{'fraction_non_alphanumeric': 0.06648065024869586, 'fraction_numerical': 0.027174572364430424, 'mean_word_length': 3.5505106265525805, 'pattern_counts': {'":': 0, '<': 12, '<?xml version=': 0, '>': 20, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 18, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The system of ionized gas dynamics

The aim of this paper is to study a system of three equations for ionized gas dynamics at high temperature, in one spatial dimension. In addition to the mass density, pressure and particle velocity, a further quantity is needed, namely, the degree of ionization. The system is supplemented by the first and second law of thermodynamics and by an equation of state; all of them involve the degree of ionization. At last, under the assumption of thermal equilibrium, the system is closed by requiring Saha's ionization equation. The geometric properties of the system are rather complicated: in particular, we prove the loss of convexity (genuine nonlinearity) for both forward and backward characteristic fields, and hence the loss of concavity of the physical entropy. This takes place in a small bounded region, which we are able to characterize by numerical estimates on the state functions. The structure of shock waves is also studied by a detailed analysis of the Hugoniot locus, which will be used in a forthcoming paper to study the shock tube problem.


Introduction
The thermodynamical variables temperature, specific entropy, specific internal energy, pressure, specific volume and the velocity of the gas are denoted in this paper by T , S, e, p, v and u, respectively. For brevity we simply refer to S and e as the entropy and internal energy, respectively. In the simplest (nonionized) thermodynamical system, an equation of state relating the variables p, v and T , for example, is assigned. Then, the knowledge of any two of them allows to fully determine the state of the system, since the other variables are determined by the first and second law of thermodynamics T dS = de + p dv. (1.1) However, if the gas is partly ionized as in this paper, further details must be taken into account. Ionization is a process where an atom or a molecule becomes charged due to the loss or gain of electrons. When a gas is heated to a high temperature, at first almost all of its molecules dissociate and it behaves like a monatomic gas. However, if the gas is heated to an even higher temperature, of the order of thousands of degrees Kelvin, some of its atoms become ionized according to the first ionization reaction X → X + + e − , where X is an atom of the monatomic gas, X + an ion and e − an electron. In the case of hydrogen, at this level the gas is made up by atoms, protons and electrons; in heavier atoms the first ionization can be followed by a second ionization and so on. For simplicity we focus on the case of a gas that undergoes only one ionization and denote the concentration (number per unit volume) of atoms, ions and electrons by n a , n i and n e , respectively.
At any given (high) temperature T, this ionization reaction reaches a state of thermal equilibrium analogous to the chemical equilibrium for ordinary chemical reaction. We may assume that the ratio n i n e n a depends only on the temperature T ; an actual formula was derived in 1920 by M. Saha [9], see also [2,4,11,12]. The state of an ionized gas depends on a further variable which measures the level of ionization, namely, the degree of ionization α = n e n a + n i .
However, by means of Saha's formula, the degree of ionization can be expressed as a function of the density ρ and T , and this makes consistent the formulation of the model. Of course, the expressions of the thermodynamical variables depend on the degree of ionization; for instance, in the case of a monatomic gas, the pressure of the ionized gas can be written as where R is the universal gas constant and m the molecular mass of the gas. In the non-ionized case α = 0 we recover the usual expression for the pressure; if α = 0 the gas is not ideal [10]. The aim of this paper is to study the mathematical properties of the gasdynamics system for an ionized gas; here, E = u 2 2 + e is the (specific) total energy. To the best of our knowledge such a study has never been done previously in spite of the fact that the system of gasdynamics attracted the interest of several researchers in the last decade, however mostly for ideal gases [10]; we quote [8] for the case of real gases. The main motivation for our study was provided by the paper [5] 1 , where the authors made laboratory experiments about the reflection at an interface of a shock wave in an ionized gas. In this paper we assume that the gas is monatomic, differently from [5], where a two-component gas is considered. This choice simplifies the analysis of the system while catching however the main features of ionization; more general cases (non-monatomic or multicomponent gases) can be considered as well but at the price of much heavier computations.
