Dynamics of electrostatic waves in relativistic electron–positron-ion degenerate plasma

Abstract

Based on quantum hydrodynamics, a rigorous two-fluid model is applied to investigate the 3-dimensional propagation characteristics of linear and nonlinear electrostatic waves in a magnetized electron–positron-ion degenerate plasma. Chandrasekhar’s equation of state (EOS) is used for the degenerate relativistic electron and positron fluids while ions are treated as fixed and uniform in space. A dispersion relation for the electronic-scale waves is obtained using the linear mode analysis. A nonlinear analysis has been performed using a reductive perturbation technique, and the corresponding Zakharov–Kuznetsov (ZK) equation is derived for the evaluation of the nonlinear model. The small$-k$ expansion perturbation method is employed to examine the instability criteria of the nonlinear waves obliquely propagating into the external magnetic field. The heading result of the present study is that the main characteristics of both linear and nonlinear modes are influenced clearly by the variations in concentrations of degenerate electrons and positrons. Also, the growth rate of the wave instability is found to increase as both the electron density and the positron concentration increase. The present results are helpful in understanding the characteristics and stability conditions of electrostatic waves in many ultra-dense systems generated in laboratory experiments of laser-irradiated solids and found in celestial environments, such as magnetar coronas, pulsar magnetospheres and black holes.

Appendices

Appendix A: Coefficients of the dispersion relation 16

$$\begin{aligned} a_{3}&=-\left[ \frac{1}{H_{e0}}+\frac{\nu _{p}}{H_{p0}}+(F_{e}+F_{p})k^{2}+\left( \frac{1}{H_{e0}^{2}}+\frac{1}{H_{p0}^{2}} \right) \Omega ^{2}\right] , \end{aligned}$$

(A1)

$$\begin{aligned} a_{2}&=\frac{\Omega ^{4}}{H_{e0}^2 H_{p0}^2} + \left[ \frac{1}{H_{e0} H_{p0}^2}+\frac{\nu _{p}}{H_{e0}^2 H_{p0}} + \frac{F_{p}H_{e0}k^{2}+k_{\parallel }^{2}\left( F_{e}H_{e0}+\frac{1}{k^{2}} \right) }{H_{e0}^3} \right. \nonumber \\&\quad \left. + \frac{F_{e}H_{p0}k^{2}+k_{\parallel }^{2}\left( F_{p}H_{p0}+\frac{\nu _{p}}{k^{2}} \right) }{H_{p0}^3} \right] \Omega ^{2} \nonumber \\&\quad +\left[ \frac{F_{p}}{H_{e0}}+\frac{\nu _{p}F_{e}}{H_{p0}}+F_{e}F_{p}k^{2}\right] k^{2}, \end{aligned}$$

(A2)

$$\begin{aligned} a_{1}&=-\frac{k_{\parallel }^{2}\Omega ^{2}}{H_{e0}^{3}H_{p0}^{3}} \left[ \left( H_{e}^{2}+H_{p}^{2}\right) \left( F_{p}H_{p0} + \nu _{p}F_{e}H_{e0} + F_{e}F_{p} H_{e0}H_{p0}k^{2} \right) \right] \nonumber \\&+ \frac{k_{\parallel }^{2}\Omega ^{4}}{k^{2}H_{e0}^{2}H_{p0}^{2}} \left( \frac{1}{H_{e0}}\right. +\,\left. \frac{\nu _{p}}{H_{p0}}+(F_{e}+F_{p})k^{2}\right) , \end{aligned}$$

(A3)

$$\begin{aligned} a_{0}&= \frac{k_{\parallel }^{4}\Omega ^{4}}{H_{e0}^{3}H_{p0}^{3}k^{2}}\left( F_{p}H_{p0} + \nu _{p}F_{e}H_{e0} + F_{e}F_{p} H_{e0}H_{p0}k^{2} \right) . \end{aligned}$$

(A4)

In the above expressions, we have introduced the quantity $F_{j}=\frac{\alpha _{j0}^{2}}{3\delta H_{j0}^{2}}+\frac{\mathcal {H}}{4 H_{j0}}k^{2}$; $H_{j0}=\sqrt{\alpha _{j0}^{2}+1}$ . The square of the wavevector is defined as $k^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2} \equiv k_{\perp }^{2}+k_{\parallel }^{2}$.

