Oblique Propagation of Electrostatic Waves in a Magnetized Electron-Positron-Ion Plasma in the Presence of Heavy Particles

1 Introduction

The perusal of electron-positron-ion (EPI) plasmas is very significant for the understanding of astrophysical as well as celestial environment. The propagation of linear and nonlinear waves in such EPI plasmas has drawn a considerable attention to many authors [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. In most astrophysical and terrestrial environments (viz. white dwarfs and Van Allen belts), positrons coexist with an admixture of electrons and ions to form EPI plasma, which was found by many researchers [13], [14], [15], [16], [17], [18]. It was noticed that plasma system with positron components behave differently than regular two-component electron-ion (EI) plasmas. The existence of EPI plasmas has been affirmed in supernovas, pulsar environments, cluster explosions, and active galactic nuclei [8], [19], [20], at the center of Milky Way galaxy [21], and in the early universe [22].

For modeling purposes, the particle distribution is mostly envisaged to be Maxwellian, when it is very near to equilibrium in a plasma system. The electrons and positrons, which are present in space and astrophysical plasma environments, are not in thermal equilibrium but highly energetic [23], [24] because of the effect of external forces or wave-particle interaction in numerous space plasma observations [25], [26] and laboratory experiments [27], [28]. These experiments substantiate the existence of accelerated, highly energetic (superthermal) particles in EPI plasmas. As the ion temperature is different from the electron and positron temperatures, so an EPI plasma system that is not in thermodynamic equilibrium does not follow a Maxwell-Boltzmann distribution. Plasmas with an excess of superthermal electrons or positrons elicit a deviation from Maxwellian equilibrium [26], [29], [30]. So the Maxwell-Boltzmann distribution is not appropriate for explicating the interaction of superthermal particles. However, the plasma system, containing higher energetic (superthermal) particle with energies greater than the energies of the particles, exists in thermal equilibrium that can be fitted more appropriately via the κ (kappa) type of Lorentzian distribution function [31], [32], [33] than via the thermal Maxwellian distribution function where the real parameter κ measures the deviation from a Maxwellian distribution (the smaller the value, the larger the deviation from a Maxwellian, in fact attained for infinite κ). Hence, we focus on a plasma system with superthermal particles modelled by a κ distribution [33].

The three-dimensional kappa (κ) velocity distribution of particles of mass m is of the form:

where F_k symbolizes the kappa distribution function, Γ is the gamma function, ω shows the most probable speed of the energetic particles, given by ω=[(2k−3/k)^1/2(k_BT/m)^1/2], with T being the characteristic kinetic temperature and ω is related to the thermal speed V_t=(k_BT/m)^1/2, and the parameter k represents the spectral index [34] that defines the strength of the superthermality. The range of this parameter is 3/2<k<∞ [35]. In the limit k→∞ [36], [37], the kappa distribution function reduces to the well-known Maxwell-Boltzmann distribution.

In the last few decades, a number of investigations have been made on the nonlinear propagation of different waves including ion-acoustic (IA), dust-ion-acoustic (DIA), and heavy ion-acoustic (HIA) waves [38], [39], [40], [41], [42], [43], [44], [45], [46]. Cairns et al. [40] studied a magnetized plasma system and observed the effects of external magnetic field, obliqueness, and ion temperature on the amplitude and width of the IA solitons. Shahmansouri and Alinejad [43] observed the linear and nonlinear excitation of arbitrary amplitude IA SWs in a magnetized plasma consisting of two-temperature electrons and cold ions. Baluku and Hellberg [47] examined the arbitrary amplitude IA SWs and double layers by using the Sagdeev potential approach in an EPI plasma containing Cairns-distributed (nonthermal) electrons, Boltzmann positrons, and cold ion. Ghosh et al. [48] studied the nonplanar IA shock waves in a homogeneous unmagnetized EPI plasma containing superthermal electrons, positrons, and singly charged hot positive ions. The existence of positively and negatively charged heavy ions and dust of opposite polarities has been shown to exist in astrophysical environments [41], [49], [50]. Baluku et al. [51] studied the behavior and existence of DIA waves in a plasma having both polarities of dust. The differences between HIA waves and DIA waves in plasmas are as follows: (i) the frequency of HIA waves is much smaller than that of DIA waves; (ii) in DIA waves, static dust grains participate only in maintaining the equilibrium charge neutrality condition, whereas in HIA waves, mobile heavy ions provide the necessary inertia; and (iii) in DIA waves, the inertia is provided by the light ions mass, whereas in HIA waves, inertia comes from the heavy ion mass. Hossen et al. [52] examined the characteristics of HIA solitary structures associated with the nonlinear electrostatic perturbations in an unmagnetized, collisionless dense plasma.

To the best of our knowledge, no theoretical investigations have been made on the propagation of HIA waves by deriving the magnetized Korteweg-de Vries (K-dV) and magnetized modified K-dV (mK-dV) equations to understand the nonlinear excitations in plasmas containing superthermal electrons and positrons and magnetized positively charged heavy ions.

