RANKINE–HUGONIOT RELATIONS IN RELATIVISTIC COMBUSTION WAVES

Based on the energy–momentum tensor in the local rest frame of a relativistic fluid

$$\begin{equation} T^{ik}=\left(\begin{array}{@{}cccc@{}}e & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{array}\right), \end{equation} \tag{ 1 }$$

momentum and energy conservations across a shock front are

$$\begin{equation} \omega _1 u_1^2 + p_1 = \omega _2 u_2^2 + p_2 \end{equation} \tag{ 2 }$$

and

$$\begin{equation} \omega _1 \gamma _1 u_1 = \omega _2 \gamma _2 u_2, \end{equation} \tag{ 3 }$$

respectively (Landau & Lifshitz 1959; Steinhardt 1982). Here, e is the fluid energy density, which includes the rest-frame energy nm₀c² of the fluid particles and the specific internal energy (see Equation (6)), p is the pressure of the fluid, ω = e + p is the specific enthalpy per unit volume, m₀ is the rest mass of one particle, and n is the particle number density which can be the number density of any charge (e.g., baryon number or lepton number) that is conserved across the shock front. The four-velocity u of the fluid has the form u = βγ, where β = v/c is the velocity in units of the speed of light and γ = 1/(1 − β²)^1/2 is the Lorentz factor. Subscripts 1 and 2 denote upstream and downstream quantities, respectively. The particle number conservation

$$\begin{equation} n_1 u_1 = n_2 u_2 \end{equation} \tag{ 4 }$$

is a complementary condition of continuity across the shock.

For analysis pertinent to the stellar interior, the ISM or intergalactic media (IGM), and the early universe, the relativistic ideal gas is described by the following EoS with the adiabatic index Γ (i.e., the Synge gas; cf. Synge 1957; Lanza et al. 1982; Cissoko 1997):

$$\begin{equation} \Gamma =1+\frac{p}{\rho \epsilon }, \end{equation} \tag{ 5 }$$

where ρ = nm₀ is the rest-frame mass density and is the specific internal energy (cf. Equation (6)). Then, the fluid energy density can be written as

$$\begin{equation} e=\rho (c^2+\epsilon)=nm_0c^2+\frac{1}{\Gamma -1}p, \end{equation} \tag{ 6 }$$

which can also be considered as the definition of the specific internal energy . Using the EoS (5), the specific enthalpy has the following form (Kennel & Coroniti 1984):

$$\begin{equation} \omega =e+p=nm_0c^2+\frac{\Gamma }{\Gamma -1}p. \end{equation} \tag{ 7 }$$

The EoS with 4/3 ⩽ Γ ⩽ 5/3 (Taub 1948; Anile 1989) is a general form for relativistic fluids. In the highly relativistic regime (Γ = 4/3), the internal energy greatly exceeds the rest-frame energy of the particle, i.e., ≫ c², and the fluid is radiation dominant with p = (1/3)ρ = (1/3)e. This is the EoS used in Steinhardt (1982) in the study of the bubble growth during the false vacuum decay in the early evolution phase of the universe. In the non-relativistic extreme (Γ = 5/3; cf. Anile 1989), ≪ c², the EoS assumes the form p = (2/3)ρ. Comparing this expression with the classical ideal gas law p = ρkT, with k being the Boltzmann constant, the internal energy is just the kinetic energy = (3/2)kT for a monatomic gas in non-relativistic fluids. Another special EoS with Γ = 2/3 accounting for the dark energy in the universe will be discussed in Section 2.3.

The sound speed in relativistic fluids is given by (cf. Landau & Lifshitz 1959)

$$\begin{equation} v_{\rm s}=c\sqrt{\frac{\partial p}{\partial e}}, \end{equation} \tag{ 8 }$$

whose four-velocity is u_s = β_sγ_s = (v_s/c)(1 − (v_s/c)²)^−1/2. Following the above discussion, we have $v_{\rm s}=c/\sqrt{3}$ in the highly relativistic regime and $v_{\rm s}=\sqrt{{\partial p}/{\partial \rho }}$ in the non-relativistic regime.

