Kappa Distribution Research Articles

Analysis of Electron Distribution Functions From the Magnetospheric Multiscale (MMS) Mission Using the Gaussian Mixture Model

AbstractVelocity distribution functions (VDFs) measured by the Magnetospheric Multiscale (MMS) mission are complex 3D data sets that can be represented as a superposition of multiple beams. Recent work proposed the use of the Gaussian Mixture Model (GMM) to identify different populations. Here we investigate the approach by considering first synthetic distributions made by synthetically creating beams of either Maxwellian distributions or kappa distributions with varying power law index. By varying the inter‐beam average difference and the beam standard deviation we evaluate the ability of the GMM in recognizing correctly the beam. We then apply the method systematically to MMS data in the tail and in the dayside of the Earth's magnetosphere. The approach relies on a GMM algorithm to detect and characterize the number of distinct clusters within a VDF, and a model selection approach based on the Bayesian Information Criterion to determine the optimal number of clusters. The conclusion of the analysis is that the GMM can estimate the complexity of a VDF in terms of the number of optimal beams provided by information theory criteria. By evaluating the complexity of VDFs, we can identify regions of interest within the plasma where significant physical phenomena may be present.

A comparative analysis of L-moments, maximum likelihood, and maximum product of spacing methods for the four-parameter kappa distribution in extreme value analysis

The study has investigated the implications of three estimation methods, namely L-moments, Maximum Likelihood, and Maximum Product of Spacing (MPS), for fitting the four-parameter Kappa Distribution (KAPD) in extreme value analysis using Monte Carlo simulations. The accuracy of the estimates has been evaluated using root mean square error (RMSE) and bias. The paper also includes an analysis of the effect of the estimation method on the estimated quantiles considering a real-life example of annual maximum peak flows and the Generalized Normal Distribution as the error distribution. Assessment metrics of the empirical analysis include standard error, L-scale, and 90% confidence limits of the estimated quantiles. The results reveal that MPS is a preferred method of estimation of parameters for KAPD, i.e. having the lowest RMSE values, especially in the presence of heavier tail and significant positive skewness for small to very large sample sizes. Secondly, the method of L-moments is recommended due to its low bias while analyzing the distribution of shape parameters having a slightly heavier tail, and slight or moderate positive skewness. The results associated with the quality of estimated quantiles using real-life data are consistent with the findings of simulation outcomes.

The subtracted-kappa distribution and its properties

We introduce a new particle distribution function in plasma physics called the subtracted-kappa (S-K) distribution. This new distribution incorporates a loss-cone feature and an enhanced high-energy tail characterized by the spectral index κ. The S-K distribution is a generalization of the well-known subtracted-Maxwellian and approaches the latter distribution in the limit as κ→∞. Further, the S-K distribution usefully includes the bi-kappa and kappa-loss-cone distributions as special cases. Two sources of free plasma energy are provided by the S-K distribution, namely, the loss cone property and the thermal anisotropy. Both free energy sources can excite wave growth. In this paper, we briefly consider the influence of the S-K distribution on three wave phenomena: (a) linear growth of whistler-mode waves generated by an injection of hot electrons into a cold plasma, (b) dispersion of R-mode and L-mode electromagnetic waves in a hot plasma, and (c) transition from linear to nonlinear growth of electromagnetic waves as determined by a critical boundary in the input-parameter space. There are many possibilities for future projects involving the S-K distribution. Charged particle distributions in space typically possess a pronounced high-energy tail that can be modeled approximately by a kappa distribution, and so the S-K distribution is an ideal tool for analyzing kinetic waves and microinstabilities in space plasmas.