After a short review of basic thermodynamics in Section 2 and a first study of the thermodynamic variables of the model in Section 3, we show in Section 4 that system (1.2) is strictly hyperbolic, as in the non-ionized case; however, differently from the latter case, the genuine non-linearity of the extreme eigenvalues of the system is lost and we study the set where this happens. We refer to [3,10] for more information on systems of conservation laws. Our analysis is completed by a detailed investigation of the Hugoniot locus of the system in Section 5 and by a study of the entropy increase in Section 6. Moreover, a study of the integral curves in provided in Section 7. At last, in Section 8 we study a related model, which is deduced by an approximation of Saha's law at very high temperatures; in this case, the system does not loose the genuine nonlinearity property.
In a second forthcoming paper we exploit the results obtained in this paper to study the shock reflection in an electromagnetic shock tube (T-tube) with a reflector. We thus provide a rigorous mathematical basis to the physical phenomena observed in [5].

Basic Thermodynamics
The equation of state of an ideal gas can be written as [4, (6)] p = ρ m p kT = n p kT, (2.1) where m p is the particle mass, n p the total number of particles per unit volume and k is Boltzmann's constant; we clearly have ρ = m p n p . The molecular mass is defined as m = N 0 m p , (2.2) where N 0 is Avogadro number. If R = kN 0 is the universal gas constant, then (2.3) becomes [4, (8)] or p = M RρT by introducing the number M = 1 m of moles. We shall often use p and T as independent thermodynamical variables; in order to compute the other variables we introduce the (specific) enthalpy H = e + pv. Using the enthalpy equation (1.1) becomes dH = T dS + vdp. (2.4) We also introduce the Gibbs function G = H − T S, see [4, (111)].By (1.1) we deduce dG = v dp − S dT , whence As usual, a subscript as T or p above means that the derivative is computed when that variable is fixed. By (2.5) we deduce the compatibility condition which is one of the so-called Maxwell reciprocity relations. In turn, by (2.5) and (2.6) we obtain Example 2.1 (Polytropic gas). In the case of a polytropic gas, i.e., an ideal gas whose specific heat is constant, we have [4, (29)] e = C v T, (2.9) where C v is the specific heat at constant volume [4, (34)]. We also define the specific heat at constant pressure C p = R m + C v . Then, e = C v T and H = C p T . The adiabatic constant is γ = Cp Cv = 1 + R mCv [4, (37)]. We conclude by computing the entropy S. By (1.1) we deduce and then S = C p log T − R m log p + Const by (2.3) and (2.9). By introducing the non-dimensional entropy η = m R S we easily find In the case of a monatomic gas, we have C v = 3 2 a 2 , C p = 5 2 a 2 and γ = 5 3 [4, (34) and (37)]. Then, Now, we briefly discuss ionization; to avoid multiple references we mainly refer to [2] but analogous expressions can be easily found in [4,11,12]. Saha's equation provides the degree of ionization of an atomic element as a function of temperature and electron density, under assumptions of thermal equilibrium. In a general form it can be written [2, (4.15)] n r+1 n e n r = G r+1 g e G r (2πm e kT ) 3 2 h 3 e − χr kT . (2.13) Here n r and n r+1 are the number densities of atoms in the ionization state r (where r electrons are missing) and the ionization state r + 1 (where r + 1 electrons are missing) of a given element, n e is the electron number density, G r and G r+1 are the partition functions of the two states, g e = 2 is the statistical weight of the electron, m e is the electron mass, h is the Planck constant, k is the Boltzmann constant, χ r is the ionization potential from state r to state r + 1. We also denote the degree of ionization by α = n e n a + n i . (2.14) As we mentioned in the Introduction, we consider the case r = 0 of a single ionization from a neutral state of a monatomic gas; then (2.13) becomes where we denoted by Z a = G 0 and Z i = G 1 the partition functions of the neutral state and 1-ionized state, respectively; T i = χ0 k is the dissociation energy expressed by the temperature. Moreover, n e = n i and then α = ni na+ni . The total number of atoms and ions is n p = n a + n i and we can assume that n p = ρ/m p . As a consequence, by (2.1) we deduce the pressure law for an ionized gas, p = (n a + n i + n e )kT = (1 + α)n p kT = (1 + α) ρ m p kT, (2.16) or better, by using the molecular mass (2.2), The equation of state (2.17) expresses α as a function of p, ρ and T . Indeed, as we show in the next Lemma 3.1, α does not depend on ρ and p independently but solely on one of these two variables. As a consequence, an ionized gas is not an ideal gas, which is characterized instead by the equation of state (2.3). The specific volume v is deduced as In the current case of an ionized gas, the specific internal energy should include not only kinetic, but also ionization energy; then e = 3 2 p ρ + χ0 mp α. But χ0 mp = kTiN0 m = R m T i and then, since the gas is monatomic, We stress that, differently from ideal gases where e = e(T ), for an ionized gas the specific internal energy e depends on both p and T . The above expressions (2.19) slightly change for non-monatomic gases; we emphasize that most numerical coefficients that occur in the following (as 5/2 or 5/3) are merely due to this assumption.