By defining a variable x = ω/n² for fluids on both sides of the wave front, we readily obtain the expressions for the particle flux (j = n₁u₁ = n₂u₂) by considering momentum conservation (2) and particle flux conservation (4)

$$\begin{equation} -j^2=\frac{p_2 - p_1}{x_2 - x_1}, \end{equation} \tag{ 9 }$$

and the so-called shock adiabat by additionally considering the energy conservation (3) (cf. Taub 1948; Landau & Lifshitz 1959; Steinhardt 1982):

$$\begin{equation} x_2\omega _2-x_1\omega _1=(p_2-p_1)(x_2+x_1). \end{equation} \tag{ 10 }$$

Referring to the form of the specific enthalpy (7) under the EoS, the variable x can be expressed as

$$\begin{equation} x=\frac{1}{n^2}\bigg (nm_0c^2 + \frac{\Gamma }{\Gamma -1}p\bigg)=\frac{m_0^2c^2}{\rho }+ \frac{\Gamma }{\Gamma -1}\frac{m_0^2 p}{\rho ^2}. \end{equation} \tag{ 11 }$$

Introducing Equation (11) to the particle flux (9), we obtain the Rayleigh relation for relativistic fluids, accounting for the conservation of number density and momentum:

$$\begin{equation} \hat{p}-1 = -u_1^2 \bigg [\hat{c}^2(\hat{V}-1)+\frac{\Gamma }{\Gamma -1}(\hat{p}\hat{V^2}-1)\bigg ]. \end{equation} \tag{ 12 }$$

Here, V = 1/ρ is the specific volume, and notations with hats are reduced variables, i.e., $\hat{p}=p_2/p_1$, $\hat{V}=V_2/V_1$, and $\hat{c}=c/\sqrt{p_1V_1}$. In the limit of $\hat{c}\rightarrow \infty$ and u₁ → 0, the Rayleigh relation degenerates to that in the non-relativistic regime (cf. Landau & Lifshitz 1959):

$$\begin{equation} M_1^2 = -\frac{\hat{p}-1}{\Gamma (\hat{V}-1)}, \end{equation} \tag{ 13 }$$

where $M=v/\sqrt{\Gamma p/\rho }$ is the non-relativistic Mach number, while in the limit of $\hat{c}\rightarrow 0$, it assumes the highly relativistic form:

$$\begin{equation} u_1^2= -\frac{(\Gamma -1)(\hat{p}-1)}{\Gamma (\hat{p}\hat{V}^2-1)}. \end{equation} \tag{ 14 }$$

By introducing x given by (11) to the shock adiabat (10), we derive the Hugoniot relation for relativistic shocks:

$$\begin{eqnarray} \hat{c}^2\bigg [\frac{\Gamma +1}{\Gamma -1}(\hat{p}\hat{V}-1)-(\hat{p}-\hat{V})\bigg ] &=&\frac{\Gamma }{\Gamma -1}\hat{p}(1-\hat{V}^2)\nonumber\\ && - \frac{\Gamma }{(\Gamma -1)^2}(\hat{p}^2\hat{V}^2-1). \nonumber\\ \end{eqnarray} \tag{ 15 }$$

The Hugoniot relation additionally takes into account of energy conservation (3) and as such describes all possible discontinuities across a shock by considering all conservation equations. In the non-relativistic extreme of $\hat{c}\rightarrow \infty$ and u₁ → 0, Equation (15) reduces to (Landau & Lifshitz 1959)

$$\begin{equation} \bigg (\hat{p}+\frac{\Gamma -1}{\Gamma +1}\bigg)\bigg (\hat{V}-\frac{\Gamma -1}{\Gamma +1}\bigg) = \frac{4\Gamma }{(\Gamma +1)^2}. \end{equation} \tag{ 16 }$$

In the highly relativistic extreme of $\hat{c}\rightarrow 0$, the right-hand side of Equation (15) vanishes, yielding

$$\begin{equation} \hat{p}\hat{V}^2=\frac{(\Gamma -1)\hat{p}+1}{(\Gamma -1)+\hat{p}}. \end{equation} \tag{ 17 }$$

The highly relativistic shock adiabat, i.e., Rayleigh and Hugoniot relations, have been derived and discussed in, e.g., Steinhardt (1982).

