real ones. We finally add noise based on the real spectra variance that we adjusted in amplitude to obtain an SN sampled with a uniform probability between 3 and 50. In that way, we have mock errors taking into account specific variations, such as the subtraction of skylines in the real extracted spectra.

The mock spectra have been fitted with the exact same methodology as outlined in Section 3.2 and $\sigma_{fit}$ which is the median of obtained posterior distribution on $\sigma$ is plotted against the true value $\sigma_{True}$ in Fig. B1. Our fit globally follows the input LOSVD parameters with a small bias that we estimate by fitting a polynomial of degree 2 on binned values of $\sigma_{fit}$. The binned data and the fitted polynomial are shown by the cyan scatter and lines in the previous graph and the polynomial expression is the following:

$$ \sigma_{\text{fit,Corr}} = -2.029 \times 10^{-4} \sigma_{\text{fit}}^2 + 1.035 \sigma_{\text{fit}} + 12.22 \quad (\text{B1}) $$

Thus, $\sigma_{fit}$ is suffering a small underestimation of around 3 per cent that is increased at low $\sigma_{True}$ value (i.e. $\sigma_{True} < 100$ km/s). It is well visible in the central plot in Fig. B1 that shows the ratio between $\Delta\sigma = \sigma_{fit} - \sigma_{True}$ and the statistical errors from the posterior distribution $\delta\sigma$ in function of $\sigma_{True}$. However, after applying equation (B1) on all the velocity dispersion posterior distribution, we obtain a $\sigma_{fit,Corr}$ and $\delta\sigma_{corr}$ that provides a better estimation of $\sigma_{True}$ as we can see on the plot on the right column of Fig. B1. Indeed, the ratios

<figure><img src="image_25.png" /><figcaption>**Figure B1.** Left-hand panel: Plot of the true velocity dispersion, $\sigma_{True}$, and the best-fitting one, $\sigma_{fit}$. Middle Panel: Plot of the ratio between the $\Delta\sigma$ (i.e. $\sigma_{True} - \sigma_{fit}$) and the statistical uncertainty measured on the posterior distribution in function of $\sigma_{True}$. Right-panel panel: Same as middle panel but $\sigma_{fit}$ has been corrected with relations given in equation (B1).</figcaption></figure>
between the true and the estimation errors are mostly contained in the [−1, 1] intervals.

In comparison, results from Bergamini et al. (2019) on mock spectra present a few per cent of overestimation on $\sigma_{fit}$ as well as an underestimation of the real errors. Thus, if we also obtain similar biases on the best-fitting solution of the velocity dispersion, our $\delta\sigma_{corr}$ are presenting a correct view of the measurement errors, which supports our use of a nested sampling method instead of the default least square from the PPXF package.

## APPENDIX C: FROM $\sigma_e$ TO $\sigma_0$

$\sigma_0$ is usually considered as a representation of the central velocity dispersion of the associated dPIE potential, but its link with the actual measurement of the LOSVD parameters is not straightforward, and we need to make some assumptions about the mass and light distribution of the considered galaxies. Following Newman et al. (2013) and Bergamini et al. (2019), the LOS velocity dispersion $\sigma_{LOS}$ is given by:

$$\sigma_{\text{LOS}}^2(R) = \frac{2G}{I(R)} \int_{R}^{\infty} \nu(r) M_{3D}(r) \mathcal{F}(r) r^{2\beta_{\text{aniso}}-2} dr \quad \text{(C1)}$$

Where $I$ and $\nu$ are the surface density and density of the luminosity distribution, respectively. $M_{3D}$ is the total mass enclosed in a sphere of radius $r$. The two lasting terms, $\mathcal{F}$ and $\beta_{aniso}$ are linked to the orbit of stars considered that in the case of isotropic orbits are expressed as follows (Cappellari 2008):

$$\mathcal{F}(r) = \sqrt{r^2 - R^2} \text{ and } \beta_{\text{aniso}} = 0 \quad \text{(C2)}$$

The actual measurements $\sigma_{ap}$ are $\sigma_{LOS}$ values averaged in an circular aperture of radius $R'$ that is defined as follows:

$$\sigma_{\text{ap}}^2(R') = \frac{2\pi}{L(R')} \int_{0}^{R'} \sigma_{\text{LOS}}^2(R) I(R) R dR \quad \text{(C3)}$$

With $L(R')$ the luminosity enclosed in a circle of radius $R'$. In the case of a dPIE potential, we have the following relations for the different terms involved in the $\sigma_{ap}$ expression:

$$\nu(r) = \frac{\sigma_0^2(r_{\text{core}} + r_{\text{cut}})}{2\Upsilon_{\odot}\pi Gr_{\text{core}}^2 r_{\text{cut}}} \frac{1}{\left(1 + \left(\frac{r}{r_{\text{core}}}\right)^2\right)\left(1 + \left(\frac{r}{r_{\text{cut}}}\right)^2\right)} \quad \text{(C4a)}$$

$$B = \frac{2\sigma_0^2 r_{\text{cut}}}{G(r_{\text{cut}} - r_{\text{core}})} \quad \text{(C4b)}$$

$$I(r) = \frac{B}{\Upsilon_{\odot}} \left( \frac{1}{\sqrt{r^2 + r_{\text{core}}^2}} - \frac{1}{\sqrt{r^2 + r_{\text{cut}}^2}} \right) \quad \text{(C4c)}$$

$$M_{3D}(r) = B \left( r_{\text{cut}} \arctan\left(\frac{r}{r_{\text{cut}}}\right) - r_{\text{core}} \arctan\left(\frac{r}{r_{\text{core}}}\right) \right) \quad \text{(C4d)}$$

$$L(r) = \frac{B}{\Upsilon_{\odot}} \left( \sqrt{r^2 + r_{\text{core}}^2} + r_{\text{cut}} - \sqrt{r^2 + r_{\text{cut}}^2} - r_{\text{core}} \right) \quad \text{(C4e)}$$

Where we define the pre-factor $B$ for clarity. To properly match $\sigma_e$ to $\sigma_0$, we have to make assumptions that fit the ones taken in Section 4.3. In particular, we assumed $\nu R_e = R_e, \text{dPIE}$ (condition 1). For the convenience of the fundamental plan implementation, we would like a relation of that type between the velocities dispersion $\sigma_e = c_p \sigma_0$

(condition 2). Thus, we make the following assumption on the light distribution and its relations to the mass one:

(i) The light distribution is well represented by a dPIE profile with parameters $\sigma_{0,\text{light}}$, $r_{\text{core},\text{light}}$ and $r_{\text{cut},\text{light}}$.

(ii) dPIEs associated with the light and the mass shared the same centre coordinates, ellipticity, and position angle.