A new generalized formulation of dielectric tensor for multi-species plasma with anisotropic drift product-bi-kappa distribution

It is well established that space plasmas often contain particle components with high-energy tails that approximately follow a power-law distribution in velocity space. Such superthermal distributions, particularly those with an excess of fast particles, are more accurately described by generalized Kappa (Lorentzian) distributions rather than Maxwellian distributions. We propose the product-bi-kappa (PBK) distribution as an alternative to the kappa-Maxwellian (KM) and bi-Maxwellian (BM) distributions, leading to a new linear transformation form of the dielectric tensor for multi-component plasmas. This method utilizes the rational form and multi-pole expansion of the plasma dispersion function to convert the dielectric tensor of multi-component PBK plasmas into a linear eigenvalue system in conjunction with Maxwell's equations. Our approach transforms the traditional numerical iteration process for solving the finite roots of the dielectric tensor from initial values into an eigenvalue problem within a new, complete linear system, thus enabling the simultaneous determination of all principal eigenvalues. Constructing the eigenvalue system for multi-component plasmas is crucial for developing new, comprehensive oblique propagation solvers for multi-species anisotropic drift plasmas, including PBK, KM, and BM plasmas. The detailed construction of the linear eigenvalue system is extensively discussed in this work.

KdV equation for kinetic Alfvén waves and ionospheric solitons

The Korteweg–de Vries (KdV) equation is derived for nonlinear kinetic Alfvén waves (KAWs) under the framework of the reductive perturbation method in single ion and bi-ion plasmas. It is pointed out that the KdV equation can be derived following the same normalization of spatial coordinates, which was used to obtain an exact solution of the equations for arbitrary amplitude KAWs [Hasegawa and Mima, Phys. Fluids 21, 87 (1978)]. The KdV equation for KAWs is derived assuming Maxwell velocity distribution for electrons to highlight the appropriate normalization procedure of the nonlinear equations for KAWs in the small amplitude limit. Then, the Kappa distribution of electrons is also considered to investigate the effects of non-thermal particles on linear and nonlinear wave dynamics. The results are applied to single ion oxygen and bi-ion oxygen–hydrogen plasmas of the upper ionosphere. It is found that the presence of 0.4% of protons in oxygen plasma of the ionosphere does not affect the shape of the soliton but the high-energy electrons reduce its amplitude. Present theoretical calculations predict the frequencies of KAWs to lie in the range of 10–30 m and widths of solitons to be larger than 100 m. These estimates are in agreement with the Freja satellite observations [Wahlund et al., Geophys. Res. Lett. 21, 1831 (1994)].

Radial Distribution of Electron Quasi-thermal Noise in the Inner Heliosphere

The electron population in the solar wind plasma can be described with three different components: a core, a halo, and a magnetic field aligned strahl. The electron quasi-thermal noise (QTN) is investigated by using an electron population model consisting of a core with a Maxwellian distribution and a halo with a kappa distribution, based on the empirical equations for electron density and temperature and the index for the kappa halo. The power spectra of the electron QTN are calculated at different heliocentric radial distances from 10 to 200 R s . The dependence of the QTN spectrum and effective Debye length on model parameters, including the ratio of the halo to the core for the density and temperature, the kappa index, and the antenna length, is further discussed. The results show that the electron QTN spectrum consists of a plateau in the low-frequency band f < f pt , a peak at the total plasma frequency f pt , and a rapidly decreasing part in the high-frequency band f > f pt . The QTN peak and plateau level continuously decrease as the radial distance increases, with the peak’s shape changing due to the variation of the kappa index. Although the model parameters are variable, the QTN plateau level presents less than an order of change with these parameters changing greatly, and only a monotonic change of the plateau is shown when the parameters are close to the practical situation. The results can provide a reference for future deep-space exploration in the inner heliosphere, and also for the design of detectors.