Hugoniot lines represented by Equation (15) are illustrated in Figure 1 for non-relativistic, relativistic, and highly relativistic fluids. Along reference line a, it is seen that the density increase is higher for relativistic fluids than for non-relativistic fluids for the same pressure enhancement ratio across the compression shock. And along reference line b, the pressure reduction is smaller in relativistic fluids than in non-relativistic fluids for the same density dilution ratio across the rarefaction shock. These differences are due to the fact that pressure assumes a larger proportion of the total energy in relativistic fluids when the flow speed gets closer to the speed of light, such as downstream of rarefaction shocks along reference line b.

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Typical shock solutions for relativistic and highly relativistic fluids are shown in Figure 2. One distinct difference from non-relativistic shocks is that the Rayleigh lines for relativistic fluids are not straight lines any more, which can be easily seen from the form of Equation (12). A more important difference between relativistic and non-relativistic shocks is that compression shocks (p₂ > p₁ and V₂ < V₁) can only be achieved with very high upstream flow speeds in relativistic fluids as indicated, for example, by the state (u₁ = 1 and $v_1=({1}/{\sqrt{2}})c$) in Figure 2. Furthermore, when the upstream flow speed is reduced by half (u₁ = 1/2 and $v_1 = ({1}/{\sqrt{5}})c$), only the rarefaction shock solution (p₂ < p₁ and V₂ > V₁) exists. The criterion distinguishing these two types of shocks is the sound speed of the upstream flow, i.e., $u_1={1}/{\sqrt{2}}$ and $v_1=({1}/{\sqrt{3}})c$, when the only tangency state between the Rayleigh and Hugoniot lines is the (1,1) point. This classification of shocks and the criterion are the same as those for the non-relativistic shocks.

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Under certain conditions of particle flux (9), the downstream pressure can assume negative values. Take the highly relativistic case as an example. Combining Equations (14) and (17), and by setting the adiabatic index to Γ = 4/3, we readily obtain the relation between the reduced downstream pressure and the upstream four-velocity:

$$\begin{equation} u_1^2=\frac{1}{8}(3\hat{p}+1). \end{equation} \tag{ 18 }$$

Then for upstream velocity in the range $0<u_1<{\sqrt{2}}/{4}$, the reduced downstream pressure is negative, i.e., $-{1}/{3} <\hat{p}<0$. Furthermore, under this condition, the reduced specific volume is an imaginary number according to the highly relativistic Hugoniot relation (17), i.e., $\hat{V}^2<0$. That is, for shock waves in highly relativistic fluids with a normal upstream flow (p₁ > 0 and V₁ > 0), whose speed u₁ is smaller than ${\sqrt{2}}/{4}$, a negative-pressure state can be achieved in the downstream flow with the specific volume (or density) being an imaginary number. The $\hat{p}\hbox{--}\hat{V}^2$ diagram showing the highly relativistic shocks with negative-pressure downstream flows (u₁ = 0.2 and 0.3) is shown in Figure 3. The physical interpretation of this type of negative-pressure fluid with imaginary density (defined as Type I) is not clear so far, although its existence can be ruled out based on entropy considerations, as will be demonstrated in the sequel.

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Another type of negative-pressure fluid (Type II) is the case of Γ = 2/3 in the EoS (5), i.e.,

$$\begin{equation} p=-\frac{1}{3}\rho \epsilon, \end{equation} \tag{ 19 }$$

which can be interpreted on the basis of dark energy in the universe. Under this EoS, the combination of highly relativistic Rayleigh (14) and Hugoniot (17) relations readily leads to the form of the upstream four-velocity:

$$\begin{equation} u_1^2=\frac{1}{8}(-3\hat{p}+1). \end{equation} \tag{ 20 }$$

For the upstream flow speed $u_1>{\sqrt{2}}/{4}$, the reduced pressure is negative, accounting for a negative-pressure downstream fluid with the upstream fluid being normal. This type of negative-pressure fluid has a real-number specific volume (or density) according to Equation (17). Shock waves involving Type II negative-pressure fluids are shown in Figure 4.

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Entropy analysis showing the availability of these two types of negative-pressure fluid is presented in the following.

