This paper reports on the properties of kappa distributions in the presence of a potential energy. The constructed phase space kappa distribution leads to a polytropic relationship between the local ...density and temperature. The kappa and polytropic indices are connected via a characteristic and universal relationship: their sum is a constant depending specifically on the potential features. Determining this constant comprises a method essential for revealing the functional form and understanding the nature of the particle potential energy. The method is applied to the solar wind proton plasma at 1 AU. It is shown that a satisfactory physical interpretation of the potential affecting solar wind protons can be given by the interplanetary radial electric field.
Key Points
Show the connection between polytropic and kappa indices
Present a method to identify the potential in space plasmas
Application to solar wind proton plasma ~ 1 AU
It is shown that the polytropic behavior-a specific power-law relationship among the thermal plasma moments-restricts the functional form of the distribution of particle velocities and energies. ...Surprisingly, the polytropic behavior requires the statistical mechanics of the plasma particles to obey the framework of kappa distributions. An already known interesting property of these distributions is that they can lead to the polytropic relationship. New results show that the reverse derivation is also true, thus, the polytropic behavior has the role of a mechanism generating kappa distributions. Ultimately, an observation of a polytropic behavior in plasma particle populations constitutes a possible indirect observation of kappa velocity or energy distributions. Finally, it is discussed how the derived equivalence between the polytropic behavior and the kappa distribution function can be used in further modeling and data analyses in space and astrophysical plasmas.
ABSTRACT This paper presents a possible generalization of the equation of state and Bernoulli's integral when a superposition of polytropic processes applies in space and astrophysical plasmas. The ...theory of polytropic thermodynamic processes for a fixed polytropic index is extended for a superposition of polytropic indices. In general, the superposition may be described by any distribution of polytropic indices, but emphasis is placed on a Gaussian distribution. The polytropic density-temperature relation has been used in numerous analyses of space plasma data. This linear relation on a log-log scale is now generalized to a concave-downward parabola that is able to describe the observations better. The model of the Gaussian superposition of polytropes is successfully applied in the proton plasma of the inner heliosheath. The estimated mean polytropic index is near zero, indicating the dominance of isobaric thermodynamic processes in the sheath, similar to other previously published analyses. By computing Bernoulli's integral and applying its conservation along the equator of the inner heliosheath, the magnetic field in the inner heliosheath is estimated, B ∼ 2.29 0.16 G. The constructed normalized histogram of the values of the magnetic field is similar to that derived from a different method that uses the concept of large-scale quantization, bringing incredible insights to this novel theory.
This paper considers the concept of wave-particle thermodynamic equilibrium in order to improve our understanding of the role of turbulent heating in the solar wind proton plasma. The thermodynamic ...equilibrium in plasmas requires the energy of a plasmon-the quantum of plasma fundamental oscillation-to be balanced by the proton-magnetized plasma energy, that is, the magnetic field and proton kinetic/thermal energy. This equilibrium has already been confirmed in several prior analyses, but also in this paper, by analyzing (i) multi-spacecraft data sets along the radial profile of the inner heliosphere, and (ii) representative data sets of a variety of 27 different space and astrophysical plasmas. Recently, it was shown that the slow mode of the near-Earth solar wind plasma is characterized by a missing energy source that is necessary for keeping the energy balance in the plasmon-proton-magnetized plasma. Here we show strong evidence that this missing energy is the turbulent energy heating the solar wind. In particular, we derive and compare the radial and velocity profiles of this missing energy and the turbulent energy in the inner heliosphere, also considering other minor contributions, such as the temperature of pickup protons. The connection of the missing plasmon-proton energy with the turbulent energy provides a new method for estimating and cross-examining the turbulent energy in space and astrophysical plasmas, while it confirms the universality of the involved new Planck-type constant that implies a large-scale quantization.
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.
In this paper we examine the physical foundations and theoretical development of the kappa distribution, which arises naturally from non-extensive Statistical Mechanics. The kappa distribution ...provides a straightforward replacement for the Maxwell distribution when dealing with systems in stationary states out of thermal equilibrium, commonly found in space and astrophysical plasmas. Prior studies have used a variety of inconsistent, and sometimes incorrect, formulations, which have led to significant confusion about these distributions. Therefore, in this study, we start from the
N
-particle phase space distribution and develop seven formulations for kappa distributions that range from the most general to several specialized versions that can be directly used with common types of space data. Collectively, these formulations and their guidelines provide a “toolbox” of useful and statistically well-grounded equations for future space physics analyses that seek to apply kappa distributions in data analysis, simulations, modeling, theory, and other work.
