Dust ion acoustic double layer in the presence of superthermal electrons

Abstract

The existence of a dust ion acoustic double layer in a collisionless, un-magnetized, multi-component plasma is reported here. The plasma model consists of ions, negatively charged dust particles and two components of superthermal electrons. By following Sagdeev potential and reductive perturbation method, the electrostatic double layer of negative polarity is shown to exist in small-amplitude regime. From the analytical study, it is observed that the amplitude of the double layer depends upon various parameters of the superthermal electrons as well as on the dust concentration. The model considered here has a good match with the data obtained from Cassini spacecraft for outer magnetosphere of Saturn (~ 14 R_s, R_s being the radius of Saturn). So, the results obtained from this study are useful for understanding the nature of the plasma waves in Saturn magnetosphere .

Appendix

In this appendix, the constants associated with the modified KdV equation (Eq. 27) are expressed.

$$ a = \left[ {\frac{{\mu_{\text{c}} \sigma_{\text{c}}^{3} \left( {\kappa_{\text{c}} - \frac{1}{2}} \right)\left( {\kappa_{\text{c}} + \frac{1}{2}} \right)\left( {\kappa_{\text{c}} + \frac{3}{2}} \right)}}{{2\left( {\kappa_{\text{c}} - \frac{3}{2}} \right)^{3} }} + \frac{{\mu_{\text{h}} \sigma_{\text{h}}^{3} \left( {\kappa_{\text{h}} - \frac{1}{2}} \right)\left( {\kappa_{\text{h}} + \frac{1}{2}} \right)\left( {\kappa_{\text{h}} + \frac{3}{2}} \right)}}{{2\left( {\kappa_{\text{h}} - \frac{3}{2}} \right)^{3} }} - \frac{15}{{2M^{6} }} - \frac{15}{{2\delta^{3} M^{6} }} + \frac{{19\gamma_{1} }}{{M^{4} }}} \right. $$

$$ {{\left. { - \frac{{3\gamma_{1}^{2} }}{{M^{2} }} - \frac{{4\gamma_{2} }}{{M^{2} }} + 3\gamma_{3} } \right]} \mathord{\left/ {\vphantom {{\left. { - \frac{{3\gamma_{1}^{2} }}{{M^{2} }} - \frac{{4\gamma_{2} }}{{M^{2} }} + 3\gamma_{3} } \right]} {\left[ { - \frac{2}{{\delta M^{3} }} - \frac{2}{{M^{3} }}} \right]}}} \right. \kern-0pt} {\left[ { - \frac{2}{{\delta M^{3} }} - \frac{2}{{M^{3} }}} \right]}} $$

$$ b = {{\left[ {\frac{{4\mu_{\text{c}} \sigma_{\text{c}}^{2} \left( {\kappa_{\text{c}} - \frac{1}{2}} \right)\left( {\kappa_{\text{c}} + \frac{1}{2}} \right)}}{{2\left( {\kappa_{\text{c}} - \frac{3}{2}} \right)^{2} }} + \frac{{\mu_{\text{h}} \sigma_{\text{h}}^{2} \left( {\kappa_{\text{h}} - \frac{1}{2}} \right)\left( {\kappa_{\text{h}} + \frac{1}{2}} \right)}}{{2\left( {\kappa_{\text{h}} - \frac{3}{2}} \right)^{2} }} + \frac{6}{{M^{4} }} - \frac{6}{{\delta^{2} M^{4} }} - \frac{{6\gamma_{1} }}{{M^{2} }} + 4\gamma_{2} } \right]} \mathord{\left/ {\vphantom {{\left[ {\frac{{4\mu_{\text{c}} \sigma_{\text{c}}^{2} \left( {\kappa_{\text{c}} - \frac{1}{2}} \right)\left( {\kappa_{\text{c}} + \frac{1}{2}} \right)}}{{2\left( {\kappa_{\text{c}} - \frac{3}{2}} \right)^{2} }} + \frac{{\mu_{\text{h}} \sigma_{\text{h}}^{2} \left( {\kappa_{\text{h}} - \frac{1}{2}} \right)\left( {\kappa_{\text{h}} + \frac{1}{2}} \right)}}{{2\left( {\kappa_{\text{h}} - \frac{3}{2}} \right)^{2} }} + \frac{6}{{M^{4} }} - \frac{6}{{\delta^{2} M^{4} }} - \frac{{6\gamma_{1} }}{{M^{2} }} + 4\gamma_{2} } \right]} {\left[ { - \frac{2}{{\delta M^{3} }} - \frac{2}{{M^{3} }}} \right]}}} \right. \kern-0pt} {\left[ { - \frac{2}{{\delta M^{3} }} - \frac{2}{{M^{3} }}} \right]}} $$

$$ c = {1 \mathord{\left/ {\vphantom {1 {\left[ {\frac{2}{{\delta M^{3} }} + \frac{2}{{M^{3} }}} \right]}}} \right. \kern-0pt} {\left[ {\frac{2}{{\delta M^{3} }} + \frac{2}{{M^{3} }}} \right]}} $$

where $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$ are constants associated with dust charge derived from current balance equation,

$$ I_{\text{ec}} + I_{\text{eh}} + I_{\text{i}} = 0,\;{\text{with}} $$

$$ I_{\text{ec}} = - e\pi r^{2} \left( {8T_{\text{ec}} /\pi m_{\text{e}} } \right)^{1/2} n_{\text{ec}} \exp \left( {\frac{e\varPhi }{{T_{\text{ec}} }}} \right) $$

$$ I_{\text{eh}} = - e\pi r^{2} \left( {8T_{\text{eh}} /\pi m_{\text{e}} } \right)^{1/2} n_{\text{eh}} \exp \left( {\frac{e\varPhi }{{T_{\text{eh}} }}} \right) $$

$$ I_{\text{i}} = - e\pi r^{2} \left( {8T_{\text{i}} /\pi m_{\text{i}} } \right)^{1/2} n_{\text{i}} \left( {1 - \frac{e\varPhi }{{T_{\text{i}} }}} \right), $$

where $\varPhi$ is the dust grain surface potential relative to the plasma potential.