All shock waves should follow the law of entropy increase. Applying the thermodynamic relation (Landau & Lifshitz 1959)

$$\begin{equation} d\bigg (\frac{\omega }{n}\bigg)=Td\bigg (\frac{\sigma }{n}\bigg)+\bigg (\frac{1}{n}\bigg)dp \end{equation} \tag{ 21 }$$

in the weak shock wave,⁴ we have the following form through Taylor expansion:

$$\begin{eqnarray} &&\frac{\omega _2}{n_2}-\frac{\omega _1}{n_1}=T_1\bigg (\frac{\sigma _2}{n_2}-\frac{\sigma _1}{n_1}\bigg)+\frac{1}{n_1}(p_2-p_1)\nonumber\\ &&\quad +\frac{1}{2}\bigg (\frac{\partial (1/n)}{\partial p_1}\bigg)_s (p_2-p_1)^2 +\frac{1}{6}\bigg (\frac{\partial ^2 (1/n)}{\partial p_1^2}\bigg)_s (p_2-p_1)^3.\nonumber\\ \end{eqnarray} \tag{ 22 }$$

Here, T is the temperature, σ is the entropy per unit proper volume, and s = σ/n is the entropy per particle. From Equation (10), we have another expression for ω₂/n₂ − ω₁/n₁, i.e.,

$$\begin{equation} \frac{\omega _2}{n_2}-\frac{\omega _1}{n_1}=\frac{1}{2}\bigg (\frac{1}{n_2}+\frac{1}{n_1}\bigg)(p_2-p_1). \end{equation} \tag{ 23 }$$

It should be noticed that in achieving Equation (23), the higher order term (ω₂/n₂ − ω₁/n₁)(p₂ − p₁) has been omitted. Combining Equations (22) and (23), we readily obtain the relation of the entropy difference across the shock front:

$$\begin{eqnarray} &&T_1\bigg (\frac{\sigma _2}{n_2}-\frac{\sigma _1}{n_1}\bigg)=\frac{1}{2}\bigg (\frac{1}{n_2}-\frac{1}{n_1}\bigg)(p_2-p_1)\nonumber\\ &&\quad -\frac{1}{2}\bigg (\frac{\partial (1/n)}{\partial p_1}\bigg)_s (p_2-p_1)^2 -\frac{1}{6}\bigg (\frac{\partial ^2 (1/n)}{\partial p_1^2}\bigg)_s (p_2-p_1)^3. \nonumber\\ \end{eqnarray} \tag{ 24 }$$

By introducing the expansion of 1/n₂ with respect to p₂ − p₁

$$\begin{equation} \frac{1}{n_2}-\frac{1}{n_1}= \bigg (\frac{\partial (1/n)}{\partial p_1}\bigg)_s (p_2-p_1) +\frac{1}{2}\bigg (\frac{\partial ^2 (1/n)}{\partial p_1^2}\bigg)_s (p_2-p_1)^2 \end{equation} \tag{ 25 }$$

to Equation (24), we have the simplified expression of the entropy difference across a relativistic shock:

$$\begin{equation} \frac{\sigma _2}{n_2}-\frac{\sigma _1}{n_1}=s_2-s_1=\frac{1}{12T_1}\bigg (\frac{\partial ^2 (1/n)}{\partial p_1^2}\bigg)_s (p_2-p_1)^3. \end{equation} \tag{ 26 }$$

The law of entropy increase requires that s₂ − s₁ > 0.

From the EoS (5), the particle number density can be expressed as

$$\begin{equation} \frac{1}{n}=\frac{(\Gamma -1)m_0\epsilon }{p}. \end{equation} \tag{ 27 }$$

By using this expression in the entropy equation (26), we have

$$\begin{equation} \frac{\sigma _2}{n_2}-\frac{\sigma _1}{n_1}=\frac{(\Gamma -1)m_0\epsilon _1}{6T_1p_1^3}(p_2-p_1), \end{equation} \tag{ 28 }$$

from which it is easily identified that for normal upstream fluids with p₁ > 0 and ₁ > 0, rarefaction waves with p₂ − p₁ < 0 only exist for the adiabatic index Γ < 1. So under the regime of weak shock, the Type I negative-pressure fluid with Γ = 4/3 does not exist due to the violation of entropy increase law, while Type II negative-pressure fluid with Γ = 2/3 is expected to be a real physical existence. However, since it is still questionable whether the entropy analysis in weak shocks can be generalized to normal strong shocks, the existence or non-existence of Type I negative-pressure fluids in strong shocks needs to be further studied.

