Rankine–Hugoniot Shock Conditions for Space and Astrophysical Plasmas Described by Kappa Distributions

This paper provides the set of Rankine–Hugoniot (R–H) jump conditions for shocks in space and astrophysical plasmas described by kappa, distributions. The characteristic result is the development of a new R–H condition that transforms the values of kappa upstream and downstream the shock. The kappa index parameterizes and labels kappa distributions, and it is necessary for characterizing the thermodynamics of space plasmas. This first approach is restricted to non-magnetized plasmas, and the whole achievement is derived by following first principles of statistical mechanics and thermodynamics. The results show that, depending on the shock strength, the kappa indices across the shock may decrease or increase, indicating cases of shock acceleration or deceleration, respectively.


Introduction
The Rankine-Hugoniot (R-H) jump conditions transform the thermodynamic properties of space and astrophysical plasmas during their passage through shock discontinuities (Rankine 1870;Hugoniot 1887Hugoniot , 1889. These conditions describe the relationships between thermodynamic variables of the plasma flow, such as, density, velocity, temperature, and magnetic field, on both sides of the shock. These relationships enable us to link the thermodynamic properties upstream and downstream of shocks and derive their characteristics, such as the shock speed and normal angles (e.g., see: Colburn & Sonett 1966;Zhuang & Russell 1981;Winterhalter et al. 1984;Szabo 1994;Petrinec & Russel 1997;Owen 2004;Zank et al. 2010;Janvier et al. 2014).
In particular, the R-H conditions connect the upstream and downstream values of (i) the density n, (ii) the bulk velocity of the flow V, (iii) the angle of the velocity flow to the shock normal a, and (iv) the temperature T, based on the following four conservations: 1. Mass conservation: = n V a n V a cos cos , 1a where the subscript notation indicates "1" for upstream and "2" for downstream; m is the particle mass (that is, the proton mass in this paper); the polytropic index γ is the exponent associated with a polytropic relationship between pressure-density µ g P n or density-temperature µ gn T 1 1 ( ) (e.g. , Parker 1963;Chandrasekhar 1967;Totten et al. 1995;Newbury et al. 1997;Kartalev et al. 2006;Nicolaou et al. 2014Nicolaou et al. , 2015; the polytropic index γ can be written in terms of an effective dimensionality, d, that is, g = + d 1 2 or g º d 2 1 ( ) (e.g., Sanderson & Uhrig 1978); thus, µ + P n d 1 2 or µ n T d 2 . Note: for the adiabatic case (only), the effective d coincides with the actual dimensionality of particles.
The solutions of the above relationships are the four explicit expressions of the downstream variables, expressed in terms of the upstream variables (e.g., Liepmann & Roshko 1957;Livadiotis 2015a