The Thermodynamic Variables of an Ionized Gas
In this section we provide several explicit expressions for the thermodynamical variables of an ionized gas. In the proofs we use from time to time the shorthand notation First, we study the properties of the degree α of ionization. By exploiting the notation introduced in the previous section, we denote We notice thatκ = mp k 1 κ = m R 1 κ . Lemma 3.1 (Degree of ionization). The degree α of ionization can be expressed as a function of ρ and T through the formula The degree of ionization can also be expressed as a function of p and T by the formula Moreover, we have Proof. We first notice that n i n e n a = n 2 i n a = n p α 2 1 − α .

Remark 3.1 (Physical values).
We report here some physical values which are commonly used in the figures below. As we wrote in the Introduction, we focus on the simple case of a hydrogen gas. In this case the specific gas constant a 2 is 8314 J kg −1 K −1 and κ = 29.9774. Moreover T i = 1.578 × 10 5 K. We frequently use the following values, where α is computed through (3.3), see [5]: , then we formally deduce T = 0 (or T = ∞, or ρ = ∞, even less physically meaningful). The reason depends on Saha's formula (2.15), which allows ionization at any positive temperature. This shows that the non-ionized case cannot be deduced as the limit for α → 0 of the ionized one.  As a consequence of Lemma 3.1, the pressure law (2.17) can be written more carefully as p = p(ρ, T ) = 1 + α(ρ, T ) R m ρT, (3.8) or as Lemma 3.2 (Pressure). We have Proof. Instead of differentiating directly (3.8), by Lemma 3.1 we understand α as a function of p and T in that expression. We denote F (ρ, T, p) = 1 + α(p, T ) R m ρT − p; by (3.4) 1 we see that By the Implicit Function Theorem and (3.4) 2 we deduce Since p ρ = −  Proof. By Lemma 3.2 we may express α as a function of ρ and T ; by (3.4) and (3.10) 2 we compute Then by (2.19) and (3.11) we deduce whence e T > 0. For an alternative proof we can directly use expression (3.2).
Proof. On the level curves η(p, T ) = c we have dη = ∂η ∂p T dp + ∂η ∂T p dT = 0; then, on these curves, where we omitted for simplicity the subscript c. We notice that F makes sense and F (p, T ) > 0 because of (3.13). Moreover, by (3.16), (3.18) and (3.4) 2 we deduce and then (3.21) follows at least locally and defines a strictly increasing function T (p). Indeed, we claim that T is defined for every p > 0. To prove this claim, we first remark that ∂T ∂p ∼ 2 5 T p both for T → 0+ and for T → +∞, because in those cases α → 0 and α → 1, respectively. This means that T ∼ Cp 2 5 for T → 0+ and T → +∞, for C an arbitrary positive constant, which proves our claim. The claim also implies that for every fixed pair (p, η) we uniquely find the relative value of T . This proves the proposition.

Properties of the System of Ionized Gasdynamics
In this section we study the main properties of system (1.2), where p and e are defined by (2.17), (2.19), respectively, with α as in (3.3).