(iii) Relations between parameters from the mass and light potentials:

(a) $\sigma_{0,\text{light}} = \sigma_0$ (condition 2)

(b) $r_{\text{core},\text{light}} = \frac{r_{\text{core}}}{\nu}$ (condition 1)

(c) $r_{\text{cut},\text{light}} = \frac{r_{\text{cut}}}{\nu}$ (condition 1)

Where, $\sigma_0$, $r_{\text{core}}$ and $r_{\text{cut}}$ are the dPIE parameters associated with the mass distribution. We can now define the projection factor $c_p$ and gives a physical interpretation to $\nu$. Injecting equation (C4a), (C4c), (C4d), and (C4e) into equation (C3) gives the following expression for the former term:

$$c_p^2(R) = \frac{6 (r_{\text{core}} + r_{\text{cut}}) \nu^2}{\pi r_{\text{core}}^2 r_{\text{cut}}} \times \left( \sqrt{\left(\frac{r_{\text{core}}}{\nu}\right)^2 + R^2} - \frac{r_{\text{core}}}{\nu} - \sqrt{\left(\frac{r_{\text{cut}}}{\nu}\right)^2 + R^2} + \frac{r_{\text{cut}}}{\nu} \right)^{-1} \times \int_0^R R' dR' \int_{R'}^{\infty} \frac{r_{\text{cut}} \tan^{-1}\left(\frac{r}{r_{\text{cut}}}\right) - r_{\text{core}} \tan^{-1}\left(\frac{r}{r_{\text{core}}}\right)}{\left(1 + \left(\frac{r_v}{r_{\text{cut}}}\right)^2\right)\left(1 + \left(\frac{r_v}{r_{\text{core}}}\right)^2\right)} \frac{\sqrt{r^2 - R^2} dr}{r^2} \quad \text{(C5)}$$

This expression is similar to the one presented by Bergamini et al. (2019) in equation C16, and we only have a scaling between the light and total mass tracer in the form of $\nu$ in addition. Thanks to the assumptions on $c_p$, $\nu$ becomes more than simply a spatial scaling between light and mass distribution, and it is linked to the ratio of the luminosity $L$ to the total mass $M_{tot}$ by the following expression:

$$L = \frac{\pi \sigma_{0,\text{light}}^2}{G} \frac{r_{\text{cut},\text{light}}^2}{r_{\text{cut},\text{light}} - r_{\text{core},\text{light}}} \quad \text{(C6)}$$

$$= \nu^{-1} M_{\text{tot}} \quad \text{(C7)}$$

## APPENDIX D: NEW MULTIPLY-IMAGED SYSTEMS

Fig. D1 present the HST images as well as the MUSE Lyman $\alpha$ narrow band and line of the new multiply imaged systems used

<table><caption>**Table D1.** Positions and redshifts of the newly identified multiply imaged systems used in this work.</caption><thead><tr><th>ID</th><th>α(J2000)</th><th>δ(J2000)</th><th>z<sub>system</sub> ± σ</th></tr></thead><tbody><tr><td>303.1</td><td>342.175814</td><td>-44.536371</td><td>3.666 ± 0.003</td></tr><tr><td>303.2</td><td>342.187884</td><td>-44.540421</td><td>3.666 ± 0.003</td></tr><tr><td>304.1</td><td>342.173801</td><td>-44.541178</td><td>3.230 ± 0.004</td></tr><tr><td>304.2</td><td>342.173909</td><td>-44.541235</td><td>3.230 ± 0.004</td></tr><tr><td>305.1</td><td>342.180303</td><td>-44.524639</td><td>3.082 ± 0.002</td></tr><tr><td>305.2</td><td>342.183869</td><td>-44.525756</td><td>3.082 ± 0.002</td></tr></tbody></table>
between the true and the estimation errors are mostly contained in the [−1, 1] intervals.

In comparison, results from Bergamini et al. (2019) on mock spectra present a few per cent of overestimation on $\sigma_{fit}$ as well as an underestimation of the real errors. Thus, if we also obtain similar biases on the best-fitting solution of the velocity dispersion, our $\delta\sigma_{corr}$ are presenting a correct view of the measurement errors, which supports our use of a nested sampling method instead of the default least square from the PPXF package.

## APPENDIX C: FROM $\sigma_e$ TO $\sigma_0$

$\sigma_0$ is usually considered as a representation of the central velocity dispersion of the associated dPIE potential, but its link with the actual measurement of the LOSVD parameters is not straightforward, and we need to make some assumptions about the mass and light distribution of the considered galaxies. Following Newman et al. (2013) and Bergamini et al. (2019), the LOS velocity dispersion $\sigma_{LOS}$ is given by:

$$\sigma_{\text{LOS}}^2(R) = \frac{2G}{I(R)} \int_R^\infty \nu(r) M_{3D}(r) \mathcal{F}(r) r^{2\beta_{\text{aniso}}-2} dr \quad \text{(C1)}$$

Where $I$ and $\nu$ are the surface density and density of the luminosity distribution, respectively. $M_{3D}$ is the total mass enclosed in a sphere of radius $r$. The two lasting terms, $\mathcal{F}$ and $\beta_{aniso}$ are linked to the orbit of stars considered that in the case of isotropic orbits are expressed as follows (Cappellari 2008):

$$\mathcal{F}(r) = \sqrt{r^2 - R^2} \text{ and } \beta_{\text{aniso}} = 0 \quad \text{(C2)}$$

The actual measurements $\sigma_{ap}$ are $\sigma_{LOS}$ values averaged in an circular aperture of radius $R'$ that is defined as follows:

$$\sigma_{\text{ap}}^2(R') = \frac{2\pi}{L(R')} \int_0^{R'} \sigma_{\text{LOS}}^2(R) I(R) R dR \quad \text{(C3)}$$

With $L(R')$ the luminosity enclosed in a circle of radius $R'$. In the case of a dPIE potential, we have the following relations for the different terms involved in the $\sigma_{ap}$ expression:

$$\nu(r) = \frac{\sigma_0^2(r_{\text{core}} + r_{\text{cut}})}{2\Upsilon_{\odot}\pi G r_{\text{core}}^2 r_{\text{cut}}} \frac{1}{\left(1 + \left(\frac{r}{r_{\text{core}}}\right)^2\right)\left(1 + \left(\frac{r}{r_{\text{cut}}}\right)^2\right)} \quad \text{(C4a)}$$

$$B = \frac{2\sigma_0^2 r_{\text{cut}}}{G (r_{\text{cut}} - r_{\text{core}})} \quad \text{(C4b)}$$

$$I(r) = \frac{B}{\Upsilon_{\odot}} \left( \frac{1}{\sqrt{r^2 + r_{\text{core}}^2}} - \frac{1}{\sqrt{r^2 + r_{\text{cut}}^2}} \right) \quad \text{(C4c)}$$

$$M_{3D}(r) = B \left( r_{\text{cut}} \arctan \left( \frac{r}{r_{\text{cut}}} \right) - r_{\text{core}} \arctan \left( \frac{r}{r_{\text{core}}} \right) \right) \quad \text{(C4d)}$$

$$L(r) = \frac{B}{\Upsilon_{\odot}} \left( \sqrt{r^2 + r_{\text{core}}^2} + r_{\text{cut}} - \sqrt{r^2 + r_{\text{cut}}^2} - r_{\text{core}} \right) \quad \text{(C4e)}$$

Where we define the pre-factor $B$ for clarity. To properly match $\sigma_e$ to $\sigma_0$, we have to make assumptions that fit the ones taken in Section 4.3. In particular, we assumed $\nu R_e = R_e, \text{dPIE}$ (condition 1). For the convenience of the fundamental plan implementation, we would like a relation of that type between the velocities dispersion $\sigma_e = c_p \sigma_0$

(condition 2). Thus, we make the following assumption on the light distribution and its relations to the mass one:

(i) The light distribution is well represented by a dPIE profile with parameters $\sigma_{0,\text{light}}$, $r_{\text{core},\text{light}}$ and $r_{\text{cut},\text{light}}$.

(ii) dPIEs associated with the light and the mass shared the same centre coordinates, ellipticity, and position angle.