( )
To date, the R-H conditions have been exclusively developed for plasmas described by the classical Maxwell-Boltzmann distributions (e.g., Cairns & Grabbe 1994;Zank et al. 1994Zank et al. , 2010Livadiotis 2015a). However, space plasmas are characterized by a non-Maxwellian behavior, typically manifested by kappa distributions. Indeed, kappa distributions have become increasingly widespread across the physics of space and astrophysical plasmas, describing particles in the heliosphere, from the solar wind and planetary magnetospheres to the distant heliospheric boundaries and beyond, to the interstellar and intergalactic plasmas (Livadiotis 2017;Ch.1).
The origin of kappa distributions in space plasmas comes from its solid connection with statistical mechanics and thermodynamics. It has been shown that the most generalized form of particle distributions that can be assigned with a temperature, and thus, be consistent with thermodynamics, is that of kappa distributions (or combinations thereof) (Livadiotis 2018a(Livadiotis , 2018b. The concept of "thermalization" is the characterization of a particle system residing in any stationary state assigned by a temperature; in these states, the particle velocities or energies are stabilized into kappa distributions. Kappa distributions are consistent with thermodynamics, but this fact alone cannot justify the generation and existence of these distributions. Once a kappa distribution of velocities or energies is generated by a certain mechanism, the preservation-or not-of this distribution is a matter of thermodynamics alone. Nevertheless, there are a number of mechanisms generating kappa distributions in space plasmas; among others, some important examples are the following: macroscopic extensivity of entropy (Livadiotis 2018c), superstatistics (Beck & Cohen 2003;Schwadron et al. 2010;Hanel et al. 2011;Livadiotis et al. 2016); local correlations among the particles induced by long-range interactions (Livadiotis & McComas 2011a); effect of shock waves (Zank et al. 2006); weak turbulence (Yoon et al. 2006;Yoon 2014Yoon , 2019; turbulence with a diffusion coefficient inversely proportional to velocity (Bian et al. 2014); effect of pickup ions (Livadiotis & McComas 2011b); pump acceleration mechanism (Fisk & Gloeckler 2014); polytropic behavior (Livadiotis 2019a(Livadiotis , 2019b(Livadiotis , 2019c(Livadiotis , 2018d(Livadiotis , 2016Nicolaou & Livadiotis 2019); and effects of Debye shielding and magnetic coupling (Livadiotis et al. 2018); for all the above, see Livadiotis (2017, Chapters 5, 6, 8, 10, 15, 16).
The kappa index is the characteristic parameter that governs the kappa distributions, and it is necessary for characterizing the thermodynamics of space plasmas. The physical meaning of the kappa index is interwoven with the particle correlations; the stronger the correlations, the smaller the kappa. The largest value of kappa, i.e., k  ¥, corresponds to the system residing at the classical thermal equilibrium, described by a Maxwell-Boltzmann distribution, and characterized by the absence of any correlations; on the other hand, the smallest possible kappa value, k  3 2 , corresponds to the furthest state from classical thermal equilibrium, a state called anti-equilibrium (Livadiotis & McComas 2013), and characterized by the highest correlations. In fact, a simple relationship exists between the correlation coefficient ρ and the kappa index κ, that is, r k = 3 2 (Livadiotis & McComas 2011a;Livadiotis 2015b, 2017; for particles with d=3 degrees of freedom).
As it was shown (Livadiotis 2018a), the kappa index is a thermodynamic quantity independent of temperature, where both the temperature and kappa are primarily defined by the zeroth law of thermodynamics and the concept of thermal equilibrium. Thereafter, the kappa index, together with density, temperature, and pressure, constitute the set of basic thermal observables for characterizing the thermodynamics of plasmas described by kappa distributions, such as space and astrophysical plasmas. It is, then, obvious that the set of R-H conditions, which is based on classical statistical mechanics and thermodynamics, needs to be modified to incorporate the concept of kappa distributions.
The set of R-H conditions incorporating kappa distributions must keep the mass, momentum, and energy conservations, i.e., Equations 1(a)-(d). As it is argued in Livadiotis (2015a), the statistical averages of mass, momentum, and energy, are independent of the kappa index, and thus, all kappa indices (infinite or finite) are physically equivalent for describing the thermodynamic nature of the plasma. The respective R-H conditions, corresponding to the conservation of mass, momentum, and energy, can be equivalently used in particle systems described either by Maxwell-Boltzmann or kappa distributions.
Therefore, for space plasmas described by kappa distributions, (i) the density n, (ii) the bulk velocity of the flow V, (iii) its angle to the shock normal a, and (iv) the temperature T, are still given by the standard R-H conditions, given by Equations 2(a)-(d); even so, the shock strength R is still given by Equation (3).
What is actually missing is an additional equation that will connect the kappa value upstream and downstream the shock, k k = f R ; 2 1 ( ). As mentioned by Livadiotis (2015a), "the values of the kappa index upstream and downstream the shock must be connected through a new R-H condition, yet to be discovered." There were several attempts to resolve the problem of incorporating kappa distributions in the R-H conditions. Unfortunately, these were based on temperature misinterpretations. In the earlier years of the theory of kappa distributions, there was the impression that the temperature was dependent on the kappa index, i.e., where t was interpreted as a temperature-related quantity that remains invariant for all kappas, frequently called the Maxwellian temperature. Given such a relationship between temperature and kappa index, k = -T T T t 3 2 ( ) ( ), and having already known the R-H condition for temperature, i.e., Equation 2(c), then it could be trivially used to develop the R-H condition for kappa index, i.e., k k = --T T t T T t 2 1 2 1 1 2 ( ) [ ( )] (e.g., Vogl et al. 2003). However, the development of the theory the previous two decades showed that the temperature and kappa index are both independent thermodynamic parameters (and that t is depended on both κ and T). (For further details, see: Livadiotis 2014Livadiotis , 2018aLivadiotis , 2018b; see also the book of kappa distributions, Livadiotis 2017, Chapter 1.) The paper attempts to develop a new set of R-H conditions incorporating kappa distributions, which is based on physical principles and consistent with thermodynamics. Specifically, the paper constructs the relationship between the upstream and downstream values of kappa indices, k k = f R ; 2 1 ( ), based on the following physical frameworks: (i) nonextensive statistical mechanics; (ii) connection of kappa distributions with statistical mechanics and thermodynamics; and (iii) mechanism of "superstatistics." The presented first approach is restricted to non-magnetized plasmas. In Section 2 we presented the method followed, which was based on the above physical concepts, in order to find the R-H condition for the kappa index. The results of our theoretical analysis are shown in Section 3, while they are discussed in detailed in Section 4. Finally, Section 5 summarizes the findings of this paper.