Eigenvalues and eigenvectors
System (1.2) can be written in Lagrangian coordinates as [10] For C 1 solutions, equation (4.1) 3 can be written as S t = 0 and we obtain We notice that equation (4.2) 3 is equivalent in Eulerian coordinates to In the variables (p, u, S) we have v t = v p p t and equation (4.2) 1 becomes p t − 1 vp u ξ = 0. In these variables, the Lagrangian characteristic speeds are λ ± = ± 1 √ −vp and λ 0 = 0, with corresponding characteristic Now, we use p, u and T as state variables.
At last, we exploit the equation of state (2.17).
We record here, for future reference, that by (4.7) we deduce By Lemma 4.1 we can introduce the notation with λ > 0. Notice that if α = 0 we recover λ = γpρ with γ = 5 3 , see [10, (18.5), (18.30)]. Under this notation system (4.5) becomes The proof of the following lemma is straightforward; for brevity it is omitted.
for λ defined in (4.9); the corresponding eigenvectors are The eigenvalue λ 0 is linearly degenerate; a pair of Riemann invariants for λ 0 is {u, p}. A Riemann invariant for both λ ± is η. The eigenvalues of system (1.2) are then u ± λ and u.
Now, we investigate the genuine nonlinearity of the eigenvalues λ ± ; we refer to [8] for more insight about the failure of this condition. Notice that, when dealing with functions that only depend on p and T , we have We have Proof. We simply compute whence (4.12).
We notice that Lemma 4.3 shows that α is not a Riemann invariant for the eigenvalues λ ± . Proof. We focus on λ + since In the following, we exploit many times identity (4.8); we split the proof into three steps.
(1) About the first three terms in the right-hand side of (4.13) we claim that and, by Lemma 4.3, we have Then (4.15) follows.
(2) About the last two terms in (4.13), we notice that so that, by Lemma 4.3 and (4.14) we have About II we have that Since qA − A − qB = Ti T [1 + φq] then, by (4.8), we deduce As a consequence (3) By (4.15) and (4.16) we deduce that The right-hand side of (4.17) is positive for α ≥ 1 2 . We claim that (4.17) is negative for suitable small values of T and p = p(T ). To prove the claim, we take T sufficiently small and fix p = p(T ) in such a way that ατ 2 = 1.
(4.18) Therefore, by using notation as in (3.3), for T → 0 and so p is also small. For T → 0 we have α ∼ 0, αq ∼ 0, A ∼ 3, B ∼ 2 and then by (4.17) and (4.18) for α < 1 2 we have Remark 4.1. The value T i /T > 124 found in (4.20) corresponds, for the value T i = 1.578 × 10 5 K, see Remark 3.1, to T 1.2 × 10 3 K. Notice that the term in braces in (4.17) tends exponentially to 1 as T → 0 for every fixed p > 0. Moreover, it also tends to 1 as p → 0 for every fixed T > 0.

The inflection locus
Now, we study in greater detail the inflection locus, which is the set Since R + ∇λ + = R − ∇λ − , either cases lead to the same result.
Lemma 4.4. The inflection locus I is the zero set of the function (4.21) Proof. Formula (4.21) follows by (4.17) with a direct calculation: the function f is the term in braces in that formula.
Formula (4.17) shows that R + ∇ log λ + > 0 if and only if f (α, T ) > 0. We refer to Figure 2 for the locus I. Notice that significant values of the temperature range from 0 to 1.5 · 10 3 K; on the other hand, the ionization degree is low, ranging from 0 to 2 · 10 −4 . Both ranges are in a physical range.
From the proof of Proposition 4.2 we deduce that ∇R ± λ ± is positive outside the bounded region of Figure 2 and negative inside it.
We now provide an explicit sufficient condition on the genuine nonlinearity of the eigenvalues λ ± . or Ti T ≤ 54.5375, then each eigenvalue λ ± is genuinely nonlinear.