In addition, the SW solutions of magnetized K-dV and mK-dV equations as well as other parameters like polarity, nonlinearity, and dispersion coefficients with amplitude and width of the solitary structures have been studied for different relevant parameters. The manuscript is organized as follows. The basic equations are provided in Section 2. Two different types of nonlinear equations, namely, K-dV and mK-dV, are derived and analyzed analytically and numerically in Section 3. A brief discussion is finally presented in Section 5.

2 Basic Equations

We consider a three-component magnetized plasma system containing positively charged heavy ions and kappa-distributed electrons and positrons with two distinct temperatures T_e and T_p. Therefore, at equilibrium condition, Z_hn_h0=n_e0−n_p0, where Z_h (n_h0) is the charge state (equilibrium number density) of the heavy ion species and n_e0 (n_p0) is the equilibrium electron (positron) number density at temperature T_e (T_p). The dynamics of the HIA waves, whose phase speed is much smaller (larger) than electron (heavy ion) thermal speed, is described by the normalized equations in the form

where n_h is the heavy ion number density normalized by n_h0; u_h is the heavy ion fluid speed normalized by C_h=(Z_hk_BT_e/m_h)^1/2 (with k_B being the Boltzmann constant and m_h being the heavy ion mass); ϕ is the electrostatic wave potential normalized by k_BT_e/e (with e being the magnitude of the charge of an electron); α=ω_hc/ω_ph (with ω_ch=Z_heB₀/m_hc being the heavy ion cyclotron frequency, B₀ being the magnitude of the external static magnetic field, c being the speed of light in vacuum, and ωph=(4πnh0Zh2e2/mh)1/2 being the heavy ion plasma frequency); μ₀=μ, μ₁=μ−1, and μ=n_e0/Z_hn_h0; n_e (n_p) is the number density of electron (positron); the time variable is normalized by ωph−1 and the space variable is normalized by λ_Dm=C_h/ω_ph. We note that the external magnetic field B₀ is acting along the z-direction (i.e. B0=z^B0, where z^ is the unit vector along the z-direction). The normalized electron and positron number densities n_e and n_p are, respectively, given by:

where σ=T_e/T_p and κ₁ (κ₂) is the spectral index of superthermal electron (positron).

3 Nonlinear Equations

To study the nonlinear propagation, we now consider different orders of nonlinearity by deriving and analyzing the K-dV and mK-dV equations to identify the basic features of HIA SWs that formed a magnetized space plasma system containing dynamical heavy ions and kappa-distributed electrons and positrons of two distinct temperatures.

4 Parametric Investigations

To identify the salient features (viz. polarity, amplitude, and width) of the ESPPs, we have numerically analyzed the solution of the K-dV equation. The results are displayed in Figures 1–7, which clearly indicate that (i) the ESPPs with ϕ⁽¹⁾>0 (ϕ⁽¹⁾<0) exist for μ>μ_c (μ<μ_c) as shown in Figures 1–5; (ii) the amplitude and width of the ESPPs (with both ϕ⁽¹⁾>0 and ϕ⁽¹⁾<0) increase (decrease) with the increase in σ (κ₂) as shown in Figures 1–4; (iii) the amplitude of the ESPPs (with ϕ⁽¹⁾<0 for μ<μ_c) decreases with the increase in μ as shown in Figure 5; (iv) the width of the ESPPs [with both ϕ⁽¹⁾>0] also decreases slightly with the gradual increase in μ and α as shown in Figure 6; (v) the width of the ESPPs (with ϕ⁽¹⁾>0) increases (decreases) with the increase in δ for its lower (upper) range, but it decreases with the increase in α as shown in Figure 7. The mK-dV equation solution (45) is also analyzed to identify the salient features (viz. polarity, amplitude, and width) of the ESPPs for different plasma parametric regimes. The results are depicted in Figures 8–10, which clearly indicates that (i) the mK-dV equation admits the SW solution with ϕ⁽¹⁾>0 only; (ii) the amplitude and width of the ESPPs decrease with the increase in (κ₂) as shown in Figure 8; (iii) the amplitude and width of the ESPPs increase with the increase in σ as shown in Figure 9; and (iv) the amplitude and width of the ESPPs decrease with the increase in μ as shown in Figure 10. The effects of different intrinsic parameters, viz. heavy ion cyclotron frequency to heavy ion plasma frequency (α), the electron-to-positron temperature ratio (σ), the electron-to-heavy ion number density ratio (μ), and the superthermality through the spectral index (κ) on the dynamical properties of HIA waves in the present plasma model, have been investigated as follows.

Figure 1:

The ESPPs with ϕ⁽¹⁾>0 for μ>μ_c, u₀=0.01, μ=0.75, κ₁=20, κ₂=3, δ=15, α=0.5, σ=0.10 (dashed curve), σ=0.15 (solid curve), and σ=0.20 (dotted curve).