Appendix B: Equations from the next-order perturbation

Collecting terms of the next higher order of $\epsilon$, we obtain the following set of equations

$$\begin{aligned}&-\lambda \frac{\partial n_{e2}}{\partial Z}+\frac{\partial u_{ez2}}{\partial Z} =-\frac{\partial n_{e1}}{\partial T}-\frac{\partial u_{ex2}}{\partial x}-\frac{\partial u_{ey2}}{\partial y}-\frac{\partial \left( n_{e1} u_{ez1}\right) }{\partial Z}+\delta \lambda u_{ez1}\frac{\partial u_{ez1}}{\partial Z}, \end{aligned}$$

(B1)

$$\begin{aligned}&-\lambda \frac{\partial n_{p2}}{\partial Z}+\frac{\partial u_{pz2}}{\partial Z} =-\frac{\partial n_{p1}}{\partial T}-\frac{\partial u_{px2}}{\partial x}-\frac{\partial u_{py2}}{\partial y}-\frac{\partial \left( n_{p1} u_{pz1}\right) }{\partial Z}+\delta \lambda u_{pz1}\frac{\partial u_{pz1}}{\partial Z}, \end{aligned}$$

(B2)

$$\begin{aligned}&-\frac{\alpha _{e0}^{2}}{3\delta H_{e0}^{2}}\frac{\partial n_{e2}}{\partial Z}+\lambda \frac{\partial u_{ez2}}{\partial Z}+H_{e0}^{-1}\frac{\partial \phi _{2}}{\partial Z} =-\frac{\lambda \alpha _{e0}^{2}}{3\delta H_{e0}^{2}}u_{ez1}\frac{\partial n_{e1}}{\partial Z}+\frac{\partial u_{ez1}}{\partial T} \nonumber \\&\qquad -\frac{\alpha _{e0}^{2}(1+3\alpha _{e0}^{2})}{9\delta H_{e0}^{4}}n_{e1}\frac{\partial n_{e1}}{\partial Z} + u_{ez1}\frac{\partial u_{ez1}}{\partial Z} + \frac{\alpha _{e0}^{2}}{3H_{e0}^{3}}n_{e1}\frac{\partial \phi _{1}}{\partial Z}-\frac{\mathcal {H}}{4H_{e0}}\frac{\partial }{\partial Z}\nabla ^{2} n_{e1}, \end{aligned}$$

(B3)

$$\begin{aligned}&-\frac{\alpha _{p0}^{2}}{3\delta H_{p0}^{2}}\frac{\partial n_{p2}}{\partial Z}+\lambda \frac{\partial u_{pz2}}{\partial Z} - H_{p0}^{-1}\frac{\partial \phi _{2}}{\partial Z} =-\frac{\lambda \alpha _{p0}^{2}}{3\delta H_{p0}^{2}}u_{pz1}\frac{\partial n_{p1}}{\partial Z} +\frac{\partial u_{pz1}}{\partial T} \nonumber \\&\qquad -\frac{\alpha _{p0}^{2}(1+3\alpha _{p0}^{2})}{9\delta H_{p0}^{4}}n_{p1}\frac{\partial n_{p1}}{\partial Z}+ u_{pz1}\frac{\partial u_{pz1}}{\partial Z} - \frac{\alpha _{p0}^{2}}{3H_{p0}^{3}}n_{p1}\frac{\partial \phi _{1}}{\partial Z}-\frac{\mathcal {H}}{4H_{p0}}\frac{\partial }{\partial Z}\nabla ^{2} n_{p1}, \end{aligned}$$

(B4)

$$\begin{aligned}&\frac{\partial n_{e2}}{\partial Z}-\nu _{p}\frac{\partial n_{p2}}{\partial Z} =\frac{\partial }{\partial Z}\nabla ^{2}\phi _{1} - \delta u_{ez1}\frac{\partial u_{ez1}}{\partial Z} + \nu _{p} \delta u_{pz1}\frac{\partial u_{pz1}}{\partial Z}, \end{aligned}$$