Representing the overall energy release per unit mass by q, which is positive and negative for exothermic and endothermic reactions, respectively, the specific enthalpies per unit volume across the reaction front⁵ are

$$\begin{eqnarray} \omega _1&=&e_1+p_1+\rho q \quad {\rm and} \nonumber \\ \omega _2&=&e_2+p_2 \end{eqnarray} \tag{ 29 }$$

for upstream and downstream flows, respectively. Following the procedure in deriving Equations (12) and (15), the Rayleigh and Hugoniot relations for relativistic reactive flows are given by

$$\begin{equation} \hat{p}-1= -u_1^2 \bigg [\hat{c}^2(\hat{V}-1)+\frac{\Gamma }{\Gamma -1}(\hat{p}\hat{V^2}-1)-\hat{q}\bigg ], \end{equation} \tag{ 30 }$$

$$\begin{eqnarray} &&\hat{c}^2\bigg [\frac{\Gamma +1}{\Gamma -1}(\hat{p}\hat{V}-1)-(\hat{p}-\hat{V}) -2\hat{q} \bigg ] = \frac{\Gamma }{\Gamma -1}\hat{p}(1-\hat{V}^2)\nonumber\\ &&\quad - \frac{\Gamma }{(\Gamma -1)^2}(\hat{p}^2\hat{V}^2-1) +\bigg (\hat{p} + \frac{\Gamma +1}{\Gamma -1}\bigg)\hat{q} + \hat{q}^2, \end{eqnarray} \tag{ 31 }$$

where the reduced heat release is $\hat{q}=({\rho _1}/{p_1})q$.

Representative Hugoniot lines described by relation (31) are shown in Figure 5. We see that while the non-reactive Hugoniot line passes through the (1,1) point in the $\hat{p}\hbox{--}\hat{V}$ diagram, the Hugoniot lines for flows with exothermic and endothermic reactions are above and below the non-reactive one, respectively. This difference leads to the possible existence of weak detonations in endothermic reactive flows, which will be discussed further when considering Figure 7.

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We next note that Rayleigh lines in relativistic reactive fluids do not pass through the (1,1) point in the $\hat{p}\hbox{--}\hat{V}$ diagram (see, e.g., Figure 6) as they do in non-reactive relativistic flows (Figure 2). This is because in relativistic reactive flows, the reaction heat release not only is part of the total energy, but it is also present in the momentum conservation equation (see Equations (2) and (29)), which then leads to the presence of the $\hat{q}$ term in the Rayleigh relation (30).

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We now separately discuss the solutions for the relativistic detonation and deflagration waves.

Detonation wave solutions for relativistic fluids with exothermic and endothermic reactions are shown in Figures 6 and 7, respectively.

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Figure 6 shows that detonation solution exists for exothermic reactive fluids with relatively large upstream speeds, i.e., $u_1 \ge \sqrt{2}$, while there is no detonation solution for fluids with relatively slow upstream speed, i.e., $u_1 \le 1/\sqrt{2}$. However, the criterion of the upstream flow speed for the existence of detonation is higher than the speed of sound,⁶ which is different from the case of non-relativistic fluids, for which this criterion is exactly the speed of sound. This is due to the existence of the heat release $\hat{q}$ in the Rayleigh relation (30) for relativistic reactive flows. By numerically exploring Rayleigh lines with different upstream speed u₁, we find that the criteria for the existence of detonation wave solutions are u₁ = 1.0 and u₁ = 0.9 for relativistic ($\hat{c}=1$ and Γ = 3/2) and highly relativistic ($\hat{c}=0$ and Γ = 4/3) gases, respectively, both with the energy release $\hat{q}=1$ (see Figure 6). Another interesting feature is that, while flows with $u_1=\sqrt{2}$ have both strong and weak detonation solutions, higher speed flows with u₁ = 4 have only weak detonation solutions. The reason for the disappearance of strong detonation for exothermic fluids with high-speed upstream flows is that the reaction heat $\hat{q}$ is greatly amplified by the large value of the upstream flow speed u₁ (this is a relativistic effect), which then leads to large pressure compression ratio $\hat{p}$ to satisfy the energy conservation requirement involved in the Hugoniot relation (31).

