ION SOLITARY PULSES IN WARM PLASMAS WITH ULTRARELATIVISTIC DEGENERATE ELECTRONS AND POSITRONS

In recent years, there has been an enormous interest in the study of electron–positron (e–p) plasmas. The latter are found in some interstellar compact objects (e.g., in neutron stars, in the interior of Jupiter, in active galactic nuclei, etc.), and in the solar atmosphere (Edwin & Murawski 1995; Tandberg-Hansen & Emslie 1998). In e–p plasmas, electrons and positrons have the same masses but opposite charges. The e–p plasma symmetry is broken in the presence of ions, and both fast and slow time scales can occur in the dynamics of electron–positron–ion (e–p–i) plasmas. Some astrophysical plasmas contain ions besides the electrons and positrons, for example, in the magnetar corona the presence of electrons and positrons is due to instability of vacuum in an ultrastrong magnetic field. The ions originate from some interior source, for example, as a result of evaporation or seismic process on the surface of a star or come from outside in the process of accretion (Beloborodov & Thompson 2007a, 2007b; Istomin & Sobyanin 2007; Thompson & Beloborodov 2005; Thompson et al. 2002). Furthermore, under certain conditions of massive white dwarfs, the plasma number densities could exceed 10²⁹ cm⁻³ (Daligault & Murillo 2005; Lallement et al. 2011). As the thermal energy of the white dwarf is slowly lost to space and the stellar material cools, the ions' behavior can be understood in the classical manner, but under certain conditions, this behavior is modified by quantum effects. Also, the presence of e–p pairs in white dwarfs can be produced during the collapse of white dwarfs to neutron stars (Woosley & Baron 1992; Koester & Chanmugam 1990; Kashiyama et al. 2011). Therefore, it is important to study the nonlinear dynamics of ion oscillations in the presence of electrons and positrons.

There has been a great deal of interest in studying the linear and nonlinear wave motions in both e–p and e–p–i plasmas (Abdelsalam et al. 2008; El-Awady et al. 2010; Akhtar & Mahmood 2011; Rizzato 1988; Lu et al. 2010; Tribeche & Boukhalfa 2011; Popel et al. 1995; Rasheed et al. 2011; Shukla et al. 1993; Moslem et al. 2007, 2008), as well as in the next generation of compressed plasma created by intense laser beams due to their interactions with dense solid materials. However, in dense astrophysical bodies, the plasma contains a very high density of electrons and positrons, which can be regarded as degenerate fluids. The number density of electrons in some compact objects is of the order of 10²⁹ cm⁻³, or even more in the core of white dwarf stars (Koester & Chanmugam 1990; Daligault & Murillo 2005; Kantor & Gusakov 2007; Kashiyama et al. 2011) and 10³⁴–10³⁵ cm⁻³ in the corona of dense strongly magnetized stars/magnetars (Beloborodov & Thompson 2007a, 2007b; Thoma 2009). So, the electron Fermi energy could be comparable to the electron mass energy and the electron speed is comparable to the speed of light in vacuum. The equation of state for the degenerate electrons in such interstellar compact objects has been obtained by Chandrasekhar (1935, 1939) for two limits, namely the nonrelativistic and ultrarelativistic limits. Chandrasekhar found that the degenerate electron equation of state is given by P_e∝n^5/3_e for nonrelativistic degenerate electrons and P_e∝n^4/3_e for ultrarelativistic degenerate electrons, where P_e is the degenerate electron pressure and n_e the electron number density. It is noted that the degenerate electron pressure depends only on the electron number density. Therefore, these interstellar compact objects provide us cosmic laboratories for studying the properties of very dense matter where quantum and relativistic effects become important (Stenflo et al. 2006; Shukla & Stenflo 2006; Shukla & Eliasson 2007, 2010, 2011; Bordin & Marklund 2007; Marklund & Bordin 2007; Shukla 2009; Marklund et al. 2007; Masood et al. 2010; Manfredi 2005; Dubinov & Dubinova 2007; Mamun & Shukla 2010a, 2010b; Rasheed et al. 2010; Masood & Eliasson 2011). In such environments, one should apply a suitable model for describing the dynamics of degenerate electrons and positrons, while the ion component can be treated as a classical plasma fluid (Dubinov & Dubinova 2007; Shukla & Eliasson 2011). Most of the investigations dealing with the ion-acoustic waves (IAWs) in dense e–p–i plasmas used different forms of distributions for electrons and positrons, such as the Thomas–Fermi approximation and nonrelativistic electron/positron equation of state (see, e.g., Abdelsalam et al. 2008; Rasheed et al. 2010). Furthermore, they neglected the thermal effect of the ions and consider the ions as a cold fluid. In the present work, we consider both effects, i.e., the ion thermal pressure and the ultrarelativistic effect of the degenerate electrons and positrons. We consider the propagation of fully nonlinear IAWs in a warm plasma with ultrarelativistic degenerate electrons and positrons. The purpose of this work is (1) to study the basic features of the ion-acoustic solitary wave profiles in a dense plasma composed of warm ions and ultrarelativistic degenerate electrons and positrons and (2) to study the behavior of the Mach number and define precisely the existence regions for the solitary pulses and their dependence on the positron-to-electron number density ratio, as well as on the ion-to-Fermi electron temperature ratio.