Method
The presented theoretical analysis is based on the following theoretical frameworks, (i) nonextensive statistical mechanics (e.g., Tsallis 1988Tsallis , 2009Tsallis et al. 1998); (ii) connection of kappa distributions with statistical mechanics and thermodynamics (e.g., Livadiotis & McComas 2009;Livadiotis 2017, Ch.1;2018a); and (iii) mechanism of "superstatistics," that is, the analysis of a kappa distribution to superposition of Maxwell-Boltzmann distributions of variant temperatures (e.g., Beck & Cohen 2003;Schwadron et al. 2010;Hanel et al. 2011;Livadiotis et al. 2016), in order to develop the evolution of kappa distributions across a shock, thus deriving the relationship between the upstream and downstream values of the kappa index, k k = f R ; 2 1 ( ). Nonextensive statistical mechanics is a generalization of the classical statistical physical theory of Boltzmann and Gibbs; among others, it describes particle systems with local correlations such as space and astrophysical plasmas. The involved entropy is a mono-parametrical function of the probability distribution. This entropic parameter q can attain any value, while q→1 recovers the Boltzmannian entropy and the Boltzmann-Gibbs statistical mechanics. The nonextensive statistical mechanics is the background of kappa distributions.
The theory of kappa distributions shows that the kappa and the entropic q indices are connected through k = q 1 1 ( ). The kappa index depends on the particle dimensionality or degrees of freedom d, as k k The physical meaning of the thermodynamic parameter kappa is better carried by its invariant value k 0 , as this is independent of the degrees of freedom (Livadiotis & McComas 2011a, 2013Livadiotis 2015bLivadiotis , 2015cLivadiotis , 2017. Therefore, throughout this paper, we use the notion of the invariant kappa index k 0 , but the typical threedimensional index can be easily retrieved, Specifically, the mechanism of "superstatistics" (e.g., Beck & Cohen 2003;Schwadron et al. 2010;Hanel et al. 2011;Livadiotis et al. 2016) deals with the analysis of the canonical distribution (derived within nonextensive statistical mechanics) as a superposition of Maxwell-Boltzmann distributions. It shows that the kappa index can be expressed in terms of thermal fluctuations, i.e., k d ). The infogram in Figure 1 shows the path of the analytical derivations.