Detonation solutions in endothermic reactive fluids (Figure 7) are quite different. Only weak detonation can be found, and its solution exists for various upstream flow speeds ranging from subsonic to supersonic. The reason for the extensive existence of weak detonations is that endothermic fluids tend to absorb the fluid kinetic energy of any strength in order to initiate the endothermic reactions (e.g., ionization of the ISM), which then behave as detonations with relatively lower pressure and density compression ratios. In Figure 7, the Rayleigh lines for all upstream flows with different speeds have single intersections with the Hugoniot line, which shows that there is no threshold for the existence of weak detonations. On the other hand, strong detonations cannot be formed for flows with any upstream speed.

Deflagration wave solutions for relativistic fluids with exothermic and endothermic reactions are shown in Figures 8 and 9, respectively.

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Figure 8 shows that exothermic reactive fluids have different kinds of deflagration waves for different upstream flow speeds. Specifically, when the upstream flow speed is higher than u₁ = 0.5, there is no deflagration solution in both relativistic and highly relativistic flows. Taking into consideration that no detonation solution exists for exothermic flows with upstream speeds $u_1\le 1/\sqrt{2}$ (see Section 3.2 and Figure 6), we conclude that for a certain range of the upstream flow speed (e.g., $1/2 \le u_1\le 1/\sqrt{2}$ in Figures 6 and 8) in relativistic exothermic reactive fluids, neither detonation nor deflagration waves can form. More specifically, numerical criteria for the existence of deflagration waves have been identified as u₁ = 0.37 and u₁ = 0.39 for the relativistic and highly relativistic gases in Figure 8, with the energy release $\hat{q}=1$. Figure 8 further shows that, for the relatively low upstream speed of u₁ = 0.3, both strong and weak deflagrations exits, but when the upstream speed becomes much lower, i.e., u₁ = 0.2, strong deflagration cannot be achieved, leaving only the weak deflagration. This can be understood by referring to the Rayleigh relation (30), which shows that as the upstream speed u₁ decreases, the pressure rarefaction ratio $\hat{p}$ (< 1) should be higher for the same reduced specific volume $\hat{V}$. Then, at a certain low value of u₁, there is no intersection between the Rayleigh and Hugoniot lines for higher $\hat{V}$, i.e., strong deflagration does not exist.

The wave response is however quite different for endothermic reactive fluids, as shown in Figure 9. No deflagration solution can be found for flows with upstream speeds of u₁ = 0.2 and 0.5. Deflagration waves emerge only when the upstream speed is as high as u₁ = 1.0 (higher than the sound speed). The numerical criteria for the existence of deflagration waves are u₁ = 0.7 and u₁ = 0.6 for relativistic and highly relativistic gases in Figure 9, respectively, with the energy release $\hat{q}=-1$. Referring to the deflagration waves with low upstream speed in exothermic reactive fluids, we expect that endothermic deflagrations need higher upstream flow speeds to propagate. Furthermore, even when the upstream flow speed is higher than the speed of sound, a shock cannot form because part of the fluid kinetic energy is absorbed by the endothermic reaction, which is exactly the case of u₁ = 1.0 in Figure 9. It is noted (see Section 3.2 and Figure 7) that the existence of weak detonations for a large range of the upstream speed is a distinct feature of endothermic flows. Consequently weak detonation waves, instead of deflagrations, should be the expected dynamic structure in endothermic reactive flows.

As a parallel analysis to the non-relativistic Chapman–Jouguet waves, the tangency point of the Rayleigh and Hugoniot lines, representing a limit case of the wave solutions, is analyzed for relativistic fluids. At the tangency point, the slopes of Rayleigh and Hugoniot lines are equal to each other, so slopes of both curves in the $\hat{p}\hbox{--}\hat{V}$ diagram are calculated here according to Equations (30) and (31):

$$\begin{equation} \bigg (\frac{d\hat{p}}{d\hat{V}}\bigg)_{\rm Rayleigh} = \frac{\hat{c}^2(\hat{p}-1)+2\frac{\Gamma }{\Gamma -1}\hat{p}\hat{V}(\hat{p}-1)}{\hat{c}^2(\hat{V}-1)+\frac{\Gamma }{\Gamma -1}(\hat{V}^2-1)-\hat{q}}, \end{equation} \tag{ 32 }$$