We consider the nonlinear propagation of the IAWs in an ultrarelativistic, collisionless, unmagnetized three-component plasma composed of warm ions, degenerate electrons, and positrons. The nonlinear dynamic of IAWs is governed by the dimensionless ion continuity equation

$$\begin{equation} \frac{\partial n_{i}}{\partial t}+\frac{\partial }{\partial x}\left(n_{i}u_{i}\right) =0, \end{equation} \tag{ 1 }$$

the ion momentum equation

$$\begin{equation} \left(\frac{\partial }{\partial t}+u_{i}\frac{\partial }{\partial x}\right) u_{i}+\sigma _{i}n_{i}\frac{\partial n_{i}}{\partial x}+\frac{\partial \phi }{\partial x}=0, \end{equation} \tag{ 2 }$$

and Poisson's equation

$$\begin{equation} \frac{\partial ^{2}\phi }{\partial x^{2}}=n_{e}-n_{p}-n_{i}, \end{equation} \tag{ 3 }$$

where n_{i, e, p} is the ion/electron/positron number density, u_i is the ion fluid velocity, and ϕ is the electric potential. Here σ_i = 3T_i/T_Fe, where T_i and T_Fe are the ion and Fermi electron temperatures, respectively.

The normalized number densities of ultrarelativistic electrons and positrons are, respectively (Chandrasekhar 1935),

$$\begin{equation} \frac{\partial \phi }{\partial x}=\frac{3 \beta _e}{4 n_e}\:\frac{\partial n_e^{4/3}}{\partial x}\:, \end{equation} \tag{ 4 }$$

and

$$\begin{equation} \frac{\partial \phi }{\partial x}={-}\frac{3 \beta _p}{4 n_p}\:\frac{\partial n_p^{4/3}}{\partial x}\:, \end{equation} \tag{ 5 }$$

