## 6 Conclusions  
Here is a brief summary of the findings of this work. We have presented the principal Hugoniots of actinium and the lanthanide promethium, which have both never been measured or calculated before, as well as those of terbium, thulium, and lutetium, the three least studied of the remaining lanthanides. We used the analytic framework established in our previous publication [22]. All five sets of the relevant parameters are summarized in Table 1. The five principal Hugoniots are compared to the available experimental data (for Tb and Tm only) and independent theoretical calculations in Figures 3-7. As can be clearly seen in these figures, in each of the five cases, the agreement between the new analytic model and independent data is very good except for the three Hugoniots from the TEFIS database, which all violate [PERSON]'s theoretical constraint on the maximum compression. Their behavior is analyzed in some detail in our paper. We anticipate that new experimental measurements could shed more light on a potential systematics of the lanthanide Hugoniots in terms of the values of the parameters \(C\) and \(B\), the maximum compression \(\eta_{\rm max}\), etc. We expect our results to serve as the initial guidance for such experiments. Our new analytic model of the principal Hugoniot [22] can be used for the validation of the \(P\)-\(V\)-\(T\) EOS by comparing the Hugoniot produced by the EOS to that given by the model. Additionally, the new model itself can be used as a basis for EOS construction. Indeed, if the Gruneisen parameter along the Hugoniot is available, e.g., using the approach discussed in this work, then it can be used in the Mie-Gruneisen-type EOS \(P-P_{\rm H}=\gamma_{\rm H}\,\rho\,(E-E_{\rm H})\)[45], where the subscript \"H\" implies that the corresponding variable is in shock-compression conditions. This EOS can then be brought in direct correspondence to the more familiar Mie-Gruneisen (M-G) EOS, \(P-P_{c}=\gamma\,\rho\,(E-E_{c})\), where the subscript \"c\" implies the cold \((T=0)\) conditions, since there exists a direct algebraic connection between \(\gamma_{\rm H}\) (of this work) and \(\gamma\) of the M-G EOS [45]. The analytic model developed in our previous study [22] and applied to five substances in this work can be used to calculate of the Hugoniots of other substances, specifically, other lanthanides and/or actinides. In this respect, the analytic knowledge of the regimes of the Hugoniot past the turnaround point is very important. In a very recent publication [46], a team of astronomers report a detailed study of a pair of shock waves produced by a collision of two clusters of galaxies that occurred roughly a billion years ago. The shocks that are associated with cluster mergers are known as radio relics and they can be used to probe the properties of the intergalactic space within the cluster, known as the intracluster medium, as well as intracluster dynamics. The study focused on a particular cluster called Abell 3667, which at least 550 galaxies are associated with and which is still coming together. It was concluded that the shock waves are propagating through it at velocities of \(\sim\)1500 km/s, which are 5-6 times larger than the velocities corresponding to turnaround points [22], \(U_{s}^{\rm max}=2c+b\sqrt{c/a}\approx b\sqrt{c/a}\sim 1.2\,U_{p}^{\rm max}\sim\)250-300 km/s (see Table 1). Moreover, at such shock velocities, many elements, especially the low-\(Z\) ones, will be under the conditions that are beyond the validity of [PERSON]'s parabolic representation (5), i.e., in the high-\(P\) regime considered in this work. Hence, to predict the properties of the intracluster medium and to describe intracluster dynamics, an analytic model of the principal Hugoniot in the high-\(P\) regime is a must.