(iii) Relations between parameters from the mass and light potentials:

(a) $\sigma_{0,\text{light}} = \sigma_0$ (condition 2)

(b) $r_{\text{core},\text{light}} = \frac{r_{\text{core}}}{\nu}$ (condition 1)

(c) $r_{\text{cut},\text{light}} = \frac{r_{\text{cut}}}{\nu}$ (condition 1)

Where, $\sigma_0$, $r_{\text{core}}$ and $r_{\text{cut}}$ are the dPIE parameters associated with the mass distribution. We can now define the projection factor $c_p$ and gives a physical interpretation to $\nu$. Injecting equation (C4a), (C4c), (C4d), and (C4e) into equation (C3) gives the following expression for the former term:

$$c_p^2(R) = \frac{6 (r_{\text{core}} + r_{\text{cut}}) \nu^2}{\pi r_{\text{core}}^2 r_{\text{cut}}} \times \left( \sqrt{\left(\frac{r_{\text{core}}}{\nu}\right)^2 + R^2} - \frac{r_{\text{core}}}{\nu} - \sqrt{\left(\frac{r_{\text{cut}}}{\nu}\right)^2 + R^2} + \frac{r_{\text{cut}}}{\nu} \right)^{-1} \times \int_0^R R' dR' \int_{R'}^\infty \frac{r_{\text{cut}} \tan^{-1}\left(\frac{r}{r_{\text{cut}}}\right) - r_{\text{core}} \tan^{-1}\left(\frac{r}{r_{\text{core}}}\right)}{\left(1 + \left(\frac{r_v}{r_{\text{cut}}}\right)^2\right)\left(1 + \left(\frac{r_v}{r_{\text{core}}}\right)^2\right)} \frac{\sqrt{r^2 - R^2} dr}{r^2} \quad \text{(C5)}$$

This expression is similar to the one presented by Bergamini et al. (2019) in equation C16, and we only have a scaling between the light and total mass tracer in the form of $\nu$ in addition. Thanks to the assumptions on $c_p$, $\nu$ becomes more than simply a spatial scaling between light and mass distribution, and it is linked to the ratio of the luminosity $L$ to the total mass $M_{tot}$ by the following expression:

$$L = \frac{\pi \sigma_{0,\text{light}}^2}{G} \frac{r_{\text{cut},\text{light}}^2}{r_{\text{cut},\text{light}} - r_{\text{core},\text{light}}} \quad \text{(C6)}$$

$$= \nu^{-1} M_{\text{tot}} \quad \text{(C7)}$$

## APPENDIX D: NEW MULTIPLY-IMAGED SYSTEMS

Fig. D1 present the HST images as well as the MUSE Lyman $\alpha$ narrow band and line of the new multiply imaged systems used

<table><caption>**Table D1.** Positions and redshifts of the newly identified multiply imaged systems used in this work.</caption><thead><tr><th>ID</th><th>α(J2000)</th><th>δ(J2000)</th><th>z<sub>system</sub> ± σ</th></tr></thead><tbody><tr><td>303.1</td><td>342.175814</td><td>-44.536371</td><td>3.666 ± 0.003</td></tr><tr><td>303.2</td><td>342.187884</td><td>-44.540421</td><td>3.666 ± 0.003</td></tr><tr><td>304.1</td><td>342.173801</td><td>-44.541178</td><td>3.230 ± 0.004</td></tr><tr><td>304.2</td><td>342.173909</td><td>-44.541235</td><td>3.230 ± 0.004</td></tr><tr><td>305.1</td><td>342.180303</td><td>-44.524639</td><td>3.082 ± 0.002</td></tr><tr><td>305.2</td><td>342.183869</td><td>-44.525756</td><td>3.082 ± 0.002</td></tr></tbody></table>
<figure><img src="image_26.png" /><figcaption>**Figure D1.** Images and spectra of the newly identified multiply imaged systems. The top row shows inverted coloured images from HST bands F814W, F606W, and F435W with magenta contours highlighting the MUSE narrow band associated with the Lyman α emission. The images are ordered by increasing the index of the multiple images (i.e. XXX.y, where XXX denotes the multiply imaged system index and y the image number in the system), and spectra are shown in the bottom row, with possible emission/absorption lines highlighted. They are centred on emission lines used to define the redshift of the system. In the label, MUSELET indicates a direct detection in the MUSE data set when candidate is for an indirect one.</figcaption></figure>

in the strong lensing analysis. Most of these images have been detected with the automated emission lines and redshift estimator MUSE Line Emission Tracker (MUSELET) provided in the MPDAF<fnref n="12" /> python package. The previous objects are denoted by the MUSELET label when the Candidate label is for indirect detection based on strong lensing model prediction and narrow-band images of the main emission lines (see Richard et al. (2021) for more details). Notably, the system 305 is the only one that has no HST counterpart as previously seen in similar clusters (Lagattuta et al. 2017). The positions and the measured MUSE redshift of these new multiply imaged systems are presented in Table D1.

## APPENDIX E: SN THRESHOLD IMPACT OF THE X-RAY FIT

To assess the possible biases behind our arbitrary choice of SN threshold used in the binning of the cluster field detailed in Section 3.1, we fit the gas distribution with only the X-ray constraints, which were made with two other SN thresholds: 6, 14. These results are then compared to the X-ray-only run used in the model discrimination of the gas-only part explained in Section 6.1. In particular, Fig. E1 presents the relative differences between the best-fitting model made with an SN of 10 (i.e. SN used in this article) and the best-fitting model associated with an SN of 6 (right-panel panel) and an SN of 14 (left-hand panel). As we can see, the differences are quite limited in the case of AS1063 as the differences in the 2D mapping of the

gas are less than 4 per cent on the constrained area and mainly below 2 per cent if we except regions on the right bottom corner of the plot. We can see that the biases on the reconstruction of the gas distribution are pretty low and are not affecting the results of our analyses.

However, if the reconstruction of the projected mass distribution is negligibly affected, it has more influence on the discrimination model of the gas distribution. Indeed, the likelihood values have changed for the best-fitting model: -33 219 (SN = 14), -33 204 (SN = 10), and -33 179 (SN = 6). The discrepancies are expected as we used different constraints, but it also affected our goodness-of-fit procedure as all the percentiles of the expected likelihood distribution have changed. With an SN of 14; the best-fitting model likelihood is outside the "5σ" CI and would not be considered good enough to model the data. In the case of a SN of 6, the likelihood is inside the "3σ" CI, which is better than what we obtained for the SN of 10 (i.e. between the bounds of the "3σ" CI and "5σ" CI).

We showed that the mass measurements provided by our techniques are only affected negligibly by the choice of SN threshold, as the constraints given by the count map are way stronger than the fluctuation of the cooling functions for different binning. These results are dependent of the considered cluster, as the amplitude of the cooling function variations will depend on the temperature gradient between two bins. The only impacting aspect is the bias in the discrimination process, as we could have needed to use more dPIEs to model the gas if we used a SN of 14. But we can balance this problem by the improvement in the likelihood when we reduce the SN threshold, which increases our confidence in our temperature measurement method. It may also be possible to choose an optimal SN by using the one minimizing the likelihood at given gas mass models.

<footnote n="12">¹²https://mpdaf.readthedocs.io/en/latest/muselet.html</footnote>
<figure><img src="image_26.png" /><figcaption>**Figure D1.** Images and spectra of the newly identified multiply imaged systems. The top row shows inverted coloured images from HST bands F814W, F606W, and F435W with magenta contours highlighting the MUSE narrow band associated with the Lyman α emission. The images are ordered by increasing the index of the multiple images (i.e. XXX.y, where XXX denotes the multiply imaged system index and y the image number in the system), and spectra are shown in the bottom row, with possible emission/absorption lines highlighted. They are centred on emission lines used to define the redshift of the system. In the label, MUSELET indicates a direct detection in the MUSE data set when candidate is for an indirect one.</figcaption></figure>

in the strong lensing analysis. Most of these images have been detected with the automated emission lines and redshift estimator MUSE Line Emission Tracker (MUSELET) provided in the MPDAF<fnref n="12" /> python package. The previous objects are denoted by the MUSELET label when the Candidate label is for indirect detection based on strong lensing model prediction and narrow-band images of the main emission lines (see Richard et al. (2021) for more details). Notably, the system 305 is the only one that has no HST counterpart as previously seen in similar clusters (Lagattuta et al. 2017). The positions and the measured MUSE redshift of these new multiply imaged systems are presented in Table D1.