where β_e = 4Kn^(0)1/3_e/3k_BT_Fe, β_p = β^1/3 β_e, K ≃ (3/4)ℏc, ℏ is the Planck constant divided by 2π, and c is the speed of light in vacuum. The variables appearing in Equations (1)–(5) have been appropriately normalized. Thus, n_i is normalized by the unperturbed electron density n⁽⁰⁾_e, u_i by the ion-acoustic speed C_s = (k_BT_Fe/m_i)^1/2, ϕ by k_BT_Fe/e, the space and time variables are in units of the Debye radius λ_D = (k_BT_Fe/4πe²n⁽⁰⁾_i)^1/2 and the ion plasma period ω⁻¹_pi = (4πe²n_i⁽⁰⁾/m_i)^−1/2, respectively, k_B is the Boltzmann constant, T_Fe = (ℏ²/2m_ek_B)(3π²n⁽⁰⁾_e)^2/3 is the Fermi electron temperature, and e is the magnitude of the electron charge. In equilibrium, the neutrality condition of the plasma is satisfied, viz., α + β = 1, where α = n⁽⁰⁾_i/n_e⁽⁰⁾ and β = n⁽⁰⁾_p/n_e⁽⁰⁾ denote the unperturbed density ratios of ions- and positrons-to-electrons, respectively.

To study the small- (but finite-) amplitude ion-acoustic solitary waves, we use the reductive perturbation method (Washimi & Taniuti 1966). According to the latter, the system's state is described by the variables F (F ≡ n_i, u_i, ϕ) at a given position x and time t. Let us consider a small deviation from the equilibrium state F⁽⁰⁾, which is given by n⁽⁰⁾_i = α and u⁽⁰⁾_i = ϕ⁽⁰⁾ = 0. Then, we can write a general expression for the independent variables as

$$\begin{equation} F=F^{(0)}+\sum _{m=1}^{\infty }\varepsilon ^{m}F^{(m)}, \end{equation} \tag{ 6 }$$

where ε is a small parameter much less than unity. The dependent variables can be stretched as

$$\begin{eqnarray} &&\xi =\varepsilon ^{1/2}(x-v t)\quad \mbox{ and }\quad \tau =\varepsilon ^{3/2}t, \end{eqnarray} \tag{ 7 }$$

where v is the wave propagation speed to be determined later. Substituting the expansion (6) and the stretching (7) into Equations (1)–(5), we obtain the lowest-order equations in ε:

$$\begin{equation} \frac{1}{\alpha }(v^{2}-\alpha ^2\sigma _{i}) n_{i}^{(1)}=\frac{1}{v}(v^{2}-\alpha ^2\sigma _{i}) u_{i}^{(1)}=\phi ^{(1)}. \end{equation} \tag{ 8 }$$

The Poisson equation gives the compatibility condition

$$\begin{equation} v=\left(\alpha ^2 \sigma _i+\frac{\alpha \: \beta _e}{1+\beta ^{1/3}}\right)^{1/2}. \end{equation} \tag{ 9 }$$

If we consider the next order in ε, we obtain a system of equations in the second-order perturbed quantities. Solving this system of equations with the aid of Equations (8) and (9), we finally obtain the Korteweg–de Vries (KdV) equation

$$\begin{equation} \frac{\partial \phi ^{(1)}}{\partial \tau }+A\phi ^{(1)}\frac{\partial \phi ^{(1)} }{\partial \xi }+B\frac{\partial ^{3}\phi ^{(1)}}{\partial \xi ^{3}}=0, \end{equation} \tag{ 10 }$$

where

$$\begin{equation} A=B\left[ \frac{-2}{3\:\beta _e^2}+\frac{2}{3\: \beta _e^2\: \beta ^{1/3}}+ \frac{(3\:\alpha \: v^2+\alpha ^3\sigma _i)}{(v^2-\alpha ^2\sigma _i)^3} \right], \end{equation} \tag{ 11 }$$

and

$$\begin{equation} B=\frac{1}{2}\left[ \frac{\alpha \:v}{(v^{2}-\alpha ^2\sigma _{i}) ^{2}}\right] ^{-1}. \end{equation} \tag{ 12 }$$

Equation (10) admits a stationary solution, given by

$$\begin{equation} \phi ^{(1)}=\phi _{m}\:\hbox{sech}^{2}\left(\frac{\chi }{\bigtriangleup }\right), \end{equation} \tag{ 13 }$$

where the amplitude ϕ_m = 3U/A and the width ▵ = (4B/U)^1/2. Here, χ is the transformed coordinate with respect to a frame moving with the velocity U (i.e., χ = ξ − Uτ). Note that the soliton amplitude ϕ_m is proportional to the soliton speed U and inversely proportional to the soliton width ▵. Hence, faster solitons will be taller and narrower, while slower ones will be shorter and wider. The qualitative features of the soliton properties are thus recovered (Dauxois & Peyrard 2005).