$$\begin{eqnarray} &&\bigg (\frac{d\hat{p}}{d\hat{V}}\bigg)_{\rm Hugoniot}\nonumber\\ &&\quad = - \frac{\hat{c}^2\left(\frac{\Gamma +1}{\Gamma -1}\hat{p}+1\right) +2\frac{\Gamma }{\Gamma -1}\hat{p}\hat{V}+2\frac{\Gamma }{(\Gamma -1)^2}\hat{p}^2\hat{V}}{\hat{c}^2\left(\frac{\Gamma +1}{\Gamma -1}\hat{V}-1\right)-\frac{\Gamma }{\Gamma -1}(1-\hat{V}^2)+2\frac{\Gamma }{(\Gamma -1)^2}\hat{p}\hat{V}^2-\hat{q}}. \nonumber\\ \end{eqnarray} \tag{ 33 }$$

And the solution to the relation

$$\begin{equation} \bigg (\frac{d\hat{p}}{d\hat{V}}\bigg)_{\rm Rayleigh}=\bigg (\frac{d\hat{p}}{d\hat{V}}\bigg)_{\rm Hugoniot} \end{equation} \tag{ 34 }$$

shows the criterion of the waves represented by the tangency point. Typical numerical solutions for the Chapman–Jouguet waves as criteria for the existence of detonation waves are given in Section 3.2 and Figure 6 caption.

It is noted that in the non-relativistic extreme of $\hat{c}\rightarrow \infty$, the tangency solution degenerates to the normal Chapman–Jouguet wave, i.e.,

$$\begin{equation} \frac{\hat{p}-1}{\hat{V}-1}= - \frac{\frac{\Gamma +1}{\Gamma -1}\hat{p}+1}{\frac{\Gamma +1}{\Gamma -1}\hat{V}-1}. \end{equation} \tag{ 35 }$$

This condition implies that the downstream flow is sonic (cf. Law 2006), which leads to the classical Chapman–Jouguet wave in which the downstream flow does not affect the the wave front so that the wave propagates steadily. However, no simple solution of the tangency point can be found for relativistic fluids as the heat release $\hat{q}$ is involved in both relations (32) and (33).

A comparison between Figures 6–9 and 2 shows that Rayleigh lines for exothermic and endothermic fluids intersect at different points from those of non-reactive fluids in the $\hat{p}\hbox{--}\hat{V}$ diagram. Specifically, in exothermic reactive fluids, the Rayleigh lines intersect at a point with $\hat{p}=1$ and $\hat{V}>1$, while for endothermic reactive fluids, the intersection is at $\hat{p}=1$ and $\hat{V}<1$. This difference between reactive and non-reactive fluids is caused by the decrease and increase of flow densities in constant-pressure exothermic and endothermic waves, respectively. The example of Figure 7 shows that because the intersection of Rayleigh lines for endothermic reactive fluids is to the right of the Hugoniot line, one can easily achieve weak detonation solutions.

Since the intersections of Rayleigh lines for reactive fluids with different reaction heats $\hat{q}$ have the same reduced pressure $\hat{p}=1$, we expect the existence of a common solution with $\hat{p}=1$ for any given reaction heat $\hat{q}$. Then, the Rayleigh relation (30) can be readily converted to the following form by considering $(\hat{p}-1$), instead of $\hat{p}$, as a common factor:

$$\begin{equation} \hat{p}-1=\frac{-u_1^2\left[\frac{\Gamma }{\Gamma -1}\hat{V}^2+\hat{c}^2\hat{V}-\left(\hat{c}^2+\hat{q}+\frac{\Gamma }{\Gamma -1}\right)\right]}{1+u_1^2\hat{V}^2\frac{\Gamma }{\Gamma -1}}. \end{equation} \tag{ 36 }$$

If we calculate the (positive) $\hat{V}$ root of the numerator on the right-hand side of this equation, the coordinate of the intersection of the Rayleigh line is

$$\begin{eqnarray} &&\hat{p}=1,{\quad} \hat{V}=\frac{\Gamma -1}{2\Gamma }\!\left[\!-\hat{c}^2+\sqrt{\hat{c}^4+ 4\frac{\Gamma }{\Gamma -1}\!\left(\hat{c}^2+\hat{q}+\frac{\Gamma }{\Gamma -1}\!\right)}\!\right]. \nonumber\\ \end{eqnarray} \tag{ 37 }$$