Auroral 3D structure retrieval from the Juno/UVS data

Context. Jovian auroras, the most powerful in the Solar System, result from the interaction between the magnetosphere and atmosphere of Jupiter. While the horizontal morphology of these phenomena has been widely studied, their vertical structure, determined by the penetration depth of the magnetospheric electron into the auroral regions, remains relatively unexplored. Previous observations, including those from the Hubble Space Telescope (HST), have addressed this question to a limited extent. Aims. In this study we aim to map the vertical structure of Jovian auroral emissions. Methods. Using observations from Juno’s UltraViolet Spectrograph (UVS), we examined the vertical structure of the auroral emissions. Building on a recent study of auroral energy mapping based on UVS observations that mapped the average energy of precipitating electrons in the Jovian auroral regions, we find a relationship between this average energy and the volume emission rate (VER) of H2 for two types of electron energy distributions: monoenergetic and a kappa distribution with κ = 2.5. Results. Using brightness maps, we derived the 3D VER structure of Jovian auroras in both northern and southern regions, across multiple spacecraft perijoves (PJs). By considering the example of PJ11, we find that the average altitude of the VER peak in the polar emission region is approximately ~250 km for the monoenergetic distribution case and ~190 km for kappa distribution case. In the main emission region, we find that the average altitude of the VER peak is approximately ~260 km for the case of monoenergetic distribution and ~197 km for kappa distribution case. For the other PJs, we obtained results that are very similar to those of PJ11. Conclusions. Our findings are, on average, consistent with measurements from the Galileo probe and the HST observations. This study contributes to a better understanding of the complexity of Jovian auroras and highlights the importance of using Juno observations when probing their vertical structure. Considering the variability in the κ parameter in the auroral region, we also studied the impact of this variability on the vertical structure of the auroral emission. This sensitivity study reveals that the influence of the κ parameter on our results was very weak. However, the impact of the κ variability on the VER amplitude shows that there is an influence on the thermal structure and chemical composition of the atmosphere in the auroral regions.

Suprathermal corrections on galactic cosmic rays driven magnetohydrodynamic waves and gravitational instability in astrophysical plasmas

The interaction of two populations of highly energetic cosmic rays (CRs) and suprathermal kappa gas in the astrophysical systems manifests exciting features of low-frequency magnetohydrodynamic (MHD) waves and instabilities. Contrary to the previous works on waves and instability analysis in Maxwellian gas, this paper investigates the effects of suprathermal corrections on the CR driven MHD waves and gravitational (Jeans) instability using the kappa distribution function. The equation of state for a kappa gas, including spectral κ− index, is considered in the CR-plasma interactions using the hydrodynamic fluid–fluid approach. The modified dispersion properties of fast, slow, and pure Alfvén waves and Jeans instability have been discussed in a suprathermal gas in astrophysical environments. The suprathermal corrections enhance the phase speed of the fast mode of MHD waves which is found to be greater in the suprathermal gas (κ&amp;gt;3/2) and smaller in the Maxwellian gas (κ→∞). In the absence of CR diffusion, the Jeans instability criterion is modified due to the simultaneous presence of CR pressure and suprathermal corrections. However, in the presence of CR diffusion, only suprathermal corrections modify the Jeans instability criterion. The suprathermal gases with higher degrees of freedom require large values of the Jeans wavenumber to produce gravitational instability and make the system more unstable. The suprathermal corrections along with modified thermal speed stabilize the growth rate of Jean instability, supporting the gravitational collapse of non-thermal gas in astrophysical systems.

Flare-accelerated Electrons in the Kappa Distribution from X-Ray Spectra with the Warm-target Model

Abstract X-ray observations provide important and valuable insights into the acceleration and propagation of nonthermal electrons during solar flares. Improved X-ray spectral analysis requires a deeper understanding of the dynamics of energetic electrons. Previous studies have demonstrated that the dynamics of accelerated electrons with a few thermal speeds are more complex than those with significantly higher speeds. To better describe the energetic electrons after injection, a model considering energy diffusion and thermalization effects in flare conditions (the warm-target model) has recently been developed for spectral analysis of hard X-rays. This model has demonstrated how the low-energy cutoff, which can hardly be constrained in cold-target modeling, can be determined. However, the power-law form may not be the most suitable representation of injected electrons. The kappa distribution, which is proposed as a physical consequence of electron acceleration, has been applied successfully in RHESSI spectral analysis. In this study, we employ the kappa-form injected electrons in the warm-target model to analyze two M-class flares, observed by RHESSI and the Spectrometer/Telescope for Imaging X-rays, respectively. The best-fit results show that the kappa-form energetic electron spectrum generates lower nonthermal energy than the power-law form when producing a similar photon spectrum in the fit range. We also demonstrated that the fit parameters associated with the kappa-form electron spectrum can be well determined with small uncertainty. Further, the kappa distribution, which covers the entire electron energy range, enables the determination of key electron properties such as total electron number density and average energy at the flare site, providing valuable information on electron acceleration processes.