## APPENDIX E: SN THRESHOLD IMPACT OF THE X-RAY FIT

To assess the possible biases behind our arbitrary choice of SN threshold used in the binning of the cluster field detailed in Section 3.1, we fit the gas distribution with only the X-ray constraints, which were made with two other SN thresholds: 6, 14. These results are then compared to the X-ray-only run used in the model discrimination of the gas-only part explained in Section 6.1. In particular, Fig. E1 presents the relative differences between the best-fitting model made with an SN of 10 (i.e. SN used in this article) and the best-fitting model associated with an SN of 6 (right-panel panel) and an SN of 14 (left-hand panel). As we can see, the differences are quite limited in the case of AS1063 as the differences in the 2D mapping of the

gas are less than 4 per cent on the constrained area and mainly below 2 per cent if we except regions on the right bottom corner of the plot. We can see that the biases on the reconstruction of the gas distribution are pretty low and are not affecting the results of our analyses.

However, if the reconstruction of the projected mass distribution is negligibly affected, it has more influence on the discrimination model of the gas distribution. Indeed, the likelihood values have changed for the best-fitting model: −33 219 (SN = 14), −33 204 (SN = 10), and −33 179 (SN = 6). The discrepancies are expected as we used different constraints, but it also affected our goodness-of-fit procedure as all the percentiles of the expected likelihood distribution have changed. With an SN of 14; the best-fitting model likelihood is outside the "5σ" CI and would not be considered good enough to model the data. In the case of a SN of 6, the likelihood is inside the "3σ" CI, which is better than what we obtained for the SN of 10 (i.e. between the bounds of the "3σ" CI and "5σ" CI).

We showed that the mass measurements provided by our techniques are only affected negligibly by the choice of SN threshold, as the constraints given by the count map are way stronger than the fluctuation of the cooling functions for different binning. These results are dependent of the considered cluster, as the amplitude of the cooling function variations will depend on the temperature gradient between two bins. The only impacting aspect is the bias in the discrimination process, as we could have needed to use more dPIEs to model the gas if we used a SN of 14. But we can balance this problem by the improvement in the likelihood when we reduce the SN threshold, which increases our confidence in our temperature measurement method. It may also be possible to choose an optimal SN by using the one minimizing the likelihood at given gas mass models.

<footnote n="12">¹²https://mpdaf.readthedocs.io/en/latest/muselet.html</footnote>
<figure><img src="image_27.png" /><figcaption>**Figure E1.** Relative differences between the best fit gas mass model made with a SN threshold of 14 and 10 (left-hand panel), and with a SN of 6 and 10 (right-panel panel).</figcaption></figure>

## APPENDIX F: MASS MODEL DPIE PARAMETERS

### F1 Mass model of AS1063

Table F1 presents statistics from the sampling of the dPIE parameters associated with each mass halo defined independently. There are two main DM haloes, one centred on the BCG positions (DM-BCG) and one on the NE clump (DM-NE). The parameters of the BCG dPIE are presented on the BCG label, while

the three potentials used for the gas distribution have the Gas prefix.

### F2 Mass models of the mock cluster

Table F2 presents statistics from the sampling of the dPIE parameters associated with each modelling tested on the mock cluster as well as the mock parameters. The dPIE labels are the same as in Table F1. The relevant parameters for the dPIE modelled according to a galaxy relation are presented in Table F3.

<table><caption>**Table F1.** From left to right: dPIE's ID, relative right ascension (Δα), relative declination (Δδ), position angle (φ), Ellipticity (ϵ), core radius ($r_{core}$), cut radius ($r_{cut}$), and velocity dispersion ($\sigma_{0,lt}$). Parameters in brackets are fixed *a priori*, and sampled parameter distributions are presented as median$^{84 \text{ per cent limit}}$ (Best-fit). The relative coordinates are calculated from the BCG centre at α(J2000) = 342.183213 and δ(J2000) = -44.530897. We add the lt label to the velocity dispersion as we refer to its definition in the LENSTOOL implementation.</caption><thead><tr><th>ID</th><th>Δα<br>arcsec</th><th>Δδ<br>arcsec</th><th>φ<br>degree</th><th>ε</th><th>r<sub>core</sub><br>kpc</th><th>r<sub>cut</sub><br>kpc</th><th>σ<sub>0,lt</sub><br>km/s</th></tr></thead><tbody><tr><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.4<sup>+0.3</sup><sub>-0.3</sub>(-37.7)</td><td>0.657<sup>+0.013</sup><sub>-0.013</sub>(0.682)</td><td>112.9<sup>+4.6</sup><sub>-4.1</sub>(118.8)</td><td>[3000]</td><td>1200<sup>+10</sup><sub>-9</sub>(1195)</td></tr><tr><td>DM-NE</td><td>-83.8<sup>+5.7</sup><sub>-5.8</sub>(-81.9)</td><td>41.8<sup>+3.8</sup><sub>-3.4</sub>(46.6)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>86<sup>+42</sup><sub>-32</sub>(61)</td><td>496<sup>+98</sup><sub>-77</sub>(459)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[-36.29]</td><td>[0.431]</td><td>7.5<sup>+3.5</sup><sub>-3.1</sub>(12.5)</td><td>355<sup>+128</sup><sub>-110</sub>(547)</td><td>363<sup>+26</sup><sub>-28</sub>(399)</td></tr><tr><td>Gas-1</td><td>9.0<sup>+3.3</sup><sub>-3.1</sub>(8.8)</td><td>-35.1<sup>+5.1</sup><sub>-6.3</sub>(-47.3)</td><td>93.0<sup>+16.8</sup><sub>-19.1</sub>(81.7)</td><td>0.091<sup>+0.046</sup><sub>-0.044</sub>(0.082)</td><td>662.3<sup>+18.1</sup><sub>-19.5</sub>(689.2)</td><td>[1250]</td><td>316<sup>+9</sup><sub>-8</sub>(312)</td></tr><tr><td>Gas-2</td><td>0.3<sup>+0.2</sup><sub>-0.2</sub>(0.4)</td><td>-1.1<sup>+0.1</sup><sub>-0.1</sub>(-1.1)</td><td>-12.2<sup>+1.3</sup><sub>-1.3</sub>(-11.1)</td><td>0.322<sup>+0.015</sup><sub>-0.017</sub>(0.336)</td><td>66.5<sup>+2.9</sup><sub>-1.9</sub>(63.7)</td><td>[1250]</td><td>212<sup>+5</sup><sub>-5</sub>(204)</td></tr><tr><td>Gas-3</td><td>-20.7<sup>+0.5</sup><sub>-0.5</sub>(-19.7)</td><td>16.6<sup>+0.5</sup><sub>-0.4</sub>(16.1)</td><td>-31.1<sup>+1.5</sup><sub>-1.6</sub>(-31.9)</td><td>0.147<sup>+0.011</sup><sub>-0.011</sub>(0.162)</td><td>169.3<sup>+3.1</sup><sub>-2.9</sub>(173.0)</td><td>[1250]</td><td>372<sup>+6</sup><sub>-6</sub>(385)</td></tr></tbody></table>
<figure><img src="image_27.png" /><figcaption>**Figure E1.** Relative differences between the best fit gas mass model made with a SN threshold of 14 and 10 (left-hand panel), and with a SN of 6 and 10 (right-panel panel).</figcaption></figure>

## APPENDIX F: MASS MODEL DPIE PARAMETERS

### F1 Mass model of AS1063

Table F1 presents statistics from the sampling of the dPIE parameters associated with each mass halo defined independently. There are two main DM haloes, one centred on the BCG positions (DM-BCG) and one on the NE clump (DM-NE). The parameters of the BCG dPIE are presented on the BCG label, while

the three potentials used for the gas distribution have the Gas prefix.

### F2 Mass models of the mock cluster

Table F2 presents statistics from the sampling of the dPIE parameters associated with each modelling tested on the mock cluster as well as the mock parameters. The dPIE labels are the same as in Table F1. The relevant parameters for the dPIE modelled according to a galaxy relation are presented in Table F3.