To study waves of arbitrary amplitude, we suppose that all the dependent variables depend only on a single variable η = x − Mt, where η is normalized by λ_D and M is the Mach number of the nonlinear wave. We assume that as ∣η∣ → ∞, the dependent variables n_e → 1, n_i → α, n_p → β, u_i → 0, and ϕ → 0.

Thus, in the stationary frame, we obtain the following expression for the ion density from Equations (1) and (2):

$$\begin{eqnarray} n_i&=&\Bigg[\left(\frac{M^2+ \alpha ^2 \sigma _i-2\phi }{2\sigma _i}\right)\nonumber\\ &&-\sqrt{\left(\frac{M^2+ \alpha ^2 \sigma _i-2\phi }{2\sigma _i}\right)^2-\frac{\alpha ^2M^2}{\sigma _i}}\Bigg]^{1/2}. \end{eqnarray} \tag{ 14 }$$

Using Equations (4), (5), and (14) in the Poisson equation (3), and multiplying both sides of the resulting equation by dϕ/dη, integrating once, and imposing the appropriate boundary conditions for localized solutions, namely ϕ  →  0 and dϕ/dη → 0 at ∣η∣ → ∞, we obtain the energy-balance-like equation

$$\begin{equation} \frac{1}{2}\left(\frac{d\phi }{d\eta }\right) ^{2}+V(\phi) =0, \end{equation} \tag{ 15 }$$

where the Sagdeev potential V(ϕ) for our purposes is

$$\begin{eqnarray} \hbox{\fontsize{9.0}{12.5}\selectfont$ V(\phi)$} &=&\hbox{\fontsize{9.0}{12.5}\selectfont$-\dsty\frac{3\beta _e}{4}\left(1+ \dsty\frac{\phi }{3\:\beta _e}\right)^4 + \dsty\frac{3 \beta _e}{4}- \dsty\frac{(-3 \beta ^{2/3} \beta _e+\phi)^4}{108 \beta \:\beta _e^{3}} + \dsty\frac{3 \beta ^{5/3} \beta _e}{4}$}\nonumber \\ &&\hbox{\fontsize{7.9}{9.9}\selectfont$-\dsty\frac{1}{3\sqrt{2}}\left(\alpha ^2-\sqrt{\dsty\frac{-4\:M^2\alpha ^2 \sigma _i+ (M^2+ \alpha ^2 \sigma _i-2\phi)^2}{\sigma _i^2}}+ \dsty\frac{M^2-2\phi }{\sigma _i}\right)^{1/2}$}\nonumber \\ &&\hbox{\fontsize{7.9}{9.9}\selectfont$\times \left(2M^2-4\phi +2\alpha ^2\sigma _i+\sigma _i\sqrt{\dsty\frac{-4\:M^2\alpha ^2 \sigma _i+ (M^2+ \alpha ^2 \sigma _i-2\phi)^2}{\sigma _i^2}}\right)$}\nonumber \\ &&\hbox{\fontsize{7.9}{9.9}\selectfont$+\:\dsty\frac{1}{3\sqrt{2}}\left(\alpha ^2+ \dsty\frac{M^2}{\sigma _i}-\sqrt{\dsty\frac{-4\:M^2\alpha ^2 \sigma _i+ (M^2+ \alpha ^2 \sigma _i)^2}{\sigma _i^2}}\right)^{1/2}$} \nonumber\\ &&\hbox{\fontsize{7.9}{9.9}\selectfont$\times \left(2M^2+2\alpha ^2\sigma _i+\sigma _i\sqrt{\dsty\frac{-4\:M^2\alpha ^2 \sigma _i+ (M^2+ \alpha ^2 \sigma _i)^2}{\sigma _i^2}}\right)$}. \end{eqnarray} \tag{ 16 }$$

The existence of the finite-amplitude localized pulses is possible, if the Sagdeev potential given by Equation (16) satisfies the following conditions.