Electron-acoustic solitary waves via suprathermal populations in Saturn's magnetosphere: Bifurcation, sensitivity, and stability analysis

Cassini and Voyager space missions observed non-thermal electron populations (with varying characteristics) in Saturn's magnetosphere, which can be correctly described using kappa distributions. Based on these observations, our objective is to inspect the evolution of electron-acoustic solitary waves (EASWs) within Saturn's magnetosphere. The propagation of weakly nonlinear (EASWs) in a collisional plasma system comprising a cold electron fluid, hot electrons following a kappa distribution, and stationary ions is investigated. By employing the reductive perturbation technique, the Korteweg–de Vries Burgers (KdV–B) equation is derived. An exact solution of the KdV–B equation, with a conformable time-derivative, is found using the unified method. It is observed that plasma current-induced collision between electrons and ions leads to remarkable dissipation, generating EASWs. Furthermore, when studying the sensitivity of the system, the appearance of a positive potential is depicted as external forces vanish, which may be due to stationary ions. Additionally, bifurcation, stability, and significant influence of plasma characteristics are considered.

Generation of Quasi-Electrostatic Slow Extraordinary Waves by Kappa Distribution with a Loss Cone

Generation of Quasi-Electrostatic Slow Extraordinary Waves by Kappa Distribution with a Loss Cone

Importance of the electron plasma parameter for excitation of chorus and formation of magnetic field irregularity in the region of their excitation

A threshold condition has been established for excitation of VLF electromagnetic radiation with a chorus structure of the dynamic spectrum in the daytime magnetosphere using the BPA (Beam Pulse Amplifier) mechanism for amplifying short noise electromagnetic pulses. The kappa distribution was used as a model function of the electron velocity distribution in the magnetosphere. Calculations performed for this distribution have shown that the threshold for excitation of chorus largely depends on the electron plasma parameter equal to the ratio of gas-kinetic pressure of electrons to magnetic pressure. This pattern is not contradicted by the dependence of the probability of excitation of chorus on the degree of magnetic field irregularity, which we derived from observations made by the Van Allen Probe spacecraft. It is sharp fluctuations in the magnetic field strength near its local minima outside the plasmasphere where the radiation under study can be excited. If there is an irregularity, the probability of detecting chorus is &gt;70 %; and if there is no or very low irregularity, the probability of the absence of any emissions is ~80 %. The results indicate a common reason for the excitation of chorus and the magnetic field irregularity — a small but finite value of the plasma parameter.

Существенность величины плазменного параметра электронов для возбуждения хоров и формирования нерегулярности магнитного поля в области их возбуждения

A threshold condition has been established for excitation of VLF electromagnetic radiation with a chorus structure of the dynamic spectrum in the daytime magnetosphere using the BPA (Beam Pulse Amplifier) mechanism for amplifying short noise electromagnetic pulses. The kappa distribution was used as a model function of the electron velocity distribution in the magnetosphere. Calculations performed for this distribution have shown that the threshold for excitation of chorus largely depends on the electron plasma parameter equal to the ratio of gas-kinetic pressure of electrons to magnetic pressure. This pattern is not contradicted by the dependence of the probability of excitation of chorus on the degree of magnetic field irregularity, which we derived from observations made by the Van Allen Probe spacecraft. It is sharp fluctuations in the magnetic field strength near its local minima outside the plasmasphere where the radiation under study can be excited. If there is an irregularity, the probability of detecting chorus is &gt;70 %; and if there is no or very low irregularity, the probability of the absence of any emissions is ~80 %. The results indicate a common reason for the excitation of chorus and the magnetic field irregularity — a small but finite value of the plasma parameter.