In the non-relativistic limit, $\hat{c}\rightarrow \infty$, the intersection is at $\hat{p}\,{=}\,1\,{\rm and }\,\hat{V}\,{=}\,1$, while in the highly relativistic limit, $\hat{c}\rightarrow 0$, the intersection is at $\hat{p}\,{=}\,1\,{\rm and }\,\hat{V}\,{=}\,\sqrt{1+(({\Gamma -1})/{\Gamma })\hat{q}}$. (See Figures 6–9 for the variation of the intersection.) In the limiting case of small reaction heat in highly relativistic fluids, i.e., $\hat{q}=({\rho _1}/{p_1})q \ll 1$, the $\hat{V}$-axis of the intersection can be expressed as $\hat{V}=1+(({\Gamma -1})/({2\Gamma }))\hat{q}$, which can be readily converted to the form of

$$\begin{equation} V_2-V_1=\frac{\Gamma -1}{2\Gamma }\frac{q}{p_1}. \end{equation} \tag{ 38 }$$

The above equation implies that in an isobaric wave, the specific volume increases and decreases for exothermic and endothermic reactions, respectively. In other words, the flow density decreases in an exothermic reaction as the fluid releases energy and increases in an endothermic reaction due to the absorption of energy, if the pressure is kept constant.

Based on entropy considerations, rarefaction shocks normally do not exist for fluids with the adiabatic index Γ > 1 (cf. Section 2.4, Landau & Lifshitz 1959; Zel'dovich & Raizer 1966), so those weak detonation as well as strong deflagration waves involving rarefaction shocks also normally do not exist. However, in non-relativistic fluids, weak detonation and strong deflagration waves can be found in systems with endothermic reactions or phase transitions, in which a rarefaction shock is not necessary (Axford 1961; Williams 1985). While these understandings still apply in exothermic relativistic detonations and deflagrations, i.e., the weak detonation solutions in Figure 6 and strong deflagration solutions in Figure 8 should normally not exist in realistic systems, there is an exception in that for very high upstream flow speeds (e.g., Rayleigh lines with u₁ = 4 in Figure 6), weak detonation waves could exist as no rarefaction shock structure is required in such detonations. Instead, these exothermic weak detonations with high upstream speeds are similar to the weak detonation in endothermic reactive fluids (Figure 7), as both reactions can be directly ignited in the shockless waves because the kinetic energy of the upstream flow is large enough to initiate the reaction. It is therefore reasonable to expect that weak detonation as well as strong deflagration waves of this kind are more common in high-speed astrophysical flows, and they are expected to have different structures from the traditional Zel'dovich–von Neumann–Döring structure in strong detonation waves (cf. Williams 1985). One such example is that of relativistic endothermic reactive fluids, for which weak detonation waves instead of weak deflagration waves are the dominant dynamics as can be seen from Figures 7 and 9.

We further note that the non-relativistic form of endothermic combustion waves has been studied for interstellar gas ionization (Axford 1961). The ionization of neutral hydrogen atoms is an endothermic reaction which forms an "ionization front" in the ISM. This front, separating the unionized (H i) and the fully ionized (H ii) regions, is dynamically identical to the detonation (or deflagration) front. Axford (1961) has shown the existence of both weak detonation and strong deflagration waves theoretically and what we discussed here is a generalization of these results to the relativistic regime. Besides the ionization of neutral hydrogen atoms around hot stars, the re-ionization (also of neutral hydrogen atoms by stars and quasars) process of the universe (Miralda-Escudé 2003) can also be considered as an endothermic combustion wave. Another endothermic fluid process in astrophysics is inverse Compton scattering, the mechanism for a large variety of high-energy X-ray and γ-ray radiations (e.g., De Jager & Harding 1992; Tavani 2011).

For relativistic fluids with Γ < 1, rarefaction shocks are allowed through entropy consideration (cf. Section 2.4), and as such lead to the formation of weak detonations. This is obviously another way of having weak detonations in astrophysical relativistic fluids.