The paper develops analytical modeling of thermal Doppler broadening of spectral profiles for particle populations described by kappa distributions, in the absence or presence of potential fields. ...The kappa distribution provides a straightforward replacement for the Maxwell distribution, that is, a generalization for describing systems characterized by local correlations among their particles, commonly found in space and astrophysical plasmas. The corresponding Voigt profiles are derived by convoluting the thermal and natural/collisional Lorentzian profiles. The kappa velocity distributions are employed to derive the thermal Doppler and Voigt profiles, while the kappa phase-space distributions in the presence of potential fields that depend on the position vector, are used to derive their respective differential profiles. We focus on attractive power-law potentials (oscillation-type, gravitational-type, and angular potentials), and study the variations of the produced Voigt differential profiles in detail. The developed formulations and guidelines provide a useful and statistically well-grounded "toolbox" for future reference in data analyses, simulations, analytical modeling, and theories of spectroscopy and related subjects of space and astrophysical plasmas.
ABSTRACT The Rankine-Hugoniot (R-H) jump conditions are the most important and frequently used equations in studies of the properties of space and astrophysical plasmas during their passage through ...shock discontinuities. This paper revisits the R-H conditions for shocks, develops the formulation of the compression ratio, and examines its range of values and properties. The analysis expresses the downstream thermodynamic variables and the compression ratio as functions of the upstream thermodynamic variables, either for equal or different polytropic indices upstream and downstream of the shock. In the general case of space plasmas with an oblique magnetic field, the compression ratio is given by a quartic polynomial, which is reduced to a cubic trinomial when the upstream/downstream polytropic indices are equal. The special cases of magnetic fields that are perpendicular or parallel to the shock normal are also examined. In any case, the compression ratio polynomial has one degree larger order, when the upstream/downstream polytropic indices are different. Emphasis is placed on the maximum value of the compression ratio, which is known to be ∼4 for adiabatic polytropic index ∼5/3. However, the compression ratio can be much larger if the upstream/downstream polytropic indices are not equal to each other and less than one. Several other issues are investigated: (i) the entropic condition, showing that statistical mechanics and thermodynamics lead to the same relation of entropy variation; (ii) the effect of kappa distributions on jump conditions; and (iii) the upper limit of the upstream temperature for a shock to exist.
Abstract
In this paper, we develop the transport equation of kappa, the fundamental thermodynamic parameter that labels kappa distributions of particle velocities. Using the recently developed ...concept of entropy defect, we are able to formulate the transport equation of kappa as a function of a general, positive or negative, rate of entropy change. Then, we derive the particular case of exchanging plasma ions with low-dimensionality, newly born pickup protons, which interact and decrease the entropy of the flow of otherwise kappa-distributed plasma protons. Finally, we apply the transport equation of kappa to the solar wind plasma protons, which leads to the radial profile of kappa values, as well as the evolution of the kappa distributions through the heliosphere. The results show that the solar wind kappa decreases with increasing heliocentric distance, corresponding to plasmas residing in stationary states far from classical thermal equilibrium. Moreover, in the outer heliosphere and the heliosheath, kappa reaches its lowest values and is spread across the far-equilibrium region of 1.5 <
κ
< 2.5, which coincides with independent observations provided by NASA’s Interstellar Boundary Explorer mission.
Abstract
The paper derives the one-to-one connecting relationships between plasma heating and its polytropic index, and addresses the consequences through the transport equation of temperature. ...Thermodynamic polytropic processes are classified in accordance to their polytropic index, the exponent of the power-law relationship of thermal pressure expressed with respect to density. These processes generalize the adiabatic one, where no heating is exchanged between the system and its environment. We show that, in addition to heating terms, the transport equation of temperature depends on the adiabatic index, instead of a general, nonadiabatic polytropic index, even when the plasma follows nonadiabatic processes. This is because all the information regarding the system's polytropic index is contained in the heating term, even for a nonconstant polytropic index. Moreover, the paper (i) defines the role of the polytropic index in the context of heating; (ii) clarifies the role of the nonadiabatic polytropic index in the transport equation of temperature; (iii) provides an alternative method for deriving the turbulent heating through the comparably simpler polytropic index path; and, finally, (iv) shows a one-component plasma proof-of-concept of this method and discusses the implications of such derived connecting relationships in the solar wind plasma in the heliosphere.
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