Proof. With the aid of the inequality (1 + a)(1 Then, both characteristic eigenvalue λ ± are genuinely nonlinear if the following inequality holds: (4.23) Now, assume that the first condition in the statement of the theorem holds; then showing that inequality (4.23) is satisfied. Next, note that In the following we slightly change notation: with reference to (5.2), we drop the + index and write 0 instead of the index −. Under this notation, we have the equation of Hugoniot locus of (α 0 , u 0 , T 0 )

The variation of T
In this section we begin the study of the Hugoniot locus by focusing first on the variation of the temperature. Then we study the thermodynamic part of this locus in the (α, T )-plane. For a contact discontinuity, the thermodynamical part of the Hugoniot locus is given by p = p 0 and α = 1 , which is excluded in the following discussion. of (α 0 , T 0 ) can be expressed as or else, by using only α and T , as Proof. By using the enthalpy, equation (5.3) 2 can be written as Now, by (2.17) and (2.19), we have pv = a 2 T (1 + α) and H = 5 2 pv + a 2 T i α. As a consequence, we deduce Then, condition (5.6) is equivalent to If we divide the equation above by a 2 T 0 we obtain (5.4). At last, by (3.9) we have We deduce (5.5) by inserting (5.7) into (5.4).  The following proposition which will be crucial in the following. Proof. We refer to Figure 3 for a graph of the function T . We split the proof into some steps.
(i) By (5.5), the thermodynamic part of the Hugoniot locus is the set defined by F (α, T ) = 0, for By differentiating (5.9) with respect to T, and then introducing p and p 0 , we compute Obviously, F T (α, T ) = 0 if and only if Φ(α, T ) = 0.
(iii) Consider the function F defined in (5.9); clearly F (α 0 , T 0 ) = 0 and F T (α 0 , T 0 ) = 0 by (5.12). Then it follows from the Implicit Function Theorem that T is a function of α in a neighbourhood of α = α 0 ; moreover, dT /dα = −F α (α, T )/F T (α, T ). By differentiating (5.9) with respect to α and introducing p and p 0 , we have Thus, by (5.10) and (5.18) we end up with the following expression for dT /dα: The numerator of the right-hand side in (5.19) is strictly positive for every 0 < α < 1 and T > 0; the denominator is strictly positive in a neighbourhood of α = α 0 by (5.12). We conclude that T (α) is a strictly increasing function in this neighbourhood.
The proposition is completely proved.
Since the function T = T (α) introduced in Proposition 5.2 is strictly increasing, then it is invertible and defines a function α = α(T ). In the following proposition we precise the asymptotic behavior of α(T ) when T is close either to 0 or ∞.
Furthermore, if T → ∞ then α → 1 and we have i.e. that it tends to a finite nonzero value for T → 0. Then, for T → 0, the left-hand side of (5.5) would be ∼ 0 while the right-hand side would be O(1), a contradiction.

The variation of p and v along the Hugoniot locus
In this section we exploit Proposition 5.2 to compute the variation of p and v along the thermodynamic part α, T (α) of the Hugoniot locus. We prove that p increases and v decreases as α increases. Proof. By (3.9) and (3.8) we deduce In the following of the proof, we work on the thermodynamic part α, T (α) of the Hugoniot locus; then T = T (α) and, with a slight abuse of notation, we denote p(α) := p α, T (α) and v(α) := v α, T (α) .
(ii) Now, we consider the variation of v. The result is a counterpart of what proved above but it is not directly deduced from it. By (5.22) and (5.19) it follows As above, D(α) is positive, while some computations lead to We claim that the term in the second bracket in (5.27) is negative for α ≥ α 0 , i.e., This would conclude the proof of the lemma: indeed, the first summand in (5.27) is strictly negative because T (α) is a strictly decreasing function in (0, 1) by Proposition 5.2.

The kinetic part of the Hugoniot locus
In this subsection we focus on the variation of the velocity u along the Hugoniot locus (5.3) of (α 0 , u 0 , T 0 ); these results are used in the forthcoming paper [?] to study the shock tube problem. By (3.8), equation (5.3) 1 can be written as Therefore, by (3.8), a point of the Hugoniot locus of (α 0 , u 0 , T 0 ) lies in the intersection of the set For simplicity, we drop the dependence of G on (α 0 , u 0 , T 0 ); of course, we have G(α 0 , u 0 , T 0 ) = 0. Since u is fixed in the following, we understand the set G(α, u, T ) as a set in the (α, T )-plane, see Figure 4.