<table><caption>**Table F1.** From left to right: dPIE's ID, relative right ascension ($\Delta\alpha$), relative declination ($\Delta\delta$), position angle ($\phi$), Ellipticity ($\epsilon$), core radius ($r_{core}$), cut radius ($r_{cut}$), and velocity dispersion ($\sigma_{0,lt}$). Parameters in brackets are fixed *a priori*, and sampled parameter distributions are presented as median$^{84 \text{ per cent limit}}$ (Best-fit). The relative coordinates are calculated from the BCG centre at $\alpha$(J2000) = 342.183213 and $\delta$(J2000) = -44.530897. We add the lt label to the velocity dispersion as we refer to its definition in the LENSTOOL implementation.</caption><thead><tr><th>ID</th><th>Δα<br>arcsec</th><th>Δδ<br>arcsec</th><th>φ<br>degree</th><th>ε</th><th>r<sub>core</sub><br>kpc</th><th>r<sub>cut</sub><br>kpc</th><th>σ<sub>0,lt</sub><br>km/s</th></tr></thead><tbody><tr><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.4<sup>+0.3</sup><sub>-0.3</sub>(-37.7)</td><td>0.657<sup>+0.013</sup><sub>-0.013</sub>(0.682)</td><td>112.9<sup>+4.6</sup><sub>-4.1</sub>(118.8)</td><td>[3000]</td><td>1200<sup>+10</sup><sub>-9</sub>(1195)</td></tr><tr><td>DM-NE</td><td>-83.8<sup>+5.7</sup><sub>-5.8</sub>(-81.9)</td><td>41.8<sup>+3.8</sup><sub>-3.4</sub>(46.6)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>86<sup>+42</sup><sub>-32</sub>(61)</td><td>496<sup>+98</sup><sub>-77</sub>(459)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[-36.29]</td><td>[0.431]</td><td>7.5<sup>+3.5</sup><sub>-3.1</sub>(12.5)</td><td>355<sup>+128</sup><sub>-110</sub>(547)</td><td>363<sup>+26</sup><sub>-28</sub>(399)</td></tr><tr><td>Gas-1</td><td>9.0<sup>+3.3</sup><sub>-3.1</sub>(8.8)</td><td>-35.1<sup>+5.1</sup><sub>-6.3</sub>(-47.3)</td><td>93.0<sup>+16.8</sup><sub>-19.1</sub>(81.7)</td><td>0.091<sup>+0.046</sup><sub>-0.044</sub>(0.082)</td><td>662.3<sup>+18.1</sup><sub>-19.5</sub>(689.2)</td><td>[1250]</td><td>316<sup>+9</sup><sub>-8</sub>(312)</td></tr><tr><td>Gas-2</td><td>0.3<sup>+0.2</sup><sub>-0.2</sub>(0.4)</td><td>-1.1<sup>+0.1</sup><sub>-0.1</sub>(-1.1)</td><td>-12.2<sup>+1.3</sup><sub>-1.3</sub>(-11.1)</td><td>0.322<sup>+0.015</sup><sub>-0.017</sub>(0.336)</td><td>66.5<sup>+2.0</sup><sub>-1.9</sub>(63.7)</td><td>[1250]</td><td>212<sup>+5</sup><sub>-5</sub>(204)</td></tr><tr><td>Gas-3</td><td>-20.7<sup>+0.5</sup><sub>-0.5</sub>(-19.7)</td><td>16.6<sup>+0.5</sup><sub>-0.4</sub>(16.1)</td><td>-31.1<sup>+1.5</sup><sub>-1.6</sub>(-31.9)</td><td>0.147<sup>+0.011</sup><sub>-0.011</sub>(0.162)</td><td>169.3<sup>+3.1</sup><sub>-2.9</sub>(173.0)</td><td>[1250]</td><td>372<sup>+6</sup><sub>-6</sub>(385)</td></tr></tbody></table>
<table><caption>**Table F2.** Same as Table F1 but for the mass model built for the mock cluster. Parameters of the mock are presented under the label *mock*. The four other labels are associated with the reference model (*Ref*.), the model with only the gas improvements (*Gas only*), with the fundamental plane only (*FP only*) and finally both with (*All*).</caption><thead><tr><th>Model</th><th>ID</th><th>Δσ<br>arcsec</th><th>Δδ<br>arcsec</th><th>φ<br>degree</th><th>ε</th><th>r<sub>core</sub><br>kpc</th><th>r<sub>cut</sub><br>kpc</th><th>σ<sub>0,lt</sub><br>km/s</th></tr></thead><tbody><tr><td rowspan="5">Mock</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.5</td><td>0.615</td><td>111.6</td><td>[3000]</td><td>1196</td></tr><tr><td>DM-NE</td><td>-84.5</td><td>37.7</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>37</td><td>736</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>9.9</td><td>273</td><td>367</td></tr><tr><td>Gas-1</td><td>4.1</td><td>-12.1</td><td>119.9</td><td>0.212</td><td>519.3</td><td>[1250]</td><td>336</td></tr><tr><td>Gas-2</td><td>0.5</td><td>-1.1</td><td>-8.3</td><td>0.292</td><td>65.3</td><td>[1250]</td><td>215</td></tr><tr><td>Gas-3</td><td>-22.8</td><td>17.1</td><td>-27.9</td><td>0.133</td><td>154.0</td><td>[1250]</td><td>338</td></tr><tr><td rowspan="3">Ref.</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.3<sup>+0.1</sup><sub>-0.1</sub>(-38.0)</td><td>0.573<sup>+0.008</sup><sub>-0.008</sub>(0.569)</td><td>105.1<sup>+4.1</sup><sub>-3.9</sub>(98.2)</td><td>[3000]</td><td>1256<sup>+9</sup><sub>-14</sub>(1259)</td></tr><tr><td>DM-NE</td><td>-80.2<sup>+1.8</sup><sub>-1.9</sub>(-77.9)</td><td>46.3<sup>+1.4</sup><sub>-1.4</sub>(48.0)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>274<sup>+18</sup><sub>-36</sub>(291)</td><td>459<sup>+23</sup><sub>-21</sub>(445)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>9.2<sup>+5.5</sup><sub>-3.9</sub>(7.2)</td><td>162<sup>+171</sup><sub>-89</sub>(64)</td><td>363<sup>+22</sup><sub>-20</sub>(340)</td></tr><tr><td rowspan="6">Gas<br>only</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.5<sup>+0.1</sup><sub>-0.1</sub>(-38.6)</td><td>0.613<sup>+0.008</sup><sub>-0.008</sub>(0.616)</td><td>107.1<sup>+3.9</sup><sub>-4.5</sub>(106.3)</td><td>[3000]</td><td>1200<sup>+10</sup><sub>-13</sub>(1190)</td></tr><tr><td>DM-NE</td><td>-81.8<sup>+1.8</sup><sub>-2.0</sub>(-79.8)</td><td>43.1<sup>+1.6</sup><sub>-1.6</sub>(42.6)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>224<sup>+50</sup><sub>-74</sub>(239)</td><td>396<sup>+46</sup><sub>-30</sub>(386)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>9.7<sup>+5.1</sup><sub>-4.1</sub>(9.1)</td><td>192<sup>+126</sup><sub>-103</sub>(260)</td><td>364<sup>+21</sup><sub>-20</sub>(352)</td></tr><tr><td>Gas-1</td><td>0.7<sup>+2.3</sup><sub>-2.4</sub>(2.9)</td><td>-8.3<sup>+2.8</sup><sub>-2.9</sub>(-9.5)</td><td>125.7<sup>+5.5</sup><sub>-5.8</sub>(126.5)</td><td>0.204<sup>+0.035</sup><sub>-0.038</sub>(0.217)</td><td>467.0<sup>+16.0</sup><sub>-19.2</sub>(487.2)</td><td>[1250]</td><td>345<sup>+9</sup><sub>-9</sub>(348)</td></tr><tr><td>Gas-2</td><td>0.3<sup>+0.2</sup><sub>-0.3</sub>(0.2)</td><td>-0.9<sup>+0.1</sup><sub>-0.1</sub>(-0.8)</td><td>-8.7<sup>+1.5</sup><sub>-1.5</sub>(-8.8)</td><td>0.307<sup>+0.017</sup><sub>-0.017</sub>(0.301)</td><td>68.1<sup>+2.2</sup><sub>-2.3</sub>(69.6)</td><td>[1250]</td><td>226<sup>+6</sup><sub>-6</sub>(229)</td></tr><tr><td>Gas-3</td><td>-24.3<sup>+0.7</sup><sub>-0.7</sub>(-24.6)</td><td>19.0<sup>+0.7</sup><sub>-0.7</sub>(18.9)</td><td>-26.0<sup>+6.7</sup><sub>-5.4</sub>(-25.9)</td><td>0.084<sup>+0.022</sup><sub>-0.023</sub>(0.067)</td><td>147.4<sup>+4.7</sup><sub>-4.8</sub>(146.8)</td><td>[1250]</td><td>314<sup>+11</sup><sub>-11</sub>(314)</td></tr><tr><td rowspan="3">FP<br>only</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.3<sup>+0.1</sup><sub>-0.1</sub>(-38.3)</td><td>0.572<sup>+0.008</sup><sub>-0.008</sub>(0.575)</td><td>109.1<sup>+3.6</sup><sub>-3.4</sub>(105.2)</td><td>[3000]</td><td>1252<sup>+12</sup><sub>-10</sub>(1265)</td></tr><tr><td>DM-NE</td><td>-82.8<sup>+1.9</sup><sub>-1.8</sub>(-79.6)</td><td>41.6<sup>+1.5</sup><sub>-1.5</sub>(39.9)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>230<sup>+50</sup><sub>-76</sub>(271)</td><td>454<sup>+51</sup><sub>-51</sub>(409)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>12.2<sup>+5.9</sup><sub>-4.4</sub>(9.3)</td><td>245<sup>+102</sup><sub>-118</sub>(107)</td><td>369<sup>+21</sup><sub>-21</sub>(343)</td></tr><tr><td rowspan="6">All</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.5<sup>+0.1</sup><sub>-0.1</sub>(-38.5)</td><td>0.609<sup>+0.009</sup><sub>-0.008</sub>(0.611)</td><td>107.7<sup>+3.8</sup><sub>-3.8</sub>(104.9)</td><td>[3000]</td><td>1194<sup>+11</sup><sub>-10</sub>(1195)</td></tr><tr><td>DM-NE</td><td>-83.0<sup>+1.7</sup><sub>-1.7</sub>(-83.7)</td><td>40.1<sup>+1.4</sup><sub>-1.4</sub>(40.8)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>97<sup>+69</sup><sub>-44</sub>(54)</td><td>512<sup>+146</sup><sub>-89</sub>(662)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>11.6<sup>+5.7</sup><sub>-4.7</sub>(4.6)</td><td>238<sup>+110</sup><sub>-113</sub>(267)</td><td>362<sup>+21</sup><sub>-20</sub>(323)</td></tr><tr><td>Gas-1</td><td>1.0<sup>+2.2</sup><sub>-2.3</sub>(3.2)</td><td>-8.3<sup>+2.9</sup><sub>-2.7</sub>(-11.3)</td><td>125.1<sup>+5.3</sup><sub>-6.0</sub>(115.9)</td><td>0.21<sup>+0.034</sup><sub>-0.037</sub>(0.171)</td><td>462.6<sup>+15.2</sup><sub>-17.1</sub>(494.1)</td><td>[1250]</td><td>344<sup>+9</sup><sub>-9</sub>(340)</td></tr><tr><td>Gas-2</td><td>0.2<sup>+0.3</sup><sub>-0.3</sub>(0.1)</td><td>-0.9<sup>+0.1</sup><sub>-0.2</sub>(-0.8)</td><td>-8.8<sup>+1.6</sup><sub>-1.6</sub>(-9.6)</td><td>0.306<sup>+0.017</sup><sub>-0.016</sub>(0.313)</td><td>68.1<sup>+2.2</sup><sub>-2.2</sub>(70.4)</td><td>[1250]</td><td>226<sup>+6</sup><sub>-6</sub>(230)</td></tr><tr><td>Gas-3</td><td>-24.4<sup>+0.7</sup><sub>-0.7</sub>(-24.3)</td><td>19.0<sup>+0.7</sup><sub>-0.7</sub>(19.0)</td><td>-24.9<sup>+6.6</sup><sub>-5.3</sub>(-31.4)</td><td>0.083<sup>+0.022</sup><sub>-0.022</sub>(0.086)</td><td>147.7<sup>+4.5</sup><sub>-4.8</sub>(149.7)</td><td>[1250]</td><td>313<sup>+11</sup><sub>-10</sub>(319)</td></tr></tbody></table>

