Dynamical models’ view on Globular Clusters and their Multiple Populations

Generally, energy equipartition refers to the exchange of energy due to gravitational encounters that bring more massive stars to a lower speed, while less massive ones result faster. Thus, the equipartition concerns the kinetic energy and is associated with the efficiency of the relaxation process. As a consequence, it depends on the radial position thus must be considered as local rather than global. 

Thanks to N-body simulations, we have several indications on the efficiency of such a process and its relation with structural and internal properties (Trenti & van der Marel 2013; Webb & Vesperini 2017; Pavlík & Vesperini 2021, 2022). One of the most important results is the Bianchini et al. (2016) fitting function for the velocity dispersion dependence on stellar mass \(\boldsymbol{\sigma(m)}\), obtained in the inner regions of simulated clusters. The relation measures the degree of kinetic energy equipartition through the equipartition mass \(\mathbf{m_\mathrm{eq}}\), showing that more massive stars are closer to equipartition and only those with \(m > m_\mathrm{eq}\) reached the complete equipartition limit \(\sigma(m) \propto m^{-1/2}\).

Our dynamical model is a continuous formulation of a multi-mass King-like model (Teodori, Straniero and Merafina 2024) as an extension of the Da Costa & Freeman (1976) model and recovering the King (1966) formalism. The distribution function (DF) is obtained as an approximated solution of a generalized expression of the Fokker-Planck equation, valid for collisional systems with a mass distribution. The DF describes the distribution in positions, velocities, and masses. Its expression reads 

\(g(r,v,m) = k(m)\,\mathrm{e}^{-m[\varphi(r)-\varphi_\mathrm{0}]/(k_\mathrm{B} \theta)}\left[\mathrm{e}^{-\varepsilon(v,m)/(k_\mathrm{B}\theta)}- \mathrm{e}^{-\varepsilon_\mathrm{c}(r,m)/(k_\mathrm{B}\theta)}\right],\)

where \(\varepsilon=mv^2/2\) and \(\varepsilon_\mathrm{c}=mv_\mathrm{e}^2/2\) are the kinetic and cut-off energy of a star with mass \(m\), with \(v_\mathrm{e}\) the escape velocity. Here \(\varphi(r)\) is the gravitational potential (with \(\varphi_\mathrm{0}\) its value in the cluster center), and \(r\) is the radial coordinate. The variable \(\theta\) is the thermodynamic temperature (Merafina 2017, 2018, 2019), and \(k_\mathrm{B}\) is the Boltzmann constant. The multiplying factor \(k(m)\) weights the DF of each mass \(m\) (as for the Da Costa & Freeman 1976 model). The mass density is obtained by integrating \(g(r,v,m)\) over velocities and masses, and the equilibrium configuration is given by solving the Poisson equation. The inputs of the model are \(\boldsymbol{\Phi_0 = (\varphi_R - \varphi_0)/k\theta }\) and the mass function, in terms of a single power law with slope \(\boldsymbol{\alpha}\) and the minimum and maximum mass \(\mathbf{m_\mathrm{min}}\) and \(\mathbf{m_\mathrm{max}}\), respectively.

We fit the velocity dispersion dependence on stellar mass \(\boldsymbol{\sigma(m)}\) by Watkins et al. 2022 (HST proper motion) through the Bianchini function and the model prediction. To obtain the velocity dispersion relation with mass for the model, we first need the \(\sigma(r,m)\) profile. This quantity is first projected in 2D and then averaged in the radial shell covered by data (see Watkins et al. 2022, Table 2). 

We obtain a good agreement between predictions and observables as well as a large compatibility between the model and the Bianchini function (Figure 1 for NGC 6397). We expect a relation between the main model parameter \(\boldsymbol{\Phi_0}\) and \(\mathbf{m_\mathrm{eq}}\), both quantifying the degree of energy equipartition (Figure 3), as we obtain. However, we must consider shell selection and projection effects for \(m_\mathrm{eq},\) that alter its prediction. 

By fitting the theoretical \(\sigma(r,m)\) radial profile with the Bianchini function as well as its projection, we can build radial profiles for \(m_\mathrm{eq}\), revealing the effect of project and radial selection effects, both reducing the predicted equipartition degree. We notice that the 3D value of \(m_\mathrm{eq}\) is more stable within the core, where the maximum degree of equipartition is reached. Furthermore, we obtain that the mass function slope \(\boldsymbol{\alpha}\) can become important at large radii (Figure 2).