Lemma 5.1 (Intersection of Hugoniot loci). Fix (α 0 , u 0 , T 0 ) and u = u 0 . Then, in the region α 0 < α < 1 there is at least one point of intersection between the set G(α, u, T ) = 0 and the thermodynamic part (5.3) 2 of the Hugoniot locus of (α 0 , T 0 ). The same result holds also in the region 0 Proof. We refer to Figure 4. By the proof of Proposition 5.5 we know that G(α 0 , u, T 0 ) < 0, while by Proposition 5.2 we have T (α 0 ) = 0. Then, in a neighborhood of (α 0 , T 0 ), the curve T = T (α) is located between the two curves T = T − (α) and T = T + (α).
In the region α 0 < α < 1, it follows by (5.31) and (5.20) that there is at least one point of intersection between the curves T = T + (α) and T = T (α).
On the other hand, Proposition 5.3 and (5.32) imply G(α, u, T ) as α, T → 0 along the thermodynamic part of the Hugoniot locus. Then, the set G(α, u, T ) = 0 and the graph of the function T (α) intersects in the region 0 At last, from the results obtained in this section we conclude with the following main result.
We notice that by (5.1) 2 we have m 2 = − p+−p− v+−v− , which shows that m is the Lagrangian shock speed . With the help of the previous lemma we now provide a result about the increase of entropy across a shock front. Proposition 6.1. Let (p − , u − , S − ) and (p + , u + , S + ) be connected by a shock wave and v pp (p ± , S ± ) > 0. If |p + − p − | is sufficiently small, then the entropy increases across the shock front.

Proof. We begin by considering an asymptotic expression of [S]
Thus, on the one hand we have Here we used the shortcut (∂v/∂p − ) S = (∂v/∂p)(p − , S − ). On the other hand, by (5.3) 2 we have Combining the above two expressions and omitting some technical details, we conclude that By (4.4), the assumption v pp (p ± , S ± ) > 0 implies that the condition of genuine nonlinearity holds and the shock is a Lax shock. Consider for example the case of a shock wave corresponding to the eigenvalue λ + , which is characterized by m > 0; for shock waves of the other family (i.e., corresponding to λ − , then m < 0) the proof is analogous. Since the shock is a Lax shock, then it is compressive, i.e., p + < p − . By Lemma 6.1, the entropy increases across the shock front if and only if S + − S − < 0. By (6.1), this inequality is implied once more by the condition v pp (p ± , S ± ) > 0. or Ti T ≤ 54.5375, then the entropy increases across a shock front whose amplitude |p + − p − | is sufficiently small.

Integral curves
We write system (1.2) as U t + F (U ) x = 0 or, for smooth solutions, where A(U ) = F ′ (U ). Let V be a set of new unknowns related to U by U = U (V ). We have Some properties of system (1.2) were already established in Lemma 4.2 as far as the coordinates (p, u, T ) were concerned. In the following, the thermodynamic variables α, T are very useful. Recalling (3.9), we consider the transformation of variables With a slight abuse of notation, we still denote by R ± and R 0 , as in Lemma 4.2, the eigenvectors of the Jacobian matrix of the flux in coordinates (α, u, T ). Proposition 7.1 (Characteristic fields). The characteristic vector fields with respect to (α, u, T ) coordinates are expressed as The result follows by what we pointed out just before the proposition. Now, we can consider integral curves. We denote with α ± (s), u ± (s), T ± (s) (7.2) an integral curve of the characteristic field R ± defined in (7.1), i.e., (α ± (s),u ± (s),Ṫ ± (s)) = R ± (α, u, T ). Indeed, we first focus on the projection of the integral curves on the (α, T )-plane. We have, with a slight abuse of notation,Ṫ (s) α(s) and We observe that the ordinary differential equation (7.3) does not depend on u and can be uniquely solved by a function T = T (α) once the initial data T (α 0 ) = T 0 is fixed. The projection on the (α, T )-plane of any such characteristic curve is the same once T 0 is fixed. By inserting T (α) in the second equatioṅ u ± = R 2 (α, u ± , T ) one finds u ± = u ± (α). Recalling (3.14), with a slight abuse of notation we still denote as a representative for the dimensionless entropy. Since η is a Riemann invariant for the characteristic fields R ± by Lemma 4.2, then it is constant along the integral curves of those fields in (α, u, T )-space [3, Theorem 7.6.2]; of course, this can be directly verified from (7.5). In other words, the projections on the (α, T )-plane of those integral curves coincide with the isentropes η(α, T ) = const.