Observations of Kappa Distributions in Solar Energetic Protons and Derived Thermodynamic Properties

In this paper, we model the high-energy tail of observed solar energetic proton energy distributions with a kappa distribution function. We employ a technique for deriving the thermodynamic parameters of solar energetic proton populations measured by the Parker Solar Probe Integrated Science Investigation of the Sun EPI-Hi high-energy telescope, over energies from 10 to 60 MeV. With this technique, we explore, for the first time, the characteristic thermodynamic properties of the solar energetic protons associated with an interplanetary coronal mass ejection (ICME) and its driven shock. We find that: (1) the spectral index or, equivalently, the thermodynamic parameter kappa of solar energetic protons (κ EP) gradually increases, starting from the pre-ICME region (upstream of the CME-driven shock), reaching a maximum in the CME ejecta (κ EP ≈ 3.5), followed by a gradual decrease throughout the trailing portion of the CME; (2) the solar energetic proton temperature and density (T EP and n EP) appear anticorrelated, a behavior consistent with subisothermal polytropic processes; and (3) values of T EP and κ EP appear to be positively correlated, indicating an increasing entropy with time. Therefore, these proton populations are characterized by a complex and evolving thermodynamic behavior, consisting of multiple subisothermal polytropic processes, and a large-scale trend of increasing temperature, kappa, and entropy. This study and its companion study by Livadiotis et al. open up a new set of procedures for investigating the thermodynamic behavior of energetic particles and their shared thermal properties.

Ion temperature gradient modes modulational stability with kappa-distribution

We investigated the modulational stability and instability of the ion temperature gradient (ITG) mode in electron-ion plasma. Ions are dynamic species, whereas electrons follow a Kappa distribution. We used the reduction perturbation approach to determine the linear dispersion relation for the fluid under consideration. The nonlinear Schrodinger equation describes nonlinear features such as nonlinear modulational stability or instability in the ITG mode. The product of dissipation and nonlinear coefficients, known as LM, contains both modulational stability and instability in the ion temperature gradient mode. Theoretical results are expanded numerically, showing the influence of various plasma parameters on modulational stability and instability, particularly the superthermality coefficient κ e . The current observations may be extended to space and laboratory plasma for modification.

Kappa-tail Technique: Modeling and Application to Solar Energetic Particles Observed by Parker Solar Probe

We develop the kappa-tail fitting technique, which analyzes observations of power-law tails of distributions and energy flux spectra, and connects them to theoretical modeling of kappa distributions, to determine the thermodynamics of the examined space plasma. In particular, we (i) construct the associated mathematical formulation; (ii) prove its decisive lead for determining whether the observed power-law is associated with kappa distributions; and (iii) provide a validation of the technique using pseudo-observations of typical input plasma parameters. Then, we apply this technique to a case study by determining the thermodynamics of solar energetic particle (SEP) protons, for an SEP event observed on 2021 April 17, by the Parker Solar Probe (PSP)/Integrated Science Investigation of the Sun instrument suite on board PSP. The results show SEP temperatures and densities of the order of ∼1 MeV and ∼5 × 10−7 cm−3, respectively.

A Transmuted Survival Kappa Distribution: Properties and Applications

A Transmuted Survival Kappa Distribution: Properties and Applications

Testing for Markovian character of transfer of fluctuations in solar wind turbulence on kinetic scales.