• 1.  
  The potential V(ϕ) has the maximum value if V''(ϕ) < 0 at ϕ = 0, where the prime denotes differentiation with respect to ϕ, so that the fixed point at the origin is unstable. This condition yields the lower limit of the Mach number M for which the soliton exists, and its inequality is

  $$\begin{equation} M>\left(\frac{\alpha \:\beta ^{1/3}\beta _e^{2}+\alpha ^2\sigma _i\:\beta ^{2/3}\beta _e+\alpha ^2\sigma _i\:\beta ^{1/3}\beta _e}{\beta ^{2/3}\beta _e+\beta ^{1/3}\beta _e}\right)^{1/2}. \end{equation} \tag{ 17 }$$
• 2.  
  The upper limit of M can be found by the condition V(ϕ_max) ⩾ 0, where the maximum potential ϕ_max is determined by $\phi _{\max }\approx ({1}/{2})(M^2-2M\alpha \sqrt{\sigma _i} +\alpha ^2 \sigma _i)$. This implies that the inequality

  $$\begin{eqnarray} &&\frac{3\beta _e}{4}+\frac{3 \beta ^{5/3}\beta _e}{4}-\frac{1}{3\sqrt{2}}\left(\alpha ^2+ \frac{2M\alpha \sqrt{\sigma _i}-\alpha ^2\sigma _i}{\sigma _i}\right)^{1/2}\nonumber\\ &&\times(2M^2+2\alpha ^2\sigma _i-2(M^2-2M\alpha \sqrt{\sigma _i}+\alpha ^2\sigma _i))\nonumber \\ &&-\frac{1}{108 \beta \:\beta _e^3}\left(-3\beta ^{2/3}\beta _e+\frac{1}{2}(M^2-2M\alpha \sqrt{\sigma _i}+\alpha ^2 \sigma _i)\right)^4\nonumber\\ &&-\frac{3\beta _e}{4}\left(1+\frac{M^2-2M\alpha \sqrt{\sigma _i}+\alpha ^2\sigma _i}{6\beta _e}\right)^4\nonumber \\ &&+\:\frac{1}{3\sqrt{2}}\left(\alpha ^2+ \frac{M^2}{\sigma _i}-\sqrt{\frac{-4\:M^2\alpha ^2 \sigma _i+ (M^2+ \alpha ^2 \sigma _i)^2}{\sigma _i^2}}\right)^{1/2}\nonumber \\ &&\times\! \left(\!2M^2+2\alpha ^2\sigma _i+\sigma _i\sqrt{\frac{-4\:M^2\alpha ^2 \sigma _i+ (M^2+ \alpha ^2 \sigma _i)^2}{\sigma _i^2}}\right)\! \ge 0\nonumber\\ \end{eqnarray} \tag{ 18 }$$

  holds.