Dynamically old clusters have experienced several relaxation times, the timescale for the relaxation process, during their life. The equipartition mass is known to relate with the number of core relaxation times \(N_\mathrm{core}\) as well as other structural properties like the concentration.

Here, we explore the relation between the dynamical model's main parameter \(\boldsymbol{\Phi_0}\) and the concentration \(\mathbf{c}\) as predicted by the model and from the literature (Figure 4).
We also present in Figures 5 and 6 the relation with \(\mathbf{N_\mathrm{core}}\) as well as the \(\mathbf{A^{+}}\) parameter that tracks the segregation of Blue Straggler Stars with respect to other stars (Ferraro et al. 2012, 2020; Alessandrini et al. 2016; Lanzoni et al. 2016). All these quantities increase with the dynamical age, as we expect and find for \(\Phi_0\). On the contrary, the core radius is smaller for dynamically older clusters, as we obtain in Figure 7.

We get an independent estimate of the parameter \(\boldsymbol{\Phi_0}\) by fitting Surface Brightness Profiles (SBPs) by Trager et al. 1995 with the prediction of the dynamical model. The analysis and fits were done by following a procedure similar to that described by McLaughlin & van der Marel (2005) and Zocchi et al. (2012), applied to single-mass models. We remove the constant mass-to-light ratio assumption and build the prediction by adopting a theoretical mass-magnitude relation. We use BaSTI isochrones (Hidalgo et al. 2018; Pietrinferni et al. 2021, 2024; Salaris et al. 2022) for an age of 13 Gyrs, with \([\alpha/\text{Fe}]=+0.4\), \(Y = 0.247\), and a different metallicity \([\text{Fe}/\text{H}]\) for each cluster, taken from the Harris catalog, as a reference case.

We summarize here our main findings.

• Our multi-mass King-like dynamical model predicts and quantifies the energy equipartition degree in GCs. It well fits internal kinematics observations (HST proper motion by Watkins et al. 2022) and shows a similar confidence with Bianchini et al. 2016 fitting function.
  
  
• The parameter \(\boldsymbol{\Phi_0}\) mainly determines the energy equipartition degree and shows a relation with the equipartition mass \(\mathbf{m_\mathrm{eq}}\). Larger equipartition comes with larger \(\Phi_0\) values and lower \(m_\mathrm{eq}\). However, complete equipartition is never reached in the analyzed clusters. In addition, \(\mathbf{m_\mathrm{eq}}\) suffers shell selection and projection effects, underestimating the equipartition degree. On the contrary, \(\Phi_0\) uniquely determines the radial and project quantities of the system without any loss of information.
  
  
• As expected, the equilibrium parameter \(\boldsymbol{\Phi_0}\) tracks the dynamical state of GCs, further supported by its relation with structural properties like concentration \(\mathbf{c}\), \(\mathbf{N_\mathrm{core}}\), \(\mathbf{A^+}\), and core radius \(\mathbf{r_\mathrm{c}}\).
  
  
• The dynamical model can successfully fit the observed SBPs (Trager et al. 1995) upon the adoption of an appropriate theoretical mass-magnitude relation. This provides an independent estimate of \(\boldsymbol{\Phi_0}\) as well as other structural properties (such as central surface brightness, core, and tidal radius). 
  The obtained \(\Phi_0\) values are compared with the ones computed by fitting \(\sigma(m)\) and result compatible at the \(\boldsymbol{2\sigma}\) level for almost all analyzed clusters.

Additional results for the analyzed clusters, such as the full list of obtained parameters, can be found in the original paper Teodori, Straniero, Merafina 2024, while additional plots, including the fits of \(\sigma(m)\) and the SBPs, are on Zenodo.

Globular Clusters stars are known to suffer mass segregation, that is, the sink of massive stars toward central regions, while low mass stars are closer to the edge of a cluster. Such a process, coupled with the evaporation due to tidal effects, can alter the shape of the mass function, as well as the structural properties of GCs.

Although mass segregation is a well-known process, a self-consistent description is still missing. In our work, we show that multi-mass King-like models are able to predict and measure the degree of mass segregation, similarly to what was done for the energy equipartition degree (Teodori, Straniero, Merafina 2024).