We apply statistical analysis to search for processes responsible for turbulence in physical systems. In our previous studies, we have shown that solar wind turbulence in the inertial range of large magnetohydrodynamic scales exhibits Markov properties. We have recently extended this approach on much smaller kinetic scales. Here we are testing for the Markovian character of stochastic processes in a kinetic regime based on magnetic field and velocity fluctuations in the solar wind, measured onboard the Magnetospheric Multiscale (MMS) mission: behind the bow shock, inside the magnetosheath, and near the magnetopause. We have verified that the Chapman-Kolmogorov necessary conditions for Markov processes is satisfied for local transfer of energy between the magnetic and velocity fields also on kinetic scales. We have confirmed that for magnetic fluctuations, the first Kramers-Moyal coefficient is linear, while the second term is quadratic, corresponding to drift and diffusion processes in the resulting Fokker-Planck equation. It means that magnetic self-similar turbulence is described by generalized Ornstein-Uhlenbeck processes. We show that for the magnetic case, the Fokker-Planck equationleads to the probability density functions of the kappa distributions, which exhibit global universal scale invariance with a linear scaling and lack of intermittency. On the contrary, for velocity fluctuations, higher order Kramers-Moyal coefficients should be taken into account and hence scale invariance is not observed. However, the nonextensity parameter in Tsallis entropy provides a robust measure of the departure of the system from equilibrium. The obtained results are important for a better understanding of the physical mechanism governing turbulent systems in space and laboratory.

An altered cell-specific subcellular distribution of translesion synthesis DNA polymerase kappa (POLK) in aging neurons.

Genomic stability is critical for cellular function, however, in the central nervous system highly metabolically active differentiated neurons are challenged to maintain their genome over the organismal lifespan without replication. DNA damage in neurons increases with chronological age and accelerates in neurodegenerative disorders, resulting in cellular and systemic dysregulation. Distinct DNA damage response strategies have evolved with a host of polymerases. The Y-family translesion synthesis (TLS) polymerases are well known for bypassing and repairing damaged DNA in dividing cells. However, their expression, dynamics, and role if any, in enduring postmitotic differentiated neurons of the brain are completely unknown. We show through systematic longitudinal studies for the first time that DNA polymerase kappa (POLK), a member of the Y-family polymerases, is highly expressed in neurons. With chronological age, there is a progressive and significant reduction of nuclear POLK with a concomitant accumulation in the cytoplasm that is predictive of brain tissue age. The reduction of nuclear POLK in old brains is congruent with an increase in DNA damage markers. The nuclear POLK colocalizes with damaged sites and DNA repair proteins. The cytoplasmic POLK accumulates with stress granules and endo/lysosomal markers. Nuclear POLK expression is significantly higher in GABAergic interneurons compared to excitatory pyramidal neurons and lowest in non-neurons, possibly reflective of the inherent biological differences such as firing rates and neuronal activity. Interneurons associated with microglia have significantly higher levels of cytoplasmic POLK in old age. Finally, we show that neuronal activity itself can lead to an increase in nuclear POLK levels and a reduction of the cytoplasmic fraction. Our findings open a new avenue in understanding how different classes of postmitotic neurons deploy TLS polymerase(s) to maintain their genomic integrity over time, which will help design strategies for longevity, healthspan, and prevention of neurodegeneration.

Zakharov–Kuznetsov Equation for Describing Low-Frequency Nonlinear Dust Acoustic Perturbations in Saturn’s Dusty Magnetosphere

A description is given of low-frequency nonlinear dust acoustic waves in Saturn’s dusty magnetosphere, which contains electrons of two types (hot and cold) obeying the kappa distribution, magnetospheric ions, and charged dust particles. For the corresponding conditions, the derivation of the Zakharov–Kuznetsov equation is given, which describes the nonlinear dynamics of dust acoustic waves in the case of low frequencies and a pancake-shaped wave packet along an external magnetic field. It is shown that under the conditions of Saturn’s magnetosphere there exist solutions of the Zakharov–Kuznetsov equation in the form of one-dimensional and three-dimensional solitons. Possible observations of the considered solitons in future space missions are discussed.