## 5 Discussion  
As Figures 3-7 clearly demonstrate, in each of the five cases, the agreement between the new analytic model and independent data is very good except for the three Hugoniot: from the TEFIS database. As a matter of fact, the three TEFIS Hugoniots appear to violate [PERSON]'s constraint \(4<\eta_{\rm max}<7\), which is based on rigorous theoretical grounds. (We have determined that this is the case for the vast majority of TEFIS Hugoniots for other substances.) It is worthwhile to dwell on [PERSON]'s constraint in some more detail. In Ref. [38], [PERSON] derives the formula for the maximum compression on the principal Hugoniot:  
\[\eta_{\rm max}\,=\,\frac{4\,(1\,+\,7\,{\cal C})}{1\,+\,4\,{\cal C}}, \tag{8}\]  
where \({\cal C}\cong 0.011\,\,A\,Z^{4.2}/[\rho_{0}\,(1\,+\,Z)^{4}]\). As can be clearly seen in Figure 4 of [22], this \(\eta_{\rm max}\) and the one given by the new analytic model are in very good agreement with each other, which lends further support to both formulations. [PERSON]'s constraint now follows directly from (8):  
\[\eta_{\rm max}\,=\,\frac{4\,+\,28\,{\cal C}}{1\,+\,4\,{\cal C}}\,<\,\frac{7\, +\,28\,{\cal C}}{1\,+\,4\,{\cal C}}\,=\,7, \tag{9}\]  
and:  
\[\eta_{\rm max}\,=\,\frac{4\,+\,28\,{\cal C}}{1\,+\,4\,{\cal C}}\,>\,\frac{4\, +\,16\,{\cal C}}{1\,+\,4\,{\cal C}}\,=\,4. \tag{10}\]  
We believe, the reason for this behavior of TEFIS Hugoniots is the way these Hugoniots are constructed; specifically, the scheme used in Refs. [39; 40] for the interpolation between the portion of the Hugoniot below and across the turnaround point described by the TFK model, and that above the turnaround point, which incorporates electron shell effects, such shell effects are clearly seen in Figures 4, 5, and 7 for Tb, Tm, and Ac, respectively. In other words, the violation of [PERSON]'s theoretical constraint by the TEFIS Hugoniots may be the artifact of the interpolation scheme used for their construction. Otherwise, for \(P\) below and above the turnaround point, \(1<\eta\stackrel{{<}}{{\sim}}4\) and \(\sim 4.5<\eta\stackrel{{<}}{{\sim}}6\), agreement between the TEFIS Hugoniots and both the new analytic model and the Thomas-Fermi model with corrections is generally good. Let us note that in a more realistic case of a shock compression of a substance beyond the corresponding turnaround point, the Hugoniot must be modeled by taking into account the well-known effects of the contribution of both the equilibrium radiation of hot plasma and relativistic effects [41; 42; 43; 44]. Indeed, at a turnaround point, \(U_{p}\sim 200\)-\(250\) km/s (see Table 1), which constitutes \(\stackrel{{<}}{{\sim}}0.1\%\) of the speed of light, and at the med-\(P\)-high-\(P\) transition point, \(U_{p}\cong 1700\pm 200\) km/s, about 0.5% of the speed of light. Hence, as the system enters the high-\(P\) regime, relativistic effects are expected to start manifesting themselves and eventually to dominate the evolution of the system at even higher \(U_{p}\). Our model can, in principle, be modified to incorporate these effects. For instance, taking into account the radiation-dominated (or the so-called strong shock) regime can be done by replacing equations for \(U_{s}(U_{p})\) and \(P(\eta)\) describing the high-\(P\) regime with their counterparts stemming from the physics of a photon gas. This goes beyond the scope of the present work, but will be undertaken in one of our subsequent studies. The new model discussed in this work does not incorporate potential electronic shell effects in the high-\(P\) regime. If present, they manifest themselves in terms of some \"irregularities\", that is, changes of the sign of \(dP/d\eta\) of the continuous line \(P=P(\eta)\) over some (small) regions of \(P\); such \"irregularities\" are seen, e.g., in the Tartarus Hugoniots of both Pm and Lu around \(10^{5}\)-\(10^{6.5}\) GPa in Figures 3 and 6, respectively, since Tartarus takes electronic shell effects into account explicitly.