The dynamical model presented in the Energy equipartition panel allows us to predict several mass-dependent quantities. Among them, we consider the half-mass radius dependence on stellar mass \(\mathbf{r_\mathrm{h}(m)}\). Such a quantity is obtained by the cumulative mass profile of a star with mass \(m\), by the condition \(M(r_\mathrm{h},m) = M_\mathrm{tot}(m)/2 \). Since the model provides dimensionless quantities, we get \(r_\mathrm{h}(m)\) normalized to the tidal radius \(r_\mathrm{t}\). The model takes as input \(\Phi_0\) and the mass function, through a slope \(\alpha\) and the minimum and maximum mass \(m_\mathrm{min}\) and \(m_\mathrm{max}\), respectively. The prediction on \(r_\mathrm{h}(m)\) depends on these parameters.

The quantity \( \mathbf{r_\mathrm{h}(m)} \) is expected to be a decreasing function of the mass for any degree of mass segregation. It should be almost flat for a completely non-segregated cluster. 

Linear relation

The slope of the non-linear prediction of the model for \(r_\mathrm{h}(m)\) can be well approximated by using a linear relation

\(\frac{r_\mathrm{h}}{r_\mathrm{t}}(m) = -\beta m +q \,.\)

Here, \(\boldsymbol{\beta}\) is estimating at first order the slope of \(\mathbf{r_\mathrm{h}(m)}\) and thus provides an empirical estimate of the segregation degree, of practical use. We can also use \(\beta\) to understand the role of model parameters in changing the mass segregation degree.

We explore a wide range of model parameters to identify how they can alter the mass segregation degree. In particular, we consider \(\Phi_0 \in [0.001, 10] M_\odot^{-1}\) and \(\alpha \in [-2, 0]\), that is, a steep and a flat mass function, respectively. We keep fixed \(m_\mathrm{min}=0.1 M_\odot\) while we vary \(m_\mathrm{max}\) from \(1.0 M_\odot\), representative of a current state GCs' mass range, and up to \(100 M_\odot\), appropriate for an initial mass function. 

In Figure 1, the green line is the predicted half-mass radius - mass relationship for a model with \(\Phi_0=0.01\,M_\odot^{-1}\), \(\alpha = 0.0\), and \(m_\mathrm{max}=100 \,M _\odot\), with the dashed black line corresponding linear fit that yields \(\beta = 1.23 \times 10^{-4} \,M_\odot^{-1}\).

As is evident from Figure 2, the linear slope \(\boldsymbol{\beta}\) of \(\mathbf{r_\mathrm{h}(m)}\) is mainly ruled by \(\boldsymbol{\Phi_0}\), while variations in the mass function (slope and mass range) have a minor role. As expected, a larger \(\Phi_0\) gives more segregation (larger \(\beta\)). 

Note that we can not arbitrarily choose any value of model parameters. A large \(m_\mathrm{max}\) provides consistent theoretical profiles only for low enough \(\Phi_0\). Furthermore, too small \(\Phi_0\) values predict a \(r_\mathrm{h}(m)\) strongly non-linear at low masses, and the linear fit is not a good approximation, yielding too large \(\beta\). Such a validity limit again changes with the maximum mass.
As a consequence, reasonable and consistent values of \(\Phi_0\) are related to the masses composing the system. As an example, for a current mass function, \(1 \lesssim \Phi (M_\odot^{-1})\lesssim 10 \) is expected, while for an initial mass function, \(0.01 \lesssim \Phi (M_\odot^{-1})\lesssim 0.1 \) is more appropriate.

The presented model offers a self-consistent approach to generate primordially segregated stellar systems as initial conditions for N-body simulations. 

To generate our clusters, we develop a customized version of the MCLUSTER code. We show in Figures 3 and 4 that the normalized mass density and mean square velocity profiles of generated stars match the theoretical prediction, as resulting from the corresponding model (\(\Phi_0=0.01 \,M_\odot^{-1}\) and a Kroupa mass function with \(m \in [0.1, 100]\,M_\odot\)).

To appreciate the differences in the mass segregation degree, we compare the generated and predicted \(\mathbf{r_\mathrm{h}(m)}\) for \(\boldsymbol{\Phi_0=0.01 \,M_\odot^{-1}}\) with \(\boldsymbol{\Phi_0=0.1 \,M_\odot^{-1}}\). For the former (Figure 5), the half-mass radius of mass bins is mostly flat and shows a spread at high masses due to statistical effects (they are less in number). This case is compatible with an almost absent segregation. The segregation effect is much more evident for the latter (Figure 6).