Derivation of the Kappa Index Expression in Terms of the Temperature Standard Deviation
First, we derive the relationship k d = F T T 0 ( ). This derivation is based on the theory of superstatistics, one of the possible mechanisms that can generate kappa distributions in space plasmas. Below, we expose in more detail the theory of superstatistics and how it is involved in the derivation Let the d-dimensional Boltzmannian distribution of kinetic energy ε, for temperature denoted by its inverse β≡k B T −1 , i.e., If the temperature is not fixed and characterized by fluctuations, then the system does not reside in the classical thermal equilibrium described by Maxwell distributions. Instead, the superposition of Maxwell distributions with different temperatures can lead to a stationary state that is described by a kappa distribution, ) denotes the Beta function. As we will see, the definition of the global temperature can be related with the inverse of the average value of β, i.e., This is the generalization of the Maxwellian distribution in Equation (6) for finite values of the kappa index κ 0 , while Equation (6) recovers for κ 0 →¥. It has been shown that particle systems with variable temperature are consistent with a particular distribution function of temperatures, that is, the gamma distribution (a generalization of the chi-square distribution; see: Abramowitz & Stegun 1972), with average b á ñ and degrees of freedom According to superstatistics, the kappa distribution can be generated by the following integral transformation which expresses the transition from the equilibrium state, described by the Maxwell-Boltzmann distribution, to the nonequilibrium state, described by the kappa distribution (of kappa index κ 0 ). Then, the integral in Equation (11) is rewritten as: that is, the kappa distribution, as shown in Equation (7), with kappa index k º d Next, we express the normalized standard deviation of temperature, dT T, as a function of the normalized standard deviation of the inverse temperature, db b á ñ. We consider the deviation of the inverse temperature b b b  + D , which corresponds to a propagated deviation to temperature, The Maxwell-Boltzmann distribution is characterized by zero temperature deviation, while the latter must be larger as the kappa index decreases. It is therefore required to have zero deviation for κ 0 →∞ and infinite deviation when κ 0 →0; hence, we keep "−1" in (24):

Propagation of the Temperature Standard Deviation
Next, we derive the error propagation relationship d d = T G T R ; 2 1 ( ). We derive the relationship that connects the thermal fluctuations upstream and downstream of the shock. This is achieved by perturbing the energy conservation condition (that is, the Bernoulli integral), in Equation 2(c), ending up with d d = T G T R ; 2 1 ( ). The energy is conserved upstream and downstream of the shock, á ñ = á ñ E E 1 2 , where E is the particle kinetic energy in the shock reference frame. Indeed, we have Then, the first term to the right becomes  We then derive the same parameter downstream of the shock: from Equations 2(b), (c). Its inverse is given by (

Returning to Equation
while in the case of r 1  , the above is approximated to ( ), as set in the infogram of Figure 1. Also, we may readily derive the condition for the temperature deviation, that is,

Connection between Upstream and Downstream Kappa Indices
At this last step, using the relationships k d =

Shock Acceleration or Deceleration
The family of curves k k = f R ; The critical curve (plotted with thick black line in Figure 2(a)) separates shocks that accelerates the plasma flow (i.e., κ 02 < κ 01 ) from those shocks that decelerate the plasma flow (i.e., κ 02 >κ 01 ). Indeed, when the kappa index decreases (κ 01 →κ 02 <κ 01 ), then the suprathermal tail at the high energy part of the distribution becomes harder (more flattened), thus, a larger number of particles correspond to higher energies and a smaller number to moderate energies (from which particles are accelerated). When the kappa index increases (κ 01 → κ 02 >κ 01 ), then the suprathermal tail becomes softer (more sharpened), thus a larger number of particles correspond to moderate energies and a smaller number to higher energies (from which particles are decelerated). Setting R=1 in Equation 43(b), we obtain the minimum possible upstream kappa index, k 01 min , for which the downstream kappa index is smaller than the upstream index, κ 02 <κ 01 ; hence, the acceleration is possible only for k k > 01 01min , where: In the case of r=d/2 and d=3, we find k = 3.486 ...