<table><caption>**Table F3.** *From left to right:* Model's labels, cut radius ($r_{cut}$), and velocity dispersion ($\sigma_{e,lt}^* / \sigma_{0,lt}^*$; Projected or central) of the reference galaxy. Parameters of the fundamental plane relations with the ratio between $R_e$ and $R_{M_{Tot}}/2$, and the fundamental plane parameters. Labels for each model are the same as in Table F2, and we use the same presentation for the parameter values as in Table F1. The only exception is the use of '-' when the parameter is not relevant to the considered relations.</caption><thead><tr><th>Model</th><th>r<sup>*</sup><sub>cut</sub><br>kpc</th><th>σ<sup>*</sup><sub>0,lt</sub>/σ<sup>*</sup><sub>e</sub><br>km/s</th><th>ν</th><th>b</th><th>a</th><th>c</th></tr></thead><tbody><tr><td>Mock</td><td>9.57</td><td>190</td><td>2.14</td><td>0.296</td><td>-0.565</td><td>1.473</td></tr><tr><td>Ref.</td><td>7.1<sup>+2.38</sup><sub>-2.38</sub>(8.19)</td><td>143<sup>+20</sup><sub>-10</sub>(137)</td><td>-</td><td>-</td><td>-</td><td>[2]</td></tr><tr><td>Gas only</td><td>5.9<sup>+2.22</sup><sub>-1.91</sub>(6.09)</td><td>147<sup>+19</sup><sub>-14</sub>(137)</td><td>-</td><td>-</td><td>-</td><td>[2]</td></tr><tr><td>FP only</td><td>4.96<sup>+3.0</sup><sub>-1.88</sub>(2.92)</td><td>193<sup>+6</sup><sub>-7</sub>(197)</td><td>2.44<sup>+0.38</sup><sub>-0.36</sub>(2.06)</td><td>0.345<sup>+0.024</sup><sub>-0.024</sub>(0.362)</td><td>-0.537<sup>+0.028</sup><sub>-0.025</sub>(-0.524)</td><td>1.357<sup>+0.11</sup><sub>-0.113</sub>(1.571)</td></tr><tr><td>All</td><td>3.88<sup>+1.91</sup><sub>-1.5</sub>(1.74)</td><td>194<sup>+6</sup><sub>-6</sub>(202)</td><td>1.86<sup>+0.35</sup><sub>-0.33</sub>(1.62)</td><td>0.329<sup>+0.024</sup><sub>-0.024</sub>(0.335)</td><td>-0.548<sup>+0.029</sup><sub>-0.028</sub>(-0.539)</td><td>1.334<sup>+0.118</sup><sub>-0.115</sub>(1.55)</td></tr></tbody></table>

This paper has been typeset from a TeX/LA TeX file prepared by the author.