We have found that

• Multi-mass dynamical models can predict the mass segregation degree through the \(r_\mathrm{h}(m)\) function.
  
  
• For each choice of the mass function shape, there exists a relation between the model parameter \(\boldsymbol{\Phi_0}\) and \(\boldsymbol{\beta}\), the slope of a linear relation between \(r_\mathrm{h}(m)\) and mass. Both \(\Phi_0\) and \(\beta\) quantify the degree of mass segregation.
  
  
• Multi-mass dynamical models can generate initial conditions for N-body simulations, allowing us to set a degree of primordial segregation self-consistently.

The dynamical evolution of Multiple Populations (MPs) is often studied through N-body simulations. The initial conditions are modeled to consider the results of SG formation studies, such as a more centrally concentrated SG. Furthermore, the current observational constraints require an early predominant loss of FG stars due to tidal effects.

Current literature works (Vesperini et al. 2018, 2021) typically assume that both populations are distributed in position and velocities according to the King single-mass model, randomly assigning masses distributed with an Initial Mass Function (IMF), often a Kroupa IMF. In addition, the early expansion driven by stellar evolution feedbacks that causes the preferential loss of FG is typically controlled by radial scaling related to the filling of the tidal radius. Stars are appropriately scaled to not reach the tidal limit by setting the ratio \(\mathbf{r_\mathrm{h}/r_\mathrm{J}}.\) Furthermore, to account for the SG formation in the inner regions, stars are scaled by setting a ratio \(\mathbf{r_\mathrm{h,FG}/r_\mathrm{h,FG}}\) larger than 1, typically from 5 up to 20.

However, the quantities described are free parameters and need exploration and constraints. In particular, if from one side the IMF of GCs and their MPs is not known, in principle a multi-mass model should be preferred for generating stars, for consistency reasons.

Several literature works have highlighted that clusters arise from their violent relaxation phase segregated in mass (Livernois et al. 2021 and refs therein). This early dynamical evolution, starting from the formation of stars and occurring due to the imbalance between potential and kinetic energy, is responsible for the monolithic structure of clusters. Violent relaxation brings to an initial collapse of the system and then adjustments toward virial equilibrium. Such an early abrupt phase has a typical timescale of the order of the free-fall timescale.

In the context of MPs, the violent relaxation phase should leave a degree of mass segregation in FG stars, a possibility poorly investigated in literature. We study the effect of primordial segregation by using the predictions of our dynamical model, as shown in the Mass segregation panel.

We set up initial conditions for MPs according to the current literature approach, replacing the choice of the King distribution function with that of our multi-mass model (Teodori, Straniero, Merafina 2024). We study the effect of primordial segregation mainly by changing the \(\boldsymbol{\Phi_0}\) parameter. Furthermore, we consider that FG and SG stars can have an age difference. We adopt the AGB scenario and account that FG stars can be up to 100 Myr old.

We show here N-body simulation results by considering an equal fraction of FG and SG stars, an underfilling condition (to regulate the tidal filling of cluster stars) given by \(\mathbf{r_\mathrm{h}/r_\mathrm{J} = 0.04}\), and a relative spatial concentration between MPs through \(\mathbf{r_\mathrm{h,FG}/r_\mathrm{h,SG}=5}.\) We consider a Kroupa IMF and simulate 12 Gyr of evolution.

As a first step, we set up a reference for the multi-mass model parameters. A choice of \(\boldsymbol{\Phi_0=0.01 M_\odot ^{-1}}\) is a good value to get an almost absent primordial segregation degree. We adopt this value for both FG and SG.

Thus, we set up both FG and SG with this value and compare the results with current knowledge approaches, namely a King single-mass model with \(\mathbf{W_0 = 7.0}\) for both FG and SG. The outcomes are mostly similar, with minor variations regarding the evolution of mass and number of stars. Indeed, in the multi-mass case (hereafter Model 1), we get a slightly smaller loss of stars with respect to the single-mass model (hereafter Model 0).