We now proceed with the presentation of our numerical results. Our plasma model is inspired as discussed in Section 1, from the white dwarf stars with number density more than 10²⁹ cm⁻³ (Koester & Chanmugam 1990; Daligault & Murillo 2005; Kantor & Gusakov 2007; Kashiyama et al. 2011) and the corona of magnetars with number density 10³⁴–10³⁵ cm⁻³ (Beloborodov & Thompson 2007a, 2007b; Thoma 2009). First, we examine the existence regions of the finite-amplitude ion-acoustic solitary pulses. The existence region is confined between the minimum and maximum Mach numbers given by Equations (17) and (18), respectively. For white dwarf stars, Figure 1 shows the minimum and maximum Mach numbers for different values of the positron-to-electron density ratio β, the electron number density n⁽⁰⁾_e, and the ion-to-Fermi electron temperature ratio σ_i. Note that the solitary pulses can propagate only in the regions between the minimum and maximum Mach numbers. It is seen that as β increases the minimum/maximum Mach number decreases. However, the increase of n⁽⁰⁾_e causes the existence regions of the solitary pulses to become narrower. Furthermore, increasing n⁽⁰⁾_e makes the subsonic pulses much more pronounced than supersonic pulses. On the other hand, for low β and n⁽⁰⁾_e, supersonic solitons can exist, but for higher values of β and n⁽⁰⁾_e only subsonic solitons propagate. The effect of the ion-to-Fermi electron temperature ratio σ_i has the same behavior as in Figure 1(a). However, it is interesting to mention that both the supersonic and subsonic solitons can exist even for higher values of σ_i, but the subsonic solitons become more pronounced.

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We have numerically solved the Sagdeev potential (16) for different values of the plasma parameters. In Figures 2–4, it is observed that the Mach number M, the positron-to-electron density ratio β, and the ion-to-Fermi electron temperature ratio σ_i change the profile of the potential well. Increasing M and β leads to an increase of both the potential depth and amplitude. However, increasing σ_i would lead to an increase of the potential depth but the amplitude would decrease. It is interesting to note that an increase of the depth of the Sagdeev potential makes the solitary pulse narrower. Therefore, the increase of M and β makes the pulses much more spiky, but the increase of σ_i makes the pulse wider and shorter. We have investigated in Figures 3 and 4 the effects of β and σ_i for subsonic solitons, i.e., we used M = 0.9. However, when we plot the same figures for supersonic solitons, the same qualitative behavior is obtained, but the soliton amplitude becomes shorter. So, we did not include those figures here. It is interesting to note that, although σ_i has a small value, it affects the pulse shape causing broadening of its profile. This behavior cannot be covered for the cold plasma case (Mamun & Shukla 2010a).

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For the corona of magnetars, we have also numerically checked the Mach number region using the Sagdeev potential (16), as well as the soliton properties. For magnetar corona parameters, the existence regions of the solitary pulses and the profile of the potential well are shown in Figures 5 and 6. It is seen that only subsonic solitons can exist (with M < 1), while the same qualitative behavior of the solitary pulses is obtained. Therefore, for magnetars with very high density only the subsonic solitons exist, but for white dwarf stars with relatively lower density, the propagating solitary pulses can be either supersonic or subsonic.

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To summarize, we have presented a study of the nonlinear ion-acoustic solitary waves propagating in a collisionless e–p–i warm plasma with ultrarelativistic degenerate electrons and positrons. Solitary pulses of both small (but finite) and arbitrary amplitudes are investigated by deriving the KdV equation and an energy-balance-like expression involving a Sagdeev-like pseudopotential, respectively. The existence regions of the solitary pulses are precisely defined. Furthermore, numerical calculations reveal that for white dwarf stars either supersonic or subsonic pulses may exist. However, for the magnetar corona only subsonic solitary waves exist. The dependence of the solitary excitation characteristics on the positron-to-electron concentration ratio, the ion-to-Fermi electron temperature ratio, and the Mach number have been investigated. The present results may be useful in understanding the basic features of the ion-acoustic solitary waves that may be of some relevance to the interior of white dwarf stars and the corona of magnetars, where the ultrarelativistic particles play a role in the wave existence.

I.Z. (Zeba Tasaduq Ali) acknowledges the financial support of the Higher Education Commission (HEC) of Pakistan under the International Research Support Initiative Program (IRSIP). W.M.M. acknowledges the financial support from the Arab Fund for Economic and Social Development (Kuwait) under the Arab Fund Fellowship Program. I.Z. thanks Prof. H. A. Shah and Prof. M. Salimullah for their assistance and useful guidance.