<footnote n="1">¹Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, CH-1290 Versoix, Switzerland</footnote>

<footnote n="2">²ESO, Alonso de Córdova 3107, Vitacura, Santiago, Chile</footnote>

<footnote n="3">³Aix Marseille Univ, CNRS, CNES, LAM, F-13388 Marseille, France</footnote>

<footnote n="4">⁴Department of Astronomy, University of Geneva, Ch. d'Ecogia 16, CH-1290 Versoix, Switzerland</footnote>

<footnote n="5">⁵Department of Astronomy, Yale University, New Haven, CT 06511, USA</footnote>

<footnote n="6">⁶Department of Physics, Yale University, New Haven, CT 06520, USA</footnote>

<footnote n="7">⁷Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK</footnote>

<footnote n="8">⁸Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK</footnote>

<footnote n="9">⁹Astrophysics Research Centre, University of KwaZulu-Natal, Westville Campus, Durban 4041, South Africa</footnote>

<footnote n="10">¹⁰School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Durban 4041, South Africa</footnote>

<footnote n="11">¹¹Instituto de Astrofísica de Canarias, c/Vía Láctea s/n, E-38205 - La Laguna, Tenerife, Spain</footnote>

<footnote n="12">¹²Departamento de Astrofísica, Universidad de La Laguna, E-38205 - La Laguna, Tenerife, Spain</footnote>

<footnote n="13">¹³The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden</footnote>

<footnote n="14">¹⁴Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astro-physique de Lyon (CRAL) UMR5574, F-69230 Saint-Genis-Laval, France</footnote>

<footnote n="15">¹⁵Space Telescope Science Institute, 3700 San Martin Dr, Baltimore, MD 21218, USA</footnote>

<footnote n="16">¹⁶Department of Physics and Astronomy, University of California Riverside, Pierce Hall, Riverside, CA 92521, USA</footnote>

<footnote n="17">¹⁷Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA</footnote>
<table><caption>**Table F2.** Same as Table F1 but for the mass model built for the mock cluster. Parameters of the mock are presented under the label *mock*. The four other labels are associated with the reference model (*Ref*.), the model with only the gas improvements (*Gas only*), with the fundamental plane only (*FP only*) and finally both with (*All*).</caption><thead><tr><th>Model</th><th>ID</th><th>Δσ<br>arcsec</th><th>Δδ<br>arcsec</th><th>φ<br>degree</th><th>ε</th><th>r<sub>core</sub><br>kpc</th><th>r<sub>cut</sub><br>kpc</th><th>σ<sub>0,lt</sub><br>km/s</th></tr></thead><tbody><tr><td rowspan="5">Mock</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.5</td><td>0.615</td><td>111.6</td><td>[3000]</td><td>1196</td></tr><tr><td>DM-NE</td><td>-84.5</td><td>37.7</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>37</td><td>736</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>9.9</td><td>273</td><td>367</td></tr><tr><td>Gas-1</td><td>4.1</td><td>-12.1</td><td>119.9</td><td>0.212</td><td>519.3</td><td>[1250]</td><td>336</td></tr><tr><td>Gas-2</td><td>0.5</td><td>-1.1</td><td>-8.3</td><td>0.292</td><td>65.3</td><td>[1250]</td><td>215</td></tr><tr><td>Gas-3</td><td>-22.8</td><td>17.1</td><td>-27.9</td><td>0.133</td><td>154.0</td><td>[1250]</td><td>338</td></tr><tr><td rowspan="3">Ref.</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.3<sup>+0.1</sup><sub>-0.1</sub>(-38.0)</td><td>0.573<sup>+0.008</sup><sub>-0.008</sub>(0.569)</td><td>105.1<sup>+4.1</sup><sub>-3.9</sub>(98.2)</td><td>[3000]</td><td>1256<sup>+9</sup><sub>-14</sub>(1259)</td></tr><tr><td>DM-NE</td><td>-80.2<sup>+1.8</sup><sub>-1.9</sub>(-77.9)</td><td>46.3<sup>+1.4</sup><sub>-1.4</sub>(48.0)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>274<sup>+18</sup><sub>-36</sub>(291)</td><td>459<sup>+23</sup><sub>-21</sub>(445)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>9.2<sup>+5.5</sup><sub>-3.9</sub>(7.2)</td><td>162<sup>+171</sup><sub>-89</sub>(64)</td><td>363<sup>+22</sup><sub>-20</sub>(340)</td></tr><tr><td rowspan="6">Gas<br>only</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.5<sup>+0.1</sup><sub>-0.1</sub>(-38.6)</td><td>0.613<sup>+0.008</sup><sub>-0.008</sub>(0.616)</td><td>107.1<sup>+3.9</sup><sub>-4.5</sub>(106.3)</td><td>[3000]</td><td>1200<sup>+10</sup><sub>-13</sub>(1190)</td></tr><tr><td>DM-NE</td><td>-81.8<sup>+1.8</sup><sub>-2.0</sub>(-79.8)</td><td>43.1<sup>+1.6</sup><sub>-1.6</sub>(42.6)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>224<sup>+50</sup><sub>-74</sub>(239)</td><td>396<sup>+46</sup><sub>-30</sub>(386)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>9.7<sup>+5.1</sup><sub>-4.1</sub>(9.1)</td><td>192<sup>+126</sup><sub>-103</sub>(260)</td><td>364<sup>+21</sup><sub>-20</sub>(352)</td></tr><tr><td>Gas-1</td><td>0.7<sup>+2.3</sup><sub>-2.4</sub>(2.9)</td><td>-8.3<sup>+2.8</sup><sub>-2.9</sub>(-9.5)</td><td>125.7<sup>+5.5</sup><sub>-5.8</sub>(126.5)</td><td>0.204<sup>+0.035</sup><sub>-0.038</sub>(0.217)</td><td>467.0<sup>+16.0</sup><sub>-19.2</sub>(487.2)</td><td>[1250]</td><td>345<sup>+9</sup><sub>-9</sub>(348)</td></tr><tr><td>Gas-2</td><td>0.3<sup>+0.2</sup><sub>-0.3</sub>(0.2)</td><td>-0.9<sup>+0.1</sup><sub>-0.1</sub>(-0.8)</td><td>-8.7<sup>+1.5</sup><sub>-1.5</sub>(-8.8)</td><td>0.307<sup>+0.017</sup><sub>-0.017</sub>(0.301)</td><td>68.1<sup>+2.2</sup><sub>-2.3</sub>(69.6)</td><td>[1250]</td><td>226<sup>+6</sup><sub>-29</sub>(229)</td></tr><tr><td>Gas-3</td><td>-24.3<sup>+0.7</sup><sub>-0.7</sub>(-24.6)</td><td>19.0<sup>+0.7</sup><sub>-0.7</sub>(18.9)</td><td>-26.0<sup>+6.7</sup><sub>-5.4</sub>(-25.9)</td><td>0.084<sup>+0.022</sup><sub>-0.023</sub>(0.067)</td><td>147.4<sup>+4.7</sup><sub>-4.8</sub>(146.8)</td><td>[1250]</td><td>314<sup>+11</sup><sub>-11</sub>(314)</td></tr><tr><td rowspan="3">FP<br>only</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.3<sup>+0.1</sup><sub>-0.1</sub>(-38.3)</td><td>0.572<sup>+0.008</sup><sub>-0.008</sub>(0.575)</td><td>109.1<sup>+3.6</sup><sub>-3.4</sub>(105.2)</td><td>[3000]</td><td>1252<sup>+12</sup><sub>-10</sub>(1265)</td></tr><tr><td>DM-NE</td><td>-82.8<sup>+1.9</sup><sub>-1.8</sub>(-79.6)</td><td>41.6<sup>+1.5</sup><sub>-1.5</sub>(39.9)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>230<sup>+50</sup><sub>-76</sub>(271)</td><td>454<sup>+51</sup><sub>-50</sub>(409)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>12.2<sup>+5.9</sup><sub>-4.4</sub>(9.3)</td><td>245<sup>+102</sup><sub>-118</sub>(107)</td><td>369<sup>+21</sup><sub>-21</sub>(343)</td></tr><tr><td rowspan="6">All</td><td>DM-BCG</td><td>[0.0]</td><td>[0.0]</td><td>-38.5<sup>+0.1</sup><sub>-0.1</sub>(-38.5)</td><td>0.609<sup>+0.009</sup><sub>-0.008</sub>(0.611)</td><td>107.7<sup>+3.8</sup><sub>-3.8</sub>(104.9)</td><td>[3000]</td><td>1194<sup>+11</sup><sub>-10</sub>(1195)</td></tr><tr><td>DM-NE</td><td>-83.0<sup>+1.7</sup><sub>-1.7</sub>(-83.7)</td><td>40.1<sup>+1.4</sup><sub>-1.4</sub>(40.8)</td><td>[0.0]</td><td>[0.0]</td><td>[2.5]</td><td>97<sup>+69</sup><sub>-44</sub>(54)</td><td>512<sup>+146</sup><sub>-89</sub>(662)</td></tr><tr><td>BCG</td><td>[0.0]</td><td>[0.0]</td><td>[ - 36.29]</td><td>[0.431]</td><td>11.6<sup>+5.7</sup><sub>-4.7</sub>(4.6)</td><td>238<sup>+110</sup><sub>-113</sub>(267)</td><td>362<sup>+21</sup><sub>-20</sub>(323)</td></tr><tr><td>Gas-1</td><td>1.0<sup>+2.2</sup><sub>-2.3</sub>(3.2)</td><td>-8.3<sup>+2.9</sup><sub>-2.7</sub>(-11.3)</td><td>125.1<sup>+5.3</sup><sub>-6.0</sub>(115.9)</td><td>0.21<sup>+0.034</sup><sub>-0.037</sub>(0.171)</td><td>462.6<sup>+15.2</sup><sub>-17.1</sub>(494.1)</td><td>[1250]</td><td>344<sup>+9</sup><sub>-9</sub>(340)</td></tr><tr><td>Gas-2</td><td>0.2<sup>+0.3</sup><sub>-0.3</sub>(0.1)</td><td>-0.9<sup>+0.1</sup><sub>-0.2</sub>(-0.8)</td><td>-8.8<sup>+1.6</sup><sub>-1.6</sub>(-9.6)</td><td>0.306<sup>+0.017</sup><sub>-0.016</sub>(0.313)</td><td>68.1<sup>+2.2</sup><sub>-2.2</sub>(70.4)</td><td>[1250]</td><td>226<sup>+6</sup><sub>-6</sub>(230)</td></tr><tr><td>Gas-3</td><td>-24.4<sup>+0.7</sup><sub>-0.7</sub>(-24.3)</td><td>19.0<sup>+0.7</sup><sub>-0.7</sub>(19.0)</td><td>-24.9<sup>+6.6</sup><sub>-5.3</sub>(-31.4)</td><td>0.083<sup>+0.022</sup><sub>-0.022</sub>(0.086)</td><td>147.7<sup>+4.5</sup><sub>-4.8</sub>(149.7)</td><td>[1250]</td><td>313<sup>+11</sup><sub>-10</sub>(319)</td></tr></tbody></table>