Furthermore, the radial mixing followed by the temporal evolution of  \(\mathbf{r_\mathrm{h,FG}/r_\mathrm{h,SG}}\) initially increases more for Model 1 with respect to Model 0, keeping a slightly higher value at the end of the evolution. This is related to a stronger early expansion of FG stars, due to the non-zero primordial segregation degree.

Note that in both cases, the final cluster shows a higher fraction of SG stars, up to 65%, while FG stars are around 35%. These values are consistent with observational evidence for most GCs (Gratton et al. 2019).

A relevant primordial segregation degree in FG can be considered by setting \(\boldsymbol{\Phi_\mathrm{0,FG}=0.1 M_\odot^{-1}}\), while keeping a non-segreated SG with \(\boldsymbol{\Phi_\mathrm{0,SG}=0.01 M_\odot^{-1}}\), hereafter Model 2.

With respect to Model 1, we obtain a larger mass loss due to evaporation for the cluster and both populations, slightly more important for FG stars (Figures 1 and 2). This also results in a slightly higher \(N_\mathrm{SG}/N_\mathrm{FG}\) (Figures 3 and 4), that is, a larger final fraction of SG stars with respect to FG stars.

The evolution of \(\mathbf{r_\mathrm{h,FG}/r_\mathrm{h,SG}}\) is similar, with an initial strong increase due to the early expansion. For the segregated case (Model 2) the maximum of the curve is higher. After 12 Gyr of evolution, we have a lower value with respect to Model 1 (Figures 5 and 6).

Our preliminary results show that a degree of primordial segregation in FG stars has mainly two effects:

• It increases the loss of stars, both for the early and long-term evolution. This is related to the spatial distribution of stellar mass, with massive stars more centrally concentrated than low-mass stars, whose evaporation is favored. This effect in naturally more evident for FG stars and favors the predominance of SG stars.
  
  
• It increases the speed toward spatial mixing between MPs, despite a stronger initial expansion as resulting from stellar evolution feedbacks.
  

The main effect of considering an age for FG stars when starting the dynamical evolution is reducing the stellar evolution feedback of massive FG stars. Indeed, at the beginning of the cluster dynamical evolution, we start from a maximum mass of FG stars \(\approx 5 M_\odot\) for an age of 100 Myr. This is expected to reduce the overall cluster mass loss due to stellar evolution.

From the dynamical model point of view, a lower maximum mass allows considering an even larger segregation, up to \(\boldsymbol{\Phi_0 = 1.0 M_\odot ^{-1}}\).

A dedicated set of simulations is required to explore the joint role of age and primordial segregation.

Our simulations show that primordial segregation plays a role in global structural properties of GCs and MPs. Future analysis can reveal the impact on other properties as well as local quantities. Among the most interesting, we have the local degree of energy equipartition and mass segregation itself, as well as anisotropic velocities.

Furthermore, it is expected that FG stars were initially much more massive than today, up to one or two orders of magnitude. However, simulations of an early predominant FG through direct N-body methods seem lacking in literature, while few cases exist for Monte Carlo-based approaches (Vesperini et al. 2021). This is probably related to the need for a strong loss of FG stars, mainly during the early phases, that likely requires a lot of computational resources, as we have found from preliminary tests. Anyway, such an aspect must be addressed for more realistic simulations in order to better constrain the free parameters of the adopted scenario and help in the comprehension of the formation and evolution of GCs and their MPs.

Orbiting around galaxies, stellar systems known as Globular Clusters (GCs) experienced a peculiar dynamical evolution from their formation up to now, lasting approximately 10 Gyr. The behavior of such objects is ruled by gravitational encounters for most of their lives, driving these systems toward relaxation. Such a process shapes the structural properties and the kinematic fingerprint of stars, mainly driving GCs toward a degree of local kinetic energy equipartition and mass segregation. The galactic tidal field is responsible for the loss of stars, a phenomenon known as evaporation.

Although such properties and processes of GCs are well known, their characterization is still discussed and unclear, in particular from the theoretical point of view. 

In this work, we outline the role of dynamical models in predicting mass-based processes like energy equipartition and mass segregation. 
We show that our dynamical model can be used to generate initial conditions for N-body simulations for primordially segregated clusters.