Proposition 7.2 (Integral curves in the (α, T )-plane). For any η 0 ∈ R the equation has a unique solution α ∞ ∈ (0, 1). The two integral curves of the characteristic fields R ± , corresponding to η 0 , have the same projection on the (α, T )-plane, which is the graph of a function T η0 = T η0 (α) for α ∈ (0, 1) \ {α ∞ }. The function T η0 is strictly increasing in both intervals (0, α ∞ ) and (α ∞ , 1); moreover, and for T = T η0 (α) we have the asymptotic expression Proof. Notice that the function η can assume any real value when α varies in (0, 1). By solving the equation η(α, T ) = η 0 for T Ti , with η given by (7.5), we have The denominator d(α) of T η0 is a strictly decreasing function of α ∈ (0, 1); moreover, lim α→0+ d(α) = +∞ and lim α→1− d(α) = −∞. Then, there is a unique value α ∞ ∈ (0, 1) satisfying (7.6). The properties of the function T η0 easily follow. Since the above asymptotic form is obtained. Remark 7.1. Notice that T η0 (α) > 0 for α ∈ (0, α ∞ ) and the value α ∞ only depends on η 0 and not on T i . Moreover, since the function T η0 increases in the interval (α ∞ , 1) and lim α→1− T η0 (α) = 0, T η0 (α) < 0 for α ∈ (α ∞ , 1), which shows that the temperature is negative and the states with α > α ∞ are not physically admissible. Moreover, we remark that, by (3.9), for α → α ∞ −, on the integral curve we have p(α, T ) → ∞ and, by (2.17) and (3.9), we also have ρ = a(1 + α) T p = a(1 + α)κ α 2 1−α 2 T − 3 2 e T i T → 0. As a consequence, from now on we understand the function T η0 to be defined only for α ∈ (0, α ∞ ); moreover, we drop the superscript η 0 when it is not strictly needed. Now, we study the convexity of the integral curves η = const. Lemma 7.1 (Convexity of the integral curves). The projection of any integral curve on the (α, T )-plane is strictly convex for Ti T ≤ 4 and strictly concave for small α. Proof. Instead of directly computing the derivatives of T η0 from (7.8) we exploit the implicit function theorem. We note that η α = 2 α(1−α) + 5 2 + Ti T , η T = − Ti T 2 (1 + α) and then Consequently, since T η0 solves the implicit equation η α, T η0 (α ∞ ) = η 0 , we find that it satisfies The denominator of the expression on the right-hand side is negative and, about the numerator, we find If Ti T ≤ 4, then the numerator is positive and this proves the former part of the statement. To prove the latter, for α → 0 we deduce T η0 (α) ∼ Ti 2 log 1 α by (7.8) and then the principal term of the numerator is −2 The lemma is completely proved. The next result deals with rarefaction waves. For brevity we call 1-rarefaction curves the rarefaction curves corresponding to the eigenvalue λ − while 2-rarefaction curves are those corresponding to the eigenvalue λ + . Consider the projections on the (α, T )-plane of the rarefaction curves corresponding to the eigenvalues λ ± through the point (α 0 , T 0 ); suppose that they are contained in the region where R ± ∇λ ± > 0. Then the projection of the 1-rarefaction curve is the integral curve through the same point defined for α ∈ (0, α 0 ] and that of the 2-rarefaction curve is the same integral curve defined for α ∈ [α 0 , α ∞ ).