<table><caption>**Table F3.** From left to right: Model's labels, cut radius ($r_{cut}$), and velocity dispersion ($\sigma_{e,lt}^* / \sigma_{0,lt}^*$; Projected or central) of the reference galaxy. Parameters of the fundamental plane relations with the ratio between $R_e$ and $R_{M_{Tot}}/2$, and the fundamental plane parameters. Labels for each model are the same as in Table F2, and we use the same presentation for the parameter values as in Table F1. The only exception is the use of '-' when the parameter is not relevant to the considered relations.</caption><thead><tr><th>Model</th><th>r<sup>*</sup><sub>cut</sub><br>kpc</th><th>σ<sup>*</sup><sub>0,lt</sub>/σ<sup>*</sup><sub>e</sub><br>km/s</th><th>ν</th><th>b</th><th>a</th><th>c</th></tr></thead><tbody><tr><td>Mock</td><td>9.57</td><td>190</td><td>2.14</td><td>0.296</td><td>-0.565</td><td>1.473</td></tr><tr><td>Ref.</td><td>7.1<sup>+2.38</sup><sub>-2.38</sub>(8.19)</td><td>143<sup>+20</sup><sub>-10</sub>(137)</td><td>-</td><td>-</td><td>-</td><td>[2]</td></tr><tr><td>Gas only</td><td>5.9<sup>+2.22</sup><sub>-1.91</sub>(6.09)</td><td>147<sup>+19</sup><sub>-14</sub>(137)</td><td>-</td><td>-</td><td>-</td><td>[2]</td></tr><tr><td>FP only</td><td>4.96<sup>+3.0</sup><sub>-1.88</sub>(2.92)</td><td>193<sup>+6</sup><sub>-7</sub>(197)</td><td>2.44<sup>+0.38</sup><sub>-0.36</sub>(2.06)</td><td>0.345<sup>+0.024</sup><sub>-0.024</sub>(0.362)</td><td>-0.537<sup>+0.028</sup><sub>-0.025</sub>(-0.524)</td><td>1.357<sup>+0.11</sup><sub>-0.113</sub>(1.571)</td></tr><tr><td>All</td><td>3.88<sup>+1.91</sup><sub>-1.5</sub>(1.74)</td><td>194<sup>+6</sup><sub>-6</sub>(202)</td><td>1.86<sup>+0.35</sup><sub>-0.33</sub>(1.62)</td><td>0.329<sup>+0.024</sup><sub>-0.024</sub>(0.335)</td><td>-0.548<sup>+0.029</sup><sub>-0.028</sub>(-0.539)</td><td>1.334<sup>+0.118</sup><sub>-0.115</sub>(1.55)</td></tr></tbody></table>

This paper has been typeset from a TeX/Latex file prepared by the author.

<footnote n="1">¹Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, CH-1290 Versoix, Switzerland</footnote>

<footnote n="2">²ESO, Alonso de Córdova 3107, Vitacura, Santiago, Chile</footnote>

<footnote n="3">³Aix Marseille Univ, CNRS, CNES, LAM, F-13388 Marseille, France</footnote>

<footnote n="4">⁴Department of Astronomy, University of Geneva, Ch. d'Ecogia 16, CH-1290 Versoix, Switzerland</footnote>

<footnote n="5">⁵Department of Astronomy, Yale University, New Haven, CT 06511, USA</footnote>

<footnote n="6">⁶Department of Physics, Yale University, New Haven, CT 06520, USA</footnote>

<footnote n="7">⁷Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK</footnote>

<footnote n="8">⁸Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK</footnote>

<footnote n="9">⁹Astrophysics Research Centre, University of KwaZulu-Natal, Westville Campus, Durban 4041, South Africa</footnote>

<footnote n="10">¹⁰School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Durban 4041, South Africa</footnote>

<footnote n="11">¹¹Instituto de Astrofísica de Canarias, c/Vía Láctea s/n, E-38205 - La Laguna, Tenerife, Spain</footnote>

<footnote n="12">¹²Departamento de Astrofísica, Universidad de La Laguna, E-38205 - La Laguna, Tenerife, Spain</footnote>

<footnote n="13">¹³The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden</footnote>

<footnote n="14">¹⁴Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astro-physique de Lyon (CRAL) UMR5574, F-69230 Saint-Genis-Laval, France</footnote>

<footnote n="15">¹⁵Space Telescope Science Institute, 3700 San Martin Dr, Baltimore, MD 21218, USA</footnote>

<footnote n="16">¹⁶Department of Physics and Astronomy, University of California Riverside, Pierce Hall, Riverside, CA 92521, USA</footnote>

<footnote n="17">¹⁷Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA</footnote>