01min
. The acceleration (A) and deceleration (D) regions are shown in Figures 2(a), (b). In panel (a) the downstream kappa index κ 02 is plotted as a function of the upstream kappa index κ 01 , for various shock strength values, R. Region D {A} is above {below} the diagonal κ 02 =κ 01 . We observe that for small upstream kappa indices, e.g., κ 01 <1, the corresponding 0iso , for shock strength R smaller than the critical R c ≈3.73205; for RR c , the super-deceleration (sD) region appears. , while for R>R c , the downstream kappa index is dramatically increasing, even for very small upstream kappa indices, a behavior we call superdeceleration (sD).
The behavior of the downstream kappa index, κ 02 , as a function of the shock strength, R, is illustrated in Figures 2(c), (d). Panel (c) plots the downstream index κ 02 against R, for various upstream kappa indices κ 01 , while panel (d) plots the corresponding ratio, k k 02 01 , against R on a semi-log scale. The strongest acceleration is achieved at the local minimum of k = f R 02 ( ), as shown in Equation 41(b); the shock strength corresponding to this minimum is given by the quartic solution of: The minimum shock strength, R min , corresponds to the smallest possible downstream kappa index, κ 02 , thus, to the strongest acceleration for a given upstream kappa index, κ 01 ; note: s 1 is expressed in terms of κ 01 , as shown in Equation (25).

Symmetry
Strictly mathematically, the two sides of the shock must be equivalent (due to mirror symmetry). The shock compresses the plasma with strength R, while on the other side rarefies the plasma with strength R −1 . There is no mathematical preference for any side of the shock, upstream or downstream; then, any R-H shock condition should obey to this symmetry: «

Connection with the Polytropic Index
In the case where the plasma particles are subject to a potential energy, the polytropic behavior has been shown to have a one-to-one relationship with kappa distributions. Namely, not just the old-known result that the kappa distributions can lead to the polytropic relationship, µ g P n or µ g- -Vernet et al. 1995;Moncuquet et al. 2002;Livadiotis 2017, Ch. 5;2018d;Nicolaou & Livadiotis 2019), but the reverse derivation has also been shown, that is, the polytropic behavior requires the particle velocities to be described by the kappa distributions. This means that the polytropic behavior has the role of a mechanism generating kappa distributions (Livadiotis 2019a(Livadiotis , 2016. Then, the developed relationship between the kappa indices upstream and downstream of the shock leads to the corresponding relationship between the polytropic indices.

What's Next: Incorporation of the Magnetic Field and Turbulence
It is our next goal to work on the generalization of the R-H conditions for particle velocities described by kappa distributions and oblique shocks in magnetized space plasmas. In this case, the temperature fluctuation dT will have to be connected with both the momentum fluctuation dV m and the magnetic fluctuation dB (Zhuang & Russell 1981;Cairns & Grabbe 1994). Finally, further investigation is necessary for understanding the role and contribution of turbulence in the fluctuations of dV m and dB, as well as of the turbulent energy (e.g., Bavvasano et al. 2000;Adhikari et al. 2015), derived from the variance of the Elsässer variables (Tu & Marsch 1995).

Summary and Conclusions
This paper developed the R-H jump conditions for shocks in space and astrophysical non-magnetized plasmas described by kappa distributions. The set of R-H conditions incorporating  kappa distributions must keep the known mass, momentum, and energy conservations. Therefore, for space plasmas described by kappa distributions, (i) the density n, (ii) the bulk velocity of the flow V, (iii) its angle to the shock normal a, and (iv) the temperature T, are still given by the standard R-H conditions. The challenge was to derive the R-H jump condition for the new thermodynamic parameter, the kappa index. This was achieved using the following theoretical frameworks: (i) nonextensive statistical mechanics; (ii) connection of kappa distributions with statistical mechanics and thermodynamics; and (iii) mechanism of "superstatistics" (that is, the analysis of a kappa distribution to superposition of Maxwell-Boltzmann distributions of variant temperatures). The result was to develop the evolution of kappa distributions across a shock, that is, to derive the relationship between the upstream and downstream values of the kappa index, k k = f R ; ( ) was derived following three steps: (i) used of the theory of superstatistics, in order to derive the kappa index expression in terms of the temperature standard deviation,  The presented theoretical analysis was restricted under the following simplifications: non-magnetized plasma; plasma flow parallel to shock normal; isotropic dimensionality and polytropic indices equal for both sides of the shock; and absence of potential energy. It is therefore straightforward to follow the method presented here in extended future theoretical analyses, in order to include oblique shocks in magnetized and anisotropic space and astrophysical plasmas.
The work was supported in part by the project NNX17AB74G of NASA's HGI Program.