Remark 7.2. The above theorem shows that the pressure is strictly increasing along the 2-rarefaction curve, which looks contradictory. However, here the state (α 0 , T 0 ) is the back state of the 2-rarefaction wave; then the pressure is strictly decreasing along the 2-rarefaction curve from the front to the back state.
Theorem 4.1 shows that R ± ∇λ ± > 0 in the region where α ≤ 60 T /T i 3 . If the point (α 0 , T 0 ) is located in this region and α 0 is sufficiently small, we conclude by (7.7) that the 1-rarefaction curve and the 2-rarefaction curve from the front to the back state stay in this region as α → 0 + .
We now continue the analysis of the integral curves for the characteristic fields R ± by considering the behavior of the component u. By (7.2) and (7.1) we have Ti T p λ .
Since Λ > 0, we deduce that u + is decreasing while u − is increasing in the interval (0, α ∞ ). We notice that by (4.9) we have

The High-Temperature-Limit model
In the case of very high temperatures one can use an approximate version of Saha's formula. More precisely, when Ti T ∼ 0, we approximate Saha's law (2.13) with n r+1 n e n r = G r+1 g e G r (2πm e kT ) We call the corresponding model as the High-Temperature-Limit (HTL) model. By using (8.1) instead of (2.13), we see that formulas (3.2) and (3.3) are replaced by respectively. Moreover, we have We rewrite now the main results we obtained in the previous sections for the HTL model. The equation of state is again (2.17). On the contrary, due to the approximation procedure, we discard the component due to the ionization energy in the internal energy; then, the internal energy, the enthalpy (2.19) and the dimensionless entropy (3.12) become e = 3 2
Notice that, by (3.14), the dimensionless entropy η now explicitly depends only on α: Of course, η depends on both T and p through α. As a consequence, if α is constant, then clearly the entropy is constant. The following result is analogous to Lemma 4.2. If we still denote by R ± the eigenvectors corresponding to λ ± as in (4.11), we have R ± ∇ log λ ± = ± 4 5p and then both λ ± are genuinely nonlinear. The eigenvalue λ 0 is linearly degenerate.
Notice that if α = 0 we formally find the eigenvalues of polytropic gas-dynamics with γ = 5 3 . Also notice that the degree of ionization is constant along the ± characteristic directions, i.e., R ± ∇α = 0. This means that both η and α are Riemann invariants for the fields associated to λ ± ; clearly, they are not independent by (8.4). A pair of Riemann invariants for λ 0 is still {u, p}.
Clearly, dα dT (T 0 ) = 0 and Φ T (α 0 , T 0 ) = 0 yields d 2 α dT 2 (T 0 ) = 0. Thus we conclude that α−α 0 = O(1)(T −T 0 ) 3 and, by (8.14), the thermodynamic part of the Hugoniot locus at (α 0 , T 0 ) stays in the region Φ (α, T ) > 0, α = α 0 , in a neighborhood of α 0 ; see Figure 7 on the right. where the forward and backward characteristic fields are not genuinely nonlinear, which also causes the decrease of the physical entropy. Nevertheless, we established precise conditions which limit the "classical" region where the forward and backward characteristic fields are genuinely nonlinear and where the physical entropy is strictly concave. In such a region, the classical theory of shock waves is valid and the compression branch is admissible. In a forthcoming paper, we will discuss the electromagnetic shock tube filled with monatomic hydrogen gas and we expect that the actual experiments only involve such a classical region.
We also proposed the High-Temperature-Limit model , an approximation of the exact model in the case that the energy (or the temperature) is extremely high when compared with the dissociation energy. In this model, the physical entropy is just a function of the degree of ionization and does not depend on the temperature. Hence, the degree of ionization is a Riemann invariant. The integral curves (rarefaction curves) are similar to those of the monatomic ( 5 3 -polytropic) gas. The shape of the thermodynamic part of the Hugoniot locus is quite different from that of the exact system considered in the previous sections.