GCs were considered as a single stellar population with low metallicity. However, the detection of chemical and photometrical anomalies revealed a more complex situation, with the presence of generations of stars, i.e., Multiple Populations (MPs). Their formation scenario is still unclear and a matter of debate, since most of the proposed explanations continue to fail the comparison with the whole observational evidence.
Indeed, a variety of processes from different astronomical fields must be considered, such as stellar formation and evolution as well as early internal dynamics, in a poorly constrained early Galactic environment.

In most of the proposed scenarios, a First Generation (FG) of stars is formed from a primordial composition. Later, at least a Second Generation (SG) is formed in situ by the material lost by FG stars and mixed to some extent with a gas with pristine composition. Nowadays, SG stars are those showing chemical and photometrical anomalies.

The formation of SG stars occurs in the framework of FG stars, whose gravitational field leads to a concentrated stellar formation. As a consequence, after their formation, SG stars are more centrally concentrated. At this time, cluster dynamical evolution proceeds, ruled by stellar evolution feedbacks (i.e., mass loss) that can bring the cluster toward an early dynamical evolution, causing the preferential loss of FG stars, more spatially extended. This can explain current observations concerning an often predominant SG. Then, dynamical evolution advances as driven by the relaxation process, mainly mixing MPs and altering their structural, spatial, and kinematic properties (D'Ercole et al. 2008).

Although the dynamical evolution is studied by existing works, it is important to highlight that the initial properties of FG and SG are poorly known, starting from their density profile as well as their initial mass function. 
In particular, many works have revealed that an early dynamical phase, known as violent relaxation, produces a segregated cluster. Such a process would affect mainly the properties of FG stars.
The role of a primordial segregation degree at the beginning of the dynamical evolution, after SG formation, is still not known.
Furthermore, an age difference between FG and SG stars is expected, and this may alter the dynamical evolution, mainly due to stellar evolution feedbacks. Such an aspect is not considered in the current state-of-the-art N-body simulations for MPs evolution.

We propose a novel and self-consistent approach to set up initial conditions for primordially segregated clusters. We also explore the role of an age difference between MPs, mainly considering an old FG. In this case, we adopt the AGB scenario, where AGB stars of the FG are responsible for the chemical enrichment of the material producing SG stars. 

Our work underlines how a multi-mass King-like model is able to predict and quantify GCs' processes related to the mass distribution, such as energy equipartition and mass segregation.

Concerning energy equipartition, we obtained a good agreement between our prediction for the velocity dispersion dependence on stellar mass and observational material (Watkins et al. 2022). Our results are also highly compatible with the Bianchini et al. 2016 fitting function. We find a strict relation between the equipartition mass \(\mathbf{m_\mathrm{eq}}\) and the equilibrium parameter of the dynamical model \(\boldsymbol{\Phi_0}\). These parameters quantify the equipartition degree and are both related to the dynamical state of GCs, as found when comparing them with other structural properties. The dynamical model also yields surface brightness profiles that well fit the observed ones (Trager et al. 1995), upon an appropriate choice of a theoretical mass-magnitude relation.

Through the relation between the half-mass radius and stellar mass \(\mathbf{r_\mathrm{h}(m)}\), the dynamical model also provides the degree of mass segregation. At first order, a linear relation results sufficient to estimate the role of input parameters. As expected, \(\boldsymbol{\Phi_0}\) is the main responsible for the segregation degree and strongly correlates with the slope of the linear relation \(\boldsymbol{\beta}\), while the mass function plays a minor role. As a consequence, our model can be used to generate stellar clusters with an initial degree of mass segregation, as we show and check.

The possibility of setting a degree of primordial segregation self-consistently allows replacing current assumptions on initial conditions for GCs and their MPs. In particular, we can consider that FG stars experienced the violent relaxation phase, arising segregated in mass. Preliminary N-body simulation results show that primordial segregation alters the cluster mass loss due to evaporation, with different strengths for FG and SG, as well as radial mixing between MPs, with respect to non-segregated cases. Further analysis can reveal the effect on other structural and kinematic properties, such as equipartition, segregation, and anisotropy. Furthermore, also an age difference between MPs must be considered in simulations set up, likely altering the feedback of stellar evolution in cluster dynamics. Finally, more realistic simulations must address the initial predominance of FG stars and their strong loss, a task currently challenging with direct N-body methods and limited computational resources.

If you are interested in sharing any comment, suggestion or have question regarding the poster, please contact me at matteo.teodori@inaf.it, or visit my website (you can scan the QR code).

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