Figure 2:

The ESPPs with ϕ⁽¹⁾<0 for μ<μ_c, u₀=0.01, μ=0.73, κ₁=20, κ₂=3, δ=15, α=0.5, σ=0.30 (dashed curve), σ=0.35 (solid curve), and σ=0.40 (dotted curve).

Figure 3:

The ESPPs with ϕ⁽¹⁾>0 for μ>μ_c, u₀=0.01, σ=0.25, μ=0.75, κ₁=20, δ=15, α=0.5, κ₂=3.25 (dashed curve), κ₂=3.50 (solid curve), and κ₂=3.75 (dotted curve).

Figure 4:

The ESPPs with ϕ⁽¹⁾<0 for μ<μ_c, u₀=0.01, σ=0.25, μ=0.73, κ₁=20, δ=15, α=0.5, κ₂=2.75 (dashed curve), κ₂=2.60 (solid curve), and κ₂=2.45 (dotted curve).

Figure 5:

The ESPPs with ϕ⁽¹⁾<0 for μ<μ_c, u₀=0.01, σ=0.25, κ₁=20, κ₂=3, δ=15, α=0.5, μ=0.65 (dashed curve), μ=0.68 (solid curve), and μ=0.70 (dotted curve).

Figure 6:

The ESPPs with ϕ⁽¹⁾>0 for μ>μ_c, u₀=0.01, δ=15, σ=0.25, κ₁=20, κ₂=3, μ=0.75 (dashed curve), μ=0.80 (solid curve), and μ=0.85 (dotted curve).

Figure 7:

The width of the ESPPs for μ>μ_c, u₀=0.01, σ=0.25, μ=0.75, κ₁=20, κ₂=3, α=0.5 (dashed curve), α=0.6 (solid curve), and α=0.7 (dotted curve).

Figure 8:

The ESPPs with ϕ⁽¹⁾>0 for u₀=0.01, σ=0.25, μ=0.75, κ₁=20, δ=15, α=0.5, κ₂=2.00 (dashed curve), κ₂=2.25 (solid curve), and κ₂=3.00 (dotted curve).

Figure 9:

The ESPPs with ϕ⁽¹⁾>0 for u₀=0.01, δ=15, μ=0.75, κ₁=20, κ₂=3, α=0.5, σ=0.10 (dashed curve), σ=0.20 (solid curve), and σ=0.30 (dotted curve).

Figure 10:

The ESPPs with ϕ⁽¹⁾>0 for u₀=0.01, δ=15, σ=0.25, κ₁=20, k₂=3, α=0.5, μ=0.75 (dashed curve), μ=0.85 (solid curve), and μ=1.00 (dotted curve).

5 Discussion

We have considered a magnetized plasma system consisting of inertial heavy ions and kappa-distributed hot electrons and hot positrons of two distinct temperatures. We have derived the magnetized K-dV and mK-dV-type partial differential equations by using the reductive perturbation method to investigate the basic features (i.e. polarity, amplitude, and width) of such a plasma system. The magnetized K-dV and mK-dV equations are solved to set out the fascinating features of HIA SWs. Then these solutions are analyzed by taking the effect of different plasma parameters. The results, which have been obtained from this theoretical investigation, can be pinpointed as follows:

1. The K-dV equation admits HIA SW solutions with either ϕ⁽¹⁾>0 (compressive) or ϕ⁽¹⁾<0 (rarefactive). The polarity of the HIA SWs depends on the critical value μ_c (where μ_c=0.73 for κ₁=20, κ₂=3, δ=15, σ=0.25, and α=0.5). On the other-hand, the mK-dV equation admits only HIA SW solution with ϕ⁽¹⁾>0 (compressive).

2. The K-dV equation is no longer valid at A₁≃0 because the amplitude of the K-dV solitons become infinitely large (for A₁=0), which has been avoided by deriving mK-dV equation to study more highly nonlinear HIA SWs.

3. The amplitude and width of both positive and negative HIA SWs (obtained from the numerical analysis of the solution of the K-dV equation) increase with the increase in T_e and Z_hn_h0 but decrease with the increase in T_p, n_e0, and κ₂.

4. The width of the K-dV solitons decreases with the increase in α and increases (decreases) with the increase in δ for its lower (upper) range.

5. The amplitude and the width of the mK-dV HIA SWs decrease with the increase in κ₂, T_p, and n_e0 but increase with the increase in T_e and Z_hn_h0.

Therefore, we hope that our present investigation would contribute to understand the prime features (i.e. polarity, amplitude, and width) of the electrostatic disturbances of HIA SWs in a magnetized plasma system. Our findings would be useful to study nonlinear structures in space (viz. peculiar velocities of galaxy clusters, cluster explosions, active galactic nuclei, pulsar magnetosphere, ionosphere [54], Saturn’s magnetosphere [37], and solar wind [55]) as well as laboratory plasma conditions [8], [56], [57], [58], [59] (viz. semiconductor plasmas [60]) containing heavy ions where the effect of two temperature superthermal electrons and positrons play a crucial role.