(B5)

where

$$\begin{aligned}&u_{ey1}=\frac{1}{\Omega H_{e0}}\left( \frac{\alpha _{e0}^{2}}{3\delta \lambda ^{2} H_{e0}^{2}-\alpha _{e0}^{2}} +1 \right) \frac{\partial \phi _{1}}{\partial X},\qquad u_{ex1}=\frac{-1}{\Omega H_{e0}}\left( \frac{\alpha _{e0}^{2}}{3\delta \lambda ^{2} H_{e0}^{2}-\alpha _{e0}^{2}} +1 \right) \frac{\partial \phi _{1}}{\partial Y},\nonumber \\&u_{py1}=\frac{1}{\Omega H_{p0}}\left( \frac{\alpha _{p0}^{2}}{3\delta \lambda ^{2} H_{p0}^{2}-\alpha _{p0}^{2}} +1 \right) \frac{\partial \phi _{1}}{\partial X},\qquad u_{px1}=\frac{-1}{\Omega H_{p0}}\left( \frac{\alpha _{p0}^{2}}{3\delta \lambda ^{2} H_{p0}^{2}-\alpha _{p0}^{2}} +1 \right) \frac{\partial \phi _{1}}{\partial Y},\nonumber \\&u_{ey2}=\frac{\lambda }{\Omega }\frac{\partial u_{ex1}}{\partial Z}, \quad \quad u_{ex2}=\frac{-\lambda }{\Omega }\frac{\partial u_{ey1}}{\partial Z}, \qquad \quad \ u_{py2}=\frac{-\lambda }{\Omega }\frac{\partial u_{px1}}{\partial Z}, \qquad \quad u_{px2}=\frac{\lambda }{\Omega }\frac{\partial u_{py1}}{\partial Z}. \end{aligned}$$

(B6)

Appendix C: Coefficients of ZK equation

$$\begin{aligned} A =&-\frac{9\delta ^{3}\lambda ^{4}H_{e0}^{2}}{\beta _{e0}^{2}}\left( H_{e0}+\frac{H_{p0}}{\nu _{p}} \right) +3\delta ^{2}\lambda ^{2}\left( \frac{9H_{e0}^{3}-H_{e0}\alpha _{e0}^{2}}{\beta _{e0}^{2}} + \frac{9H_{p0}^{3}-2H_{p0}\alpha _{p0}^{2}}{\beta _{p0}^{2}} + \frac{H_{e0}^{2}\alpha _{p0}^{2}}{\beta _{e0}^{2}H_{p0}\nu _{p}} \right) \nonumber \\&- \delta \left( \frac{\alpha _{e0}^{2} (3\alpha _{e0}^{2}+\beta _{e0}+1)}{H_{e0}\beta _{e0}^{2}} + \frac{\alpha _{p0}^{2} (3\alpha _{p0}^{2}+\beta _{p0}+1)}{H_{p0}\beta _{p0}^{2}} \right) , \end{aligned}$$

(C1)

$$\begin{aligned} B =&\frac{3}{4}\delta \mathcal {H}\left( \frac{H_{e0}}{\beta _{e0}} - \frac{H_{p0}}{\beta _{p0}}\right) + \frac{\beta _{p0}}{3\delta \nu _{p} H_{p0}}, \end{aligned}$$

(C2)

$$\begin{aligned} C =&B - \frac{\lambda ^{2}}{\Omega ^{2}}\left( \frac{\alpha _{e0}^{2}}{\beta _{e0}} - \frac{\alpha _{p0}^{2}}{\beta _{p0}} \right) , \qquad D = 6\delta \lambda \left( \frac{H_{p0}^{2}}{\beta _{p0}} - \frac{H_{e0}^{2}}{\beta _{e0}} \right) , \end{aligned}$$

(C3)

where $\beta _{j0} = 3\delta \lambda ^{2} H_{j0}^{2} - \alpha _{j0}^{2}$.